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diff --git a/doc/RecTutorial/RecTutorial.tex b/doc/RecTutorial/RecTutorial.tex deleted file mode 100644 index f2cb383e..00000000 --- a/doc/RecTutorial/RecTutorial.tex +++ /dev/null @@ -1,3690 +0,0 @@ -\documentclass[11pt]{article} -\title{A Tutorial on [Co-]Inductive Types in Coq} -\author{Eduardo Gim\'enez\thanks{Eduardo.Gimenez@inria.fr}, -Pierre Cast\'eran\thanks{Pierre.Casteran@labri.fr}} -\date{May 1998 --- \today} - -\usepackage{multirow} -% \usepackage{aeguill} -% \externaldocument{RefMan-gal.v} -% \externaldocument{RefMan-ext.v} -% \externaldocument{RefMan-tac.v} -% \externaldocument{RefMan-oth} -% \externaldocument{RefMan-tus.v} -% \externaldocument{RefMan-syn.v} -% \externaldocument{Extraction.v} -\input{recmacros} -\input{coqartmacros} -\newcommand{\refmancite}[1]{{}} -% \newcommand{\refmancite}[1]{\cite{coqrefman}} -% \newcommand{\refmancite}[1]{\cite[#1] {]{coqrefman}} - -\usepackage[latin1]{inputenc} -\usepackage[T1]{fontenc} -\usepackage{makeidx} -% \usepackage{multind} -\usepackage{alltt} -\usepackage{verbatim} -\usepackage{amssymb} -\usepackage{amsmath} -\usepackage{theorem} -\usepackage[dvips]{epsfig} -\usepackage{epic} -\usepackage{eepic} -% \usepackage{ecltree} -\usepackage{moreverb} -\usepackage{color} -\usepackage{pifont} -\usepackage{xr} -\usepackage{url} - -\usepackage{alltt} -\renewcommand{\familydefault}{ptm} -\renewcommand{\seriesdefault}{m} -\renewcommand{\shapedefault}{n} -\newtheorem{exercise}{Exercise}[section] -\makeindex -\begin{document} -\maketitle - -\begin{abstract} -This document\footnote{The first versions of this document were entirely written by Eduardo Gimenez. -Pierre Cast\'eran wrote the 2004 and 2006 revisions.} is an introduction to the definition and -use of inductive and co-inductive types in the {\coq} proof environment. It explains how types like natural numbers and infinite streams are defined -in {\coq}, and the kind of proof techniques that can be used to reason -about them (case analysis, induction, inversion of predicates, -co-induction, etc). Each technique is illustrated through an -executable and self-contained {\coq} script. -\end{abstract} -%\RRkeyword{Proof environments, recursive types.} -%\makeRT - -\addtocontents{toc}{\protect \thispagestyle{empty}} -\pagenumbering{arabic} - -\cleardoublepage -\tableofcontents -\clearpage - -\section{About this document} - -This document is an introduction to the definition and use of -inductive and co-inductive types in the {\coq} proof environment. It was born from the -notes written for the course about the version V5.10 of {\coq}, given -by Eduardo Gimenez at -the Ecole Normale Sup\'erieure de Lyon in March 1996. This article is -a revised and improved version of these notes for the version V8.0 of -the system. - - -We assume that the reader has some familiarity with the -proofs-as-programs paradigm of Logic \cite{Coquand:metamathematical} and the generalities -of the {\coq} system \cite{coqrefman}. You would take a greater advantage of -this document if you first read the general tutorial about {\coq} and -{\coq}'s FAQ, both available on \cite{coqsite}. -A text book \cite{coqart}, accompanied with a lot of -examples and exercises \cite{Booksite}, presents a detailed description -of the {\coq} system and its underlying -formalism: the Calculus of Inductive Construction. -Finally, the complete description of {\coq} is given in the reference manual -\cite{coqrefman}. Most of the tactics and commands we describe have -several options, which we do not present exhaustively. -If some script herein uses a non described feature, please refer to -the Reference Manual. - - -If you are familiar with other proof environments -based on type theory and the LCF style ---like PVS, LEGO, Isabelle, -etc--- then you will find not difficulty to guess the unexplained -details. - -The better way to read this document is to start up the {\coq} system, -type by yourself the examples and exercises, and observe the -behavior of the system. All the examples proposed in this tutorial -can be downloaded from the same site as the present document. - - -The tutorial is organised as follows. The next section describes how -inductive types are defined in {\coq}, and introduces some useful ones, -like natural numbers, the empty type, the propositional equality type, -and the logical connectives. Section \ref{CaseAnalysis} explains -definitions by pattern-matching and their connection with the -principle of case analysis. This principle is the most basic -elimination rule associated with inductive or co-inductive types - and follows a -general scheme that we illustrate for some of the types introduced in -Section \ref{Introduction}. Section \ref{CaseTechniques} illustrates -the pragmatics of this principle, showing different proof techniques -based on it. Section \ref{StructuralInduction} introduces definitions -by structural recursion and proofs by induction. -Section~\ref{CaseStudy} presents some elaborate techniques -about dependent case analysis. Finally, Section -\ref{CoInduction} is a brief introduction to co-inductive types ---i.e., types containing infinite objects-- and the principle of -co-induction. - - -Thanks to Bruno Barras, Yves Bertot, Hugo Herbelin, Jean-Fran\c{c}ois Monin -and Michel L\'evy for their help. - -\subsection*{Lexical conventions} -The \texttt{typewriter} font is used to represent text -input by the user, while the \textit{italic} font is used to represent -the text output by the system as answers. - - -Moreover, the mathematical symbols \coqle{}, \coqdiff, \(\exists\), -\(\forall\), \arrow{}, $\rightarrow{}$ \coqor{}, \coqand{}, and \funarrow{} -stand for the character strings \citecoq{<=}, \citecoq{<>}, -\citecoq{exists}, \citecoq{forall}, \citecoq{->}, \citecoq{<-}, -\texttt{\char'134/}, \texttt{/\char'134}, and \citecoq{=>}, -respectively. For instance, the \coq{} statement -%V8 A prendre -% inclusion numero 1 -% traduction numero 1 -\begin{alltt} -\hide{Open Scope nat_scope. Check (}forall A:Type,(exists x : A, forall (y:A), x <> y) -> 2 = 3\hide{).} -\end{alltt} -is written as follows in this tutorial: -%V8 A prendre -% inclusion numero 2 -% traduction numero 2 -\begin{alltt} -\hide{Check (}{\prodsym}A:Type,(\exsym{}x:A, {\prodsym}y:A, x {\coqdiff} y) \arrow{} 2 = 3\hide{).} -\end{alltt} - -When a fragment of \coq{} input text appears in the middle of -regular text, we often place this fragment between double quotes -``\dots.'' These double quotes do not belong to the \coq{} syntax. - -Finally, any -string enclosed between \texttt{(*} and \texttt{*)} is a comment and -is ignored by the \coq{} system. - -\section{Introducing Inductive Types} -\label{Introduction} - -Inductive types are types closed with respect to their introduction -rules. These rules explain the most basic or \textsl{canonical} ways -of constructing an element of the type. In this sense, they -characterize the recursive type. Different rules must be considered as -introducing different objects. In order to fix ideas, let us introduce -in {\coq} the most well-known example of a recursive type: the type of -natural numbers. - -%V8 A prendre -\begin{alltt} -Inductive nat : Set := - | O : nat - | S : nat\arrow{}nat. -\end{alltt} - -The definition of a recursive type has two main parts. First, we -establish what kind of recursive type we will characterize (a set, in -this case). Second, we present the introduction rules that define the -type ({\Z} and {\SUCC}), also called its {\sl constructors}. The constructors -{\Z} and {\SUCC} determine all the elements of this type. In other -words, if $n\mbox{:}\nat$, then $n$ must have been introduced either -by the rule {\Z} or by an application of the rule {\SUCC} to a -previously constructed natural number. In this sense, we can say -that {\nat} is \emph{closed}. On the contrary, the type -$\Set$ is an {\it open} type, since we do not know {\it a priori} all -the possible ways of introducing an object of type \texttt{Set}. - -After entering this command, the constants {\nat}, {\Z} and {\SUCC} are -available in the current context. We can see their types using the -\texttt{Check} command \refmancite{Section \ref{Check}}: - -%V8 A prendre -\begin{alltt} -Check nat. -\it{}nat : Set -\tt{}Check O. -\it{}O : nat -\tt{}Check S. -\it{}S : nat {\arrow} nat -\end{alltt} - -Moreover, {\coq} adds to the context three constants named - $\natind$, $\natrec$ and $\natrect$, which - correspond to different principles of structural induction on -natural numbers that {\coq} infers automatically from the definition. We -will come back to them in Section \ref{StructuralInduction}. - - -In fact, the type of natural numbers as well as several useful -theorems about them are already defined in the basic library of {\coq}, -so there is no need to introduce them. Therefore, let us throw away -our (re)definition of {\nat}, using the command \texttt{Reset}. - -%V8 A prendre -\begin{alltt} -Reset nat. -Print nat. -\it{}Inductive nat : Set := O : nat | S : nat \arrow{} nat -For S: Argument scope is [nat_scope] -\end{alltt} - -Notice that \coq{}'s \emph{interpretation scope} for natural numbers -(called \texttt{nat\_scope}) -allows us to read and write natural numbers in decimal form (see \cite{coqrefman}). For instance, the constructor \texttt{O} can be read or written -as the digit $0$, and the term ``~\texttt{S (S (S O))}~'' as $3$. - -%V8 A prendre -\begin{alltt} -Check O. -\it 0 : nat. -\tt -Check (S (S (S O))). -\it 3 : nat -\end{alltt} - -Let us now take a look to some other -recursive types contained in the standard library of {\coq}. - -\subsection{Lists} -Lists are defined in library \citecoq{List}\footnote{Notice that in versions of -{\coq} -prior to 8.1, the parameter $A$ had sort \citecoq{Set} instead of \citecoq{Type}; -the constant \citecoq{list} was thus of type \citecoq{Set\arrow{} Set}.} - - -\begin{alltt} -Require Import List. -Print list. -\it -Inductive list (A : Type) : Type:= - nil : list A | cons : A {\arrow} list A {\arrow} list A -For nil: Argument A is implicit -For cons: Argument A is implicit -For list: Argument scope is [type_scope] -For nil: Argument scope is [type_scope] -For cons: Argument scopes are [type_scope _ _] -\end{alltt} - -In this definition, \citecoq{A} is a \emph{general parameter}, global -to both constructors. -This kind of definition allows us to build a whole family of -inductive types, indexed over the sort \citecoq{Type}. -This can be observed if we consider the type of identifiers -\citecoq{list}, \citecoq{cons} and \citecoq{nil}. -Notice the notation \citecoq{(A := \dots)} which must be used -when {\coq}'s type inference algorithm cannot infer the implicit -parameter \citecoq{A}. -\begin{alltt} -Check list. -\it list - : Type {\arrow} Type - -\tt Check (nil (A:=nat)). -\it nil - : list nat - -\tt Check (nil (A:= nat {\arrow} nat)). -\it nil - : list (nat {\arrow} nat) - -\tt Check (fun A: Type {\funarrow} (cons (A:=A))). -\it fun A : Type {\funarrow} cons (A:=A) - : {\prodsym} A : Type, A {\arrow} list A {\arrow} list A - -\tt Check (cons 3 (cons 2 nil)). -\it 3 :: 2 :: nil - : list nat - -\tt Check (nat :: bool ::nil). -\it nat :: bool :: nil - : list Set - -\tt Check ((3<=4) :: True ::nil). -\it (3<=4) :: True :: nil - : list Prop - -\tt Check (Prop::Set::nil). -\it Prop::Set::nil - : list Type -\end{alltt} - -\subsection{Vectors.} -\label{vectors} - -Like \texttt{list}, \citecoq{vector} is a polymorphic type: -if $A$ is a type, and $n$ a natural number, ``~\citecoq{vector $A$ $n$}~'' -is the type of vectors of elements of $A$ and size $n$. - - -\begin{alltt} -Require Import Bvector. - -Print vector. -\it -Inductive vector (A : Type) : nat {\arrow} Type := - Vnil : vector A 0 - | Vcons : A {\arrow} {\prodsym} n : nat, vector A n {\arrow} vector A (S n) -For vector: Argument scopes are [type_scope nat_scope] -For Vnil: Argument scope is [type_scope] -For Vcons: Argument scopes are [type_scope _ nat_scope _] -\end{alltt} - - -Remark the difference between the two parameters $A$ and $n$: -The first one is a \textsl{general parameter}, global to all the -introduction rules,while the second one is an \textsl{index}, which is -instantiated differently in the introduction rules. -Such types parameterized by regular -values are called \emph{dependent types}. - -\begin{alltt} -Check (Vnil nat). -\it Vnil nat - : vector nat 0 - -\tt Check (fun (A:Type)(a:A){\funarrow} Vcons _ a _ (Vnil _)). -\it fun (A : Type) (a : A) {\funarrow} Vcons A a 0 (Vnil A) - : {\prodsym} A : Type, A {\arrow} vector A 1 - - -\tt Check (Vcons _ 5 _ (Vcons _ 3 _ (Vnil _))). -\it Vcons nat 5 1 (Vcons nat 3 0 (Vnil nat)) - : vector nat 2 -\end{alltt} - -\subsection{The contradictory proposition.} -Another example of an inductive type is the contradictory proposition. -This type inhabits the universe of propositions, and has no element -at all. -%V8 A prendre -\begin{alltt} -Print False. -\it{} Inductive False : Prop := -\end{alltt} - -\noindent Notice that no constructor is given in this definition. - -\subsection{The tautological proposition.} -Similarly, the -tautological proposition {\True} is defined as an inductive type -with only one element {\I}: - -%V8 A prendre -\begin{alltt} -Print True. -\it{}Inductive True : Prop := I : True -\end{alltt} - -\subsection{Relations as inductive types.} -Some relations can also be introduced in a smart way as an inductive family -of propositions. Let us take as example the order $n \leq m$ on natural -numbers, called \citecoq{le} in {\coq}. - This relation is introduced through -the following definition, quoted from the standard library\footnote{In the interpretation scope -for Peano arithmetic: -\citecoq{nat\_scope}, ``~\citecoq{n <= m}~'' is equivalent to -``~\citecoq{le n m}~'' .}: - - - - -%V8 A prendre -\begin{alltt} -Print le. \it -Inductive le (n:nat) : nat\arrow{}Prop := -| le_n: n {\coqle} n -| le_S: {\prodsym} m, n {\coqle} m \arrow{} n {\coqle} S m. -\end{alltt} - -Notice that in this definition $n$ is a general parameter, -while the second argument of \citecoq{le} is an index (see section -~\ref{vectors}). - This definition -introduces the binary relation $n {\leq} m$ as the family of unary predicates -``\textsl{to be greater or equal than a given $n$}'', parameterized by $n$. - -The introduction rules of this type can be seen as a sort of Prolog -rules for proving that a given integer $n$ is less or equal than another one. -In fact, an object of type $n{\leq} m$ is nothing but a proof -built up using the constructors \textsl{le\_n} and -\textsl{le\_S} of this type. As an example, let us construct -a proof that zero is less or equal than three using {\coq}'s interactive -proof mode. -Such an object can be obtained applying three times the second -introduction rule of \citecoq{le}, to a proof that zero is less or equal -than itself, -which is provided by the first constructor of \citecoq{le}: - -%V8 A prendre -\begin{alltt} -Theorem zero_leq_three: 0 {\coqle} 3. -Proof. -\it{} 1 subgoal - -============================ - 0 {\coqle} 3 - -\tt{}Proof. - constructor 2. - -\it{} 1 subgoal -============================ - 0 {\coqle} 2 - -\tt{} constructor 2. -\it{} 1 subgoal -============================ - 0 {\coqle} 1 - -\tt{} constructor 2 -\it{} 1 subgoal -============================ - 0 {\coqle} 0 - -\tt{} constructor 1. - -\it{}Proof completed -\tt{}Qed. -\end{alltt} - -\noindent When -the current goal is an inductive type, the tactic -``~\citecoq{constructor $i$}~'' \refmancite{Section \ref{constructor}} applies the $i$-th constructor in the -definition of the type. We can take a look at the proof constructed -using the command \texttt{Print}: - -%V8 A prendre -\begin{alltt} -Print Print zero_leq_three. -\it{}zero_leq_three = -zero_leq_three = le_S 0 2 (le_S 0 1 (le_S 0 0 (le_n 0))) - : 0 {\coqle} 3 -\end{alltt} - -When the parameter $i$ is not supplied, the tactic \texttt{constructor} -tries to apply ``~\texttt{constructor $1$}~'', ``~\texttt{constructor $2$}~'',\dots, -``~\texttt{constructor $n$}~'' where $n$ is the number of constructors -of the inductive type (2 in our example) of the conclusion of the goal. -Our little proof can thus be obtained iterating the tactic -\texttt{constructor} until it fails: - -%V8 A prendre -\begin{alltt} -Lemma zero_leq_three': 0 {\coqle} 3. - repeat constructor. -Qed. -\end{alltt} - -Notice that the strict order on \texttt{nat}, called \citecoq{lt} -is not inductively defined: the proposition $n<p$ (notation for \citecoq{lt $n$ $p$}) -is reducible to \citecoq{(S $n$) $\leq$ p}. - -\begin{alltt} -Print lt. -\it -lt = fun n m : nat {\funarrow} S n {\coqle} m - : nat {\arrow} nat {\arrow} Prop -\tt -Lemma zero_lt_three : 0 < 3. -Proof. - repeat constructor. -Qed. - -Print zero_lt_three. -\it zero_lt_three = le_S 1 2 (le_S 1 1 (le_n 1)) - : 0 < 3 -\end{alltt} - - - -\subsection{About general parameters (\coq{} version $\geq$ 8.1)} -\label{parameterstuff} - -Since version $8.1$, it is possible to write more compact inductive definitions -than in earlier versions. - -Consider the following alternative definition of the relation $\leq$ on -type \citecoq{nat}: - -\begin{alltt} -Inductive le'(n:nat):nat -> Prop := - | le'_n : le' n n - | le'_S : forall p, le' (S n) p -> le' n p. - -Hint Constructors le'. -\end{alltt} - -We notice that the type of the second constructor of \citecoq{le'} -has an argument whose type is \citecoq{le' (S n) p}. -This constrasts with earlier versions -of {\coq}, in which a general parameter $a$ of an inductive -type $I$ had to appear only in applications of the form $I\,\dots\,a$. - -Since version $8.1$, if $a$ is a general parameter of an inductive -type $I$, the type of an argument of a constructor of $I$ may be -of the form $I\,\dots\,t_a$ , where $t_a$ is any term. -Notice that the final type of the constructors must be of the form -$I\,\dots\,a$, since these constructors describe how to form -inhabitants of type $I\,\dots\,a$ (this is the role of parameter $a$). - -Another example of this new feature is {\coq}'s definition of accessibility -(see Section~\ref{WellFoundedRecursion}), which has a general parameter -$x$; the constructor for the predicate -``$x$ is accessible'' takes an argument of type ``$y$ is accessible''. - - - -In earlier versions of {\coq}, a relation like \citecoq{le'} would have to be -defined without $n$ being a general parameter. - -\begin{alltt} -Reset le'. - -Inductive le': nat-> nat -> Prop := - | le'_n : forall n, le' n n - | le'_S : forall n p, le' (S n) p -> le' n p. -\end{alltt} - - - - -\subsection{The propositional equality type.} \label{equality} -In {\coq}, the propositional equality between two inhabitants $a$ and -$b$ of -the same type $A$ , -noted $a=b$, is introduced as a family of recursive predicates -``~\textsl{to be equal to $a$}~'', parameterised by both $a$ and its type -$A$. This family of types has only one introduction rule, which -corresponds to reflexivity. -Notice that the syntax ``\citecoq{$a$ = $b$}~'' is an abbreviation -for ``\citecoq{eq $a$ $b$}~'', and that the parameter $A$ is \emph{implicit}, -as it can be infered from $a$. -%V8 A prendre -\begin{alltt} -Print eq. -\it{} Inductive eq (A : Type) (x : A) : A \arrow{} Prop := - refl_equal : x = x -For eq: Argument A is implicit -For refl_equal: Argument A is implicit -For eq: Argument scopes are [type_scope _ _] -For refl_equal: Argument scopes are [type_scope _] -\end{alltt} - -Notice also that the first parameter $A$ of \texttt{eq} has type -\texttt{Type}. The type system of {\coq} allows us to consider equality between -various kinds of terms: elements of a set, proofs, propositions, -types, and so on. -Look at \cite{coqrefman, coqart} to get more details on {\coq}'s type -system, as well as implicit arguments and argument scopes. - - -\begin{alltt} -Lemma eq_3_3 : 2 + 1 = 3. -Proof. - reflexivity. -Qed. - -Lemma eq_proof_proof : refl_equal (2*6) = refl_equal (3*4). -Proof. - reflexivity. -Qed. - -Print eq_proof_proof. -\it eq_proof_proof = -refl_equal (refl_equal (3 * 4)) - : refl_equal (2 * 6) = refl_equal (3 * 4) -\tt - -Lemma eq_lt_le : ( 2 < 4) = (3 {\coqle} 4). -Proof. - reflexivity. -Qed. - -Lemma eq_nat_nat : nat = nat. -Proof. - reflexivity. -Qed. - -Lemma eq_Set_Set : Set = Set. -Proof. - reflexivity. -Qed. -\end{alltt} - -\subsection{Logical connectives.} \label{LogicalConnectives} -The conjunction and disjunction of two propositions are also examples -of recursive types: - -\begin{alltt} -Inductive or (A B : Prop) : Prop := - or_introl : A \arrow{} A {\coqor} B | or_intror : B \arrow{} A {\coqor} B - -Inductive and (A B : Prop) : Prop := - conj : A \arrow{} B \arrow{} A {\coqand} B - -\end{alltt} - -The propositions $A$ and $B$ are general parameters of these -connectives. Choosing different universes for -$A$ and $B$ and for the inductive type itself gives rise to different -type constructors. For example, the type \textsl{sumbool} is a -disjunction but with computational contents. - -\begin{alltt} -Inductive sumbool (A B : Prop) : Set := - left : A \arrow{} \{A\} + \{B\} | right : B \arrow{} \{A\} + \{B\} -\end{alltt} - - - -This type --noted \texttt{\{$A$\}+\{$B$\}} in {\coq}-- can be used in {\coq} -programs as a sort of boolean type, to check whether it is $A$ or $B$ -that is true. The values ``~\citecoq{left $p$}~'' and -``~\citecoq{right $q$}~'' replace the boolean values \textsl{true} and -\textsl{false}, respectively. The advantage of this type over -\textsl{bool} is that it makes available the proofs $p$ of $A$ or $q$ -of $B$, which could be necessary to construct a verification proof -about the program. -For instance, let us consider the certified program \citecoq{le\_lt\_dec} -of the Standard Library. - -\begin{alltt} -Require Import Compare_dec. -Check le_lt_dec. -\it -le_lt_dec - : {\prodsym} n m : nat, \{n {\coqle} m\} + \{m < n\} - -\end{alltt} - -We use \citecoq{le\_lt\_dec} to build a function for computing -the max of two natural numbers: - -\begin{alltt} -Definition max (n p :nat) := match le_lt_dec n p with - | left _ {\funarrow} p - | right _ {\funarrow} n - end. -\end{alltt} - -In the following proof, the case analysis on the term -``~\citecoq{le\_lt\_dec n p}~'' gives us an access to proofs -of $n\leq p$ in the first case, $p<n$ in the other. - -\begin{alltt} -Theorem le_max : {\prodsym} n p, n {\coqle} p {\arrow} max n p = p. -Proof. - intros n p ; unfold max ; case (le_lt_dec n p); simpl. -\it -2 subgoals - - n : nat - p : nat - ============================ - n {\coqle} p {\arrow} n {\coqle} p {\arrow} p = p - -subgoal 2 is: - p < n {\arrow} n {\coqle} p {\arrow} n = p -\tt - trivial. - intros; absurd (p < p); eauto with arith. -Qed. -\end{alltt} - - - Once the program verified, the proofs are -erased by the extraction procedure: - -\begin{alltt} -Extraction max. -\it -(** val max : nat {\arrow} nat {\arrow} nat **) - -let max n p = - match le_lt_dec n p with - | Left {\arrow} p - | Right {\arrow} n -\end{alltt} - -Another example of use of \citecoq{sumbool} is given in Section -\ref{WellFoundedRecursion}: the theorem \citecoq{eq\_nat\_dec} of -library \citecoq{Coq.Arith.Peano\_dec} is used in an euclidean division -algorithm. - -\subsection{The existential quantifier.}\label{ex-def} -The existential quantifier is yet another example of a logical -connective introduced as an inductive type. - -\begin{alltt} -Inductive ex (A : Type) (P : A \arrow{} Prop) : Prop := - ex_intro : {\prodsym} x : A, P x \arrow{} ex P -\end{alltt} - -Notice that {\coq} uses the abreviation ``~\citecoq{\exsym\,$x$:$A$, $B$}~'' -for \linebreak ``~\citecoq{ex (fun $x$:$A$ \funarrow{} $B$)}~''. - - -\noindent The former quantifier inhabits the universe of propositions. -As for the conjunction and disjunction connectives, there is also another -version of existential quantification inhabiting the universes $\Type_i$, -which is written \texttt{sig $P$}. The syntax -``~\citecoq{\{$x$:$A$ | $B$\}}~'' is an abreviation for ``~\citecoq{sig (fun $x$:$A$ {\funarrow} $B$)}~''. - - - -%\paragraph{The logical connectives.} Conjuction and disjuction are -%also introduced as recursive types: -%\begin{alltt} -%Print or. -%\end{alltt} -%begin{alltt} -%Print and. -%\end{alltt} - - -\subsection{Mutually Dependent Definitions} -\label{MutuallyDependent} - -Mutually dependent definitions of recursive types are also allowed in -{\coq}. A typical example of these kind of declaration is the -introduction of the trees of unbounded (but finite) width: -\label{Forest} -\begin{alltt} -Inductive tree(A:Type) : Type := - node : A {\arrow} forest A \arrow{} tree A -with forest (A: Set) : Type := - nochild : forest A | - addchild : tree A \arrow{} forest A \arrow{} forest A. -\end{alltt} -\noindent Yet another example of mutually dependent types are the -predicates \texttt{even} and \texttt{odd} on natural numbers: -\label{Even} -\begin{alltt} -Inductive - even : nat\arrow{}Prop := - evenO : even O | - evenS : {\prodsym} n, odd n \arrow{} even (S n) -with - odd : nat\arrow{}Prop := - oddS : {\prodsym} n, even n \arrow{} odd (S n). -\end{alltt} - -\begin{alltt} -Lemma odd_49 : odd (7 * 7). - simpl; repeat constructor. -Qed. -\end{alltt} - - - -\section{Case Analysis and Pattern-matching} -\label{CaseAnalysis} -\subsection{Non-dependent Case Analysis} -An \textsl{elimination rule} for the type $A$ is some way to use an -object $a:A$ in order to define an object in some type $B$. -A natural elimination rule for an inductive type is \emph{case analysis}. - - -For instance, any value of type {\nat} is built using either \texttt{O} or \texttt{S}. -Thus, a systematic way of building a value of type $B$ from any -value of type {\nat} is to associate to \texttt{O} a constant $t_O:B$ and -to every term of the form ``~\texttt{S $p$}~'' a term $t_S:B$. The following -construction has type $B$: -\begin{alltt} -match \(n\) return \(B\) with O \funarrow \(t\sb{O}\) | S p \funarrow \(t\sb{S}\) end -\end{alltt} - - -In most of the cases, {\coq} is able to infer the type $B$ of the object -defined, so the ``\texttt{return $B$}'' part can be omitted. - -The computing rules associated with this construct are the expected ones -(the notation $t_S\{q/\texttt{p}\}$ stands for the substitution of $p$ by -$q$ in $t_S$ :) - -\begin{eqnarray*} -\texttt{match $O$ return $b$ with O {\funarrow} $t_O$ | S p {\funarrow} $t_S$ end} &\Longrightarrow& t_O\\ -\texttt{match $S\;q$ return $b$ with O {\funarrow} $t_O$ | S p {\funarrow} $t_S$ end} &\Longrightarrow& t_S\{q/\texttt{p}\} -\end{eqnarray*} - - -\subsubsection{Example: the predecessor function.}\label{firstpred} -An example of a definition by case analysis is the function which -computes the predecessor of any given natural number: -\begin{alltt} -Definition pred (n:nat) := match n with - | O {\funarrow} O - | S m {\funarrow} m - end. - -Eval simpl in pred 56. -\it{} = 55 - : nat -\tt -Eval simpl in pred 0. -\it{} = 0 - : nat - -\tt{}Eval simpl in fun p {\funarrow} pred (S p). -\it{} = fun p : nat {\funarrow} p - : nat {\arrow} nat -\end{alltt} - -As in functional programming, tuples and wild-cards can be used in -patterns \refmancite{Section \ref{ExtensionsOfCases}}. Such -definitions are automatically compiled by {\coq} into an expression which -may contain several nested case expressions. For example, the -exclusive \emph{or} on booleans can be defined as follows: -\begin{alltt} -Definition xorb (b1 b2:bool) := - match b1, b2 with - | false, true {\funarrow} true - | true, false {\funarrow} true - | _ , _ {\funarrow} false - end. -\end{alltt} - -This kind of definition is compiled in {\coq} as follows\footnote{{\coq} uses -the conditional ``~\citecoq{if $b$ then $a$ else $b$}~'' as an abreviation to -``~\citecoq{match $b$ with true \funarrow{} $a$ | false \funarrow{} $b$ end}~''.}: - -\begin{alltt} -Print xorb. -xorb = -fun b1 b2 : bool {\funarrow} -if b1 then if b2 then false else true - else if b2 then true else false - : bool {\arrow} bool {\arrow} bool -\end{alltt} - -\subsection{Dependent Case Analysis} -\label{DependentCase} - -For a pattern matching construct of the form -``~\citecoq{match n with \dots end}~'' a more general typing rule -is obtained considering that the type of the whole expression -may also depend on \texttt{n}. - For instance, let us consider some function -$Q:\texttt{nat}\arrow{}\texttt{Type}$, and $n:\citecoq{nat}$. -In order to build a term of type $Q\;n$, we can associate -to the constructor \texttt{O} some term $t_O: Q\;\texttt{O}$ and to -the pattern ``~\texttt{S p}~'' some term $t_S : Q\;(S\;p)$. -Notice that the terms $t_O$ and $t_S$ do not have the same type. - -The syntax of the \emph{dependent case analysis} and its -associated typing rule make precise how the resulting -type depends on the argument of the pattern matching, and -which constraint holds on the branches of the pattern matching: - -\label{Prod-sup-rule} -\[ -\begin{array}[t]{l} -Q: \texttt{nat}{\arrow}\texttt{Type}\quad{t_O}:{{Q\;\texttt{O}}} \quad -\smalljuge{p:\texttt{nat}}{t_p}{{Q\;(\texttt{S}\;p)}} \quad n:\texttt{nat} \\ -\hline -{\texttt{match \(n\) as \(n\sb{0}\) return \(Q\;n\sb{0}\) with | O \funarrow \(t\sb{O}\) | S p \funarrow \(t\sb{S}\) end}}:{{Q\;n}} -\end{array} -\] - - -The interest of this rule of \textsl{dependent} pattern-matching is -that it can also be read as the following logical principle (when $Q$ has type \citecoq{nat\arrow{}Prop} -by \texttt{Prop} in the type of $Q$): in order to prove -that a property $Q$ holds for all $n$, it is sufficient to prove that -$Q$ holds for {\Z} and that for all $p:\nat$, $Q$ holds for -$(\SUCC\;p)$. The former, non-dependent version of case analysis can -be obtained from this latter rule just taking $Q$ as a constant -function on $n$. - -Notice that destructuring $n$ into \citecoq{O} or ``~\citecoq{S p}~'' - doesn't -make appear in the goal the equalities ``~$n=\citecoq{O}$~'' - and ``~$n=\citecoq{S p}$~''. -They are ``internalized'' in the rules above (see section~\ref{inversion}.) - -\subsubsection{Example: strong specification of the predecessor function.} - -In Section~\ref{firstpred}, the predecessor function was defined directly -as a function from \texttt{nat} to \texttt{nat}. It remains to prove -that this function has some desired properties. Another way to proceed -is to, first introduce a specification of what is the predecessor of a -natural number, under the form of a {\coq} type, then build an inhabitant -of this type: in other words, a realization of this specification. This way, the correctness -of this realization is ensured by {\coq}'s type system. - -A reasonable specification for $\pred$ is to say that for all $n$ -there exists another $m$ such that either $m=n=0$, or $(\SUCC\;m)$ -is equal to $n$. The function $\pred$ should be just the way to -compute such an $m$. - -\begin{alltt} -Definition pred_spec (n:nat) := - \{m:nat | n=0{\coqand} m=0 {\coqor} n = S m\}. - -Definition predecessor : {\prodsym} n:nat, pred_spec n. - intro n; case n. -\it{} - n : nat - ============================ - pred_spec 0 - -\tt{} unfold pred_spec;exists 0;auto. -\it{} - ========================================= - {\prodsym} n0 : nat, pred_spec (S n0) -\tt{} - unfold pred_spec; intro n0; exists n0; auto. -Defined. -\end{alltt} - -If we print the term built by {\coq}, its dependent pattern-matching structure can be observed: - -\begin{alltt} -predecessor = fun n : nat {\funarrow} -\textbf{match n as n0 return (pred_spec n0) with} -\textbf{| O {\funarrow}} - exist (fun m : nat {\funarrow} 0 = 0 {\coqand} m = 0 {\coqor} 0 = S m) 0 - (or_introl (0 = 1) - (conj (refl_equal 0) (refl_equal 0))) -\textbf{| S n0 {\funarrow}} - exist (fun m : nat {\funarrow} S n0 = 0 {\coqand} m = 0 {\coqor} S n0 = S m) n0 - (or_intror (S n0 = 0 {\coqand} n0 = 0) (refl_equal (S n0))) -\textbf{end} : {\prodsym} n : nat, \textbf{pred_spec n} -\end{alltt} - - -Notice that there are many variants to the pattern ``~\texttt{intros \dots; case \dots}~''. Look at for tactics -``~\texttt{destruct}~'', ``~\texttt{intro \emph{pattern}}~'', etc. in -the reference manual and/or the book. - -\noindent The command \texttt{Extraction} \refmancite{Section -\ref{ExtractionIdent}} can be used to see the computational -contents associated to the \emph{certified} function \texttt{predecessor}: -\begin{alltt} -Extraction predecessor. -\it -(** val predecessor : nat {\arrow} pred_spec **) - -let predecessor = function - | O {\arrow} O - | S n0 {\arrow} n0 -\end{alltt} - - -\begin{exercise} \label{expand} -Prove the following theorem: -\begin{alltt} -Theorem nat_expand : {\prodsym} n:nat, - n = match n with - | 0 {\funarrow} 0 - | S p {\funarrow} S p - end. -\end{alltt} -\end{exercise} - -\subsection{Some Examples of Case Analysis} -\label{CaseScheme} -The reader will find in the Reference manual all details about -typing case analysis (chapter 4: Calculus of Inductive Constructions, -and chapter 15: Extended Pattern-Matching). - -The following commented examples will show the different situations to consider. - - -%\subsubsection{General Scheme} - -%Case analysis is then the most basic elimination rule that {\coq} -%provides for inductive types. This rule follows a general schema, -%valid for any inductive type $I$. First, if $I$ has type -%``~$\forall\,(z_1:A_1)\ldots(z_r:A_r),S$~'', with $S$ either $\Set$, $\Prop$ or -%$\Type$, then a case expression on $p$ of type ``~$R\;a_1\ldots a_r$~'' -% inhabits ``~$Q\;a_1\ldots a_r\;p$~''. The types of the branches of the case expression -%are obtained from the definition of the type in this way: if the type -%of the $i$-th constructor $c_i$ of $R$ is -%``~$\forall\, (x_1:T_1)\ldots -%(x_n:T_n),(R\;q_1\ldots q_r)$~'', then the $i-th$ branch must have the -%form ``~$c_i\; x_1\; \ldots \;x_n\; \funarrow{}\; t_i$~'' where -%$$(x_1:T_1),\ldots, (x_n:T_n) \vdash t_i : Q\;q_1\ldots q_r)$$ -% for non-dependent case -%analysis, and $$(x_1:T_1)\ldots (x_n:T_n)\vdash t_i :Q\;q_1\ldots -%q_r\;({c}_i\;x_1\;\ldots x_n)$$ for dependent one. In the -%following section, we illustrate this general scheme for different -%recursive types. -%%\textbf{A vérifier} - -\subsubsection{The Empty Type} - -In a definition by case analysis, there is one branch for each -introduction rule of the type. Hence, in a definition by case analysis -on $p:\False$ there are no cases to be considered. In other words, the -rule of (non-dependent) case analysis for the type $\False$ is -(for $s$ in \texttt{Prop}, \texttt{Set} or \texttt{Type}): - -\begin{center} -\snregla {\JM{Q}{s}\;\;\;\;\; - \JM{p}{\False}} - {\JM{\texttt{match $p$ return $Q$ with end}}{Q}} -\end{center} - -As a corollary, if we could construct an object in $\False$, then it -could be possible to define an object in any type. The tactic -\texttt{contradiction} \refmancite{Section \ref{Contradiction}} -corresponds to the application of the elimination rule above. It -searches in the context for an absurd hypothesis (this is, a -hypothesis whose type is $\False$) and then proves the goal by a case -analysis of it. - -\begin{alltt} -Theorem fromFalse : False \arrow{} 0=1. -Proof. - intro H. - contradiction. -Qed. -\end{alltt} - - -In {\coq} the negation is defined as follows : - -\begin{alltt} -Definition not (P:Prop) := P {\arrow} False -\end{alltt} - -The proposition ``~\citecoq{not $A$}~'' is also written ``~$\neg A$~''. - -If $A$ and $B$ are propositions, $a$ is a proof of $A$ and -$H$ is a proof of $\neg A$, -the term ``~\citecoq{match $H\;a$ return $B$ with end}~'' is a proof term of -$B$. -Thus, if your goal is $B$ and you have some hypothesis $H:\neg A$, -the tactic ``~\citecoq{case $H$}~'' generates a new subgoal with -statement $A$, as shown by the following example\footnote{Notice that -$a\coqdiff b$ is just an abreviation for ``~\coqnot a= b~''}. - -\begin{alltt} -Fact Nosense : 0 {\coqdiff} 0 {\arrow} 2 = 3. -Proof. - intro H; case H. -\it -=========================== - 0 = 0 -\tt - reflexivity. -Qed. -\end{alltt} - -The tactic ``~\texttt{absurd $A$}~'' (where $A$ is any proposition), -is based on the same principle, but -generates two subgoals: $A$ and $\neg A$, for solving $B$. - -\subsubsection{The Equality Type} - -Let $A:\Type$, $a$, $b$ of type $A$, and $\pi$ a proof of -$a=b$. Non dependent case analysis of $\pi$ allows us to -associate to any proof of ``~$Q\;a$~'' a proof of ``~$Q\;b$~'', -where $Q:A\arrow{} s$ (where $s\in\{\Prop, \Set, \Type\}$). -The following term is a proof of ``~$Q\;a\, \arrow{}\, Q\;b$~''. - -\begin{alltt} -fun H : Q a {\funarrow} - match \(\pi\) in (_ = y) return Q y with - refl_equal {\funarrow} H - end -\end{alltt} -Notice the header of the \texttt{match} construct. -It expresses how the resulting type ``~\citecoq{Q y}~'' depends on -the \emph{type} of \texttt{p}. -Notice also that in the pattern introduced by the keyword \texttt{in}, -the parameter \texttt{a} in the type ``~\texttt{a = y}~'' must be -implicit, and replaced by a wildcard '\texttt{\_}'. - - -Therefore, case analysis on a proof of the equality $a=b$ -amounts to replacing all the occurrences of the term $b$ with the term -$a$ in the goal to be proven. Let us illustrate this through an -example: the transitivity property of this equality. -\begin{alltt} -Theorem trans : {\prodsym} n m p:nat, n=m \arrow{} m=p \arrow{} n=p. -Proof. - intros n m p eqnm. -\it{} - n : nat - m : nat - p : nat - eqnm : n = m - ============================ - m = p {\arrow} n = p -\tt{} case eqnm. -\it{} - n : nat - m : nat - p : nat - eqnm : n = m - ============================ - n = p {\arrow} n = p -\tt{} trivial. -Qed. -\end{alltt} - -%\noindent The case analysis on the hypothesis $H:n=m$ yields the -%tautological subgoal $n=p\rightarrow n=p$, that is directly proven by -%the tactic \texttt{Trivial}. - -\begin{exercise} -Prove the symmetry property of equality. -\end{exercise} - -Instead of using \texttt{case}, we can use the tactic -\texttt{rewrite} \refmancite{Section \ref{Rewrite}}. If $H$ is a proof -of $a=b$, then -``~\citecoq{rewrite $H$}~'' - performs a case analysis on a proof of $b=a$, obtained by applying a -symmetry theorem to $H$. This application of symmetry allows us to rewrite -the equality from left to right, which looks more natural. An optional -parameter (either \texttt{\arrow{}} or \texttt{$\leftarrow$}) can be used to precise -in which sense the equality must be rewritten. By default, -``~\texttt{rewrite} $H$~'' corresponds to ``~\texttt{rewrite \arrow{}} $H$~'' -\begin{alltt} -Lemma Rw : {\prodsym} x y: nat, y = y * x {\arrow} y * x * x = y. - intros x y e; do 2 rewrite <- e. -\it -1 subgoal - - x : nat - y : nat - e : y = y * x - ============================ - y = y -\tt - reflexivity. -Qed. -\end{alltt} - -Notice that, if $H:a=b$, then the tactic ``~\texttt{rewrite $H$}~'' - replaces \textsl{all} the -occurrences of $a$ by $b$. However, in certain situations we could be -interested in rewriting some of the occurrences, but not all of them. -This can be done using the tactic \texttt{pattern} \refmancite{Section -\ref{Pattern}}. Let us consider yet another example to -illustrate this. - -Let us start with some simple theorems of arithmetic; two of them -are already proven in the Standard Library, the last is left as an exercise. - -\begin{alltt} -\it -mult_1_l - : {\prodsym} n : nat, 1 * n = n - -mult_plus_distr_r - : {\prodsym} n m p : nat, (n + m) * p = n * p + m * p - -mult_distr_S : {\prodsym} n p : nat, n * p + p = (S n)* p. -\end{alltt} - -Let us now prove a simple result: - -\begin{alltt} -Lemma four_n : {\prodsym} n:nat, n+n+n+n = 4*n. -Proof. - intro n;rewrite <- (mult_1_l n). -\it - n : nat - ============================ - 1 * n + 1 * n + 1 * n + 1 * n = 4 * (1 * n) -\end{alltt} - -We can see that the \texttt{rewrite} tactic call replaced \emph{all} -the occurrences of \texttt{n} by the term ``~\citecoq{1 * n}~''. -If we want to do the rewriting ony on the leftmost occurrence of -\texttt{n}, we can mark this occurrence using the \texttt{pattern} -tactic: - - -\begin{alltt} - Undo. - intro n; pattern n at 1. - \it - n : nat - ============================ - (fun n0 : nat {\funarrow} n0 + n + n + n = 4 * n) n -\end{alltt} -Applying the tactic ``~\citecoq{pattern n at 1}~'' allowed us -to explicitly abstract the first occurrence of \texttt{n} from the -goal, putting this goal under the form ``~\citecoq{$Q$ n}~'', -thus pointing to \texttt{rewrite} the particular predicate on $n$ -that we search to prove. - - -\begin{alltt} - rewrite <- mult_1_l. -\it -1 subgoal - - n : nat - ============================ - 1 * n + n + n + n = 4 * n -\tt - repeat rewrite mult_distr_S. -\it - n : nat - ============================ - 4 * n = 4 * n -\tt - trivial. -Qed. -\end{alltt} - -\subsubsection{The Predicate $n {\leq} m$} - - -The last but one instance of the elimination schema that we will illustrate is -case analysis for the predicate $n {\leq} m$: - -Let $n$ and $p$ be terms of type \citecoq{nat}, and $Q$ a predicate -of type $\citecoq{nat}\arrow{}\Prop$. -If $H$ is a proof of ``~\texttt{n {\coqle} p}~'', -$H_0$ a proof of ``~\texttt{$Q$ n}~'' and -$H_S$ a proof of the statement ``~\citecoq{{\prodsym}m:nat, n {\coqle} m {\arrow} Q (S m)}~'', -then the term -\begin{alltt} -match H in (_ {\coqle} q) return (Q q) with - | le_n {\funarrow} H0 - | le_S m Hm {\funarrow} HS m Hm -end -\end{alltt} - is a proof term of ``~\citecoq{$Q$ $p$}~''. - - -The two patterns of this \texttt{match} construct describe -all possible forms of proofs of ``~\citecoq{n {\coqle} m}~'' (notice -again that the general parameter \texttt{n} is implicit in - the ``~\texttt{in \dots}~'' -clause and is absent from the match patterns. - - -Notice that the choice of introducing some of the arguments of the -predicate as being general parameters in its definition has -consequences on the rule of case analysis that is derived. In -particular, the type $Q$ of the object defined by the case expression -only depends on the indexes of the predicate, and not on the general -parameters. In the definition of the predicate $\leq$, the first -argument of this relation is a general parameter of the -definition. Hence, the predicate $Q$ to be proven only depends on the -second argument of the relation. In other words, the integer $n$ is -also a general parameter of the rule of case analysis. - -An example of an application of this rule is the following theorem, -showing that any integer greater or equal than $1$ is the successor of another -natural number: - -\begin{alltt} -Lemma predecessor_of_positive : - {\prodsym} n, 1 {\coqle} n {\arrow} {\exsym} p:nat, n = S p. -Proof. - intros n H;case H. -\it - n : nat - H : 1 {\coqle} n - ============================ - {\exsym} p : nat, 1 = S p -\tt - exists 0; trivial. -\it - - n : nat - H : 1 {\coqle} n - ============================ - {\prodsym} m : nat, 0 {\coqle} m {\arrow} {\exsym} p : nat, S m = S p -\tt - intros m _ . - exists m. - trivial. -Qed. -\end{alltt} - - -\subsubsection{Vectors} - -The \texttt{vector} polymorphic and dependent family of types will -give an idea of the most general scheme of pattern-matching. - -For instance, let us define a function for computing the tail of -any vector. Notice that we shall build a \emph{total} function, -by considering that the tail of an empty vector is this vector itself. -In that sense, it will be slightly different from the \texttt{Vtail} -function of the Standard Library, which is defined only for vectors -of type ``~\citecoq{vector $A$ (S $n$)}~''. - -The header of the function we want to build is the following: - -\begin{verbatim} -Definition Vtail_total - (A : Type) (n : nat) (v : vector A n) : vector A (pred n):= -\end{verbatim} - -Since the branches will not have the same type -(depending on the parameter \texttt{n}), -the body of this function is a dependent pattern matching on -\citecoq{v}. -So we will have : -\begin{verbatim} -match v in (vector _ n0) return (vector A (pred n0)) with -\end{verbatim} - -The first branch deals with the constructor \texttt{Vnil} and must -return a value in ``~\citecoq{vector A (pred 0)}~'', convertible -to ``~\citecoq{vector A 0}~''. So, we propose: -\begin{alltt} -| Vnil {\funarrow} Vnil A -\end{alltt} - -The second branch considers a vector in ``~\citecoq{vector A (S n0)}~'' -of the form -``~\citecoq{Vcons A n0 v0}~'', with ``~\citecoq{v0:vector A n0}~'', -and must return a value of type ``~\citecoq{vector A (pred (S n0))}~'', -which is convertible to ``~\citecoq{vector A n0}~''. -This second branch is thus : -\begin{alltt} -| Vcons _ n0 v0 {\funarrow} v0 -\end{alltt} - -Here is the full definition: - -\begin{alltt} -Definition Vtail_total - (A : Type) (n : nat) (v : vector A n) : vector A (pred n):= -match v in (vector _ n0) return (vector A (pred n0)) with -| Vnil {\funarrow} Vnil A -| Vcons _ n0 v0 {\funarrow} v0 -end. -\end{alltt} - - -\subsection{Case Analysis and Logical Paradoxes} - -In the previous section we have illustrated the general scheme for -generating the rule of case analysis associated to some recursive type -from the definition of the type. However, if the logical soundness is -to be preserved, certain restrictions to this schema are -necessary. This section provides a brief explanation of these -restrictions. - - -\subsubsection{The Positivity Condition} -\label{postypes} - -In order to make sense of recursive types as types closed under their -introduction rules, a constraint has to be imposed on the possible -forms of such rules. This constraint, known as the -\textsl{positivity condition}, is necessary to prevent the user from -naively introducing some recursive types which would open the door to -logical paradoxes. An example of such a dangerous type is the -``inductive type'' \citecoq{Lambda}, whose only constructor is -\citecoq{lambda} of type \citecoq{(Lambda\arrow False)\arrow Lambda}. - Following the pattern -given in Section \ref{CaseScheme}, the rule of (non dependent) case -analysis for \citecoq{Lambda} would be the following: - -\begin{center} -\snregla {\JM{Q}{\Prop}\;\;\;\;\; - \JM{p}{\texttt{Lambda}}\;\;\;\;\; - {h : {\texttt{Lambda}}\arrow\False\; \vdash\; t\,:\,Q}} - {\JM{\citecoq{match $p$ return $Q$ with lambda h {\funarrow} $t$ end}}{Q}} -\end{center} - -In order to avoid paradoxes, it is impossible to construct -the type \citecoq{Lambda} in {\coq}: - -\begin{alltt} -Inductive Lambda : Set := - lambda : (Lambda {\arrow} False) {\arrow} Lambda. -\it -Error: Non strictly positive occurrence of "Lambda" in - "(Lambda {\arrow} False) {\arrow} Lambda" -\end{alltt} - -In order to explain this danger, we -will declare some constants for simulating the construction of -\texttt{Lambda} as an inductive type. - -Let us open some section, and declare two variables, the first one for -\texttt{Lambda}, the other for the constructor \texttt{lambda}. - -\begin{alltt} -Section Paradox. -Variable Lambda : Set. -Variable lambda : (Lambda {\arrow} False) {\arrow}Lambda. -\end{alltt} - -Since \texttt{Lambda} is not a truely inductive type, we can't use -the \texttt{match} construct. Nevertheless, we can simulate it by a -variable \texttt{matchL} such that the term -``~\citecoq{matchL $l$ $Q$ (fun $h$ : Lambda {\arrow} False {\funarrow} $t$)}~'' -should be understood as -``~\citecoq{match $l$ return $Q$ with | lambda h {\funarrow} $t$)}~'' - - -\begin{alltt} -Variable matchL : Lambda {\arrow} - {\prodsym} Q:Prop, ((Lambda {\arrow}False) {\arrow} Q) {\arrow} - Q. -\end{alltt} - ->From these constants, it is possible to define application by case -analysis. Then, through auto-application, the well-known looping term -$(\lambda x.(x\;x)\;\lambda x.(x\;x))$ provides a proof of falsehood. - -\begin{alltt} -Definition application (f x: Lambda) :False := - matchL f False (fun h {\funarrow} h x). - -Definition Delta : Lambda := - lambda (fun x : Lambda {\funarrow} application x x). - -Definition loop : False := application Delta Delta. - -Theorem two_is_three : 2 = 3. -Proof. - elim loop. -Qed. - -End Paradox. -\end{alltt} - -\noindent This example can be seen as a formulation of Russell's -paradox in type theory associating $(\textsl{application}\;x\;x)$ to the -formula $x\not\in x$, and \textsl{Delta} to the set $\{ x \mid -x\not\in x\}$. If \texttt{matchL} would satisfy the reduction rule -associated to case analysis, that is, -$$ \citecoq{matchL (lambda $f$) $Q$ $h$} \Longrightarrow h\;f$$ -then the term \texttt{loop} -would compute into itself. This is not actually surprising, since the -proof of the logical soundness of {\coq} strongly lays on the property -that any well-typed term must terminate. Hence, non-termination is -usually a synonymous of inconsistency. - -%\paragraph{} In this case, the construction of a non-terminating -%program comes from the so-called \textsl{negative occurrence} of -%$\Lambda$ in the type of the constructor $\lambda$. In order to be -%admissible for {\coq}, all the occurrences of the recursive type in its -%own introduction rules must be positive, in the sense on the following -%definition: -% -%\begin{enumerate} -%\item $R$ is positive in $(R\;\vec{t})$; -%\item $R$ is positive in $(x: A)C$ if it does not -%occur in $A$ and $R$ is positive in $C$; -%\item if $P\equiv (\vec{x}:\vec{T})Q$, then $R$ is positive in $(P -%\rightarrow C)$ if $R$ does not occur in $\vec{T}$, $R$ is positive -%in $C$, and either -%\begin{enumerate} -%\item $Q\equiv (R\;\vec{q})$ or -%\item $Q\equiv (J\;\vec{t})$, \label{relax} -% where $J$ is a recursive type, and for any term $t_i$ either : -% \begin{enumerate} -% \item $R$ does not occur in $t_i$, or -% \item $t_i\equiv (z:\vec{Z})(R\;\vec{q})$, $R$ does not occur -% in $\vec{Z}$, $t_i$ instantiates a general -% parameter of $J$, and this parameter is positive in the -% arguments of the constructors of $J$. -% \end{enumerate} -%\end{enumerate} -%\end{enumerate} -%\noindent Those types obtained by erasing option (\ref{relax}) in the -%definition above are called \textsl{strictly positive} types. - - -\subsubsection*{Remark} In this case, the construction of a non-terminating -program comes from the so-called \textsl{negative occurrence} of -\texttt{Lambda} in the argument of the constructor \texttt{lambda}. - -The reader will find in the Reference Manual a complete formal -definition of the notions of \emph{positivity condition} and -\emph{strict positivity} that an inductive definition must satisfy. - - -%In order to be -%admissible for {\coq}, the type $R$ must be positive in the types of the -%arguments of its own introduction rules, in the sense on the following -%definition: - -%\textbf{La définition du manuel de référence est plus complexe: -%la recopier ou donner seulement des exemples? -%} -%\begin{enumerate} -%\item $R$ is positive in $T$ if $R$ does not occur in $T$; -%\item $R$ is positive in $(R\;\vec{t})$ if $R$ does not occur in $\vec{t}$; -%\item $R$ is positive in $(x:A)C$ if it does not -% occur in $A$ and $R$ is positive in $C$; -%\item $R$ is positive in $(J\;\vec{t})$, \label{relax} -% if $J$ is a recursive type, and for any term $t_i$ either : -% \begin{enumerate} -% \item $R$ does not occur in $t_i$, or -% \item $R$ is positive in $t_i$, $t_i$ instantiates a general -% parameter of $J$, and this parameter is positive in the -% arguments of the constructors of $J$. -% \end{enumerate} -%\end{enumerate} - -%\noindent When we can show that $R$ is positive without using the item -%(\ref{relax}) of the definition above, then we say that $R$ is -%\textsl{strictly positive}. - -%\textbf{Changer le discours sur les ordinaux} - -Notice that the positivity condition does not forbid us to -put functional recursive -arguments in the constructors. - -For instance, let us consider the type of infinitely branching trees, -with labels in \texttt{Z}. -\begin{alltt} -Require Import ZArith. - -Inductive itree : Set := -| ileaf : itree -| inode : Z {\arrow} (nat {\arrow} itree) {\arrow} itree. -\end{alltt} - -In this representation, the $i$-th child of a tree -represented by ``~\texttt{inode $z$ $s$}~'' is obtained by applying -the function $s$ to $i$. -The following definitions show how to construct a tree with a single -node, a tree of height 1 and a tree of height 2: - -\begin{alltt} -Definition isingle l := inode l (fun i {\funarrow} ileaf). - -Definition t1 := inode 0 (fun n {\funarrow} isingle (Z_of_nat n)). - -Definition t2 := - inode 0 - (fun n : nat {\funarrow} - inode (Z_of_nat n) - (fun p {\funarrow} isingle (Z_of_nat (n*p)))). -\end{alltt} - - -Let us define a preorder on infinitely branching trees. - In order to compare two non-leaf trees, -it is necessary to compare each of their children - without taking care of the order in which they -appear: - -\begin{alltt} -Inductive itree_le : itree{\arrow} itree {\arrow} Prop := - | le_leaf : {\prodsym} t, itree_le ileaf t - | le_node : {\prodsym} l l' s s', - Zle l l' {\arrow} - ({\prodsym} i, {\exsym} j:nat, itree_le (s i) (s' j)){\arrow} - itree_le (inode l s) (inode l' s'). - -\end{alltt} - -Notice that a call to the predicate \texttt{itree\_le} appears as -a general parameter of the inductive type \texttt{ex} (see Sect.\ref{ex-def}). -This kind of definition is accepted by {\coq}, but may lead to some -difficulties, since the induction principle automatically -generated by the system -is not the most appropriate (see chapter 14 of~\cite{coqart} for a detailed -explanation). - - -The following definition, obtained by -skolemising the -proposition \linebreak $\forall\, i,\exists\, j,(\texttt{itree\_le}\;(s\;i)\;(s'\;j))$ in -the type of \texttt{itree\_le}, does not present this problem: - - -\begin{alltt} -Inductive itree_le' : itree{\arrow} itree {\arrow} Prop := - | le_leaf' : {\prodsym} t, itree_le' ileaf t - | le_node' : {\prodsym} l l' s s' g, - Zle l l' {\arrow} - ({\prodsym} i, itree_le' (s i) (s' (g i))) {\arrow} - itree_le' (inode l s) (inode l' s'). - -\end{alltt} -\iffalse -\begin{alltt} -Lemma t1_le'_t2 : itree_le' t1 t2. -Proof. - unfold t1, t2. - constructor 2 with (fun i : nat {\funarrow} 2 * i). - auto with zarith. - unfold isingle; - intro i ; constructor 2 with (fun i :nat {\funarrow} i). - auto with zarith. - constructor . -Qed. -\end{alltt} -\fi - -%In general, strictly positive definitions are preferable to only -%positive ones. The reason is that it is sometimes difficult to derive -%structural induction combinators for the latter ones. Such combinators -%are automatically generated for strictly positive types, but not for -%the only positive ones. Nevertheless, sometimes non-strictly positive -%definitions provide a smarter or shorter way of declaring a recursive -%type. - -Another example is the type of trees - of unbounded width, in which a recursive subterm -\texttt{(ltree A)} instantiates the type of polymorphic lists: - -\begin{alltt} -Require Import List. - -Inductive ltree (A:Set) : Set := - lnode : A {\arrow} list (ltree A) {\arrow} ltree A. -\end{alltt} - -This declaration can be transformed -adding an extra type to the definition, as was done in Section -\ref{MutuallyDependent}. - - -\subsubsection{Impredicative Inductive Types} - -An inductive type $I$ inhabiting a universe $U$ is \textsl{predicative} -if the introduction rules of $I$ do not make a universal -quantification on a universe containing $U$. All the recursive types -previously introduced are examples of predicative types. An example of -an impredicative one is the following type: -%\textsl{exT}, the dependent product -%of a certain set (or proposition) $x$, and a proof of a property $P$ -%about $x$. - -%\begin{alltt} -%Print exT. -%\end{alltt} -%\textbf{ttention, EXT c'est ex!} -%\begin{alltt} -%Check (exists P:Prop, P {\arrow} not P). -%\end{alltt} - -%This type is useful for expressing existential quantification over -%types, like ``there exists a proposition $x$ such that $(P\;x)$'' -%---written $(\textsl{EXT}\; x:Prop \mid (P\;x))$ in {\coq}. However, - -\begin{alltt} -Inductive prop : Prop := - prop_intro : Prop {\arrow} prop. -\end{alltt} - -Notice -that the constructor of this type can be used to inject any -proposition --even itself!-- into the type. - -\begin{alltt} -Check (prop_intro prop).\it -prop_intro prop - : prop -\end{alltt} - -A careless use of such a -self-contained objects may lead to a variant of Burali-Forti's -paradox. The construction of Burali-Forti's paradox is more -complicated than Russel's one, so we will not describe it here, and -point the interested reader to \cite{Bar98,Coq86}. - - -Another example is the second order existential quantifier for propositions: - -\begin{alltt} -Inductive ex_Prop (P : Prop {\arrow} Prop) : Prop := - exP_intro : {\prodsym} X : Prop, P X {\arrow} ex_Prop P. -\end{alltt} - -%\begin{alltt} -%(* -%Check (match prop_inject with (prop_intro p _) {\funarrow} p end). - -%Error: Incorrect elimination of "prop_inject" in the inductive type -% ex -%The elimination predicate ""fun _ : prop {\funarrow} Prop" has type -% "prop {\arrow} Type" -%It should be one of : -% "Prop" - -%Elimination of an inductive object of sort : "Prop" -%is not allowed on a predicate in sort : "Type" -%because non-informative objects may not construct informative ones. - -%*) -%Print prop_inject. - -%(* -%prop_inject = -%prop_inject = prop_intro prop (fun H : prop {\funarrow} H) -% : prop -%*) -%\end{alltt} - -% \textbf{Et par ça? -%} - -Notice that predicativity on sort \citecoq{Set} forbids us to build -the following definitions. - - -\begin{alltt} -Inductive aSet : Set := - aSet_intro: Set {\arrow} aSet. - -\it{}User error: Large non-propositional inductive types must be in Type -\tt -Inductive ex_Set (P : Set {\arrow} Prop) : Set := - exS_intro : {\prodsym} X : Set, P X {\arrow} ex_Set P. - -\it{}User error: Large non-propositional inductive types must be in Type -\end{alltt} - -Nevertheless, one can define types like \citecoq{aSet} and \citecoq{ex\_Set}, as inhabitants of \citecoq{Type}. - -\begin{alltt} -Inductive ex_Set (P : Set {\arrow} Prop) : Type := - exS_intro : {\prodsym} X : Set, P X {\arrow} ex_Set P. -\end{alltt} - -In the following example, the inductive type \texttt{typ} can be defined, -but the term associated with the interactive Definition of -\citecoq{typ\_inject} is incompatible with {\coq}'s hierarchy of universes: - - -\begin{alltt} -Inductive typ : Type := - typ_intro : Type {\arrow} typ. - -Definition typ_inject: typ. - split; exact typ. -\it Proof completed - -\tt{}Defined. -\it Error: Universe Inconsistency. -\tt -Abort. -\end{alltt} - -One possible way of avoiding this new source of paradoxes is to -restrict the kind of eliminations by case analysis that can be done on -impredicative types. In particular, projections on those universes -equal or bigger than the one inhabited by the impredicative type must -be forbidden \cite{Coq86}. A consequence of this restriction is that it -is not possible to define the first projection of the type -``~\citecoq{ex\_Prop $P$}~'': -\begin{alltt} -Check (fun (P:Prop{\arrow}Prop)(p: ex_Prop P) {\funarrow} - match p with exP_intro X HX {\funarrow} X end). -\it -Error: -Incorrect elimination of "p" in the inductive type -"ex_Prop", the return type has sort "Type" while it should be -"Prop" - -Elimination of an inductive object of sort "Prop" -is not allowed on a predicate in sort "Type" -because proofs can be eliminated only to build proofs. -\end{alltt} - -%In order to explain why, let us consider for example the following -%impredicative type \texttt{ALambda}. -%\begin{alltt} -%Inductive ALambda : Set := -% alambda : (A:Set)(A\arrow{}False)\arrow{}ALambda. -% -%Definition Lambda : Set := ALambda. -%Definition lambda : (ALambda\arrow{}False)\arrow{}ALambda := (alambda ALambda). -%Lemma CaseAL : (Q:Prop)ALambda\arrow{}((ALambda\arrow{}False)\arrow{}Q)\arrow{}Q. -%\end{alltt} -% -%This type contains all the elements of the dangerous type $\Lambda$ -%described at the beginning of this section. Try to construct the -%non-ending term $(\Delta\;\Delta)$ as an object of -%\texttt{ALambda}. Why is it not possible? - -\subsubsection{Extraction Constraints} - -There is a final constraint on case analysis that is not motivated by -the potential introduction of paradoxes, but for compatibility reasons -with {\coq}'s extraction mechanism \refmancite{Appendix -\ref{CamlHaskellExtraction}}. This mechanism is based on the -classification of basic types into the universe $\Set$ of sets and the -universe $\Prop$ of propositions. The objects of a type in the -universe $\Set$ are considered as relevant for computation -purposes. The objects of a type in $\Prop$ are considered just as -formalised comments, not necessary for execution. The extraction -mechanism consists in erasing such formal comments in order to obtain -an executable program. Hence, in general, it is not possible to define -an object in a set (that should be kept by the extraction mechanism) -by case analysis of a proof (which will be thrown away). - -Nevertheless, this general rule has an exception which is important in -practice: if the definition proceeds by case analysis on a proof of a -\textsl{singleton proposition} or an empty type (\emph{e.g.} \texttt{False}), - then it is allowed. A singleton -proposition is a non-recursive proposition with a single constructor -$c$, all whose arguments are proofs. For example, the propositional -equality and the conjunction of two propositions are examples of -singleton propositions. - -%From the point of view of the extraction -%mechanism, such types are isomorphic to a type containing a single -%object $c$, so a definition $\Case{x}{c \Rightarrow b}$ is -%directly replaced by $b$ as an extra optimisation. - -\subsubsection{Strong Case Analysis on Proofs} - -One could consider allowing - to define a proposition $Q$ by case -analysis on the proofs of another recursive proposition $R$. As we -will see in Section \ref{Discrimination}, this would enable one to prove that -different introduction rules of $R$ construct different -objects. However, this property would be in contradiction with the principle -of excluded middle of classical logic, because this principle entails -that the proofs of a proposition cannot be distinguished. This -principle is not provable in {\coq}, but it is frequently introduced by -the users as an axiom, for reasoning in classical logic. For this -reason, the definition of propositions by case analysis on proofs is - not allowed in {\coq}. - -\begin{alltt} - -Definition comes_from_the_left (P Q:Prop)(H:P{\coqor}Q): Prop := - match H with - | or_introl p {\funarrow} True - | or_intror q {\funarrow} False - end. -\it -Error: -Incorrect elimination of "H" in the inductive type -"or", the return type has sort "Type" while it should be -"Prop" - -Elimination of an inductive object of sort "Prop" -is not allowed on a predicate in sort "Type" -because proofs can be eliminated only to build proofs. - -\end{alltt} - -On the other hand, if we replace the proposition $P {\coqor} Q$ with -the informative type $\{P\}+\{Q\}$, the elimination is accepted: - -\begin{alltt} -Definition comes_from_the_left_sumbool - (P Q:Prop)(x:\{P\} + \{Q\}): Prop := - match x with - | left p {\funarrow} True - | right q {\funarrow} False - end. -\end{alltt} - - -\subsubsection{Summary of Constraints} - -To end with this section, the following table summarizes which -universe $U_1$ may inhabit an object of type $Q$ defined by case -analysis on $x:R$, depending on the universe $U_2$ inhabited by the -inductive types $R$.\footnote{In the box indexed by $U_1=\citecoq{Type}$ -and $U_2=\citecoq{Set}$, the answer ``yes'' takes into account the -predicativity of sort \citecoq{Set}. If you are working with the -option ``impredicative-set'', you must put in this box the -condition ``if $R$ is predicative''.} - - -\begin{center} -%%% displease hevea less by using * in multirow rather than \LL -\renewcommand{\multirowsetup}{\centering} -%\newlength{\LL} -%\settowidth{\LL}{$x : R : U_2$} -\begin{tabular}{|c|c|c|c|c|} -\hline -\multirow{5}*{$x : R : U_2$} & -\multicolumn{4}{|c|}{$Q : U_1$}\\ -\hline -& &\textsl{Set} & \textsl{Prop} & \textsl{Type}\\ -\cline{2-5} -&\textsl{Set} & yes & yes & yes\\ -\cline{2-5} -&\textsl{Prop} & if $R$ singleton & yes & no\\ -\cline{2-5} -&\textsl{Type} & yes & yes & yes\\ -\hline -\end{tabular} -\end{center} - -\section{Some Proof Techniques Based on Case Analysis} -\label{CaseTechniques} - -In this section we illustrate the use of case analysis as a proof -principle, explaining the proof techniques behind three very useful -{\coq} tactics, called \texttt{discriminate}, \texttt{injection} and -\texttt{inversion}. - -\subsection{Discrimination of introduction rules} -\label{Discrimination} - -In the informal semantics of recursive types described in Section -\ref{Introduction} it was said that each of the introduction rules of a -recursive type is considered as being different from all the others. -It is possible to capture this fact inside the logical system using -the propositional equality. We take as example the following theorem, -stating that \textsl{O} constructs a natural number different -from any of those constructed with \texttt{S}. - -\begin{alltt} -Theorem S_is_not_O : {\prodsym} n, S n {\coqdiff} 0. -\end{alltt} - -In order to prove this theorem, we first define a proposition by case -analysis on natural numbers, so that the proposition is true for {\Z} -and false for any natural number constructed with {\SUCC}. This uses -the empty and singleton type introduced in Sections \ref{Introduction}. - -\begin{alltt} -Definition Is_zero (x:nat):= match x with - | 0 {\funarrow} True - | _ {\funarrow} False - end. -\end{alltt} - -\noindent Then, we prove the following lemma: - -\begin{alltt} -Lemma O_is_zero : {\prodsym} m, m = 0 {\arrow} Is_zero m. -Proof. - intros m H; subst m. -\it{} -================ - Is_zero 0 -\tt{} -simpl;trivial. -Qed. -\end{alltt} - -\noindent Finally, the proof of \texttt{S\_is\_not\_O} follows by the -application of the previous lemma to $S\;n$. - - -\begin{alltt} - - red; intros n Hn. - \it{} - n : nat - Hn : S n = 0 - ============================ - False \tt - - apply O_is_zero with (m := S n). - assumption. -Qed. -\end{alltt} - - -The tactic \texttt{discriminate} \refmancite{Section \ref{Discriminate}} is -a special-purpose tactic for proving disequalities between two -elements of a recursive type introduced by different constructors. It -generalizes the proof method described here for natural numbers to any -[co]-inductive type. This tactic is also capable of proving disequalities -where the difference is not in the constructors at the head of the -terms, but deeper inside them. For example, it can be used to prove -the following theorem: - -\begin{alltt} -Theorem disc2 : {\prodsym} n, S (S n) {\coqdiff} 1. -Proof. - intros n Hn; discriminate. -Qed. -\end{alltt} - -When there is an assumption $H$ in the context stating a false -equality $t_1=t_2$, \texttt{discriminate} solves the goal by first -proving $(t_1\not =t_2)$ and then reasoning by absurdity with respect -to $H$: - -\begin{alltt} -Theorem disc3 : {\prodsym} n, S (S n) = 0 {\arrow} {\prodsym} Q:Prop, Q. -Proof. - intros n Hn Q. - discriminate. -Qed. -\end{alltt} - -\noindent In this case, the proof proceeds by absurdity with respect -to the false equality assumed, whose negation is proved by -discrimination. - -\subsection{Injectiveness of introduction rules} - -Another useful property about recursive types is the -\textsl{injectiveness} of introduction rules, i.e., that whenever two -objects were built using the same introduction rule, then this rule -should have been applied to the same element. This can be stated -formally using the propositional equality: - -\begin{alltt} -Theorem inj : {\prodsym} n m, S n = S m {\arrow} n = m. -Proof. -\end{alltt} - -\noindent This theorem is just a corollary of a lemma about the -predecessor function: - -\begin{alltt} - Lemma inj_pred : {\prodsym} n m, n = m {\arrow} pred n = pred m. - Proof. - intros n m eq_n_m. - rewrite eq_n_m. - trivial. - Qed. -\end{alltt} -\noindent Once this lemma is proven, the theorem follows directly -from it: -\begin{alltt} - intros n m eq_Sn_Sm. - apply inj_pred with (n:= S n) (m := S m); assumption. -Qed. -\end{alltt} - -This proof method is implemented by the tactic \texttt{injection} -\refmancite{Section \ref{injection}}. This tactic is applied to -a term $t$ of type ``~$c\;{t_1}\;\dots\;t_n = c\;t'_1\;\dots\;t'_n$~'', where $c$ is some constructor of -an inductive type. The tactic \texttt{injection} is applied as deep as -possible to derive the equality of all pairs of subterms of $t_i$ and $t'_i$ -placed in the same position. All these equalities are put as antecedents -of the current goal. - - - -Like \texttt{discriminate}, the tactic \citecoq{injection} -can be also applied if $x$ does not -occur in a direct sub-term, but somewhere deeper inside it. Its -application may leave some trivial goals that can be easily solved -using the tactic \texttt{trivial}. - -\begin{alltt} - - Lemma list_inject : {\prodsym} (A:Type)(a b :A)(l l':list A), - a :: b :: l = b :: a :: l' {\arrow} a = b {\coqand} l = l'. -Proof. - intros A a b l l' e. - - -\it - e : a :: b :: l = b :: a :: l' - ============================ - a = b {\coqand} l = l' -\tt - injection e. -\it - ============================ - l = l' {\arrow} b = a {\arrow} a = b {\arrow} a = b {\coqand} l = l' - -\tt{} auto. -Qed. -\end{alltt} - -\subsection{Inversion Techniques}\label{inversion} - -In section \ref{DependentCase}, we motivated the rule of dependent case -analysis as a way of internalizing the informal equalities $n=O$ and -$n=\SUCC\;p$ associated to each case. This internalisation -consisted in instantiating $n$ with the corresponding term in the type -of each branch. However, sometimes it could be better to internalise -these equalities as extra hypotheses --for example, in order to use -the tactics \texttt{rewrite}, \texttt{discriminate} or -\texttt{injection} presented in the previous sections. This is -frequently the case when the element analysed is denoted by a term -which is not a variable, or when it is an object of a particular -instance of a recursive family of types. Consider for example the -following theorem: - -\begin{alltt} -Theorem not_le_Sn_0 : {\prodsym} n:nat, ~ (S n {\coqle} 0). -\end{alltt} - -\noindent Intuitively, this theorem should follow by case analysis on -the hypothesis $H:(S\;n\;\leq\;\Z)$, because no introduction rule allows -to instantiate the arguments of \citecoq{le} with respectively a successor -and zero. However, there -is no way of capturing this with the typing rule for case analysis -presented in section \ref{Introduction}, because it does not take into -account what particular instance of the family the type of $H$ is. -Let us try it: -\begin{alltt} -Proof. - red; intros n H; case H. -\it 2 subgoals - - n : nat - H : S n {\coqle} 0 - ============================ - False - -subgoal 2 is: - {\prodsym} m : nat, S n {\coqle} m {\arrow} False -\tt -Undo. -\end{alltt} - -\noindent What is necessary here is to make available the equalities -``~$\SUCC\;n = \Z$~'' and ``~$\SUCC\;m = \Z$~'' - as extra hypotheses of the -branches, so that the goal can be solved using the -\texttt{Discriminate} tactic. In order to obtain the desired -equalities as hypotheses, let us prove an auxiliary lemma, that our -theorem is a corollary of: - -\begin{alltt} - Lemma not_le_Sn_0_with_constraints : - {\prodsym} n p , S n {\coqle} p {\arrow} p = 0 {\arrow} False. - Proof. - intros n p H; case H . -\it -2 subgoals - - n : nat - p : nat - H : S n {\coqle} p - ============================ - S n = 0 {\arrow} False - -subgoal 2 is: - {\prodsym} m : nat, S n {\coqle} m {\arrow} S m = 0 {\arrow} False -\tt - intros;discriminate. - intros;discriminate. -Qed. -\end{alltt} -\noindent Our main theorem can now be solved by an application of this lemma: -\begin{alltt} -Show. -\it -2 subgoals - - n : nat - p : nat - H : S n {\coqle} p - ============================ - S n = 0 {\arrow} False - -subgoal 2 is: - {\prodsym} m : nat, S n {\coqle} m {\arrow} S m = 0 {\arrow} False -\tt - eapply not_le_Sn_0_with_constraints; eauto. -Qed. -\end{alltt} - - -The general method to address such situations consists in changing the -goal to be proven into an implication, introducing as preconditions -the equalities needed to eliminate the cases that make no -sense. This proof technique is implemented by the tactic -\texttt{inversion} \refmancite{Section \ref{Inversion}}. In order -to prove a goal $G\;\vec{q}$ from an object of type $R\;\vec{t}$, -this tactic automatically generates a lemma $\forall, \vec{x}. -(R\;\vec{x}) \rightarrow \vec{x}=\vec{t}\rightarrow \vec{B}\rightarrow -(G\;\vec{q})$, where the list of propositions $\vec{B}$ correspond to -the subgoals that cannot be directly proven using -\texttt{discriminate}. This lemma can either be saved for later -use, or generated interactively. In this latter case, the subgoals -yielded by the tactic are the hypotheses $\vec{B}$ of the lemma. If the -lemma has been stored, then the tactic \linebreak - ``~\citecoq{inversion \dots using \dots}~'' can be -used to apply it. - -Let us show both techniques on our previous example: - -\subsubsection{Interactive mode} - -\begin{alltt} -Theorem not_le_Sn_0' : {\prodsym} n:nat, ~ (S n {\coqle} 0). -Proof. - red; intros n H ; inversion H. -Qed. -\end{alltt} - - -\subsubsection{Static mode} - -\begin{alltt} - -Derive Inversion le_Sn_0_inv with ({\prodsym} n :nat, S n {\coqle} 0). -Theorem le_Sn_0'' : {\prodsym} n p : nat, ~ S n {\coqle} 0 . -Proof. - intros n p H; - inversion H using le_Sn_0_inv. -Qed. -\end{alltt} - - -In the example above, all the cases are solved using discriminate, so -there remains no subgoal to be proven (i.e. the list $\vec{B}$ is -empty). Let us present a second example, where this list is not empty: - - -\begin{alltt} -TTheorem le_reverse_rules : - {\prodsym} n m:nat, n {\coqle} m {\arrow} - n = m {\coqor} - {\exsym} p, n {\coqle} p {\coqand} m = S p. -Proof. - intros n m H; inversion H. -\it -2 subgoals - - - - - n : nat - m : nat - H : n {\coqle} m - H0 : n = m - ============================ - m = m {\coqor} ({\exsym} p : nat, m {\coqle} p {\coqand} m = S p) - -subgoal 2 is: - n = S m0 {\coqor} ({\exsym} p : nat, n {\coqle} p {\coqand} S m0 = S p) -\tt - left;trivial. - right; exists m0; split; trivial. -\it -Proof completed -\end{alltt} - -This example shows how this tactic can be used to ``reverse'' the -introduction rules of a recursive type, deriving the possible premises -that could lead to prove a given instance of the predicate. This is -why these tactics are called \texttt{inversion} tactics: they go back -from conclusions to premises. - -The hypotheses corresponding to the propositional equalities are not -needed in this example, since the tactic does the necessary rewriting -to solve the subgoals. When the equalities are no longer needed after -the inversion, it is better to use the tactic -\texttt{Inversion\_clear}. This variant of the tactic clears from the -context all the equalities introduced. - -\begin{alltt} -Restart. - intros n m H; inversion_clear H. -\it -\it - - n : nat - m : nat - ============================ - m = m {\coqor} ({\exsym} p : nat, m {\coqle} p {\coqand} m = S p) -\tt - left;trivial. -\it - n : nat - m : nat - m0 : nat - H0 : n {\coqle} m0 - ============================ - n = S m0 {\coqor} ({\exsym} p : nat, n {\coqle} p {\coqand} S m0 = S p) -\tt - right; exists m0; split; trivial. -Qed. -\end{alltt} - - -%This proof technique works in most of the cases, but not always. In -%particular, it could not if the list $\vec{t}$ contains a term $t_j$ -%whose type $T$ depends on a previous term $t_i$, with $i<j$. Remark -%that if this is the case, the propositional equality $x_j=t_j$ is not -%well-typed, since $x_j:T(x_i)$ but $t_j:T(t_i)$, and both types are -%not convertible (otherwise, the problem could be solved using the -%tactic \texttt{Case}). - - - -\begin{exercise} -Consider the following language of arithmetic expression, and -its operational semantics, described by a set of rewriting rules. -%\textbf{J'ai enlevé une règle de commutativité de l'addition qui -%me paraissait bizarre du point de vue de la sémantique opérationnelle} - -\begin{alltt} -Inductive ArithExp : Set := - | Zero : ArithExp - | Succ : ArithExp {\arrow} ArithExp - | Plus : ArithExp {\arrow} ArithExp {\arrow} ArithExp. - -Inductive RewriteRel : ArithExp {\arrow} ArithExp {\arrow} Prop := - | RewSucc : {\prodsym} e1 e2 :ArithExp, - RewriteRel e1 e2 {\arrow} - RewriteRel (Succ e1) (Succ e2) - | RewPlus0 : {\prodsym} e:ArithExp, - RewriteRel (Plus Zero e) e - | RewPlusS : {\prodsym} e1 e2:ArithExp, - RewriteRel e1 e2 {\arrow} - RewriteRel (Plus (Succ e1) e2) - (Succ (Plus e1 e2)). - -\end{alltt} -\begin{enumerate} -\item Prove that \texttt{Zero} cannot be rewritten any further. -\item Prove that an expression of the form ``~$\texttt{Succ}\;e$~'' is always -rewritten -into an expression of the same form. -\end{enumerate} -\end{exercise} - -%Theorem zeroNotCompute : (e:ArithExp)~(RewriteRel Zero e). -%Intro e. -%Red. -%Intro H. -%Inversion_clear H. -%Defined. -%Theorem evalPlus : -% (e1,e2:ArithExp) -% (RewriteRel (Succ e1) e2)\arrow{}(EX e3 : ArithExp | e2=(Succ e3)). -%Intros e1 e2 H. -%Inversion_clear H. -%Exists e3;Reflexivity. -%Qed. - - -\section{Inductive Types and Structural Induction} -\label{StructuralInduction} - -Elements of inductive types are well-founded with -respect to the structural order induced by the constructors of the -type. In addition to case analysis, this extra hypothesis about -well-foundedness justifies a stronger elimination rule for them, called -\textsl{structural induction}. This form of elimination consists in -defining a value ``~$f\;x$~'' from some element $x$ of the inductive type -$I$, assuming that values have been already associated in the same way -to the sub-parts of $x$ of type $I$. - - -Definitions by structural induction are expressed through the -\texttt{Fixpoint} command \refmancite{Section -\ref{Fixpoint}}. This command is quite close to the -\texttt{let-rec} construction of functional programming languages. -For example, the following definition introduces the addition of two -natural numbers (already defined in the Standard Library:) - -\begin{alltt} -Fixpoint plus (n p:nat) \{struct n\} : nat := - match n with - | 0 {\funarrow} p - | S m {\funarrow} S (plus m p) - end. -\end{alltt} - -The definition is by structural induction on the first argument of the -function. This is indicated by the ``~\citecoq{\{struct n\}}~'' -directive in the function's header\footnote{This directive is optional -in the case of a function of a single argument}. - In -order to be accepted, the definition must satisfy a syntactical -condition, called the \textsl{guardedness condition}. Roughly -speaking, this condition constrains the arguments of a recursive call -to be pattern variables, issued from a case analysis of the formal -argument of the function pointed by the \texttt{struct} directive. - In the case of the -function \texttt{plus}, the argument \texttt{m} in the recursive call is a -pattern variable issued from a case analysis of \texttt{n}. Therefore, the -definition is accepted. - -Notice that we could have defined the addition with structural induction -on its second argument: -\begin{alltt} -Fixpoint plus' (n p:nat) \{struct p\} : nat := - match p with - | 0 {\funarrow} n - | S q {\funarrow} S (plus' n q) - end. -\end{alltt} - -%This notation is useful when defining a function whose decreasing -%argument has a dependent type. As an example, consider the following -%recursivly defined proof of the theorem -%$(n,m:\texttt{nat})n<m \rightarrow (S\;n)<(S\;m)$: -%\begin{alltt} -%Fixpoint lt_n_S [n,m:nat;p:(lt n m)] : (lt (S n) (S m)) := -% <[n0:nat](lt (S n) (S n0))> -% Cases p of -% lt_intro1 {\funarrow} (lt_intro1 (S n)) -% | (lt_intro2 m1 p2) {\funarrow} (lt_intro2 (S n) (S m1) (lt_n_S n m1 p2)) -% end. -%\end{alltt} - -%The guardedness condition must be satisfied only by the last argument -%of the enclosed list. For example, the following declaration is an -%alternative way of defining addition: - -%\begin{alltt} -%Reset add. -%Fixpoint add [n:nat] : nat\arrow{}nat := -% Cases n of -% O {\funarrow} [x:nat]x -% | (S m) {\funarrow} [x:nat](add m (S x)) -% end. -%\end{alltt} - -In the following definition of addition, -the second argument of {\tt plus{'}{'}} grows at each -recursive call. However, as the first one always decreases, the -definition is sound. -\begin{alltt} -Fixpoint plus'' (n p:nat) \{struct n\} : nat := - match n with - | 0 {\funarrow} p - | S m {\funarrow} plus'' m (S p) - end. -\end{alltt} - - Moreover, the argument in the recursive call -could be a deeper component of $n$. This is the case in the following -definition of a boolean function determining whether a number is even -or odd: - -\begin{alltt} -Fixpoint even_test (n:nat) : bool := - match n - with 0 {\funarrow} true - | 1 {\funarrow} false - | S (S p) {\funarrow} even_test p - end. -\end{alltt} - -Mutually dependent definitions by structural induction are also -allowed. For example, the previous function \textsl{even} could alternatively -be defined using an auxiliary function \textsl{odd}: - -\begin{alltt} -Reset even_test. - - - -Fixpoint even_test (n:nat) : bool := - match n - with - | 0 {\funarrow} true - | S p {\funarrow} odd_test p - end -with odd_test (n:nat) : bool := - match n - with - | 0 {\funarrow} false - | S p {\funarrow} even_test p - end. -\end{alltt} - -%\begin{exercise} -%Define a function by structural induction that computes the number of -%nodes of a tree structure defined in page \pageref{Forest}. -%\end{exercise} - -Definitions by structural induction are computed - only when they are applied, and the decreasing argument -is a term having a constructor at the head. We can check this using -the \texttt{Eval} command, which computes the normal form of a well -typed term. - -\begin{alltt} -Eval simpl in even_test. -\it - = even_test - : nat {\arrow} bool -\tt -Eval simpl in (fun x : nat {\funarrow} even x). -\it - = fun x : nat {\funarrow} even x - : nat {\arrow} Prop -\tt -Eval simpl in (fun x : nat => plus 5 x). -\it - = fun x : nat {\funarrow} S (S (S (S (S x)))) - -\tt -Eval simpl in (fun x : nat {\funarrow} even_test (plus 5 x)). -\it - = fun x : nat {\funarrow} odd_test x - : nat {\arrow} bool -\tt -Eval simpl in (fun x : nat {\funarrow} even_test (plus x 5)). -\it - = fun x : nat {\funarrow} even_test (x + 5) - : nat {\arrow} bool -\end{alltt} - - -%\begin{exercise} -%Prove that the second definition of even satisfies the following -%theorem: -%\begin{verbatim} -%Theorem unfold_even : -% (x:nat) -% (even x)= (Cases x of -% O {\funarrow} true -% | (S O) {\funarrow} false -% | (S (S m)) {\funarrow} (even m) -% end). -%\end{verbatim} -%\end{exercise} - -\subsection{Proofs by Structural Induction} - -The principle of structural induction can be also used in order to -define proofs, that is, to prove theorems. Let us call an -\textsl{elimination combinator} any function that, given a predicate -$P$, defines a proof of ``~$P\;x$~'' by structural induction on $x$. In -{\coq}, the principle of proof by induction on natural numbers is a -particular case of an elimination combinator. The definition of this -combinator depends on three general parameters: the predicate to be -proven, the base case, and the inductive step: - -\begin{alltt} -Section Principle_of_Induction. -Variable P : nat {\arrow} Prop. -Hypothesis base_case : P 0. -Hypothesis inductive_step : {\prodsym} n:nat, P n {\arrow} P (S n). -Fixpoint nat_ind (n:nat) : (P n) := - match n return P n with - | 0 {\funarrow} base_case - | S m {\funarrow} inductive_step m (nat_ind m) - end. - -End Principle_of_Induction. -\end{alltt} - -As this proof principle is used very often, {\coq} automatically generates it -when an inductive type is introduced. Similar principles -\texttt{nat\_rec} and \texttt{nat\_rect} for defining objects in the -universes $\Set$ and $\Type$ are also automatically generated -\footnote{In fact, whenever possible, {\coq} generates the -principle \texttt{$I$\_rect}, then derives from it the -weaker principles \texttt{$I$\_ind} and \texttt{$I$\_rec}. -If some principle has to be defined by hand, the user may try -to build \texttt{$I$\_rect} (if possible). Thanks to {\coq}'s conversion -rule, this principle can be used directly to build proofs and/or -programs.}. The -command \texttt{Scheme} \refmancite{Section \ref{Scheme}} can be -used to generate an elimination combinator from certain parameters, -like the universe that the defined objects must inhabit, whether the -case analysis in the definitions must be dependent or not, etc. For -example, it can be used to generate an elimination combinator for -reasoning on even natural numbers from the mutually dependent -predicates introduced in page \pageref{Even}. We do not display the -combinators here by lack of space, but you can see them using the -\texttt{Print} command. - -\begin{alltt} -Scheme Even_induction := Minimality for even Sort Prop -with Odd_induction := Minimality for odd Sort Prop. -\end{alltt} - -\begin{alltt} -Theorem even_plus_four : {\prodsym} n:nat, even n {\arrow} even (4+n). -Proof. - intros n H. - elim H using Even_induction with (P0 := fun n {\funarrow} odd (4+n)); - simpl;repeat constructor;assumption. -Qed. -\end{alltt} - -Another example of an elimination combinator is the principle -of double induction on natural numbers, introduced by the following -definition: - -\begin{alltt} -Section Principle_of_Double_Induction. -Variable P : nat {\arrow} nat {\arrow}Prop. -Hypothesis base_case1 : {\prodsym} m:nat, P 0 m. -Hypothesis base_case2 : {\prodsym} n:nat, P (S n) 0. -Hypothesis inductive_step : {\prodsym} n m:nat, P n m {\arrow} - \,\, P (S n) (S m). - -Fixpoint nat_double_ind (n m:nat)\{struct n\} : P n m := - match n, m return P n m with - | 0 , x {\funarrow} base_case1 x - | (S x), 0 {\funarrow} base_case2 x - | (S x), (S y) {\funarrow} inductive_step x y (nat_double_ind x y) - end. -End Principle_of_Double_Induction. -\end{alltt} - -Changing the type of $P$ into $\nat\rightarrow\nat\rightarrow\Type$, -another combinator for constructing -(certified) programs, \texttt{nat\_double\_rect}, can be defined in exactly the same way. -This definition is left as an exercise.\label{natdoublerect} - -\iffalse -\begin{alltt} -Section Principle_of_Double_Recursion. -Variable P : nat {\arrow} nat {\arrow} Type. -Hypothesis base_case1 : {\prodsym} x:nat, P 0 x. -Hypothesis base_case2 : {\prodsym} x:nat, P (S x) 0. -Hypothesis inductive_step : {\prodsym} n m:nat, P n m {\arrow} P (S n) (S m). -Fixpoint nat_double_rect (n m:nat)\{struct n\} : P n m := - match n, m return P n m with - 0 , x {\funarrow} base_case1 x - | (S x), 0 {\funarrow} base_case2 x - | (S x), (S y) {\funarrow} inductive_step x y (nat_double_rect x y) - end. -End Principle_of_Double_Recursion. -\end{alltt} -\fi -For instance the function computing the minimum of two natural -numbers can be defined in the following way: - -\begin{alltt} -Definition min : nat {\arrow} nat {\arrow} nat := - nat_double_rect (fun (x y:nat) {\funarrow} nat) - (fun (x:nat) {\funarrow} 0) - (fun (y:nat) {\funarrow} 0) - (fun (x y r:nat) {\funarrow} S r). -Eval compute in (min 5 8). -\it -= 5 : nat -\end{alltt} - - -%\begin{exercise} -% -%Define the combinator \texttt{nat\_double\_rec}, and apply it -%to give another definition of \citecoq{le\_lt\_dec} (using the theorems -%of the \texttt{Arith} library). -%\end{exercise} - -\subsection{Using Elimination Combinators.} -The tactic \texttt{apply} can be used to apply one of these proof -principles during the development of a proof. - -\begin{alltt} -Lemma not_circular : {\prodsym} n:nat, n {\coqdiff} S n. -Proof. - intro n. - apply nat_ind with (P:= fun n {\funarrow} n {\coqdiff} S n). -\it - - - -2 subgoals - - n : nat - ============================ - 0 {\coqdiff} 1 - - -subgoal 2 is: - {\prodsym} n0 : nat, n0 {\coqdiff} S n0 {\arrow} S n0 {\coqdiff} S (S n0) - -\tt - discriminate. - red; intros n0 Hn0 eqn0Sn0;injection eqn0Sn0;trivial. -Qed. -\end{alltt} - -The tactic \texttt{elim} \refmancite{Section \ref{Elim}} is a -refinement of \texttt{apply}, specially designed for the application -of elimination combinators. If $t$ is an object of an inductive type -$I$, then ``~\citecoq{elim $t$}~'' tries to find an abstraction $P$ of the -current goal $G$ such that $(P\;t)\equiv G$. Then it solves the goal -applying ``~$I\texttt{\_ind}\;P$~'', where $I$\texttt{\_ind} is the -combinator associated to $I$. The different cases of the induction -then appear as subgoals that remain to be solved. -In the previous proof, the tactic call ``~\citecoq{apply nat\_ind with (P:= fun n {\funarrow} n {\coqdiff} S n)}~'' can simply be replaced with ``~\citecoq{elim n}~''. - -The option ``~\citecoq{\texttt{elim} $t$ \texttt{using} $C$}~'' - allows to use a -derived combinator $C$ instead of the default one. Consider the -following theorem, stating that equality is decidable on natural -numbers: - -\label{iseqpage} -\begin{alltt} -Lemma eq_nat_dec : {\prodsym} n p:nat, \{n=p\}+\{n {\coqdiff} p\}. -Proof. - intros n p. -\end{alltt} - -Let us prove this theorem using the combinator \texttt{nat\_double\_rect} -of section~\ref{natdoublerect}. The example also illustrates how -\texttt{elim} may sometimes fail in finding a suitable abstraction $P$ -of the goal. Note that if ``~\texttt{elim n}~'' - is used directly on the -goal, the result is not the expected one. - -\vspace{12pt} - -%\pagebreak -\begin{alltt} - elim n using nat_double_rect. -\it -4 subgoals - - n : nat - p : nat - ============================ - {\prodsym} x : nat, \{x = p\} + \{x {\coqdiff} p\} - -subgoal 2 is: - nat {\arrow} \{0 = p\} + \{0 {\coqdiff} p\} - -subgoal 3 is: - nat {\arrow} {\prodsym} m : nat, \{m = p\} + \{m {\coqdiff} p\} {\arrow} \{S m = p\} + \{S m {\coqdiff} p\} - -subgoal 4 is: - nat -\end{alltt} - -The four sub-goals obtained do not correspond to the premises that -would be expected for the principle \texttt{nat\_double\_rec}. The -problem comes from the fact that -this principle for eliminating $n$ -has a universally quantified formula as conclusion, which confuses -\texttt{elim} about the right way of abstracting the goal. - -%In effect, let us consider the type of the goal before the call to -%\citecoq{elim}: ``~\citecoq{\{n = p\} + \{n {\coqdiff} p\}}~''. - -%Among all the abstractions that can be built by ``~\citecoq{elim n}~'' -%let us consider this one -%$P=$\citecoq{fun n :nat {\funarrow} fun q : nat {\funarrow} {\{q= p\} + \{q {\coqdiff} p\}}}. -%It is easy to verify that -%$P$ has type \citecoq{nat {\arrow} nat {\arrow} Set}, and that, if some -%$q:\citecoq{nat}$ is given, then $P\;q\;$ matches the current goal. -%Then applying \citecoq{nat\_double\_rec} with $P$ generates -%four goals, corresponding to - - - - -Therefore, -in this case the abstraction must be explicited using the -\texttt{pattern} tactic. Once the right abstraction is provided, the rest of -the proof is immediate: - -\begin{alltt} -Undo. - pattern p,n. -\it - n : nat - p : nat - ============================ - (fun n0 n1 : nat {\funarrow} \{n1 = n0\} + \{n1 {\coqdiff} n0\}) p n -\tt - elim n using nat_double_rec. -\it -3 subgoals - - n : nat - p : nat - ============================ - {\prodsym} x : nat, \{x = 0\} + \{x {\coqdiff} 0\} - -subgoal 2 is: - {\prodsym} x : nat, \{0 = S x\} + \{0 {\coqdiff} S x\} -subgoal 3 is: - {\prodsym} n0 m : nat, \{m = n0\} + \{m {\coqdiff} n0\} {\arrow} \{S m = S n0\} + \{S m {\coqdiff} S n0\} - -\tt - destruct x; auto. - destruct x; auto. - intros n0 m H; case H. - intro eq; rewrite eq ; auto. - intro neg; right; red ; injection 1; auto. -Defined. -\end{alltt} - - -Notice that the tactic ``~\texttt{decide equality}~'' -\refmancite{Section\ref{DecideEquality}} generalises the proof -above to a large class of inductive types. It can be used for proving -a proposition of the form -$\forall\,(x,y:R),\{x=y\}+\{x{\coqdiff}y\}$, where $R$ is an inductive datatype -all whose constructors take informative arguments ---like for example -the type {\nat}: - -\begin{alltt} -Definition eq_nat_dec' : {\prodsym} n p:nat, \{n=p\} + \{n{\coqdiff}p\}. - decide equality. -Defined. -\end{alltt} - -\begin{exercise} -\begin{enumerate} -\item Define a recursive function of name \emph{nat2itree} -that maps any natural number $n$ into an infinitely branching -tree of height $n$. -\item Provide an elimination combinator for these trees. -\item Prove that the relation \citecoq{itree\_le} is a preorder -(i.e. reflexive and transitive). -\end{enumerate} -\end{exercise} - -\begin{exercise} \label{zeroton} -Define the type of lists, and a predicate ``being an ordered list'' -using an inductive family. Then, define the function -$(from\;n)=0::1\;\ldots\; n::\texttt{nil}$ and prove that it always generates an -ordered list. -\end{exercise} - -\begin{exercise} -Prove that \citecoq{le' n p} and \citecoq{n $\leq$ p} are logically equivalent -for all n and p. (\citecoq{le'} is defined in section \ref{parameterstuff}). -\end{exercise} - - -\subsection{Well-founded Recursion} -\label{WellFoundedRecursion} - -Structural induction is a strong elimination rule for inductive types. -This method can be used to define any function whose termination is -a consequence of the well-foundedness of a certain order relation $R$ decreasing -at each recursive call. What makes this principle so strong is the -possibility of reasoning by structural induction on the proof that -certain $R$ is well-founded. In order to illustrate this we have -first to introduce the predicate of accessibility. - -\begin{alltt} -Print Acc. -\it -Inductive Acc (A : Type) (R : A {\arrow} A {\arrow} Prop) (x:A) : Prop := - Acc_intro : ({\prodsym} y : A, R y x {\arrow} Acc R y) {\arrow} Acc R x -For Acc: Argument A is implicit -For Acc_intro: Arguments A, R are implicit - -\dots -\end{alltt} - -\noindent This inductive predicate characterizes those elements $x$ of -$A$ such that any descending $R$-chain $\ldots x_2\;R\;x_1\;R\;x$ -starting from $x$ is finite. A well-founded relation is a relation -such that all the elements of $A$ are accessible. -\emph{Notice the use of parameter $x$ (see Section~\ref{parameterstuff}, page -\pageref{parameterstuff}).} - -Consider now the problem of representing in {\coq} the following ML -function $\textsl{div}(x,y)$ on natural numbers, which computes -$\lceil\frac{x}{y}\rceil$ if $y>0$ and yields $x$ otherwise. - -\begin{verbatim} -let rec div x y = - if x = 0 then 0 - else if y = 0 then x - else (div (x-y) y)+1;; -\end{verbatim} - - -The equality test on natural numbers can be implemented using the -function \textsl{eq\_nat\_dec} that is defined page \pageref{iseqpage}. Giving $x$ and -$y$, this function yields either the value $(\textsl{left}\;p)$ if -there exists a proof $p:x=y$, or the value $(\textsl{right}\;q)$ if -there exists $q:a\not = b$. The subtraction function is already -defined in the library \citecoq{Minus}. - -Hence, direct translation of the ML function \textsl{div} would be: - -\begin{alltt} -Require Import Minus. - -Fixpoint div (x y:nat)\{struct x\}: nat := - if eq_nat_dec x 0 - then 0 - else if eq_nat_dec y 0 - then x - else S (div (x-y) y). - -\it Error: -Recursive definition of div is ill-formed. -In environment -div : nat {\arrow} nat {\arrow} nat -x : nat -y : nat -_ : x {\coqdiff} 0 -_ : y {\coqdiff} 0 - -Recursive call to div has principal argument equal to -"x - y" -instead of a subterm of x -\end{alltt} - - -The program \texttt{div} is rejected by {\coq} because it does not verify -the syntactical condition to ensure termination. In particular, the -argument of the recursive call is not a pattern variable issued from a -case analysis on $x$. -We would have the same problem if we had the directive -``~\citecoq{\{struct y\}}~'' instead of ``~\citecoq{\{struct x\}}~''. -However, we know that this program always -stops. One way to justify its termination is to define it by -structural induction on a proof that $x$ is accessible trough the -relation $<$. Notice that any natural number $x$ is accessible -for this relation. In order to do this, it is first necessary to prove -some auxiliary lemmas, justifying that the first argument of -\texttt{div} decreases at each recursive call. - -\begin{alltt} -Lemma minus_smaller_S : {\prodsym} x y:nat, x - y < S x. -Proof. - intros x y; pattern y, x; - elim x using nat_double_ind. - destruct x0; auto with arith. - simpl; auto with arith. - simpl; auto with arith. -Qed. - - -Lemma minus_smaller_positive : - {\prodsym} x y:nat, x {\coqdiff}0 {\arrow} y {\coqdiff} 0 {\arrow} x - y < x. -Proof. - destruct x; destruct y; - ( simpl;intros; apply minus_smaller || - intros; absurd (0=0); auto). -Qed. -\end{alltt} - -\noindent The last two lemmas are necessary to prove that for any pair -of positive natural numbers $x$ and $y$, if $x$ is accessible with -respect to \citecoq{lt}, then so is $x-y$. - -\begin{alltt} -Definition minus_decrease : {\prodsym} x y:nat, Acc lt x {\arrow} - x {\coqdiff} 0 {\arrow} - y {\coqdiff} 0 {\arrow} - Acc lt (x-y). -Proof. - intros x y H; case H. - intros Hz posz posy. - apply Hz; apply minus_smaller_positive; assumption. -Defined. -\end{alltt} - -Let us take a look at the proof of the lemma \textsl{minus\_decrease}, since -the way in which it has been proven is crucial for what follows. -\begin{alltt} -Print minus_decrease. -\it -minus_decrease = -fun (x y : nat) (H : Acc lt x) {\funarrow} -match H in (Acc _ y0) return (y0 {\coqdiff} 0 {\arrow} y {\coqdiff} 0 {\arrow} Acc lt (y0 - y)) with -| Acc_intro z Hz {\funarrow} - fun (posz : z {\coqdiff} 0) (posy : y {\coqdiff} 0) {\funarrow} - Hz (z - y) (minus_smaller_positive z y posz posy) -end - : {\prodsym} x y : nat, Acc lt x {\arrow} x {\coqdiff} 0 {\arrow} y {\coqdiff} 0 {\arrow} Acc lt (x - y) - -\end{alltt} -\noindent Notice that the function call -$(\texttt{minus\_decrease}\;n\;m\;H)$ -indeed yields an accessibility proof that is \textsl{structurally -smaller} than its argument $H$, because it is (an application of) its -recursive component $Hz$. This enables to justify the following -definition of \textsl{div\_aux}: - -\begin{alltt} -Definition div_aux (x y:nat)(H: Acc lt x):nat. - fix 3. - intros. - refine (if eq_nat_dec x 0 - then 0 - else if eq_nat_dec y 0 - then y - else div_aux (x-y) y _). -\it - div_aux : {\prodsym} x : nat, nat {\arrow} Acc lt x {\arrow} nat - x : nat - y : nat - H : Acc lt x - _ : x {\coqdiff} 0 - _0 : y {\coqdiff} 0 - ============================ - Acc lt (x - y) - -\tt - apply (minus_decrease x y H);auto. -Defined. -\end{alltt} - -The main division function is easily defined, using the theorem -\citecoq{lt\_wf} of the library \citecoq{Wf\_nat}. This theorem asserts that -\citecoq{nat} is well founded w.r.t. \citecoq{lt}, thus any natural number -is accessible. -\begin{alltt} -Definition div x y := div_aux x y (lt_wf x). -\end{alltt} - -Let us explain the proof above. In the definition of \citecoq{div\_aux}, -what decreases is not $x$ but the \textsl{proof} of the accessibility -of $x$. The tactic ``~\texttt{fix 3}~'' is used to indicate that the proof -proceeds by structural induction on the third argument of the theorem ---that is, on the accessibility proof. It also introduces a new -hypothesis in the context, named as the current theorem, and with the -same type as the goal. Then, the proof is refined with an incomplete -proof term, containing a hole \texttt{\_}. This hole corresponds to the proof -of accessibility for $x-y$, and is filled up with the (smaller!) -accessibility proof provided by the function \texttt{minus\_decrease}. - - -\noindent Let us take a look to the term \textsl{div\_aux} defined: - -\pagebreak -\begin{alltt} -Print div_aux. -\it -div_aux = -(fix div_aux (x y : nat) (H : Acc lt x) \{struct H\} : nat := - match eq_nat_dec x 0 with - | left _ {\funarrow} 0 - | right _ {\funarrow} - match eq_nat_dec y 0 with - | left _ {\funarrow} y - | right _0 {\funarrow} div_aux (x - y) y (minus_decrease x y H _ _0) - end - end) - : {\prodsym} x : nat, nat {\arrow} Acc lt x {\arrow} nat - -\end{alltt} - -If the non-informative parts from this proof --that is, the -accessibility proof-- are erased, then we obtain exactly the program -that we were looking for. -\begin{alltt} - -Extraction div. - -\it -let div x y = - div_aux x y -\tt - -Extraction div_aux. - -\it -let rec div_aux x y = - match eq_nat_dec x O with - | Left {\arrow} O - | Right {\arrow} - (match eq_nat_dec y O with - | Left {\arrow} y - | Right {\arrow} div_aux (minus x y) y) -\end{alltt} - -This methodology enables the representation -of any program whose termination can be proved in {\coq}. Once the -expected properties from this program have been verified, the -justification of its termination can be thrown away, keeping just the -desired computational behavior for it. - -\section{A case study in dependent elimination}\label{CaseStudy} - -Dependent types are very expressive, but ignoring some useful -techniques can cause some problems to the beginner. -Let us consider again the type of vectors (see section~\ref{vectors}). -We want to prove a quite trivial property: the only value of type -``~\citecoq{vector A 0}~'' is ``~\citecoq{Vnil $A$}~''. - -Our first naive attempt leads to a \emph{cul-de-sac}. -\begin{alltt} -Lemma vector0_is_vnil : - {\prodsym} (A:Type)(v:vector A 0), v = Vnil A. -Proof. - intros A v;inversion v. -\it -1 subgoal - - A : Set - v : vector A 0 - ============================ - v = Vnil A -\tt -Abort. -\end{alltt} - -Another attempt is to do a case analysis on a vector of any length -$n$, under an explicit hypothesis $n=0$. The tactic -\texttt{discriminate} will help us to get rid of the case -$n=\texttt{S $p$}$. -Unfortunately, even the statement of our lemma is refused! - -\begin{alltt} - Lemma vector0_is_vnil_aux : - {\prodsym} (A:Type)(n:nat)(v:vector A n), n = 0 {\arrow} v = Vnil A. - -\it -Error: In environment -A : Type -n : nat -v : vector A n -e : n = 0 -The term "Vnil A" has type "vector A 0" while it is expected to have type - "vector A n" -\end{alltt} - -In effect, the equality ``~\citecoq{v = Vnil A}~'' is ill-typed and this is -because the type ``~\citecoq{vector A n}~'' is not \emph{convertible} -with ``~\citecoq{vector A 0}~''. - -This problem can be solved if we consider the heterogeneous -equality \citecoq{JMeq} \cite{conor:motive} -which allows us to consider terms of different types, even if this -equality can only be proven for terms in the same type. -The axiom \citecoq{JMeq\_eq}, from the library \citecoq{JMeq} allows us to convert a -heterogeneous equality to a standard one. - -\begin{alltt} -Lemma vector0_is_vnil_aux : - {\prodsym} (A:Type)(n:nat)(v:vector A n), - n= 0 {\arrow} JMeq v (Vnil A). -Proof. - destruct v. - auto. - intro; discriminate. -Qed. -\end{alltt} - -Our property of vectors of null length can be easily proven: - -\begin{alltt} -Lemma vector0_is_vnil : {\prodsym} (A:Type)(v:vector A 0), v = Vnil A. - intros a v;apply JMeq_eq. - apply vector0_is_vnil_aux. - trivial. -Qed. -\end{alltt} - -It is interesting to look at another proof of -\citecoq{vector0\_is\_vnil}, which illustrates a technique developed -and used by various people (consult in the \emph{Coq-club} mailing -list archive the contributions by Yves Bertot, Pierre Letouzey, Laurent Théry, -Jean Duprat, and Nicolas Magaud, Venanzio Capretta and Conor McBride). -This technique is also used for unfolding infinite list definitions -(see chapter13 of~\cite{coqart}). -Notice that this definition does not rely on any axiom (\emph{e.g.} \texttt{JMeq\_eq}). - -We first give a new definition of the identity on vectors. Before that, -we make the use of constructors and selectors lighter thanks to -the implicit arguments feature: - -\begin{alltt} -Implicit Arguments Vcons [A n]. -Implicit Arguments Vnil [A]. -Implicit Arguments Vhead [A n]. -Implicit Arguments Vtail [A n]. - -Definition Vid : {\prodsym} (A : Type)(n:nat), vector A n {\arrow} vector A n. -Proof. - destruct n; intro v. - exact Vnil. - exact (Vcons (Vhead v) (Vtail v)). -Defined. -\end{alltt} - - -Then we prove that \citecoq{Vid} is the identity on vectors: - -\begin{alltt} -Lemma Vid_eq : {\prodsym} (n:nat) (A:Type)(v:vector A n), v=(Vid _ n v). -Proof. - destruct v. - -\it - A : Type - ============================ - Vnil = Vid A 0 Vnil - -subgoal 2 is: - Vcons a v = Vid A (S n) (Vcons a v) -\tt - reflexivity. - reflexivity. -Defined. -\end{alltt} - -Why defining a new identity function on vectors? The following -dialogue shows that \citecoq{Vid} has some interesting computational -properties: - -\begin{alltt} -Eval simpl in (fun (A:Type)(v:vector A 0) {\funarrow} (Vid _ _ v)). -\it = fun (A : Type) (_ : vector A 0) {\funarrow} Vnil - : {\prodsym} A : Type, vector A 0 {\arrow} vector A 0 - -\end{alltt} - -Notice that the plain identity on vectors doesn't convert \citecoq{v} -into \citecoq{Vnil}. -\begin{alltt} -Eval simpl in (fun (A:Type)(v:vector A 0) {\funarrow} v). -\it = fun (A : Type) (v : vector A 0) {\funarrow} v - : {\prodsym} A : Type, vector A 0 {\arrow} vector A 0 -\end{alltt} - -Then we prove easily that any vector of length 0 is \citecoq{Vnil}: - -\begin{alltt} -Theorem zero_nil : {\prodsym} A (v:vector A 0), v = Vnil. -Proof. - intros. - change (Vnil (A:=A)) with (Vid _ 0 v). -\it -1 subgoal - - A : Type - v : vector A 0 - ============================ - v = Vid A 0 v -\tt - apply Vid_eq. -Defined. -\end{alltt} - -A similar result can be proven about vectors of strictly positive -length\footnote{As for \citecoq{Vid} and \citecoq{Vid\_eq}, this definition -is from Jean Duprat.}. - -\begin{alltt} - - -Theorem decomp : - {\prodsym} (A : Type) (n : nat) (v : vector A (S n)), - v = Vcons (Vhead v) (Vtail v). -Proof. - intros. - change (Vcons (Vhead v) (Vtail v)) with (Vid _ (S n) v). -\it - 1 subgoal - - A : Type - n : nat - v : vector A (S n) - ============================ - v = Vid A (S n) v - -\tt{} apply Vid_eq. -Defined. -\end{alltt} - - -Both lemmas: \citecoq{zero\_nil} and \citecoq{decomp}, -can be used to easily derive a double recursion principle -on vectors of same length: - - -\begin{alltt} -Definition vector_double_rect : - {\prodsym} (A:Type) (P: {\prodsym} (n:nat),(vector A n){\arrow}(vector A n) {\arrow} Type), - P 0 Vnil Vnil {\arrow} - ({\prodsym} n (v1 v2 : vector A n) a b, P n v1 v2 {\arrow} - P (S n) (Vcons a v1) (Vcons b v2)) {\arrow} - {\prodsym} n (v1 v2 : vector A n), P n v1 v2. - induction n. - intros; rewrite (zero_nil _ v1); rewrite (zero_nil _ v2). - auto. - intros v1 v2; rewrite (decomp _ _ v1);rewrite (decomp _ _ v2). - apply X0; auto. -Defined. -\end{alltt} - -Notice that, due to the conversion rule of {\coq}'s type system, -this function can be used directly with \citecoq{Prop} or \citecoq{Type} -instead of type (thus it is useless to build -\citecoq{vector\_double\_ind} and \citecoq{vector\_double\_rec}) from scratch. - -We finish this example with showing how to define the bitwise -\emph{or} on boolean vectors of the same length, -and proving a little property about this -operation. - -\begin{alltt} -Definition bitwise_or n v1 v2 : vector bool n := - vector_double_rect - bool - (fun n v1 v2 {\funarrow} vector bool n) - Vnil - (fun n v1 v2 a b r {\funarrow} Vcons (orb a b) r) n v1 v2. -\end{alltt} - -Let us define recursively the $n$-th element of a vector. Notice -that it must be a partial function, in case $n$ is greater or equal -than the length of the vector. Since {\coq} only considers total -functions, the function returns a value in an \emph{option} type. - -\begin{alltt} -Fixpoint vector_nth (A:Type)(n:nat)(p:nat)(v:vector A p) - \{struct v\} - : option A := - match n,v with - _ , Vnil {\funarrow} None - | 0 , Vcons b _ _ {\funarrow} Some b - | S n', Vcons _ p' v' {\funarrow} vector_nth A n' p' v' - end. -Implicit Arguments vector_nth [A p]. -\end{alltt} - -We can now prove --- using the double induction combinator --- -a simple property relying \citecoq{vector\_nth} and \citecoq{bitwise\_or}: - -\begin{alltt} -Lemma nth_bitwise : - {\prodsym} (n:nat) (v1 v2: vector bool n) i a b, - vector_nth i v1 = Some a {\arrow} - vector_nth i v2 = Some b {\arrow} - vector_nth i (bitwise_or _ v1 v2) = Some (orb a b). -Proof. - intros n v1 v2; pattern n,v1,v2. - apply vector_double_rect. - simpl. - destruct i; discriminate 1. - destruct i; simpl;auto. - injection 1; injection 2;intros; subst a; subst b; auto. -Qed. -\end{alltt} - - -\section{Co-inductive Types and Non-ending Constructions} -\label{CoInduction} - -The objects of an inductive type are well-founded with respect to -the constructors of the type. In other words, these objects are built -by applying \emph{a finite number of times} the constructors of the type. -Co-inductive types are obtained by relaxing this condition, -and may contain non-well-founded objects \cite{EG96,EG95a}. An -example of a co-inductive type is the type of infinite -sequences formed with elements of type $A$, also called streams. This -type can be introduced through the following definition: - -\begin{alltt} - CoInductive Stream (A: Type) :Type := - | Cons : A\arrow{}Stream A\arrow{}Stream A. -\end{alltt} - -If we are interested in finite or infinite sequences, we consider the type -of \emph{lazy lists}: - -\begin{alltt} -CoInductive LList (A: Type) : Type := - | LNil : LList A - | LCons : A {\arrow} LList A {\arrow} LList A. -\end{alltt} - - -It is also possible to define co-inductive types for the -trees with infinitely-many branches (see Chapter 13 of~\cite{coqart}). - -Structural induction is the way of expressing that inductive types -only contain well-founded objects. Hence, this elimination principle -is not valid for co-inductive types, and the only elimination rule for -streams is case analysis. This principle can be used, for example, to -define the destructors \textsl{head} and \textsl{tail}. - -\begin{alltt} - Definition head (A:Type)(s : Stream A) := - match s with Cons a s' {\funarrow} a end. - - Definition tail (A : Type)(s : Stream A) := - match s with Cons a s' {\funarrow} s' end. -\end{alltt} - -Infinite objects are defined by means of (non-ending) methods of -construction, like in lazy functional programming languages. Such -methods can be defined using the \texttt{CoFixpoint} command -\refmancite{Section \ref{CoFixpoint}}. For example, the following -definition introduces the infinite list $[a,a,a,\ldots]$: - -\begin{alltt} - CoFixpoint repeat (A:Type)(a:A) : Stream A := - Cons a (repeat a). -\end{alltt} - - -However, not every co-recursive definition is an admissible method of -construction. Similarly to the case of structural induction, the -definition must verify a \textsl{guardedness} condition to be -accepted. This condition states that any recursive call in the -definition must be protected --i.e, be an argument of-- some -constructor, and only an argument of constructors \cite{EG94a}. The -following definitions are examples of valid methods of construction: - -\begin{alltt} -CoFixpoint iterate (A: Type)(f: A {\arrow} A)(a : A) : Stream A:= - Cons a (iterate f (f a)). - -CoFixpoint map - (A B:Type)(f: A {\arrow} B)(s : Stream A) : Stream B:= - match s with Cons a tl {\funarrow} Cons (f a) (map f tl) end. -\end{alltt} - -\begin{exercise} -Define two different methods for constructing the stream which -infinitely alternates the values \citecoq{true} and \citecoq{false}. -\end{exercise} -\begin{exercise} -Using the destructors \texttt{head} and \texttt{tail}, define a function -which takes the n-th element of an infinite stream. -\end{exercise} - -A non-ending method of construction is computed lazily. This means -that its definition is unfolded only when the object that it -introduces is eliminated, that is, when it appears as the argument of -a case expression. We can check this using the command -\texttt{Eval}. - -\begin{alltt} -Eval simpl in (fun (A:Type)(a:A) {\funarrow} repeat a). -\it = fun (A : Type) (a : A) {\funarrow} repeat a - : {\prodsym} A : Type, A {\arrow} Stream A -\tt -Eval simpl in (fun (A:Type)(a:A) {\funarrow} head (repeat a)). -\it = fun (A : Type) (a : A) {\funarrow} a - : {\prodsym} A : Type, A {\arrow} A -\end{alltt} - -%\begin{exercise} -%Prove the following theorem: -%\begin{verbatim} -%Theorem expand_repeat : (a:A)(repeat a)=(Cons a (repeat a)). -%\end{verbatim} -%Hint: Prove first the streams version of the lemma in exercise -%\ref{expand}. -%\end{exercise} - -\subsection{Extensional Properties} - -Case analysis is also a valid proof principle for infinite -objects. However, this principle is not sufficient to prove -\textsl{extensional} properties, that is, properties concerning the -whole infinite object \cite{EG95a}. A typical example of an -extensional property is the predicate expressing that two streams have -the same elements. In many cases, the minimal reflexive relation $a=b$ -that is used as equality for inductive types is too small to capture -equality between streams. Consider for example the streams -$\texttt{iterate}\;f\;(f\;x)$ and -$(\texttt{map}\;f\;(\texttt{iterate}\;f\;x))$. Even though these two streams have -the same elements, no finite expansion of their definitions lead to -equal terms. In other words, in order to deal with extensional -properties, it is necessary to construct infinite proofs. The type of -infinite proofs of equality can be introduced as a co-inductive -predicate, as follows: -\begin{alltt} -CoInductive EqSt (A: Type) : Stream A {\arrow} Stream A {\arrow} Prop := - eqst : {\prodsym} s1 s2: Stream A, - head s1 = head s2 {\arrow} - EqSt (tail s1) (tail s2) {\arrow} - EqSt s1 s2. -\end{alltt} - -It is possible to introduce proof principles for reasoning about -infinite objects as combinators defined through -\texttt{CoFixpoint}. However, oppositely to the case of inductive -types, proof principles associated to co-inductive types are not -elimination but \textsl{introduction} combinators. An example of such -a combinator is Park's principle for proving the equality of two -streams, usually called the \textsl{principle of co-induction}. It -states that two streams are equal if they satisfy a -\textit{bisimulation}. A bisimulation is a binary relation $R$ such -that any pair of streams $s_1$ ad $s_2$ satisfying $R$ have equal -heads, and tails also satisfying $R$. This principle is in fact a -method for constructing an infinite proof: - -\begin{alltt} -Section Parks_Principle. -Variable A : Type. -Variable R : Stream A {\arrow} Stream A {\arrow} Prop. -Hypothesis bisim1 : {\prodsym} s1 s2:Stream A, - R s1 s2 {\arrow} head s1 = head s2. - -Hypothesis bisim2 : {\prodsym} s1 s2:Stream A, - R s1 s2 {\arrow} R (tail s1) (tail s2). - -CoFixpoint park_ppl : - {\prodsym} s1 s2:Stream A, R s1 s2 {\arrow} EqSt s1 s2 := - fun s1 s2 (p : R s1 s2) {\funarrow} - eqst s1 s2 (bisim1 s1 s2 p) - (park_ppl (tail s1) - (tail s2) - (bisim2 s1 s2 p)). -End Parks_Principle. -\end{alltt} - -Let us use the principle of co-induction to prove the extensional -equality mentioned above. -\begin{alltt} -Theorem map_iterate : {\prodsym} (A:Type)(f:A{\arrow}A)(x:A), - EqSt (iterate f (f x)) - (map f (iterate f x)). -Proof. - intros A f x. - apply park_ppl with - (R:= fun s1 s2 {\funarrow} - {\exsym} x: A, s1 = iterate f (f x) {\coqand} - s2 = map f (iterate f x)). - - intros s1 s2 (x0,(eqs1,eqs2)); - rewrite eqs1; rewrite eqs2; reflexivity. - intros s1 s2 (x0,(eqs1,eqs2)). - exists (f x0);split; - [rewrite eqs1|rewrite eqs2]; reflexivity. - exists x;split; reflexivity. -Qed. -\end{alltt} - -The use of Park's principle is sometimes annoying, because it requires -to find an invariant relation and prove that it is indeed a -bisimulation. In many cases, a shorter proof can be obtained trying -to construct an ad-hoc infinite proof, defined by a guarded -declaration. The tactic ``~``\texttt{Cofix $f$}~'' can be used to do -that. Similarly to the tactic \texttt{fix} indicated in Section -\ref{WellFoundedRecursion}, this tactic introduces an extra hypothesis -$f$ into the context, whose type is the same as the current goal. Note -that the applications of $f$ in the proof \textsl{must be guarded}. In -order to prevent us from doing unguarded calls, we can define a tactic -that always apply a constructor before using $f$ \refmancite{Chapter -\ref{WritingTactics}} : - -\begin{alltt} -Ltac infiniteproof f := - cofix f; - constructor; - [clear f| simpl; try (apply f; clear f)]. -\end{alltt} - - -In the example above, this tactic produces a much simpler proof -that the former one: - -\begin{alltt} -Theorem map_iterate' : {\prodsym} ((A:Type)f:A{\arrow}A)(x:A), - EqSt (iterate f (f x)) - (map f (iterate f x)). -Proof. - infiniteproof map_iterate'. - reflexivity. -Qed. -\end{alltt} - -\begin{exercise} -Define a co-inductive type of name $Nat$ that contains non-standard -natural numbers --this is, verifying - -$$\exists m \in \mbox{\texttt{Nat}}, \forall\, n \in \mbox{\texttt{Nat}}, n<m$$. -\end{exercise} - -\begin{exercise} -Prove that the extensional equality of streams is an equivalence relation -using Park's co-induction principle. -\end{exercise} - - -\begin{exercise} -Provide a suitable definition of ``being an ordered list'' for infinite lists -and define a principle for proving that an infinite list is ordered. Apply -this method to the list $[0,1,\ldots ]$. Compare the result with -exercise \ref{zeroton}. -\end{exercise} - -\subsection{About injection, discriminate, and inversion} -Since co-inductive types are closed w.r.t. their constructors, -the techniques shown in Section~\ref{CaseTechniques} work also -with these types. - -Let us consider the type of lazy lists, introduced on page~\pageref{CoInduction}. -The following lemmas are straightforward applications - of \texttt{discriminate} and \citecoq{injection}: - -\begin{alltt} -Lemma Lnil_not_Lcons : {\prodsym} (A:Type)(a:A)(l:LList A), - LNil {\coqdiff} (LCons a l). -Proof. - intros;discriminate. -Qed. - -Lemma injection_demo : {\prodsym} (A:Type)(a b : A)(l l': LList A), - LCons a (LCons b l) = LCons b (LCons a l') {\arrow} - a = b {\coqand} l = l'. -Proof. - intros A a b l l' e; injection e; auto. -Qed. - -\end{alltt} - -In order to show \citecoq{inversion} at work, let us define -two predicates on lazy lists: - -\begin{alltt} -Inductive Finite (A:Type) : LList A {\arrow} Prop := -| Lnil_fin : Finite (LNil (A:=A)) -| Lcons_fin : {\prodsym} a l, Finite l {\arrow} Finite (LCons a l). - -CoInductive Infinite (A:Type) : LList A {\arrow} Prop := -| LCons_inf : {\prodsym} a l, Infinite l {\arrow} Infinite (LCons a l). -\end{alltt} - -\noindent -First, two easy theorems: -\begin{alltt} -Lemma LNil_not_Infinite : {\prodsym} (A:Type), ~ Infinite (LNil (A:=A)). -Proof. - intros A H;inversion H. -Qed. - -Lemma Finite_not_Infinite : {\prodsym} (A:Type)(l:LList A), - Finite l {\arrow} ~ Infinite l. -Proof. - intros A l H; elim H. - apply LNil_not_Infinite. - intros a l0 F0 I0' I1. - case I0'; inversion_clear I1. - trivial. -Qed. -\end{alltt} - - -On the other hand, the next proof uses the \citecoq{cofix} tactic. -Notice the destructuration of \citecoq{l}, which allows us to -apply the constructor \texttt{LCons\_inf}, thus satisfying - the guard condition: -\begin{alltt} -Lemma Not_Finite_Infinite : {\prodsym} (A:Type)(l:LList A), - ~ Finite l {\arrow} Infinite l. -Proof. - cofix H. - destruct l. - intro; - absurd (Finite (LNil (A:=A))); - [auto|constructor]. -\it - - - - -1 subgoal - - H : forall (A : Type) (l : LList A), ~ Finite l -> Infinite l - A : Type - a : A - l : LList A - H0 : ~ Finite (LCons a l) - ============================ - Infinite l -\end{alltt} -At this point, one must not apply \citecoq{H}! . It would be possible -to solve the current goal by an inversion of ``~\citecoq{Finite (LCons a l)}~'', but, since the guard condition would be violated, the user -would get an error message after typing \citecoq{Qed}. -In order to satisfy the guard condition, we apply the constructor of -\citecoq{Infinite}, \emph{then} apply \citecoq{H}. - -\begin{alltt} - constructor. - apply H. - red; intro H1;case H0. - constructor. - trivial. -Qed. -\end{alltt} - - - - -The reader is invited to replay this proof and understand each of its steps. - - -\bibliographystyle{abbrv} -\bibliography{manbiblio,morebib} - -\end{document} - diff --git a/doc/RecTutorial/RecTutorial.v b/doc/RecTutorial/RecTutorial.v deleted file mode 100644 index 28aaf752..00000000 --- a/doc/RecTutorial/RecTutorial.v +++ /dev/null @@ -1,1232 +0,0 @@ -Check (forall A:Type, (exists x:A, forall (y:A), x <> y) -> 2 = 3). - - - -Inductive nat : Set := - | O : nat - | S : nat->nat. -Check nat. -Check O. -Check S. - -Reset nat. -Print nat. - - -Print le. - -Theorem zero_leq_three: 0 <= 3. - -Proof. - constructor 2. - constructor 2. - constructor 2. - constructor 1. - -Qed. - -Print zero_leq_three. - - -Lemma zero_leq_three': 0 <= 3. - repeat constructor. -Qed. - - -Lemma zero_lt_three : 0 < 3. -Proof. - repeat constructor. -Qed. - -Print zero_lt_three. - -Inductive le'(n:nat):nat -> Prop := - | le'_n : le' n n - | le'_S : forall p, le' (S n) p -> le' n p. - -Hint Constructors le'. - - -Require Import List. - -Print list. - -Check list. - -Check (nil (A:=nat)). - -Check (nil (A:= nat -> nat)). - -Check (fun A: Type => (cons (A:=A))). - -Check (cons 3 (cons 2 nil)). - -Check (nat :: bool ::nil). - -Check ((3<=4) :: True ::nil). - -Check (Prop::Set::nil). - -Require Import Bvector. - -Print vector. - -Check (Vnil nat). - -Check (fun (A:Type)(a:A)=> Vcons _ a _ (Vnil _)). - -Check (Vcons _ 5 _ (Vcons _ 3 _ (Vnil _))). - -Lemma eq_3_3 : 2 + 1 = 3. -Proof. - reflexivity. -Qed. -Print eq_3_3. - -Lemma eq_proof_proof : refl_equal (2*6) = refl_equal (3*4). -Proof. - reflexivity. -Qed. -Print eq_proof_proof. - -Lemma eq_lt_le : ( 2 < 4) = (3 <= 4). -Proof. - reflexivity. -Qed. - -Lemma eq_nat_nat : nat = nat. -Proof. - reflexivity. -Qed. - -Lemma eq_Set_Set : Set = Set. -Proof. - reflexivity. -Qed. - -Lemma eq_Type_Type : Type = Type. -Proof. - reflexivity. -Qed. - - -Check (2 + 1 = 3). - - -Check (Type = Type). - -Goal Type = Type. -reflexivity. -Qed. - - -Print or. - -Print and. - - -Print sumbool. - -Print ex. - -Require Import ZArith. -Require Import Compare_dec. - -Check le_lt_dec. - -Definition max (n p :nat) := match le_lt_dec n p with - | left _ => p - | right _ => n - end. - -Theorem le_max : forall n p, n <= p -> max n p = p. -Proof. - intros n p ; unfold max ; case (le_lt_dec n p); simpl. - trivial. - intros; absurd (p < p); eauto with arith. -Qed. - -Extraction max. - - - - - - -Inductive tree(A:Type) : Type := - node : A -> forest A -> tree A -with - forest (A: Type) : Type := - nochild : forest A | - addchild : tree A -> forest A -> forest A. - - - - - -Inductive - even : nat->Prop := - evenO : even O | - evenS : forall n, odd n -> even (S n) -with - odd : nat->Prop := - oddS : forall n, even n -> odd (S n). - -Lemma odd_49 : odd (7 * 7). - simpl; repeat constructor. -Qed. - - - -Definition nat_case := - fun (Q : Type)(g0 : Q)(g1 : nat -> Q)(n:nat) => - match n return Q with - | 0 => g0 - | S p => g1 p - end. - -Eval simpl in (nat_case nat 0 (fun p => p) 34). - -Eval simpl in (fun g0 g1 => nat_case nat g0 g1 34). - -Eval simpl in (fun g0 g1 => nat_case nat g0 g1 0). - - -Definition pred (n:nat) := match n with O => O | S m => m end. - -Eval simpl in pred 56. - -Eval simpl in pred 0. - -Eval simpl in fun p => pred (S p). - - -Definition xorb (b1 b2:bool) := -match b1, b2 with - | false, true => true - | true, false => true - | _ , _ => false -end. - - - Definition pred_spec (n:nat) := {m:nat | n=0 /\ m=0 \/ n = S m}. - - - Definition predecessor : forall n:nat, pred_spec n. - intro n;case n. - unfold pred_spec;exists 0;auto. - unfold pred_spec; intro n0;exists n0; auto. - Defined. - -Print predecessor. - -Extraction predecessor. - -Theorem nat_expand : - forall n:nat, n = match n with 0 => 0 | S p => S p end. - intro n;case n;simpl;auto. -Qed. - -Check (fun p:False => match p return 2=3 with end). - -Theorem fromFalse : False -> 0=1. - intro absurd. - contradiction. -Qed. - -Section equality_elimination. - Variables (A: Type) - (a b : A) - (p : a = b) - (Q : A -> Type). - Check (fun H : Q a => - match p in (eq _ y) return Q y with - refl_equal => H - end). - -End equality_elimination. - - -Theorem trans : forall n m p:nat, n=m -> m=p -> n=p. -Proof. - intros n m p eqnm. - case eqnm. - trivial. -Qed. - -Lemma Rw : forall x y: nat, y = y * x -> y * x * x = y. - intros x y e; do 2 rewrite <- e. - reflexivity. -Qed. - - -Require Import Arith. - -Check mult_1_l. -(* -mult_1_l - : forall n : nat, 1 * n = n -*) - -Check mult_plus_distr_r. -(* -mult_plus_distr_r - : forall n m p : nat, (n + m) * p = n * p + m * p - -*) - -Lemma mult_distr_S : forall n p : nat, n * p + p = (S n)* p. - simpl;auto with arith. -Qed. - -Lemma four_n : forall n:nat, n+n+n+n = 4*n. - intro n;rewrite <- (mult_1_l n). - - Undo. - intro n; pattern n at 1. - - - rewrite <- mult_1_l. - repeat rewrite mult_distr_S. - trivial. -Qed. - - -Section Le_case_analysis. - Variables (n p : nat) - (H : n <= p) - (Q : nat -> Prop) - (H0 : Q n) - (HS : forall m, n <= m -> Q (S m)). - Check ( - match H in (_ <= q) return (Q q) with - | le_n => H0 - | le_S m Hm => HS m Hm - end - ). - - -End Le_case_analysis. - - -Lemma predecessor_of_positive : forall n, 1 <= n -> exists p:nat, n = S p. -Proof. - intros n H; case H. - exists 0; trivial. - intros m Hm; exists m;trivial. -Qed. - -Definition Vtail_total - (A : Type) (n : nat) (v : vector A n) : vector A (pred n):= -match v in (vector _ n0) return (vector A (pred n0)) with -| Vnil => Vnil A -| Vcons _ n0 v0 => v0 -end. - -Definition Vtail' (A:Type)(n:nat)(v:vector A n) : vector A (pred n). - intros A n v; case v. - simpl. - exact (Vnil A). - simpl. - auto. -Defined. - -(* -Inductive Lambda : Set := - lambda : (Lambda -> False) -> Lambda. - - -Error: Non strictly positive occurrence of "Lambda" in - "(Lambda -> False) -> Lambda" - -*) - -Section Paradox. - Variable Lambda : Set. - Variable lambda : (Lambda -> False) ->Lambda. - - Variable matchL : Lambda -> forall Q:Prop, ((Lambda ->False) -> Q) -> Q. - (* - understand matchL Q l (fun h : Lambda -> False => t) - - as match l return Q with lambda h => t end - *) - - Definition application (f x: Lambda) :False := - matchL f False (fun h => h x). - - Definition Delta : Lambda := lambda (fun x : Lambda => application x x). - - Definition loop : False := application Delta Delta. - - Theorem two_is_three : 2 = 3. - Proof. - elim loop. - Qed. - -End Paradox. - - -Require Import ZArith. - - - -Inductive itree : Set := -| ileaf : itree -| inode : Z-> (nat -> itree) -> itree. - -Definition isingle l := inode l (fun i => ileaf). - -Definition t1 := inode 0 (fun n => isingle (Z_of_nat (2*n))). - -Definition t2 := inode 0 - (fun n : nat => - inode (Z_of_nat n) - (fun p => isingle (Z_of_nat (n*p)))). - - -Inductive itree_le : itree-> itree -> Prop := - | le_leaf : forall t, itree_le ileaf t - | le_node : forall l l' s s', - Zle l l' -> - (forall i, exists j:nat, itree_le (s i) (s' j)) -> - itree_le (inode l s) (inode l' s'). - - -Theorem itree_le_trans : - forall t t', itree_le t t' -> - forall t'', itree_le t' t'' -> itree_le t t''. - induction t. - constructor 1. - - intros t'; case t'. - inversion 1. - intros z0 i0 H0. - intro t'';case t''. - inversion 1. - intros. - inversion_clear H1. - constructor 2. - inversion_clear H0;eauto with zarith. - inversion_clear H0. - intro i2; case (H4 i2). - intros. - generalize (H i2 _ H0). - intros. - case (H3 x);intros. - generalize (H5 _ H6). - exists x0;auto. -Qed. - - - -Inductive itree_le' : itree-> itree -> Prop := - | le_leaf' : forall t, itree_le' ileaf t - | le_node' : forall l l' s s' g, - Zle l l' -> - (forall i, itree_le' (s i) (s' (g i))) -> - itree_le' (inode l s) (inode l' s'). - - - - - -Lemma t1_le_t2 : itree_le t1 t2. - unfold t1, t2. - constructor. - auto with zarith. - intro i; exists (2 * i). - unfold isingle. - constructor. - auto with zarith. - exists i;constructor. -Qed. - - - -Lemma t1_le'_t2 : itree_le' t1 t2. - unfold t1, t2. - constructor 2 with (fun i : nat => 2 * i). - auto with zarith. - unfold isingle; - intro i ; constructor 2 with (fun i :nat => i). - auto with zarith. - constructor . -Qed. - - -Require Import List. - -Inductive ltree (A:Set) : Set := - lnode : A -> list (ltree A) -> ltree A. - -Inductive prop : Prop := - prop_intro : Prop -> prop. - -Check (prop_intro prop). - -Inductive ex_Prop (P : Prop -> Prop) : Prop := - exP_intro : forall X : Prop, P X -> ex_Prop P. - -Lemma ex_Prop_inhabitant : ex_Prop (fun P => P -> P). -Proof. - exists (ex_Prop (fun P => P -> P)). - trivial. -Qed. - - - - -(* - -Check (fun (P:Prop->Prop)(p: ex_Prop P) => - match p with exP_intro X HX => X end). -Error: -Incorrect elimination of "p" in the inductive type -"ex_Prop", the return type has sort "Type" while it should be -"Prop" - -Elimination of an inductive object of sort "Prop" -is not allowed on a predicate in sort "Type" -because proofs can be eliminated only to build proofs - -*) - - -Inductive typ : Type := - typ_intro : Type -> typ. - -Definition typ_inject: typ. -split. -exact typ. -(* -Defined. - -Error: Universe Inconsistency. -*) -Abort. -(* - -Inductive aSet : Set := - aSet_intro: Set -> aSet. - - -User error: Large non-propositional inductive types must be in Type - -*) - -Inductive ex_Set (P : Set -> Prop) : Type := - exS_intro : forall X : Set, P X -> ex_Set P. - - -Inductive comes_from_the_left (P Q:Prop): P \/ Q -> Prop := - c1 : forall p, comes_from_the_left P Q (or_introl (A:=P) Q p). - -Goal (comes_from_the_left _ _ (or_introl True I)). -split. -Qed. - -Goal ~(comes_from_the_left _ _ (or_intror True I)). - red;inversion 1. - (* discriminate H0. - *) -Abort. - -Reset comes_from_the_left. - -(* - - - - - - - Definition comes_from_the_left (P Q:Prop)(H:P \/ Q): Prop := - match H with - | or_introl p => True - | or_intror q => False - end. - -Error: -Incorrect elimination of "H" in the inductive type -"or", the return type has sort "Type" while it should be -"Prop" - -Elimination of an inductive object of sort "Prop" -is not allowed on a predicate in sort "Type" -because proofs can be eliminated only to build proofs - -*) - -Definition comes_from_the_left_sumbool - (P Q:Prop)(x:{P}+{Q}): Prop := - match x with - | left p => True - | right q => False - end. - - - - -Close Scope Z_scope. - - - - - -Theorem S_is_not_O : forall n, S n <> 0. - -Definition Is_zero (x:nat):= match x with - | 0 => True - | _ => False - end. - Lemma O_is_zero : forall m, m = 0 -> Is_zero m. - Proof. - intros m H; subst m. - (* - ============================ - Is_zero 0 - *) - simpl;trivial. - Qed. - - red; intros n Hn. - apply O_is_zero with (m := S n). - assumption. -Qed. - -Theorem disc2 : forall n, S (S n) <> 1. -Proof. - intros n Hn; discriminate. -Qed. - - -Theorem disc3 : forall n, S (S n) = 0 -> forall Q:Prop, Q. -Proof. - intros n Hn Q. - discriminate. -Qed. - - - -Theorem inj_succ : forall n m, S n = S m -> n = m. -Proof. - - -Lemma inj_pred : forall n m, n = m -> pred n = pred m. -Proof. - intros n m eq_n_m. - rewrite eq_n_m. - trivial. -Qed. - - intros n m eq_Sn_Sm. - apply inj_pred with (n:= S n) (m := S m); assumption. -Qed. - -Lemma list_inject : forall (A:Type)(a b :A)(l l':list A), - a :: b :: l = b :: a :: l' -> a = b /\ l = l'. -Proof. - intros A a b l l' e. - injection e. - auto. -Qed. - - -Theorem not_le_Sn_0 : forall n:nat, ~ (S n <= 0). -Proof. - red; intros n H. - case H. -Undo. - -Lemma not_le_Sn_0_with_constraints : - forall n p , S n <= p -> p = 0 -> False. -Proof. - intros n p H; case H ; - intros; discriminate. -Qed. - -eapply not_le_Sn_0_with_constraints; eauto. -Qed. - - -Theorem not_le_Sn_0' : forall n:nat, ~ (S n <= 0). -Proof. - red; intros n H ; inversion H. -Qed. - -Derive Inversion le_Sn_0_inv with (forall n :nat, S n <= 0). -Check le_Sn_0_inv. - -Theorem le_Sn_0'' : forall n p : nat, ~ S n <= 0 . -Proof. - intros n p H; - inversion H using le_Sn_0_inv. -Qed. - -Derive Inversion_clear le_Sn_0_inv' with (forall n :nat, S n <= 0). -Check le_Sn_0_inv'. - - -Theorem le_reverse_rules : - forall n m:nat, n <= m -> - n = m \/ - exists p, n <= p /\ m = S p. -Proof. - intros n m H; inversion H. - left;trivial. - right; exists m0; split; trivial. -Restart. - intros n m H; inversion_clear H. - left;trivial. - right; exists m0; split; trivial. -Qed. - -Inductive ArithExp : Set := - Zero : ArithExp - | Succ : ArithExp -> ArithExp - | Plus : ArithExp -> ArithExp -> ArithExp. - -Inductive RewriteRel : ArithExp -> ArithExp -> Prop := - RewSucc : forall e1 e2 :ArithExp, - RewriteRel e1 e2 -> RewriteRel (Succ e1) (Succ e2) - | RewPlus0 : forall e:ArithExp, - RewriteRel (Plus Zero e) e - | RewPlusS : forall e1 e2:ArithExp, - RewriteRel e1 e2 -> - RewriteRel (Plus (Succ e1) e2) (Succ (Plus e1 e2)). - - - -Fixpoint plus (n p:nat) {struct n} : nat := - match n with - | 0 => p - | S m => S (plus m p) - end. - -Fixpoint plus' (n p:nat) {struct p} : nat := - match p with - | 0 => n - | S q => S (plus' n q) - end. - -Fixpoint plus'' (n p:nat) {struct n} : nat := - match n with - | 0 => p - | S m => plus'' m (S p) - end. - - -Fixpoint even_test (n:nat) : bool := - match n - with 0 => true - | 1 => false - | S (S p) => even_test p - end. - - -Reset even_test. - -Fixpoint even_test (n:nat) : bool := - match n - with - | 0 => true - | S p => odd_test p - end -with odd_test (n:nat) : bool := - match n - with - | 0 => false - | S p => even_test p - end. - - - -Eval simpl in even_test. - - - -Eval simpl in (fun x : nat => even_test x). - -Eval simpl in (fun x : nat => plus 5 x). -Eval simpl in (fun x : nat => even_test (plus 5 x)). - -Eval simpl in (fun x : nat => even_test (plus x 5)). - - -Section Principle_of_Induction. -Variable P : nat -> Prop. -Hypothesis base_case : P 0. -Hypothesis inductive_step : forall n:nat, P n -> P (S n). -Fixpoint nat_ind (n:nat) : (P n) := - match n return P n with - | 0 => base_case - | S m => inductive_step m (nat_ind m) - end. - -End Principle_of_Induction. - -Scheme Even_induction := Minimality for even Sort Prop -with Odd_induction := Minimality for odd Sort Prop. - -Theorem even_plus_four : forall n:nat, even n -> even (4+n). -Proof. - intros n H. - elim H using Even_induction with (P0 := fun n => odd (4+n)); - simpl;repeat constructor;assumption. -Qed. - - -Section Principle_of_Double_Induction. -Variable P : nat -> nat ->Prop. -Hypothesis base_case1 : forall x:nat, P 0 x. -Hypothesis base_case2 : forall x:nat, P (S x) 0. -Hypothesis inductive_step : forall n m:nat, P n m -> P (S n) (S m). -Fixpoint nat_double_ind (n m:nat){struct n} : P n m := - match n, m return P n m with - | 0 , x => base_case1 x - | (S x), 0 => base_case2 x - | (S x), (S y) => inductive_step x y (nat_double_ind x y) - end. -End Principle_of_Double_Induction. - -Section Principle_of_Double_Recursion. -Variable P : nat -> nat -> Type. -Hypothesis base_case1 : forall x:nat, P 0 x. -Hypothesis base_case2 : forall x:nat, P (S x) 0. -Hypothesis inductive_step : forall n m:nat, P n m -> P (S n) (S m). -Fixpoint nat_double_rect (n m:nat){struct n} : P n m := - match n, m return P n m with - | 0 , x => base_case1 x - | (S x), 0 => base_case2 x - | (S x), (S y) => inductive_step x y (nat_double_rect x y) - end. -End Principle_of_Double_Recursion. - -Definition min : nat -> nat -> nat := - nat_double_rect (fun (x y:nat) => nat) - (fun (x:nat) => 0) - (fun (y:nat) => 0) - (fun (x y r:nat) => S r). - -Eval compute in (min 5 8). -Eval compute in (min 8 5). - - - -Lemma not_circular : forall n:nat, n <> S n. -Proof. - intro n. - apply nat_ind with (P:= fun n => n <> S n). - discriminate. - red; intros n0 Hn0 eqn0Sn0;injection eqn0Sn0;trivial. -Qed. - -Definition eq_nat_dec : forall n p:nat , {n=p}+{n <> p}. -Proof. - intros n p. - apply nat_double_rect with (P:= fun (n q:nat) => {q=p}+{q <> p}). -Undo. - pattern p,n. - elim n using nat_double_rect. - destruct x; auto. - destruct x; auto. - intros n0 m H; case H. - intro eq; rewrite eq ; auto. - intro neg; right; red ; injection 1; auto. -Defined. - -Definition eq_nat_dec' : forall n p:nat, {n=p}+{n <> p}. - decide equality. -Defined. - - - -Require Import Le. -Lemma le'_le : forall n p, le' n p -> n <= p. -Proof. - induction 1;auto with arith. -Qed. - -Lemma le'_n_Sp : forall n p, le' n p -> le' n (S p). -Proof. - induction 1;auto. -Qed. - -Hint Resolve le'_n_Sp. - - -Lemma le_le' : forall n p, n<=p -> le' n p. -Proof. - induction 1;auto with arith. -Qed. - - -Print Acc. - - -Require Import Minus. - -(* -Fixpoint div (x y:nat){struct x}: nat := - if eq_nat_dec x 0 - then 0 - else if eq_nat_dec y 0 - then x - else S (div (x-y) y). - -Error: -Recursive definition of div is ill-formed. -In environment -div : nat -> nat -> nat -x : nat -y : nat -_ : x <> 0 -_ : y <> 0 - -Recursive call to div has principal argument equal to -"x - y" -instead of a subterm of x - -*) - -Lemma minus_smaller_S: forall x y:nat, x - y < S x. -Proof. - intros x y; pattern y, x; - elim x using nat_double_ind. - destruct x0; auto with arith. - simpl; auto with arith. - simpl; auto with arith. -Qed. - -Lemma minus_smaller_positive : forall x y:nat, x <>0 -> y <> 0 -> - x - y < x. -Proof. - destruct x; destruct y; - ( simpl;intros; apply minus_smaller_S || - intros; absurd (0=0); auto). -Qed. - -Definition minus_decrease : forall x y:nat, Acc lt x -> - x <> 0 -> - y <> 0 -> - Acc lt (x-y). -Proof. - intros x y H; case H. - intros Hz posz posy. - apply Hz; apply minus_smaller_positive; assumption. -Defined. - -Print minus_decrease. - - - -Definition div_aux (x y:nat)(H: Acc lt x):nat. - fix 3. - intros. - refine (if eq_nat_dec x 0 - then 0 - else if eq_nat_dec y 0 - then y - else div_aux (x-y) y _). - apply (minus_decrease x y H);assumption. -Defined. - - -Print div_aux. -(* -div_aux = -(fix div_aux (x y : nat) (H : Acc lt x) {struct H} : nat := - match eq_nat_dec x 0 with - | left _ => 0 - | right _ => - match eq_nat_dec y 0 with - | left _ => y - | right _0 => div_aux (x - y) y (minus_decrease x y H _ _0) - end - end) - : forall x : nat, nat -> Acc lt x -> nat -*) - -Require Import Wf_nat. -Definition div x y := div_aux x y (lt_wf x). - -Extraction div. -(* -let div x y = - div_aux x y -*) - -Extraction div_aux. - -(* -let rec div_aux x y = - match eq_nat_dec x O with - | Left -> O - | Right -> - (match eq_nat_dec y O with - | Left -> y - | Right -> div_aux (minus x y) y) -*) - -Lemma vector0_is_vnil : forall (A:Type)(v:vector A 0), v = Vnil A. -Proof. - intros A v;inversion v. -Abort. - -(* - Lemma vector0_is_vnil_aux : forall (A:Type)(n:nat)(v:vector A n), - n= 0 -> v = Vnil A. - -Toplevel input, characters 40281-40287 -> Lemma vector0_is_vnil_aux : forall (A:Set)(n:nat)(v:vector A n), n= 0 -> v = Vnil A. -> ^^^^^^ -Error: In environment -A : Set -n : nat -v : vector A n -e : n = 0 -The term "Vnil A" has type "vector A 0" while it is expected to have type - "vector A n" -*) - Require Import JMeq. - - -(* On devrait changer Set en Type ? *) - -Lemma vector0_is_vnil_aux : forall (A:Type)(n:nat)(v:vector A n), - n= 0 -> JMeq v (Vnil A). -Proof. - destruct v. - auto. - intro; discriminate. -Qed. - -Lemma vector0_is_vnil : forall (A:Type)(v:vector A 0), v = Vnil A. -Proof. - intros a v;apply JMeq_eq. - apply vector0_is_vnil_aux. - trivial. -Qed. - - -Implicit Arguments Vcons [A n]. -Implicit Arguments Vnil [A]. -Implicit Arguments Vhead [A n]. -Implicit Arguments Vtail [A n]. - -Definition Vid : forall (A : Type)(n:nat), vector A n -> vector A n. -Proof. - destruct n; intro v. - exact Vnil. - exact (Vcons (Vhead v) (Vtail v)). -Defined. - -Eval simpl in (fun (A:Type)(v:vector A 0) => (Vid _ _ v)). - -Eval simpl in (fun (A:Type)(v:vector A 0) => v). - - - -Lemma Vid_eq : forall (n:nat) (A:Type)(v:vector A n), v=(Vid _ n v). -Proof. - destruct v. - reflexivity. - reflexivity. -Defined. - -Theorem zero_nil : forall A (v:vector A 0), v = Vnil. -Proof. - intros. - change (Vnil (A:=A)) with (Vid _ 0 v). - apply Vid_eq. -Defined. - - -Theorem decomp : - forall (A : Type) (n : nat) (v : vector A (S n)), - v = Vcons (Vhead v) (Vtail v). -Proof. - intros. - change (Vcons (Vhead v) (Vtail v)) with (Vid _ (S n) v). - apply Vid_eq. -Defined. - - - -Definition vector_double_rect : - forall (A:Type) (P: forall (n:nat),(vector A n)->(vector A n) -> Type), - P 0 Vnil Vnil -> - (forall n (v1 v2 : vector A n) a b, P n v1 v2 -> - P (S n) (Vcons a v1) (Vcons b v2)) -> - forall n (v1 v2 : vector A n), P n v1 v2. - induction n. - intros; rewrite (zero_nil _ v1); rewrite (zero_nil _ v2). - auto. - intros v1 v2; rewrite (decomp _ _ v1);rewrite (decomp _ _ v2). - apply X0; auto. -Defined. - -Require Import Bool. - -Definition bitwise_or n v1 v2 : vector bool n := - vector_double_rect bool (fun n v1 v2 => vector bool n) - Vnil - (fun n v1 v2 a b r => Vcons (orb a b) r) n v1 v2. - - -Fixpoint vector_nth (A:Type)(n:nat)(p:nat)(v:vector A p){struct v} - : option A := - match n,v with - _ , Vnil => None - | 0 , Vcons b _ _ => Some b - | S n', Vcons _ p' v' => vector_nth A n' p' v' - end. - -Implicit Arguments vector_nth [A p]. - - -Lemma nth_bitwise : forall (n:nat) (v1 v2: vector bool n) i a b, - vector_nth i v1 = Some a -> - vector_nth i v2 = Some b -> - vector_nth i (bitwise_or _ v1 v2) = Some (orb a b). -Proof. - intros n v1 v2; pattern n,v1,v2. - apply vector_double_rect. - simpl. - destruct i; discriminate 1. - destruct i; simpl;auto. - injection 1; injection 2;intros; subst a; subst b; auto. -Qed. - - Set Implicit Arguments. - - CoInductive Stream (A:Type) : Type := - | Cons : A -> Stream A -> Stream A. - - CoInductive LList (A: Type) : Type := - | LNil : LList A - | LCons : A -> LList A -> LList A. - - - - - - Definition head (A:Type)(s : Stream A) := match s with Cons a s' => a end. - - Definition tail (A : Type)(s : Stream A) := - match s with Cons a s' => s' end. - - CoFixpoint repeat (A:Type)(a:A) : Stream A := Cons a (repeat a). - - CoFixpoint iterate (A: Type)(f: A -> A)(a : A) : Stream A:= - Cons a (iterate f (f a)). - - CoFixpoint map (A B:Type)(f: A -> B)(s : Stream A) : Stream B:= - match s with Cons a tl => Cons (f a) (map f tl) end. - -Eval simpl in (fun (A:Type)(a:A) => repeat a). - -Eval simpl in (fun (A:Type)(a:A) => head (repeat a)). - - -CoInductive EqSt (A: Type) : Stream A -> Stream A -> Prop := - eqst : forall s1 s2: Stream A, - head s1 = head s2 -> - EqSt (tail s1) (tail s2) -> - EqSt s1 s2. - - -Section Parks_Principle. -Variable A : Type. -Variable R : Stream A -> Stream A -> Prop. -Hypothesis bisim1 : forall s1 s2:Stream A, R s1 s2 -> - head s1 = head s2. -Hypothesis bisim2 : forall s1 s2:Stream A, R s1 s2 -> - R (tail s1) (tail s2). - -CoFixpoint park_ppl : forall s1 s2:Stream A, R s1 s2 -> - EqSt s1 s2 := - fun s1 s2 (p : R s1 s2) => - eqst s1 s2 (bisim1 p) - (park_ppl (bisim2 p)). -End Parks_Principle. - - -Theorem map_iterate : forall (A:Type)(f:A->A)(x:A), - EqSt (iterate f (f x)) (map f (iterate f x)). -Proof. - intros A f x. - apply park_ppl with - (R:= fun s1 s2 => exists x: A, - s1 = iterate f (f x) /\ s2 = map f (iterate f x)). - - intros s1 s2 (x0,(eqs1,eqs2));rewrite eqs1;rewrite eqs2;reflexivity. - intros s1 s2 (x0,(eqs1,eqs2)). - exists (f x0);split;[rewrite eqs1|rewrite eqs2]; reflexivity. - exists x;split; reflexivity. -Qed. - -Ltac infiniteproof f := - cofix f; constructor; [clear f| simpl; try (apply f; clear f)]. - - -Theorem map_iterate' : forall (A:Type)(f:A->A)(x:A), - EqSt (iterate f (f x)) (map f (iterate f x)). -infiniteproof map_iterate'. - reflexivity. -Qed. - - -Implicit Arguments LNil [A]. - -Lemma Lnil_not_Lcons : forall (A:Type)(a:A)(l:LList A), - LNil <> (LCons a l). - intros;discriminate. -Qed. - -Lemma injection_demo : forall (A:Type)(a b : A)(l l': LList A), - LCons a (LCons b l) = LCons b (LCons a l') -> - a = b /\ l = l'. -Proof. - intros A a b l l' e; injection e; auto. -Qed. - - -Inductive Finite (A:Type) : LList A -> Prop := -| Lnil_fin : Finite (LNil (A:=A)) -| Lcons_fin : forall a l, Finite l -> Finite (LCons a l). - -CoInductive Infinite (A:Type) : LList A -> Prop := -| LCons_inf : forall a l, Infinite l -> Infinite (LCons a l). - -Lemma LNil_not_Infinite : forall (A:Type), ~ Infinite (LNil (A:=A)). -Proof. - intros A H;inversion H. -Qed. - -Lemma Finite_not_Infinite : forall (A:Type)(l:LList A), - Finite l -> ~ Infinite l. -Proof. - intros A l H; elim H. - apply LNil_not_Infinite. - intros a l0 F0 I0' I1. - case I0'; inversion_clear I1. - trivial. -Qed. - -Lemma Not_Finite_Infinite : forall (A:Type)(l:LList A), - ~ Finite l -> Infinite l. -Proof. - cofix H. - destruct l. - intro; absurd (Finite (LNil (A:=A)));[auto|constructor]. - constructor. - apply H. - red; intro H1;case H0. - constructor. - trivial. -Qed. - - - diff --git a/doc/RecTutorial/coqartmacros.tex b/doc/RecTutorial/coqartmacros.tex deleted file mode 100644 index 6fb7534d..00000000 --- a/doc/RecTutorial/coqartmacros.tex +++ /dev/null @@ -1,180 +0,0 @@ -\usepackage{url} - -\newcommand{\variantspringer}[1]{#1} -\newcommand{\marginok}[1]{\marginpar{\raggedright OK:#1}} -\newcommand{\tab}{{\null\hskip1cm}} -\newcommand{\Ltac}{\mbox{\emph{$\cal L$}tac}} -\newcommand{\coq}{\mbox{\emph{Coq}}} -\newcommand{\lcf}{\mbox{\emph{LCF}}} -\newcommand{\hol}{\mbox{\emph{HOL}}} -\newcommand{\pvs}{\mbox{\emph{PVS}}} -\newcommand{\isabelle}{\mbox{\emph{Isabelle}}} -\newcommand{\prolog}{\mbox{\emph{Prolog}}} -\newcommand{\goalbar}{\tt{}============================\it} -\newcommand{\gallina}{\mbox{\emph{Gallina}}} -\newcommand{\joker}{\texttt{\_}} -\newcommand{\eprime}{\(\e^{\prime}\)} -\newcommand{\Ztype}{\citecoq{Z}} -\newcommand{\propsort}{\citecoq{Prop}} -\newcommand{\setsort}{\citecoq{Set}} -\newcommand{\typesort}{\citecoq{Type}} -\newcommand{\ocaml}{\mbox{\emph{OCAML}}} -\newcommand{\haskell}{\mbox{\emph{Haskell}}} -\newcommand{\why}{\mbox{\emph{Why}}} -\newcommand{\Pascal}{\mbox{\emph{Pascal}}} - 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-%%% logique minimale -\newcommand{\propzero}{\mbox{$P_0$}} % types de Fzero - -%%% logique propositionnelle classique -\newcommand {\ff}{\boldsymbol{f}} % faux -\newcommand {\vv}{\boldsymbol{t}} % vrai - -\newcommand{\verite}{\mbox{$\cal{B}$}} % {\ff,\vv} -\newcommand{\sequ}[2]{\mbox{$#1 \vdash #2 $}} % sequent -\newcommand{\strip}[1]{#1^o} % enlever les variables d'un contexte - - - -%%% tactiques -\newcommand{\decomp}{\delta} % decomposition -\newcommand{\recomp}{\rho} % recomposition - -%%% divers -\newcommand{\cqfd}{\mbox{\textbf{cqfd}}} -\newcommand{\fail}{\mbox{\textbf{F}}} -\newcommand{\succes}{\mbox{$\blacksquare$}} -%%% Environnements - - -%% Fzero -\newcommand{\con}{\mbox{$\cal C$}} -\newcommand{\var}{\mbox{$\cal V$}} - -\newcommand{\atomzero}{\mbox{${\cal A}_0$}} % types de base de Fzero -\newcommand{\typezero}{\mbox{${\cal T}_0$}} % types de Fzero -\newcommand{\termzero}{\mbox{$\Lambda_0$}} % termes de Fzero -\newcommand{\conzero}{\mbox{$\cal C_0$}} % contextes de Fzero - -\newcommand{\buts}{\mbox{$\cal B$}} % buts - -%%% for drawing terms -% abstraction [x:t]e -\newcommand{\PicAbst}[3]{\begin{bundle}{\bf abst}\chunk{#1}\chunk{#2}\chunk{#3}% - \end{bundle}} - -% the same in DeBruijn form -\newcommand{\PicDbj}[2]{\begin{bundle}{\bf abst}\chunk{#1}\chunk{#2} - \end{bundle}} - - -% applications -\newcommand{\PicAppl}[2]{\begin{bundle}{\bf appl}\chunk{#1}\chunk{#2}% - \end{bundle}} - -% variables -\newcommand{\PicVar}[1]{\begin{bundle}{\bf var}\chunk{#1} - \end{bundle}} - -% constantes -\newcommand{\PicCon}[1]{\begin{bundle}{\bf const}\chunk{#1}\end{bundle}} - -% arrows -\newcommand{\PicImpl}[2]{\begin{bundle}{\impl}\chunk{#1}\chunk{#2}% - \end{bundle}} - - - -%%%% scripts coq -\newcommand{\prompt}{\mbox{\sl Coq $<\;$}} -\newcommand{\natquicksort}{\texttt{nat\_quicksort}} -\newcommand{\citecoq}[1]{\mbox{\texttt{#1}}} -\newcommand{\safeit}{\it} -\newtheorem{remarque}{Remark}[section] -%\newtheorem{definition}{Definition}[chapter] diff --git a/doc/RecTutorial/manbiblio.bib b/doc/RecTutorial/manbiblio.bib deleted file mode 100644 index 26064e86..00000000 --- a/doc/RecTutorial/manbiblio.bib +++ /dev/null @@ -1,870 +0,0 @@ - -@STRING{toappear="To appear"} -@STRING{lncs="Lecture Notes in Computer Science"} - -@TECHREPORT{RefManCoq, - AUTHOR = {Bruno~Barras, Samuel~Boutin, - Cristina~Cornes, Judicaël~Courant, Yann~Coscoy, David~Delahaye, - Daniel~de~Rauglaudre, Jean-Christophe~Filliâtre, Eduardo~Giménez, - Hugo~Herbelin, Gérard~Huet, Henri~Laulhère, César~Muñoz, - Chetan~Murthy, Catherine~Parent-Vigouroux, Patrick~Loiseleur, - Christine~Paulin-Mohring, Amokrane~Saïbi, Benjamin~Werner}, - INSTITUTION = {INRIA}, - TITLE = {{The Coq Proof Assistant Reference Manual -- Version V6.2}}, - YEAR = {1998} -} - -@INPROCEEDINGS{Aud91, - AUTHOR = {Ph. Audebaud}, - BOOKTITLE = {Proceedings of the sixth Conf. on Logic in Computer Science.}, - PUBLISHER = {IEEE}, - TITLE = {Partial {Objects} in the {Calculus of Constructions}}, - YEAR = {1991} -} - -@PHDTHESIS{Aud92, - AUTHOR = {Ph. Audebaud}, - SCHOOL = {{Universit\'e} Bordeaux I}, - TITLE = {Extension du Calcul des Constructions par Points fixes}, - YEAR = {1992} -} - -@INPROCEEDINGS{Audebaud92b, - AUTHOR = {Ph. Audebaud}, - BOOKTITLE = {{Proceedings of the 1992 Workshop on Types for Proofs and Programs}}, - EDITOR = {{B. Nordstr\"om and K. Petersson and G. Plotkin}}, - NOTE = {Also Research Report LIP-ENS-Lyon}, - PAGES = {pp 21--34}, - TITLE = {{CC+ : an extension of the Calculus of Constructions with fixpoints}}, - YEAR = {1992} -} - -@INPROCEEDINGS{Augustsson85, - AUTHOR = {L. Augustsson}, - TITLE = {{Compiling Pattern Matching}}, - BOOKTITLE = {Conference Functional Programming and -Computer Architecture}, - YEAR = {1985} -} - -@INPROCEEDINGS{EG94a, - AUTHOR = {E. Gim\'enez}, - EDITORS = {P. Dybjer and B. Nordstr\"om and J. Smith}, - BOOKTITLE = {Workshop on Types for Proofs and Programs}, - PAGES = {39-59}, - SERIES = {LNCS}, - NUMBER = {996}, - TITLE = {{Codifying guarded definitions with recursive schemes}}, - YEAR = {1994}, - PUBLISHER = {Springer-Verlag}, -} - -@INPROCEEDINGS{EG95a, - AUTHOR = {E. Gim\'enez}, - BOOKTITLE = {Workshop on Types for Proofs and Programs}, - SERIES = {LNCS}, - NUMBER = {1158}, - PAGES = {135-152}, - TITLE = {An application of co-Inductive types in Coq: - verification of the Alternating Bit Protocol}, - EDITORS = {S. Berardi and M. Coppo}, - PUBLISHER = {Springer-Verlag}, - YEAR = {1995} -} - -@PhdThesis{EG96, - author = {E. Gim\'enez}, - title = {A Calculus of Infinite Constructions and its - application to the verification of communicating systems}, - school = {Ecole Normale Sup\'erieure de Lyon}, - year = {1996} -} - -@ARTICLE{BaCo85, - AUTHOR = {J.L. Bates and R.L. Constable}, - JOURNAL = {ACM transactions on Programming Languages and Systems}, - TITLE = {Proofs as {Programs}}, - VOLUME = {7}, - YEAR = {1985} -} - -@BOOK{Bar81, - AUTHOR = {H.P. Barendregt}, - PUBLISHER = {North-Holland}, - TITLE = {The Lambda Calculus its Syntax and Semantics}, - YEAR = {1981} -} - -@TECHREPORT{Bar91, - AUTHOR = {H. Barendregt}, - INSTITUTION = {Catholic University Nijmegen}, - NOTE = {In Handbook of Logic in Computer Science, Vol II}, - NUMBER = {91-19}, - TITLE = {Lambda {Calculi with Types}}, - YEAR = {1991} -} - -@BOOK{Bastad92, - EDITOR = {B. Nordstr\"om and K. Petersson and G. Plotkin}, - PUBLISHER = {Available by ftp at site ftp.inria.fr}, - TITLE = {Proceedings of the 1992 Workshop on Types for Proofs and Programs}, - YEAR = {1992} -} - -@BOOK{Bee85, - AUTHOR = {M.J. Beeson}, - PUBLISHER = {Springer-Verlag}, - TITLE = {Foundations of Constructive Mathematics, Metamathematical Studies}, - YEAR = {1985} -} - -@ARTICLE{BeKe92, - AUTHOR = {G. Bellin and J. Ketonen}, - JOURNAL = {Theoretical Computer Science}, - PAGES = {115--142}, - TITLE = {A decision procedure revisited : Notes on direct logic, linear logic and its implementation}, - VOLUME = {95}, - YEAR = {1992} -} - -@BOOK{Bis67, - AUTHOR = {E. Bishop}, - PUBLISHER = {McGraw-Hill}, - TITLE = {Foundations of Constructive Analysis}, - YEAR = {1967} -} - -@BOOK{BoMo79, - AUTHOR = {R.S. Boyer and J.S. Moore}, - KEY = {BoMo79}, - PUBLISHER = {Academic Press}, - SERIES = {ACM Monograph}, - TITLE = {A computational logic}, - YEAR = {1979} -} - -@MASTERSTHESIS{Bou92, - AUTHOR = {S. Boutin}, - MONTH = sep, - SCHOOL = {{Universit\'e Paris 7}}, - TITLE = {Certification d'un compilateur {ML en Coq}}, - YEAR = {1992} -} - -@ARTICLE{Bru72, - AUTHOR = {N.J. de Bruijn}, - JOURNAL = {Indag. Math.}, - TITLE = {{Lambda-Calculus Notation with Nameless Dummies, a Tool for Automatic Formula Manipulation, with Application to the Church-Rosser Theorem}}, - VOLUME = {34}, - YEAR = {1972} -} - -@INCOLLECTION{Bru80, - AUTHOR = {N.J. de Bruijn}, - BOOKTITLE = {to H.B. Curry : Essays on Combinatory Logic, Lambda Calculus and Formalism.}, - EDITOR = {J.P. Seldin and J.R. Hindley}, - PUBLISHER = {Academic Press}, - TITLE = {A survey of the project {Automath}}, - YEAR = {1980} -} - -@TECHREPORT{Leroy90, - AUTHOR = {X. Leroy}, - TITLE = {The {ZINC} experiment: an economical implementation -of the {ML} language}, - INSTITUTION = {INRIA}, - NUMBER = {117}, - YEAR = {1990} -} - -@BOOK{Caml, - AUTHOR = {P. Weis and X. Leroy}, - PUBLISHER = {InterEditions}, - TITLE = {Le langage Caml}, - YEAR = {1993} -} - -@TECHREPORT{CoC89, - AUTHOR = {Projet Formel}, - INSTITUTION = {INRIA}, - NUMBER = {110}, - TITLE = {{The Calculus of Constructions. Documentation and user's guide, Version 4.10}}, - YEAR = {1989} -} - -@INPROCEEDINGS{CoHu85a, - AUTHOR = {Th. Coquand and G. Huet}, - ADDRESS = {Linz}, - BOOKTITLE = {EUROCAL'85}, - PUBLISHER = {Springer-Verlag}, - SERIES = {LNCS}, - TITLE = {{Constructions : A Higher Order Proof System for Mechanizing Mathematics}}, - VOLUME = {203}, - YEAR = {1985} -} - -@Misc{Bar98, - author = {B. Barras}, - title = {A formalisation of - \uppercase{B}urali-\uppercase{F}orti's paradox in Coq}, - howpublished = {Distributed within the bunch of contribution to the - Coq system}, - year = {1998}, - month = {March}, - note = {\texttt{http://pauillac.inria.fr/coq}} -} - - -@INPROCEEDINGS{CoHu85b, - AUTHOR = {Th. Coquand and G. Huet}, - BOOKTITLE = {Logic Colloquium'85}, - EDITOR = {The Paris Logic Group}, - PUBLISHER = {North-Holland}, - TITLE = {{Concepts Math\'ematiques et Informatiques formalis\'es dans le Calcul des Constructions}}, - YEAR = {1987} -} - -@ARTICLE{CoHu86, - AUTHOR = {Th. Coquand and G. Huet}, - JOURNAL = {Information and Computation}, - NUMBER = {2/3}, - TITLE = {The {Calculus of Constructions}}, - VOLUME = {76}, - YEAR = {1988} -} - -@BOOK{Con86, - AUTHOR = {R.L. {Constable et al.}}, - PUBLISHER = {Prentice-Hall}, - TITLE = {{Implementing Mathematics with the Nuprl Proof Development System}}, - YEAR = {1986} -} - -@INPROCEEDINGS{CoPa89, - AUTHOR = {Th. Coquand and C. Paulin-Mohring}, - BOOKTITLE = {Proceedings of Colog'88}, - EDITOR = {P. Martin-L{\"o}f and G. Mints}, - PUBLISHER = {Springer-Verlag}, - SERIES = {LNCS}, - TITLE = {Inductively defined types}, - VOLUME = {417}, - YEAR = {1990} -} - -@PHDTHESIS{Coq85, - AUTHOR = {Th. Coquand}, - MONTH = jan, - SCHOOL = {Universit\'e Paris~7}, - TITLE = {Une Th\'eorie des Constructions}, - YEAR = {1985} -} - -@INPROCEEDINGS{Coq86, - AUTHOR = {Th. Coquand}, - ADDRESS = {Cambridge, MA}, - BOOKTITLE = {Symposium on Logic in Computer Science}, - PUBLISHER = {IEEE Computer Society Press}, - TITLE = {{An Analysis of Girard's Paradox}}, - YEAR = {1986} -} - -@INPROCEEDINGS{Coq90, - AUTHOR = {Th. Coquand}, - BOOKTITLE = {Logic and Computer Science}, - EDITOR = {P. Oddifredi}, - NOTE = {INRIA Research Report 1088, also in~\cite{CoC89}}, - PUBLISHER = {Academic Press}, - TITLE = {{Metamathematical Investigations of a Calculus of Constructions}}, - YEAR = {1990} -} - -@INPROCEEDINGS{Coq92, - AUTHOR = {Th. Coquand}, - BOOKTITLE = {in \cite{Bastad92}}, - TITLE = {{Pattern Matching with Dependent Types}}, - YEAR = {1992}, - crossref = {Bastad92} -} - -@TECHREPORT{COQ93, - AUTHOR = {G. Dowek and A. Felty and H. Herbelin and G. Huet and C. Murthy and C. Parent and C. Paulin-Mohring and B. Werner}, - INSTITUTION = {INRIA}, - MONTH = may, - NUMBER = {154}, - TITLE = {{The Coq Proof Assistant User's Guide Version 5.8}}, - YEAR = {1993} -} - -@INPROCEEDINGS{Coquand93, - AUTHOR = {Th. Coquand}, - BOOKTITLE = {in \cite{Nijmegen93}}, - TITLE = {{Infinite Objects in Type Theory}}, - YEAR = {1993}, - crossref = {Nijmegen93} -} - -@MASTERSTHESIS{Cou94a, - AUTHOR = {J. Courant}, - MONTH = sep, - SCHOOL = {DEA d'Informatique, ENS Lyon}, - TITLE = {Explicitation de preuves par r\'ecurrence implicite}, - YEAR = {1994} -} - -@TECHREPORT{CPar93, - AUTHOR = {C. Parent}, - INSTITUTION = {Ecole {Normale} {Sup\'erieure} de {Lyon}}, - MONTH = oct, - NOTE = {Also in~\cite{Nijmegen93}}, - NUMBER = {93-29}, - TITLE = {Developing certified programs in the system {Coq}- {The} {Program} tactic}, - YEAR = {1993} -} - -@PHDTHESIS{CPar95, - AUTHOR = {C. Parent}, - SCHOOL = {Ecole {Normale} {Sup\'erieure} de {Lyon}}, - TITLE = {{Synth\`ese de preuves de programmes dans le Calcul des Constructions Inductives}}, - YEAR = {1995} -} - -@TECHREPORT{Dow90, - AUTHOR = {G. Dowek}, - INSTITUTION = {INRIA}, - NUMBER = {1283}, - TITLE = {{Naming and Scoping in a Mathematical Vernacular}}, - TYPE = {Research Report}, - YEAR = {1990} -} - -@ARTICLE{Dow91a, - AUTHOR = {G. Dowek}, - JOURNAL = {{Compte Rendu de l'Acad\'emie des Sciences}}, - NOTE = {(The undecidability of Third Order Pattern Matching in Calculi with Dependent Types or Type Constructors)}, - NUMBER = {12}, - PAGES = {951--956}, - TITLE = {{L'Ind\'ecidabilit\'e du Filtrage du Troisi\`eme Ordre dans les Calculs avec Types D\'ependants ou Constructeurs de Types}}, - VOLUME = {I, 312}, - YEAR = {1991} -} - -@INPROCEEDINGS{Dow91b, - AUTHOR = {G. Dowek}, - BOOKTITLE = {Proceedings of Mathematical Foundation of Computer Science}, - NOTE = {Also INRIA Research Report}, - PAGES = {151--160}, - PUBLISHER = {Springer-Verlag}, - SERIES = {LNCS}, - TITLE = {{A Second Order Pattern Matching Algorithm in the Cube of Typed {$\lambda$}-calculi}}, - VOLUME = {520}, - YEAR = {1991} -} - -@PHDTHESIS{Dow91c, - AUTHOR = {G. Dowek}, - MONTH = dec, - SCHOOL = {{Universit\'e Paris 7}}, - TITLE = {{D\'emonstration automatique dans le Calcul des Constructions}}, - YEAR = {1991} -} - -@ARTICLE{dowek93, - AUTHOR = {G. Dowek}, - TITLE = {{A Complete Proof Synthesis Method for the Cube of Type Systems}}, - JOURNAL = {Journal Logic Computation}, - VOLUME = {3}, - NUMBER = {3}, - PAGES = {287--315}, - MONTH = {June}, - YEAR = {1993} -} - -@UNPUBLISHED{Dow92a, - AUTHOR = {G. Dowek}, - NOTE = {To appear in Theoretical Computer Science}, - TITLE = {{The Undecidability of Pattern Matching in Calculi where Primitive Recursive Functions are Representable}}, - YEAR = {1992} -} - -@ARTICLE{Dow94a, - AUTHOR = {G. Dowek}, - JOURNAL = {Annals of Pure and Applied Logic}, - VOLUME = {69}, - PAGES = {135--155}, - TITLE = {Third order matching is decidable}, - YEAR = {1994} -} - -@INPROCEEDINGS{Dow94b, - AUTHOR = {G. Dowek}, - BOOKTITLE = {Proceedings of the second international conference on typed lambda calculus and applications}, - TITLE = {{Lambda-calculus, Combinators and the Comprehension Schema}}, - YEAR = {1995} -} - -@INPROCEEDINGS{Dyb91, - AUTHOR = {P. Dybjer}, - BOOKTITLE = {Logical Frameworks}, - EDITOR = {G. Huet and G. Plotkin}, - PAGES = {59--79}, - PUBLISHER = {Cambridge University Press}, - TITLE = {{Inductive sets and families in {Martin-L{\"o}f's Type Theory} and their set-theoretic semantics : An inversion principle for {Martin-L\"of's} type theory}}, - VOLUME = {14}, - YEAR = {1991} -} - -@ARTICLE{Dyc92, - AUTHOR = {Roy Dyckhoff}, - JOURNAL = {The Journal of Symbolic Logic}, - MONTH = sep, - NUMBER = {3}, - TITLE = {Contraction-free sequent calculi for intuitionistic logic}, - VOLUME = {57}, - YEAR = {1992} -} - -@MASTERSTHESIS{Fil94, - AUTHOR = {J.-C. Filli\^atre}, - MONTH = sep, - SCHOOL = {DEA d'Informatique, ENS Lyon}, - TITLE = {Une proc\'edure de d\'ecision pour le {C}alcul des {P}r\'edicats {D}irect. {E}tude et impl\'ementation dans le syst\`eme {C}oq}, - YEAR = {1994} -} - -@TECHREPORT{Filliatre95, - AUTHOR = {J.-C. Filli\^atre}, - INSTITUTION = {LIP-ENS-Lyon}, - TITLE = {{A decision procedure for Direct Predicate Calculus}}, - TYPE = {Research report}, - NUMBER = {96--25}, - YEAR = {1995} -} - -@UNPUBLISHED{Fle90, - AUTHOR = {E. Fleury}, - MONTH = jul, - NOTE = {Rapport de Stage}, - TITLE = {Implantation des algorithmes de {Floyd et de Dijkstra} dans le {Calcul des Constructions}}, - YEAR = {1990} -} - - -@TechReport{Gim98, - author = {E. Gim\'nez}, - title = {A Tutorial on Recursive Types in Coq}, - institution = {INRIA}, - year = {1998} -} - -@TECHREPORT{HKP97, - author = {G. Huet and G. Kahn and Ch. Paulin-Mohring}, - title = {The {Coq} Proof Assistant - A tutorial, Version 6.1}, - institution = {INRIA}, - type = {rapport technique}, - month = {Août}, - year = {1997}, - note = {Version révisée distribuée avec {Coq}}, - number = {204}, -} - -@INPROCEEDINGS{Gir70, - AUTHOR = {J.-Y. Girard}, - BOOKTITLE = {Proceedings of the 2nd Scandinavian Logic Symposium}, - PUBLISHER = {North-Holland}, - TITLE = {Une extension de l'interpr\'etation de {G\"odel} \`a l'analyse, et son application \`a l'\'elimination des coupures dans l'analyse et la th\'eorie des types}, - YEAR = {1970} -} - -@PHDTHESIS{Gir72, - AUTHOR = {J.-Y. Girard}, - SCHOOL = {Universit\'e Paris~7}, - TITLE = {Interpr\'etation fonctionnelle et \'elimination des coupures de l'arithm\'etique d'ordre sup\'erieur}, - YEAR = {1972} -} - -@BOOK{Gir89, - AUTHOR = {J.-Y. Girard and Y. Lafont and P. Taylor}, - PUBLISHER = {Cambridge University Press}, - SERIES = {Cambridge Tracts in Theoretical Computer Science 7}, - TITLE = {Proofs and Types}, - YEAR = {1989} -} - -@MASTERSTHESIS{Hir94, - AUTHOR = {D. Hirschkoff}, - MONTH = sep, - SCHOOL = {DEA IARFA, Ecole des Ponts et Chauss\'ees, Paris}, - TITLE = {{Ecriture d'une tactique arithm\'etique pour le syst\`eme Coq}}, - YEAR = {1994} -} - -@INCOLLECTION{How80, - AUTHOR = {W.A. Howard}, - BOOKTITLE = {to H.B. Curry : Essays on Combinatory Logic, Lambda Calculus and Formalism.}, - EDITOR = {J.P. Seldin and J.R. Hindley}, - NOTE = {Unpublished 1969 Manuscript}, - PUBLISHER = {Academic Press}, - TITLE = {The Formulae-as-Types Notion of Constructions}, - YEAR = {1980} -} - -@INCOLLECTION{HuetLevy79, - AUTHOR = {G. Huet and J.-J. L\'{e}vy}, - TITLE = {Call by Need Computations in Non-Ambigous -Linear Term Rewriting Systems}, - NOTE = {Also research report 359, INRIA, 1979}, - BOOKTITLE = {Computational Logic, Essays in Honor of -Alan Robinson}, - EDITOR = {J.-L. Lassez and G. Plotkin}, - PUBLISHER = {The MIT press}, - YEAR = {1991} -} - -@INPROCEEDINGS{Hue87, - AUTHOR = {G. Huet}, - BOOKTITLE = {Programming of Future Generation Computers}, - EDITOR = {K. Fuchi and M. Nivat}, - NOTE = {Also in Proceedings of TAPSOFT87, LNCS 249, Springer-Verlag, 1987, pp 276--286}, - PUBLISHER = {Elsevier Science}, - TITLE = {Induction Principles Formalized in the {Calculus of Constructions}}, - YEAR = {1988} -} - -@INPROCEEDINGS{Hue88, - AUTHOR = {G. Huet}, - BOOKTITLE = {A perspective in Theoretical Computer Science. Commemorative Volume for Gift Siromoney}, - EDITOR = {R. Narasimhan}, - NOTE = {Also in~\cite{CoC89}}, - PUBLISHER = {World Scientific Publishing}, - TITLE = {{The Constructive Engine}}, - YEAR = {1989} -} - -@BOOK{Hue89, - EDITOR = {G. Huet}, - PUBLISHER = {Addison-Wesley}, - SERIES = {The UT Year of Programming Series}, - TITLE = {Logical Foundations of Functional Programming}, - YEAR = {1989} -} - -@INPROCEEDINGS{Hue92, - AUTHOR = {G. Huet}, - BOOKTITLE = {Proceedings of 12th FST/TCS Conference, New Delhi}, - PAGES = {229--240}, - PUBLISHER = {Springer Verlag}, - SERIES = {LNCS}, - TITLE = {{The Gallina Specification Language : A case study}}, - VOLUME = {652}, - YEAR = {1992} -} - -@ARTICLE{Hue94, - AUTHOR = {G. Huet}, - JOURNAL = {J. Functional Programming}, - PAGES = {371--394}, - PUBLISHER = {Cambridge University Press}, - TITLE = {Residual theory in $\lambda$-calculus: a formal development}, - VOLUME = {4,3}, - YEAR = {1994} -} - -@ARTICLE{KeWe84, - AUTHOR = {J. Ketonen and R. Weyhrauch}, - JOURNAL = {Theoretical Computer Science}, - PAGES = {297--307}, - TITLE = {A decidable fragment of {P}redicate {C}alculus}, - VOLUME = {32}, - YEAR = {1984} -} - -@BOOK{Kle52, - AUTHOR = {S.C. Kleene}, - PUBLISHER = {North-Holland}, - SERIES = {Bibliotheca Mathematica}, - TITLE = {Introduction to Metamathematics}, - YEAR = {1952} -} - -@BOOK{Kri90, - AUTHOR = {J.-L. Krivine}, - PUBLISHER = {Masson}, - SERIES = {Etudes et recherche en informatique}, - TITLE = {Lambda-calcul {types et mod\`eles}}, - YEAR = {1990} -} - -@ARTICLE{Laville91, - AUTHOR = {A. Laville}, - TITLE = {Comparison of Priority Rules in Pattern -Matching and Term Rewriting}, - JOURNAL = {Journal of Symbolic Computation}, - VOLUME = {11}, - PAGES = {321--347}, - YEAR = {1991} -} - -@BOOK{LE92, - EDITOR = {G. Huet and G. Plotkin}, - PUBLISHER = {Cambridge University Press}, - TITLE = {Logical Environments}, - YEAR = {1992} -} - -@INPROCEEDINGS{LePa94, - AUTHOR = {F. Leclerc and C. Paulin-Mohring}, - BOOKTITLE = {{Types for Proofs and Programs, Types' 93}}, - EDITOR = {H. Barendregt and T. Nipkow}, - PUBLISHER = {Springer-Verlag}, - SERIES = {LNCS}, - TITLE = {{Programming with Streams in Coq. A case study : The Sieve of Eratosthenes}}, - VOLUME = {806}, - YEAR = {1994} -} - -@BOOK{LF91, - EDITOR = {G. Huet and G. Plotkin}, - PUBLISHER = {Cambridge University Press}, - TITLE = {Logical Frameworks}, - YEAR = {1991} -} - -@BOOK{MaL84, - AUTHOR = {{P. Martin-L\"of}}, - PUBLISHER = {Bibliopolis}, - SERIES = {Studies in Proof Theory}, - TITLE = {Intuitionistic Type Theory}, - YEAR = {1984} -} - -@INPROCEEDINGS{manoury94, - AUTHOR = {P. Manoury}, - TITLE = {{A User's Friendly Syntax to Define -Recursive Functions as Typed $\lambda-$Terms}}, - BOOKTITLE = {{Types for Proofs and Programs, TYPES'94}}, - SERIES = {LNCS}, - VOLUME = {996}, - MONTH = jun, - YEAR = {1994} -} - -@ARTICLE{MaSi94, - AUTHOR = {P. Manoury and M. Simonot}, - JOURNAL = {TCS}, - TITLE = {Automatizing termination proof of recursively defined function}, - YEAR = {To appear} -} - -@TECHREPORT{maranget94, - AUTHOR = {L. Maranget}, - INSTITUTION = {INRIA}, - NUMBER = {2385}, - TITLE = {{Two Techniques for Compiling Lazy Pattern Matching}}, - YEAR = {1994} -} - -@INPROCEEDINGS{Moh89a, - AUTHOR = {C. Paulin-Mohring}, - ADDRESS = {Austin}, - BOOKTITLE = {Sixteenth Annual ACM Symposium on Principles of Programming Languages}, - MONTH = jan, - PUBLISHER = {ACM}, - TITLE = {Extracting ${F}_{\omega}$'s programs from proofs in the {Calculus of Constructions}}, - YEAR = {1989} -} - -@PHDTHESIS{Moh89b, - AUTHOR = {C. Paulin-Mohring}, - MONTH = jan, - SCHOOL = {{Universit\'e Paris 7}}, - TITLE = {Extraction de programmes dans le {Calcul des Constructions}}, - YEAR = {1989} -} - -@INPROCEEDINGS{Moh93, - AUTHOR = {C. Paulin-Mohring}, - BOOKTITLE = {Proceedings of the conference Typed Lambda Calculi and Applications}, - EDITOR = {M. Bezem and J.-F. Groote}, - NOTE = {Also LIP research report 92-49, ENS Lyon}, - NUMBER = {664}, - PUBLISHER = {Springer-Verlag}, - SERIES = {LNCS}, - TITLE = {{Inductive Definitions in the System Coq - Rules and Properties}}, - YEAR = {1993} -} - -@MASTERSTHESIS{Mun94, - AUTHOR = {C. Mu\~noz}, - MONTH = sep, - SCHOOL = {DEA d'Informatique Fondamentale, Universit\'e Paris 7}, - TITLE = {D\'emonstration automatique dans la logique propositionnelle intuitionniste}, - YEAR = {1994} -} - -@BOOK{Nijmegen93, - EDITOR = {H. Barendregt and T. Nipkow}, - PUBLISHER = {Springer-Verlag}, - SERIES = {LNCS}, - TITLE = {Types for Proofs and Programs}, - VOLUME = {806}, - YEAR = {1994} -} - -@BOOK{NoPS90, - AUTHOR = {B. {Nordstr\"om} and K. Peterson and J. Smith}, - BOOKTITLE = {Information Processing 83}, - PUBLISHER = {Oxford Science Publications}, - SERIES = {International Series of Monographs on Computer Science}, - TITLE = {Programming in {Martin-L\"of's} Type Theory}, - YEAR = {1990} -} - -@ARTICLE{Nor88, - AUTHOR = {B. {Nordstr\"om}}, - JOURNAL = {BIT}, - TITLE = {Terminating General Recursion}, - VOLUME = {28}, - YEAR = {1988} -} - -@BOOK{Odi90, - EDITOR = {P. Odifreddi}, - PUBLISHER = {Academic Press}, - TITLE = {Logic and Computer Science}, - YEAR = {1990} -} - -@INPROCEEDINGS{PaMS92, - AUTHOR = {M. Parigot and P. Manoury and M. Simonot}, - ADDRESS = {St. Petersburg, Russia}, - BOOKTITLE = {Logic Programming and automated reasoning}, - EDITOR = {A. Voronkov}, - MONTH = jul, - NUMBER = {624}, - PUBLISHER = {Springer-Verlag}, - SERIES = {LNCS}, - TITLE = {{ProPre : A Programming language with proofs}}, - YEAR = {1992} -} - -@ARTICLE{Par92, - AUTHOR = {M. Parigot}, - JOURNAL = {Theoretical Computer Science}, - NUMBER = {2}, - PAGES = {335--356}, - TITLE = {{Recursive Programming with Proofs}}, - VOLUME = {94}, - YEAR = {1992} -} - -@INPROCEEDINGS{Parent95b, - AUTHOR = {C. Parent}, - BOOKTITLE = {{Mathematics of Program Construction'95}}, - PUBLISHER = {Springer-Verlag}, - SERIES = {LNCS}, - TITLE = {{Synthesizing proofs from programs in -the Calculus of Inductive Constructions}}, - VOLUME = {947}, - YEAR = {1995} -} - -@ARTICLE{PaWe92, - AUTHOR = {C. Paulin-Mohring and B. Werner}, - JOURNAL = {Journal of Symbolic Computation}, - PAGES = {607--640}, - TITLE = {{Synthesis of ML programs in the system Coq}}, - VOLUME = {15}, - YEAR = {1993} -} - -@INPROCEEDINGS{Prasad93, - AUTHOR = {K.V. Prasad}, - BOOKTITLE = {{Proceedings of CONCUR'93}}, - PUBLISHER = {Springer-Verlag}, - SERIES = {LNCS}, - TITLE = {{Programming with broadcasts}}, - VOLUME = {715}, - YEAR = {1993} -} - -@INPROCEEDINGS{puel-suarez90, - AUTHOR = {L.Puel and A. Su\'arez}, - BOOKTITLE = {{Conference Lisp and Functional Programming}}, - SERIES = {ACM}, - PUBLISHER = {Springer-Verlag}, - TITLE = {{Compiling Pattern Matching by Term -Decomposition}}, - YEAR = {1990} -} - -@UNPUBLISHED{Rou92, - AUTHOR = {J. Rouyer}, - MONTH = aug, - NOTE = {To appear as a technical report}, - TITLE = {{D\'eveloppement de l'Algorithme d'Unification dans le Calcul des Constructions}}, - YEAR = {1992} -} - -@TECHREPORT{Saibi94, - AUTHOR = {A. Sa\"{\i}bi}, - INSTITUTION = {INRIA}, - MONTH = dec, - NUMBER = {2345}, - TITLE = {{Axiomatization of a lambda-calculus with explicit-substitutions in the Coq System}}, - YEAR = {1994} -} - -@MASTERSTHESIS{saidi94, - AUTHOR = {H. Saidi}, - MONTH = sep, - SCHOOL = {DEA d'Informatique Fondamentale, Universit\'e Paris 7}, - TITLE = {R\'esolution d'\'equations dans le syst\`eme T - de G\"odel}, - YEAR = {1994} -} - -@MASTERSTHESIS{Ter92, - AUTHOR = {D. Terrasse}, - MONTH = sep, - SCHOOL = {IARFA}, - TITLE = {{Traduction de TYPOL en COQ. Application \`a Mini ML}}, - YEAR = {1992} -} - -@TECHREPORT{ThBeKa92, - AUTHOR = {L. Th\'ery and Y. Bertot and G. Kahn}, - INSTITUTION = {INRIA Sophia}, - MONTH = may, - NUMBER = {1684}, - TITLE = {Real theorem provers deserve real user-interfaces}, - TYPE = {Research Report}, - YEAR = {1992} -} - -@BOOK{TrDa89, - AUTHOR = {A.S. Troelstra and D. van Dalen}, - PUBLISHER = {North-Holland}, - SERIES = {Studies in Logic and the foundations of Mathematics, volumes 121 and 123}, - TITLE = {Constructivism in Mathematics, an introduction}, - YEAR = {1988} -} - -@INCOLLECTION{wadler87, - AUTHOR = {P. Wadler}, - TITLE = {Efficient Compilation of Pattern Matching}, - BOOKTITLE = {The Implementation of Functional Programming -Languages}, - EDITOR = {S.L. Peyton Jones}, - PUBLISHER = {Prentice-Hall}, - YEAR = {1987} -} - -@PHDTHESIS{Wer94, - AUTHOR = {B. Werner}, - SCHOOL = {Universit\'e Paris 7}, - TITLE = {Une th\'eorie des constructions inductives}, - TYPE = {Th\`ese de Doctorat}, - YEAR = {1994} -} - - diff --git a/doc/RecTutorial/morebib.bib b/doc/RecTutorial/morebib.bib deleted file mode 100644 index 11dde2cd..00000000 --- a/doc/RecTutorial/morebib.bib +++ /dev/null @@ -1,55 +0,0 @@ -@book{coqart, - title = "Interactive Theorem Proving and Program Development. - Coq'Art: The Calculus of Inductive Constructions", - author = "Yves Bertot and Pierre Castéran", - publisher = "Springer Verlag", - series = "Texts in Theoretical Computer Science. An EATCS series", - year = 2004 -} - -@Article{Coquand:Huet, - author = {Thierry Coquand and Gérard Huet}, - title = {The Calculus of Constructions}, - journal = {Information and Computation}, - year = {1988}, - volume = {76}, -} - -@INcollection{Coquand:metamathematical, - author = "Thierry Coquand", - title = "Metamathematical Investigations on a Calculus of Constructions", - booktitle="Logic and Computer Science", - year = {1990}, - editor="P. Odifreddi", - publisher = "Academic Press", -} - -@Misc{coqrefman, - title = {The {C}oq reference manual}, - author={{C}oq {D}evelopment Team}, - note= {LogiCal Project, \texttt{http://coq.inria.fr/}} - } - -@Misc{coqsite, - author= {{C}oq {D}evelopment Team}, - title = {The \emph{Coq} proof assistant}, - note = {Documentation, system download. {C}ontact: \texttt{http://coq.inria.fr/}} -} - - - -@Misc{Booksite, - author = {Yves Bertot and Pierre Cast\'eran}, - title = {Coq'{A}rt: examples and exercises}, - note = {\url{http://www.labri.fr/Perso/~casteran/CoqArt}} -} - - -@InProceedings{conor:motive, - author ="Conor McBride", - title = "Elimination with a motive", - booktitle = "Types for Proofs and Programs'2000", - volume = 2277, - pages = "197-217", - year = "2002", -} diff --git a/doc/RecTutorial/recmacros.tex b/doc/RecTutorial/recmacros.tex deleted file mode 100644 index 0334553f..00000000 --- a/doc/RecTutorial/recmacros.tex +++ /dev/null @@ -1,75 +0,0 @@ -%=================================== -% Style of the document -%=================================== -%\newtheorem{example}{Example}[section] -%\newtheorem{exercise}{Exercise}[section] - - -\newcommand{\comentario}[1]{\texttt{#1}} - -%=================================== -% Keywords -%=================================== - -\newcommand{\Prop}{\texttt{Prop}} -\newcommand{\Set}{\texttt{Set}} -\newcommand{\Type}{\texttt{Type}} -\newcommand{\true}{\texttt{true}} -\newcommand{\false}{\texttt{false}} -\newcommand{\Lth}{\texttt{Lth}} - -\newcommand{\Nat}{\texttt{nat}} -\newcommand{\nat}{\texttt{nat}} -\newcommand{\Z} {\texttt{O}} -\newcommand{\SUCC}{\texttt{S}} -\newcommand{\pred}{\texttt{pred}} - -\newcommand{\False}{\texttt{False}} -\newcommand{\True}{\texttt{True}} -\newcommand{\I}{\texttt{I}} - -\newcommand{\natind}{\texttt{nat\_ind}} -\newcommand{\natrec}{\texttt{nat\_rec}} -\newcommand{\natrect}{\texttt{nat\_rect}} - -\newcommand{\eqT}{\texttt{eqT}} -\newcommand{\identityT}{\texttt{identityT}} - -\newcommand{\map}{\texttt{map}} -\newcommand{\iterates}{\texttt{iterates}} - - -%=================================== -% Numbering -%=================================== - - -\newtheorem{definition}{Definition}[section] -\newtheorem{example}{Example}[section] - - -%=================================== -% Judgements -%=================================== - - -\newcommand{\JM}[2]{\ensuremath{#1 : #2}} - -%=================================== -% Expressions -%=================================== - -\newcommand{\Case}[3][]{\ensuremath{#1\textsf{Case}~#2~\textsf of}~#3~\textsf{end}} - -%======================================= - -\newcommand{\snreglados} [3] {\begin{tabular}{c} \ensuremath{#1} \\[2pt] - \ensuremath{#2}\\ \hline \ensuremath{#3} \end{tabular}} - - -\newcommand{\snregla} [2] {\begin{tabular}{c} - \ensuremath{#1}\\ \hline \ensuremath{#2} \end{tabular}} - - -%======================================= - diff --git a/doc/common/macros.tex b/doc/common/macros.tex deleted file mode 100755 index d745f34a..00000000 --- a/doc/common/macros.tex +++ /dev/null @@ -1,529 +0,0 @@ -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -% MACROS FOR THE REFERENCE MANUAL OF COQ % -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -% For commentaries (define \com as {} for the release manual) -%\newcommand{\com}[1]{{\it(* #1 *)}} -%\newcommand{\com}[1]{} - -%%OPTIONS for HACHA -%\renewcommand{\cuttingunit}{section} - - -%BEGIN LATEX -\newenvironment{centerframe}% -{\bgroup -\dimen0=\textwidth -\advance\dimen0 by -2\fboxrule -\advance\dimen0 by -2\fboxsep -\setbox0=\hbox\bgroup -\begin{minipage}{\dimen0}% -\begin{center}}% -{\end{center}% -\end{minipage}\egroup -\centerline{\fbox{\box0}}\egroup -} -%END LATEX -%HEVEA \newenvironment{centerframe}{\begin{center}}{\end{center}} - -%HEVEA \renewcommand{\vec}[1]{\mathbf{#1}} -%\renewcommand{\ominus}{-} % Hevea does a good job translating these commands -%\renewcommand{\oplus}{+} -%\renewcommand{\otimes}{\times} -%\newcommand{\land}{\wedge} -%\newcommand{\lor}{\vee} -%HEVEA \renewcommand{\k}[1]{#1} % \k{a} is supposed to produce a with a little stroke -%HEVEA \newcommand{\phantom}[1]{\qquad} - -%%%%%%%%%%%%%%%%%%%%%%% -% Formatting commands % -%%%%%%%%%%%%%%%%%%%%%%% - -\newcommand{\ErrMsg}{\medskip \noindent {\bf Error message: }} -\newcommand{\ErrMsgx}{\medskip \noindent {\bf Error messages: }} -\newcommand{\variant}{\medskip \noindent {\bf Variant: }} -\newcommand{\variants}{\medskip \noindent {\bf Variants: }} -\newcommand{\SeeAlso}{\medskip \noindent {\bf See also: }} -\newcommand{\Rem}{\medskip \noindent {\bf Remark: }} -\newcommand{\Rems}{\medskip \noindent {\bf Remarks: }} -\newcommand{\Example}{\medskip \noindent {\bf Example: }} -\newcommand{\examples}{\medskip \noindent {\bf Examples: }} -\newcommand{\Warning}{\medskip \noindent {\bf Warning: }} -\newcommand{\Warns}{\medskip \noindent {\bf Warnings: }} -\newcounter{ex} -\newcommand{\firstexample}{\setcounter{ex}{1}} -\newcommand{\example}[1]{ -\medskip \noindent \textbf{Example \arabic{ex}: }\textit{#1} -\addtocounter{ex}{1}} - 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-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -% \mbox{\sf } series for roman text in maths formulas % -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\newcommand{\alors}{\mbox{\textsf{then}}} -\newcommand{\alter}{\mbox{\textsf{alter}}} -\newcommand{\bool}{\mbox{\textsf{bool}}} -\newcommand{\conc}{\mbox{\textsf{conc}}} -\newcommand{\cons}{\mbox{\textsf{cons}}} -\newcommand{\consf}{\mbox{\textsf{consf}}} -\newcommand{\emptyf}{\mbox{\textsf{emptyf}}} -\newcommand{\EqSt}{\mbox{\textsf{EqSt}}} -\newcommand{\false}{\mbox{\textsf{false}}} -\newcommand{\filter}{\mbox{\textsf{filter}}} -\newcommand{\forest}{\mbox{\textsf{forest}}} -\newcommand{\from}{\mbox{\textsf{from}}} -\newcommand{\hd}{\mbox{\textsf{hd}}} -\newcommand{\Length}{\mbox{\textsf{Length}}} -\newcommand{\length}{\mbox{\textsf{length}}} -\newcommand{\LengthA}{\mbox {\textsf{Length\_A}}} -\newcommand{\List}{\mbox{\textsf{List}}} -\newcommand{\ListA}{\mbox{\textsf{List\_A}}} -\newcommand{\LNil}{\mbox{\textsf{Lnil}}} -\newcommand{\LCons}{\mbox{\textsf{Lcons}}} -\newcommand{\nat}{\mbox{\textsf{nat}}} -\newcommand{\nO}{\mbox{\textsf{O}}} -\newcommand{\nS}{\mbox{\textsf{S}}} -\newcommand{\node}{\mbox{\textsf{node}}} -\newcommand{\Nil}{\mbox{\textsf{nil}}} -\newcommand{\Prop}{\mbox{\textsf{Prop}}} -\newcommand{\Set}{\mbox{\textsf{Set}}} -\newcommand{\si}{\mbox{\textsf{if}}} -\newcommand{\sinon}{\mbox{\textsf{else}}} -\newcommand{\Str}{\mbox{\textsf{Stream}}} -\newcommand{\tl}{\mbox{\textsf{tl}}} -\newcommand{\tree}{\mbox{\textsf{tree}}} -\newcommand{\true}{\mbox{\textsf{true}}} -\newcommand{\Type}{\mbox{\textsf{Type}}} -\newcommand{\unfold}{\mbox{\textsf{unfold}}} -\newcommand{\zeros}{\mbox{\textsf{zeros}}} - 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#3}$\hss}\vss}}\cr -% \noalign{\hrule} -% \noalign{\vskip\the\lineskip} -% \mud{\copy\tempa}\cr}} -% \tempc=\wd\tempb -% \advance\tempc by \wd\tempa -% \divide\tempc by 2 } -% \newcommand{\irule}[3]{{\irulehelp{#1}{#2}{#3} -% \hbox to \wd\tempa{\hss \box\tempb \hss}}} - -\newcommand{\sverb}[1]{{\tt #1}} -\newcommand{\mover}[2]{{#1\over #2}} -\newcommand{\jd}[2]{#1 \vdash #2} -\newcommand{\mathline}[1]{\[#1\]} -\newcommand{\zrule}[2]{#2: #1} -\newcommand{\orule}[3]{#3: {\mover{#1}{#2}}} -\newcommand{\trule}[4]{#4: \mover{#1 \qquad #2} {#3}} -\newcommand{\thrule}[5]{#5: {\mover{#1 \qquad #2 \qquad #3}{#4}}} - - - -% placement of figures - -%BEGIN LATEX -\renewcommand{\topfraction}{.99} -\renewcommand{\bottomfraction}{.99} -\renewcommand{\textfraction}{.01} -\renewcommand{\floatpagefraction}{.9} -%END LATEX - -% Macros Bruno pour description de la syntaxe - -\def\bfbar{\ensuremath{|\hskip -0.22em{}|\hskip -0.24em{}|}} -\def\TERMbar{\bfbar} -\def\TERMbarbar{\bfbar\bfbar} - - -%% Macros pour les grammaires -\def\GR#1{\text{\large(}#1\text{\large)}} -\def\NT#1{\langle\textit{#1}\rangle} -\def\NTL#1#2{\langle\textit{#1}\rangle_{#2}} -\def\TERM#1{{\bf\textrm{\bf #1}}} -%\def\TERM#1{{\bf\textsf{#1}}} -\def\KWD#1{\TERM{#1}} -\def\ETERM#1{\TERM{#1}} -\def\CHAR#1{\TERM{#1}} - -\def\STAR#1{#1*} -\def\STARGR#1{\GR{#1}*} -\def\PLUS#1{#1+} -\def\PLUSGR#1{\GR{#1}+} -\def\OPT#1{#1?} -\def\OPTGR#1{\GR{#1}?} -%% Tableaux de definition de non-terminaux -\newenvironment{cadre} - {\begin{array}{|c|}\hline\\} - {\\\\\hline\end{array}} -\newenvironment{rulebox} - {$$\begin{cadre}\begin{array}{r@{~}c@{~}l@{}l@{}r}} - {\end{array}\end{cadre}$$} -\def\DEFNT#1{\NT{#1} & ::= &} -\def\EXTNT#1{\NT{#1} & ::= & ... \\&|&} -\def\RNAME#1{(\textsc{#1})} -\def\SEPDEF{\\\\} -\def\nlsep{\\&|&} -\def\nlcont{\\&&} -\newenvironment{rules} - {\begin{center}\begin{rulebox}} - {\end{rulebox}\end{center}} - -% $Id: macros.tex 13091 2010-06-08 13:56:19Z herbelin $ - - -%%% Local Variables: -%%% mode: latex -%%% TeX-master: "Reference-Manual" -%%% End: diff --git a/doc/common/styles/html/coqremote/cover.html b/doc/common/styles/html/coqremote/cover.html deleted file mode 100644 index c3091b4e..00000000 --- a/doc/common/styles/html/coqremote/cover.html +++ /dev/null @@ -1,131 +0,0 @@ -<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" - "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> -<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"> - -<head> - -<meta http-equiv="Content-Type" content="text/html; 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font-weight:bold; font-size: 300%; line-height: 2ex">Reference Manual</h1> - -<h2 style="text-align:center; font-size: 120%"> - Version 8.3<a name="text1"></a><sup><a href="#note1"><span style="font-size: 80%">1</span></a></sup> - -<br/><br/><br/><br/><br/><br/> -<span style="text-align: center; font-size: 120%; ">The Coq Development Team</span> -<br/><br/><br/><br/><br/><br/> -</h2> - -<div style="text-align: left; font-size: 80%; text-indent: 0pt"> -<ul style="list-style: none; margin-left: 0pt"> - <li>V7.x © INRIA 1999-2004</li> - <li>V8.x © INRIA 2004-2010</li> -</ul> - -<p style="text-indent: 0pt">This material may be distributed only subject to the terms and conditions set forth in the Open Publication License, v1.0 or later (the latest version is presently available at <a href="http://www.opencontent.org/openpub">http://www.opencontent.org/openpub</a>). Options A and B are not elected.</p> -</div> -<br/> - -<hr style="width: 75%"/> -<div style="text-align: left; font-size: 80%; text-indent: 0pt"> -<dl> - <dt><a name="note1" href="toc.html#text1">1</a></dt> - <dd>This research was partly supported by IST working group ``Types''</dd> -</dl> -</div> - -</div> - -</div> - -<div id="sidebarWrapper"> -<div id="sidebar"> - -<div class="block"> -<h2 class="title">Navigation</h2> -<div class="content"> - -<ul class="menu"> - -<li class="leaf"><a href="index.html">Cover</a></li> - -<li class="leaf"><a href="toc.html">Table of contents</a></li> -<li class="leaf">Index - <ul class="menu"> - <li><a href="general-index.html">General</a></li> - <li><a href="command-index.html">Commands</a></li> - <li><a href="tactic-index.html">Tactics</a></li> - <li><a href="error-index.html">Errors</a></li> - </ul> - -</li> - -</ul> - -</div> -</div> - -</div> -</div> - - -<div id="footer"> -<div id="nav-footer"> - <ul class="links-menu-footer"> - <li><a href="mailto:www-coq at lix.polytechnique.fr">webmaster</a></li> - <li><a href="http://validator.w3.org/check?uri=referer">xhtml valid</a></li> - <li><a href="http://jigsaw.w3.org/css-validator/check/referer">CSS valid</a></li> - </ul> - -</div> - -</div> - -</body> - -</html> diff --git a/doc/common/styles/html/coqremote/hevea.css b/doc/common/styles/html/coqremote/hevea.css deleted file mode 100644 index 5f4edef6..00000000 --- a/doc/common/styles/html/coqremote/hevea.css +++ /dev/null @@ -1,36 +0,0 @@ - -.li-itemize{margin:1ex 0ex;} -.li-enumerate{margin:1ex 0ex;} -.dd-description{margin:0ex 0ex 1ex 4ex;} -.dt-description{margin:0ex;} -.toc{list-style:none;} -.thefootnotes{text-align:left;margin:0ex;} -.dt-thefootnotes{margin:0em;} -.dd-thefootnotes{margin:0em 0em 0em 2em;} -.footnoterule{margin:1em auto 1em 0px;width:50%;} -.caption{padding-left:2ex; 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-<link rel="stylesheet" type="text/css" href="style.css"> -<link rel="stylesheet" type="text/css" href="coqdoc.css"> -<link rel="stylesheet" type="text/css" href="hevea.css"> - -\end{rawhtml}} - -% for HeVeA - -\htmlhead{\begin{rawhtml} - -<div id="container"> - -<div id="header"> -<h1>Coq Reference Manual</h1> -</div> - -<div id="content"> - -\end{rawhtml}} - -\htmlfoot{\begin{rawhtml} - -<div id="footer"> - -<div class="content"> -<ul class="menu"> - <li><a href="index.html">Cover</a></li> - <li><a href="toc.html">Table of contents</a></li> - <li><a href="general-index.html">General index</a></li> - <li><a href="command-index.html">Commands index</a></li> - <li><a href="tactic-index.html">Tactics index</a></li> - <li><a href="error-index.html">Errors index</a></li> -</ul> - -</div> -</div> - -</div> -</div> -\end{rawhtml}} - diff --git a/doc/common/title.tex b/doc/common/title.tex deleted file mode 100755 index 38ec0e55..00000000 --- a/doc/common/title.tex +++ /dev/null @@ -1,73 +0,0 @@ -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -% File title.tex -% Page formatting commands -% Macro \coverpage -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -%\setlength{\marginparwidth}{0pt} -%\setlength{\oddsidemargin}{0pt} -%\setlength{\evensidemargin}{0pt} -%\setlength{\marginparsep}{0pt} -%\setlength{\topmargin}{0pt} -%\setlength{\textwidth}{16.9cm} -%\setlength{\textheight}{22cm} -%\usepackage{fullpage} - -%\newcommand{\printingdate}{\today} -%\newcommand{\isdraft}{\Large\bf\today\\[20pt]} -%\newcommand{\isdraft}{\vspace{20pt}} - -\newcommand{\coverpage}[3]{ -\thispagestyle{empty} -\begin{center} -\bfseries % for the rest of this page, until \end{center} -\Huge -The Coq Proof Assistant\\[12pt] -#1\\[20pt] -\Large\today\\[20pt] -Version \coqversion%\footnote[1]{This research was partly supported by IST working group ``Types''} - -\vspace{0pt plus .5fill} -#2 -\par\vfill -The Coq Development Team - -\vspace*{15pt} -\end{center} -\newpage - -\thispagestyle{empty} -\hbox{}\vfill % without \hbox \vfill does not work at the top of the page -\begin{flushleft} -%BEGIN LATEX -V\coqversion, \today -\par\vspace{20pt} -%END LATEX -\copyright INRIA 1999-2004 ({\Coq} versions 7.x) - -\copyright INRIA 2004-2011 ({\Coq} versions 8.x) - -#3 -\end{flushleft} -} % end of \coverpage definition - - -% \newcommand{\shorttitle}[1]{ -% \begin{center} -% \begin{huge} -% \begin{bf} -% The Coq Proof Assistant\\ -% \vspace{10pt} -% #1\\ -% \end{bf} -% \end{huge} -% \end{center} -% \vspace{5pt} -% } - -% Local Variables: -% mode: LaTeX -% TeX-master: "" -% End: - -% $Id: title.tex 14641 2011-11-06 11:59:10Z herbelin $ diff --git a/doc/faq/FAQ.tex b/doc/faq/FAQ.tex deleted file mode 100644 index bd6f7dbf..00000000 --- a/doc/faq/FAQ.tex +++ /dev/null @@ -1,2557 +0,0 @@ -\RequirePackage{ifpdf} -\ifpdf % si on est en pdflatex -\documentclass[a4paper,pdftex]{article} -\else -\documentclass[a4paper]{article} -\fi -\pagestyle{plain} - -% yay les symboles -\usepackage{stmaryrd} -\usepackage{amssymb} -\usepackage{url} -%\usepackage{multicol} -\usepackage{hevea} -\usepackage{fullpage} -\usepackage[latin1]{inputenc} -\usepackage[english]{babel} - -\ifpdf % si on est en pdflatex - \usepackage[pdftex]{graphicx} -\else - \usepackage[dvips]{graphicx} -\fi - -\input{../common/version.tex} -%\input{../macros.tex} - -% Making hevea happy -%HEVEA \renewcommand{\textbar}{|} -%HEVEA \renewcommand{\textunderscore}{\_} - -\def\Question#1{\stepcounter{question}\subsubsection{#1}} - -% version et date -\def\faqversion{0.1} - -% les macros d'amour -\def\Coq{\textsc{Coq}} -\def\Why{\textsc{Why}} -\def\Caduceus{\textsc{Caduceus}} -\def\Krakatoa{\textsc{Krakatoa}} -\def\Ltac{\textsc{Ltac}} -\def\CoqIde{\textsc{CoqIde}} - -\newcommand{\coqtt}[1]{{\tt #1}} -\newcommand{\coqimp}{{\mbox{\tt ->}}} -\newcommand{\coqequiv}{{\mbox{\tt <->}}} - - -% macro pour les tactics -\def\split{{\tt split}} -\def\assumption{{\tt assumption}} -\def\auto{{\tt auto}} -\def\trivial{{\tt trivial}} -\def\tauto{{\tt tauto}} -\def\left{{\tt left}} -\def\right{{\tt right}} -\def\decompose{{\tt decompose}} -\def\intro{{\tt intro}} -\def\intros{{\tt intros}} -\def\field{{\tt field}} -\def\ring{{\tt ring}} -\def\apply{{\tt apply}} -\def\exact{{\tt exact}} -\def\cut{{\tt cut}} -\def\assert{{\tt assert}} -\def\solve{{\tt solve}} -\def\idtac{{\tt idtac}} -\def\fail{{\tt fail}} -\def\existstac{{\tt exists}} -\def\firstorder{{\tt firstorder}} -\def\congruence{{\tt congruence}} -\def\gb{{\tt gb}} -\def\generalize{{\tt generalize}} -\def\abstracttac{{\tt abstract}} -\def\eapply{{\tt eapply}} -\def\unfold{{\tt unfold}} -\def\rewrite{{\tt rewrite}} -\def\replace{{\tt replace}} -\def\simpl{{\tt simpl}} -\def\elim{{\tt elim}} -\def\set{{\tt set}} -\def\pose{{\tt pose}} -\def\case{{\tt case}} -\def\destruct{{\tt destruct}} -\def\reflexivity{{\tt reflexivity}} -\def\transitivity{{\tt transitivity}} -\def\symmetry{{\tt symmetry}} -\def\Focus{{\tt Focus}} -\def\discriminate{{\tt discriminate}} -\def\contradiction{{\tt contradiction}} -\def\intuition{{\tt intuition}} -\def\try{{\tt try}} -\def\repeat{{\tt repeat}} -\def\eauto{{\tt eauto}} -\def\subst{{\tt subst}} -\def\symmetryin{{\tt symmetryin}} -\def\instantiate{{\tt instantiate}} -\def\inversion{{\tt inversion}} -\def\specialize{{\tt specialize}} -\def\Defined{{\tt Defined}} -\def\Qed{{\tt Qed}} -\def\pattern{{\tt pattern}} -\def\Type{{\tt Type}} -\def\Prop{{\tt Prop}} -\def\Set{{\tt Set}} - - -\newcommand\vfile[2]{\ahref{#1}{\tt {#2}.v}} -\urldef{\InitWf}\url - {http://coq.inria.fr/library/Coq.Init.Wf.html} -\urldef{\LogicBerardi}\url - {http://coq.inria.fr/library/Coq.Logic.Berardi.html} -\urldef{\LogicClassical}\url - {http://coq.inria.fr/library/Coq.Logic.Classical.html} -\urldef{\LogicClassicalFacts}\url - {http://coq.inria.fr/library/Coq.Logic.ClassicalFacts.html} -\urldef{\LogicClassicalDescription}\url - {http://coq.inria.fr/library/Coq.Logic.ClassicalDescription.html} -\urldef{\LogicProofIrrelevance}\url - {http://coq.inria.fr/library/Coq.Logic.ProofIrrelevance.html} -\urldef{\LogicEqdep}\url - {http://coq.inria.fr/library/Coq.Logic.Eqdep.html} -\urldef{\LogicEqdepDec}\url - {http://coq.inria.fr/library/Coq.Logic.Eqdep_dec.html} - - - - -\begin{document} -\bibliographystyle{plain} -\newcounter{question} -\renewcommand{\thesubsubsection}{\arabic{question}} - -%%%%%%% Coq pour les nuls %%%%%%% - -\title{Coq Version {\coqversion} for the Clueless\\ - \large(\protect\ref{lastquestion} - \ Hints) -} -\author{Pierre Castéran \and Hugo Herbelin \and Florent Kirchner \and Benjamin Monate \and Julien Narboux} -\maketitle - -%%%%%%% - -\begin{abstract} -This note intends to provide an easy way to get acquainted with the -{\Coq} theorem prover. It tries to formulate appropriate answers -to some of the questions any newcomers will face, and to give -pointers to other references when possible. -\end{abstract} - -%%%%%%% - -%\begin{multicols}{2} -\tableofcontents -%\end{multicols} - -%%%%%%% - -\newpage - -\section{Introduction} -This FAQ is the sum of the questions that came to mind as we developed -proofs in \Coq. Since we are singularly short-minded, we wrote the -answers we found on bits of papers to have them at hand whenever the -situation occurs again. This is pretty much the result of that: a -collection of tips one can refer to when proofs become intricate. Yes, -this means we won't take the blame for the shortcomings of this -FAQ. But if you want to contribute and send in your own question and -answers, feel free to write to us\ldots - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\section{Presentation} - -\Question{What is {\Coq}?}\label{whatiscoq} -The {\Coq} tool is a formal proof management system: a proof done with {\Coq} is mechanically checked by the machine. -In particular, {\Coq} allows: -\begin{itemize} - \item the definition of mathematical objects and programming objects, - \item to state mathematical theorems and software specifications, - \item to interactively develop formal proofs of these theorems, - \item to check these proofs by a small certification ``kernel''. -\end{itemize} -{\Coq} is based on a logical framework called ``Calculus of Inductive -Constructions'' extended by a modular development system for theories. - -\Question{Did you really need to name it like that?} -Some French computer scientists have a tradition of naming their -software as animal species: Caml, Elan, Foc or Phox are examples -of this tacit convention. In French, ``coq'' means rooster, and it -sounds like the initials of the Calculus of Constructions CoC on which -it is based. - -\Question{Is {\Coq} a theorem prover?} - -{\Coq} comes with decision and semi-decision procedures ( -propositional calculus, Presburger's arithmetic, ring and field -simplification, resolution, ...) but the main style for proving -theorems is interactively by using LCF-style tactics. - - -\Question{What are the other theorem provers?} -Many other theorem provers are available for use nowadays. -Isabelle, HOL, HOL Light, Lego, Nuprl, PVS are examples of provers that are fairly similar -to {\Coq} by the way they interact with the user. Other relatives of -{\Coq} are ACL2, Agda/Alfa, Twelf, Kiv, Mizar, NqThm, -\begin{htmlonly}% -Omega\ldots -\end{htmlonly} -\begin{latexonly}% -{$\Omega$}mega\ldots -\end{latexonly} - -\Question{What do I have to trust when I see a proof checked by Coq?} - -You have to trust: - -\begin{description} -\item[The theory behind Coq] The theory of {\Coq} version 8.0 is -generally admitted to be consistent wrt Zermelo-Fraenkel set theory + -inaccessible cardinals. Proofs of consistency of subsystems of the -theory of Coq can be found in the literature. -\item[The Coq kernel implementation] You have to trust that the -implementation of the {\Coq} kernel mirrors the theory behind {\Coq}. The -kernel is intentionally small to limit the risk of conceptual or -accidental implementation bugs. -\item[The Objective Caml compiler] The {\Coq} kernel is written using the -Objective Caml language but it uses only the most standard features -(no object, no label ...), so that it is highly unprobable that an -Objective Caml bug breaks the consistency of {\Coq} without breaking all -other kinds of features of {\Coq} or of other software compiled with -Objective Caml. -\item[Your hardware] In theory, if your hardware does not work -properly, it can accidentally be the case that False becomes -provable. But it is more likely the case that the whole {\Coq} system -will be unusable. You can check your proof using different computers -if you feel the need to. -\item[Your axioms] Your axioms must be consistent with the theory -behind {\Coq}. -\end{description} - - -\Question{Where can I find information about the theory behind {\Coq}?} -\begin{description} -\item[The Calculus of Inductive Constructions] The -\ahref{http://coq.inria.fr/doc/Reference-Manual006.html}{corresponding} -chapter and the chapter on -\ahref{http://coq.inria.fr/doc/Reference-Manual007.html}{modules} in -the {\Coq} Reference Manual. -\item[Type theory] A book~\cite{ProofsTypes} or some lecture -notes~\cite{Types:Dowek}. -\item[Inductive types] -Christine Paulin-Mohring's habilitation thesis~\cite{Pau96b}. -\item[Co-Inductive types] -Eduardo Giménez' thesis~\cite{EGThese}. -\item[Miscellaneous] A -\ahref{http://coq.inria.fr/doc/biblio.html}{bibliography} about Coq -\end{description} - - -\Question{How can I use {\Coq} to prove programs?} - -You can either extract a program from a proof by using the extraction -mechanism or use dedicated tools, such as -\ahref{http://why.lri.fr}{\Why}, -\ahref{http://krakatoa.lri.fr}{\Krakatoa}, -\ahref{http://why.lri.fr/caduceus/index.en.html}{\Caduceus}, to prove -annotated programs written in other languages. - -%\Question{How many {\Coq} users are there?} -% -%An estimation is about 100 regular users. - -\Question{How old is {\Coq}?} - -The first implementation is from 1985 (it was named {\sf CoC} which is -the acronym of the name of the logic it implemented: the Calculus of -Constructions). The first official release of {\Coq} (version 4.10) -was distributed in 1989. - -\Question{What are the \Coq-related tools?} - -There are graphical user interfaces: -\begin{description} -\item[Coqide] A GTK based GUI for \Coq. -\item[Pcoq] A GUI for {\Coq} with proof by pointing and pretty printing. -\item[coqwc] A tool similar to {\tt wc} to count lines in {\Coq} files. -\item[Proof General] A emacs mode for {\Coq} and many other proof assistants. -\item[ProofWeb] The ProofWeb online web interface for {\Coq} (and other proof assistants), with a focus on teaching. -\item[ProverEditor] is an experimental Eclipse plugin with support for {\Coq}. -\end{description} - -There are documentation and browsing tools: - -\begin{description} -\item[Helm/Mowgli] A rendering, searching and publishing tool. -\item[coq-tex] A tool to insert {\Coq} examples within .tex files. -\item[coqdoc] A documentation tool for \Coq. -\item[coqgraph] A tool to generate a dependency graph from {\Coq} sources. -\end{description} - -There are front-ends for specific languages: - -\begin{description} -\item[Why] A back-end generator of verification conditions. -\item[Krakatoa] A Java code certification tool that uses both {\Coq} and {\Why} to verify the soundness of implementations with regards to the specifications. -\item[Caduceus] A C code certification tool that uses both {\Coq} and \Why. -\item[Zenon] A first-order theorem prover. -\item[Focal] The \ahref{http://focal.inria.fr}{Focal} project aims at building an environment to develop certified computer algebra libraries. -\item[Concoqtion] is a dependently-typed extension of Objective Caml (and of MetaOCaml) with specifications expressed and proved in Coq. -\item[Ynot] is an extension of Coq providing a "Hoare Type Theory" for specifying higher-order, imperative and concurrent programs. -\item[Ott]is a tool to translate the descriptions of the syntax and semantics of programming languages to the syntax of Coq, or of other provers. -\end{description} - -\Question{What are the high-level tactics of \Coq} - -\begin{itemize} -\item Decision of quantifier-free Presburger's Arithmetic -\item Simplification of expressions on rings and fields -\item Decision of closed systems of equations -\item Semi-decision of first-order logic -\item Prolog-style proof search, possibly involving equalities -\end{itemize} - -\Question{What are the main libraries available for \Coq} - -\begin{itemize} -\item Basic Peano's arithmetic, binary integer numbers, rational numbers, -\item Real analysis, -\item Libraries for lists, boolean, maps, floating-point numbers, -\item Libraries for relations, sets and constructive algebra, -\item Geometry -\end{itemize} - - -\Question{What are the mathematical applications for {\Coq}?} - -{\Coq} is used for formalizing mathematical theories, for teaching, -and for proving properties of algorithms or programs libraries. - -The largest mathematical formalization has been done at the University -of Nijmegen (see the -\ahref{http://c-corn.cs.ru.nl}{Constructive Coq -Repository at Nijmegen}). - -A symbolic step has also been obtained by formalizing in full a proof -of the Four Color Theorem. - -\Question{What are the industrial applications for {\Coq}?} - -{\Coq} is used e.g. to prove properties of the JavaCard system -(especially by Schlumberger and Trusted Logic). It has -also been used to formalize the semantics of the Lucid-Synchrone -data-flow synchronous calculus used by Esterel-Technologies. - -\iffalse -todo christine compilo lustre? -\fi - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\section{Documentation} - -\Question{Where can I find documentation about {\Coq}?} -All the documentation about \Coq, from the reference manual~\cite{Coq:manual} to -friendly tutorials~\cite{Coq:Tutorial} and documentation of the standard library, is available -\ahref{http://coq.inria.fr/doc-eng.html}{online}. -All these documents are viewable either in browsable HTML, or as -downloadable postscripts. - -\Question{Where can I find this FAQ on the web?} - -This FAQ is available online at \ahref{http://coq.inria.fr/faq}{\url{http://coq.inria.fr/faq}}. - -\Question{How can I submit suggestions / improvements / additions for this FAQ?} - -This FAQ is unfinished (in the sense that there are some obvious -sections that are missing). Please send contributions to Coq-Club. - -\Question{Is there any mailing list about {\Coq}?} -The main {\Coq} mailing list is \url{coq-club@inria.fr}, which -broadcasts questions and suggestions about the implementation, the -logical formalism or proof developments. See -\ahref{http://coq.inria.fr/mailman/listinfo/coq-club}{\url{https://sympa-roc.inria.fr/wws/info/coq-club}} for -subscription. For bugs reports see question \ref{coqbug}. - -\Question{Where can I find an archive of the list?} -The archives of the {\Coq} mailing list are available at -\ahref{http://pauillac.inria.fr/pipermail/coq-club}{\url{https://sympa-roc.inria.fr/wws/arc/coq-club}}. - - -\Question{How can I be kept informed of new releases of {\Coq}?} - -New versions of {\Coq} are announced on the coq-club mailing list. - - -\Question{Is there any book about {\Coq}?} - -The first book on \Coq, Yves Bertot and Pierre Castéran's Coq'Art has been published by Springer-Verlag in 2004: -\begin{quote} -``This book provides a pragmatic introduction to the development of -proofs and certified programs using \Coq. With its large collection of -examples and exercises it is an invaluable tool for researchers, -students, and engineers interested in formal methods and the -development of zero-default software.'' -\end{quote} - -\Question{Where can I find some {\Coq} examples?} - -There are examples in the manual~\cite{Coq:manual} and in the -Coq'Art~\cite{Coq:coqart} exercises \ahref{\url{http://www.labri.fr/Perso/~casteran/CoqArt/index.html}}{\url{http://www.labri.fr/Perso/~casteran/CoqArt/index.html}}. -You can also find large developments using -{\Coq} in the {\Coq} user contributions: -\ahref{http://coq.inria.fr/contribs}{\url{http://coq.inria.fr/contribs}}. - -\Question{How can I report a bug?}\label{coqbug} - -You can use the web interface accessible at \ahref{http://coq.inria.fr/bugs}{\url{http://coq.inria.fr/bugs}}. - - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\section{Installation} - -\Question{What is the license of {\Coq}?} -{\Coq} is distributed under the GNU Lesser General License -(LGPL). - -\Question{Where can I find the sources of {\Coq}?} -The sources of {\Coq} can be found online in the tar.gz'ed packages -(\ahref{http://coq.inria.fr}{\url{http://coq.inria.fr}}, link -``download''). Development sources can be accessed at -\ahref{https://gforge.inria.fr/scm/?group_id=269}{\url{https://gforge.inria.fr/scm/?group_id=269}} - -\Question{On which platform is {\Coq} available?} -Compiled binaries are available for Linux, MacOS X, and Windows. The -sources can be easily compiled on all platforms supporting Objective -Caml. - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\section{The logic of {\Coq}} - -\subsection{General} - -\Question{What is the logic of \Coq?} - -{\Coq} is based on an axiom-free type theory called -the Calculus of Inductive Constructions (see Coquand \cite{CoHu86}, -Luo~\cite{Luo90} -and Coquand--Paulin-Mohring \cite{CoPa89}). It includes higher-order -functions and predicates, inductive and co-inductive datatypes and -predicates, and a stratified hierarchy of sets. - -\Question{Is \Coq's logic intuitionistic or classical?} - -{\Coq}'s logic is modular. The core logic is intuitionistic -(i.e. excluded-middle $A\vee\neg A$ is not granted by default). It can -be extended to classical logic on demand by requiring an -optional module stating $A\vee\neg A$. - -\Question{Can I define non-terminating programs in \Coq?} - -All programs in {\Coq} are terminating. Especially, loops -must come with an evidence of their termination. - -Non-terminating programs can be simulated by passing around a -bound on how long the program is allowed to run before dying. - -\Question{How is equational reasoning working in {\Coq}?} - - {\Coq} comes with an internal notion of computation called -{\em conversion} (e.g. $(x+1)+y$ is internally equivalent to -$(x+y)+1$; similarly applying argument $a$ to a function mapping $x$ -to some expression $t$ converts to the expression $t$ where $x$ is -replaced by $a$). This notion of conversion (which is decidable -because {\Coq} programs are terminating) covers a certain part of -equational reasoning but is limited to sequential evaluation of -expressions of (not necessarily closed) programs. Besides conversion, -equations have to be treated by hand or using specialised tactics. - -\subsection{Axioms} - -\Question{What axioms can be safely added to {\Coq}?} - -There are a few typical useful axioms that are independent from the -Calculus of Inductive Constructions and that are considered consistent with -the theory of {\Coq}. -Most of these axioms are stated in the directory {\tt Logic} of the -standard library of {\Coq}. The most interesting ones are - -\begin{itemize} -\item Excluded-middle: $\forall A:Prop, A \vee \neg A$ -\item Proof-irrelevance: $\forall A:Prop \forall p_1 p_2:A, p_1=p_2$ -\item Unicity of equality proofs (or equivalently Streicher's axiom $K$): -$\forall A \forall x y:A \forall p_1 p_2:x=y, p_1=p_2$ -\item Hilbert's $\epsilon$ operator: if $A \neq \emptyset$, then there is $\epsilon_P$ such that $\exists x P(x) \rightarrow P(\epsilon_P)$ -\item Church's $\iota$ operator: if $A \neq \emptyset$, then there is $\iota_P$ such that $\exists! x P(x) \rightarrow P(\iota_P)$ -\item The axiom of unique choice: $\forall x \exists! y R(x,y) \rightarrow \exists f \forall x R(x,f(x))$ -\item The functional axiom of choice: $\forall x \exists y R(x,y) \rightarrow \exists f \forall x R(x,f(x))$ -\item Extensionality of predicates: $\forall P Q:A\rightarrow Prop, (\forall x, P(x) \leftrightarrow Q(x)) \rightarrow P=Q$ -\item Extensionality of functions: $\forall f g:A\rightarrow B, (\forall x, f(x)=g(x)) \rightarrow f=g$ -\end{itemize} - -Here is a summary of the relative strength of these axioms, most -proofs can be found in directory {\tt Logic} of the standard library. -The justification of their validity relies on the interpretability in -set theory. - -%HEVEA\imgsrc{axioms.png} -%BEGIN LATEX -\ifpdf % si on est en pdflatex -\includegraphics[width=1.0\textwidth]{axioms.png} -\else -\includegraphics[width=1.0\textwidth]{axioms.eps} -\fi -%END LATEX - -\Question{What standard axioms are inconsistent with {\Coq}?} - -The axiom of unique choice together with classical logic -(e.g. excluded-middle) are inconsistent in the variant of the Calculus -of Inductive Constructions where {\Set} is impredicative. - -As a consequence, the functional form of the axiom of choice and -excluded-middle, or any form of the axiom of choice together with -predicate extensionality are inconsistent in the {\Set}-impredicative -version of the Calculus of Inductive Constructions. - -The main purpose of the \Set-predicative restriction of the Calculus -of Inductive Constructions is precisely to accommodate these axioms -which are quite standard in mathematical usage. - -The $\Set$-predicative system is commonly considered consistent by -interpreting it in a standard set-theoretic boolean model, even with -classical logic, axiom of choice and predicate extensionality added. - -\Question{What is Streicher's axiom $K$} -\label{Streicher} - -Streicher's axiom $K$~\cite{HofStr98} is an axiom that asserts -dependent elimination of reflexive equality proofs. - -\begin{coq_example*} -Axiom Streicher_K : - forall (A:Type) (x:A) (P: x=x -> Prop), - P (refl_equal x) -> forall p: x=x, P p. -\end{coq_example*} - -In the general case, axiom $K$ is an independent statement of the -Calculus of Inductive Constructions. However, it is true on decidable -domains (see file \vfile{\LogicEqdepDec}{Eqdep\_dec}). It is also -trivially a consequence of proof-irrelevance (see -\ref{proof-irrelevance}) hence of classical logic. - -Axiom $K$ is equivalent to {\em Uniqueness of Identity Proofs} \cite{HofStr98} - -\begin{coq_example*} -Axiom UIP : forall (A:Set) (x y:A) (p1 p2: x=y), p1 = p2. -\end{coq_example*} - -Axiom $K$ is also equivalent to {\em Uniqueness of Reflexive Identity Proofs} \cite{HofStr98} - -\begin{coq_example*} -Axiom UIP_refl : forall (A:Set) (x:A) (p: x=x), p = refl_equal x. -\end{coq_example*} - -Axiom $K$ is also equivalent to - -\begin{coq_example*} -Axiom - eq_rec_eq : - forall (A:Set) (x:A) (P: A->Set) (p:P x) (h: x=x), - p = eq_rect x P p x h. -\end{coq_example*} - -It is also equivalent to the injectivity of dependent equality (dependent equality is itself equivalent to equality of dependent pairs). - -\begin{coq_example*} -Inductive eq_dep (U:Set) (P:U -> Set) (p:U) (x:P p) : -forall q:U, P q -> Prop := - eq_dep_intro : eq_dep U P p x p x. -Axiom - eq_dep_eq : - forall (U:Set) (u:U) (P:U -> Set) (p1 p2:P u), - eq_dep U P u p1 u p2 -> p1 = p2. -\end{coq_example*} - -\Question{What is proof-irrelevance} -\label{proof-irrelevance} - -A specificity of the Calculus of Inductive Constructions is to permit -statements about proofs. This leads to the question of comparing two -proofs of the same proposition. Identifying all proofs of the same -proposition is called {\em proof-irrelevance}: -$$ -\forall A:\Prop, \forall p q:A, p=q -$$ - -Proof-irrelevance (in {\Prop}) can be assumed without contradiction in -{\Coq}. It expresses that only provability matters, whatever the exact -form of the proof is. This is in harmony with the common purely -logical interpretation of {\Prop}. Contrastingly, proof-irrelevance is -inconsistent in {\Set} since there are types in {\Set}, such as the -type of booleans, that provably have at least two distinct elements. - -Proof-irrelevance (in {\Prop}) is a consequence of classical logic -(see proofs in file \vfile{\LogicClassical}{Classical} and -\vfile{\LogicBerardi}{Berardi}). Proof-irrelevance is also a -consequence of propositional extensionality (i.e. \coqtt{(A {\coqequiv} B) -{\coqimp} A=B}, see the proof in file -\vfile{\LogicClassicalFacts}{ClassicalFacts}). - -Proof-irrelevance directly implies Streicher's axiom $K$. - -\Question{What about functional extensionality?} - -Extensionality of functions is admittedly consistent with the -Set-predicative Calculus of Inductive Constructions. - -%\begin{coq_example*} -% Axiom extensionality : (A,B:Set)(f,g:(A->B))(x:A)(f x)=(g x)->f=g. -%\end{coq_example*} - -Let {\tt A}, {\tt B} be types. To deal with extensionality on -\verb=A->B= without relying on a general extensionality axiom, -a possible approach is to define one's own extensional equality on -\verb=A->B=. - -\begin{coq_eval} -Variables A B : Set. -\end{coq_eval} - -\begin{coq_example*} -Definition ext_eq (f g: A->B) := forall x:A, f x = g x. -\end{coq_example*} - -and to reason on \verb=A->B= as a setoid (see the Chapter on -Setoids in the Reference Manual). - -\Question{Is {\Prop} impredicative?} - -Yes, the sort {\Prop} of propositions is {\em -impredicative}. Otherwise said, a statement of the form $\forall -A:Prop, P(A)$ can be instantiated by itself: if $\forall A:\Prop, P(A)$ -is provable, then $P(\forall A:\Prop, P(A))$ is. - -\Question{Is {\Set} impredicative?} - -No, the sort {\Set} lying at the bottom of the hierarchy of -computational types is {\em predicative} in the basic {\Coq} system. -This means that a family of types in {\Set}, e.g. $\forall A:\Set, A -\rightarrow A$, is not a type in {\Set} and it cannot be applied on -itself. - -However, the sort {\Set} was impredicative in the original versions of -{\Coq}. For backward compatibility, or for experiments by -knowledgeable users, the logic of {\Coq} can be set impredicative for -{\Set} by calling {\Coq} with the option {\tt -impredicative-set}. - -{\Set} has been made predicative from version 8.0 of {\Coq}. The main -reason is to interact smoothly with a classical mathematical world -where both excluded-middle and the axiom of description are valid (see -file \vfile{\LogicClassicalDescription}{ClassicalDescription} for a -proof that excluded-middle and description implies the double negation -of excluded-middle in {\Set} and file {\tt Hurkens\_Set.v} from the -user contribution {\tt Rocq/PARADOXES} for a proof that -impredicativity of {\Set} implies the simple negation of -excluded-middle in {\Set}). - -\Question{Is {\Type} impredicative?} - -No, {\Type} is stratified. This is hidden for the -user, but {\Coq} internally maintains a set of constraints ensuring -stratification. - -If {\Type} were impredicative then it would be possible to encode -Girard's systems $U-$ and $U$ in {\Coq} and it is known from Girard, -Coquand, Hurkens and Miquel that systems $U-$ and $U$ are inconsistent -[Girard 1972, Coquand 1991, Hurkens 1993, Miquel 2001]. This encoding -can be found in file {\tt Logic/Hurkens.v} of {\Coq} standard library. - -For instance, when the user see {\tt $\forall$ X:Type, X->X : Type}, each -occurrence of {\Type} is implicitly bound to a different level, say -$\alpha$ and $\beta$ and the actual statement is {\tt -forall X:Type($\alpha$), X->X : Type($\beta$)} with the constraint -$\alpha<\beta$. - -When a statement violates a constraint, the message {\tt Universe -inconsistency} appears. Example: {\tt fun (x:Type) (y:$\forall$ X:Type, X -{\coqimp} X) => y x x}. - -\Question{I have two proofs of the same proposition. Can I prove they are equal?} - -In the base {\Coq} system, the answer is generally no. However, if -classical logic is set, the answer is yes for propositions in {\Prop}. -The answer is also yes if proof irrelevance holds (see question -\ref{proof-irrelevance}). - -There are also ``simple enough'' propositions for which you can prove -the equality without requiring any extra axioms. This is typically -the case for propositions defined deterministically as a first-order -inductive predicate on decidable sets. See for instance in question -\ref{le-uniqueness} an axiom-free proof of the unicity of the proofs of -the proposition {\tt le m n} (less or equal on {\tt nat}). - -% It is an ongoing work of research to natively include proof -% irrelevance in {\Coq}. - -\Question{I have two proofs of an equality statement. Can I prove they are -equal?} - - Yes, if equality is decidable on the domain considered (which -is the case for {\tt nat}, {\tt bool}, etc): see {\Coq} file -\verb=Eqdep_dec.v=). No otherwise, unless -assuming Streicher's axiom $K$ (see \cite{HofStr98}) or a more general -assumption such as proof-irrelevance (see \ref{proof-irrelevance}) or -classical logic. - -All of these statements can be found in file \vfile{\LogicEqdep}{Eqdep}. - -\Question{Can I prove that the second components of equal dependent -pairs are equal?} - - The answer is the same as for proofs of equality -statements. It is provable if equality on the domain of the first -component is decidable (look at \verb=inj_right_pair= from file -\vfile{\LogicEqdepDec}{Eqdep\_dec}), but not provable in the general -case. However, it is consistent (with the Calculus of Constructions) -to assume it is true. The file \vfile{\LogicEqdep}{Eqdep} actually -provides an axiom (equivalent to Streicher's axiom $K$) which entails -the result (look at \verb=inj_pair2= in \vfile{\LogicEqdep}{Eqdep}). - -\subsection{Impredicativity} - -\Question{Why {\tt injection} does not work on impredicative {\tt Set}?} - - E.g. in this case (this occurs only in the {\tt Set}-impredicative - variant of \Coq): - -\begin{coq_eval} -Reset Initial. -\end{coq_eval} - -\begin{coq_example*} -Inductive I : Type := - intro : forall k:Set, k -> I. -Lemma eq_jdef : - forall x y:nat, intro _ x = intro _ y -> x = y. -Proof. - intros x y H; injection H. -\end{coq_example*} - - Injectivity of constructors is restricted to predicative types. If -injectivity on large inductive types were not restricted, we would be -allowed to derive an inconsistency (e.g. following the lines of -Burali-Forti paradox). The question remains open whether injectivity -is consistent on some large inductive types not expressive enough to -encode known paradoxes (such as type I above). - - -\Question{What is a ``large inductive definition''?} - -An inductive definition in {\Prop} or {\Set} is called large -if its constructors embed sets or propositions. As an example, here is -a large inductive type: - -\begin{coq_example*} -Inductive sigST (P:Set -> Set) : Type := - existST : forall X:Set, P X -> sigST P. -\end{coq_example*} - -In the {\tt Set} impredicative variant of {\Coq}, large inductive -definitions in {\tt Set} have restricted elimination schemes to -prevent inconsistencies. Especially, projecting the set or the -proposition content of a large inductive definition is forbidden. If -it were allowed, it would be possible to encode e.g. Burali-Forti -paradox \cite{Gir70,Coq85}. - - -\Question{Is Coq's logic conservative over Coquand's Calculus of -Constructions?} - -Yes for the non Set-impredicative version of the Calculus of Inductive -Constructions. Indeed, the impredicative sort of the Calculus of -Constructions can only be interpreted as the sort {\Prop} since {\Set} -is predicative. But {\Prop} can be - - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\section{Talkin' with the Rooster} - - -%%%%%%% -\subsection{My goal is ..., how can I prove it?} - - -\Question{My goal is a conjunction, how can I prove it?} - -Use some theorem or assumption or use the {\split} tactic. -\begin{coq_example} -Goal forall A B:Prop, A->B-> A/\B. -intros. -split. -assumption. -assumption. -Qed. -\end{coq_example} - -\Question{My goal contains a conjunction as an hypothesis, how can I use it?} - -If you want to decompose your hypothesis into other hypothesis you can use the {\decompose} tactic: - -\begin{coq_example} -Goal forall A B:Prop, A/\B-> B. -intros. -decompose [and] H. -assumption. -Qed. -\end{coq_example} - - -\Question{My goal is a disjunction, how can I prove it?} - -You can prove the left part or the right part of the disjunction using -{\left} or {\right} tactics. If you want to do a classical -reasoning step, use the {\tt classic} axiom to prove the right part with the assumption -that the left part of the disjunction is false. - -\begin{coq_example} -Goal forall A B:Prop, A-> A\/B. -intros. -left. -assumption. -Qed. -\end{coq_example} - -An example using classical reasoning: - -\begin{coq_example} -Require Import Classical. - -Ltac classical_right := -match goal with -| _:_ |-?X1 \/ _ => (elim (classic X1);intro;[left;trivial|right]) -end. - -Ltac classical_left := -match goal with -| _:_ |- _ \/?X1 => (elim (classic X1);intro;[right;trivial|left]) -end. - - -Goal forall A B:Prop, (~A -> B) -> A\/B. -intros. -classical_right. -auto. -Qed. -\end{coq_example} - -\Question{My goal is an universally quantified statement, how can I prove it?} - -Use some theorem or assumption or introduce the quantified variable in -the context using the {\intro} tactic. If there are several -variables you can use the {\intros} tactic. A good habit is to -provide names for these variables: {\Coq} will do it anyway, but such -automatic naming decreases legibility and robustness. - - -\Question{My goal contains an universally quantified statement, how can I use it?} - -If the universally quantified assumption matches the goal you can -use the {\apply} tactic. If it is an equation you can use the -{\rewrite} tactic. Otherwise you can use the {\specialize} tactic -to instantiate the quantified variables with terms. The variant -{\tt assert(Ht := H t)} makes a copy of assumption {\tt H} before -instantiating it. - - -\Question{My goal is an existential, how can I prove it?} - -Use some theorem or assumption or exhibit the witness using the {\existstac} tactic. -\begin{coq_example} -Goal exists x:nat, forall y, x+y=y. -exists 0. -intros. -auto. -Qed. -\end{coq_example} - - -\Question{My goal is solvable by some lemma, how can I prove it?} - -Just use the {\apply} tactic. - -\begin{coq_eval} -Reset Initial. -\end{coq_eval} - -\begin{coq_example} -Lemma mylemma : forall x, x+0 = x. -auto. -Qed. - -Goal 3+0 = 3. -apply mylemma. -Qed. -\end{coq_example} - - - -\Question{My goal contains False as an hypothesis, how can I prove it?} - -You can use the {\contradiction} or {\intuition} tactics. - - -\Question{My goal is an equality of two convertible terms, how can I prove it?} - -Just use the {\reflexivity} tactic. - -\begin{coq_example} -Goal forall x, 0+x = x. -intros. -reflexivity. -Qed. -\end{coq_example} - -\Question{My goal is a {\tt let x := a in ...}, how can I prove it?} - -Just use the {\intro} tactic. - - -\Question{My goal is a {\tt let (a, ..., b) := c in}, how can I prove it?} - -Just use the {\destruct} c as (a,...,b) tactic. - - -\Question{My goal contains some existential hypotheses, how can I use it?} - -You can use the tactic {\elim} with you hypotheses as an argument. - -\Question{My goal contains some existential hypotheses, how can I use it and decompose my knowledge about this new thing into different hypotheses?} - -\begin{verbatim} -Ltac DecompEx H P := elim H;intro P;intro TO;decompose [and] TO;clear TO;clear H. -\end{verbatim} - - -\Question{My goal is an equality, how can I swap the left and right hand terms?} - -Just use the {\symmetry} tactic. -\begin{coq_example} -Goal forall x y : nat, x=y -> y=x. -intros. -symmetry. -assumption. -Qed. -\end{coq_example} - -\Question{My hypothesis is an equality, how can I swap the left and right hand terms?} - -Just use the {\symmetryin} tactic. - -\begin{coq_example} -Goal forall x y : nat, x=y -> y=x. -intros. -symmetry in H. -assumption. -Qed. -\end{coq_example} - - -\Question{My goal is an equality, how can I prove it by transitivity?} - -Just use the {\transitivity} tactic. -\begin{coq_example} -Goal forall x y z : nat, x=y -> y=z -> x=z. -intros. -transitivity y. -assumption. -assumption. -Qed. -\end{coq_example} - - -\Question{My goal would be solvable using {\tt apply;assumption} if it would not create meta-variables, how can I prove it?} - -You can use {\tt eapply yourtheorem;eauto} but it won't work in all cases ! (for example if more than one hypothesis match one of the subgoals generated by \eapply) so you should rather use {\tt try solve [eapply yourtheorem;eauto]}, otherwise some metavariables may be incorrectly instantiated. - -\begin{coq_example} -Lemma trans : forall x y z : nat, x=y -> y=z -> x=z. -intros. -transitivity y;assumption. -Qed. - -Goal forall x y z : nat, x=y -> y=z -> x=z. -intros. -eapply trans;eauto. -Qed. - -Goal forall x y z t : nat, x=y -> x=t -> y=z -> x=z. -intros. -eapply trans;eauto. -Undo. -eapply trans. -apply H. -auto. -Qed. - -Goal forall x y z t : nat, x=y -> x=t -> y=z -> x=z. -intros. -eapply trans;eauto. -Undo. -try solve [eapply trans;eauto]. -eapply trans. -apply H. -auto. -Qed. - -\end{coq_example} - -\Question{My goal is solvable by some lemma within a set of lemmas and I don't want to remember which one, how can I prove it?} - -You can use a what is called a hints' base. - -\begin{coq_example} -Require Import ZArith. -Require Ring. -Open Local Scope Z_scope. -Lemma toto1 : 1+1 = 2. -ring. -Qed. -Lemma toto2 : 2+2 = 4. -ring. -Qed. -Lemma toto3 : 2+1 = 3. -ring. -Qed. - -Hint Resolve toto1 toto2 toto3 : mybase. - -Goal 2+(1+1)=4. -auto with mybase. -Qed. -\end{coq_example} - - -\Question{My goal is one of the hypotheses, how can I prove it?} - -Use the {\assumption} tactic. - -\begin{coq_example} -Goal 1=1 -> 1=1. -intro. -assumption. -Qed. -\end{coq_example} - - -\Question{My goal appears twice in the hypotheses and I want to choose which one is used, how can I do it?} - -Use the {\exact} tactic. -\begin{coq_example} -Goal 1=1 -> 1=1 -> 1=1. -intros. -exact H0. -Qed. -\end{coq_example} - -\Question{What can be the difference between applying one hypothesis or another in the context of the last question?} - -From a proof point of view it is equivalent but if you want to extract -a program from your proof, the two hypotheses can lead to different -programs. - - -\Question{My goal is a propositional tautology, how can I prove it?} - -Just use the {\tauto} tactic. -\begin{coq_example} -Goal forall A B:Prop, A-> (A\/B) /\ A. -intros. -tauto. -Qed. -\end{coq_example} - -\Question{My goal is a first order formula, how can I prove it?} - -Just use the semi-decision tactic: \firstorder. - -\iffalse -todo: demander un exemple à Pierre -\fi - -\Question{My goal is solvable by a sequence of rewrites, how can I prove it?} - -Just use the {\congruence} tactic. -\begin{coq_example} -Goal forall a b c d e, a=d -> b=e -> c+b=d -> c+e=a. -intros. -congruence. -Qed. -\end{coq_example} - - -\Question{My goal is a disequality solvable by a sequence of rewrites, how can I prove it?} - -Just use the {\congruence} tactic. - -\begin{coq_example} -Goal forall a b c d, a<>d -> b=a -> d=c+b -> b<>c+b. -intros. -congruence. -Qed. -\end{coq_example} - - -\Question{My goal is an equality on some ring (e.g. natural numbers), how can I prove it?} - -Just use the {\ring} tactic. - -\begin{coq_example} -Require Import ZArith. -Require Ring. -Open Local Scope Z_scope. -Goal forall a b : Z, (a+b)*(a+b) = a*a + 2*a*b + b*b. -intros. -ring. -Qed. -\end{coq_example} - -\Question{My goal is an equality on some field (e.g. real numbers), how can I prove it?} - -Just use the {\field} tactic. - -\begin{coq_example} -Require Import Reals. -Require Ring. -Open Local Scope R_scope. -Goal forall a b : R, b*a<>0 -> (a/b) * (b/a) = 1. -intros. -field. -cut (b*a <>0 -> a<>0). -cut (b*a <>0 -> b<>0). -auto. -auto with real. -auto with real. -Qed. -\end{coq_example} - - -\Question{My goal is an inequality on integers in Presburger's arithmetic (an expression build from +,-,constants and variables), how can I prove it?} - - -\begin{coq_example} -Require Import ZArith. -Require Omega. -Open Local Scope Z_scope. -Goal forall a : Z, a>0 -> a+a > a. -intros. -omega. -Qed. -\end{coq_example} - - -\Question{My goal is an equation solvable using equational hypothesis on some ring (e.g. natural numbers), how can I prove it?} - -You need the {\gb} tactic (see Loïc Pottier's homepage). - -\subsection{Tactics usage} - -\Question{I want to state a fact that I will use later as an hypothesis, how can I do it?} - -If you want to use forward reasoning (first proving the fact and then -using it) you just need to use the {\assert} tactic. If you want to use -backward reasoning (proving your goal using an assumption and then -proving the assumption) use the {\cut} tactic. - -\begin{coq_example} -Goal forall A B C D : Prop, (A -> B) -> (B->C) -> A -> C. -intros. -assert (A->C). -intro;apply H0;apply H;assumption. -apply H2. -assumption. -Qed. - -Goal forall A B C D : Prop, (A -> B) -> (B->C) -> A -> C. -intros. -cut (A->C). -intro. -apply H2;assumption. -intro;apply H0;apply H;assumption. -Qed. -\end{coq_example} - - - - -\Question{I want to state a fact that I will use later as an hypothesis and prove it later, how can I do it?} - -You can use {\cut} followed by {\intro} or you can use the following {\Ltac} command: -\begin{verbatim} -Ltac assert_later t := cut t;[intro|idtac]. -\end{verbatim} - -\Question{What is the difference between {\Qed} and {\Defined}?} - -These two commands perform type checking, but when {\Defined} is used the new definition is set as transparent, otherwise it is defined as opaque (see \ref{opaque}). - - -\Question{How can I know what a tactic does?} - -You can use the {\tt info} command. - - - -\Question{Why {\auto} does not work? How can I fix it?} - -You can increase the depth of the proof search or add some lemmas in the base of hints. -Perhaps you may need to use \eauto. - -\Question{What is {\eauto}?} - -This is the same tactic as \auto, but it relies on {\eapply} instead of \apply. - -\Question{How can I speed up {\auto}?} - -You can use \texttt{info }{\auto} to replace {\auto} by the tactics it generates. -You can split your hint bases into smaller ones. - - -\Question{What is the equivalent of {\tauto} for classical logic?} - -Currently there are no equivalent tactic for classical logic. You can use Gödel's ``not not'' translation. - - -\Question{I want to replace some term with another in the goal, how can I do it?} - -If one of your hypothesis (say {\tt H}) states that the terms are equal you can use the {\rewrite} tactic. Otherwise you can use the {\replace} {\tt with} tactic. - -\Question{I want to replace some term with another in an hypothesis, how can I do it?} - -You can use the {\rewrite} {\tt in} tactic. - -\Question{I want to replace some symbol with its definition, how can I do it?} - -You can use the {\unfold} tactic. - -\Question{How can I reduce some term?} - -You can use the {\simpl} tactic. - -\Question{How can I declare a shortcut for some term?} - -You can use the {\set} or {\pose} tactics. - -\Question{How can I perform case analysis?} - -You can use the {\case} or {\destruct} tactics. - -\Question{How can I prevent the case tactic from losing information ?} - -You may want to use the (now standard) {\tt case\_eq} tactic. See the Coq'Art page 159. - -\Question{Why should I name my intros?} - -When you use the {\intro} tactic you don't have to give a name to your -hypothesis. If you do so the name will be generated by {\Coq} but your -scripts may be less robust. If you add some hypothesis to your theorem -(or change their order), you will have to change your proof to adapt -to the new names. - -\Question{How can I automatize the naming?} - -You can use the {\tt Show Intro.} or {\tt Show Intros.} commands to generate the names and use your editor to generate a fully named {\intro} tactic. -This can be automatized within {\tt xemacs}. - -\begin{coq_example} -Goal forall A B C : Prop, A -> B -> C -> A/\B/\C. -Show Intros. -(* -A B C H H0 -H1 -*) -intros A B C H H0 H1. -repeat split;assumption. -Qed. -\end{coq_example} - -\Question{I want to automatize the use of some tactic, how can I do it?} - -You need to use the {\tt proof with T} command and add {\ldots} at the -end of your sentences. - -For instance: -\begin{coq_example} -Goal forall A B C : Prop, A -> B/\C -> A/\B/\C. -Proof with assumption. -intros. -split... -Qed. -\end{coq_example} - -\Question{I want to execute the {\texttt proof with} tactic only if it solves the goal, how can I do it?} - -You need to use the {\try} and {\solve} tactics. For instance: -\begin{coq_example} -Require Import ZArith. -Require Ring. -Open Local Scope Z_scope. -Goal forall a b c : Z, a+b=b+a. -Proof with try solve [ring]. -intros... -Qed. -\end{coq_example} - -\Question{How can I do the opposite of the {\intro} tactic?} - -You can use the {\generalize} tactic. - -\begin{coq_example} -Goal forall A B : Prop, A->B-> A/\B. -intros. -generalize H. -intro. -auto. -Qed. -\end{coq_example} - -\Question{One of the hypothesis is an equality between a variable and some term, I want to get rid of this variable, how can I do it?} - -You can use the {\subst} tactic. This will rewrite the equality everywhere and clear the assumption. - -\Question{What can I do if I get ``{\tt generated subgoal term has metavariables in it }''?} - -You should use the {\eapply} tactic, this will generate some goals containing metavariables. - -\Question{How can I instantiate some metavariable?} - -Just use the {\instantiate} tactic. - - -\Question{What is the use of the {\pattern} tactic?} - -The {\pattern} tactic transforms the current goal, performing -beta-expansion on all the applications featuring this tactic's -argument. For instance, if the current goal includes a subterm {\tt -phi(t)}, then {\tt pattern t} transforms the subterm into {\tt (fun -x:A => phi(x)) t}. This can be useful when {\apply} fails on matching, -to abstract the appropriate terms. - -\Question{What is the difference between assert, cut and generalize?} - -PS: Notice for people that are interested in proof rendering that \assert -and {\pose} (and \cut) are not rendered the same as {\generalize} (see the -HELM experimental rendering tool at \ahref{http://helm.cs.unibo.it/library.html}{\url{http://helm.cs.unibo.it}}, link -HELM, link COQ Online). Indeed {\generalize} builds a beta-expanded term -while \assert, {\pose} and {\cut} uses a let-in. - -\begin{verbatim} - (* Goal is T *) - generalize (H1 H2). - (* Goal is A->T *) - ... a proof of A->T ... -\end{verbatim} - -is rendered into something like -\begin{verbatim} - (h) ... the proof of A->T ... - we proved A->T - (h0) by (H1 H2) we proved A - by (h h0) we proved T -\end{verbatim} -while -\begin{verbatim} - (* Goal is T *) - assert q := (H1 H2). - (* Goal is A *) - ... a proof of A ... - (* Goal is A |- T *) - ... a proof of T ... -\end{verbatim} -is rendered into something like -\begin{verbatim} - (q) ... the proof of A ... - we proved A - ... the proof of T ... - we proved T -\end{verbatim} -Otherwise said, {\generalize} is not rendered in a forward-reasoning way, -while {\assert} is. - -\Question{What can I do if \Coq can not infer some implicit argument ?} - -You can state explicitely what this implicit argument is. See \ref{implicit}. - -\Question{How can I explicit some implicit argument ?}\label{implicit} - -Just use \texttt{A:=term} where \texttt{A} is the argument. - -For instance if you want to use the existence of ``nil'' on nat*nat lists: -\begin{verbatim} -exists (nil (A:=(nat*nat))). -\end{verbatim} - -\iffalse -\Question{Is there anyway to do pattern matching with dependent types?} - -todo -\fi - -\subsection{Proof management} - - -\Question{How can I change the order of the subgoals?} - -You can use the {\Focus} command to concentrate on some goal. When the goal is proved you will see the remaining goals. - -\Question{How can I change the order of the hypothesis?} - -You can use the {\tt Move ... after} command. - -\Question{How can I change the name of an hypothesis?} - -You can use the {\tt Rename ... into} command. - -\Question{How can I delete some hypothesis?} - -You can use the {\tt Clear} command. - -\Question{How can use a proof which is not finished?} - -You can use the {\tt Admitted} command to state your current proof as an axiom. -You can use the {\tt admit} tactic to omit a portion of a proof. - -\Question{How can I state a conjecture?} - -You can use the {\tt Admitted} command to state your current proof as an axiom. - -\Question{What is the difference between a lemma, a fact and a theorem?} - -From {\Coq} point of view there are no difference. But some tools can -have a different behavior when you use a lemma rather than a -theorem. For instance {\tt coqdoc} will not generate documentation for -the lemmas within your development. - -\Question{How can I organize my proofs?} - -You can organize your proofs using the section mechanism of \Coq. Have -a look at the manual for further information. - - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\section{Inductive and Co-inductive types} - -\subsection{General} - -\Question{How can I prove that two constructors are different?} - -You can use the {\discriminate} tactic. - -\begin{coq_example} -Inductive toto : Set := | C1 : toto | C2 : toto. -Goal C1 <> C2. -discriminate. -Qed. -\end{coq_example} - -\Question{During an inductive proof, how to get rid of impossible cases of an inductive definition?} - -Use the {\inversion} tactic. - - -\Question{How can I prove that 2 terms in an inductive set are equal? Or different?} - -Have a look at \coqtt{decide equality} and \coqtt{discriminate} in the \ahref{http://coq.inria.fr/doc/main.html}{Reference Manual}. - -\Question{Why is the proof of \coqtt{0+n=n} on natural numbers -trivial but the proof of \coqtt{n+0=n} is not?} - - Since \coqtt{+} (\coqtt{plus}) on natural numbers is defined by analysis on its first argument - -\begin{coq_example} -Print plus. -\end{coq_example} - -{\noindent} The expression \coqtt{0+n} evaluates to \coqtt{n}. As {\Coq} reasons -modulo evaluation of expressions, \coqtt{0+n} and \coqtt{n} are -considered equal and the theorem \coqtt{0+n=n} is an instance of the -reflexivity of equality. On the other side, \coqtt{n+0} does not -evaluate to \coqtt{n} and a proof by induction on \coqtt{n} is -necessary to trigger the evaluation of \coqtt{+}. - -\Question{Why is dependent elimination in Prop not -available by default?} - - -This is just because most of the time it is not needed. To derive a -dependent elimination principle in {\tt Prop}, use the command {\tt Scheme} and -apply the elimination scheme using the \verb=using= option of -\verb=elim=, \verb=destruct= or \verb=induction=. - - -\Question{Argh! I cannot write expressions like ``~{\tt if n <= p then p else n}~'', as in any programming language} -\label{minmax} - -The short answer : You should use {\texttt le\_lt\_dec n p} instead.\\ - -That's right, you can't. -If you type for instance the following ``definition'': -\begin{coq_eval} -Reset Initial. -\end{coq_eval} -\begin{coq_example} -Definition max (n p : nat) := if n <= p then p else n. -\end{coq_example} - -As \Coq~ says, the term ``~\texttt{n <= p}~'' is a proposition, i.e. a -statement that belongs to the mathematical world. There are many ways to -prove such a proposition, either by some computation, or using some already -proven theoremas. For instance, proving $3-2 \leq 2^{45503}$ is very easy, -using some theorems on arithmetical operations. If you compute both numbers -before comparing them, you risk to use a lot of time and space. - - -On the contrary, a function for computing the greatest of two natural numbers -is an algorithm which, called on two natural numbers -$n$ and $p$, determines wether $n\leq p$ or $p < n$. -Such a function is a \emph{decision procedure} for the inequality of - \texttt{nat}. The possibility of writing such a procedure comes -directly from de decidability of the order $\leq$ on natural numbers. - - -When you write a piece of code like -``~\texttt{if n <= p then \dots{} else \dots}~'' -in a -programming language like \emph{ML} or \emph{Java}, a call to such a -decision procedure is generated. The decision procedure is in general -a primitive function, written in a low-level language, in the correctness -of which you have to trust. - -The standard Library of the system \emph{Coq} contains a -(constructive) proof of decidability of the order $\leq$ on -\texttt{nat} : the function \texttt{le\_lt\_dec} of -the module \texttt{Compare\_dec} of library \texttt{Arith}. - -The following code shows how to define correctly \texttt{min} and -\texttt{max}, and prove some properties of these functions. - -\begin{coq_example} -Require Import Compare_dec. - -Definition max (n p : nat) := if le_lt_dec n p then p else n. - -Definition min (n p : nat) := if le_lt_dec n p then n else p. - -Eval compute in (min 4 7). - -Theorem min_plus_max : forall n p, min n p + max n p = n + p. -Proof. - intros n p; - unfold min, max; - case (le_lt_dec n p); - simpl; auto with arith. -Qed. - -Theorem max_equiv : forall n p, max n p = p <-> n <= p. -Proof. - unfold max; intros n p; case (le_lt_dec n p);simpl; auto. - intuition auto with arith. - split. - intro e; rewrite e; auto with arith. - intro H; absurd (p < p); eauto with arith. -Qed. -\end{coq_example} - -\Question{I wrote my own decision procedure for $\leq$, which -is much faster than yours, but proving such theorems as - \texttt{max\_equiv} seems to be quite difficult} - -Your code is probably the following one: - -\begin{coq_example} -Fixpoint my_le_lt_dec (n p :nat) {struct n}: bool := - match n, p with 0, _ => true - | S n', S p' => my_le_lt_dec n' p' - | _ , _ => false - end. - -Definition my_max (n p:nat) := if my_le_lt_dec n p then p else n. - -Definition my_min (n p:nat) := if my_le_lt_dec n p then n else p. -\end{coq_example} - - -For instance, the computation of \texttt{my\_max 567 321} is almost -immediate, whereas one can't wait for the result of -\texttt{max 56 32}, using \emph{Coq's} \texttt{le\_lt\_dec}. - -This is normal. Your definition is a simple recursive function which -returns a boolean value. Coq's \texttt{le\_lt\_dec} is a \emph{certified -function}, i.e. a complex object, able not only to tell wether $n\leq p$ -or $p<n$, but also of building a complete proof of the correct inequality. -What make \texttt{le\_lt\_dec} inefficient for computing \texttt{min} -and \texttt{max} is the building of a huge proof term. - -Nevertheless, \texttt{le\_lt\_dec} is very useful. Its type -is a strong specification, using the -\texttt{sumbool} type (look at the reference manual or chapter 9 of -\cite{coqart}). Eliminations of the form -``~\texttt{case (le\_lt\_dec n p)}~'' provide proofs of -either $n \leq p$ or $p < n$, allowing to prove easily theorems as in -question~\ref{minmax}. Unfortunately, this not the case of your -\texttt{my\_le\_lt\_dec}, which returns a quite non-informative boolean -value. - - -\begin{coq_example} -Check le_lt_dec. -\end{coq_example} - -You should keep in mind that \texttt{le\_lt\_dec} is useful to build -certified programs which need to compare natural numbers, and is not -designed to compare quickly two numbers. - -Nevertheless, the \emph{extraction} of \texttt{le\_lt\_dec} towards -\emph{Ocaml} or \emph{Haskell}, is a reasonable program for comparing two -natural numbers in Peano form in linear time. - -It is also possible to keep your boolean function as a decision procedure, -but you have to establish yourself the relationship between \texttt{my\_le\_lt\_dec} and the propositions $n\leq p$ and $p<n$: - -\begin{coq_example*} -Theorem my_le_lt_dec_true : - forall n p, my_le_lt_dec n p = true <-> n <= p. - -Theorem my_le_lt_dec_false : - forall n p, my_le_lt_dec n p = false <-> p < n. -\end{coq_example*} - - -\subsection{Recursion} - -\Question{Why can't I define a non terminating program?} - - Because otherwise the decidability of the type-checking -algorithm (which involves evaluation of programs) is not ensured. On -another side, if non terminating proofs were allowed, we could get a -proof of {\tt False}: - -\begin{coq_example*} -(* This is fortunately not allowed! *) -Fixpoint InfiniteProof (n:nat) : False := InfiniteProof n. -Theorem Paradox : False. -Proof (InfiniteProof O). -\end{coq_example*} - - -\Question{Why only structurally well-founded loops are allowed?} - - The structural order on inductive types is a simple and -powerful notion of termination. The consistency of the Calculus of -Inductive Constructions relies on it and another consistency proof -would have to be made for stronger termination arguments (such -as the termination of the evaluation of CIC programs themselves!). - -In spite of this, all non-pathological termination orders can be mapped -to a structural order. Tools to do this are provided in the file -\vfile{\InitWf}{Wf} of the standard library of {\Coq}. - -\Question{How to define loops based on non structurally smaller -recursive calls?} - - The procedure is as follows (we consider the definition of {\tt -mergesort} as an example). - -\begin{itemize} - -\item Define the termination order, say {\tt R} on the type {\tt A} of -the arguments of the loop. - -\begin{coq_eval} -Open Scope R_scope. -Require Import List. -\end{coq_eval} - -\begin{coq_example*} -Definition R (a b:list nat) := length a < length b. -\end{coq_example*} - -\item Prove that this order is well-founded (in fact that all elements in {\tt A} are accessible along {\tt R}). - -\begin{coq_example*} -Lemma Rwf : well_founded R. -\end{coq_example*} - -\item Define the step function (which needs proofs that recursive -calls are on smaller arguments). - -\begin{coq_example*} -Definition split (l : list nat) - : {l1: list nat | R l1 l} * {l2 : list nat | R l2 l} - := (* ... *) . -Definition concat (l1 l2 : list nat) : list nat := (* ... *) . -Definition merge_step (l : list nat) (f: forall l':list nat, R l' l -> list nat) := - let (lH1,lH2) := (split l) in - let (l1,H1) := lH1 in - let (l2,H2) := lH2 in - concat (f l1 H1) (f l2 H2). -\end{coq_example*} - -\item Define the recursive function by fixpoint on the step function. - -\begin{coq_example*} -Definition merge := Fix Rwf (fun _ => list nat) merge_step. -\end{coq_example*} - -\end{itemize} - -\Question{What is behind the accessibility and well-foundedness proofs?} - - Well-foundedness of some relation {\tt R} on some type {\tt A} -is defined as the accessibility of all elements of {\tt A} along {\tt R}. - -\begin{coq_example} -Print well_founded. -Print Acc. -\end{coq_example} - -The structure of the accessibility predicate is a well-founded tree -branching at each node {\tt x} in {\tt A} along all the nodes {\tt x'} -less than {\tt x} along {\tt R}. Any sequence of elements of {\tt A} -decreasing along the order {\tt R} are branches in the accessibility -tree. Hence any decreasing along {\tt R} is mapped into a structural -decreasing in the accessibility tree of {\tt R}. This is emphasised in -the definition of {\tt fix} which recurs not on its argument {\tt x:A} -but on the accessibility of this argument along {\tt R}. - -See file \vfile{\InitWf}{Wf}. - -\Question{How to perform simultaneous double induction?} - - In general a (simultaneous) double induction is simply solved by an -induction on the first hypothesis followed by an inversion over the -second hypothesis. Here is an example - -\begin{coq_eval} -Reset Initial. -\end{coq_eval} - -\begin{coq_example} -Inductive even : nat -> Prop := - | even_O : even 0 - | even_S : forall n:nat, even n -> even (S (S n)). - -Inductive odd : nat -> Prop := - | odd_SO : odd 1 - | odd_S : forall n:nat, odd n -> odd (S (S n)). - -Lemma not_even_and_odd : forall n:nat, even n -> odd n -> False. -induction 1. - inversion 1. - inversion 1. apply IHeven; trivial. -\end{coq_example} -\begin{coq_eval} -Qed. -\end{coq_eval} - -In case the type of the second induction hypothesis is not -dependent, {\tt inversion} can just be replaced by {\tt destruct}. - -\Question{How to define a function by simultaneous double recursion?} - - The same trick applies, you can even use the pattern-matching -compilation algorithm to do the work for you. Here is an example: - -\begin{coq_example} -Fixpoint minus (n m:nat) {struct n} : nat := - match n, m with - | O, _ => 0 - | S k, O => S k - | S k, S l => minus k l - end. -Print minus. -\end{coq_example} - -In case of dependencies in the type of the induction objects -$t_1$ and $t_2$, an extra argument stating $t_1=t_2$ must be given to -the fixpoint definition - -\Question{How to perform nested and double induction?} - - To reason by nested (i.e. lexicographic) induction, just reason by -induction on the successive components. - -\smallskip - -Double induction (or induction on pairs) is a restriction of the -lexicographic induction. Here is an example of double induction. - -\begin{coq_example} -Lemma nat_double_ind : -forall P : nat -> nat -> Prop, P 0 0 -> - (forall m n, P m n -> P m (S n)) -> - (forall m n, P m n -> P (S m) n) -> - forall m n, P m n. -intros P H00 HmS HSn; induction m. -(* case 0 *) -induction n; [assumption | apply HmS; apply IHn]. -(* case Sm *) -intro n; apply HSn; apply IHm. -\end{coq_example} -\begin{coq_eval} -Qed. -\end{coq_eval} - -\Question{How to define a function by nested recursion?} - - The same trick applies. Here is the example of Ackermann -function. - -\begin{coq_example} -Fixpoint ack (n:nat) : nat -> nat := - match n with - | O => S - | S n' => - (fix ack' (m:nat) : nat := - match m with - | O => ack n' 1 - | S m' => ack n' (ack' m') - end) - end. -\end{coq_example} - - -\subsection{Co-inductive types} - -\Question{I have a cofixpoint $t:=F(t)$ and I want to prove $t=F(t)$. How to do it?} - -Just case-expand $F({\tt t})$ then complete by a trivial case analysis. -Here is what it gives on e.g. the type of streams on naturals - -\begin{coq_eval} -Set Implicit Arguments. -\end{coq_eval} -\begin{coq_example} -CoInductive Stream (A:Set) : Set := - Cons : A -> Stream A -> Stream A. -CoFixpoint nats (n:nat) : Stream nat := Cons n (nats (S n)). -Lemma Stream_unfold : - forall n:nat, nats n = Cons n (nats (S n)). -Proof. - intro; - change (nats n = match nats n with - | Cons x s => Cons x s - end). - case (nats n); reflexivity. -Qed. -\end{coq_example} - - - -\section{Syntax and notations} - -\Question{I do not want to type ``forall'' because it is too long, what can I do?} - -You can define your own notation for forall: -\begin{verbatim} -Notation "fa x : t, P" := (forall x:t, P) (at level 200, x ident). -\end{verbatim} -or if your are using {\CoqIde} you can define a pretty symbol for for all and an input method (see \ref{forallcoqide}). - - - -\Question{How can I define a notation for square?} - -You can use for instance: -\begin{verbatim} -Notation "x ^2" := (Rmult x x) (at level 20). -\end{verbatim} -Note that you can not use: -\begin{tt} -Notation "x $^²$" := (Rmult x x) (at level 20). -\end{tt} -because ``$^2$'' is an iso-latin character. If you really want this kind of notation you should use UTF-8. - - -\Question{Why ``no associativity'' and ``left associativity'' at the same level does not work?} - -Because we relie on camlp4 for syntactical analysis and camlp4 does not really implement no associativity. By default, non associative operators are defined as right associative. - - - -\Question{How can I know the associativity associated with a level?} - -You can do ``Print Grammar constr'', and decode the output from camlp4, good luck ! - -\section{Modules} - - - - -%%%%%%% -\section{\Ltac} - -\Question{What is {\Ltac}?} - -{\Ltac} is the tactic language for \Coq. It provides the user with a -high-level ``toolbox'' for tactic creation. - -\Question{Is there any printing command in {\Ltac}?} - -You can use the {\idtac} tactic with a string argument. This string -will be printed out. The same applies to the {\fail} tactic - -\Question{What is the syntax for let in {\Ltac}?} - -If $x_i$ are identifiers and $e_i$ and $expr$ are tactic expressions, then let reads: -\begin{center} -{\tt let $x_1$:=$e_1$ with $x_2$:=$e_2$\ldots with $x_n$:=$e_n$ in -$expr$}. -\end{center} -Beware that if $expr$ is complex (i.e. features at least a sequence) parenthesis -should be added around it. For example: -\begin{coq_example} -Ltac twoIntro := let x:=intro in (x;x). -\end{coq_example} - -\Question{What is the syntax for pattern matching in {\Ltac}?} - -Pattern matching on a term $expr$ (non-linear first order unification) -with patterns $p_i$ and tactic expressions $e_i$ reads: -\begin{center} -\hspace{10ex} -{\tt match $expr$ with -\hspace*{2ex}$p_1$ => $e_1$ -\hspace*{1ex}\textbar$p_2$ => $e_2$ -\hspace*{1ex}\ldots -\hspace*{1ex}\textbar$p_n$ => $e_n$ -\hspace*{1ex}\textbar\ \textunderscore\ => $e_{n+1}$ -end. -} -\end{center} -Underscore matches all terms. - -\Question{What is the semantics for ``match goal''?} - -The semantics of {\tt match goal} depends on whether it returns -tactics or not. The {\tt match goal} expression matches the current -goal against a series of patterns: {$hyp_1 {\ldots} hyp_n$ \textbar- -$ccl$}. It uses a first-order unification algorithm and in case of -success, if the right-hand-side is an expression, it tries to type it -while if the right-hand-side is a tactic, it tries to apply it. If the -typing or the tactic application fails, the {\tt match goal} tries all -the possible combinations of $hyp_i$ before dropping the branch and -moving to the next one. Underscore matches all terms. - -\Question{Why can't I use a ``match goal'' returning a tactic in a non -tail-recursive position?} - -This is precisely because the semantics of {\tt match goal} is to -apply the tactic on the right as soon as a pattern unifies what is -meaningful only in tail-recursive uses. - -The semantics in non tail-recursive call could have been the one used -for terms (i.e. fail if the tactic expression is not typable, but -don't try to apply it). For uniformity of semantics though, this has -been rejected. - -\Question{How can I generate a new name?} - -You can use the following syntax: -{\tt let id:=fresh in \ldots}\\ -For example: -\begin{coq_example} -Ltac introIdGen := let id:=fresh in intro id. -\end{coq_example} - - -\iffalse -\Question{How can I access the type of a term?} - -You can use typeof. -todo -\fi - -\iffalse -\Question{How can I define static and dynamic code?} -\fi - -\section{Tactics written in Ocaml} - -\Question{Can you show me an example of a tactic written in OCaml?} - -You have some examples of tactics written in Ocaml in the ``plugins'' directory of {\Coq} sources. - - - - -\section{Case studies} - -\iffalse -\Question{How can I define vectors or lists of size n?} -\fi - - -\Question{How to prove that 2 sets are different?} - - You need to find a property true on one set and false on the -other one. As an example we show how to prove that {\tt bool} and {\tt -nat} are discriminable. As discrimination property we take the -property to have no more than 2 elements. - -\begin{coq_example*} -Theorem nat_bool_discr : bool <> nat. -Proof. - pose (discr := - fun X:Set => - ~ (forall a b:X, ~ (forall x:X, x <> a -> x <> b -> False))). - intro Heq; assert (H: discr bool). - intro H; apply (H true false); destruct x; auto. - rewrite Heq in H; apply H; clear H. - destruct a; destruct b as [|n]; intro H0; eauto. - destruct n; [ apply (H0 2); discriminate | eauto ]. -Qed. -\end{coq_example*} - -\Question{Is there an axiom-free proof of Streicher's axiom $K$ for -the equality on {\tt nat}?} -\label{K-nat} - -Yes, because equality is decidable on {\tt nat}. Here is the proof. - -\begin{coq_example*} -Require Import Eqdep_dec. -Require Import Peano_dec. -Theorem K_nat : - forall (x:nat) (P:x = x -> Prop), P (refl_equal x) -> forall p:x = x, P p. -Proof. -intros; apply K_dec_set with (p := p). -apply eq_nat_dec. -assumption. -Qed. -\end{coq_example*} - -Similarly, we have - -\begin{coq_example*} -Theorem eq_rect_eq_nat : - forall (p:nat) (Q:nat->Type) (x:Q p) (h:p=p), x = eq_rect p Q x p h. -Proof. -intros; apply K_nat with (p := h); reflexivity. -Qed. -\end{coq_example*} - -\Question{How to prove that two proofs of {\tt n<=m} on {\tt nat} are equal?} -\label{le-uniqueness} - -This is provable without requiring any axiom because axiom $K$ -directly holds on {\tt nat}. Here is a proof using question \ref{K-nat}. - -\begin{coq_example*} -Require Import Arith. -Scheme le_ind' := Induction for le Sort Prop. -Theorem le_uniqueness_proof : forall (n m : nat) (p q : n <= m), p = q. -Proof. -induction p using le_ind'; intro q. - replace (le_n n) with - (eq_rect _ (fun n0 => n <= n0) (le_n n) _ (refl_equal n)). - 2:reflexivity. - generalize (refl_equal n). - pattern n at 2 4 6 10, q; case q; [intro | intros m l e]. - rewrite <- eq_rect_eq_nat; trivial. - contradiction (le_Sn_n m); rewrite <- e; assumption. - replace (le_S n m p) with - (eq_rect _ (fun n0 => n <= n0) (le_S n m p) _ (refl_equal (S m))). - 2:reflexivity. - generalize (refl_equal (S m)). - pattern (S m) at 1 3 4 6, q; case q; [intro Heq | intros m0 l HeqS]. - contradiction (le_Sn_n m); rewrite Heq; assumption. - injection HeqS; intro Heq; generalize l HeqS. - rewrite <- Heq; intros; rewrite <- eq_rect_eq_nat. - rewrite (IHp l0); reflexivity. -Qed. -\end{coq_example*} - -\Question{How to exploit equalities on sets} - -To extract information from an equality on sets, you need to -find a predicate of sets satisfied by the elements of the sets. As an -example, let's consider the following theorem. - -\begin{coq_example*} -Theorem interval_discr : - forall m n:nat, - {x : nat | x <= m} = {x : nat | x <= n} -> m = n. -\end{coq_example*} - -We have a proof requiring the axiom of proof-irrelevance. We -conjecture that proof-irrelevance can be circumvented by introducing a -primitive definition of discrimination of the proofs of -\verb!{x : nat | x <= m}!. - -\begin{latexonly}% -The proof can be found in file {\tt interval$\_$discr.v} in this directory. -%Here is the proof -%\begin{small} -%\begin{flushleft} -%\begin{texttt} -%\def_{\ifmmode\sb\else\subscr\fi} -%\include{interval_discr.v} -%%% WARNING semantics of \_ has changed ! -%\end{texttt} -%$a\_b\_c$ -%\end{flushleft} -%\end{small} -\end{latexonly}% -\begin{htmlonly}% -\ahref{./interval_discr.v}{Here} is the proof. -\end{htmlonly} - -\Question{I have a problem of dependent elimination on -proofs, how to solve it?} - -\begin{coq_eval} -Reset Initial. -\end{coq_eval} - -\begin{coq_example*} -Inductive Def1 : Set := c1 : Def1. -Inductive DefProp : Def1 -> Prop := - c2 : forall d:Def1, DefProp d. -Inductive Comb : Set := - c3 : forall d:Def1, DefProp d -> Comb. -Lemma eq_comb : - forall (d1 d1':Def1) (d2:DefProp d1) (d2':DefProp d1'), - d1 = d1' -> c3 d1 d2 = c3 d1' d2'. -\end{coq_example*} - - You need to derive the dependent elimination -scheme for DefProp by hand using {\coqtt Scheme}. - -\begin{coq_eval} -Abort. -\end{coq_eval} - -\begin{coq_example*} -Scheme DefProp_elim := Induction for DefProp Sort Prop. -Lemma eq_comb : - forall d1 d1':Def1, - d1 = d1' -> - forall (d2:DefProp d1) (d2':DefProp d1'), c3 d1 d2 = c3 d1' d2'. -intros. -destruct H. -destruct d2 using DefProp_elim. -destruct d2' using DefProp_elim. -reflexivity. -Qed. -\end{coq_example*} - - -\Question{And what if I want to prove the following?} - -\begin{coq_example*} -Inductive natProp : nat -> Prop := - | p0 : natProp 0 - | pS : forall n:nat, natProp n -> natProp (S n). -Inductive package : Set := - pack : forall n:nat, natProp n -> package. -Lemma eq_pack : - forall n n':nat, - n = n' -> - forall (np:natProp n) (np':natProp n'), pack n np = pack n' np'. -\end{coq_example*} - - - -\begin{coq_eval} -Abort. -\end{coq_eval} -\begin{coq_example*} -Scheme natProp_elim := Induction for natProp Sort Prop. -Definition pack_S : package -> package. -destruct 1. -apply (pack (S n)). -apply pS; assumption. -Defined. -Lemma eq_pack : - forall n n':nat, - n = n' -> - forall (np:natProp n) (np':natProp n'), pack n np = pack n' np'. -intros n n' Heq np np'. -generalize dependent n'. -induction np using natProp_elim. -induction np' using natProp_elim; intros; auto. - discriminate Heq. -induction np' using natProp_elim; intros; auto. - discriminate Heq. -change (pack_S (pack n np) = pack_S (pack n0 np')). -apply (f_equal (A:=package)). -apply IHnp. -auto. -Qed. -\end{coq_example*} - - - - - - - -\section{Publishing tools} - -\Question{How can I generate some latex from my development?} - -You can use {\tt coqdoc}. - -\Question{How can I generate some HTML from my development?} - -You can use {\tt coqdoc}. - -\Question{How can I generate some dependency graph from my development?} - -You can use the tool \verb|coqgraph| developped by Philippe Audebaud in 2002. -This tool transforms dependancies generated by \verb|coqdep| into 'dot' files which can be visualized using the Graphviz software (http://www.graphviz.org/). - -\Question{How can I cite some {\Coq} in my latex document?} - -You can use {\tt coq\_tex}. - -\Question{How can I cite the {\Coq} reference manual?} - -You can use this bibtex entry: -\begin{verbatim} -@Manual{Coq:manual, - title = {The Coq proof assistant reference manual}, - author = {\mbox{The Coq development team}}, - note = {Version 8.3}, - year = {2009}, - url = "http://coq.inria.fr" -} -\end{verbatim} - -\Question{Where can I publish my developments in {\Coq}?} - -You can submit your developments as a user contribution to the {\Coq} -development team. This ensures its liveness along the evolution and -possible changes of {\Coq}. - -You can also submit your developments to the HELM/MoWGLI repository at -the University of Bologna (see -\ahref{http://mowgli.cs.unibo.it}{\url{http://mowgli.cs.unibo.it}}). For -developments submitted in this database, it is possible to visualize -the developments in natural language and execute various retrieving -requests. - -\Question{How can I read my proof in natural language?} - -You can submit your proof to the HELM/MoWGLI repository and use the -rendering tool provided by the server (see -\ahref{http://mowgli.cs.unibo.it}{\url{http://mowgli.cs.unibo.it}}). - -\section{\CoqIde} - -\Question{What is {\CoqIde}?} - -{\CoqIde} is a gtk based GUI for \Coq. - -\Question{How to enable Emacs keybindings?} - -Depending on your configuration, use either one of these two methods - -\begin{itemize} - -\item Insert \texttt{gtk-key-theme-name = "Emacs"} - in your \texttt{.coqide-gtk2rc} file. It may be in the current dir - or in \verb#$HOME# dir. This is done by default. - -\item If in Gnome, run the gnome configuration editor (\texttt{gconf-editor}) -and set key \texttt{gtk-key-theme} to \texttt{Emacs} in the category -\texttt{desktop/gnome/interface}. - -\end{itemize} - - - -%$ juste pour que la coloration emacs marche - -\Question{How to enable antialiased fonts?} - - Set the \verb#GDK_USE_XFT# variable to \verb#1#. This is by default - with \verb#Gtk >= 2.2#. If some of your fonts are not available, - set \verb#GDK_USE_XFT# to \verb#0#. - -\Question{How to use those Forall and Exists pretty symbols?}\label{forallcoqide} - Thanks to the notation features in \Coq, you just need to insert these -lines in your {\Coq} buffer:\\ -\begin{tt} -Notation "$\forall$ x : t, P" := (forall x:t, P) (at level 200, x ident). -\end{tt}\\ -\begin{tt} -Notation "$\exists$ x : t, P" := (exists x:t, P) (at level 200, x ident). -\end{tt} - -Copy/Paste of these lines from this file will not work outside of \CoqIde. -You need to load a file containing these lines or to enter the $\forall$ -using an input method (see \ref{inputmeth}). To try it just use \verb#Require Import utf8# from inside -\CoqIde. -To enable these notations automatically start coqide with -\begin{verbatim} - coqide -l utf8 -\end{verbatim} -In the ide subdir of {\Coq} library, you will find a sample utf8.v with some -pretty simple notations. - -\Question{How to define an input method for non ASCII symbols?}\label{inputmeth} - -\begin{itemize} -\item First solution: type \verb#<CONTROL><SHIFT>2200# to enter a forall in the script widow. - 2200 is the hexadecimal code for forall in unicode charts and is encoded as - in UTF-8. - 2203 is for exists. See \ahref{http://www.unicode.org}{\url{http://www.unicode.org}} for more codes. -\item Second solution: rebind \verb#<AltGr>a# to forall and \verb#<AltGr>e# to exists. - Under X11, you need to use something like -\begin{verbatim} - xmodmap -e "keycode 24 = a A F13 F13" - xmodmap -e "keycode 26 = e E F14 F14" -\end{verbatim} - and then to add -\begin{verbatim} - bind "F13" {"insert-at-cursor" ("")} - bind "F14" {"insert-at-cursor" ("")} -\end{verbatim} - to your "binding "text"" section in \verb#.coqiderc-gtk2rc.# - The strange ("") argument is the UTF-8 encoding for - 0x2200. - You can compute these encodings using the lablgtk2 toplevel with -\begin{verbatim} -Glib.Utf8.from_unichar 0x2200;; -\end{verbatim} - Further symbols can be bound on higher Fxx keys or on even on other keys you - do not need . -\end{itemize} - -\Question{How to build a custom {\CoqIde} with user ml code?} - Use - coqmktop -ide -byte m1.cmo...mi.cmo - or - coqmktop -ide -opt m1.cmx...mi.cmx - -\Question{How to customize the shortcuts for menus?} - Two solutions are offered: -\begin{itemize} -\item Edit \$HOME/.coqide.keys by hand or -\item Add "gtk-can-change-accels = 1" in your .coqide-gtk2rc file. Then - from \CoqIde, you may select a menu entry and press the desired - shortcut. -\end{itemize} - -\Question{What encoding should I use? What is this $\backslash$x\{iiii\} in my file?} - The encoding option is related to the way files are saved. - Keep it as UTF-8 until it becomes important for you to exchange files - with non UTF-8 aware applications. - If you choose something else than UTF-8, then missing characters will - be encoded by $\backslash$x\{....\} or $\backslash$x\{........\} - where each dot is an hex. digit. - The number between braces is the hexadecimal UNICODE index for the - missing character. - -\Question{How to get rid of annoying unwanted automatic templates?} - -Some users may experiment problems with unwanted automatic -templates while using Coqide. This is due to a change in the -modifiers keys available through GTK. The straightest way to get -rid of the problem is to edit by hand your .coqiderc (either -\verb|/home/<user>/.coqiderc| under Linux, or \\ -\verb|C:\Documents and Settings\<user>\.coqiderc| under Windows) - and replace any occurence of \texttt{MOD4} by \texttt{MOD1}. - - - -\section{Extraction} - -\Question{What is program extraction?} - -Program extraction consist in generating a program from a constructive proof. - -\Question{Which language can I extract to?} - -You can extract your programs to Objective Caml and Haskell. - -\Question{How can I extract an incomplete proof?} - -You can provide programs for your axioms. - - - -%%%%%%% -\section{Glossary} - -\Question{Can you explain me what an evaluable constant is?} - -An evaluable constant is a constant which is unfoldable. - -\Question{What is a goal?} - -The goal is the statement to be proved. - -\Question{What is a meta variable?} - -A meta variable in {\Coq} represents a ``hole'', i.e. a part of a proof -that is still unknown. - -\Question{What is Gallina?} - -Gallina is the specification language of \Coq. Complete documentation -of this language can be found in the Reference Manual. - -\Question{What is The Vernacular?} - -It is the language of commands of Gallina i.e. definitions, lemmas, {\ldots} - - -\Question{What is a dependent type?} - -A dependant type is a type which depends on some term. For instance -``vector of size n'' is a dependant type representing all the vectors -of size $n$. Its type depends on $n$ - -\Question{What is a proof by reflection?} - -This is a proof generated by some computation which is done using the -internal reduction of {\Coq} (not using the tactic language of {\Coq} -(\Ltac) nor the implementation language for \Coq). An example of -tactic using the reflection mechanism is the {\ring} tactic. The -reflection method consist in reflecting a subset of {\Coq} language (for -example the arithmetical expressions) into an object of the {\Coq} -language itself (in this case an inductive type denoting arithmetical -expressions). For more information see~\cite{howe,harrison,boutin} -and the last chapter of the Coq'Art. - -\Question{What is intuitionistic logic?} - -This is any logic which does not assume that ``A or not A''. - - -\Question{What is proof-irrelevance?} - -See question \ref{proof-irrelevance} - - -\Question{What is the difference between opaque and transparent?}{\label{opaque}} - -Opaque definitions can not be unfolded but transparent ones can. - - -\section{Troubleshooting} - -\Question{What can I do when {\tt Qed.} is slow?} - -Sometime you can use the {\abstracttac} tactic, which makes as if you had -stated some local lemma, this speeds up the typing process. - -\Question{Why \texttt{Reset Initial.} does not work when using \texttt{coqc}?} - -The initial state corresponds to the state of \texttt{coqtop} when the interactive -session began. It does not make sense in files to compile. - - -\Question{What can I do if I get ``No more subgoals but non-instantiated existential variables''?} - -This means that {\eauto} or {\eapply} didn't instantiate an -existential variable which eventually got erased by some computation. -You may backtrack to the faulty occurrence of {\eauto} or {\eapply} -and give the missing argument an explicit value. Alternatively, you -can use the commands \texttt{Show Existentials.} and -\texttt{Existential.} to display and instantiate the remainig -existential variables. - - -\begin{coq_example} -Lemma example_show_existentials : forall a b c:nat, a=b -> b=c -> a=c. -Proof. -intros. -eapply trans_equal. -Show Existentials. -eassumption. -assumption. -Qed. -\end{coq_example} - - -\Question{What can I do if I get ``Cannot solve a second-order unification problem''?} - -You can help {\Coq} using the {\pattern} tactic. - -\Question{Why does {\Coq} tell me that \texttt{\{x:A|(P x)\}} is not convertible with \texttt{(sig A P)}?} - - This is because \texttt{\{x:A|P x\}} is a notation for -\texttt{sig (fun x:A => P x)}. Since {\Coq} does not reason up to -$\eta$-conversion, this is different from \texttt{sig P}. - - -\Question{I copy-paste a term and {\Coq} says it is not convertible - to the original term. Sometimes it even says the copied term is not -well-typed.} - - This is probably due to invisible implicit information (implicit -arguments, coercions and Cases annotations) in the printed term, which -is not re-synthesised from the copied-pasted term in the same way as -it is in the original term. - - Consider for instance {\tt (@eq Type True True)}. This term is -printed as {\tt True=True} and re-parsed as {\tt (@eq Prop True -True)}. The two terms are not convertible (hence they fool tactics -like {\tt pattern}). - - There is currently no satisfactory answer to the problem. However, -the command {\tt Set Printing All} is useful for diagnosing the -problem. - - Due to coercions, one may even face type-checking errors. In some -rare cases, the criterion to hide coercions is a bit too loose, which -may result in a typing error message if the parser is not able to find -again the missing coercion. - - - -\section{Conclusion and Farewell.} -\label{ccl} - -\Question{What if my question isn't answered here?} -\label{lastquestion} - -Don't panic \verb+:-)+. You can try the {\Coq} manual~\cite{Coq:manual} for a technical -description of the prover. The Coq'Art~\cite{Coq:coqart} is the first -book written on {\Coq} and provides a comprehensive review of the -theorem prover as well as a number of example and exercises. Finally, -the tutorial~\cite{Coq:Tutorial} provides a smooth introduction to -theorem proving in \Coq. - - -%%%%%%% -\newpage -\nocite{LaTeX:intro} -\nocite{LaTeX:symb} -\bibliography{fk} - -%%%%%%% -\typeout{*********************************************} -\typeout{********* That makes {\thequestion} questions **********} -\typeout{*********************************************} - -\end{document} diff --git a/doc/faq/axioms.eps b/doc/faq/axioms.eps deleted file mode 100644 index 3f3c01c4..00000000 --- a/doc/faq/axioms.eps +++ /dev/null @@ -1,378 +0,0 @@ -%!PS-Adobe-2.0 EPSF-2.0 -%%Title: axioms.fig -%%Creator: fig2dev Version 3.2 Patchlevel 4 -%%CreationDate: Wed May 5 18:30:03 2004 -%%For: herbelin@limoux.polytechnique.fr (Hugo Herbelin) -%%BoundingBox: 0 0 437 372 -%%Magnification: 1.0000 -%%EndComments -/$F2psDict 200 dict def -$F2psDict begin -$F2psDict /mtrx matrix put -/col-1 {0 setgray} bind def -/col0 {0.000 0.000 0.000 srgb} bind def -/col1 {0.000 0.000 1.000 srgb} bind def -/col2 {0.000 1.000 0.000 srgb} bind def -/col3 {0.000 1.000 1.000 srgb} bind def -/col4 {1.000 0.000 0.000 srgb} bind def -/col5 {1.000 0.000 1.000 srgb} bind def -/col6 {1.000 1.000 0.000 srgb} bind def -/col7 {1.000 1.000 1.000 srgb} bind def -/col8 {0.000 0.000 0.560 srgb} bind def -/col9 {0.000 0.000 0.690 srgb} bind def -/col10 {0.000 0.000 0.820 srgb} bind def -/col11 {0.530 0.810 1.000 srgb} bind def -/col12 {0.000 0.560 0.000 srgb} bind def -/col13 {0.000 0.690 0.000 srgb} bind def -/col14 {0.000 0.820 0.000 srgb} bind def -/col15 {0.000 0.560 0.560 srgb} bind def -/col16 {0.000 0.690 0.690 srgb} bind def -/col17 {0.000 0.820 0.820 srgb} bind def -/col18 {0.560 0.000 0.000 srgb} bind def -/col19 {0.690 0.000 0.000 srgb} bind def -/col20 {0.820 0.000 0.000 srgb} bind def -/col21 {0.560 0.000 0.560 srgb} bind def -/col22 {0.690 0.000 0.690 srgb} bind def -/col23 {0.820 0.000 0.820 srgb} bind def -/col24 {0.500 0.190 0.000 srgb} bind def -/col25 {0.630 0.250 0.000 srgb} bind def -/col26 {0.750 0.380 0.000 srgb} bind def -/col27 {1.000 0.500 0.500 srgb} bind def -/col28 {1.000 0.630 0.630 srgb} bind def -/col29 {1.000 0.750 0.750 srgb} bind def -/col30 {1.000 0.880 0.880 srgb} bind def -/col31 {1.000 0.840 0.000 srgb} bind def - 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1049 1950 1052 2250 -2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 2 - 1 1 1.00 60.00 120.00 - 1048 1125 1051 1425 -2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 2 - 1 1 1.00 60.00 120.00 - 3075 975 1575 975 -2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 2 - 1 1 1.00 60.00 120.00 - 3075 1725 2025 1725 -2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 4 - 8025 5925 8250 5925 9000 4950 9150 4950 -2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 2 - 1 1 1.00 60.00 120.00 - 8625 5400 8250 3900 -2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 2 - 1 1 1.00 60.00 120.00 - 7050 7350 4575 7950 -2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 2 - 1 1 1.00 60.00 120.00 - 4200 7500 4200 7950 -2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 2 - 1 1 1.00 60.00 120.00 - 4714 6255 7039 7080 -2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 2 - 1 1 1.00 60.00 120.00 - 1500 3900 2175 6000 -2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 1 2 - 1 1 1.00 60.00 120.00 - 1 1 1.00 60.00 120.00 - 1139 2771 1364 3521 -2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 2 - 4425 4875 7350 3825 -3 0 0 1 0 7 50 -1 -1 0.000 0 0 0 4 - 6450 7050 4050 6675 3750 6825 3750 7050 - 0.000 1.000 1.000 0.000 -4 0 0 50 -1 0 12 0.0000 4 180 1830 6900 3750 Predicate extensionality\001 -4 0 0 50 -1 0 12 0.0000 4 135 1800 3825 3750 Relational choice axiom\001 -4 0 0 50 -1 0 12 0.0000 4 180 2100 6150 6150 Propositional extensionality\001 -4 0 0 50 -1 0 12 0.0000 4 180 1005 450 1050 Operator iota\001 -4 0 0 50 -1 2 12 0.0000 4 135 1020 450 1650 Constructive\001 -4 0 0 50 -1 2 12 0.0000 4 180 1530 450 1875 definite description\001 -4 0 0 50 -1 2 12 0.0000 4 180 2010 9000 5175 Functional extensionality\001 -4 0 0 50 -1 0 12 0.0000 4 180 4740 150 10050 Statements in boldface are the most "interesting" ones for Coq\001 -4 0 0 50 -1 0 12 0.0000 4 180 4800 3600 9375 Invariance by substitution of reflexivity proofs for equality on A\001 -4 0 0 50 -1 0 12 0.0000 4 180 3735 3600 8475 Uniqueness of reflexivity proofs for equality on A\001 -4 0 0 50 -1 0 12 0.0000 4 180 2670 3600 8775 Uniqueness of equality proofs on A\001 -4 0 0 50 -1 0 12 0.0000 4 135 1080 3600 8175 Axiom K on A\001 -4 0 0 50 -1 2 12 0.0000 4 135 1455 6450 7275 Proof-irrelevance\001 -4 0 0 50 -1 2 12 0.0000 4 180 3360 3600 9075 Injectivity of equality on Sigma-types on A\001 -4 0 0 50 -1 0 12 0.0000 4 180 2175 4950 6525 (needs Prop-impredicativity)\001 -4 0 0 50 -1 0 12 0.0000 4 180 705 6000 6750 (Berardi)\001 -4 0 0 50 -1 2 12 0.0000 4 135 1290 3675 6225 Excluded-middle\001 -4 0 0 50 -1 0 12 0.0000 4 180 1905 4950 5550 Propositional degeneracy\001 -4 0 0 50 -1 0 12 0.0000 4 180 1050 3750 5250 (Diaconescu)\001 -4 0 0 50 -1 0 12 0.0000 4 180 2475 3375 7425 Decidability of equality on any A\001 -4 0 0 50 -1 0 12 0.0000 4 180 1620 1275 5025 (if Set impredicative)\001 -4 0 0 50 -1 0 12 0.0000 4 135 1560 1575 6225 Not excluded-middle\001 -4 0 0 50 -1 0 12 0.0000 4 180 1770 450 2625 in propositional context\001 -4 0 0 50 -1 2 12 0.0000 4 135 1020 3150 1650 Constructive\001 -4 0 0 50 -1 2 12 0.0000 4 180 1665 3150 1875 indefinite description\001 -4 0 0 50 -1 0 12 0.0000 4 180 2610 3150 2400 Constructive indefinite description\001 -4 0 0 50 -1 0 12 0.0000 4 180 1770 3150 2625 in propositional context\001 -4 0 0 50 -1 2 12 0.0000 4 135 1935 3150 3150 Functional choice axiom\001 -4 0 0 50 -1 0 12 0.0000 4 135 2100 450 2400 Constructive definite descr.\001 -4 0 0 50 -1 2 12 0.0000 4 180 1845 450 3750 Axiom of unique choice\001 -4 0 0 50 -1 2 12 0.0000 4 180 1365 3150 1050 Operator epsilon\001 diff --git a/doc/faq/axioms.png b/doc/faq/axioms.png Binary files differdeleted file mode 100644 index 2aee0916..00000000 --- a/doc/faq/axioms.png +++ /dev/null diff --git a/doc/faq/fk.bib b/doc/faq/fk.bib deleted file mode 100644 index 976b36b0..00000000 --- a/doc/faq/fk.bib +++ /dev/null @@ -1,2220 +0,0 @@ -%%%%%%% FAQ %%%%%%% - -@book{ProofsTypes, - Author="Girard, Jean-Yves and Yves Lafont and Paul Taylor", - Title="Proofs and Types", - Publisher="Cambrige Tracts in Theoretical Computer Science, Cambridge University Press", - Year="1989" -} - -@misc{Types:Dowek, - author = "Gilles Dowek", - title = "Th{\'e}orie des types", - year = 2002, - howpublished = "Lecture notes", - url= "http://www.lix.polytechnique.fr/~dowek/Cours/theories_des_types.ps.gz" -} - -@PHDTHESIS{EGThese, - author = {Eduardo Giménez}, - title = {Un Calcul de Constructions Infinies et son application -a la vérification de systèmes communicants}, - type = {thèse d'Université}, - school = {Ecole Normale Supérieure de Lyon}, - month = {December}, - year = {1996}, -} - - -%%%%%%% Semantique %%%%%%% - -@misc{Sem:cours, - author = "François Pottier", - title = "{Typage et Programmation}", - year = "2002", - howpublished = "Lecture notes", - note = "DEA PSPL" -} - -@inproceedings{Sem:Dubois, - author = {Catherine Dubois}, - editor = {Mark Aagaard and - John Harrison}, - title = "{Proving ML Type Soundness Within Coq}", - pages = {126-144}, - booktitle = {TPHOLs}, - publisher = {Springer}, - series = {Lecture Notes in Computer Science}, - volume = {1869}, - year = {2000}, - isbn = {3-540-67863-8}, - bibsource = {DBLP, http://dblp.uni-trier.de} -} - -@techreport{Sem:Plotkin, -author = {Gordon D. Plotkin}, -institution = {Aarhus University}, -number = {{DAIMI FN-19}}, -title = {{A structural approach to operational semantics}}, -year = {1981} -} - -@article{Sem:RemyV98, - author = "Didier R{\'e}my and J{\'e}r{\^o}me Vouillon", - title = "Objective {ML}: - An effective object-oriented extension to {ML}", - journal = "Theory And Practice of Object Systems", - year = 1998, - volume = "4", - number = "1", - pages = "27--50", - note = {A preliminary version appeared in the proceedings - of the 24th ACM Conference on Principles - of Programming Languages, 1997} -} - -@book{Sem:Winskel, - AUTHOR = {Winskel, Glynn}, - TITLE = {The Formal Semantics of Programming Languages}, - NOTE = {WIN g2 93:1 P-Ex}, - YEAR = {1993}, - PUBLISHER = {The MIT Press}, - SERIES = {Foundations of Computing}, - } - -@Article{Sem:WrightFelleisen, - refkey = "C1210", - title = "A Syntactic Approach to Type Soundness", - author = "Andrew K. Wright and Matthias Felleisen", - pages = "38--94", - journal = "Information and Computation", - month = "15~" # nov, - year = "1994", - volume = "115", - number = "1" -} - -@inproceedings{Sem:Nipkow-MOD, - author={Tobias Nipkow}, - title={Jinja: Towards a Comprehensive Formal Semantics for a - {J}ava-like Language}, - booktitle={Proc.\ Marktobderdorf Summer School 2003}, - publisher={IOS Press},editor={H. Schwichtenberg and K. Spies}, - year=2003, - note={To appear} -} - -%%%%%%% Coq %%%%%%% - -@book{Coq:coqart, - title = "Interactive Theorem Proving and Program Development, - Coq'Art: The Calculus of Inductive Constructions", - author = "Yves Bertot and Pierre Castéran", - publisher = "Springer Verlag", - series = "Texts in Theoretical Computer Science. An - EATCS series", - year = 2004 -} - -@phdthesis{Coq:Del01, - AUTHOR = "David Delahaye", - TITLE = "Conception de langages pour décrire les preuves et les - automatisations dans les outils d'aide à la preuve", - SCHOOL = {Universit\'e Paris~6}, - YEAR = "2001", - Type = {Th\`ese de Doctorat} -} - -@techreport{Coq:gimenez-tut, - author = "Eduardo Gim\'enez", - title = "A Tutorial on Recursive Types in Coq", - number = "RT-0221", - pages = "42 p.", - url = "citeseer.nj.nec.com/gimenez98tutorial.html" } - -@phdthesis{Coq:Mun97, - AUTHOR = "César Mu{\~{n}}oz", - TITLE = "Un calcul de substitutions pour la repr\'esentation - de preuves partielles en th\'eorie de types", - SCHOOL = {Universit\'e Paris~7}, - Number = {Unit\'e de recherche INRIA-Rocquencourt, TU-0488}, - YEAR = "1997", - Note = {English version available as INRIA research report RR-3309}, - Type = {Th\`ese de Doctorat} -} - -@PHDTHESIS{Coq:Filliatre99, - AUTHOR = {J.-C. Filli\^atre}, - TITLE = {{Preuve de programmes imp\'eratifs en th\'eorie des types}}, - TYPE = {Th{\`e}se de Doctorat}, - SCHOOL = {Universit\'e Paris-Sud}, - YEAR = 1999, - MONTH = {July}, -} - -@manual{Coq:Tutorial, - AUTHOR = {G\'erard Huet and Gilles Kahn and Christine Paulin-Mohring}, - TITLE = {{The Coq Proof Assistant A Tutorial}}, - YEAR = 2004 -} - -%%%%%%% PVS %%%%%%% - -@manual{PVS:prover, - title = "{PVS} Prover Guide", - author = "N. Shankar and S. Owre and J. M. Rushby and D. W. J. - Stringer-Calvert", - month = sep, - year = "1999", - organization = "Computer Science Laboratory, SRI International", - address = "Menlo Park, CA", -} - -@techreport{PVS-Semantics:TR, - TITLE = {The Formal Semantics of {PVS}}, - AUTHOR = {Sam Owre and Natarajan Shankar}, - NUMBER = {CR-1999-209321}, - INSTITUTION = {Computer Science Laboratory, SRI International}, - ADDRESS = {Menlo Park, CA}, - MONTH = may, - YEAR = 1999, -} - -@techreport{PVS-Tactics:DiVito, - TITLE = {A {PVS} Prover Strategy Package for Common Manipulations}, - AUTHOR = {Ben L. Di Vito}, - NUMBER = {TM-2002-211647}, - INSTITUTION = {Langley Research Center}, - ADDRESS = {Hampton, VA}, - MONTH = apr, - YEAR = 2002, -} - -@misc{PVS-Tactics:cours, - author = "César Muñoz", - title = "Strategies in {PVS}", - howpublished = "Lecture notes", - note = "National Institute of Aerospace", - year = 2002 -} - -@techreport{PVS-Tactics:field, - author = "C. Mu{\~n}oz and M. Mayero", - title = "Real Automation in the Field", - institution = "ICASE-NASA Langley", - number = "NASA/CR-2001-211271 Interim ICASE Report No. 39", - month = "dec", - year = "2001" -} - -%%%%%%% Autres Prouveurs %%%%%%% - -@misc{ACL2:repNuPrl, - author = "James L. Caldwell and John Cowles", - title = "{Representing Nuprl Proof Objects in ACL2: toward a proof checker for Nuprl}", - url = "http://www.cs.uwyo.edu/~jlc/papers/proof_checking.ps" } - -@inproceedings{Elan:ckl-strat, - author = {H. Cirstea and C. Kirchner and L. Liquori}, - title = "{Rewrite Strategies in the Rewriting Calculus}", - booktitle = {WRLA'02}, - publisher = "{Elsevier Science B.V.}", - series = {Electronic Notes in Theoretical Computer Science}, - volume = {71}, - year = {2003}, -} - -@book{LCF:GMW, - author = {M. Gordon and R. Milner and C. Wadsworth}, - publisher = {sv}, - series = {lncs}, - volume = 78, - title = {Edinburgh {LCF}: A Mechanized Logic of Computation}, - year = 1979 -} - -%%%%%%% LaTeX %%%%%%% - -@manual{LaTeX:symb, - title = "The Great, Big List of \LaTeX\ Symbols", - author = "David Carlisle and Scott Pakin and Alexander Holt", - month = feb, - year = 2001, -} - -@manual{LaTeX:intro, - title = "The Not So Short Introduction to \LaTeX2e", - author = "Tobias Oetiker", - month = jan, - year = 1999, -} - -@MANUAL{CoqManualV7, - AUTHOR = {{The {Coq} Development Team}}, - TITLE = {{The Coq Proof Assistant Reference Manual -- Version - V7.1}}, - YEAR = {2001}, - MONTH = OCT, - NOTE = {http://coq.inria.fr} -} - -@MANUAL{CoqManual96, - TITLE = {The {Coq Proof Assistant Reference Manual} Version 6.1}, - AUTHOR = {B. Barras and S. Boutin and C. Cornes and J. Courant and - J.-C. Filli\^atre and - H. Herbelin and G. Huet and P. Manoury and C. Mu{\~{n}}oz and - C. Murthy and C. Parent and C. Paulin-Mohring and - A. Sa{\"\i}bi and B. Werner}, - ORGANIZATION = {{INRIA-Rocquencourt}-{CNRS-ENS Lyon}}, - URL = {ftp://ftp.inria.fr/INRIA/coq/V6.1/doc/Reference-Manual.dvi.gz}, - YEAR = 1996, - MONTH = DEC -} - -@MANUAL{CoqTutorial99, - AUTHOR = {G.~Huet and G.~Kahn and Ch.~Paulin-Mohring}, - TITLE = {The {\sf Coq} Proof Assistant - A tutorial - Version 6.3}, - MONTH = JUL, - YEAR = {1999}, - ABSTRACT = {http://coq.inria.fr/doc/tutorial.html} -} - -@MANUAL{CoqTutorialV7, - AUTHOR = {G.~Huet and G.~Kahn and Ch.~Paulin-Mohring}, - TITLE = {The {\sf Coq} Proof Assistant - A tutorial - Version 7.1}, - MONTH = OCT, - YEAR = {2001}, - NOTE = {http://coq.inria.fr} -} - -@TECHREPORT{modelpa2000, - AUTHOR = {B. Bérard and P. Castéran and E. Fleury and L. Fribourg - and J.-F. Monin and C. Paulin and A. Petit and D. Rouillard}, - TITLE = {Automates temporisés CALIFE}, - INSTITUTION = {Calife}, - YEAR = 2000, - URL = {http://www.loria.fr/projets/calife/WebCalifePublic/FOURNITURES/F1.1.ps.gz}, - TYPE = {Fourniture {F1.1}} -} - -@TECHREPORT{CaFrPaRo2000, - AUTHOR = {P. Castéran and E. Freund and C. Paulin and D. Rouillard}, - TITLE = {Bibliothèques Coq et Isabelle-HOL pour les systèmes de transitions et les p-automates}, - INSTITUTION = {Calife}, - YEAR = 2000, - URL = {http://www.loria.fr/projets/calife/WebCalifePublic/FOURNITURES/F5.4.ps.gz}, - TYPE = {Fourniture {F5.4}} -} - -@PROCEEDINGS{TPHOLs99, - TITLE = {International Conference on - Theorem Proving in Higher Order Logics (TPHOLs'99)}, - YEAR = 1999, - EDITOR = {Y. Bertot and G. Dowek and C. Paulin-Mohring and L. Th{\'e}ry}, - SERIES = {Lecture Notes in Computer Science}, - MONTH = SEP, - PUBLISHER = {{Sprin\-ger-Verlag}}, - ADDRESS = {Nice}, - TYPE_PUBLI = {editeur} -} - -@INPROCEEDINGS{Pau01, - AUTHOR = {Christine Paulin-Mohring}, - TITLE = {Modelisation of Timed Automata in {Coq}}, - BOOKTITLE = {Theoretical Aspects of Computer Software (TACS'2001)}, - PAGES = {298--315}, - YEAR = 2001, - EDITOR = {N. Kobayashi and B. Pierce}, - VOLUME = 2215, - SERIES = {Lecture Notes in Computer Science}, - PUBLISHER = {Springer-Verlag} -} - -@PHDTHESIS{Moh89b, - AUTHOR = {C. Paulin-Mohring}, - MONTH = JAN, - SCHOOL = {{Paris 7}}, - TITLE = {Extraction de programmes dans le {Calcul des Constructions}}, - TYPE = {Thèse d'université}, - YEAR = {1989}, - URL = {http://www.lri.fr/~paulin/these.ps.gz} -} - -@ARTICLE{HuMo92, - AUTHOR = {G. Huet and C. Paulin-Mohring}, - EDITION = {INRIA}, - JOURNAL = {Courrier du CNRS - Informatique}, - TITLE = {Preuves et Construction de Programmes}, - YEAR = {1992}, - CATEGORY = {national} -} - -@INPROCEEDINGS{LePa94, - AUTHOR = {F. Leclerc and C. Paulin-Mohring}, - TITLE = {Programming with Streams in {Coq}. A case study : The Sieve of Eratosthenes}, - EDITOR = {H. Barendregt and T. Nipkow}, - VOLUME = 806, - SERIES = {Lecture Notes in Computer Science}, - BOOKTITLE = {{Types for Proofs and Programs, Types' 93}}, - YEAR = 1994, - PUBLISHER = {Springer-Verlag} -} - -@INPROCEEDINGS{Moh86, - AUTHOR = {C. Mohring}, - ADDRESS = {Cambridge, MA}, - BOOKTITLE = {Symposium on Logic in Computer Science}, - PUBLISHER = {IEEE Computer Society Press}, - TITLE = {Algorithm Development in the {Calculus of Constructions}}, - YEAR = {1986} -} - -@INPROCEEDINGS{Moh89a, - AUTHOR = {C. Paulin-Mohring}, - ADDRESS = {Austin}, - BOOKTITLE = {Sixteenth Annual ACM Symposium on Principles of Programming Languages}, - MONTH = JAN, - PUBLISHER = {ACM}, - TITLE = {Extracting ${F}_{\omega}$'s programs from proofs in the {Calculus of Constructions}}, - YEAR = {1989} -} - -@INCOLLECTION{Moh89c, - AUTHOR = {C. Paulin-Mohring}, - TITLE = {{R\'ealisabilit\'e et extraction de programmes}}, - BOOKTITLE = {Logique et Informatique : une introduction}, - PUBLISHER = {INRIA}, - YEAR = 1991, - EDITOR = {B. Courcelle}, - VOLUME = 8, - SERIES = {Collection Didactique}, - PAGES = {163-180}, - CATEGORY = {national} -} - -@INPROCEEDINGS{Moh93, - AUTHOR = {C. Paulin-Mohring}, - BOOKTITLE = {Proceedings of the conference Typed Lambda Calculi a -nd Applications}, - EDITOR = {M. Bezem and J.-F. Groote}, - INSTITUTION = {LIP-ENS Lyon}, - NOTE = {LIP research report 92-49}, - NUMBER = 664, - SERIES = {Lecture Notes in Computer Science}, - TITLE = {{Inductive Definitions in the System {Coq} - Rules and Properties}}, - TYPE = {research report}, - YEAR = 1993 -} - -@ARTICLE{PaWe92, - AUTHOR = {C. Paulin-Mohring and B. Werner}, - JOURNAL = {Journal of Symbolic Computation}, - TITLE = {{Synthesis of ML programs in the system Coq}}, - VOLUME = {15}, - YEAR = {1993}, - PAGES = {607--640} -} - -@INPROCEEDINGS{Pau96, - AUTHOR = {C. Paulin-Mohring}, - TITLE = {Circuits as streams in {Coq} : Verification of a sequential multiplier}, - BOOKTITLE = {Types for Proofs and Programs, TYPES'95}, - EDITOR = {S. Berardi and M. Coppo}, - SERIES = {Lecture Notes in Computer Science}, - YEAR = 1996, - VOLUME = 1158 -} - -@PHDTHESIS{Pau96b, - AUTHOR = {Christine Paulin-Mohring}, - TITLE = {Définitions Inductives en Théorie des Types d'Ordre Supérieur}, - SCHOOL = {Université Claude Bernard Lyon I}, - YEAR = 1996, - MONTH = DEC, - TYPE = {Habilitation à diriger les recherches}, - URL = {http://www.lri.fr/~paulin/habilitation.ps.gz} -} - -@INPROCEEDINGS{PfPa89, - AUTHOR = {F. Pfenning and C. Paulin-Mohring}, - BOOKTITLE = {Proceedings of Mathematical Foundations of Programming Semantics}, - NOTE = {technical report CMU-CS-89-209}, - PUBLISHER = {Springer-Verlag}, - SERIES = {Lecture Notes in Computer Science}, - VOLUME = 442, - TITLE = {Inductively defined types in the {Calculus of Constructions}}, - YEAR = {1990} -} - -@MISC{krakatoa02, - AUTHOR = {Claude March\'e and Christine Paulin and Xavier Urbain}, - TITLE = {The \textsc{Krakatoa} proof tool}, - YEAR = 2002, - NOTE = {\url{http://krakatoa.lri.fr/}} -} - -@ARTICLE{marche03jlap, - AUTHOR = {Claude March{\'e} and Christine Paulin-Mohring and Xavier Urbain}, - TITLE = {The \textsc{Krakatoa} Tool for Certification of \textsc{Java/JavaCard} Programs annotated in \textsc{JML}}, - JOURNAL = {Journal of Logic and Algebraic Programming}, - YEAR = 2003, - NOTE = {To appear}, - URL = {http://krakatoa.lri.fr}, - TOPICS = {team} -} -@ARTICLE{marche04jlap, - AUTHOR = {Claude March{\'e} and Christine Paulin-Mohring and Xavier Urbain}, - TITLE = {The \textsc{Krakatoa} Tool for Certification of \textsc{Java/JavaCard} Programs annotated in \textsc{JML}}, - JOURNAL = {Journal of Logic and Algebraic Programming}, - YEAR = 2004, - VOLUME = 58, - NUMBER = {1--2}, - PAGES = {89--106}, - URL = {http://krakatoa.lri.fr}, - TOPICS = {team} -} - -@TECHREPORT{catano03deliv, - AUTHOR = {N{\'e}stor Cata{\~n}o and Marek Gawkowski and -Marieke Huisman and Bart Jacobs and Claude March{\'e} and Christine Paulin -and Erik Poll and Nicole Rauch and Xavier Urbain}, - TITLE = {Logical Techniques for Applet Verification}, - INSTITUTION = {VerifiCard Project}, - YEAR = 2003, - TYPE = {Deliverable}, - NUMBER = {5.2}, - TOPICS = {team}, - NOTE = {Available from \url{http://www.verificard.org}} -} - -@TECHREPORT{kmu2002rr, - AUTHOR = {Keiichirou Kusakari and Claude Marché and Xavier Urbain}, - TITLE = {Termination of Associative-Commutative Rewriting using Dependency Pairs Criteria}, - INSTITUTION = {LRI}, - YEAR = 2002, - TYPE = {Research Report}, - NUMBER = 1304, - TYPE_PUBLI = {interne}, - TOPICS = {team}, - NOTE = {\url{http://www.lri.fr/~urbain/textes/rr1304.ps.gz}}, - URL = {http://www.lri.fr/~urbain/textes/rr1304.ps.gz} -} - -@ARTICLE{marche2004jsc, - AUTHOR = {Claude March\'e and Xavier Urbain}, - TITLE = {Modular {\&} Incremental Proofs of {AC}-Termination}, - JOURNAL = {Journal of Symbolic Computation}, - YEAR = 2004, - TOPICS = {team} -} - -@INPROCEEDINGS{contejean03wst, - AUTHOR = {Evelyne Contejean and Claude Marché and Benjamin Monate and Xavier Urbain}, - TITLE = {{Proving Termination of Rewriting with {\sc C\textit{i}ME}}}, - CROSSREF = {wst03}, - PAGES = {71--73}, - NOTE = {\url{http://cime.lri.fr/}}, - URL = {http://cime.lri.fr/}, - YEAR = 2003, - TYPE_PUBLI = {icolcomlec}, - TOPICS = {team} -} - -@TECHREPORT{contejean04rr, - AUTHOR = {Evelyne Contejean and Claude March{\'e} and Ana-Paula Tom{\'a}s and Xavier Urbain}, - TITLE = {Mechanically proving termination using polynomial interpretations}, - INSTITUTION = {LRI}, - YEAR = {2004}, - TYPE = {Research Report}, - NUMBER = {1382}, - TYPE_PUBLI = {interne}, - TOPICS = {team}, - URL = {http://www.lri.fr/~urbain/textes/rr1382.ps.gz} -} - -@UNPUBLISHED{duran_sub, - AUTHOR = {Francisco Duran and Salvador Lucas and - Claude {March\'e} and {Jos\'e} Meseguer and Xavier Urbain}, - TITLE = {Termination of Membership Equational Programs}, - NOTE = {Submitted} -} - -@PROCEEDINGS{comon95lncs, - TITLE = {Term Rewriting}, - BOOKTITLE = {Term Rewriting}, - TOPICS = {team, cclserver}, - YEAR = 1995, - EDITOR = {Hubert Comon and Jean-Pierre Jouannaud}, - SERIES = {Lecture Notes in Computer Science}, - VOLUME = {909}, - PUBLISHER = {{Sprin\-ger-Verlag}}, - ORGANIZATION = {French Spring School of Theoretical Computer - Science}, - TYPE_PUBLI = {editeur}, - CLEF_LABO = {CJ95} -} - -@PROCEEDINGS{lics94, - TITLE = {Proceedings of the Ninth Annual IEEE Symposium on Logic - in Computer Science}, - BOOKTITLE = {Proceedings of the Ninth Annual IEEE Symposium on Logic - in Computer Science}, - YEAR = 1994, - MONTH = JUL, - ADDRESS = {Paris, France}, - ORGANIZATION = {{IEEE} Comp. Soc. Press} -} - -@PROCEEDINGS{rta91, - TITLE = {4th International Conference on Rewriting Techniques and - Applications}, - BOOKTITLE = {4th International Conference on Rewriting Techniques and - Applications}, - EDITOR = {Ronald. V. Book}, - YEAR = 1991, - MONTH = APR, - ADDRESS = {Como, Italy}, - PUBLISHER = {{Sprin\-ger-Verlag}}, - SERIES = {Lecture Notes in Computer Science}, - VOLUME = 488 -} - -@PROCEEDINGS{rta96, - TITLE = {7th International Conference on Rewriting Techniques and - Applications}, - BOOKTITLE = {7th International Conference on Rewriting Techniques and - Applications}, - EDITOR = {Harald Ganzinger}, - PUBLISHER = {{Sprin\-ger-Verlag}}, - YEAR = 1996, - MONTH = JUL, - ADDRESS = {New Brunswick, NJ, USA}, - SERIES = {Lecture Notes in Computer Science}, - VOLUME = 1103 -} - -@PROCEEDINGS{rta97, - TITLE = {8th International Conference on Rewriting Techniques and - Applications}, - BOOKTITLE = {8th International Conference on Rewriting Techniques and - Applications}, - EDITOR = {Hubert Comon}, - PUBLISHER = {{Sprin\-ger-Verlag}}, - YEAR = 1997, - MONTH = JUN, - ADDRESS = {Barcelona, Spain}, - SERIES = {Lecture Notes in Computer Science}, - VOLUME = {1232} -} - -@PROCEEDINGS{rta98, - TITLE = {9th International Conference on Rewriting Techniques and - Applications}, - BOOKTITLE = {9th International Conference on Rewriting Techniques and - Applications}, - EDITOR = {Tobias Nipkow}, - PUBLISHER = {{Sprin\-ger-Verlag}}, - YEAR = 1998, - MONTH = APR, - ADDRESS = {Tsukuba, Japan}, - SERIES = {Lecture Notes in Computer Science}, - VOLUME = {1379} -} - -@PROCEEDINGS{rta00, - TITLE = {11th International Conference on Rewriting Techniques and Applications}, - BOOKTITLE = {11th International Conference on Rewriting Techniques and Applications}, - EDITOR = {Leo Bachmair}, - PUBLISHER = {{Sprin\-ger-Verlag}}, - SERIES = {Lecture Notes in Computer Science}, - VOLUME = 1833, - MONTH = JUL, - YEAR = 2000, - ADDRESS = {Norwich, UK} -} - -@PROCEEDINGS{srt95, - TITLE = {Proceedings of the Conference on Symbolic Rewriting - Techniques}, - BOOKTITLE = {Proceedings of the Conference on Symbolic Rewriting - Techniques}, - YEAR = 1995, - EDITOR = {Manuel Bronstein and Volker Weispfenning}, - ADDRESS = {Monte Verita, Switzerland} -} - -@BOOK{comon01cclbook, - BOOKTITLE = {Constraints in Computational Logics}, - TITLE = {Constraints in Computational Logics}, - EDITOR = {Hubert Comon and Claude March{\'e} and Ralf Treinen}, - YEAR = 2001, - PUBLISHER = {{Sprin\-ger-Verlag}}, - SERIES = {Lecture Notes in Computer Science}, - VOLUME = 2002, - TOPICS = {team}, - TYPE_PUBLI = {editeur} -} - -@PROCEEDINGS{wst03, - BOOKTITLE = {{Extended Abstracts of the 6th International Workshop on Termination, WST'03}}, - TITLE = {{Extended Abstracts of the 6th International Workshop on Termination, WST'03}}, - YEAR = {2003}, - EDITOR = {Albert Rubio}, - MONTH = JUN, - NOTE = {Technical Report DSIC II/15/03, Universidad Politécnica de Valencia, Spain} -} - -@INPROCEEDINGS{FilliatreLetouzey03, - AUTHOR = {J.-C. Filli\^atre and P. Letouzey}, - TITLE = {{Functors for Proofs and Programs}}, - BOOKTITLE = {Proceedings of The European Symposium on Programming}, - YEAR = 2004, - ADDRESS = {Barcelona, Spain}, - MONTH = {March 29-April 2}, - NOTE = {To appear}, - URL = {http://www.lri.fr/~filliatr/ftp/publis/fpp.ps.gz} -} - -@TECHREPORT{Filliatre03, - AUTHOR = {J.-C. Filli\^atre}, - TITLE = {{Why: a multi-language multi-prover verification tool}}, - INSTITUTION = {{LRI, Universit\'e Paris Sud}}, - TYPE = {{Research Report}}, - NUMBER = {1366}, - MONTH = {March}, - YEAR = 2003, - URL = {http://www.lri.fr/~filliatr/ftp/publis/why-tool.ps.gz} -} - -@ARTICLE{FilliatrePottier02, - AUTHOR = {J.-C. Filli{\^a}tre and F. Pottier}, - TITLE = {{Producing All Ideals of a Forest, Functionally}}, - JOURNAL = {Journal of Functional Programming}, - VOLUME = 13, - NUMBER = 5, - PAGES = {945--956}, - MONTH = {September}, - YEAR = 2003, - URL = {http://www.lri.fr/~filliatr/ftp/publis/kr-fp.ps.gz}, - ABSTRACT = { - We present a functional implementation of Koda and Ruskey's - algorithm for generating all ideals of a forest poset as a Gray - code. Using a continuation-based approach, we give an extremely - concise formulation of the algorithm's core. Then, in a number of - steps, we derive a first-order version whose efficiency is - comparable to a C implementation given by Knuth.} -} - -@UNPUBLISHED{FORS01, - AUTHOR = {J.-C. Filli{\^a}tre and S. Owre and H. Rue{\ss} and N. Shankar}, - TITLE = {Deciding Propositional Combinations of Equalities and Inequalities}, - NOTE = {Unpublished}, - MONTH = OCT, - YEAR = 2001, - URL = {http://www.lri.fr/~filliatr/ftp/publis/ics.ps}, - ABSTRACT = { - We address the problem of combining individual decision procedures - into a single decision procedure. Our combination approach is based - on using the canonizer obtained from Shostak's combination algorithm - for equality. We illustrate our approach with a combination - algorithm for equality, disequality, arithmetic inequality, and - propositional logic. Unlike the Nelson--Oppen combination where the - processing of equalities is distributed across different closed - decision procedures, our combination involves the centralized - processing of equalities in a single procedure. The termination - argument for the combination is based on that for Shostak's - algorithm. We also give soundness and completeness arguments.} -} - -@INPROCEEDINGS{ICS, - AUTHOR = {J.-C. Filli{\^a}tre and S. Owre and H. Rue{\ss} and N. Shankar}, - TITLE = {{ICS: Integrated Canonization and Solving (Tool presentation)}}, - BOOKTITLE = {Proceedings of CAV'2001}, - EDITOR = {G. Berry and H. Comon and A. Finkel}, - PUBLISHER = {Springer-Verlag}, - SERIES = {Lecture Notes in Computer Science}, - VOLUME = 2102, - PAGES = {246--249}, - YEAR = 2001 -} - -@INPROCEEDINGS{Filliatre01a, - AUTHOR = {J.-C. Filli\^atre}, - TITLE = {La supériorité de l'ordre supérieur}, - BOOKTITLE = {Journées Francophones des Langages Applicatifs}, - PAGES = {15--26}, - MONTH = {Janvier}, - YEAR = 2002, - ADDRESS = {Anglet, France}, - URL = {http://www.lri.fr/~filliatr/ftp/publis/sos.ps.gz}, - CODE = {http://www.lri.fr/~filliatr/ftp/ocaml/misc/koda-ruskey.ps}, - ABSTRACT = { - Nous présentons ici une écriture fonctionnelle de l'algorithme de - Koda-Ruskey, un algorithme pour engendrer une large famille - de codes de Gray. En s'inspirant de techniques de programmation par - continuation, nous aboutissons à un code de neuf lignes seulement, - bien plus élégant que les implantations purement impératives - proposées jusqu'ici, notamment par Knuth. Dans un second temps, - nous montrons comment notre code peut être légèrement modifié pour - aboutir à une version de complexité optimale. - Notre implantation en Objective Caml rivalise d'efficacité avec les - meilleurs codes C. Nous détaillons les calculs de complexité, - un exercice intéressant en présence d'ordre supérieur et d'effets de - bord combinés.} -} - -@TECHREPORT{Filliatre00c, - AUTHOR = {J.-C. Filli\^atre}, - TITLE = {{Design of a proof assistant: Coq version 7}}, - INSTITUTION = {{LRI, Universit\'e Paris Sud}}, - TYPE = {{Research Report}}, - NUMBER = {1369}, - MONTH = {October}, - YEAR = 2000, - URL = {http://www.lri.fr/~filliatr/ftp/publis/coqv7.ps.gz}, - ABSTRACT = { - We present the design and implementation of the new version of the - Coq proof assistant. The main novelty is the isolation of the - critical part of the system, which consists in a type checker for - the Calculus of Inductive Constructions. This kernel is now - completely independent of the rest of the system and has been - rewritten in a purely functional way. This leads to greater clarity - and safety, without compromising efficiency. It also opens the way to - the ``bootstrap'' of the Coq system, where the kernel will be - certified using Coq itself.} -} - -@TECHREPORT{Filliatre00b, - AUTHOR = {J.-C. Filli\^atre}, - TITLE = {{Hash consing in an ML framework}}, - INSTITUTION = {{LRI, Universit\'e Paris Sud}}, - TYPE = {{Research Report}}, - NUMBER = {1368}, - MONTH = {September}, - YEAR = 2000, - URL = {http://www.lri.fr/~filliatr/ftp/publis/hash-consing.ps.gz}, - ABSTRACT = { - Hash consing is a technique to share values that are structurally - equal. Beyond the obvious advantage of saving memory blocks, hash - consing may also be used to gain speed in several operations (like - equality test) and data structures (like sets or maps) when sharing is - maximal. However, physical adresses cannot be used directly for this - purpose when the garbage collector is likely to move blocks - underneath. We present an easy solution in such a framework, with - many practical benefits.} -} - -@MISC{ocamlweb, - AUTHOR = {J.-C. Filli\^atre and C. March\'e}, - TITLE = {{ocamlweb, a literate programming tool for Objective Caml}}, - NOTE = {Available at \url{http://www.lri.fr/~filliatr/ocamlweb/}}, - URL = {http://www.lri.fr/~filliatr/ocamlweb/} -} - -@ARTICLE{Filliatre00a, - AUTHOR = {J.-C. Filli\^atre}, - TITLE = {{Verification of Non-Functional Programs - using Interpretations in Type Theory}}, - JOURNAL = {Journal of Functional Programming}, - VOLUME = 13, - NUMBER = 4, - PAGES = {709--745}, - MONTH = {July}, - YEAR = 2003, - NOTE = {English translation of~\cite{Filliatre99}.}, - URL = {http://www.lri.fr/~filliatr/ftp/publis/jphd.ps.gz}, - ABSTRACT = {We study the problem of certifying programs combining imperative and - functional features within the general framework of type theory. - - Type theory constitutes a powerful specification language, which is - naturally suited for the proof of purely functional programs. To - deal with imperative programs, we propose a logical interpretation - of an annotated program as a partial proof of its specification. The - construction of the corresponding partial proof term is based on a - static analysis of the effects of the program, and on the use of - monads. The usual notion of monads is refined in order to account - for the notion of effect. The missing subterms in the partial proof - term are seen as proof obligations, whose actual proofs are left to - the user. We show that the validity of those proof obligations - implies the total correctness of the program. - We also establish a result of partial completeness. - - This work has been implemented in the Coq proof assistant. - It appears as a tactic taking an annotated program as argument and - generating a set of proof obligations. Several nontrivial - algorithms have been certified using this tactic.} -} - -@ARTICLE{Filliatre99c, - AUTHOR = {J.-C. Filli\^atre}, - TITLE = {{Formal Proof of a Program: Find}}, - JOURNAL = {Science of Computer Programming}, - YEAR = 2001, - NOTE = {To appear}, - URL = {http://www.lri.fr/~filliatr/ftp/publis/find.ps.gz}, - ABSTRACT = {In 1971, C.~A.~R.~Hoare gave the proof of correctness and termination of a - rather complex algorithm, in a paper entitled \emph{Proof of a - program: Find}. It is a hand-made proof, where the - program is given together with its formal specification and where - each step is fully - justified by a mathematical reasoning. We present here a formal - proof of the same program in the system Coq, using the - recent tactic of the system developed to establishing the total - correctness of - imperative programs. We follow Hoare's paper as close as - possible, keeping the same program and the same specification. We - show that we get exactly the same proof obligations, which are - proved in a straightforward way, following the original paper. - We also explain how more informal reasonings of Hoare's proof are - formalized in the system Coq. - This demonstrates the adequacy of the system Coq in the - process of certifying imperative programs.} -} - -@TECHREPORT{Filliatre99b, - AUTHOR = {J.-C. Filli\^atre}, - TITLE = {{A theory of monads parameterized by effects}}, - INSTITUTION = {{LRI, Universit\'e Paris Sud}}, - TYPE = {{Research Report}}, - NUMBER = {1367}, - MONTH = {November}, - YEAR = 1999, - URL = {http://www.lri.fr/~filliatr/ftp/publis/monads.ps.gz}, - ABSTRACT = {Monads were introduced in computer science to express the semantics - of programs with computational effects, while type and effect - inference was introduced to mark out those effects. - In this article, we propose a combination of the notions of effects - and monads, where the monadic operators are parameterized by effects. - We establish some relationships between those generalized monads and - the classical ones. - Then we use a generalized monad to translate imperative programs - into purely functional ones. We establish the correctness of that - translation. This work has been put into practice in the Coq proof - assistant to establish the correctness of imperative programs.} -} - -@PHDTHESIS{Filliatre99, - AUTHOR = {J.-C. Filli\^atre}, - TITLE = {{Preuve de programmes imp\'eratifs en th\'eorie des types}}, - TYPE = {Th{\`e}se de Doctorat}, - SCHOOL = {Universit\'e Paris-Sud}, - YEAR = 1999, - MONTH = {July}, - URL = {http://www.lri.fr/~filliatr/ftp/publis/these.ps.gz}, - ABSTRACT = {Nous étudions le problème de la certification de programmes mêlant - traits impératifs et fonctionnels dans le cadre de la théorie des - types. - - La théorie des types constitue un puissant langage de spécification, - naturellement adapté à la preuve de programmes purement - fonctionnels. Pour y certifier également des programmes impératifs, - nous commençons par exprimer leur sémantique de manière purement - fonctionnelle. Cette traduction repose sur une analyse statique des - effets de bord des programmes, et sur l'utilisation de la notion de - monade, notion que nous raffinons en l'associant à la notion d'effet - de manière générale. Nous montrons que cette traduction est - sémantiquement correcte. - - Puis, à partir d'un programme annoté, nous construisons une preuve - de sa spécification, traduite de manière fonctionnelle. Cette preuve - est bâtie sur la traduction fonctionnelle précédemment - introduite. Elle est presque toujours incomplète, les parties - manquantes étant autant d'obligations de preuve qui seront laissées - à la charge de l'utilisateur. Nous montrons que la validité de ces - obligations entraîne la correction totale du programme. - - Nous avons implanté notre travail dans l'assistant de preuve - Coq, avec lequel il est dès à présent distribué. Cette - implantation se présente sous la forme d'une tactique prenant en - argument un programme annoté et engendrant les obligations de - preuve. Plusieurs algorithmes non triviaux ont été certifiés à - l'aide de cet outil (Find, Quicksort, Heapsort, algorithme de - Knuth-Morris-Pratt).} -} - -@INPROCEEDINGS{FilliatreMagaud99, - AUTHOR = {J.-C. Filli\^atre and N. Magaud}, - TITLE = {{Certification of sorting algorithms in the system Coq}}, - BOOKTITLE = {Theorem Proving in Higher Order Logics: - Emerging Trends}, - YEAR = 1999, - ABSTRACT = {We present the formal proofs of total correctness of three sorting - algorithms in the system Coq, namely \textit{insertion sort}, - \textit{quicksort} and \textit{heapsort}. The implementations are - imperative programs working in-place on a given array. Those - developments demonstrate the usefulness of inductive types and higher-order - logic in the process of software certification. They also - show that the proof of rather complex algorithms may be done in a - small amount of time --- only a few days for each development --- - and without great difficulty.}, - URL = {http://www.lri.fr/~filliatr/ftp/publis/Filliatre-Magaud.ps.gz} -} - -@INPROCEEDINGS{Filliatre98, - AUTHOR = {J.-C. Filli\^atre}, - TITLE = {{Proof of Imperative Programs in Type Theory}}, - BOOKTITLE = {International Workshop, TYPES '98, Kloster Irsee, Germany}, - PUBLISHER = {Springer-Verlag}, - VOLUME = 1657, - SERIES = {Lecture Notes in Computer Science}, - MONTH = MAR, - YEAR = {1998}, - ABSTRACT = {We present a new approach to certifying imperative programs, - in the context of Type Theory. - The key is a functional translation of imperative programs, which is - made possible by an analysis of their effects. - On sequential imperative programs, we get the same proof - obligations as those given by Floyd-Hoare logic, - but our approach also includes functional constructions. - As a side-effect, we propose a way to eradicate the use of auxiliary - variables in specifications. - This work has been implemented in the Coq Proof Assistant and applied - on non-trivial examples.}, - URL = {http://www.lri.fr/~filliatr/ftp/publis/types98.ps.gz} -} - -@TECHREPORT{Filliatre97, - AUTHOR = {J.-C. Filli\^atre}, - INSTITUTION = {LIP - ENS Lyon}, - NUMBER = {97--04}, - TITLE = {{Finite Automata Theory in Coq: - A constructive proof of Kleene's theorem}}, - TYPE = {Research Report}, - MONTH = {February}, - YEAR = {1997}, - ABSTRACT = {We describe here a development in the system Coq - of a piece of Finite Automata Theory. The main result is the Kleene's - theorem, expressing that regular expressions and finite automata - define the same languages. From a constructive proof of this result, - we automatically obtain a functional program that compiles any - regular expression into a finite automata, which constitutes the main - part of the implementation of {\tt grep}-like programs. This - functional program is obtained by the automatic method of {\em - extraction} which removes the logical parts of the proof to keep only - its informative contents. Starting with an idea of what we would - have written in ML, we write the specification and do the proofs in - such a way that we obtain the expected program, which is therefore - efficient.}, - URL = {ftp://ftp.ens-lyon.fr/pub/LIP/Rapports/RR/RR97/RR97-04.ps.Z} -} - -@TECHREPORT{Filliatre95, - AUTHOR = {J.-C. Filli\^atre}, - INSTITUTION = {LIP - ENS Lyon}, - NUMBER = {96--25}, - TITLE = {{A decision procedure for Direct Predicate - Calculus: study and implementation in - the Coq system}}, - TYPE = {Research Report}, - MONTH = {February}, - YEAR = {1995}, - ABSTRACT = {The paper of J. Ketonen and R. Weyhrauch \emph{A - decidable fragment of Predicate Calculus} defines a decidable - fragment of first-order predicate logic - Direct Predicate Calculus - - as the subset which is provable in Gentzen sequent calculus - without the contraction rule, and gives an effective decision - procedure for it. This report is a detailed study of this - procedure. We extend the decidability to non-prenex formulas. We - prove that the intuitionnistic fragment is still decidable, with a - refinement of the same procedure. An intuitionnistic version has - been implemented in the Coq system using a translation into - natural deduction.}, - URL = {ftp://ftp.ens-lyon.fr/pub/LIP/Rapports/RR/RR96/RR96-25.ps.Z} -} - -@TECHREPORT{Filliatre94, - AUTHOR = {J.-C. Filli\^atre}, - MONTH = {Juillet}, - INSTITUTION = {Ecole Normale Sup\'erieure}, - TITLE = {{Une proc\'edure de d\'ecision pour le Calcul des Pr\'edicats Direct~: \'etude et impl\'ementation dans le syst\`eme Coq}}, - TYPE = {Rapport de {DEA}}, - YEAR = {1994}, - URL = {ftp://ftp.lri.fr/LRI/articles/filliatr/memoire.dvi.gz} -} - -@TECHREPORT{CourantFilliatre93, - AUTHOR = {J. Courant et J.-C. Filli\^atre}, - MONTH = {Septembre}, - INSTITUTION = {Ecole Normale Sup\'erieure}, - TITLE = {{Formalisation de la th\'eorie des langages - formels en Coq}}, - TYPE = {Rapport de ma\^{\i}trise}, - YEAR = {1993}, - URL = {http://www.ens-lyon.fr/~jcourant/stage_maitrise.dvi.gz}, - URL2 = {http://www.ens-lyon.fr/~jcourant/stage_maitrise.ps.gz} -} - -@INPROCEEDINGS{tphols2000-Letouzey, - crossref = "tphols2000", - title = "Formalizing {S}t{\aa}lmarck's algorithm in {C}oq", - author = "Pierre Letouzey and Laurent Th{\'e}ry", - pages = "387--404"} - -@PROCEEDINGS{tphols2000, - editor = "J. Harrison and M. Aagaard", - booktitle = "Theorem Proving in Higher Order Logics: - 13th International Conference, TPHOLs 2000", - series = "Lecture Notes in Computer Science", - volume = 1869, - year = 2000, - publisher = "Springer-Verlag"} - -@InCollection{howe, - author = {Doug Howe}, - title = {Computation Meta theory in Nuprl}, - booktitle = {The Proceedings of the Ninth International Conference of Autom -ated Deduction}, - volume = {310}, - editor = {E. Lusk and R. Overbeek}, - publisher = {Springer-Verlag}, - pages = {238--257}, - year = {1988} -} - -@TechReport{harrison, - author = {John Harrison}, - title = {Meta theory and Reflection in Theorem Proving:a Survey and Cri -tique}, - institution = {SRI International Cambridge Computer Science Research Center}, - year = {1995}, - number = {CRC-053} -} - -@InCollection{cc, - author = {Thierry Coquand and Gérard Huet}, - title = {The Calculus of Constructions}, - booktitle = {Information and Computation}, - year = {1988}, - volume = {76}, - number = {2/3} -} - - -@InProceedings{coquandcci, - author = {Thierry Coquand and Christine Paulin-Mohring}, - title = {Inductively defined types}, - booktitle = {Proceedings of Colog'88}, - year = {1990}, - editor = {P. Martin-Löf and G. Mints}, - volume = {417}, - series = {LNCS}, - publisher = {Springer-Verlag} -} - - -@InProceedings{boutin, - author = {Samuel Boutin}, - title = {Using reflection to build efficient and certified decision pro -cedures.}, - booktitle = {Proceedings of TACS'97}, - year = {1997}, - editor = {M. Abadi and T. Ito}, - volume = {1281}, - series = {LNCS}, - publisher = {Springer-Verlag} -} - -@Manual{Coq:manual, - title = {The Coq proof assistant reference manual}, - author = {\mbox{The Coq development team}}, - note = {Version 8.3}, - year = {2010}, - url = "http://coq.inria.fr/doc" -} - -@string{jfp = "Journal of Functional Programming"} -@STRING{lncs="Lecture Notes in Computer Science"} -@STRING{lnai="Lecture Notes in Artificial Intelligence"} -@string{SV = "{Sprin\-ger-Verlag}"} - -@INPROCEEDINGS{Aud91, - AUTHOR = {Ph. Audebaud}, - BOOKTITLE = {Proceedings of the sixth Conf. on Logic in Computer Science.}, - PUBLISHER = {IEEE}, - TITLE = {Partial {Objects} in the {Calculus of Constructions}}, - YEAR = {1991} -} - -@PHDTHESIS{Aud92, - AUTHOR = {Ph. Audebaud}, - SCHOOL = {{Universit\'e} Bordeaux I}, - TITLE = {Extension du Calcul des Constructions par Points fixes}, - YEAR = {1992} -} - -@INPROCEEDINGS{Audebaud92b, - AUTHOR = {Ph. Audebaud}, - BOOKTITLE = {{Proceedings of the 1992 Workshop on Types for Proofs and Programs}}, - EDITOR = {{B. Nordstr\"om and K. Petersson and G. Plotkin}}, - NOTE = {Also Research Report LIP-ENS-Lyon}, - PAGES = {pp 21--34}, - TITLE = {{CC+ : an extension of the Calculus of Constructions with fixpoints}}, - YEAR = {1992} -} - -@INPROCEEDINGS{Augustsson85, - AUTHOR = {L. Augustsson}, - TITLE = {{Compiling Pattern Matching}}, - BOOKTITLE = {Conference Functional Programming and -Computer Architecture}, - YEAR = {1985} -} - -@ARTICLE{BaCo85, - AUTHOR = {J.L. Bates and R.L. Constable}, - JOURNAL = {ACM transactions on Programming Languages and Systems}, - TITLE = {Proofs as {Programs}}, - VOLUME = {7}, - YEAR = {1985} -} - -@BOOK{Bar81, - AUTHOR = {H.P. Barendregt}, - PUBLISHER = {North-Holland}, - TITLE = {The Lambda Calculus its Syntax and Semantics}, - YEAR = {1981} -} - -@TECHREPORT{Bar91, - AUTHOR = {H. Barendregt}, - INSTITUTION = {Catholic University Nijmegen}, - NOTE = {In Handbook of Logic in Computer Science, Vol II}, - NUMBER = {91-19}, - TITLE = {Lambda {Calculi with Types}}, - YEAR = {1991} -} - -@ARTICLE{BeKe92, - AUTHOR = {G. Bellin and J. Ketonen}, - JOURNAL = {Theoretical Computer Science}, - PAGES = {115--142}, - TITLE = {A decision procedure revisited : Notes on direct logic, linear logic and its implementation}, - VOLUME = {95}, - YEAR = {1992} -} - -@BOOK{Bee85, - AUTHOR = {M.J. Beeson}, - PUBLISHER = SV, - TITLE = {Foundations of Constructive Mathematics, Metamathematical Studies}, - YEAR = {1985} -} - -@BOOK{Bis67, - AUTHOR = {E. 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Werner}, - INSTITUTION = {INRIA}, - MONTH = may, - NUMBER = {154}, - TITLE = {{The Coq Proof Assistant User's Guide Version 5.8}}, - YEAR = {1993} -} - -@TECHREPORT{CPar93, - AUTHOR = {C. Parent}, - INSTITUTION = {Ecole {Normale} {Sup\'erieure} de {Lyon}}, - MONTH = oct, - NOTE = {Also in~\cite{Nijmegen93}}, - NUMBER = {93-29}, - TITLE = {Developing certified programs in the system {Coq}- {The} {Program} tactic}, - YEAR = {1993} -} - -@PHDTHESIS{CPar95, - AUTHOR = {C. Parent}, - SCHOOL = {Ecole {Normale} {Sup\'erieure} de {Lyon}}, - TITLE = {{Synth\`ese de preuves de programmes dans le Calcul des Constructions Inductives}}, - YEAR = {1995} -} - -@BOOK{Caml, - AUTHOR = {P. Weis and X. 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Documentation and user's guide, Version 4.10}}, - YEAR = {1989} -} - -@INPROCEEDINGS{CoHu85a, - AUTHOR = {Thierry Coquand and Gérard Huet}, - ADDRESS = {Linz}, - BOOKTITLE = {EUROCAL'85}, - PUBLISHER = SV, - SERIES = LNCS, - TITLE = {{Constructions : A Higher Order Proof System for Mechanizing Mathematics}}, - VOLUME = {203}, - YEAR = {1985} -} - -@INPROCEEDINGS{CoHu85b, - AUTHOR = {Thierry Coquand and Gérard Huet}, - BOOKTITLE = {Logic Colloquium'85}, - EDITOR = {The Paris Logic Group}, - PUBLISHER = {North-Holland}, - TITLE = {{Concepts Math\'ematiques et Informatiques formalis\'es dans le Calcul des Constructions}}, - YEAR = {1987} -} - -@ARTICLE{CoHu86, - AUTHOR = {Thierry Coquand and Gérard Huet}, - JOURNAL = {Information and Computation}, - NUMBER = {2/3}, - TITLE = {The {Calculus of Constructions}}, - VOLUME = {76}, - YEAR = {1988} -} - -@INPROCEEDINGS{CoPa89, - AUTHOR = {Thierry Coquand and Christine Paulin-Mohring}, - BOOKTITLE = {Proceedings of Colog'88}, - EDITOR = {P. Martin-L\"of and G. Mints}, - PUBLISHER = SV, - SERIES = LNCS, - TITLE = {Inductively defined types}, - VOLUME = {417}, - YEAR = {1990} -} - -@BOOK{Con86, - AUTHOR = {R.L. {Constable et al.}}, - PUBLISHER = {Prentice-Hall}, - TITLE = {{Implementing Mathematics with the Nuprl Proof Development System}}, - YEAR = {1986} -} - -@PHDTHESIS{Coq85, - AUTHOR = {Thierry Coquand}, - MONTH = jan, - SCHOOL = {Universit\'e Paris~7}, - TITLE = {Une Th\'eorie des Constructions}, - YEAR = {1985} -} - -@INPROCEEDINGS{Coq86, - AUTHOR = {Thierry Coquand}, - ADDRESS = {Cambridge, MA}, - BOOKTITLE = {Symposium on Logic in Computer Science}, - PUBLISHER = {IEEE Computer Society Press}, - TITLE = {{An Analysis of Girard's Paradox}}, - YEAR = {1986} -} - -@INPROCEEDINGS{Coq90, - AUTHOR = {Thierry Coquand}, - BOOKTITLE = {Logic and Computer Science}, - EDITOR = {P. Oddifredi}, - NOTE = {INRIA Research Report 1088, also in~\cite{CoC89}}, - PUBLISHER = {Academic Press}, - TITLE = {{Metamathematical Investigations of a Calculus of Constructions}}, - YEAR = {1990} -} - -@INPROCEEDINGS{Coq91, - AUTHOR = {Thierry Coquand}, - BOOKTITLE = {Proceedings 9th Int. Congress of Logic, Methodology and Philosophy of Science}, - TITLE = {{A New Paradox in Type Theory}}, - MONTH = {August}, - YEAR = {1991} -} - -@INPROCEEDINGS{Coq92, - AUTHOR = {Thierry Coquand}, - TITLE = {{Pattern Matching with Dependent Types}}, - YEAR = {1992}, - crossref = {Bastad92} -} - -@INPROCEEDINGS{Coquand93, - AUTHOR = {Thierry Coquand}, - TITLE = {{Infinite Objects in Type Theory}}, - YEAR = {1993}, - crossref = {Nijmegen93} -} - -@MASTERSTHESIS{Cou94a, - AUTHOR = {J. Courant}, - MONTH = sep, - SCHOOL = {DEA d'Informatique, ENS Lyon}, - TITLE = {Explicitation de preuves par r\'ecurrence implicite}, - YEAR = {1994} -} - -@INPROCEEDINGS{Del99, - author = "Delahaye, D.", - title = "Information Retrieval in a Coq Proof Library using - Type Isomorphisms", - booktitle = {Proceedings of TYPES'99, L\"okeberg}, - publisher = SV, - series = lncs, - year = "1999", - url = - "\\{\sf ftp://ftp.inria.fr/INRIA/Projects/coq/David.Delahaye/papers/}"# - "{\sf TYPES99-SIsos.ps.gz}" -} - -@INPROCEEDINGS{Del00, - author = "Delahaye, D.", - title = "A {T}actic {L}anguage for the {S}ystem {{\sf Coq}}", - booktitle = "Proceedings of Logic for Programming and Automated Reasoning - (LPAR), Reunion Island", - publisher = SV, - series = LNCS, - volume = "1955", - pages = "85--95", - month = "November", - year = "2000", - url = - "{\sf ftp://ftp.inria.fr/INRIA/Projects/coq/David.Delahaye/papers/}"# - "{\sf LPAR2000-ltac.ps.gz}" -} - -@INPROCEEDINGS{DelMay01, - author = "Delahaye, D. and Mayero, M.", - title = {{\tt Field}: une proc\'edure de d\'ecision pour les nombres r\'eels - en {\Coq}}, - booktitle = "Journ\'ees Francophones des Langages Applicatifs, Pontarlier", - publisher = "INRIA", - month = "Janvier", - year = "2001", - url = - "\\{\sf ftp://ftp.inria.fr/INRIA/Projects/coq/David.Delahaye/papers/}"# - "{\sf JFLA2000-Field.ps.gz}" -} - -@TECHREPORT{Dow90, - AUTHOR = {G. 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Dowek}, - MONTH = dec, - SCHOOL = {Universit\'e Paris 7}, - TITLE = {D\'emonstration automatique dans le Calcul des Constructions}, - YEAR = {1991} -} - -@article{Dow92a, - AUTHOR = {G. Dowek}, - TITLE = {The Undecidability of Pattern Matching in Calculi where Primitive Recursive Functions are Representable}, - YEAR = 1993, - journal = tcs, - volume = 107, - number = 2, - pages = {349-356} -} - - -@ARTICLE{Dow94a, - AUTHOR = {G. Dowek}, - JOURNAL = {Annals of Pure and Applied Logic}, - VOLUME = {69}, - PAGES = {135--155}, - TITLE = {Third order matching is decidable}, - YEAR = {1994} -} - -@INPROCEEDINGS{Dow94b, - AUTHOR = {G. Dowek}, - BOOKTITLE = {Proceedings of the second international conference on typed lambda calculus and applications}, - TITLE = {Lambda-calculus, Combinators and the Comprehension Schema}, - YEAR = {1995} -} - -@INPROCEEDINGS{Dyb91, - AUTHOR = {P. Dybjer}, - BOOKTITLE = {Logical Frameworks}, - EDITOR = {G. Huet and G. 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Plotkin}, - PUBLISHER = {Cambridge University Press}, - TITLE = {Logical Environments}, - YEAR = {1992} -} - -@BOOK{LF91, - EDITOR = {G. Huet and G. Plotkin}, - PUBLISHER = {Cambridge University Press}, - TITLE = {Logical Frameworks}, - YEAR = {1991} -} - -@ARTICLE{Laville91, - AUTHOR = {A. Laville}, - TITLE = {Comparison of Priority Rules in Pattern -Matching and Term Rewriting}, - JOURNAL = {Journal of Symbolic Computation}, - VOLUME = {11}, - PAGES = {321--347}, - YEAR = {1991} -} - -@INPROCEEDINGS{LePa94, - AUTHOR = {F. Leclerc and C. Paulin-Mohring}, - BOOKTITLE = {{Types for Proofs and Programs, Types' 93}}, - EDITOR = {H. Barendregt and T. Nipkow}, - PUBLISHER = SV, - SERIES = {LNCS}, - TITLE = {{Programming with Streams in Coq. A case study : The Sieve of Eratosthenes}}, - VOLUME = {806}, - YEAR = {1994} -} - -@TECHREPORT{Leroy90, - AUTHOR = {X. Leroy}, - TITLE = {The {ZINC} experiment: an economical implementation -of the {ML} language}, - INSTITUTION = {INRIA}, - NUMBER = {117}, - YEAR = {1990} -} - -@INPROCEEDINGS{Let02, - author = {P. Letouzey}, - title = {A New Extraction for Coq}, - booktitle = {Proceedings of the TYPES'2002 workshop}, - year = 2002, - note = {to appear}, - url = {draft at \url{http://www.lri.fr/~letouzey/download/extraction2002.ps.gz}} -} - -@BOOK{MaL84, - AUTHOR = {{P. Martin-L\"of}}, - PUBLISHER = {Bibliopolis}, - SERIES = {Studies in Proof Theory}, - TITLE = {Intuitionistic Type Theory}, - YEAR = {1984} -} - -@ARTICLE{MaSi94, - AUTHOR = {P. Manoury and M. Simonot}, - JOURNAL = {TCS}, - TITLE = {Automatizing termination proof of recursively defined function}, - YEAR = {To appear} -} - -@INPROCEEDINGS{Moh89a, - AUTHOR = {Christine Paulin-Mohring}, - ADDRESS = {Austin}, - BOOKTITLE = {Sixteenth Annual ACM Symposium on Principles of Programming Languages}, - MONTH = jan, - PUBLISHER = {ACM}, - TITLE = {Extracting ${F}_{\omega}$'s programs from proofs in the {Calculus of Constructions}}, - YEAR = {1989} -} - -@PHDTHESIS{Moh89b, - AUTHOR = {Christine Paulin-Mohring}, - MONTH = jan, - SCHOOL = {{Universit\'e Paris 7}}, - TITLE = {Extraction de programmes dans le {Calcul des Constructions}}, - YEAR = {1989} -} - -@INPROCEEDINGS{Moh93, - AUTHOR = {Christine Paulin-Mohring}, - BOOKTITLE = {Proceedings of the conference Typed Lambda Calculi and Applications}, - EDITOR = {M. Bezem and J.-F. Groote}, - NOTE = {Also LIP research report 92-49, ENS Lyon}, - NUMBER = {664}, - PUBLISHER = SV, - SERIES = {LNCS}, - TITLE = {{Inductive Definitions in the System Coq - Rules and Properties}}, - YEAR = {1993} -} - -@BOOK{Moh97, - AUTHOR = {Christine Paulin-Mohring}, - MONTH = jan, - PUBLISHER = {{ENS Lyon}}, - TITLE = {{Le syst\`eme Coq. \mbox{Th\`ese d'habilitation}}}, - YEAR = {1997} -} - -@MASTERSTHESIS{Mun94, - AUTHOR = {C. Mu{\~n}oz}, - MONTH = sep, - SCHOOL = {DEA d'Informatique Fondamentale, Universit\'e Paris 7}, - TITLE = {D\'emonstration automatique dans la logique propositionnelle intuitionniste}, - YEAR = {1994} -} - -@PHDTHESIS{Mun97d, - AUTHOR = "C. Mu{\~{n}}oz", - TITLE = "Un calcul de substitutions pour la repr\'esentation - de preuves partielles en th\'eorie de types", - SCHOOL = {Universit\'e Paris 7}, - YEAR = "1997", - Note = {Version en anglais disponible comme rapport de - recherche INRIA RR-3309}, - Type = {Th\`ese de Doctorat} -} - -@BOOK{NoPS90, - AUTHOR = {B. {Nordstr\"om} and K. Peterson and J. Smith}, - BOOKTITLE = {Information Processing 83}, - PUBLISHER = {Oxford Science Publications}, - SERIES = {International Series of Monographs on Computer Science}, - TITLE = {Programming in {Martin-L\"of's} Type Theory}, - YEAR = {1990} -} - -@ARTICLE{Nor88, - AUTHOR = {B. {Nordstr\"om}}, - JOURNAL = {BIT}, - TITLE = {Terminating General Recursion}, - VOLUME = {28}, - YEAR = {1988} -} - -@BOOK{Odi90, - EDITOR = {P. Odifreddi}, - PUBLISHER = {Academic Press}, - TITLE = {Logic and Computer Science}, - YEAR = {1990} -} - -@INPROCEEDINGS{PaMS92, - AUTHOR = {M. Parigot and P. Manoury and M. Simonot}, - ADDRESS = {St. Petersburg, Russia}, - BOOKTITLE = {Logic Programming and automated reasoning}, - EDITOR = {A. Voronkov}, - MONTH = jul, - NUMBER = {624}, - PUBLISHER = SV, - SERIES = {LNCS}, - TITLE = {{ProPre : A Programming language with proofs}}, - YEAR = {1992} -} - -@ARTICLE{PaWe92, - AUTHOR = {Christine Paulin-Mohring and Benjamin Werner}, - JOURNAL = {Journal of Symbolic Computation}, - PAGES = {607--640}, - TITLE = {{Synthesis of ML programs in the system Coq}}, - VOLUME = {15}, - YEAR = {1993} -} - -@ARTICLE{Par92, - AUTHOR = {M. Parigot}, - JOURNAL = {Theoretical Computer Science}, - NUMBER = {2}, - PAGES = {335--356}, - TITLE = {{Recursive Programming with Proofs}}, - VOLUME = {94}, - YEAR = {1992} -} - -@INPROCEEDINGS{Parent95b, - AUTHOR = {C. Parent}, - BOOKTITLE = {{Mathematics of Program Construction'95}}, - PUBLISHER = SV, - SERIES = {LNCS}, - TITLE = {{Synthesizing proofs from programs in -the Calculus of Inductive Constructions}}, - VOLUME = {947}, - YEAR = {1995} -} - -@INPROCEEDINGS{Prasad93, - AUTHOR = {K.V. Prasad}, - BOOKTITLE = {{Proceedings of CONCUR'93}}, - PUBLISHER = SV, - SERIES = {LNCS}, - TITLE = {{Programming with broadcasts}}, - VOLUME = {715}, - YEAR = {1993} -} - -@BOOK{RC95, - author = "di~Cosmo, R.", - title = "Isomorphisms of Types: from $\lambda$-calculus to information - retrieval and language design", - series = "Progress in Theoretical Computer Science", - publisher = "Birkhauser", - year = "1995", - note = "ISBN-0-8176-3763-X" -} - -@TECHREPORT{Rou92, - AUTHOR = {J. Rouyer}, - INSTITUTION = {INRIA}, - MONTH = nov, - NUMBER = {1795}, - TITLE = {{D{\'e}veloppement de l'Algorithme d'Unification dans le Calcul des Constructions}}, - YEAR = {1992} -} - -@TECHREPORT{Saibi94, - AUTHOR = {A. Sa\"{\i}bi}, - INSTITUTION = {INRIA}, - MONTH = dec, - NUMBER = {2345}, - TITLE = {{Axiomatization of a lambda-calculus with explicit-substitutions in the Coq System}}, - YEAR = {1994} -} - - -@MASTERSTHESIS{Ter92, - AUTHOR = {D. Terrasse}, - MONTH = sep, - SCHOOL = {IARFA}, - TITLE = {{Traduction de TYPOL en COQ. Application \`a Mini ML}}, - YEAR = {1992} -} - -@TECHREPORT{ThBeKa92, - AUTHOR = {L. Th\'ery and Y. Bertot and G. Kahn}, - INSTITUTION = {INRIA Sophia}, - MONTH = may, - NUMBER = {1684}, - TITLE = {Real theorem provers deserve real user-interfaces}, - TYPE = {Research Report}, - YEAR = {1992} -} - -@BOOK{TrDa89, - AUTHOR = {A.S. Troelstra and D. van Dalen}, - PUBLISHER = {North-Holland}, - SERIES = {Studies in Logic and the foundations of Mathematics, volumes 121 and 123}, - TITLE = {Constructivism in Mathematics, an introduction}, - YEAR = {1988} -} - -@PHDTHESIS{Wer94, - AUTHOR = {B. Werner}, - SCHOOL = {Universit\'e Paris 7}, - TITLE = {Une th\'eorie des constructions inductives}, - TYPE = {Th\`ese de Doctorat}, - YEAR = {1994} -} - -@PHDTHESIS{Bar99, - AUTHOR = {B. Barras}, - SCHOOL = {Universit\'e Paris 7}, - TITLE = {Auto-validation d'un système de preuves avec familles inductives}, - TYPE = {Th\`ese de Doctorat}, - YEAR = {1999} -} - -@UNPUBLISHED{ddr98, - AUTHOR = {D. de Rauglaudre}, - TITLE = {Camlp4 version 1.07.2}, - YEAR = {1998}, - NOTE = {In Camlp4 distribution} -} - -@ARTICLE{dowek93, - AUTHOR = {G. Dowek}, - TITLE = {{A Complete Proof Synthesis Method for the Cube of Type Systems}}, - JOURNAL = {Journal Logic Computation}, - VOLUME = {3}, - NUMBER = {3}, - PAGES = {287--315}, - MONTH = {June}, - YEAR = {1993} -} - -@INPROCEEDINGS{manoury94, - AUTHOR = {P. Manoury}, - TITLE = {{A User's Friendly Syntax to Define -Recursive Functions as Typed $\lambda-$Terms}}, - BOOKTITLE = {{Types for Proofs and Programs, TYPES'94}}, - SERIES = {LNCS}, - VOLUME = {996}, - MONTH = jun, - YEAR = {1994} -} - -@TECHREPORT{maranget94, - AUTHOR = {L. Maranget}, - INSTITUTION = {INRIA}, - NUMBER = {2385}, - TITLE = {{Two Techniques for Compiling Lazy Pattern Matching}}, - YEAR = {1994} -} - -@INPROCEEDINGS{puel-suarez90, - AUTHOR = {L.Puel and A. Su\'arez}, - BOOKTITLE = {{Conference Lisp and Functional Programming}}, - SERIES = {ACM}, - PUBLISHER = SV, - TITLE = {{Compiling Pattern Matching by Term -Decomposition}}, - YEAR = {1990} -} - -@MASTERSTHESIS{saidi94, - AUTHOR = {H. Saidi}, - MONTH = sep, - SCHOOL = {DEA d'Informatique Fondamentale, Universit\'e Paris 7}, - TITLE = {R\'esolution d'\'equations dans le syst\`eme T - de G\"odel}, - YEAR = {1994} -} - -@misc{streicher93semantical, - author = "T. Streicher", - title = "Semantical Investigations into Intensional Type Theory", - note = "Habilitationsschrift, LMU Munchen.", - year = "1993" } - - - -@Misc{Pcoq, - author = {Lemme Team}, - title = {Pcoq a graphical user-interface for {Coq}}, - note = {\url{http://www-sop.inria.fr/lemme/pcoq/}} -} - - -@Misc{ProofGeneral, - author = {David Aspinall}, - title = {Proof General}, - note = {\url{http://proofgeneral.inf.ed.ac.uk/}} -} - - - -@Book{CoqArt, - author = {Yves bertot and Pierre Castéran}, - title = {Coq'Art}, - publisher = {Springer-Verlag}, - year = 2004, - note = {To appear} -} - -@INCOLLECTION{wadler87, - AUTHOR = {P. Wadler}, - TITLE = {Efficient Compilation of Pattern Matching}, - BOOKTITLE = {The Implementation of Functional Programming -Languages}, - EDITOR = {S.L. Peyton Jones}, - PUBLISHER = {Prentice-Hall}, - YEAR = {1987} -} - - -@COMMENT{cross-references, must be at end} - -@BOOK{Bastad92, - EDITOR = {B. Nordstr\"om and K. Petersson and G. Plotkin}, - PUBLISHER = {Available by ftp at site ftp.inria.fr}, - TITLE = {Proceedings of the 1992 Workshop on Types for Proofs and Programs}, - YEAR = {1992} -} - -@BOOK{Nijmegen93, - EDITOR = {H. Barendregt and T. Nipkow}, - PUBLISHER = SV, - SERIES = LNCS, - TITLE = {Types for Proofs and Programs}, - VOLUME = {806}, - YEAR = {1994} -} - -@PHDTHESIS{Luo90, - AUTHOR = {Z. Luo}, - TITLE = {An Extended Calculus of Constructions}, - SCHOOL = {University of Edinburgh}, - YEAR = {1990} -} diff --git a/doc/faq/hevea.sty b/doc/faq/hevea.sty deleted file mode 100644 index 6d49aa8c..00000000 --- a/doc/faq/hevea.sty +++ /dev/null @@ -1,78 +0,0 @@ -% hevea : hevea.sty -% This is a very basic style file for latex document to be processed -% with hevea. It contains definitions of LaTeX environment which are -% processed in a special way by the translator. -% Mostly : -% - latexonly, not processed by hevea, processed by latex. -% - htmlonly , the reverse. -% - rawhtml, to include raw HTML in hevea output. -% - toimage, to send text to the image file. -% The package also provides hevea logos, html related commands (ahref -% etc.), void cutting and image commands. -\NeedsTeXFormat{LaTeX2e} -\ProvidesPackage{hevea}[2002/01/11] -\RequirePackage{comment} -\newif\ifhevea\heveafalse -\@ifundefined{ifimagen}{\newif\ifimagen\imagenfalse} -\makeatletter% -\newcommand{\heveasmup}[2]{% -\raise #1\hbox{$\m@th$% - \csname S@\f@size\endcsname - \fontsize\sf@size 0% - \math@fontsfalse\selectfont -#2% -}}% -\DeclareRobustCommand{\hevea}{H\kern-.15em\heveasmup{.2ex}{E}\kern-.15emV\kern-.15em\heveasmup{.2ex}{E}\kern-.15emA}% -\DeclareRobustCommand{\hacha}{H\kern-.15em\heveasmup{.2ex}{A}\kern-.15emC\kern-.1em\heveasmup{.2ex}{H}\kern-.15emA}% -\DeclareRobustCommand{\html}{\protect\heveasmup{0.ex}{HTML}} -%%%%%%%%% Hyperlinks hevea style -\newcommand{\ahref}[2]{{#2}} -\newcommand{\ahrefloc}[2]{{#2}} -\newcommand{\aname}[2]{{#2}} -\newcommand{\ahrefurl}[1]{\texttt{#1}} -\newcommand{\footahref}[2]{#2\footnote{\texttt{#1}}} -\newcommand{\mailto}[1]{\texttt{#1}} -\newcommand{\imgsrc}[2][]{} -\newcommand{\home}[1]{\protect\raisebox{-.75ex}{\char126}#1} -\AtBeginDocument -{\@ifundefined{url} -{%url package is not loaded -\let\url\ahref\let\oneurl\ahrefurl\let\footurl\footahref} -{}} -%% Void cutting instructions -\newcounter{cuttingdepth} -\newcommand{\tocnumber}{} -\newcommand{\notocnumber}{} -\newcommand{\cuttingunit}{} -\newcommand{\cutdef}[2][]{} -\newcommand{\cuthere}[2]{} -\newcommand{\cutend}{} -\newcommand{\htmlhead}[1]{} -\newcommand{\htmlfoot}[1]{} -\newcommand{\htmlprefix}[1]{} -\newenvironment{cutflow}[1]{}{} -\newcommand{\cutname}[1]{} -\newcommand{\toplinks}[3]{} -%%%% Html only -\excludecomment{rawhtml} -\newcommand{\rawhtmlinput}[1]{} -\excludecomment{htmlonly} -%%%% Latex only -\newenvironment{latexonly}{}{} -\newenvironment{verblatex}{}{} -%%%% Image file stuff -\def\toimage{\endgroup} -\def\endtoimage{\begingroup\def\@currenvir{toimage}} -\def\verbimage{\endgroup} -\def\endverbimage{\begingroup\def\@currenvir{verbimage}} -\newcommand{\imageflush}[1][]{} -%%% Bgcolor definition -\newsavebox{\@bgcolorbin} -\newenvironment{bgcolor}[2][] - {\newcommand{\@mycolor}{#2}\begin{lrbox}{\@bgcolorbin}\vbox\bgroup} - {\egroup\end{lrbox}% - \begin{flushleft}% - \colorbox{\@mycolor}{\usebox{\@bgcolorbin}}% - \end{flushleft}} -%%% Postlude -\makeatother diff --git a/doc/faq/interval_discr.v b/doc/faq/interval_discr.v deleted file mode 100644 index ed2c0e37..00000000 --- a/doc/faq/interval_discr.v +++ /dev/null @@ -1,419 +0,0 @@ -(** Sketch of the proof of {p:nat|p<=n} = {p:nat|p<=m} -> n=m - - - preliminary results on the irrelevance of boundedness proofs - - introduce the notion of finite cardinal |A| - - prove that |{p:nat|p<=n}| = n - - prove that |A| = n /\ |A| = m -> n = m if equality is decidable on A - - prove that equality is decidable on A - - conclude -*) - -(** * Preliminary results on [nat] and [le] *) - -(** Proving axiom K on [nat] *) - -Require Import Eqdep_dec. -Require Import Arith. - -Theorem eq_rect_eq_nat : - forall (p:nat) (Q:nat->Type) (x:Q p) (h:p=p), x = eq_rect p Q x p h. -Proof. -intros. -apply K_dec_set with (p := h). -apply eq_nat_dec. -reflexivity. -Qed. - -(** Proving unicity of proofs of [(n<=m)%nat] *) - -Scheme le_ind' := Induction for le Sort Prop. - -Theorem le_uniqueness_proof : forall (n m : nat) (p q : n <= m), p = q. -Proof. -induction p using le_ind'; intro q. - replace (le_n n) with - (eq_rect _ (fun n0 => n <= n0) (le_n n) _ (refl_equal n)). - 2:reflexivity. - generalize (refl_equal n). - pattern n at 2 4 6 10, q; case q; [intro | intros m l e]. - rewrite <- eq_rect_eq_nat; trivial. - contradiction (le_Sn_n m); rewrite <- e; assumption. - replace (le_S n m p) with - (eq_rect _ (fun n0 => n <= n0) (le_S n m p) _ (refl_equal (S m))). - 2:reflexivity. - generalize (refl_equal (S m)). - pattern (S m) at 1 3 4 6, q; case q; [intro Heq | intros m0 l HeqS]. - contradiction (le_Sn_n m); rewrite Heq; assumption. - injection HeqS; intro Heq; generalize l HeqS. - rewrite <- Heq; intros; rewrite <- eq_rect_eq_nat. - rewrite (IHp l0); reflexivity. -Qed. - -(** Proving irrelevance of boundedness proofs while building - elements of interval *) - -Lemma dep_pair_intro : - forall (n x y:nat) (Hx : x<=n) (Hy : y<=n), x=y -> - exist (fun x => x <= n) x Hx = exist (fun x => x <= n) y Hy. -Proof. -intros n x y Hx Hy Heq. -generalize Hy. -rewrite <- Heq. -intros. -rewrite (le_uniqueness_proof x n Hx Hy0). -reflexivity. -Qed. - -(** * Proving that {p:nat|p<=n} = {p:nat|p<=m} -> n=m *) - -(** Definition of having finite cardinality [n+1] for a set [A] *) - -Definition card (A:Set) n := - exists f, - (forall x:A, f x <= n) /\ - (forall x y:A, f x = f y -> x = y) /\ - (forall m, m <= n -> exists x:A, f x = m). - -Require Import Arith. - -(** Showing that the interval [0;n] has cardinality [n+1] *) - -Theorem card_interval : forall n, card {x:nat|x<=n} n. -Proof. -intro n. -exists (fun x:{x:nat|x<=n} => proj1_sig x). -split. -(* bounded *) -intro x; apply (proj2_sig x). -split. -(* injectivity *) -intros (p,Hp) (q,Hq). -simpl. -intro Hpq. -apply dep_pair_intro; assumption. -(* surjectivity *) -intros m Hmn. -exists (exist (fun x : nat => x <= n) m Hmn). -reflexivity. -Qed. - -(** Showing that equality on the interval [0;n] is decidable *) - -Lemma interval_dec : - forall n (x y : {m:nat|m<=n}), {x=y}+{x<>y}. -Proof. -intros n (p,Hp). -induction p; intros ([|q],Hq). -left. - apply dep_pair_intro. - reflexivity. -right. - intro H; discriminate H. -right. - intro H; discriminate H. -assert (Hp' : p <= n). - apply le_Sn_le; assumption. -assert (Hq' : q <= n). - apply le_Sn_le; assumption. -destruct (IHp Hp' (exist (fun m => m <= n) q Hq')) - as [Heq|Hneq]. -left. - injection Heq; intro Heq'. - apply dep_pair_intro. - apply eq_S. - assumption. -right. - intro HeqS. - injection HeqS; intro Heq. - apply Hneq. - apply dep_pair_intro. - assumption. -Qed. - -(** Showing that the cardinality relation is functional on decidable sets *) - -Lemma card_inj_aux : - forall (A:Type) f g n, - (forall x:A, f x <= 0) -> - (forall x y:A, f x = f y -> x = y) -> - (forall m, m <= S n -> exists x:A, g x = m) - -> False. -Proof. -intros A f g n Hfbound Hfinj Hgsurj. -destruct (Hgsurj (S n) (le_n _)) as (x,Hx). -destruct (Hgsurj n (le_S _ _ (le_n _))) as (x',Hx'). -assert (Hfx : 0 = f x). -apply le_n_O_eq. -apply Hfbound. -assert (Hfx' : 0 = f x'). -apply le_n_O_eq. -apply Hfbound. -assert (x=x'). -apply Hfinj. -rewrite <- Hfx. -rewrite <- Hfx'. -reflexivity. -rewrite H in Hx. -rewrite Hx' in Hx. -apply (n_Sn _ Hx). -Qed. - -(** For [dec_restrict], we use a lemma on the negation of equality -that requires proof-irrelevance. It should be possible to avoid this -lemma by generalizing over a first-order definition of [x<>y], say -[neq] such that [{x=y}+{neq x y}] and [~(x=y /\ neq x y)]; for such -[neq], unicity of proofs could be proven *) - - Require Import Classical. - Lemma neq_dep_intro : - forall (A:Set) (z x y:A) (p:x<>z) (q:y<>z), x=y -> - exist (fun x => x <> z) x p = exist (fun x => x <> z) y q. - Proof. - intros A z x y p q Heq. - generalize q; clear q; rewrite <- Heq; intro q. - rewrite (proof_irrelevance _ p q); reflexivity. - Qed. - -Lemma dec_restrict : - forall (A:Set), - (forall x y :A, {x=y}+{x<>y}) -> - forall z (x y :{a:A|a<>z}), {x=y}+{x<>y}. -Proof. -intros A Hdec z (x,Hx) (y,Hy). -destruct (Hdec x y) as [Heq|Hneq]. -left; apply neq_dep_intro; assumption. -right; intro Heq; injection Heq; exact Hneq. -Qed. - -Lemma pred_inj : forall n m, - 0 <> n -> 0 <> m -> pred m = pred n -> m = n. -Proof. -destruct n. -intros m H; destruct H; reflexivity. -destruct m. -intros _ H; destruct H; reflexivity. -simpl; intros _ _ H. -rewrite H. -reflexivity. -Qed. - -Lemma le_neq_lt : forall n m, n <= m -> n<>m -> n < m. -Proof. -intros n m Hle Hneq. -destruct (le_lt_eq_dec n m Hle). -assumption. -contradiction. -Qed. - -Lemma inj_restrict : - forall (A:Set) (f:A->nat) x y z, - (forall x y : A, f x = f y -> x = y) - -> x <> z -> f y < f z -> f z <= f x - -> pred (f x) = f y - -> False. - -(* Search error sans le type de f !! *) -Proof. -intros A f x y z Hfinj Hneqx Hfy Hfx Heq. -assert (f z <> f x). - apply sym_not_eq. - intro Heqf. - apply Hneqx. - apply Hfinj. - assumption. -assert (f x = S (f y)). - assert (0 < f x). - apply le_lt_trans with (f z). - apply le_O_n. - apply le_neq_lt; assumption. - apply pred_inj. - apply O_S. - apply lt_O_neq; assumption. - exact Heq. -assert (f z <= f y). -destruct (le_lt_or_eq _ _ Hfx). - apply lt_n_Sm_le. - rewrite <- H0. - assumption. - contradiction Hneqx. - symmetry. - apply Hfinj. - assumption. -contradiction (lt_not_le (f y) (f z)). -Qed. - -Theorem card_inj : forall m n (A:Set), - (forall x y :A, {x=y}+{x<>y}) -> - card A m -> card A n -> m = n. -Proof. -induction m; destruct n; -intros A Hdec - (f,(Hfbound,(Hfinj,Hfsurj))) - (g,(Hgbound,(Hginj,Hgsurj))). -(* 0/0 *) -reflexivity. -(* 0/Sm *) -destruct (card_inj_aux _ _ _ _ Hfbound Hfinj Hgsurj). -(* Sn/0 *) -destruct (card_inj_aux _ _ _ _ Hgbound Hginj Hfsurj). -(* Sn/Sm *) -destruct (Hgsurj (S n) (le_n _)) as (xSn,HSnx). -rewrite IHm with (n:=n) (A := {x:A|x<>xSn}). -reflexivity. -(* decidability of eq on {x:A|x<>xSm} *) -apply dec_restrict. -assumption. -(* cardinality of {x:A|x<>xSn} is m *) -pose (f' := fun x' : {x:A|x<>xSn} => - let (x,Hneq) := x' in - if le_lt_dec (f xSn) (f x) - then pred (f x) - else f x). -exists f'. -split. -(* f' is bounded *) -unfold f'. -intros (x,_). -destruct (le_lt_dec (f xSn) (f x)) as [Hle|Hge]. -change m with (pred (S m)). -apply le_pred. -apply Hfbound. -apply le_S_n. -apply le_trans with (f xSn). -exact Hge. -apply Hfbound. -split. -(* f' is injective *) -unfold f'. -intros (x,Hneqx) (y,Hneqy) Heqf'. -destruct (le_lt_dec (f xSn) (f x)) as [Hlefx|Hgefx]; -destruct (le_lt_dec (f xSn) (f y)) as [Hlefy|Hgefy]. -(* f xSn <= f x et f xSn <= f y *) -assert (Heq : x = y). - apply Hfinj. - assert (f xSn <> f y). - apply sym_not_eq. - intro Heqf. - apply Hneqy. - apply Hfinj. - assumption. - assert (0 < f y). - apply le_lt_trans with (f xSn). - apply le_O_n. - apply le_neq_lt; assumption. - assert (f xSn <> f x). - apply sym_not_eq. - intro Heqf. - apply Hneqx. - apply Hfinj. - assumption. - assert (0 < f x). - apply le_lt_trans with (f xSn). - apply le_O_n. - apply le_neq_lt; assumption. - apply pred_inj. - apply lt_O_neq; assumption. - apply lt_O_neq; assumption. - assumption. -apply neq_dep_intro; assumption. -(* f y < f xSn <= f x *) -destruct (inj_restrict A f x y xSn); assumption. -(* f x < f xSn <= f y *) -symmetry in Heqf'. -destruct (inj_restrict A f y x xSn); assumption. -(* f x < f xSn et f y < f xSn *) -assert (Heq : x=y). - apply Hfinj; assumption. -apply neq_dep_intro; assumption. -(* f' is surjective *) -intros p Hlep. -destruct (le_lt_dec (f xSn) p) as [Hle|Hlt]. -(* case f xSn <= p *) -destruct (Hfsurj (S p) (le_n_S _ _ Hlep)) as (x,Hx). -assert (Hneq : x <> xSn). - intro Heqx. - rewrite Heqx in Hx. - rewrite Hx in Hle. - apply le_Sn_n with p; assumption. -exists (exist (fun a => a<>xSn) x Hneq). -unfold f'. -destruct (le_lt_dec (f xSn) (f x)) as [Hle'|Hlt']. -rewrite Hx; reflexivity. -rewrite Hx in Hlt'. -contradiction (le_not_lt (f xSn) p). -apply lt_trans with (S p). -apply lt_n_Sn. -assumption. -(* case p < f xSn *) -destruct (Hfsurj p (le_S _ _ Hlep)) as (x,Hx). -assert (Hneq : x <> xSn). - intro Heqx. - rewrite Heqx in Hx. - rewrite Hx in Hlt. - apply (lt_irrefl p). - assumption. -exists (exist (fun a => a<>xSn) x Hneq). -unfold f'. -destruct (le_lt_dec (f xSn) (f x)) as [Hle'|Hlt']. - rewrite Hx in Hle'. - contradiction (lt_irrefl p). - apply lt_le_trans with (f xSn); assumption. - assumption. -(* cardinality of {x:A|x<>xSn} is n *) -pose (g' := fun x' : {x:A|x<>xSn} => - let (x,Hneq) := x' in - if Hdec x xSn then 0 else g x). -exists g'. -split. -(* g is bounded *) -unfold g'. -intros (x,_). -destruct (Hdec x xSn) as [_|Hneq]. -apply le_O_n. -assert (Hle_gx:=Hgbound x). -destruct (le_lt_or_eq _ _ Hle_gx). -apply lt_n_Sm_le. -assumption. -contradiction Hneq. -apply Hginj. -rewrite HSnx. -assumption. -split. -(* g is injective *) -unfold g'. -intros (x,Hneqx) (y,Hneqy) Heqg'. -destruct (Hdec x xSn) as [Heqx|_]. -contradiction Hneqx. -destruct (Hdec y xSn) as [Heqy|_]. -contradiction Hneqy. -assert (Heq : x=y). - apply Hginj; assumption. -apply neq_dep_intro; assumption. -(* g is surjective *) -intros p Hlep. -destruct (Hgsurj p (le_S _ _ Hlep)) as (x,Hx). -assert (Hneq : x<>xSn). - intro Heq. - rewrite Heq in Hx. - rewrite Hx in HSnx. - rewrite HSnx in Hlep. - contradiction (le_Sn_n _ Hlep). -exists (exist (fun a => a<>xSn) x Hneq). -simpl. -destruct (Hdec x xSn) as [Heqx|_]. -contradiction Hneq. -assumption. -Qed. - -(** Conclusion *) - -Theorem interval_discr : - forall n m, {p:nat|p<=n} = {p:nat|p<=m} -> n=m. -Proof. -intros n m Heq. -apply card_inj with (A := {p:nat|p<=n}). -apply interval_dec. -apply card_interval. -rewrite Heq. -apply card_interval. -Qed. diff --git a/doc/refman/AddRefMan-pre.tex b/doc/refman/AddRefMan-pre.tex deleted file mode 100644 index 461e8e6d..00000000 --- a/doc/refman/AddRefMan-pre.tex +++ /dev/null @@ -1,62 +0,0 @@ -%\coverpage{Addendum to the Reference Manual}{\ } -%\addcontentsline{toc}{part}{Additional documentation} -%BEGIN LATEX -\setheaders{Presentation of the Addendum} -%END LATEX -\chapter*{Presentation of the Addendum} - -Here you will find several pieces of additional documentation for the -\Coq\ Reference Manual. Each of this chapters is concentrated on a -particular topic, that should interest only a fraction of the \Coq\ -users: that's the reason why they are apart from the Reference -Manual. - -\begin{description} - -\item[Extended pattern-matching] This chapter details the use of - generalized pattern-matching. It is contributed by Cristina Cornes - and Hugo Herbelin. - -\item[Implicit coercions] This chapter details the use of the coercion - mechanism. It is contributed by Amokrane Saïbi. - -%\item[Proof of imperative programs] This chapter explains how to -% prove properties of annotated programs with imperative features. -% It is contributed by Jean-Christophe Filliâtre - -\item[Program extraction] This chapter explains how to extract in practice ML - files from $\FW$ terms. It is contributed by Jean-Christophe - Filliâtre and Pierre Letouzey. - -\item[Program] This chapter explains the use of the \texttt{Program} - vernacular which allows the development of certified - programs in \Coq. It is contributed by Matthieu Sozeau and replaces - the previous \texttt{Program} tactic by Catherine Parent. - -%\item[Natural] This chapter is due to Yann Coscoy. It is the user -% manual of the tools he wrote for printing proofs in natural -% language. At this time, French and English languages are supported. - -\item[omega] \texttt{omega}, written by Pierre Crégut, solves a whole - class of arithmetic problems. - -\item[The {\tt ring} tactic] This is a tactic to do AC rewriting. This - chapter explains how to use it and how it works. - The chapter is contributed by Patrick Loiseleur. - -\item[The {\tt Setoid\_replace} tactic] This is a - tactic to do rewriting on types equipped with specific (only partially - substitutive) equality. The chapter is contributed by Clément Renard. - -\item[Calling external provers] This chapter describes several tactics - which call external provers. - -\end{description} - -\atableofcontents - - -%%% Local Variables: -%%% mode: latex -%%% TeX-master: "Reference-Manual" -%%% End: diff --git a/doc/refman/Cases.tex b/doc/refman/Cases.tex deleted file mode 100644 index 6f58269f..00000000 --- a/doc/refman/Cases.tex +++ /dev/null @@ -1,750 +0,0 @@ -\achapter{Extended pattern-matching} -%BEGIN LATEX -\defaultheaders -%END LATEX -\aauthor{Cristina Cornes and Hugo Herbelin} - -\label{Mult-match-full} -\ttindex{Cases} -\index{ML-like patterns} - -This section describes the full form of pattern-matching in {\Coq} terms. - -\asection{Patterns}\label{implementation} The full syntax of {\tt -match} is presented in Figures~\ref{term-syntax} -and~\ref{term-syntax-aux}. Identifiers in patterns are either -constructor names or variables. Any identifier that is not the -constructor of an inductive or coinductive type is considered to be a -variable. A variable name cannot occur more than once in a given -pattern. It is recommended to start variable names by a lowercase -letter. - -If a pattern has the form $(c~\vec{x})$ where $c$ is a constructor -symbol and $\vec{x}$ is a linear vector of (distinct) variables, it is -called {\em simple}: it is the kind of pattern recognized by the basic -version of {\tt match}. On the opposite, if it is a variable $x$ or -has the form $(c~\vec{p})$ with $p$ not only made of variables, the -pattern is called {\em nested}. - -A variable pattern matches any value, and the identifier is bound to -that value. The pattern ``\texttt{\_}'' (called ``don't care'' or -``wildcard'' symbol) also matches any value, but does not bind -anything. It may occur an arbitrary number of times in a -pattern. Alias patterns written \texttt{(}{\sl pattern} \texttt{as} -{\sl identifier}\texttt{)} are also accepted. This pattern matches the -same values as {\sl pattern} does and {\sl identifier} is bound to the -matched value. -A pattern of the form {\pattern}{\tt |}{\pattern} is called -disjunctive. A list of patterns separated with commas is also -considered as a pattern and is called {\em multiple pattern}. However -multiple patterns can only occur at the root of pattern-matching -equations. Disjunctions of {\em multiple pattern} are allowed though. - -Since extended {\tt match} expressions are compiled into the primitive -ones, the expressiveness of the theory remains the same. Once the -stage of parsing has finished only simple patterns remain. Re-nesting -of pattern is performed at printing time. An easy way to see the -result of the expansion is to toggle off the nesting performed at -printing (use here {\tt Set Printing Matching}), then by printing the term -with \texttt{Print} if the term is a constant, or using the command -\texttt{Check}. - -The extended \texttt{match} still accepts an optional {\em elimination -predicate} given after the keyword \texttt{return}. Given a pattern -matching expression, if all the right-hand-sides of \texttt{=>} ({\em -rhs} in short) have the same type, then this type can be sometimes -synthesized, and so we can omit the \texttt{return} part. Otherwise -the predicate after \texttt{return} has to be provided, like for the basic -\texttt{match}. - -Let us illustrate through examples the different aspects of extended -pattern matching. Consider for example the function that computes the -maximum of two natural numbers. We can write it in primitive syntax -by: - -\begin{coq_example} -Fixpoint max (n m:nat) {struct m} : nat := - match n with - | O => m - | S n' => match m with - | O => S n' - | S m' => S (max n' m') - end - end. -\end{coq_example} - -\paragraph{Multiple patterns} - -Using multiple patterns in the definition of {\tt max} allows to write: - -\begin{coq_example} -Reset max. -Fixpoint max (n m:nat) {struct m} : nat := - match n, m with - | O, _ => m - | S n', O => S n' - | S n', S m' => S (max n' m') - end. -\end{coq_example} - -which will be compiled into the previous form. - -The pattern-matching compilation strategy examines patterns from left -to right. A \texttt{match} expression is generated {\bf only} when -there is at least one constructor in the column of patterns. E.g. the -following example does not build a \texttt{match} expression. - -\begin{coq_example} -Check (fun x:nat => match x return nat with - | y => y - end). -\end{coq_example} - -\paragraph{Aliasing subpatterns} - -We can also use ``\texttt{as} {\ident}'' to associate a name to a -sub-pattern: - -\begin{coq_example} -Reset max. -Fixpoint max (n m:nat) {struct n} : nat := - match n, m with - | O, _ => m - | S n' as p, O => p - | S n', S m' => S (max n' m') - end. -\end{coq_example} - -\paragraph{Nested patterns} - -Here is now an example of nested patterns: - -\begin{coq_example} -Fixpoint even (n:nat) : bool := - match n with - | O => true - | S O => false - | S (S n') => even n' - end. -\end{coq_example} - -This is compiled into: - -\begin{coq_example} -Print even. -\end{coq_example} - -In the previous examples patterns do not conflict with, but -sometimes it is comfortable to write patterns that admit a non -trivial superposition. Consider -the boolean function \texttt{lef} that given two natural numbers -yields \texttt{true} if the first one is less or equal than the second -one and \texttt{false} otherwise. We can write it as follows: - -\begin{coq_example} -Fixpoint lef (n m:nat) {struct m} : bool := - match n, m with - | O, x => true - | x, O => false - | S n, S m => lef n m - end. -\end{coq_example} - -Note that the first and the second multiple pattern superpose because -the couple of values \texttt{O O} matches both. Thus, what is the result -of the function on those values? To eliminate ambiguity we use the -{\em textual priority rule}: we consider patterns ordered from top to -bottom, then a value is matched by the pattern at the $ith$ row if and -only if it is not matched by some pattern of a previous row. Thus in the -example, -\texttt{O O} is matched by the first pattern, and so \texttt{(lef O O)} -yields \texttt{true}. - -Another way to write this function is: - -\begin{coq_example} -Reset lef. -Fixpoint lef (n m:nat) {struct m} : bool := - match n, m with - | O, x => true - | S n, S m => lef n m - | _, _ => false - end. -\end{coq_example} - -Here the last pattern superposes with the first two. Because -of the priority rule, the last pattern -will be used only for values that do not match neither the first nor -the second one. - -Terms with useless patterns are not accepted by the -system. Here is an example: -% Test failure -\begin{coq_eval} -Set Printing Depth 50. - (********** The following is not correct and should produce **********) - (**************** Error: This clause is redundant ********************) -\end{coq_eval} -\begin{coq_example} -Check (fun x:nat => - match x with - | O => true - | S _ => false - | x => true - end). -\end{coq_example} - -\paragraph{Disjunctive patterns} - -Multiple patterns that share the same right-hand-side can be -factorized using the notation \nelist{\multpattern}{\tt |}. For instance, -{\tt max} can be rewritten as follows: - -\begin{coq_eval} -Reset max. -\end{coq_eval} -\begin{coq_example} -Fixpoint max (n m:nat) {struct m} : nat := - match n, m with - | S n', S m' => S (max n' m') - | 0, p | p, 0 => p - end. -\end{coq_example} - -Similarly, factorization of (non necessary multiple) patterns -that share the same variables is possible by using the notation -\nelist{\pattern}{\tt |}. Here is an example: - -\begin{coq_example} -Definition filter_2_4 (n:nat) : nat := - match n with - | 2 as m | 4 as m => m - | _ => 0 - end. -\end{coq_example} - -Here is another example using disjunctive subpatterns. - -\begin{coq_example} -Definition filter_some_square_corners (p:nat*nat) : nat*nat := - match p with - | ((2 as m | 4 as m), (3 as n | 5 as n)) => (m,n) - | _ => (0,0) - end. -\end{coq_example} - -\asection{About patterns of parametric types} -When matching objects of a parametric type, constructors in patterns -{\em do not expect} the parameter arguments. Their value is deduced -during expansion. -Consider for example the type of polymorphic lists: - -\begin{coq_example} -Inductive List (A:Set) : Set := - | nil : List A - | cons : A -> List A -> List A. -\end{coq_example} - -We can check the function {\em tail}: - -\begin{coq_example} -Check - (fun l:List nat => - match l with - | nil => nil nat - | cons _ l' => l' - end). -\end{coq_example} - - -When we use parameters in patterns there is an error message: -% Test failure -\begin{coq_eval} -Set Printing Depth 50. -(********** The following is not correct and should produce **********) -(******** Error: The constructor cons expects 2 arguments ************) -\end{coq_eval} -\begin{coq_example} -Check - (fun l:List nat => - match l with - | nil A => nil nat - | cons A _ l' => l' - end). -\end{coq_example} - - - -\asection{Matching objects of dependent types} -The previous examples illustrate pattern matching on objects of -non-dependent types, but we can also -use the expansion strategy to destructure objects of dependent type. -Consider the type \texttt{listn} of lists of a certain length: -\label{listn} - -\begin{coq_example} -Inductive listn : nat -> Set := - | niln : listn 0 - | consn : forall n:nat, nat -> listn n -> listn (S n). -\end{coq_example} - -\asubsection{Understanding dependencies in patterns} -We can define the function \texttt{length} over \texttt{listn} by: - -\begin{coq_example} -Definition length (n:nat) (l:listn n) := n. -\end{coq_example} - -Just for illustrating pattern matching, -we can define it by case analysis: - -\begin{coq_example} -Reset length. -Definition length (n:nat) (l:listn n) := - match l with - | niln => 0 - | consn n _ _ => S n - end. -\end{coq_example} - -We can understand the meaning of this definition using the -same notions of usual pattern matching. - -% -% Constraining of dependencies is not longer valid in V7 -% -\iffalse -Now suppose we split the second pattern of \texttt{length} into two -cases so to give an -alternative definition using nested patterns: -\begin{coq_example} -Definition length1 (n:nat) (l:listn n) := - match l with - | niln => 0 - | consn n _ niln => S n - | consn n _ (consn _ _ _) => S n - end. -\end{coq_example} - -It is obvious that \texttt{length1} is another version of -\texttt{length}. We can also give the following definition: -\begin{coq_example} -Definition length2 (n:nat) (l:listn n) := - match l with - | niln => 0 - | consn n _ niln => 1 - | consn n _ (consn m _ _) => S (S m) - end. -\end{coq_example} - -If we forget that \texttt{listn} is a dependent type and we read these -definitions using the usual semantics of pattern matching, we can conclude -that \texttt{length1} -and \texttt{length2} are different functions. -In fact, they are equivalent -because the pattern \texttt{niln} implies that \texttt{n} can only match -the value $0$ and analogously the pattern \texttt{consn} determines that \texttt{n} can -only match values of the form $(S~v)$ where $v$ is the value matched by -\texttt{m}. - -The converse is also true. If -we destructure the length value with the pattern \texttt{O} then the list -value should be $niln$. -Thus, the following term \texttt{length3} corresponds to the function -\texttt{length} but this time defined by case analysis on the dependencies instead of on the list: - -\begin{coq_example} -Definition length3 (n:nat) (l:listn n) := - match l with - | niln => 0 - | consn O _ _ => 1 - | consn (S n) _ _ => S (S n) - end. -\end{coq_example} - -When we have nested patterns of dependent types, the semantics of -pattern matching becomes a little more difficult because -the set of values that are matched by a sub-pattern may be conditioned by the -values matched by another sub-pattern. Dependent nested patterns are -somehow constrained patterns. -In the examples, the expansion of -\texttt{length1} and \texttt{length2} yields exactly the same term - but the -expansion of \texttt{length3} is completely different. \texttt{length1} and -\texttt{length2} are expanded into two nested case analysis on -\texttt{listn} while \texttt{length3} is expanded into a case analysis on -\texttt{listn} containing a case analysis on natural numbers inside. - - -In practice the user can think about the patterns as independent and -it is the expansion algorithm that cares to relate them. \\ -\fi -% -% -% - -\asubsection{When the elimination predicate must be provided} -The examples given so far do not need an explicit elimination predicate - because all the rhs have the same type and the -strategy succeeds to synthesize it. -Unfortunately when dealing with dependent patterns it often happens -that we need to write cases where the type of the rhs are -different instances of the elimination predicate. -The function \texttt{concat} for \texttt{listn} -is an example where the branches have different type -and we need to provide the elimination predicate: - -\begin{coq_example} -Fixpoint concat (n:nat) (l:listn n) (m:nat) (l':listn m) {struct l} : - listn (n + m) := - match l in listn n return listn (n + m) with - | niln => l' - | consn n' a y => consn (n' + m) a (concat n' y m l') - end. -\end{coq_example} -The elimination predicate is {\tt fun (n:nat) (l:listn n) => listn~(n+m)}. -In general if $m$ has type $(I~q_1\ldots q_r~t_1\ldots t_s)$ where -$q_1\ldots q_r$ are parameters, the elimination predicate should be of -the form~: -{\tt fun $y_1$\ldots $y_s$ $x$:($I$~$q_1$\ldots $q_r$~$y_1$\ldots - $y_s$) => Q}. - -In the concrete syntax, it should be written~: -\[ \kw{match}~m~\kw{as}~x~\kw{in}~(I~\_\ldots \_~y_1\ldots y_s)~\kw{return}~Q~\kw{with}~\ldots~\kw{end}\] - -The variables which appear in the \kw{in} and \kw{as} clause are new -and bounded in the property $Q$ in the \kw{return} clause. The -parameters of the inductive definitions should not be mentioned and -are replaced by \kw{\_}. - -Recall that a list of patterns is also a pattern. So, when -we destructure several terms at the same time and the branches have -different type we need to provide -the elimination predicate for this multiple pattern. -It is done using the same scheme, each term may be associated to an -\kw{as} and \kw{in} clause in order to introduce a dependent product. - -For example, an equivalent definition for \texttt{concat} (even though the matching on the second term is trivial) would have -been: - -\begin{coq_example} -Reset concat. -Fixpoint concat (n:nat) (l:listn n) (m:nat) (l':listn m) {struct l} : - listn (n + m) := - match l in listn n, l' return listn (n + m) with - | niln, x => x - | consn n' a y, x => consn (n' + m) a (concat n' y m x) - end. -\end{coq_example} - -% Notice that this time, the predicate \texttt{[n,\_:nat](listn (plus n -% m))} is binary because we -% destructure both \texttt{l} and \texttt{l'} whose types have arity one. -% In general, if we destructure the terms $e_1\ldots e_n$ -% the predicate will be of arity $m$ where $m$ is the sum of the -% number of dependencies of the type of $e_1, e_2,\ldots e_n$ -% (the $\lambda$-abstractions -% should correspond from left to right to each dependent argument of the -% type of $e_1\ldots e_n$). -When the arity of the predicate (i.e. number of abstractions) is not -correct Coq raises an error message. For example: - -% Test failure -\begin{coq_eval} -Reset concat. -Set Printing Depth 50. -(********** The following is not correct and should produce ***********) -(** Error: the term l' has type listn m while it is expected to have **) -(** type listn (?31 + ?32) **) -\end{coq_eval} -\begin{coq_example} -Fixpoint concat - (n:nat) (l:listn n) (m:nat) - (l':listn m) {struct l} : listn (n + m) := - match l, l' with - | niln, x => x - | consn n' a y, x => consn (n' + m) a (concat n' y m x) - end. -\end{coq_example} - -\asection{Using pattern matching to write proofs} -In all the previous examples the elimination predicate does not depend -on the object(s) matched. But it may depend and the typical case -is when we write a proof by induction or a function that yields an -object of dependent type. An example of proof using \texttt{match} in -given in Section~\ref{refine-example}. - -For example, we can write -the function \texttt{buildlist} that given a natural number -$n$ builds a list of length $n$ containing zeros as follows: - -\begin{coq_example} -Fixpoint buildlist (n:nat) : listn n := - match n return listn n with - | O => niln - | S n => consn n 0 (buildlist n) - end. -\end{coq_example} - -We can also use multiple patterns. -Consider the following definition of the predicate less-equal -\texttt{Le}: - -\begin{coq_example} -Inductive LE : nat -> nat -> Prop := - | LEO : forall n:nat, LE 0 n - | LES : forall n m:nat, LE n m -> LE (S n) (S m). -\end{coq_example} - -We can use multiple patterns to write the proof of the lemma - \texttt{forall (n m:nat), (LE n m)}\verb=\/=\texttt{(LE m n)}: - -\begin{coq_example} -Fixpoint dec (n m:nat) {struct n} : LE n m \/ LE m n := - match n, m return LE n m \/ LE m n with - | O, x => or_introl (LE x 0) (LEO x) - | x, O => or_intror (LE x 0) (LEO x) - | S n as n', S m as m' => - match dec n m with - | or_introl h => or_introl (LE m' n') (LES n m h) - | or_intror h => or_intror (LE n' m') (LES m n h) - end - end. -\end{coq_example} -In the example of \texttt{dec}, -the first \texttt{match} is dependent while -the second is not. - -% In general, consider the terms $e_1\ldots e_n$, -% where the type of $e_i$ is an instance of a family type -% $\lb (\vec{d_i}:\vec{D_i}) \mto T_i$ ($1\leq i -% \leq n$). Then, in expression \texttt{match} $e_1,\ldots, -% e_n$ \texttt{of} \ldots \texttt{end}, the -% elimination predicate ${\cal P}$ should be of the form: -% $[\vec{d_1}:\vec{D_1}][x_1:T_1]\ldots [\vec{d_n}:\vec{D_n}][x_n:T_n]Q.$ - -The user can also use \texttt{match} in combination with the tactic -\texttt{refine} (see Section~\ref{refine}) to build incomplete proofs -beginning with a \texttt{match} construction. - -\asection{Pattern-matching on inductive objects involving local -definitions} - -If local definitions occur in the type of a constructor, then there -are two ways to match on this constructor. Either the local -definitions are skipped and matching is done only on the true arguments -of the constructors, or the bindings for local definitions can also -be caught in the matching. - -Example. - -\begin{coq_eval} -Reset Initial. -Require Import Arith. -\end{coq_eval} - -\begin{coq_example*} -Inductive list : nat -> Set := - | nil : list 0 - | cons : forall n:nat, let m := (2 * n) in list m -> list (S (S m)). -\end{coq_example*} - -In the next example, the local definition is not caught. - -\begin{coq_example} -Fixpoint length n (l:list n) {struct l} : nat := - match l with - | nil => 0 - | cons n l0 => S (length (2 * n) l0) - end. -\end{coq_example} - -But in this example, it is. - -\begin{coq_example} -Fixpoint length' n (l:list n) {struct l} : nat := - match l with - | nil => 0 - | cons _ m l0 => S (length' m l0) - end. -\end{coq_example} - -\Rem for a given matching clause, either none of the local -definitions or all of them can be caught. - -\asection{Pattern-matching and coercions} - -If a mismatch occurs between the expected type of a pattern and its -actual type, a coercion made from constructors is sought. If such a -coercion can be found, it is automatically inserted around the -pattern. - -Example: - -\begin{coq_example} -Inductive I : Set := - | C1 : nat -> I - | C2 : I -> I. -Coercion C1 : nat >-> I. -Check (fun x => match x with - | C2 O => 0 - | _ => 0 - end). -\end{coq_example} - - -\asection{When does the expansion strategy fail ?}\label{limitations} -The strategy works very like in ML languages when treating -patterns of non-dependent type. -But there are new cases of failure that are due to the presence of -dependencies. - -The error messages of the current implementation may be sometimes -confusing. When the tactic fails because patterns are somehow -incorrect then error messages refer to the initial expression. But the -strategy may succeed to build an expression whose sub-expressions are -well typed when the whole expression is not. In this situation the -message makes reference to the expanded expression. We encourage -users, when they have patterns with the same outer constructor in -different equations, to name the variable patterns in the same -positions with the same name. -E.g. to write {\small\texttt{(cons n O x) => e1}} -and {\small\texttt{(cons n \_ x) => e2}} instead of -{\small\texttt{(cons n O x) => e1}} and -{\small\texttt{(cons n' \_ x') => e2}}. -This helps to maintain certain name correspondence between the -generated expression and the original. - -Here is a summary of the error messages corresponding to each situation: - -\begin{ErrMsgs} -\item \sverb{The constructor } {\sl - ident} \sverb{ expects } {\sl num} \sverb{ arguments} - - \sverb{The variable } {\sl ident} \sverb{ is bound several times - in pattern } {\sl term} - - \sverb{Found a constructor of inductive type } {\term} - \sverb{ while a constructor of } {\term} \sverb{ is expected} - - Patterns are incorrect (because constructors are not applied to - the correct number of the arguments, because they are not linear or - they are wrongly typed). - -\item \errindex{Non exhaustive pattern-matching} - -The pattern matching is not exhaustive. - -\item \sverb{The elimination predicate } {\sl term} \sverb{ should be - of arity } {\sl num} \sverb{ (for non dependent case) or } {\sl - num} \sverb{ (for dependent case)} - -The elimination predicate provided to \texttt{match} has not the - expected arity. - - -%\item the whole expression is wrongly typed - -% CADUC ? -% , or the synthesis of -% implicit arguments fails (for example to find the elimination -% predicate or to resolve implicit arguments in the rhs). - -% There are {\em nested patterns of dependent type}, the elimination -% predicate corresponds to non-dependent case and has the form -% $[x_1:T_1]...[x_n:T_n]T$ and {\bf some} $x_i$ occurs {\bf free} in -% $T$. Then, the strategy may fail to find out a correct elimination -% predicate during some step of compilation. In this situation we -% recommend the user to rewrite the nested dependent patterns into -% several \texttt{match} with {\em simple patterns}. - -\item {\tt Unable to infer a match predicate\\ - Either there is a type incompatiblity or the problem involves\\ - dependencies} - - There is a type mismatch between the different branches. - The user should provide an elimination predicate. - -% Obsolete ? -% \item because of nested patterns, it may happen that even though all -% the rhs have the same type, the strategy needs dependent elimination -% and so an elimination predicate must be provided. The system warns -% about this situation, trying to compile anyway with the -% non-dependent strategy. The risen message is: - -% \begin{itemize} -% \item {\tt Warning: This pattern matching may need dependent -% elimination to be compiled. I will try, but if fails try again -% giving dependent elimination predicate.} -% \end{itemize} - - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -% % LA PROPAGATION DES CONTRAINTES ARRIERE N'EST PAS FAITE DANS LA V7 -% TODO -% \item there are {\em nested patterns of dependent type} and the -% strategy builds a term that is well typed but recursive calls in fix -% point are reported as illegal: -% \begin{itemize} -% \item {\tt Error: Recursive call applied to an illegal term ...} -% \end{itemize} - -% This is because the strategy generates a term that is correct w.r.t. -% the initial term but which does not pass the guard condition. In -% this situation we recommend the user to transform the nested dependent -% patterns into {\em several \texttt{match} of simple patterns}. Let us -% explain this with an example. Consider the following definition of a -% function that yields the last element of a list and \texttt{O} if it is -% empty: - -% \begin{coq_example} -% Fixpoint last [n:nat; l:(listn n)] : nat := -% match l of -% (consn _ a niln) => a -% | (consn m _ x) => (last m x) | niln => O -% end. -% \end{coq_example} - -% It fails because of the priority between patterns, we know that this -% definition is equivalent to the following more explicit one (which -% fails too): - -% \begin{coq_example*} -% Fixpoint last [n:nat; l:(listn n)] : nat := -% match l of -% (consn _ a niln) => a -% | (consn n _ (consn m b x)) => (last n (consn m b x)) -% | niln => O -% end. -% \end{coq_example*} - -% Note that the recursive call {\tt (last n (consn m b x))} is not -% guarded. When treating with patterns of dependent types the strategy -% interprets the first definition of \texttt{last} as the second -% one\footnote{In languages of the ML family the first definition would -% be translated into a term where the variable \texttt{x} is shared in -% the expression. When patterns are of non-dependent types, Coq -% compiles as in ML languages using sharing. When patterns are of -% dependent types the compilation reconstructs the term as in the -% second definition of \texttt{last} so to ensure the result of -% expansion is well typed.}. Thus it generates a term where the -% recursive call is rejected by the guard condition. - -% You can get rid of this problem by writing the definition with -% \emph{simple patterns}: - -% \begin{coq_example} -% Fixpoint last [n:nat; l:(listn n)] : nat := -% <[_:nat]nat>match l of -% (consn m a x) => Cases x of niln => a | _ => (last m x) end -% | niln => O -% end. -% \end{coq_example} - -\end{ErrMsgs} - - -%%% Local Variables: -%%% mode: latex -%%% TeX-master: "Reference-Manual" -%%% End: diff --git a/doc/refman/Classes.tex b/doc/refman/Classes.tex deleted file mode 100644 index f427609f..00000000 --- a/doc/refman/Classes.tex +++ /dev/null @@ -1,423 +0,0 @@ -\def\Haskell{\textsc{Haskell}\xspace} -\def\eol{\setlength\parskip{0pt}\par} -\def\indent#1{\noindent\kern#1} -\def\cst#1{\textsf{#1}} - -\newcommand\tele[1]{\overrightarrow{#1}} - -\achapter{\protect{Type Classes}} -\aauthor{Matthieu Sozeau} -\label{typeclasses} - -\begin{flushleft} - \em The status of Type Classes is experimental. -\end{flushleft} - -This chapter presents a quick reference of the commands related to type -classes. For an actual introduction to type classes, there is a -description of the system \cite{sozeau08} and the literature on type -classes in \Haskell which also applies. - -\asection{Class and Instance declarations} -\label{ClassesInstances} - -The syntax for class and instance declarations is the same as -record syntax of \Coq: -\def\kw{\texttt} -\def\classid{\texttt} - -\begin{center} -\[\begin{array}{l} -\kw{Class}~\classid{Id}~(\alpha_1 : \tau_1) \cdots (\alpha_n : \tau_n) -[: \sort] := \{\\ -\begin{array}{p{0em}lcl} - & \cst{f}_1 & : & \type_1 ; \\ - & \vdots & & \\ - & \cst{f}_m & : & \type_m \}. -\end{array}\end{array}\] -\end{center} -\begin{center} -\[\begin{array}{l} -\kw{Instance}~\ident~:~\classid{Id}~\term_1 \cdots \term_n := \{\\ -\begin{array}{p{0em}lcl} - & \cst{f}_1 & := & \term_{f_1} ; \\ - & \vdots & & \\ - & \cst{f}_m & := & \term_{f_m} \}. -\end{array}\end{array}\] -\end{center} -\begin{coq_eval} - Reset Initial. - Generalizable All Variables. -\end{coq_eval} - -The $\tele{\alpha_i : \tau_i}$ variables are called the \emph{parameters} -of the class and the $\tele{f_k : \type_k}$ are called the -\emph{methods}. Each class definition gives rise to a corresponding -record declaration and each instance is a regular definition whose name -is given by $\ident$ and type is an instantiation of the record type. - -We'll use the following example class in the rest of the chapter: - -\begin{coq_example*} -Class EqDec (A : Type) := { - eqb : A -> A -> bool ; - eqb_leibniz : forall x y, eqb x y = true -> x = y }. -\end{coq_example*} - -This class implements a boolean equality test which is compatible with -Leibniz equality on some type. An example implementation is: - -\begin{coq_example*} -Instance unit_EqDec : EqDec unit := -{ eqb x y := true ; - eqb_leibniz x y H := - match x, y return x = y with tt, tt => refl_equal tt end }. -\end{coq_example*} - -If one does not give all the members in the \texttt{Instance} -declaration, Coq enters the proof-mode and the user is asked to build -inhabitants of the remaining fields, e.g.: - -\begin{coq_example*} -Instance eq_bool : EqDec bool := -{ eqb x y := if x then y else negb y }. -\end{coq_example*} -\begin{coq_example} -Proof. intros x y H. - destruct x ; destruct y ; (discriminate || reflexivity). -Defined. -\end{coq_example} - -One has to take care that the transparency of every field is determined -by the transparency of the \texttt{Instance} proof. One can use -alternatively the \texttt{Program} \texttt{Instance} \comindex{Program Instance} variant which has -richer facilities for dealing with obligations. - -\asection{Binding classes} - -Once a type class is declared, one can use it in class binders: -\begin{coq_example} -Definition neqb {A} {eqa : EqDec A} (x y : A) := negb (eqb x y). -\end{coq_example} - -When one calls a class method, a constraint is generated that is -satisfied only in contexts where the appropriate instances can be -found. In the example above, a constraint \texttt{EqDec A} is generated and -satisfied by \texttt{{eqa : EqDec A}}. In case no satisfying constraint can be -found, an error is raised: - -\begin{coq_example} -Definition neqb' (A : Type) (x y : A) := negb (eqb x y). -\end{coq_example} - -The algorithm used to solve constraints is a variant of the eauto tactic -that does proof search with a set of lemmas (the instances). It will use -local hypotheses as well as declared lemmas in the -\texttt{typeclass\_instances} database. Hence the example can also be -written: - -\begin{coq_example} -Definition neqb' A (eqa : EqDec A) (x y : A) := negb (eqb x y). -\end{coq_example} - -However, the generalizing binders should be used instead as they have -particular support for type classes: -\begin{itemize} -\item They automatically set the maximally implicit status for type - class arguments, making derived functions as easy to use as class - methods. In the example above, \texttt{A} and \texttt{eqa} should be - set maximally implicit. -\item They support implicit quantification on partialy applied type - classes. -\item They also support implicit quantification on superclasses - (\S \ref{classes:superclasses}) -\end{itemize} - -Following the previous example, one can write: -\begin{coq_example} -Definition neqb_impl `{eqa : EqDec A} (x y : A) := negb (eqb x y). -\end{coq_example} - -Here \texttt{A} is implicitly generalized, and the resulting function -is equivalent to the one above. - -The parsing of generalized type-class binders is different from regular -binders: -\begin{itemize} -\item Implicit arguments of the class type are ignored. -\item Superclasses arguments are automatically generalized. -\item Any remaining arguments not given as part of a type class binder - will be automatically generalized. In other words, the rightmost - parameters are automatically generalized if not mentionned. -\end{itemize} - -One can switch off this special treatment using the $!$ mark in front of -the class name (see example below). - -\asection{Parameterized Instances} - -One can declare parameterized instances as in \Haskell simply by giving -the constraints as a binding context before the instance, e.g.: - -\begin{coq_example} -Instance prod_eqb `(EA : EqDec A, EB : EqDec B) : EqDec (A * B) := -{ eqb x y := match x, y with - | (la, ra), (lb, rb) => andb (eqb la lb) (eqb ra rb) - end }. -\end{coq_example} -\begin{coq_eval} -Admitted. -\end{coq_eval} - -These instances are used just as well as lemmas in the instance hint database. - -\asection{Sections and contexts} -\label{SectionContext} -To ease the parametrization of developments by type classes, we provide -a new way to introduce variables into section contexts, compatible with -the implicit argument mechanism. -The new command works similarly to the \texttt{Variables} vernacular -(see \ref{Variable}), except it accepts any binding context as argument. -For example: - -\begin{coq_example} -Section EqDec_defs. - Context `{EA : EqDec A}. -\end{coq_example} - -\begin{coq_example*} - Global Instance option_eqb : EqDec (option A) := - { eqb x y := match x, y with - | Some x, Some y => eqb x y - | None, None => true - | _, _ => false - end }. -\end{coq_example*} -\begin{coq_eval} -Proof. -intros x y ; destruct x ; destruct y ; intros H ; -try discriminate ; try apply eqb_leibniz in H ; -subst ; auto. -Defined. -\end{coq_eval} - -\begin{coq_example} -End EqDec_defs. -About option_eqb. -\end{coq_example} - -Here the \texttt{Global} modifier redeclares the instance at the end of -the section, once it has been generalized by the context variables it uses. - -\asection{Building hierarchies} - -\subsection{Superclasses} -\label{classes:superclasses} -One can also parameterize classes by other classes, generating a -hierarchy of classes and superclasses. In the same way, we give the -superclasses as a binding context: - -\begin{coq_example*} -Class Ord `(E : EqDec A) := - { le : A -> A -> bool }. -\end{coq_example*} - -Contrary to \Haskell, we have no special syntax for superclasses, but -this declaration is morally equivalent to: -\begin{verbatim} -Class `(E : EqDec A) => Ord A := - { le : A -> A -> bool }. -\end{verbatim} - -This declaration means that any instance of the \texttt{Ord} class must -have an instance of \texttt{EqDec}. The parameters of the subclass contain -at least all the parameters of its superclasses in their order of -appearance (here \texttt{A} is the only one). -As we have seen, \texttt{Ord} is encoded as a record type with two parameters: -a type \texttt{A} and an \texttt{E} of type \texttt{EqDec A}. However, one can -still use it as if it had a single parameter inside generalizing binders: the -generalization of superclasses will be done automatically. -\begin{coq_example*} -Definition le_eqb `{Ord A} (x y : A) := andb (le x y) (le y x). -\end{coq_example*} - -In some cases, to be able to specify sharing of structures, one may want to give -explicitly the superclasses. It is possible to do it directly in regular -generalized binders, and using the \texttt{!} modifier in class -binders. For example: -\begin{coq_example*} -Definition lt `{eqa : EqDec A, ! Ord eqa} (x y : A) := - andb (le x y) (neqb x y). -\end{coq_example*} - -The \texttt{!} modifier switches the way a binder is parsed back to the -regular interpretation of Coq. In particular, it uses the implicit -arguments mechanism if available, as shown in the example. - -\subsection{Substructures} - -Substructures are components of a class which are instances of a class -themselves. They often arise when using classes for logical properties, -e.g.: - -\begin{coq_eval} -Require Import Relations. -\end{coq_eval} -\begin{coq_example*} -Class Reflexive (A : Type) (R : relation A) := - reflexivity : forall x, R x x. -Class Transitive (A : Type) (R : relation A) := - transitivity : forall x y z, R x y -> R y z -> R x z. -\end{coq_example*} - -This declares singleton classes for reflexive and transitive relations, -(see \ref{SingletonClass} for an explanation). -These may be used as part of other classes: - -\begin{coq_example*} -Class PreOrder (A : Type) (R : relation A) := -{ PreOrder_Reflexive :> Reflexive A R ; - PreOrder_Transitive :> Transitive A R }. -\end{coq_example*} - -The syntax \texttt{:>} indicates that each \texttt{PreOrder} can be seen -as a \texttt{Reflexive} relation. So each time a reflexive relation is -needed, a preorder can be used instead. This is very similar to the -coercion mechanism of \texttt{Structure} declarations. -The implementation simply declares each projection as an instance. - -One can also declare existing objects or structure -projections using the \texttt{Existing Instance} command to achieve the -same effect. - -\section{Summary of the commands -\label{TypeClassCommands}} - -\subsection{\tt Class {\ident} {\binder$_1$ \ldots~\binder$_n$} - : \sort := \{ field$_1$ ; \ldots ; field$_k$ \}.} -\comindex{Class} -\label{Class} - -The \texttt{Class} command is used to declare a type class with -parameters {\binder$_1$} to {\binder$_n$} and fields {\tt field$_1$} to -{\tt field$_k$}. - -\begin{Variants} -\item \label{SingletonClass} {\tt Class {\ident} {\binder$_1$ \ldots \binder$_n$} - : \sort := \ident$_1$ : \type$_1$.} - This variant declares a \emph{singleton} class whose only method is - {\tt \ident$_1$}. This singleton class is a so-called definitional - class, represented simply as a definition - {\tt {\ident} \binder$_1$ \ldots \binder$_n$ := \type$_1$} and whose - instances are themselves objects of this type. Definitional classes - are not wrapped inside records, and the trivial projection of an - instance of such a class is convertible to the instance itself. This can - be useful to make instances of existing objects easily and to reduce - proof size by not inserting useless projections. The class - constant itself is declared rigid during resolution so that the class - abstraction is maintained. - -\item \label{ExistingClass} {\tt Existing Class {\ident}.\comindex{Existing Class}} - This variant declares a class a posteriori from a constant or - inductive definition. No methods or instances are defined. -\end{Variants} - -\subsection{\tt Instance {\ident} {\binder$_1$ \ldots \binder$_n$} : - {Class} {t$_1$ \ldots t$_n$} [| \textit{priority}] - := \{ field$_1$ := b$_1$ ; \ldots ; field$_i$ := b$_i$ \}} -\comindex{Instance} -\label{Instance} - -The \texttt{Instance} command is used to declare a type class instance -named {\ident} of the class \emph{Class} with parameters {t$_1$} to {t$_n$} and -fields {\tt b$_1$} to {\tt b$_i$}, where each field must be a declared -field of the class. Missing fields must be filled in interactive proof mode. - -An arbitrary context of the form {\tt \binder$_1$ \ldots \binder$_n$} -can be put after the name of the instance and before the colon to -declare a parameterized instance. -An optional \textit{priority} can be declared, 0 being the highest -priority as for auto hints. - -\begin{Variants} -\item {\tt Instance {\ident} {\binder$_1$ \ldots \binder$_n$} : - forall {\binder$_{n+1}$ \ldots \binder$_m$}, - {Class} {t$_1$ \ldots t$_n$} [| \textit{priority}] := \term} - This syntax is used for declaration of singleton class instances or - for directly giving an explicit term of type - {\tt forall {\binder$_{n+1}$ \ldots \binder$_m$}, {Class} {t$_1$ \ldots t$_n$}}. - One need not even mention the unique field name for singleton classes. - -\item {\tt Global Instance} One can use the \texttt{Global} modifier on - instances declared in a section so that their generalization is automatically - redeclared after the section is closed. - -\item {\tt Program Instance} \comindex{Program Instance} - Switches the type-checking to \Program~(chapter \ref{Program}) - and uses the obligation mechanism to manage missing fields. - -\item {\tt Declare Instance} \comindex{Declare Instance} - In a {\tt Module Type}, this command states that a corresponding - concrete instance should exist in any implementation of this - {\tt Module Type}. This is similar to the distinction between - {\tt Parameter} vs. {\tt Definition}, or between {\tt Declare Module} - and {\tt Module}. - -\end{Variants} - -Besides the {\tt Class} and {\tt Instance} vernacular commands, there -are a few other commands related to type classes. - -\subsection{\tt Existing Instance {\ident}} -\comindex{Existing Instance} -\label{ExistingInstance} - -This commands adds an arbitrary constant whose type ends with an applied -type class to the instance database. It can be used for redeclaring -instances at the end of sections, or declaring structure projections as -instances. This is almost equivalent to {\tt Hint Resolve {\ident} : - typeclass\_instances}. - -\subsection{\tt Context {\binder$_1$ \ldots \binder$_n$}} -\comindex{Context} -\label{Context} - -Declares variables according to the given binding context, which might -use implicit generalization (see \ref{SectionContext}). - -\subsection{\tt Typeclasses Transparent, Opaque {\ident$_1$ \ldots \ident$_n$}} -\comindex{Typeclasses Transparent} -\comindex{Typeclasses Opaque} -\label{TypeclassesTransparency} - -This commands defines the transparency of {\ident$_1$ \ldots \ident$_n$} -during type class resolution. It is useful when some constants prevent some -unifications and make resolution fail. It is also useful to declare -constants which should never be unfolded during proof-search, like -fixpoints or anything which does not look like an abbreviation. This can -additionally speed up proof search as the typeclass map can be indexed -by such rigid constants (see \ref{HintTransparency}). -By default, all constants and local variables are considered transparent. -One should take care not to make opaque any constant that is used to -abbreviate a type, like {\tt relation A := A -> A -> Prop}. - -This is equivalent to {\tt Hint Transparent,Opaque} {\ident} {\tt: typeclass\_instances}. - -\subsection{\tt Typeclasses eauto := [debug] [dfs | bfs] [\emph{depth}]} -\comindex{Typeclasses eauto} -\label{TypeclassesEauto} - -This commands allows to customize the type class resolution tactic, -based on a variant of eauto. The flags semantics are: -\begin{itemize} -\item {\tt debug} In debug mode, the trace of successfully applied - tactics is printed. -\item {\tt dfs, bfs} This sets the search strategy to depth-first search - (the default) or breadth-first search. -\item {\emph{depth}} This sets the depth of the search (the default is 100). -\end{itemize} - -%%% Local Variables: -%%% mode: latex -%%% TeX-master: "Reference-Manual" -%%% End: diff --git a/doc/refman/Coercion.tex b/doc/refman/Coercion.tex deleted file mode 100644 index 3b6c949b..00000000 --- a/doc/refman/Coercion.tex +++ /dev/null @@ -1,564 +0,0 @@ -\achapter{Implicit Coercions} -\aauthor{Amokrane Saïbi} - -\label{Coercions-full} -\index{Coercions!presentation} - -\asection{General Presentation} - -This section describes the inheritance mechanism of {\Coq}. In {\Coq} with -inheritance, we are not interested in adding any expressive power to -our theory, but only convenience. Given a term, possibly not typable, -we are interested in the problem of determining if it can be well -typed modulo insertion of appropriate coercions. We allow to write: - -\begin{itemize} -\item $f~a$ where $f:forall~ x:A, B$ and $a:A'$ when $A'$ can - be seen in some sense as a subtype of $A$. -\item $x:A$ when $A$ is not a type, but can be seen in - a certain sense as a type: set, group, category etc. -\item $f~a$ when $f$ is not a function, but can be seen in a certain sense - as a function: bijection, functor, any structure morphism etc. -\end{itemize} - -\asection{Classes} -\index{Coercions!classes} - A class with $n$ parameters is any defined name with a type -$forall~ (x_1:A_1)..(x_n:A_n), s$ where $s$ is a sort. Thus a class with -parameters is considered as a single class and not as a family of -classes. An object of a class $C$ is any term of type $C~t_1 -.. t_n$. In addition to these user-classes, we have two abstract -classes: - -\begin{itemize} -\item {\tt Sortclass}, the class of sorts; - its objects are the terms whose type is a sort. -\item {\tt Funclass}, the class of functions; - its objects are all the terms with a functional - type, i.e. of form $forall~ x:A, B$. -\end{itemize} - -Formally, the syntax of a classes is defined on Figure~\ref{fig:classes}. -\begin{figure} -\begin{centerframe} -\begin{tabular}{lcl} -{\class} & ::= & {\qualid} \\ - & $|$ & {\tt Sortclass} \\ - & $|$ & {\tt Funclass} -\end{tabular} -\end{centerframe} -\caption{Syntax of classes} -\label{fig:classes} -\end{figure} - -\asection{Coercions} -\index{Coercions!Funclass} -\index{Coercions!Sortclass} - A name $f$ can be declared as a coercion between a source user-class -$C$ with $n$ parameters and a target class $D$ if one of these -conditions holds: - -\newcommand{\oftype}{\!:\!} - -\begin{itemize} -\item $D$ is a user-class, then the type of $f$ must have the form - $forall~ (x_1 \oftype A_1)..(x_n \oftype A_n)(y\oftype C~x_1..x_n), D~u_1..u_m$ where $m$ - is the number of parameters of $D$. -\item $D$ is {\tt Funclass}, then the type of $f$ must have the form - $forall~ (x_1\oftype A_1)..(x_n\oftype A_n)(y\oftype C~x_1..x_n)(x:A), B$. -\item $D$ is {\tt Sortclass}, then the type of $f$ must have the form - $forall~ (x_1\oftype A_1)..(x_n\oftype A_n)(y\oftype C~x_1..x_n), s$ with $s$ a sort. -\end{itemize} - -We then write $f:C \mbox{\texttt{>->}} D$. The restriction on the type -of coercions is called {\em the uniform inheritance condition}. -Remark that the abstract classes {\tt Funclass} and {\tt Sortclass} -cannot be source classes. - -To coerce an object $t:C~t_1..t_n$ of $C$ towards $D$, we have to -apply the coercion $f$ to it; the obtained term $f~t_1..t_n~t$ is -then an object of $D$. - -\asection{Identity Coercions} -\index{Coercions!identity} - - Identity coercions are special cases of coercions used to go around -the uniform inheritance condition. Let $C$ and $D$ be two classes -with respectively $n$ and $m$ parameters and -$f:forall~(x_1:T_1)..(x_k:T_k)(y:C~u_1..u_n), D~v_1..v_m$ a function which -does not verify the uniform inheritance condition. To declare $f$ as -coercion, one has first to declare a subclass $C'$ of $C$: - -$$C' := fun~ (x_1:T_1)..(x_k:T_k) => C~u_1..u_n$$ - -\noindent We then define an {\em identity coercion} between $C'$ and $C$: -\begin{eqnarray*} -Id\_C'\_C & := & fun~ (x_1:T_1)..(x_k:T_k)(y:C'~x_1..x_k) => (y:C~u_1..u_n)\\ -\end{eqnarray*} - -We can now declare $f$ as coercion from $C'$ to $D$, since we can -``cast'' its type as -$forall~ (x_1:T_1)..(x_k:T_k)(y:C'~x_1..x_k),D~v_1..v_m$.\\ The identity -coercions have a special status: to coerce an object $t:C'~t_1..t_k$ -of $C'$ towards $C$, we does not have to insert explicitly $Id\_C'\_C$ -since $Id\_C'\_C~t_1..t_k~t$ is convertible with $t$. However we -``rewrite'' the type of $t$ to become an object of $C$; in this case, -it becomes $C~u_1^*..u_k^*$ where each $u_i^*$ is the result of the -substitution in $u_i$ of the variables $x_j$ by $t_j$. - - -\asection{Inheritance Graph} -\index{Coercions!inheritance graph} -Coercions form an inheritance graph with classes as nodes. We call -{\em coercion path} an ordered list of coercions between two nodes of -the graph. A class $C$ is said to be a subclass of $D$ if there is a -coercion path in the graph from $C$ to $D$; we also say that $C$ -inherits from $D$. Our mechanism supports multiple inheritance since a -class may inherit from several classes, contrary to simple inheritance -where a class inherits from at most one class. However there must be -at most one path between two classes. If this is not the case, only -the {\em oldest} one is valid and the others are ignored. So the order -of declaration of coercions is important. - -We extend notations for coercions to coercion paths. For instance -$[f_1;..;f_k]:C \mbox{\texttt{>->}} D$ is the coercion path composed -by the coercions $f_1..f_k$. The application of a coercion path to a -term consists of the successive application of its coercions. - -\asection{Declaration of Coercions} - -%%%%% "Class" is useless, since classes are implicitely defined via coercions. - -% \asubsection{\tt Class {\qualid}.}\comindex{Class} -% Declares {\qualid} as a new class. - -% \begin{ErrMsgs} -% \item {\qualid} \errindex{not declared} -% \item {\qualid} \errindex{is already a class} -% \item \errindex{Type of {\qualid} does not end with a sort} -% \end{ErrMsgs} - -% \begin{Variant} -% \item {\tt Class Local {\qualid}.} \\ -% Declares the construction denoted by {\qualid} as a new local class to -% the current section. -% \end{Variant} - -% END "Class" is useless - -\asubsection{\tt Coercion {\qualid} : {\class$_1$} >-> {\class$_2$}.} -\comindex{Coercion} - -Declares the construction denoted by {\qualid} as a coercion between -{\class$_1$} and {\class$_2$}. - -% Useless information -% The classes {\class$_1$} and {\class$_2$} are first declared if necessary. - -\begin{ErrMsgs} -\item {\qualid} \errindex{not declared} -\item {\qualid} \errindex{is already a coercion} -\item \errindex{Funclass cannot be a source class} -\item \errindex{Sortclass cannot be a source class} -\item {\qualid} \errindex{is not a function} -\item \errindex{Cannot find the source class of {\qualid}} -\item \errindex{Cannot recognize {\class$_1$} as a source class of {\qualid}} -\item {\qualid} \errindex{does not respect the uniform inheritance condition} -\item \errindex{Found target class {\class} instead of {\class$_2$}} - -\end{ErrMsgs} - -When the coercion {\qualid} is added to the inheritance graph, non -valid coercion paths are ignored; they are signaled by a warning. -\\[0.3cm] -\noindent {\bf Warning :} -\begin{enumerate} -\item \begin{tabbing} -{\tt Ambiguous paths: }\= $[f_1^1;..;f_{n_1}^1] : C_1\mbox{\tt >->}D_1$\\ - \> ... \\ - \>$[f_1^m;..;f_{n_m}^m] : C_m\mbox{\tt >->}D_m$ - \end{tabbing} -\end{enumerate} - -\begin{Variants} -\item {\tt Local Coercion {\qualid} : {\class$_1$} >-> {\class$_2$}.} -\comindex{Local Coercion}\\ - Declares the construction denoted by {\qualid} as a coercion local to - the current section. - -\item {\tt Coercion {\ident} := {\term}}\comindex{Coercion}\\ - This defines {\ident} just like \texttt{Definition {\ident} := - {\term}}, and then declares {\ident} as a coercion between it - source and its target. - -\item {\tt Coercion {\ident} := {\term} : {\type}}\\ - This defines {\ident} just like - \texttt{Definition {\ident} : {\type} := {\term}}, and then - declares {\ident} as a coercion between it source and its target. - -\item {\tt Local Coercion {\ident} := {\term}}\comindex{Local Coercion}\\ - This defines {\ident} just like \texttt{Let {\ident} := - {\term}}, and then declares {\ident} as a coercion between it - source and its target. - -\item Assumptions can be declared as coercions at declaration -time. This extends the grammar of assumptions from -Figure~\ref{sentences-syntax} as follows: -\comindex{Variable \mbox{\rm (and coercions)}} -\comindex{Axiom \mbox{\rm (and coercions)}} -\comindex{Parameter \mbox{\rm (and coercions)}} -\comindex{Hypothesis \mbox{\rm (and coercions)}} - -\begin{tabular}{lcl} -%% Declarations -{\assumption} & ::= & {\assumptionkeyword} {\assums} {\tt .} \\ -&&\\ -{\assums} & ::= & {\simpleassums} \\ - & $|$ & \nelist{{\tt (} \simpleassums {\tt )}}{} \\ -&&\\ -{\simpleassums} & ::= & \nelist{\ident}{} {\tt :}\zeroone{{\tt >}} {\term}\\ -\end{tabular} - -If the extra {\tt >} is present before the type of some assumptions, these -assumptions are declared as coercions. - -\item Constructors of inductive types can be declared as coercions at -definition time of the inductive type. This extends and modifies the -grammar of inductive types from Figure \ref{sentences-syntax} as follows: -\comindex{Inductive \mbox{\rm (and coercions)}} -\comindex{CoInductive \mbox{\rm (and coercions)}} - -\begin{center} -\begin{tabular}{lcl} -%% Inductives -{\inductive} & ::= & - {\tt Inductive} \nelist{\inductivebody}{with} {\tt .} \\ - & $|$ & {\tt CoInductive} \nelist{\inductivebody}{with} {\tt .} \\ - & & \\ -{\inductivebody} & ::= & - {\ident} \zeroone{\binders} {\tt :} {\term} {\tt :=} \\ - && ~~~\zeroone{\zeroone{\tt |} \nelist{\constructor}{|}} \\ - & & \\ -{\constructor} & ::= & {\ident} \zeroone{\binders} \zeroone{{\tt :}\zeroone{\tt >} {\term}} \\ -\end{tabular} -\end{center} - -Especially, if the extra {\tt >} is present in a constructor -declaration, this constructor is declared as a coercion. -\end{Variants} - -\asubsection{\tt Identity Coercion {\ident}:{\class$_1$} >-> {\class$_2$}.} -\comindex{Identity Coercion} - -We check that {\class$_1$} is a constant with a value of the form -$fun~ (x_1:T_1)..(x_n:T_n) => (\mbox{\class}_2~t_1..t_m)$ where $m$ is the -number of parameters of \class$_2$. Then we define an identity -function with the type -$forall~ (x_1:T_1)..(x_n:T_n)(y:\mbox{\class}_1~x_1..x_n), -{\mbox{\class}_2}~t_1..t_m$, and we declare it as an identity -coercion between {\class$_1$} and {\class$_2$}. - -\begin{ErrMsgs} -\item {\class$_1$} \errindex{must be a transparent constant} -\end{ErrMsgs} - -\begin{Variants} -\item {\tt Local Identity Coercion {\ident}:{\ident$_1$} >-> {\ident$_2$}.} \\ -Idem but locally to the current section. - -\item {\tt SubClass {\ident} := {\type}.} \\ -\comindex{SubClass} - If {\type} is a class -{\ident'} applied to some arguments then {\ident} is defined and an -identity coercion of name {\tt Id\_{\ident}\_{\ident'}} is -declared. Otherwise said, this is an abbreviation for - -{\tt Definition {\ident} := {\type}.} - - followed by - -{\tt Identity Coercion Id\_{\ident}\_{\ident'}:{\ident} >-> {\ident'}}. - -\item {\tt Local SubClass {\ident} := {\type}.} \\ -Same as before but locally to the current section. - -\end{Variants} - -\asection{Displaying Available Coercions} - -\asubsection{\tt Print Classes.} -\comindex{Print Classes} -Print the list of declared classes in the current context. - -\asubsection{\tt Print Coercions.} -\comindex{Print Coercions} -Print the list of declared coercions in the current context. - -\asubsection{\tt Print Graph.} -\comindex{Print Graph} -Print the list of valid coercion paths in the current context. - -\asubsection{\tt Print Coercion Paths {\class$_1$} {\class$_2$}.} -\comindex{Print Coercion Paths} -Print the list of valid coercion paths from {\class$_1$} to {\class$_2$}. - -\asection{Activating the Printing of Coercions} - -\asubsection{\tt Set Printing Coercions.} -\comindex{Set Printing Coercions} -\comindex{Unset Printing Coercions} - -This command forces all the coercions to be printed. -Conversely, to skip the printing of coercions, use - {\tt Unset Printing Coercions}. -By default, coercions are not printed. - -\asubsection{\tt Set Printing Coercion {\qualid}.} -\comindex{Set Printing Coercion} -\comindex{Unset Printing Coercion} - -This command forces coercion denoted by {\qualid} to be printed. -To skip the printing of coercion {\qualid}, use - {\tt Unset Printing Coercion {\qualid}}. -By default, a coercion is never printed. - -\asection{Classes as Records} -\label{Coercions-and-records} -\index{Coercions!and records} -We allow the definition of {\em Structures with Inheritance} (or -classes as records) by extending the existing {\tt Record} macro -(see Section~\ref{Record}). Its new syntax is: - -\begin{center} -\begin{tabular}{l} -{\tt Record \zeroone{>}~{\ident} \zeroone{\binders} : {\sort} := \zeroone{\ident$_0$} \verb+{+} \\ -~~~~\begin{tabular}{l} - {\tt \ident$_1$ $[$:$|$:>$]$ \term$_1$ ;} \\ - ... \\ - {\tt \ident$_n$ $[$:$|$:>$]$ \term$_n$ \verb+}+. } - \end{tabular} -\end{tabular} -\end{center} -The identifier {\ident} is the name of the defined record and {\sort} -is its type. The identifier {\ident$_0$} is the name of its -constructor. The identifiers {\ident$_1$}, .., {\ident$_n$} are the -names of its fields and {\term$_1$}, .., {\term$_n$} their respective -types. The alternative {\tt $[$:$|$:>$]$} is ``{\tt :}'' or ``{\tt -:>}''. If {\tt {\ident$_i$}:>{\term$_i$}}, then {\ident$_i$} is -automatically declared as coercion from {\ident} to the class of -{\term$_i$}. Remark that {\ident$_i$} always verifies the uniform -inheritance condition. If the optional ``{\tt >}'' before {\ident} is -present, then {\ident$_0$} (or the default name {\tt Build\_{\ident}} -if {\ident$_0$} is omitted) is automatically declared as a coercion -from the class of {\term$_n$} to {\ident} (this may fail if the -uniform inheritance condition is not satisfied). - -\Rem The keyword {\tt Structure}\comindex{Structure} is a synonym of {\tt -Record}. - -\asection{Coercions and Sections} -\index{Coercions!and sections} - The inheritance mechanism is compatible with the section -mechanism. The global classes and coercions defined inside a section -are redefined after its closing, using their new value and new -type. The classes and coercions which are local to the section are -simply forgotten. -Coercions with a local source class or a local target class, and -coercions which do not verify the uniform inheritance condition any longer -are also forgotten. - -\asection{Coercions and Modules} -\index{Coercions!and modules} - -From Coq version 8.3, the coercions present in a module are activated -only when the module is explicitly imported. Formerly, the coercions -were activated as soon as the module was required, whatever it was -imported or not. - -To recover the behavior of the versions of Coq prior to 8.3, use the -following command: - -\comindex{Set Automatic Coercions Import} -\comindex{Unset Automatic Coercions Import} -\begin{verbatim} -Set Automatic Coercions Import. -\end{verbatim} - -To cancel the effect of the option, use instead: - -\begin{verbatim} -Unset Automatic Coercions Import. -\end{verbatim} - -\asection{Examples} - - There are three situations: - -\begin{itemize} -\item $f~a$ is ill-typed where $f:forall~x:A,B$ and $a:A'$. If there is a - coercion path between $A'$ and $A$, $f~a$ is transformed into - $f~a'$ where $a'$ is the result of the application of this - coercion path to $a$. - -We first give an example of coercion between atomic inductive types - -%\begin{\small} -\begin{coq_example} -Definition bool_in_nat (b:bool) := if b then 0 else 1. -Coercion bool_in_nat : bool >-> nat. -Check (0 = true). -Set Printing Coercions. -Check (0 = true). -\end{coq_example} -%\end{small} - -\begin{coq_eval} -Unset Printing Coercions. -\end{coq_eval} - -\Warning ``\verb|Check true=O.|'' fails. This is ``normal'' behaviour of -coercions. To validate \verb|true=O|, the coercion is searched from -\verb=nat= to \verb=bool=. There is none. - -We give an example of coercion between classes with parameters. - -%\begin{\small} -\begin{coq_example} -Parameters - (C : nat -> Set) (D : nat -> bool -> Set) (E : bool -> Set). -Parameter f : forall n:nat, C n -> D (S n) true. -Coercion f : C >-> D. -Parameter g : forall (n:nat) (b:bool), D n b -> E b. -Coercion g : D >-> E. -Parameter c : C 0. -Parameter T : E true -> nat. -Check (T c). -Set Printing Coercions. -Check (T c). -\end{coq_example} -%\end{small} - -\begin{coq_eval} -Unset Printing Coercions. -\end{coq_eval} - -We give now an example using identity coercions. - -%\begin{small} -\begin{coq_example} -Definition D' (b:bool) := D 1 b. -Identity Coercion IdD'D : D' >-> D. -Print IdD'D. -Parameter d' : D' true. -Check (T d'). -Set Printing Coercions. -Check (T d'). -\end{coq_example} -%\end{small} - -\begin{coq_eval} -Unset Printing Coercions. -\end{coq_eval} - - - In the case of functional arguments, we use the monotonic rule of -sub-typing. Approximatively, to coerce $t:forall~x:A, B$ towards -$forall~x:A',B'$, one have to coerce $A'$ towards $A$ and $B$ towards -$B'$. An example is given below: - -%\begin{small} -\begin{coq_example} -Parameters (A B : Set) (h : A -> B). -Coercion h : A >-> B. -Parameter U : (A -> E true) -> nat. -Parameter t : B -> C 0. -Check (U t). -Set Printing Coercions. -Check (U t). -\end{coq_example} -%\end{small} - -\begin{coq_eval} -Unset Printing Coercions. -\end{coq_eval} - - Remark the changes in the result following the modification of the -previous example. - -%\begin{small} -\begin{coq_example} -Parameter U' : (C 0 -> B) -> nat. -Parameter t' : E true -> A. -Check (U' t'). -Set Printing Coercions. -Check (U' t'). -\end{coq_example} -%\end{small} - -\begin{coq_eval} -Unset Printing Coercions. -\end{coq_eval} - -\item An assumption $x:A$ when $A$ is not a type, is ill-typed. It is - replaced by $x:A'$ where $A'$ is the result of the application - to $A$ of the coercion path between the class of $A$ and {\tt - Sortclass} if it exists. This case occurs in the abstraction - $fun~ x:A => t$, universal quantification $forall~x:A, B$, - global variables and parameters of (co-)inductive definitions - and functions. In $forall~x:A, B$, such a coercion path may be - applied to $B$ also if necessary. - -%\begin{small} -\begin{coq_example} -Parameter Graph : Type. -Parameter Node : Graph -> Type. -Coercion Node : Graph >-> Sortclass. -Parameter G : Graph. -Parameter Arrows : G -> G -> Type. -Check Arrows. -Parameter fg : G -> G. -Check fg. -Set Printing Coercions. -Check fg. -\end{coq_example} -%\end{small} - -\begin{coq_eval} -Unset Printing Coercions. -\end{coq_eval} - -\item $f~a$ is ill-typed because $f:A$ is not a function. The term - $f$ is replaced by the term obtained by applying to $f$ the - coercion path between $A$ and {\tt Funclass} if it exists. - -%\begin{small} -\begin{coq_example} -Parameter bij : Set -> Set -> Set. -Parameter ap : forall A B:Set, bij A B -> A -> B. -Coercion ap : bij >-> Funclass. -Parameter b : bij nat nat. -Check (b 0). -Set Printing Coercions. -Check (b 0). -\end{coq_example} -%\end{small} - -\begin{coq_eval} -Unset Printing Coercions. -\end{coq_eval} - -Let us see the resulting graph of this session. - -%\begin{small} -\begin{coq_example} -Print Graph. -\end{coq_example} -%\end{small} - -\end{itemize} - - -%%% Local Variables: -%%% mode: latex -%%% TeX-master: "Reference-Manual" -%%% End: diff --git a/doc/refman/Extraction.tex b/doc/refman/Extraction.tex deleted file mode 100644 index cc9cf5c8..00000000 --- a/doc/refman/Extraction.tex +++ /dev/null @@ -1,551 +0,0 @@ -\achapter{Extraction of programs in Objective Caml and Haskell} -\label{Extraction} -\aauthor{Jean-Christophe Filliâtre and Pierre Letouzey} -\index{Extraction} - -We present here the \Coq\ extraction commands, used to build certified -and relatively efficient functional programs, extracting them from -either \Coq\ functions or \Coq\ proofs of specifications. The -functional languages available as output are currently \ocaml{}, -\textsc{Haskell} and \textsc{Scheme}. In the following, ``ML'' will -be used (abusively) to refer to any of the three. - -\paragraph{Differences with old versions.} -The current extraction mechanism is new for version 7.0 of {\Coq}. -In particular, the \FW\ toplevel used as an intermediate step between -\Coq\ and ML has been withdrawn. It is also not possible -any more to import ML objects in this \FW\ toplevel. -The current mechanism also differs from -the one in previous versions of \Coq: there is no more -an explicit toplevel for the language (formerly called \textsc{Fml}). - -\asection{Generating ML code} -\comindex{Extraction} -\comindex{Recursive Extraction} -\comindex{Extraction Module} -\comindex{Recursive Extraction Module} - -The next two commands are meant to be used for rapid preview of -extraction. They both display extracted term(s) inside \Coq. - -\begin{description} -\item {\tt Extraction \qualid.} ~\par - Extracts one constant or module in the \Coq\ toplevel. - -\item {\tt Recursive Extraction \qualid$_1$ \dots\ \qualid$_n$.} ~\par - Recursive extraction of all the globals (or modules) \qualid$_1$ \dots\ - \qualid$_n$ and all their dependencies in the \Coq\ toplevel. -\end{description} - -%% TODO error messages - -All the following commands produce real ML files. User can choose to produce -one monolithic file or one file per \Coq\ library. - -\begin{description} -\item {\tt Extraction "{\em file}"} - \qualid$_1$ \dots\ \qualid$_n$. ~\par - Recursive extraction of all the globals (or modules) \qualid$_1$ \dots\ - \qualid$_n$ and all their dependencies in one monolithic file {\em file}. - Global and local identifiers are renamed according to the chosen ML - language to fulfill its syntactic conventions, keeping original - names as much as possible. - -\item {\tt Extraction Library} \ident. ~\par - Extraction of the whole \Coq\ library {\tt\ident.v} to an ML module - {\tt\ident.ml}. In case of name clash, identifiers are here renamed - using prefixes \verb!coq_! or \verb!Coq_! to ensure a - session-independent renaming. - -\item {\tt Recursive Extraction Library} \ident. ~\par - Extraction of the \Coq\ library {\tt\ident.v} and all other modules - {\tt\ident.v} depends on. -\end{description} - -The list of globals \qualid$_i$ does not need to be -exhaustive: it is automatically completed into a complete and minimal -environment. - -\asection{Extraction options} - -\asubsection{Setting the target language} -\comindex{Extraction Language} - -The ability to fix target language is the first and more important -of the extraction options. Default is Ocaml. -\begin{description} -\item {\tt Extraction Language Ocaml}. -\item {\tt Extraction Language Haskell}. -\item {\tt Extraction Language Scheme}. -\end{description} - -\asubsection{Inlining and optimizations} - -Since Objective Caml is a strict language, the extracted -code has to be optimized in order to be efficient (for instance, when -using induction principles we do not want to compute all the recursive -calls but only the needed ones). So the extraction mechanism provides -an automatic optimization routine that will be -called each time the user want to generate Ocaml programs. Essentially, -it performs constants inlining and reductions. Therefore some -constants may not appear in resulting monolithic Ocaml program. -In the case of modular extraction, even if some inlining is done, the -inlined constant are nevertheless printed, to ensure -session-independent programs. - -Concerning Haskell, such optimizations are less useful because of -lazyness. We still make some optimizations, for example in order to -produce more readable code. - -All these optimizations are controled by the following \Coq\ options: - -\begin{description} - -\item \comindex{Set Extraction Optimize} -{\tt Set Extraction Optimize.} - -\item \comindex{Unset Extraction Optimize} -{\tt Unset Extraction Optimize.} - -Default is Set. This control all optimizations made on the ML terms -(mostly reduction of dummy beta/iota redexes, but also simplifications on -Cases, etc). Put this option to Unset if you want a ML term as close as -possible to the Coq term. - -\item \comindex{Set Extraction AutoInline} -{\tt Set Extraction AutoInline.} - -\item \comindex{Unset Extraction AutoInline} -{\tt Unset Extraction AutoInline.} - -Default is Set, so by default, the extraction mechanism feels free to -inline the bodies of some defined constants, according to some heuristics -like size of bodies, useness of some arguments, etc. Those heuristics are -not always perfect, you may want to disable this feature, do it by Unset. - -\item \comindex{Extraction Inline} -{\tt Extraction Inline} \qualid$_1$ \dots\ \qualid$_n$. - -\item \comindex{Extraction NoInline} -{\tt Extraction NoInline} \qualid$_1$ \dots\ \qualid$_n$. - -In addition to the automatic inline feature, you can now tell precisely to -inline some more constants by the {\tt Extraction Inline} command. Conversely, -you can forbid the automatic inlining of some specific constants by -the {\tt Extraction NoInline} command. -Those two commands enable a precise control of what is inlined and what is not. - -\item \comindex{Print Extraction Inline} -{\tt Print Extraction Inline}. - -Prints the current state of the table recording the custom inlinings -declared by the two previous commands. - -\item \comindex{Reset Extraction Inline} -{\tt Reset Extraction Inline}. - -Puts the table recording the custom inlinings back to empty. - -\end{description} - - -\paragraph{Inlining and printing of a constant declaration.} - -A user can explicitly ask for a constant to be extracted by two means: -\begin{itemize} -\item by mentioning it on the extraction command line -\item by extracting the whole \Coq\ module of this constant. -\end{itemize} -In both cases, the declaration of this constant will be present in the -produced file. -But this same constant may or may not be inlined in the following -terms, depending on the automatic/custom inlining mechanism. - - -For the constants non-explicitly required but needed for dependency -reasons, there are two cases: -\begin{itemize} -\item If an inlining decision is taken, whether automatically or not, -all occurrences of this constant are replaced by its extracted body, and -this constant is not declared in the generated file. -\item If no inlining decision is taken, the constant is normally - declared in the produced file. -\end{itemize} - -\asubsection{Extra elimination of useless arguments} - -\begin{description} -\item \comindex{Extraction Implicit} - {\tt Extraction Implicit} \qualid\ [ \ident$_1$ \dots\ \ident$_n$ ]. - -This experimental command allows to declare some arguments of -\qualid\ as implicit, i.e. useless in extracted code and hence to -be removed by extraction. Here \qualid\ can be any function or -inductive constructor, and \ident$_i$ are the names of the concerned -arguments. In fact, an argument can also be referred by a number -indicating its position, starting from 1. When an actual extraction -takes place, an error is raised if the {\tt Extraction Implicit} -declarations cannot be honored, that is if any of the implicited -variables still occurs in the final code. This declaration of useless -arguments is independent but complementary to the main elimination -principles of extraction (logical parts and types). -\end{description} - -\asubsection{Realizing axioms}\label{extraction:axioms} - -Extraction will fail if it encounters an informative -axiom not realized (see Section~\ref{extraction:axioms}). -A warning will be issued if it encounters an logical axiom, to remind -user that inconsistent logical axioms may lead to incorrect or -non-terminating extracted terms. - -It is possible to assume some axioms while developing a proof. Since -these axioms can be any kind of proposition or object or type, they may -perfectly well have some computational content. But a program must be -a closed term, and of course the system cannot guess the program which -realizes an axiom. Therefore, it is possible to tell the system -what ML term corresponds to a given axiom. - -\comindex{Extract Constant} -\begin{description} -\item{\tt Extract Constant \qualid\ => \str.} ~\par - Give an ML extraction for the given constant. - The \str\ may be an identifier or a quoted string. -\item{\tt Extract Inlined Constant \qualid\ => \str.} ~\par - Same as the previous one, except that the given ML terms will - be inlined everywhere instead of being declared via a let. -\end{description} - -Note that the {\tt Extract Inlined Constant} command is sugar -for an {\tt Extract Constant} followed by a {\tt Extraction Inline}. -Hence a {\tt Reset Extraction Inline} will have an effect on the -realized and inlined axiom. - -Of course, it is the responsibility of the user to ensure that the ML -terms given to realize the axioms do have the expected types. In -fact, the strings containing realizing code are just copied in the -extracted files. The extraction recognizes whether the realized axiom -should become a ML type constant or a ML object declaration. - -\Example -\begin{coq_example} -Axiom X:Set. -Axiom x:X. -Extract Constant X => "int". -Extract Constant x => "0". -\end{coq_example} - -Notice that in the case of type scheme axiom (i.e. whose type is an -arity, that is a sequence of product finished by a sort), then some type -variables has to be given. The syntax is then: - -\begin{description} -\item{\tt Extract Constant \qualid\ \str$_1$ \ldots \str$_n$ => \str.} ~\par -\end{description} - -The number of type variables is checked by the system. - -\Example -\begin{coq_example} -Axiom Y : Set -> Set -> Set. -Extract Constant Y "'a" "'b" => " 'a*'b ". -\end{coq_example} - -Realizing an axiom via {\tt Extract Constant} is only useful in the -case of an informative axiom (of sort Type or Set). A logical axiom -have no computational content and hence will not appears in extracted -terms. But a warning is nonetheless issued if extraction encounters a -logical axiom. This warning reminds user that inconsistent logical -axioms may lead to incorrect or non-terminating extracted terms. - -If an informative axiom has not been realized before an extraction, a -warning is also issued and the definition of the axiom is filled with -an exception labeled {\tt AXIOM TO BE REALIZED}. The user must then -search these exceptions inside the extracted file and replace them by -real code. - -\comindex{Extract Inductive} - -The system also provides a mechanism to specify ML terms for inductive -types and constructors. For instance, the user may want to use the ML -native boolean type instead of \Coq\ one. The syntax is the following: - -\begin{description} -\item{\tt Extract Inductive \qualid\ => \str\ [ \str\ \dots \str\ ]\ -{\it optstring}.} ~\par - Give an ML extraction for the given inductive type. You must specify - extractions for the type itself (first \str) and all its - constructors (between square brackets). If given, the final optional - string should contain a function emulating pattern-matching over this - inductive type. If this optional string is not given, the ML - extraction must be an ML inductive datatype, and the native - pattern-matching of the language will be used. -\end{description} - -For an inductive type with $k$ constructor, the function used to -emulate the match should expect $(k+1)$ arguments, first the $k$ -branches in functional form, and then the inductive element to -destruct. For instance, the match branch \verb$| S n => foo$ gives the -functional form \verb$(fun n -> foo)$. Note that a constructor with no -argument is considered to have one unit argument, in order to block -early evaluation of the branch: \verb$| O => bar$ leads to the functional -form \verb$(fun () -> bar)$. For instance, when extracting {\tt nat} -into {\tt int}, the code to provide has type: -{\tt (unit->'a)->(int->'a)->int->'a}. - -As for {\tt Extract Inductive}, this command should be used with care: -\begin{itemize} -\item The ML code provided by the user is currently \emph{not} checked at all by - extraction, even for syntax errors. - -\item Extracting an inductive type to a pre-existing ML inductive type -is quite sound. But extracting to a general type (by providing an -ad-hoc pattern-matching) will often \emph{not} be fully rigorously -correct. For instance, when extracting {\tt nat} to Ocaml's {\tt -int}, it is theoretically possible to build {\tt nat} values that are -larger than Ocaml's {\tt max\_int}. It is the user's responsability to -be sure that no overflow or other bad events occur in practice. - -\item Translating an inductive type to an ML type does \emph{not} -magically improve the asymptotic complexity of functions, even if the -ML type is an efficient representation. For instance, when extracting -{\tt nat} to Ocaml's {\tt int}, the function {\tt mult} stays -quadratic. It might be interesting to associate this translation with -some specific {\tt Extract Constant} when primitive counterparts exist. -\end{itemize} - -\Example -Typical examples are the following: -\begin{coq_example} -Extract Inductive unit => "unit" [ "()" ]. -Extract Inductive bool => "bool" [ "true" "false" ]. -Extract Inductive sumbool => "bool" [ "true" "false" ]. -\end{coq_example} - -If an inductive constructor or type has arity 2 and the corresponding -string is enclosed by parenthesis, then the rest of the string is used -as infix constructor or type. -\begin{coq_example} -Extract Inductive list => "list" [ "[]" "(::)" ]. -Extract Inductive prod => "(*)" [ "(,)" ]. -\end{coq_example} - -As an example of translation to a non-inductive datatype, let's turn -{\tt nat} into Ocaml's {\tt int} (see caveat above): -\begin{coq_example} -Extract Inductive nat => int [ "0" "succ" ] - "(fun fO fS n -> if n=0 then fO () else fS (n-1))". -\end{coq_example} - -\asubsection{Avoiding conflicts with existing filenames} - -\comindex{Extraction Blacklist} - -When using {\tt Extraction Library}, the names of the extracted files -directly depends from the names of the \Coq\ files. It may happen that -these filenames are in conflict with already existing files, -either in the standard library of the target language or in other -code that is meant to be linked with the extracted code. -For instance the module {\tt List} exists both in \Coq\ and in Ocaml. -It is possible to instruct the extraction not to use particular filenames. - -\begin{description} -\item{\tt Extraction Blacklist \ident \ldots \ident.} ~\par - Instruct the extraction to avoid using these names as filenames - for extracted code. -\item{\tt Print Extraction Blacklist.} ~\par - Show the current list of filenames the extraction should avoid. -\item{\tt Reset Extraction Blacklist.} ~\par - Allow the extraction to use any filename. -\end{description} - -For Ocaml, a typical use of these commands is -{\tt Extraction Blacklist String List}. - -\asection{Differences between \Coq\ and ML type systems} - - -Due to differences between \Coq\ and ML type systems, -some extracted programs are not directly typable in ML. -We now solve this problem (at least in Ocaml) by adding -when needed some unsafe casting {\tt Obj.magic}, which give -a generic type {\tt 'a} to any term. - -For example, here are two kinds of problem that can occur: - -\begin{itemize} - \item If some part of the program is {\em very} polymorphic, there - may be no ML type for it. In that case the extraction to ML works - all right but the generated code may be refused by the ML - type-checker. A very well known example is the {\em distr-pair} - function: -\begin{verbatim} -Definition dp := - fun (A B:Set)(x:A)(y:B)(f:forall C:Set, C->C) => (f A x, f B y). -\end{verbatim} - -In Ocaml, for instance, the direct extracted term would be: - -\begin{verbatim} -let dp x y f = Pair((f () x),(f () y)) -\end{verbatim} - -and would have type: -\begin{verbatim} -dp : 'a -> 'a -> (unit -> 'a -> 'b) -> ('b,'b) prod -\end{verbatim} - -which is not its original type, but a restriction. - -We now produce the following correct version: -\begin{verbatim} -let dp x y f = Pair ((Obj.magic f () x), (Obj.magic f () y)) -\end{verbatim} - - \item Some definitions of \Coq\ may have no counterpart in ML. This - happens when there is a quantification over types inside the type - of a constructor; for example: -\begin{verbatim} -Inductive anything : Set := dummy : forall A:Set, A -> anything. -\end{verbatim} - -which corresponds to the definition of an ML dynamic type. -In Ocaml, we must cast any argument of the constructor dummy. - -\end{itemize} - -Even with those unsafe castings, you should never get error like -``segmentation fault''. In fact even if your program may seem -ill-typed to the Ocaml type-checker, it can't go wrong: it comes -from a Coq well-typed terms, so for example inductives will always -have the correct number of arguments, etc. - -More details about the correctness of the extracted programs can be -found in \cite{Let02}. - -We have to say, though, that in most ``realistic'' programs, these -problems do not occur. For example all the programs of Coq library are -accepted by Caml type-checker without any {\tt Obj.magic} (see examples below). - - - -\asection{Some examples} - -We present here two examples of extractions, taken from the -\Coq\ Standard Library. We choose \ocaml\ as target language, -but all can be done in the other dialects with slight modifications. -We then indicate where to find other examples and tests of Extraction. - -\asubsection{A detailed example: Euclidean division} - -The file {\tt Euclid} contains the proof of Euclidean division -(theorem {\tt eucl\_dev}). The natural numbers defined in the example -files are unary integers defined by two constructors $O$ and $S$: -\begin{coq_example*} -Inductive nat : Set := - | O : nat - | S : nat -> nat. -\end{coq_example*} - -This module contains a theorem {\tt eucl\_dev}, whose type is: -\begin{verbatim} -forall b:nat, b > 0 -> forall a:nat, diveucl a b -\end{verbatim} -where {\tt diveucl} is a type for the pair of the quotient and the -modulo, plus some logical assertions that disappear during extraction. -We can now extract this program to \ocaml: - -\begin{coq_eval} -Reset Initial. -\end{coq_eval} -\begin{coq_example} -Require Import Euclid Wf_nat. -Extraction Inline gt_wf_rec lt_wf_rec induction_ltof2. -Recursive Extraction eucl_dev. -\end{coq_example} - -The inlining of {\tt gt\_wf\_rec} and others is not -mandatory. It only enhances readability of extracted code. -You can then copy-paste the output to a file {\tt euclid.ml} or let -\Coq\ do it for you with the following command: - -\begin{coq_example} -Extraction "euclid" eucl_dev. -\end{coq_example} - -Let us play the resulting program: - -\begin{verbatim} -# #use "euclid.ml";; -type nat = O | S of nat -type sumbool = Left | Right -val minus : nat -> nat -> nat = <fun> -val le_lt_dec : nat -> nat -> sumbool = <fun> -val le_gt_dec : nat -> nat -> sumbool = <fun> -type diveucl = Divex of nat * nat -val eucl_dev : nat -> nat -> diveucl = <fun> -# eucl_dev (S (S O)) (S (S (S (S (S O)))));; -- : diveucl = Divex (S (S O), S O) -\end{verbatim} -It is easier to test on \ocaml\ integers: -\begin{verbatim} -# let rec nat_of_int = function 0 -> O | n -> S (nat_of_int (n-1));; -val i2n : int -> nat = <fun> -# let rec int_of_nat = function O -> 0 | S p -> 1+(int_of_nat p);; -val n2i : nat -> int = <fun> -# let div a b = - let Divex (q,r) = eucl_dev (nat_of_int b) (nat_of_int a) - in (int_of_nat q, int_of_nat r);; -val div : int -> int -> int * int = <fun> -# div 173 15;; -- : int * int = (11, 8) -\end{verbatim} - -Note that these {\tt nat\_of\_int} and {\tt int\_of\_nat} are now -available via a mere {\tt Require Import ExtrOcamlIntConv} and then -adding these functions to the list of functions to extract. This file -{\tt ExtrOcamlIntConv.v} and some others in {\tt plugins/extraction/} -are meant to help building concrete program via extraction. - -\asubsection{Extraction's horror museum} - -Some pathological examples of extraction are grouped in the file -{\tt test-suite/success/extraction.v} of the sources of \Coq. - -\asubsection{Users' Contributions} - - Several of the \Coq\ Users' Contributions use extraction to produce - certified programs. In particular the following ones have an automatic - extraction test (just run {\tt make} in those directories): - - \begin{itemize} - \item Bordeaux/Additions - \item Bordeaux/EXCEPTIONS - \item Bordeaux/SearchTrees - \item Dyade/BDDS - \item Lannion - \item Lyon/CIRCUITS - \item Lyon/FIRING-SQUAD - \item Marseille/CIRCUITS - \item Muenchen/Higman - \item Nancy/FOUnify - \item Rocq/ARITH/Chinese - \item Rocq/COC - \item Rocq/GRAPHS - \item Rocq/HIGMAN - \item Sophia-Antipolis/Stalmarck - \item Suresnes/BDD - \end{itemize} - - Lannion, Rocq/HIGMAN and Lyon/CIRCUITS are a bit particular. They are - examples of developments where {\tt Obj.magic} are needed. - This is probably due to an heavy use of impredicativity. - After compilation those two examples run nonetheless, - thanks to the correction of the extraction~\cite{Let02}. - -% $Id: Extraction.tex 14575 2011-10-18 15:49:01Z letouzey $ - -%%% Local Variables: -%%% mode: latex -%%% TeX-master: "Reference-Manual" -%%% End: diff --git a/doc/refman/Helm.tex b/doc/refman/Helm.tex deleted file mode 100644 index ed94dfc5..00000000 --- a/doc/refman/Helm.tex +++ /dev/null @@ -1,313 +0,0 @@ -\label{Helm} -\index{XML exportation} -\index{Proof rendering} - -This section describes the exportation of {\Coq} theories to XML that -has been contributed by Claudio Sacerdoti Coen. Currently, the main -applications are the rendering and searching tool -developed within the HELM\footnote{Hypertextual Electronic Library of -Mathematics} and MoWGLI\footnote{Mathematics on the Web, Get it by -Logic and Interfaces} projects mainly at the University of Bologna and -partly at INRIA-Sophia Antipolis. - -\subsection{Practical use of the XML exportation tool} - -The basic way to export the logical content of a file into XML format -is to use {\tt coqc} with option {\tt -xml}. -When the {\tt -xml} flag is set, every definition or declaration is -immediately exported to XML once concluded. -The system environment variable {\tt COQ\_XML\_LIBRARY\_ROOT} must be -previously set to a directory in which the logical structure of the -exported objects is reflected. - - For {\tt Makefile} files generated by \verb+coq_makefile+ (see section - \ref{Makefile}), it is sufficient to compile the files using - \begin{quotation} - \verb+make COQ_XML=-xml+ - \end{quotation} - - To export a development to XML, the suggested procedure is then: - - \begin{enumerate} - \item add to your own contribution a valid \verb+Make+ file and use - \verb+coq_makefile+ to generate the \verb+Makefile+ from the \verb+Make+ - file. - - \Warning Since logical names are used to structure the XML - hierarchy, always add to the \verb+Make+ file at least one \verb+"-R"+ - option to map physical file names to logical module paths. - \item set the \verb+COQ_XML_LIBRARY_ROOT+ environment variable to - the directory where the XML file hierarchy must be physically - rooted. - \item compile your contribution with \verb+"make COQ_XML=-xml"+ - \end{enumerate} - -\Rem In case the system variable {\tt COQ\_XML\_LIBRARY\_ROOT} is not set, -the output is done on the standard output. Also, the files are -compressed using {\tt gzip} after creation. This is to save disk space -since the XML format is very verbose. - -\subsection{Reflection of the logical structure into the file system} - -For each {\Coq} logical object, several independent files associated -to this object are created. The structure of the long name of the -object is reflected in the directory structure of the file system. -E.g. an object of long name {\tt -{\ident$_1$}.{\ldots}.{\ident$_n$}.{\ident}} is exported to files in the -subdirectory {{\ident$_1$}/{\ldots}/{\ident$_n$}} of the directory -bound to the environment variable {\tt COQ\_XML\_LIBRARY\_ROOT}. - -\subsection{What is exported?} - -The XML exportation tool exports the logical content of {\Coq} -theories. This covers global definitions (including lemmas, theorems, -...), global assumptions (parameters and axioms), local assumptions or -definitions, and inductive definitions. - -Vernacular files are exported to {\tt .theory.xml} files. -%Variables, -%definitions, theorems, axioms and proofs are exported to individual -%files whose suffixes range from {\tt .var.xml}, {\tt .con.xml}, {\tt -%.con.body.xml}, {\tt .con.types.xml} to {\tt .con.proof_tree.xml}. -Comments are pre-processed with {\sf coqdoc} (see section -\ref{coqdoc}). Especially, they have to be enclosed within {\tt (**} -and {\tt *)} to be exported. - -For each inductive definition of name -{\ident$_1$}.{\ldots}.{\ident$_n$}.{\ident}, a file named {\tt -{\ident}.ind.xml} is created in the subdirectory {\tt -{\ident$_1$}/{\ldots}/{\ident$_n$}} of the xml library root -directory. It contains the arities and constructors of the type. For mutual inductive definitions, the file is named after the -name of the first inductive type of the block. - -For each global definition of base name {\tt -{\ident$_1$}.{\ldots}.{\ident$_n$}.{\ident}}, files named -{\tt {\ident}.con.body.xml} and {\tt {\ident}.con.xml} are created in the -subdirectory {\tt {\ident$_1$}/{\ldots}/{\ident$_n$}}. They -respectively contain the body and the type of the definition. - -For each global assumption of base name {\tt -{\ident$_1$}.{\ident$_2$}.{\ldots}.{\ident$_n$}.{\ident}}, a file -named {\tt {\ident}.con.xml} is created in the subdirectory {\tt -{\ident$_1$}/{\ldots}/{\ident$_n$}}. It contains the type of the -global assumption. - -For each local assumption or definition of base name {\ident} located -in sections {\ident$'_1$}, {\ldots}, {\ident$'_p$} of the module {\tt -{\ident$_1$}.{\ident$_2$}.{\ldots}.{\ident$_n$}.{\ident}}, a file -named {\tt {\ident}.var.xml} is created in the subdirectory {\tt -{\ident$_1$}/{\ldots}/{\ident$_n$}/{\ident$'_1$}/\ldots/{\ident$'_p$}}. -It contains its type and, if a definition, its body. - -In order to do proof-rendering (for example in natural language), some -redundant typing information is required, i.e. the type of at least -some of the subterms of the bodies and types of the CIC objects. These -types are called inner types and are exported to files of suffix {\tt -.types.xml} by the exportation tool. - - -% Deactivated -%% \subsection{Proof trees} - -%% For each definition or theorem that has been built with tactics, an -%% extra file of suffix {\tt proof\_tree.xml} is created. It contains the -%% proof scripts and is used for rendering the proof. - -\subsection[Inner types]{Inner types\label{inner-types}} - -The type of a subterm of a construction is called an {\em inner type} -if it respects the following conditions. - -\begin{enumerate} - \item Its sort is \verb+Prop+\footnote{or {\tt CProp} which is the - "sort"-like definition used in C-CoRN (see - \url{http://vacuumcleaner.cs.kun.nl/c-corn}) to type - computationally relevant predicative propositions.}. - \item It is not a type cast nor an atomic term (variable, constructor or constant). - \item If it's root is an abstraction, then the root's parent node is - not an abstraction, i.e. only the type of the outer abstraction of - a block of nested abstractions is printed. -\end{enumerate} - -The rationale for the 3$^{rd}$ condition is that the type of the inner -abstractions could be easily computed starting from the type of the -outer ones; moreover, the types of the inner abstractions requires a -lot of disk/memory space: removing the 3$^{rd}$ condition leads to XML -file that are two times as big as the ones exported applying the 3$^{rd}$ -condition. - -\subsection{Interactive exportation commands} - -There are also commands to be used interactively in {\tt coqtop}. - -\subsubsection[\tt Print XML {\qualid}]{\tt Print XML {\qualid}\comindex{Print XML}} - -If the variable {\tt COQ\_XML\_LIBRARY\_ROOT} is set, this command creates -files containing the logical content in XML format of {\qualid}. If -the variable is not set, the result is displayed on the standard -output. - -\begin{Variants} -\item {\tt Print XML File {\str} {\qualid}}\\ -This writes the logical content of {\qualid} in XML format to files -whose prefix is {\str}. -\end{Variants} - -\subsubsection[{\tt Show XML Proof}]{{\tt Show XML Proof}\comindex{Show XML Proof}} - -If the variable {\tt COQ\_XML\_LIBRARY\_ROOT} is set, this command creates -files containing the current proof in progress in XML format. It -writes also an XML file made of inner types. If the variable is not -set, the result is displayed on the standard output. - -\begin{Variants} -\item {\tt Show XML File {\str} Proof}\\ This writes the -logical content of {\qualid} in XML format to files whose prefix is -{\str}. -\end{Variants} - -\subsection{Applications: rendering, searching and publishing} - -The HELM team at the University of Bologna has developed tools -exploiting the XML exportation of {\Coq} libraries. This covers -rendering, searching and publishing tools. - -All these tools require a running http server and, if possible, a -MathML compliant browser. The procedure to install the suite of tools -ultimately allowing rendering and searching can be found on the HELM -web site \url{http://helm.cs.unibo.it/library.html}. - -It may be easier though to upload your developments on the HELM http -server and to re-use the infrastructure running on it. This requires -publishing your development. To this aim, follow the instructions on -\url{http://mowgli.cs.unibo.it}. - -Notice that the HELM server already hosts a copy of the standard -library of {\Coq} and of the {\Coq} user contributions. - -\subsection{Technical informations} - -\subsubsection{CIC with Explicit Named Substitutions} - -The exported files are XML encoding of the lambda-terms used by the -\Coq\ system. The implementative details of the \Coq\ system are hidden as much -as possible, so that the XML DTD is a straightforward encoding of the -Calculus of (Co)Inductive Constructions. - -Nevertheless, there is a feature of the \Coq\ system that can not be -hidden in a completely satisfactory way: discharging (see Sect.\ref{Section}). -In \Coq\ it is possible -to open a section, declare variables and use them in the rest of the section -as if they were axiom declarations. Once the section is closed, every definition and theorem in the section is discharged by abstracting it over the section -variables. Variable declarations as well as section declarations are entirely -dropped. Since we are interested in an XML encoding of definitions and -theorems as close as possible to those directly provided the user, we -do not want to export discharged forms. Exporting non-discharged theorem -and definitions together with theorems that rely on the discharged forms -obliges the tools that work on the XML encoding to implement discharging to -achieve logical consistency. Moreover, the rendering of the files can be -misleading, since hyperlinks can be shown between occurrences of the discharge -form of a definition and the non-discharged definition, that are different -objects. - -To overcome the previous limitations, Claudio Sacerdoti Coen developed in his -PhD. thesis an extension of CIC, called Calculus of (Co)Inductive Constructions -with Explicit Named Substitutions, that is a slight extension of CIC where -discharging is not necessary. The DTD of the exported XML files describes -constants, inductive types and variables of the Calculus of (Co)Inductive -Constructions with Explicit Named Substitutions. The conversion to the new -calculus is performed during the exportation phase. - -The following example shows a very small \Coq\ development together with its -version in CIC with Explicit Named Substitutions. - -\begin{verbatim} -# CIC version: # -Section S. - Variable A : Prop. - - Definition impl := A -> A. - - Theorem t : impl. (* uses the undischarged form of impl *) - Proof. - exact (fun (a:A) => a). - Qed. - -End S. - -Theorem t' : (impl False). (* uses the discharged form of impl *) - Proof. - exact (t False). (* uses the discharged form of t *) - Qed. -\end{verbatim} - -\begin{verbatim} -# Corresponding CIC with Explicit Named Substitutions version: # -Section S. - Variable A : Prop. - - Definition impl(A) := A -> A. (* theorems and definitions are - explicitly abstracted over the - variables. The name is sufficient to - completely describe the abstraction *) - - Theorem t(A) : impl. (* impl where A is not instantiated *) - Proof. - exact (fun (a:A) => a). - Qed. - -End S. - -Theorem t'() : impl{False/A}. (* impl where A is instantiated with False - Notice that t' does not depend on A *) - Proof. - exact t{False/A}. (* t where A is instantiated with False *) - Qed. -\end{verbatim} - -Further details on the typing and reduction rules of the calculus can be -found in Claudio Sacerdoti Coen PhD. dissertation, where the consistency -of the calculus is also proved. - -\subsubsection{The CIC with Explicit Named Substitutions XML DTD} - -A copy of the DTD can be found in the file ``\verb+cic.dtd+'' in the -\verb+plugins/xml+ source directory of \Coq. -The following is a very brief overview of the elements described in the DTD. - -\begin{description} - \item[]\texttt{<ConstantType>} - is the root element of the files that correspond to constant types. - \item[]\texttt{<ConstantBody>} - is the root element of the files that correspond to constant bodies. - It is used only for closed definitions and theorems (i.e. when no - metavariable occurs in the body or type of the constant) - \item[]\texttt{<CurrentProof>} - is the root element of the file that correspond to the body of a constant - that depends on metavariables (e.g. unfinished proofs) - \item[]\texttt{<Variable>} - is the root element of the files that correspond to variables - \item[]\texttt{<InductiveTypes>} - is the root element of the files that correspond to blocks - of mutually defined inductive definitions -\end{description} - -The elements - \verb+<LAMBDA>+, \verb+<CAST>+, \verb+<PROD>+, \verb+<REL>+, \verb+<SORT>+, - \verb+<APPLY>+, \verb+<VAR>+, \verb+<META>+, \verb+<IMPLICIT>+, \verb+<CONST>+, \verb+<LETIN>+, \verb+<MUTIND>+, \verb+<MUTCONSTRUCT>+, \verb+<MUTCASE>+, - \verb+<FIX>+ and \verb+<COFIX>+ are used to encode the constructors of CIC. - The \verb+sort+ or \verb+type+ attribute of the element, if present, is - respectively the sort or the type of the term, that is a sort because of the - typing rules of CIC. - -The element \verb+<instantiate>+ correspond to the application of an explicit -named substitution to its first argument, that is a reference to a definition -or declaration in the environment. - -All the other elements are just syntactic sugar. - - -%%% Local Variables: -%%% mode: latex -%%% TeX-master: "Reference-Manual" -%%% End: diff --git a/doc/refman/Micromega.tex b/doc/refman/Micromega.tex deleted file mode 100644 index 2fe7c2f7..00000000 --- a/doc/refman/Micromega.tex +++ /dev/null @@ -1,198 +0,0 @@ -\achapter{Micromega : tactics for solving arithmetics goals over ordered rings} -\aauthor{Frédéric Besson and Evgeny Makarov} -\newtheorem{theorem}{Theorem} - -For using the tactics out-of-the-box, read Section~\ref{sec:psatz-hurry}. -% -Section~\ref{sec:psatz-back} presents some background explaining the proof principle for solving polynomials goals. -% -Section~\ref{sec:lia} explains how to get a complete procedure for linear integer arithmetic. - -\asection{The {\tt psatz} tactic in a hurry} -\tacindex{psatz} -\label{sec:psatz-hurry} -Load the {\tt Psatz} module ({\tt Require Psatz}.). This module defines the tactics: -{\tt lia}, {\tt psatzl D}, %{\tt sos D} -and {\tt psatz D n} where {\tt D} is {\tt Z}, {\tt Q} or {\tt R} and {\tt n} is an optional integer limiting the proof search depth. - % - \begin{itemize} - \item The {\tt psatzl} tactic solves linear goals using an embedded (naive) linear programming prover \emph{i.e.}, - fourier elimination. - \item The {\tt psatz} tactic solves polynomial goals using John Harrison's Hol light driver to the external prover {\tt cspd}\footnote{Sources and binaries can be found at \url{https://projects.coin-or.org/Csdp}}. Note that the {\tt csdp} driver is generating - a \emph{proof cache} thus allowing to rerun scripts even without {\tt csdp}. - \item The {\tt lia} (linear integer arithmetic) tactic is specialised to solve linear goals over $\mathbb{Z}$. - It extends {\tt psatzl Z} and exploits the discreetness of $\mathbb{Z}$. -%% \item The {\tt sos} tactic is another Hol light driver to the {\tt csdp} prover. In theory, it is less general than -%% {\tt psatz}. In practice, even when {\tt psatz} fails, it can be worth a try -- see -%% Section~\ref{sec:psatz-back} for details. - \end{itemize} - -These tactics solve propositional formulas parameterised by atomic arithmetics expressions -interpreted over a domain $D \in \{\mathbb{Z}, \mathbb{Q}, \mathbb{R} \}$. -The syntax of the formulas is the following: -\[ -\begin{array}{lcl} - F &::=& A \mid P \mid \mathit{True} \mid \mathit{False} \mid F_1 \land F_2 \mid F_1 \lor F_2 \mid F_1 \leftrightarrow F_2 \mid F_1 \to F_2 \mid \sim F\\ - A &::=& p_1 = p_2 \mid p_1 > p_2 \mid p_1 < p_2 \mid p_1 \ge p_2 \mid p_1 \le p_2 \\ - p &::=& c \mid x \mid {-}p \mid p_1 - p_2 \mid p_1 + p_2 \mid p_1 \times p_2 \mid p \verb!^! n - \end{array} - \] - where $c$ is a numeric constant, $x\in D$ is a numeric variable and the operators $-$, $+$, $\times$, are - respectively subtraction, addition, product, $p \verb!^!n $ is exponentiation by a constant $n$, $P$ is an - arbitrary proposition. %that is mostly ignored. -%% -%% Over $\mathbb{Z}$, $c$ is an integer ($c \in \mathtt{Z}$), over $\mathbb{Q}$, $c$ is -The following table details for each domain $D \in \{\mathbb{Z},\mathbb{Q},\mathbb{R}\}$ the range of constants $c$ and exponent $n$. -\[ -\begin{array}{|c|c|c|c|} - \hline - &\mathbb{Z} & \mathbb{Q} & \mathbb{R} \\ - \hline - c &\mathtt{Z} & \mathtt{Q} & \{R1, R0\} \\ - \hline - n &\mathtt{Z} & \mathtt{Z} & \mathtt{nat}\\ - \hline -\end{array} -\] - -\asection{\emph{Positivstellensatz} refutations} -\label{sec:psatz-back} - -The name {\tt psatz} is an abbreviation for \emph{positivstellensatz} -- literally positivity theorem -- which -generalises Hilbert's \emph{nullstellensatz}. -% -It relies on the notion of $\mathit{Cone}$. Given a (finite) set of polynomials $S$, $Cone(S)$ is -inductively defined as the smallest set of polynomials closed under the following rules: -\[ -\begin{array}{l} -\dfrac{p \in S}{p \in Cone(S)} \quad -\dfrac{}{p^2 \in Cone(S)} \quad -\dfrac{p_1 \in Cone(S) \quad p_2 \in Cone(S) \quad \Join \in \{+,*\}} {p_1 \Join p_2 \in Cone(S)}\\ -\end{array} -\] -The following theorem provides a proof principle for checking that a set of polynomial inequalities do not have solutions\footnote{Variants deal with equalities and strict inequalities.}: -\begin{theorem} - \label{thm:psatz} - Let $S$ be a set of polynomials.\\ - If ${-}1$ belongs to $Cone(S)$ then the conjunction $\bigwedge_{p \in S} p\ge 0$ is unsatisfiable. -\end{theorem} -A proof based on this theorem is called a \emph{positivstellensatz} refutation. -% -The tactics work as follows. Formulas are normalised into conjonctive normal form $\bigwedge_i C_i$ where -$C_i$ has the general form $(\bigwedge_{j\in S_i} p_j \Join 0) \to \mathit{False})$ and $\Join \in \{>,\ge,=\}$ for $D\in -\{\mathbb{Q},\mathbb{R}\}$ and $\Join \in \{\ge, =\}$ for $\mathbb{Z}$. -% -For each conjunct $C_i$, the tactic calls a oracle which searches for $-1$ within the cone. -% -Upon success, the oracle returns a \emph{cone expression} that is normalised by the {\tt ring} tactic (see chapter~\ref{ring}) and checked to be -$-1$. - -To illustrate the working of the tactic, consider we wish to prove the following Coq goal.\\ -\begin{coq_eval} - Require Import ZArith Psatz. - Open Scope Z_scope. -\end{coq_eval} -\begin{coq_example*} - Goal forall x, -x^2 >= 0 -> x - 1 >= 0 -> False. -\end{coq_example*} -\begin{coq_eval} -intro x; psatz Z 2. -\end{coq_eval} -Such a goal is solved by {\tt intro x; psatz Z 2}. The oracle returns the cone expression $2 \times -(\mathbf{x-1}) + \mathbf{x-1}\times\mathbf{x-1} + \mathbf{-x^2}$ (polynomial hypotheses are printed in bold). By construction, this -expression belongs to $Cone(\{-x^2, x -1\})$. Moreover, by running {\tt ring} we obtain $-1$. By -Theorem~\ref{thm:psatz}, the goal is valid. -% - -\paragraph{The {\tt psatzl} tactic} is searching for \emph{linear} refutations using a fourier -elimination\footnote{More efficient linear programming techniques could equally be employed}. -As a result, this tactic explore a subset of the $Cone$ defined as: -\[ -LinCone(S) =\left\{ \left. \sum_{p \in S} \alpha_p \times p\ \right|\ \alpha_p \mbox{ are positive constants} \right\} -\] -Basically, the deductive power of {\tt psatzl} is the combined deductive power of {\tt ring\_simplify} and {\tt fourier}. - -\paragraph{The {\tt psatz} tactic} explores the $Cone$ by increasing degrees -- hence the depth parameter $n$. -In theory, such a proof search is complete -- if the goal is provable the search eventually stops. -Unfortunately, the external oracle is using numeric (approximate) optimisation techniques that might miss a -refutation. - -%% \paragraph{The {\tt sos} tactic} -- where {\tt sos} stands for \emph{sum of squares} -- tries to prove that a -%% single polynomial $p$ is positive by expressing it as a sum of squares \emph{i.e.,} $\sum_{i\in S} p_i^2$. -%% This amounts to searching for $p$ in the cone without generators \emph{i.e.}, $Cone(\{\})$. -% - -\asection{ {\tt lia} : the linear integer arithmetic tactic } -\tacindex{lia} -\label{sec:lia} - -The tactic {\tt lia} offers an alternative to the {\tt omega} and {\tt romega} tactic (see -Chapter~\ref{OmegaChapter}). It solves goals that {\tt omega} and {\tt romega} do not solve, such as the -following so-called \emph{omega nightmare}~\cite{TheOmegaPaper}. -\begin{coq_example*} - Goal forall x y, - 27 <= 11 * x + 13 * y <= 45 -> - -10 <= 7 * x - 9 * y <= 4 -> False. -\end{coq_example*} -\begin{coq_eval} -intro x; lia; -\end{coq_eval} -The estimation of the relative efficiency of lia \emph{vs} {\tt omega} -and {\tt romega} is under evaluation. - -\paragraph{High level view of {\tt lia}.} -Over $\mathbb{R}$, \emph{positivstellensatz} refutations are a complete proof principle\footnote{In practice, the oracle might fail to produce such a refutation.}. -% -However, this is not the case over $\mathbb{Z}$. -% -Actually, \emph{positivstellensatz} refutations are not even sufficient to decide linear \emph{integer} -arithmetics. -% -The canonical exemple is {\tt 2 * x = 1 -> False} which is a theorem of $\mathbb{Z}$ but not a theorem of $\mathbb{R}$. -% -To remedy this weakness, the {\tt lia} tactic is using recursively a combination of: -% -\begin{itemize} -\item linear \emph{positivstellensatz} refutations \emph{i.e.}, {\tt psatzl Z}; -\item cutting plane proofs; -\item case split. -\end{itemize} - -\paragraph{Cutting plane proofs} are a way to take into account the discreetness of $\mathbb{Z}$ by rounding up -(rational) constants up-to the closest integer. -% -\begin{theorem} - Let $p$ be an integer and $c$ a rational constant. - \[ - p \ge c \Rightarrow p \ge \lceil c \rceil - \] -\end{theorem} -For instance, from $2 * x = 1$ we can deduce -\begin{itemize} -\item $x \ge 1/2$ which cut plane is $ x \ge \lceil 1/2 \rceil = 1$; -\item $ x \le 1/2$ which cut plane is $ x \le \lfloor 1/2 \rfloor = 0$. -\end{itemize} -By combining these two facts (in normal form) $x - 1 \ge 0$ and $-x \ge 0$, we conclude by exhibiting a -\emph{positivstellensatz} refutation ($-1 \equiv \mathbf{x-1} + \mathbf{-x} \in Cone(\{x-1,x\})$). - -Cutting plane proofs and linear \emph{positivstellensatz} refutations are a complete proof principle for integer linear arithmetic. - -\paragraph{Case split} allow to enumerate over the possible values of an expression. -\begin{theorem} - Let $p$ be an integer and $c_1$ and $c_2$ integer constants. - \[ - c_1 \le p \le c_2 \Rightarrow \bigvee_{x \in [c_1,c_2]} p = x - \] -\end{theorem} -Our current oracle tries to find an expression $e$ with a small range $[c_1,c_2]$. -% -We generate $c_2 - c_1$ subgoals which contexts are enriched with an equation $e = i$ for $i \in [c_1,c_2]$ and -recursively search for a proof. - -% This technique is used to solve so-called \emph{Omega nightmare} - - -%%% Local Variables: -%%% mode: latex -%%% TeX-master: "Reference-Manual" -%%% End: diff --git a/doc/refman/Natural.tex b/doc/refman/Natural.tex deleted file mode 100644 index 9a9abe5d..00000000 --- a/doc/refman/Natural.tex +++ /dev/null @@ -1,425 +0,0 @@ -\achapter{\texttt{Natural} : proofs in natural language} -\aauthor{Yann Coscoy} - -\asection{Introduction} - -\Natural~ is a package allowing the writing of proofs in natural -language. For instance, the proof in \Coq~of the induction principle on pairs -of natural numbers looks like this: - -\begin{coq_example*} -Require Natural. -\end{coq_example*} -\begin{coq_example} -Print nat_double_ind. -\end{coq_example} - -Piping it through the \Natural~pretty-printer gives: - -\comindex{Print Natural} -\begin{coq_example} -Print Natural nat_double_ind. -\end{coq_example} - -\asection{Activating \Natural} - -To enable the printing of proofs in natural language, you should -type under \texttt{coqtop} or \texttt{coqtop -full} the command - -\begin{coq_example*} -Require Natural. -\end{coq_example*} - -By default, proofs are transcripted in english. If you wish to print them -in French, set the French option by - -\comindex{Set Natural} -\begin{coq_example*} -Set Natural French. -\end{coq_example*} - -If you want to go back to English, type in - -\begin{coq_example*} -Set Natural English. -\end{coq_example*} - -Currently, only \verb=French= and \verb=English= are available. - -You may see for example the natural transcription of the proof of -the induction principle on pairs of natural numbers: - -\begin{coq_example*} -Print Natural nat_double_ind. -\end{coq_example*} - -You may also show in natural language the current proof in progress: - -\comindex{Show Natural} -\begin{coq_example} -Goal (n:nat)(le O n). -Induction n. -Show Natural Proof. -\end{coq_example} - -\subsection*{Restrictions} - -For \Natural, a proof is an object of type a proposition (i.e. an -object of type something of type {\tt Prop}). Only proofs are written -in natural language when typing {\tt Print Natural \ident}. All other -objects (the objects of type something which is of type {\tt Set} or -{\tt Type}) are written as usual $\lambda$-terms. - -\asection{Customizing \Natural} - -The transcription of proofs in natural language is mainly a paraphrase of -the formal proofs, but some specific hints in the transcription -can be given. -Three kinds of customization are available. - -\asubsection{Implicit proof steps} - -\subsubsection*{Implicit lemmas} - -Applying a given lemma or theorem \verb=lem1= of statement, say $A -\Rightarrow B$, to an hypothesis, say $H$ (assuming $A$) produces the -following kind of output translation: - -\begin{verbatim} -... -Using lem1 with H we get B. -... -\end{verbatim} - -But sometimes, you may prefer not to see the explicit invocation to -the lemma. You may prefer to see: - -\begin{verbatim} -... -With H we have A. -... -\end{verbatim} - -This is possible by declaring the lemma as implicit. You should type: - -\comindex{Add Natural} -\begin{coq_example*} -Add Natural Implicit lem1. -\end{coq_example*} - -By default, the lemmas \verb=proj1=, \verb=proj2=, \verb=sym_equal= -and \verb=sym_eqT= are declared implicit. To remove a lemma or a theorem -previously declared as implicit, say \verb=lem1=, use the command - -\comindex{Remove Natural} -\begin{coq_example*} -Remove Natural Implicit lem1. -\end{coq_example*} - -To test if the lemma or theorem \verb=lem1= is, or is not, -declared as implicit, type - -\comindex{Test Natural} -\begin{coq_example*} -Test Natural Implicit for lem1. -\end{coq_example*} - -\subsubsection*{Implicit proof constructors} - -Let \verb=constr1= be a proof constructor of a given inductive -proposition (or predicate) -\verb=Q= (of type \verb=Prop=). Assume \verb=constr1= proves -\verb=(x:A)(P x)->(Q x)=. Then, applying \verb=constr1= to an hypothesis, -say \verb=H= (assuming \verb=(P a)=) produces the following kind of output: - -\begin{verbatim} -... -By the definition of Q, with H we have (Q a). -... -\end{verbatim} - -But sometimes, you may prefer not to see the explicit invocation to -this constructor. You may prefer to see: - -\begin{verbatim} -... -With H we have (Q a). -... -\end{verbatim} - -This is possible by declaring the constructor as implicit. You should -type, as before: - -\comindex{Add Natural Implicit} -\begin{coq_example*} -Add Natural Implicit constr1. -\end{coq_example*} - -By default, the proposition (or predicate) constructors - -\verb=conj=, \verb=or_introl=, \verb=or_intror=, \verb=ex_intro=, -\verb=exT_intro=, \verb=refl_equal=, \verb=refl_eqT= and \verb=exist= - -\noindent are declared implicit. Note that declaring implicit the -constructor of a datatype (i.e. an inductive type of type \verb=Set=) -has no effect. - -As above, you can remove or test a constant declared implicit. - -\subsubsection*{Implicit inductive constants} - -Let \verb=Ind= be an inductive type (either a proposition (or a -predicate) -- on \verb=Prop= --, or a datatype -- on \verb=Set=). -Suppose the proof proceeds by induction on an hypothesis \verb=h= -proving \verb=Ind= (or more generally \verb=(Ind A1 ... An)=). The -following kind of output is produced: - -\begin{verbatim} -... -With H, we will prove A by induction on the definition of Ind. -Case 1. ... -Case 2. ... -... -\end{verbatim} - -But sometimes, you may prefer not to see the explicit invocation to -\verb=Ind=. You may prefer to see: - -\begin{verbatim} -... -We will prove A by induction on H. -Case 1. ... -Case 2. ... -... -\end{verbatim} - -This is possible by declaring the inductive type as implicit. You should -type, as before: - -\comindex{Add Natural Implicit} -\begin{coq_example*} -Add Natural Implicit Ind. -\end{coq_example*} - -This kind of parameterization works for any inductively defined -proposition (or predicate) or datatype. Especially, it works whatever -the definition is recursive or purely by cases. - -By default, the data type \verb=nat= and the inductive connectives -\verb=and=, \verb=or=, \verb=sig=, \verb=False=, \verb=eq=, -\verb=eqT=, \verb=ex= and \verb=exT= are declared implicit. - -As above, you can remove or test a constant declared implicit. Use -{\tt Remove Natural Contractible $id$} or {\tt Test Natural -Contractible for $id$}. - -\asubsection{Contractible proof steps} - -\subsubsection*{Contractible lemmas or constructors} - -Some lemmas, theorems or proof constructors of inductive predicates are -often applied in a row and you obtain an output of this kind: - -\begin{verbatim} -... -Using T with H1 and H2 we get P. - * By H3 we have Q. - Using T with theses results we get R. -... -\end{verbatim} - -where \verb=T=, \verb=H1=, \verb=H2= and \verb=H3= prove statements -of the form \verb=(X,Y:Prop)X->Y->(L X Y)=, \verb=A=, \verb=B= and \verb=C= -respectively (and thus \verb=R= is \verb=(L (L A B) C)=). - -You may obtain a condensed output of the form - -\begin{verbatim} -... -Using T with H1, H2, and H3 we get R. -... -\end{verbatim} - -by declaring \verb=T= as contractible: - -\comindex{Add Natural Contractible} -\begin{coq_example*} -Add Natural Contractible T. -\end{coq_example*} - -By default, the lemmas \verb=proj1=, \verb=proj2= and the proof -constructors \verb=conj=, \verb=or_introl=, \verb=or_intror= are -declared contractible. As for implicit notions, you can remove or -test a lemma or constructor declared contractible. - -\subsubsection*{Contractible induction steps} - -Let \verb=Ind= be an inductive type. When the proof proceeds by -induction in a row, you may obtain an output of this kind: - -\begin{verbatim} -... -We have (Ind A (Ind B C)). -We use definition of Ind in a study in two cases. -Case 1: We have A. -Case 2: We have (Ind B C). - We use definition of Ind in a study of two cases. - Case 2.1: We have B. - Case 2.2: We have C. -... -\end{verbatim} - -You may prefer to see - -\begin{verbatim} -... -We have (Ind A (Ind B C)). -We use definition of Ind in a study in three cases. -Case 1: We have A. -Case 2: We have B. -Case 3: We have C. -... -\end{verbatim} - -This is possible by declaring \verb=Ind= as contractible: - -\begin{coq_example*} -Add Natural Contractible T. -\end{coq_example*} - -By default, only \verb=or= is declared as a contractible inductive -constant. -As for implicit notions, you can remove or test an inductive notion declared -contractible. - -\asubsection{Transparent definitions} - -``Normal'' definitions are all constructions except proofs and proof constructors. - -\subsubsection*{Transparent non inductive normal definitions} - -When using the definition of a non inductive constant, say \verb=D=, the -following kind of output is produced: - -\begin{verbatim} -... -We have proved C which is equivalent to D. -... -\end{verbatim} - -But you may prefer to hide that D comes from the definition of C as -follows: - -\begin{verbatim} -... -We have prove D. -... -\end{verbatim} - -This is possible by declaring \verb=C= as transparent: - -\comindex{Add Natural Transparent} -\begin{coq_example*} -Add Natural Transparent D. -\end{coq_example*} - -By default, only \verb=not= (normally written \verb=~=) is declared as -a non inductive transparent definition. -As for implicit and contractible definitions, you can remove or test a -non inductive definition declared transparent. -Use \texttt{Remove Natural Transparent} \ident or -\texttt{Test Natural Transparent for} \ident. - -\subsubsection*{Transparent inductive definitions} - -Let \verb=Ind= be an inductive proposition (more generally: a -predicate \verb=(Ind x1 ... xn)=). Suppose the definition of -\verb=Ind= is non recursive and built with just -one constructor proving something like \verb=A -> B -> Ind=. -When coming back to the definition of \verb=Ind= the -following kind of output is produced: - -\begin{verbatim} -... -Assume Ind (H). - We use H with definition of Ind. - We have A and B. - ... -\end{verbatim} - -When \verb=H= is not used a second time in the proof, you may prefer -to hide that \verb=A= and \verb=B= comes from the definition of -\verb=Ind=. You may prefer to get directly: - -\begin{verbatim} -... -Assume A and B. -... -\end{verbatim} - -This is possible by declaring \verb=Ind= as transparent: - -\begin{coq_example*} -Add Natural Transparent Ind. -\end{coq_example*} - -By default, \verb=and=, \verb=or=, \verb=ex=, \verb=exT=, \verb=sig= -are declared as inductive transparent constants. As for implicit and -contractible constants, you can remove or test an inductive -constant declared transparent. - -As for implicit and contractible constants, you can remove or test an -inductive constant declared transparent. - -\asubsection{Extending the maximal depth of nested text} - -The depth of nested text is limited. To know the current depth, do: - -\comindex{Set Natural Depth} -\begin{coq_example} -Set Natural Depth. -\end{coq_example} - -To change the maximal depth of nested text (for instance to 125) do: - -\begin{coq_example} -Set Natural Depth 125. -\end{coq_example} - -\asubsection{Restoring the default parameterization} - -The command \verb=Set Natural Default= sets back the parameterization tables of -\Natural~ to their default values, as listed in the above sections. -Moreover, the language is set back to English and the max depth of -nested text is set back to its initial value. - -\asubsection{Printing the current parameterization} - -The commands {\tt Print Natural Implicit}, {\tt Print Natural -Contractible} and {\tt Print \\ Natural Transparent} print the list of -constructions declared {\tt Implicit}, {\tt Contractible}, -{\tt Transparent} respectively. - -\asubsection{Interferences with \texttt{Reset}} - -The customization of \texttt{Natural} is dependent of the \texttt{Reset} -command. If you reset the environment back to a point preceding an -\verb=Add Natural ...= command, the effect of the command will be -erased. Similarly, a reset back to a point before a -\verb=Remove Natural ... = command invalidates the removal. - -\asection{Error messages} - -An error occurs when trying to \verb=Print=, to \verb=Add=, to -\verb=Test=, or to \verb=remove= an undefined ident. Similarly, an -error occurs when trying to set a language unknown from \Natural. -Errors may also occur when trying to parameterize the printing of -proofs: some parameterization are effectively forbidden. -Note that to \verb=Remove= an ident absent from a table or to -\verb=Add= to a table an already present ident does not lead to an -error. - -%%% Local Variables: -%%% mode: latex -%%% TeX-master: "Reference-Manual" -%%% End: diff --git a/doc/refman/Nsatz.tex b/doc/refman/Nsatz.tex deleted file mode 100644 index 794e461f..00000000 --- a/doc/refman/Nsatz.tex +++ /dev/null @@ -1,110 +0,0 @@ -\achapter{Nsatz: tactics for proving equalities in integral domains} -\aauthor{Loïc Pottier} - -The tactic \texttt{nsatz} proves goals of the form - -\[ \begin{array}{l} - \forall X_1,\ldots,X_n \in A,\\ - P_1(X_1,\ldots,X_n) = Q_1(X_1,\ldots,X_n) , \ldots , P_s(X_1,\ldots,X_n) =Q_s(X_1,\ldots,X_n)\\ - \vdash P(X_1,\ldots,X_n) = Q(X_1,\ldots,X_n)\\ - \end{array} -\] -where $P,Q, P_1,Q_1,\ldots,P_s,Q_s$ are polynomials and A is an integral -domain, i.e. a commutative ring with no zero divisor. For example, A can be -$\mathbb{R}$, $\mathbb{Z}$, of $\mathbb{Q}$. Note that the equality $=$ used in these -goals can be any setoid equality -(see \ref{setoidtactics}) -, not only Leibnitz equality. - -It also proves formulas -\[ \begin{array}{l} - \forall X_1,\ldots,X_n \in A,\\ - P_1(X_1,\ldots,X_n) = Q_1(X_1,\ldots,X_n) \wedge \ldots \wedge P_s(X_1,\ldots,X_n) =Q_s(X_1,\ldots,X_n)\\ - \rightarrow P(X_1,\ldots,X_n) = Q(X_1,\ldots,X_n)\\ - \end{array} -\] doing automatic introductions. - -\asection{Using the basic tactic \texttt{nsatz}} -\tacindex{nsatz} - -Load the -\texttt{Nsatz} module: \texttt{Require Import Nsatz}.\\ - and use the tactic \texttt{nsatz}. - -\asection{More about \texttt{nsatz}} - -Hilbert's Nullstellensatz theorem shows how to reduce proofs of equalities on -polynomials on a commutative ring A with no zero divisor to algebraic computations: it is easy to see that if a polynomial -$P$ in $A[X_1,\ldots,X_n]$ verifies $c P^r = \sum_{i=1}^{s} S_i P_i$, with $c -\in A$, $c \not = 0$, $r$ a positive integer, and the $S_i$s in -$A[X_1,\ldots,X_n]$, then $P$ is zero whenever polynomials $P_1,...,P_s$ are -zero (the converse is also true when A is an algebraic closed field: -the method is complete). - -So, proving our initial problem can reduce into finding $S_1,\ldots,S_s$, $c$ -and $r$ such that $c (P-Q)^r = \sum_{i} S_i (P_i-Q_i)$, which will be proved by the -tactic \texttt{ring}. - -This is achieved by the computation of a Groebner basis of the -ideal generated by $P_1-Q_1,...,P_s-Q_s$, with an adapted version of the Buchberger -algorithm. - -This computation is done after a step of {\em reification}, which is -performed using {\em Type Classes} -(see \ref{typeclasses}) -. - -The \texttt{Nsatz} module defines the generic tactic -\texttt{nsatz}, which uses the low-level tactic \texttt{nsatz\_domainpv}: \\ -\vspace*{3mm} -\texttt{nsatz\_domainpv pretac rmax strategy lparam lvar simpltac domain} - -where: - -\begin{itemize} - \item \texttt{pretac} is a tactic depending on the ring A; its goal is to -make apparent the generic operations of a domain (ring\_eq, ring\_plus, etc), -both in the goal and the hypotheses; it is executed first. By default it is \texttt{ltac:idtac}. - - \item \texttt{rmax} is a bound when for searching r s.t.$c (P-Q)^r = -\sum_{i=1..s} S_i (P_i - Q_i)$ - - \item \texttt{strategy} gives the order on variables $X_1,...X_n$ and -the strategy used in Buchberger algorithm (see -\cite{sugar} for details): - - \begin{itemize} - \item strategy = 0: reverse lexicographic order and newest s-polynomial. - \item strategy = 1: reverse lexicographic order and sugar strategy. - \item strategy = 2: pure lexicographic order and newest s-polynomial. - \item strategy = 3: pure lexicographic order and sugar strategy. - \end{itemize} - - \item \texttt{lparam} is the list of variables -$X_{i_1},\ldots,X_{i_k}$ among $X_1,...,X_n$ which are considered as - parameters: computation will be performed with rational fractions in these - variables, i.e. polynomials are considered with coefficients in -$R(X_{i_1},\ldots,X_{i_k})$. In this case, the coefficient $c$ can be a non -constant polynomial in $X_{i_1},\ldots,X_{i_k}$, and the tactic produces a goal -which states that $c$ is not zero. - - \item \texttt{lvar} is the list of the variables -in the decreasing order in which they will be used in Buchberger algorithm. If \texttt{lvar} = {(@nil -R)}, then \texttt{lvar} is replaced by all the variables which are not in -lparam. - - \item \texttt{simpltac} is a tactic depending on the ring A; its goal is to -simplify goals and make apparent the generic operations of a domain after -simplifications. By default it is \texttt{ltac:simpl}. - - \item \texttt{domain} is the object of type Domain representing A, its -operations and properties of integral domain. - -\end{itemize} - -See file \texttt{Nsatz.v} for examples. - -%%% Local Variables: -%%% mode: latex -%%% TeX-master: "Reference-Manual" -%%% End: diff --git a/doc/refman/Omega.tex b/doc/refman/Omega.tex deleted file mode 100644 index b9e899ce..00000000 --- a/doc/refman/Omega.tex +++ /dev/null @@ -1,226 +0,0 @@ -\achapter{Omega: a solver of quantifier-free problems in -Presburger Arithmetic} -\aauthor{Pierre Crégut} -\label{OmegaChapter} - -\asection{Description of {\tt omega}} -\tacindex{omega} -\label{description} - -{\tt omega} solves a goal in Presburger arithmetic, i.e. a universally -quantified formula made of equations and inequations. Equations may -be specified either on the type \verb=nat= of natural numbers or on -the type \verb=Z= of binary-encoded integer numbers. Formulas on -\verb=nat= are automatically injected into \verb=Z=. The procedure -may use any hypothesis of the current proof session to solve the goal. - -Multiplication is handled by {\tt omega} but only goals where at -least one of the two multiplicands of products is a constant are -solvable. This is the restriction meant by ``Presburger arithmetic''. - -If the tactic cannot solve the goal, it fails with an error message. -In any case, the computation eventually stops. - -\asubsection{Arithmetical goals recognized by {\tt omega}} - -{\tt omega} applied only to quantifier-free formulas built from the -connectors - -\begin{quote} -\verb=/\, \/, ~, ->= -\end{quote} - -on atomic formulas. Atomic formulas are built from the predicates - -\begin{quote} -\verb!=, le, lt, gt, ge! -\end{quote} - - on \verb=nat= or from the predicates - -\begin{quote} -\verb!=, <, <=, >, >=! -\end{quote} - - on \verb=Z=. In expressions of type \verb=nat=, {\tt omega} recognizes - -\begin{quote} -\verb!plus, minus, mult, pred, S, O! -\end{quote} - -and in expressions of type \verb=Z=, {\tt omega} recognizes - -\begin{quote} -\verb!+, -, *, Zsucc!, and constants. -\end{quote} - -All expressions of type \verb=nat= or \verb=Z= not built on these -operators are considered abstractly as if they -were arbitrary variables of type \verb=nat= or \verb=Z=. - -\asubsection{Messages from {\tt omega}} -\label{errors} - -When {\tt omega} does not solve the goal, one of the following errors -is generated: - -\begin{ErrMsgs} - -\item \errindex{omega can't solve this system} - - This may happen if your goal is not quantifier-free (if it is - universally quantified, try {\tt intros} first; if it contains - existentials quantifiers too, {\tt omega} is not strong enough to solve your - goal). This may happen also if your goal contains arithmetical - operators unknown from {\tt omega}. Finally, your goal may be really - wrong! - -\item \errindex{omega: Not a quantifier-free goal} - - If your goal is universally quantified, you should first apply {\tt - intro} as many time as needed. - -\item \errindex{omega: Unrecognized predicate or connective: {\sl ident}} - -\item \errindex{omega: Unrecognized atomic proposition: {\sl prop}} - -\item \errindex{omega: Can't solve a goal with proposition variables} - -\item \errindex{omega: Unrecognized proposition} - -\item \errindex{omega: Can't solve a goal with non-linear products} - -\item \errindex{omega: Can't solve a goal with equality on {\sl type}} - -\end{ErrMsgs} - -%% Ce code est débranché pour l'instant -%% -% \asubsection{Control over the output} -% There are some flags that can be set to get more information on the procedure - -% \begin{itemize} -% \item \verb=Time= to get the time used by the procedure -% \item \verb=System= to visualize the normalized systems. -% \item \verb=Action= to visualize the actions performed by the OMEGA -% procedure (see \ref{technical}). -% \end{itemize} - -% \comindex{Set omega Time} -% \comindex{UnSet omega Time} -% \comindex{Switch omega Time} -% \comindex{Set omega System} -% \comindex{UnSet omega System} -% \comindex{Switch omega System} -% \comindex{Set omega Action} -% \comindex{UnSet omega Action} -% \comindex{Switch omega Action} - -% Use {\tt Set omega {\rm\sl flag}} to set the flag -% {\rm\sl flag}. Use {\tt Unset omega {\rm\sl flag}} to unset it and -% {\tt Switch omega {\rm\sl flag}} to toggle it. - -\section{Using {\tt omega}} - -The {\tt omega} tactic does not belong to the core system. It should be -loaded by -\begin{coq_example*} -Require Import Omega. -Open Scope Z_scope. -\end{coq_example*} - -\example{} - -\begin{coq_example} -Goal forall m n:Z, 1 + 2 * m <> 2 * n. -intros; omega. -\end{coq_example} -\begin{coq_eval} -Abort. -\end{coq_eval} - -\example{} - -\begin{coq_example} -Goal forall z:Z, z > 0 -> 2 * z + 1 > z. -intro; omega. -\end{coq_example} - -% Other examples can be found in \verb+$COQLIB/theories/DEMOS/OMEGA+. - -\asection{Technical data} -\label{technical} - -\asubsection{Overview of the tactic} -\begin{itemize} - -\item The goal is negated twice and the first negation is introduced as an - hypothesis. -\item Hypothesis are decomposed in simple equations or inequations. Multiple - goals may result from this phase. -\item Equations and inequations over \verb=nat= are translated over - \verb=Z=, multiple goals may result from the translation of - substraction. -\item Equations and inequations are normalized. -\item Goals are solved by the {\it OMEGA} decision procedure. -\item The script of the solution is replayed. - -\end{itemize} - -\asubsection{Overview of the {\it OMEGA} decision procedure} - -The {\it OMEGA} decision procedure involved in the {\tt omega} tactic uses -a small subset of the decision procedure presented in - -\begin{quote} - "The Omega Test: a fast and practical integer programming -algorithm for dependence analysis", William Pugh, Communication of the -ACM , 1992, p 102-114. -\end{quote} - -Here is an overview, look at the original paper for more information. - -\begin{itemize} - -\item Equations and inequations are normalized by division by the GCD of their - coefficients. -\item Equations are eliminated, using the Banerjee test to get a coefficient - equal to one. -\item Note that each inequation defines a half space in the space of real value - of the variables. - \item Inequations are solved by projecting on the hyperspace - defined by cancelling one of the variable. They are partitioned - according to the sign of the coefficient of the eliminated - variable. Pairs of inequations from different classes define a - new edge in the projection. - \item Redundant inequations are eliminated or merged in new - equations that can be eliminated by the Banerjee test. -\item The last two steps are iterated until a contradiction is reached - (success) or there is no more variable to eliminate (failure). - -\end{itemize} - -It may happen that there is a real solution and no integer one. The last -steps of the Omega procedure (dark shadow) are not implemented, so the -decision procedure is only partial. - -\asection{Bugs} - -\begin{itemize} -\item The simplification procedure is very dumb and this results in - many redundant cases to explore. - -\item Much too slow. - -\item Certainly other bugs! You can report them to - -\begin{quote} - \url{Pierre.Cregut@cnet.francetelecom.fr} -\end{quote} - -\end{itemize} - -%%% Local Variables: -%%% mode: latex -%%% TeX-master: "Reference-Manual" -%%% End: diff --git a/doc/refman/Polynom.tex b/doc/refman/Polynom.tex deleted file mode 100644 index 3898bf4c..00000000 --- a/doc/refman/Polynom.tex +++ /dev/null @@ -1,1000 +0,0 @@ -\achapter{The \texttt{ring} and \texttt{field} tactic families} -\aauthor{Bruno Barras, Benjamin Gr\'egoire, Assia - Mahboubi, Laurent Th\'ery\footnote{based on previous work from - Patrick Loiseleur and Samuel Boutin}} -\label{ring} -\tacindex{ring} - -This chapter presents the tactics dedicated to deal with ring and -field equations. - -\asection{What does this tactic do?} - -\texttt{ring} does associative-commutative rewriting in ring and semi-ring -structures. Assume you have two binary functions $\oplus$ and $\otimes$ -that are associative and commutative, with $\oplus$ distributive on -$\otimes$, and two constants 0 and 1 that are unities for $\oplus$ and -$\otimes$. A \textit{polynomial} is an expression built on variables $V_0, V_1, -\dots$ and constants by application of $\oplus$ and $\otimes$. - -Let an {\it ordered product} be a product of variables $V_{i_1} -\otimes \ldots \otimes V_{i_n}$ verifying $i_1 \le i_2 \le \dots \le -i_n$. Let a \textit{monomial} be the product of a constant and an -ordered product. We can order the monomials by the lexicographic -order on products of variables. Let a \textit{canonical sum} be an -ordered sum of monomials that are all different, i.e. each monomial in -the sum is strictly less than the following monomial according to the -lexicographic order. It is an easy theorem to show that every -polynomial is equivalent (modulo the ring properties) to exactly one -canonical sum. This canonical sum is called the \textit{normal form} -of the polynomial. In fact, the actual representation shares monomials -with same prefixes. So what does \texttt{ring}? It normalizes -polynomials over any ring or semi-ring structure. The basic use of -\texttt{ring} is to simplify ring expressions, so that the user does -not have to deal manually with the theorems of associativity and -commutativity. - -\begin{Examples} -\item In the ring of integers, the normal form of -$x (3 + yx + 25(1 - z)) + zx$ is $28x + (-24)xz + xxy$. -\end{Examples} - -\texttt{ring} is also able to compute a normal form modulo monomial -equalities. For example, under the hypothesis that $2x^2 = yz+1$, - the normal form of $2(x + 1)x - x - zy$ is $x+1$. - -\asection{The variables map} - -It is frequent to have an expression built with + and - $\times$, but rarely on variables only. -Let us associate a number to each subterm of a ring -expression in the \gallina\ language. For example in the ring -\texttt{nat}, consider the expression: - -\begin{quotation} -\begin{verbatim} -(plus (mult (plus (f (5)) x) x) - (mult (if b then (4) else (f (3))) (2))) -\end{verbatim} -\end{quotation} - -\noindent As a ring expression, it has 3 subterms. Give each subterm a -number in an arbitrary order: - -\begin{tabular}{ccl} -0 & $\mapsto$ & \verb|if b then (4) else (f (3))| \\ -1 & $\mapsto$ & \verb|(f (5))| \\ -2 & $\mapsto$ & \verb|x| \\ -\end{tabular} - -\noindent Then normalize the ``abstract'' polynomial - -$$((V_1 \otimes V_2) \oplus V_2) \oplus (V_0 \otimes 2) $$ - -\noindent In our example the normal form is: - -$$(2 \otimes V_0) \oplus (V_1 \otimes V_2) \oplus (V_2 \otimes V_2)$$ - -\noindent Then substitute the variables by their values in the variables map to -get the concrete normal polynomial: - -\begin{quotation} -\begin{verbatim} -(plus (mult (2) (if b then (4) else (f (3)))) - (plus (mult (f (5)) x) (mult x x))) -\end{verbatim} -\end{quotation} - -\asection{Is it automatic?} - -Yes, building the variables map and doing the substitution after -normalizing is automatically done by the tactic. So you can just forget -this paragraph and use the tactic according to your intuition. - -\asection{Concrete usage in \Coq -\tacindex{ring} -\tacindex{ring\_simplify}} - -The {\tt ring} tactic solves equations upon polynomial expressions of -a ring (or semi-ring) structure. It proceeds by normalizing both hand -sides of the equation (w.r.t. associativity, commutativity and -distributivity, constant propagation, rewriting of monomials) -and comparing syntactically the results. - -{\tt ring\_simplify} applies the normalization procedure described -above to the terms given. The tactic then replaces all occurrences of -the terms given in the conclusion of the goal by their normal -forms. If no term is given, then the conclusion should be an equation -and both hand sides are normalized. -The tactic can also be applied in a hypothesis. - -The tactic must be loaded by \texttt{Require Import Ring}. The ring -structures must be declared with the \texttt{Add Ring} command (see -below). The ring of booleans is predefined; if one wants to use the -tactic on \texttt{nat} one must first require the module -\texttt{ArithRing} (exported by \texttt{Arith}); -for \texttt{Z}, do \texttt{Require Import -ZArithRing} or simply \texttt{Require Import ZArith}; -for \texttt{N}, do \texttt{Require Import NArithRing} or -\texttt{Require Import NArith}. - -\Example -\begin{coq_eval} -Reset Initial. -\end{coq_eval} -\begin{coq_example} -Require Import ZArith. -Open Scope Z_scope. -Goal forall a b c:Z, - (a + b + c)^2 = - a * a + b^2 + c * c + 2 * a * b + 2 * a * c + 2 * b * c. -\end{coq_example} -\begin{coq_example} -intros; ring. -\end{coq_example} -\begin{coq_eval} -Abort. -\end{coq_eval} -\begin{coq_example} -Goal forall a b:Z, 2*a*b = 30 -> - (a+b)^2 = a^2 + b^2 + 30. -\end{coq_example} -\begin{coq_example} -intros a b H; ring [H]. -\end{coq_example} -\begin{coq_eval} -Reset Initial. -\end{coq_eval} - -\begin{Variants} - \item {\tt ring [\term$_1$ {\ldots} \term$_n$]} decides the equality of two - terms modulo ring operations and rewriting of the equalities - defined by \term$_1$ {\ldots} \term$_n$. Each of \term$_1$ - {\ldots} \term$_n$ has to be a proof of some equality $m = p$, - where $m$ is a monomial (after ``abstraction''), - $p$ a polynomial and $=$ the corresponding equality of the ring structure. - - \item {\tt ring\_simplify [\term$_1$ {\ldots} \term$_n$] $t_1 \ldots t_m$ in -{\ident}} - performs the simplification in the hypothesis named {\tt ident}. -\end{Variants} - -\Warning \texttt{ring\_simplify \term$_1$; ring\_simplify \term$_2$} is -not equivalent to \texttt{ring\_simplify \term$_1$ \term$_2$}. In the -latter case the variables map is shared between the two terms, and -common subterm $t$ of \term$_1$ and \term$_2$ will have the same -associated variable number. So the first alternative should be -avoided for terms belonging to the same ring theory. - - -\begin{ErrMsgs} -\item \errindex{not a valid ring equation} - The conclusion of the goal is not provable in the corresponding ring - theory. -\item \errindex{arguments of ring\_simplify do not have all the same type} - {\tt ring\_simplify} cannot simplify terms of several rings at the - same time. Invoke the tactic once per ring structure. -\item \errindex{cannot find a declared ring structure over {\tt term}} - No ring has been declared for the type of the terms to be - simplified. Use {\tt Add Ring} first. -\item \errindex{cannot find a declared ring structure for equality - {\tt term}} - Same as above is the case of the {\tt ring} tactic. -\end{ErrMsgs} - -\asection{Adding a ring structure -\comindex{Add Ring}} - -Declaring a new ring consists in proving that a ring signature (a -carrier set, an equality, and ring operations: {\tt -Ring\_theory.ring\_theory} and {\tt Ring\_theory.semi\_ring\_theory}) -satisfies the ring axioms. Semi-rings (rings without $+$ inverse) are -also supported. The equality can be either Leibniz equality, or any -relation declared as a setoid (see~\ref{setoidtactics}). The definition -of ring and semi-rings (see module {\tt Ring\_theory}) is: -\begin{verbatim} - Record ring_theory : Prop := mk_rt { - Radd_0_l : forall x, 0 + x == x; - Radd_sym : forall x y, x + y == y + x; - Radd_assoc : forall x y z, x + (y + z) == (x + y) + z; - Rmul_1_l : forall x, 1 * x == x; - Rmul_sym : forall x y, x * y == y * x; - Rmul_assoc : forall x y z, x * (y * z) == (x * y) * z; - Rdistr_l : forall x y z, (x + y) * z == (x * z) + (y * z); - Rsub_def : forall x y, x - y == x + -y; - Ropp_def : forall x, x + (- x) == 0 - }. - -Record semi_ring_theory : Prop := mk_srt { - SRadd_0_l : forall n, 0 + n == n; - SRadd_sym : forall n m, n + m == m + n ; - SRadd_assoc : forall n m p, n + (m + p) == (n + m) + p; - SRmul_1_l : forall n, 1*n == n; - SRmul_0_l : forall n, 0*n == 0; - SRmul_sym : forall n m, n*m == m*n; - SRmul_assoc : forall n m p, n*(m*p) == (n*m)*p; - SRdistr_l : forall n m p, (n + m)*p == n*p + m*p - }. -\end{verbatim} - -This implementation of {\tt ring} also features a notion of constant -that can be parameterized. This can be used to improve the handling of -closed expressions when operations are effective. It consists in -introducing a type of \emph{coefficients} and an implementation of the -ring operations, and a morphism from the coefficient type to the ring -carrier type. The morphism needs not be injective, nor surjective. As -an example, one can consider the real numbers. The set of coefficients -could be the rational numbers, upon which the ring operations can be -implemented. The fact that there exists a morphism is defined by the -following properties: -\begin{verbatim} - Record ring_morph : Prop := mkmorph { - morph0 : [cO] == 0; - morph1 : [cI] == 1; - morph_add : forall x y, [x +! y] == [x]+[y]; - morph_sub : forall x y, [x -! y] == [x]-[y]; - morph_mul : forall x y, [x *! y] == [x]*[y]; - morph_opp : forall x, [-!x] == -[x]; - morph_eq : forall x y, x?=!y = true -> [x] == [y] - }. - - Record semi_morph : Prop := mkRmorph { - Smorph0 : [cO] == 0; - Smorph1 : [cI] == 1; - Smorph_add : forall x y, [x +! y] == [x]+[y]; - Smorph_mul : forall x y, [x *! y] == [x]*[y]; - Smorph_eq : forall x y, x?=!y = true -> [x] == [y] - }. -\end{verbatim} -where {\tt c0} and {\tt cI} denote the 0 and 1 of the coefficient set, -{\tt +!}, {\tt *!}, {\tt -!} are the implementations of the ring -operations, {\tt ==} is the equality of the coefficients, {\tt ?+!} is -an implementation of this equality, and {\tt [x]} is a notation for -the image of {\tt x} by the ring morphism. - -Since {\tt Z} is an initial ring (and {\tt N} is an initial -semi-ring), it can always be considered as a set of -coefficients. There are basically three kinds of (semi-)rings: -\begin{description} -\item[abstract rings] to be used when operations are not - effective. The set of coefficients is {\tt Z} (or {\tt N} for - semi-rings). -\item[computational rings] to be used when operations are - effective. The set of coefficients is the ring itself. The user only - has to provide an implementation for the equality. -\item[customized ring] for other cases. The user has to provide the - coefficient set and the morphism. -\end{description} - -This implementation of ring can also recognize simple -power expressions as ring expressions. A power function is specified by -the following property: -\begin{verbatim} - Section POWER. - Variable Cpow : Set. - Variable Cp_phi : N -> Cpow. - Variable rpow : R -> Cpow -> R. - - Record power_theory : Prop := mkpow_th { - rpow_pow_N : forall r n, req (rpow r (Cp_phi n)) (pow_N rI rmul r n) - }. - - End POWER. -\end{verbatim} - - -The syntax for adding a new ring is {\tt Add Ring $name$ : $ring$ -($mod_1$,\dots,$mod_2$)}. The name is not relevent. It is just used -for error messages. The term $ring$ is a proof that the ring signature -satisfies the (semi-)ring axioms. The optional list of modifiers is -used to tailor the behavior of the tactic. The following list -describes their syntax and effects: -\begin{description} -\item[abstract] declares the ring as abstract. This is the default. -\item[decidable \term] declares the ring as computational. The expression - \term{} is - the correctness proof of an equality test {\tt ?=!} (which should be - evaluable). Its type should be of - the form {\tt forall x y, x?=!y = true $\rightarrow$ x == y}. -\item[morphism \term] declares the ring as a customized one. The expression - \term{} is - a proof that there exists a morphism between a set of coefficient - and the ring carrier (see {\tt Ring\_theory.ring\_morph} and {\tt - Ring\_theory.semi\_morph}). -\item[setoid \term$_1$ \term$_2$] forces the use of given setoid. The - expression \term$_1$ is a proof that the equality is indeed a setoid - (see {\tt Setoid.Setoid\_Theory}), and \term$_2$ a proof that the - ring operations are morphisms (see {\tt Ring\_theory.ring\_eq\_ext} and - {\tt Ring\_theory.sring\_eq\_ext}). This modifier needs not be used if the - setoid and morphisms have been declared. -\item[constants [\ltac]] specifies a tactic expression that, given a term, - returns either an object of the coefficient set that is mapped to - the expression via the morphism, or returns {\tt - InitialRing.NotConstant}. The default behaviour is to map only 0 and - 1 to their counterpart in the coefficient set. This is generally not - desirable for non trivial computational rings. -\item[preprocess [\ltac]] - specifies a tactic that is applied as a preliminary step for {\tt - ring} and {\tt ring\_simplify}. It can be used to transform a goal - so that it is better recognized. For instance, {\tt S n} can be - changed to {\tt plus 1 n}. -\item[postprocess [\ltac]] specifies a tactic that is applied as a final step - for {\tt ring\_simplify}. For instance, it can be used to undo - modifications of the preprocessor. -\item[power\_tac {\term} [\ltac]] allows {\tt ring} and {\tt ring\_simplify} to - recognize power expressions with a constant positive integer exponent - (example: $x^2$). The term {\term} is a proof that a given power function - satisfies the specification of a power function ({\term} has to be a - proof of {\tt Ring\_theory.power\_theory}) and {\ltac} specifies a - tactic expression that, given a term, ``abstracts'' it into an - object of type {\tt N} whose interpretation via {\tt Cp\_phi} (the - evaluation function of power coefficient) is the original term, or - returns {\tt InitialRing.NotConstant} if not a constant coefficient - (i.e. {\ltac} is the inverse function of {\tt Cp\_phi}). - See files {\tt plugins/setoid\_ring/ZArithRing.v} and - {\tt plugins/setoid\_ring/RealField.v} for examples. - By default the tactic does not recognize power expressions as ring - expressions. -\item[sign {\term}] allows {\tt ring\_simplify} to use a minus operation - when outputing its normal form, i.e writing $x - y$ instead of $x + (-y)$. - The term {\term} is a proof that a given sign function indicates expressions - that are signed ({\term} has to be a - proof of {\tt Ring\_theory.get\_sign}). See {\tt plugins/setoid\_ring/IntialRing.v} for examples of sign function. -\item[div {\term}] allows {\tt ring} and {\tt ring\_simplify} to use moniomals -with coefficient other than 1 in the rewriting. The term {\term} is a proof that a given division function satisfies the specification of an euclidean - division function ({\term} has to be a - proof of {\tt Ring\_theory.div\_theory}). For example, this function is - called when trying to rewrite $7x$ by $2x = z$ to tell that $7 = 3 * 2 + 1$. - See {\tt plugins/setoid\_ring/IntialRing.v} for examples of div function. - -\end{description} - - -\begin{ErrMsgs} -\item \errindex{bad ring structure} - The proof of the ring structure provided is not of the expected type. -\item \errindex{bad lemma for decidability of equality} - The equality function provided in the case of a computational ring - has not the expected type. -\item \errindex{ring {\it operation} should be declared as a morphism} - A setoid associated to the carrier of the ring structure as been - found, but the ring operation should be declared as - morphism. See~\ref{setoidtactics}. -\end{ErrMsgs} - -\asection{How does it work?} - -The code of \texttt{ring} is a good example of tactic written using -\textit{reflection}. What is reflection? Basically, it is writing -\Coq{} tactics in \Coq, rather than in \ocaml. From the philosophical -point of view, it is using the ability of the Calculus of -Constructions to speak and reason about itself. For the \texttt{ring} -tactic we used \Coq\ as a programming language and also as a proof -environment to build a tactic and to prove it correctness. - -The interested reader is strongly advised to have a look at the file -\texttt{Ring\_polynom.v}. Here a type for polynomials is defined: - -\begin{small} -\begin{flushleft} -\begin{verbatim} -Inductive PExpr : Type := - | PEc : C -> PExpr - | PEX : positive -> PExpr - | PEadd : PExpr -> PExpr -> PExpr - | PEsub : PExpr -> PExpr -> PExpr - | PEmul : PExpr -> PExpr -> PExpr - | PEopp : PExpr -> PExpr - | PEpow : PExpr -> N -> PExpr. -\end{verbatim} -\end{flushleft} -\end{small} - -Polynomials in normal form are defined as: -\begin{small} -\begin{flushleft} -\begin{verbatim} - Inductive Pol : Type := - | Pc : C -> Pol - | Pinj : positive -> Pol -> Pol - | PX : Pol -> positive -> Pol -> Pol. -\end{verbatim} -\end{flushleft} -\end{small} -where {\tt Pinj n P} denotes $P$ in which $V_i$ is replaced by -$V_{i+n}$, and {\tt PX P n Q} denotes $P \otimes V_1^{n} \oplus Q'$, -$Q'$ being $Q$ where $V_i$ is replaced by $V_{i+1}$. - - -Variables maps are represented by list of ring elements, and two -interpretation functions, one that maps a variables map and a -polynomial to an element of the concrete ring, and the second one that -does the same for normal forms: -\begin{small} -\begin{flushleft} -\begin{verbatim} -Definition PEeval : list R -> PExpr -> R := [...]. -Definition Pphi_dev : list R -> Pol -> R := [...]. -\end{verbatim} -\end{flushleft} -\end{small} - -A function to normalize polynomials is defined, and the big theorem is -its correctness w.r.t interpretation, that is: - -\begin{small} -\begin{flushleft} -\begin{verbatim} -Definition norm : PExpr -> Pol := [...]. -Lemma Pphi_dev_ok : - forall l pe npe, norm pe = npe -> PEeval l pe == Pphi_dev l npe. -\end{verbatim} -\end{flushleft} -\end{small} - -So now, what is the scheme for a normalization proof? Let \texttt{p} -be the polynomial expression that the user wants to normalize. First a -little piece of ML code guesses the type of \texttt{p}, the ring -theory \texttt{T} to use, an abstract polynomial \texttt{ap} and a -variables map \texttt{v} such that \texttt{p} is -$\beta\delta\iota$-equivalent to \verb|(PEeval v ap)|. Then we -replace it by \verb|(Pphi_dev v (norm ap))|, using the -main correctness theorem and we reduce it to a concrete expression -\texttt{p'}, which is the concrete normal form of -\texttt{p}. This is summarized in this diagram: -\begin{center} -\begin{tabular}{rcl} -\texttt{p} & $\rightarrow_{\beta\delta\iota}$ - & \texttt{(PEeval v ap)} \\ - & & $=_{\mathrm{(by\ the\ main\ correctness\ theorem)}}$ \\ -\texttt{p'} - & $\leftarrow_{\beta\delta\iota}$ - & \texttt{(Pphi\_dev v (norm ap))} -\end{tabular} -\end{center} -The user do not see the right part of the diagram. -From outside, the tactic behaves like a -$\beta\delta\iota$ simplification extended with AC rewriting rules. -Basically, the proof is only the application of the main -correctness theorem to well-chosen arguments. - - -\asection{Dealing with fields -\tacindex{field} -\tacindex{field\_simplify} -\tacindex{field\_simplify\_eq}} - - -The {\tt field} tactic is an extension of the {\tt ring} to deal with -rational expresision. Given a rational expression $F=0$. It first reduces the expression $F$ to a common denominator $N/D= 0$ where $N$ and $D$ are two ring -expressions. -For example, if we take $F = (1 - 1/x) x - x + 1$, this gives -$ N= (x -1) x - x^2 + x$ and $D= x$. It then calls {\tt ring} -to solve $N=0$. Note that {\tt field} also generates non-zero conditions -for all the denominators it encounters in the reduction. -In our example, it generates the condition $x \neq 0$. These -conditions appear as one subgoal which is a conjunction if there are -several denominators. -Non-zero conditions are {\it always} polynomial expressions. For example -when reducing the expression $1/(1 + 1/x)$, two side conditions are -generated: $x\neq 0$ and $x + 1 \neq 0$. Factorized expressions are -broken since a field is an integral domain, and when the equality test -on coefficients is complete w.r.t. the equality of the target field, -constants can be proven different from zero automatically. - -The tactic must be loaded by \texttt{Require Import Field}. New field -structures can be declared to the system with the \texttt{Add Field} -command (see below). The field of real numbers is defined in module -\texttt{RealField} (in texttt{plugins/setoid\_ring}). It is exported -by module \texttt{Rbase}, so that requiring \texttt{Rbase} or -\texttt{Reals} is enough to use the field tactics on real -numbers. Rational numbers in canonical form are also declared as a -field in module \texttt{Qcanon}. - - -\Example -\begin{coq_eval} -Reset Initial. -\end{coq_eval} -\begin{coq_example} -Require Import Reals. -Open Scope R_scope. -Goal forall x, x <> 0 -> - (1 - 1/x) * x - x + 1 = 0. -\end{coq_example} -\begin{coq_example} -intros; field; auto. -\end{coq_example} -\begin{coq_eval} -Abort. -\end{coq_eval} -\begin{coq_example} -Goal forall x y, y <> 0 -> y = x -> x/y = 1. -\end{coq_example} -\begin{coq_example} -intros x y H H1; field [H1]; auto. -\end{coq_example} -\begin{coq_eval} -Reset Initial. -\end{coq_eval} - -\begin{Variants} - \item {\tt field [\term$_1$ {\ldots} \term$_n$]} decides the equality of two - terms modulo field operations and rewriting of the equalities - defined by \term$_1$ {\ldots} \term$_n$. Each of \term$_1$ - {\ldots} \term$_n$ has to be a proof of some equality $m = p$, - where $m$ is a monomial (after ``abstraction''), - $p$ a polynomial and $=$ the corresponding equality of the field structure. - Beware that rewriting works with the equality $m=p$ only if $p$ is a - polynomial since rewriting is handled by the underlying {\tt ring} - tactic. - \item {\tt field\_simplify} - performs the simplification in the conclusion of the goal, $F_1 = F_2$ - becomes $N_1/D_1 = N_2/D_2$. A normalization step (the same as the - one for rings) is then applied to $N_1$, $D_1$, $N_2$ and - $D_2$. This way, polynomials remain in factorized form during the - fraction simplifications. This yields smaller expressions when - reducing to the same denominator since common factors can be - cancelled. - - \item {\tt field\_simplify [\term$_1$ {\ldots} \term$_n$]} - performs the simplification in the conclusion of the goal using - the equalities - defined by \term$_1$ {\ldots} \term$_n$. - - \item {\tt field\_simplify [\term$_1$ {\ldots} \term$_n$] $t_1$ \ldots -$t_m$} - performs the simplification in the terms $t_1$ \ldots $t_m$ - of the conclusion of the goal using - the equalities - defined by \term$_1$ {\ldots} \term$_n$. - - \item {\tt field\_simplify in $H$} - performs the simplification in the assumption $H$. - - \item {\tt field\_simplify [\term$_1$ {\ldots} \term$_n$] in $H$} - performs the simplification in the assumption $H$ using - the equalities - defined by \term$_1$ {\ldots} \term$_n$. - - \item {\tt field\_simplify [\term$_1$ {\ldots} \term$_n$] $t_1$ \ldots -$t_m$ in $H$} - performs the simplification in the terms $t_1$ \ldots $t_n$ - of the assumption $H$ using - the equalities - defined by \term$_1$ {\ldots} \term$_m$. - - \item {\tt field\_simplify\_eq} - performs the simplification in the conclusion of the goal removing - the denominator. $F_1 = F_2$ - becomes $N_1 D_2 = N_2 D_1$. - - \item {\tt field\_simplify\_eq [\term$_1$ {\ldots} \term$_n$]} - performs the simplification in the conclusion of the goal using - the equalities - defined by \term$_1$ {\ldots} \term$_n$. - - \item {\tt field\_simplify\_eq} in $H$ - performs the simplification in the assumption $H$. - - \item {\tt field\_simplify\_eq [\term$_1$ {\ldots} \term$_n$] in $H$} - performs the simplification in the assumption $H$ using - the equalities - defined by \term$_1$ {\ldots} \term$_n$. -\end{Variants} - -\asection{Adding a new field structure -\comindex{Add Field}} - -Declaring a new field consists in proving that a field signature (a -carrier set, an equality, and field operations: {\tt -Field\_theory.field\_theory} and {\tt Field\_theory.semi\_field\_theory}) -satisfies the field axioms. Semi-fields (fields without $+$ inverse) are -also supported. The equality can be either Leibniz equality, or any -relation declared as a setoid (see~\ref{setoidtactics}). The definition -of fields and semi-fields is: -\begin{verbatim} -Record field_theory : Prop := mk_field { - F_R : ring_theory rO rI radd rmul rsub ropp req; - F_1_neq_0 : ~ 1 == 0; - Fdiv_def : forall p q, p / q == p * / q; - Finv_l : forall p, ~ p == 0 -> / p * p == 1 -}. - -Record semi_field_theory : Prop := mk_sfield { - SF_SR : semi_ring_theory rO rI radd rmul req; - SF_1_neq_0 : ~ 1 == 0; - SFdiv_def : forall p q, p / q == p * / q; - SFinv_l : forall p, ~ p == 0 -> / p * p == 1 -}. -\end{verbatim} - -The result of the normalization process is a fraction represented by -the following type: -\begin{verbatim} -Record linear : Type := mk_linear { - num : PExpr C; - denum : PExpr C; - condition : list (PExpr C) }. -\end{verbatim} -where {\tt num} and {\tt denum} are the numerator and denominator; -{\tt condition} is a list of expressions that have appeared as a -denominator during the normalization process. These expressions must -be proven different from zero for the correctness of the algorithm. - -The syntax for adding a new field is {\tt Add Field $name$ : $field$ -($mod_1$,\dots,$mod_2$)}. The name is not relevent. It is just used -for error messages. $field$ is a proof that the field signature -satisfies the (semi-)field axioms. The optional list of modifiers is -used to tailor the behaviour of the tactic. Since field tactics are -built upon ring tactics, all mofifiers of the {\tt Add Ring} -apply. There is only one specific modifier: -\begin{description} -\item[completeness \term] allows the field tactic to prove - automatically that the image of non-zero coefficients are mapped to - non-zero elements of the field. \term is a proof of {\tt forall x y, - [x] == [y] -> x?=!y = true}, which is the completeness of equality - on coefficients w.r.t. the field equality. -\end{description} - -\asection{Legacy implementation} - -\Warning This tactic is the {\tt ring} tactic of previous versions of -\Coq{} and it should be considered as deprecated. It will probably be -removed in future releases. It has been kept only for compatibility -reasons and in order to help moving existing code to the newer -implementation described above. For more details, please refer to the -Coq Reference Manual, version 8.0. - - -\subsection{\tt legacy ring \term$_1$ \dots\ \term$_n$ -\tacindex{legacy ring} -\comindex{Add Legacy Ring} -\comindex{Add Legacy Semi Ring}} - -This tactic, written by Samuel Boutin and Patrick Loiseleur, applies -associative commutative rewriting on every ring. The tactic must be -loaded by \texttt{Require Import LegacyRing}. The ring must be declared in -the \texttt{Add Ring} command. The ring of booleans (with \texttt{andb} -as multiplication and \texttt{xorb} as addition) -is predefined; if one wants to use the tactic on \texttt{nat} one must -first require the module \texttt{LegacyArithRing}; for \texttt{Z}, do -\texttt{Require Import LegacyZArithRing}; for \texttt{N}, do \texttt{Require -Import LegacyNArithRing}. - -The terms \term$_1$, \dots, \term$_n$ must be subterms of the goal -conclusion. The tactic \texttt{ring} normalizes these terms -w.r.t. associativity and commutativity and replace them by their -normal form. - -\begin{Variants} -\item \texttt{legacy ring} When the goal is an equality $t_1=t_2$, it - acts like \texttt{ring\_simplify} $t_1$ $t_2$ and then - solves the equality by reflexivity. - -\item \texttt{ring\_nat} is a tactic macro for \texttt{repeat rewrite - S\_to\_plus\_one; ring}. The theorem \texttt{S\_to\_plus\_one} is a - proof that \texttt{forall (n:nat), S n = plus (S O) n}. - -\end{Variants} - -You can have a look at the files \texttt{LegacyRing.v}, -\texttt{ArithRing.v}, \texttt{ZArithRing.v} to see examples of the -\texttt{Add Ring} command. - -\subsection{Add a ring structure} - -It can be done in the \Coq toplevel (No ML file to edit and to link -with \Coq). First, \texttt{ring} can handle two kinds of structure: -rings and semi-rings. Semi-rings are like rings without an opposite to -addition. Their precise specification (in \gallina) can be found in -the file - -\begin{quotation} -\begin{verbatim} -plugins/ring/Ring_theory.v -\end{verbatim} -\end{quotation} - -The typical example of ring is \texttt{Z}, the typical -example of semi-ring is \texttt{nat}. - -The specification of a -ring is divided in two parts: first the record of constants -($\oplus$, $\otimes$, 1, 0, $\ominus$) and then the theorems -(associativity, commutativity, etc.). - -\begin{small} -\begin{flushleft} -\begin{verbatim} -Section Theory_of_semi_rings. - -Variable A : Type. -Variable Aplus : A -> A -> A. -Variable Amult : A -> A -> A. -Variable Aone : A. -Variable Azero : A. -(* There is also a "weakly decidable" equality on A. That means - that if (A_eq x y)=true then x=y but x=y can arise when - (A_eq x y)=false. On an abstract ring the function [x,y:A]false - is a good choice. The proof of A_eq_prop is in this case easy. *) -Variable Aeq : A -> A -> bool. - -Record Semi_Ring_Theory : Prop := -{ SR_plus_sym : (n,m:A)[| n + m == m + n |]; - SR_plus_assoc : (n,m,p:A)[| n + (m + p) == (n + m) + p |]; - - SR_mult_sym : (n,m:A)[| n*m == m*n |]; - SR_mult_assoc : (n,m,p:A)[| n*(m*p) == (n*m)*p |]; - SR_plus_zero_left :(n:A)[| 0 + n == n|]; - SR_mult_one_left : (n:A)[| 1*n == n |]; - SR_mult_zero_left : (n:A)[| 0*n == 0 |]; - SR_distr_left : (n,m,p:A) [| (n + m)*p == n*p + m*p |]; - SR_plus_reg_left : (n,m,p:A)[| n + m == n + p |] -> m==p; - SR_eq_prop : (x,y:A) (Is_true (Aeq x y)) -> x==y -}. -\end{verbatim} -\end{flushleft} -\end{small} - -\begin{small} -\begin{flushleft} -\begin{verbatim} -Section Theory_of_rings. - -Variable A : Type. - -Variable Aplus : A -> A -> A. -Variable Amult : A -> A -> A. -Variable Aone : A. -Variable Azero : A. -Variable Aopp : A -> A. -Variable Aeq : A -> A -> bool. - - -Record Ring_Theory : Prop := -{ Th_plus_sym : (n,m:A)[| n + m == m + n |]; - Th_plus_assoc : (n,m,p:A)[| n + (m + p) == (n + m) + p |]; - Th_mult_sym : (n,m:A)[| n*m == m*n |]; - Th_mult_assoc : (n,m,p:A)[| n*(m*p) == (n*m)*p |]; - Th_plus_zero_left :(n:A)[| 0 + n == n|]; - Th_mult_one_left : (n:A)[| 1*n == n |]; - Th_opp_def : (n:A) [| n + (-n) == 0 |]; - Th_distr_left : (n,m,p:A) [| (n + m)*p == n*p + m*p |]; - Th_eq_prop : (x,y:A) (Is_true (Aeq x y)) -> x==y -}. -\end{verbatim} -\end{flushleft} -\end{small} - -To define a ring structure on A, you must provide an addition, a -multiplication, an opposite function and two unities 0 and 1. - -You must then prove all theorems that make -(A,Aplus,Amult,Aone,Azero,Aeq) -a ring structure, and pack them with the \verb|Build_Ring_Theory| -constructor. - -Finally to register a ring the syntax is: - -\comindex{Add Legacy Ring} -\begin{quotation} - \texttt{Add Legacy Ring} \textit{A Aplus Amult Aone Azero Ainv Aeq T} - \texttt{[} \textit{c1 \dots cn} \texttt{].} -\end{quotation} - -\noindent where \textit{A} is a term of type \texttt{Set}, -\textit{Aplus} is a term of type \texttt{A->A->A}, -\textit{Amult} is a term of type \texttt{A->A->A}, -\textit{Aone} is a term of type \texttt{A}, -\textit{Azero} is a term of type \texttt{A}, -\textit{Ainv} is a term of type \texttt{A->A}, -\textit{Aeq} is a term of type \texttt{A->bool}, -\textit{T} is a term of type -\texttt{(Ring\_Theory }\textit{A Aplus Amult Aone Azero Ainv - Aeq}\texttt{)}. -The arguments \textit{c1 \dots cn}, -are the names of constructors which define closed terms: a -subterm will be considered as a constant if it is either one of the -terms \textit{c1 \dots cn} or the application of one of these terms to -closed terms. For \texttt{nat}, the given constructors are \texttt{S} -and \texttt{O}, and the closed terms are \texttt{O}, \texttt{(S O)}, -\texttt{(S (S O))}, \ldots - -\begin{Variants} -\item \texttt{Add Legacy Semi Ring} \textit{A Aplus Amult Aone Azero Aeq T} - \texttt{[} \textit{c1 \dots\ cn} \texttt{].}\comindex{Add Legacy Semi - Ring} - - There are two differences with the \texttt{Add Ring} command: there - is no inverse function and the term $T$ must be of type - \texttt{(Semi\_Ring\_Theory }\textit{A Aplus Amult Aone Azero - Aeq}\texttt{)}. - -\item \texttt{Add Legacy Abstract Ring} \textit{A Aplus Amult Aone Azero Ainv - Aeq T}\texttt{.}\comindex{Add Legacy Abstract Ring} - - This command should be used for when the operations of rings are not - computable; for example the real numbers of - \texttt{theories/REALS/}. Here $0+1$ is not beta-reduced to $1$ but - you still may want to \textit{rewrite} it to $1$ using the ring - axioms. The argument \texttt{Aeq} is not used; a good choice for - that function is \verb+[x:A]false+. - -\item \texttt{Add Legacy Abstract Semi Ring} \textit{A Aplus Amult Aone Azero - Aeq T}\texttt{.}\comindex{Add Legacy Abstract Semi Ring} - -\end{Variants} - -\begin{ErrMsgs} -\item \errindex{Not a valid (semi)ring theory}. - - That happens when the typing condition does not hold. -\end{ErrMsgs} - -Currently, the hypothesis is made than no more than one ring structure -may be declared for a given type in \texttt{Set} or \texttt{Type}. -This allows automatic detection of the theory used to achieve the -normalization. On popular demand, we can change that and allow several -ring structures on the same set. - -The table of ring theories is compatible with the \Coq\ -sectioning mechanism. If you declare a ring inside a section, the -declaration will be thrown away when closing the section. -And when you load a compiled file, all the \texttt{Add Ring} -commands of this file that are not inside a section will be loaded. - -The typical example of ring is \texttt{Z}, and the typical example of -semi-ring is \texttt{nat}. Another ring structure is defined on the -booleans. - -\Warning Only the ring of booleans is loaded by default with the -\texttt{Ring} module. To load the ring structure for \texttt{nat}, -load the module \texttt{ArithRing}, and for \texttt{Z}, -load the module \texttt{ZArithRing}. - -\subsection{\tt legacy field -\tacindex{legacy field}} - -This tactic written by David~Delahaye and Micaela~Mayero solves equalities -using commutative field theory. Denominators have to be non equal to zero and, -as this is not decidable in general, this tactic may generate side conditions -requiring some expressions to be non equal to zero. This tactic must be loaded -by {\tt Require Import LegacyField}. Field theories are declared (as for -{\tt legacy ring}) with -the {\tt Add Legacy Field} command. - -\subsection{\tt Add Legacy Field -\comindex{Add Legacy Field}} - -This vernacular command adds a commutative field theory to the database for the -tactic {\tt field}. You must provide this theory as follows: -\begin{flushleft} -{\tt Add Legacy Field {\it A} {\it Aplus} {\it Amult} {\it Aone} {\it Azero} {\it -Aopp} {\it Aeq} {\it Ainv} {\it Rth} {\it Tinvl}} -\end{flushleft} -where {\tt {\it A}} is a term of type {\tt Type}, {\tt {\it Aplus}} is -a term of type {\tt A->A->A}, {\tt {\it Amult}} is a term of type {\tt - A->A->A}, {\tt {\it Aone}} is a term of type {\tt A}, {\tt {\it - Azero}} is a term of type {\tt A}, {\tt {\it Aopp}} is a term of -type {\tt A->A}, {\tt {\it Aeq}} is a term of type {\tt A->bool}, {\tt - {\it Ainv}} is a term of type {\tt A->A}, {\tt {\it Rth}} is a term -of type {\tt (Ring\_Theory {\it A Aplus Amult Aone Azero Ainv Aeq})}, -and {\tt {\it Tinvl}} is a term of type {\tt forall n:{\it A}, - {\~{}}(n={\it Azero})->({\it Amult} ({\it Ainv} n) n)={\it Aone}}. -To build a ring theory, refer to Chapter~\ref{ring} for more details. - -This command adds also an entry in the ring theory table if this theory is not -already declared. So, it is useless to keep, for a given type, the {\tt Add -Ring} command if you declare a theory with {\tt Add Field}, except if you plan -to use specific features of {\tt ring} (see Chapter~\ref{ring}). However, the -module {\tt ring} is not loaded by {\tt Add Field} and you have to make a {\tt -Require Import Ring} if you want to call the {\tt ring} tactic. - -\begin{Variants} - -\item {\tt Add Legacy Field {\it A} {\it Aplus} {\it Amult} {\it Aone} {\it Azero} -{\it Aopp} {\it Aeq} {\it Ainv} {\it Rth} {\it Tinvl}}\\ -{\tt \phantom{Add Field }with minus:={\it Aminus}} - -Adds also the term {\it Aminus} which must be a constant expressed by -means of {\it Aopp}. - -\item {\tt Add Legacy Field {\it A} {\it Aplus} {\it Amult} {\it Aone} {\it Azero} -{\it Aopp} {\it Aeq} {\it Ainv} {\it Rth} {\it Tinvl}}\\ -{\tt \phantom{Add Legacy Field }with div:={\it Adiv}} - -Adds also the term {\it Adiv} which must be a constant expressed by -means of {\it Ainv}. - -\end{Variants} - -\SeeAlso \cite{DelMay01} for more details regarding the implementation of {\tt -legacy field}. - -\asection{History of \texttt{ring}} - -First Samuel Boutin designed the tactic \texttt{ACDSimpl}. -This tactic did lot of rewriting. But the proofs -terms generated by rewriting were too big for \Coq's type-checker. -Let us see why: - -\begin{coq_eval} -Require Import ZArith. -Open Scope Z_scope. -\end{coq_eval} -\begin{coq_example} -Goal forall x y z:Z, x + 3 + y + y * z = x + 3 + y + z * y. -\end{coq_example} -\begin{coq_example*} -intros; rewrite (Zmult_comm y z); reflexivity. -Save toto. -\end{coq_example*} -\begin{coq_example} -Print toto. -\end{coq_example} - -At each step of rewriting, the whole context is duplicated in the proof -term. Then, a tactic that does hundreds of rewriting generates huge proof -terms. Since \texttt{ACDSimpl} was too slow, Samuel Boutin rewrote it -using reflection (see his article in TACS'97 \cite{Bou97}). Later, the -stuff was rewritten by Patrick -Loiseleur: the new tactic does not any more require \texttt{ACDSimpl} -to compile and it makes use of $\beta\delta\iota$-reduction -not only to replace the rewriting steps, but also to achieve the -interleaving of computation and -reasoning (see \ref{DiscussReflection}). He also wrote a -few ML code for the \texttt{Add Ring} command, that allow to register -new rings dynamically. - -Proofs terms generated by \texttt{ring} are quite small, they are -linear in the number of $\oplus$ and $\otimes$ operations in the -normalized terms. Type-checking those terms requires some time because it -makes a large use of the conversion rule, but -memory requirements are much smaller. - -\asection{Discussion} -\label{DiscussReflection} - -Efficiency is not the only motivation to use reflection -here. \texttt{ring} also deals with constants, it rewrites for example the -expression $34 + 2*x -x + 12$ to the expected result $x + 46$. For the -tactic \texttt{ACDSimpl}, the only constants were 0 and 1. So the -expression $34 + 2*(x - 1) + 12$ is interpreted as -$V_0 \oplus V_1 \otimes (V_2 \ominus 1) \oplus V_3$, -with the variables mapping -$\{V_0 \mt 34; V_1 \mt 2; V_2 \mt x; V_3 \mt 12 \}$. Then it is -rewritten to $34 - x + 2*x + 12$, very far from the expected -result. Here rewriting is not sufficient: you have to do some kind of -reduction (some kind of \textit{computation}) to achieve the -normalization. - -The tactic \texttt{ring} is not only faster than a classical one: -using reflection, we get for free integration of computation and -reasoning that would be very complex to implement in the classic fashion. - -Is it the ultimate way to write tactics? The answer is: yes and -no. The \texttt{ring} tactic uses intensively the conversion rule of -\CIC, that is replaces proof by computation the most as it is -possible. It can be useful in all situations where a classical tactic -generates huge proof terms. Symbolic Processing and Tautologies are in -that case. But there are also tactics like \texttt{auto} or -\texttt{linear} that do many complex computations, using side-effects -and backtracking, and generate a small proof term. Clearly, it would -be significantly less efficient to replace them by tactics using -reflection. - -Another idea suggested by Benjamin Werner: reflection could be used to -couple an external tool (a rewriting program or a model checker) with -\Coq. We define (in \Coq) a type of terms, a type of \emph{traces}, -and prove a correction theorem that states that \emph{replaying -traces} is safe w.r.t some interpretation. Then we let the external -tool do every computation (using side-effects, backtracking, -exception, or others features that are not available in pure lambda -calculus) to produce the trace: now we can check in Coq{} that the -trace has the expected semantic by applying the correction lemma. - -%%% Local Variables: -%%% mode: latex -%%% TeX-master: "Reference-Manual" -%%% End: diff --git a/doc/refman/Program.tex b/doc/refman/Program.tex deleted file mode 100644 index b41014ab..00000000 --- a/doc/refman/Program.tex +++ /dev/null @@ -1,295 +0,0 @@ -\achapter{\Program{}} -\label{Program} -\aauthor{Matthieu Sozeau} -\index{Program} - -\begin{flushleft} - \em The status of \Program\ is experimental. -\end{flushleft} - -We present here the new \Program\ tactic commands, used to build certified -\Coq\ programs, elaborating them from their algorithmic skeleton and a -rich specification \cite{Sozeau06}. It can be sought of as a dual of extraction -(see Chapter~\ref{Extraction}). The goal of \Program~is to program as in a regular -functional programming language whilst using as rich a specification as -desired and proving that the code meets the specification using the whole \Coq{} proof -apparatus. This is done using a technique originating from the -``Predicate subtyping'' mechanism of \PVS \cite{Rushby98}, which generates type-checking -conditions while typing a term constrained to a particular type. -Here we insert existential variables in the term, which must be filled -with proofs to get a complete \Coq\ term. \Program\ replaces the -\Program\ tactic by Catherine Parent \cite{Parent95b} which had a similar goal but is no longer -maintained. - -The languages available as input are currently restricted to \Coq's term -language, but may be extended to \ocaml{}, \textsc{Haskell} and others -in the future. We use the same syntax as \Coq\ and permit to use implicit -arguments and the existing coercion mechanism. -Input terms and types are typed in an extended system (\Russell) and -interpreted into \Coq\ terms. The interpretation process may produce -some proof obligations which need to be resolved to create the final term. - -\asection{Elaborating programs} -The main difference from \Coq\ is that an object in a type $T : \Set$ -can be considered as an object of type $\{ x : T~|~P\}$ for any -wellformed $P : \Prop$. -If we go from $T$ to the subset of $T$ verifying property $P$, we must -prove that the object under consideration verifies it. \Russell\ will -generate an obligation for every such coercion. In the other direction, -\Russell\ will automatically insert a projection. - -Another distinction is the treatment of pattern-matching. Apart from the -following differences, it is equivalent to the standard {\tt match} -operation (see Section~\ref{Caseexpr}). -\begin{itemize} -\item Generation of equalities. A {\tt match} expression is always - generalized by the corresponding equality. As an example, - the expression: - -\begin{coq_example*} - match x with - | 0 => t - | S n => u - end. -\end{coq_example*} -will be first rewrote to: -\begin{coq_example*} - (match x as y return (x = y -> _) with - | 0 => fun H : x = 0 -> t - | S n => fun H : x = S n -> u - end) (refl_equal n). -\end{coq_example*} - - This permits to get the proper equalities in the context of proof - obligations inside clauses, without which reasoning is very limited. - -\item Generation of inequalities. If a pattern intersects with a - previous one, an inequality is added in the context of the second - branch. See for example the definition of {\tt div2} below, where the second - branch is typed in a context where $\forall p, \_ <> S (S p)$. - -\item Coercion. If the object being matched is coercible to an inductive - type, the corresponding coercion will be automatically inserted. This also - works with the previous mechanism. -\end{itemize} - -To give more control over the generation of equalities, the typechecker will -fall back directly to \Coq's usual typing of dependent pattern-matching -if a {\tt return} or {\tt in} clause is specified. Likewise, -the {\tt if} construct is not treated specially by \Program{} so boolean -tests in the code are not automatically reflected in the obligations. -One can use the {\tt dec} combinator to get the correct hypotheses as in: - -\begin{coq_eval} -Require Import Program Arith. -\end{coq_eval} -\begin{coq_example} -Program Definition id (n : nat) : { x : nat | x = n } := - if dec (leb n 0) then 0 - else S (pred n). -\end{coq_example} - -Finally, the let tupling construct {\tt let (x1, ..., xn) := t in b} -does not produce an equality, contrary to the let pattern construct -{\tt let '(x1, ..., xn) := t in b}. - -The next two commands are similar to their standard counterparts -Definition (see Section~\ref{Basic-definitions}) and Fixpoint (see Section~\ref{Fixpoint}) in that -they define constants. However, they may require the user to prove some -goals to construct the final definitions. - -\subsection{\tt Program Definition {\ident} := {\term}. - \comindex{Program Definition}\label{ProgramDefinition}} - -This command types the value {\term} in \Russell\ and generate proof -obligations. Once solved using the commands shown below, it binds the final -\Coq\ term to the name {\ident} in the environment. - -\begin{ErrMsgs} -\item \errindex{{\ident} already exists} -\end{ErrMsgs} - -\begin{Variants} -\item {\tt Program Definition {\ident} {\tt :}{\term$_1$} := - {\term$_2$}.}\\ - It interprets the type {\term$_1$}, potentially generating proof - obligations to be resolved. Once done with them, we have a \Coq\ type - {\term$_1'$}. It then checks that the type of the interpretation of - {\term$_2$} is coercible to {\term$_1'$}, and registers {\ident} as - being of type {\term$_1'$} once the set of obligations generated - during the interpretation of {\term$_2$} and the aforementioned - coercion derivation are solved. -\item {\tt Program Definition {\ident} {\binder$_1$}\ldots{\binder$_n$} - {\tt :}\term$_1$ {\tt :=} {\term$_2$}.}\\ - This is equivalent to \\ - {\tt Program Definition\,{\ident}\,{\tt :\,forall} % - {\binder$_1$}\ldots{\binder$_n$}{\tt ,}\,\term$_1$\,{\tt :=}} \\ - \qquad {\tt fun}\,{\binder$_1$}\ldots{\binder$_n$}\,{\tt =>}\,{\term$_2$}\,% - {\tt .} -\end{Variants} - -\begin{ErrMsgs} -\item \errindex{In environment {\dots} the term: {\term$_2$} does not have type - {\term$_1$}}.\\ - \texttt{Actually, it has type {\term$_3$}}. -\end{ErrMsgs} - -\SeeAlso Sections \ref{Opaque}, \ref{Transparent}, \ref{unfold} - -\subsection{\tt Program Fixpoint {\ident} {\params} {\tt \{order\}} : type := \term - \comindex{Program Fixpoint} - \label{ProgramFixpoint}} - -The structural fixpoint operator behaves just like the one of Coq -(see Section~\ref{Fixpoint}), except it may also generate obligations. -It works with mutually recursive definitions too. - -\begin{coq_eval} -Admit Obligations. -\end{coq_eval} -\begin{coq_example} -Program Fixpoint div2 (n : nat) : { x : nat | n = 2 * x \/ n = 2 * x + 1 } := - match n with - | S (S p) => S (div2 p) - | _ => O - end. -\end{coq_example} - -Here we have one obligation for each branch (branches for \verb:0: and \verb:(S 0): are -automatically generated by the pattern-matching compilation algorithm). -\begin{coq_example} - Obligation 1. -\end{coq_example} - -One can use a well-founded order or a measure as termination orders using the syntax: -\begin{coq_eval} -Reset Initial. -Require Import Arith. -Require Import Program. -\end{coq_eval} -\begin{coq_example*} -Program Fixpoint div2 (n : nat) {measure n} : - { x : nat | n = 2 * x \/ n = 2 * x + 1 } := - match n with - | S (S p) => S (div2 p) - | _ => O - end. -\end{coq_example*} - -The order annotation can be either: -\begin{itemize} -\item {\tt measure f (R)?} where {\tt f} is a value of type {\tt X} - computed on any subset of the arguments and the optional - (parenthesised) term {\tt (R)} is a relation - on {\tt X}. By default {\tt X} defaults to {\tt nat} and {\tt R} to - {\tt lt}. -\item {\tt wf R x} which is equivalent to {\tt measure x (R)}. -\end{itemize} - -\paragraph{Caution} -When defining structurally recursive functions, the -generated obligations should have the prototype of the currently defined functional -in their context. In this case, the obligations should be transparent -(e.g. defined using {\tt Defined}) so that the guardedness condition on -recursive calls can be checked by the -kernel's type-checker. There is an optimization in the generation of -obligations which gets rid of the hypothesis corresponding to the -functionnal when it is not necessary, so that the obligation can be -declared opaque (e.g. using {\tt Qed}). However, as soon as it appears in the -context, the proof of the obligation is \emph{required} to be declared transparent. - -No such problems arise when using measures or well-founded recursion. - -\subsection{\tt Program Lemma {\ident} : type. - \comindex{Program Lemma} - \label{ProgramLemma}} - -The \Russell\ language can also be used to type statements of logical -properties. It will generate obligations, try to solve them -automatically and fail if some unsolved obligations remain. -In this case, one can first define the lemma's -statement using {\tt Program Definition} and use it as the goal afterwards. -Otherwise the proof will be started with the elobarted version as a goal. -The {\tt Program} prefix can similarly be used as a prefix for {\tt Variable}, {\tt - Hypothesis}, {\tt Axiom} etc... - -\section{Solving obligations} -The following commands are available to manipulate obligations. The -optional identifier is used when multiple functions have unsolved -obligations (e.g. when defining mutually recursive blocks). The optional -tactic is replaced by the default one if not specified. - -\begin{itemize} -\item {\tt [Local|Global] Obligation Tactic := \tacexpr}\comindex{Obligation Tactic} - Sets the default obligation - solving tactic applied to all obligations automatically, whether to - solve them or when starting to prove one, e.g. using {\tt Next}. - Local makes the setting last only for the current module. Inside - sections, local is the default. -\item {\tt Show Obligation Tactic}\comindex{Show Obligation Tactic} - Displays the current default tactic. -\item {\tt Obligations [of \ident]}\comindex{Obligations} Displays all remaining - obligations. -\item {\tt Obligation num [of \ident]}\comindex{Obligation} Start the proof of - obligation {\tt num}. -\item {\tt Next Obligation [of \ident]}\comindex{Next Obligation} Start the proof of the next - unsolved obligation. -\item {\tt Solve Obligations [of \ident] [using - \tacexpr]}\comindex{Solve Obligations} - Tries to solve - each obligation of \ident using the given tactic or the default one. -\item {\tt Solve All Obligations [using \tacexpr]} Tries to solve - each obligation of every program using the given tactic or the default - one (useful for mutually recursive definitions). -\item {\tt Admit Obligations [of \ident]}\comindex{Admit Obligations} - Admits all obligations (does not work with structurally recursive programs). -\item {\tt Preterm [of \ident]}\comindex{Preterm} - Shows the term that will be fed to - the kernel once the obligations are solved. Useful for debugging. -\item {\tt Set Transparent Obligations}\comindex{Set Transparent Obligations} - Control whether all obligations should be declared as transparent (the - default), or if the system should infer which obligations can be declared opaque. -\end{itemize} - -The module {\tt Coq.Program.Tactics} defines the default tactic for solving -obligations called {\tt program\_simpl}. Importing -{\tt Coq.Program.Program} also adds some useful notations, as documented in the file itself. - -\section{Frequently Asked Questions - \label{ProgramFAQ}} - -\begin{itemize} -\item {Ill-formed recursive definitions} - This error can happen when one tries to define a - function by structural recursion on a subset object, which means the Coq - function looks like: - - \verb$Program Fixpoint f (x : A | P) := match x with A b => f b end.$ - - Supposing $b : A$, the argument at the recursive call to f is not a - direct subterm of x as b is wrapped inside an {\tt exist} constructor to build - an object of type \verb${x : A | P}$. Hence the definition is rejected - by the guardedness condition checker. However one can use - wellfounded recursion on subset objects like this: - -\begin{verbatim} -Program Fixpoint f (x : A | P) { measure (size x) } := - match x with A b => f b end. -\end{verbatim} - - One will then just have to prove that the measure decreases at each recursive - call. There are three drawbacks though: - \begin{enumerate} - \item A measure function has to be defined; - \item The reduction is a little more involved, although it works well - using lazy evaluation; - \item Mutual recursion on the underlying inductive type isn't possible - anymore, but nested mutual recursion is always possible. - \end{enumerate} -\end{itemize} - -%%% Local Variables: -%%% mode: latex -%%% TeX-master: "Reference-Manual" -%%% compile-command: "BIBINPUTS=\".\" make QUICK=1 -C ../.. doc/refman/Reference-Manual.pdf" -%%% End: diff --git a/doc/refman/RefMan-add.tex b/doc/refman/RefMan-add.tex deleted file mode 100644 index 9d7ca7b1..00000000 --- a/doc/refman/RefMan-add.tex +++ /dev/null @@ -1,60 +0,0 @@ -\chapter[List of additional documentation]{List of additional documentation\label{Addoc}} - -\section[Tutorials]{Tutorials\label{Tutorial}} -A companion volume to this reference manual, the \Coq\ Tutorial, is -aimed at gently introducing new users to developing proofs in \Coq\ -without assuming prior knowledge of type theory. In a second step, the -user can read also the tutorial on recursive types (document {\tt -RecTutorial.ps}). - -\section[The \Coq\ standard library]{The \Coq\ standard library\label{Addoc-library}} -A brief description of the \Coq\ standard library is given in the additional -document {\tt Library.dvi}. - -\section[Installation and un-installation procedures]{Installation and un-installation procedures\label{Addoc-install}} -A \verb!INSTALL! file in the distribution explains how to install -\Coq. - -\section[{\tt Extraction} of programs]{{\tt Extraction} of programs\label{Addoc-extract}} -{\tt Extraction} is a package offering some special facilities to -extract ML program files. It is described in the separate document -{\tt Extraction.dvi} -\index{Extraction of programs} - -\section[{\tt Program}]{A tool for {\tt Program}-ing\label{Addoc-program}} -{\tt Program} is a package offering some special facilities to -extract ML program files. It is described in the separate document -{\tt Program.dvi} -\index{Program-ing} - -\section[Proof printing in {\tt Natural} language]{Proof printing in {\tt Natural} language\label{Addoc-natural}} -{\tt Natural} is a tool to print proofs in natural language. -It is described in the separate document {\tt Natural.dvi}. -\index{Natural@{\tt Print Natural}} -\index{Printing in natural language} - -\section[The {\tt Omega} decision tactic]{The {\tt Omega} decision tactic\label{Addoc-omega}} -{\bf Omega} is a tactic to automatically solve arithmetical goals in -Presburger arithmetic (i.e. arithmetic without multiplication). -It is described in the separate document {\tt Omega.dvi}. -\index{Omega@{\tt Omega}} - -\section[Simplification on rings]{Simplification on rings\label{Addoc-polynom}} -A documentation of the package {\tt polynom} (simplification on rings) -can be found in the document {\tt Polynom.dvi} -\index{Polynom@{\tt Polynom}} -\index{Simplification on rings} - -%\section[Anomalies]{Anomalies\label{Addoc-anomalies}} -%The separate document {\tt Anomalies.*} gives a list of known -%anomalies and bugs of the system. Before communicating us an -%anomalous behavior, please check first whether it has been already -%reported in this document. - -% $Id: RefMan-add.tex 10061 2007-08-08 13:14:05Z msozeau $ - - -%%% Local Variables: -%%% mode: latex -%%% TeX-master: "Reference-Manual" -%%% End: diff --git a/doc/refman/RefMan-cic.tex b/doc/refman/RefMan-cic.tex deleted file mode 100644 index dab164e7..00000000 --- a/doc/refman/RefMan-cic.tex +++ /dev/null @@ -1,1716 +0,0 @@ -\chapter[Calculus of Inductive Constructions]{Calculus of Inductive Constructions -\label{Cic} -\index{Cic@\textsc{CIC}} -\index{pCic@p\textsc{CIC}} -\index{Calculus of (Co)Inductive Constructions}} - -The underlying formal language of {\Coq} is a {\em Calculus of - Constructions} with {\em Inductive Definitions}. It is presented in -this chapter. -For {\Coq} version V7, this Calculus was known as the -{\em Calculus of (Co)Inductive Constructions}\index{Calculus of - (Co)Inductive Constructions} (\iCIC\ in short). -The underlying calculus of {\Coq} version V8.0 and up is a weaker - calculus where the sort \Set{} satisfies predicative rules. -We call this calculus the -{\em Predicative Calculus of (Co)Inductive - Constructions}\index{Predicative Calculus of - (Co)Inductive Constructions} (\pCIC\ in short). -In Section~\ref{impredicativity} we give the extra-rules for \iCIC. A - compiling option of \Coq{} allows to type-check theories in this - extended system. - -In \CIC\, all objects have a {\em type}. There are types for functions (or -programs), there are atomic types (especially datatypes)... but also -types for proofs and types for the types themselves. -Especially, any object handled in the formalism must belong to a -type. For instance, the statement {\it ``for all x, P''} is not -allowed in type theory; you must say instead: {\it ``for all x -belonging to T, P''}. The expression {\it ``x belonging to T''} is -written {\it ``x:T''}. One also says: {\it ``x has type T''}. -The terms of {\CIC} are detailed in Section~\ref{Terms}. - -In \CIC\, there is an internal reduction mechanism. In particular, it -allows to decide if two programs are {\em intentionally} equal (one -says {\em convertible}). Convertibility is presented in section -\ref{convertibility}. - -The remaining sections are concerned with the type-checking of terms. -The beginner can skip them. - -The reader seeking a background on the Calculus of Inductive -Constructions may read several papers. Giménez and Castéran~\cite{GimCas05} -provide -an introduction to inductive and coinductive definitions in Coq. In -their book~\cite{CoqArt}, Bertot and Castéran give a precise -description of the \CIC{} based on numerous practical examples. -Barras~\cite{Bar99}, Werner~\cite{Wer94} and -Paulin-Mohring~\cite{Moh97} are the most recent theses dealing with -Inductive Definitions. Coquand-Huet~\cite{CoHu85a,CoHu85b,CoHu86} -introduces the Calculus of Constructions. Coquand-Paulin~\cite{CoPa89} -extended this calculus to inductive definitions. The {\CIC} is a -formulation of type theory including the possibility of inductive -constructions, Barendregt~\cite{Bar91} studies the modern form of type -theory. - -\section[The terms]{The terms\label{Terms}} - -In most type theories, one usually makes a syntactic distinction -between types and terms. This is not the case for \CIC\ which defines -both types and terms in the same syntactical structure. This is -because the type-theory itself forces terms and types to be defined in -a mutual recursive way and also because similar constructions can be -applied to both terms and types and consequently can share the same -syntactic structure. - -Consider for instance the $\ra$ constructor and assume \nat\ is the -type of natural numbers. Then $\ra$ is used both to denote -$\nat\ra\nat$ which is the type of functions from \nat\ to \nat, and -to denote $\nat \ra \Prop$ which is the type of unary predicates over -the natural numbers. Consider abstraction which builds functions. It -serves to build ``ordinary'' functions as $\kw{fun}~x:\nat \Ra ({\tt mult} ~x~x)$ (assuming {\tt mult} is already defined) but may build also -predicates over the natural numbers. For instance $\kw{fun}~x:\nat \Ra -(x=x)$ will -represent a predicate $P$, informally written in mathematics -$P(x)\equiv x=x$. If $P$ has type $\nat \ra \Prop$, $(P~x)$ is a -proposition, furthermore $\kw{forall}~x:\nat,(P~x)$ will represent the type of -functions which associate to each natural number $n$ an object of -type $(P~n)$ and consequently represent proofs of the formula -``$\forall x.P(x)$''. - -\subsection[Sorts]{Sorts\label{Sorts} -\index{Sorts}} -Types are seen as terms of the language and then should belong to -another type. The type of a type is always a constant of the language -called a {\em sort}. - -The two basic sorts in the language of \CIC\ are \Set\ and \Prop. - -The sort \Prop\ intends to be the type of logical propositions. If -$M$ is a logical proposition then it denotes a class, namely the class -of terms representing proofs of $M$. An object $m$ belonging to $M$ -witnesses the fact that $M$ is true. An object of type \Prop\ is -called a {\em proposition}. - -The sort \Set\ intends to be the type of specifications. This includes -programs and the usual sets such as booleans, naturals, lists -etc. - -These sorts themselves can be manipulated as ordinary terms. -Consequently sorts also should be given a type. Because assuming -simply that \Set\ has type \Set\ leads to an inconsistent theory, we -have infinitely many sorts in the language of \CIC. These are, in -addition to \Set\ and \Prop\, a hierarchy of universes \Type$(i)$ -for any integer $i$. We call \Sort\ the set of sorts -which is defined by: -\[\Sort \equiv \{\Prop,\Set,\Type(i)| i \in \NN\} \] -\index{Type@{\Type}} -\index{Prop@{\Prop}} -\index{Set@{\Set}} -The sorts enjoy the following properties: {\Prop:\Type(0)}, {\Set:\Type(0)} and - {\Type$(i)$:\Type$(i+1)$}. - -The user will never mention explicitly the index $i$ when referring to -the universe \Type$(i)$. One only writes \Type. The -system itself generates for each instance of \Type\ a new -index for the universe and checks that the constraints between these -indexes can be solved. From the user point of view we consequently -have {\sf Type :Type}. - -We shall make precise in the typing rules the constraints between the -indexes. - -\paragraph{Implementation issues} -In practice, the {\Type} hierarchy is implemented using algebraic -universes. An algebraic universe $u$ is either a variable (a qualified -identifier with a number) or a successor of an algebraic universe (an -expression $u+1$), or an upper bound of algebraic universes (an -expression $max(u_1,...,u_n)$), or the base universe (the expression -$0$) which corresponds, in the arity of sort-polymorphic inductive -types, to the predicative sort {\Set}. A graph of constraints between -the universe variables is maintained globally. To ensure the existence -of a mapping of the universes to the positive integers, the graph of -constraints must remain acyclic. Typing expressions that violate the -acyclicity of the graph of constraints results in a \errindex{Universe -inconsistency} error (see also Section~\ref{PrintingUniverses}). - -\subsection{Constants} -Besides the sorts, the language also contains constants denoting -objects in the environment. These constants may denote previously -defined objects but also objects related to inductive definitions -(either the type itself or one of its constructors or destructors). - -\medskip\noindent {\bf Remark. } In other presentations of \CIC, -the inductive objects are not seen as -external declarations but as first-class terms. Usually the -definitions are also completely ignored. This is a nice theoretical -point of view but not so practical. An inductive definition is -specified by a possibly huge set of declarations, clearly we want to -share this specification among the various inductive objects and not -to duplicate it. So the specification should exist somewhere and the -various objects should refer to it. We choose one more level of -indirection where the objects are just represented as constants and -the environment gives the information on the kind of object the -constant refers to. - -\medskip -Our inductive objects will be manipulated as constants declared in the -environment. This roughly corresponds to the way they are actually -implemented in the \Coq\ system. It is simple to map this presentation -in a theory where inductive objects are represented by terms. - -\subsection{Terms} - -Terms are built from variables, global names, constructors, -abstraction, application, local declarations bindings (``let-in'' -expressions) and product. - -From a syntactic point of view, types cannot be distinguished from terms, -except that they cannot start by an abstraction, and that if a term is -a sort or a product, it should be a type. - -More precisely the language of the {\em Calculus of Inductive - Constructions} is built from the following rules: - -\begin{enumerate} -\item the sorts {\sf Set, Prop, Type} are terms. -\item names for global constants of the environment are terms. -\item variables are terms. -\item if $x$ is a variable and $T$, $U$ are terms then $\forall~x:T,U$ - ($\kw{forall}~x:T,U$ in \Coq{} concrete syntax) is a term. If $x$ - occurs in $U$, $\forall~x:T,U$ reads as {\it ``for all x of type T, - U''}. As $U$ depends on $x$, one says that $\forall~x:T,U$ is a - {\em dependent product}. If $x$ doesn't occurs in $U$ then - $\forall~x:T,U$ reads as {\it ``if T then U''}. A non dependent - product can be written: $T \rightarrow U$. -\item if $x$ is a variable and $T$, $U$ are terms then $\lb~x:T \mto U$ - ($\kw{fun}~x:T\Ra U$ in \Coq{} concrete syntax) is a term. This is a - notation for the $\lambda$-abstraction of - $\lambda$-calculus\index{lambda-calculus@$\lambda$-calculus} - \cite{Bar81}. The term $\lb~x:T \mto U$ is a function which maps - elements of $T$ to $U$. -\item if $T$ and $U$ are terms then $(T\ U)$ is a term - ($T~U$ in \Coq{} concrete syntax). The term $(T\ - U)$ reads as {\it ``T applied to U''}. -\item if $x$ is a variable, and $T$, $U$ are terms then - $\kw{let}~x:=T~\kw{in}~U$ is a - term which denotes the term $U$ where the variable $x$ is locally - bound to $T$. This stands for the common ``let-in'' construction of - functional programs such as ML or Scheme. -\end{enumerate} - -\paragraph{Notations.} Application associates to the left such that -$(t~t_1\ldots t_n)$ represents $(\ldots (t~t_1)\ldots t_n)$. The -products and arrows associate to the right such that $\forall~x:A,B\ra C\ra -D$ represents $\forall~x:A,(B\ra (C\ra D))$. One uses sometimes -$\forall~x~y:A,B$ or -$\lb~x~y:A\mto B$ to denote the abstraction or product of several variables -of the same type. The equivalent formulation is $\forall~x:A, \forall y:A,B$ or -$\lb~x:A \mto \lb y:A \mto B$ - -\paragraph{Free variables.} -The notion of free variables is defined as usual. In the expressions -$\lb~x:T\mto U$ and $\forall x:T, U$ the occurrences of $x$ in $U$ -are bound. They are represented by de Bruijn indexes in the internal -structure of terms. - -\paragraph[Substitution.]{Substitution.\index{Substitution}} -The notion of substituting a term $t$ to free occurrences of a -variable $x$ in a term $u$ is defined as usual. The resulting term -is written $\subst{u}{x}{t}$. - - -\section[Typed terms]{Typed terms\label{Typed-terms}} - -As objects of type theory, terms are subjected to {\em type -discipline}. The well typing of a term depends on an environment which -consists in a global environment (see below) and a local context. - -\paragraph{Local context.} -A {\em local context} (or shortly context) is an ordered list of -declarations of variables. The declaration of some variable $x$ is -either an assumption, written $x:T$ ($T$ is a type) or a definition, -written $x:=t:T$. We use brackets to write contexts. A -typical example is $[x:T;y:=u:U;z:V]$. Notice that the variables -declared in a context must be distinct. If $\Gamma$ declares some $x$, -we write $x \in \Gamma$. By writing $(x:T) \in \Gamma$ we mean that -either $x:T$ is an assumption in $\Gamma$ or that there exists some $t$ such -that $x:=t:T$ is a definition in $\Gamma$. If $\Gamma$ defines some -$x:=t:T$, we also write $(x:=t:T) \in \Gamma$. Contexts must be -themselves {\em well formed}. For the rest of the chapter, the -notation $\Gamma::(y:T)$ (resp. $\Gamma::(y:=t:T)$) denotes the context -$\Gamma$ enriched with the declaration $y:T$ (resp. $y:=t:T$). The -notation $[]$ denotes the empty context. \index{Context} - -% Does not seem to be used further... -% Si dans l'explication WF(E)[Gamma] concernant les constantes -% definies ds un contexte - -We define the inclusion of two contexts $\Gamma$ and $\Delta$ (written -as $\Gamma \subset \Delta$) as the property, for all variable $x$, -type $T$ and term $t$, if $(x:T) \in \Gamma$ then $(x:T) \in \Delta$ -and if $(x:=t:T) \in \Gamma$ then $(x:=t:T) \in \Delta$. -%We write -% $|\Delta|$ for the length of the context $\Delta$, that is for the number -% of declarations (assumptions or definitions) in $\Delta$. - -A variable $x$ is said to be free in $\Gamma$ if $\Gamma$ contains a -declaration $y:T$ such that $x$ is free in $T$. - -\paragraph[Environment.]{Environment.\index{Environment}} -Because we are manipulating global declarations (constants and global -assumptions), we also need to consider a global environment $E$. - -An environment is an ordered list of declarations of global -names. Declarations are either assumptions or ``standard'' -definitions, that is abbreviations for well-formed terms -but also definitions of inductive objects. In the latter -case, an object in the environment will define one or more constants -(that is types and constructors, see Section~\ref{Cic-inductive-definitions}). - -An assumption will be represented in the environment as -\Assum{\Gamma}{c}{T} which means that $c$ is assumed of some type $T$ -well-defined in some context $\Gamma$. An (ordinary) definition will -be represented in the environment as \Def{\Gamma}{c}{t}{T} which means -that $c$ is a constant which is valid in some context $\Gamma$ whose -value is $t$ and type is $T$. - -The rules for inductive definitions (see section -\ref{Cic-inductive-definitions}) have to be considered as assumption -rules to which the following definitions apply: if the name $c$ is -declared in $E$, we write $c \in E$ and if $c:T$ or $c:=t:T$ is -declared in $E$, we write $(c : T) \in E$. - -\paragraph[Typing rules.]{Typing rules.\label{Typing-rules}\index{Typing rules}} -In the following, we assume $E$ is a valid environment w.r.t. -inductive definitions. We define simultaneously two -judgments. The first one \WTEG{t}{T} means the term $t$ is well-typed -and has type $T$ in the environment $E$ and context $\Gamma$. The -second judgment \WFE{\Gamma} means that the environment $E$ is -well-formed and the context $\Gamma$ is a valid context in this -environment. It also means a third property which makes sure that any -constant in $E$ was defined in an environment which is included in -$\Gamma$ -\footnote{This requirement could be relaxed if we instead introduced - an explicit mechanism for instantiating constants. At the external - level, the Coq engine works accordingly to this view that all the - definitions in the environment were built in a sub-context of the - current context.}. - -A term $t$ is well typed in an environment $E$ iff there exists a -context $\Gamma$ and a term $T$ such that the judgment \WTEG{t}{T} can -be derived from the following rules. -\begin{description} -\item[W-E] \inference{\WF{[]}{[]}} -\item[W-S] % Ce n'est pas vrai : x peut apparaitre plusieurs fois dans Gamma -\inference{\frac{\WTEG{T}{s}~~~~s \in \Sort~~~~x \not\in - \Gamma % \cup E - } - {\WFE{\Gamma::(x:T)}}~~~~~ - \frac{\WTEG{t}{T}~~~~x \not\in - \Gamma % \cup E - }{\WFE{\Gamma::(x:=t:T)}}} -\item[Def] \inference{\frac{\WTEG{t}{T}~~~c \notin E \cup \Gamma} - {\WF{E;\Def{\Gamma}{c}{t}{T}}{\Gamma}}} -\item[Assum] \inference{\frac{\WTEG{T}{s}~~~~s \in \Sort~~~~c \notin E \cup \Gamma} - {\WF{E;\Assum{\Gamma}{c}{T}}{\Gamma}}} -\item[Ax] \index{Typing rules!Ax} -\inference{\frac{\WFE{\Gamma}}{\WTEG{\Prop}{\Type(p)}}~~~~~ -\frac{\WFE{\Gamma}}{\WTEG{\Set}{\Type(q)}}} -\inference{\frac{\WFE{\Gamma}~~~~i<j}{\WTEG{\Type(i)}{\Type(j)}}} -\item[Var]\index{Typing rules!Var} - \inference{\frac{ \WFE{\Gamma}~~~~~(x:T) \in \Gamma~~\mbox{or}~~(x:=t:T) \in \Gamma~\mbox{for some $t$}}{\WTEG{x}{T}}} -\item[Const] \index{Typing rules!Const} -\inference{\frac{\WFE{\Gamma}~~~~(c:T) \in E~~\mbox{or}~~(c:=t:T) \in E~\mbox{for some $t$} }{\WTEG{c}{T}}} -\item[Prod] \index{Typing rules!Prod} -\inference{\frac{\WTEG{T}{s}~~~~s \in \Sort~~~ - \WTE{\Gamma::(x:T)}{U}{\Prop}} - { \WTEG{\forall~x:T,U}{\Prop}}} -\inference{\frac{\WTEG{T}{s}~~~~s \in\{\Prop, \Set\}~~~~~~ - \WTE{\Gamma::(x:T)}{U}{\Set}} - { \WTEG{\forall~x:T,U}{\Set}}} -\inference{\frac{\WTEG{T}{\Type(i)}~~~~i\leq k~~~ - \WTE{\Gamma::(x:T)}{U}{\Type(j)}~~~j \leq k} - {\WTEG{\forall~x:T,U}{\Type(k)}}} -\item[Lam]\index{Typing rules!Lam} -\inference{\frac{\WTEG{\forall~x:T,U}{s}~~~~ \WTE{\Gamma::(x:T)}{t}{U}} - {\WTEG{\lb~x:T\mto t}{\forall x:T, U}}} -\item[App]\index{Typing rules!App} - \inference{\frac{\WTEG{t}{\forall~x:U,T}~~~~\WTEG{u}{U}} - {\WTEG{(t\ u)}{\subst{T}{x}{u}}}} -\item[Let]\index{Typing rules!Let} -\inference{\frac{\WTEG{t}{T}~~~~ \WTE{\Gamma::(x:=t:T)}{u}{U}} - {\WTEG{\kw{let}~x:=t~\kw{in}~u}{\subst{U}{x}{t}}}} -\end{description} - -\Rem We may have $\kw{let}~x:=t~\kw{in}~u$ -well-typed without having $((\lb~x:T\mto u)~t)$ well-typed (where -$T$ is a type of $t$). This is because the value $t$ associated to $x$ -may be used in a conversion rule (see Section~\ref{conv-rules}). - -\section[Conversion rules]{Conversion rules\index{Conversion rules} -\label{conv-rules}} -\paragraph[$\beta$-reduction.]{$\beta$-reduction.\label{beta}\index{beta-reduction@$\beta$-reduction}} - -We want to be able to identify some terms as we can identify the -application of a function to a given argument with its result. For -instance the identity function over a given type $T$ can be written -$\lb~x:T\mto x$. In any environment $E$ and context $\Gamma$, we want to identify any object $a$ (of type $T$) with the -application $((\lb~x:T\mto x)~a)$. We define for this a {\em reduction} (or a -{\em conversion}) rule we call $\beta$: -\[ \WTEGRED{((\lb~x:T\mto - t)~u)}{\triangleright_{\beta}}{\subst{t}{x}{u}} \] -We say that $\subst{t}{x}{u}$ is the {\em $\beta$-contraction} of -$((\lb~x:T\mto t)~u)$ and, conversely, that $((\lb~x:T\mto t)~u)$ -is the {\em $\beta$-expansion} of $\subst{t}{x}{u}$. - -According to $\beta$-reduction, terms of the {\em Calculus of - Inductive Constructions} enjoy some fundamental properties such as -confluence, strong normalization, subject reduction. These results are -theoretically of great importance but we will not detail them here and -refer the interested reader to \cite{Coq85}. - -\paragraph[$\iota$-reduction.]{$\iota$-reduction.\label{iota}\index{iota-reduction@$\iota$-reduction}} -A specific conversion rule is associated to the inductive objects in -the environment. We shall give later on (see Section~\ref{iotared}) the -precise rules but it just says that a destructor applied to an object -built from a constructor behaves as expected. This reduction is -called $\iota$-reduction and is more precisely studied in -\cite{Moh93,Wer94}. - - -\paragraph[$\delta$-reduction.]{$\delta$-reduction.\label{delta}\index{delta-reduction@$\delta$-reduction}} - -We may have defined variables in contexts or constants in the global -environment. It is legal to identify such a reference with its value, -that is to expand (or unfold) it into its value. This -reduction is called $\delta$-reduction and shows as follows. - -$$\WTEGRED{x}{\triangleright_{\delta}}{t}~~~~~\mbox{if $(x:=t:T) \in \Gamma$}~~~~~~~~~\WTEGRED{c}{\triangleright_{\delta}}{t}~~~~~\mbox{if $(c:=t:T) \in E$}$$ - - -\paragraph[$\zeta$-reduction.]{$\zeta$-reduction.\label{zeta}\index{zeta-reduction@$\zeta$-reduction}} - -Coq allows also to remove local definitions occurring in terms by -replacing the defined variable by its value. The declaration being -destroyed, this reduction differs from $\delta$-reduction. It is -called $\zeta$-reduction and shows as follows. - -$$\WTEGRED{\kw{let}~x:=u~\kw{in}~t}{\triangleright_{\zeta}}{\subst{t}{x}{u}}$$ - -\paragraph[Convertibility.]{Convertibility.\label{convertibility} -\index{beta-reduction@$\beta$-reduction}\index{iota-reduction@$\iota$-reduction}\index{delta-reduction@$\delta$-reduction}\index{zeta-reduction@$\zeta$-reduction}} - -Let us write $\WTEGRED{t}{\triangleright}{u}$ for the contextual closure of the relation $t$ reduces to $u$ in the environment $E$ and context $\Gamma$ with one of the previous reduction $\beta$, $\iota$, $\delta$ or $\zeta$. - -We say that two terms $t_1$ and $t_2$ are {\em convertible} (or {\em - equivalent)} in the environment $E$ and context $\Gamma$ iff there exists a term $u$ such that $\WTEGRED{t_1}{\triangleright \ldots \triangleright}{u}$ -and $\WTEGRED{t_2}{\triangleright \ldots \triangleright}{u}$. -We then write $\WTEGCONV{t_1}{t_2}$. - -The convertibility relation allows to introduce a new typing rule -which says that two convertible well-formed types have the same -inhabitants. - -At the moment, we did not take into account one rule between universes -which says that any term in a universe of index $i$ is also a term in -the universe of index $i+1$. This property is included into the -conversion rule by extending the equivalence relation of -convertibility into an order inductively defined by: -\begin{enumerate} -\item if $\WTEGCONV{t}{u}$ then $\WTEGLECONV{t}{u}$, -\item if $i \leq j$ then $\WTEGLECONV{\Type(i)}{\Type(j)}$, -\item for any $i$, $\WTEGLECONV{\Prop}{\Type(i)}$, -\item for any $i$, $\WTEGLECONV{\Set}{\Type(i)}$, -\item $\WTEGLECONV{\Prop}{\Set}$, -\item if $\WTEGCONV{T}{U}$ and $\WTELECONV{\Gamma::(x:T)}{T'}{U'}$ then $\WTEGLECONV{\forall~x:T,T'}{\forall~x:U,U'}$. -\end{enumerate} - -The conversion rule is now exactly: - -\begin{description}\label{Conv} -\item[Conv]\index{Typing rules!Conv} - \inference{ - \frac{\WTEG{U}{s}~~~~\WTEG{t}{T}~~~~\WTEGLECONV{T}{U}}{\WTEG{t}{U}}} - \end{description} - - -\paragraph{$\eta$-conversion. -\label{eta} -\index{eta-conversion@$\eta$-conversion} -\index{eta-reduction@$\eta$-reduction}} - -An other important rule is the $\eta$-conversion. It is to identify -terms over a dummy abstraction of a variable followed by an -application of this variable. Let $T$ be a type, $t$ be a term in -which the variable $x$ doesn't occurs free. We have -\[ \WTEGRED{\lb~x:T\mto (t\ x)}{\triangleright}{t} \] -Indeed, as $x$ doesn't occur free in $t$, for any $u$ one -applies to $\lb~x:T\mto (t\ x)$, it $\beta$-reduces to $(t\ u)$. So -$\lb~x:T\mto (t\ x)$ and $t$ can be identified. - -\Rem The $\eta$-reduction is not taken into account in the -convertibility rule of \Coq. - -\paragraph[Normal form.]{Normal form.\index{Normal form}\label{Normal-form}\label{Head-normal-form}\index{Head normal form}} -A term which cannot be any more reduced is said to be in {\em normal - form}. There are several ways (or strategies) to apply the reduction -rule. Among them, we have to mention the {\em head reduction} which -will play an important role (see Chapter~\ref{Tactics}). Any term can -be written as $\lb~x_1:T_1\mto \ldots \lb x_k:T_k \mto -(t_0\ t_1\ldots t_n)$ where -$t_0$ is not an application. We say then that $t_0$ is the {\em head - of $t$}. If we assume that $t_0$ is $\lb~x:T\mto u_0$ then one step of -$\beta$-head reduction of $t$ is: -\[\lb~x_1:T_1\mto \ldots \lb x_k:T_k\mto (\lb~x:T\mto u_0\ t_1\ldots t_n) -~\triangleright ~ \lb~(x_1:T_1)\ldots(x_k:T_k)\mto -(\subst{u_0}{x}{t_1}\ t_2 \ldots t_n)\] -Iterating the process of head reduction until the head of the reduced -term is no more an abstraction leads to the {\em $\beta$-head normal - form} of $t$: -\[ t \triangleright \ldots \triangleright -\lb~x_1:T_1\mto \ldots\lb x_k:T_k\mto (v\ u_1 -\ldots u_m)\] -where $v$ is not an abstraction (nor an application). Note that the -head normal form must not be confused with the normal form since some -$u_i$ can be reducible. - -Similar notions of head-normal forms involving $\delta$, $\iota$ and $\zeta$ -reductions or any combination of those can also be defined. - -\section{Derived rules for environments} - -From the original rules of the type system, one can derive new rules -which change the context of definition of objects in the environment. -Because these rules correspond to elementary operations in the \Coq\ -engine used in the discharge mechanism at the end of a section, we -state them explicitly. - -\paragraph{Mechanism of substitution.} - -One rule which can be proved valid, is to replace a term $c$ by its -value in the environment. As we defined the substitution of a term for -a variable in a term, one can define the substitution of a term for a -constant. One easily extends this substitution to contexts and -environments. - -\paragraph{Substitution Property:} -\inference{\frac{\WF{E;\Def{\Gamma}{c}{t}{T}; F}{\Delta}} - {\WF{E; \subst{F}{c}{t}}{\subst{\Delta}{c}{t}}}} - - -\paragraph{Abstraction.} - -One can modify the context of definition of a constant $c$ by -abstracting a constant with respect to the last variable $x$ of its -defining context. For doing that, we need to check that the constants -appearing in the body of the declaration do not depend on $x$, we need -also to modify the reference to the constant $c$ in the environment -and context by explicitly applying this constant to the variable $x$. -Because of the rules for building environments and terms we know the -variable $x$ is available at each stage where $c$ is mentioned. - -\paragraph{Abstracting property:} - \inference{\frac{\WF{E; \Def{\Gamma::(x:U)}{c}{t}{T}; - F}{\Delta}~~~~\WFE{\Gamma}} - {\WF{E;\Def{\Gamma}{c}{\lb~x:U\mto t}{\forall~x:U,T}; - \subst{F}{c}{(c~x)}}{\subst{\Delta}{c}{(c~x)}}}} - -\paragraph{Pruning the context.} -We said the judgment \WFE{\Gamma} means that the defining contexts of -constants in $E$ are included in $\Gamma$. If one abstracts or -substitutes the constants with the above rules then it may happen -that the context $\Gamma$ is now bigger than the one needed for -defining the constants in $E$. Because defining contexts are growing -in $E$, the minimum context needed for defining the constants in $E$ -is the same as the one for the last constant. One can consequently -derive the following property. - -\paragraph{Pruning property:} -\inference{\frac{\WF{E; \Def{\Delta}{c}{t}{T}}{\Gamma}} - {\WF{E;\Def{\Delta}{c}{t}{T}}{\Delta}}} - - -\section[Inductive Definitions]{Inductive Definitions\label{Cic-inductive-definitions}} - -A (possibly mutual) inductive definition is specified by giving the -names and the type of the inductive sets or families to be -defined and the names and types of the constructors of the inductive -predicates. An inductive declaration in the environment can -consequently be represented with two contexts (one for inductive -definitions, one for constructors). - -Stating the rules for inductive definitions in their general form -needs quite tedious definitions. We shall try to give a concrete -understanding of the rules by precising them on running examples. We -take as examples the type of natural numbers, the type of -parameterized lists over a type $A$, the relation which states that -a list has some given length and the mutual inductive definition of trees and -forests. - -\subsection{Representing an inductive definition} -\subsubsection{Inductive definitions without parameters} -As for constants, inductive definitions can be defined in a non-empty -context. \\ -We write \NInd{\Gamma}{\Gamma_I}{\Gamma_C} an inductive -definition valid in a context $\Gamma$, a -context of definitions $\Gamma_I$ and a context of constructors -$\Gamma_C$. -\paragraph{Examples.} -The inductive declaration for the type of natural numbers will be: -\[\NInd{}{\nat:\Set}{\nO:\nat,\nS:\nat\ra\nat}\] -In a context with a variable $A:\Set$, the lists of elements in $A$ are -represented by: -\[\NInd{A:\Set}{\List:\Set}{\Nil:\List,\cons : A \ra \List \ra - \List}\] - Assuming - $\Gamma_I$ is $[I_1:A_1;\ldots;I_k:A_k]$, and $\Gamma_C$ is - $[c_1:C_1;\ldots;c_n:C_n]$, the general typing rules are, - for $1\leq j\leq k$ and $1\leq i\leq n$: - -\bigskip -\inference{\frac{\NInd{\Gamma}{\Gamma_I}{\Gamma_C} \in E}{(I_j:A_j) \in E}} - -\inference{\frac{\NInd{\Gamma}{\Gamma_I}{\Gamma_C} \in E}{(c_i:C_i) \in E}} - -\subsubsection{Inductive definitions with parameters} - -We have to slightly complicate the representation above in order to handle -the delicate problem of parameters. -Let us explain that on the example of \List. With the above definition, -the type \List\ can only be used in an environment where we -have a variable $A:\Set$. Generally one want to consider lists of -elements in different types. For constants this is easily done by abstracting -the value over the parameter. In the case of inductive definitions we -have to handle the abstraction over several objects. - -One possible way to do that would be to define the type \List\ -inductively as being an inductive family of type $\Set\ra\Set$: -\[\NInd{}{\List:\Set\ra\Set}{\Nil:(\forall A:\Set,\List~A), - \cons : (\forall A:\Set, A \ra \List~A \ra \List~A)}\] -There are drawbacks to this point of view. The -information which says that for any $A$, $(\List~A)$ is an inductively defined -\Set\ has been lost. -So we introduce two important definitions. - -\paragraph{Inductive parameters, real arguments.} -An inductive definition $\NInd{\Gamma}{\Gamma_I}{\Gamma_C}$ admits -$r$ inductive parameters if each type of constructors $(c:C)$ in -$\Gamma_C$ is such that -\[C\equiv \forall -p_1:P_1,\ldots,\forall p_r:P_r,\forall a_1:A_1, \ldots \forall a_n:A_n, -(I~p_1~\ldots p_r~t_1\ldots t_q)\] -with $I$ one of the inductive definitions in $\Gamma_I$. -We say that $q$ is the number of real arguments of the constructor -$c$. -\paragraph{Context of parameters.} -If an inductive definition $\NInd{\Gamma}{\Gamma_I}{\Gamma_C}$ admits -$r$ inductive parameters, then there exists a context $\Gamma_P$ of -size $r$, such that $\Gamma_P=[p_1:P_1;\ldots;p_r:P_r]$ and -if $(t:A) \in \Gamma_I,\Gamma_C$ then $A$ can be written as -$\forall p_1:P_1,\ldots \forall p_r:P_r,A'$. -We call $\Gamma_P$ the context of parameters of the inductive -definition and use the notation $\forall \Gamma_P,A'$ for the term $A$. -\paragraph{Remark.} -If we have a term $t$ in an instance of an -inductive definition $I$ which starts with a constructor $c$, then the -$r$ first arguments of $c$ (the parameters) can be deduced from the -type $T$ of $t$: these are exactly the $r$ first arguments of $I$ in -the head normal form of $T$. -\paragraph{Examples.} -The \List{} definition has $1$ parameter: -\[\NInd{}{\List:\Set\ra\Set}{\Nil:(\forall A:\Set, \List~A), - \cons : (\forall A:\Set, A \ra \List~A \ra \List~A)}\] -This is also the case for this more complex definition where there is -a recursive argument on a different instance of \List: -\[\NInd{}{\List:\Set\ra\Set}{\Nil:(\forall A:\Set, \List~A), - \cons : (\forall A:\Set, A \ra \List~(A \ra A) \ra \List~A)}\] -But the following definition has $0$ parameters: -\[\NInd{}{\List:\Set\ra\Set}{\Nil:(\forall A:\Set, \List~A), - \cons : (\forall A:\Set, A \ra \List~A \ra \List~(A*A))}\] - -%\footnote{ -%The interested reader may compare the above definition with the two -%following ones which have very different logical meaning:\\ -%$\NInd{}{\List:\Set}{\Nil:\List,\cons : (A:\Set)A -% \ra \List \ra \List}$ \\ -%$\NInd{}{\List:\Set\ra\Set}{\Nil:(A:\Set)(\List~A),\cons : (A:\Set)A -% \ra (\List~A\ra A) \ra (\List~A)}$.} -\paragraph{Concrete syntax.} -In the Coq system, the context of parameters is given explicitly -after the name of the inductive definitions and is shared between the -arities and the type of constructors. -% The vernacular declaration of polymorphic trees and forests will be:\\ -% \begin{coq_example*} -% Inductive Tree (A:Set) : Set := -% Node : A -> Forest A -> Tree A -% with Forest (A : Set) : Set := -% Empty : Forest A -% | Cons : Tree A -> Forest A -> Forest A -% \end{coq_example*} -% will correspond in our formalism to: -% \[\NInd{}{{\tt Tree}:\Set\ra\Set;{\tt Forest}:\Set\ra \Set} -% {{\tt Node} : \forall A:\Set, A \ra {\tt Forest}~A \ra {\tt Tree}~A, -% {\tt Empty} : \forall A:\Set, {\tt Forest}~A, -% {\tt Cons} : \forall A:\Set, {\tt Tree}~A \ra {\tt Forest}~A \ra -% {\tt Forest}~A}\] -We keep track in the syntax of the number of -parameters. - -Formally the representation of an inductive declaration -will be -\Ind{\Gamma}{p}{\Gamma_I}{\Gamma_C} for an inductive -definition valid in a context $\Gamma$ with $p$ parameters, a -context of definitions $\Gamma_I$ and a context of constructors -$\Gamma_C$. - -The definition \Ind{\Gamma}{p}{\Gamma_I}{\Gamma_C} will be -well-formed exactly when \NInd{\Gamma}{\Gamma_I}{\Gamma_C} is and -when $p$ is (less or equal than) the number of parameters in -\NInd{\Gamma}{\Gamma_I}{\Gamma_C}. - -\paragraph{Examples} -The declaration for parameterized lists is: -\[\Ind{}{1}{\List:\Set\ra\Set}{\Nil:(\forall A:\Set,\List~A),\cons : - (\forall A:\Set, A \ra \List~A \ra \List~A)}\] - -The declaration for the length of lists is: -\[\Ind{}{1}{\Length:\forall A:\Set, (\List~A)\ra \nat\ra\Prop} - {\LNil:\forall A:\Set, \Length~A~(\Nil~A)~\nO,\\ - \LCons :\forall A:\Set,\forall a:A, \forall l:(\List~A),\forall n:\nat, (\Length~A~l~n)\ra (\Length~A~(\cons~A~a~l)~(\nS~n))}\] - -The declaration for a mutual inductive definition of forests and trees is: -\[\NInd{}{\tree:\Set,\forest:\Set} - {\\~~\node:\forest \ra \tree, - \emptyf:\forest,\consf:\tree \ra \forest \ra \forest\-}\] - -These representations are the ones obtained as the result of the \Coq\ -declaration: -\begin{coq_example*} -Inductive nat : Set := - | O : nat - | S : nat -> nat. -Inductive list (A:Set) : Set := - | nil : list A - | cons : A -> list A -> list A. -\end{coq_example*} -\begin{coq_example*} -Inductive Length (A:Set) : list A -> nat -> Prop := - | Lnil : Length A (nil A) O - | Lcons : - forall (a:A) (l:list A) (n:nat), - Length A l n -> Length A (cons A a l) (S n). -Inductive tree : Set := - node : forest -> tree -with forest : Set := - | emptyf : forest - | consf : tree -> forest -> forest. -\end{coq_example*} -% The inductive declaration in \Coq\ is slightly different from the one -% we described theoretically. The difference is that in the type of -% constructors the inductive definition is explicitly applied to the -% parameters variables. -The \Coq\ type-checker verifies that all -parameters are applied in the correct manner in the conclusion of the -type of each constructors~: - -In particular, the following definition will not be accepted because -there is an occurrence of \List\ which is not applied to the parameter -variable in the conclusion of the type of {\tt cons'}: -\begin{coq_eval} -Set Printing Depth 50. -(********** The following is not correct and should produce **********) -(********* Error: The 1st argument of list' must be A in ... *********) -\end{coq_eval} -\begin{coq_example} -Inductive list' (A:Set) : Set := - | nil' : list' A - | cons' : A -> list' A -> list' (A*A). -\end{coq_example} -Since \Coq{} version 8.1, there is no restriction about parameters in -the types of arguments of constructors. The following definition is -valid: -\begin{coq_example} -Inductive list' (A:Set) : Set := - | nil' : list' A - | cons' : A -> list' (A->A) -> list' A. -\end{coq_example} - - -\subsection{Types of inductive objects} -We have to give the type of constants in an environment $E$ which -contains an inductive declaration. - -\begin{description} -\item[Ind-Const] Assuming - $\Gamma_I$ is $[I_1:A_1;\ldots;I_k:A_k]$, and $\Gamma_C$ is - $[c_1:C_1;\ldots;c_n:C_n]$, - -\inference{\frac{\Ind{\Gamma}{p}{\Gamma_I}{\Gamma_C} \in E - ~~j=1\ldots k}{(I_j:A_j) \in E}} - -\inference{\frac{\Ind{\Gamma}{p}{\Gamma_I}{\Gamma_C} \in E - ~~~~i=1.. n} - {(c_i:C_i) \in E}} -\end{description} - -\paragraph{Example.} -We have $(\List:\Set \ra \Set), (\cons:\forall~A:\Set,A\ra(\List~A)\ra -(\List~A))$, \\ -$(\Length:\forall~A:\Set, (\List~A)\ra\nat\ra\Prop)$, $\tree:\Set$ and $\forest:\Set$. - -From now on, we write $\ListA$ instead of $(\List~A)$ and $\LengthA$ -for $(\Length~A)$. - -%\paragraph{Parameters.} -%%The parameters introduce a distortion between the inside specification -%%of the inductive declaration where parameters are supposed to be -%%instantiated (this representation is appropriate for checking the -%%correctness or deriving the destructor principle) and the outside -%%typing rules where the inductive objects are seen as objects -%%abstracted with respect to the parameters. - -%In the definition of \List\ or \Length\, $A$ is a parameter because -%what is effectively inductively defined is $\ListA$ or $\LengthA$ for -%a given $A$ which is constant in the type of constructors. But when -%we define $(\LengthA~l~n)$, $l$ and $n$ are not parameters because the -%constructors manipulate different instances of this family. - -\subsection{Well-formed inductive definitions} -We cannot accept any inductive declaration because some of them lead -to inconsistent systems. We restrict ourselves to definitions which -satisfy a syntactic criterion of positivity. Before giving the formal -rules, we need a few definitions: - -\paragraph[Definitions]{Definitions\index{Positivity}\label{Positivity}} - -A type $T$ is an {\em arity of sort $s$}\index{Arity} if it converts -to the sort $s$ or to a product $\forall~x:T,U$ with $U$ an arity -of sort $s$. (For instance $A\ra \Set$ or $\forall~A:\Prop,A\ra -\Prop$ are arities of sort respectively \Set\ and \Prop). A {\em type - of constructor of $I$}\index{Type of constructor} is either a term -$(I~t_1\ldots ~t_n)$ or $\fa x:T,C$ with $C$ recursively -a {\em type of constructor of $I$}. - -\smallskip - -The type of constructor $T$ will be said to {\em satisfy the positivity -condition} for a constant $X$ in the following cases: - -\begin{itemize} -\item $T=(X~t_1\ldots ~t_n)$ and $X$ does not occur free in -any $t_i$ -\item $T=\forall~x:U,V$ and $X$ occurs only strictly positively in $U$ and -the type $V$ satisfies the positivity condition for $X$ -\end{itemize} - -The constant $X$ {\em occurs strictly positively} in $T$ in the -following cases: - -\begin{itemize} -\item $X$ does not occur in $T$ -\item $T$ converts to $(X~t_1 \ldots ~t_n)$ and $X$ does not occur in - any of $t_i$ -\item $T$ converts to $\forall~x:U,V$ and $X$ does not occur in - type $U$ but occurs strictly positively in type $V$ -\item $T$ converts to $(I~a_1 \ldots ~a_m ~ t_1 \ldots ~t_p)$ where - $I$ is the name of an inductive declaration of the form - $\Ind{\Gamma}{m}{I:A}{c_1:\forall p_1:P_1,\ldots \forall - p_m:P_m,C_1;\ldots;c_n:\forall p_1:P_1,\ldots \forall - p_m:P_m,C_n}$ - (in particular, it is not mutually defined and it has $m$ - parameters) and $X$ does not occur in any of the $t_i$, and the - (instantiated) types of constructor $C_i\{p_j/a_j\}_{j=1\ldots m}$ - of $I$ satisfy - the nested positivity condition for $X$ -%\item more generally, when $T$ is not a type, $X$ occurs strictly -%positively in $T[x:U]u$ if $X$ does not occur in $U$ but occurs -%strictly positively in $u$ -\end{itemize} - -The type of constructor $T$ of $I$ {\em satisfies the nested -positivity condition} for a constant $X$ in the following -cases: - -\begin{itemize} -\item $T=(I~b_1\ldots b_m~u_1\ldots ~u_{p})$, $I$ is an inductive - definition with $m$ parameters and $X$ does not occur in -any $u_i$ -\item $T=\forall~x:U,V$ and $X$ occurs only strictly positively in $U$ and -the type $V$ satisfies the nested positivity condition for $X$ -\end{itemize} - -\paragraph{Example} - -$X$ occurs strictly positively in $A\ra X$ or $X*A$ or $({\tt list}~ -X)$ but not in $X \ra A$ or $(X \ra A)\ra A$ nor $({\tt neg}~X)$ -assuming the notion of product and lists were already defined and {\tt - neg} is an inductive definition with declaration \Ind{}{A:\Set}{{\tt - neg}:\Set}{{\tt neg}:(A\ra{\tt False}) \ra {\tt neg}}. Assuming -$X$ has arity ${\tt nat \ra Prop}$ and {\tt ex} is the inductively -defined existential quantifier, the occurrence of $X$ in ${\tt (ex~ - nat~ \lb~n:nat\mto (X~ n))}$ is also strictly positive. - -\paragraph{Correctness rules.} -We shall now describe the rules allowing the introduction of a new -inductive definition. - -\begin{description} -\item[W-Ind] Let $E$ be an environment and - $\Gamma,\Gamma_P,\Gamma_I,\Gamma_C$ are contexts such that - $\Gamma_I$ is $[I_1:\forall \Gamma_P,A_1;\ldots;I_k:\forall - \Gamma_P,A_k]$ and $\Gamma_C$ is - $[c_1:\forall \Gamma_P,C_1;\ldots;c_n:\forall \Gamma_P,C_n]$. -\inference{ - \frac{ - (\WTE{\Gamma;\Gamma_P}{A_j}{s'_j})_{j=1\ldots k} - ~~ (\WTE{\Gamma;\Gamma_I;\Gamma_P}{C_i}{s_{q_i}})_{i=1\ldots n} -} - {\WF{E;\Ind{\Gamma}{p}{\Gamma_I}{\Gamma_C}}{\Gamma}}} -provided that the following side conditions hold: -\begin{itemize} -\item $k>0$ and all of $I_j$ and $c_i$ are distinct names for $j=1\ldots k$ and $i=1\ldots n$, -\item $p$ is the number of parameters of \NInd{\Gamma}{\Gamma_I}{\Gamma_C} - and $\Gamma_P$ is the context of parameters, -\item for $j=1\ldots k$ we have that $A_j$ is an arity of sort $s_j$ and $I_j - \notin \Gamma \cup E$, -\item for $i=1\ldots n$ we have that $C_i$ is a type of constructor of - $I_{q_i}$ which satisfies the positivity condition for $I_1 \ldots I_k$ - and $c_i \notin \Gamma \cup E$. -\end{itemize} -\end{description} -One can remark that there is a constraint between the sort of the -arity of the inductive type and the sort of the type of its -constructors which will always be satisfied for the impredicative sort -(\Prop) but may fail to define inductive definition -on sort \Set{} and generate constraints between universes for -inductive definitions in the {\Type} hierarchy. - -\paragraph{Examples.} -It is well known that existential quantifier can be encoded as an -inductive definition. -The following declaration introduces the second-order existential -quantifier $\exists X.P(X)$. -\begin{coq_example*} -Inductive exProp (P:Prop->Prop) : Prop - := exP_intro : forall X:Prop, P X -> exProp P. -\end{coq_example*} -The same definition on \Set{} is not allowed and fails~: -\begin{coq_eval} -(********** The following is not correct and should produce **********) -(*** Error: Large non-propositional inductive types must be in Type***) -\end{coq_eval} -\begin{coq_example} -Inductive exSet (P:Set->Prop) : Set - := exS_intro : forall X:Set, P X -> exSet P. -\end{coq_example} -It is possible to declare the same inductive definition in the -universe \Type. -The \texttt{exType} inductive definition has type $(\Type_i \ra\Prop)\ra -\Type_j$ with the constraint that the parameter \texttt{X} of \texttt{exT\_intro} has type $\Type_k$ with $k<j$ and $k\leq i$. -\begin{coq_example*} -Inductive exType (P:Type->Prop) : Type - := exT_intro : forall X:Type, P X -> exType P. -\end{coq_example*} -%We shall assume for the following definitions that, if necessary, we -%annotated the type of constructors such that we know if the argument -%is recursive or not. We shall write the type $(x:_R T)C$ if it is -%a recursive argument and $(x:_P T)C$ if the argument is not recursive. - -\paragraph[Sort-polymorphism of inductive families.]{Sort-polymorphism of inductive families.\index{Sort-polymorphism of inductive families}} - -From {\Coq} version 8.1, inductive families declared in {\Type} are -polymorphic over their arguments in {\Type}. - -If $A$ is an arity and $s$ a sort, we write $A_{/s}$ for the arity -obtained from $A$ by replacing its sort with $s$. Especially, if $A$ -is well-typed in some environment and context, then $A_{/s}$ is typable -by typability of all products in the Calculus of Inductive Constructions. -The following typing rule is added to the theory. - -\begin{description} -\item[Ind-Family] Let $\Ind{\Gamma}{p}{\Gamma_I}{\Gamma_C}$ be an - inductive definition. Let $\Gamma_P = [p_1:P_1;\ldots;p_{p}:P_{p}]$ - be its context of parameters, $\Gamma_I = [I_1:\forall - \Gamma_P,A_1;\ldots;I_k:\forall \Gamma_P,A_k]$ its context of - definitions and $\Gamma_C = [c_1:\forall - \Gamma_P,C_1;\ldots;c_n:\forall \Gamma_P,C_n]$ its context of - constructors, with $c_i$ a constructor of $I_{q_i}$. - - Let $m \leq p$ be the length of the longest prefix of parameters - such that the $m$ first arguments of all occurrences of all $I_j$ in - all $C_k$ (even the occurrences in the hypotheses of $C_k$) are - exactly applied to $p_1~\ldots~p_m$ ($m$ is the number of {\em - recursively uniform parameters} and the $p-m$ remaining parameters - are the {\em recursively non-uniform parameters}). Let $q_1$, - \ldots, $q_r$, with $0\leq r\leq m$, be a (possibly) partial - instantiation of the recursively uniform parameters of - $\Gamma_P$. We have: - -\inference{\frac -{\left\{\begin{array}{l} -\Ind{\Gamma}{p}{\Gamma_I}{\Gamma_C} \in E\\ -(E[\Gamma] \vdash q_l : P'_l)_{l=1\ldots r}\\ -(\WTEGLECONV{P'_l}{\subst{P_l}{p_u}{q_u}_{u=1\ldots l-1}})_{l=1\ldots r}\\ -1 \leq j \leq k -\end{array} -\right.} -{E[\Gamma] \vdash (I_j\,q_1\,\ldots\,q_r:\forall [p_{r+1}:P_{r+1};\ldots;p_{p}:P_{p}], (A_j)_{/s_j})} -} - -provided that the following side conditions hold: - -\begin{itemize} -\item $\Gamma_{P'}$ is the context obtained from $\Gamma_P$ by -replacing each $P_l$ that is an arity with $P'_l$ for $1\leq l \leq r$ (notice that -$P_l$ arity implies $P'_l$ arity since $\WTEGLECONV{P'_l}{ \subst{P_l}{p_u}{q_u}_{u=1\ldots l-1}}$); -\item there are sorts $s_i$, for $1 \leq i \leq k$ such that, for - $\Gamma_{I'} = [I_1:\forall - \Gamma_{P'},(A_1)_{/s_1};\ldots;I_k:\forall \Gamma_{P'},(A_k)_{/s_k}]$ -we have $(\WTE{\Gamma;\Gamma_{I'};\Gamma_{P'}}{C_i}{s_{q_i}})_{i=1\ldots n}$; -\item the sorts are such that all eliminations, to {\Prop}, {\Set} and - $\Type(j)$, are allowed (see section~\ref{elimdep}). -\end{itemize} -\end{description} - -Notice that if $I_j\,q_1\,\ldots\,q_r$ is typable using the rules {\bf -Ind-Const} and {\bf App}, then it is typable using the rule {\bf -Ind-Family}. Conversely, the extended theory is not stronger than the -theory without {\bf Ind-Family}. We get an equiconsistency result by -mapping each $\Ind{\Gamma}{p}{\Gamma_I}{\Gamma_C}$ occurring into a -given derivation into as many different inductive types and constructors -as the number of different (partial) replacements of sorts, needed for -this derivation, in the parameters that are arities (this is possible -because $\Ind{\Gamma}{p}{\Gamma_I}{\Gamma_C}$ well-formed implies -that $\Ind{\Gamma}{p}{\Gamma_{I'}}{\Gamma_{C'}}$ is well-formed and -has the same allowed eliminations, where -$\Gamma_{I'}$ is defined as above and $\Gamma_{C'} = [c_1:\forall -\Gamma_{P'},C_1;\ldots;c_n:\forall \Gamma_{P'},C_n]$). That is, -the changes in the types of each partial instance -$q_1\,\ldots\,q_r$ can be characterized by the ordered sets of arity -sorts among the types of parameters, and to each signature is -associated a new inductive definition with fresh names. Conversion is -preserved as any (partial) instance $I_j\,q_1\,\ldots\,q_r$ or -$C_i\,q_1\,\ldots\,q_r$ is mapped to the names chosen in the specific -instance of $\Ind{\Gamma}{p}{\Gamma_I}{\Gamma_C}$. - -\newcommand{\Single}{\mbox{\textsf{Set}}} - -In practice, the rule {\bf Ind-Family} is used by {\Coq} only when all the -inductive types of the inductive definition are declared with an arity whose -sort is in the $\Type$ -hierarchy. Then, the polymorphism is over the parameters whose -type is an arity of sort in the {\Type} hierarchy. -The sort $s_j$ are -chosen canonically so that each $s_j$ is minimal with respect to the -hierarchy ${\Prop}\subset{\Set_p}\subset\Type$ where $\Set_p$ is -predicative {\Set}. -%and ${\Prop_u}$ is the sort of small singleton -%inductive types (i.e. of inductive types with one single constructor -%and that contains either proofs or inhabitants of singleton types -%only). -More precisely, an empty or small singleton inductive definition -(i.e. an inductive definition of which all inductive types are -singleton -- see paragraph~\ref{singleton}) is set in -{\Prop}, a small non-singleton inductive family is set in {\Set} (even -in case {\Set} is impredicative -- see Section~\ref{impredicativity}), -and otherwise in the {\Type} hierarchy. -% TODO: clarify the case of a partial application ?? - -Note that the side-condition about allowed elimination sorts in the -rule~{\bf Ind-Family} is just to avoid to recompute the allowed -elimination sorts at each instance of a pattern-matching (see -section~\ref{elimdep}). - -As an example, let us consider the following definition: -\begin{coq_example*} -Inductive option (A:Type) : Type := -| None : option A -| Some : A -> option A. -\end{coq_example*} - -As the definition is set in the {\Type} hierarchy, it is used -polymorphically over its parameters whose types are arities of a sort -in the {\Type} hierarchy. Here, the parameter $A$ has this property, -hence, if \texttt{option} is applied to a type in {\Set}, the result is -in {\Set}. Note that if \texttt{option} is applied to a type in {\Prop}, -then, the result is not set in \texttt{Prop} but in \texttt{Set} -still. This is because \texttt{option} is not a singleton type (see -section~\ref{singleton}) and it would loose the elimination to {\Set} and -{\Type} if set in {\Prop}. - -\begin{coq_example} -Check (fun A:Set => option A). -Check (fun A:Prop => option A). -\end{coq_example} - -Here is another example. - -\begin{coq_example*} -Inductive prod (A B:Type) : Type := pair : A -> B -> prod A B. -\end{coq_example*} - -As \texttt{prod} is a singleton type, it will be in {\Prop} if applied -twice to propositions, in {\Set} if applied twice to at least one type -in {\Set} and none in {\Type}, and in {\Type} otherwise. In all cases, -the three kind of eliminations schemes are allowed. - -\begin{coq_example} -Check (fun A:Set => prod A). -Check (fun A:Prop => prod A A). -Check (fun (A:Prop) (B:Set) => prod A B). -Check (fun (A:Type) (B:Prop) => prod A B). -\end{coq_example} - -\subsection{Destructors} -The specification of inductive definitions with arities and -constructors is quite natural. But we still have to say how to use an -object in an inductive type. - -This problem is rather delicate. There are actually several different -ways to do that. Some of them are logically equivalent but not always -equivalent from the computational point of view or from the user point -of view. - -From the computational point of view, we want to be able to define a -function whose domain is an inductively defined type by using a -combination of case analysis over the possible constructors of the -object and recursion. - -Because we need to keep a consistent theory and also we prefer to keep -a strongly normalizing reduction, we cannot accept any sort of -recursion (even terminating). So the basic idea is to restrict -ourselves to primitive recursive functions and functionals. - -For instance, assuming a parameter $A:\Set$ exists in the context, we -want to build a function \length\ of type $\ListA\ra \nat$ which -computes the length of the list, so such that $(\length~(\Nil~A)) = \nO$ -and $(\length~(\cons~A~a~l)) = (\nS~(\length~l))$. We want these -equalities to be recognized implicitly and taken into account in the -conversion rule. - -From the logical point of view, we have built a type family by giving -a set of constructors. We want to capture the fact that we do not -have any other way to build an object in this type. So when trying to -prove a property $(P~m)$ for $m$ in an inductive definition it is -enough to enumerate all the cases where $m$ starts with a different -constructor. - -In case the inductive definition is effectively a recursive one, we -want to capture the extra property that we have built the smallest -fixed point of this recursive equation. This says that we are only -manipulating finite objects. This analysis provides induction -principles. - -For instance, in order to prove $\forall l:\ListA,(\LengthA~l~(\length~l))$ -it is enough to prove: - -\noindent $(\LengthA~(\Nil~A)~(\length~(\Nil~A)))$ and - -\smallskip -$\forall a:A, \forall l:\ListA, (\LengthA~l~(\length~l)) \ra -(\LengthA~(\cons~A~a~l)~(\length~(\cons~A~a~l)))$. -\smallskip - -\noindent which given the conversion equalities satisfied by \length\ is the -same as proving: -$(\LengthA~(\Nil~A)~\nO)$ and $\forall a:A, \forall l:\ListA, -(\LengthA~l~(\length~l)) \ra -(\LengthA~(\cons~A~a~l)~(\nS~(\length~l)))$. - -One conceptually simple way to do that, following the basic scheme -proposed by Martin-L\"of in his Intuitionistic Type Theory, is to -introduce for each inductive definition an elimination operator. At -the logical level it is a proof of the usual induction principle and -at the computational level it implements a generic operator for doing -primitive recursion over the structure. - -But this operator is rather tedious to implement and use. We choose in -this version of Coq to factorize the operator for primitive recursion -into two more primitive operations as was first suggested by Th. Coquand -in~\cite{Coq92}. One is the definition by pattern-matching. The second one is a definition by guarded fixpoints. - -\subsubsection[The {\tt match\ldots with \ldots end} construction.]{The {\tt match\ldots with \ldots end} construction.\label{Caseexpr} -\index{match@{\tt match\ldots with\ldots end}}} - -The basic idea of this destructor operation is that we have an object -$m$ in an inductive type $I$ and we want to prove a property $(P~m)$ -which in general depends on $m$. For this, it is enough to prove the -property for $m = (c_i~u_1\ldots u_{p_i})$ for each constructor of $I$. - -The \Coq{} term for this proof will be written~: -\[\kw{match}~m~\kw{with}~ (c_1~x_{11}~...~x_{1p_1}) \Ra f_1 ~|~\ldots~|~ - (c_n~x_{n1}...x_{np_n}) \Ra f_n~ \kw{end}\] -In this expression, if -$m$ is a term built from a constructor $(c_i~u_1\ldots u_{p_i})$ then -the expression will behave as it is specified with $i$-th branch and -will reduce to $f_i$ where the $x_{i1}$\ldots $x_{ip_i}$ are replaced -by the $u_1\ldots u_p$ according to the $\iota$-reduction. - -Actually, for type-checking a \kw{match\ldots with\ldots end} -expression we also need to know the predicate $P$ to be proved by case -analysis. In the general case where $I$ is an inductively defined -$n$-ary relation, $P$ is a $n+1$-ary relation: the $n$ first arguments -correspond to the arguments of $I$ (parameters excluded), and the last -one corresponds to object $m$. \Coq{} can sometimes infer this -predicate but sometimes not. The concrete syntax for describing this -predicate uses the \kw{as\ldots in\ldots return} construction. For -instance, let us assume that $I$ is an unary predicate with one -parameter. The predicate is made explicit using the syntax~: -\[\kw{match}~m~\kw{as}~ x~ \kw{in}~ I~\verb!_!~a~ \kw{return}~ (P~ x) - ~\kw{with}~ (c_1~x_{11}~...~x_{1p_1}) \Ra f_1 ~|~\ldots~|~ - (c_n~x_{n1}...x_{np_n}) \Ra f_n \kw{end}\] -The \kw{as} part can be omitted if either the result type does not -depend on $m$ (non-dependent elimination) or $m$ is a variable (in -this case, the result type can depend on $m$). The \kw{in} part can be -omitted if the result type does not depend on the arguments of -$I$. Note that the arguments of $I$ corresponding to parameters -\emph{must} be \verb!_!, because the result type is not generalized to -all possible values of the parameters. The expression after \kw{in} -must be seen as an \emph{inductive type pattern}. As a final remark, -expansion of implicit arguments and notations apply to this pattern. - -For the purpose of presenting the inference rules, we use a more -compact notation~: -\[ \Case{(\lb a x \mto P)}{m}{ \lb x_{11}~...~x_{1p_1} \mto f_1 ~|~\ldots~|~ - \lb x_{n1}...x_{np_n} \mto f_n}\] - -%% CP 06/06 Obsolete avec la nouvelle syntaxe et incompatible avec la -%% presentation theorique qui suit -% \paragraph{Non-dependent elimination.} -% -% When defining a function of codomain $C$ by case analysis over an -% object in an inductive type $I$, we build an object of type $I -% \ra C$. The minimality principle on an inductively defined logical -% predicate $I$ of type $A \ra \Prop$ is often used to prove a property -% $\forall x:A,(I~x)\ra (C~x)$. These are particular cases of the dependent -% principle that we stated before with a predicate which does not depend -% explicitly on the object in the inductive definition. - -% For instance, a function testing whether a list is empty -% can be -% defined as: -% \[\kw{fun} l:\ListA \Ra \kw{match}~l~\kw{with}~ \Nil \Ra \true~ -% |~(\cons~a~m) \Ra \false \kw{end}\] -% represented by -% \[\lb~l:\ListA \mto\Case{\bool}{l}{\true~ |~ \lb a~m,~\false}\] -%\noindent {\bf Remark. } - -% In the system \Coq\ the expression above, can be -% written without mentioning -% the dummy abstraction: -% \Case{\bool}{l}{\Nil~ \mbox{\tt =>}~\true~ |~ (\cons~a~m)~ -% \mbox{\tt =>}~ \false} - -\paragraph[Allowed elimination sorts.]{Allowed elimination sorts.\index{Elimination sorts}} - -An important question for building the typing rule for \kw{match} is -what can be the type of $P$ with respect to the type of the inductive -definitions. - -We define now a relation \compat{I:A}{B} between an inductive -definition $I$ of type $A$ and an arity $B$. This relation states that -an object in the inductive definition $I$ can be eliminated for -proving a property $P$ of type $B$. - -The case of inductive definitions in sorts \Set\ or \Type{} is simple. -There is no restriction on the sort of the predicate to be -eliminated. - -\paragraph{Notations.} -The \compat{I:A}{B} is defined as the smallest relation satisfying the -following rules: -We write \compat{I}{B} for \compat{I:A}{B} where $A$ is the type of -$I$. - -\begin{description} -\item[Prod] \inference{\frac{\compat{(I~x):A'}{B'}} - {\compat{I:\forall x:A, A'}{\forall x:A, B'}}} -\item[{\Set} \& \Type] \inference{\frac{ - s_1 \in \{\Set,\Type(j)\}, - s_2 \in \Sort}{\compat{I:s_1}{I\ra s_2}}} -\end{description} - -The case of Inductive definitions of sort \Prop{} is a bit more -complicated, because of our interpretation of this sort. The only -harmless allowed elimination, is the one when predicate $P$ is also of -sort \Prop. -\begin{description} -\item[\Prop] \inference{\compat{I:\Prop}{I\ra\Prop}} -\end{description} -\Prop{} is the type of logical propositions, the proofs of properties -$P$ in \Prop{} could not be used for computation and are consequently -ignored by the extraction mechanism. -Assume $A$ and $B$ are two propositions, and the logical disjunction -$A\vee B$ is defined inductively by~: -\begin{coq_example*} -Inductive or (A B:Prop) : Prop := - lintro : A -> or A B | rintro : B -> or A B. -\end{coq_example*} -The following definition which computes a boolean value by case over -the proof of \texttt{or A B} is not accepted~: -\begin{coq_eval} -(***************************************************************) -(*** This example should fail with ``Incorrect elimination'' ***) -\end{coq_eval} -\begin{coq_example} -Definition choice (A B: Prop) (x:or A B) := - match x with lintro a => true | rintro b => false end. -\end{coq_example} -From the computational point of view, the structure of the proof of -\texttt{(or A B)} in this term is needed for computing the boolean -value. - -In general, if $I$ has type \Prop\ then $P$ cannot have type $I\ra -\Set$, because it will mean to build an informative proof of type -$(P~m)$ doing a case analysis over a non-computational object that -will disappear in the extracted program. But the other way is safe -with respect to our interpretation we can have $I$ a computational -object and $P$ a non-computational one, it just corresponds to proving -a logical property of a computational object. - -% Also if $I$ is in one of the sorts \{\Prop, \Set\}, one cannot in -% general allow an elimination over a bigger sort such as \Type. But -% this operation is safe whenever $I$ is a {\em small inductive} type, -% which means that all the types of constructors of -% $I$ are small with the following definition:\\ -% $(I~t_1\ldots t_s)$ is a {\em small type of constructor} and -% $\forall~x:T,C$ is a small type of constructor if $C$ is and if $T$ -% has type \Prop\ or \Set. \index{Small inductive type} - -% We call this particular elimination which gives the possibility to -% compute a type by induction on the structure of a term, a {\em strong -% elimination}\index{Strong elimination}. - -In the same spirit, elimination on $P$ of type $I\ra -\Type$ cannot be allowed because it trivially implies the elimination -on $P$ of type $I\ra \Set$ by cumulativity. It also implies that there -is two proofs of the same property which are provably different, -contradicting the proof-irrelevance property which is sometimes a -useful axiom~: -\begin{coq_example} -Axiom proof_irrelevance : forall (P : Prop) (x y : P), x=y. -\end{coq_example} -\begin{coq_eval} -Reset proof_irrelevance. -\end{coq_eval} -The elimination of an inductive definition of type \Prop\ on a -predicate $P$ of type $I\ra \Type$ leads to a paradox when applied to -impredicative inductive definition like the second-order existential -quantifier \texttt{exProp} defined above, because it give access to -the two projections on this type. - -%\paragraph{Warning: strong elimination} -%\index{Elimination!Strong elimination} -%In previous versions of Coq, for a small inductive definition, only the -%non-informative strong elimination on \Type\ was allowed, because -%strong elimination on \Typeset\ was not compatible with the current -%extraction procedure. In this version, strong elimination on \Typeset\ -%is accepted but a dummy element is extracted from it and may generate -%problems if extracted terms are explicitly used such as in the -%{\tt Program} tactic or when extracting ML programs. - -\paragraph[Empty and singleton elimination]{Empty and singleton elimination\label{singleton} -\index{Elimination!Singleton elimination} -\index{Elimination!Empty elimination}} - -There are special inductive definitions in \Prop\ for which more -eliminations are allowed. -\begin{description} -\item[\Prop-extended] -\inference{ - \frac{I \mbox{~is an empty or singleton - definition}~~~s \in \Sort}{\compat{I:\Prop}{I\ra s}} -} -\end{description} - -% A {\em singleton definition} has always an informative content, -% even if it is a proposition. - -A {\em singleton -definition} has only one constructor and all the arguments of this -constructor have type \Prop. In that case, there is a canonical -way to interpret the informative extraction on an object in that type, -such that the elimination on any sort $s$ is legal. Typical examples are -the conjunction of non-informative propositions and the equality. -If there is an hypothesis $h:a=b$ in the context, it can be used for -rewriting not only in logical propositions but also in any type. -% In that case, the term \verb!eq_rec! which was defined as an axiom, is -% now a term of the calculus. -\begin{coq_example} -Print eq_rec. -Extraction eq_rec. -\end{coq_example} -An empty definition has no constructors, in that case also, -elimination on any sort is allowed. - -\paragraph{Type of branches.} -Let $c$ be a term of type $C$, we assume $C$ is a type of constructor -for an inductive definition $I$. Let $P$ be a term that represents the -property to be proved. -We assume $r$ is the number of parameters. - -We define a new type \CI{c:C}{P} which represents the type of the -branch corresponding to the $c:C$ constructor. -\[ -\begin{array}{ll} -\CI{c:(I_i~p_1\ldots p_r\ t_1 \ldots t_p)}{P} &\equiv (P~t_1\ldots ~t_p~c) \\[2mm] -\CI{c:\forall~x:T,C}{P} &\equiv \forall~x:T,\CI{(c~x):C}{P} -\end{array} -\] -We write \CI{c}{P} for \CI{c:C}{P} with $C$ the type of $c$. - -\paragraph{Examples.} -For $\ListA$ the type of $P$ will be $\ListA\ra s$ for $s \in \Sort$. \\ -$ \CI{(\cons~A)}{P} \equiv -\forall a:A, \forall l:\ListA,(P~(\cons~A~a~l))$. - -For $\LengthA$, the type of $P$ will be -$\forall l:\ListA,\forall n:\nat, (\LengthA~l~n)\ra \Prop$ and the expression -\CI{(\LCons~A)}{P} is defined as:\\ -$\forall a:A, \forall l:\ListA, \forall n:\nat, \forall -h:(\LengthA~l~n), (P~(\cons~A~a~l)~(\nS~n)~(\LCons~A~a~l~n~l))$.\\ -If $P$ does not depend on its third argument, we find the more natural -expression:\\ -$\forall a:A, \forall l:\ListA, \forall n:\nat, -(\LengthA~l~n)\ra(P~(\cons~A~a~l)~(\nS~n))$. - -\paragraph{Typing rule.} - -Our very general destructor for inductive definition enjoys the -following typing rule -% , where we write -% \[ -% \Case{P}{c}{[x_{11}:T_{11}]\ldots[x_{1p_1}:T_{1p_1}]g_1\ldots -% [x_{n1}:T_{n1}]\ldots[x_{np_n}:T_{np_n}]g_n} -% \] -% for -% \[ -% \Case{P}{c}{(c_1~x_{11}~...~x_{1p_1}) \Ra g_1 ~|~\ldots~|~ -% (c_n~x_{n1}...x_{np_n}) \Ra g_n } -% \] - -\begin{description} -\item[match] \label{elimdep} \index{Typing rules!match} -\inference{ -\frac{\WTEG{c}{(I~q_1\ldots q_r~t_1\ldots t_s)}~~ - \WTEG{P}{B}~~\compat{(I~q_1\ldots q_r)}{B} - ~~ -(\WTEG{f_i}{\CI{(c_{p_i}~q_1\ldots q_r)}{P}})_{i=1\ldots l}} -{\WTEG{\Case{P}{c}{f_1|\ldots |f_l}}{(P\ t_1\ldots t_s\ c)}}}%\\[3mm] - -provided $I$ is an inductive type in a declaration -\Ind{\Delta}{r}{\Gamma_I}{\Gamma_C} with -$\Gamma_C = [c_1:C_1;\ldots;c_n:C_n]$ and $c_{p_1}\ldots c_{p_l}$ are the -only constructors of $I$. -\end{description} - -\paragraph{Example.} -For \List\ and \Length\ the typing rules for the {\tt match} expression -are (writing just $t:M$ instead of \WTEG{t}{M}, the environment and -context being the same in all the judgments). - -\[\frac{l:\ListA~~P:\ListA\ra s~~~f_1:(P~(\Nil~A))~~ - f_2:\forall a:A, \forall l:\ListA, (P~(\cons~A~a~l))} - {\Case{P}{l}{f_1~|~f_2}:(P~l)}\] - -\[\frac{ - \begin{array}[b]{@{}c@{}} -H:(\LengthA~L~N) \\ P:\forall l:\ListA, \forall n:\nat, (\LengthA~l~n)\ra - \Prop\\ - f_1:(P~(\Nil~A)~\nO~\LNil) \\ - f_2:\forall a:A, \forall l:\ListA, \forall n:\nat, \forall - h:(\LengthA~l~n), (P~(\cons~A~a~n)~(\nS~n)~(\LCons~A~a~l~n~h)) - \end{array}} - {\Case{P}{H}{f_1~|~f_2}:(P~L~N~H)}\] - -\paragraph[Definition of $\iota$-reduction.]{Definition of $\iota$-reduction.\label{iotared} -\index{iota-reduction@$\iota$-reduction}} -We still have to define the $\iota$-reduction in the general case. - -A $\iota$-redex is a term of the following form: -\[\Case{P}{(c_{p_i}~q_1\ldots q_r~a_1\ldots a_m)}{f_1|\ldots | - f_l}\] -with $c_{p_i}$ the $i$-th constructor of the inductive type $I$ with $r$ -parameters. - -The $\iota$-contraction of this term is $(f_i~a_1\ldots a_m)$ leading -to the general reduction rule: -\[ \Case{P}{(c_{p_i}~q_1\ldots q_r~a_1\ldots a_m)}{f_1|\ldots | - f_n} \triangleright_{\iota} (f_i~a_1\ldots a_m) \] - -\subsection[Fixpoint definitions]{Fixpoint definitions\label{Fix-term} \index{Fix@{\tt Fix}}} -The second operator for elimination is fixpoint definition. -This fixpoint may involve several mutually recursive definitions. -The basic concrete syntax for a recursive set of mutually recursive -declarations is (with $\Gamma_i$ contexts)~: -\[\kw{fix}~f_1 (\Gamma_1) :A_1:=t_1~\kw{with} \ldots \kw{with}~ f_n -(\Gamma_n) :A_n:=t_n\] -The terms are obtained by projections from this set of declarations -and are written -\[\kw{fix}~f_1 (\Gamma_1) :A_1:=t_1~\kw{with} \ldots \kw{with}~ f_n -(\Gamma_n) :A_n:=t_n~\kw{for}~f_i\] -In the inference rules, we represent such a -term by -\[\Fix{f_i}{f_1:A_1':=t_1' \ldots f_n:A_n':=t_n'}\] -with $t_i'$ (resp. $A_i'$) representing the term $t_i$ abstracted -(resp. generalized) with -respect to the bindings in the context $\Gamma_i$, namely -$t_i'=\lb \Gamma_i \mto t_i$ and $A_i'=\forall \Gamma_i, A_i$. - -\subsubsection{Typing rule} -The typing rule is the expected one for a fixpoint. - -\begin{description} -\item[Fix] \index{Typing rules!Fix} -\inference{\frac{(\WTEG{A_i}{s_i})_{i=1\ldots n}~~~~ - (\WTE{\Gamma,f_1:A_1,\ldots,f_n:A_n}{t_i}{A_i})_{i=1\ldots n}} - {\WTEG{\Fix{f_i}{f_1:A_1:=t_1 \ldots f_n:A_n:=t_n}}{A_i}}} -\end{description} - -Any fixpoint definition cannot be accepted because non-normalizing terms -will lead to proofs of absurdity. - -The basic scheme of recursion that should be allowed is the one needed for -defining primitive -recursive functionals. In that case the fixpoint enjoys a special -syntactic restriction, namely one of the arguments belongs to an -inductive type, the function starts with a case analysis and recursive -calls are done on variables coming from patterns and representing subterms. - -For instance in the case of natural numbers, a proof of the induction -principle of type -\[\forall P:\nat\ra\Prop, (P~\nO)\ra(\forall n:\nat, (P~n)\ra(P~(\nS~n)))\ra -\forall n:\nat, (P~n)\] -can be represented by the term: -\[\begin{array}{l} -\lb P:\nat\ra\Prop\mto\lb f:(P~\nO)\mto \lb g:(\forall n:\nat, -(P~n)\ra(P~(\nS~n))) \mto\\ -\Fix{h}{h:\forall n:\nat, (P~n):=\lb n:\nat\mto \Case{P}{n}{f~|~\lb - p:\nat\mto (g~p~(h~p))}} -\end{array} -\] - -Before accepting a fixpoint definition as being correctly typed, we -check that the definition is ``guarded''. A precise analysis of this -notion can be found in~\cite{Gim94}. - -The first stage is to precise on which argument the fixpoint will be -decreasing. The type of this argument should be an inductive -definition. - -For doing this the syntax of fixpoints is extended and becomes - \[\Fix{f_i}{f_1/k_1:A_1:=t_1 \ldots f_n/k_n:A_n:=t_n}\] -where $k_i$ are positive integers. -Each $A_i$ should be a type (reducible to a term) starting with at least -$k_i$ products $\forall y_1:B_1,\ldots \forall y_{k_i}:B_{k_i}, A'_i$ -and $B_{k_i}$ -being an instance of an inductive definition. - -Now in the definition $t_i$, if $f_j$ occurs then it should be applied -to at least $k_j$ arguments and the $k_j$-th argument should be -syntactically recognized as structurally smaller than $y_{k_i}$ - - -The definition of being structurally smaller is a bit technical. -One needs first to define the notion of -{\em recursive arguments of a constructor}\index{Recursive arguments}. -For an inductive definition \Ind{\Gamma}{r}{\Gamma_I}{\Gamma_C}, -the type of a constructor $c$ has the form -$\forall p_1:P_1,\ldots \forall p_r:P_r, -\forall x_1:T_1, \ldots \forall x_r:T_r, (I_j~p_1\ldots -p_r~t_1\ldots t_s)$ the recursive arguments will correspond to $T_i$ in -which one of the $I_l$ occurs. - - -The main rules for being structurally smaller are the following:\\ -Given a variable $y$ of type an inductive -definition in a declaration -\Ind{\Gamma}{r}{\Gamma_I}{\Gamma_C} -where $\Gamma_I$ is $[I_1:A_1;\ldots;I_k:A_k]$, and $\Gamma_C$ is - $[c_1:C_1;\ldots;c_n:C_n]$. -The terms structurally smaller than $y$ are: -\begin{itemize} -\item $(t~u), \lb x:u \mto t$ when $t$ is structurally smaller than $y$ . -\item \Case{P}{c}{f_1\ldots f_n} when each $f_i$ is structurally - smaller than $y$. \\ - If $c$ is $y$ or is structurally smaller than $y$, its type is an inductive - definition $I_p$ part of the inductive - declaration corresponding to $y$. - Each $f_i$ corresponds to a type of constructor $C_q \equiv - \forall p_1:P_1,\ldots,\forall p_r:P_r, \forall y_1:B_1, \ldots \forall y_k:B_k, (I~a_1\ldots a_k)$ - and can consequently be - written $\lb y_1:B'_1\mto \ldots \lb y_k:B'_k\mto g_i$. - ($B'_i$ is obtained from $B_i$ by substituting parameters variables) - the variables $y_j$ occurring - in $g_i$ corresponding to recursive arguments $B_i$ (the ones in - which one of the $I_l$ occurs) are structurally smaller than $y$. -\end{itemize} -The following definitions are correct, we enter them using the -{\tt Fixpoint} command as described in Section~\ref{Fixpoint} and show -the internal representation. -\begin{coq_example} -Fixpoint plus (n m:nat) {struct n} : nat := - match n with - | O => m - | S p => S (plus p m) - end. -Print plus. -Fixpoint lgth (A:Set) (l:list A) {struct l} : nat := - match l with - | nil => O - | cons a l' => S (lgth A l') - end. -Print lgth. -Fixpoint sizet (t:tree) : nat := let (f) := t in S (sizef f) - with sizef (f:forest) : nat := - match f with - | emptyf => O - | consf t f => plus (sizet t) (sizef f) - end. -Print sizet. -\end{coq_example} - - -\subsubsection[Reduction rule]{Reduction rule\index{iota-reduction@$\iota$-reduction}} -Let $F$ be the set of declarations: $f_1/k_1:A_1:=t_1 \ldots -f_n/k_n:A_n:=t_n$. -The reduction for fixpoints is: -\[ (\Fix{f_i}{F}~a_1\ldots -a_{k_i}) \triangleright_{\iota} \substs{t_i}{f_k}{\Fix{f_k}{F}}{k=1\ldots n} -~a_1\ldots a_{k_i}\] -when $a_{k_i}$ starts with a constructor. -This last restriction is needed in order to keep strong normalization -and corresponds to the reduction for primitive recursive operators. - -We can illustrate this behavior on examples. -\begin{coq_example} -Goal forall n m:nat, plus (S n) m = S (plus n m). -reflexivity. -Abort. -Goal forall f:forest, sizet (node f) = S (sizef f). -reflexivity. -Abort. -\end{coq_example} -But assuming the definition of a son function from \tree\ to \forest: -\begin{coq_example} -Definition sont (t:tree) : forest - := let (f) := t in f. -\end{coq_example} -The following is not a conversion but can be proved after a case analysis. -\begin{coq_eval} -(******************************************************************) -(** Error: Impossible to unify .... **) -\end{coq_eval} -\begin{coq_example} -Goal forall t:tree, sizet t = S (sizef (sont t)). -reflexivity. (** this one fails **) -destruct t. -reflexivity. -\end{coq_example} -\begin{coq_eval} -Abort. -\end{coq_eval} - -% La disparition de Program devrait rendre la construction Match obsolete -% \subsubsection{The {\tt Match \ldots with \ldots end} expression} -% \label{Matchexpr} -% %\paragraph{A unary {\tt Match\ldots with \ldots end}.} -% \index{Match...with...end@{\tt Match \ldots with \ldots end}} -% The {\tt Match} operator which was a primitive notion in older -% presentations of the Calculus of Inductive Constructions is now just a -% macro definition which generates the good combination of {\tt Case} -% and {\tt Fix} operators in order to generate an operator for primitive -% recursive definitions. It always considers an inductive definition as -% a single inductive definition. - -% The following examples illustrates this feature. -% \begin{coq_example} -% Definition nat_pr : (C:Set)C->(nat->C->C)->nat->C -% :=[C,x,g,n]Match n with x g end. -% Print nat_pr. -% \end{coq_example} -% \begin{coq_example} -% Definition forest_pr -% : (C:Set)C->(tree->forest->C->C)->forest->C -% := [C,x,g,n]Match n with x g end. -% \end{coq_example} - -% Cet exemple faisait error (HH le 12/12/96), j'ai change pour une -% version plus simple -%\begin{coq_example} -%Definition forest_pr -% : (P:forest->Set)(P emptyf)->((t:tree)(f:forest)(P f)->(P (consf t f))) -% ->(f:forest)(P f) -% := [C,x,g,n]Match n with x g end. -%\end{coq_example} - -\subsubsection{Mutual induction} - -The principles of mutual induction can be automatically generated -using the {\tt Scheme} command described in Section~\ref{Scheme}. - -\section{Coinductive types} -The implementation contains also coinductive definitions, which are -types inhabited by infinite objects. -More information on coinductive definitions can be found -in~\cite{Gimenez95b,Gim98,GimCas05}. -%They are described in Chapter~\ref{Coinductives}. - -\section[\iCIC : the Calculus of Inductive Construction with - impredicative \Set]{\iCIC : the Calculus of Inductive Construction with - impredicative \Set\label{impredicativity}} - -\Coq{} can be used as a type-checker for \iCIC{}, the original -Calculus of Inductive Constructions with an impredicative sort \Set{} -by using the compiler option \texttt{-impredicative-set}. - -For example, using the ordinary \texttt{coqtop} command, the following -is rejected. -\begin{coq_eval} -(** This example should fail ******************************* - Error: The term forall X:Set, X -> X has type Type - while it is expected to have type Set -***) -\end{coq_eval} -\begin{coq_example} -Definition id: Set := forall X:Set,X->X. -\end{coq_example} -while it will type-check, if one use instead the \texttt{coqtop - -impredicative-set} command. - -The major change in the theory concerns the rule for product formation -in the sort \Set, which is extended to a domain in any sort~: -\begin{description} -\item [Prod] \index{Typing rules!Prod (impredicative Set)} -\inference{\frac{\WTEG{T}{s}~~~~s \in \Sort~~~~~~ - \WTE{\Gamma::(x:T)}{U}{\Set}} - { \WTEG{\forall~x:T,U}{\Set}}} -\end{description} -This extension has consequences on the inductive definitions which are -allowed. -In the impredicative system, one can build so-called {\em large inductive - definitions} like the example of second-order existential -quantifier (\texttt{exSet}). - -There should be restrictions on the eliminations which can be -performed on such definitions. The eliminations rules in the -impredicative system for sort \Set{} become~: -\begin{description} -\item[\Set] \inference{\frac{s \in - \{\Prop, \Set\}}{\compat{I:\Set}{I\ra s}} -~~~~\frac{I \mbox{~is a small inductive definition}~~~~s \in - \{\Type(i)\}} - {\compat{I:\Set}{I\ra s}}} -\end{description} - - - -% $Id: RefMan-cic.tex 13029 2010-05-28 11:49:12Z herbelin $ - -%%% Local Variables: -%%% mode: latex -%%% TeX-master: "Reference-Manual" -%%% End: - - diff --git a/doc/refman/RefMan-coi.tex b/doc/refman/RefMan-coi.tex deleted file mode 100644 index b0f0212e..00000000 --- a/doc/refman/RefMan-coi.tex +++ /dev/null @@ -1,406 +0,0 @@ -%\documentstyle[11pt,../tools/coq-tex/coq]{article} -%\input{title} - -%\include{macros} -%\begin{document} - -%\coverpage{Co-inductive types in Coq}{Eduardo Gim\'enez} -\chapter[Co-inductive types in Coq]{Co-inductive types in Coq\label{Coinductives}} - -%\begin{abstract} -{\it Co-inductive} types are types whose elements may not be well-founded. -A formal study of the Calculus of Constructions extended by -co-inductive types has been presented -in \cite{Gim94}. It is based on the notion of -{\it guarded definitions} introduced by Th. Coquand -in \cite{Coquand93}. The implementation is by E. Gim\'enez. -%\end{abstract} - -\section{A short introduction to co-inductive types} - -We assume that the reader is rather familiar with inductive types. -These types are characterized by their {\it constructors}, which can be -regarded as the basic methods from which the elements -of the type can be built up. It is implicit in the definition -of an inductive type that -its elements are the result of a {\it finite} number of -applications of its constructors. Co-inductive types arise from -relaxing this implicit condition and admitting that an element of -the type can also be introduced by a non-ending (but effective) process -of construction defined in terms of the basic methods which characterize the -type. So we could think in the wider notion of types defined by -constructors (let us call them {\it recursive types}) and classify -them into inductive and co-inductive ones, depending on whether or not -we consider non-ending methods as admissible for constructing elements -of the type. Note that in both cases we obtain a ``closed type'', all whose -elements are pre-determined in advance (by the constructors). When we -know that $a$ is an element of a recursive type (no matter if it is -inductive or co-inductive) what we know is that it is the result of applying -one of the basic forms of construction allowed for the type. -So the more primitive way of eliminating an element of a recursive type is -by case analysis, i.e. by considering through which constructor it could have -been introduced. In the case of inductive sets, the additional knowledge that -constructors can be applied only a finite number of times provide -us with a more powerful way of eliminating their elements, say, -the principle of -induction. This principle is obviously not valid for co-inductive types, -since it is just the expression of this extra knowledge attached to inductive -types. - - -An example of a co-inductive type is the type of infinite sequences formed with -elements of type $A$, or streams for shorter. In Coq, -it can be introduced using the \verb!CoInductive! command~: -\begin{coq_example} -CoInductive Stream (A:Set) : Set := - cons : A -> Stream A -> Stream A. -\end{coq_example} - -The syntax of this command is the same as the -command \verb!Inductive! (cf. section -\ref{gal_Inductive_Definitions}). -Definition of mutually coinductive types are possible. - -As was already said, there are not principles of -induction for co-inductive sets, the only way of eliminating these -elements is by case analysis. -In the example of streams, this elimination principle can be -used for instance to define the well known -destructors on streams $\hd : (\Str\;A)\rightarrow A$ -and $\tl: (\Str\;A)\rightarrow (\Str\;A)$ : -\begin{coq_example} -Section Destructors. -Variable A : Set. -Definition hd (x:Stream A) := match x with - | cons a s => a - end. -Definition tl (x:Stream A) := match x with - | cons a s => s - end. -\end{coq_example} -\begin{coq_example*} -End Destructors. -\end{coq_example*} - -\subsection{Non-ending methods of construction} - -At this point the reader should have realized that we have left unexplained -what is a ``non-ending but effective process of -construction'' of a stream. In the widest sense, a -method is a non-ending process of construction if we can eliminate the -stream that it introduces, in other words, if we can reduce -any case analysis on it. In this sense, the following ways of -introducing a stream are not acceptable. -\begin{center} -$\zeros = (\cons\;\nat\;\nO\;(\tl\;\zeros))\;\;:\;\;(\Str\;\nat)$\\[12pt] -$\filter\;(\cons\;A\;a\;s) = \si\;\;(P\;a)\;\;\alors\;\;(\cons\;A\;a\;(\filter\;s))\;\;\sinon\;\;(\filter\;s) )\;\;:\;\;(\Str\;A)$ -\end{center} -\noindent The former it is not valid since the stream can not be eliminated -to obtain its tail. In the latter, a stream is naively defined as -the result of erasing from another (arbitrary) stream -all the elements which does not verify a certain property $P$. This -does not always makes sense, for example it does not when all the elements -of the stream verify $P$, in which case we can not eliminate it to -obtain its head\footnote{Note that there is no notion of ``the empty -stream'', a stream is always infinite and build by a \texttt{cons}.}. -On the contrary, the following definitions are acceptable methods for -constructing a stream~: -\begin{center} -$\zeros = (\cons\;\nat\;\nO\;\zeros)\;\;:\;\;(\Str\;\nat)\;\;\;(*)$\\[12pt] -$(\from\;n) = (\cons\;\nat\;n\;(\from\;(\nS\;n)))\;:\;(\Str\;\nat)$\\[12pt] -$\alter = (\cons\;\bool\;\true\;(\cons\;\bool\;\false\;\alter))\;:\;(\Str\;\bool)$. -\end{center} -\noindent The first one introduces a stream containing all the natural numbers -greater than a given one, and the second the stream which infinitely -alternates the booleans true and false. - -In general it is not evident to realise when a definition can -be accepted or not. However, there is a class of definitions that -can be easily recognised as being valid : those -where (1) all the recursive calls of the method are done -after having explicitly mentioned which is (at least) the first constructor -to start building the element, and (2) no other -functions apart from constructors are applied to recursive calls. -This class of definitions is usually -referred as {\it guarded-by-constructors} -definitions \cite{Coquand93,Gim94}. -The methods $\from$ -and $\alter$ are examples of definitions which are guarded by constructors. -The definition of function $\filter$ is not, because there is no -constructor to guard -the recursive call in the {\it else} branch. Neither is the one of -$\zeros$, since there is function applied to the recursive call -which is not a constructor. However, there is a difference between -the definition of $\zeros$ and $\filter$. The former may be seen as a -wrong way of characterising an object which makes sense, and it can -be reformulated in an admissible way using the equation (*). On the contrary, -the definition of -$\filter$ can not be patched, since is the idea itself -of traversing an infinite -construction searching for an element whose existence is not ensured -which does not make sense. - - - -Guarded definitions are exactly the kind of non-ending process of -construction which are allowed in Coq. The way of introducing -a guarded definition in Coq is using the special command -{\tt CoFixpoint}. This command verifies that the definition introduces an -element of a co-inductive type, and checks if it is guarded by constructors. -If we try to -introduce the definitions above, $\from$ and $\alter$ will be accepted, -while $\zeros$ and $\filter$ will be rejected giving some explanation -about why. -\begin{coq_example} -CoFixpoint zeros : Stream nat := cons nat 0%N (tl nat zeros). -CoFixpoint zeros : Stream nat := cons nat 0%N zeros. -CoFixpoint from (n:nat) : Stream nat := cons nat n (from (S n)). -\end{coq_example} - -As in the \verb!Fixpoint! command (see Section~\ref{Fixpoint}), it is possible -to introduce a block of mutually dependent methods. The general syntax -for this case is : - -{\tt CoFixpoint {\ident$_1$} :{\term$_1$} := {\term$_1'$}\\ - with\\ - \mbox{}\hspace{0.1cm} $\ldots$ \\ - with {\ident$_m$} : {\term$_m$} := {\term$_m'$}} - - -\subsection{Non-ending methods and reduction} - -The elimination of a stream introduced by a \verb!CoFixpoint! definition -is done lazily, i.e. its definition can be expanded only when it occurs -at the head of an application which is the argument of a case expression. -Isolately it is considered as a canonical expression which -is completely evaluated. We can test this using the command \verb!compute! -to calculate the normal forms of some terms~: -\begin{coq_example} -Eval compute in (from 0). -Eval compute in (hd nat (from 0)). -Eval compute in (tl nat (from 0)). -\end{coq_example} -\noindent Thus, the equality -$(\from\;n)\equiv(\cons\;\nat\;n\;(\from \; (\S\;n)))$ -does not hold as definitional one. Nevertheless, it can be proved -as a propositional equality, in the sense of Leibniz's equality. -The version {\it à la Leibniz} of the equality above follows from -a general lemma stating that eliminating and then re-introducing a stream -yields the same stream. -\begin{coq_example} -Lemma unfold_Stream : - forall x:Stream nat, x = match x with - | cons a s => cons nat a s - end. -\end{coq_example} - -\noindent The proof is immediate from the analysis of -the possible cases for $x$, which transforms -the equality in a trivial one. - -\begin{coq_example} -olddestruct x. -trivial. -\end{coq_example} -\begin{coq_eval} -Qed. -\end{coq_eval} -The application of this lemma to $(\from\;n)$ puts this -constant at the head of an application which is an argument -of a case analysis, forcing its expansion. -We can test the type of this application using Coq's command \verb!Check!, -which infers the type of a given term. -\begin{coq_example} -Check (fun n:nat => unfold_Stream (from n)). -\end{coq_example} - \noindent Actually, The elimination of $(\from\;n)$ has actually -no effect, because it is followed by a re-introduction, -so the type of this application is in fact -definitionally equal to the -desired proposition. We can test this computing -the normal form of the application above to see its type. -\begin{coq_example} -Transparent unfold_Stream. -Eval compute in (fun n:nat => unfold_Stream (from n)). -\end{coq_example} - - -\section{Reasoning about infinite objects} - -At a first sight, it might seem that -case analysis does not provide a very powerful way -of reasoning about infinite objects. In fact, what we can prove about -an infinite object using -only case analysis is just what we can prove unfolding its method -of construction a finite number of times, which is not always -enough. Consider for example the following method for appending -two streams~: -\begin{coq_example} -Variable A : Set. -CoFixpoint conc (s1 s2:Stream A) : Stream A := - cons A (hd A s1) (conc (tl A s1) s2). -\end{coq_example} - -Informally speaking, we expect that for all pair of streams $s_1$ and $s_2$, -$(\conc\;s_1\;s_2)$ -defines the ``the same'' stream as $s_1$, -in the sense that if we would be able to unfold the definition -``up to the infinite'', we would obtain definitionally equal normal forms. -However, no finite unfolding of the definitions gives definitionally -equal terms. Their equality can not be proved just using case analysis. - - -The weakness of the elimination principle proposed for infinite objects -contrast with the power provided by the inductive -elimination principles, but it is not actually surprising. It just means -that we can not expect to prove very interesting things about infinite -objects doing finite proofs. To take advantage of infinite objects we -have to consider infinite proofs as well. For example, -if we want to catch up the equality between $(\conc\;s_1\;s_2)$ and -$s_1$ we have to introduce first the type of the infinite proofs -of equality between streams. This is a -co-inductive type, whose elements are build up from a -unique constructor, requiring a proof of the equality of the -heads of the streams, and an (infinite) proof of the equality -of their tails. - -\begin{coq_example} -CoInductive EqSt : Stream A -> Stream A -> Prop := - eqst : - forall s1 s2:Stream A, - hd A s1 = hd A s2 -> EqSt (tl A s1) (tl A s2) -> EqSt s1 s2. -\end{coq_example} -\noindent Now the equality of both streams can be proved introducing -an infinite object of type - -\noindent $(\EqSt\;s_1\;(\conc\;s_1\;s_2))$ by a \verb!CoFixpoint! -definition. -\begin{coq_example} -CoFixpoint eqproof (s1 s2:Stream A) : EqSt s1 (conc s1 s2) := - eqst s1 (conc s1 s2) (refl_equal (hd A (conc s1 s2))) - (eqproof (tl A s1) s2). -\end{coq_example} -\begin{coq_eval} -Reset eqproof. -\end{coq_eval} -\noindent Instead of giving an explicit definition, -we can use the proof editor of Coq to help us in -the construction of the proof. -A tactic \verb!Cofix! allows to place a \verb!CoFixpoint! definition -inside a proof. -This tactic introduces a variable in the context which has -the same type as the current goal, and its application stands -for a recursive call in the construction of the proof. If no name is -specified for this variable, the name of the lemma is chosen by -default. -%\pagebreak - -\begin{coq_example} -Lemma eqproof : forall s1 s2:Stream A, EqSt s1 (conc s1 s2). -cofix. -\end{coq_example} - -\noindent An easy (and wrong!) way of finishing the proof is just to apply the -variable \verb!eqproof!, which has the same type as the goal. - -\begin{coq_example} -intros. -apply eqproof. -\end{coq_example} - -\noindent The ``proof'' constructed in this way -would correspond to the \verb!CoFixpoint! definition -\begin{coq_example*} -CoFixpoint eqproof : forall s1 s2:Stream A, EqSt s1 (conc s1 s2) := - eqproof. -\end{coq_example*} - -\noindent which is obviously non-guarded. This means that -we can use the proof editor to -define a method of construction which does not make sense. However, -the system will never accept to include it as part of the theory, -because the guard condition is always verified before saving the proof. - -\begin{coq_example} -Qed. -\end{coq_example} - -\noindent Thus, the user must be careful in the -construction of infinite proofs -with the tactic \verb!Cofix!. Remark that once it has been used -the application of tactics performing automatic proof search in -the environment (like for example \verb!Auto!) -could introduce unguarded recursive calls in the proof. -The command \verb!Guarded! allows to verify -if the guarded condition has been violated -during the construction of the proof. This command can be -applied even if the proof term is not complete. - - - -\begin{coq_example} -Restart. -cofix. -auto. -Guarded. -Undo. -Guarded. -\end{coq_example} - -\noindent To finish with this example, let us restart from the -beginning and show how to construct an admissible proof~: - -\begin{coq_example} -Restart. - cofix. -\end{coq_example} - -%\pagebreak - -\begin{coq_example} -intros. -apply eqst. -trivial. -simpl. -apply eqproof. -Qed. -\end{coq_example} - - -\section{Experiments with co-inductive types} - -Some examples involving co-inductive types are available with -the distributed system, in the theories library and in the contributions -of the Lyon site. Here we present a short description of their contents~: -\begin{itemize} -\item Directory \verb!theories/LISTS! : - \begin{itemize} - \item File \verb!Streams.v! : The type of streams and the -extensional equality between streams. - \end{itemize} - -\item Directory \verb!contrib/Lyon/COINDUCTIVES! : - \begin{itemize} - \item Directory \verb!ARITH! : An arithmetic where $\infty$ -is an explicit constant of the language instead of a metatheoretical notion. - \item Directory \verb!STREAM! : - \begin{itemize} - \item File \verb!Examples! : -Several examples of guarded definitions, as well as -of frequent errors in the introduction of a stream. A different -way of defining the extensional equality of two streams, -and the proofs showing that it is equivalent to the one in \verb!theories!. - \item File \verb!Alter.v! : An example showing how -an infinite proof introduced by a guarded definition can be also described -using an operator of co-recursion \cite{Gimenez95b}. - \end{itemize} -\item Directory \verb!PROCESSES! : A proof of the alternating -bit protocol based on Pra\-sad's Calculus of Broadcasting Systems \cite{Prasad93}, -and the verification of an interpreter for this calculus. -See \cite{Gimenez95b} for a complete description about this development. - \end{itemize} -\end{itemize} - -%\end{document} - -% $Id: RefMan-coi.tex 10421 2008-01-05 14:06:51Z herbelin $ diff --git a/doc/refman/RefMan-com.tex b/doc/refman/RefMan-com.tex deleted file mode 100644 index 13a4219a..00000000 --- a/doc/refman/RefMan-com.tex +++ /dev/null @@ -1,384 +0,0 @@ -\chapter[The \Coq~commands]{The \Coq~commands\label{Addoc-coqc} -\ttindex{coqtop} -\ttindex{coqc}} - -There are three \Coq~commands: -\begin{itemize} -\item {\tt coqtop}: The \Coq\ toplevel (interactive mode) ; -\item {\tt coqc} : The \Coq\ compiler (batch compilation). -\item {\tt coqchk} : The \Coq\ checker (validation of compiled libraries) -\end{itemize} -The options are (basically) the same for the first two commands, and -roughly described below. You can also look at the \verb!man! pages of -\verb!coqtop! and \verb!coqc! for more details. - - -\section{Interactive use ({\tt coqtop})} - -In the interactive mode, also known as the \Coq~toplevel, the user can -develop his theories and proofs step by step. The \Coq~toplevel is -run by the command {\tt coqtop}. - -\index{byte-code} -\index{native code} -\label{binary-images} -They are two different binary images of \Coq: the byte-code one and -the native-code one (if Objective Caml provides a native-code compiler -for your platform, which is supposed in the following). When invoking -\verb!coqtop! or \verb!coqc!, the native-code version of the system is -used. The command-line options \verb!-byte! and \verb!-opt! explicitly -select the byte-code and the native-code versions, respectively. - -The byte-code toplevel is based on a Caml -toplevel (to allow the dynamic link of tactics). You can switch to -the Caml toplevel with the command \verb!Drop.!, and come back to the -\Coq~toplevel with the command \verb!Toplevel.loop();;!. - -% The command \verb!coqtop -searchisos! runs the search tool {\sf -% Coq\_SearchIsos} (see Section~\ref{coqsearchisos}, -% page~\pageref{coqsearchisos}) and, as the \Coq~system, can be combined -% with the option \verb!-opt!. - -\section{Batch compilation ({\tt coqc})} -The {\tt coqc} command takes a name {\em file} as argument. Then it -looks for a vernacular file named {\em file}{\tt .v}, and tries to -compile it into a {\em file}{\tt .vo} file (See ~\ref{compiled}). - -\Warning The name {\em file} must be a regular {\Coq} identifier, as -defined in the Section~\ref{lexical}. It -must only contain letters, digits or underscores -(\_). Thus it can be \verb+/bar/foo/toto.v+ but cannot be -\verb+/bar/foo/to-to.v+ . - -Notice that the \verb!-byte! and \verb!-opt! options are still -available with \verb!coqc! and allow you to select the byte-code or -native-code versions of the system. - - -\section[Resource file]{Resource file\index{Resource file}} - -When \Coq\ is launched, with either {\tt coqtop} or {\tt coqc}, the -resource file \verb:$HOME/.coqrc.7.0: is loaded, where \verb:$HOME: is -the home directory of the user. If this file is not found, then the -file \verb:$HOME/.coqrc: is searched. You can also specify an -arbitrary name for the resource file (see option \verb:-init-file: -below), or the name of another user to load the resource file of -someone else (see option \verb:-user:). - -This file may contain, for instance, \verb:Add LoadPath: commands to add -directories to the load path of \Coq. -It is possible to skip the loading of the resource file with the -option \verb:-q:. - -\section[Environment variables]{Environment variables\label{EnvVariables} -\index{Environment variables}} - -There are three environment variables used by the \Coq\ system. -\verb:$COQBIN: for the directory where the binaries are, -\verb:$COQLIB: for the directory where the standard library is, and -\verb:$COQTOP: for the directory of the sources. The latter is useful -only for developers that are writing their own tactics and are using -\texttt{coq\_makefile} (see \ref{Makefile}). If \verb:$COQBIN: or -\verb:$COQLIB: are not defined, \Coq\ will use the default values -(defined at installation time). So these variables are useful only if -you move the \Coq\ binaries and library after installation. - -\section[Options]{Options\index{Options of the command line} -\label{vmoption} -\label{coqoptions}} - -The following command-line options are recognized by the commands {\tt - coqc} and {\tt coqtop}, unless stated otherwise: - -\begin{description} -\item[{\tt -byte}]\ - - Run the byte-code version of \Coq{}. - -\item[{\tt -opt}]\ - - Run the native-code version of \Coq{}. - -\item[{\tt -I} {\em directory}, {\tt -include} {\em directory}]\ - - Add physical path {\em directory} to the list of directories where to - look for a file and bind it to the empty logical directory. The - subdirectory structure of {\em directory} is recursively available - from {\Coq} using absolute names (see Section~\ref{LongNames}). - -\item[{\tt -I} {\em directory} {\tt -as} {\em dirpath}]\ - - Add physical path {\em directory} to the list of directories where to - look for a file and bind it to the logical directory {\dirpath}. The - subdirectory structure of {\em directory} is recursively available - from {\Coq} using absolute names extending the {\dirpath} prefix. - - \SeeAlso {\tt Add LoadPath} in Section~\ref{AddLoadPath} and logical - paths in Section~\ref{Libraries}. - -\item[{\tt -R} {\em directory} {\dirpath}, {\tt -R} {\em directory} {\tt -as} {\dirpath}]\ - - Do as {\tt -I} {\em directory} {\tt -as} {\dirpath} but make the - subdirectory structure of {\em directory} recursively visible so - that the recursive contents of physical {\em directory} is available - from {\Coq} using short or partially qualified names. - - \SeeAlso {\tt Add Rec LoadPath} in Section~\ref{AddRecLoadPath} and logical - paths in Section~\ref{Libraries}. - -\item[{\tt -top} {\dirpath}]\ - - This sets the toplevel module name to {\dirpath} instead of {\tt - Top}. Not valid for {\tt coqc}. - -\item[{\tt -notop} {\dirpath}]\ - - This sets the toplevel module name to the empty logical dirpath. Not - valid for {\tt coqc}. - -\item[{\tt -exclude-dir} {\em subdirectory}]\ - - This tells to exclude any subdirectory named {\em subdirectory} - while processing option {\tt -R}. Without this option only the - conventional version control management subdirectories named {\tt - CVS} and {\tt \_darcs} are excluded. - -\item[{\tt -is} {\em file}, {\tt -inputstate} {\em file}]\ - - Cause \Coq~to use the state put in the file {\em file} as its input - state. The default state is {\em initial.coq}. - Mainly useful to build the standard input state. - -\item[{\tt -outputstate} {\em file}]\ - - Cause \Coq~to dump its state to file {\em file}.coq just after finishing - parsing and evaluating all the arguments from the command line. - -\item[{\tt -nois}]\ - - Cause \Coq~to begin with an empty state. Mainly useful to build the - standard input state. - -%Obsolete? -% -%\item[{\tt -notactics}]\ -% -% Forbid the dynamic loading of tactics in the bytecode version of {\Coq}. - -\item[{\tt -init-file} {\em file}]\ - - Take {\em file} as the resource file. - -\item[{\tt -q}]\ - - Cause \Coq~not to load the resource file. - -\item[{\tt -user} {\em username}]\ - - Take resource file of user {\em username} (that is - \verb+~+{\em username}{\tt /.coqrc.7.0}) instead of yours. - -\item[{\tt -load-ml-source} {\em file}]\ - - Load the Caml source file {\em file}. - -\item[{\tt -load-ml-object} {\em file}]\ - - Load the Caml object file {\em file}. - -\item[{\tt -l} {\em file}, {\tt -load-vernac-source} {\em file}]\ - - Load \Coq~file {\em file}{\tt .v} - -\item[{\tt -lv} {\em file}, {\tt -load-vernac-source-verbose} {\em file}]\ - - Load \Coq~file {\em file}{\tt .v} with - a copy of the contents of the file on standard input. - -\item[{\tt -load-vernac-object} {\em file}]\ - - Load \Coq~compiled file {\em file}{\tt .vo} - -%\item[{\tt -preload} {\em file}]\ \\ -%Add {\em file}{\tt .vo} to the files to be loaded and opened -%before making the initial state. -% -\item[{\tt -require} {\em file}]\ - - Load \Coq~compiled file {\em file}{\tt .vo} and import it ({\tt - Require} {\em file}). - -\item[{\tt -compile} {\em file}]\ - - This compiles file {\em file}{\tt .v} into {\em file}{\tt .vo}. - This option implies options {\tt -batch} and {\tt -silent}. It is - only available for {\tt coqtop}. - -\item[{\tt -compile-verbose} {\em file}]\ - - This compiles file {\em file}{\tt .v} into {\em file}{\tt .vo} with - a copy of the contents of the file on standard input. - This option implies options {\tt -batch} and {\tt -silent}. It is - only available for {\tt coqtop}. - -\item[{\tt -verbose}]\ - - This option is only for {\tt coqc}. It tells to compile the file with - a copy of its contents on standard input. - -\item[{\tt -batch}]\ - - Batch mode : exit just after arguments parsing. This option is only - used by {\tt coqc}. - -%Mostly unused in the code -%\item[{\tt -debug}]\ -% -% Switch on the debug flag. - -\item[{\tt -xml}]\ - - This option is for use with {\tt coqc}. It tells \Coq\ to export on - the standard output the content of the compiled file into XML format. - -\item[{\tt -quality}] - - Improve the legibility of the proof terms produced by some tactics. - -\item[{\tt -emacs}]\ - - Tells \Coq\ it is executed under Emacs. - -\item[{\tt -impredicative-set}]\ - - Change the logical theory of {\Coq} by declaring the sort {\tt Set} - impredicative; warning: this is known to be inconsistent with - some standard axioms of classical mathematics such as the functional - axiom of choice or the principle of description - -\item[{\tt -dump-glob} {\em file}]\ - - This dumps references for global names in file {\em file} - (to be used by coqdoc, see~\ref{coqdoc}) - -\item[{\tt -dont-load-proofs}]\ - - This avoids loading in memory the proofs of opaque theorems - resulting in a smaller memory requirement and faster compilation; - warning: this invalidates some features such as the extraction tool. - -\item[{\tt -vm}]\ - - This activates the use of the bytecode-based conversion algorithm - for the current session (see Section~\ref{SetVirtualMachine}). - -\item[{\tt -image} {\em file}]\ - - This option sets the binary image to be used to be {\em file} - instead of the standard one. Not of general use. - -\item[{\tt -bindir} {\em directory}]\ - - Set for {\tt coqc} the directory containing \Coq\ binaries. - It is equivalent to do \texttt{export COQBIN=}{\em directory} - before lauching {\tt coqc}. - -\item[{\tt -where}]\ - - Print the \Coq's standard library location and exit. - -\item[{\tt -v}]\ - - Print the \Coq's version and exit. - -\item[{\tt -h}, {\tt --help}]\ - - Print a short usage and exit. - -\end{description} - - -\section{Compiled libraries checker ({\tt coqchk})} - -The {\tt coqchk} command takes a list of library paths as argument. -The corresponding compiled libraries (.vo files) are searched in the -path, recursively processing the libraries they depend on. The content -of all these libraries is then type-checked. The effect of {\tt - coqchk} is only to return with normal exit code in case of success, -and with positive exit code if an error has been found. Error messages -are not deemed to help the user understand what is wrong. In the -current version, it does not modify the compiled libraries to mark -them as successfully checked. - -Note that non-logical information is not checked. By logical -information, we mean the type and optional body associated to names. -It excludes for instance anything related to the concrete syntax of -objects (customized syntax rules, association between short and long -names), implicit arguments, etc. - -This tool can be used for several purposes. One is to check that a -compiled library provided by a third-party has not been forged and -that loading it cannot introduce inconsistencies.\footnote{Ill-formed - non-logical information might for instance bind {\tt - Coq.Init.Logic.True} to short name {\tt False}, so apparently {\tt - False} is inhabited, but using fully qualified names, {\tt - Coq.Init.Logic.False} will always refer to the absurd proposition, - what we guarantee is that there is no proof of this latter - constant.} -Another point is to get an even higher level of security. Since {\tt - coqtop} can be extended with custom tactics, possibly ill-typed -code, it cannot be guaranteed that the produced compiled libraries are -correct. {\tt coqchk} is a standalone verifier, and thus it cannot be -tainted by such malicious code. - -Command-line options {\tt -I}, {\tt -R}, {\tt -where} and -{\tt -impredicative-set} are supported by {\tt coqchk} and have the -same meaning as for {\tt coqtop}. Extra options are: -\begin{description} -\item[{\tt -norec} $module$]\ - - Check $module$ but do not force check of its dependencies. -\item[{\tt -admit} $module$] \ - - Do not check $module$ and any of its dependencies, unless - explicitly required. -\item[{\tt -o}]\ - - At exit, print a summary about the context. List the names of all - assumptions and variables (constants without body). -\item[{\tt -silent}]\ - - Do not write progress information in standard output. -\end{description} - -Environment variable \verb:$COQLIB: can be set to override the -location of the standard library. - -The algorithm for deciding which modules are checked or admitted is -the following: assuming that {\tt coqchk} is called with argument $M$, -option {\tt -norec} $N$, and {\tt -admit} $A$. Let us write -$\overline{S}$ the set of reflexive transitive dependencies of set -$S$. Then: -\begin{itemize} -\item Modules $C=\overline{M}\backslash\overline{A}\cup M\cup N$ are - loaded and type-checked before being added to the context. -\item And $\overline{M}\cup\overline{N}\backslash C$ is the set of - modules that are loaded and added to the context without - type-checking. Basic integrity checks (checksums) are nonetheless - performed. -\end{itemize} - -As a rule of thumb, the {\tt -admit} can be used to tell that some -libraries have already been checked. So {\tt coqchk A B} can be split -in {\tt coqchk A \&\& coqchk B -admit A} without type-checking any -definition twice. Of course, the latter is slightly slower since it -makes more disk access. It is also less secure since an attacker might -have replaced the compiled library $A$ after it has been read by the -first command, but before it has been read by the second command. - -% $Id: RefMan-com.tex 12443 2009-10-29 16:17:29Z gmelquio $ - -%%% Local Variables: -%%% mode: latex -%%% TeX-master: "Reference-Manual" -%%% End: diff --git a/doc/refman/RefMan-decl.tex b/doc/refman/RefMan-decl.tex deleted file mode 100644 index ba8a5ac6..00000000 --- a/doc/refman/RefMan-decl.tex +++ /dev/null @@ -1,808 +0,0 @@ -\newcommand{\DPL}{Mathematical Proof Language} - -\chapter{The \DPL\label{DPL}\index{DPL}} - -\section{Introduction} - -\subsection{Foreword} - -In this chapter, we describe an alternative language that may be used -to do proofs using the Coq proof assistant. The language described -here uses the same objects (proof-terms) as Coq, but it differs in the -way proofs are described. This language was created by Pierre -Corbineau at the Radboud University of Nijmegen, The Netherlands. - -The intent is to provide language where proofs are less formalism-{} -and implementation-{}sensitive, and in the process to ease a bit the -learning of computer-{}aided proof verification. - -\subsection{What is a declarative proof ?{}} -In vanilla Coq, proofs are written in the imperative style: the user -issues commands that transform a so called proof state until it -reaches a state where the proof is completed. In the process, the user -mostly described the transitions of this system rather than the -intermediate states it goes through. - -The purpose of a declarative proof language is to take the opposite -approach where intermediate states are always given by the user, but -the transitions of the system are automated as much as possible. - -\subsection{Well-formedness and Completeness} - -The \DPL{} introduces a notion of well-formed -proofs which are weaker than correct (and complete) -proofs. Well-formed proofs are actually proof script where only the -reasoning is incomplete. All the other aspects of the proof are -correct: -\begin{itemize} -\item All objects referred to exist where they are used -\item Conclusion steps actually prove something related to the - conclusion of the theorem (the {\tt thesis}. -\item Hypothesis introduction steps are done when the goal is an - implication with a corresponding assumption. -\item Sub-objects in the elimination steps for tuples are correct - sub-objects of the tuple being decomposed. -\item Patterns in case analysis are type-correct, and induction is well guarded. -\end{itemize} - -\subsection{Note for tactics users} - -This section explain what differences the casual Coq user will -experience using the \DPL . -\begin{enumerate} -\item The focusing mechanism is constrained so that only one goal at - a time is visible. -\item Giving a statement that Coq cannot prove does not produce an - error, only a warning: this allows to go on with the proof and fill - the gap later. -\item Tactics can still be used for justifications and after -{\texttt{escape}}. -\end{enumerate} - -\subsection{Compatibility} - -The \DPL{} is available for all Coq interfaces that use -text-based interaction, including: -\begin{itemize} -\item the command-{}line toplevel {\texttt{coqtop}} -\item the native GUI {\texttt{coqide}} -\item the Proof-{}General emacs mode -\item Cezary Kaliszyk'{}s Web interface -\item L.E. Mamane'{}s tmEgg TeXmacs plugin -\end{itemize} - -However it is not supported by structured editors such as PCoq. - - - -\section{Syntax} - -Here is a complete formal description of the syntax for DPL commands. - -\begin{figure}[htbp] -\begin{centerframe} -\begin{tabular}{lcl@{\qquad}r} - instruction & ::= & {\tt proof} \\ - & $|$ & {\tt assume } \nelist{statement}{\tt and} - \zeroone{[{\tt and } \{{\tt we have}\}-clause]} \\ - & $|$ & \{{\tt let},{\tt be}\}-clause \\ - & $|$ & \{{\tt given}\}-clause \\ - & $|$ & \{{\tt consider}\}-clause {\tt from} term \\ - & $|$ & ({\tt have} $|$ {\tt then} $|$ {\tt thus} $|$ {\tt hence}]) statement - justification \\ - & $|$ & \zeroone{\tt thus} ($\sim${\tt =}|{\tt =}$\sim$) \zeroone{\ident{\tt :}}\term\relax justification \\ & $|$ & {\tt suffices} (\{{\tt to have}\}-clause $|$ - \nelist{statement}{\tt and } \zeroone{{\tt and} \{{\tt to have}\}-clause})\\ - & & {\tt to show} statement justification \\ - & $|$ & ({\tt claim} $|$ {\tt focus on}) statement \\ - & $|$ & {\tt take} \term \\ - & $|$ & {\tt define} \ident \sequence{var}{,} {\tt as} \term\\ - & $|$ & {\tt reconsider} (\ident $|$ {\tt thesis}) {\tt as} type\\ - & $|$ & - {\tt per} ({\tt cases}$|${\tt induction}) {\tt on} \term \\ - & $|$ & {\tt per cases of} type justification \\ - & $|$ & {\tt suppose} \zeroone{\nelist{ident}{,} {\tt and}}~ - {\tt it is }pattern\\ - & & \zeroone{{\tt such that} \nelist{statement} {\tt and} \zeroone{{\tt and} \{{\tt we have}\}-clause}} \\ - & $|$ & {\tt end} - ({\tt proof} $|$ {\tt claim} $|$ {\tt focus} $|$ {\tt cases} $|$ {\tt induction}) \\ - & $|$ & {\tt escape} \\ - & $|$ & {\tt return} \medskip \\ - \{$\alpha,\beta$\}-clause & ::=& $\alpha$ \nelist{var}{,}~ - $\beta$ {\tt such that} \nelist{statement}{\tt and } \\ - & & \zeroone{{\tt and } \{$\alpha,\beta$\}-clause} \medskip\\ - statement & ::= & \zeroone{\ident {\tt :}} type \\ - & $|$ & {\tt thesis} \\ - & $|$ & {\tt thesis for} \ident \medskip \\ - var & ::= & \ident \zeroone{{\tt :} type} \medskip \\ - justification & ::= & - \zeroone{{\tt by} ({\tt *} | \nelist{\term}{,})} - ~\zeroone{{\tt using} tactic} \\ -\end{tabular} -\end{centerframe} -\caption{Syntax of mathematical proof commands} -\end{figure} - -The lexical conventions used here follows those of section \ref{lexical}. - - -Conventions:\begin{itemize} - - \item {\texttt{<{}tactic>{}}} stands for an Coq tactic. - - \end{itemize} - -\subsection{Temporary names} - -In proof commands where an optional name is asked for, omitting the -name will trigger the creation of a fresh temporary name (e.g. for a -hypothesis). Temporary names always start with an undescore '\_' -character (e.g. {\tt \_hyp0}). Temporary names have a lifespan of one -command: they get erased after the next command. They can however be safely in the step after their creation. - -\section{Language description} - -\subsection{Starting and Ending a mathematical proof} - - The standard way to use the \DPL is to first state a {\texttt{Lemma/Theorem/Definition}} and then use the {\texttt{proof}} command to switch the current subgoal to mathematical mode. After the proof is completed, the {\texttt{end proof}} command will close the mathematical proof. If any subgoal remains to be proved, they will be displayed using the usual Coq display. - -\begin{coq_example} -Theorem this_is_trivial: True. -proof. - thus thesis. -end proof. -Qed. -\end{coq_example} - -The {\texttt{proof}} command only applies to \emph{one subgoal}, thus -if several sub-goals are already present, the {\texttt{proof .. end - proof}} sequence has to be used several times. - -\begin{coq_eval} -Theorem T: (True /\ True) /\ True. - split. split. -\end{coq_eval} -\begin{coq_example} - Show. - proof. (* first subgoal *) - thus thesis. - end proof. - trivial. (* second subgoal *) - proof. (* third subgoal *) - thus thesis. - end proof. -\end{coq_example} -\begin{coq_eval} -Abort. -\end{coq_eval} - -As with all other block structures, the {\texttt{end proof}} command -assumes that your proof is complete. If not, executing it will be -equivalent to admitting that the statement is proved: A warning will -be issued and you will not be able to run the {\texttt{Qed}} -command. Instead, you can run {\texttt{Admitted}} if you wish to start -another theorem and come back -later. - -\begin{coq_example} -Theorem this_is_not_so_trivial: False. -proof. -end proof. (* here a warning is issued *) -Qed. (* fails : the proof in incomplete *) -Admitted. (* Oops! *) -\end{coq_example} -\begin{coq_eval} -Reset this_is_not_so_trivial. -\end{coq_eval} - -\subsection{Switching modes} - -When writing a mathematical proof, you may wish to use procedural -tactics at some point. One way to do so is to write a using-{}phrase -in a deduction step (see section~\ref{justifications}). The other way -is to use an {\texttt{escape...return}} block. - -\begin{coq_eval} -Theorem T: True. -proof. -\end{coq_eval} -\begin{coq_example} - Show. - escape. - auto. - return. -\end{coq_example} -\begin{coq_eval} -Abort. -\end{coq_eval} - -The return statement expects all subgoals to be closed, otherwise a -warning is issued and the proof cannot be saved anymore. - -It is possible to use the {\texttt{proof}} command inside an -{\texttt{escape...return}} block, thus nesting a mathematical proof -inside a procedural proof inside a mathematical proof ... - -\subsection{Computation steps} - -The {\tt reconsider ... as} command allows to change the type of a hypothesis or of {\tt thesis} to a convertible one. - -\begin{coq_eval} -Theorem T: let a:=false in let b:= true in ( if a then True else False -> if b then True else False). -intros a b. -proof. -assume H:(if a then True else False). -\end{coq_eval} -\begin{coq_example} - Show. - reconsider H as False. - reconsider thesis as True. -\end{coq_example} -\begin{coq_eval} -Abort. -\end{coq_eval} - - -\subsection{Deduction steps} - -The most common instruction in a mathematical proof is the deduction step: - it asserts a new statement (a formula/type of the \CIC) and tries to prove it using a user-provided indication : the justification. The asserted statement is then added as a hypothesis to the proof context. - -\begin{coq_eval} -Theorem T: forall x, x=2 -> 2+x=4. -proof. -let x be such that H:(x=2). -\end{coq_eval} -\begin{coq_example} - Show. - have H':(2+x=2+2) by H. -\end{coq_example} -\begin{coq_eval} -Abort. -\end{coq_eval} - -It is very often the case that the justifications uses the last hypothesis introduced in the context, so the {\tt then} keyword can be used as a shortcut, e.g. if we want to do the same as the last example : - -\begin{coq_eval} -Theorem T: forall x, x=2 -> 2+x=4. -proof. -let x be such that H:(x=2). -\end{coq_eval} -\begin{coq_example} - Show. - then (2+x=2+2). -\end{coq_example} -\begin{coq_eval} -Abort. -\end{coq_eval} - -In this example, you can also see the creation of a temporary name {\tt \_fact}. - -\subsection{Iterated equalities} - -A common proof pattern when doing a chain of deductions, is to do -multiple rewriting steps over the same term, thus proving the -corresponding equalities. The iterated equalities are a syntactic -support for this kind of reasoning: - -\begin{coq_eval} -Theorem T: forall x, x=2 -> x + x = x * x. -proof. -let x be such that H:(x=2). -\end{coq_eval} -\begin{coq_example} - Show. - have (4 = 4). - ~= (2 * 2). - ~= (x * x) by H. - =~ (2 + 2). - =~ H':(x + x) by H. -\end{coq_example} -\begin{coq_eval} -Abort. -\end{coq_eval} - -Notice that here we use temporary names heavily. - -\subsection{Subproofs} - -When an intermediate step in a proof gets too complicated or involves a well contained set of intermediate deductions, it can be useful to insert its proof as a subproof of the current proof. this is done by using the {\tt claim ... end claim} pair of commands. - -\begin{coq_eval} -Theorem T: forall x, x + x = x * x -> x = 0 \/ x = 2. -proof. -let x be such that H:(x + x = x * x). -\end{coq_eval} -\begin{coq_example} -Show. -claim H':((x - 2) * x = 0). -\end{coq_example} - -A few steps later ... - -\begin{coq_example} -thus thesis. -end claim. -\end{coq_example} - -Now the rest of the proof can happen. - -\begin{coq_eval} -Abort. -\end{coq_eval} - -\subsection{Conclusion steps} - -The commands described above have a conclusion counterpart, where the -new hypothesis is used to refine the conclusion. - -\begin{figure}[b] - \centering -\begin{tabular}{c|c|c|c|c|} - X & \,simple\, & \,with previous step\, & - \,opens sub-proof\, & \,iterated equality\, \\ -\hline -intermediate step & {\tt have} & {\tt then} & - {\tt claim} & {\tt $\sim$=/=$\sim$}\\ -conclusion step & {\tt thus} & {\tt hence} & - {\tt focus on} & {\tt thus $\sim$=/=$\sim$}\\ -\hline -\end{tabular} -\caption{Correspondence between basic forward steps and conclusion steps} -\end{figure} - -Let us begin with simple examples : - -\begin{coq_eval} -Theorem T: forall (A B:Prop), A -> B -> A /\ B. -intros A B HA HB. -proof. -\end{coq_eval} -\begin{coq_example} -Show. -hence B. -\end{coq_example} -\begin{coq_eval} -Abort. -\end{coq_eval} - -In the next example, we have to use {\tt thus} because {\tt HB} is no longer -the last hypothesis. - -\begin{coq_eval} -Theorem T: forall (A B C:Prop), A -> B -> C -> A /\ B /\ C. -intros A B C HA HB HC. -proof. -\end{coq_eval} -\begin{coq_example} -Show. -thus B by HB. -\end{coq_example} -\begin{coq_eval} -Abort. -\end{coq_eval} - -The command fails the refinement process cannot find a place to fit -the object in a proof of the conclusion. - - -\begin{coq_eval} -Theorem T: forall (A B C:Prop), A -> B -> C -> A /\ B. -intros A B C HA HB HC. -proof. -\end{coq_eval} -\begin{coq_example} -Show. -hence C. (* fails *) -\end{coq_example} -\begin{coq_eval} -Abort. -\end{coq_eval} - -The refinement process may induce non -reversible choices, e.g. when proving a disjunction it may {\it - choose} one side of the disjunction. - -\begin{coq_eval} -Theorem T: forall (A B:Prop), B -> A \/ B. -intros A B HB. -proof. -\end{coq_eval} -\begin{coq_example} -Show. -hence B. -\end{coq_example} -\begin{coq_eval} -Abort. -\end{coq_eval} - -In this example you can see that the right branch was chosen since {\tt D} remains to be proved. - -\begin{coq_eval} -Theorem T: forall (A B C D:Prop), C -> D -> (A /\ B) \/ (C /\ D). -intros A B C D HC HD. -proof. -\end{coq_eval} -\begin{coq_example} -Show. -thus C by HC. -\end{coq_example} -\begin{coq_eval} -Abort. -\end{coq_eval} - -Now for existential statements, we can use the {\tt take} command to -choose {\tt 2} as an explicit witness of existence. - -\begin{coq_eval} -Theorem T: forall (P:nat -> Prop), P 2 -> exists x,P x. -intros P HP. -proof. -\end{coq_eval} -\begin{coq_example} -Show. -take 2. -\end{coq_example} -\begin{coq_eval} -Abort. -\end{coq_eval} - -It is also possible to prove the existence directly. - -\begin{coq_eval} -Theorem T: forall (P:nat -> Prop), P 2 -> exists x,P x. -intros P HP. -proof. -\end{coq_eval} -\begin{coq_example} -Show. -hence (P 2). -\end{coq_example} -\begin{coq_eval} -Abort. -\end{coq_eval} - -Here a more involved example where the choice of {\tt P 2} propagates -the choice of {\tt 2} to another part of the formula. - -\begin{coq_eval} -Theorem T: forall (P:nat -> Prop) (R:nat -> nat -> Prop), P 2 -> R 0 2 -> exists x, exists y, P y /\ R x y. -intros P R HP HR. -proof. -\end{coq_eval} -\begin{coq_example} -Show. -thus (P 2) by HP. -\end{coq_example} -\begin{coq_eval} -Abort. -\end{coq_eval} - -Now, an example with the {\tt suffices} command. {\tt suffices} -is a sort of dual for {\tt have}: it allows to replace the conclusion -(or part of it) by a sufficient condition. - -\begin{coq_eval} -Theorem T: forall (A B:Prop) (P:nat -> Prop), (forall x, P x -> B) -> A -> A /\ B. -intros A B P HP HA. -proof. -\end{coq_eval} -\begin{coq_example} -Show. -suffices to have x such that HP':(P x) to show B by HP,HP'. -\end{coq_example} -\begin{coq_eval} -Abort. -\end{coq_eval} - -Finally, an example where {\tt focus} is handy : local assumptions. - -\begin{coq_eval} -Theorem T: forall (A:Prop) (P:nat -> Prop), P 2 -> A -> A /\ (forall x, x = 2 -> P x). -intros A P HP HA. -proof. -\end{coq_eval} -\begin{coq_example} -Show. -focus on (forall x, x = 2 -> P x). -let x be such that (x = 2). -hence thesis by HP. -end focus. -\end{coq_example} -\begin{coq_eval} -Abort. -\end{coq_eval} - -\subsection{Declaring an Abbreviation} - -In order to shorten long expressions, it is possible to use the {\tt - define ... as ...} command to give a name to recurring expressions. - -\begin{coq_eval} -Theorem T: forall x, x = 0 -> x + x = x * x . -proof. -let x be such that H:(x = 0). -\end{coq_eval} -\begin{coq_example} -Show. -define sqr x as (x * x). -reconsider thesis as (x + x = sqr x). -\end{coq_example} -\begin{coq_eval} -Abort. -\end{coq_eval} - -\subsection{Introduction steps} - -When the {\tt thesis} consists of a hypothetical formula (implication -or universal quantification (e.g. \verb+A -> B+) , it is possible to -assume the hypothetical part {\tt A} and then prove {\tt B}. In the -\DPL{}, this comes in two syntactic flavors that are semantically -equivalent : {\tt let} and {\tt assume}. Their syntax is designed so that {\tt let} works better for universal quantifiers and {\tt assume} for implications. - -\begin{coq_eval} -Theorem T: forall (P:nat -> Prop), forall x, P x -> P x. -proof. -let P:(nat -> Prop). -\end{coq_eval} -\begin{coq_example} -Show. -let x:nat. -assume HP:(P x). -\end{coq_example} -\begin{coq_eval} -Abort. -\end{coq_eval} - -In the {\tt let} variant, the type of the assumed object is optional -provided it can be deduced from the command. The objects introduced by -let can be followed by assumptions using {\tt such that}. - -\begin{coq_eval} -Theorem T: forall (P:nat -> Prop), forall x, P x -> P x. -proof. -let P:(nat -> Prop). -\end{coq_eval} -\begin{coq_example} -Show. -let x. (* fails because x's type is not clear *) -let x be such that HP:(P x). (* here x's type is inferred from (P x) *) -\end{coq_example} -\begin{coq_eval} -Abort. -\end{coq_eval} - -In the {\tt assume } variant, the type of the assumed object is mandatory but the name is optional : - -\begin{coq_eval} -Theorem T: forall (P:nat -> Prop), forall x, P x -> P x -> P x. -proof. -let P:(nat -> Prop). -let x:nat. -\end{coq_eval} -\begin{coq_example} -Show. -assume (P x). (* temporary name created *) -\end{coq_example} -\begin{coq_eval} -Abort. -\end{coq_eval} - -After {\tt such that}, it is also the case : - -\begin{coq_eval} -Theorem T: forall (P:nat -> Prop), forall x, P x -> P x. -proof. -let P:(nat -> Prop). -\end{coq_eval} -\begin{coq_example} -Show. -let x be such that (P x). (* temporary name created *) -\end{coq_example} -\begin{coq_eval} -Abort. -\end{coq_eval} - -\subsection{Tuple elimination steps} - -In the \CIC, many objects dealt with in simple proofs are tuples : -pairs , records, existentially quantified formulas. These are so -common that the \DPL{} provides a mechanism to extract members of -those tuples, and also objects in tuples within tuples within -tuples... - -\begin{coq_eval} -Theorem T: forall (P:nat -> Prop) (A:Prop), (exists x, (P x /\ A)) -> A. -proof. -let P:(nat -> Prop),A:Prop be such that H:(exists x, P x /\ A) . -\end{coq_eval} -\begin{coq_example} -Show. -consider x such that HP:(P x) and HA:A from H. -\end{coq_example} -\begin{coq_eval} -Abort. -\end{coq_eval} - -Here is an example with pairs: - -\begin{coq_eval} -Theorem T: forall p:(nat * nat)%type, (fst p >= snd p) \/ (fst p < snd p). -proof. -let p:(nat * nat)%type. -\end{coq_eval} -\begin{coq_example} -Show. -consider x:nat,y:nat from p. -reconsider thesis as (x >= y \/ x < y). -\end{coq_example} -\begin{coq_eval} -Abort. -\end{coq_eval} - -It is sometimes desirable to combine assumption and tuple -decomposition. This can be done using the {\tt given} command. - -\begin{coq_eval} -Theorem T: forall P:(nat -> Prop), (forall n , P n -> P (n - 1)) -> -(exists m, P m) -> P 0. -proof. -let P:(nat -> Prop) be such that HP:(forall n , P n -> P (n - 1)). -\end{coq_eval} -\begin{coq_example} -Show. -given m such that Hm:(P m). -\end{coq_example} -\begin{coq_eval} -Abort. -\end{coq_eval} - -\subsection{Disjunctive reasoning} - -In some proofs (most of them usually) one has to consider several -cases and prove that the {\tt thesis} holds in all the cases. This is -done by first specifying which object will be subject to case -distinction (usually a disjunction) using {\tt per cases}, and then specifying which case is being proved by using {\tt suppose}. - - -\begin{coq_eval} -Theorem T: forall (A B C:Prop), (A -> C) -> (B -> C) -> (A \/ B) -> C. -proof. -let A:Prop,B:Prop,C:Prop be such that HAC:(A -> C) and HBC:(B -> C). -assume HAB:(A \/ B). -\end{coq_eval} -\begin{coq_example} -per cases on HAB. -suppose A. - hence thesis by HAC. -suppose HB:B. - thus thesis by HB,HBC. -end cases. -\end{coq_example} -\begin{coq_eval} -Abort. -\end{coq_eval} - -The proof is well formed (but incomplete) even if you type {\tt end - cases} or the next {\tt suppose} before the previous case is proved. - -If the disjunction is derived from a more general principle, e.g. the -excluded middle axiom), it is desirable to just specify which instance -of it is being used : - -\begin{coq_eval} -Section Coq. -\end{coq_eval} -\begin{coq_example} -Hypothesis EM : forall P:Prop, P \/ ~ P. -\end{coq_example} -\begin{coq_eval} -Theorem T: forall (A C:Prop), (A -> C) -> (~A -> C) -> C. -proof. -let A:Prop,C:Prop be such that HAC:(A -> C) and HNAC:(~A -> C). -\end{coq_eval} -\begin{coq_example} -per cases of (A \/ ~A) by EM. -suppose (~A). - hence thesis by HNAC. -suppose A. - hence thesis by HAC. -end cases. -\end{coq_example} -\begin{coq_eval} -Abort. -\end{coq_eval} - -\subsection{Proofs per cases} - -If the case analysis is to be made on a particular object, the script -is very similar: it starts with {\tt per cases on }\emph{object} instead. - -\begin{coq_eval} -Theorem T: forall (A C:Prop), (A -> C) -> (~A -> C) -> C. -proof. -let A:Prop,C:Prop be such that HAC:(A -> C) and HNAC:(~A -> C). -\end{coq_eval} -\begin{coq_example} -per cases on (EM A). -suppose (~A). -\end{coq_example} -\begin{coq_eval} -Abort. -End Coq. -\end{coq_eval} - -If the object on which a case analysis occurs in the statement to be -proved, the command {\tt suppose it is }\emph{pattern} is better -suited than {\tt suppose}. \emph{pattern} may contain nested patterns -with {\tt as} clauses. A detailed description of patterns is to be -found in figure \ref{term-syntax-aux}. here is an example. - -\begin{coq_eval} -Theorem T: forall (A B:Prop) (x:bool), (if x then A else B) -> A \/ B. -proof. -let A:Prop,B:Prop,x:bool. -\end{coq_eval} -\begin{coq_example} -per cases on x. -suppose it is true. - assume A. - hence A. -suppose it is false. - assume B. - hence B. -end cases. -\end{coq_example} -\begin{coq_eval} -Abort. -\end{coq_eval} - -\subsection{Proofs by induction} - -Proofs by induction are very similar to proofs per cases: they start -with {\tt per induction on }{\tt object} and proceed with {\tt suppose - it is }\emph{pattern}{\tt and }\emph{induction hypothesis}. The -induction hypothesis can be given explicitly or identified by the -sub-object $m$ it refers to using {\tt thesis for }\emph{m}. - -\begin{coq_eval} -Theorem T: forall (n:nat), n + 0 = n. -proof. -let n:nat. -\end{coq_eval} -\begin{coq_example} -per induction on n. -suppose it is 0. - thus (0 + 0 = 0). -suppose it is (S m) and H:thesis for m. - then (S (m + 0) = S m). - thus =~ (S m + 0). -end induction. -\end{coq_example} -\begin{coq_eval} -Abort. -\end{coq_eval} - -\subsection{Justifications}\label{justifications} - - -Intuitively, justifications are hints for the system to understand how -to prove the statements the user types in. In the case of this -language justifications are made of two components: - -Justification objects : {\texttt{by}} followed by a comma-{}separated -list of objects that will be used by a selected tactic to prove the -statement. This defaults to the empty list (the statement should then -be tautological). The * wildcard provides the usual tactics behavior: -use all statements in local context. However, this wildcard should be -avoided since it reduces the robustness of the script. - -Justification tactic : {\texttt{using}} followed by a Coq tactic that -is executed to prove the statement. The default is a solver for -(intuitionistic) first-{}order with equality. - -\section{More details and Formal Semantics} - -I suggest the users looking for more information have a look at the -paper \cite{corbineau08types}. They will find in that paper a formal -semantics of the proof state transition induces by mathematical -commands. diff --git a/doc/refman/RefMan-ext.tex b/doc/refman/RefMan-ext.tex deleted file mode 100644 index c0e578fe..00000000 --- a/doc/refman/RefMan-ext.tex +++ /dev/null @@ -1,1756 +0,0 @@ -\chapter[Extensions of \Gallina{}]{Extensions of \Gallina{}\label{Gallina-extension}\index{Gallina}} - -{\gallina} is the kernel language of {\Coq}. We describe here extensions of -the Gallina's syntax. - -\section{Record types -\comindex{Record} -\label{Record}} - -The \verb+Record+ construction is a macro allowing the definition of -records as is done in many programming languages. Its syntax is -described on Figure~\ref{record-syntax}. In fact, the \verb+Record+ -macro is more general than the usual record types, since it allows -also for ``manifest'' expressions. In this sense, the \verb+Record+ -construction allows to define ``signatures''. - -\begin{figure}[h] -\begin{centerframe} -\begin{tabular}{lcl} -{\sentence} & ++= & {\record}\\ - & & \\ -{\record} & ::= & - {\tt Record} {\ident} \zeroone{\binders} \zeroone{{\tt :} {\sort}} \verb.:=. \\ -&& ~~~~\zeroone{\ident} - \verb!{! \zeroone{\nelist{\field}{;}} \verb!}! \verb:.:\\ - & & \\ -{\field} & ::= & {\name} \zeroone{\binders} : {\type} [ {\tt where} {\it notation} ] \\ - & $|$ & {\name} \zeroone{\binders} {\typecstr} := {\term} -\end{tabular} -\end{centerframe} -\caption{Syntax for the definition of {\tt Record}} -\label{record-syntax} -\end{figure} - -\noindent In the expression - -\smallskip -{\tt Record} {\ident} {\params} \texttt{:} - {\sort} := {\ident$_0$} \verb+{+ - {\ident$_1$} \binders$_1$ \texttt{:} {\term$_1$}; - \dots - {\ident$_n$} \binders$_n$ \texttt{:} {\term$_n$} \verb+}+. -\smallskip - -\noindent the identifier {\ident} is the name of the defined record -and {\sort} is its type. The identifier {\ident$_0$} is the name of -its constructor. If {\ident$_0$} is omitted, the default name {\tt -Build\_{\ident}} is used. If {\sort} is omitted, the default sort is ``{\Type}''. -The identifiers {\ident$_1$}, .., -{\ident$_n$} are the names of fields and {\tt forall} \binders$_1${\tt ,} {\term$_1$}, ..., {\tt forall} \binders$_n${\tt ,} {\term$_n$} -their respective types. Remark that the type of {\ident$_i$} may -depend on the previous {\ident$_j$} (for $j<i$). Thus the order of the -fields is important. Finally, {\params} are the parameters of the -record. - -More generally, a record may have explicitly defined (a.k.a. -manifest) fields. For instance, {\tt Record} {\ident} {\tt [} -{\params} {\tt ]} \texttt{:} {\sort} := \verb+{+ {\ident$_1$} -\texttt{:} {\type$_1$} \verb+;+ {\ident$_2$} \texttt{:=} {\term$_2$} -\verb+;+ {\ident$_3$} \texttt{:} {\type$_3$} \verb+}+ in which case -the correctness of {\type$_3$} may rely on the instance {\term$_2$} of -{\ident$_2$} and {\term$_2$} in turn may depend on {\ident$_1$}. - - -\Example -The set of rational numbers may be defined as: -\begin{coq_eval} -Reset Initial. -\end{coq_eval} -\begin{coq_example} -Record Rat : Set := mkRat - {sign : bool; - top : nat; - bottom : nat; - Rat_bottom_cond : 0 <> bottom; - Rat_irred_cond : - forall x y z:nat, (x * y) = top /\ (x * z) = bottom -> x = 1}. -\end{coq_example} - -Remark here that the field -\verb+Rat_cond+ depends on the field \verb+bottom+. - -%Let us now see the work done by the {\tt Record} macro. -%First the macro generates an inductive definition -%with just one constructor: -% -%\medskip -%\noindent -%{\tt Inductive {\ident} \zeroone{\binders} : {\sort} := \\ -%\mbox{}\hspace{0.4cm} {\ident$_0$} : forall ({\ident$_1$}:{\term$_1$}) .. -%({\ident$_n$}:{\term$_n$}), {\ident} {\rm\sl params}.} -%\medskip - -Let us now see the work done by the {\tt Record} macro. First the -macro generates an inductive definition with just one constructor: -\begin{quote} -{\tt Inductive {\ident} {\params} :{\sort} :=} \\ -\qquad {\tt - {\ident$_0$} ({\ident$_1$}:{\term$_1$}) .. ({\ident$_n$}:{\term$_n$}).} -\end{quote} -To build an object of type {\ident}, one should provide the -constructor {\ident$_0$} with $n$ terms filling the fields of -the record. - -As an example, let us define the rational $1/2$: -\begin{coq_example*} -Require Import Arith. -Theorem one_two_irred : - forall x y z:nat, x * y = 1 /\ x * z = 2 -> x = 1. -\end{coq_example*} -\begin{coq_eval} -Lemma mult_m_n_eq_m_1 : forall m n:nat, m * n = 1 -> m = 1. -destruct m; trivial. -intros; apply f_equal with (f := S). -destruct m; trivial. -destruct n; simpl in H. - rewrite <- mult_n_O in H. - discriminate. - rewrite <- plus_n_Sm in H. - discriminate. -Qed. - -intros x y z [H1 H2]. - apply mult_m_n_eq_m_1 with (n := y); trivial. -\end{coq_eval} -\ldots -\begin{coq_example*} -Qed. -\end{coq_example*} -\begin{coq_example} -Definition half := mkRat true 1 2 (O_S 1) one_two_irred. -\end{coq_example} -\begin{coq_example} -Check half. -\end{coq_example} - -The macro generates also, when it is possible, the projection -functions for destructuring an object of type {\ident}. These -projection functions have the same name that the corresponding -fields. If a field is named ``\verb=_='' then no projection is built -for it. In our example: - -\begin{coq_example} -Eval compute in half.(top). -Eval compute in half.(bottom). -Eval compute in half.(Rat_bottom_cond). -\end{coq_example} -\begin{coq_eval} -Reset Initial. -\end{coq_eval} - -\begin{Warnings} -\item {\tt Warning: {\ident$_i$} cannot be defined.} - - It can happen that the definition of a projection is impossible. - This message is followed by an explanation of this impossibility. - There may be three reasons: - \begin{enumerate} - \item The name {\ident$_i$} already exists in the environment (see - Section~\ref{Axiom}). - \item The body of {\ident$_i$} uses an incorrect elimination for - {\ident} (see Sections~\ref{Fixpoint} and~\ref{Caseexpr}). - \item The type of the projections {\ident$_i$} depends on previous - projections which themselves could not be defined. - \end{enumerate} -\end{Warnings} - -\begin{ErrMsgs} - -\item \errindex{A record cannot be recursive} - - The record name {\ident} appears in the type of its fields. - -\item During the definition of the one-constructor inductive - definition, all the errors of inductive definitions, as described in - Section~\ref{gal_Inductive_Definitions}, may also occur. - -\end{ErrMsgs} - -\SeeAlso Coercions and records in Section~\ref{Coercions-and-records} -of the chapter devoted to coercions. - -\Rem {\tt Structure} is a synonym of the keyword {\tt Record}. - -\Rem Creation of an object of record type can be done by calling {\ident$_0$} -and passing arguments in the correct order. - -\begin{coq_example} -Record point := { x : nat; y : nat }. -Definition a := Build_point 5 3. -\end{coq_example} - -The following syntax allows to create objects by using named fields. The -fields do not have to be in any particular order, nor do they have to be all -present if the missing ones can be inferred or prompted for (see -Section~\ref{Program}). - -\begin{coq_example} -Definition b := {| x := 5; y := 3 |}. -Definition c := {| y := 3; x := 5 |}. -\end{coq_example} - -This syntax can also be used for pattern matching. - -\begin{coq_example} -Eval compute in ( - match b with - | {| y := S n |} => n - | _ => 0 - end). -\end{coq_example} - -\begin{coq_eval} -Reset Initial. -\end{coq_eval} - -\Rem An experimental syntax for projections based on a dot notation is -available. The command to activate it is -\begin{quote} -{\tt Set Printing Projections.} -\end{quote} - -\begin{figure}[t] -\begin{centerframe} -\begin{tabular}{lcl} -{\term} & ++= & {\term} {\tt .(} {\qualid} {\tt )}\\ - & $|$ & {\term} {\tt .(} {\qualid} \nelist{\termarg}{} {\tt )}\\ - & $|$ & {\term} {\tt .(} {@}{\qualid} \nelist{\term}{} {\tt )} -\end{tabular} -\end{centerframe} -\caption{Syntax of \texttt{Record} projections} -\label{fig:projsyntax} -\end{figure} - -The corresponding grammar rules are given Figure~\ref{fig:projsyntax}. -When {\qualid} denotes a projection, the syntax {\tt - {\term}.({\qualid})} is equivalent to {\qualid~\term}, the syntax -{\tt {\term}.({\qualid}~{\termarg}$_1$~ \ldots~ {\termarg}$_n$)} to -{\qualid~{\termarg}$_1$ \ldots {\termarg}$_n$~\term}, and the syntax -{\tt {\term}.(@{\qualid}~{\term}$_1$~\ldots~{\term}$_n$)} to -{@\qualid~{\term}$_1$ \ldots {\term}$_n$~\term}. In each case, {\term} -is the object projected and the other arguments are the parameters of -the inductive type. - -To deactivate the printing of projections, use -{\tt Unset Printing Projections}. - - -\section{Variants and extensions of {\mbox{\tt match}} -\label{Extensions-of-match} -\index{match@{\tt match\ldots with\ldots end}}} - -\subsection{Multiple and nested pattern-matching -\index{ML-like patterns} -\label{Mult-match}} - -The basic version of \verb+match+ allows pattern-matching on simple -patterns. As an extension, multiple nested patterns or disjunction of -patterns are allowed, as in ML-like languages. - -The extension just acts as a macro that is expanded during parsing -into a sequence of {\tt match} on simple patterns. Especially, a -construction defined using the extended {\tt match} is generally -printed under its expanded form (see~\texttt{Set Printing Matching} in -section~\ref{SetPrintingMatching}). - -\SeeAlso Chapter~\ref{Mult-match-full}. - -\subsection{Pattern-matching on boolean values: the {\tt if} expression -\label{if-then-else} -\index{if@{\tt if ... then ... else}}} - -For inductive types with exactly two constructors and for -pattern-matchings expressions which do not depend on the arguments of -the constructors, it is possible to use a {\tt if ... then ... else} -notation. For instance, the definition - -\begin{coq_example} -Definition not (b:bool) := - match b with - | true => false - | false => true - end. -\end{coq_example} - -\noindent can be alternatively written - -\begin{coq_eval} -Reset not. -\end{coq_eval} -\begin{coq_example} -Definition not (b:bool) := if b then false else true. -\end{coq_example} - -More generally, for an inductive type with constructors {\tt C$_1$} -and {\tt C$_2$}, we have the following equivalence - -\smallskip - -{\tt if {\term} \zeroone{\ifitem} then {\term}$_1$ else {\term}$_2$} $\equiv$ -\begin{tabular}[c]{l} -{\tt match {\term} \zeroone{\ifitem} with}\\ -{\tt \verb!|! C$_1$ \_ {\ldots} \_ \verb!=>! {\term}$_1$} \\ -{\tt \verb!|! C$_2$ \_ {\ldots} \_ \verb!=>! {\term}$_2$} \\ -{\tt end} -\end{tabular} - -Here is an example. - -\begin{coq_example} -Check (fun x (H:{x=0}+{x<>0}) => - match H with - | left _ => true - | right _ => false - end). -\end{coq_example} - -Notice that the printing uses the {\tt if} syntax because {\tt sumbool} is -declared as such (see Section~\ref{printing-options}). - -\subsection{Irrefutable patterns: the destructuring {\tt let} variants -\index{let in@{\tt let ... in}} -\label{Letin}} - -Pattern-matching on terms inhabiting inductive type having only one -constructor can be alternatively written using {\tt let ... in ...} -constructions. There are two variants of them. - -\subsubsection{First destructuring {\tt let} syntax} -The expression {\tt let -(}~{\ident$_1$},\ldots,{\ident$_n$}~{\tt ) :=}~{\term$_0$}~{\tt -in}~{\term$_1$} performs case analysis on a {\term$_0$} which must be in -an inductive type with one constructor having itself $n$ arguments. Variables -{\ident$_1$}\ldots{\ident$_n$} are bound to the $n$ arguments of the -constructor in expression {\term$_1$}. For instance, the definition - -\begin{coq_example} -Definition fst (A B:Set) (H:A * B) := match H with - | pair x y => x - end. -\end{coq_example} - -can be alternatively written - -\begin{coq_eval} -Reset fst. -\end{coq_eval} -\begin{coq_example} -Definition fst (A B:Set) (p:A * B) := let (x, _) := p in x. -\end{coq_example} -Notice that reduction is different from regular {\tt let ... in ...} -construction since it happens only if {\term$_0$} is in constructor -form. Otherwise, the reduction is blocked. - -The pretty-printing of a definition by matching on a -irrefutable pattern can either be done using {\tt match} or the {\tt -let} construction (see Section~\ref{printing-options}). - -If {\term} inhabits an inductive type with one constructor {\tt C}, -we have an equivalence between - -{\tt let ({\ident}$_1$,\ldots,{\ident}$_n$) \zeroone{\ifitem} := {\term} in {\term}'} - -\noindent and - -{\tt match {\term} \zeroone{\ifitem} with C {\ident}$_1$ {\ldots} {\ident}$_n$ \verb!=>! {\term}' end} - - -\subsubsection{Second destructuring {\tt let} syntax\index{let '... in}} - -Another destructuring {\tt let} syntax is available for inductive types with -one constructor by giving an arbitrary pattern instead of just a tuple -for all the arguments. For example, the preceding example can be written: -\begin{coq_eval} -Reset fst. -\end{coq_eval} -\begin{coq_example} -Definition fst (A B:Set) (p:A*B) := let 'pair x _ := p in x. -\end{coq_example} - -This is useful to match deeper inside tuples and also to use notations -for the pattern, as the syntax {\tt let 'p := t in b} allows arbitrary -patterns to do the deconstruction. For example: - -\begin{coq_example} -Definition deep_tuple (A:Set) (x:(A*A)*(A*A)) : A*A*A*A := - let '((a,b), (c, d)) := x in (a,b,c,d). -Notation " x 'with' p " := (exist _ x p) (at level 20). -Definition proj1_sig' (A:Set) (P:A->Prop) (t:{ x:A | P x }) : A := - let 'x with p := t in x. -\end{coq_example} - -When printing definitions which are written using this construct it -takes precedence over {\tt let} printing directives for the datatype -under consideration (see Section~\ref{printing-options}). - -\subsection{Controlling pretty-printing of {\tt match} expressions -\label{printing-options}} - -The following commands give some control over the pretty-printing of -{\tt match} expressions. - -\subsubsection{Printing nested patterns -\label{SetPrintingMatching} -\comindex{Set Printing Matching} -\comindex{Unset Printing Matching} -\comindex{Test Printing Matching}} - -The Calculus of Inductive Constructions knows pattern-matching only -over simple patterns. It is however convenient to re-factorize nested -pattern-matching into a single pattern-matching over a nested pattern. -{\Coq}'s printer try to do such limited re-factorization. - -\begin{quote} -{\tt Set Printing Matching.} -\end{quote} -This tells {\Coq} to try to use nested patterns. This is the default -behavior. - -\begin{quote} -{\tt Unset Printing Matching.} -\end{quote} -This tells {\Coq} to print only simple pattern-matching problems in -the same way as the {\Coq} kernel handles them. - -\begin{quote} -{\tt Test Printing Matching.} -\end{quote} -This tells if the printing matching mode is on or off. The default is -on. - -\subsubsection{Printing of wildcard pattern -\comindex{Set Printing Wildcard} -\comindex{Unset Printing Wildcard} -\comindex{Test Printing Wildcard}} - -Some variables in a pattern may not occur in the right-hand side of -the pattern-matching clause. There are options to control the -display of these variables. - -\begin{quote} -{\tt Set Printing Wildcard.} -\end{quote} -The variables having no occurrences in the right-hand side of the -pattern-matching clause are just printed using the wildcard symbol -``{\tt \_}''. - -\begin{quote} -{\tt Unset Printing Wildcard.} -\end{quote} -The variables, even useless, are printed using their usual name. But some -non dependent variables have no name. These ones are still printed -using a ``{\tt \_}''. - -\begin{quote} -{\tt Test Printing Wildcard.} -\end{quote} -This tells if the wildcard printing mode is on or off. The default is -to print wildcard for useless variables. - -\subsubsection{Printing of the elimination predicate -\comindex{Set Printing Synth} -\comindex{Unset Printing Synth} -\comindex{Test Printing Synth}} - -In most of the cases, the type of the result of a matched term is -mechanically synthesizable. Especially, if the result type does not -depend of the matched term. - -\begin{quote} -{\tt Set Printing Synth.} -\end{quote} -The result type is not printed when {\Coq} knows that it can -re-synthesize it. - -\begin{quote} -{\tt Unset Printing Synth.} -\end{quote} -This forces the result type to be always printed. - -\begin{quote} -{\tt Test Printing Synth.} -\end{quote} -This tells if the non-printing of synthesizable types is on or off. -The default is to not print synthesizable types. - -\subsubsection{Printing matching on irrefutable pattern -\comindex{Add Printing Let {\ident}} -\comindex{Remove Printing Let {\ident}} -\comindex{Test Printing Let for {\ident}} -\comindex{Print Table Printing Let}} - -If an inductive type has just one constructor, -pattern-matching can be written using {\tt let} ... {\tt :=} -... {\tt in}~... - -\begin{quote} -{\tt Add Printing Let {\ident}.} -\end{quote} -This adds {\ident} to the list of inductive types for which -pattern-matching is written using a {\tt let} expression. - -\begin{quote} -{\tt Remove Printing Let {\ident}.} -\end{quote} -This removes {\ident} from this list. - -\begin{quote} -{\tt Test Printing Let for {\ident}.} -\end{quote} -This tells if {\ident} belongs to the list. - -\begin{quote} -{\tt Print Table Printing Let.} -\end{quote} -This prints the list of inductive types for which pattern-matching is -written using a {\tt let} expression. - -The list of inductive types for which pattern-matching is written -using a {\tt let} expression is managed synchronously. This means that -it is sensible to the command {\tt Reset}. - -\subsubsection{Printing matching on booleans -\comindex{Add Printing If {\ident}} -\comindex{Remove Printing If {\ident}} -\comindex{Test Printing If for {\ident}} -\comindex{Print Table Printing If}} - -If an inductive type is isomorphic to the boolean type, -pattern-matching can be written using {\tt if} ... {\tt then} ... {\tt - else} ... - -\begin{quote} -{\tt Add Printing If {\ident}.} -\end{quote} -This adds {\ident} to the list of inductive types for which -pattern-matching is written using an {\tt if} expression. - -\begin{quote} -{\tt Remove Printing If {\ident}.} -\end{quote} -This removes {\ident} from this list. - -\begin{quote} -{\tt Test Printing If for {\ident}.} -\end{quote} -This tells if {\ident} belongs to the list. - -\begin{quote} -{\tt Print Table Printing If.} -\end{quote} -This prints the list of inductive types for which pattern-matching is -written using an {\tt if} expression. - -The list of inductive types for which pattern-matching is written -using an {\tt if} expression is managed synchronously. This means that -it is sensible to the command {\tt Reset}. - -\subsubsection{Example} - -This example emphasizes what the printing options offer. - -\begin{coq_example} -Test Printing Let for prod. -Print fst. -Remove Printing Let prod. -Unset Printing Synth. -Unset Printing Wildcard. -Print fst. -\end{coq_example} - -% \subsection{Still not dead old notations} - -% The following variant of {\tt match} is inherited from older version -% of {\Coq}. - -% \medskip -% \begin{tabular}{lcl} -% {\term} & ::= & {\annotation} {\tt Match} {\term} {\tt with} {\terms} {\tt end}\\ -% \end{tabular} -% \medskip - -% This syntax is a macro generating a combination of {\tt match} with {\tt -% Fix} implementing a combinator for primitive recursion equivalent to -% the {\tt Match} construction of \Coq\ V5.8. It is provided only for -% sake of compatibility with \Coq\ V5.8. It is recommended to avoid it. -% (see Section~\ref{Matchexpr}). - -% There is also a notation \texttt{Case} that is the -% ancestor of \texttt{match}. Again, it is still in the code for -% compatibility with old versions but the user should not use it. - -% Explained in RefMan-gal.tex -%% \section{Forced type} - -%% In some cases, one may wish to assign a particular type to a term. The -%% syntax to force the type of a term is the following: - -%% \medskip -%% \begin{tabular}{lcl} -%% {\term} & ++= & {\term} {\tt :} {\term}\\ -%% \end{tabular} -%% \medskip - -%% It forces the first term to be of type the second term. The -%% type must be compatible with -%% the term. More precisely it must be either a type convertible to -%% the automatically inferred type (see Chapter~\ref{Cic}) or a type -%% coercible to it, (see \ref{Coercions}). When the type of a -%% whole expression is forced, it is usually not necessary to give the types of -%% the variables involved in the term. - -%% Example: - -%% \begin{coq_example} -%% Definition ID := forall X:Set, X -> X. -%% Definition id := (fun X x => x):ID. -%% Check id. -%% \end{coq_example} - -\section{Advanced recursive functions} - -The \emph{experimental} command -\begin{center} - \texttt{Function {\ident} {\binder$_1$}\ldots{\binder$_n$} - \{decrease\_annot\} : type$_0$ := \term$_0$} - \comindex{Function} - \label{Function} -\end{center} -can be seen as a generalization of {\tt Fixpoint}. It is actually a -wrapper for several ways of defining a function \emph{and other useful - related objects}, namely: an induction principle that reflects the -recursive structure of the function (see \ref{FunInduction}), and its -fixpoint equality. The meaning of this -declaration is to define a function {\it ident}, similarly to {\tt - Fixpoint}. Like in {\tt Fixpoint}, the decreasing argument must be -given (unless the function is not recursive), but it must not -necessary be \emph{structurally} decreasing. The point of the {\tt - \{\}} annotation is to name the decreasing argument \emph{and} to -describe which kind of decreasing criteria must be used to ensure -termination of recursive calls. - -The {\tt Function} construction enjoys also the {\tt with} extension -to define mutually recursive definitions. However, this feature does -not work for non structural recursive functions. % VRAI?? - -See the documentation of {\tt functional induction} -(see Section~\ref{FunInduction}) and {\tt Functional Scheme} -(see Section~\ref{FunScheme} and \ref{FunScheme-examples}) for how to use the -induction principle to easily reason about the function. - -\noindent {\bf Remark: } To obtain the right principle, it is better -to put rigid parameters of the function as first arguments. For -example it is better to define plus like this: - -\begin{coq_example*} -Function plus (m n : nat) {struct n} : nat := - match n with - | 0 => m - | S p => S (plus m p) - end. -\end{coq_example*} -\noindent than like this: -\begin{coq_eval} -Reset plus. -\end{coq_eval} -\begin{coq_example*} -Function plus (n m : nat) {struct n} : nat := - match n with - | 0 => m - | S p => S (plus p m) - end. -\end{coq_example*} - -\paragraph[Limitations]{Limitations\label{sec:Function-limitations}} -\term$_0$ must be build as a \emph{pure pattern-matching tree} -(\texttt{match...with}) with applications only \emph{at the end} of -each branch. For now dependent cases are not treated. - - - -\begin{ErrMsgs} -\item \errindex{The recursive argument must be specified} -\item \errindex{No argument name \ident} -\item \errindex{Cannot use mutual definition with well-founded - recursion or measure} - -\item \errindex{Cannot define graph for \ident\dots} (warning) - - The generation of the graph relation \texttt{(R\_\ident)} used to - compute the induction scheme of \ident\ raised a typing error. Only - the ident is defined, the induction scheme will not be generated. - - This error happens generally when: - - \begin{itemize} - \item the definition uses pattern matching on dependent types, which - \texttt{Function} cannot deal with yet. - \item the definition is not a \emph{pattern-matching tree} as - explained above. - \end{itemize} - -\item \errindex{Cannot define principle(s) for \ident\dots} (warning) - - The generation of the graph relation \texttt{(R\_\ident)} succeeded - but the induction principle could not be built. Only the ident is - defined. Please report. - -\item \errindex{Cannot build functional inversion principle} (warning) - - \texttt{functional inversion} will not be available for the - function. -\end{ErrMsgs} - - -\SeeAlso{\ref{FunScheme}, \ref{FunScheme-examples}, \ref{FunInduction}} - -Depending on the {\tt \{$\ldots$\}} annotation, different definition -mechanisms are used by {\tt Function}. More precise description -given below. - -\begin{Variants} -\item \texttt{ Function {\ident} {\binder$_1$}\ldots{\binder$_n$} - : type$_0$ := \term$_0$} - - Defines the not recursive function \ident\ as if declared with - \texttt{Definition}. Moreover the following are defined: - - \begin{itemize} - \item {\tt\ident\_rect}, {\tt\ident\_rec} and {\tt\ident\_ind}, - which reflect the pattern matching structure of \term$_0$ (see the - documentation of {\tt Inductive} \ref{Inductive}); - \item The inductive \texttt{R\_\ident} corresponding to the graph of - \ident\ (silently); - \item \texttt{\ident\_complete} and \texttt{\ident\_correct} which are - inversion information linking the function and its graph. - \end{itemize} -\item \texttt{Function {\ident} {\binder$_1$}\ldots{\binder$_n$} - {\tt \{}{\tt struct} \ident$_0${\tt\}} : type$_0$ := \term$_0$} - - Defines the structural recursive function \ident\ as if declared - with \texttt{Fixpoint}. Moreover the following are defined: - - \begin{itemize} - \item The same objects as above; - \item The fixpoint equation of \ident: \texttt{\ident\_equation}. - \end{itemize} - -\item \texttt{Function {\ident} {\binder$_1$}\ldots{\binder$_n$} {\tt - \{}{\tt measure \term$_1$} \ident$_0${\tt\}} : type$_0$ := - \term$_0$} -\item \texttt{Function {\ident} {\binder$_1$}\ldots{\binder$_n$} - {\tt \{}{\tt wf \term$_1$} \ident$_0${\tt\}} : type$_0$ := \term$_0$} - -Defines a recursive function by well founded recursion. \textbf{The -module \texttt{Recdef} of the standard library must be loaded for this -feature}. The {\tt \{\}} annotation is mandatory and must be one of -the following: -\begin{itemize} -\item {\tt \{measure} \term$_1$ \ident$_0${\tt\}} with \ident$_0$ - being the decreasing argument and \term$_1$ being a function - from type of \ident$_0$ to \texttt{nat} for which value on the - decreasing argument decreases (for the {\tt lt} order on {\tt - nat}) at each recursive call of \term$_0$, parameters of the - function are bound in \term$_0$; -\item {\tt \{wf} \term$_1$ \ident$_0${\tt\}} with \ident$_0$ being - the decreasing argument and \term$_1$ an ordering relation on - the type of \ident$_0$ (i.e. of type T$_{\ident_0}$ - $\to$ T$_{\ident_0}$ $\to$ {\tt Prop}) for which - the decreasing argument decreases at each recursive call of - \term$_0$. The order must be well founded. parameters of the - function are bound in \term$_0$. -\end{itemize} - -Depending on the annotation, the user is left with some proof -obligations that will be used to define the function. These proofs -are: proofs that each recursive call is actually decreasing with -respect to the given criteria, and (if the criteria is \texttt{wf}) a -proof that the ordering relation is well founded. - -%Completer sur measure et wf - -Once proof obligations are discharged, the following objects are -defined: - -\begin{itemize} -\item The same objects as with the \texttt{struct}; -\item The lemma \texttt{\ident\_tcc} which collects all proof - obligations in one property; -\item The lemmas \texttt{\ident\_terminate} and \texttt{\ident\_F} - which is needed to be inlined during extraction of \ident. -\end{itemize} - - - -%Complete!! -The way this recursive function is defined is the subject of several -papers by Yves Bertot and Antonia Balaa on the one hand, and Gilles Barthe, -Julien Forest, David Pichardie, and Vlad Rusu on the other hand. - -%Exemples ok ici - -\bigskip - -\noindent {\bf Remark: } Proof obligations are presented as several -subgoals belonging to a Lemma {\ident}{\tt\_tcc}. % These subgoals are independent which means that in order to -% abort them you will have to abort each separately. - - - -%The decreasing argument cannot be dependent of another?? - -%Exemples faux ici -\end{Variants} - - -\section{Section mechanism -\index{Sections} -\label{Section}} - -The sectioning mechanism allows to organize a proof in structured -sections. Then local declarations become available (see -Section~\ref{Basic-definitions}). - -\subsection{\tt Section {\ident}\comindex{Section}} - -This command is used to open a section named {\ident}. - -%% Discontinued ? -%% \begin{Variants} -%% \comindex{Chapter} -%% \item{\tt Chapter {\ident}}\\ -%% Same as {\tt Section {\ident}} -%% \end{Variants} - -\subsection{\tt End {\ident} -\comindex{End}} - -This command closes the section named {\ident}. After closing of the -section, the local declarations (variables and local definitions) get -{\em discharged}, meaning that they stop being visible and that all -global objects defined in the section are generalized with respect to -the variables and local definitions they each depended on in the -section. - - -Here is an example : -\begin{coq_example} -Section s1. -Variables x y : nat. -Let y' := y. -Definition x' := S x. -Definition x'' := x' + y'. -Print x'. -End s1. -Print x'. -Print x''. -\end{coq_example} -Notice the difference between the value of {\tt x'} and {\tt x''} -inside section {\tt s1} and outside. - -\begin{ErrMsgs} -\item \errindex{This is not the last opened section} -\end{ErrMsgs} - -\begin{Remarks} -\item Most commands, like {\tt Hint}, {\tt Notation}, option management, ... -which appear inside a section are canceled when the -section is closed. -% see Section~\ref{LongNames} -%\item Usually all identifiers must be distinct. -%However, a name already used in a closed section (see \ref{Section}) -%can be reused. In this case, the old name is no longer accessible. - -% Obsolète -%\item A module implicitly open a section. Be careful not to name a -%module with an identifier already used in the module (see \ref{compiled}). -\end{Remarks} - -\input{RefMan-mod.v} - -\section{Libraries and qualified names} - -\subsection{Names of libraries and files -\label{Libraries} -\index{Libraries} -\index{Physical paths} -\index{Logical paths}} - -\paragraph{Libraries} - -The theories developed in {\Coq} are stored in {\em library files} -which are hierarchically classified into {\em libraries} and {\em -sublibraries}. To express this hierarchy, library names are -represented by qualified identifiers {\qualid}, i.e. as list of -identifiers separated by dots (see Section~\ref{qualid}). For -instance, the library file {\tt Mult} of the standard {\Coq} library -{\tt Arith} has name {\tt Coq.Arith.Mult}. The identifier -that starts the name of a library is called a {\em library root}. -All library files of the standard library of {\Coq} have reserved root -{\tt Coq} but library file names based on other roots can be obtained -by using {\tt coqc} options {\tt -I} or {\tt -R} (see -Section~\ref{coqoptions}). Also, when an interactive {\Coq} session -starts, a library of root {\tt Top} is started, unless option {\tt --top} or {\tt -notop} is set (see Section~\ref{coqoptions}). - -As library files are stored on the file system of the underlying -operating system, a translation from file-system names to {\Coq} names -is needed. In this translation, names in the file system are called -{\em physical} paths while {\Coq} names are contrastingly called {\em -logical} names. Logical names are mapped to physical paths using the -commands {\tt Add LoadPath} or {\tt Add Rec LoadPath} (see -Sections~\ref{AddLoadPath} and~\ref{AddRecLoadPath}). - -\subsection{Qualified names -\label{LongNames} -\index{Qualified identifiers} -\index{Absolute names}} - -Library files are modules which possibly contain submodules which -eventually contain constructions (axioms, parameters, definitions, -lemmas, theorems, remarks or facts). The {\em absolute name}, or {\em -full name}, of a construction in some library file is a qualified -identifier starting with the logical name of the library file, -followed by the sequence of submodules names encapsulating the -construction and ended by the proper name of the construction. -Typically, the absolute name {\tt Coq.Init.Logic.eq} denotes Leibniz' -equality defined in the module {\tt Logic} in the sublibrary {\tt -Init} of the standard library of \Coq. - -The proper name that ends the name of a construction is the {\it short -name} (or sometimes {\it base name}) of the construction (for -instance, the short name of {\tt Coq.Init.Logic.eq} is {\tt eq}). Any -partial suffix of the absolute name is a {\em partially qualified name} -(e.g. {\tt Logic.eq} is a partially qualified name for {\tt -Coq.Init.Logic.eq}). Especially, the short name of a construction is -its shortest partially qualified name. - -{\Coq} does not accept two constructions (definition, theorem, ...) -with the same absolute name but different constructions can have the -same short name (or even same partially qualified names as soon as the -full names are different). - -Notice that the notion of absolute, partially qualified and -short names also applies to library file names. - -\paragraph{Visibility} - -{\Coq} maintains a table called {\it name table} which maps partially -qualified names of constructions to absolute names. This table is -updated by the commands {\tt Require} (see \ref{Require}), {\tt -Import} and {\tt Export} (see \ref{Import}) and also each time a new -declaration is added to the context. An absolute name is called {\it -visible} from a given short or partially qualified name when this -latter name is enough to denote it. This means that the short or -partially qualified name is mapped to the absolute name in {\Coq} name -table. - -A similar table exists for library file names. It is updated by the -vernacular commands {\tt Add LoadPath} and {\tt Add Rec LoadPath} (or -their equivalent as options of the {\Coq} executables, {\tt -I} and -{\tt -R}). - -It may happen that a visible name is hidden by the short name or a -qualified name of another construction. In this case, the name that -has been hidden must be referred to using one more level of -qualification. To ensure that a construction always remains -accessible, absolute names can never be hidden. - -Examples: -\begin{coq_eval} -Reset Initial. -\end{coq_eval} -\begin{coq_example} -Check 0. -Definition nat := bool. -Check 0. -Check Datatypes.nat. -Locate nat. -\end{coq_example} - -\SeeAlso Command {\tt Locate} in Section~\ref{Locate} and {\tt Locate -Library} in Section~\ref{Locate Library}. - -%% \paragraph{The special case of remarks and facts} -%% -%% In contrast with definitions, lemmas, theorems, axioms and parameters, -%% the absolute name of remarks includes the segment of sections in which -%% it is defined. Concretely, if a remark {\tt R} is defined in -%% subsection {\tt S2} of section {\tt S1} in module {\tt M}, then its -%% absolute name is {\tt M.S1.S2.R}. The same for facts, except that the -%% name of the innermost section is dropped from the full name. Then, if -%% a fact {\tt F} is defined in subsection {\tt S2} of section {\tt S1} -%% in module {\tt M}, then its absolute name is {\tt M.S1.F}. - -\section{Implicit arguments -\index{Implicit arguments} -\label{Implicit Arguments}} - -An implicit argument of a function is an argument which can be -inferred from contextual knowledge. There are different kinds of -implicit arguments that can be considered implicit in different -ways. There are also various commands to control the setting or the -inference of implicit arguments. - -\subsection{The different kinds of implicit arguments} - -\subsubsection{Implicit arguments inferable from the knowledge of other -arguments of a function} - -The first kind of implicit arguments covers the arguments that are -inferable from the knowledge of the type of other arguments of the -function, or of the type of the surrounding context of the -application. Especially, such implicit arguments correspond to -parameters dependent in the type of the function. Typical implicit -arguments are the type arguments in polymorphic functions. -There are several kinds of such implicit arguments. - -\paragraph{Strict Implicit Arguments.} -An implicit argument can be either strict or non strict. An implicit -argument is said {\em strict} if, whatever the other arguments of the -function are, it is still inferable from the type of some other -argument. Technically, an implicit argument is strict if it -corresponds to a parameter which is not applied to a variable which -itself is another parameter of the function (since this parameter -may erase its arguments), not in the body of a {\tt match}, and not -itself applied or matched against patterns (since the original -form of the argument can be lost by reduction). - -For instance, the first argument of -\begin{quote} -\verb|cons: forall A:Set, A -> list A -> list A| -\end{quote} -in module {\tt List.v} is strict because {\tt list} is an inductive -type and {\tt A} will always be inferable from the type {\tt -list A} of the third argument of {\tt cons}. -On the contrary, the second argument of a term of type -\begin{quote} -\verb|forall P:nat->Prop, forall n:nat, P n -> ex nat P| -\end{quote} -is implicit but not strict, since it can only be inferred from the -type {\tt P n} of the third argument and if {\tt P} is, e.g., {\tt -fun \_ => True}, it reduces to an expression where {\tt n} does not -occur any longer. The first argument {\tt P} is implicit but not -strict either because it can only be inferred from {\tt P n} and {\tt -P} is not canonically inferable from an arbitrary {\tt n} and the -normal form of {\tt P n} (consider e.g. that {\tt n} is {\tt 0} and -the third argument has type {\tt True}, then any {\tt P} of the form -{\tt fun n => match n with 0 => True | \_ => \mbox{\em anything} end} would -be a solution of the inference problem). - -\paragraph{Contextual Implicit Arguments.} -An implicit argument can be {\em contextual} or not. An implicit -argument is said {\em contextual} if it can be inferred only from the -knowledge of the type of the context of the current expression. For -instance, the only argument of -\begin{quote} -\verb|nil : forall A:Set, list A| -\end{quote} -is contextual. Similarly, both arguments of a term of type -\begin{quote} -\verb|forall P:nat->Prop, forall n:nat, P n \/ n = 0| -\end{quote} -are contextual (moreover, {\tt n} is strict and {\tt P} is not). - -\paragraph{Reversible-Pattern Implicit Arguments.} -There is another class of implicit arguments that can be reinferred -unambiguously if all the types of the remaining arguments are -known. This is the class of implicit arguments occurring in the type -of another argument in position of reversible pattern, which means it -is at the head of an application but applied only to uninstantiated -distinct variables. Such an implicit argument is called {\em -reversible-pattern implicit argument}. A typical example is the -argument {\tt P} of {\tt nat\_rec} in -\begin{quote} -{\tt nat\_rec : forall P : nat -> Set, - P 0 -> (forall n : nat, P n -> P (S n)) -> forall x : nat, P x}. -\end{quote} -({\tt P} is reinferable by abstracting over {\tt n} in the type {\tt P n}). - -See Section~\ref{SetReversiblePatternImplicit} for the automatic declaration -of reversible-pattern implicit arguments. - -\subsubsection{Implicit arguments inferable by resolution} - -This corresponds to a class of non dependent implicit arguments that -are solved based on the structure of their type only. - -\subsection{Maximal or non maximal insertion of implicit arguments} - -In case a function is partially applied, and the next argument to be -applied is an implicit argument, two disciplines are applicable. In the -first case, the function is considered to have no arguments furtherly: -one says that the implicit argument is not maximally inserted. In -the second case, the function is considered to be implicitly applied -to the implicit arguments it is waiting for: one says that the -implicit argument is maximally inserted. - -Each implicit argument can be declared to have to be inserted -maximally or non maximally. This can be governed argument per argument -by the command {\tt Implicit Arguments} (see~\ref{ImplicitArguments}) -or globally by the command {\tt Set Maximal Implicit Insertion} -(see~\ref{SetMaximalImplicitInsertion}). See also -Section~\ref{PrintImplicit}. - -\subsection{Casual use of implicit arguments} - -In a given expression, if it is clear that some argument of a function -can be inferred from the type of the other arguments, the user can -force the given argument to be guessed by replacing it by ``{\tt \_}''. If -possible, the correct argument will be automatically generated. - -\begin{ErrMsgs} - -\item \errindex{Cannot infer a term for this placeholder} - - {\Coq} was not able to deduce an instantiation of a ``{\tt \_}''. - -\end{ErrMsgs} - -\subsection{Declaration of implicit arguments for a constant -\comindex{Implicit Arguments}} -\label{ImplicitArguments} - -In case one wants that some arguments of a given object (constant, -inductive types, constructors, assumptions, local or not) are always -inferred by Coq, one may declare once and for all which are the expected -implicit arguments of this object. There are two ways to do this, -a-priori and a-posteriori. - -\subsubsection{Implicit Argument Binders} - -In the first setting, one wants to explicitly give the implicit -arguments of a constant as part of its definition. To do this, one has -to surround the bindings of implicit arguments by curly braces: -\begin{coq_eval} -Reset Initial. -\end{coq_eval} -\begin{coq_example} -Definition id {A : Type} (x : A) : A := x. -\end{coq_example} - -This automatically declares the argument {\tt A} of {\tt id} as a -maximally inserted implicit argument. One can then do as-if the argument -was absent in every situation but still be able to specify it if needed: -\begin{coq_example} -Definition compose {A B C} (g : B -> C) (f : A -> B) := - fun x => g (f x). -Goal forall A, compose id id = id (A:=A). -\end{coq_example} - -The syntax is supported in all top-level definitions: {\tt Definition}, -{\tt Fixpoint}, {\tt Lemma} and so on. For (co-)inductive datatype -declarations, the semantics are the following: an inductive parameter -declared as an implicit argument need not be repeated in the inductive -definition but will become implicit for the constructors of the -inductive only, not the inductive type itself. For example: - -\begin{coq_example} -Inductive list {A : Type} : Type := -| nil : list -| cons : A -> list -> list. -Print list. -\end{coq_example} - -One can always specify the parameter if it is not uniform using the -usual implicit arguments disambiguation syntax. - -\subsubsection{The Implicit Arguments Vernacular Command} - -To set implicit arguments for a constant a-posteriori, one can use the -command: -\begin{quote} -\tt Implicit Arguments {\qualid} [ \nelist{\possiblybracketedident}{} ] -\end{quote} -where the list of {\possiblybracketedident} is the list of parameters -to be declared implicit, each of the identifier of the list being -optionally surrounded by square brackets, then meaning that this -parameter has to be maximally inserted. - -After the above declaration is issued, implicit arguments can just (and -have to) be skipped in any expression involving an application of -{\qualid}. - -\begin{Variants} -\item {\tt Global Implicit Arguments {\qualid} [ \nelist{\possiblybracketedident}{} ] -\comindex{Global Implicit Arguments}} - -Tells to recompute the implicit arguments of {\qualid} after ending of -the current section if any, enforcing the implicit arguments known -from inside the section to be the ones declared by the command. - -\item {\tt Local Implicit Arguments {\qualid} [ \nelist{\possiblybracketedident}{} ] -\comindex{Local Implicit Arguments}} - -When in a module, tells not to activate the implicit arguments of -{\qualid} declared by this commands to contexts that requires the -module. - -\item {\tt \zeroone{Global {\sl |} Local} Implicit Arguments {\qualid} \sequence{[ \nelist{\possiblybracketedident}{} ]}{}} - -For names of constants, inductive types, constructors, lemmas which -can only be applied to a fixed number of arguments (this excludes for -instance constants whose type is polymorphic), multiple lists -of implicit arguments can be given. These lists must be of different -length, and, depending on the number of arguments {\qualid} is applied -to in practice, the longest applicable list of implicit arguments is -used to select which implicit arguments are inserted. - -For printing, the omitted arguments are the ones of the longest list -of implicit arguments of the sequence. - -\end{Variants} - -\Example -\begin{coq_eval} -Reset Initial. -\end{coq_eval} -\begin{coq_example*} -Inductive list (A:Type) : Type := - | nil : list A - | cons : A -> list A -> list A. -\end{coq_example*} -\begin{coq_example} -Check (cons nat 3 (nil nat)). -Implicit Arguments cons [A]. -Implicit Arguments nil [A]. -Check (cons 3 nil). -Fixpoint map (A B:Type) (f:A->B) (l:list A) : list B := - match l with nil => nil | cons a t => cons (f a) (map A B f t) end. -Fixpoint length (A:Type) (l:list A) : nat := - match l with nil => 0 | cons _ m => S (length A m) end. -Implicit Arguments map [A B]. -Implicit Arguments length [[A]]. (* A has to be maximally inserted *) -Check (fun l:list (list nat) => map length l). -Implicit Arguments map [A B] [A] []. -Check (fun l => map length l = map (list nat) nat length l). -\end{coq_example} - -\Rem To know which are the implicit arguments of an object, use the command -{\tt Print Implicit} (see \ref{PrintImplicit}). - -\Rem If the list of arguments is empty, the command removes the -implicit arguments of {\qualid}. - -\subsection{Automatic declaration of implicit arguments for a constant} - -{\Coq} can also automatically detect what are the implicit arguments -of a defined object. The command is just -\begin{quote} -{\tt Implicit Arguments {\qualid} -\comindex{Implicit Arguments}} -\end{quote} -The auto-detection is governed by options telling if strict, -contextual, or reversible-pattern implicit arguments must be -considered or not (see -Sections~\ref{SetStrictImplicit},~\ref{SetContextualImplicit},~\ref{SetReversiblePatternImplicit} -and also~\ref{SetMaximalImplicitInsertion}). - -\begin{Variants} -\item {\tt Global Implicit Arguments {\qualid} -\comindex{Global Implicit Arguments}} - -Tells to recompute the implicit arguments of {\qualid} after ending of -the current section if any. - -\item {\tt Local Implicit Arguments {\qualid} -\comindex{Local Implicit Arguments}} - -When in a module, tells not to activate the implicit arguments of -{\qualid} computed by this declaration to contexts that requires the -module. - -\end{Variants} - -\Example -\begin{coq_eval} -Reset Initial. -\end{coq_eval} -\begin{coq_example*} -Inductive list (A:Set) : Set := - | nil : list A - | cons : A -> list A -> list A. -\end{coq_example*} -\begin{coq_example} -Implicit Arguments cons. -Print Implicit cons. -Implicit Arguments nil. -Print Implicit nil. -Set Contextual Implicit. -Implicit Arguments nil. -Print Implicit nil. -\end{coq_example} - -The computation of implicit arguments takes account of the -unfolding of constants. For instance, the variable {\tt p} below has -type {\tt (Transitivity R)} which is reducible to {\tt forall x,y:U, R x -y -> forall z:U, R y z -> R x z}. As the variables {\tt x}, {\tt y} and -{\tt z} appear strictly in body of the type, they are implicit. - -\begin{coq_example*} -Variable X : Type. -Definition Relation := X -> X -> Prop. -Definition Transitivity (R:Relation) := - forall x y:X, R x y -> forall z:X, R y z -> R x z. -Variables (R : Relation) (p : Transitivity R). -Implicit Arguments p. -\end{coq_example*} -\begin{coq_example} -Print p. -Print Implicit p. -\end{coq_example} -\begin{coq_example*} -Variables (a b c : X) (r1 : R a b) (r2 : R b c). -\end{coq_example*} -\begin{coq_example} -Check (p r1 r2). -\end{coq_example} - -\subsection{Mode for automatic declaration of implicit arguments -\label{Auto-implicit} -\comindex{Set Implicit Arguments} -\comindex{Unset Implicit Arguments}} - -In case one wants to systematically declare implicit the arguments -detectable as such, one may switch to the automatic declaration of -implicit arguments mode by using the command -\begin{quote} -\tt Set Implicit Arguments. -\end{quote} -Conversely, one may unset the mode by using {\tt Unset Implicit -Arguments}. The mode is off by default. Auto-detection of implicit -arguments is governed by options controlling whether strict and -contextual implicit arguments have to be considered or not. - -\subsection{Controlling strict implicit arguments -\comindex{Set Strict Implicit} -\comindex{Unset Strict Implicit} -\label{SetStrictImplicit}} - -When the mode for automatic declaration of implicit arguments is on, -the default is to automatically set implicit only the strict implicit -arguments plus, for historical reasons, a small subset of the non -strict implicit arguments. To relax this constraint and to -set implicit all non strict implicit arguments by default, use the command -\begin{quote} -\tt Unset Strict Implicit. -\end{quote} -Conversely, use the command {\tt Set Strict Implicit} to -restore the original mode that declares implicit only the strict implicit arguments plus a small subset of the non strict implicit arguments. - -In the other way round, to capture exactly the strict implicit arguments and no more than the strict implicit arguments, use the command: -\comindex{Set Strongly Strict Implicit} -\comindex{Unset Strongly Strict Implicit} -\begin{quote} -\tt Set Strongly Strict Implicit. -\end{quote} -Conversely, use the command {\tt Unset Strongly Strict Implicit} to -let the option ``{\tt Strict Implicit}'' decide what to do. - -\Rem In versions of {\Coq} prior to version 8.0, the default was to -declare the strict implicit arguments as implicit. - -\subsection{Controlling contextual implicit arguments -\comindex{Set Contextual Implicit} -\comindex{Unset Contextual Implicit} -\label{SetContextualImplicit}} - -By default, {\Coq} does not automatically set implicit the contextual -implicit arguments. To tell {\Coq} to infer also contextual implicit -argument, use command -\begin{quote} -\tt Set Contextual Implicit. -\end{quote} -Conversely, use command {\tt Unset Contextual Implicit} to -unset the contextual implicit mode. - -\subsection{Controlling reversible-pattern implicit arguments -\comindex{Set Reversible Pattern Implicit} -\comindex{Unset Reversible Pattern Implicit} -\label{SetReversiblePatternImplicit}} - -By default, {\Coq} does not automatically set implicit the reversible-pattern -implicit arguments. To tell {\Coq} to infer also reversible-pattern implicit -argument, use command -\begin{quote} -\tt Set Reversible Pattern Implicit. -\end{quote} -Conversely, use command {\tt Unset Reversible Pattern Implicit} to -unset the reversible-pattern implicit mode. - -\subsection{Controlling the insertion of implicit arguments not followed by explicit arguments -\comindex{Set Maximal Implicit Insertion} -\comindex{Unset Maximal Implicit Insertion} -\label{SetMaximalImplicitInsertion}} - -Implicit arguments can be declared to be automatically inserted when a -function is partially applied and the next argument of the function is -an implicit one. In case the implicit arguments are automatically -declared (with the command {\tt Set Implicit Arguments}), the command -\begin{quote} -\tt Set Maximal Implicit Insertion. -\end{quote} -is used to tell to declare the implicit arguments with a maximal -insertion status. By default, automatically declared implicit -arguments are not declared to be insertable maximally. To restore the -default mode for maximal insertion, use command {\tt Unset Maximal -Implicit Insertion}. - -\subsection{Explicit applications -\index{Explicitly given implicit arguments} -\label{Implicits-explicitation} -\index{qualid@{\qualid}}} - -In presence of non strict or contextual argument, or in presence of -partial applications, the synthesis of implicit arguments may fail, so -one may have to give explicitly certain implicit arguments of an -application. The syntax for this is {\tt (\ident:=\term)} where {\ident} -is the name of the implicit argument and {\term} is its corresponding -explicit term. Alternatively, one can locally deactivate the hiding of -implicit arguments of a function by using the notation -{\tt @{\qualid}~{\term}$_1$..{\term}$_n$}. This syntax extension is -given Figure~\ref{fig:explicitations}. -\begin{figure} -\begin{centerframe} -\begin{tabular}{lcl} -{\term} & ++= & @ {\qualid} \nelist{\term}{}\\ -& $|$ & @ {\qualid}\\ -& $|$ & {\qualid} \nelist{\textrm{\textsl{argument}}}{}\\ -\\ -{\textrm{\textsl{argument}}} & ::= & {\term} \\ -& $|$ & {\tt ({\ident}:={\term})}\\ -\end{tabular} -\end{centerframe} -\caption{Syntax for explicitly giving implicit arguments} -\label{fig:explicitations} -\end{figure} - -\noindent {\bf Example (continued): } -\begin{coq_example} -Check (p r1 (z:=c)). -Check (p (x:=a) (y:=b) r1 (z:=c) r2). -\end{coq_example} - -\subsection{Displaying what the implicit arguments are -\comindex{Print Implicit} -\label{PrintImplicit}} - -To display the implicit arguments associated to an object, and to know -if each of them is to be used maximally or not, use the command -\begin{quote} -\tt Print Implicit {\qualid}. -\end{quote} - -\subsection{Explicit displaying of implicit arguments for pretty-printing -\comindex{Set Printing Implicit} -\comindex{Unset Printing Implicit} -\comindex{Set Printing Implicit Defensive} -\comindex{Unset Printing Implicit Defensive}} - -By default the basic pretty-printing rules hide the inferable implicit -arguments of an application. To force printing all implicit arguments, -use command -\begin{quote} -{\tt Set Printing Implicit.} -\end{quote} -Conversely, to restore the hiding of implicit arguments, use command -\begin{quote} -{\tt Unset Printing Implicit.} -\end{quote} - -By default the basic pretty-printing rules display the implicit arguments that are not detected as strict implicit arguments. This ``defensive'' mode can quickly make the display cumbersome so this can be deactivated by using the command -\begin{quote} -{\tt Unset Printing Implicit Defensive.} -\end{quote} -Conversely, to force the display of non strict arguments, use command -\begin{quote} -{\tt Set Printing Implicit Defensive.} -\end{quote} - -\SeeAlso {\tt Set Printing All} in Section~\ref{SetPrintingAll}. - -\subsection{Interaction with subtyping} - -When an implicit argument can be inferred from the type of more than -one of the other arguments, then only the type of the first of these -arguments is taken into account, and not an upper type of all of -them. As a consequence, the inference of the implicit argument of -``='' fails in - -\begin{coq_example*} -Check nat = Prop. -\end{coq_example*} - -but succeeds in - -\begin{coq_example*} -Check Prop = nat. -\end{coq_example*} - - - - -\subsection{Canonical structures -\comindex{Canonical Structure}} - -A canonical structure is an instance of a record/structure type that -can be used to solve equations involving implicit arguments. Assume -that {\qualid} denotes an object $(Build\_struc~ c_1~ \ldots~ c_n)$ in the -structure {\em struct} of which the fields are $x_1$, ..., -$x_n$. Assume that {\qualid} is declared as a canonical structure -using the command -\begin{quote} -{\tt Canonical Structure {\qualid}.} -\end{quote} -Then, each time an equation of the form $(x_i~ -\_)=_{\beta\delta\iota\zeta}c_i$ has to be solved during the -type-checking process, {\qualid} is used as a solution. Otherwise -said, {\qualid} is canonically used to extend the field $c_i$ into a -complete structure built on $c_i$. - -Canonical structures are particularly useful when mixed with -coercions and strict implicit arguments. Here is an example. -\begin{coq_example*} -Require Import Relations. -Require Import EqNat. -Set Implicit Arguments. -Unset Strict Implicit. -Structure Setoid : Type := - {Carrier :> Set; - Equal : relation Carrier; - Prf_equiv : equivalence Carrier Equal}. -Definition is_law (A B:Setoid) (f:A -> B) := - forall x y:A, Equal x y -> Equal (f x) (f y). -Axiom eq_nat_equiv : equivalence nat eq_nat. -Definition nat_setoid : Setoid := Build_Setoid eq_nat_equiv. -Canonical Structure nat_setoid. -\end{coq_example*} - -Thanks to \texttt{nat\_setoid} declared as canonical, the implicit -arguments {\tt A} and {\tt B} can be synthesized in the next statement. -\begin{coq_example} -Lemma is_law_S : is_law S. -\end{coq_example} - -\Rem If a same field occurs in several canonical structure, then -only the structure declared first as canonical is considered. - -\begin{Variants} -\item {\tt Canonical Structure {\ident} := {\term} : {\type}.}\\ - {\tt Canonical Structure {\ident} := {\term}.}\\ - {\tt Canonical Structure {\ident} : {\type} := {\term}.} - -These are equivalent to a regular definition of {\ident} followed by -the declaration - -{\tt Canonical Structure {\ident}}. -\end{Variants} - -\SeeAlso more examples in user contribution \texttt{category} -(\texttt{Rocq/ALGEBRA}). - -\subsubsection{Print Canonical Projections. -\comindex{Print Canonical Projections}} - -This displays the list of global names that are components of some -canonical structure. For each of them, the canonical structure of -which it is a projection is indicated. For instance, the above example -gives the following output: - -\begin{coq_example} -Print Canonical Projections. -\end{coq_example} - -\subsection{Implicit types of variables} -\comindex{Implicit Types} - -It is possible to bind variable names to a given type (e.g. in a -development using arithmetic, it may be convenient to bind the names -{\tt n} or {\tt m} to the type {\tt nat} of natural numbers). The -command for that is -\begin{quote} -\tt Implicit Types \nelist{\ident}{} : {\type} -\end{quote} -The effect of the command is to automatically set the type of bound -variables starting with {\ident} (either {\ident} itself or -{\ident} followed by one or more single quotes, underscore or digits) -to be {\type} (unless the bound variable is already declared with an -explicit type in which case, this latter type is considered). - -\Example -\begin{coq_example} -Require Import List. -Implicit Types m n : nat. -Lemma cons_inj_nat : forall m n l, n :: l = m :: l -> n = m. -intros m n. -Lemma cons_inj_bool : forall (m n:bool) l, n :: l = m :: l -> n = m. -\end{coq_example} - -\begin{Variants} -\item {\tt Implicit Type {\ident} : {\type}}\\ -This is useful for declaring the implicit type of a single variable. -\item - {\tt Implicit Types\,% -(\,{\ident$_{1,1}$}\ldots{\ident$_{1,k_1}$}\,{\tt :}\,{\term$_1$} {\tt )}\,% -\ldots\,{\tt (}\,{\ident$_{n,1}$}\ldots{\ident$_{n,k_n}$}\,{\tt :}\,% -{\term$_n$} {\tt )}.}\\ - Adds $n$ blocks of implicit types with different specifications. -\end{Variants} - - -\subsection{Implicit generalization -\label{implicit-generalization} -\comindex{Generalizable Variables}} - -Implicit generalization is an automatic elaboration of a statement with -free variables into a closed statement where these variables are -quantified explicitly. Implicit generalization is done inside binders -starting with a \verb|`| and terms delimited by \verb|`{ }| and -\verb|`( )|, always introducing maximally inserted implicit arguments for -the generalized variables. Inside implicit generalization -delimiters, free variables in the current context are automatically -quantified using a product or a lambda abstraction to generate a closed -term. In the following statement for example, the variables \texttt{n} -and \texttt{m} are automatically generalized and become explicit -arguments of the lemma as we are using \verb|`( )|: - -\begin{coq_example} -Generalizable All Variables. -Lemma nat_comm : `(n = n + 0). -\end{coq_example} -\begin{coq_eval} -Abort. -\end{coq_eval} -One can control the set of generalizable identifiers with the -\texttt{Generalizable} vernacular command to avoid unexpected -generalizations when mistyping identifiers. There are three variants of -the command: - -\begin{quote} -{\tt (Global)? Generalizable (All|No) Variable(s)? ({\ident$_1$ \ident$_n$})?.} -\end{quote} - -\begin{Variants} -\item {\tt Generalizable All Variables.} All variables are candidate for - generalization if they appear free in the context under a - generalization delimiter. This may result in confusing errors in - case of typos. In such cases, the context will probably contain some - unexpected generalized variable. - -\item {\tt Generalizable No Variables.} Disable implicit generalization - entirely. This is the default behavior. - -\item {\tt Generalizable Variable(s)? {\ident$_1$ \ident$_n$}.} - Allow generalization of the given identifiers only. Calling this - command multiple times adds to the allowed identifiers. - -\item {\tt Global Generalizable} Allows to export the choice of - generalizable variables. -\end{Variants} - -One can also use implicit generalization for binders, in which case the -generalized variables are added as binders and set maximally implicit. -\begin{coq_example*} -Definition id `(x : A) : A := x. -\end{coq_example*} -\begin{coq_example} -Print id. -\end{coq_example} - -The generalizing binders \verb|`{ }| and \verb|`( )| work similarly to -their explicit counterparts, only binding the generalized variables -implicitly, as maximally-inserted arguments. In these binders, the -binding name for the bound object is optional, whereas the type is -mandatory, dually to regular binders. - -\section{Coercions -\label{Coercions} -\index{Coercions}} - -Coercions can be used to implicitly inject terms from one {\em class} in -which they reside into another one. A {\em class} is either a sort -(denoted by the keyword {\tt Sortclass}), a product type (denoted by the -keyword {\tt Funclass}), or a type constructor (denoted by its name), -e.g. an inductive type or any constant with a type of the form -\texttt{forall} $(x_1:A_1) .. (x_n:A_n),~s$ where $s$ is a sort. - -Then the user is able to apply an -object that is not a function, but can be coerced to a function, and -more generally to consider that a term of type A is of type B provided -that there is a declared coercion between A and B. The main command is -\comindex{Coercion} -\begin{quote} -\tt Coercion {\qualid} : {\class$_1$} >-> {\class$_2$}. -\end{quote} -which declares the construction denoted by {\qualid} as a -coercion between {\class$_1$} and {\class$_2$}. - -More details and examples, and a description of the commands related -to coercions are provided in Chapter~\ref{Coercions-full}. - -\section[Printing constructions in full]{Printing constructions in full\label{SetPrintingAll} -\comindex{Set Printing All} -\comindex{Unset Printing All}} - -Coercions, implicit arguments, the type of pattern-matching, but also -notations (see Chapter~\ref{Addoc-syntax}) can obfuscate the behavior -of some tactics (typically the tactics applying to occurrences of -subterms are sensitive to the implicit arguments). The command -\begin{quote} -{\tt Set Printing All.} -\end{quote} -deactivates all high-level printing features such as coercions, -implicit arguments, returned type of pattern-matching, notations and -various syntactic sugar for pattern-matching or record projections. -Otherwise said, {\tt Set Printing All} includes the effects -of the commands {\tt Set Printing Implicit}, {\tt Set Printing -Coercions}, {\tt Set Printing Synth}, {\tt Unset Printing Projections} -and {\tt Unset Printing Notations}. To reactivate the high-level -printing features, use the command -\begin{quote} -{\tt Unset Printing All.} -\end{quote} - -\section[Printing universes]{Printing universes\label{PrintingUniverses} -\comindex{Set Printing Universes} -\comindex{Unset Printing Universes}} - -The following command: -\begin{quote} -{\tt Set Printing Universes} -\end{quote} -activates the display of the actual level of each occurrence of -{\Type}. See Section~\ref{Sorts} for details. This wizard option, in -combination with \texttt{Set Printing All} (see -section~\ref{SetPrintingAll}) can help to diagnose failures to unify -terms apparently identical but internally different in the Calculus of -Inductive Constructions. To reactivate the display of the actual level -of the occurrences of {\Type}, use -\begin{quote} -{\tt Unset Printing Universes.} -\end{quote} - -\comindex{Print Universes} - -The constraints on the internal level of the occurrences of {\Type} -(see Section~\ref{Sorts}) can be printed using the command -\begin{quote} -{\tt Print Universes.} -\end{quote} - - -%%% Local Variables: -%%% mode: latex -%%% TeX-master: "Reference-Manual" -%%% End: diff --git a/doc/refman/RefMan-gal.tex b/doc/refman/RefMan-gal.tex deleted file mode 100644 index d1489591..00000000 --- a/doc/refman/RefMan-gal.tex +++ /dev/null @@ -1,1696 +0,0 @@ -\chapter{The \gallina{} specification language -\label{Gallina}\index{Gallina}} -\label{BNF-syntax} % Used referred to as a chapter label - -This chapter describes \gallina, the specification language of {\Coq}. -It allows to develop mathematical theories and to prove specifications -of programs. The theories are built from axioms, hypotheses, -parameters, lemmas, theorems and definitions of constants, functions, -predicates and sets. The syntax of logical objects involved in -theories is described in Section~\ref{term}. The language of -commands, called {\em The Vernacular} is described in section -\ref{Vernacular}. - -In {\Coq}, logical objects are typed to ensure their logical -correctness. The rules implemented by the typing algorithm are described in -Chapter \ref{Cic}. - -\subsection*{About the grammars in the manual -\index{BNF metasyntax}} - -Grammars are presented in Backus-Naur form (BNF). Terminal symbols are -set in {\tt typewriter font}. In addition, there are special -notations for regular expressions. - -An expression enclosed in square brackets \zeroone{\ldots} means at -most one occurrence of this expression (this corresponds to an -optional component). - -The notation ``\nelist{\entry}{sep}'' stands for a non empty -sequence of expressions parsed by {\entry} and -separated by the literal ``{\tt sep}''\footnote{This is similar to the -expression ``{\entry} $\{$ {\tt sep} {\entry} $\}$'' in -standard BNF, or ``{\entry}~{$($} {\tt sep} {\entry} {$)$*}'' in -the syntax of regular expressions.}. - -Similarly, the notation ``\nelist{\entry}{}'' stands for a non -empty sequence of expressions parsed by the ``{\entry}'' entry, -without any separator between. - -At the end, the notation ``\sequence{\entry}{\tt sep}'' stands for a -possibly empty sequence of expressions parsed by the ``{\entry}'' entry, -separated by the literal ``{\tt sep}''. - -\section{Lexical conventions -\label{lexical}\index{Lexical conventions}} - -\paragraph{Blanks} -Space, newline and horizontal tabulation are considered as blanks. -Blanks are ignored but they separate tokens. - -\paragraph{Comments} - -Comments in {\Coq} are enclosed between {\tt (*} and {\tt - *)}\index{Comments}, and can be nested. They can contain any -character. However, string literals must be correctly closed. Comments -are treated as blanks. - -\paragraph{Identifiers and access identifiers} - -Identifiers, written {\ident}, are sequences of letters, digits, -\verb!_! and \verb!'!, that do not start with a digit or \verb!'!. -That is, they are recognized by the following lexical class: - -\index{ident@\ident} -\begin{center} -\begin{tabular}{rcl} -{\firstletter} & ::= & {\tt a..z} $\mid$ {\tt A..Z} $\mid$ {\tt \_} -$\mid$ {\tt unicode-letter} -\\ -{\subsequentletter} & ::= & {\tt a..z} $\mid$ {\tt A..Z} $\mid$ {\tt 0..9} -$\mid$ {\tt \_} % $\mid$ {\tt \$} -$\mid$ {\tt '} -$\mid$ {\tt unicode-letter} -$\mid$ {\tt unicode-id-part} \\ -{\ident} & ::= & {\firstletter} \sequencewithoutblank{\subsequentletter}{} -\end{tabular} -\end{center} -All characters are meaningful. In particular, identifiers are -case-sensitive. The entry {\tt unicode-letter} non-exhaustively -includes Latin, Greek, Gothic, Cyrillic, Arabic, Hebrew, Georgian, -Hangul, Hiragana and Katakana characters, CJK ideographs, mathematical -letter-like symbols, hyphens, non-breaking space, {\ldots} The entry -{\tt unicode-id-part} non-exhaustively includes symbols for prime -letters and subscripts. - -Access identifiers, written {\accessident}, are identifiers prefixed -by \verb!.! (dot) without blank. They are used in the syntax of qualified -identifiers. - -\paragraph{Natural numbers and integers} -Numerals are sequences of digits. Integers are numerals optionally preceded by a minus sign. - -\index{num@{\num}} -\index{integer@{\integer}} -\begin{center} -\begin{tabular}{r@{\quad::=\quad}l} -{\digit} & {\tt 0..9} \\ -{\num} & \nelistwithoutblank{\digit}{} \\ -{\integer} & \zeroone{\tt -}{\num} \\ -\end{tabular} -\end{center} - -\paragraph[Strings]{Strings\label{strings} -\index{string@{\qstring}}} -Strings are delimited by \verb!"! (double quote), and enclose a -sequence of any characters different from \verb!"! or the sequence -\verb!""! to denote the double quote character. In grammars, the -entry for quoted strings is {\qstring}. - -\paragraph{Keywords} -The following identifiers are reserved keywords, and cannot be -employed otherwise: -\begin{center} -\begin{tabular}{llllll} -\verb!_! & -\verb!as! & -\verb!at! & -\verb!cofix! & -\verb!else! & -\verb!end! \\ -% -\verb!exists! & -\verb!exists2! & -\verb!fix! & -\verb!for! & -\verb!forall! & -\verb!fun! \\ -% -\verb!if! & -\verb!IF! & -\verb!in! & -\verb!let! & -\verb!match! & -\verb!mod! \\ -% -\verb!Prop! & -\verb!return! & -\verb!Set! & -\verb!then! & -\verb!Type! & -\verb!using! \\ -% -\verb!where! & -\verb!with! & -\end{tabular} -\end{center} - - -\paragraph{Special tokens} -The following sequences of characters are special tokens: -\begin{center} -\begin{tabular}{lllllll} -\verb/!/ & -\verb!%! & -\verb!&! & -\verb!&&! & -\verb!(! & -\verb!()! & -\verb!)! \\ -% -\verb!*! & -\verb!+! & -\verb!++! & -\verb!,! & -\verb!-! & -\verb!->! & -\verb!.! \\ -% -\verb!.(! & -\verb!..! & -\verb!/! & -\verb!/\! & -\verb!:! & -\verb!::! & -\verb!:<! \\ -% -\verb!:=! & -\verb!:>! & -\verb!;! & -\verb!<! & -\verb!<-! & -\verb!<->! & -\verb!<:! \\ -% -\verb!<=! & -\verb!<>! & -\verb!=! & -\verb!=>! & -\verb!=_D! & -\verb!>! & -\verb!>->! \\ -% -\verb!>=! & -\verb!?! & -\verb!?=! & -\verb!@! & -\verb![! & -\verb!\/! & -\verb!]! \\ -% -\verb!^! & -\verb!{! & -\verb!|! & -\verb!|-! & -\verb!||! & -\verb!}! & -\verb!~! \\ -\end{tabular} -\end{center} - -Lexical ambiguities are resolved according to the ``longest match'' -rule: when a sequence of non alphanumerical characters can be decomposed -into several different ways, then the first token is the longest -possible one (among all tokens defined at this moment), and so on. - -\section{Terms \label{term}\index{Terms}} - -\subsection{Syntax of terms} - -Figures \ref{term-syntax} and \ref{term-syntax-aux} describe the basic -set of terms which form the {\em Calculus of Inductive Constructions} -(also called \CIC). The formal presentation of {\CIC} is given in -Chapter \ref{Cic}. Extensions of this syntax are given in chapter -\ref{Gallina-extension}. How to customize the syntax is described in -Chapter \ref{Addoc-syntax}. - -\begin{figure}[htbp] -\begin{centerframe} -\begin{tabular}{lcl@{\quad~}r} % warning: page width exceeded with \qquad -{\term} & ::= & - {\tt forall} {\binders} {\tt ,} {\term} &(\ref{products})\\ - & $|$ & {\tt fun} {\binders} {\tt =>} {\term} &(\ref{abstractions})\\ - & $|$ & {\tt fix} {\fixpointbodies} &(\ref{fixpoints})\\ - & $|$ & {\tt cofix} {\cofixpointbodies} &(\ref{fixpoints})\\ - & $|$ & {\tt let} {\ident} \zeroone{\binders} {\typecstr} {\tt :=} {\term} - {\tt in} {\term} &(\ref{let-in})\\ - & $|$ & {\tt let fix} {\fixpointbody} {\tt in} {\term} &(\ref{fixpoints})\\ - & $|$ & {\tt let cofix} {\cofixpointbody} - {\tt in} {\term} &(\ref{fixpoints})\\ - & $|$ & {\tt let} {\tt (} \sequence{\name}{,} {\tt )} \zeroone{\ifitem} - {\tt :=} {\term} - {\tt in} {\term} &(\ref{caseanalysis}, \ref{Mult-match})\\ - & $|$ & {\tt if} {\term} \zeroone{\ifitem} {\tt then} {\term} - {\tt else} {\term} &(\ref{caseanalysis}, \ref{Mult-match})\\ - & $|$ & {\term} {\tt :} {\term} &(\ref{typecast})\\ - & $|$ & {\term} {\tt ->} {\term} &(\ref{products})\\ - & $|$ & {\term} \nelist{\termarg}{}&(\ref{applications})\\ - & $|$ & {\tt @} {\qualid} \sequence{\term}{} - &(\ref{Implicits-explicitation})\\ - & $|$ & {\term} {\tt \%} {\ident} &(\ref{scopechange})\\ - & $|$ & {\tt match} \nelist{\caseitem}{\tt ,} - \zeroone{\returntype} {\tt with} &\\ - && ~~~\zeroone{\zeroone{\tt |} \nelist{\eqn}{|}} {\tt end} - &(\ref{caseanalysis})\\ - & $|$ & {\qualid} &(\ref{qualid})\\ - & $|$ & {\sort} &(\ref{Gallina-sorts})\\ - & $|$ & {\num} &(\ref{numerals})\\ - & $|$ & {\_} &(\ref{hole})\\ - & & &\\ -{\termarg} & ::= & {\term} &\\ - & $|$ & {\tt (} {\ident} {\tt :=} {\term} {\tt )} - &(\ref{Implicits-explicitation})\\ -%% & $|$ & {\tt (} {\num} {\tt :=} {\term} {\tt )} -%% &(\ref{Implicits-explicitation})\\ -&&&\\ -{\binders} & ::= & \nelist{\binder}{} \\ -&&&\\ -{\binder} & ::= & {\name} & (\ref{Binders}) \\ - & $|$ & {\tt (} \nelist{\name}{} {\tt :} {\term} {\tt )} &\\ - & $|$ & {\tt (} {\name} {\typecstr} {\tt :=} {\term} {\tt )} &\\ -& & &\\ -{\name} & ::= & {\ident} &\\ - & $|$ & {\tt \_} &\\ -&&&\\ -{\qualid} & ::= & {\ident} & \\ - & $|$ & {\qualid} {\accessident} &\\ - & & &\\ -{\sort} & ::= & {\tt Prop} ~$|$~ {\tt Set} ~$|$~ {\tt Type} & -\end{tabular} -\end{centerframe} -\caption{Syntax of terms} -\label{term-syntax} -\index{term@{\term}} -\index{sort@{\sort}} -\end{figure} - - - -\begin{figure}[htb] -\begin{centerframe} -\begin{tabular}{lcl} -{\fixpointbodies} & ::= & - {\fixpointbody} \\ - & $|$ & {\fixpointbody} {\tt with} \nelist{\fixpointbody}{{\tt with}} - {\tt for} {\ident} \\ -{\cofixpointbodies} & ::= & - {\cofixpointbody} \\ - & $|$ & {\cofixpointbody} {\tt with} \nelist{\cofixpointbody}{{\tt with}} - {\tt for} {\ident} \\ -&&\\ -{\fixpointbody} & ::= & - {\ident} {\binders} \zeroone{\annotation} {\typecstr} - {\tt :=} {\term} \\ -{\cofixpointbody} & ::= & {\ident} \zeroone{\binders} {\typecstr} {\tt :=} {\term} \\ - & &\\ -{\annotation} & ::= & {\tt \{ struct} {\ident} {\tt \}} \\ -&&\\ -{\caseitem} & ::= & {\term} \zeroone{{\tt as} \name} - \zeroone{{\tt in} \term} \\ -&&\\ -{\ifitem} & ::= & \zeroone{{\tt as} {\name}} {\returntype} \\ -&&\\ -{\returntype} & ::= & {\tt return} {\term} \\ -&&\\ -{\eqn} & ::= & \nelist{\multpattern}{\tt |} {\tt =>} {\term}\\ -&&\\ -{\multpattern} & ::= & \nelist{\pattern}{\tt ,}\\ -&&\\ -{\pattern} & ::= & {\qualid} \nelist{\pattern}{} \\ - & $|$ & {\pattern} {\tt as} {\ident} \\ - & $|$ & {\pattern} {\tt \%} {\ident} \\ - & $|$ & {\qualid} \\ - & $|$ & {\tt \_} \\ - & $|$ & {\num} \\ - & $|$ & {\tt (} \nelist{\orpattern}{,} {\tt )} \\ -\\ -{\orpattern} & ::= & \nelist{\pattern}{\tt |}\\ -\end{tabular} -\end{centerframe} -\caption{Syntax of terms (continued)} -\label{term-syntax-aux} -\end{figure} - - -%%%%%%% - -\subsection{Types} - -{\Coq} terms are typed. {\Coq} types are recognized by the same -syntactic class as {\term}. We denote by {\type} the semantic subclass -of types inside the syntactic class {\term}. -\index{type@{\type}} - - -\subsection{Qualified identifiers and simple identifiers -\label{qualid} -\label{ident}} - -{\em Qualified identifiers} ({\qualid}) denote {\em global constants} -(definitions, lemmas, theorems, remarks or facts), {\em global -variables} (parameters or axioms), {\em inductive -types} or {\em constructors of inductive types}. -{\em Simple identifiers} (or shortly {\ident}) are a -syntactic subset of qualified identifiers. Identifiers may also -denote local {\em variables}, what qualified identifiers do not. - -\subsection{Numerals -\label{numerals}} - -Numerals have no definite semantics in the calculus. They are mere -notations that can be bound to objects through the notation mechanism -(see Chapter~\ref{Addoc-syntax} for details). Initially, numerals are -bound to Peano's representation of natural numbers -(see~\ref{libnats}). - -Note: negative integers are not at the same level as {\num}, for this -would make precedence unnatural. - -\subsection{Sorts -\index{Sorts} -\index{Type@{\Type}} -\index{Set@{\Set}} -\index{Prop@{\Prop}} -\index{Sorts} -\label{Gallina-sorts}} - -There are three sorts \Set, \Prop\ and \Type. -\begin{itemize} -\item \Prop\ is the universe of {\em logical propositions}. -The logical propositions themselves are typing the proofs. -We denote propositions by {\form}. This constitutes a semantic -subclass of the syntactic class {\term}. -\index{form@{\form}} -\item \Set\ is is the universe of {\em program -types} or {\em specifications}. -The specifications themselves are typing the programs. -We denote specifications by {\specif}. This constitutes a semantic -subclass of the syntactic class {\term}. -\index{specif@{\specif}} -\item {\Type} is the type of {\Set} and {\Prop} -\end{itemize} -\noindent More on sorts can be found in Section~\ref{Sorts}. - -\bigskip - -{\Coq} terms are typed. {\Coq} types are recognized by the same -syntactic class as {\term}. We denote by {\type} the semantic subclass -of types inside the syntactic class {\term}. -\index{type@{\type}} - -\subsection{Binders -\label{Binders} -\index{binders}} - -Various constructions such as {\tt fun}, {\tt forall}, {\tt fix} and -{\tt cofix} {\em bind} variables. A binding is represented by an -identifier. If the binding variable is not used in the expression, the -identifier can be replaced by the symbol {\tt \_}. When the type of a -bound variable cannot be synthesized by the system, it can be -specified with the notation {\tt (}\,{\ident}\,{\tt :}\,{\type}\,{\tt -)}. There is also a notation for a sequence of binding variables -sharing the same type: {\tt (}\,{\ident$_1$}\ldots{\ident$_n$}\,{\tt -:}\,{\type}\,{\tt )}. - -Some constructions allow the binding of a variable to value. This is -called a ``let-binder''. The entry {\binder} of the grammar accepts -either an assumption binder as defined above or a let-binder. -The notation in the -latter case is {\tt (}\,{\ident}\,{\tt :=}\,{\term}\,{\tt )}. In a -let-binder, only one variable can be introduced at the same -time. It is also possible to give the type of the variable as follows: -{\tt (}\,{\ident}\,{\tt :}\,{\term}\,{\tt :=}\,{\term}\,{\tt )}. - -Lists of {\binder} are allowed. In the case of {\tt fun} and {\tt - forall}, it is intended that at least one binder of the list is an -assumption otherwise {\tt fun} and {\tt forall} gets identical. Moreover, -parentheses can be omitted in the case of a single sequence of -bindings sharing the same type (e.g.: {\tt fun~(x~y~z~:~A)~=>~t} can -be shortened in {\tt fun~x~y~z~:~A~=>~t}). - -\subsection{Abstractions -\label{abstractions} -\index{abstractions}} - -The expression ``{\tt fun} {\ident} {\tt :} {\type} {\tt =>}~{\term}'' -defines the {\em abstraction} of the variable {\ident}, of type -{\type}, over the term {\term}. It denotes a function of the variable -{\ident} that evaluates to the expression {\term} (e.g. {\tt fun x:$A$ -=> x} denotes the identity function on type $A$). -% The variable {\ident} is called the {\em parameter} of the function -% (we sometimes say the {\em formal parameter}). -The keyword {\tt fun} can be followed by several binders as given in -Section~\ref{Binders}. Functions over several variables are -equivalent to an iteration of one-variable functions. For instance the -expression ``{\tt fun}~{\ident$_{1}$}~{\ldots}~{\ident$_{n}$}~{\tt -:}~\type~{\tt =>}~{\term}'' denotes the same function as ``{\tt -fun}~{\ident$_{1}$}~{\tt :}~\type~{\tt =>}~{\ldots}~{\tt -fun}~{\ident$_{n}$}~{\tt :}~\type~{\tt =>}~{\term}''. If a let-binder -occurs in the list of binders, it is expanded to a local definition -(see Section~\ref{let-in}). - -\subsection{Products -\label{products} -\index{products}} - -The expression ``{\tt forall}~{\ident}~{\tt :}~{\type}{\tt -,}~{\term}'' denotes the {\em product} of the variable {\ident} of -type {\type}, over the term {\term}. As for abstractions, {\tt forall} -is followed by a binder list, and products over several variables are -equivalent to an iteration of one-variable products. -Note that {\term} is intended to be a type. - -If the variable {\ident} occurs in {\term}, the product is called {\em -dependent product}. The intention behind a dependent product {\tt -forall}~$x$~{\tt :}~{$A$}{\tt ,}~{$B$} is twofold. It denotes either -the universal quantification of the variable $x$ of type $A$ in the -proposition $B$ or the functional dependent product from $A$ to $B$ (a -construction usually written $\Pi_{x:A}.B$ in set theory). - -Non dependent product types have a special notation: ``$A$ {\tt ->} -$B$'' stands for ``{\tt forall \_:}$A${\tt ,}~$B$''. The non dependent -product is used both to denote the propositional implication and -function types. - -\subsection{Applications -\label{applications} -\index{applications}} - -The expression \term$_0$ \term$_1$ denotes the application of -\term$_0$ to \term$_1$. - -The expression {\tt }\term$_0$ \term$_1$ ... \term$_n${\tt} -denotes the application of the term \term$_0$ to the arguments -\term$_1$ ... then \term$_n$. It is equivalent to {\tt (} {\ldots} -{\tt (} {\term$_0$} {\term$_1$} {\tt )} {\ldots} {\tt )} {\term$_n$} {\tt }: -associativity is to the left. - -The notation {\tt (}\,{\ident}\,{\tt :=}\,{\term}\,{\tt )} for -arguments is used for making explicit the value of implicit arguments -(see Section~\ref{Implicits-explicitation}). - -\subsection{Type cast -\label{typecast} -\index{Cast}} - -The expression ``{\term}~{\tt :}~{\type}'' is a type cast -expression. It enforces the type of {\term} to be {\type}. - -\subsection{Inferable subterms -\label{hole} -\index{\_}} - -Expressions often contain redundant pieces of information. Subterms that -can be automatically inferred by {\Coq} can be replaced by the -symbol ``\_'' and {\Coq} will guess the missing piece of information. - -\subsection{Local definitions (let-in) -\label{let-in} -\index{Local definitions} -\index{let-in}} - - -{\tt let}~{\ident}~{\tt :=}~{\term$_1$}~{\tt in}~{\term$_2$} denotes -the local binding of \term$_1$ to the variable $\ident$ in -\term$_2$. -There is a syntactic sugar for local definition of functions: {\tt -let} {\ident} {\binder$_1$} {\ldots} {\binder$_n$} {\tt :=} {\term$_1$} -{\tt in} {\term$_2$} stands for {\tt let} {\ident} {\tt := fun} -{\binder$_1$} {\ldots} {\binder$_n$} {\tt =>} {\term$_2$} {\tt in} -{\term$_2$}. - -\subsection{Definition by case analysis -\label{caseanalysis} -\index{match@{\tt match\ldots with\ldots end}}} - -Objects of inductive types can be destructurated by a case-analysis -construction called {\em pattern-matching} expression. A -pattern-matching expression is used to analyze the structure of an -inductive objects and to apply specific treatments accordingly. - -This paragraph describes the basic form of pattern-matching. See -Section~\ref{Mult-match} and Chapter~\ref{Mult-match-full} for the -description of the general form. The basic form of pattern-matching is -characterized by a single {\caseitem} expression, a {\multpattern} -restricted to a single {\pattern} and {\pattern} restricted to the -form {\qualid} \nelist{\ident}{}. - -The expression {\tt match} {\term$_0$} {\returntype} {\tt with} -{\pattern$_1$} {\tt =>} {\term$_1$} {\tt $|$} {\ldots} {\tt $|$} -{\pattern$_n$} {\tt =>} {\term$_n$} {\tt end}, denotes a {\em -pattern-matching} over the term {\term$_0$} (expected to be of an -inductive type $I$). The terms {\term$_1$}\ldots{\term$_n$} are the -{\em branches} of the pattern-matching expression. Each of -{\pattern$_i$} has a form \qualid~\nelist{\ident}{} where {\qualid} -must denote a constructor. There should be exactly one branch for -every constructor of $I$. - -The {\returntype} expresses the type returned by the whole {\tt match} -expression. There are several cases. In the {\em non dependent} case, -all branches have the same type, and the {\returntype} is the common -type of branches. In this case, {\returntype} can usually be omitted -as it can be inferred from the type of the branches\footnote{Except if -the inductive type is empty in which case there is no equation that can be -used to infer the return type.}. - -In the {\em dependent} case, there are three subcases. In the first -subcase, the type in each branch may depend on the exact value being -matched in the branch. In this case, the whole pattern-matching itself -depends on the term being matched. This dependency of the term being -matched in the return type is expressed with an ``{\tt as {\ident}}'' -clause where {\ident} is dependent in the return type. -For instance, in the following example: -\begin{coq_example*} -Inductive bool : Type := true : bool | false : bool. -Inductive eq (A:Type) (x:A) : A -> Prop := refl_equal : eq A x x. -Inductive or (A:Prop) (B:Prop) : Prop := -| or_introl : A -> or A B -| or_intror : B -> or A B. -Definition bool_case (b:bool) : or (eq bool b true) (eq bool b false) -:= match b as x return or (eq bool x true) (eq bool x false) with - | true => or_introl (eq bool true true) (eq bool true false) - (refl_equal bool true) - | false => or_intror (eq bool false true) (eq bool false false) - (refl_equal bool false) - end. -\end{coq_example*} -the branches have respective types {\tt or (eq bool true true) (eq -bool true false)} and {\tt or (eq bool false true) (eq bool false -false)} while the whole pattern-matching expression has type {\tt or -(eq bool b true) (eq bool b false)}, the identifier {\tt x} being used -to represent the dependency. Remark that when the term being matched -is a variable, the {\tt as} clause can be omitted and the term being -matched can serve itself as binding name in the return type. For -instance, the following alternative definition is accepted and has the -same meaning as the previous one. -\begin{coq_example*} -Definition bool_case (b:bool) : or (eq bool b true) (eq bool b false) -:= match b return or (eq bool b true) (eq bool b false) with - | true => or_introl (eq bool true true) (eq bool true false) - (refl_equal bool true) - | false => or_intror (eq bool false true) (eq bool false false) - (refl_equal bool false) - end. -\end{coq_example*} - -The second subcase is only relevant for annotated inductive types such -as the equality predicate (see Section~\ref{Equality}), the order -predicate on natural numbers % (see Section~\ref{le}) % undefined reference -or the type of -lists of a given length (see Section~\ref{listn}). In this configuration, -the type of each branch can depend on the type dependencies specific -to the branch and the whole pattern-matching expression has a type -determined by the specific dependencies in the type of the term being -matched. This dependency of the return type in the annotations of the -inductive type is expressed using a {\tt -``in~I~\_~$\ldots$~\_~\ident$_1$~$\ldots$~\ident$_n$}'' clause, where -\begin{itemize} -\item $I$ is the inductive type of the term being matched; - -\item the names \ident$_i$'s correspond to the arguments of the -inductive type that carry the annotations: the return type is dependent -on them; - -\item the {\_}'s denote the family parameters of the inductive type: -the return type is not dependent on them. -\end{itemize} - -For instance, in the following example: -\begin{coq_example*} -Definition sym_equal (A:Type) (x y:A) (H:eq A x y) : eq A y x := - match H in eq _ _ z return eq A z x with - | refl_equal => refl_equal A x - end. -\end{coq_example*} -the type of the branch has type {\tt eq~A~x~x} because the third -argument of {\tt eq} is {\tt x} in the type of the pattern {\tt -refl\_equal}. On the contrary, the type of the whole pattern-matching -expression has type {\tt eq~A~y~x} because the third argument of {\tt -eq} is {\tt y} in the type of {\tt H}. This dependency of the case -analysis in the third argument of {\tt eq} is expressed by the -identifier {\tt z} in the return type. - -Finally, the third subcase is a combination of the first and second -subcase. In particular, it only applies to pattern-matching on terms -in a type with annotations. For this third subcase, both -the clauses {\tt as} and {\tt in} are available. - -There are specific notations for case analysis on types with one or -two constructors: ``{\tt if {\ldots} then {\ldots} else {\ldots}}'' -and ``{\tt let (}\nelist{\ldots}{,}{\tt ) := } {\ldots} {\tt in} -{\ldots}'' (see Sections~\ref{if-then-else} and~\ref{Letin}). - -%\SeeAlso Section~\ref{Mult-match} for convenient extensions of pattern-matching. - -\subsection{Recursive functions -\label{fixpoints} -\index{fix@{fix \ident$_i$\{\dots\}}}} - -The expression ``{\tt fix} \ident$_1$ \binder$_1$ {\tt :} {\type$_1$} -\texttt{:=} \term$_1$ {\tt with} {\ldots} {\tt with} \ident$_n$ -\binder$_n$~{\tt :} {\type$_n$} \texttt{:=} \term$_n$ {\tt for} -{\ident$_i$}'' denotes the $i$\nth component of a block of functions -defined by mutual well-founded recursion. It is the local counterpart -of the {\tt Fixpoint} command. See Section~\ref{Fixpoint} for more -details. When $n=1$, the ``{\tt for}~{\ident$_i$}'' clause is omitted. - -The expression ``{\tt cofix} \ident$_1$~\binder$_1$ {\tt :} -{\type$_1$} {\tt with} {\ldots} {\tt with} \ident$_n$ \binder$_n$ {\tt -:} {\type$_n$}~{\tt for} {\ident$_i$}'' denotes the $i$\nth component of -a block of terms defined by a mutual guarded co-recursion. It is the -local counterpart of the {\tt CoFixpoint} command. See -Section~\ref{CoFixpoint} for more details. When $n=1$, the ``{\tt -for}~{\ident$_i$}'' clause is omitted. - -The association of a single fixpoint and a local -definition have a special syntax: ``{\tt let fix}~$f$~{\ldots}~{\tt - :=}~{\ldots}~{\tt in}~{\ldots}'' stands for ``{\tt let}~$f$~{\tt := - fix}~$f$~\ldots~{\tt :=}~{\ldots}~{\tt in}~{\ldots}''. The same - applies for co-fixpoints. - - -\section{The Vernacular -\label{Vernacular}} - -\begin{figure}[tbp] -\begin{centerframe} -\begin{tabular}{lcl} -{\sentence} & ::= & {\assumption} \\ - & $|$ & {\definition} \\ - & $|$ & {\inductive} \\ - & $|$ & {\fixpoint} \\ - & $|$ & {\assertion} {\proof} \\ -&&\\ -%% Assumptions -{\assumption} & ::= & {\assumptionkeyword} {\assums} {\tt .} \\ -&&\\ -{\assumptionkeyword} & $\!\!$ ::= & {\tt Axiom} $|$ {\tt Conjecture} \\ - & $|$ & {\tt Parameter} $|$ {\tt Parameters} \\ - & $|$ & {\tt Variable} $|$ {\tt Variables} \\ - & $|$ & {\tt Hypothesis} $|$ {\tt Hypotheses}\\ -&&\\ -{\assums} & ::= & \nelist{\ident}{} {\tt :} {\term} \\ - & $|$ & \nelist{{\tt (} \nelist{\ident}{} {\tt :} {\term} {\tt )}}{} \\ -&&\\ -%% Definitions -{\definition} & ::= & - {\tt Definition} {\ident} \zeroone{\binders} {\typecstr} {\tt :=} {\term} {\tt .} \\ - & $|$ & {\tt Let} {\ident} \zeroone{\binders} {\typecstr} {\tt :=} {\term} {\tt .} \\ -&&\\ -%% Inductives -{\inductive} & ::= & - {\tt Inductive} \nelist{\inductivebody}{with} {\tt .} \\ - & $|$ & {\tt CoInductive} \nelist{\inductivebody}{with} {\tt .} \\ - & & \\ -{\inductivebody} & ::= & - {\ident} \zeroone{\binders} {\tt :} {\term} {\tt :=} \\ - && ~~\zeroone{\zeroone{\tt |} \nelist{$\!${\ident}$\!$ \zeroone{\binders} {\typecstrwithoutblank}}{|}} \\ - & & \\ %% TODO: where ... -%% Fixpoints -{\fixpoint} & ::= & {\tt Fixpoint} \nelist{\fixpointbody}{with} {\tt .} \\ - & $|$ & {\tt CoFixpoint} \nelist{\cofixpointbody}{with} {\tt .} \\ -&&\\ -%% Lemmas & proofs -{\assertion} & ::= & - {\statkwd} {\ident} \zeroone{\binders} {\tt :} {\term} {\tt .} \\ -&&\\ - {\statkwd} & ::= & {\tt Theorem} $|$ {\tt Lemma} \\ - & $|$ & {\tt Remark} $|$ {\tt Fact}\\ - & $|$ & {\tt Corollary} $|$ {\tt Proposition} \\ - & $|$ & {\tt Definition} $|$ {\tt Example} \\\\ -&&\\ -{\proof} & ::= & {\tt Proof} {\tt .} {\dots} {\tt Qed} {\tt .}\\ - & $|$ & {\tt Proof} {\tt .} {\dots} {\tt Defined} {\tt .}\\ - & $|$ & {\tt Proof} {\tt .} {\dots} {\tt Admitted} {\tt .}\\ -\end{tabular} -\end{centerframe} -\caption{Syntax of sentences} -\label{sentences-syntax} -\end{figure} - -Figure \ref{sentences-syntax} describes {\em The Vernacular} which is the -language of commands of \gallina. A sentence of the vernacular -language, like in many natural languages, begins with a capital letter -and ends with a dot. - -The different kinds of command are described hereafter. They all suppose -that the terms occurring in the sentences are well-typed. - -%% -%% Axioms and Parameters -%% -\subsection{Assumptions -\index{Declarations} -\label{Declarations}} - -Assumptions extend the environment\index{Environment} with axioms, -parameters, hypotheses or variables. An assumption binds an {\ident} -to a {\type}. It is accepted by {\Coq} if and only if this {\type} is -a correct type in the environment preexisting the declaration and if -{\ident} was not previously defined in the same module. This {\type} -is considered to be the type (or specification, or statement) assumed -by {\ident} and we say that {\ident} has type {\type}. - -\subsubsection{{\tt Axiom {\ident} :{\term} .} -\comindex{Axiom} -\label{Axiom}} - -This command links {\term} to the name {\ident} as its specification -in the global context. The fact asserted by {\term} is thus assumed as -a postulate. - -\begin{ErrMsgs} -\item \errindex{{\ident} already exists} -\end{ErrMsgs} - -\begin{Variants} -\item \comindex{Parameter}\comindex{Parameters} - {\tt Parameter {\ident} :{\term}.} \\ - Is equivalent to {\tt Axiom {\ident} : {\term}} - -\item {\tt Parameter {\ident$_1$}\ldots{\ident$_n$} {\tt :}{\term}.}\\ - Adds $n$ parameters with specification {\term} - -\item - {\tt Parameter\,% -(\,{\ident$_{1,1}$}\ldots{\ident$_{1,k_1}$}\,{\tt :}\,{\term$_1$} {\tt )}\,% -\ldots\,{\tt (}\,{\ident$_{n,1}$}\ldots{\ident$_{n,k_n}$}\,{\tt :}\,% -{\term$_n$} {\tt )}.}\\ - Adds $n$ blocks of parameters with different specifications. - -\item \comindex{Conjecture} - {\tt Conjecture {\ident} :{\term}.}\\ - Is equivalent to {\tt Axiom {\ident} : {\term}}. -\end{Variants} - -\noindent {\bf Remark: } It is possible to replace {\tt Parameter} by -{\tt Parameters}. - - -\subsubsection{{\tt Variable {\ident} :{\term}}. -\comindex{Variable} -\comindex{Variables} -\label{Variable}} - -This command links {\term} to the name {\ident} in the context of the -current section (see Section~\ref{Section} for a description of the section -mechanism). When the current section is closed, name {\ident} will be -unknown and every object using this variable will be explicitly -parametrized (the variable is {\em discharged}). Using the {\tt -Variable} command out of any section is equivalent to using {\tt Parameter}. - -\begin{ErrMsgs} -\item \errindex{{\ident} already exists} -\end{ErrMsgs} - -\begin{Variants} -\item {\tt Variable {\ident$_1$}\ldots{\ident$_n$} {\tt :}{\term}.}\\ - Links {\term} to names {\ident$_1$}\ldots{\ident$_n$}. -\item - {\tt Variable\,% -(\,{\ident$_{1,1}$}\ldots{\ident$_{1,k_1}$}\,{\tt :}\,{\term$_1$} {\tt )}\,% -\ldots\,{\tt (}\,{\ident$_{n,1}$}\ldots{\ident$_{n,k_n}$}\,{\tt :}\,% -{\term$_n$} {\tt )}.}\\ - Adds $n$ blocks of variables with different specifications. -\item \comindex{Hypothesis} - \comindex{Hypotheses} - {\tt Hypothesis {\ident} {\tt :}{\term}.} \\ - \texttt{Hypothesis} is a synonymous of \texttt{Variable} -\end{Variants} - -\noindent {\bf Remark: } It is possible to replace {\tt Variable} by -{\tt Variables} and {\tt Hypothesis} by {\tt Hypotheses}. - -It is advised to use the keywords \verb:Axiom: and \verb:Hypothesis: -for logical postulates (i.e. when the assertion {\term} is of sort -\verb:Prop:), and to use the keywords \verb:Parameter: and -\verb:Variable: in other cases (corresponding to the declaration of an -abstract mathematical entity). - -%% -%% Definitions -%% -\subsection{Definitions -\index{Definitions} -\label{Basic-definitions}} - -Definitions extend the environment\index{Environment} with -associations of names to terms. A definition can be seen as a way to -give a meaning to a name or as a way to abbreviate a term. In any -case, the name can later be replaced at any time by its definition. - -The operation of unfolding a name into its definition is called -$\delta$-conversion\index{delta-reduction@$\delta$-reduction} (see -Section~\ref{delta}). A definition is accepted by the system if and -only if the defined term is well-typed in the current context of the -definition and if the name is not already used. The name defined by -the definition is called a {\em constant}\index{Constant} and the term -it refers to is its {\em body}. A definition has a type which is the -type of its body. - -A formal presentation of constants and environments is given in -Section~\ref{Typed-terms}. - -\subsubsection{\tt Definition {\ident} := {\term}. -\comindex{Definition}} - -This command binds {\term} to the name {\ident} in the -environment, provided that {\term} is well-typed. - -\begin{ErrMsgs} -\item \errindex{{\ident} already exists} -\end{ErrMsgs} - -\begin{Variants} -\item {\tt Definition {\ident} {\tt :}{\term$_1$} := {\term$_2$}.}\\ - It checks that the type of {\term$_2$} is definitionally equal to - {\term$_1$}, and registers {\ident} as being of type {\term$_1$}, - and bound to value {\term$_2$}. -\item {\tt Definition {\ident} {\binder$_1$}\ldots{\binder$_n$} - {\tt :}\term$_1$ {\tt :=} {\term$_2$}.}\\ - This is equivalent to \\ - {\tt Definition\,{\ident}\,{\tt :\,forall}\,% - {\binder$_1$}\ldots{\binder$_n$}{\tt ,}\,\term$_1$\,{\tt :=}}\,% - {\tt fun}\,{\binder$_1$}\ldots{\binder$_n$}\,{\tt =>}\,{\term$_2$}\,% - {\tt .} - -\item {\tt Example {\ident} := {\term}.}\\ -{\tt Example {\ident} {\tt :}{\term$_1$} := {\term$_2$}.}\\ -{\tt Example {\ident} {\binder$_1$}\ldots{\binder$_n$} - {\tt :}\term$_1$ {\tt :=} {\term$_2$}.}\\ -\comindex{Example} -These are synonyms of the {\tt Definition} forms. -\end{Variants} - -\begin{ErrMsgs} -\item \errindex{Error: The term {\term} has type {\type} while it is expected to have type {\type}} -\end{ErrMsgs} - -\SeeAlso Sections \ref{Opaque}, \ref{Transparent}, \ref{unfold}. - -\subsubsection{\tt Let {\ident} := {\term}. -\comindex{Let}} - -This command binds the value {\term} to the name {\ident} in the -environment of the current section. The name {\ident} disappears -when the current section is eventually closed, and, all -persistent objects (such as theorems) defined within the -section and depending on {\ident} are prefixed by the local definition -{\tt let {\ident} := {\term} in}. - -\begin{ErrMsgs} -\item \errindex{{\ident} already exists} -\end{ErrMsgs} - -\begin{Variants} -\item {\tt Let {\ident} : {\term$_1$} := {\term$_2$}.} -\end{Variants} - -\SeeAlso Sections \ref{Section} (section mechanism), \ref{Opaque}, -\ref{Transparent} (opaque/transparent constants), \ref{unfold} (tactic - {\tt unfold}). - -%% -%% Inductive Types -%% -\subsection{Inductive definitions -\index{Inductive definitions} -\label{gal_Inductive_Definitions} -\comindex{Inductive} -\label{Inductive}} - -We gradually explain simple inductive types, simple -annotated inductive types, simple parametric inductive types, -mutually inductive types. We explain also co-inductive types. - -\subsubsection{Simple inductive types} - -The definition of a simple inductive type has the following form: - -\medskip -{\tt -\begin{tabular}{l} -Inductive {\ident} : {\sort} := \\ -\begin{tabular}{clcl} - & {\ident$_1$} &:& {\type$_1$} \\ - | & {\ldots} && \\ - | & {\ident$_n$} &:& {\type$_n$} -\end{tabular} -\end{tabular} -} -\medskip - -The name {\ident} is the name of the inductively defined type and -{\sort} is the universes where it lives. -The names {\ident$_1$}, {\ldots}, {\ident$_n$} -are the names of its constructors and {\type$_1$}, {\ldots}, -{\type$_n$} their respective types. The types of the constructors have -to satisfy a {\em positivity condition} (see Section~\ref{Positivity}) -for {\ident}. This condition ensures the soundness of the inductive -definition. If this is the case, the constants {\ident}, -{\ident$_1$}, {\ldots}, {\ident$_n$} are added to the environment with -their respective types. Accordingly to the universe where -the inductive type lives ({\it e.g.} its type {\sort}), {\Coq} provides a -number of destructors for {\ident}. Destructors are named -{\ident}{\tt\_ind}, {\ident}{\tt \_rec} or {\ident}{\tt \_rect} which -respectively correspond to elimination principles on {\tt Prop}, {\tt -Set} and {\tt Type}. The type of the destructors expresses structural -induction/recursion principles over objects of {\ident}. We give below -two examples of the use of the {\tt Inductive} definitions. - -The set of natural numbers is defined as: -\begin{coq_example} -Inductive nat : Set := - | O : nat - | S : nat -> nat. -\end{coq_example} - -The type {\tt nat} is defined as the least \verb:Set: containing {\tt - O} and closed by the {\tt S} constructor. The constants {\tt nat}, -{\tt O} and {\tt S} are added to the environment. - -Now let us have a look at the elimination principles. They are three -of them: -{\tt nat\_ind}, {\tt nat\_rec} and {\tt nat\_rect}. The type of {\tt - nat\_ind} is: -\begin{coq_example} -Check nat_ind. -\end{coq_example} - -This is the well known structural induction principle over natural -numbers, i.e. the second-order form of Peano's induction principle. -It allows to prove some universal property of natural numbers ({\tt -forall n:nat, P n}) by induction on {\tt n}. - -The types of {\tt nat\_rec} and {\tt nat\_rect} are similar, except -that they pertain to {\tt (P:nat->Set)} and {\tt (P:nat->Type)} -respectively . They correspond to primitive induction principles -(allowing dependent types) respectively over sorts \verb:Set: and -\verb:Type:. The constant {\ident}{\tt \_ind} is always provided, -whereas {\ident}{\tt \_rec} and {\ident}{\tt \_rect} can be impossible -to derive (for example, when {\ident} is a proposition). - -\begin{coq_eval} -Reset Initial. -\end{coq_eval} -\begin{Variants} -\item -\begin{coq_example*} -Inductive nat : Set := O | S (_:nat). -\end{coq_example*} -In the case where inductive types have no annotations (next section -gives an example of such annotations), -%the positivity condition implies that -a constructor can be defined by only giving the type of -its arguments. -\end{Variants} - -\subsubsection{Simple annotated inductive types} - -In an annotated inductive types, the universe where the inductive -type is defined is no longer a simple sort, but what is called an -arity, which is a type whose conclusion is a sort. - -As an example of annotated inductive types, let us define the -$even$ predicate: - -\begin{coq_example} -Inductive even : nat -> Prop := - | even_0 : even O - | even_SS : forall n:nat, even n -> even (S (S n)). -\end{coq_example} - -The type {\tt nat->Prop} means that {\tt even} is a unary predicate -(inductively defined) over natural numbers. The type of its two -constructors are the defining clauses of the predicate {\tt even}. The -type of {\tt even\_ind} is: - -\begin{coq_example} -Check even_ind. -\end{coq_example} - -From a mathematical point of view it asserts that the natural numbers -satisfying the predicate {\tt even} are exactly in the smallest set of -naturals satisfying the clauses {\tt even\_0} or {\tt even\_SS}. This -is why, when we want to prove any predicate {\tt P} over elements of -{\tt even}, it is enough to prove it for {\tt O} and to prove that if -any natural number {\tt n} satisfies {\tt P} its double successor {\tt - (S (S n))} satisfies also {\tt P}. This is indeed analogous to the -structural induction principle we got for {\tt nat}. - -\begin{ErrMsgs} -\item \errindex{Non strictly positive occurrence of {\ident} in {\type}} -\item \errindex{The conclusion of {\type} is not valid; it must be -built from {\ident}} -\end{ErrMsgs} - -\subsubsection{Parametrized inductive types} -In the previous example, each constructor introduces a -different instance of the predicate {\tt even}. In some cases, -all the constructors introduces the same generic instance of the -inductive definition, in which case, instead of an annotation, we use -a context of parameters which are binders shared by all the -constructors of the definition. - -% Inductive types may be parameterized. Parameters differ from inductive -% type annotations in the fact that recursive invokations of inductive -% types must always be done with the same values of parameters as its -% specification. - -The general scheme is: -\begin{center} -{\tt Inductive} {\ident} {\binder$_1$}\ldots{\binder$_k$} : {\term} := - {\ident$_1$}: {\term$_1$} | {\ldots} | {\ident$_n$}: \term$_n$ -{\tt .} -\end{center} -Parameters differ from inductive type annotations in the fact that the -conclusion of each type of constructor {\term$_i$} invoke the inductive -type with the same values of parameters as its specification. - - - -A typical example is the definition of polymorphic lists: -\begin{coq_example*} -Inductive list (A:Set) : Set := - | nil : list A - | cons : A -> list A -> list A. -\end{coq_example*} - -Note that in the type of {\tt nil} and {\tt cons}, we write {\tt - (list A)} and not just {\tt list}.\\ The constants {\tt nil} and -{\tt cons} will have respectively types: - -\begin{coq_example} -Check nil. -Check cons. -\end{coq_example} - -Types of destructors are also quantified with {\tt (A:Set)}. - -\begin{coq_eval} -Reset Initial. -\end{coq_eval} -\begin{Variants} -\item -\begin{coq_example*} -Inductive list (A:Set) : Set := nil | cons (_:A) (_:list A). -\end{coq_example*} -This is an alternative definition of lists where we specify the -arguments of the constructors rather than their full type. -\end{Variants} - -\begin{ErrMsgs} -\item \errindex{The {\num}th argument of {\ident} must be {\ident'} in -{\type}} -\end{ErrMsgs} - -\paragraph{New from \Coq{} V8.1} The condition on parameters for -inductive definitions has been relaxed since \Coq{} V8.1. It is now -possible in the type of a constructor, to invoke recursively the -inductive definition on an argument which is not the parameter itself. - -One can define~: -\begin{coq_example} -Inductive list2 (A:Set) : Set := - | nil2 : list2 A - | cons2 : A -> list2 (A*A) -> list2 A. -\end{coq_example} -\begin{coq_eval} -Reset list2. -\end{coq_eval} -that can also be written by specifying only the type of the arguments: -\begin{coq_example*} -Inductive list2 (A:Set) : Set := nil2 | cons2 (_:A) (_:list2 (A*A)). -\end{coq_example*} -But the following definition will give an error: -\begin{coq_example} -Inductive listw (A:Set) : Set := - | nilw : listw (A*A) - | consw : A -> listw (A*A) -> listw (A*A). -\end{coq_example} -Because the conclusion of the type of constructors should be {\tt - listw A} in both cases. - -A parametrized inductive definition can be defined using -annotations instead of parameters but it will sometimes give a -different (bigger) sort for the inductive definition and will produce -a less convenient rule for case elimination. - -\SeeAlso Sections~\ref{Cic-inductive-definitions} and~\ref{Tac-induction}. - - -\subsubsection{Mutually defined inductive types -\comindex{Inductive} -\label{Mutual-Inductive}} - -The definition of a block of mutually inductive types has the form: - -\medskip -{\tt -\begin{tabular}{l} -Inductive {\ident$_1$} : {\type$_1$} := \\ -\begin{tabular}{clcl} - & {\ident$_1^1$} &:& {\type$_1^1$} \\ - | & {\ldots} && \\ - | & {\ident$_{n_1}^1$} &:& {\type$_{n_1}^1$} -\end{tabular} \\ -with\\ -~{\ldots} \\ -with {\ident$_m$} : {\type$_m$} := \\ -\begin{tabular}{clcl} - & {\ident$_1^m$} &:& {\type$_1^m$} \\ - | & {\ldots} \\ - | & {\ident$_{n_m}^m$} &:& {\type$_{n_m}^m$}. -\end{tabular} -\end{tabular} -} -\medskip - -\noindent It has the same semantics as the above {\tt Inductive} -definition for each \ident$_1$, {\ldots}, \ident$_m$. All names -\ident$_1$, {\ldots}, \ident$_m$ and \ident$_1^1$, \dots, -\ident$_{n_m}^m$ are simultaneously added to the environment. Then -well-typing of constructors can be checked. Each one of the -\ident$_1$, {\ldots}, \ident$_m$ can be used on its own. - -It is also possible to parametrize these inductive definitions. -However, parameters correspond to a local -context in which the whole set of inductive declarations is done. For -this reason, the parameters must be strictly the same for each -inductive types The extended syntax is: - -\medskip -{\tt -\begin{tabular}{l} -Inductive {\ident$_1$} {\params} : {\type$_1$} := \\ -\begin{tabular}{clcl} - & {\ident$_1^1$} &:& {\type$_1^1$} \\ - | & {\ldots} && \\ - | & {\ident$_{n_1}^1$} &:& {\type$_{n_1}^1$} -\end{tabular} \\ -with\\ -~{\ldots} \\ -with {\ident$_m$} {\params} : {\type$_m$} := \\ -\begin{tabular}{clcl} - & {\ident$_1^m$} &:& {\type$_1^m$} \\ - | & {\ldots} \\ - | & {\ident$_{n_m}^m$} &:& {\type$_{n_m}^m$}. -\end{tabular} -\end{tabular} -} -\medskip - -\Example -The typical example of a mutual inductive data type is the one for -trees and forests. We assume given two types $A$ and $B$ as variables. -It can be declared the following way. - -\begin{coq_eval} -Reset Initial. -\end{coq_eval} -\begin{coq_example*} -Variables A B : Set. -Inductive tree : Set := - node : A -> forest -> tree -with forest : Set := - | leaf : B -> forest - | cons : tree -> forest -> forest. -\end{coq_example*} - -This declaration generates automatically six induction -principles. They are respectively -called {\tt tree\_rec}, {\tt tree\_ind}, {\tt - tree\_rect}, {\tt forest\_rec}, {\tt forest\_ind}, {\tt - forest\_rect}. These ones are not the most general ones but are -just the induction principles corresponding to each inductive part -seen as a single inductive definition. - -To illustrate this point on our example, we give the types of {\tt - tree\_rec} and {\tt forest\_rec}. - -\begin{coq_example} -Check tree_rec. -Check forest_rec. -\end{coq_example} - -Assume we want to parametrize our mutual inductive definitions with -the two type variables $A$ and $B$, the declaration should be done the -following way: - -\begin{coq_eval} -Reset tree. -\end{coq_eval} -\begin{coq_example*} -Inductive tree (A B:Set) : Set := - node : A -> forest A B -> tree A B -with forest (A B:Set) : Set := - | leaf : B -> forest A B - | cons : tree A B -> forest A B -> forest A B. -\end{coq_example*} - -Assume we define an inductive definition inside a section. When the -section is closed, the variables declared in the section and occurring -free in the declaration are added as parameters to the inductive -definition. - -\SeeAlso Section~\ref{Section}. - -\subsubsection{Co-inductive types -\label{CoInductiveTypes} -\comindex{CoInductive}} - -The objects of an inductive type are well-founded with respect to the -constructors of the type. In other words, such objects contain only a -{\it finite} number of constructors. Co-inductive types arise from -relaxing this condition, and admitting types whose objects contain an -infinity of constructors. Infinite objects are introduced by a -non-ending (but effective) process of construction, defined in terms -of the constructors of the type. - -An example of a co-inductive type is the type of infinite sequences of -natural numbers, usually called streams. It can be introduced in \Coq\ -using the \texttt{CoInductive} command: -\begin{coq_example} -CoInductive Stream : Set := - Seq : nat -> Stream -> Stream. -\end{coq_example} - -The syntax of this command is the same as the command \texttt{Inductive} -(see Section~\ref{gal_Inductive_Definitions}). Notice that no -principle of induction is derived from the definition of a -co-inductive type, since such principles only make sense for inductive -ones. For co-inductive ones, the only elimination principle is case -analysis. For example, the usual destructors on streams -\texttt{hd:Stream->nat} and \texttt{tl:Str->Str} can be defined as -follows: -\begin{coq_example} -Definition hd (x:Stream) := let (a,s) := x in a. -Definition tl (x:Stream) := let (a,s) := x in s. -\end{coq_example} - -Definition of co-inductive predicates and blocks of mutually -co-inductive definitions are also allowed. An example of a -co-inductive predicate is the extensional equality on streams: - -\begin{coq_example} -CoInductive EqSt : Stream -> Stream -> Prop := - eqst : - forall s1 s2:Stream, - hd s1 = hd s2 -> EqSt (tl s1) (tl s2) -> EqSt s1 s2. -\end{coq_example} - -In order to prove the extensionally equality of two streams $s_1$ and -$s_2$ we have to construct an infinite proof of equality, that is, -an infinite object of type $(\texttt{EqSt}\;s_1\;s_2)$. We will see -how to introduce infinite objects in Section~\ref{CoFixpoint}. - -%% -%% (Co-)Fixpoints -%% -\subsection{Definition of recursive functions} - -\subsubsection{Definition of functions by recursion over inductive objects} - -This section describes the primitive form of definition by recursion -over inductive objects. See Section~\ref{Function} for more advanced -constructions. The command: -\begin{center} - \texttt{Fixpoint {\ident} {\params} {\tt \{struct} - \ident$_0$ {\tt \}} : type$_0$ := \term$_0$ - \comindex{Fixpoint}\label{Fixpoint}} -\end{center} -allows to define functions by pattern-matching over inductive objects -using a fixed point construction. -The meaning of this declaration is to define {\it ident} a recursive -function with arguments specified by the binders in {\params} such -that {\it ident} applied to arguments corresponding to these binders -has type \type$_0$, and is equivalent to the expression \term$_0$. The -type of the {\ident} is consequently {\tt forall {\params} {\tt,} - \type$_0$} and the value is equivalent to {\tt fun {\params} {\tt - =>} \term$_0$}. - -To be accepted, a {\tt Fixpoint} definition has to satisfy some -syntactical constraints on a special argument called the decreasing -argument. They are needed to ensure that the {\tt Fixpoint} definition -always terminates. The point of the {\tt \{struct \ident {\tt \}}} -annotation is to let the user tell the system which argument decreases -along the recursive calls. For instance, one can define the addition -function as : - -\begin{coq_example} -Fixpoint add (n m:nat) {struct n} : nat := - match n with - | O => m - | S p => S (add p m) - end. -\end{coq_example} - -The {\tt \{struct \ident {\tt \}}} annotation may be left implicit, in -this case the system try successively arguments from left to right -until it finds one that satisfies the decreasing condition. Note that -some fixpoints may have several arguments that fit as decreasing -arguments, and this choice influences the reduction of the -fixpoint. Hence an explicit annotation must be used if the leftmost -decreasing argument is not the desired one. Writing explicit -annotations can also speed up type-checking of large mutual fixpoints. - -The {\tt match} operator matches a value (here \verb:n:) with the -various constructors of its (inductive) type. The remaining arguments -give the respective values to be returned, as functions of the -parameters of the corresponding constructor. Thus here when \verb:n: -equals \verb:O: we return \verb:m:, and when \verb:n: equals -\verb:(S p): we return \verb:(S (add p m)):. - -The {\tt match} operator is formally described -in detail in Section~\ref{Caseexpr}. The system recognizes that in -the inductive call {\tt (add p m)} the first argument actually -decreases because it is a {\em pattern variable} coming from {\tt match - n with}. - -\Example The following definition is not correct and generates an -error message: - -\begin{coq_eval} -Set Printing Depth 50. -(********** The following is not correct and should produce **********) -(********* Error: Recursive call to wrongplus ... **********) -\end{coq_eval} -\begin{coq_example} -Fixpoint wrongplus (n m:nat) {struct n} : nat := - match m with - | O => n - | S p => S (wrongplus n p) - end. -\end{coq_example} - -because the declared decreasing argument {\tt n} actually does not -decrease in the recursive call. The function computing the addition -over the second argument should rather be written: - -\begin{coq_example*} -Fixpoint plus (n m:nat) {struct m} : nat := - match m with - | O => n - | S p => S (plus n p) - end. -\end{coq_example*} - -The ordinary match operation on natural numbers can be mimicked in the -following way. -\begin{coq_example*} -Fixpoint nat_match - (C:Set) (f0:C) (fS:nat -> C -> C) (n:nat) {struct n} : C := - match n with - | O => f0 - | S p => fS p (nat_match C f0 fS p) - end. -\end{coq_example*} -The recursive call may not only be on direct subterms of the recursive -variable {\tt n} but also on a deeper subterm and we can directly -write the function {\tt mod2} which gives the remainder modulo 2 of a -natural number. -\begin{coq_example*} -Fixpoint mod2 (n:nat) : nat := - match n with - | O => O - | S p => match p with - | O => S O - | S q => mod2 q - end - end. -\end{coq_example*} -In order to keep the strong normalization property, the fixed point -reduction will only be performed when the argument in position of the -decreasing argument (which type should be in an inductive definition) -starts with a constructor. - -The {\tt Fixpoint} construction enjoys also the {\tt with} extension -to define functions over mutually defined inductive types or more -generally any mutually recursive definitions. - -\begin{Variants} -\item {\tt Fixpoint {\ident$_1$} {\params$_1$} :{\type$_1$} := {\term$_1$}\\ - with {\ldots} \\ - with {\ident$_m$} {\params$_m$} :{\type$_m$} := {\term$_m$}}\\ - Allows to define simultaneously {\ident$_1$}, {\ldots}, - {\ident$_m$}. -\end{Variants} - -\Example -The size of trees and forests can be defined the following way: -\begin{coq_eval} -Reset Initial. -Variables A B : Set. -Inductive tree : Set := - node : A -> forest -> tree -with forest : Set := - | leaf : B -> forest - | cons : tree -> forest -> forest. -\end{coq_eval} -\begin{coq_example*} -Fixpoint tree_size (t:tree) : nat := - match t with - | node a f => S (forest_size f) - end - with forest_size (f:forest) : nat := - match f with - | leaf b => 1 - | cons t f' => (tree_size t + forest_size f') - end. -\end{coq_example*} -A generic command {\tt Scheme} is useful to build automatically various -mutual induction principles. It is described in Section~\ref{Scheme}. - -\subsubsection{Definitions of recursive objects in co-inductive types} - -The command: -\begin{center} - \texttt{CoFixpoint {\ident} : \type$_0$ := \term$_0$} - \comindex{CoFixpoint}\label{CoFixpoint} -\end{center} -introduces a method for constructing an infinite object of a -coinduc\-tive type. For example, the stream containing all natural -numbers can be introduced applying the following method to the number -\texttt{O} (see Section~\ref{CoInductiveTypes} for the definition of -{\tt Stream}, {\tt hd} and {\tt tl}): -\begin{coq_eval} -Reset Initial. -CoInductive Stream : Set := - Seq : nat -> Stream -> Stream. -Definition hd (x:Stream) := match x with - | Seq a s => a - end. -Definition tl (x:Stream) := match x with - | Seq a s => s - end. -\end{coq_eval} -\begin{coq_example} -CoFixpoint from (n:nat) : Stream := Seq n (from (S n)). -\end{coq_example} - -Oppositely to recursive ones, there is no decreasing argument in a -co-recursive definition. To be admissible, a method of construction -must provide at least one extra constructor of the infinite object for -each iteration. A syntactical guard condition is imposed on -co-recursive definitions in order to ensure this: each recursive call -in the definition must be protected by at least one constructor, and -only by constructors. That is the case in the former definition, where -the single recursive call of \texttt{from} is guarded by an -application of \texttt{Seq}. On the contrary, the following recursive -function does not satisfy the guard condition: - -\begin{coq_eval} -Set Printing Depth 50. -(********** The following is not correct and should produce **********) -(***************** Error: Unguarded recursive call *******************) -\end{coq_eval} -\begin{coq_example} -CoFixpoint filter (p:nat -> bool) (s:Stream) : Stream := - if p (hd s) then Seq (hd s) (filter p (tl s)) else filter p (tl s). -\end{coq_example} - -The elimination of co-recursive definition is done lazily, i.e. the -definition is expanded only when it occurs at the head of an -application which is the argument of a case analysis expression. In -any other context, it is considered as a canonical expression which is -completely evaluated. We can test this using the command -\texttt{Eval}, which computes the normal forms of a term: - -\begin{coq_example} -Eval compute in (from 0). -Eval compute in (hd (from 0)). -Eval compute in (tl (from 0)). -\end{coq_example} - -\begin{Variants} -\item{\tt CoFixpoint {\ident$_1$} {\params} :{\type$_1$} := - {\term$_1$}}\\ As for most constructions, arguments of co-fixpoints - expressions can be introduced before the {\tt :=} sign. -\item{\tt CoFixpoint {\ident$_1$} :{\type$_1$} := {\term$_1$}\\ - with\\ - \mbox{}\hspace{0.1cm} $\ldots$ \\ - with {\ident$_m$} : {\type$_m$} := {\term$_m$}}\\ -As in the \texttt{Fixpoint} command (see Section~\ref{Fixpoint}), it -is possible to introduce a block of mutually dependent methods. -\end{Variants} - -%% -%% Theorems & Lemmas -%% -\subsection{Assertions and proofs} -\label{Assertions} - -An assertion states a proposition (or a type) of which the proof (or -an inhabitant of the type) is interactively built using tactics. The -interactive proof mode is described in -Chapter~\ref{Proof-handling} and the tactics in Chapter~\ref{Tactics}. -The basic assertion command is: - -\subsubsection{\tt Theorem {\ident} \zeroone{\binders} : {\type}. -\comindex{Theorem}} - -After the statement is asserted, {\Coq} needs a proof. Once a proof of -{\type} under the assumptions represented by {\binders} is given and -validated, the proof is generalized into a proof of {\tt forall - \zeroone{\binders}, {\type}} and the theorem is bound to the name -{\ident} in the environment. - -\begin{ErrMsgs} - -\item \errindex{The term {\form} has type {\ldots} which should be Set, - Prop or Type} - -\item \errindexbis{{\ident} already exists}{already exists} - - The name you provided is already defined. You have then to choose - another name. - -\end{ErrMsgs} - -\begin{Variants} -\item {\tt Lemma {\ident} \zeroone{\binders} : {\type}.}\comindex{Lemma}\\ - {\tt Remark {\ident} \zeroone{\binders} : {\type}.}\comindex{Remark}\\ - {\tt Fact {\ident} \zeroone{\binders} : {\type}.}\comindex{Fact}\\ - {\tt Corollary {\ident} \zeroone{\binders} : {\type}.}\comindex{Corollary}\\ - {\tt Proposition {\ident} \zeroone{\binders} : {\type}.}\comindex{Proposition} - -These commands are synonyms of \texttt{Theorem {\ident} \zeroone{\binders} : {\type}}. - -\item {\tt Theorem \nelist{{\ident} \zeroone{\binders}: {\type}}{with}.} - -This command is useful for theorems that are proved by simultaneous -induction over a mutually inductive assumption, or that assert mutually -dependent statements in some mutual coinductive type. It is equivalent -to {\tt Fixpoint} or {\tt CoFixpoint} -(see Section~\ref{CoFixpoint}) but using tactics to build the proof of -the statements (or the body of the specification, depending on the -point of view). The inductive or coinductive types on which the -induction or coinduction has to be done is assumed to be non ambiguous -and is guessed by the system. - -Like in a {\tt Fixpoint} or {\tt CoFixpoint} definition, the induction -hypotheses have to be used on {\em structurally smaller} arguments -(for a {\tt Fixpoint}) or be {\em guarded by a constructor} (for a {\tt - CoFixpoint}). The verification that recursive proof arguments are -correct is done only at the time of registering the lemma in the -environment. To know if the use of induction hypotheses is correct at -some time of the interactive development of a proof, use the command -{\tt Guarded} (see Section~\ref{Guarded}). - -The command can be used also with {\tt Lemma}, -{\tt Remark}, etc. instead of {\tt Theorem}. - -\item {\tt Definition {\ident} \zeroone{\binders} : {\type}.} - -This allows to define a term of type {\type} using the proof editing mode. It -behaves as {\tt Theorem} but is intended to be used in conjunction with - {\tt Defined} (see \ref{Defined}) in order to define a - constant of which the computational behavior is relevant. - -The command can be used also with {\tt Example} instead -of {\tt Definition}. - -\SeeAlso Sections~\ref{Opaque} and~\ref{Transparent} ({\tt Opaque} -and {\tt Transparent}) and~\ref{unfold} (tactic {\tt unfold}). - -\item {\tt Let {\ident} \zeroone{\binders} : {\type}.} - -Like {\tt Definition {\ident} \zeroone{\binders} : {\type}.} except -that the definition is turned into a local definition generalized over -the declarations depending on it after closing the current section. - -\item {\tt Fixpoint \nelist{{\ident} {\binders} \zeroone{\annotation} {\typecstr} \zeroone{{\tt :=} {\term}}}{with}.} -\comindex{Fixpoint} - -This generalizes the syntax of {\tt Fixpoint} so that one or more -bodies can be defined interactively using the proof editing mode (when -a body is omitted, its type is mandatory in the syntax). When the -block of proofs is completed, it is intended to be ended by {\tt - Defined}. - -\item {\tt CoFixpoint \nelist{{\ident} \zeroone{\binders} {\typecstr} \zeroone{{\tt :=} {\term}}}{with}.} -\comindex{CoFixpoint} - -This generalizes the syntax of {\tt CoFixpoint} so that one or more bodies -can be defined interactively using the proof editing mode. - -\end{Variants} - -\subsubsection{{\tt Proof.} {\dots} {\tt Qed.} -\comindex{Proof} -\comindex{Qed}} - -A proof starts by the keyword {\tt Proof}. Then {\Coq} enters the -proof editing mode until the proof is completed. The proof editing -mode essentially contains tactics that are described in chapter -\ref{Tactics}. Besides tactics, there are commands to manage the proof -editing mode. They are described in Chapter~\ref{Proof-handling}. When -the proof is completed it should be validated and put in the -environment using the keyword {\tt Qed}. -\medskip - -\ErrMsg -\begin{enumerate} -\item \errindex{{\ident} already exists} -\end{enumerate} - -\begin{Remarks} -\item Several statements can be simultaneously asserted. -\item Not only other assertions but any vernacular command can be given -while in the process of proving a given assertion. In this case, the command is -understood as if it would have been given before the statements still to be -proved. -\item {\tt Proof} is recommended but can currently be omitted. On the -opposite side, {\tt Qed} (or {\tt Defined}, see below) is mandatory to -validate a proof. -\item Proofs ended by {\tt Qed} are declared opaque. Their content - cannot be unfolded (see \ref{Conversion-tactics}), thus realizing - some form of {\em proof-irrelevance}. To be able to unfold a proof, - the proof should be ended by {\tt Defined} (see below). -\end{Remarks} - -\begin{Variants} -\item \comindex{Defined} - {\tt Proof.} {\dots} {\tt Defined.}\\ - Same as {\tt Proof.} {\dots} {\tt Qed.} but the proof is - then declared transparent, which means that its - content can be explicitly used for type-checking and that it - can be unfolded in conversion tactics (see - \ref{Conversion-tactics}, \ref{Opaque}, \ref{Transparent}). -%Not claimed to be part of Gallina... -%\item {\tt Proof.} {\dots} {\tt Save.}\\ -% Same as {\tt Proof.} {\dots} {\tt Qed.} -%\item {\tt Goal} \type {\dots} {\tt Save} \ident \\ -% Same as {\tt Lemma} \ident {\tt :} \type \dots {\tt Save.} -% This is intended to be used in the interactive mode. -\item \comindex{Admitted} - {\tt Proof.} {\dots} {\tt Admitted.}\\ - Turns the current asserted statement into an axiom and exits the - proof mode. -\end{Variants} - -% Local Variables: -% mode: LaTeX -% TeX-master: "Reference-Manual" -% End: - diff --git a/doc/refman/RefMan-ide.tex b/doc/refman/RefMan-ide.tex deleted file mode 100644 index 5d9c8c16..00000000 --- a/doc/refman/RefMan-ide.tex +++ /dev/null @@ -1,328 +0,0 @@ -\chapter[\Coq{} Integrated Development Environment]{\Coq{} Integrated Development Environment\label{Addoc-coqide} -\ttindex{coqide}} - -The \Coq{} Integrated Development Environment is a graphical tool, to -be used as a user-friendly replacement to \texttt{coqtop}. Its main -purpose is to allow the user to navigate forward and backward into a -\Coq{} vernacular file, executing corresponding commands or undoing -them respectively. % CREDITS ? Proof general, lablgtk, ... - -\CoqIDE{} is run by typing the command \verb|coqide| on the command -line. Without argument, the main screen is displayed with an ``unnamed -buffer'', and with a file name as argument, another buffer displaying -the contents of that file. Additionally, \verb|coqide| accepts the same -options as \verb|coqtop|, given in Chapter~\ref{Addoc-coqc}, the ones having -obviously no meaning for \CoqIDE{} being ignored. Additionally, \verb|coqide| accepts the option \verb|-enable-geoproof| to enable the support for \emph{GeoProof} \footnote{\emph{GeoProof} is dynamic geometry software which can be used in conjunction with \CoqIDE{} to interactively build a Coq statement corresponding to a geometric figure. More information about \emph{GeoProof} can be found here: \url{http://home.gna.org/geoproof/} }. - - -\begin{figure}[t] -\begin{center} -%HEVEA\imgsrc{coqide.png} -%BEGIN LATEX -\ifpdf % si on est en pdflatex -\includegraphics[width=1.0\textwidth]{coqide.png} -\else -\includegraphics[width=1.0\textwidth]{coqide.eps} -\fi -%END LATEX -\end{center} -\caption{\CoqIDE{} main screen} -\label{fig:coqide} -\end{figure} - -A sample \CoqIDE{} main screen, while navigating into a file -\verb|Fermat.v|, is shown on Figure~\ref{fig:coqide}. At -the top is a menu bar, and a tool bar below it. The large window on -the left is displaying the various \emph{script buffers}. The upper right -window is the \emph{goal window}, where goals to -prove are displayed. The lower right window is the \emph{message window}, -where various messages resulting from commands are displayed. At the -bottom is the status bar. - -\section{Managing files and buffers, basic edition} - -In the script window, you may open arbitrarily many buffers to -edit. The \emph{File} menu allows you to open files or create some, -save them, print or export them into various formats. Among all these -buffers, there is always one which is the current \emph{running - buffer}, whose name is displayed on a green background, which is the -one where Coq commands are currently executed. - -Buffers may be edited as in any text editor, and classical basic -editing commands (Copy/Paste, \ldots) are available in the \emph{Edit} -menu. \CoqIDE{} offers only basic editing commands, so if you need -more complex editing commands, you may launch your favorite text -editor on the current buffer, using the \emph{Edit/External Editor} -menu. - -\section{Interactive navigation into \Coq{} scripts} - -The running buffer is the one where navigation takes place. The -toolbar proposes five basic commands for this. The first one, -represented by a down arrow icon, is for going forward executing one -command. If that command is successful, the part of the script that -has been executed is displayed on a green background. If that command -fails, the error message is displayed in the message window, and the -location of the error is emphasized by a red underline. - -On Figure~\ref{fig:coqide}, the running buffer is \verb|Fermat.v|, all -commands until the \verb|Theorem| have been already executed, and the -user tried to go forward executing \verb|Induction n|. That command -failed because no such tactic exist (tactics are now in -lowercase\ldots), and the wrong word is underlined. - -Notice that the green part of the running buffer is not editable. If -you ever want to modify something you have to go backward using the up -arrow tool, or even better, put the cursor where you want to go back -and use the \textsf{goto} button. Unlike with \verb|coqtop|, you -should never use \verb|Undo| to go backward. - -Two additional tool buttons exist, one to go directly to the end and -one to go back to the beginning. If you try to go to the end, or in -general to run several commands using the \textsf{goto} button, the - execution will stop whenever an error is found. - -If you ever try to execute a command which happens to run during a -long time, and would like to abort it before its -termination, you may use the interrupt button (the white cross on a red circle). - -Finally, notice that these navigation buttons are also available in -the menu, where their keyboard shortcuts are given. - -\section[Try tactics automatically]{Try tactics automatically\label{sec:trytactics}} - -The menu \texttt{Try Tactics} provides some features for automatically -trying to solve the current goal using simple tactics. If such a -tactic succeeds in solving the goal, then its text is automatically -inserted into the script. There is finally a combination of these -tactics, called the \emph{proof wizard} which will try each of them in -turn. This wizard is also available as a tool button (the light -bulb). The set of tactics tried by the wizard is customizable in -the preferences. - -These tactics are general ones, in particular they do not refer to -particular hypotheses. You may also try specific tactics related to -the goal or one of the hypotheses, by clicking with the right mouse -button on the goal or the considered hypothesis. This is the -``contextual menu on goals'' feature, that may be disabled in the -preferences if undesirable. - -\section{Proof folding} - -As your script grows bigger and bigger, it might be useful to hide the proofs -of your theorems and lemmas. - -This feature is toggled via the \texttt{Hide} entry of the \texttt{Navigation} -menu. The proof shall be enclosed between \texttt{Proof.} and \texttt{Qed.}, -both with their final dots. The proof that shall be hidden or revealed is the -first one whose beginning statement (such as \texttt{Theorem}) precedes the -insertion cursor. - -\section{Vernacular commands, templates} - -The \texttt{Templates} menu allows to use shortcuts to insert -vernacular commands. This is a nice way to proceed if you are not sure -of the spelling of the command you want. - -Moreover, this menu offers some \emph{templates} which will automatic -insert a complex command like Fixpoint with a convenient shape for its -arguments. - -\section{Queries} - -\begin{figure}[t] -\begin{center} -%HEVEA\imgsrc{coqide-queries.png} -%BEGIN LATEX -\ifpdf % si on est en pdflatex -\includegraphics[width=1.0\textwidth]{coqide-queries.png} -\else -\includegraphics[width=1.0\textwidth]{coqide-queries.eps} -\fi -%END LATEX -\end{center} -\caption{\CoqIDE{}: the query window} -\label{fig:querywindow} -\end{figure} - - -We call \emph{query} any vernacular command that do not change the -current state, such as \verb|Check|, \verb|SearchAbout|, etc. Those -commands are of course useless during compilation of a file, hence -should not be included in scripts. To run such commands without -writing them in the script, \CoqIDE{} offers another input window -called the \emph{query window}. This window can be displayed on -demand, either by using the \texttt{Window} menu, or directly using -shortcuts given in the \texttt{Queries} menu. Indeed, with \CoqIDE{} -the simplest way to perform a \texttt{SearchAbout} on some identifier -is to select it using the mouse, and pressing \verb|F2|. This will -both make appear the query window and run the \texttt{SearchAbout} in -it, displaying the result. Shortcuts \verb|F3| and \verb|F4| are for -\verb|Check| and \verb|Print| respectively. -Figure~\ref{fig:querywindow} displays the query window after selection -of the word ``mult'' in the script windows, and pressing \verb|F4| to -print its definition. - -\section{Compilation} - -The \verb|Compile| menu offers direct commands to: -\begin{itemize} -\item compile the current buffer -\item run a compilation using \verb|make| -\item go to the last compilation error -\item create a \verb|makefile| using \verb|coq_makefile|. -\end{itemize} - -\section{Customizations} - -You may customize your environment using menu -\texttt{Edit/Preferences}. A new window will be displayed, with -several customization sections presented as a notebook. - -The first section is for selecting the text font used for scripts, goal -and message windows. - -The second section is devoted to file management: you may -configure automatic saving of files, by periodically saving the -contents into files named \verb|#f#| for each opened file -\verb|f|. You may also activate the \emph{revert} feature: in case a -opened file is modified on the disk by a third party, \CoqIDE{} may read -it again for you. Note that in the case you edited that same file, you -will be prompt to choose to either discard your changes or not. The -\texttt{File charset encoding} choice is described below in -Section~\ref{sec:coqidecharencoding} - - -The \verb|Externals| section allows to customize the external commands -for compilation, printing, web browsing. In the browser command, you -may use \verb|%s| to denote the URL to open, for example: % -\verb|mozilla -remote "OpenURL(%s)"|. - -The \verb|Tactics Wizard| section allows to defined the set of tactics -that should be tried, in sequence, to solve the current goal. - -The last section is for miscellaneous boolean settings, such as the -``contextual menu on goals'' feature presented in -Section~\ref{sec:trytactics}. - -Notice that these settings are saved in the file \verb|.coqiderc| of -your home directory. - -A gtk2 accelerator keymap is saved under the name \verb|.coqide.keys|. -This file should not be edited manually: to modify a given menu -shortcut, go to the corresponding menu item without releasing the -mouse button, press the key you want for the new shortcut, and release -the mouse button afterwards. - -For experts: it is also possible to set up a specific gtk resource -file, under the name \verb|.coqide-gtk2rc|, following the gtk2 -resources syntax -\url{http://developer.gnome.org/doc/API/2.0/gtk/gtk-Resource-Files.html}. -Such a default resource file can be found in the subdirectory -\verb=lib/coq/ide= of the root installation directory of \Coq{} -(alternatively, it can be found in the subdirectory \verb=ide= of the -source archive of \Coq{}). You may -copy this file into your home directory, and edit it using any text -editor, \CoqIDE{} itself for example. - -\section{Using unicode symbols} - -\CoqIDE{} supports unicode character encoding in its text windows, -consequently a large set of symbols is available for notations. - -\subsection{Displaying unicode symbols} - -You just need to define suitable notations as described in -Chapter~\ref{Addoc-syntax}. For example, to use the mathematical symbols -$\forall$ and $\exists$, you may define -\begin{quote}\tt -Notation "$\forall$ x : t, P" := \\ -\qquad (forall x:t, P) (at level 200, x ident).\\ -Notation "$\exists$ x : t, P" := \\ -\qquad (exists x:t, P) (at level 200, x ident). -\end{quote} -There exists a small set of such notations already defined, in the -file \verb|utf8.v| of \Coq{} library, so you may enable them just by -\verb|Require utf8| inside \CoqIDE{}, or equivalently, by starting -\CoqIDE{} with \verb|coqide -l utf8|. - -However, there are some issues when using such unicode symbols: you of -course need to use a character font which supports them. In the Fonts -section of the preferences, the Preview line displays some unicode symbols, so -you could figure out if the selected font is OK. Related to this, one -thing you may need to do is choose whether Gtk should use antialiased -fonts or not, by setting the environment variable \verb|GDK_USE_XFT| -to 1 or 0 respectively. - -\subsection{Defining an input method for non ASCII symbols} - -To input an Unicode symbol, a general method is to press both the -CONTROL and the SHIFT keys, and type the hexadecimal code of the -symbol required, for example \verb|2200| for the $\forall$ symbol. -A list of symbol codes is available at \url{http://www.unicode.org}. - -This method obviously doesn't scale, that's why the preferred -alternative is to use an Input Method Editor. On POSIX systems (Linux -distributions, BSD variants and MacOS X), you can use \texttt{uim} to -support \LaTeX{}-style edition and, if using X Windows, you can also use -the XCompose method (XIM). How to use \texttt{uim} is documented below. - -If you have \texttt{uim} 1.5.x, first manually copy the files located -in directory {\tt ide/uim} of the Coq source distribution to the -{\tt uim} directory of input methods which typically is {\tt - /usr/share/uim}. Execute {\tt uim-module-manager -{-}register - coqide}. Then, execute \texttt{uim-pref-gtk} as regular user and set -the default Input Method to "CoqIDE" in the "Global Settings" group -(don't forget to tick the checkbox "Specify default IM"; you may also -have to first edit the set of "Enabled input method" to add CoqIDE to -the set). You can now execute CoqIDE with the following commands -(assuming you use a Bourne-style shell): - -\begin{verbatim} -$ export GTK_IM_MODULE=uim -$ coqide -\end{verbatim} - -If you then type the sequence "\verb=\Gamma=", you will see the sequence being -replaced by $\Gamma$ as soon as you type the second "a". - -If you have \texttt{uim} 1.6.x, the \LaTeX{} input method is built-in. You just have to execute \texttt{uim-pref-gtk} and choose "ELatin" as default Input Method in the "Global Settings". Then, in the "ELatin" group set the layout to "TeX", and remember the content of the "[ELatin] on" field (by default "<Control>$\backslash$"). Proceed then as above. - -\subsection[Character encoding for saved files]{Character encoding for saved files\label{sec:coqidecharencoding}} - -In the \texttt{Files} section of the preferences, the encoding option -is related to the way files are saved. - -If you have no need to exchange files with non UTF-8 aware -applications, it is better to choose the UTF-8 encoding, since it -guarantees that your files will be read again without problems. (This -is because when \CoqIDE{} reads a file, it tries to automatically -detect its character encoding.) - -If you choose something else than UTF-8, then missing characters will -be written encoded by \verb|\x{....}| or \verb|\x{........}| where -each dot is an hexadecimal digit: the number between braces is the -hexadecimal UNICODE index for the missing character. - - -\section{Building a custom \CoqIDE{} with user \textsc{ML} code} - -You can do this as described in Section~\ref{Coqmktop} for a -custom coq text toplevel, simply by adding -option \verb|-ide| to \verb|coqmktop|, that is something like -\begin{quote} -\texttt{coqmktop -ide -byte $m_1$.cmo \ldots{} $m_n$.cmo} -\end{quote} -or -\begin{quote} -\texttt{coqmktop -ide -opt $m_1$.cmx \ldots{} $m_n$.cmx} -\end{quote} - - - -% $Id: RefMan-ide.tex 13701 2010-12-10 07:48:30Z herbelin $ - -%%% Local Variables: -%%% mode: latex -%%% TeX-master: "Reference-Manual" -%%% End: diff --git a/doc/refman/RefMan-ind.tex b/doc/refman/RefMan-ind.tex deleted file mode 100644 index af4257ca..00000000 --- a/doc/refman/RefMan-ind.tex +++ /dev/null @@ -1,511 +0,0 @@ - -%\documentstyle[11pt]{article} -%\input{title} - -%\include{macros} -%\makeindex - -%\begin{document} -%\coverpage{The module {\tt Equality}}{Cristina CORNES} - -%\tableofcontents - -\chapter[Tactics for inductive types and families]{Tactics for inductive types and families\label{Addoc-equality}} - -This chapter details a few special tactics useful for inferring facts -from inductive hypotheses. They can be considered as tools that -macro-generate complicated uses of the basic elimination tactics for -inductive types. - -Sections \ref{inversion_introduction} to \ref{inversion_using} present -inversion tactics and Section~\ref{scheme} describes -a command {\tt Scheme} for automatic generation of induction schemes -for mutual inductive types. - -%\end{document} -%\documentstyle[11pt]{article} -%\input{title} - -%\begin{document} -%\coverpage{Module Inv: Inversion Tactics}{Cristina CORNES} - -\section[Generalities about inversion]{Generalities about inversion\label{inversion_introduction}} -When working with (co)inductive predicates, we are very often faced to -some of these situations: -\begin{itemize} -\item we have an inconsistent instance of an inductive predicate in the - local context of hypotheses. Thus, the current goal can be trivially - proved by absurdity. - -\item we have a hypothesis that is an instance of an inductive - predicate, and the instance has some variables whose constraints we - would like to derive. -\end{itemize} - -The inversion tactics are very useful to simplify the work in these -cases. Inversion tools can be classified in three groups: -\begin{enumerate} -\item tactics for inverting an instance without stocking the inversion - lemma in the context: - (\texttt{Dependent}) \texttt{Inversion} and - (\texttt{Dependent}) \texttt{Inversion\_clear}. -\item commands for generating and stocking in the context the inversion - lemma corresponding to an instance: \texttt{Derive} - (\texttt{Dependent}) \texttt{Inversion}, \texttt{Derive} - (\texttt{Dependent}) \texttt{Inversion\_clear}. -\item tactics for inverting an instance using an already defined - inversion lemma: \texttt{Inversion \ldots using}. -\end{enumerate} - -These tactics work for inductive types of arity $(\vec{x}:\vec{T})s$ -where $s \in \{Prop,Set,Type\}$. Sections \ref{inversion_primitive}, -\ref{inversion_derivation} and \ref{inversion_using} -describe respectively each group of tools. - -As inversion proofs may be large in size, we recommend the user to -stock the lemmas whenever the same instance needs to be inverted -several times.\\ - -Let's consider the relation \texttt{Le} over natural numbers and the -following variables: - -\begin{coq_eval} -Restore State "Initial". -\end{coq_eval} - -\begin{coq_example*} -Inductive Le : nat -> nat -> Set := - | LeO : forall n:nat, Le 0%N n - | LeS : forall n m:nat, Le n m -> Le (S n) (S m). -Variable P : nat -> nat -> Prop. -Variable Q : forall n m:nat, Le n m -> Prop. -\end{coq_example*} - -For example purposes we defined \verb+Le: nat->nat->Set+ - but we may have defined -it \texttt{Le} of type \verb+nat->nat->Prop+ or \verb+nat->nat->Type+. - - -\section[Inverting an instance]{Inverting an instance\label{inversion_primitive}} -\subsection{The non dependent case} -\begin{itemize} - -\item \texttt{Inversion\_clear} \ident~\\ -\index{Inversion-clear@{\tt Inversion\_clear}} - Let the type of \ident~ in the local context be $(I~\vec{t})$, - where $I$ is a (co)inductive predicate. Then, - \texttt{Inversion} applied to \ident~ derives for each possible - constructor $c_i$ of $(I~\vec{t})$, {\bf all} the necessary - conditions that should hold for the instance $(I~\vec{t})$ to be - proved by $c_i$. Finally it erases \ident~ from the context. - - - -For example, consider the goal: -\begin{coq_eval} -Lemma ex : forall n m:nat, Le (S n) m -> P n m. -intros. -\end{coq_eval} - -\begin{coq_example} -Show. -\end{coq_example} - -To prove the goal we may need to reason by cases on \texttt{H} and to - derive that \texttt{m} is necessarily of -the form $(S~m_0)$ for certain $m_0$ and that $(Le~n~m_0)$. -Deriving these conditions corresponds to prove that the -only possible constructor of \texttt{(Le (S n) m)} is -\texttt{LeS} and that we can invert the -\texttt{->} in the type of \texttt{LeS}. -This inversion is possible because \texttt{Le} is the smallest set closed by -the constructors \texttt{LeO} and \texttt{LeS}. - - -\begin{coq_example} -inversion_clear H. -\end{coq_example} - -Note that \texttt{m} has been substituted in the goal for \texttt{(S m0)} -and that the hypothesis \texttt{(Le n m0)} has been added to the -context. - -\item \texttt{Inversion} \ident~\\ -\index{Inversion@{\tt Inversion}} - This tactic differs from {\tt Inversion\_clear} in the fact that - it adds the equality constraints in the context and - it does not erase the hypothesis \ident. - - -In the previous example, {\tt Inversion\_clear} -has substituted \texttt{m} by \texttt{(S m0)}. Sometimes it is -interesting to have the equality \texttt{m=(S m0)} in the -context to use it after. In that case we can use \texttt{Inversion} that -does not clear the equalities: - -\begin{coq_example*} -Undo. -\end{coq_example*} -\begin{coq_example} -inversion H. -\end{coq_example} - -\begin{coq_eval} -Undo. -\end{coq_eval} - -Note that the hypothesis \texttt{(S m0)=m} has been deduced and -\texttt{H} has not been cleared from the context. - -\end{itemize} - -\begin{Variants} - -\item \texttt{Inversion\_clear } \ident~ \texttt{in} \ident$_1$ \ldots - \ident$_n$\\ -\index{Inversion_clear...in@{\tt Inversion\_clear...in}} - Let \ident$_1$ \ldots \ident$_n$, be identifiers in the local context. This - tactic behaves as generalizing \ident$_1$ \ldots \ident$_n$, and then performing - {\tt Inversion\_clear}. - -\item \texttt{Inversion } \ident~ \texttt{in} \ident$_1$ \ldots \ident$_n$\\ -\index{Inversion ... in@{\tt Inversion ... in}} - Let \ident$_1$ \ldots \ident$_n$, be identifiers in the local context. This - tactic behaves as generalizing \ident$_1$ \ldots \ident$_n$, and then performing - \texttt{Inversion}. - - -\item \texttt{Simple Inversion} \ident~ \\ -\index{Simple Inversion@{\tt Simple Inversion}} - It is a very primitive inversion tactic that derives all the necessary - equalities but it does not simplify - the constraints as \texttt{Inversion} and - {\tt Inversion\_clear} do. - -\end{Variants} - - -\subsection{The dependent case} -\begin{itemize} -\item \texttt{Dependent Inversion\_clear} \ident~\\ -\index{Dependent Inversion-clear@{\tt Dependent Inversion\_clear}} - Let the type of \ident~ in the local context be $(I~\vec{t})$, - where $I$ is a (co)inductive predicate, and let the goal depend both on - $\vec{t}$ and \ident. Then, - \texttt{Dependent Inversion\_clear} applied to \ident~ derives - for each possible constructor $c_i$ of $(I~\vec{t})$, {\bf all} the - necessary conditions that should hold for the instance $(I~\vec{t})$ to be - proved by $c_i$. It also substitutes \ident~ for the corresponding - term in the goal and it erases \ident~ from the context. - - -For example, consider the goal: -\begin{coq_eval} -Lemma ex_dep : forall (n m:nat) (H:Le (S n) m), Q (S n) m H. -intros. -\end{coq_eval} - -\begin{coq_example} -Show. -\end{coq_example} - -As \texttt{H} occurs in the goal, we may want to reason by cases on its -structure and so, we would like inversion tactics to -substitute \texttt{H} by the corresponding term in constructor form. -Neither \texttt{Inversion} nor {\tt Inversion\_clear} make such a -substitution. To have such a behavior we use the dependent inversion tactics: - -\begin{coq_example} -dependent inversion_clear H. -\end{coq_example} - -Note that \texttt{H} has been substituted by \texttt{(LeS n m0 l)} and -\texttt{m} by \texttt{(S m0)}. - - -\end{itemize} - -\begin{Variants} - -\item \texttt{Dependent Inversion\_clear } \ident~ \texttt{ with } \term\\ -\index{Dependent Inversion_clear...with@{\tt Dependent Inversion\_clear...with}} - \noindent Behaves as \texttt{Dependent Inversion\_clear} but allows to give - explicitly the good generalization of the goal. It is useful when - the system fails to generalize the goal automatically. If - \ident~ has type $(I~\vec{t})$ and $I$ has type - $(\vec{x}:\vec{T})s$, then \term~ must be of type - $I:(\vec{x}:\vec{T})(I~\vec{x})\rightarrow s'$ where $s'$ is the - type of the goal. - - - -\item \texttt{Dependent Inversion} \ident~\\ -\index{Dependent Inversion@{\tt Dependent Inversion}} - This tactic differs from \texttt{Dependent Inversion\_clear} in the fact that - it also adds the equality constraints in the context and - it does not erase the hypothesis \ident~. - -\item \texttt{Dependent Inversion } \ident~ \texttt{ with } \term \\ -\index{Dependent Inversion...with@{\tt Dependent Inversion...with}} - Analogous to \texttt{Dependent Inversion\_clear .. with..} above. -\end{Variants} - - - -\section[Deriving the inversion lemmas]{Deriving the inversion lemmas\label{inversion_derivation}} -\subsection{The non dependent case} - -The tactics (\texttt{Dependent}) \texttt{Inversion} and (\texttt{Dependent}) -{\tt Inversion\_clear} work on a -certain instance $(I~\vec{t})$ of an inductive predicate. At each -application, they inspect the given instance and derive the -corresponding inversion lemma. If we have to invert the same -instance several times it is recommended to stock the lemma in the -context and to reuse it whenever we need it. - -The families of commands \texttt{Derive Inversion}, \texttt{Derive -Dependent Inversion}, \texttt{Derive} \\ {\tt Inversion\_clear} and \texttt{Derive Dependent Inversion\_clear} -allow to generate inversion lemmas for given instances and sorts. Next -section describes the tactic \texttt{Inversion}$\ldots$\texttt{using} that refines the -goal with a specified inversion lemma. - -\begin{itemize} - -\item \texttt{Derive Inversion\_clear} \ident~ \texttt{with} - $(\vec{x}:\vec{T})(I~\vec{t})$ \texttt{Sort} \sort~ \\ -\index{Derive Inversion_clear...with@{\tt Derive Inversion\_clear...with}} - Let $I$ be an inductive predicate and $\vec{x}$ the variables - occurring in $\vec{t}$. This command generates and stocks - the inversion lemma for the sort \sort~ corresponding to the instance - $(\vec{x}:\vec{T})(I~\vec{t})$ with the name \ident~ in the {\bf - global} environment. When applied it is equivalent to have - inverted the instance with the tactic {\tt Inversion\_clear}. - - - For example, to generate the inversion lemma for the instance - \texttt{(Le (S n) m)} and the sort \texttt{Prop} we do: -\begin{coq_example} -Derive Inversion_clear leminv with (forall n m:nat, Le (S n) m) Sort - Prop. -\end{coq_example} - -Let us inspect the type of the generated lemma: -\begin{coq_example} -Check leminv. -\end{coq_example} - - - -\end{itemize} - -%\variants -%\begin{enumerate} -%\item \verb+Derive Inversion_clear+ \ident$_1$ \ident$_2$ \\ -%\index{Derive Inversion_clear@{\tt Derive Inversion\_clear}} -% Let \ident$_1$ have type $(I~\vec{t})$ in the local context ($I$ -% an inductive predicate). Then, this command has the same semantics -% as \verb+Derive Inversion_clear+ \ident$_2$~ \verb+with+ -% $(\vec{x}:\vec{T})(I~\vec{t})$ \verb+Sort Prop+ where $\vec{x}$ are the free -% variables of $(I~\vec{t})$ declared in the local context (variables -% of the global context are considered as constants). -%\item \verb+Derive Inversion+ \ident$_1$~ \ident$_2$~\\ -%\index{Derive Inversion@{\tt Derive Inversion}} -% Analogous to the previous command. -%\item \verb+Derive Inversion+ $num$ \ident~ \ident~ \\ -%\index{Derive Inversion@{\tt Derive Inversion}} -% This command behaves as \verb+Derive Inversion+ \ident~ {\it -% namehyp} performed on the goal number $num$. -% -%\item \verb+Derive Inversion_clear+ $num$ \ident~ \ident~ \\ -%\index{Derive Inversion_clear@{\tt Derive Inversion\_clear}} -% This command behaves as \verb+Derive Inversion_clear+ \ident~ -% \ident~ performed on the goal number $num$. -%\end{enumerate} - - - -A derived inversion lemma is adequate for inverting the instance -with which it was generated, \texttt{Derive} applied to -different instances yields different lemmas. In general, if we generate -the inversion lemma with -an instance $(\vec{x}:\vec{T})(I~\vec{t})$ and a sort $s$, the inversion lemma will -expect a predicate of type $(\vec{x}:\vec{T})s$ as first argument. \\ - -\begin{Variant} -\item \texttt{Derive Inversion} \ident~ \texttt{with} - $(\vec{x}:\vec{T})(I~\vec{t})$ \texttt{Sort} \sort\\ -\index{Derive Inversion...with@{\tt Derive Inversion...with}} - Analogous of \texttt{Derive Inversion\_clear .. with ..} but - when applied it is equivalent to having - inverted the instance with the tactic \texttt{Inversion}. -\end{Variant} - -\subsection{The dependent case} -\begin{itemize} -\item \texttt{Derive Dependent Inversion\_clear} \ident~ \texttt{with} - $(\vec{x}:\vec{T})(I~\vec{t})$ \texttt{Sort} \sort~ \\ -\index{Derive Dependent Inversion\_clear...with@{\tt Derive Dependent Inversion\_clear...with}} - Let $I$ be an inductive predicate. This command generates and stocks - the dependent inversion lemma for the sort \sort~ corresponding to the instance - $(\vec{x}:\vec{T})(I~\vec{t})$ with the name \ident~ in the {\bf - global} environment. When applied it is equivalent to having - inverted the instance with the tactic \texttt{Dependent Inversion\_clear}. -\end{itemize} - -\begin{coq_example} -Derive Dependent Inversion_clear leminv_dep with - (forall n m:nat, Le (S n) m) Sort Prop. -\end{coq_example} - -\begin{coq_example} -Check leminv_dep. -\end{coq_example} - -\begin{Variants} -\item \texttt{Derive Dependent Inversion} \ident~ \texttt{with} - $(\vec{x}:\vec{T})(I~\vec{t})$ \texttt{Sort} \sort~ \\ -\index{Derive Dependent Inversion...with@{\tt Derive Dependent Inversion...with}} - Analogous to \texttt{Derive Dependent Inversion\_clear}, but when - applied it is equivalent to having - inverted the instance with the tactic \texttt{Dependent Inversion}. - -\end{Variants} - -\section[Using already defined inversion lemmas]{Using already defined inversion lemmas\label{inversion_using}} -\begin{itemize} -\item \texttt{Inversion} \ident \texttt{ using} \ident$'$ \\ -\index{Inversion...using@{\tt Inversion...using}} - Let \ident~ have type $(I~\vec{t})$ ($I$ an inductive - predicate) in the local context, and \ident$'$ be a (dependent) inversion - lemma. Then, this tactic refines the current goal with the specified - lemma. - - -\begin{coq_eval} -Abort. -\end{coq_eval} - -\begin{coq_example} -Show. -\end{coq_example} -\begin{coq_example} -inversion H using leminv. -\end{coq_example} - - -\end{itemize} -\variant -\begin{enumerate} -\item \texttt{Inversion} \ident~ \texttt{using} \ident$'$ \texttt{in} \ident$_1$\ldots \ident$_n$\\ -\index{Inversion...using...in@{\tt Inversion...using...in}} -This tactic behaves as generalizing \ident$_1$\ldots \ident$_n$, -then doing \texttt{Use Inversion} \ident~\ident$'$. -\end{enumerate} - -\section[\tt Scheme ...]{\tt Scheme ...\index{Scheme@{\tt Scheme}}\label{Scheme} -\label{scheme}} -The {\tt Scheme} command is a high-level tool for generating -automatically (possibly mutual) induction principles for given types -and sorts. Its syntax follows the schema : - -\noindent -{\tt Scheme {\ident$_1$} := Induction for \term$_1$ Sort {\sort$_1$} \\ - with\\ - \mbox{}\hspace{0.1cm} .. \\ - with {\ident$_m$} := Induction for {\term$_m$} Sort - {\sort$_m$}}\\ -\term$_1$ \ldots \term$_m$ are different inductive types belonging to -the same package of mutual inductive definitions. This command -generates {\ident$_1$}\ldots{\ident$_m$} to be mutually recursive -definitions. Each term {\ident$_i$} proves a general principle -of mutual induction for objects in type {\term$_i$}. - -\Example -The definition of principle of mutual induction for {\tt tree} and -{\tt forest} over the sort {\tt Set} is defined by the command: -\begin{coq_eval} -Restore State "Initial". -Variables A B : Set. -Inductive tree : Set := - node : A -> forest -> tree -with forest : Set := - | leaf : B -> forest - | cons : tree -> forest -> forest. -\end{coq_eval} -\begin{coq_example*} -Scheme tree_forest_rec := Induction for tree - Sort Set - with forest_tree_rec := Induction for forest Sort Set. -\end{coq_example*} -You may now look at the type of {\tt tree\_forest\_rec} : -\begin{coq_example} -Check tree_forest_rec. -\end{coq_example} -This principle involves two different predicates for {\tt trees} and -{\tt forests}; it also has three premises each one corresponding to a -constructor of one of the inductive definitions. - -The principle {\tt tree\_forest\_rec} shares exactly the same -premises, only the conclusion now refers to the property of forests. -\begin{coq_example} -Check forest_tree_rec. -\end{coq_example} - -\begin{Variant} -\item {\tt Scheme {\ident$_1$} := Minimality for \term$_1$ Sort {\sort$_1$} \\ - with\\ - \mbox{}\hspace{0.1cm} .. \\ - with {\ident$_m$} := Minimality for {\term$_m$} Sort - {\sort$_m$}}\\ -Same as before but defines a non-dependent elimination principle more -natural in case of inductively defined relations. -\end{Variant} - -\Example -With the predicates {\tt odd} and {\tt even} inductively defined as: -% \begin{coq_eval} -% Restore State "Initial". -% \end{coq_eval} -\begin{coq_example*} -Inductive odd : nat -> Prop := - oddS : forall n:nat, even n -> odd (S n) -with even : nat -> Prop := - | evenO : even 0%N - | evenS : forall n:nat, odd n -> even (S n). -\end{coq_example*} -The following command generates a powerful elimination -principle: -\begin{coq_example*} -Scheme odd_even := Minimality for odd Sort Prop - with even_odd := Minimality for even Sort Prop. -\end{coq_example*} -The type of {\tt odd\_even} for instance will be: -\begin{coq_example} -Check odd_even. -\end{coq_example} -The type of {\tt even\_odd} shares the same premises but the -conclusion is {\tt (n:nat)(even n)->(Q n)}. - -\subsection[\tt Combined Scheme ...]{\tt Combined Scheme ...\index{CombinedScheme@{\tt Combined Scheme}}\label{CombinedScheme} -\label{combinedscheme}} -The {\tt Combined Scheme} command is a tool for combining -induction principles generated by the {\tt Scheme} command. -Its syntax follows the schema : - -\noindent -{\tt Combined Scheme {\ident$_0$} from {\ident$_1$}, .., {\ident$_n$}}\\ -\ident$_1$ \ldots \ident$_n$ are different inductive principles that must belong to -the same package of mutual inductive principle definitions. This command -generates {\ident$_0$} to be the conjunction of the principles: it is -build from the common premises of the principles and concluded by the -conjunction of their conclusions. For exemple, we can combine the -induction principles for trees and forests: - -\begin{coq_example*} -Combined Scheme tree_forest_mutind from tree_ind, forest_ind. -Check tree_forest_mutind. -\end{coq_example*} - -%\end{document} - -% $Id: RefMan-ind.tex 10421 2008-01-05 14:06:51Z herbelin $ diff --git a/doc/refman/RefMan-int.tex b/doc/refman/RefMan-int.tex deleted file mode 100644 index 7b531409..00000000 --- a/doc/refman/RefMan-int.tex +++ /dev/null @@ -1,148 +0,0 @@ -%BEGIN LATEX -\setheaders{Introduction} -%END LATEX -\chapter*{Introduction} - -This document is the Reference Manual of version \coqversion{} of the \Coq\ -proof assistant. A companion volume, the \Coq\ Tutorial, is provided -for the beginners. It is advised to read the Tutorial first. -A book~\cite{CoqArt} on practical uses of the \Coq{} system was published in 2004 and is a good support for both the beginner and -the advanced user. - -%The system \Coq\ is designed to develop mathematical proofs. It can be -%used by mathematicians to develop mathematical theories and by -%computer scientists to write formal specifications, -The \Coq{} system is designed to develop mathematical proofs, and -especially to write formal specifications, programs and to verify that -programs are correct with respect to their specification. It provides -a specification language named \gallina. Terms of \gallina\ can -represent programs as well as properties of these programs and proofs -of these properties. Using the so-called \textit{Curry-Howard - isomorphism}, programs, properties and proofs are formalized in the -same language called \textit{Calculus of Inductive Constructions}, -that is a $\lambda$-calculus with a rich type system. All logical -judgments in \Coq\ are typing judgments. The very heart of the Coq -system is the type-checking algorithm that checks the correctness of -proofs, in other words that checks that a program complies to its -specification. \Coq\ also provides an interactive proof assistant to -build proofs using specific programs called \textit{tactics}. - -All services of the \Coq\ proof assistant are accessible by -interpretation of a command language called \textit{the vernacular}. - -\Coq\ has an interactive mode in which commands are interpreted as the -user types them in from the keyboard and a compiled mode where -commands are processed from a file. - -\begin{itemize} -\item The interactive mode may be used as a debugging mode in which - the user can develop his theories and proofs step by step, - backtracking if needed and so on. The interactive mode is run with - the {\tt coqtop} command from the operating system (which we shall - assume to be some variety of UNIX in the rest of this document). -\item The compiled mode acts as a proof checker taking a file - containing a whole development in order to ensure its correctness. - Moreover, \Coq's compiler provides an output file containing a - compact representation of its input. The compiled mode is run with - the {\tt coqc} command from the operating system. - -\end{itemize} -These two modes are documented in Chapter~\ref{Addoc-coqc}. - -Other modes of interaction with \Coq{} are possible: through an emacs -shell window, an emacs generic user-interface for proof assistant -(ProofGeneral~\cite{ProofGeneral}) or through a customized interface -(PCoq~\cite{Pcoq}). These facilities are not documented here. There -is also a \Coq{} Integrated Development Environment described in -Chapter~\ref{Addoc-coqide}. - -\section*{How to read this book} - -This is a Reference Manual, not a User Manual, then it is not made for a -continuous reading. However, it has some structure that is explained -below. - -\begin{itemize} -\item The first part describes the specification language, - Gallina. Chapters~\ref{Gallina} and~\ref{Gallina-extension} - describe the concrete syntax as well as the meaning of programs, - theorems and proofs in the Calculus of Inductive - Constructions. Chapter~\ref{Theories} describes the standard library - of \Coq. Chapter~\ref{Cic} is a mathematical description of the - formalism. Chapter~\ref{chapter:Modules} describes the module system. - -\item The second part describes the proof engine. It is divided in - five chapters. Chapter~\ref{Vernacular-commands} presents all - commands (we call them \emph{vernacular commands}) that are not - directly related to interactive proving: requests to the - environment, complete or partial evaluation, loading and compiling - files. How to start and stop proofs, do multiple proofs in parallel - is explained in Chapter~\ref{Proof-handling}. In - Chapter~\ref{Tactics}, all commands that realize one or more steps - of the proof are presented: we call them \emph{tactics}. The - language to combine these tactics into complex proof strategies is - given in Chapter~\ref{TacticLanguage}. Examples of tactics are - described in Chapter~\ref{Tactics-examples}. - -%\item The third part describes how to extend the system in two ways: -% adding parsing and pretty-printing rules -% (Chapter~\ref{Addoc-syntax}) and writing new tactics -% (Chapter~\ref{TacticLanguage}). - -\item The third part describes how to extend the syntax of \Coq. It -corresponds to the Chapter~\ref{Addoc-syntax}. - -\item In the fourth part more practical tools are documented. First in - Chapter~\ref{Addoc-coqc}, the usage of \texttt{coqc} (batch mode) - and \texttt{coqtop} (interactive mode) with their options is - described. Then, in Chapter~\ref{Utilities}, - various utilities that come with the \Coq\ distribution are - presented. - Finally, Chapter~\ref{Addoc-coqide} describes the \Coq{} integrated - development environment. -\end{itemize} - -At the end of the document, after the global index, the user can find -specific indexes for tactics, vernacular commands, and error -messages. - -\section*{List of additional documentation} - -This manual does not contain all the documentation the user may need -about \Coq{}. Various informations can be found in the following -documents: -\begin{description} - -\item[Tutorial] - A companion volume to this reference manual, the \Coq{} Tutorial, is - aimed at gently introducing new users to developing proofs in \Coq{} - without assuming prior knowledge of type theory. In a second step, the - user can read also the tutorial on recursive types (document {\tt - RecTutorial.ps}). - -\item[Addendum] The fifth part (the Addendum) of the Reference Manual - is distributed as a separate document. It contains more - detailed documentation and examples about some specific aspects of the - system that may interest only certain users. It shares the indexes, - the page numbers and - the bibliography with the Reference Manual. If you see in one of the - indexes a page number that is outside the Reference Manual, it refers - to the Addendum. - -\item[Installation] A text file INSTALL that comes with the sources - explains how to install \Coq{}. - -\item[The \Coq{} standard library] -A commented version of sources of the \Coq{} standard library -(including only the specifications, the proofs are removed) -is given in the additional document {\tt Library.ps}. - -\end{description} - - -% $Id: RefMan-int.tex 11307 2008-08-06 08:38:57Z jnarboux $ - -%%% Local Variables: -%%% mode: latex -%%% TeX-master: "Reference-Manual" -%%% End: diff --git a/doc/refman/RefMan-lib.tex b/doc/refman/RefMan-lib.tex deleted file mode 100644 index 2c8abc88..00000000 --- a/doc/refman/RefMan-lib.tex +++ /dev/null @@ -1,1102 +0,0 @@ -\chapter[The {\Coq} library]{The {\Coq} library\index{Theories}\label{Theories}} - -The \Coq\ library is structured into two parts: - -\begin{description} -\item[The initial library:] it contains - elementary logical notions and data-types. It constitutes the - basic state of the system directly available when running - \Coq; - -\item[The standard library:] general-purpose libraries containing - various developments of \Coq\ axiomatizations about sets, lists, - sorting, arithmetic, etc. This library comes with the system and its - modules are directly accessible through the \verb!Require! command - (see Section~\ref{Require}); -\end{description} - -In addition, user-provided libraries or developments are provided by -\Coq\ users' community. These libraries and developments are available -for download at \texttt{http://coq.inria.fr} (see -Section~\ref{Contributions}). - -The chapter briefly reviews the \Coq\ libraries. - -\section[The basic library]{The basic library\label{Prelude}} - -This section lists the basic notions and results which are directly -available in the standard \Coq\ system\footnote{Most -of these constructions are defined in the -{\tt Prelude} module in directory {\tt theories/Init} at the {\Coq} -root directory; this includes the modules -{\tt Notations}, -{\tt Logic}, -{\tt Datatypes}, -{\tt Specif}, -{\tt Peano}, -{\tt Wf} and -{\tt Tactics}. -Module {\tt Logic\_Type} also makes it in the initial state}. - -\subsection[Notations]{Notations\label{Notations}} - -This module defines the parsing and pretty-printing of many symbols -(infixes, prefixes, etc.). However, it does not assign a meaning to -these notations. The purpose of this is to define and fix once for all -the precedence and associativity of very common notations. The main -notations fixed in the initial state are listed on -Figure~\ref{init-notations}. - -\begin{figure} -\begin{center} -\begin{tabular}{|cll|} -\hline -Notation & Precedence & Associativity \\ -\hline -\verb!_ <-> _! & 95 & no \\ -\verb!_ \/ _! & 85 & right \\ -\verb!_ /\ _! & 80 & right \\ -\verb!~ _! & 75 & right \\ -\verb!_ = _! & 70 & no \\ -\verb!_ = _ = _! & 70 & no \\ -\verb!_ = _ :> _! & 70 & no \\ -\verb!_ <> _! & 70 & no \\ -\verb!_ <> _ :> _! & 70 & no \\ -\verb!_ < _! & 70 & no \\ -\verb!_ > _! & 70 & no \\ -\verb!_ <= _! & 70 & no \\ -\verb!_ >= _! & 70 & no \\ -\verb!_ < _ < _! & 70 & no \\ -\verb!_ < _ <= _! & 70 & no \\ -\verb!_ <= _ < _! & 70 & no \\ -\verb!_ <= _ <= _! & 70 & no \\ -\verb!_ + _! & 50 & left \\ -\verb!_ || _! & 50 & left \\ -\verb!_ - _! & 50 & left \\ -\verb!_ * _! & 40 & left \\ -\verb!_ && _! & 40 & left \\ -\verb!_ / _! & 40 & left \\ -\verb!- _! & 35 & right \\ -\verb!/ _! & 35 & right \\ -\verb!_ ^ _! & 30 & right \\ -\hline -\end{tabular} -\end{center} -\caption{Notations in the initial state} -\label{init-notations} -\end{figure} - -\subsection[Logic]{Logic\label{Logic}} - -\begin{figure} -\begin{centerframe} -\begin{tabular}{lclr} -{\form} & ::= & {\tt True} & ({\tt True})\\ - & $|$ & {\tt False} & ({\tt False})\\ - & $|$ & {\tt\char'176} {\form} & ({\tt not})\\ - & $|$ & {\form} {\tt /$\backslash$} {\form} & ({\tt and})\\ - & $|$ & {\form} {\tt $\backslash$/} {\form} & ({\tt or})\\ - & $|$ & {\form} {\tt ->} {\form} & (\em{primitive implication})\\ - & $|$ & {\form} {\tt <->} {\form} & ({\tt iff})\\ - & $|$ & {\tt forall} {\ident} {\tt :} {\type} {\tt ,} - {\form} & (\em{primitive for all})\\ - & $|$ & {\tt exists} {\ident} \zeroone{{\tt :} {\specif}} {\tt - ,} {\form} & ({\tt ex})\\ - & $|$ & {\tt exists2} {\ident} \zeroone{{\tt :} {\specif}} {\tt - ,} {\form} {\tt \&} {\form} & ({\tt ex2})\\ - & $|$ & {\term} {\tt =} {\term} & ({\tt eq})\\ - & $|$ & {\term} {\tt =} {\term} {\tt :>} {\specif} & ({\tt eq}) -\end{tabular} -\end{centerframe} -\caption{Syntax of formulas} -\label{formulas-syntax} -\end{figure} - -The basic library of {\Coq} comes with the definitions of standard -(intuitionistic) logical connectives (they are defined as inductive -constructions). They are equipped with an appealing syntax enriching the -(subclass {\form}) of the syntactic class {\term}. The syntax -extension is shown on Figure~\ref{formulas-syntax}. - -% The basic library of {\Coq} comes with the definitions of standard -% (intuitionistic) logical connectives (they are defined as inductive -% constructions). They are equipped with an appealing syntax enriching -% the (subclass {\form}) of the syntactic class {\term}. The syntax -% extension \footnote{This syntax is defined in module {\tt -% LogicSyntax}} is shown on Figure~\ref{formulas-syntax}. - -\Rem Implication is not defined but primitive (it is a non-dependent -product of a proposition over another proposition). There is also a -primitive universal quantification (it is a dependent product over a -proposition). The primitive universal quantification allows both -first-order and higher-order quantification. - -\subsubsection[Propositional Connectives]{Propositional Connectives\label{Connectives} -\index{Connectives}} - -First, we find propositional calculus connectives: -\ttindex{True} -\ttindex{I} -\ttindex{False} -\ttindex{not} -\ttindex{and} -\ttindex{conj} -\ttindex{proj1} -\ttindex{proj2} - -\begin{coq_eval} -Set Printing Depth 50. -\end{coq_eval} -\begin{coq_example*} -Inductive True : Prop := I. -Inductive False : Prop := . -Definition not (A: Prop) := A -> False. -Inductive and (A B:Prop) : Prop := conj (_:A) (_:B). -Section Projections. -Variables A B : Prop. -Theorem proj1 : A /\ B -> A. -Theorem proj2 : A /\ B -> B. -End Projections. -\end{coq_example*} -\begin{coq_eval} -Abort All. -\end{coq_eval} -\ttindex{or} -\ttindex{or\_introl} -\ttindex{or\_intror} -\ttindex{iff} -\ttindex{IF\_then\_else} -\begin{coq_example*} -Inductive or (A B:Prop) : Prop := - | or_introl (_:A) - | or_intror (_:B). -Definition iff (P Q:Prop) := (P -> Q) /\ (Q -> P). -Definition IF_then_else (P Q R:Prop) := P /\ Q \/ ~ P /\ R. -\end{coq_example*} - -\subsubsection[Quantifiers]{Quantifiers\label{Quantifiers} -\index{Quantifiers}} - -Then we find first-order quantifiers: -\ttindex{all} -\ttindex{ex} -\ttindex{exists} -\ttindex{ex\_intro} -\ttindex{ex2} -\ttindex{exists2} -\ttindex{ex\_intro2} - -\begin{coq_example*} -Definition all (A:Set) (P:A -> Prop) := forall x:A, P x. -Inductive ex (A: Set) (P:A -> Prop) : Prop := - ex_intro (x:A) (_:P x). -Inductive ex2 (A:Set) (P Q:A -> Prop) : Prop := - ex_intro2 (x:A) (_:P x) (_:Q x). -\end{coq_example*} - -The following abbreviations are allowed: -\begin{center} - \begin{tabular}[h]{|l|l|} - \hline - \verb+exists x:A, P+ & \verb+ex A (fun x:A => P)+ \\ - \verb+exists x, P+ & \verb+ex _ (fun x => P)+ \\ - \verb+exists2 x:A, P & Q+ & \verb+ex2 A (fun x:A => P) (fun x:A => Q)+ \\ - \verb+exists2 x, P & Q+ & \verb+ex2 _ (fun x => P) (fun x => Q)+ \\ - \hline - \end{tabular} -\end{center} - -The type annotation ``\texttt{:A}'' can be omitted when \texttt{A} can be -synthesized by the system. - -\subsubsection[Equality]{Equality\label{Equality} -\index{Equality}} - -Then, we find equality, defined as an inductive relation. That is, -given a type \verb:A: and an \verb:x: of type \verb:A:, the -predicate \verb:(eq A x): is the smallest one which contains \verb:x:. -This definition, due to Christine Paulin-Mohring, is equivalent to -define \verb:eq: as the smallest reflexive relation, and it is also -equivalent to Leibniz' equality. - -\ttindex{eq} -\ttindex{refl\_equal} - -\begin{coq_example*} -Inductive eq (A:Type) (x:A) : A -> Prop := - refl_equal : eq A x x. -\end{coq_example*} - -\subsubsection[Lemmas]{Lemmas\label{PreludeLemmas}} - -Finally, a few easy lemmas are provided. - -\ttindex{absurd} - -\begin{coq_example*} -Theorem absurd : forall A C:Prop, A -> ~ A -> C. -\end{coq_example*} -\begin{coq_eval} -Abort. -\end{coq_eval} -\ttindex{sym\_eq} -\ttindex{trans\_eq} -\ttindex{f\_equal} -\ttindex{sym\_not\_eq} -\begin{coq_example*} -Section equality. -Variables A B : Type. -Variable f : A -> B. -Variables x y z : A. -Theorem sym_eq : x = y -> y = x. -Theorem trans_eq : x = y -> y = z -> x = z. -Theorem f_equal : x = y -> f x = f y. -Theorem sym_not_eq : x <> y -> y <> x. -\end{coq_example*} -\begin{coq_eval} -Abort. -Abort. -Abort. -Abort. -\end{coq_eval} -\ttindex{eq\_ind\_r} -\ttindex{eq\_rec\_r} -\ttindex{eq\_rect} -\ttindex{eq\_rect\_r} -%Definition eq_rect: (A:Set)(x:A)(P:A->Type)(P x)->(y:A)(x=y)->(P y). -\begin{coq_example*} -End equality. -Definition eq_ind_r : - forall (A:Type) (x:A) (P:A->Prop), P x -> forall y:A, y = x -> P y. -Definition eq_rec_r : - forall (A:Type) (x:A) (P:A->Set), P x -> forall y:A, y = x -> P y. -Definition eq_rect_r : - forall (A:Type) (x:A) (P:A->Type), P x -> forall y:A, y = x -> P y. -\end{coq_example*} -\begin{coq_eval} -Abort. -Abort. -Abort. -\end{coq_eval} -%Abort (for now predefined eq_rect) -\begin{coq_example*} -Hint Immediate sym_eq sym_not_eq : core. -\end{coq_example*} -\ttindex{f\_equal$i$} - -The theorem {\tt f\_equal} is extended to functions with two to five -arguments. The theorem are names {\tt f\_equal2}, {\tt f\_equal3}, -{\tt f\_equal4} and {\tt f\_equal5}. -For instance {\tt f\_equal3} is defined the following way. -\begin{coq_example*} -Theorem f_equal3 : - forall (A1 A2 A3 B:Type) (f:A1 -> A2 -> A3 -> B) - (x1 y1:A1) (x2 y2:A2) (x3 y3:A3), - x1 = y1 -> x2 = y2 -> x3 = y3 -> f x1 x2 x3 = f y1 y2 y3. -\end{coq_example*} -\begin{coq_eval} -Abort. -\end{coq_eval} - -\subsection[Datatypes]{Datatypes\label{Datatypes} -\index{Datatypes}} - -\begin{figure} -\begin{centerframe} -\begin{tabular}{rclr} -{\specif} & ::= & {\specif} {\tt *} {\specif} & ({\tt prod})\\ - & $|$ & {\specif} {\tt +} {\specif} & ({\tt sum})\\ - & $|$ & {\specif} {\tt + \{} {\specif} {\tt \}} & ({\tt sumor})\\ - & $|$ & {\tt \{} {\specif} {\tt \} + \{} {\specif} {\tt \}} & - ({\tt sumbool})\\ - & $|$ & {\tt \{} {\ident} {\tt :} {\specif} {\tt |} {\form} {\tt \}} - & ({\tt sig})\\ - & $|$ & {\tt \{} {\ident} {\tt :} {\specif} {\tt |} {\form} {\tt \&} - {\form} {\tt \}} & ({\tt sig2})\\ - & $|$ & {\tt \{} {\ident} {\tt :} {\specif} {\tt \&} {\specif} {\tt - \}} & ({\tt sigT})\\ - & $|$ & {\tt \{} {\ident} {\tt :} {\specif} {\tt \&} {\specif} {\tt - \&} {\specif} {\tt \}} & ({\tt sigT2})\\ - & & & \\ -{\term} & ::= & {\tt (} {\term} {\tt ,} {\term} {\tt )} & ({\tt pair}) -\end{tabular} -\end{centerframe} -\caption{Syntax of data-types and specifications} -\label{specif-syntax} -\end{figure} - - -In the basic library, we find the definition\footnote{They are in {\tt - Datatypes.v}} of the basic data-types of programming, again -defined as inductive constructions over the sort \verb:Set:. Some of -them come with a special syntax shown on Figure~\ref{specif-syntax}. - -\subsubsection[Programming]{Programming\label{Programming} -\index{Programming} -\label{libnats} -\ttindex{unit} -\ttindex{tt} -\ttindex{bool} -\ttindex{true} -\ttindex{false} -\ttindex{nat} -\ttindex{O} -\ttindex{S} -\ttindex{option} -\ttindex{Some} -\ttindex{None} -\ttindex{identity} -\ttindex{refl\_identity}} - -\begin{coq_example*} -Inductive unit : Set := tt. -Inductive bool : Set := true | false. -Inductive nat : Set := O | S (n:nat). -Inductive option (A:Set) : Set := Some (_:A) | None. -Inductive identity (A:Type) (a:A) : A -> Type := - refl_identity : identity A a a. -\end{coq_example*} - -Note that zero is the letter \verb:O:, and {\sl not} the numeral -\verb:0:. - -The predicate {\tt identity} is logically -equivalent to equality but it lives in sort {\tt - Type}. It is mainly maintained for compatibility. - -We then define the disjoint sum of \verb:A+B: of two sets \verb:A: and -\verb:B:, and their product \verb:A*B:. -\ttindex{sum} -\ttindex{A+B} -\ttindex{+} -\ttindex{inl} -\ttindex{inr} -\ttindex{prod} -\ttindex{A*B} -\ttindex{*} -\ttindex{pair} -\ttindex{fst} -\ttindex{snd} - -\begin{coq_example*} -Inductive sum (A B:Set) : Set := inl (_:A) | inr (_:B). -Inductive prod (A B:Set) : Set := pair (_:A) (_:B). -Section projections. -Variables A B : Set. -Definition fst (H: prod A B) := match H with - | pair x y => x - end. -Definition snd (H: prod A B) := match H with - | pair x y => y - end. -End projections. -\end{coq_example*} - -Some operations on {\tt bool} are also provided: {\tt andb} (with -infix notation {\tt \&\&}), {\tt orb} (with -infix notation {\tt ||}), {\tt xorb}, {\tt implb} and {\tt negb}. - -\subsection{Specification} - -The following notions\footnote{They are defined in module {\tt -Specif.v}} allow to build new data-types and specifications. -They are available with the syntax shown on -Figure~\ref{specif-syntax}. - -For instance, given \verb|A:Type| and \verb|P:A->Prop|, the construct -\verb+{x:A | P x}+ (in abstract syntax \verb+(sig A P)+) is a -\verb:Type:. We may build elements of this set as \verb:(exist x p): -whenever we have a witness \verb|x:A| with its justification -\verb|p:P x|. - -From such a \verb:(exist x p): we may in turn extract its witness -\verb|x:A| (using an elimination construct such as \verb:match:) but -{\sl not} its justification, which stays hidden, like in an abstract -data-type. In technical terms, one says that \verb:sig: is a ``weak -(dependent) sum''. A variant \verb:sig2: with two predicates is also -provided. - -\ttindex{\{x:A $\mid$ (P x)\}} -\ttindex{sig} -\ttindex{exist} -\ttindex{sig2} -\ttindex{exist2} - -\begin{coq_example*} -Inductive sig (A:Set) (P:A -> Prop) : Set := exist (x:A) (_:P x). -Inductive sig2 (A:Set) (P Q:A -> Prop) : Set := - exist2 (x:A) (_:P x) (_:Q x). -\end{coq_example*} - -A ``strong (dependent) sum'' \verb+{x:A & P x}+ may be also defined, -when the predicate \verb:P: is now defined as a -constructor of types in \verb:Type:. - -\ttindex{\{x:A \& (P x)\}} -\ttindex{\&} -\ttindex{sigT} -\ttindex{existT} -\ttindex{projT1} -\ttindex{projT2} -\ttindex{sigT2} -\ttindex{existT2} - -\begin{coq_example*} -Inductive sigT (A:Type) (P:A -> Type) : Type := existT (x:A) (_:P x). -Section Projections. -Variable A : Type. -Variable P : A -> Type. -Definition projT1 (H:sigT A P) := let (x, h) := H in x. -Definition projT2 (H:sigT A P) := - match H return P (projT1 H) with - existT x h => h - end. -End Projections. -Inductive sigT2 (A: Type) (P Q:A -> Type) : Type := - existT2 (x:A) (_:P x) (_:Q x). -\end{coq_example*} - -A related non-dependent construct is the constructive sum -\verb"{A}+{B}" of two propositions \verb:A: and \verb:B:. -\label{sumbool} -\ttindex{sumbool} -\ttindex{left} -\ttindex{right} -\ttindex{\{A\}+\{B\}} - -\begin{coq_example*} -Inductive sumbool (A B:Prop) : Set := left (_:A) | right (_:B). -\end{coq_example*} - -This \verb"sumbool" construct may be used as a kind of indexed boolean -data-type. An intermediate between \verb"sumbool" and \verb"sum" is -the mixed \verb"sumor" which combines \verb"A:Set" and \verb"B:Prop" -in the \verb"Set" \verb"A+{B}". -\ttindex{sumor} -\ttindex{inleft} -\ttindex{inright} -\ttindex{A+\{B\}} - -\begin{coq_example*} -Inductive sumor (A:Set) (B:Prop) : Set := -| inleft (_:A) -| inright (_:B). -\end{coq_example*} - -We may define variants of the axiom of choice, like in Martin-Löf's -Intuitionistic Type Theory. -\ttindex{Choice} -\ttindex{Choice2} -\ttindex{bool\_choice} - -\begin{coq_example*} -Lemma Choice : - forall (S S':Set) (R:S -> S' -> Prop), - (forall x:S, {y : S' | R x y}) -> - {f : S -> S' | forall z:S, R z (f z)}. -Lemma Choice2 : - forall (S S':Set) (R:S -> S' -> Set), - (forall x:S, {y : S' & R x y}) -> - {f : S -> S' & forall z:S, R z (f z)}. -Lemma bool_choice : - forall (S:Set) (R1 R2:S -> Prop), - (forall x:S, {R1 x} + {R2 x}) -> - {f : S -> bool | - forall x:S, f x = true /\ R1 x \/ f x = false /\ R2 x}. -\end{coq_example*} -\begin{coq_eval} -Abort. -Abort. -Abort. -\end{coq_eval} - -The next construct builds a sum between a data-type \verb|A:Type| and -an exceptional value encoding errors: - -\ttindex{Exc} -\ttindex{value} -\ttindex{error} - -\begin{coq_example*} -Definition Exc := option. -Definition value := Some. -Definition error := None. -\end{coq_example*} - - -This module ends with theorems, -relating the sorts \verb:Set: or \verb:Type: and -\verb:Prop: in a way which is consistent with the realizability -interpretation. -\ttindex{False\_rect} -\ttindex{False\_rec} -\ttindex{eq\_rect} -\ttindex{absurd\_set} -\ttindex{and\_rect} - -\begin{coq_example*} -Definition except := False_rec. -Theorem absurd_set : forall (A:Prop) (C:Set), A -> ~ A -> C. -Theorem and_rect : - forall (A B:Prop) (P:Type), (A -> B -> P) -> A /\ B -> P. -\end{coq_example*} -%\begin{coq_eval} -%Abort. -%Abort. -%\end{coq_eval} - -\subsection{Basic Arithmetics} - -The basic library includes a few elementary properties of natural -numbers, together with the definitions of predecessor, addition and -multiplication\footnote{This is in module {\tt Peano.v}}. It also -provides a scope {\tt nat\_scope} gathering standard notations for -common operations (+, *) and a decimal notation for numbers. That is he -can write \texttt{3} for \texttt{(S (S (S O)))}. This also works on -the left hand side of a \texttt{match} expression (see for example -section~\ref{refine-example}). This scope is opened by default. - -%Remove the redefinition of nat -\begin{coq_eval} -Reset Initial. -\end{coq_eval} - -The following example is not part of the standard library, but it -shows the usage of the notations: - -\begin{coq_example*} -Fixpoint even (n:nat) : bool := - match n with - | 0 => true - | 1 => false - | S (S n) => even n - end. -\end{coq_example*} - - -\ttindex{eq\_S} -\ttindex{pred} -\ttindex{pred\_Sn} -\ttindex{eq\_add\_S} -\ttindex{not\_eq\_S} -\ttindex{IsSucc} -\ttindex{O\_S} -\ttindex{n\_Sn} -\ttindex{plus} -\ttindex{plus\_n\_O} -\ttindex{plus\_n\_Sm} -\ttindex{mult} -\ttindex{mult\_n\_O} -\ttindex{mult\_n\_Sm} - -\begin{coq_example*} -Theorem eq_S : forall x y:nat, x = y -> S x = S y. -\end{coq_example*} -\begin{coq_eval} -Abort. -\end{coq_eval} -\begin{coq_example*} -Definition pred (n:nat) : nat := - match n with - | 0 => 0 - | S u => u - end. -Theorem pred_Sn : forall m:nat, m = pred (S m). -Theorem eq_add_S : forall n m:nat, S n = S m -> n = m. -Hint Immediate eq_add_S : core. -Theorem not_eq_S : forall n m:nat, n <> m -> S n <> S m. -\end{coq_example*} -\begin{coq_eval} -Abort All. -\end{coq_eval} -\begin{coq_example*} -Definition IsSucc (n:nat) : Prop := - match n with - | 0 => False - | S p => True - end. -Theorem O_S : forall n:nat, 0 <> S n. -Theorem n_Sn : forall n:nat, n <> S n. -\end{coq_example*} -\begin{coq_eval} -Abort All. -\end{coq_eval} -\begin{coq_example*} -Fixpoint plus (n m:nat) {struct n} : nat := - match n with - | 0 => m - | S p => S (p + m) - end. -where "n + m" := (plus n m) : nat_scope. -Lemma plus_n_O : forall n:nat, n = n + 0. -Lemma plus_n_Sm : forall n m:nat, S (n + m) = n + S m. -\end{coq_example*} -\begin{coq_eval} -Abort All. -\end{coq_eval} -\begin{coq_example*} -Fixpoint mult (n m:nat) {struct n} : nat := - match n with - | 0 => 0 - | S p => m + p * m - end. -where "n * m" := (mult n m) : nat_scope. -Lemma mult_n_O : forall n:nat, 0 = n * 0. -Lemma mult_n_Sm : forall n m:nat, n * m + n = n * (S m). -\end{coq_example*} -\begin{coq_eval} -Abort All. -\end{coq_eval} - -Finally, it gives the definition of the usual orderings \verb:le:, -\verb:lt:, \verb:ge:, and \verb:gt:. -\ttindex{le} -\ttindex{le\_n} -\ttindex{le\_S} -\ttindex{lt} -\ttindex{ge} -\ttindex{gt} - -\begin{coq_example*} -Inductive le (n:nat) : nat -> Prop := - | le_n : le n n - | le_S : forall m:nat, n <= m -> n <= (S m). -where "n <= m" := (le n m) : nat_scope. -Definition lt (n m:nat) := S n <= m. -Definition ge (n m:nat) := m <= n. -Definition gt (n m:nat) := m < n. -\end{coq_example*} - -Properties of these relations are not initially known, but may be -required by the user from modules \verb:Le: and \verb:Lt:. Finally, -\verb:Peano: gives some lemmas allowing pattern-matching, and a double -induction principle. - -\ttindex{nat\_case} -\ttindex{nat\_double\_ind} - -\begin{coq_example*} -Theorem nat_case : - forall (n:nat) (P:nat -> Prop), - P 0 -> (forall m:nat, P (S m)) -> P n. -\end{coq_example*} -\begin{coq_eval} -Abort All. -\end{coq_eval} -\begin{coq_example*} -Theorem nat_double_ind : - forall R:nat -> nat -> Prop, - (forall n:nat, R 0 n) -> - (forall n:nat, R (S n) 0) -> - (forall n m:nat, R n m -> R (S n) (S m)) -> forall n m:nat, R n m. -\end{coq_example*} -\begin{coq_eval} -Abort All. -\end{coq_eval} - -\subsection{Well-founded recursion} - -The basic library contains the basics of well-founded recursion and -well-founded induction\footnote{This is defined in module {\tt Wf.v}}. -\index{Well foundedness} -\index{Recursion} -\index{Well founded induction} -\ttindex{Acc} -\ttindex{Acc\_inv} -\ttindex{Acc\_rect} -\ttindex{well\_founded} - -\begin{coq_example*} -Section Well_founded. -Variable A : Type. -Variable R : A -> A -> Prop. -Inductive Acc (x:A) : Prop := - Acc_intro : (forall y:A, R y x -> Acc y) -> Acc x. -Lemma Acc_inv : Acc x -> forall y:A, R y x -> Acc y. -\end{coq_example*} -\begin{coq_eval} -destruct 1; trivial. -Defined. -\end{coq_eval} -%% Acc_rect now primitively defined -%% Section AccRec. -%% Variable P : A -> Set. -%% Variable F : -%% forall x:A, -%% (forall y:A, R y x -> Acc y) -> (forall y:A, R y x -> P y) -> P x. -%% Fixpoint Acc_rec (x:A) (a:Acc x) {struct a} : P x := -%% F x (Acc_inv x a) -%% (fun (y:A) (h:R y x) => Acc_rec y (Acc_inv x a y h)). -%% End AccRec. -\begin{coq_example*} -Definition well_founded := forall a:A, Acc a. -Hypothesis Rwf : well_founded. -Theorem well_founded_induction : - forall P:A -> Set, - (forall x:A, (forall y:A, R y x -> P y) -> P x) -> forall a:A, P a. -Theorem well_founded_ind : - forall P:A -> Prop, - (forall x:A, (forall y:A, R y x -> P y) -> P x) -> forall a:A, P a. -\end{coq_example*} -\begin{coq_eval} -Abort All. -\end{coq_eval} -The automatically generated scheme {\tt Acc\_rect} -can be used to define functions by fixpoints using -well-founded relations to justify termination. Assuming -extensionality of the functional used for the recursive call, the -fixpoint equation can be proved. -\ttindex{Fix\_F} -\ttindex{fix\_eq} -\ttindex{Fix\_F\_inv} -\ttindex{Fix\_F\_eq} -\begin{coq_example*} -Section FixPoint. -Variable P : A -> Type. -Variable F : forall x:A, (forall y:A, R y x -> P y) -> P x. -Fixpoint Fix_F (x:A) (r:Acc x) {struct r} : P x := - F x (fun (y:A) (p:R y x) => Fix_F y (Acc_inv x r y p)). -Definition Fix (x:A) := Fix_F x (Rwf x). -Hypothesis F_ext : - forall (x:A) (f g:forall y:A, R y x -> P y), - (forall (y:A) (p:R y x), f y p = g y p) -> F x f = F x g. -Lemma Fix_F_eq : - forall (x:A) (r:Acc x), - F x (fun (y:A) (p:R y x) => Fix_F y (Acc_inv x r y p)) = Fix_F x r. -Lemma Fix_F_inv : forall (x:A) (r s:Acc x), Fix_F x r = Fix_F x s. -Lemma fix_eq : forall x:A, Fix x = F x (fun (y:A) (p:R y x) => Fix y). -\end{coq_example*} -\begin{coq_eval} -Abort All. -\end{coq_eval} -\begin{coq_example*} -End FixPoint. -End Well_founded. -\end{coq_example*} - -\subsection{Accessing the {\Type} level} - -The basic library includes the definitions\footnote{This is in module -{\tt Logic\_Type.v}} of the counterparts of some data-types and logical -quantifiers at the \verb:Type: level: negation, pair, and properties -of {\tt identity}. - -\ttindex{notT} -\ttindex{prodT} -\ttindex{pairT} -\begin{coq_eval} -Reset Initial. -\end{coq_eval} -\begin{coq_example*} -Definition notT (A:Type) := A -> False. -Inductive prodT (A B:Type) : Type := pairT (_:A) (_:B). -\end{coq_example*} - -At the end, it defines data-types at the {\Type} level. - -\subsection{Tactics} - -A few tactics defined at the user level are provided in the initial -state\footnote{This is in module {\tt Tactics.v}}. - -\section{The standard library} - -\subsection{Survey} - -The rest of the standard library is structured into the following -subdirectories: - -\begin{tabular}{lp{12cm}} - {\bf Logic} & Classical logic and dependent equality \\ - {\bf Arith} & Basic Peano arithmetic \\ - {\bf NArith} & Basic positive integer arithmetic \\ - {\bf ZArith} & Basic relative integer arithmetic \\ - {\bf Numbers} & Various approaches to natural, integer and cyclic numbers (currently axiomatically and on top of 2$^{31}$ binary words) \\ - {\bf Bool} & Booleans (basic functions and results) \\ - {\bf Lists} & Monomorphic and polymorphic lists (basic functions and - results), Streams (infinite sequences defined with co-inductive - types) \\ - {\bf Sets} & Sets (classical, constructive, finite, infinite, power set, - etc.) \\ - {\bf FSets} & Specification and implementations of finite sets and finite - maps (by lists and by AVL trees)\\ - {\bf Reals} & Axiomatization of real numbers (classical, basic functions, - integer part, fractional part, limit, derivative, Cauchy - series, power series and results,...)\\ - {\bf Relations} & Relations (definitions and basic results) \\ - {\bf Sorting} & Sorted list (basic definitions and heapsort correctness) \\ - {\bf Strings} & 8-bits characters and strings\\ - {\bf Wellfounded} & Well-founded relations (basic results) \\ - -\end{tabular} -\medskip - -These directories belong to the initial load path of the system, and -the modules they provide are compiled at installation time. So they -are directly accessible with the command \verb!Require! (see -Chapter~\ref{Other-commands}). - -The different modules of the \Coq\ standard library are described in the -additional document \verb!Library.dvi!. They are also accessible on the WWW -through the \Coq\ homepage -\footnote{\texttt{http://coq.inria.fr}}. - -\subsection[Notations for integer arithmetics]{Notations for integer arithmetics\index{Arithmetical notations}} - -On Figure~\ref{zarith-syntax} is described the syntax of expressions -for integer arithmetics. It is provided by requiring and opening the -module {\tt ZArith} and opening scope {\tt Z\_scope}. - -\ttindex{+} -\ttindex{*} -\ttindex{-} -\ttindex{/} -\ttindex{<=} -\ttindex{>=} -\ttindex{<} -\ttindex{>} -\ttindex{?=} -\ttindex{mod} - -\begin{figure} -\begin{center} -\begin{tabular}{l|l|l|l} -Notation & Interpretation & Precedence & Associativity\\ -\hline -\verb!_ < _! & {\tt Zlt} &&\\ -\verb!x <= y! & {\tt Zle} &&\\ -\verb!_ > _! & {\tt Zgt} &&\\ -\verb!x >= y! & {\tt Zge} &&\\ -\verb!x < y < z! & {\tt x < y \verb!/\! y < z} &&\\ -\verb!x < y <= z! & {\tt x < y \verb!/\! y <= z} &&\\ -\verb!x <= y < z! & {\tt x <= y \verb!/\! y < z} &&\\ -\verb!x <= y <= z! & {\tt x <= y \verb!/\! y <= z} &&\\ -\verb!_ ?= _! & {\tt Zcompare} & 70 & no\\ -\verb!_ + _! & {\tt Zplus} &&\\ -\verb!_ - _! & {\tt Zminus} &&\\ -\verb!_ * _! & {\tt Zmult} &&\\ -\verb!_ / _! & {\tt Zdiv} &&\\ -\verb!_ mod _! & {\tt Zmod} & 40 & no \\ -\verb!- _! & {\tt Zopp} &&\\ -\verb!_ ^ _! & {\tt Zpower} &&\\ -\end{tabular} -\end{center} -\caption{Definition of the scope for integer arithmetics ({\tt Z\_scope})} -\label{zarith-syntax} -\end{figure} - -Figure~\ref{zarith-syntax} shows the notations provided by {\tt -Z\_scope}. It specifies how notations are interpreted and, when not -already reserved, the precedence and associativity. - -\begin{coq_example} -Require Import ZArith. -Check (2 + 3)%Z. -Open Scope Z_scope. -Check 2 + 3. -\end{coq_example} - -\subsection[Peano's arithmetic (\texttt{nat})]{Peano's arithmetic (\texttt{nat})\index{Peano's arithmetic} -\ttindex{nat\_scope}} - -While in the initial state, many operations and predicates of Peano's -arithmetic are defined, further operations and results belong to other -modules. For instance, the decidability of the basic predicates are -defined here. This is provided by requiring the module {\tt Arith}. - -Figure~\ref{nat-syntax} describes notation available in scope {\tt -nat\_scope}. - -\begin{figure} -\begin{center} -\begin{tabular}{l|l} -Notation & Interpretation \\ -\hline -\verb!_ < _! & {\tt lt} \\ -\verb!x <= y! & {\tt le} \\ -\verb!_ > _! & {\tt gt} \\ -\verb!x >= y! & {\tt ge} \\ -\verb!x < y < z! & {\tt x < y \verb!/\! y < z} \\ -\verb!x < y <= z! & {\tt x < y \verb!/\! y <= z} \\ -\verb!x <= y < z! & {\tt x <= y \verb!/\! y < z} \\ -\verb!x <= y <= z! & {\tt x <= y \verb!/\! y <= z} \\ -\verb!_ + _! & {\tt plus} \\ -\verb!_ - _! & {\tt minus} \\ -\verb!_ * _! & {\tt mult} \\ -\end{tabular} -\end{center} -\caption{Definition of the scope for natural numbers ({\tt nat\_scope})} -\label{nat-syntax} -\end{figure} - -\subsection{Real numbers library} - -\subsubsection[Notations for real numbers]{Notations for real numbers\index{Notations for real numbers}} - -This is provided by requiring and opening the module {\tt Reals} and -opening scope {\tt R\_scope}. This set of notations is very similar to -the notation for integer arithmetics. The inverse function was added. -\begin{figure} -\begin{center} -\begin{tabular}{l|l} -Notation & Interpretation \\ -\hline -\verb!_ < _! & {\tt Rlt} \\ -\verb!x <= y! & {\tt Rle} \\ -\verb!_ > _! & {\tt Rgt} \\ -\verb!x >= y! & {\tt Rge} \\ -\verb!x < y < z! & {\tt x < y \verb!/\! y < z} \\ -\verb!x < y <= z! & {\tt x < y \verb!/\! y <= z} \\ -\verb!x <= y < z! & {\tt x <= y \verb!/\! y < z} \\ -\verb!x <= y <= z! & {\tt x <= y \verb!/\! y <= z} \\ -\verb!_ + _! & {\tt Rplus} \\ -\verb!_ - _! & {\tt Rminus} \\ -\verb!_ * _! & {\tt Rmult} \\ -\verb!_ / _! & {\tt Rdiv} \\ -\verb!- _! & {\tt Ropp} \\ -\verb!/ _! & {\tt Rinv} \\ -\verb!_ ^ _! & {\tt pow} \\ -\end{tabular} -\end{center} -\label{reals-syntax} -\caption{Definition of the scope for real arithmetics ({\tt R\_scope})} -\end{figure} - -\begin{coq_eval} -Reset Initial. -\end{coq_eval} -\begin{coq_example} -Require Import Reals. -Check (2 + 3)%R. -Open Scope R_scope. -Check 2 + 3. -\end{coq_example} - -\subsubsection{Some tactics} - -In addition to the \verb|ring|, \verb|field| and \verb|fourier| -tactics (see Chapter~\ref{Tactics}) there are: -\begin{itemize} -\item {\tt discrR} \tacindex{discrR} - - Proves that a real integer constant $c_1$ is different from another - real integer constant $c_2$. - -\begin{coq_example*} -Require Import DiscrR. -Goal 5 <> 0. -\end{coq_example*} - -\begin{coq_example} -discrR. -\end{coq_example} - -\begin{coq_eval} -Abort. -\end{coq_eval} - -\item {\tt split\_Rabs} allows to unfold {\tt Rabs} constant and splits -corresponding conjonctions. -\tacindex{split\_Rabs} - -\begin{coq_example*} -Require Import SplitAbsolu. -Goal forall x:R, x <= Rabs x. -\end{coq_example*} - -\begin{coq_example} -intro; split_Rabs. -\end{coq_example} - -\begin{coq_eval} -Abort. -\end{coq_eval} - -\item {\tt split\_Rmult} allows to split a condition that a product is - non null into subgoals corresponding to the condition on each - operand of the product. -\tacindex{split\_Rmult} - -\begin{coq_example*} -Require Import SplitRmult. -Goal forall x y z:R, x * y * z <> 0. -\end{coq_example*} - -\begin{coq_example} -intros; split_Rmult. -\end{coq_example} - -\end{itemize} - -All this tactics has been written with the tactic language Ltac -described in Chapter~\ref{TacticLanguage}. - -\begin{coq_eval} -Reset Initial. -\end{coq_eval} - -\subsection[List library]{List library\index{Notations for lists} -\ttindex{length} -\ttindex{head} -\ttindex{tail} -\ttindex{app} -\ttindex{rev} -\ttindex{nth} -\ttindex{map} -\ttindex{flat\_map} -\ttindex{fold\_left} -\ttindex{fold\_right}} - -Some elementary operations on polymorphic lists are defined here. They -can be accessed by requiring module {\tt List}. - -It defines the following notions: -\begin{center} -\begin{tabular}{l|l} -\hline -{\tt length} & length \\ -{\tt head} & first element (with default) \\ -{\tt tail} & all but first element \\ -{\tt app} & concatenation \\ -{\tt rev} & reverse \\ -{\tt nth} & accessing $n$-th element (with default) \\ -{\tt map} & applying a function \\ -{\tt flat\_map} & applying a function returning lists \\ -{\tt fold\_left} & iterator (from head to tail) \\ -{\tt fold\_right} & iterator (from tail to head) \\ -\hline -\end{tabular} -\end{center} - -Table show notations available when opening scope {\tt list\_scope}. - -\begin{figure} -\begin{center} -\begin{tabular}{l|l|l|l} -Notation & Interpretation & Precedence & Associativity\\ -\hline -\verb!_ ++ _! & {\tt app} & 60 & right \\ -\verb!_ :: _! & {\tt cons} & 60 & right \\ -\end{tabular} -\end{center} -\label{list-syntax} -\caption{Definition of the scope for lists ({\tt list\_scope})} -\end{figure} - - -\section[Users' contributions]{Users' contributions\index{Contributions} -\label{Contributions}} - -Numerous users' contributions have been collected and are available at -URL \url{http://coq.inria.fr/contribs/}. On this web page, you have a list -of all contributions with informations (author, institution, quick -description, etc.) and the possibility to download them one by one. -You will also find informations on how to submit a new -contribution. - -% $Id: RefMan-lib.tex 13091 2010-06-08 13:56:19Z herbelin $ - -%%% Local Variables: -%%% mode: latex -%%% TeX-master: "Reference-Manual" -%%% End: diff --git a/doc/refman/RefMan-ltac.tex b/doc/refman/RefMan-ltac.tex deleted file mode 100644 index d8747254..00000000 --- a/doc/refman/RefMan-ltac.tex +++ /dev/null @@ -1,1308 +0,0 @@ -\chapter[The tactic language]{The tactic language\label{TacticLanguage}} - -%\geometry{a4paper,body={5in,8in}} - -This chapter gives a compact documentation of Ltac, the tactic -language available in {\Coq}. We start by giving the syntax, and next, -we present the informal semantics. If you want to know more regarding -this language and especially about its foundations, you can refer -to~\cite{Del00}. Chapter~\ref{Tactics-examples} is devoted to giving -examples of use of this language on small but also with non-trivial -problems. - - -\section{Syntax} - -\def\tacexpr{\textrm{\textsl{expr}}} -\def\tacexprlow{\textrm{\textsl{tacexpr$_1$}}} -\def\tacexprinf{\textrm{\textsl{tacexpr$_2$}}} -\def\tacexprpref{\textrm{\textsl{tacexpr$_3$}}} -\def\atom{\textrm{\textsl{atom}}} -%%\def\recclause{\textrm{\textsl{rec\_clause}}} -\def\letclause{\textrm{\textsl{let\_clause}}} -\def\matchrule{\textrm{\textsl{match\_rule}}} -\def\contextrule{\textrm{\textsl{context\_rule}}} -\def\contexthyp{\textrm{\textsl{context\_hyp}}} -\def\tacarg{\nterm{tacarg}} -\def\cpattern{\nterm{cpattern}} - -The syntax of the tactic language is given Figures~\ref{ltac} -and~\ref{ltac_aux}. See Chapter~\ref{BNF-syntax} for a description of -the BNF metasyntax used in these grammar rules. Various already -defined entries will be used in this chapter: entries -{\naturalnumber}, {\integer}, {\ident}, {\qualid}, {\term}, -{\cpattern} and {\atomictac} represent respectively the natural and -integer numbers, the authorized identificators and qualified names, -{\Coq}'s terms and patterns and all the atomic tactics described in -Chapter~\ref{Tactics}. The syntax of {\cpattern} is the same as that -of terms, but it is extended with pattern matching metavariables. In -{\cpattern}, a pattern-matching metavariable is represented with the -syntax {\tt ?id} where {\tt id} is an {\ident}. The notation {\tt \_} -can also be used to denote metavariable whose instance is -irrelevant. In the notation {\tt ?id}, the identifier allows us to -keep instantiations and to make constraints whereas {\tt \_} shows -that we are not interested in what will be matched. On the right hand -side of pattern-matching clauses, the named metavariable are used -without the question mark prefix. There is also a special notation for -second-order pattern-matching problems: in an applicative pattern of -the form {\tt @?id id$_1$ \ldots id$_n$}, the variable {\tt id} -matches any complex expression with (possible) dependencies in the -variables {\tt id$_1$ \ldots id$_n$} and returns a functional term of -the form {\tt fun id$_1$ \ldots id$_n$ => {\term}}. - - -The main entry of the grammar is {\tacexpr}. This language is used in -proof mode but it can also be used in toplevel definitions as shown in -Figure~\ref{ltactop}. - -\begin{Remarks} -\item The infix tacticals ``\dots\ {\tt ||} \dots'' and ``\dots\ {\tt - ;} \dots'' are associative. - -\item In {\tacarg}, there is an overlap between {\qualid} as a -direct tactic argument and {\qualid} as a particular case of -{\term}. The resolution is done by first looking for a reference of -the tactic language and if it fails, for a reference to a term. To -force the resolution as a reference of the tactic language, use the -form {\tt ltac :} {\qualid}. To force the resolution as a reference to -a term, use the syntax {\tt ({\qualid})}. - -\item As shown by the figure, tactical {\tt ||} binds more than the -prefix tacticals {\tt try}, {\tt repeat}, {\tt do}, {\tt info} and -{\tt abstract} which themselves bind more than the postfix tactical -``{\tt \dots\ ;[ \dots\ ]}'' which binds more than ``\dots\ {\tt ;} -\dots''. - -For instance -\begin{quote} -{\tt try repeat \tac$_1$ || - \tac$_2$;\tac$_3$;[\tac$_{31}$|\dots|\tac$_{3n}$];\tac$_4$.} -\end{quote} -is understood as -\begin{quote} -{\tt (try (repeat (\tac$_1$ || \tac$_2$)));} \\ -{\tt ((\tac$_3$;[\tac$_{31}$|\dots|\tac$_{3n}$]);\tac$_4$).} -\end{quote} -\end{Remarks} - - -\begin{figure}[htbp] -\begin{centerframe} -\begin{tabular}{lcl} -{\tacexpr} & ::= & - {\tacexpr} {\tt ;} {\tacexpr}\\ -& | & {\tacexpr} {\tt ; [} \nelist{\tacexpr}{|} {\tt ]}\\ -& | & {\tacexprpref}\\ -\\ -{\tacexprpref} & ::= & - {\tt do} {\it (}{\naturalnumber} {\it |} {\ident}{\it )} {\tacexprpref}\\ -& | & {\tt info} {\tacexprpref}\\ -& | & {\tt progress} {\tacexprpref}\\ -& | & {\tt repeat} {\tacexprpref}\\ -& | & {\tt try} {\tacexprpref}\\ -& | & {\tacexprinf} \\ -\\ -{\tacexprinf} & ::= & - {\tacexprlow} {\tt ||} {\tacexprpref}\\ -& | & {\tacexprlow}\\ -\\ -{\tacexprlow} & ::= & -{\tt fun} \nelist{\name}{} {\tt =>} {\atom}\\ -& | & -{\tt let} \zeroone{\tt rec} \nelist{\letclause}{\tt with} {\tt in} -{\atom}\\ -& | & -{\tt match goal with} \nelist{\contextrule}{\tt |} {\tt end}\\ -& | & -{\tt match reverse goal with} \nelist{\contextrule}{\tt |} {\tt end}\\ -& | & -{\tt match} {\tacexpr} {\tt with} \nelist{\matchrule}{\tt |} {\tt end}\\ -& | & -{\tt lazymatch goal with} \nelist{\contextrule}{\tt |} {\tt end}\\ -& | & -{\tt lazymatch reverse goal with} \nelist{\contextrule}{\tt |} {\tt end}\\ -& | & -{\tt lazymatch} {\tacexpr} {\tt with} \nelist{\matchrule}{\tt |} {\tt end}\\ -& | & {\tt abstract} {\atom}\\ -& | & {\tt abstract} {\atom} {\tt using} {\ident} \\ -& | & {\tt first [} \nelist{\tacexpr}{\tt |} {\tt ]}\\ -& | & {\tt solve [} \nelist{\tacexpr}{\tt |} {\tt ]}\\ -& | & {\tt idtac} \sequence{\messagetoken}{}\\ -& | & {\tt fail} \zeroone{\naturalnumber} \sequence{\messagetoken}{}\\ -& | & {\tt fresh} ~|~ {\tt fresh} {\qstring}\\ -& | & {\tt context} {\ident} {\tt [} {\term} {\tt ]}\\ -& | & {\tt eval} {\nterm{redexpr}} {\tt in} {\term}\\ -& | & {\tt type of} {\term}\\ -& | & {\tt external} {\qstring} {\qstring} \nelist{\tacarg}{}\\ -& | & {\tt constr :} {\term}\\ -& | & \atomictac\\ -& | & {\qualid} \nelist{\tacarg}{}\\ -& | & {\atom}\\ -\\ -{\atom} & ::= & - {\qualid} \\ -& | & ()\\ -& | & {\integer}\\ -& | & {\tt (} {\tacexpr} {\tt )}\\ -\\ -{\messagetoken}\!\!\!\!\!\! & ::= & {\qstring} ~|~ {\ident} ~|~ {\integer} \\ -\end{tabular} -\end{centerframe} -\caption{Syntax of the tactic language} -\label{ltac} -\end{figure} - - - -\begin{figure}[htbp] -\begin{centerframe} -\begin{tabular}{lcl} -\tacarg & ::= & - {\qualid}\\ -& $|$ & {\tt ()} \\ -& $|$ & {\tt ltac :} {\atom}\\ -& $|$ & {\term}\\ -\\ -\letclause & ::= & {\ident} \sequence{\name}{} {\tt :=} {\tacexpr}\\ -\\ -\contextrule & ::= & - \nelist{\contexthyp}{\tt ,} {\tt |-}{\cpattern} {\tt =>} {\tacexpr}\\ -& $|$ & {\tt |-} {\cpattern} {\tt =>} {\tacexpr}\\ -& $|$ & {\tt \_ =>} {\tacexpr}\\ -\\ -\contexthyp & ::= & {\name} {\tt :} {\cpattern}\\ - & $|$ & {\name} {\tt :=} {\cpattern} \zeroone{{\tt :} {\cpattern}}\\ -\\ -\matchrule & ::= & - {\cpattern} {\tt =>} {\tacexpr}\\ -& $|$ & {\tt context} {\zeroone{\ident}} {\tt [} {\cpattern} {\tt ]} - {\tt =>} {\tacexpr}\\ -& $|$ & {\tt appcontext} {\zeroone{\ident}} {\tt [} {\cpattern} {\tt ]} - {\tt =>} {\tacexpr}\\ -& $|$ & {\tt \_ =>} {\tacexpr}\\ -\end{tabular} -\end{centerframe} -\caption{Syntax of the tactic language (continued)} -\label{ltac_aux} -\end{figure} - -\begin{figure}[ht] -\begin{centerframe} -\begin{tabular}{lcl} -\nterm{top} & ::= & \zeroone{\tt Local} {\tt Ltac} \nelist{\nterm{ltac\_def}} {\tt with} \\ -\\ -\nterm{ltac\_def} & ::= & {\ident} \sequence{\ident}{} {\tt :=} -{\tacexpr}\\ -& $|$ &{\qualid} \sequence{\ident}{} {\tt ::=}{\tacexpr} -\end{tabular} -\end{centerframe} -\caption{Tactic toplevel definitions} -\label{ltactop} -\end{figure} - - -%% -%% Semantics -%% -\section{Semantics} -%\index[tactic]{Tacticals} -\index{Tacticals} -%\label{Tacticals} - -Tactic expressions can only be applied in the context of a goal. The -evaluation yields either a term, an integer or a tactic. Intermediary -results can be terms or integers but the final result must be a tactic -which is then applied to the current goal. - -There is a special case for {\tt match goal} expressions of which -the clauses evaluate to tactics. Such expressions can only be used as -end result of a tactic expression (never as argument of a non recursive local -definition or of an application). - -The rest of this section explains the semantics of every construction -of Ltac. - - -%% \subsection{Values} - -%% Values are given by Figure~\ref{ltacval}. All these values are tactic values, -%% i.e. to be applied to a goal, except {\tt Fun}, {\tt Rec} and $arg$ values. - -%% \begin{figure}[ht] -%% \noindent{}\framebox[6in][l] -%% {\parbox{6in} -%% {\begin{center} -%% \begin{tabular}{lp{0.1in}l} -%% $vexpr$ & ::= & $vexpr$ {\tt ;} $vexpr$\\ -%% & | & $vexpr$ {\tt ; [} {\it (}$vexpr$ {\tt |}{\it )}$^*$ $vexpr$ {\tt -%% ]}\\ -%% & | & $vatom$\\ -%% \\ -%% $vatom$ & ::= & {\tt Fun} \nelist{\inputfun}{} {\tt ->} {\tacexpr}\\ -%% %& | & {\tt Rec} \recclause\\ -%% & | & -%% {\tt Rec} \nelist{\recclause}{\tt And} {\tt In} -%% {\tacexpr}\\ -%% & | & -%% {\tt Match Context With} {\it (}$context\_rule$ {\tt |}{\it )}$^*$ -%% $context\_rule$\\ -%% & | & {\tt (} $vexpr$ {\tt )}\\ -%% & | & $vatom$ {\tt Orelse} $vatom$\\ -%% & | & {\tt Do} {\it (}{\naturalnumber} {\it |} {\ident}{\it )} $vatom$\\ -%% & | & {\tt Repeat} $vatom$\\ -%% & | & {\tt Try} $vatom$\\ -%% & | & {\tt First [} {\it (}$vexpr$ {\tt |}{\it )}$^*$ $vexpr$ {\tt ]}\\ -%% & | & {\tt Solve [} {\it (}$vexpr$ {\tt |}{\it )}$^*$ $vexpr$ {\tt ]}\\ -%% & | & {\tt Idtac}\\ -%% & | & {\tt Fail}\\ -%% & | & {\primitivetactic}\\ -%% & | & $arg$ -%% \end{tabular} -%% \end{center}}} -%% \caption{Values of ${\cal L}_{tac}$} -%% \label{ltacval} -%% \end{figure} - -%% \subsection{Evaluation} - -\subsubsection[Sequence]{Sequence\tacindex{;} -\index{Tacticals!;@{\tt {\tac$_1$};\tac$_2$}}} - -A sequence is an expression of the following form: -\begin{quote} -{\tacexpr}$_1$ {\tt ;} {\tacexpr}$_2$ -\end{quote} -The expressions {\tacexpr}$_1$ and {\tacexpr}$_2$ are evaluated -to $v_1$ and $v_2$ which have to be tactic values. The tactic $v_1$ is -then applied and $v_2$ is applied to every subgoal generated by the -application of $v_1$. Sequence is left-associative. - -\subsubsection[General sequence]{General sequence\tacindex{;[\ldots$\mid$\ldots$\mid$\ldots]}} -%\tacindex{; [ | ]} -%\index{; [ | ]@{\tt ;[\ldots$\mid$\ldots$\mid$\ldots]}} -\index{Tacticals!; [ \mid ]@{\tt {\tac$_0$};[{\tac$_1$}$\mid$\ldots$\mid$\tac$_n$]}} - -A general sequence has the following form: -\begin{quote} -{\tacexpr}$_0$ {\tt ; [} {\tacexpr}$_1$ {\tt |} $...$ {\tt |} -{\tacexpr}$_n$ {\tt ]} -\end{quote} -The expressions {\tacexpr}$_i$ are evaluated to $v_i$, for $i=0,...,n$ -and all have to be tactics. The tactic $v_0$ is applied and $v_i$ is -applied to the $i$-th generated subgoal by the application of $v_0$, -for $=1,...,n$. It fails if the application of $v_0$ does not generate -exactly $n$ subgoals. - -\begin{Variants} - \item If no tactic is given for the $i$-th generated subgoal, it -behaves as if the tactic {\tt idtac} were given. For instance, {\tt -split ; [ | auto ]} is a shortcut for -{\tt split ; [ idtac | auto ]}. - - \item {\tacexpr}$_0$ {\tt ; [} {\tacexpr}$_1$ {\tt |} $...$ {\tt |} - {\tacexpr}$_i$ {\tt |} {\tt ..} {\tt |} {\tacexpr}$_{i+1+j}$ {\tt |} - $...$ {\tt |} {\tacexpr}$_n$ {\tt ]} - - In this variant, {\tt idtac} is used for the subgoals numbered from - $i+1$ to $i+j$ (assuming $n$ is the number of subgoals). - - Note that {\tt ..} is part of the syntax, while $...$ is the meta-symbol used - to describe a list of {\tacexpr} of arbitrary length. - - \item {\tacexpr}$_0$ {\tt ; [} {\tacexpr}$_1$ {\tt |} $...$ {\tt |} - {\tacexpr}$_i$ {\tt |} {\tacexpr} {\tt ..} {\tt |} - {\tacexpr}$_{i+1+j}$ {\tt |} $...$ {\tt |} {\tacexpr}$_n$ {\tt ]} - - In this variant, {\tacexpr} is used instead of {\tt idtac} for the - subgoals numbered from $i+1$ to $i+j$. - -\end{Variants} - - - -\subsubsection[For loop]{For loop\tacindex{do} -\index{Tacticals!do@{\tt do}}} - -There is a for loop that repeats a tactic {\num} times: -\begin{quote} -{\tt do} {\num} {\tacexpr} -\end{quote} -{\tacexpr} is evaluated to $v$. $v$ must be a tactic value. $v$ is -applied {\num} times. Supposing ${\num}>1$, after the first -application of $v$, $v$ is applied, at least once, to the generated -subgoals and so on. It fails if the application of $v$ fails before -the {\num} applications have been completed. - -\subsubsection[Repeat loop]{Repeat loop\tacindex{repeat} -\index{Tacticals!repeat@{\tt repeat}}} - -We have a repeat loop with: -\begin{quote} -{\tt repeat} {\tacexpr} -\end{quote} -{\tacexpr} is evaluated to $v$. If $v$ denotes a tactic, this tactic -is applied to the goal. If the application fails, the tactic is -applied recursively to all the generated subgoals until it eventually -fails. The recursion stops in a subgoal when the tactic has failed. -The tactic {\tt repeat {\tacexpr}} itself never fails. - -\subsubsection[Error catching]{Error catching\tacindex{try} -\index{Tacticals!try@{\tt try}}} - -We can catch the tactic errors with: -\begin{quote} -{\tt try} {\tacexpr} -\end{quote} -{\tacexpr} is evaluated to $v$. $v$ must be a tactic value. $v$ is -applied. If the application of $v$ fails, it catches the error and -leaves the goal unchanged. If the level of the exception is positive, -then the exception is re-raised with its level decremented. - -\subsubsection[Detecting progress]{Detecting progress\tacindex{progress}} - -We can check if a tactic made progress with: -\begin{quote} -{\tt progress} {\tacexpr} -\end{quote} -{\tacexpr} is evaluated to $v$. $v$ must be a tactic value. $v$ is -applied. If the application of $v$ produced one subgoal equal to the -initial goal (up to syntactical equality), then an error of level 0 is -raised. - -\ErrMsg \errindex{Failed to progress} - -\subsubsection[Branching]{Branching\tacindex{$\mid\mid$} -\index{Tacticals!orelse@{\tt $\mid\mid$}}} - -We can easily branch with the following structure: -\begin{quote} -{\tacexpr}$_1$ {\tt ||} {\tacexpr}$_2$ -\end{quote} -{\tacexpr}$_1$ and {\tacexpr}$_2$ are evaluated to $v_1$ and -$v_2$. $v_1$ and $v_2$ must be tactic values. $v_1$ is applied and if -it fails to progress then $v_2$ is applied. Branching is left-associative. - -\subsubsection[First tactic to work]{First tactic to work\tacindex{first} -\index{Tacticals!first@{\tt first}}} - -We may consider the first tactic to work (i.e. which does not fail) among a -panel of tactics: -\begin{quote} -{\tt first [} {\tacexpr}$_1$ {\tt |} $...$ {\tt |} {\tacexpr}$_n$ {\tt ]} -\end{quote} -{\tacexpr}$_i$ are evaluated to $v_i$ and $v_i$ must be tactic values, for -$i=1,...,n$. Supposing $n>1$, it applies $v_1$, if it works, it stops else it -tries to apply $v_2$ and so on. It fails when there is no applicable tactic. - -\ErrMsg \errindex{No applicable tactic} - -\subsubsection[Solving]{Solving\tacindex{solve} -\index{Tacticals!solve@{\tt solve}}} - -We may consider the first to solve (i.e. which generates no subgoal) among a -panel of tactics: -\begin{quote} -{\tt solve [} {\tacexpr}$_1$ {\tt |} $...$ {\tt |} {\tacexpr}$_n$ {\tt ]} -\end{quote} -{\tacexpr}$_i$ are evaluated to $v_i$ and $v_i$ must be tactic values, for -$i=1,...,n$. Supposing $n>1$, it applies $v_1$, if it solves, it stops else it -tries to apply $v_2$ and so on. It fails if there is no solving tactic. - -\ErrMsg \errindex{Cannot solve the goal} - -\subsubsection[Identity]{Identity\tacindex{idtac} -\index{Tacticals!idtac@{\tt idtac}}} - -The constant {\tt idtac} is the identity tactic: it leaves any goal -unchanged but it appears in the proof script. - -\variant {\tt idtac \nelist{\messagetoken}{}} - -This prints the given tokens. Strings and integers are printed -literally. If a (term) variable is given, its contents are printed. - - -\subsubsection[Failing]{Failing\tacindex{fail} -\index{Tacticals!fail@{\tt fail}}} - -The tactic {\tt fail} is the always-failing tactic: it does not solve -any goal. It is useful for defining other tacticals since it can be -catched by {\tt try} or {\tt match goal}. - -\begin{Variants} -\item {\tt fail $n$}\\ -The number $n$ is the failure level. If no level is specified, it -defaults to $0$. The level is used by {\tt try} and {\tt match goal}. -If $0$, it makes {\tt match goal} considering the next clause -(backtracking). If non zero, the current {\tt match goal} block or -{\tt try} command is aborted and the level is decremented. - -\item {\tt fail \nelist{\messagetoken}{}}\\ -The given tokens are used for printing the failure message. - -\item {\tt fail $n$ \nelist{\messagetoken}{}}\\ -This is a combination of the previous variants. -\end{Variants} - -\ErrMsg \errindex{Tactic Failure {\it message} (level $n$)}. - -\subsubsection[Local definitions]{Local definitions\index{Ltac!let} -\index{Ltac!let rec} -\index{let!in Ltac} -\index{let rec!in Ltac}} - -Local definitions can be done as follows: -\begin{quote} -{\tt let} {\ident}$_1$ {\tt :=} {\tacexpr}$_1$\\ -{\tt with} {\ident}$_2$ {\tt :=} {\tacexpr}$_2$\\ -...\\ -{\tt with} {\ident}$_n$ {\tt :=} {\tacexpr}$_n$ {\tt in}\\ -{\tacexpr} -\end{quote} -each {\tacexpr}$_i$ is evaluated to $v_i$, then, {\tacexpr} is -evaluated by substituting $v_i$ to each occurrence of {\ident}$_i$, -for $i=1,...,n$. There is no dependencies between the {\tacexpr}$_i$ -and the {\ident}$_i$. - -Local definitions can be recursive by using {\tt let rec} instead of -{\tt let}. In this latter case, the definitions are evaluated lazily -so that the {\tt rec} keyword can be used also in non recursive cases -so as to avoid the eager evaluation of local definitions. - -\subsubsection{Application} - -An application is an expression of the following form: -\begin{quote} -{\qualid} {\tacarg}$_1$ ... {\tacarg}$_n$ -\end{quote} -The reference {\qualid} must be bound to some defined tactic -definition expecting at least $n$ arguments. The expressions -{\tacexpr}$_i$ are evaluated to $v_i$, for $i=1,...,n$. -%If {\tacexpr} is a {\tt Fun} or {\tt Rec} value then the body is evaluated by -%substituting $v_i$ to the formal parameters, for $i=1,...,n$. For recursive -%clauses, the bodies are lazily substituted (when an identifier to be evaluated -%is the name of a recursive clause). - -%\subsection{Application of tactic values} - -\subsubsection[Function construction]{Function construction\index{fun!in Ltac} -\index{Ltac!fun}} - -A parameterized tactic can be built anonymously (without resorting to -local definitions) with: -\begin{quote} -{\tt fun} {\ident${}_1$} ... {\ident${}_n$} {\tt =>} {\tacexpr} -\end{quote} -Indeed, local definitions of functions are a syntactic sugar for -binding a {\tt fun} tactic to an identifier. - -\subsubsection[Pattern matching on terms]{Pattern matching on terms\index{Ltac!match} -\index{match!in Ltac}} - -We can carry out pattern matching on terms with: -\begin{quote} -{\tt match} {\tacexpr} {\tt with}\\ -~~~{\cpattern}$_1$ {\tt =>} {\tacexpr}$_1$\\ -~{\tt |} {\cpattern}$_2$ {\tt =>} {\tacexpr}$_2$\\ -~...\\ -~{\tt |} {\cpattern}$_n$ {\tt =>} {\tacexpr}$_n$\\ -~{\tt |} {\tt \_} {\tt =>} {\tacexpr}$_{n+1}$\\ -{\tt end} -\end{quote} -The expression {\tacexpr} is evaluated and should yield a term which -is matched against {\cpattern}$_1$. The matching is non-linear: if a -metavariable occurs more than once, it should match the same -expression every time. It is first-order except on the -variables of the form {\tt @?id} that occur in head position of an -application. For these variables, the matching is second-order and -returns a functional term. - -If the matching with {\cpattern}$_1$ succeeds, then {\tacexpr}$_1$ is -evaluated into some value by substituting the pattern matching -instantiations to the metavariables. If {\tacexpr}$_1$ evaluates to a -tactic and the {\tt match} expression is in position to be applied to -a goal (e.g. it is not bound to a variable by a {\tt let in}), then -this tactic is applied. If the tactic succeeds, the list of resulting -subgoals is the result of the {\tt match} expression. If -{\tacexpr}$_1$ does not evaluate to a tactic or if the {\tt match} -expression is not in position to be applied to a goal, then the result -of the evaluation of {\tacexpr}$_1$ is the result of the {\tt match} -expression. - -If the matching with {\cpattern}$_1$ fails, or if it succeeds but the -evaluation of {\tacexpr}$_1$ fails, or if the evaluation of -{\tacexpr}$_1$ succeeds but returns a tactic in execution position -whose execution fails, then {\cpattern}$_2$ is used and so on. The -pattern {\_} matches any term and shunts all remaining patterns if -any. If all clauses fail (in particular, there is no pattern {\_}) -then a no-matching-clause error is raised. - -\begin{ErrMsgs} - -\item \errindex{No matching clauses for match} - - No pattern can be used and, in particular, there is no {\tt \_} pattern. - -\item \errindex{Argument of match does not evaluate to a term} - - This happens when {\tacexpr} does not denote a term. - -\end{ErrMsgs} - -\begin{Variants} - -\item \index{lazymatch!in Ltac} -\index{Ltac!lazymatch} -Using {\tt lazymatch} instead of {\tt match} has an effect if the -right-hand-side of a clause returns a tactic. With {\tt match}, the -tactic is applied to the current goal (and the next clause is tried if -it fails). With {\tt lazymatch}, the tactic is directly returned as -the result of the whole {\tt lazymatch} block without being first -tried to be applied to the goal. Typically, if the {\tt lazymatch} -block is bound to some variable $x$ in a {\tt let in}, then tactic -expression gets bound to the variable $x$. - -\item \index{context!in pattern} -There is a special form of patterns to match a subterm against the -pattern: -\begin{quote} -{\tt context} {\ident} {\tt [} {\cpattern} {\tt ]} -\end{quote} -It matches any term which one subterm matches {\cpattern}. If there is -a match, the optional {\ident} is assign the ``matched context'', that -is the initial term where the matched subterm is replaced by a -hole. The definition of {\tt context} in expressions below will show -how to use such term contexts. - -If the evaluation of the right-hand-side of a valid match fails, the -next matching subterm is tried. If no further subterm matches, the -next clause is tried. Matching subterms are considered top-bottom and -from left to right (with respect to the raw printing obtained by -setting option {\tt Printing All}, see Section~\ref{SetPrintingAll}). - -\begin{coq_example} -Ltac f x := - match x with - context f [S ?X] => - idtac X; (* To display the evaluation order *) - assert (p := refl_equal 1 : X=1); (* To filter the case X=1 *) - let x:= context f[O] in assert (x=O) (* To observe the context *) - end. -Goal True. -f (3+4). -\end{coq_example} - -\item \index{appcontext!in pattern} -For historical reasons, {\tt context} considers $n$-ary applications -such as {\tt (f 1 2)} as a whole, and not as a sequence of unary -applications {\tt ((f 1) 2)}. Hence {\tt context [f ?x]} will fail -to find a matching subterm in {\tt (f 1 2)}: if the pattern is a partial -application, the matched subterms will be necessarily be -applications with exactly the same number of arguments. -Alternatively, one may now use the following variant of {\tt context}: -\begin{quote} -{\tt appcontext} {\ident} {\tt [} {\cpattern} {\tt ]} -\end{quote} -The behavior of {\tt appcontext} is the same as the one of {\tt - context}, except that a matching subterm could be a partial -part of a longer application. For instance, in {\tt (f 1 2)}, -an {\tt appcontext [f ?x]} will find the matching subterm {\tt (f 1)}. - -\end{Variants} - -\subsubsection[Pattern matching on goals]{Pattern matching on goals\index{Ltac!match goal} -\index{Ltac!match reverse goal} -\index{match goal!in Ltac} -\index{match reverse goal!in Ltac}} - -We can make pattern matching on goals using the following expression: -\begin{quote} -\begin{tabbing} -{\tt match goal with}\\ -~~\={\tt |} $hyp_{1,1}${\tt ,}...{\tt ,}$hyp_{1,m_1}$ - ~~{\tt |-}{\cpattern}$_1${\tt =>} {\tacexpr}$_1$\\ - \>{\tt |} $hyp_{2,1}${\tt ,}...{\tt ,}$hyp_{2,m_2}$ - ~~{\tt |-}{\cpattern}$_2${\tt =>} {\tacexpr}$_2$\\ -~~...\\ - \>{\tt |} $hyp_{n,1}${\tt ,}...{\tt ,}$hyp_{n,m_n}$ - ~~{\tt |-}{\cpattern}$_n${\tt =>} {\tacexpr}$_n$\\ - \>{\tt |\_}~~~~{\tt =>} {\tacexpr}$_{n+1}$\\ -{\tt end} -\end{tabbing} -\end{quote} - -If each hypothesis pattern $hyp_{1,i}$, with $i=1,...,m_1$ -is matched (non-linear first-order unification) by an hypothesis of -the goal and if {\cpattern}$_1$ is matched by the conclusion of the -goal, then {\tacexpr}$_1$ is evaluated to $v_1$ by substituting the -pattern matching to the metavariables and the real hypothesis names -bound to the possible hypothesis names occurring in the hypothesis -patterns. If $v_1$ is a tactic value, then it is applied to the -goal. If this application fails, then another combination of -hypotheses is tried with the same proof context pattern. If there is -no other combination of hypotheses then the second proof context -pattern is tried and so on. If the next to last proof context pattern -fails then {\tacexpr}$_{n+1}$ is evaluated to $v_{n+1}$ and $v_{n+1}$ -is applied. Note also that matching against subterms (using the {\tt -context} {\ident} {\tt [} {\cpattern} {\tt ]}) is available and may -itself induce extra backtrackings. - -\ErrMsg \errindex{No matching clauses for match goal} - -No clause succeeds, i.e. all matching patterns, if any, -fail at the application of the right-hand-side. - -\medskip - -It is important to know that each hypothesis of the goal can be -matched by at most one hypothesis pattern. The order of matching is -the following: hypothesis patterns are examined from the right to the -left (i.e. $hyp_{i,m_i}$ before $hyp_{i,1}$). For each hypothesis -pattern, the goal hypothesis are matched in order (fresher hypothesis -first), but it possible to reverse this order (older first) with -the {\tt match reverse goal with} variant. - -\variant -\index{lazymatch goal!in Ltac} -\index{Ltac!lazymatch goal} -\index{lazymatch reverse goal!in Ltac} -\index{Ltac!lazymatch reverse goal} -Using {\tt lazymatch} instead of {\tt match} has an effect if the -right-hand-side of a clause returns a tactic. With {\tt match}, the -tactic is applied to the current goal (and the next clause is tried if -it fails). With {\tt lazymatch}, the tactic is directly returned as -the result of the whole {\tt lazymatch} block without being first -tried to be applied to the goal. Typically, if the {\tt lazymatch} -block is bound to some variable $x$ in a {\tt let in}, then tactic -expression gets bound to the variable $x$. - -\begin{coq_example} -Ltac test_lazy := - lazymatch goal with - | _ => idtac "here"; fail - | _ => idtac "wasn't lazy"; trivial - end. -Ltac test_eager := - match goal with - | _ => idtac "here"; fail - | _ => idtac "wasn't lazy"; trivial - end. -Goal True. -test_lazy || idtac "was lazy". -test_eager || idtac "was lazy". -\end{coq_example} - -\subsubsection[Filling a term context]{Filling a term context\index{context!in expression}} - -The following expression is not a tactic in the sense that it does not -produce subgoals but generates a term to be used in tactic -expressions: -\begin{quote} -{\tt context} {\ident} {\tt [} {\tacexpr} {\tt ]} -\end{quote} -{\ident} must denote a context variable bound by a {\tt context} -pattern of a {\tt match} expression. This expression evaluates -replaces the hole of the value of {\ident} by the value of -{\tacexpr}. - -\ErrMsg \errindex{not a context variable} - - -\subsubsection[Generating fresh hypothesis names]{Generating fresh hypothesis names\index{Ltac!fresh} -\index{fresh!in Ltac}} - -Tactics sometimes have to generate new names for hypothesis. Letting -the system decide a name with the {\tt intro} tactic is not so good -since it is very awkward to retrieve the name the system gave. -The following expression returns an identifier: -\begin{quote} -{\tt fresh} \nelist{\textrm{\textsl{component}}}{} -\end{quote} -It evaluates to an identifier unbound in the goal. This fresh -identifier is obtained by concatenating the value of the -\textrm{\textsl{component}}'s (each of them is, either an {\ident} which -has to refer to a name, or directly a name denoted by a -{\qstring}). If the resulting name is already used, it is padded -with a number so that it becomes fresh. If no component is -given, the name is a fresh derivative of the name {\tt H}. - -\subsubsection[Computing in a constr]{Computing in a constr\index{Ltac!eval} -\index{eval!in Ltac}} - -Evaluation of a term can be performed with: -\begin{quote} -{\tt eval} {\nterm{redexpr}} {\tt in} {\term} -\end{quote} -where \nterm{redexpr} is a reduction tactic among {\tt red}, {\tt -hnf}, {\tt compute}, {\tt simpl}, {\tt cbv}, {\tt lazy}, {\tt unfold}, -{\tt fold}, {\tt pattern}. - -\subsubsection{Type-checking a term} -%\tacindex{type of} -\index{Ltac!type of} -\index{type of!in Ltac} - -The following returns the type of {\term}: - -\begin{quote} -{\tt type of} {\term} -\end{quote} - -\subsubsection[Accessing tactic decomposition]{Accessing tactic decomposition\tacindex{info} -\index{Tacticals!info@{\tt info}}} - -Tactical ``{\tt info} {\tacexpr}'' is not really a tactical. For -elementary tactics, this is equivalent to \tacexpr. For complex tactic -like \texttt{auto}, it displays the operations performed by the -tactic. - -\subsubsection[Proving a subgoal as a separate lemma]{Proving a subgoal as a separate lemma\tacindex{abstract} -\index{Tacticals!abstract@{\tt abstract}}} - -From the outside ``\texttt{abstract \tacexpr}'' is the same as -{\tt solve \tacexpr}. Internally it saves an auxiliary lemma called -{\ident}\texttt{\_subproof}\textit{n} where {\ident} is the name of the -current goal and \textit{n} is chosen so that this is a fresh name. - -This tactical is useful with tactics such as \texttt{omega} or -\texttt{discriminate} that generate huge proof terms. With that tool -the user can avoid the explosion at time of the \texttt{Save} command -without having to cut manually the proof in smaller lemmas. - -\begin{Variants} -\item \texttt{abstract {\tacexpr} using {\ident}}.\\ - Give explicitly the name of the auxiliary lemma. -\end{Variants} - -\ErrMsg \errindex{Proof is not complete} - -\subsubsection[Calling an external tactic]{Calling an external tactic\index{Ltac!external}} - -The tactic {\tt external} allows to run an executable outside the -{\Coq} executable. The communication is done via an XML encoding of -constructions. The syntax of the command is - -\begin{quote} -{\tt external} "\textsl{command}" "\textsl{request}" \nelist{\tacarg}{} -\end{quote} - -The string \textsl{command}, to be interpreted in the default -execution path of the operating system, is the name of the external -command. The string \textsl{request} is the name of a request to be -sent to the external command. Finally the list of tactic arguments -have to evaluate to terms. An XML tree of the following form is sent -to the standard input of the external command. -\medskip - -\begin{tabular}{l} -\texttt{<REQUEST req="}\textsl{request}\texttt{">}\\ -the XML tree of the first argument\\ -{\ldots}\\ -the XML tree of the last argument\\ -\texttt{</REQUEST>}\\ -\end{tabular} -\medskip - -Conversely, the external command must send on its standard output an -XML tree of the following forms: - -\medskip -\begin{tabular}{l} -\texttt{<TERM>}\\ -the XML tree of a term\\ -\texttt{</TERM>}\\ -\end{tabular} -\medskip - -\noindent or - -\medskip -\begin{tabular}{l} -\texttt{<CALL uri="}\textsl{ltac\_qualified\_ident}\texttt{">}\\ -the XML tree of a first argument\\ -{\ldots}\\ -the XML tree of a last argument\\ -\texttt{</CALL>}\\ -\end{tabular} - -\medskip -\noindent where \textsl{ltac\_qualified\_ident} is the name of a -defined {\ltac} function and each subsequent XML tree is recursively a -\texttt{CALL} or a \texttt{TERM} node. - -The Document Type Definition (DTD) for terms of the Calculus of -Inductive Constructions is the one developed as part of the MoWGLI -European project. It can be found in the file {\tt dev/doc/cic.dtd} of -the {\Coq} source archive. - -An example of parser for this DTD, written in the Objective Caml - -Camlp4 language, can be found in the file {\tt parsing/g\_xml.ml4} of -the {\Coq} source archive. - -\section[Tactic toplevel definitions]{Tactic toplevel definitions\comindex{Ltac}} - -\subsection{Defining {\ltac} functions} - -Basically, {\ltac} toplevel definitions are made as follows: -%{\tt Tactic Definition} {\ident} {\tt :=} {\tacexpr}\\ -% -%{\tacexpr} is evaluated to $v$ and $v$ is associated to {\ident}. Next, every -%script is evaluated by substituting $v$ to {\ident}. -% -%We can define functional definitions by:\\ -\begin{quote} -{\tt Ltac} {\ident} {\ident}$_1$ ... {\ident}$_n$ {\tt :=} -{\tacexpr} -\end{quote} -This defines a new {\ltac} function that can be used in any tactic -script or new {\ltac} toplevel definition. - -\Rem The preceding definition can equivalently be written: -\begin{quote} -{\tt Ltac} {\ident} {\tt := fun} {\ident}$_1$ ... {\ident}$_n$ -{\tt =>} {\tacexpr} -\end{quote} -Recursive and mutual recursive function definitions are also -possible with the syntax: -\begin{quote} -{\tt Ltac} {\ident}$_1$ {\ident}$_{1,1}$ ... -{\ident}$_{1,m_1}$~~{\tt :=} {\tacexpr}$_1$\\ -{\tt with} {\ident}$_2$ {\ident}$_{2,1}$ ... {\ident}$_{2,m_2}$~~{\tt :=} -{\tacexpr}$_2$\\ -...\\ -{\tt with} {\ident}$_n$ {\ident}$_{n,1}$ ... {\ident}$_{n,m_n}$~~{\tt :=} -{\tacexpr}$_n$ -\end{quote} -\medskip -It is also possible to \emph{redefine} an existing user-defined tactic -using the syntax: -\begin{quote} -{\tt Ltac} {\qualid} {\ident}$_1$ ... {\ident}$_n$ {\tt ::=} -{\tacexpr} -\end{quote} -A previous definition of \qualid must exist in the environment. -The new definition will always be used instead of the old one and -it goes accross module boundaries. - -If preceded by the keyword {\tt Local} the tactic definition will not -be exported outside the current module. - -\subsection[Printing {\ltac} tactics]{Printing {\ltac} tactics\comindex{Print Ltac}} - -Defined {\ltac} functions can be displayed using the command - -\begin{quote} -{\tt Print Ltac {\qualid}.} -\end{quote} - -\section[Debugging {\ltac} tactics]{Debugging {\ltac} tactics\comindex{Set Ltac Debug} -\comindex{Unset Ltac Debug} -\comindex{Test Ltac Debug}} - -The {\ltac} interpreter comes with a step-by-step debugger. The -debugger can be activated using the command - -\begin{quote} -{\tt Set Ltac Debug.} -\end{quote} - -\noindent and deactivated using the command - -\begin{quote} -{\tt Unset Ltac Debug.} -\end{quote} - -To know if the debugger is on, use the command \texttt{Test Ltac Debug}. -When the debugger is activated, it stops at every step of the -evaluation of the current {\ltac} expression and it prints information -on what it is doing. The debugger stops, prompting for a command which -can be one of the following: - -\medskip -\begin{tabular}{ll} -simple newline: & go to the next step\\ -h: & get help\\ -x: & exit current evaluation\\ -s: & continue current evaluation without stopping\\ -r$n$: & advance $n$ steps further\\ -\end{tabular} -\endinput - -\subsection{Permutation on closed lists} - -\begin{figure}[b] -\begin{center} -\fbox{\begin{minipage}{0.95\textwidth} -\begin{coq_example*} -Require Import List. -Section Sort. -Variable A : Set. -Inductive permut : list A -> list A -> Prop := - | permut_refl : forall l, permut l l - | permut_cons : - forall a l0 l1, permut l0 l1 -> permut (a :: l0) (a :: l1) - | permut_append : forall a l, permut (a :: l) (l ++ a :: nil) - | permut_trans : - forall l0 l1 l2, permut l0 l1 -> permut l1 l2 -> permut l0 l2. -End Sort. -\end{coq_example*} -\end{center} -\caption{Definition of the permutation predicate} -\label{permutpred} -\end{figure} - - -Another more complex example is the problem of permutation on closed -lists. The aim is to show that a closed list is a permutation of -another one. First, we define the permutation predicate as shown on -Figure~\ref{permutpred}. - -\begin{figure}[p] -\begin{center} -\fbox{\begin{minipage}{0.95\textwidth} -\begin{coq_example} -Ltac Permut n := - match goal with - | |- (permut _ ?l ?l) => apply permut_refl - | |- (permut _ (?a :: ?l1) (?a :: ?l2)) => - let newn := eval compute in (length l1) in - (apply permut_cons; Permut newn) - | |- (permut ?A (?a :: ?l1) ?l2) => - match eval compute in n with - | 1 => fail - | _ => - let l1' := constr:(l1 ++ a :: nil) in - (apply (permut_trans A (a :: l1) l1' l2); - [ apply permut_append | compute; Permut (pred n) ]) - end - end. -Ltac PermutProve := - match goal with - | |- (permut _ ?l1 ?l2) => - match eval compute in (length l1 = length l2) with - | (?n = ?n) => Permut n - end - end. -\end{coq_example} -\end{minipage}} -\end{center} -\caption{Permutation tactic} -\label{permutltac} -\end{figure} - -\begin{figure}[p] -\begin{center} -\fbox{\begin{minipage}{0.95\textwidth} -\begin{coq_example*} -Lemma permut_ex1 : - permut nat (1 :: 2 :: 3 :: nil) (3 :: 2 :: 1 :: nil). -Proof. -PermutProve. -Qed. - -Lemma permut_ex2 : - permut nat - (0 :: 1 :: 2 :: 3 :: 4 :: 5 :: 6 :: 7 :: 8 :: 9 :: nil) - (0 :: 2 :: 4 :: 6 :: 8 :: 9 :: 7 :: 5 :: 3 :: 1 :: nil). -Proof. -PermutProve. -Qed. -\end{coq_example*} -\end{minipage}} -\end{center} -\caption{Examples of {\tt PermutProve} use} -\label{permutlem} -\end{figure} - -Next, we can write naturally the tactic and the result can be seen on -Figure~\ref{permutltac}. We can notice that we use two toplevel -definitions {\tt PermutProve} and {\tt Permut}. The function to be -called is {\tt PermutProve} which computes the lengths of the two -lists and calls {\tt Permut} with the length if the two lists have the -same length. {\tt Permut} works as expected. If the two lists are -equal, it concludes. Otherwise, if the lists have identical first -elements, it applies {\tt Permut} on the tail of the lists. Finally, -if the lists have different first elements, it puts the first element -of one of the lists (here the second one which appears in the {\tt - permut} predicate) at the end if that is possible, i.e., if the new -first element has been at this place previously. To verify that all -rotations have been done for a list, we use the length of the list as -an argument for {\tt Permut} and this length is decremented for each -rotation down to, but not including, 1 because for a list of length -$n$, we can make exactly $n-1$ rotations to generate at most $n$ -distinct lists. Here, it must be noticed that we use the natural -numbers of {\Coq} for the rotation counter. On Figure~\ref{ltac}, we -can see that it is possible to use usual natural numbers but they are -only used as arguments for primitive tactics and they cannot be -handled, in particular, we cannot make computations with them. So, a -natural choice is to use {\Coq} data structures so that {\Coq} makes -the computations (reductions) by {\tt eval compute in} and we can get -the terms back by {\tt match}. - -With {\tt PermutProve}, we can now prove lemmas such those shown on -Figure~\ref{permutlem}. - - -\subsection{Deciding intuitionistic propositional logic} - -\begin{figure}[tbp] -\begin{center} -\fbox{\begin{minipage}{0.95\textwidth} -\begin{coq_example} -Ltac Axioms := - match goal with - | |- True => trivial - | _:False |- _ => elimtype False; assumption - | _:?A |- ?A => auto - end. -Ltac DSimplif := - repeat - (intros; - match goal with - | id:(~ _) |- _ => red in id - | id:(_ /\ _) |- _ => - elim id; do 2 intro; clear id - | id:(_ \/ _) |- _ => - elim id; intro; clear id - | id:(?A /\ ?B -> ?C) |- _ => - cut (A -> B -> C); - [ intro | intros; apply id; split; assumption ] - | id:(?A \/ ?B -> ?C) |- _ => - cut (B -> C); - [ cut (A -> C); - [ intros; clear id - | intro; apply id; left; assumption ] - | intro; apply id; right; assumption ] - | id0:(?A -> ?B),id1:?A |- _ => - cut B; [ intro; clear id0 | apply id0; assumption ] - | |- (_ /\ _) => split - | |- (~ _) => red - end). -\end{coq_example} -\end{minipage}} -\end{center} -\caption{Deciding intuitionistic propositions (1)} -\label{tautoltaca} -\end{figure} - -\begin{figure} -\begin{center} -\fbox{\begin{minipage}{0.95\textwidth} -\begin{coq_example} -Ltac TautoProp := - DSimplif; - Axioms || - match goal with - | id:((?A -> ?B) -> ?C) |- _ => - cut (B -> C); - [ intro; cut (A -> B); - [ intro; cut C; - [ intro; clear id | apply id; assumption ] - | clear id ] - | intro; apply id; intro; assumption ]; TautoProp - | id:(~ ?A -> ?B) |- _ => - cut (False -> B); - [ intro; cut (A -> False); - [ intro; cut B; - [ intro; clear id | apply id; assumption ] - | clear id ] - | intro; apply id; red; intro; assumption ]; TautoProp - | |- (_ \/ _) => (left; TautoProp) || (right; TautoProp) - end. -\end{coq_example} -\end{minipage}} -\end{center} -\caption{Deciding intuitionistic propositions (2)} -\label{tautoltacb} -\end{figure} - -The pattern matching on goals allows a complete and so a powerful -backtracking when returning tactic values. An interesting application -is the problem of deciding intuitionistic propositional logic. -Considering the contraction-free sequent calculi {\tt LJT*} of -Roy~Dyckhoff (\cite{Dyc92}), it is quite natural to code such a tactic -using the tactic language. On Figure~\ref{tautoltaca}, the tactic {\tt - Axioms} tries to conclude using usual axioms. The {\tt DSimplif} -tactic applies all the reversible rules of Dyckhoff's system. -Finally, on Figure~\ref{tautoltacb}, the {\tt TautoProp} tactic (the -main tactic to be called) simplifies with {\tt DSimplif}, tries to -conclude with {\tt Axioms} and tries several paths using the -backtracking rules (one of the four Dyckhoff's rules for the left -implication to get rid of the contraction and the right or). - -\begin{figure}[tb] -\begin{center} -\fbox{\begin{minipage}{0.95\textwidth} -\begin{coq_example*} -Lemma tauto_ex1 : forall A B:Prop, A /\ B -> A \/ B. -Proof. -TautoProp. -Qed. - -Lemma tauto_ex2 : - forall A B:Prop, (~ ~ B -> B) -> (A -> B) -> ~ ~ A -> B. -Proof. -TautoProp. -Qed. -\end{coq_example*} -\end{minipage}} -\end{center} -\caption{Proofs of tautologies with {\tt TautoProp}} -\label{tautolem} -\end{figure} - -For example, with {\tt TautoProp}, we can prove tautologies like those of -Figure~\ref{tautolem}. - - -\subsection{Deciding type isomorphisms} - -A more tricky problem is to decide equalities between types and modulo -isomorphisms. Here, we choose to use the isomorphisms of the simply typed -$\lb{}$-calculus with Cartesian product and $unit$ type (see, for example, -\cite{RC95}). The axioms of this $\lb{}$-calculus are given by -Figure~\ref{isosax}. - -\begin{figure} -\begin{center} -\fbox{\begin{minipage}{0.95\textwidth} -\begin{coq_eval} -Reset Initial. -\end{coq_eval} -\begin{coq_example*} -Open Scope type_scope. -Section Iso_axioms. -Variables A B C : Set. -Axiom Com : A * B = B * A. -Axiom Ass : A * (B * C) = A * B * C. -Axiom Cur : (A * B -> C) = (A -> B -> C). -Axiom Dis : (A -> B * C) = (A -> B) * (A -> C). -Axiom P_unit : A * unit = A. -Axiom AR_unit : (A -> unit) = unit. -Axiom AL_unit : (unit -> A) = A. -Lemma Cons : B = C -> A * B = A * C. -Proof. -intro Heq; rewrite Heq; apply refl_equal. -Qed. -End Iso_axioms. -\end{coq_example*} -\end{minipage}} -\end{center} -\caption{Type isomorphism axioms} -\label{isosax} -\end{figure} - -The tactic to judge equalities modulo this axiomatization can be written as -shown on Figures~\ref{isosltac1} and~\ref{isosltac2}. The algorithm is quite -simple. Types are reduced using axioms that can be oriented (this done by {\tt -MainSimplif}). The normal forms are sequences of Cartesian -products without Cartesian product in the left component. These normal forms -are then compared modulo permutation of the components (this is done by {\tt -CompareStruct}). The main tactic to be called and realizing this algorithm is -{\tt IsoProve}. - -\begin{figure} -\begin{center} -\fbox{\begin{minipage}{0.95\textwidth} -\begin{coq_example} -Ltac DSimplif trm := - match trm with - | (?A * ?B * ?C) => - rewrite <- (Ass A B C); try MainSimplif - | (?A * ?B -> ?C) => - rewrite (Cur A B C); try MainSimplif - | (?A -> ?B * ?C) => - rewrite (Dis A B C); try MainSimplif - | (?A * unit) => - rewrite (P_unit A); try MainSimplif - | (unit * ?B) => - rewrite (Com unit B); try MainSimplif - | (?A -> unit) => - rewrite (AR_unit A); try MainSimplif - | (unit -> ?B) => - rewrite (AL_unit B); try MainSimplif - | (?A * ?B) => - (DSimplif A; try MainSimplif) || (DSimplif B; try MainSimplif) - | (?A -> ?B) => - (DSimplif A; try MainSimplif) || (DSimplif B; try MainSimplif) - end - with MainSimplif := - match goal with - | |- (?A = ?B) => try DSimplif A; try DSimplif B - end. -Ltac Length trm := - match trm with - | (_ * ?B) => let succ := Length B in constr:(S succ) - | _ => constr:1 - end. -Ltac assoc := repeat rewrite <- Ass. -\end{coq_example} -\end{minipage}} -\end{center} -\caption{Type isomorphism tactic (1)} -\label{isosltac1} -\end{figure} - -\begin{figure} -\begin{center} -\fbox{\begin{minipage}{0.95\textwidth} -\begin{coq_example} -Ltac DoCompare n := - match goal with - | [ |- (?A = ?A) ] => apply refl_equal - | [ |- (?A * ?B = ?A * ?C) ] => - apply Cons; let newn := Length B in DoCompare newn - | [ |- (?A * ?B = ?C) ] => - match eval compute in n with - | 1 => fail - | _ => - pattern (A * B) at 1; rewrite Com; assoc; DoCompare (pred n) - end - end. -Ltac CompareStruct := - match goal with - | [ |- (?A = ?B) ] => - let l1 := Length A - with l2 := Length B in - match eval compute in (l1 = l2) with - | (?n = ?n) => DoCompare n - end - end. -Ltac IsoProve := MainSimplif; CompareStruct. -\end{coq_example} -\end{minipage}} -\end{center} -\caption{Type isomorphism tactic (2)} -\label{isosltac2} -\end{figure} - -Figure~\ref{isoslem} gives examples of what can be solved by {\tt IsoProve}. - -\begin{figure} -\begin{center} -\fbox{\begin{minipage}{0.95\textwidth} -\begin{coq_example*} -Lemma isos_ex1 : - forall A B:Set, A * unit * B = B * (unit * A). -Proof. -intros; IsoProve. -Qed. - -Lemma isos_ex2 : - forall A B C:Set, - (A * unit -> B * (C * unit)) = - (A * unit -> (C -> unit) * C) * (unit -> A -> B). -Proof. -intros; IsoProve. -Qed. -\end{coq_example*} -\end{minipage}} -\end{center} -\caption{Type equalities solved by {\tt IsoProve}} -\label{isoslem} -\end{figure} - -%%% Local Variables: -%%% mode: latex -%%% TeX-master: "Reference-Manual" -%%% End: diff --git a/doc/refman/RefMan-mod.tex b/doc/refman/RefMan-mod.tex deleted file mode 100644 index 68d57226..00000000 --- a/doc/refman/RefMan-mod.tex +++ /dev/null @@ -1,411 +0,0 @@ -\section{Module system -\index{Modules} -\label{section:Modules}} - -The module system provides a way of packaging related elements -together, as well as a mean of massive abstraction. - -\begin{figure}[t] -\begin{centerframe} -\begin{tabular}{rcl} -{\modtype} & ::= & {\qualid} \\ - & $|$ & {\modtype} \texttt{ with Definition }{\qualid} := {\term} \\ - & $|$ & {\modtype} \texttt{ with Module }{\qualid} := {\qualid} \\ - & $|$ & {\qualid} \nelist{\qualid}{}\\ - & $|$ & $!${\qualid} \nelist{\qualid}{}\\ - &&\\ - -{\onemodbinding} & ::= & {\tt ( [Import|Export] \nelist{\ident}{} : {\modtype} )}\\ - &&\\ - -{\modbindings} & ::= & \nelist{\onemodbinding}{}\\ - &&\\ - -{\modexpr} & ::= & \nelist{\qualid}{} \\ - & $|$ & $!$\nelist{\qualid}{} -\end{tabular} -\end{centerframe} -\caption{Syntax of modules} -\end{figure} - -In the syntax of module application, the $!$ prefix indicates that -any {\tt Inline} directive in the type of the functor arguments -will be ignored (see \ref{Inline} below). - -\subsection{\tt Module {\ident} -\comindex{Module}} - -This command is used to start an interactive module named {\ident}. - -\begin{Variants} - -\item{\tt Module {\ident} {\modbindings}} - - Starts an interactive functor with parameters given by {\modbindings}. - -\item{\tt Module {\ident} \verb.:. {\modtype}} - - Starts an interactive module specifying its module type. - -\item{\tt Module {\ident} {\modbindings} \verb.:. {\modtype}} - - Starts an interactive functor with parameters given by - {\modbindings}, and output module type {\modtype}. - -\item{\tt Module {\ident} \verb.<:. {\modtype$_1$} \verb.<:. $\ldots$ \verb.<:.{ \modtype$_n$}} - - Starts an interactive module satisfying each {\modtype$_i$}. - -\item{\tt Module {\ident} {\modbindings} \verb.<:. {\modtype$_1$} \verb.<:. $\ldots$ \verb.<:. {\modtype$_n$}} - - Starts an interactive functor with parameters given by - {\modbindings}. The output module type is verified against each - module type {\modtype$_i$}. - -\item\texttt{Module [Import|Export]} - - Behaves like \texttt{Module}, but automatically imports or exports - the module. - -\end{Variants} -\subsubsection{Reserved commands inside an interactive module: -\comindex{Include}} -\begin{enumerate} -\item {\tt Include {\module}} - - Includes the content of {\module} in the current interactive - module. Here {\module} can be a module expresssion or a module type - expression. If {\module} is a high-order module or module type - expression then the system tries to instanciate {\module} - by the current interactive module. - -\item {\tt Include {\module$_1$} \verb.<+. $\ldots$ \verb.<+. {\module$_n$}} - -is a shortcut for {\tt Include {\module$_1$}} $\ldots$ {\tt Include {\module$_n$}} -\end{enumerate} -\subsection{\tt End {\ident} -\comindex{End}} - -This command closes the interactive module {\ident}. If the module type -was given the content of the module is matched against it and an error -is signaled if the matching fails. If the module is basic (is not a -functor) its components (constants, inductive types, submodules etc) are -now available through the dot notation. - -\begin{ErrMsgs} -\item \errindex{No such label {\ident}} -\item \errindex{Signature components for label {\ident} do not match} -\item \errindex{This is not the last opened module} -\end{ErrMsgs} - - -\subsection{\tt Module {\ident} := {\modexpr} -\comindex{Module}} - -This command defines the module identifier {\ident} to be equal to -{\modexpr}. - -\begin{Variants} -\item{\tt Module {\ident} {\modbindings} := {\modexpr}} - - Defines a functor with parameters given by {\modbindings} and body {\modexpr}. - -% Particular cases of the next 2 items -%\item{\tt Module {\ident} \verb.:. {\modtype} := {\modexpr}} -% -% Defines a module with body {\modexpr} and interface {\modtype}. -%\item{\tt Module {\ident} \verb.<:. {\modtype} := {\modexpr}} -% -% Defines a module with body {\modexpr}, satisfying {\modtype}. - -\item{\tt Module {\ident} {\modbindings} \verb.:. {\modtype} := - {\modexpr}} - - Defines a functor with parameters given by {\modbindings} (possibly none), - and output module type {\modtype}, with body {\modexpr}. - -\item{\tt Module {\ident} {\modbindings} \verb.<:. {\modtype$_1$} \verb.<:. $\ldots$ \verb.<:. {\modtype$_n$}:= - {\modexpr}} - - Defines a functor with parameters given by {\modbindings} (possibly none) - with body {\modexpr}. The body is checked against each {\modtype$_i$}. - -\item{\tt Module {\ident} {\modbindings} := {\modexpr$_1$} \verb.<+. $\ldots$ \verb.<+. {\modexpr$_n$}} - - is equivalent to an interactive module where each {\modexpr$_i$} are included. - -\end{Variants} - -\subsection{\tt Module Type {\ident} -\comindex{Module Type}} - -This command is used to start an interactive module type {\ident}. - -\begin{Variants} - -\item{\tt Module Type {\ident} {\modbindings}} - - Starts an interactive functor type with parameters given by {\modbindings}. - -\end{Variants} -\subsubsection{Reserved commands inside an interactive module type: -\comindex{Include}\comindex{Inline}} -\label{Inline} -\begin{enumerate} -\item {\tt Include {\module}} - - Same as {\tt Include} inside a module. - -\item {\tt Include {\module$_1$} \verb.<+. $\ldots$ \verb.<+. {\module$_n$}} - -is a shortcut for {\tt Include {\module$_1$}} $\ldots$ {\tt Include {\module$_n$}} - -\item {\tt {\assumptionkeyword} Inline {\assums} } - - The instance of this assumption will be automatically expanded at functor - application, except when this functor application is prefixed by a $!$ annotation. -\end{enumerate} -\subsection{\tt End {\ident} -\comindex{End}} - -This command closes the interactive module type {\ident}. - -\begin{ErrMsgs} -\item \errindex{This is not the last opened module type} -\end{ErrMsgs} - -\subsection{\tt Module Type {\ident} := {\modtype}} - -Defines a module type {\ident} equal to {\modtype}. - -\begin{Variants} -\item {\tt Module Type {\ident} {\modbindings} := {\modtype}} - - Defines a functor type {\ident} specifying functors taking arguments - {\modbindings} and returning {\modtype}. - -\item{\tt Module Type {\ident} {\modbindings} := {\modtype$_1$} \verb.<+. $\ldots$ \verb.<+. {\modtype$_n$}} - - is equivalent to an interactive module type were each {\modtype$_i$} are included. - -\end{Variants} - -\subsection{\tt Declare Module {\ident} : {\modtype}} - -Declares a module {\ident} of type {\modtype}. - -\begin{Variants} - -\item{\tt Declare Module {\ident} {\modbindings} \verb.:. {\modtype}} - - Declares a functor with parameters {\modbindings} and output module - type {\modtype}. - - -\end{Variants} - - -\subsubsection{Example} - -Let us define a simple module. -\begin{coq_example} -Module M. - Definition T := nat. - Definition x := 0. - Definition y : bool. - exact true. - Defined. -End M. -\end{coq_example} -Inside a module one can define constants, prove theorems and do any -other things that can be done in the toplevel. Components of a closed -module can be accessed using the dot notation: -\begin{coq_example} -Print M.x. -\end{coq_example} -A simple module type: -\begin{coq_example} -Module Type SIG. - Parameter T : Set. - Parameter x : T. -End SIG. -\end{coq_example} - -Now we can create a new module from \texttt{M}, giving it a less -precise specification: the \texttt{y} component is dropped as well -as the body of \texttt{x}. - -\begin{coq_eval} -Set Printing Depth 50. -(********** The following is not correct and should produce **********) -(***************** Error: N.y not a defined object *******************) -\end{coq_eval} -\begin{coq_example} -Module N : SIG with Definition T := nat := M. -Print N.T. -Print N.x. -Print N.y. -\end{coq_example} -\begin{coq_eval} -Reset N. -\end{coq_eval} - -\noindent -The definition of \texttt{N} using the module type expression -\texttt{SIG with Definition T:=nat} is equivalent to the following -one: - -\begin{coq_example*} -Module Type SIG'. - Definition T : Set := nat. - Parameter x : T. -End SIG'. -Module N : SIG' := M. -\end{coq_example*} -If we just want to be sure that the our implementation satisfies a -given module type without restricting the interface, we can use a -transparent constraint -\begin{coq_example} -Module P <: SIG := M. -Print P.y. -\end{coq_example} -Now let us create a functor, i.e. a parametric module -\begin{coq_example} -Module Two (X Y: SIG). -\end{coq_example} -\begin{coq_example*} - Definition T := (X.T * Y.T)%type. - Definition x := (X.x, Y.x). -\end{coq_example*} -\begin{coq_example} -End Two. -\end{coq_example} -and apply it to our modules and do some computations -\begin{coq_example} -Module Q := Two M N. -Eval compute in (fst Q.x + snd Q.x). -\end{coq_example} -In the end, let us define a module type with two sub-modules, sharing -some of the fields and give one of its possible implementations: -\begin{coq_example} -Module Type SIG2. - Declare Module M1 : SIG. - Module M2 <: SIG. - Definition T := M1.T. - Parameter x : T. - End M2. -End SIG2. -\end{coq_example} -\begin{coq_example*} -Module Mod <: SIG2. - Module M1. - Definition T := nat. - Definition x := 1. - End M1. - Module M2 := M. -\end{coq_example*} -\begin{coq_example} -End Mod. -\end{coq_example} -Notice that \texttt{M} is a correct body for the component \texttt{M2} -since its \texttt{T} component is equal \texttt{nat} and hence -\texttt{M1.T} as specified. -\begin{coq_eval} -Reset Initial. -\end{coq_eval} - -\begin{Remarks} -\item Modules and module types can be nested components of each other. -\item One can have sections inside a module or a module type, but - not a module or a module type inside a section. -\item Commands like \texttt{Hint} or \texttt{Notation} can - also appear inside modules and module types. Note that in case of a - module definition like: - - \smallskip - \noindent - {\tt Module N : SIG := M.} - \smallskip - - or - - \smallskip - {\tt Module N : SIG.\\ - \ \ \dots\\ - End N.} - \smallskip - - hints and the like valid for \texttt{N} are not those defined in - \texttt{M} (or the module body) but the ones defined in - \texttt{SIG}. - -\end{Remarks} - -\subsection{\tt Import {\qualid} -\comindex{Import} -\label{Import}} - -If {\qualid} denotes a valid basic module (i.e. its module type is a -signature), makes its components available by their short names. - -Example: - -\begin{coq_example} -Module Mod. -\end{coq_example} -\begin{coq_example} - Definition T:=nat. - Check T. -\end{coq_example} -\begin{coq_example} -End Mod. -Check Mod.T. -Check T. (* Incorrect ! *) -Import Mod. -Check T. (* Now correct *) -\end{coq_example} -\begin{coq_eval} -Reset Mod. -\end{coq_eval} - -Some features defined in modules are activated only when a module is -imported. This is for instance the case of notations (see -Section~\ref{Notation}). - -\begin{Variants} -\item{\tt Export {\qualid}}\comindex{Export} - - When the module containing the command {\tt Export {\qualid}} is - imported, {\qualid} is imported as well. -\end{Variants} - -\begin{ErrMsgs} - \item \errindexbis{{\qualid} is not a module}{is not a module} -% this error is impossible in the import command -% \item \errindex{Cannot mask the absolute name {\qualid} !} -\end{ErrMsgs} - -\begin{Warnings} - \item Warning: Trying to mask the absolute name {\qualid} ! -\end{Warnings} - -\subsection{\tt Print Module {\ident} -\comindex{Print Module}} - -Prints the module type and (optionally) the body of the module {\ident}. - -\subsection{\tt Print Module Type {\ident} -\comindex{Print Module Type}} - -Prints the module type corresponding to {\ident}. - -\subsection{\tt Locate Module {\qualid} -\comindex{Locate Module}} - -Prints the full name of the module {\qualid}. - - -%%% Local Variables: -%%% mode: latex -%%% TeX-master: "Reference-Manual" -%%% End: diff --git a/doc/refman/RefMan-modr.tex b/doc/refman/RefMan-modr.tex deleted file mode 100644 index b2ea232c..00000000 --- a/doc/refman/RefMan-modr.tex +++ /dev/null @@ -1,565 +0,0 @@ -\chapter[The Module System]{The Module System\label{chapter:Modules}} - -The module system extends the Calculus of Inductive Constructions -providing a convenient way to structure large developments as well as -a mean of massive abstraction. -%It is described in details in Judicael's thesis and Jacek's thesis - -\section{Modules and module types} - -\paragraph{Access path.} It is denoted by $p$, it can be either a module -variable $X$ or, if $p'$ is an access path and $id$ an identifier, then -$p'.id$ is an access path. - -\paragraph{Structure element.} It is denoted by \elem\ and is either a -definition of a constant, an assumption, a definition of an inductive, - a definition of a module, an alias of module or a module type abbreviation. - -\paragraph{Structure expression.} It is denoted by $S$ and can be: -\begin{itemize} -\item an access path $p$ -\item a plain structure $\struct{\nelist{\elem}{;}}$ -\item a functor $\functor{X}{S}{S'}$, where $X$ is a module variable, - $S$ and $S'$ are structure expression -\item an application $S\,p$, where $S$ is a structure expression and $p$ -an access path -\item a refined structure $\with{S}{p}{p'}$ or $\with{S}{p}{t:T}$ where $S$ -is a structure expression, $p$ and $p'$ are access paths, $t$ is a term -and $T$ is the type of $t$. -\end{itemize} - -\paragraph{Module definition,} is written $\Mod{X}{S}{S'}$ and - consists of a module variable $X$, a module type -$S$ which can be any structure expression and optionally a module implementation $S'$ - which can be any structure expression except a refined structure. - -\paragraph{Module alias,} is written $\ModA{X}{p}$ and - consists of a module variable $X$ and a module path $p$. - -\paragraph{Module type abbreviation,} is written $\ModType{Y}{S}$, where -$Y$ is an identifier and $S$ is any structure expression . - - -\section{Typing Modules} - -In order to introduce the typing system we first slightly extend -the syntactic class of terms and environments given in -section~\ref{Terms}. The environments, apart from definitions of -constants and inductive types now also hold any other structure elements. -Terms, apart from variables, constants and complex terms, -include also access paths. - -We also need additional typing judgments: -\begin{itemize} -\item \WFT{E}{S}, denoting that a structure $S$ is well-formed, - -\item \WTM{E}{p}{S}, denoting that the module pointed by $p$ has type $S$ in -environment $E$. - -\item \WEV{E}{S}{\overline{S}}, denoting that a structure $S$ is evaluated to -a structure $\overline{S}$ in weak head normal form. - -\item \WS{E}{S_1}{S_2}, denoting that a structure $S_1$ is a subtype of a -structure $S_2$. - -\item \WS{E}{\elem_1}{\elem_2}, denoting that a structure element - $\elem_1$ is more precise that a structure element $\elem_2$. -\end{itemize} -The rules for forming structures are the following: -\begin{description} -\item[WF-STR] -\inference{% - \frac{ - \WF{E;E'}{} - }{%%%%%%%%%%%%%%%%%%%%% - \WFT{E}{\struct{E'}} - } -} -\item[WF-FUN] -\inference{% - \frac{ - \WFT{E;\ModS{X}{S}}{\overline{S'}} - }{%%%%%%%%%%%%%%%%%%%%%%%%%% - \WFT{E}{\functor{X}{S}{S'}} - } -} -\end{description} -Evaluation of structures to weak head normal form: -\begin{description} -\item[WEVAL-APP] -\inference{% - \frac{ - \begin{array}{c} - \WEV{E}{S}{\functor{X}{S_1}{S_2}}~~~~~\WEV{E}{S_1}{\overline{S_1}}\\ - \WTM{E}{p}{S_3}\qquad \WS{E}{S_3}{\overline{S_1}} - \end{array} - }{%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \WEV{E}{S\,p}{S_2\{p/X,t_1/p_1.c_1,\ldots,t_n/p_n.c_n\}} - } -} -\end{description} -In the last rule, $\{t_1/p_1.c_1,\ldots,t_n/p_n.c_n\}$ is the resulting - substitution from the inlining mechanism. We substitute in $S$ the - inlined fields $p_i.c_i$ form $\ModS{X}{S_1}$ by the corresponding delta-reduced term $t_i$ in $p$. -\begin{description} -\item[WEVAL-WITH-MOD] -\inference{% - \frac{ - \begin{array}{c} - \WEV{E}{S}{\structe{\ModS{X}{S_1}}}~~~~~\WEV{E;\elem_1;\ldots;\elem_i}{S_1}{\overline{S_1}}\\ - \WTM{E}{p}{S_2}\qquad \WS{E;\elem_1;\ldots;\elem_i}{S_2}{\overline{S_1}} - \end{array} - }{%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \begin{array}{c} - \WEVT{E}{\with{S}{x}{p}}{\structes{\ModA{X}{p}}{p/X}} - \end{array} - } -} -\item[WEVAL-WITH-MOD-REC] -\inference{% - \frac{ - \begin{array}{c} - \WEV{E}{S}{\structe{\ModS{X_1}{S_1}}}\\ - \WEV{E;\elem_1;\ldots;\elem_i}{\with{S_1}{p}{p_1}}{\overline{S_2}} - \end{array} - }{%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \begin{array}{c} - \WEVT{E}{\with{S}{X_1.p}{p_1}}{\structes{\ModS{X}{\overline{S_2}}}{p_1/X_1.p}} - \end{array} - } -} -\item[WEVAL-WITH-DEF] -\inference{% - \frac{ - \begin{array}{c} - \WEV{E}{S}{\structe{\Assum{}{c}{T_1}}}\\ - \WS{E;\elem_1;\ldots;\elem_i}{\Def{}{c}{t}{T}}{\Assum{}{c}{T_1}} - \end{array} - }{%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \begin{array}{c} - \WEVT{E}{\with{S}{c}{t:T}}{\structe{\Def{}{c}{t}{T}}} - \end{array} - } -} -\item[WEVAL-WITH-DEF-REC] -\inference{% - \frac{ - \begin{array}{c} - \WEV{E}{S}{\structe{\ModS{X_1}{S_1}}}\\ - \WEV{E;\elem_1;\ldots;\elem_i}{\with{S_1}{p}{p_1}}{\overline{S_2}} - \end{array} - }{%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \begin{array}{c} - \WEVT{E}{\with{S}{X_1.p}{t:T}}{\structe{\ModS{X}{\overline{S_2}}}} - \end{array} - } -} - -\item[WEVAL-PATH-MOD] -\inference{% - \frac{ - \begin{array}{c} - \WEV{E}{p}{\structe{ \Mod{X}{S}{S_1}}}\\ - \WEV{E;\elem_1;\ldots;\elem_i}{S}{\overline{S}} - \end{array} - }{%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \WEV{E}{p.X}{\overline{S}} - } -} -\inference{% - \frac{ - \begin{array}{c} - \WF{E}{}~~~~~~\Mod{X}{S}{S_1}\in E\\ - \WEV{E}{S}{\overline{S}} - \end{array} - }{%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \WEV{E}{X}{\overline{S}} - } -} -\item[WEVAL-PATH-ALIAS] -\inference{% - \frac{ - \begin{array}{c} - \WEV{E}{p}{\structe{\ModA{X}{p_1}}}\\ - \WEV{E;\elem_1;\ldots;\elem_i}{p_1}{\overline{S}} - \end{array} - }{%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \WEV{E}{p.X}{\overline{S}} - } -} -\inference{% - \frac{ - \begin{array}{c} - \WF{E}{}~~~~~~~\ModA{X}{p_1}\in E\\ - \WEV{E}{p_1}{\overline{S}} - \end{array} - }{%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \WEV{E}{X}{\overline{S}} - } -} -\item[WEVAL-PATH-TYPE] -\inference{% - \frac{ - \begin{array}{c} - \WEV{E}{p}{\structe{\ModType{Y}{S}}}\\ - \WEV{E;\elem_1;\ldots;\elem_i}{S}{\overline{S}} - \end{array} - }{%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \WEV{E}{p.Y}{\overline{S}} - } -} -\item[WEVAL-PATH-TYPE] -\inference{% - \frac{ - \begin{array}{c} - \WF{E}{}~~~~~~~\ModType{Y}{S}\in E\\ - \WEV{E}{S}{\overline{S}} - \end{array} - }{%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \WEV{E}{Y}{\overline{S}} - } -} -\end{description} - Rules for typing module: -\begin{description} -\item[MT-EVAL] -\inference{% - \frac{ - \WEV{E}{p}{\overline{S}} - }{%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \WTM{E}{p}{\overline{S}} - } -} -\item[MT-STR] -\inference{% - \frac{ - \WTM{E}{p}{S} - }{%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \WTM{E}{p}{S/p} - } -} -\end{description} -The last rule, called strengthening is used to make all module fields -manifestly equal to themselves. The notation $S/p$ has the following -meaning: -\begin{itemize} -\item if $S\lra\struct{\elem_1;\dots;\elem_n}$ then - $S/p=\struct{\elem_1/p;\dots;\elem_n/p}$ where $\elem/p$ is defined as - follows: - \begin{itemize} - \item $\Def{}{c}{t}{T}/p\footnote{Opaque definitions are processed as assumptions.} ~=~ \Def{}{c}{t}{T}$ - \item $\Assum{}{c}{U}/p ~=~ \Def{}{c}{p.c}{U}$ - \item $\ModS{X}{S}/p ~=~ \ModA{X}{p.X}$ - \item $\ModA{X}{p'}/p ~=~ \ModA{X}{p'}$ - \item $\Ind{}{\Gamma_P}{\Gamma_C}{\Gamma_I}/p ~=~ \Indp{}{\Gamma_P}{\Gamma_C}{\Gamma_I}{p}$ - \item $\Indpstr{}{\Gamma_P}{\Gamma_C}{\Gamma_I}{p'}{p} ~=~ \Indp{}{\Gamma_P}{\Gamma_C}{\Gamma_I}{p'}$ - \end{itemize} -\item if $S\lra\functor{X}{S'}{S''}$ then $S/p=S$ -\end{itemize} -The notation $\Indp{}{\Gamma_P}{\Gamma_C}{\Gamma_I}{p}$ denotes an -inductive definition that is definitionally equal to the inductive -definition in the module denoted by the path $p$. All rules which have -$\Ind{}{\Gamma_P}{\Gamma_C}{\Gamma_I}$ as premises are also valid for -$\Indp{}{\Gamma_P}{\Gamma_C}{\Gamma_I}{p}$. We give the formation rule -for $\Indp{}{\Gamma_P}{\Gamma_C}{\Gamma_I}{p}$ below as well as -the equality rules on inductive types and constructors. \\ - -The module subtyping rules: -\begin{description} -\item[MSUB-STR] -\inference{% - \frac{ - \begin{array}{c} - \WS{E;\elem_1;\dots;\elem_n}{\elem_{\sigma(i)}}{\elem'_i} - \textrm{ \ for } i=1..m \\ - \sigma : \{1\dots m\} \ra \{1\dots n\} \textrm{ \ injective} - \end{array} - }{ - \WS{E}{\struct{\elem_1;\dots;\elem_n}}{\struct{\elem'_1;\dots;\elem'_m}} - } -} -\item[MSUB-FUN] -\inference{% T_1 -> T_2 <: T_1' -> T_2' - \frac{ - \WS{E}{\overline{S_1'}}{\overline{S_1}}~~~~~~~~~~\WS{E;\ModS{X}{S_1'}}{\overline{S_2}}{\overline{S_2'}} - }{%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \WS{E}{\functor{X}{S_1}{S_2}}{\functor{X}{S_1'}{S_2'}} - } -} -% these are derived rules -% \item[MSUB-EQ] -% \inference{% -% \frac{ -% \WS{E}{T_1}{T_2}~~~~~~~~~~\WTERED{}{T_1}{=}{T_1'}~~~~~~~~~~\WTERED{}{T_2}{=}{T_2'} -% }{%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -% \WS{E}{T_1'}{T_2'} -% } -% } -% \item[MSUB-REFL] -% \inference{% -% \frac{ -% \WFT{E}{T} -% }{ -% \WS{E}{T}{T} -% } -% } -\end{description} -Structure element subtyping rules: -\begin{description} -\item[ASSUM-ASSUM] -\inference{% - \frac{ - \WTELECONV{}{T_1}{T_2} - }{ - \WSE{\Assum{}{c}{T_1}}{\Assum{}{c}{T_2}} - } -} -\item[DEF-ASSUM] -\inference{% - \frac{ - \WTELECONV{}{T_1}{T_2} - }{ - \WSE{\Def{}{c}{t}{T_1}}{\Assum{}{c}{T_2}} - } -} -\item[ASSUM-DEF] -\inference{% - \frac{ - \WTELECONV{}{T_1}{T_2}~~~~~~~~\WTECONV{}{c}{t_2} - }{ - \WSE{\Assum{}{c}{T_1}}{\Def{}{c}{t_2}{T_2}} - } -} -\item[DEF-DEF] -\inference{% - \frac{ - \WTELECONV{}{T_1}{T_2}~~~~~~~~\WTECONV{}{t_1}{t_2} - }{ - \WSE{\Def{}{c}{t_1}{T_1}}{\Def{}{c}{t_2}{T_2}} - } -} -\item[IND-IND] -\inference{% - \frac{ - \WTECONV{}{\Gamma_P}{\Gamma_P'}% - ~~~~~~~~\WTECONV{\Gamma_P}{\Gamma_C}{\Gamma_C'}% - ~~~~~~~~\WTECONV{\Gamma_P;\Gamma_C}{\Gamma_I}{\Gamma_I'}% - }{ - \WSE{\Ind{}{\Gamma_P}{\Gamma_C}{\Gamma_I}}% - {\Ind{}{\Gamma_P'}{\Gamma_C'}{\Gamma_I'}} - } -} -\item[INDP-IND] -\inference{% - \frac{ - \WTECONV{}{\Gamma_P}{\Gamma_P'}% - ~~~~~~~~\WTECONV{\Gamma_P}{\Gamma_C}{\Gamma_C'}% - ~~~~~~~~\WTECONV{\Gamma_P;\Gamma_C}{\Gamma_I}{\Gamma_I'}% - }{ - \WSE{\Indp{}{\Gamma_P}{\Gamma_C}{\Gamma_I}{p}}% - {\Ind{}{\Gamma_P'}{\Gamma_C'}{\Gamma_I'}} - } -} -\item[INDP-INDP] -\inference{% - \frac{ - \WTECONV{}{\Gamma_P}{\Gamma_P'}% - ~~~~~~\WTECONV{\Gamma_P}{\Gamma_C}{\Gamma_C'}% - ~~~~~~\WTECONV{\Gamma_P;\Gamma_C}{\Gamma_I}{\Gamma_I'}% - ~~~~~~\WTECONV{}{p}{p'} - }{ - \WSE{\Indp{}{\Gamma_P}{\Gamma_C}{\Gamma_I}{p}}% - {\Indp{}{\Gamma_P'}{\Gamma_C'}{\Gamma_I'}{p'}} - } -} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\item[MOD-MOD] -\inference{% - \frac{ - \WSE{S_1}{S_2} - }{ - \WSE{\ModS{X}{S_1}}{\ModS{X}{S_2}} - } -} -\item[ALIAS-MOD] -\inference{% - \frac{ - \WTM{E}{p}{S_1}~~~~~~~~\WSE{S_1}{S_2} - }{ - \WSE{\ModA{X}{p}}{\ModS{X}{S_2}} - } -} -\item[MOD-ALIAS] -\inference{% - \frac{ - \WTM{E}{p}{S_2}~~~~~~~~ - \WSE{S_1}{S_2}~~~~~~~~\WTECONV{}{X}{p} - }{ - \WSE{\ModS{X}{S_1}}{\ModA{X}{p}} - } -} -\item[ALIAS-ALIAS] -\inference{% - \frac{ - \WTECONV{}{p_1}{p_2} - }{ - \WSE{\ModA{X}{p_1}}{\ModA{X}{p_2}} - } -} -\item[MODTYPE-MODTYPE] -\inference{% - \frac{ - \WSE{S_1}{S_2}~~~~~~~~\WSE{S_2}{S_1} - }{ - \WSE{\ModType{Y}{S_1}}{\ModType{Y}{S_2}} - } -} -\end{description} -New environment formation rules -\begin{description} -\item[WF-MOD] -\inference{% - \frac{ - \WF{E}{}~~~~~~~~\WFT{E}{S} - }{ - \WF{E;\ModS{X}{S}}{} - } -} -\item[WF-MOD] -\inference{% - \frac{ -\begin{array}{c} - \WS{E}{S_2}{S_1}\\ - \WF{E}{}~~~~~\WFT{E}{S_1}~~~~~\WFT{E}{S_2} -\end{array} - }{ - \WF{E;\Mod{X}{S_1}{S_2}}{} - } -} - -\item[WF-ALIAS] -\inference{% - \frac{ - \WF{E}{}~~~~~~~~~~~\WTE{}{p}{S} - }{ - \WF{E,\ModA{X}{p}}{} - } -} -\item[WF-MODTYPE] -\inference{% - \frac{ - \WF{E}{}~~~~~~~~~~~\WFT{E}{S} - }{ - \WF{E,\ModType{Y}{S}}{} - } -} -\item[WF-IND] -\inference{% - \frac{ - \begin{array}{c} - \WF{E;\Ind{}{\Gamma_P}{\Gamma_C}{\Gamma_I}}{}\\ - \WT{E}{}{p:\struct{\elem_1;\dots;\elem_n;\Ind{}{\Gamma_P'}{\Gamma_C'}{\Gamma_I'};\dots}}\\ - \WS{E}{\Ind{}{\Gamma_P'}{\Gamma_C'}{\Gamma_I'}}{\Ind{}{\Gamma_P}{\Gamma_C}{\Gamma_I}} - \end{array} - }{%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \WF{E;\Indp{}{\Gamma_P}{\Gamma_C}{\Gamma_I}{p}}{} - } -} -\end{description} -Component access rules -\begin{description} -\item[ACC-TYPE] -\inference{% - \frac{ - \WTEG{p}{\struct{\elem_1;\dots;\elem_i;\Assum{}{c}{T};\dots}} - }{ - \WTEG{p.c}{T} - } -} -\\ -\inference{% - \frac{ - \WTEG{p}{\struct{\elem_1;\dots;\elem_i;\Def{}{c}{t}{T};\dots}} - }{ - \WTEG{p.c}{T} - } -} -\item[ACC-DELTA] -Notice that the following rule extends the delta rule defined in -section~\ref{delta} -\inference{% - \frac{ - \WTEG{p}{\struct{\elem_1;\dots;\elem_i;\Def{}{c}{t}{U};\dots}} - }{ - \WTEGRED{p.c}{\triangleright_\delta}{t} - } -} -\\ -In the rules below we assume $\Gamma_P$ is $[p_1:P_1;\ldots;p_r:P_r]$, - $\Gamma_I$ is $[I_1:A_1;\ldots;I_k:A_k]$, and $\Gamma_C$ is - $[c_1:C_1;\ldots;c_n:C_n]$ -\item[ACC-IND] -\inference{% - \frac{ - \WTEG{p}{\struct{\elem_1;\dots;\elem_i;\Ind{}{\Gamma_P}{\Gamma_C}{\Gamma_I};\dots}} - }{ - \WTEG{p.I_j}{(p_1:P_1)\ldots(p_r:P_r)A_j} - } -} -\inference{% - \frac{ - \WTEG{p}{\struct{\elem_1;\dots;\elem_i;\Ind{}{\Gamma_P}{\Gamma_C}{\Gamma_I};\dots}} - }{ - \WTEG{p.c_m}{(p_1:P_1)\ldots(p_r:P_r){C_m}{I_j}{(I_j~p_1\ldots - p_r)}_{j=1\ldots k}} - } -} -\item[ACC-INDP] -\inference{% - \frac{ - \WT{E}{}{p}{\struct{\elem_1;\dots;\elem_i;\Indp{}{\Gamma_P}{\Gamma_C}{\Gamma_I}{p'};\dots}} - }{ - \WTRED{E}{}{p.I_i}{\triangleright_\delta}{p'.I_i} - } -} -\inference{% - \frac{ - \WT{E}{}{p}{\struct{\elem_1;\dots;\elem_i;\Indp{}{\Gamma_P}{\Gamma_C}{\Gamma_I}{p'};\dots}} - }{ - \WTRED{E}{}{p.c_i}{\triangleright_\delta}{p'.c_i} - } -} - -\end{description} - -% %%% replaced by \triangle_\delta -% Module path equality is a transitive and reflexive closure of the -% relation generated by ACC-MODEQ and ENV-MODEQ. -% \begin{itemize} -% \item []MP-EQ-REFL -% \inference{% -% \frac{ -% \WTEG{p}{T} -% }{ -% \WTEG{p}{p} -% } -% } -% \item []MP-EQ-TRANS -% \inference{% -% \frac{ -% \WTEGRED{p}{=}{p'}~~~~~~\WTEGRED{p'}{=}{p''} -% }{ -% \WTEGRED{p'}{=}{p''} -% } -% } - -% \end{itemize} - - -% $Id: RefMan-modr.tex 11197 2008-07-01 13:05:41Z soubiran $ - -%%% Local Variables: -%%% mode: latex -%%% TeX-master: "Reference-Manual" -%%% End: - diff --git a/doc/refman/RefMan-oth.tex b/doc/refman/RefMan-oth.tex deleted file mode 100644 index 025788f5..00000000 --- a/doc/refman/RefMan-oth.tex +++ /dev/null @@ -1,1196 +0,0 @@ -\chapter[Vernacular commands]{Vernacular commands\label{Vernacular-commands} -\label{Other-commands}} - -\section{Displaying} - -\subsection[\tt Print {\qualid}.]{\tt Print {\qualid}.\comindex{Print}} -This command displays on the screen informations about the declared or -defined object referred by {\qualid}. - -\begin{ErrMsgs} -\item {\qualid} \errindex{not a defined object} -\end{ErrMsgs} - -\begin{Variants} -\item {\tt Print Term {\qualid}.} -\comindex{Print Term}\\ -This is a synonym to {\tt Print {\qualid}} when {\qualid} denotes a -global constant. - -\item {\tt About {\qualid}.} -\label{About} -\comindex{About}\\ -This displays various informations about the object denoted by {\qualid}: -its kind (module, constant, assumption, inductive, -constructor, abbreviation\ldots), long name, type, implicit -arguments and argument scopes. It does not print the body of -definitions or proofs. - -%\item {\tt Print Proof {\qualid}.}\comindex{Print Proof}\\ -%In case \qualid\ denotes an opaque theorem defined in a section, -%it is stored on a special unprintable form and displayed as -%{\tt <recipe>}. {\tt Print Proof} forces the printable form of \qualid\ -%to be computed and displays it. -\end{Variants} - -\subsection[\tt Print All.]{\tt Print All.\comindex{Print All}} -This command displays informations about the current state of the -environment, including sections and modules. - -\begin{Variants} -\item {\tt Inspect \num.}\comindex{Inspect}\\ -This command displays the {\num} last objects of the current -environment, including sections and modules. -\item {\tt Print Section {\ident}.}\comindex{Print Section}\\ -should correspond to a currently open section, this command -displays the objects defined since the beginning of this section. -% Discontinued -%% \item {\tt Print.}\comindex{Print}\\ -%% This command displays the axioms and variables declarations in the -%% environment as well as the constants defined since the last variable -%% was introduced. -\end{Variants} - -\section{Options and Flags} -\subsection[\tt Set {\rm\sl option} {\rm\sl value}.]{\tt Set {\rm\sl option} {\rm\sl value}.\comindex{Set}} -This command sets {\rm\sl option} to {\rm\sl value}. The original value of -{\rm\sl option} is restored when the current module ends. - -\begin{Variants} -\item {\tt Set {\rm\sl flag}.}\\ -This command switches {\rm\sl flag} on. The original state of -{\rm\sl flag} is restored when the current module ends. -\item {\tt Local Set {\rm\sl option} {\rm\sl value}.\comindex{Local Set}} -This command sets {\rm\sl option} to {\rm\sl value}. The original value of -{\rm\sl option} is restored when the current \emph{section} ends. -\item {\tt Local Set {\rm\sl flag}.}\\ -This command switches {\rm\sl flag} on. The original state of -{\rm\sl flag} is restored when the current \emph{section} ends. -\item {\tt Global Set {\rm\sl option} {\rm\sl value}.\comindex{Global Set}} -This command sets {\rm\sl option} to {\rm\sl value}. The original value of -{\rm\sl option} is \emph{not} restored at the end of the module. Additionally, -if set in a file, {\rm\sl option} is set to {\rm\sl value} when the file is -{\tt Require}-d. -\item {\tt Global Set {\rm\sl flag}.}\\ -This command switches {\rm\sl flag} on. The original state of -{\rm\sl flag} is \emph{not} restored at the end of the module. Additionally, -if set in a file, {\rm\sl flag} is switched on when the file is -{\tt Require}-d. -\end{Variants} - -\subsection[\tt Unset {\rm\sl flag}.]{\tt Unset {\rm\sl flag}.\comindex{Unset}} -This command switches {\rm\sl flag} off. The original state of {\rm\sl flag} -is restored when the current module ends. - -\begin{Variants} -\item {\tt Local Unset {\rm\sl flag}.\comindex{Local Unset}}\\ -This command switches {\rm\sl flag} off. The original state of {\rm\sl flag} -is restored when the current \emph{section} ends. -\item {\tt Global Unset {\rm\sl flag}.\comindex{Global Unset}}\\ -This command switches {\rm\sl flag} off. The original state of -{\rm\sl flag} is \emph{not} restored at the end of the module. Additionally, -if set in a file, {\rm\sl flag} is switched on when the file is -{\tt Require}-d. -\end{Variants} - -\subsection[\tt Test {\rm\sl option}.]{\tt Test {\rm\sl option}.\comindex{Test}} -This command prints the current value of {\rm\sl option}. - -\begin{Variants} -\item {\tt Test {\rm\sl flag}.}\\ -This command prints whether {\rm\sl flag} is on or off. -\end{Variants} - -\section{Requests to the environment} - -\subsection[\tt Check {\term}.]{\tt Check {\term}.\label{Check} -\comindex{Check}} -This command displays the type of {\term}. When called in proof mode, -the term is checked in the local context of the current subgoal. - -\subsection[\tt Eval {\rm\sl convtactic} in {\term}.]{\tt Eval {\rm\sl convtactic} in {\term}.\comindex{Eval}} - -This command performs the specified reduction on {\term}, and displays -the resulting term with its type. The term to be reduced may depend on -hypothesis introduced in the first subgoal (if a proof is in -progress). - -\SeeAlso Section~\ref{Conversion-tactics}. - -\subsection[\tt Compute {\term}.]{\tt Compute {\term}.\comindex{Compute}} - -This command performs a call-by-value evaluation of {\term} by using -the bytecode-based virtual machine. It is a shortcut for -{\tt Eval vm\_compute in {\term}}. - -\SeeAlso Section~\ref{Conversion-tactics}. - -\subsection[\tt Extraction \term.]{\tt Extraction \term.\label{ExtractionTerm} -\comindex{Extraction}} -This command displays the extracted term from -{\term}. The extraction is processed according to the distinction -between {\Set} and {\Prop}; that is to say, between logical and -computational content (see Section~\ref{Sorts}). The extracted term is -displayed in Objective Caml syntax, where global identifiers are still -displayed as in \Coq\ terms. - -\begin{Variants} -\item \texttt{Recursive Extraction {\qualid$_1$} \ldots{} {\qualid$_n$}.}\\ - Recursively extracts all the material needed for the extraction of - globals {\qualid$_1$} \ldots{} {\qualid$_n$}. -\end{Variants} - -\SeeAlso Chapter~\ref{Extraction}. - -\subsection[\tt Print Assumptions {\qualid}.]{\tt Print Assumptions {\qualid}.\comindex{Print Assumptions}} -\label{PrintAssumptions} - -This commands display all the assumptions (axioms, parameters and -variables) a theorem or definition depends on. Especially, it informs -on the assumptions with respect to which the validity of a theorem -relies. - -\begin{Variants} -\item \texttt{\tt Print Opaque Dependencies {\qualid}. - \comindex{Print Opaque Dependencies}}\\ - Displays the set of opaque constants {\qualid} relies on in addition - to the assumptions. -\end{Variants} - -\subsection[\tt Search {\term}.]{\tt Search {\term}.\comindex{Search}} -This command displays the name and type of all theorems of the current -context whose statement's conclusion has the form {\tt ({\term} t1 .. - tn)}. This command is useful to remind the user of the name of -library lemmas. - -\begin{coq_example} -Search le. -Search (@eq bool). -\end{coq_example} - -\begin{Variants} -\item -{\tt Search {\term} inside {\module$_1$} \ldots{} {\module$_n$}.} - -This restricts the search to constructions defined in modules -{\module$_1$} \ldots{} {\module$_n$}. - -\item {\tt Search {\term} outside {\module$_1$} \ldots{} {\module$_n$}.} - -This restricts the search to constructions not defined in modules -{\module$_1$} \ldots{} {\module$_n$}. - -\begin{ErrMsgs} -\item \errindex{Module/section \module{} not found} -No module \module{} has been required (see Section~\ref{Require}). -\end{ErrMsgs} - -\end{Variants} - -\subsection[\tt SearchAbout {\qualid}.]{\tt SearchAbout {\qualid}.\comindex{SearchAbout}} -This command displays the name and type of all objects (theorems, -axioms, etc) of the current context whose statement contains \qualid. -This command is useful to remind the user of the name of library -lemmas. - -\begin{ErrMsgs} -\item \errindex{The reference \qualid\ was not found in the current -environment}\\ - There is no constant in the environment named \qualid. -\end{ErrMsgs} - -\newcommand{\termpatternorstr}{{\termpattern}\textrm{\textsl{-}}{\str}} - -\begin{Variants} -\item {\tt SearchAbout {\str}.} - -If {\str} is a valid identifier, this command displays the name and type -of all objects (theorems, axioms, etc) of the current context whose -name contains {\str}. If {\str} is a notation's string denoting some -reference {\qualid} (referred to by its main symbol as in \verb="+"= -or by its notation's string as in \verb="_ + _"= or \verb="_ 'U' _"=, see -Section~\ref{Notation}), the command works like {\tt SearchAbout -{\qualid}}. - -\item {\tt SearchAbout {\str}\%{\delimkey}.} - -The string {\str} must be a notation or the main symbol of a notation -which is then interpreted in the scope bound to the delimiting key -{\delimkey} (see Section~\ref{scopechange}). - -\item {\tt SearchAbout {\termpattern}.} - -This searches for all statements or types of definition that contains -a subterm that matches the pattern {\termpattern} (holes of the -pattern are either denoted by ``{\texttt \_}'' or -by ``{\texttt ?{\ident}}'' when non linear patterns are expected). - -\item {\tt SearchAbout [ \nelist{\zeroone{-}{\termpatternorstr}}{} -].}\\ - -\noindent where {\termpatternorstr} is a -{\termpattern} or a {\str}, or a {\str} followed by a scope -delimiting key {\tt \%{\delimkey}}. - -This generalization of {\tt SearchAbout} searches for all objects -whose statement or type contains a subterm matching {\termpattern} (or -{\qualid} if {\str} is the notation for a reference {\qualid}) and -whose name contains all {\str} of the request that correspond to valid -identifiers. If a {\termpattern} or a {\str} is prefixed by ``-'', the -search excludes the objects that mention that {\termpattern} or that -{\str}. - -\item -\begin{tabular}[t]{@{}l} - {\tt SearchAbout {\termpatternorstr} inside {\module$_1$} \ldots{} {\module$_n$}.} \\ - {\tt SearchAbout [ \nelist{{\termpatternorstr}}{} ] - inside {\module$_1$} \ldots{} {\module$_n$}.} -\end{tabular} - -This restricts the search to constructions defined in modules -{\module$_1$} \ldots{} {\module$_n$}. - -\item -\begin{tabular}[t]{@{}l} - {\tt SearchAbout {\termpatternorstr} outside {\module$_1$}...{\module$_n$}.} \\ - {\tt SearchAbout [ \nelist{{\termpatternorstr}}{} ] - outside {\module$_1$}...{\module$_n$}.} -\end{tabular} - -This restricts the search to constructions not defined in modules -{\module$_1$} \ldots{} {\module$_n$}. - -\end{Variants} - -\examples - -\begin{coq_example*} -Require Import ZArith. -\end{coq_example*} -\begin{coq_example} -SearchAbout [ Zmult Zplus "distr" ]. -SearchAbout [ "+"%Z "*"%Z "distr" -positive -Prop]. -SearchAbout (?x * _ + ?x * _)%Z outside OmegaLemmas. -\end{coq_example} - -\subsection[\tt SearchPattern {\termpattern}.]{\tt SearchPattern {\term}.\comindex{SearchPattern}} - -This command displays the name and type of all theorems of the current -context whose statement's conclusion or last hypothesis and conclusion -matches the expression {\term} where holes in the latter are denoted -by ``{\texttt \_}''. It is a variant of {\tt SearchAbout - {\termpattern}} that does not look for subterms but searches for -statements whose conclusion has exactly the expected form, or whose -statement finishes by the given series of hypothesis/conclusion. - -\begin{coq_example} -Require Import Arith. -SearchPattern (_ + _ = _ + _). -SearchPattern (nat -> bool). -SearchPattern (forall l : list _, _ l l). -\end{coq_example} - -Patterns need not be linear: you can express that the same expression -must occur in two places by using pattern variables `{\texttt -?{\ident}}''. - -\begin{coq_example} -Require Import Arith. -SearchPattern (?X1 + _ = _ + ?X1). -\end{coq_example} - -\begin{Variants} -\item {\tt SearchPattern {\term} inside -{\module$_1$} \ldots{} {\module$_n$}.} - -This restricts the search to constructions defined in modules -{\module$_1$} \ldots{} {\module$_n$}. - -\item {\tt SearchPattern {\term} outside {\module$_1$} \ldots{} {\module$_n$}.} - -This restricts the search to constructions not defined in modules -{\module$_1$} \ldots{} {\module$_n$}. - -\end{Variants} - -\subsection[\tt SearchRewrite {\term}.]{\tt SearchRewrite {\term}.\comindex{SearchRewrite}} - -This command displays the name and type of all theorems of the current -context whose statement's conclusion is an equality of which one side matches -the expression {\term}. Holes in {\term} are denoted by ``{\texttt \_}''. - -\begin{coq_example} -Require Import Arith. -SearchRewrite (_ + _ + _). -\end{coq_example} - -\begin{Variants} -\item {\tt SearchRewrite {\term} inside -{\module$_1$} \ldots{} {\module$_n$}.} - -This restricts the search to constructions defined in modules -{\module$_1$} \ldots{} {\module$_n$}. - -\item {\tt SearchRewrite {\term} outside {\module$_1$} \ldots{} {\module$_n$}.} - -This restricts the search to constructions not defined in modules -{\module$_1$} \ldots{} {\module$_n$}. - -\end{Variants} - -% \subsection[\tt SearchIsos {\term}.]{\tt SearchIsos {\term}.\comindex{SearchIsos}} -% \label{searchisos} -% \texttt{SearchIsos} searches terms by their type modulo isomorphism. -% This command displays the full name of all constants, variables, -% inductive types, and inductive constructors of the current -% context whose type is isomorphic to {\term} modulo the contextual part of the -% following axiomatization (the mutual inductive types with one constructor, -% without implicit arguments, and for which projections exist, are regarded as a -% sequence of $\sa{}$): - - -% \begin{tabbing} -% \ \ \ \ \=11.\ \=\kill -% \>1.\>$A=B\mx{ if }A\stackrel{\bt{}\io{}}{\lra{}}B$\\ -% \>2.\>$\sa{}x:A.B=\sa{}y:A.B[x\la{}y]\mx{ if }y\not\in{}FV(\sa{}x:A.B)$\\ -% \>3.\>$\Pi{}x:A.B=\Pi{}y:A.B[x\la{}y]\mx{ if }y\not\in{}FV(\Pi{}x:A.B)$\\ -% \>4.\>$\sa{}x:A.B=\sa{}x:B.A\mx{ if }x\not\in{}FV(A,B)$\\ -% \>5.\>$\sa{}x:(\sa{}y:A.B).C=\sa{}x:A.\sa{}y:B[y\la{}x].C[x\la{}(x,y)]$\\ -% \>6.\>$\Pi{}x:(\sa{}y:A.B).C=\Pi{}x:A.\Pi{}y:B[y\la{}x].C[x\la{}(x,y)]$\\ -% \>7.\>$\Pi{}x:A.\sa{}y:B.C=\sa{}y:(\Pi{}x:A.B).(\Pi{}x:A.C[y\la{}(y\sm{}x)]$\\ -% \>8.\>$\sa{}x:A.unit=A$\\ -% \>9.\>$\sa{}x:unit.A=A[x\la{}tt]$\\ -% \>10.\>$\Pi{}x:A.unit=unit$\\ -% \>11.\>$\Pi{}x:unit.A=A[x\la{}tt]$ -% \end{tabbing} - -% For more informations about the exact working of this command, see -% \cite{Del97}. - -\subsection[\tt Locate {\qualid}.]{\tt Locate {\qualid}.\comindex{Locate} -\label{Locate}} -This command displays the full name of the qualified identifier {\qualid} -and consequently the \Coq\ module in which it is defined. - -\begin{coq_eval} -(*************** The last line should produce **************************) -(*********** Error: I.Dont.Exist not a defined object ******************) -\end{coq_eval} -\begin{coq_eval} -Set Printing Depth 50. -\end{coq_eval} -\begin{coq_example} -Locate nat. -Locate Datatypes.O. -Locate Init.Datatypes.O. -Locate Coq.Init.Datatypes.O. -Locate I.Dont.Exist. -\end{coq_example} - -\SeeAlso Section \ref{LocateSymbol} - -\subsection{The {\sc Whelp} searching tool -\label{Whelp}} - -{\sc Whelp} is an experimental searching and browsing tool for the -whole {\Coq} library and the whole set of {\Coq} user contributions. -{\sc Whelp} requires a browser to work. {\sc Whelp} has been developed -at the University of Bologna as part of the HELM\footnote{Hypertextual -Electronic Library of Mathematics} and MoWGLI\footnote{Mathematics on -the Web, Get it by Logics and Interfaces} projects. It can be invoked -directly from the {\Coq} toplevel or from {\CoqIDE}, assuming a -graphical environment is also running. The browser to use can be -selected by setting the environment variable {\tt -COQREMOTEBROWSER}. If not explicitly set, it defaults to -\verb!firefox -remote \"OpenURL(%s,new-tab)\" || firefox %s &"! or -\verb!C:\\PROGRA~1\\INTERN~1\\IEXPLORE %s!, depending on the -underlying operating system (in the command, the string \verb!%s! -serves as metavariable for the url to open). -The Whelp tool relies on a dedicated Whelp server and on another server -called Getter that retrieves formal documents. The default Whelp server name -can be obtained using the command {\tt Test Whelp Server} -\comindex{Test Whelp Server} and the default Getter can be obtained -using the command: {\tt Test Whelp Getter} \comindex{Test Whelp -Getter}. The Whelp server name can be changed using the command: - -\smallskip -\noindent {\tt Set Whelp Server {\str}}.\\ -where {\str} is a URL (e.g. {\tt http://mowgli.cs.unibo.it:58080}). -\comindex{Set Whelp Server} -\smallskip - -\noindent The Getter can be changed using the command: -\smallskip - -\noindent {\tt Set Whelp Getter {\str}}.\\ -where {\str} is a URL (e.g. {\tt http://mowgli.cs.unibo.it:58081}). -\comindex{Set Whelp Getter} - -\bigskip - -The {\sc Whelp} commands are: - -\subsubsection{\tt Whelp Locate "{\sl reg\_expr}". -\comindex{Whelp Locate}} - -This command opens a browser window and displays the result of seeking -for all names that match the regular expression {\sl reg\_expr} in the -{\Coq} library and user contributions. The regular expression can -contain the special operators are * and ? that respectively stand for -an arbitrary substring and for exactly one character. - -\variant {\tt Whelp Locate {\ident}.}\\ -This is equivalent to {\tt Whelp Locate "{\ident}"}. - -\subsubsection{\tt Whelp Match {\pattern}. -\comindex{Whelp Match}} - -This command opens a browser window and displays the result of seeking -for all statements that match the pattern {\pattern}. Holes in the -pattern are represented by the wildcard character ``\_''. - -\subsubsection[\tt Whelp Instance {\pattern}.]{\tt Whelp Instance {\pattern}.\comindex{Whelp Instance}} - -This command opens a browser window and displays the result of seeking -for all statements that are instances of the pattern {\pattern}. The -pattern is here assumed to be an universally quantified expression. - -\subsubsection[\tt Whelp Elim {\qualid}.]{\tt Whelp Elim {\qualid}.\comindex{Whelp Elim}} - -This command opens a browser window and displays the result of seeking -for all statements that have the ``form'' of an elimination scheme -over the type denoted by {\qualid}. - -\subsubsection[\tt Whelp Hint {\term}.]{\tt Whelp Hint {\term}.\comindex{Whelp Hint}} - -This command opens a browser window and displays the result of seeking -for all statements that can be instantiated so that to prove the -statement {\term}. - -\variant {\tt Whelp Hint.}\\ This is equivalent to {\tt Whelp Hint -{\sl goal}} where {\sl goal} is the current goal to prove. Notice that -{\Coq} does not send the local environment of definitions to the {\sc -Whelp} tool so that it only works on requests strictly based on, only, -definitions of the standard library and user contributions. - -\section{Loading files} - -\Coq\ offers the possibility of loading different -parts of a whole development stored in separate files. Their contents -will be loaded as if they were entered from the keyboard. This means -that the loaded files are ASCII files containing sequences of commands -for \Coq's toplevel. This kind of file is called a {\em script} for -\Coq\index{Script file}. The standard (and default) extension of -\Coq's script files is {\tt .v}. - -\subsection[\tt Load {\ident}.]{\tt Load {\ident}.\comindex{Load}\label{Load}} -This command loads the file named {\ident}{\tt .v}, searching -successively in each of the directories specified in the {\em - loadpath}. (see Section~\ref{loadpath}) - -\begin{Variants} -\item {\tt Load {\str}.}\label{Load-str}\\ - Loads the file denoted by the string {\str}, where {\str} is any - complete filename. Then the \verb.~. and {\tt ..} - abbreviations are allowed as well as shell variables. If no - extension is specified, \Coq\ will use the default extension {\tt - .v} -\item {\tt Load Verbose {\ident}.}, - {\tt Load Verbose {\str}}\\ - \comindex{Load Verbose} - Display, while loading, the answers of \Coq\ to each command - (including tactics) contained in the loaded file - \SeeAlso Section~\ref{Begin-Silent} -\end{Variants} - -\begin{ErrMsgs} -\item \errindex{Can't find file {\ident} on loadpath} -\end{ErrMsgs} - -\section[Compiled files]{Compiled files\label{compiled}\index{Compiled files}} - -This section describes the commands used to load compiled files (see -Chapter~\ref{Addoc-coqc} for documentation on how to compile a file). -A compiled file is a particular case of module called {\em library file}. - -%%%%%%%%%%%% -% Import and Export described in RefMan-mod.tex -% the minor difference (to avoid multiple Exporting of libraries) in -% the treatment of normal modules and libraries by Export omitted - -\subsection[\tt Require {\qualid}.]{\tt Require {\qualid}.\label{Require} -\comindex{Require}} - -This command looks in the loadpath for a file containing -module {\qualid} and adds the corresponding module to the environment -of {\Coq}. As library files have dependencies in other library files, -the command {\tt Require {\qualid}} recursively requires all library -files the module {\qualid} depends on and adds the corresponding modules to the -environment of {\Coq} too. {\Coq} assumes that the compiled files have -been produced by a valid {\Coq} compiler and their contents are then not -replayed nor rechecked. - -To locate the file in the file system, {\qualid} is decomposed under -the form {\dirpath}{\tt .}{\textsl{ident}} and the file {\ident}{\tt -.vo} is searched in the physical directory of the file system that is -mapped in {\Coq} loadpath to the logical path {\dirpath} (see -Section~\ref{loadpath}). The mapping between physical directories and -logical names at the time of requiring the file must be consistent -with the mapping used to compile the file. - -\begin{Variants} -\item {\tt Require Import {\qualid}.} \comindex{Require} - - This loads and declares the module {\qualid} and its dependencies - then imports the contents of {\qualid} as described in - Section~\ref{Import}. - - It does not import the modules on which {\qualid} depends unless - these modules were itself required in module {\qualid} using {\tt - Require Export}, as described below, or recursively required through - a sequence of {\tt Require Export}. - - If the module required has already been loaded, {\tt Require Import - {\qualid}} simply imports it, as {\tt Import {\qualid}} would. - -\item {\tt Require Export {\qualid}.} - \comindex{Require Export} - - This command acts as {\tt Require Import} {\qualid}, but if a - further module, say {\it A}, contains a command {\tt Require - Export} {\it B}, then the command {\tt Require Import} {\it A} - also imports the module {\it B}. - -\item {\tt Require \zeroone{Import {\sl |} Export} {\qualid}$_1$ \ldots {\qualid}$_n$.} - - This loads the modules {\qualid}$_1$, \ldots, {\qualid}$_n$ and - their recursive dependencies. If {\tt Import} or {\tt Export} is - given, it also imports {\qualid}$_1$, \ldots, {\qualid}$_n$ and all - the recursive dependencies that were marked or transitively marked - as {\tt Export}. - -\item {\tt Require \zeroone{Import {\sl |} Export} {\str}.} - - This shortcuts the resolution of the qualified name into a library - file name by directly requiring the module to be found in file - {\str}.vo. -\end{Variants} - -\begin{ErrMsgs} - -\item \errindex{Cannot load {\qualid}: no physical path bound to {\dirpath}} - -\item \errindex{Cannot find library foo in loadpath} - - The command did not find the file {\tt foo.vo}. Either {\tt - foo.v} exists but is not compiled or {\tt foo.vo} is in a directory - which is not in your {\tt LoadPath} (see Section~\ref{loadpath}). - -\item \errindex{Compiled library {\ident}.vo makes inconsistent assumptions over library {\qualid}} - - The command tried to load library file {\ident}.vo that depends on - some specific version of library {\qualid} which is not the one - already loaded in the current {\Coq} session. Probably {\ident}.v - was not properly recompiled with the last version of the file - containing module {\qualid}. - -\item \errindex{Bad magic number} - - \index{Bad-magic-number@{\tt Bad Magic Number}} - The file {\tt{\ident}.vo} was found but either it is not a \Coq\ - compiled module, or it was compiled with an older and incompatible - version of \Coq. - -\item \errindex{The file {\ident}.vo contains library {\dirpath} and not - library {\dirpath'}} - - The library file {\dirpath'} is indirectly required by the {\tt - Require} command but it is bound in the current loadpath to the file - {\ident}.vo which was bound to a different library name {\dirpath} - at the time it was compiled. - -\end{ErrMsgs} - -\SeeAlso Chapter~\ref{Addoc-coqc} - -\subsection[\tt Print Libraries.]{\tt Print Libraries.\comindex{Print Libraries}} - -This command displays the list of library files loaded in the current -{\Coq} session. For each of these libraries, it also tells if it is -imported. - -\subsection[\tt Declare ML Module {\str$_1$} .. {\str$_n$}.]{\tt Declare ML Module {\str$_1$} .. {\str$_n$}.\comindex{Declare ML Module}} -This commands loads the Objective Caml compiled files {\str$_1$} {\dots} -{\str$_n$} (dynamic link). It is mainly used to load tactics -dynamically. -% (see Chapter~\ref{WritingTactics}). - The files are -searched into the current Objective Caml loadpath (see the command {\tt -Add ML Path} in the Section~\ref{loadpath}). Loading of Objective Caml -files is only possible under the bytecode version of {\tt coqtop} -(i.e. {\tt coqtop} called with options {\tt -byte}, see chapter -\ref{Addoc-coqc}), or when Coq has been compiled with a version of -Objective Caml that supports native {\tt Dynlink} ($\ge$ 3.11). - -\begin{ErrMsgs} -\item \errindex{File not found on loadpath : \str} -\item \errindex{Loading of ML object file forbidden in a native Coq} -\end{ErrMsgs} - -\subsection[\tt Print ML Modules.]{\tt Print ML Modules.\comindex{Print ML Modules}} -This print the name of all \ocaml{} modules loaded with \texttt{Declare - ML Module}. To know from where these module were loaded, the user -should use the command \texttt{Locate File} (see Section~\ref{Locate File}) - -\section[Loadpath]{Loadpath\label{loadpath}\index{Loadpath}} - -There are currently two loadpaths in \Coq. A loadpath where seeking -{\Coq} files (extensions {\tt .v} or {\tt .vo} or {\tt .vi}) and one where -seeking Objective Caml files. The default loadpath contains the -directory ``\texttt{.}'' denoting the current directory and mapped to the empty logical path (see Section~\ref{LongNames}). - -\subsection[\tt Pwd.]{\tt Pwd.\comindex{Pwd}\label{Pwd}} -This command displays the current working directory. - -\subsection[\tt Cd {\str}.]{\tt Cd {\str}.\comindex{Cd}} -This command changes the current directory according to {\str} -which can be any valid path. - -\begin{Variants} -\item {\tt Cd.}\\ - Is equivalent to {\tt Pwd.} -\end{Variants} - -\subsection[\tt Add LoadPath {\str} as {\dirpath}.]{\tt Add LoadPath {\str} as {\dirpath}.\comindex{Add LoadPath}\label{AddLoadPath}} - -This command adds the physical directory {\str} to the current {\Coq} -loadpath and maps it to the logical directory {\dirpath}, which means -that every file \textrm{\textsl{dirname}}/\textrm{\textsl{basename.v}} -physically lying in subdirectory {\str}/\textrm{\textsl{dirname}} -becomes accessible in {\Coq} through absolute logical name -{\dirpath}{\tt .}\textrm{\textsl{dirname}}{\tt -.}\textrm{\textsl{basename}}. - -\Rem {\tt Add LoadPath} also adds {\str} to the current ML loadpath. - -\begin{Variants} -\item {\tt Add LoadPath {\str}.}\\ -Performs as {\tt Add LoadPath {\str} as {\dirpath}} but for the empty directory path. -\end{Variants} - -\subsection[\tt Add Rec LoadPath {\str} as {\dirpath}.]{\tt Add Rec LoadPath {\str} as {\dirpath}.\comindex{Add Rec LoadPath}\label{AddRecLoadPath}} -This command adds the physical directory {\str} and all its subdirectories to -the current \Coq\ loadpath. The top directory {\str} is mapped to the -logical directory {\dirpath} and any subdirectory {\textsl{pdir}} of it is -mapped to logical name {\dirpath}{\tt .}\textsl{pdir} and -recursively. Subdirectories corresponding to invalid {\Coq} -identifiers are skipped, and, by convention, subdirectories named {\tt -CVS} or {\tt \_darcs} are skipped too. - -Otherwise, said, {\tt Add Rec LoadPath {\str} as {\dirpath}} behaves -as {\tt Add LoadPath {\str} as {\dirpath}} excepts that files lying in -validly named subdirectories of {\str} need not be qualified to be -found. - -In case of files with identical base name, files lying in most recently -declared {\dirpath} are found first and explicit qualification is -required to refer to the other files of same base name. - -If several files with identical base name are present in different -subdirectories of a recursive loadpath declared via a single instance of -{\tt Add Rec LoadPath}, which of these files is found first is -system-dependent and explicit qualification is recommended. - -\Rem {\tt Add Rec LoadPath} also recursively adds {\str} to the current ML loadpath. - -\begin{Variants} -\item {\tt Add Rec LoadPath {\str}.}\\ -Works as {\tt Add Rec LoadPath {\str} as {\dirpath}} but for the empty logical directory path. -\end{Variants} - -\subsection[\tt Remove LoadPath {\str}.]{\tt Remove LoadPath {\str}.\comindex{Remove LoadPath}} -This command removes the path {\str} from the current \Coq\ loadpath. - -\subsection[\tt Print LoadPath.]{\tt Print LoadPath.\comindex{Print LoadPath}} -This command displays the current \Coq\ loadpath. - -\begin{Variants} -\item {\tt Print LoadPath {\dirpath}.}\\ -Works as {\tt Print LoadPath} but displays only the paths that extend the {\dirpath} prefix. -\end{Variants} - -\subsection[\tt Add ML Path {\str}.]{\tt Add ML Path {\str}.\comindex{Add ML Path}} -This command adds the path {\str} to the current Objective Caml loadpath (see -the command {\tt Declare ML Module} in the Section~\ref{compiled}). - -\Rem This command is implied by {\tt Add LoadPath {\str} as {\dirpath}}. - -\subsection[\tt Add Rec ML Path {\str}.]{\tt Add Rec ML Path {\str}.\comindex{Add Rec ML Path}} -This command adds the directory {\str} and all its subdirectories -to the current Objective Caml loadpath (see -the command {\tt Declare ML Module} in the Section~\ref{compiled}). - -\Rem This command is implied by {\tt Add Rec LoadPath {\str} as {\dirpath}}. - -\subsection[\tt Print ML Path {\str}.]{\tt Print ML Path {\str}.\comindex{Print ML Path}} -This command displays the current Objective Caml loadpath. -This command makes sense only under the bytecode version of {\tt -coqtop}, i.e. using option {\tt -byte} (see the -command {\tt Declare ML Module} in the section -\ref{compiled}). - -\subsection[\tt Locate File {\str}.]{\tt Locate File {\str}.\comindex{Locate - File}\label{Locate File}} -This command displays the location of file {\str} in the current loadpath. -Typically, {\str} is a \texttt{.cmo} or \texttt{.vo} or \texttt{.v} file. - -\subsection[\tt Locate Library {\dirpath}.]{\tt Locate Library {\dirpath}.\comindex{Locate Library}\label{Locate Library}} -This command gives the status of the \Coq\ module {\dirpath}. It tells if the -module is loaded and if not searches in the load path for a module -of logical name {\dirpath}. - -\section{States and Reset} - -\subsection[\tt Reset \ident.]{\tt Reset \ident.\comindex{Reset}} -This command removes all the objects in the environment since \ident\ -was introduced, including \ident. \ident\ may be the name of a defined -or declared object as well as the name of a section. One cannot reset -over the name of a module or of an object inside a module. - -\begin{ErrMsgs} -\item \ident: \errindex{no such entry} -\end{ErrMsgs} - -\subsection[\tt Back.]{\tt Back.\comindex{Back}} - -This commands undoes all the effects of the last vernacular -command. This does not include commands that only access to the -environment like those described in the previous sections of this -chapter (for instance {\tt Require} and {\tt Load} can be undone, but -not {\tt Check} and {\tt Locate}). Commands read from a vernacular -file are considered as a single command. - -\begin{Variants} -\item {\tt Back $n$} \\ - Undoes $n$ vernacular commands. -\end{Variants} - -\begin{ErrMsgs} -\item \errindex{Reached begin of command history} \\ - Happens when there is vernacular command to undo. -\end{ErrMsgs} - -\subsection[\tt Backtrack $\num_1$ $\num_2$ $\num_3$.]{\tt Backtrack $\num_1$ $\num_2$ $\num_3$.\comindex{Backtrack}} - -This command is dedicated for the use in graphical interfaces. It -allows to backtrack to a particular \emph{global} state, i.e. -typically a state corresponding to a previous line in a script. A -global state includes declaration environment but also proof -environment (see Chapter~\ref{Proof-handling}). The three numbers -$\num_1$, $\num_2$ and $\num_3$ represent the following: -\begin{itemize} -\item $\num_3$: Number of \texttt{Abort} to perform, i.e. the number - of currently opened nested proofs that must be canceled (see - Chapter~\ref{Proof-handling}). -\item $\num_2$: \emph{Proof state number} to unbury once aborts have - been done. Coq will compute the number of \texttt{Undo} to perform - (see Chapter~\ref{Proof-handling}). -\item $\num_1$: Environment state number to unbury, Coq will compute - the number of \texttt{Back} to perform. -\end{itemize} - - -\subsubsection{How to get state numbers?} -\label{sec:statenums} - - -Notice that when in \texttt{-emacs} mode, \Coq\ displays the current -proof and environment state numbers in the prompt. More precisely the -prompt in \texttt{-emacs} mode is the following: - -\verb!<prompt>! \emph{$id_i$} \verb!<! $\num_1$ -\verb!|! $id_1$\verb!|!$id_2$\verb!|!\dots\verb!|!$id_n$ -\verb!|! $\num_2$ \verb!< </prompt>! - -Where: - -\begin{itemize} -\item \emph{$id_i$} is the name of the current proof (if there is - one, otherwise \texttt{Coq} is displayed, see -Chapter~\ref{Proof-handling}). -\item $\num_1$ is the environment state number after the last - command. -\item $\num_2$ is the proof state number after the last - command. -\item $id_1$ $id_2$ {\dots} $id_n$ are the currently opened proof names - (order not significant). -\end{itemize} - -It is then possible to compute the \texttt{Backtrack} command to -unbury the state corresponding to a particular prompt. For example, -suppose the current prompt is: - -\verb!<! goal4 \verb!<! 35 -\verb!|!goal1\verb!|!goal4\verb!|!goal3\verb!|!goal2\verb!|! -\verb!|!8 \verb!< </prompt>! - -and we want to backtrack to a state labeled by: - -\verb!<! goal2 \verb!<! 32 -\verb!|!goal1\verb!|!goal2 -\verb!|!12 \verb!< </prompt>! - -We have to perform \verb!Backtrack 32 12 2! , i.e. perform 2 -\texttt{Abort}s (to cancel goal4 and goal3), then rewind proof until -state 12 and finally go back to environment state 32. Notice that this -supposes that proofs are nested in a regular way (no \texttt{Resume} or -\texttt{Suspend} commands). - -\begin{Variants} -\item {\tt BackTo n}. \comindex{BackTo}\\ - Is a more basic form of \texttt{Backtrack} where only the first - argument (global environment number) is given, no \texttt{abort} and - no \texttt{Undo} is performed. -\end{Variants} - -\subsection[\tt Restore State \str.]{\tt Restore State \str.\comindex{Restore State}} - Restores the state contained in the file \str. - -\begin{Variants} -\item {\tt Restore State \ident}\\ - Equivalent to {\tt Restore State "}{\ident}{\tt .coq"}. -\item {\tt Reset Initial.}\comindex{Reset Initial}\\ - Goes back to the initial state (like after the command {\tt coqtop}, - when the interactive session began). This command is only available - interactively. -\end{Variants} - -\subsection[\tt Write State \str.]{\tt Write State \str.\comindex{Write State}} -Writes the current state into a file \str{} for -use in a further session. This file can be given as the {\tt - inputstate} argument of the commands {\tt coqtop} and {\tt coqc}. - -\begin{Variants} -\item {\tt Write State \ident}\\ - Equivalent to {\tt Write State "}{\ident}{\tt .coq"}. - The state is saved in the current directory (see Section~\ref{Pwd}). -\end{Variants} - -\section{Quitting and debugging} - -\subsection[\tt Quit.]{\tt Quit.\comindex{Quit}} -This command permits to quit \Coq. - -\subsection[\tt Drop.]{\tt Drop.\comindex{Drop}\label{Drop}} - -This is used mostly as a debug facility by \Coq's implementors -and does not concern the casual user. -This command permits to leave {\Coq} temporarily and enter the -Objective Caml toplevel. The Objective Caml command: - -\begin{flushleft} -\begin{verbatim} -#use "include";; -\end{verbatim} -\end{flushleft} - -\noindent add the right loadpaths and loads some toplevel printers for -all abstract types of \Coq - section\_path, identifiers, terms, judgments, -\dots. You can also use the file \texttt{base\_include} instead, -that loads only the pretty-printers for section\_paths and -identifiers. -% See Section~\ref{test-and-debug} more information on the -% usage of the toplevel. -You can return back to \Coq{} with the command: - -\begin{flushleft} -\begin{verbatim} -go();; -\end{verbatim} -\end{flushleft} - -\begin{Warnings} -\item It only works with the bytecode version of {\Coq} (i.e. {\tt coqtop} called with option {\tt -byte}, see the contents of Section~\ref{binary-images}). -\item You must have compiled {\Coq} from the source package and set the - environment variable \texttt{COQTOP} to the root of your copy of the sources (see Section~\ref{EnvVariables}). -\end{Warnings} - -\subsection[\tt Time \textrm{\textsl{command}}.]{\tt Time \textrm{\textsl{command}}.\comindex{Time} -\label{time}} -This command executes the vernacular command \textrm{\textsl{command}} -and display the time needed to execute it. - - -\subsection[\tt Timeout \textrm{\textsl{int}} \textrm{\textsl{command}}.]{\tt Timeout \textrm{\textsl{int}} \textrm{\textsl{command}}.\comindex{Timeout} -\label{timeout}} - -This command executes the vernacular command \textrm{\textsl{command}}. If -the command has not terminated after the time specified by the integer -(time expressed in seconds), then it is interrupted and an error message -is displayed. - -\subsection[\tt Set Default Timeout \textrm{\textsl{int}}.]{\tt Set - Default Timeout \textrm{\textsl{int}}.\comindex{Set Default Timeout}} - -After using this command, all subsequent commands behave as if they -were passed to a {\tt Timeout} command. Commands already starting by -a {\tt Timeout} are unaffected. - -\subsection[\tt Unset Default Timeout.]{\tt Unset Default Timeout.\comindex{Unset Default Timeout}} - -This command turns off the use of a defaut timeout. - -\subsection[\tt Test Default Timeout.]{\tt Test Default Timeout.\comindex{Test Default Timeout}} - -This command displays whether some default timeout has be set or not. - -\section{Controlling display} - -\subsection[\tt Set Silent.]{\tt Set Silent.\comindex{Set Silent} -\label{Begin-Silent} -\index{Silent mode}} -This command turns off the normal displaying. - -\subsection[\tt Unset Silent.]{\tt Unset Silent.\comindex{Unset Silent}} -This command turns the normal display on. - -\subsection[\tt Set Printing Width {\integer}.]{\tt Set Printing Width {\integer}.\comindex{Set Printing Width}} -This command sets which left-aligned part of the width of the screen -is used for display. - -\subsection[\tt Unset Printing Width.]{\tt Unset Printing Width.\comindex{Unset Printing Width}} -This command resets the width of the screen used for display to its -default value (which is 78 at the time of writing this documentation). - -\subsection[\tt Test Printing Width.]{\tt Test Printing Width.\comindex{Test Printing Width}} -This command displays the current screen width used for display. - -\subsection[\tt Set Printing Depth {\integer}.]{\tt Set Printing Depth {\integer}.\comindex{Set Printing Depth}} -This command sets the nesting depth of the formatter used for -pretty-printing. Beyond this depth, display of subterms is replaced by -dots. - -\subsection[\tt Unset Printing Depth.]{\tt Unset Printing Depth.\comindex{Unset Printing Depth}} -This command resets the nesting depth of the formatter used for -pretty-printing to its default value (at the -time of writing this documentation, the default value is 50). - -\subsection[\tt Test Printing Depth.]{\tt Test Printing Depth.\comindex{Test Printing Depth}} -This command displays the current nesting depth used for display. - -%\subsection{\tt Explain ...} -%Not yet documented. - -%\subsection{\tt Go ...} -%Not yet documented. - -%\subsection{\tt Abstraction ...} -%Not yet documented. - -\section{Controlling the reduction strategies and the conversion algorithm} -\label{Controlling reduction strategy} - -{\Coq} provides reduction strategies that the tactics can invoke and -two different algorithms to check the convertibility of types. -The first conversion algorithm lazily -compares applicative terms while the other is a brute-force but efficient -algorithm that first normalizes the terms before comparing them. The -second algorithm is based on a bytecode representation of terms -similar to the bytecode representation used in the ZINC virtual -machine~\cite{Leroy90}. It is specially useful for intensive -computation of algebraic values, such as numbers, and for reflexion-based -tactics. The commands to fine-tune the reduction strategies and the -lazy conversion algorithm are described first. - -\subsection[\tt Opaque \qualid$_1$ {\dots} \qualid$_n$.]{\tt Opaque \qualid$_1$ {\dots} \qualid$_n$.\comindex{Opaque}\label{Opaque}} -This command has an effect on unfoldable constants, i.e. -on constants defined by {\tt Definition} or {\tt Let} (with an explicit -body), or by a command assimilated to a definition such as {\tt -Fixpoint}, {\tt Program Definition}, etc, or by a proof ended by {\tt -Defined}. The command tells not to unfold -the constants {\qualid$_1$} {\dots} {\qualid$_n$} in tactics using -$\delta$-conversion (unfolding a constant is replacing it by its -definition). - -{\tt Opaque} has also on effect on the conversion algorithm of {\Coq}, -telling to delay the unfolding of a constant as later as possible in -case {\Coq} has to check the conversion (see Section~\ref{conv-rules}) -of two distinct applied constants. - -The scope of {\tt Opaque} is limited to the current section, or -current file, unless the variant {\tt Global Opaque \qualid$_1$ {\dots} -\qualid$_n$} is used. - -\SeeAlso sections \ref{Conversion-tactics}, \ref{Automatizing}, -\ref{Theorem} - -\begin{ErrMsgs} -\item \errindex{The reference \qualid\ was not found in the current -environment}\\ - There is no constant referred by {\qualid} in the environment. - Nevertheless, if you asked \texttt{Opaque foo bar} - and if \texttt{bar} does not exist, \texttt{foo} is set opaque. -\end{ErrMsgs} - -\subsection[\tt Transparent \qualid$_1$ {\dots} \qualid$_n$.]{\tt Transparent \qualid$_1$ {\dots} \qualid$_n$.\comindex{Transparent}\label{Transparent}} -This command is the converse of {\tt Opaque} and it applies on -unfoldable constants to restore their unfoldability after an {\tt -Opaque} command. - -Note in particular that constants defined by a proof ended by {\tt -Qed} are not unfoldable and {\tt Transparent} has no effect on -them. This is to keep with the usual mathematical practice of {\em -proof irrelevance}: what matters in a mathematical development is the -sequence of lemma statements, not their actual proofs. This -distinguishes lemmas from the usual defined constants, whose actual -values are of course relevant in general. - -The scope of {\tt Transparent} is limited to the current section, or -current file, unless the variant {\tt Global Transparent \qualid$_1$ -\dots \qualid$_n$} is used. - -\begin{ErrMsgs} -% \item \errindex{Can not set transparent.}\\ -% It is a constant from a required module or a parameter. -\item \errindex{The reference \qualid\ was not found in the current -environment}\\ - There is no constant referred by {\qualid} in the environment. -\end{ErrMsgs} - -\SeeAlso sections \ref{Conversion-tactics}, \ref{Automatizing}, -\ref{Theorem} - -\subsection{\tt Strategy {\it level} [ \qualid$_1$ {\dots} \qualid$_n$ - ].\comindex{Strategy}\comindex{Local Strategy}\label{Strategy}} -This command generalizes the behavior of {\tt Opaque} and {\tt - Transparent} commands. It is used to fine-tune the strategy for -unfolding constants, both at the tactic level and at the kernel -level. This command associates a level to \qualid$_1$ {\dots} -\qualid$_n$. Whenever two expressions with two distinct head -constants are compared (for instance, this comparison can be triggered -by a type cast), the one with lower level is expanded first. In case -of a tie, the second one (appearing in the cast type) is expanded. - -Levels can be one of the following (higher to lower): -\begin{description} -\item[opaque]: level of opaque constants. They cannot be expanded by - tactics (behaves like $+\infty$, see next item). -\item[\num]: levels indexed by an integer. Level $0$ corresponds - to the default behavior, which corresponds to transparent - constants. This level can also be referred to as {\bf transparent}. - Negative levels correspond to constants to be expanded before normal - transparent constants, while positive levels correspond to constants - to be expanded after normal transparent constants. -\item[expand]: level of constants that should be expanded first - (behaves like $-\infty$) -\end{description} - -These directives survive section and module closure, unless the -command is prefixed by {\tt Local}. In the latter case, the behavior -regarding sections and modules is the same as for the {\tt - Transparent} and {\tt Opaque} commands. - -\subsection{\tt Declare Reduction \ident\ := {\rm\sl convtactic}.} - -This command allows to give a short name to a reduction expression, -for instance {\tt lazy beta delta [foo bar]}. This short name can -then be used in {\tt Eval \ident\ in ...} or {\tt eval} directives. -This command accepts the {\tt Local} modifier, for discarding -this reduction name at the end of the file or module. For the moment -the name cannot be qualified. In particular declaring the same name -in several modules or in several functor applications will be refused -if these declarations are not local. The name \ident\ cannot be used -directly as an Ltac tactic, but nothing prevent the user to also -perform a {\tt Ltac \ident\ := {\rm\sl convtactic}}. - -\SeeAlso sections \ref{Conversion-tactics} - -\subsection{\tt Set Virtual Machine -\label{SetVirtualMachine} -\comindex{Set Virtual Machine}} - -This activates the bytecode-based conversion algorithm. - -\subsection{\tt Unset Virtual Machine -\comindex{Unset Virtual Machine}} - -This deactivates the bytecode-based conversion algorithm. - -\subsection{\tt Test Virtual Machine -\comindex{Test Virtual Machine}} - -This tells if the bytecode-based conversion algorithm is -activated. The default behavior is to have the bytecode-based -conversion algorithm deactivated. - -\SeeAlso sections~\ref{vmcompute} and~\ref{vmoption}. - -\section{Controlling the locality of commands} - -\subsection{{\tt Local}, {\tt Global} -\comindex{Local} -\comindex{Global} -} - -Some commands support a {\tt Local} or {\tt Global} prefix modifier to -control the scope of their effect. There are four kinds of commands: - -\begin{itemize} -\item Commands whose default is to extend their effect both outside the - section and the module or library file they occur in. - - For these commands, the {\tt Local} modifier limits the effect of - the command to the current section or module it occurs in. - - As an example, the {\tt Coercion} (see Section~\ref{Coercions}) - and {\tt Strategy} (see Section~\ref{Strategy}) - commands belong to this category. - -\item Commands whose default behavior is to stop their effect at the - end of the section they occur in but to extent their effect outside - the module or library file they occur in. - - For these commands, the {\tt Local} modifier limits the effect of - the command to the current module if the command does not occur in a - section and the {\tt Global} modifier extends the effect outside the - current sections and current module if the command occurs in a - section. - - As an example, the {\tt Implicit Arguments} (see - Section~\ref{Implicit Arguments}), {\tt Ltac} (see - Chapter~\ref{TacticLanguage}) or {\tt Notation} (see - Section~\ref{Notation}) commands belong to this category. - - Notice that a subclass of these commands do not support extension of - their scope outside sections at all and the {\tt Global} is not - applicable to them. - -\item Commands whose default behavior is to stop their effect at the - end of the section or module they occur in. - - For these commands, the {\tt Global} modifier extends their effect - outside the sections and modules they occurs in. - - The {\tt Transparent} and {\tt Opaque} (see - Section~\ref{Controlling reduction strategy}) commands belong to - this category. - -\item Commands whose default behavior is to extend their effect - outside sections but not outside modules when they occur in a - section and to extend their effect outside the module or library - file they occur in when no section contains them. - - For these commands, the {\tt Local} modifier limits the effect to - the current section or module while the {\tt Global} modifier extends - the effect outside the module even when the command occurs in a section. - - The {\tt Set} and {\tt Unset} commands belong to this category. -\end{itemize} - - -% $Id: RefMan-oth.tex 13923 2011-03-21 16:25:20Z letouzey $ - -%%% Local Variables: -%%% mode: latex -%%% TeX-master: "Reference-Manual" -%%% End: diff --git a/doc/refman/RefMan-pre.tex b/doc/refman/RefMan-pre.tex deleted file mode 100644 index a2cdb5ec..00000000 --- a/doc/refman/RefMan-pre.tex +++ /dev/null @@ -1,783 +0,0 @@ -%BEGIN LATEX -\setheaders{Credits} -%END LATEX -\chapter*{Credits} -%\addcontentsline{toc}{section}{Credits} - -\Coq{}~ is a proof assistant for higher-order logic, allowing the -development of computer programs consistent with their formal -specification. It is the result of about ten years of research of the -Coq project. We shall briefly survey here three main aspects: the -\emph{logical language} in which we write our axiomatizations and -specifications, the \emph{proof assistant} which allows the development -of verified mathematical proofs, and the \emph{program extractor} which -synthesizes computer programs obeying their formal specifications, -written as logical assertions in the language. - -The logical language used by {\Coq} is a variety of type theory, -called the \emph{Calculus of Inductive Constructions}. Without going -back to Leibniz and Boole, we can date the creation of what is now -called mathematical logic to the work of Frege and Peano at the turn -of the century. The discovery of antinomies in the free use of -predicates or comprehension principles prompted Russell to restrict -predicate calculus with a stratification of \emph{types}. This effort -culminated with \emph{Principia Mathematica}, the first systematic -attempt at a formal foundation of mathematics. A simplification of -this system along the lines of simply typed $\lambda$-calculus -occurred with Church's \emph{Simple Theory of Types}. The -$\lambda$-calculus notation, originally used for expressing -functionality, could also be used as an encoding of natural deduction -proofs. This Curry-Howard isomorphism was used by N. de Bruijn in the -\emph{Automath} project, the first full-scale attempt to develop and -mechanically verify mathematical proofs. This effort culminated with -Jutting's verification of Landau's \emph{Grundlagen} in the 1970's. -Exploiting this Curry-Howard isomorphism, notable achievements in -proof theory saw the emergence of two type-theoretic frameworks; the -first one, Martin-L\"of's \emph{Intuitionistic Theory of Types}, -attempts a new foundation of mathematics on constructive principles. -The second one, Girard's polymorphic $\lambda$-calculus $F_\omega$, is -a very strong functional system in which we may represent higher-order -logic proof structures. Combining both systems in a higher-order -extension of the Automath languages, T. Coquand presented in 1985 the -first version of the \emph{Calculus of Constructions}, CoC. This strong -logical system allowed powerful axiomatizations, but direct inductive -definitions were not possible, and inductive notions had to be defined -indirectly through functional encodings, which introduced -inefficiencies and awkwardness. The formalism was extended in 1989 by -T. Coquand and C. Paulin with primitive inductive definitions, leading -to the current \emph{Calculus of Inductive Constructions}. This -extended formalism is not rigorously defined here. Rather, numerous -concrete examples are discussed. We refer the interested reader to -relevant research papers for more information about the formalism, its -meta-theoretic properties, and semantics. However, it should not be -necessary to understand this theoretical material in order to write -specifications. It is possible to understand the Calculus of Inductive -Constructions at a higher level, as a mixture of predicate calculus, -inductive predicate definitions presented as typed PROLOG, and -recursive function definitions close to the language ML. - -Automated theorem-proving was pioneered in the 1960's by Davis and -Putnam in propositional calculus. A complete mechanization (in the -sense of a semi-decision procedure) of classical first-order logic was -proposed in 1965 by J.A. Robinson, with a single uniform inference -rule called \emph{resolution}. Resolution relies on solving equations -in free algebras (i.e. term structures), using the \emph{unification - algorithm}. Many refinements of resolution were studied in the -1970's, but few convincing implementations were realized, except of -course that PROLOG is in some sense issued from this effort. A less -ambitious approach to proof development is computer-aided -proof-checking. The most notable proof-checkers developed in the -1970's were LCF, designed by R. Milner and his colleagues at U. -Edinburgh, specialized in proving properties about denotational -semantics recursion equations, and the Boyer and Moore theorem-prover, -an automation of primitive recursion over inductive data types. While -the Boyer-Moore theorem-prover attempted to synthesize proofs by a -combination of automated methods, LCF constructed its proofs through -the programming of \emph{tactics}, written in a high-level functional -meta-language, ML. - -The salient feature which clearly distinguishes our proof assistant -from say LCF or Boyer and Moore's, is its possibility to extract -programs from the constructive contents of proofs. This computational -interpretation of proof objects, in the tradition of Bishop's -constructive mathematics, is based on a realizability interpretation, -in the sense of Kleene, due to C. Paulin. The user must just mark his -intention by separating in the logical statements the assertions -stating the existence of a computational object from the logical -assertions which specify its properties, but which may be considered -as just comments in the corresponding program. Given this information, -the system automatically extracts a functional term from a consistency -proof of its specifications. This functional term may be in turn -compiled into an actual computer program. This methodology of -extracting programs from proofs is a revolutionary paradigm for -software engineering. Program synthesis has long been a theme of -research in artificial intelligence, pioneered by R. Waldinger. The -Tablog system of Z. Manna and R. Waldinger allows the deductive -synthesis of functional programs from proofs in tableau form of their -specifications, written in a variety of first-order logic. Development -of a systematic \emph{programming logic}, based on extensions of -Martin-L\"of's type theory, was undertaken at Cornell U. by the Nuprl -team, headed by R. Constable. The first actual program extractor, PX, -was designed and implemented around 1985 by S. Hayashi from Kyoto -University. It allows the extraction of a LISP program from a proof -in a logical system inspired by the logical formalisms of S. Feferman. -Interest in this methodology is growing in the theoretical computer -science community. We can foresee the day when actual computer systems -used in applications will contain certified modules, automatically -generated from a consistency proof of their formal specifications. We -are however still far from being able to use this methodology in a -smooth interaction with the standard tools from software engineering, -i.e. compilers, linkers, run-time systems taking advantage of special -hardware, debuggers, and the like. We hope that {\Coq} can be of use -to researchers interested in experimenting with this new methodology. - -A first implementation of CoC was started in 1984 by G. Huet and T. -Coquand. Its implementation language was CAML, a functional -programming language from the ML family designed at INRIA in -Rocquencourt. The core of this system was a proof-checker for CoC seen -as a typed $\lambda$-calculus, called the \emph{Constructive Engine}. -This engine was operated through a high-level notation permitting the -declaration of axioms and parameters, the definition of mathematical -types and objects, and the explicit construction of proof objects -encoded as $\lambda$-terms. A section mechanism, designed and -implemented by G. Dowek, allowed hierarchical developments of -mathematical theories. This high-level language was called the -\emph{Mathematical Vernacular}. Furthermore, an interactive -\emph{Theorem Prover} permitted the incremental construction of proof -trees in a top-down manner, subgoaling recursively and backtracking -from dead-alleys. The theorem prover executed tactics written in CAML, -in the LCF fashion. A basic set of tactics was predefined, which the -user could extend by his own specific tactics. This system (Version -4.10) was released in 1989. Then, the system was extended to deal -with the new calculus with inductive types by C. Paulin, with -corresponding new tactics for proofs by induction. A new standard set -of tactics was streamlined, and the vernacular extended for tactics -execution. A package to compile programs extracted from proofs to -actual computer programs in CAML or some other functional language was -designed and implemented by B. Werner. A new user-interface, relying -on a CAML-X interface by D. de Rauglaudre, was designed and -implemented by A. Felty. It allowed operation of the theorem-prover -through the manipulation of windows, menus, mouse-sensitive buttons, -and other widgets. This system (Version 5.6) was released in 1991. - -\Coq{} was ported to the new implementation Caml-light of X. Leroy and -D. Doligez by D. de Rauglaudre (Version 5.7) in 1992. A new version -of \Coq{} was then coordinated by C. Murthy, with new tools designed -by C. Parent to prove properties of ML programs (this methodology is -dual to program extraction) and a new user-interaction loop. This -system (Version 5.8) was released in May 1993. A Centaur interface -\textsc{CTCoq} was then developed by Y. Bertot from the Croap project -from INRIA-Sophia-Antipolis. - -In parallel, G. Dowek and H. Herbelin developed a new proof engine, -allowing the general manipulation of existential variables -consistently with dependent types in an experimental version of \Coq{} -(V5.9). - -The version V5.10 of \Coq{} is based on a generic system for -manipulating terms with binding operators due to Chet Murthy. A new -proof engine allows the parallel development of partial proofs for -independent subgoals. The structure of these proof trees is a mixed -representation of derivation trees for the Calculus of Inductive -Constructions with abstract syntax trees for the tactics scripts, -allowing the navigation in a proof at various levels of details. The -proof engine allows generic environment items managed in an -object-oriented way. This new architecture, due to C. Murthy, -supports several new facilities which make the system easier to extend -and to scale up: - -\begin{itemize} -\item User-programmable tactics are allowed -\item It is possible to separately verify development modules, and to - load their compiled images without verifying them again - a quick - relocation process allows their fast loading -\item A generic parsing scheme allows user-definable notations, with a - symmetric table-driven pretty-printer -\item Syntactic definitions allow convenient abbreviations -\item A limited facility of meta-variables allows the automatic - synthesis of certain type expressions, allowing generic notations - for e.g. equality, pairing, and existential quantification. -\end{itemize} - -In the Fall of 1994, C. Paulin-Mohring replaced the structure of -inductively defined types and families by a new structure, allowing -the mutually recursive definitions. P. Manoury implemented a -translation of recursive definitions into the primitive recursive -style imposed by the internal recursion operators, in the style of the -ProPre system. C. Mu{\~n}oz implemented a decision procedure for -intuitionistic propositional logic, based on results of R. Dyckhoff. -J.C. Filli{\^a}tre implemented a decision procedure for first-order -logic without contraction, based on results of J. Ketonen and R. -Weyhrauch. Finally C. Murthy implemented a library of inversion -tactics, relieving the user from tedious definitions of ``inversion -predicates''. - -\begin{flushright} -Rocquencourt, Feb. 1st 1995\\ -Gérard Huet -\end{flushright} - -\section*{Credits: addendum for version 6.1} -%\addcontentsline{toc}{section}{Credits: addendum for version V6.1} - -The present version 6.1 of \Coq{} is based on the V5.10 architecture. It -was ported to the new language Objective Caml by Bruno Barras. The -underlying framework has slightly changed and allows more conversions -between sorts. - -The new version provides powerful tools for easier developments. - -Cristina Cornes designed an extension of the \Coq{} syntax to allow -definition of terms using a powerful pattern-matching analysis in the -style of ML programs. - -Amokrane Saïbi wrote a mechanism to simulate -inheritance between types families extending a proposal by Peter -Aczel. He also developed a mechanism to automatically compute which -arguments of a constant may be inferred by the system and consequently -do not need to be explicitly written. - -Yann Coscoy designed a command which explains a proof term using -natural language. Pierre Cr{\'e}gut built a new tactic which solves -problems in quantifier-free Presburger Arithmetic. Both -functionalities have been integrated to the \Coq{} system by Hugo -Herbelin. - -Samuel Boutin designed a tactic for simplification of commutative -rings using a canonical set of rewriting rules and equality modulo -associativity and commutativity. - -Finally the organisation of the \Coq{} distribution has been supervised -by Jean-Christophe Filliâtre with the help of Judicaël Courant -and Bruno Barras. - -\begin{flushright} -Lyon, Nov. 18th 1996\\ -Christine Paulin -\end{flushright} - -\section*{Credits: addendum for version 6.2} -%\addcontentsline{toc}{section}{Credits: addendum for version V6.2} - -In version 6.2 of \Coq{}, the parsing is done using camlp4, a -preprocessor and pretty-printer for CAML designed by Daniel de -Rauglaudre at INRIA. Daniel de Rauglaudre made the first adaptation -of \Coq{} for camlp4, this work was continued by Bruno Barras who also -changed the structure of \Coq{} abstract syntax trees and the primitives -to manipulate them. The result of -these changes is a faster parsing procedure with greatly improved -syntax-error messages. The user-interface to introduce grammar or -pretty-printing rules has also changed. - -Eduardo Giménez redesigned the internal -tactic libraries, giving uniform names -to Caml functions corresponding to \Coq{} tactic names. - -Bruno Barras wrote new more efficient reductions functions. - -Hugo Herbelin introduced more uniform notations in the \Coq{} -specification language: the definitions by fixpoints and -pattern-matching have a more readable syntax. Patrick Loiseleur -introduced user-friendly notations for arithmetic expressions. - -New tactics were introduced: Eduardo Giménez improved a mechanism to -introduce macros for tactics, and designed special tactics for -(co)inductive definitions; Patrick Loiseleur designed a tactic to -simplify polynomial expressions in an arbitrary commutative ring which -generalizes the previous tactic implemented by Samuel Boutin. -Jean-Christophe Filli\^atre introduced a tactic for refining a goal, -using a proof term with holes as a proof scheme. - -David Delahaye designed the \textsf{SearchIsos} tool to search an -object in the library given its type (up to isomorphism). - -Henri Laulhère produced the \Coq{} distribution for the Windows environment. - -Finally, Hugo Herbelin was the main coordinator of the \Coq{} -documentation with principal contributions by Bruno Barras, David Delahaye, -Jean-Christophe Filli\^atre, Eduardo -Giménez, Hugo Herbelin and Patrick Loiseleur. - -\begin{flushright} -Orsay, May 4th 1998\\ -Christine Paulin -\end{flushright} - -\section*{Credits: addendum for version 6.3} -The main changes in version V6.3 was the introduction of a few new tactics -and the extension of the guard condition for fixpoint definitions. - - -B. Barras extended the unification algorithm to complete partial terms -and solved various tricky bugs related to universes.\\ -D. Delahaye developed the \texttt{AutoRewrite} tactic. He also designed the new -behavior of \texttt{Intro} and provided the tacticals \texttt{First} and -\texttt{Solve}.\\ -J.-C. Filli\^atre developed the \texttt{Correctness} tactic.\\ -E. Gim\'enez extended the guard condition in fixpoints.\\ -H. Herbelin designed the new syntax for definitions and extended the -\texttt{Induction} tactic.\\ -P. Loiseleur developed the \texttt{Quote} tactic and -the new design of the \texttt{Auto} -tactic, he also introduced the index of -errors in the documentation.\\ -C. Paulin wrote the \texttt{Focus} command and introduced -the reduction functions in definitions, this last feature -was proposed by J.-F. Monin from CNET Lannion. - -\begin{flushright} -Orsay, Dec. 1999\\ -Christine Paulin -\end{flushright} - -%\newpage - -\section*{Credits: versions 7} - -The version V7 is a new implementation started in September 1999 by -Jean-Christophe Filliâtre. This is a major revision with respect to -the internal architecture of the system. The \Coq{} version 7.0 was -distributed in March 2001, version 7.1 in September 2001, version -7.2 in January 2002, version 7.3 in May 2002 and version 7.4 in -February 2003. - -Jean-Christophe Filliâtre designed the architecture of the new system, he -introduced a new representation for environments and wrote a new kernel -for type-checking terms. His approach was to use functional -data-structures in order to get more sharing, to prepare the addition -of modules and also to get closer to a certified kernel. - -Hugo Herbelin introduced a new structure of terms with local -definitions. He introduced ``qualified'' names, wrote a new -pattern-matching compilation algorithm and designed a more compact -algorithm for checking the logical consistency of universes. He -contributed to the simplification of {\Coq} internal structures and the -optimisation of the system. He added basic tactics for forward -reasoning and coercions in patterns. - -David Delahaye introduced a new language for tactics. General tactics -using pattern-matching on goals and context can directly be written -from the {\Coq} toplevel. He also provided primitives for the design -of user-defined tactics in \textsc{Caml}. - -Micaela Mayero contributed the library on real numbers. -Olivier Desmettre extended this library with axiomatic -trigonometric functions, square, square roots, finite sums, Chasles -property and basic plane geometry. - -Jean-Christophe Filliâtre and Pierre Letouzey redesigned a new -extraction procedure from \Coq{} terms to \textsc{Caml} or -\textsc{Haskell} programs. This new -extraction procedure, unlike the one implemented in previous version -of \Coq{} is able to handle all terms in the Calculus of Inductive -Constructions, even involving universes and strong elimination. P. -Letouzey adapted user contributions to extract ML programs when it was -sensible. -Jean-Christophe Filliâtre wrote \verb=coqdoc=, a documentation -tool for {\Coq} libraries usable from version 7.2. - -Bruno Barras improved the reduction algorithms efficiency and -the confidence level in the correctness of {\Coq} critical type-checking -algorithm. - -Yves Bertot designed the \texttt{SearchPattern} and -\texttt{SearchRewrite} tools and the support for the \textsc{pcoq} interface -(\url{http://www-sop.inria.fr/lemme/pcoq/}). - -Micaela Mayero and David Delahaye introduced {\tt Field}, a decision tactic for commutative fields. - -Christine Paulin changed the elimination rules for empty and singleton -propositional inductive types. - -Loïc Pottier developed {\tt Fourier}, a tactic solving linear inequalities on real numbers. - -Pierre Crégut developed a new version based on reflexion of the {\tt Omega} -decision tactic. - -Claudio Sacerdoti Coen designed an XML output for the {\Coq} -modules to be used in the Hypertextual Electronic Library of -Mathematics (HELM cf \url{http://www.cs.unibo.it/helm}). - -A library for efficient representation of finite maps using binary trees -contributed by Jean Goubault was integrated in the basic theories. - -Pierre Courtieu developed a command and a tactic to reason on the -inductive structure of recursively defined functions. - -Jacek Chrz\k{a}szcz designed and implemented the module system of -{\Coq} whose foundations are in Judicaël Courant's PhD thesis. - -\bigskip - -The development was coordinated by C. Paulin. - -Many discussions within the Démons team and the LogiCal project -influenced significantly the design of {\Coq} especially with -%J. Chrz\k{a}szcz, P. Courtieu, -J. Courant, J. Duprat, J. Goubault, A. Miquel, -C. Marché, B. Monate and B. Werner. - -Intensive users suggested improvements of the system : -Y. Bertot, L. Pottier, L. Théry, P. Zimmerman from INRIA, -C. Alvarado, P. Crégut, J.-F. Monin from France Telecom R \& D. -\begin{flushright} -Orsay, May. 2002\\ -Hugo Herbelin \& Christine Paulin -\end{flushright} - -\section*{Credits: version 8.0} - -{\Coq} version 8 is a major revision of the {\Coq} proof assistant. -First, the underlying logic is slightly different. The so-called {\em -impredicativity} of the sort {\tt Set} has been dropped. The main -reason is that it is inconsistent with the principle of description -which is quite a useful principle for formalizing %classical -mathematics within classical logic. Moreover, even in an constructive -setting, the impredicativity of {\tt Set} does not add so much in -practice and is even subject of criticism from a large part of the -intuitionistic mathematician community. Nevertheless, the -impredicativity of {\tt Set} remains optional for users interested in -investigating mathematical developments which rely on it. - -Secondly, the concrete syntax of terms has been completely -revised. The main motivations were - -\begin{itemize} -\item a more uniform, purified style: all constructions are now lowercase, - with a functional programming perfume (e.g. abstraction is now - written {\tt fun}), and more directly accessible to the novice - (e.g. dependent product is now written {\tt forall} and allows - omission of types). Also, parentheses and are no longer mandatory - for function application. -\item extensibility: some standard notations (e.g. ``<'' and ``>'') were - incompatible with the previous syntax. Now all standard arithmetic - notations (=, +, *, /, <, <=, ... and more) are directly part of the - syntax. -\end{itemize} - -Together with the revision of the concrete syntax, a new mechanism of -{\em interpretation scopes} permits to reuse the same symbols -(typically +, -, *, /, <, <=) in various mathematical theories without -any ambiguities for {\Coq}, leading to a largely improved readability of -{\Coq} scripts. New commands to easily add new symbols are also -provided. - -Coming with the new syntax of terms, a slight reform of the tactic -language and of the language of commands has been carried out. The -purpose here is a better uniformity making the tactics and commands -easier to use and to remember. - -Thirdly, a restructuration and uniformisation of the standard library -of {\Coq} has been performed. There is now just one Leibniz' equality -usable for all the different kinds of {\Coq} objects. Also, the set of -real numbers now lies at the same level as the sets of natural and -integer numbers. Finally, the names of the standard properties of -numbers now follow a standard pattern and the symbolic -notations for the standard definitions as well. - -The fourth point is the release of \CoqIDE{}, a new graphical -gtk2-based interface fully integrated to {\Coq}. Close in style from -the Proof General Emacs interface, it is faster and its integration -with {\Coq} makes interactive developments more friendly. All -mathematical Unicode symbols are usable within \CoqIDE{}. - -Finally, the module system of {\Coq} completes the picture of {\Coq} -version 8.0. Though released with an experimental status in the previous -version 7.4, it should be considered as a salient feature of the new -version. - -Besides, {\Coq} comes with its load of novelties and improvements: new -or improved tactics (including a new tactic for solving first-order -statements), new management commands, extended libraries. - -\bigskip - -Bruno Barras and Hugo Herbelin have been the main contributors of the -reflexion and the implementation of the new syntax. The smart -automatic translator from old to new syntax released with {\Coq} is also -their work with contributions by Olivier Desmettre. - -Hugo Herbelin is the main designer and implementor of the notion of -interpretation scopes and of the commands for easily adding new notations. - -Hugo Herbelin is the main implementor of the restructuration of the -standard library. - -Pierre Corbineau is the main designer and implementor of the new -tactic for solving first-order statements in presence of inductive -types. He is also the maintainer of the non-domain specific automation -tactics. - -Benjamin Monate is the developer of the \CoqIDE{} graphical -interface with contributions by Jean-Christophe Filliâtre, Pierre -Letouzey, Claude Marché and Bruno Barras. - -Claude Marché coordinated the edition of the Reference Manual for - \Coq{} V8.0. - -Pierre Letouzey and Jacek Chrz\k{a}szcz respectively maintained the -extraction tool and module system of {\Coq}. - -Jean-Christophe Filliâtre, Pierre Letouzey, Hugo Herbelin and -contributors from Sophia-Antipolis and Nijmegen participated to the -extension of the library. - -Julien Narboux built a NSIS-based automatic {\Coq} installation tool for -the Windows platform. - -Hugo Herbelin and Christine Paulin coordinated the development which -was under the responsability of Christine Paulin. - -\begin{flushright} -Palaiseau \& Orsay, Apr. 2004\\ -Hugo Herbelin \& Christine Paulin\\ -(updated Apr. 2006) -\end{flushright} - -\section*{Credits: version 8.1} - -{\Coq} version 8.1 adds various new functionalities. - -Benjamin Grégoire implemented an alternative algorithm to check the -convertibility of terms in the {\Coq} type-checker. This alternative -algorithm works by compilation to an efficient bytecode that is -interpreted in an abstract machine similar to Xavier Leroy's ZINC -machine. Convertibility is performed by comparing the normal -forms. This alternative algorithm is specifically interesting for -proofs by reflection. More generally, it is convenient in case of -intensive computations. - -Christine Paulin implemented an extension of inductive types allowing -recursively non uniform parameters. Hugo Herbelin implemented -sort-polymorphism for inductive types. - -Claudio Sacerdoti Coen improved the tactics for rewriting on arbitrary -compatible equivalence relations. He also generalized rewriting to -arbitrary transition systems. - -Claudio Sacerdoti Coen added new features to the module system. - -Benjamin Grégoire, Assia Mahboubi and Bruno Barras developed a new -more efficient and more general simplification algorithm on rings and -semi-rings. - -Laurent Théry and Bruno Barras developed a new significantly more efficient -simplification algorithm on fields. - -Hugo Herbelin, Pierre Letouzey, Julien Forest, Julien Narboux and -Claudio Sacerdoti Coen added new tactic features. - -Hugo Herbelin implemented matching on disjunctive patterns. - -New mechanisms made easier the communication between {\Coq} and external -provers. Nicolas Ayache and Jean-Christophe Filliâtre implemented -connections with the provers {\sc cvcl}, {\sc Simplify} and {\sc -zenon}. Hugo Herbelin implemented an experimental protocol for calling -external tools from the tactic language. - -Matthieu Sozeau developed \textsc{Russell}, an experimental language -to specify the behavior of programs with subtypes. - -A mechanism to automatically use some specific tactic to solve -unresolved implicit has been implemented by Hugo Herbelin. - -Laurent Théry's contribution on strings and Pierre Letouzey and -Jean-Christophe Filliâtre's contribution on finite maps have been -integrated to the {\Coq} standard library. Pierre Letouzey developed a -library about finite sets ``à la Objective Caml''. With Jean-Marc -Notin, he extended the library on lists. Pierre Letouzey's -contribution on rational numbers has been integrated and extended.. - -Pierre Corbineau extended his tactic for solving first-order -statements. He wrote a reflection-based intuitionistic tautology -solver. - -Pierre Courtieu, Julien Forest and Yves Bertot added extra support to -reason on the inductive structure of recursively defined functions. - -Jean-Marc Notin significantly contributed to the general maintenance -of the system. He also took care of {\textsf{coqdoc}}. - -Pierre Castéran contributed to the documentation of (co-)inductive -types and suggested improvements to the libraries. - -Pierre Corbineau implemented a declarative mathematical proof -language, usable in combination with the tactic-based style of proof. - -Finally, many users suggested improvements of the system through the -Coq-Club mailing list and bug-tracker systems, especially user groups -from INRIA Rocquencourt, Radbout University, University of -Pennsylvania and Yale University. - -\enlargethispage{\baselineskip} -\begin{flushright} -Palaiseau, July 2006\\ -Hugo Herbelin -\end{flushright} - -\section*{Credits: version 8.2} - -{\Coq} version 8.2 adds new features, new libraries and -improves on many various aspects. - -Regarding the language of Coq, the main novelty is the introduction by -Matthieu Sozeau of a package of commands providing Haskell-style -type classes. Type classes, that come with a few convenient features -such as type-based resolution of implicit arguments, plays a new role -of landmark in the architecture of Coq with respect to automatization. -For instance, thanks to type classes support, Matthieu Sozeau could -implement a new resolution-based version of the tactics dedicated to -rewriting on arbitrary transitive relations. - -Another major improvement of Coq 8.2 is the evolution of the -arithmetic libraries and of the tools associated to them. Benjamin -Grégoire and Laurent Théry contributed a modular library for building -arbitrarily large integers from bounded integers while Evgeny Makarov -contributed a modular library of abstract natural and integer -arithmetics together with a few convenient tactics. On his side, -Pierre Letouzey made numerous extensions to the arithmetic libraries on -$\mathbb{Z}$ and $\mathbb{Q}$, including extra support for -automatization in presence of various number-theory concepts. - -Frédéric Besson contributed a reflexive tactic based on -Krivine-Stengle Positivstellensatz (the easy way) for validating -provability of systems of inequalities. The platform is flexible enough -to support the validation of any algorithm able to produce a -``certificate'' for the Positivstellensatz and this covers the case of -Fourier-Motzkin (for linear systems in $\mathbb{Q}$ and $\mathbb{R}$), -Fourier-Motzkin with cutting planes (for linear systems in -$\mathbb{Z}$) and sum-of-squares (for non-linear systems). Evgeny -Makarov made the platform generic over arbitrary ordered rings. - -Arnaud Spiwack developed a library of 31-bits machine integers and, -relying on Benjamin Grégoire and Laurent Théry's library, delivered a -library of unbounded integers in base $2^{31}$. As importantly, he -developed a notion of ``retro-knowledge'' so as to safely extend the -kernel-located bytecode-based efficient evaluation algorithm of Coq -version 8.1 to use 31-bits machine arithmetics for efficiently -computing with the library of integers he developed. - -Beside the libraries, various improvements contributed to provide a -more comfortable end-user language and more expressive tactic -language. Hugo Herbelin and Matthieu Sozeau improved the -pattern-matching compilation algorithm (detection of impossible -clauses in pattern-matching, automatic inference of the return -type). Hugo Herbelin, Pierre Letouzey and Matthieu Sozeau contributed -various new convenient syntactic constructs and new tactics or tactic -features: more inference of redundant information, better unification, -better support for proof or definition by fixpoint, more expressive -rewriting tactics, better support for meta-variables, more convenient -notations, ... - -Élie Soubiran improved the module system, adding new features (such as -an ``include'' command) and making it more flexible and more -general. He and Pierre Letouzey improved the support for modules in -the extraction mechanism. - -Matthieu Sozeau extended the \textsc{Russell} language, ending in an -convenient way to write programs of given specifications, Pierre -Corbineau extended the Mathematical Proof Language and the -automatization tools that accompany it, Pierre Letouzey supervised and -extended various parts the standard library, Stéphane Glondu -contributed a few tactics and improvements, Jean-Marc Notin provided -help in debugging, general maintenance and {\tt coqdoc} support, -Vincent Siles contributed extensions of the {\tt Scheme} command and -of {\tt injection}. - -Bruno Barras implemented the {\tt coqchk} tool: this is a stand-alone -type-checker that can be used to certify {\tt .vo} files. Especially, -as this verifier runs in a separate process, it is granted not to be -``hijacked'' by virtually malicious extensions added to {\Coq}. - -Yves Bertot, Jean-Christophe Filliâtre, Pierre Courtieu and -Julien Forest acted as maintainers of features they implemented in -previous versions of Coq. - -Julien Narboux contributed to CoqIDE. -Nicolas Tabareau made the adaptation of the interface of the old -``setoid rewrite'' tactic to the new version. Lionel Mamane worked on -the interaction between Coq and its external interfaces. With Samuel -Mimram, he also helped making Coq compatible with recent software -tools. Russell O'Connor, Cezary Kaliscyk, Milad Niqui contributed to -improved the libraries of integers, rational, and real numbers. We -also thank many users and partners for suggestions and feedback, in -particular Pierre Castéran and Arthur Charguéraud, the INRIA Marelle -team, Georges Gonthier and the INRIA-Microsoft Mathematical Components team, -the Foundations group at Radbout university in Nijmegen, reporters of bugs -and participants to the Coq-Club mailing list. - -\begin{flushright} -Palaiseau, June 2008\\ -Hugo Herbelin\\ -\end{flushright} - -\section*{Credits: version 8.3} - -{\Coq} version 8.3 is before all a transition version with refinements -or extensions of the existing features and libraries and a new tactic -{\tt nsatz} based on Hilbert's Nullstellensatz for deciding systems of -equations over rings. - -With respect to libraries, the main evolutions are due to Pierre -Letouzey with a rewriting of the library of finite sets {\tt FSets} -and a new round of evolutions in the modular development of arithmetic -(library {\tt Numbers}). The reason for making {\tt FSets} evolve is -that the computational and logical contents were quite intertwined in -the original implementation, leading in some cases to longer -computations than expected and this problem is solved in the new {\tt - MSets} implementation. As for the modular arithmetic library, it was -only dealing with the basic arithmetic operators in the former version -and its current extension adds the standard theory of the division, -min and max functions, all made available for free to any -implementation of $\mathbb{N}$, $\mathbb{Z}$ or -$\mathbb{Z}/n\mathbb{Z}$. - -The main other evolutions of the library are due to Hugo Herbelin who -made a revision of the sorting library (includingh a certified -merge-sort) and to Guillaume Melquiond who slightly revised and -cleaned up the library of reals. - -The module system evolved significantly. Besides the resolution of -some efficiency issues and a more flexible construction of module -types, Élie Soubiran brought a new model of name equivalence, the -$\Delta$-equivalence, which respects as much as possible the names -given by the users. He also designed with Pierre Letouzey a new -convenient operator \verb!<+! for nesting functor application, what -provides a light notation for inheriting the properties of cascading -modules. - -The new tactic {\tt nsatz} is due to Loïc Pottier. It works by -computing Gr\"obner bases. Regarding the existing tactics, various -improvements have been done by Matthieu Sozeau, Hugo Herbelin and -Pierre Letouzey. - -Matthieu Sozeau extended and refined the type classes and {\tt - Program} features (the {\sc Russell} language). Pierre Letouzey -maintained and improved the extraction mechanism. Bruno Barras and -\'Elie Soubiran maintained the Coq checker, Julien Forest maintained -the {\tt Function} mechanism for reasoning over recursively defined -functions. Matthieu Sozeau, Hugo Herbelin and Jean-Marc Notin -maintained {\tt coqdoc}. Frédéric Besson maintained the {\sc - Micromega} plateform for deciding systems of inequalities. Pierre -Courtieu maintained the support for the Proof General Emacs -interface. Claude Marché maintained the plugin for calling external -provers ({\tt dp}). Yves Bertot made some improvements to the -libraries of lists and integers. Matthias Puech improved the search -functions. Guillaume Melquiond usefully contributed here and -there. Yann Régis-Gianas grounded the support for Unicode on a more -standard and more robust basis. - -Though invisible from outside, Arnaud Spiwack improved the general -process of management of existential variables. Pierre Letouzey and -Stéphane Glondu improved the compilation scheme of the Coq archive. -Vincent Gross provided support to CoqIDE. Jean-Marc Notin provided -support for benchmarking and archiving. - -Many users helped by reporting problems, providing patches, suggesting -improvements or making useful comments, either on the bug tracker or -on the Coq-club mailing list. This includes but not exhaustively -Cédric Auger, Arthur Charguéraud, François Garillot, Georges Gonthier, -Robin Green, Stéphane Lescuyer, Eelis van der Weegen,~... - -Though not directly related to the implementation, special thanks are -going to Yves Bertot, Pierre Castéran, Adam Chlipala, and Benjamin -Pierce for the excellent teaching materials they provided. - -\begin{flushright} -Paris, April 2010\\ -Hugo Herbelin\\ -\end{flushright} - -%new Makefile - -%\newpage - -% Integration of ZArith lemmas from Sophia and Nijmegen. - - -% $Id: RefMan-pre.tex 13271 2010-07-08 18:10:54Z herbelin $ - -%%% Local Variables: -%%% mode: latex -%%% TeX-master: "Reference-Manual" -%%% End: diff --git a/doc/refman/RefMan-pro.tex b/doc/refman/RefMan-pro.tex deleted file mode 100644 index 3a8936eb..00000000 --- a/doc/refman/RefMan-pro.tex +++ /dev/null @@ -1,385 +0,0 @@ -\chapter[Proof handling]{Proof handling\index{Proof editing} -\label{Proof-handling}} - -In \Coq's proof editing mode all top-level commands documented in -Chapter~\ref{Vernacular-commands} remain available -and the user has access to specialized commands dealing with proof -development pragmas documented in this section. He can also use some -other specialized commands called {\em tactics}. They are the very -tools allowing the user to deal with logical reasoning. They are -documented in Chapter~\ref{Tactics}.\\ -When switching in editing proof mode, the prompt -\index{Prompt} -{\tt Coq <} is changed into {\tt {\ident} <} where {\ident} is the -declared name of the theorem currently edited. - -At each stage of a proof development, one has a list of goals to -prove. Initially, the list consists only in the theorem itself. After -having applied some tactics, the list of goals contains the subgoals -generated by the tactics. - -To each subgoal is associated a number of -hypotheses called the {\em \index*{local context}} of the goal. -Initially, the local context contains the local variables and -hypotheses of the current section (see Section~\ref{Variable}) and the -local variables and hypotheses of the theorem statement. It is -enriched by the use of certain tactics (see e.g. {\tt intro} in -Section~\ref{intro}). - -When a proof is completed, the message {\tt Proof completed} is -displayed. One can then register this proof as a defined constant in the -environment. Because there exists a correspondence between proofs and -terms of $\lambda$-calculus, known as the {\em Curry-Howard -isomorphism} \cite{How80,Bar91,Gir89,Hue89}, \Coq~ stores proofs as -terms of {\sc Cic}. Those terms are called {\em proof - terms}\index{Proof term}. - -It is possible to edit several proofs in parallel: see Section -\ref{Resume}. - -\ErrMsg When one attempts to use a proof editing command out of the -proof editing mode, \Coq~ raises the error message : \errindex{No focused - proof}. - -\section{Switching on/off the proof editing mode} - -The proof editing mode is entered by asserting a statement, which -typically is the assertion of a theorem: - -\begin{quote} -{\tt Theorem {\ident} \zeroone{\binders} : {\form}.\comindex{Theorem} -\label{Theorem}} -\end{quote} - -The list of assertion commands is given in -Section~\ref{Assertions}. The command {\tt Goal} can also be used. - -\subsection[Goal {\form}.]{\tt Goal {\form}.\comindex{Goal}\label{Goal}} - -This is intended for quick assertion of statements, without knowing in -advance which name to give to the assertion, typically for quick -testing of the provability of a statement. If the proof of the -statement is eventually completed and validated, the statement is then -bound to the name {\tt Unnamed\_thm} (or a variant of this name not -already used for another statement). - -\subsection[\tt Qed.]{\tt Qed.\comindex{Qed}\label{Qed}} -This command is available in interactive editing proof mode when the -proof is completed. Then {\tt Qed} extracts a proof term from the -proof script, switches back to {\Coq} top-level and attaches the -extracted proof term to the declared name of the original goal. This -name is added to the environment as an {\tt Opaque} constant. - -\begin{ErrMsgs} -\item \errindex{Attempt to save an incomplete proof} -%\item \ident\ \errindex{already exists}\\ -% The implicit name is already defined. You have then to provide -% explicitly a new name (see variant 3 below). -\item Sometimes an error occurs when building the proof term, -because tactics do not enforce completely the term construction -constraints. - -The user should also be aware of the fact that since the proof term is -completely rechecked at this point, one may have to wait a while when -the proof is large. In some exceptional cases one may even incur a -memory overflow. -\end{ErrMsgs} - -\begin{Variants} - -\item {\tt Defined.} -\comindex{Defined} -\label{Defined} - - Defines the proved term as a transparent constant. - -\item {\tt Save.} -\comindex{Save} - - This is a deprecated equivalent to {\tt Qed}. - -\item {\tt Save {\ident}.} - - Forces the name of the original goal to be {\ident}. This command - (and the following ones) can only be used if the original goal has - been opened using the {\tt Goal} command. - -\item {\tt Save Theorem {\ident}.} \\ - {\tt Save Lemma {\ident}.} \\ - {\tt Save Remark {\ident}.}\\ - {\tt Save Fact {\ident}.} - {\tt Save Corollary {\ident}.} - {\tt Save Proposition {\ident}.} - - Are equivalent to {\tt Save {\ident}.} -\end{Variants} - -\subsection[\tt Admitted.]{\tt Admitted.\comindex{Admitted}\label{Admitted}} -This command is available in interactive editing proof mode to give up -the current proof and declare the initial goal as an axiom. - -\subsection[\tt Proof {\term}.]{\tt Proof {\term}.\comindex{Proof} -\label{BeginProof}} -This command applies in proof editing mode. It is equivalent to {\tt - exact {\term}; Save.} That is, you have to give the full proof in -one gulp, as a proof term (see Section~\ref{exact}). - -\variant {\tt Proof.} - - Is a noop which is useful to delimit the sequence of tactic commands - which start a proof, after a {\tt Theorem} command. It is a good - practice to use {\tt Proof.} as an opening parenthesis, closed in - the script with a closing {\tt Qed.} - -\SeeAlso {\tt Proof with {\tac}.} in Section~\ref{ProofWith}. - -\subsection[\tt Abort.]{\tt Abort.\comindex{Abort}} - -This command cancels the current proof development, switching back to -the previous proof development, or to the \Coq\ toplevel if no other -proof was edited. - -\begin{ErrMsgs} -\item \errindex{No focused proof (No proof-editing in progress)} -\end{ErrMsgs} - -\begin{Variants} - -\item {\tt Abort {\ident}.} - - Aborts the editing of the proof named {\ident}. - -\item {\tt Abort All.} - - Aborts all current goals, switching back to the \Coq\ toplevel. - -\end{Variants} - -%%%% -\subsection[\tt Suspend.]{\tt Suspend.\comindex{Suspend}} - -This command applies in proof editing mode. It switches back to the -\Coq\ toplevel, but without canceling the current proofs. - -%%%% -\subsection[\tt Resume.]{\tt Resume.\comindex{Resume}\label{Resume}} - -This commands switches back to the editing of the last edited proof. - -\begin{ErrMsgs} -\item \errindex{No proof-editing in progress} -\end{ErrMsgs} - -\begin{Variants} - -\item {\tt Resume {\ident}.} - - Restarts the editing of the proof named {\ident}. This can be used - to navigate between currently edited proofs. - -\end{Variants} - -\begin{ErrMsgs} -\item \errindex{No such proof} -\end{ErrMsgs} - - -%%%% -\subsection[\tt Existential {\num} := {\term}.]{\tt Existential {\num} := {\term}.\comindex{Existential} -\label{Existential}} - -This command allows to instantiate an existential variable. {\tt \num} -is an index in the list of uninstantiated existential variables -displayed by {\tt Show Existentials.} (described in Section~\ref{Show}) - -This command is intended to be used to instantiate existential -variables when the proof is completed but some uninstantiated -existential variables remain. To instantiate existential variables -during proof edition, you should use the tactic {\tt instantiate}. - -\SeeAlso {\tt instantiate (\num:= \term).} in Section~\ref{instantiate}. - - -%%%%%%%% -\section{Navigation in the proof tree} -%%%%%%%% - -\subsection[\tt Undo.]{\tt Undo.\comindex{Undo}} - -This command cancels the effect of the last tactic command. Thus, it -backtracks one step. - -\begin{ErrMsgs} -\item \errindex{No focused proof (No proof-editing in progress)} -\item \errindex{Undo stack would be exhausted} -\end{ErrMsgs} - -\begin{Variants} - -\item {\tt Undo {\num}.} - - Repeats {\tt Undo} {\num} times. - -\end{Variants} - -\subsection[\tt Set Undo {\num}.]{\tt Set Undo {\num}.\comindex{Set Undo}} - -This command changes the maximum number of {\tt Undo}'s that will be -possible when doing a proof. It only affects proofs started after -this command, such that if you want to change the current undo limit -inside a proof, you should first restart this proof. - -\subsection[\tt Unset Undo.]{\tt Unset Undo.\comindex{Unset Undo}} - -This command resets the default number of possible {\tt Undo} commands -(which is currently 12). - -\subsection[\tt Restart.]{\tt Restart.\comindex{Restart}} -This command restores the proof editing process to the original goal. - -\begin{ErrMsgs} -\item \errindex{No focused proof to restart} -\end{ErrMsgs} - -\subsection[\tt Focus.]{\tt Focus.\comindex{Focus}} -This focuses the attention on the first subgoal to prove and the printing -of the other subgoals is suspended until the focused subgoal is -solved or unfocused. This is useful when there are many current -subgoals which clutter your screen. - -\begin{Variant} -\item {\tt Focus {\num}.}\\ -This focuses the attention on the $\num^{th}$ subgoal to prove. - -\end{Variant} - -\subsection[\tt Unfocus.]{\tt Unfocus.\comindex{Unfocus}} -Turns off the focus mode. - - -\section{Requesting information} - -\subsection[\tt Show.]{\tt Show.\comindex{Show}\label{Show}} -This command displays the current goals. - -\begin{Variants} -\item {\tt Show {\num}.}\\ - Displays only the {\num}-th subgoal.\\ -\begin{ErrMsgs} -\item \errindex{No such goal} -\item \errindex{No focused proof} -\end{ErrMsgs} - -\item {\tt Show Implicits.}\comindex{Show Implicits}\\ - Displays the current goals, printing the implicit arguments of - constants. - -\item {\tt Show Implicits {\num}.}\\ - Same as above, only displaying the {\num}-th subgoal. - -\item {\tt Show Script.}\comindex{Show Script}\\ - Displays the whole list of tactics applied from the beginning - of the current proof. - This tactics script may contain some holes (subgoals not yet proved). - They are printed under the form \verb!<Your Tactic Text here>!. - -\item {\tt Show Tree.}\comindex{Show Tree}\\ -This command can be seen as a more structured way of -displaying the state of the proof than that -provided by {\tt Show Script}. Instead of just giving -the list of tactics that have been applied, it -shows the derivation tree constructed by then. -Each node of the tree contains the conclusion -of the corresponding sub-derivation (i.e. a -goal with its corresponding local context) and -the tactic that has generated all the -sub-derivations. The leaves of this tree are -the goals which still remain to be proved. - -%\item {\tt Show Node}\comindex{Show Node}\\ -% Not yet documented - -\item {\tt Show Proof.}\comindex{Show Proof}\\ -It displays the proof term generated by the -tactics that have been applied. -If the proof is not completed, this term contain holes, -which correspond to the sub-terms which are still to be -constructed. These holes appear as a question mark indexed -by an integer, and applied to the list of variables in -the context, since it may depend on them. -The types obtained by abstracting away the context from the -type of each hole-placer are also printed. - -\item {\tt Show Conjectures.}\comindex{Show Conjectures}\\ -It prints the list of the names of all the theorems that -are currently being proved. -As it is possible to start proving a previous lemma during -the proof of a theorem, this list may contain several -names. - -\item{\tt Show Intro.}\comindex{Show Intro}\\ -If the current goal begins by at least one product, this command -prints the name of the first product, as it would be generated by -an anonymous {\tt Intro}. The aim of this command is to ease the -writing of more robust scripts. For example, with an appropriate -Proof General macro, it is possible to transform any anonymous {\tt - Intro} into a qualified one such as {\tt Intro y13}. -In the case of a non-product goal, it prints nothing. - -\item{\tt Show Intros.}\comindex{Show Intros}\\ -This command is similar to the previous one, it simulates the naming -process of an {\tt Intros}. - -\item{\tt Show Existentials}\comindex{Show Existentials}\\ It displays -the set of all uninstantiated existential variables in the current proof tree, -along with the type and the context of each variable. - -\end{Variants} - - -\subsection[\tt Guarded.]{\tt Guarded.\comindex{Guarded}\label{Guarded}} - -Some tactics (e.g. refine \ref{refine}) allow to build proofs using -fixpoint or co-fixpoint constructions. Due to the incremental nature -of interactive proof construction, the check of the termination (or -guardedness) of the recursive calls in the fixpoint or cofixpoint -constructions is postponed to the time of the completion of the proof. - -The command \verb!Guarded! allows to verify if the guard condition for -fixpoint and cofixpoint is violated at some time of the construction -of the proof without having to wait the completion of the proof." - - -\section{Controlling the effect of proof editing commands} - -\subsection[\tt Set Hyps Limit {\num}.]{\tt Set Hyps Limit {\num}.\comindex{Set Hyps Limit}} -This command sets the maximum number of hypotheses displayed in -goals after the application of a tactic. -All the hypotheses remains usable in the proof development. - - -\subsection[\tt Unset Hyps Limit.]{\tt Unset Hyps Limit.\comindex{Unset Hyps Limit}} -This command goes back to the default mode which is to print all -available hypotheses. - -% $Id: RefMan-pro.tex 13091 2010-06-08 13:56:19Z herbelin $ - - -\subsection[\tt Set Automatic Introduction.]{\tt Set Automatic Introduction.\comindex{Set Automatic Introduction}\comindex{Unset Automatic Introduction}\label{Set Automatic Introduction}} - -The option {\tt Automatic Introduction} controls the way binders are -handled in assertion commands such as {\tt Theorem {\ident} - \zeroone{\binders} : {\form}}. When the option is set, which is the -default, {\binders} are automatically put in the local context of the -goal to prove. - -The option can be unset by issuing {\tt Unset Automatic Introduction}. -When the option is unset, {\binders} are discharged on the statement -to be proved and a tactic such as {\tt intro} (see -Section~\ref{intro}) has to be used to move the assumptions to the -local context. - -%%% Local Variables: -%%% mode: latex -%%% TeX-master: "Reference-Manual" -%%% End: diff --git a/doc/refman/RefMan-syn.tex b/doc/refman/RefMan-syn.tex deleted file mode 100644 index 2b0d636e..00000000 --- a/doc/refman/RefMan-syn.tex +++ /dev/null @@ -1,1148 +0,0 @@ -\chapter[Syntax extensions and interpretation scopes]{Syntax extensions and interpretation scopes\label{Addoc-syntax}} - -In this chapter, we introduce advanced commands to modify the way -{\Coq} parses and prints objects, i.e. the translations between the -concrete and internal representations of terms and commands. The main -commands are {\tt Notation} and {\tt Infix} which are described in -section \ref{Notation}. It also happens that the same symbolic -notation is expected in different contexts. To achieve this form of -overloading, {\Coq} offers a notion of interpretation scope. This is -described in Section~\ref{scopes}. - -\Rem The commands {\tt Grammar}, {\tt Syntax} and {\tt Distfix} which -were present for a while in {\Coq} are no longer available from {\Coq} -version 8.0. The underlying AST structure is also no longer available. -The functionalities of the command {\tt Syntactic Definition} are -still available, see Section~\ref{Abbreviations}. - -\section[Notations]{Notations\label{Notation} -\comindex{Notation}} - -\subsection{Basic notations} - -A {\em notation} is a symbolic abbreviation denoting some term -or term pattern. - -A typical notation is the use of the infix symbol \verb=/\= to denote -the logical conjunction (\texttt{and}). Such a notation is declared -by - -\begin{coq_example*} -Notation "A /\ B" := (and A B). -\end{coq_example*} - -The expression \texttt{(and A B)} is the abbreviated term and the -string \verb="A /\ B"= (called a {\em notation}) tells how it is -symbolically written. - -A notation is always surrounded by double quotes (excepted when the -abbreviation is a single identifier, see \ref{Abbreviations}). The -notation is composed of {\em tokens} separated by spaces. Identifiers -in the string (such as \texttt{A} and \texttt{B}) are the {\em -parameters} of the notation. They must occur at least once each in the -denoted term. The other elements of the string (such as \verb=/\=) are -the {\em symbols}. - -An identifier can be used as a symbol but it must be surrounded by -simple quotes to avoid the confusion with a parameter. Similarly, -every symbol of at least 3 characters and starting with a simple quote -must be quoted (then it starts by two single quotes). Here is an example. - -\begin{coq_example*} -Notation "'IF' c1 'then' c2 'else' c3" := (IF_then_else c1 c2 c3). -\end{coq_example*} - -%TODO quote the identifier when not in front, not a keyword, as in "x 'U' y" ? - -A notation binds a syntactic expression to a term. Unless the parser -and pretty-printer of {\Coq} already know how to deal with the -syntactic expression (see \ref{ReservedNotation}), explicit precedences and -associativity rules have to be given. - -\subsection[Precedences and associativity]{Precedences and associativity\index{Precedences} -\index{Associativity}} - -Mixing different symbolic notations in a same text may cause serious -parsing ambiguity. To deal with the ambiguity of notations, {\Coq} -uses precedence levels ranging from 0 to 100 (plus one extra level -numbered 200) and associativity rules. - -Consider for example the new notation - -\begin{coq_example*} -Notation "A \/ B" := (or A B). -\end{coq_example*} - -Clearly, an expression such as {\tt forall A:Prop, True \verb=/\= A \verb=\/= -A \verb=\/= False} is ambiguous. To tell the {\Coq} parser how to -interpret the expression, a priority between the symbols \verb=/\= and -\verb=\/= has to be given. Assume for instance that we want conjunction -to bind more than disjunction. This is expressed by assigning a -precedence level to each notation, knowing that a lower level binds -more than a higher level. Hence the level for disjunction must be -higher than the level for conjunction. - -Since connectives are the less tight articulation points of a text, it -is reasonable to choose levels not so far from the higher level which -is 100, for example 85 for disjunction and 80 for -conjunction\footnote{which are the levels effectively chosen in the -current implementation of {\Coq}}. - -Similarly, an associativity is needed to decide whether {\tt True \verb=/\= -False \verb=/\= False} defaults to {\tt True \verb=/\= (False -\verb=/\= False)} (right associativity) or to {\tt (True -\verb=/\= False) \verb=/\= False} (left associativity). We may -even consider that the expression is not well-formed and that -parentheses are mandatory (this is a ``no associativity'')\footnote{ -{\Coq} accepts notations declared as no associative but the parser on -which {\Coq} is built, namely {\camlpppp}, currently does not implement the -no-associativity and replace it by a left associativity; hence it is -the same for {\Coq}: no-associativity is in fact left associativity}. -We don't know of a special convention of the associativity of -disjunction and conjunction, let's apply for instance a right -associativity (which is the choice of {\Coq}). - -Precedence levels and associativity rules of notations have to be -given between parentheses in a list of modifiers that the -\texttt{Notation} command understands. Here is how the previous -examples refine. - -\begin{coq_example*} -Notation "A /\ B" := (and A B) (at level 80, right associativity). -Notation "A \/ B" := (or A B) (at level 85, right associativity). -\end{coq_example*} - -By default, a notation is considered non associative, but the -precedence level is mandatory (except for special cases whose level is -canonical). The level is either a number or the mention {\tt next -level} whose meaning is obvious. The list of levels already assigned -is on Figure~\ref{init-notations}. - -\subsection{Complex notations} - -Notations can be made from arbitraly complex symbols. One can for -instance define prefix notations. - -\begin{coq_example*} -Notation "~ x" := (not x) (at level 75, right associativity). -\end{coq_example*} - -One can also define notations for incomplete terms, with the hole -expected to be inferred at typing time. - -\begin{coq_example*} -Notation "x = y" := (@eq _ x y) (at level 70, no associativity). -\end{coq_example*} - -One can define {\em closed} notations whose both sides are symbols. In -this case, the default precedence level for inner subexpression is 200. - -\begin{coq_eval} -Set Printing Depth 50. -(********** The following is correct but produces **********) -(**** an incompatibility with the reserved notation ********) -\end{coq_eval} -\begin{coq_example*} -Notation "( x , y )" := (@pair _ _ x y) (at level 0). -\end{coq_example*} - -One can also define notations for binders. - -\begin{coq_eval} -Set Printing Depth 50. -(********** The following is correct but produces **********) -(**** an incompatibility with the reserved notation ********) -\end{coq_eval} -\begin{coq_example*} -Notation "{ x : A | P }" := (sig A (fun x => P)) (at level 0). -\end{coq_example*} - -In the last case though, there is a conflict with the notation for -type casts. This last notation, as shown by the command {\tt Print Grammar -constr} is at level 100. To avoid \verb=x : A= being parsed as a type cast, -it is necessary to put {\tt x} at a level below 100, typically 99. Hence, a -correct definition is - -\begin{coq_example*} -Notation "{ x : A | P }" := (sig A (fun x => P)) (at level 0, x at level 99). -\end{coq_example*} - -%This change has retrospectively an effect on the notation for notation -%{\tt "{ A } + { B }"}. For the sake of factorization, {\tt A} must be -%put at level 99 too, which gives -% -%\begin{coq_example*} -%Notation "{ A } + { B }" := (sumbool A B) (at level 0, A at level 99). -%\end{coq_example*} - -See the next section for more about factorization. - -\subsection{Simple factorization rules} - -{\Coq} extensible parsing is performed by Camlp5 which is essentially a -LL1 parser. Hence, some care has to be taken not to hide already -existing rules by new rules. Some simple left factorization work has -to be done. Here is an example. - -\begin{coq_eval} -(********** The next rule for notation _ < _ < _ produces **********) -(*** Error: Notation _ < _ < _ is already defined at level 70 ... ***) -\end{coq_eval} -\begin{coq_example*} -Notation "x < y" := (lt x y) (at level 70). -Notation "x < y < z" := (x < y /\ y < z) (at level 70). -\end{coq_example*} - -In order to factorize the left part of the rules, the subexpression -referred by {\tt y} has to be at the same level in both rules. However -the default behavior puts {\tt y} at the next level below 70 -in the first rule (no associativity is the default), and at the level -200 in the second rule (level 200 is the default for inner expressions). -To fix this, we need to force the parsing level of {\tt y}, -as follows. - -\begin{coq_example*} -Notation "x < y" := (lt x y) (at level 70). -Notation "x < y < z" := (x < y /\ y < z) (at level 70, y at next level). -\end{coq_example*} - -For the sake of factorization with {\Coq} predefined rules, simple -rules have to be observed for notations starting with a symbol: -e.g. rules starting with ``\{'' or ``('' should be put at level 0. The -list of {\Coq} predefined notations can be found in Chapter~\ref{Theories}. - -The command to display the current state of the {\Coq} term parser is -\comindex{Print Grammar constr} - -\begin{quote} -\tt Print Grammar constr. -\end{quote} - -\variant - -\comindex{Print Grammar pattern} -{\tt Print Grammar pattern.}\\ - -This displays the state of the subparser of patterns (the parser -used in the grammar of the {\tt match} {\tt with} constructions). - -\subsection{Displaying symbolic notations} - -The command \texttt{Notation} has an effect both on the {\Coq} parser and -on the {\Coq} printer. For example: - -\begin{coq_example} -Check (and True True). -\end{coq_example} - -However, printing, especially pretty-printing, requires -more care than parsing. We may want specific indentations, -line breaks, alignment if on several lines, etc. - -The default printing of notations is very rudimentary. For printing a -notation, a {\em formatting box} is opened in such a way that if the -notation and its arguments cannot fit on a single line, a line break -is inserted before the symbols of the notation and the arguments on -the next lines are aligned with the argument on the first line. - -A first, simple control that a user can have on the printing of a -notation is the insertion of spaces at some places of the -notation. This is performed by adding extra spaces between the symbols -and parameters: each extra space (other than the single space needed -to separate the components) is interpreted as a space to be inserted -by the printer. Here is an example showing how to add spaces around -the bar of the notation. - -\begin{coq_example} -Notation "{{ x : A | P }}" := (sig (fun x : A => P)) - (at level 0, x at level 99). -Check (sig (fun x : nat => x=x)). -\end{coq_example} - -The second, more powerful control on printing is by using the {\tt -format} modifier. Here is an example - -\begin{small} -\begin{coq_example} -Notation "'If' c1 'then' c2 'else' c3" := (IF_then_else c1 c2 c3) -(at level 200, right associativity, format -"'[v ' 'If' c1 '/' '[' 'then' c2 ']' '/' '[' 'else' c3 ']' ']'"). -\end{coq_example} -\end{small} - -A {\em format} is an extension of the string denoting the notation with -the possible following elements delimited by single quotes: - -\begin{itemize} -\item extra spaces are translated into simple spaces -\item tokens of the form \verb='/ '= are translated into breaking point, - in case a line break occurs, an indentation of the number of spaces - after the ``\verb=/='' is applied (2 spaces in the given example) -\item token of the form \verb='//'= force writing on a new line -\item well-bracketed pairs of tokens of the form \verb='[ '= and \verb=']'= - are translated into printing boxes; in case a line break occurs, - an extra indentation of the number of spaces given after the ``\verb=[='' - is applied (4 spaces in the example) -\item well-bracketed pairs of tokens of the form \verb='[hv '= and \verb=']'= - are translated into horizontal-orelse-vertical printing boxes; - if the content of the box does not fit on a single line, then every breaking - point forces a newline and an extra indentation of the number of spaces - given after the ``\verb=[='' is applied at the beginning of each newline - (3 spaces in the example) -\item well-bracketed pairs of tokens of the form \verb='[v '= and - \verb=']'= are translated into vertical printing boxes; every - breaking point forces a newline, even if the line is large enough to - display the whole content of the box, and an extra indentation of the - number of spaces given after the ``\verb=[='' is applied at the beginning - of each newline -\end{itemize} - -Thus, for the previous example, we get -%\footnote{The ``@'' is here to shunt -%the notation "'IF' A 'then' B 'else' C" which is defined in {\Coq} -%initial state}: - -Notations do not survive the end of sections. No typing of the denoted -expression is performed at definition time. Type-checking is done only -at the time of use of the notation. - -\begin{coq_example} -Check - (IF_then_else (IF_then_else True False True) - (IF_then_else True False True) - (IF_then_else True False True)). -\end{coq_example} - -\Rem -Sometimes, a notation is expected only for the parser. -%(e.g. because -%the underlying parser of {\Coq}, namely {\camlpppp}, is LL1 and some extra -%rules are needed to circumvent the absence of factorization). -To do so, the option {\em only parsing} is allowed in the list of modifiers of -\texttt{Notation}. - -\subsection{The \texttt{Infix} command -\comindex{Infix}} - -The \texttt{Infix} command is a shortening for declaring notations of -infix symbols. Its syntax is - -\begin{quote} -\noindent\texttt{Infix "{\symbolentry}" :=} {\qualid} {\tt (} \nelist{\em modifier}{,} {\tt )}. -\end{quote} - -and it is equivalent to - -\begin{quote} -\noindent\texttt{Notation "x {\symbolentry} y" := ({\qualid} x y) (} \nelist{\em modifier}{,} {\tt )}. -\end{quote} - -where {\tt x} and {\tt y} are fresh names distinct from {\qualid}. Here is an example. - -\begin{coq_example*} -Infix "/\" := and (at level 80, right associativity). -\end{coq_example*} - -\subsection{Reserving notations -\label{ReservedNotation} -\comindex{ReservedNotation}} - -A given notation may be used in different contexts. {\Coq} expects all -uses of the notation to be defined at the same precedence and with the -same associativity. To avoid giving the precedence and associativity -every time, it is possible to declare a parsing rule in advance -without giving its interpretation. Here is an example from the initial -state of {\Coq}. - -\begin{coq_example} -Reserved Notation "x = y" (at level 70, no associativity). -\end{coq_example} - -Reserving a notation is also useful for simultaneously defined an -inductive type or a recursive constant and a notation for it. - -\Rem The notations mentioned on Figure~\ref{init-notations} are -reserved. Hence their precedence and associativity cannot be changed. - -\subsection{Simultaneous definition of terms and notations -\comindex{Fixpoint {\ldots} where {\ldots}} -\comindex{CoFixpoint {\ldots} where {\ldots}} -\comindex{Inductive {\ldots} where {\ldots}}} - -Thanks to reserved notations, the inductive, coinductive, recursive -and corecursive definitions can benefit of customized notations. To do -this, insert a {\tt where} notation clause after the definition of the -(co)inductive type or (co)recursive term (or after the definition of -each of them in case of mutual definitions). The exact syntax is given -on Figure~\ref{notation-syntax}. Here are examples: - -\begin{coq_eval} -Set Printing Depth 50. -(********** The following is correct but produces an error **********) -(********** because the symbol /\ is already bound **********) -(**** Error: The conclusion of A -> B -> A /\ B is not valid *****) -\end{coq_eval} - -\begin{coq_example*} -Inductive and (A B:Prop) : Prop := conj : A -> B -> A /\ B -where "A /\ B" := (and A B). -\end{coq_example*} - -\begin{coq_eval} -Set Printing Depth 50. -(********** The following is correct but produces an error **********) -(********** because the symbol + is already bound **********) -(**** Error: no recursive definition *****) -\end{coq_eval} - -\begin{coq_example*} -Fixpoint plus (n m:nat) {struct n} : nat := - match n with - | O => m - | S p => S (p+m) - end -where "n + m" := (plus n m). -\end{coq_example*} - -\subsection{Displaying informations about notations -\comindex{Set Printing Notations} -\comindex{Unset Printing Notations}} - -To deactivate the printing of all notations, use the command -\begin{quote} -\tt Unset Printing Notations. -\end{quote} -To reactivate it, use the command -\begin{quote} -\tt Set Printing Notations. -\end{quote} -The default is to use notations for printing terms wherever possible. - -\SeeAlso {\tt Set Printing All} in Section~\ref{SetPrintingAll}. - -\subsection{Locating notations -\comindex{Locate} -\label{LocateSymbol}} - -To know to which notations a given symbol belongs to, use the command -\begin{quote} -\tt Locate {\symbolentry} -\end{quote} -where symbol is any (composite) symbol surrounded by quotes. To locate -a particular notation, use a string where the variables of the -notation are replaced by ``\_''. - -\Example -\begin{coq_example} -Locate "exists". -Locate "'exists' _ , _". -\end{coq_example} - -\SeeAlso Section \ref{Locate}. - -\begin{figure} -\begin{small} -\begin{centerframe} -\begin{tabular}{lcl} -{\sentence} & ::= & - \zeroone{\tt Local} \texttt{Notation} {\str} \texttt{:=} {\term} - \zeroone{\modifiers} \zeroone{:{\scope}} .\\ - & $|$ & - \zeroone{\tt Local} \texttt{Infix} {\str} \texttt{:=} {\qualid} - \zeroone{\modifiers} \zeroone{:{\scope}} .\\ - & $|$ & - \zeroone{\tt Local} \texttt{Reserved Notation} {\str} - \zeroone{\modifiers} .\\ - & $|$ & {\tt Inductive} - \nelist{{\inductivebody} \zeroone{\declnotation}}{with}{\tt .}\\ - & $|$ & {\tt CoInductive} - \nelist{{\inductivebody} \zeroone{\declnotation}}{with}{\tt .}\\ - & $|$ & {\tt Fixpoint} - \nelist{{\fixpointbody} \zeroone{\declnotation}}{with} {\tt .} \\ - & $|$ & {\tt CoFixpoint} - \nelist{{\cofixpointbody} \zeroone{\declnotation}}{with} {\tt .} \\ -\\ -{\declnotation} & ::= & - \zeroone{{\tt where} \nelist{{\str} {\tt :=} {\term} \zeroone{:{\scope}}}{\tt and}}. -\\ -\\ -{\modifiers} - & ::= & \nelist{\ident}{,} {\tt at level} {\naturalnumber} \\ - & $|$ & \nelist{\ident}{,} {\tt at next level} \\ - & $|$ & {\tt at level} {\naturalnumber} \\ - & $|$ & {\tt left associativity} \\ - & $|$ & {\tt right associativity} \\ - & $|$ & {\tt no associativity} \\ - & $|$ & {\ident} {\tt ident} \\ - & $|$ & {\ident} {\tt binder} \\ - & $|$ & {\ident} {\tt closed binder} \\ - & $|$ & {\ident} {\tt global} \\ - & $|$ & {\ident} {\tt bigint} \\ - & $|$ & {\tt only parsing} \\ - & $|$ & {\tt format} {\str} -\end{tabular} -\end{centerframe} -\end{small} -\caption{Syntax of the variants of {\tt Notation}} -\label{notation-syntax} -\end{figure} - -\subsection{Notations and simple binders} - -Notations can be defined for binders as in the example: - -\begin{coq_eval} -Set Printing Depth 50. -(********** The following is correct but produces **********) -(**** an incompatibility with the reserved notation ********) -\end{coq_eval} -\begin{coq_example*} -Notation "{ x : A | P }" := (sig (fun x : A => P)) (at level 0). -\end{coq_example*} - -The binding variables in the left-hand-side that occur as a parameter -of the notation naturally bind all their occurrences appearing in -their respective scope after instantiation of the parameters of the -notation. - -Contrastingly, the binding variables that are not a parameter of the -notation do not capture the variables of same name that -could appear in their scope after instantiation of the -notation. E.g., for the notation - -\begin{coq_example*} -Notation "'exists_different' n" := (exists p:nat, p<>n) (at level 200). -\end{coq_example*} -the next command fails because {\tt p} does not bind in -the instance of {\tt n}. -\begin{coq_eval} -Set Printing Depth 50. -(********** The following produces **********) -(**** The reference p was not found in the current environment ********) -\end{coq_eval} -\begin{coq_example} -Check (exists_different p). -\end{coq_example} - -\Rem Binding variables must not necessarily be parsed using the -{\tt ident} entry. For factorization purposes, they can be said to be -parsed at another level (e.g. {\tt x} in \verb="{ x : A | P }"= must be -parsed at level 99 to be factorized with the notation -\verb="{ A } + { B }"= for which {\tt A} can be any term). -However, even if parsed as a term, this term must at the end be effectively -a single identifier. - -\subsection{Notations with recursive patterns} -\label{RecursiveNotations} - -A mechanism is provided for declaring elementary notations with -recursive patterns. The basic example is: - -\begin{coq_example*} -Notation "[ x ; .. ; y ]" := (cons x .. (cons y nil) ..). -\end{coq_example*} - -On the right-hand side, an extra construction of the form {\tt ..} $t$ -{\tt ..} can be used. Notice that {\tt ..} is part of the {\Coq} -syntax and it must not be confused with the three-dots notation -$\ldots$ used in this manual to denote a sequence of arbitrary size. - -On the left-hand side, the part ``$x$ $s$ {\tt ..} $s$ $y$'' of the -notation parses any number of time (but at least one time) a sequence -of expressions separated by the sequence of tokens $s$ (in the -example, $s$ is just ``{\tt ;}''). - -In the right-hand side, the term enclosed within {\tt ..} must be a -pattern with two holes of the form $\phi([~]_E,[~]_I)$ where the first -hole is occupied either by $x$ or by $y$ and the second hole is -occupied by an arbitrary term $t$ called the {\it terminating} -expression of the recursive notation. The subterm {\tt ..} $\phi(x,t)$ -{\tt ..} (or {\tt ..} $\phi(y,t)$ {\tt ..}) must itself occur at -second position of the same pattern where the first hole is occupied -by the other variable, $y$ or $x$. Otherwise said, the right-hand side -must contain a subterm of the form either $\phi(x,${\tt ..} -$\phi(y,t)$ {\tt ..}$)$ or $\phi(y,${\tt ..} $\phi(x,t)$ {\tt ..}$)$. -The pattern $\phi$ is the {\em iterator} of the recursive notation -and, of course, the name $x$ and $y$ can be chosen arbitrarily. - -The parsing phase produces a list of expressions which are used to -fill in order the first hole of the iterating pattern which is -repeatedly nested as many times as the length of the list, the second -hole being the nesting point. In the innermost occurrence of the -nested iterating pattern, the second hole is finally filled with the -terminating expression. - -In the example above, the iterator $\phi([~]_E,[~]_I)$ is {\tt cons - $[~]_E$ $[~]_I$} and the terminating expression is {\tt nil}. Here are -other examples: -\begin{coq_example*} -Notation "( x , y , .. , z )" := (pair .. (pair x y) .. z) (at level 0). -Notation "[| t * ( x , y , .. , z ) ; ( a , b , .. , c ) * u |]" := - (pair (pair .. (pair (pair t x) (pair t y)) .. (pair t z)) - (pair .. (pair (pair a u) (pair b u)) .. (pair c u))) - (t at level 39). -\end{coq_example*} - -Notations with recursive patterns can be reserved like standard -notations, they can also be declared within interpretation scopes (see -section \ref{scopes}). - -\subsection{Notations with recursive patterns involving binders} - -Recursive notations can also be used with binders. The basic example is: - -\begin{coq_example*} -Notation "'exists' x .. y , p" := (ex (fun x => .. (ex (fun y => p)) ..)) - (at level 200, x binder, y binder, right associativity). -\end{coq_example*} - -The principle is the same as in Section~\ref{RecursiveNotations} -except that in the iterator $\phi([~]_E,[~]_I)$, the first hole is a -placeholder occurring at the position of the binding variable of a {\tt - fun} or a {\tt forall}. - -To specify that the part ``$x$ {\tt ..} $y$'' of the notation -parses a sequence of binders, $x$ and $y$ must be marked as {\tt - binder} in the list of modifiers of the notation. Then, the list of -binders produced at the parsing phase are used to fill in the first -hole of the iterating pattern which is repeatedly nested as many times -as the number of binders generated. If ever the generalization -operator {\tt `} (see Section~\ref{implicit-generalization}) is used -in the binding list, the added binders are taken into account too. - -Binders parsing exist in two flavors. If $x$ and $y$ are marked as -{\tt binder}, then a sequence such as {\tt a b c : T} will be accepted -and interpreted as the sequence of binders {\tt (a:T) (b:T) - (c:T)}. For instance, in the notation above, the syntax {\tt exists - a b : nat, a = b} is provided. - -The variables $x$ and $y$ can also be marked as {\tt closed binder} in -which case only well-bracketed binders of the form {\tt (a b c:T)} or -{\tt \{a b c:T\}} etc. are accepted. - -With closed binders, the recursive sequence in the left-hand side can -be of the general form $x$ $s$ {\tt ..} $s$ $y$ where $s$ is an -arbitrary sequence of tokens. With open binders though, $s$ has to be -empty. Here is an example of recursive notation with closed binders: - -\begin{coq_example*} -Notation "'mylet' f x .. y := t 'in' u":= - (let f := fun x => .. (fun y => t) .. in u) - (x closed binder, y closed binder, at level 200, right associativity). -\end{coq_example*} - -\subsection{Summary} - -\paragraph{Syntax of notations} - -The different syntactic variants of the command \texttt{Notation} are -given on Figure~\ref{notation-syntax}. The optional {\tt :{\scope}} is -described in the Section~\ref{scopes}. - -\Rem No typing of the denoted expression is performed at definition -time. Type-checking is done only at the time of use of the notation. - -\Rem Many examples of {\tt Notation} may be found in the files -composing the initial state of {\Coq} (see directory {\tt -\$COQLIB/theories/Init}). - -\Rem The notation \verb="{ x }"= has a special status in such a way -that complex notations of the form \verb="x + { y }"= or -\verb="x * { y }"= can be nested with correct precedences. Especially, -every notation involving a pattern of the form \verb="{ x }"= is -parsed as a notation where the pattern \verb="{ x }"= has been simply -replaced by \verb="x"= and the curly brackets are parsed separately. -E.g. \verb="y + { z }"= is not parsed as a term of the given form but -as a term of the form \verb="y + z"= where \verb=z= has been parsed -using the rule parsing \verb="{ x }"=. Especially, level and -precedences for a rule including patterns of the form \verb="{ x }"= -are relative not to the textual notation but to the notation where the -curly brackets have been removed (e.g. the level and the associativity -given to some notation, say \verb="{ y } & { z }"= in fact applies to -the underlying \verb="{ x }"=-free rule which is \verb="y & z"=). - -\paragraph{Persistence of notations} - -Notations do not survive the end of sections. They survive modules -unless the command {\tt Local Notation} is used instead of {\tt -Notation}. - -\section[Interpretation scopes]{Interpretation scopes\index{Interpretation scopes} -\label{scopes}} -% Introduction - -An {\em interpretation scope} is a set of notations for terms with -their interpretation. Interpretation scopes provides with a weak, -purely syntactical form of notations overloading: a same notation, for -instance the infix symbol \verb=+= can be used to denote distinct -definitions of an additive operator. Depending on which interpretation -scopes is currently open, the interpretation is different. -Interpretation scopes can include an interpretation for -numerals and strings. However, this is only made possible at the -{\ocaml} level. - -See Figure \ref{notation-syntax} for the syntax of notations including -the possibility to declare them in a given scope. Here is a typical -example which declares the notation for conjunction in the scope {\tt -type\_scope}. - -\begin{verbatim} -Notation "A /\ B" := (and A B) : type_scope. -\end{verbatim} - -\Rem A notation not defined in a scope is called a {\em lonely} notation. - -\subsection{Global interpretation rules for notations} - -At any time, the interpretation of a notation for term is done within -a {\em stack} of interpretation scopes and lonely notations. In case a -notation has several interpretations, the actual interpretation is the -one defined by (or in) the more recently declared (or open) lonely -notation (or interpretation scope) which defines this notation. -Typically if a given notation is defined in some scope {\scope} but -has also an interpretation not assigned to a scope, then, if {\scope} -is open before the lonely interpretation is declared, then the lonely -interpretation is used (and this is the case even if the -interpretation of the notation in {\scope} is given after the lonely -interpretation: otherwise said, only the order of lonely -interpretations and opening of scopes matters, and not the declaration -of interpretations within a scope). - -The initial state of {\Coq} declares three interpretation scopes and -no lonely notations. These scopes, in opening order, are {\tt -core\_scope}, {\tt type\_scope} and {\tt nat\_scope}. - -The command to add a scope to the interpretation scope stack is -\comindex{Open Scope} -\comindex{Close Scope} -\begin{quote} -{\tt Open Scope} {\scope}. -\end{quote} -It is also possible to remove a scope from the interpretation scope -stack by using the command -\begin{quote} -{\tt Close Scope} {\scope}. -\end{quote} -Notice that this command does not only cancel the last {\tt Open Scope -{\scope}} but all the invocation of it. - -\Rem {\tt Open Scope} and {\tt Close Scope} do not survive the end of -sections where they occur. When defined outside of a section, they are -exported to the modules that import the module where they occur. - -\begin{Variants} - -\item {\tt Local Open Scope} {\scope}. - -\item {\tt Local Close Scope} {\scope}. - -These variants are not exported to the modules that import the module -where they occur, even if outside a section. - -\item {\tt Global Open Scope} {\scope}. - -\item {\tt Global Close Scope} {\scope}. - -These variants survive sections. They behave as if {\tt Global} were -absent when not inside a section. - -\end{Variants} - -\subsection{Local interpretation rules for notations} - -In addition to the global rules of interpretation of notations, some -ways to change the interpretation of subterms are available. - -\subsubsection{Local opening of an interpretation scope -\label{scopechange} -\index{\%} -\comindex{Delimit Scope}} - -It is possible to locally extend the interpretation scope stack using -the syntax ({\term})\%{\delimkey} (or simply {\term}\%{\delimkey} -for atomic terms), where {\delimkey} is a special identifier called -{\em delimiting key} and bound to a given scope. - -In such a situation, the term {\term}, and all its subterms, are -interpreted in the scope stack extended with the scope bound to -{\delimkey}. - -To bind a delimiting key to a scope, use the command - -\begin{quote} -\texttt{Delimit Scope} {\scope} \texttt{with} {\ident} -\end{quote} - -\subsubsection{Binding arguments of a constant to an interpretation scope -\comindex{Arguments Scope}} - -It is possible to set in advance that some arguments of a given -constant have to be interpreted in a given scope. The command is -\begin{quote} -{\tt Arguments Scope} {\qualid} {\tt [ \nelist{\optscope}{} ]} -\end{quote} -where the list is a list made either of {\tt \_} or of a scope name. -Each scope in the list is bound to the corresponding parameter of -{\qualid} in order. When interpreting a term, if some of the -arguments of {\qualid} are built from a notation, then this notation -is interpreted in the scope stack extended by the scopes bound (if any) -to these arguments. - -\begin{Variants} -\item {\tt Global Arguments Scope} {\qualid} {\tt [ \nelist{\optscope}{} ]} - -This behaves like {\tt Arguments Scope} {\qualid} {\tt [ -\nelist{\optscope}{} ]} but survives when a section is closed instead -of stopping working at section closing. Without the {\tt Global} modifier, -the effect of the command stops when the section it belongs to ends. - -\item {\tt Local Arguments Scope} {\qualid} {\tt [ \nelist{\optscope}{} ]} - -This behaves like {\tt Arguments Scope} {\qualid} {\tt [ - \nelist{\optscope}{} ]} but does not survive modules and files. -Without the {\tt Local} modifier, the effect of the command is -visible from within other modules or files. - -\end{Variants} - - -\SeeAlso The command to show the scopes bound to the arguments of a -function is described in Section~\ref{About}. - -\subsubsection{Binding types of arguments to an interpretation scope} - -When an interpretation scope is naturally associated to a type -(e.g. the scope of operations on the natural numbers), it may be -convenient to bind it to this type. The effect of this is that any -argument of a function that syntactically expects a parameter of this -type is interpreted using scope. More precisely, it applies only if -this argument is built from a notation, and if so, this notation is -interpreted in the scope stack extended by this particular scope. It -does not apply to the subterms of this notation (unless the -interpretation of the notation itself expects arguments of the same -type that would trigger the same scope). - -\comindex{Bind Scope} -More generally, any {\class} (see Chapter~\ref{Coercions-full}) can be -bound to an interpretation scope. The command to do it is -\begin{quote} -{\tt Bind Scope} {\scope} \texttt{with} {\class} -\end{quote} - -\Example -\begin{coq_example} -Parameter U : Set. -Bind Scope U_scope with U. -Parameter Uplus : U -> U -> U. -Parameter P : forall T:Set, T -> U -> Prop. -Parameter f : forall T:Set, T -> U. -Infix "+" := Uplus : U_scope. -Unset Printing Notations. -Open Scope nat_scope. (* Define + on the nat as the default for + *) -Check (fun x y1 y2 z t => P _ (x + t) ((f _ (y1 + y2) + z))). -\end{coq_example} - -\Rem The scope {\tt type\_scope} has also a local effect on -interpretation. See the next section. - -\SeeAlso The command to show the scopes bound to the arguments of a -function is described in Section~\ref{About}. - -\subsection[The {\tt type\_scope} interpretation scope]{The {\tt type\_scope} interpretation scope\index{type\_scope}} - -The scope {\tt type\_scope} has a special status. It is a primitive -interpretation scope which is temporarily activated each time a -subterm of an expression is expected to be a type. This includes goals -and statements, types of binders, domain and codomain of implication, -codomain of products, and more generally any type argument of a -declared or defined constant. - -\subsection{Interpretation scopes used in the standard library of {\Coq}} - -We give an overview of the scopes used in the standard library of -{\Coq}. For a complete list of notations in each scope, use the -commands {\tt Print Scopes} or {\tt Print Scopes {\scope}}. - -\subsubsection{\tt type\_scope} - -This includes infix {\tt *} for product types and infix {\tt +} for -sum types. It is delimited by key {\tt type}. - -\subsubsection{\tt nat\_scope} - -This includes the standard arithmetical operators and relations on -type {\tt nat}. Positive numerals in this scope are mapped to their -canonical representent built from {\tt O} and {\tt S}. The scope is -delimited by key {\tt nat}. - -\subsubsection{\tt N\_scope} - -This includes the standard arithmetical operators and relations on -type {\tt N} (binary natural numbers). It is delimited by key {\tt N} -and comes with an interpretation for numerals as closed term of type {\tt Z}. - -\subsubsection{\tt Z\_scope} - -This includes the standard arithmetical operators and relations on -type {\tt Z} (binary integer numbers). It is delimited by key {\tt Z} -and comes with an interpretation for numerals as closed term of type {\tt Z}. - -\subsubsection{\tt positive\_scope} - -This includes the standard arithmetical operators and relations on -type {\tt positive} (binary strictly positive numbers). It is -delimited by key {\tt positive} and comes with an interpretation for -numerals as closed term of type {\tt positive}. - -\subsubsection{\tt Q\_scope} - -This includes the standard arithmetical operators and relations on -type {\tt Q} (rational numbers defined as fractions of an integer and -a strictly positive integer modulo the equality of the -numerator-denominator cross-product). As for numerals, only $0$ and -$1$ have an interpretation in scope {\tt Q\_scope} (their -interpretations are $\frac{0}{1}$ and $\frac{1}{1}$ respectively). - -\subsubsection{\tt Qc\_scope} - -This includes the standard arithmetical operators and relations on the -type {\tt Qc} of rational numbers defined as the type of irreducible -fractions of an integer and a strictly positive integer. - -\subsubsection{\tt real\_scope} - -This includes the standard arithmetical operators and relations on -type {\tt R} (axiomatic real numbers). It is delimited by key {\tt R} -and comes with an interpretation for numerals as term of type {\tt -R}. The interpretation is based on the binary decomposition. The -numeral 2 is represented by $1+1$. The interpretation $\phi(n)$ of an -odd positive numerals greater $n$ than 3 is {\tt 1+(1+1)*$\phi((n-1)/2)$}. -The interpretation $\phi(n)$ of an even positive numerals greater $n$ -than 4 is {\tt (1+1)*$\phi(n/2)$}. Negative numerals are represented as the -opposite of the interpretation of their absolute value. E.g. the -syntactic object {\tt -11} is interpreted as {\tt --(1+(1+1)*((1+1)*(1+(1+1))))} where the unit $1$ and all the operations are -those of {\tt R}. - -\subsubsection{\tt bool\_scope} - -This includes notations for the boolean operators. It is -delimited by key {\tt bool}. - -\subsubsection{\tt list\_scope} - -This includes notations for the list operators. It is -delimited by key {\tt list}. - -\subsubsection{\tt core\_scope} - -This includes the notation for pairs. It is delimited by key {\tt core}. - -\subsubsection{\tt string\_scope} - -This includes notation for strings as elements of the type {\tt -string}. Special characters and escaping follow {\Coq} conventions -on strings (see Section~\ref{strings}). Especially, there is no -convention to visualize non printable characters of a string. The -file {\tt String.v} shows an example that contains quotes, a newline -and a beep (i.e. the ascii character of code 7). - -\subsubsection{\tt char\_scope} - -This includes interpretation for all strings of the form -\verb!"!$c$\verb!"! where $c$ is an ascii character, or of the form -\verb!"!$nnn$\verb!"! where $nnn$ is a three-digits number (possibly -with leading 0's), or of the form \verb!""""!. Their respective -denotations are the ascii code of $c$, the decimal ascii code $nnn$, -or the ascii code of the character \verb!"! (i.e. the ascii code -34), all of them being represented in the type {\tt ascii}. - -\subsection{Displaying informations about scopes} - -\subsubsection{\tt Print Visibility\comindex{Print Visibility}} - -This displays the current stack of notations in scopes and lonely -notations that is used to interpret a notation. The top of the stack -is displayed last. Notations in scopes whose interpretation is hidden -by the same notation in a more recently open scope are not -displayed. Hence each notation is displayed only once. - -\variant - -{\tt Print Visibility {\scope}}\\ - -This displays the current stack of notations in scopes and lonely -notations assuming that {\scope} is pushed on top of the stack. This -is useful to know how a subterm locally occurring in the scope of -{\scope} is interpreted. - -\subsubsection{\tt Print Scope {\scope}\comindex{Print Scope}} - -This displays all the notations defined in interpretation scope -{\scope}. It also displays the delimiting key if any and the class to -which the scope is bound, if any. - -\subsubsection{\tt Print Scopes\comindex{Print Scopes}} - -This displays all the notations, delimiting keys and corresponding -class of all the existing interpretation scopes. -It also displays the lonely notations. - -\section[Abbreviations]{Abbreviations\index{Abbreviations} -\label{Abbreviations} -\comindex{Notation}} - -An {\em abbreviation} is a name, possibly applied to arguments, that -denotes a (presumably) more complex expression. Here are examples: - -\begin{coq_eval} -Require Import List. -Require Import Relations. -Set Printing Notations. -\end{coq_eval} -\begin{coq_example} -Notation Nlist := (list nat). -Check 1 :: 2 :: 3 :: nil. -Notation reflexive R := (forall x, R x x). -Check forall A:Prop, A <-> A. -Check reflexive iff. -\end{coq_example} - -An abbreviation expects no precedence nor associativity, since it -follows the usual syntax of application. Abbreviations are used as -much as possible by the {\Coq} printers unless the modifier -\verb=(only parsing)= is given. - -Abbreviations are bound to an absolute name as an ordinary -definition is, and they can be referred by qualified names too. - -Abbreviations are syntactic in the sense that they are bound to -expressions which are not typed at the time of the definition of the -abbreviation but at the time it is used. Especially, abbreviations can -be bound to terms with holes (i.e. with ``\_''). The general syntax -for abbreviations is -\begin{quote} -\zeroone{{\tt Local}} \texttt{Notation} {\ident} \sequence{\ident} {\ident} \texttt{:=} {\term} - \zeroone{{\tt (only parsing)}}~\verb=.= -\end{quote} - -\Example -\begin{coq_eval} -Set Strict Implicit. -Reset Initial. -\end{coq_eval} -\begin{coq_example} -Definition explicit_id (A:Set) (a:A) := a. -Notation id := (explicit_id _). -Check (id 0). -\end{coq_example} - -Abbreviations do not survive the end of sections. No typing of the denoted -expression is performed at definition time. Type-checking is done only -at the time of use of the abbreviation. - -%\Rem \index{Syntactic Definition} % -%Abbreviations are similar to the {\em syntactic -%definitions} available in versions of {\Coq} prior to version 8.0, -%except that abbreviations are used for printing (unless the modifier -%\verb=(only parsing)= is given) while syntactic definitions were not. - -\section{Tactic Notations} - -Tactic notations allow to customize the syntax of the tactics of the -tactic language\footnote{Tactic notations are just a simplification of -the {\tt Grammar tactic simple\_tactic} command that existed in -versions prior to version 8.0.}. Tactic notations obey the following -syntax -\medskip - -\noindent -\begin{tabular}{lcl} -{\sentence} & ::= & \texttt{Tactic Notation} \zeroone{\taclevel} \nelist{\proditem}{} \\ -& & \texttt{:= {\tac} .}\\ -{\proditem} & ::= & {\str} $|$ {\tacargtype}{\tt ({\ident})} \\ -{\taclevel} & ::= & {\tt (at level} {\naturalnumber}{\tt )} \\ -{\tacargtype} & ::= & -%{\tt preident} $|$ -{\tt ident} $|$ -{\tt simple\_intropattern} $|$ -{\tt reference} \\ & $|$ & -{\tt hyp} $|$ -{\tt hyp\_list} $|$ -{\tt ne\_hyp\_list} \\ & $|$ & -% {\tt quantified\_hypothesis} \\ & $|$ & -{\tt constr} $|$ -{\tt constr\_list} $|$ -{\tt ne\_constr\_list} \\ & $|$ & -%{\tt castedopenconstr} $|$ -{\tt integer} $|$ -{\tt integer\_list} $|$ -{\tt ne\_integer\_list} \\ & $|$ & -{\tt int\_or\_var} $|$ -{\tt int\_or\_var\_list} $|$ -{\tt ne\_int\_or\_var\_list} \\ & $|$ & -{\tt tactic} $|$ {\tt tactic$n$} \qquad\mbox{(for $0\leq n\leq 5$)} - -\end{tabular} -\medskip - -A tactic notation {\tt Tactic Notation {\taclevel} -{\sequence{\proditem}{}} := {\tac}} extends the parser and -pretty-printer of tactics with a new rule made of the list of -production items. It then evaluates into the tactic expression -{\tac}. For simple tactics, it is recommended to use a terminal -symbol, i.e. a {\str}, for the first production item. The tactic -level indicates the parsing precedence of the tactic notation. This -information is particularly relevant for notations of tacticals. -Levels 0 to 5 are available (default is 0). -To know the parsing precedences of the -existing tacticals, use the command {\tt Print Grammar tactic.} - -Each type of tactic argument has a specific semantic regarding how it -is parsed and how it is interpreted. The semantic is described in the -following table. The last command gives examples of tactics which -use the corresponding kind of argument. - -\medskip -\noindent -\begin{tabular}{l|l|l|l} -Tactic argument type & parsed as & interpreted as & as in tactic \\ -\hline & & & \\ -{\tt\small ident} & identifier & a user-given name & {\tt intro} \\ -{\tt\small simple\_intropattern} & intro\_pattern & an intro\_pattern & {\tt intros}\\ -{\tt\small hyp} & identifier & an hypothesis defined in context & {\tt clear}\\ -%% quantified_hypothesis actually not supported -%%{\tt\small quantified\_hypothesis} & identifier or integer & a named or non dep. hyp. of the goal & {\tt intros until}\\ -{\tt\small reference} & qualified identifier & a global reference of term & {\tt unfold}\\ -{\tt\small constr} & term & a term & {\tt exact} \\ -%% castedopenconstr actually not supported -%%{\tt\small castedopenconstr} & term & a term with its sign. of exist. var. & {\tt refine}\\ -{\tt\small integer} & integer & an integer & \\ -{\tt\small int\_or\_var} & identifier or integer & an integer & {\tt do} \\ -{\tt\small tactic} & tactic at level 5 & a tactic & \\ -{\tt\small tactic$n$} & tactic at level $n$ & a tactic & \\ -{\tt\small {\nterm{entry}}\_list} & list of {\nterm{entry}} & a list of how {\nterm{entry}} is interpreted & \\ -{\tt\small ne\_{\nterm{entry}}\_list} & non-empty list of {\nterm{entry}} & a list of how {\nterm{entry}} is interpreted& \\ -\end{tabular} - -\Rem In order to be bound in tactic definitions, each syntactic entry -for argument type must include the case of simple {\ltac} identifier -as part of what it parses. This is naturally the case for {\tt ident}, -{\tt simple\_intropattern}, {\tt reference}, {\tt constr}, ... but not -for {\tt integer}. This is the reason for introducing a special entry -{\tt int\_or\_var} which evaluates to integers only but which -syntactically includes identifiers in order to be usable in tactic -definitions. - -\Rem The {\tt {\nterm{entry}}\_list} and {\tt ne\_{\nterm{entry}}\_list} -entries can be used in primitive tactics or in other notations at -places where a list of the underlying entry can be used: {\nterm{entry}} is -either {\tt\small constr}, {\tt\small hyp}, {\tt\small integer} or -{\tt\small int\_or\_var}. - -% $Id: RefMan-syn.tex 13329 2010-07-26 11:05:39Z herbelin $ - -%%% Local Variables: -%%% mode: latex -%%% TeX-master: "Reference-Manual" -%%% End: diff --git a/doc/refman/RefMan-tac.tex b/doc/refman/RefMan-tac.tex deleted file mode 100644 index 14d95ab8..00000000 --- a/doc/refman/RefMan-tac.tex +++ /dev/null @@ -1,4341 +0,0 @@ -% TODO: unify the use of \form and \type to mean a type -% or use \form specifically for a type of type Prop -\chapter{Tactics -\index{Tactics} -\label{Tactics}} - -A deduction rule is a link between some (unique) formula, that we call -the {\em conclusion} and (several) formulas that we call the {\em -premises}. Indeed, a deduction rule can be read in two ways. The first -one has the shape: {\it ``if I know this and this then I can deduce -this''}. For instance, if I have a proof of $A$ and a proof of $B$ -then I have a proof of $A \land B$. This is forward reasoning from -premises to conclusion. The other way says: {\it ``to prove this I -have to prove this and this''}. For instance, to prove $A \land B$, I -have to prove $A$ and I have to prove $B$. This is backward reasoning -which proceeds from conclusion to premises. We say that the conclusion -is {\em the goal}\index{goal} to prove and premises are {\em the -subgoals}\index{subgoal}. The tactics implement {\em backward -reasoning}. When applied to a goal, a tactic replaces this goal with -the subgoals it generates. We say that a tactic reduces a goal to its -subgoal(s). - -Each (sub)goal is denoted with a number. The current goal is numbered -1. By default, a tactic is applied to the current goal, but one can -address a particular goal in the list by writing {\sl n:\tac} which -means {\it ``apply tactic {\tac} to goal number {\sl n}''}. -We can show the list of subgoals by typing {\tt Show} (see -Section~\ref{Show}). - -Since not every rule applies to a given statement, every tactic cannot be -used to reduce any goal. In other words, before applying a tactic to a -given goal, the system checks that some {\em preconditions} are -satisfied. If it is not the case, the tactic raises an error message. - -Tactics are build from atomic tactics and tactic expressions (which -extends the folklore notion of tactical) to combine those atomic -tactics. This chapter is devoted to atomic tactics. The tactic -language will be described in Chapter~\ref{TacticLanguage}. - -There are, at least, three levels of atomic tactics. The simplest one -implements basic rules of the logical framework. The second level is -the one of {\em derived rules} which are built by combination of other -tactics. The third one implements heuristics or decision procedures to -build a complete proof of a goal. - -\section{Invocation of tactics -\label{tactic-syntax} -\index{tactic@{\tac}}} - -A tactic is applied as an ordinary command. If the tactic does not -address the first subgoal, the command may be preceded by the wished -subgoal number as shown below: - -\begin{tabular}{lcl} -{\commandtac} & ::= & {\num} {\tt :} {\tac} {\tt .}\\ - & $|$ & {\tac} {\tt .} -\end{tabular} - -\section{Explicit proof as a term} - -\subsection{\tt exact \term -\tacindex{exact} -\label{exact}} - -This tactic applies to any goal. It gives directly the exact proof -term of the goal. Let {\T} be our goal, let {\tt p} be a term of type -{\tt U} then {\tt exact p} succeeds iff {\tt T} and {\tt U} are -convertible (see Section~\ref{conv-rules}). - -\begin{ErrMsgs} -\item \errindex{Not an exact proof} -\end{ErrMsgs} - -\begin{Variants} - \item \texttt{eexact \term}\tacindex{eexact} - - This tactic behaves like \texttt{exact} but is able to handle terms with meta-variables. - -\end{Variants} - - -\subsection{\tt refine \term -\tacindex{refine} -\label{refine} -\index{?@{\texttt{?}}}} - -This tactic allows to give an exact proof but still with some -holes. The holes are noted ``\texttt{\_}''. - -\begin{ErrMsgs} -\item \errindex{invalid argument}: - the tactic \texttt{refine} doesn't know what to do - with the term you gave. -\item \texttt{Refine passed ill-formed term}: the term you gave is not - a valid proof (not easy to debug in general). - This message may also occur in higher-level tactics, which call - \texttt{refine} internally. -\item \errindex{Cannot infer a term for this placeholder} - there is a hole in the term you gave - which type cannot be inferred. Put a cast around it. -\end{ErrMsgs} - -An example of use is given in Section~\ref{refine-example}. - -\section{Basics -\index{Typing rules}} - -Tactics presented in this section implement the basic typing rules of -{\CIC} given in Chapter~\ref{Cic}. - -\subsection{{\tt assumption} -\tacindex{assumption}} - -This tactic applies to any goal. It implements the -``Var''\index{Typing rules!Var} rule given in -Section~\ref{Typed-terms}. It looks in the local context for an -hypothesis which type is equal to the goal. If it is the case, the -subgoal is proved. Otherwise, it fails. - -\begin{ErrMsgs} -\item \errindex{No such assumption} -\end{ErrMsgs} - -\begin{Variants} -\tacindex{eassumption} - \item \texttt{eassumption} - - This tactic behaves like \texttt{assumption} but is able to handle - goals with meta-variables. - -\end{Variants} - - -\subsection{\tt clear {\ident} -\tacindex{clear} -\label{clear}} - -This tactic erases the hypothesis named {\ident} in the local context -of the current goal. Then {\ident} is no more displayed and no more -usable in the proof development. - -\begin{Variants} - -\item {\tt clear {\ident$_1$} {\ldots} {\ident$_n$}} - - This is equivalent to {\tt clear {\ident$_1$}. {\ldots} clear - {\ident$_n$}.} - -\item {\tt clearbody {\ident}}\tacindex{clearbody} - - This tactic expects {\ident} to be a local definition then clears - its body. Otherwise said, this tactic turns a definition into an - assumption. - -\item \texttt{clear - {\ident$_1$} {\ldots} {\ident$_n$}} - - This tactic clears all hypotheses except the ones depending in - the hypotheses named {\ident$_1$} {\ldots} {\ident$_n$} and in the - goal. - -\item \texttt{clear} - - This tactic clears all hypotheses except the ones depending in - goal. - -\item {\tt clear dependent \ident \tacindex{clear dependent}} - - This clears the hypothesis \ident\ and all hypotheses - which depend on it. - -\end{Variants} - -\begin{ErrMsgs} -\item \errindex{{\ident} not found} -\item \errindexbis{{\ident} is used in the conclusion}{is used in the - conclusion} -\item \errindexbis{{\ident} is used in the hypothesis {\ident'}}{is - used in the hypothesis} -\end{ErrMsgs} - -\subsection{\tt move {\ident$_1$} after {\ident$_2$} -\tacindex{move} -\label{move}} - -This moves the hypothesis named {\ident$_1$} in the local context -after the hypothesis named {\ident$_2$}. - -If {\ident$_1$} comes before {\ident$_2$} in the order of dependences, -then all hypotheses between {\ident$_1$} and {\ident$_2$} which -(possibly indirectly) depend on {\ident$_1$} are moved also. - -If {\ident$_1$} comes after {\ident$_2$} in the order of dependences, -then all hypotheses between {\ident$_1$} and {\ident$_2$} which -(possibly indirectly) occur in {\ident$_1$} are moved also. - -\begin{Variants} - -\item {\tt move {\ident$_1$} before {\ident$_2$}} - -This moves {\ident$_1$} towards and just before the hypothesis named {\ident$_2$}. - -\item {\tt move {\ident} at top} - -This moves {\ident} at the top of the local context (at the beginning of the context). - -\item {\tt move {\ident} at bottom} - -This moves {\ident} at the bottom of the local context (at the end of the context). - -\end{Variants} - -\begin{ErrMsgs} - -\item \errindex{{\ident$_i$} not found} - -\item \errindex{Cannot move {\ident$_1$} after {\ident$_2$}: - it occurs in {\ident$_2$}} - -\item \errindex{Cannot move {\ident$_1$} after {\ident$_2$}: - it depends on {\ident$_2$}} - -\end{ErrMsgs} - -\subsection{\tt rename {\ident$_1$} into {\ident$_2$} -\tacindex{rename}} - -This renames hypothesis {\ident$_1$} into {\ident$_2$} in the current -context\footnote{but it does not rename the hypothesis in the - proof-term...} - -\begin{Variants} - -\item {\tt rename {\ident$_1$} into {\ident$_2$}, \ldots, - {\ident$_{2k-1}$} into {\ident$_{2k}$}} - - Is equivalent to the sequence of the corresponding atomic {\tt rename}. - -\end{Variants} - -\begin{ErrMsgs} - -\item \errindex{{\ident$_1$} not found} - -\item \errindexbis{{\ident$_2$} is already used}{is already used} - -\end{ErrMsgs} - -\subsection{\tt intro -\tacindex{intro} -\label{intro}} - -This tactic applies to a goal which is either a product or starts with -a let binder. If the goal is a product, the tactic implements the -``Lam''\index{Typing rules!Lam} rule given in -Section~\ref{Typed-terms}\footnote{Actually, only the second subgoal will be -generated since the other one can be automatically checked.}. If the -goal starts with a let binder then the tactic implements a mix of the -``Let''\index{Typing rules!Let} and ``Conv''\index{Typing rules!Conv}. - -If the current goal is a dependent product {\tt forall $x$:$T$, $U$} (resp {\tt -let $x$:=$t$ in $U$}) then {\tt intro} puts {\tt $x$:$T$} (resp {\tt $x$:=$t$}) - in the local context. -% Obsolete (quantified names already avoid hypotheses names): -% Otherwise, it puts -% {\tt x}{\it n}{\tt :T} where {\it n} is such that {\tt x}{\it n} is a -%fresh name. -The new subgoal is $U$. -% If the {\tt x} has been renamed {\tt x}{\it n} then it is replaced -% by {\tt x}{\it n} in {\tt U}. - -If the goal is a non dependent product {\tt $T$ -> $U$}, then it puts -in the local context either {\tt H}{\it n}{\tt :$T$} (if $T$ is of -type {\tt Set} or {\tt Prop}) or {\tt X}{\it n}{\tt :$T$} (if the type -of $T$ is {\tt Type}). The optional index {\it n} is such that {\tt -H}{\it n} or {\tt X}{\it n} is a fresh identifier. -In both cases the new subgoal is $U$. - -If the goal is neither a product nor starting with a let definition, -the tactic {\tt intro} applies the tactic {\tt red} until the tactic -{\tt intro} can be applied or the goal is not reducible. - -\begin{ErrMsgs} -\item \errindex{No product even after head-reduction} -\item \errindexbis{{\ident} is already used}{is already used} -\end{ErrMsgs} - -\begin{Variants} - -\item {\tt intros}\tacindex{intros} - - Repeats {\tt intro} until it meets the head-constant. It never reduces - head-constants and it never fails. - -\item {\tt intro {\ident}} - - Applies {\tt intro} but forces {\ident} to be the name of the - introduced hypothesis. - - \ErrMsg \errindex{name {\ident} is already used} - - \Rem If a name used by {\tt intro} hides the base name of a global - constant then the latter can still be referred to by a qualified name - (see \ref{LongNames}). - -\item {\tt intros \ident$_1$ \dots\ \ident$_n$} - - Is equivalent to the composed tactic {\tt intro \ident$_1$; \dots\ ; - intro \ident$_n$}. - - More generally, the \texttt{intros} tactic takes a pattern as - argument in order to introduce names for components of an inductive - definition or to clear introduced hypotheses; This is explained - in~\ref{intros-pattern}. - -\item {\tt intros until {\ident}} \tacindex{intros until} - - Repeats {\tt intro} until it meets a premise of the goal having form - {\tt (} {\ident}~{\tt :}~{\term} {\tt )} and discharges the variable - named {\ident} of the current goal. - - \ErrMsg \errindex{No such hypothesis in current goal} - -\item {\tt intros until {\num}} \tacindex{intros until} - - Repeats {\tt intro} until the {\num}-th non-dependent product. For - instance, on the subgoal % - \verb+forall x y:nat, x=y -> y=x+ the tactic \texttt{intros until 1} - is equivalent to \texttt{intros x y H}, as \verb+x=y -> y=x+ is the - first non-dependent product. And on the subgoal % - \verb+forall x y z:nat, x=y -> y=x+ the tactic \texttt{intros until 1} - is equivalent to \texttt{intros x y z} as the product on \texttt{z} - can be rewritten as a non-dependent product: % - \verb+forall x y:nat, nat -> x=y -> y=x+ - - - \ErrMsg \errindex{No such hypothesis in current goal} - - Happens when {\num} is 0 or is greater than the number of non-dependent - products of the goal. - -\item {\tt intro after \ident} \tacindex{intro after}\\ - {\tt intro before \ident} \tacindex{intro before}\\ - {\tt intro at top} \tacindex{intro at top}\\ - {\tt intro at bottom} \tacindex{intro at bottom} - - Applies {\tt intro} and moves the freshly introduced hypothesis - respectively after the hypothesis \ident{}, before the hypothesis - \ident{}, at the top of the local context, or at the bottom of the - local context. All hypotheses on which the new hypothesis depends - are moved too so as to respect the order of dependencies between - hypotheses. Note that {\tt intro at bottom} is a synonym for {\tt - intro} with no argument. - -\begin{ErrMsgs} -\item \errindex{No product even after head-reduction} -\item \errindex{No such hypothesis} : {\ident} -\end{ErrMsgs} - -\item {\tt intro \ident$_1$ after \ident$_2$}\\ - {\tt intro \ident$_1$ before \ident$_2$}\\ - {\tt intro \ident$_1$ at top}\\ - {\tt intro \ident$_1$ at bottom} - - Behaves as previously but naming the introduced hypothesis - \ident$_1$. It is equivalent to {\tt intro \ident$_1$} followed by - the appropriate call to {\tt move}~(see Section~\ref{move}). - -\begin{ErrMsgs} -\item \errindex{No product even after head-reduction} -\item \errindex{No such hypothesis} : {\ident} -\end{ErrMsgs} - -\end{Variants} - -\subsection{\tt apply \term -\tacindex{apply} -\label{apply}} - -This tactic applies to any goal. The argument {\term} is a term -well-formed in the local context. The tactic {\tt apply} tries to -match the current goal against the conclusion of the type of {\term}. -If it succeeds, then the tactic returns as many subgoals as the number -of non dependent premises of the type of {\term}. If the conclusion of -the type of {\term} does not match the goal {\em and} the conclusion -is an inductive type isomorphic to a tuple type, then each component -of the tuple is recursively matched to the goal in the left-to-right -order. - -The tactic {\tt apply} relies on first-order unification with -dependent types unless the conclusion of the type of {\term} is of the -form {\tt ($P$~ $t_1$~\ldots ~$t_n$)} with $P$ to be instantiated. In -the latter case, the behavior depends on the form of the goal. If the -goal is of the form {\tt (fun $x$ => $Q$)~$u_1$~\ldots~$u_n$} and the -$t_i$ and $u_i$ unifies, then $P$ is taken to be (fun $x$ => $Q$). -Otherwise, {\tt apply} tries to define $P$ by abstracting over -$t_1$~\ldots ~$t_n$ in the goal. See {\tt pattern} in -Section~\ref{pattern} to transform the goal so that it gets the form -{\tt (fun $x$ => $Q$)~$u_1$~\ldots~$u_n$}. - -\begin{ErrMsgs} -\item \errindex{Impossible to unify \dots\ with \dots} - - The {\tt apply} - tactic failed to match the conclusion of {\term} and the current goal. - You can help the {\tt apply} tactic by transforming your - goal with the {\tt change} or {\tt pattern} tactics (see - sections~\ref{pattern},~\ref{change}). - -\item \errindex{Unable to find an instance for the variables -{\ident} \ldots {\ident}} - - This occurs when some instantiations of the premises of {\term} are not - deducible from the unification. This is the case, for instance, when - you want to apply a transitivity property. In this case, you have to - use one of the variants below: - -\end{ErrMsgs} - -\begin{Variants} - -\item{\tt apply {\term} with {\term$_1$} \dots\ {\term$_n$}} - \tacindex{apply \dots\ with} - - Provides {\tt apply} with explicit instantiations for all dependent - premises of the type of {\term} which do not occur in the conclusion - and consequently cannot be found by unification. Notice that - {\term$_1$} \dots\ {\term$_n$} must be given according to the order - of these dependent premises of the type of {\term}. - - \ErrMsg \errindex{Not the right number of missing arguments} - -\item{\tt apply {\term} with ({\vref$_1$} := {\term$_1$}) \dots\ ({\vref$_n$} - := {\term$_n$})} - - This also provides {\tt apply} with values for instantiating - premises. Here, variables are referred by names and non-dependent - products by increasing numbers (see syntax in Section~\ref{Binding-list}). - -\item {\tt apply} {\term$_1$} {\tt ,} \ldots {\tt ,} {\term$_n$} - - This is a shortcut for {\tt apply} {\term$_1$} {\tt ; [ ..~|} - \ldots~{\tt ; [ ..~| {\tt apply} {\term$_n$} ]} \ldots~{\tt ]}, i.e. for the - successive applications of {\term$_{i+1}$} on the last subgoal - generated by {\tt apply} {\term$_i$}, starting from the application - of {\term$_1$}. - -\item {\tt eapply \term}\tacindex{eapply}\label{eapply} - - The tactic {\tt eapply} behaves as {\tt apply} but does not fail - when no instantiation are deducible for some variables in the - premises. Rather, it turns these variables into so-called - existential variables which are variables still to instantiate. An - existential variable is identified by a name of the form {\tt ?$n$} - where $n$ is a number. The instantiation is intended to be found - later in the proof. - - An example of use of {\tt eapply} is given in - Section~\ref{eapply-example}. - -\item {\tt simple apply {\term}} \tacindex{simple apply} - - This behaves like {\tt apply} but it reasons modulo conversion only - on subterms that contain no variables to instantiate. For instance, - if {\tt id := fun x:nat => x} and {\tt H : forall y, id y = y} then - {\tt simple apply H} on goal {\tt O = O} does not succeed because it - would require the conversion of {\tt f ?y} and {\tt O} where {\tt - ?y} is a variable to instantiate. Tactic {\tt simple apply} does not - either traverse tuples as {\tt apply} does. - - Because it reasons modulo a limited amount of conversion, {\tt - simple apply} fails quicker than {\tt apply} and it is then - well-suited for uses in used-defined tactics that backtrack often. - -\item \zeroone{{\tt simple}} {\tt apply} {\term$_1$} \zeroone{{\tt with} - {\bindinglist$_1$}} {\tt ,} \ldots {\tt ,} {\term$_n$} \zeroone{{\tt with} - {\bindinglist$_n$}}\\ - \zeroone{{\tt simple}} {\tt eapply} {\term$_1$} \zeroone{{\tt with} - {\bindinglist$_1$}} {\tt ,} \ldots {\tt ,} {\term$_n$} \zeroone{{\tt with} - {\bindinglist$_n$}} - - This summarizes the different syntaxes for {\tt apply} and {\tt eapply}. - -\item {\tt lapply {\term}} \tacindex{lapply} - - This tactic applies to any goal, say {\tt G}. The argument {\term} - has to be well-formed in the current context, its type being - reducible to a non-dependent product {\tt A -> B} with {\tt B} - possibly containing products. Then it generates two subgoals {\tt - B->G} and {\tt A}. Applying {\tt lapply H} (where {\tt H} has type - {\tt A->B} and {\tt B} does not start with a product) does the same - as giving the sequence {\tt cut B. 2:apply H.} where {\tt cut} is - described below. - - \Warning When {\term} contains more than one non - dependent product the tactic {\tt lapply} only takes into account the - first product. - -\end{Variants} - -\subsection{{\tt set ( {\ident} {\tt :=} {\term} \tt )} -\label{tactic:set} -\tacindex{set} -\tacindex{pose} -\tacindex{remember}} - -This replaces {\term} by {\ident} in the conclusion or in the -hypotheses of the current goal and adds the new definition {\ident -{\tt :=} \term} to the local context. The default is to make this -replacement only in the conclusion. - -\begin{Variants} - -\item {\tt set (} {\ident} {\tt :=} {\term} {\tt ) in {\occgoalset}} - -This notation allows to specify which occurrences of {\term} have to -be substituted in the context. The {\tt in {\occgoalset}} clause is an -occurrence clause whose syntax and behavior is described in -Section~\ref{Occurrences clauses}. - -\item {\tt set (} {\ident} \nelist{\binder}{} {\tt :=} {\term} {\tt )} - - This is equivalent to {\tt set (} {\ident} {\tt :=} {\tt fun} - \nelist{\binder}{} {\tt =>} {\term} {\tt )}. - -\item {\tt set } {\term} - - This behaves as {\tt set (} {\ident} := {\term} {\tt )} but {\ident} - is generated by {\Coq}. This variant also supports an occurrence clause. - -\item {\tt set (} {\ident$_0$} \nelist{\binder}{} {\tt :=} {\term} - {\tt ) in {\occgoalset}}\\ - {\tt set {\term} in {\occgoalset}} - - These are the general forms which combine the previous possibilities. - -\item {\tt remember {\term} {\tt as} {\ident}} - - This behaves as {\tt set (} {\ident} := {\term} {\tt ) in *} and using a - logical (Leibniz's) equality instead of a local definition. - -\item {\tt remember {\term} {\tt as} {\ident} in {\occgoalset}} - - This is a more general form of {\tt remember} that remembers the - occurrences of {\term} specified by an occurrences set. - -\item {\tt pose ( {\ident} {\tt :=} {\term} {\tt )}} - - This adds the local definition {\ident} := {\term} to the current - context without performing any replacement in the goal or in the - hypotheses. It is equivalent to {\tt set ( {\ident} {\tt :=} - {\term} {\tt ) in |-}}. - -\item {\tt pose (} {\ident} \nelist{\binder}{} {\tt :=} {\term} {\tt )} - - This is equivalent to {\tt pose (} {\ident} {\tt :=} {\tt fun} - \nelist{\binder}{} {\tt =>} {\term} {\tt )}. - -\item{\tt pose {\term}} - - This behaves as {\tt pose (} {\ident} := {\term} {\tt )} but - {\ident} is generated by {\Coq}. - -\end{Variants} - -\subsection{{\tt assert ( {\ident} : {\form} \tt )} -\tacindex{assert}} - -This tactic applies to any goal. {\tt assert (H : U)} adds a new -hypothesis of name \texttt{H} asserting \texttt{U} to the current goal -and opens a new subgoal \texttt{U}\footnote{This corresponds to the - cut rule of sequent calculus.}. The subgoal {\texttt U} comes first -in the list of subgoals remaining to prove. - -\begin{ErrMsgs} -\item \errindex{Not a proposition or a type} - - Arises when the argument {\form} is neither of type {\tt Prop}, {\tt - Set} nor {\tt Type}. - -\end{ErrMsgs} - -\begin{Variants} - -\item{\tt assert {\form}} - - This behaves as {\tt assert (} {\ident} : {\form} {\tt )} but - {\ident} is generated by {\Coq}. - -\item{\tt assert (} {\ident} := {\term} {\tt )} - - This behaves as {\tt assert ({\ident} : {\type});[exact - {\term}|idtac]} where {\type} is the type of {\term}. - -\item {\tt cut {\form}}\tacindex{cut} - - This tactic applies to any goal. It implements the non dependent - case of the ``App''\index{Typing rules!App} rule given in - Section~\ref{Typed-terms}. (This is Modus Ponens inference rule.) - {\tt cut U} transforms the current goal \texttt{T} into the two - following subgoals: {\tt U -> T} and \texttt{U}. The subgoal {\tt U - -> T} comes first in the list of remaining subgoal to prove. - -\item \texttt{assert {\form} by {\tac}}\tacindex{assert by} - - This tactic behaves like \texttt{assert} but applies {\tac} - to solve the subgoals generated by \texttt{assert}. - -\item \texttt{assert {\form} as {\intropattern}\tacindex{assert as}} - - If {\intropattern} is a naming introduction pattern (see - Section~\ref{intros-pattern}), the hypothesis is named after this - introduction pattern (in particular, if {\intropattern} is {\ident}, - the tactic behaves like \texttt{assert ({\ident} : {\form})}). - - If {\intropattern} is a disjunctive/conjunctive introduction - pattern, the tactic behaves like \texttt{assert {\form}} then destructing the - resulting hypothesis using the given introduction pattern. - -\item \texttt{assert {\form} as {\intropattern} by {\tac}} - - This combines the two previous variants of {\tt assert}. - -\item \texttt{pose proof {\term} as {\intropattern}\tacindex{pose proof}} - - This tactic behaves like \texttt{assert T as {\intropattern} by - exact {\term}} where \texttt{T} is the type of {\term}. - - In particular, \texttt{pose proof {\term} as {\ident}} behaves as - \texttt{assert ({\ident}:T) by exact {\term}} (where \texttt{T} is - the type of {\term}) and \texttt{pose proof {\term} as - {\disjconjintropattern}\tacindex{pose proof}} behaves - like \texttt{destruct {\term} as {\disjconjintropattern}}. - -\item {\tt specialize ({\ident} \term$_1$ {\ldots} \term$_n$)\tacindex{specialize}} \\ - {\tt specialize {\ident} with \bindinglist} - - The tactic {\tt specialize} works on local hypothesis \ident. - The premises of this hypothesis (either universal - quantifications or non-dependent implications) are instantiated - by concrete terms coming either from arguments \term$_1$ - $\ldots$ \term$_n$ or from a bindings list (see - Section~\ref{Binding-list} for more about bindings lists). In the - second form, all instantiation elements must be given, whereas - in the first form the application to \term$_1$ {\ldots} - \term$_n$ can be partial. The first form is equivalent to - {\tt assert (\ident':=\ident \term$_1$ {\ldots} \term$_n$); - clear \ident; rename \ident' into \ident}. - - The name {\ident} can also refer to a global lemma or - hypothesis. In this case, for compatibility reasons, the - behavior of {\tt specialize} is close to that of {\tt - generalize}: the instantiated statement becomes an additional - premise of the goal. - -%% Moreover, the old syntax allows the use of a number after {\tt specialize} -%% for controlling the number of premises to instantiate. Giving this -%% number should not be mandatory anymore (automatic detection of how -%% many premises can be eaten without leaving meta-variables). Hence -%% no documentation for this integer optional argument of specialize - -\end{Variants} - -\subsection{{\tt apply {\term} in {\ident}} -\tacindex{apply \ldots\ in}} - -This tactic applies to any goal. The argument {\term} is a term -well-formed in the local context and the argument {\ident} is an -hypothesis of the context. The tactic {\tt apply {\term} in {\ident}} -tries to match the conclusion of the type of {\ident} against a non -dependent premise of the type of {\term}, trying them from right to -left. If it succeeds, the statement of hypothesis {\ident} is -replaced by the conclusion of the type of {\term}. The tactic also -returns as many subgoals as the number of other non dependent premises -in the type of {\term} and of the non dependent premises of the type -of {\ident}. If the conclusion of the type of {\term} does not match -the goal {\em and} the conclusion is an inductive type isomorphic to a -tuple type, then the tuple is (recursively) decomposed and the first -component of the tuple of which a non dependent premise matches the -conclusion of the type of {\ident}. Tuples are decomposed in a -width-first left-to-right order (for instance if the type of {\tt H1} -is a \verb=A <-> B= statement, and the type of {\tt H2} is \verb=A= -then {\tt apply H1 in H2} transforms the type of {\tt H2} into {\tt - B}). The tactic {\tt apply} relies on first-order pattern-matching -with dependent types. - -\begin{ErrMsgs} -\item \errindex{Statement without assumptions} - -This happens if the type of {\term} has no non dependent premise. - -\item \errindex{Unable to apply} - -This happens if the conclusion of {\ident} does not match any of the -non dependent premises of the type of {\term}. -\end{ErrMsgs} - -\begin{Variants} -\item {\tt apply \nelist{\term}{,} in {\ident}} - -This applies each of {\term} in sequence in {\ident}. - -\item {\tt apply \nelist{{\term} with {\bindinglist}}{,} in {\ident}} - -This does the same but uses the bindings in each {\bindinglist} to -instantiate the parameters of the corresponding type of {\term} -(see syntax of bindings in Section~\ref{Binding-list}). - -\item {\tt eapply \nelist{{\term} with {\bindinglist}}{,} in {\ident}} -\tacindex{eapply {\ldots} in} - -This works as {\tt apply \nelist{{\term} with {\bindinglist}}{,} in -{\ident}} but turns unresolved bindings into existential variables, if -any, instead of failing. - -\item {\tt apply \nelist{{\term}{,} with {\bindinglist}}{,} in {\ident} as {\disjconjintropattern}} - -This works as {\tt apply \nelist{{\term}{,} with {\bindinglist}}{,} in -{\ident}} then destructs the hypothesis {\ident} along -{\disjconjintropattern} as {\tt destruct {\ident} as -{\disjconjintropattern}} would. - -\item {\tt eapply \nelist{{\term}{,} with {\bindinglist}}{,} in {\ident} as {\disjconjintropattern}} - -This works as {\tt apply \nelist{{\term}{,} with {\bindinglist}}{,} in {\ident} as {\disjconjintropattern}} but using {\tt eapply}. - -\item {\tt simple apply {\term} in {\ident}} -\tacindex{simple apply {\ldots} in} -\tacindex{simple eapply {\ldots} in} - -This behaves like {\tt apply {\term} in {\ident}} but it reasons -modulo conversion only on subterms that contain no variables to -instantiate. For instance, if {\tt id := fun x:nat => x} and {\tt H : - forall y, id y = y -> True} and {\tt H0 : O = O} then {\tt simple - apply H in H0} does not succeed because it would require the -conversion of {\tt f ?y} and {\tt O} where {\tt ?y} is a variable to -instantiate. Tactic {\tt simple apply {\term} in {\ident}} does not -either traverse tuples as {\tt apply {\term} in {\ident}} does. - -\item {\tt \zeroone{simple} apply \nelist{{\term} \zeroone{with {\bindinglist}}}{,} in {\ident} \zeroone{as {\disjconjintropattern}}}\\ -{\tt \zeroone{simple} eapply \nelist{{\term} \zeroone{with {\bindinglist}}}{,} in {\ident} \zeroone{as {\disjconjintropattern}}} - -This summarizes the different syntactic variants of {\tt apply {\term} - in {\ident}} and {\tt eapply {\term} in {\ident}}. -\end{Variants} - -\subsection{\tt generalize \term -\tacindex{generalize} -\label{generalize}} - -This tactic applies to any goal. It generalizes the conclusion w.r.t. -one subterm of it. For example: - -\begin{coq_eval} -Goal forall x y:nat, (0 <= x + y + y). -intros. -\end{coq_eval} -\begin{coq_example} -Show. -generalize (x + y + y). -\end{coq_example} - -\begin{coq_eval} -Abort. -\end{coq_eval} - -If the goal is $G$ and $t$ is a subterm of type $T$ in the goal, then -{\tt generalize} \textit{t} replaces the goal by {\tt forall (x:$T$), $G'$} -where $G'$ is obtained from $G$ by replacing all occurrences of $t$ by -{\tt x}. The name of the variable (here {\tt n}) is chosen based on $T$. - -\begin{Variants} -\item {\tt generalize {\term$_1$ , \dots\ , \term$_n$}} - - Is equivalent to {\tt generalize \term$_n$; \dots\ ; generalize - \term$_1$}. Note that the sequence of \term$_i$'s are processed - from $n$ to $1$. - -\item {\tt generalize {\term} at {\num$_1$ \dots\ \num$_i$}} - - Is equivalent to {\tt generalize \term} but generalizing only over - the specified occurrences of {\term} (counting from left to right on the - expression printed using option {\tt Set Printing All}). - -\item {\tt generalize {\term} as {\ident}} - - Is equivalent to {\tt generalize \term} but use {\ident} to name the - generalized hypothesis. - -\item {\tt generalize {\term$_1$} at {\num$_{11}$ \dots\ \num$_{1i_1}$} - as {\ident$_1$} - , {\ldots} , - {\term$_n$} at {\num$_{n1}$ \dots\ \num$_{ni_n}$} - as {\ident$_2$}} - - This is the most general form of {\tt generalize} that combines the - previous behaviors. - -\item {\tt generalize dependent \term} \tacindex{generalize dependent} - - This generalizes {\term} but also {\em all} hypotheses which depend - on {\term}. It clears the generalized hypotheses. - -\end{Variants} - - -\subsection{\tt revert \ident$_1$ \dots\ \ident$_n$ -\tacindex{revert} -\label{revert}} - -This applies to any goal with variables \ident$_1$ \dots\ \ident$_n$. -It moves the hypotheses (possibly defined) to the goal, if this respects -dependencies. This tactic is the inverse of {\tt intro}. - -\begin{ErrMsgs} -\item \errindexbis{{\ident} is used in the hypothesis {\ident'}}{is - used in the hypothesis} -\end{ErrMsgs} - -\begin{Variants} -\item {\tt revert dependent \ident \tacindex{revert dependent}} - - This moves to the goal the hypothesis \ident\ and all hypotheses - which depend on it. - -\end{Variants} - -\subsection{\tt change \term -\tacindex{change} -\label{change}} - -This tactic applies to any goal. It implements the rule -``Conv''\index{Typing rules!Conv} given in Section~\ref{Conv}. {\tt - change U} replaces the current goal \T\ with \U\ providing that -\U\ is well-formed and that \T\ and \U\ are convertible. - -\begin{ErrMsgs} -\item \errindex{Not convertible} -\end{ErrMsgs} - -\tacindex{change \dots\ in} -\begin{Variants} -\item {\tt change \term$_1$ with \term$_2$} - - This replaces the occurrences of \term$_1$ by \term$_2$ in the - current goal. The terms \term$_1$ and \term$_2$ must be - convertible. - -\item {\tt change \term$_1$ at \num$_1$ \dots\ \num$_i$ with \term$_2$} - - This replaces the occurrences numbered \num$_1$ \dots\ \num$_i$ of - \term$_1$ by \term$_2$ in the current goal. - The terms \term$_1$ and \term$_2$ must be convertible. - - \ErrMsg {\tt Too few occurrences} - -\item {\tt change {\term} in {\ident}} - -\item {\tt change \term$_1$ with \term$_2$ in {\ident}} - -\item {\tt change \term$_1$ at \num$_1$ \dots\ \num$_i$ with \term$_2$ in - {\ident}} - - This applies the {\tt change} tactic not to the goal but to the - hypothesis {\ident}. - -\end{Variants} - -\SeeAlso \ref{Conversion-tactics} - -\subsection{\tt fix {\ident} {\num} -\tacindex{fix} -\label{tactic:fix}} - -This tactic is a primitive tactic to start a proof by induction. In -general, it is easier to rely on higher-level induction tactics such -as the ones described in Section~\ref{Tac-induction}. - -In the syntax of the tactic, the identifier {\ident} is the name given -to the induction hypothesis. The natural number {\num} tells on which -premise of the current goal the induction acts, starting -from 1 and counting both dependent and non dependent -products. Especially, the current lemma must be composed of at least -{\num} products. - -Like in a {\tt fix} expression, the induction -hypotheses have to be used on structurally smaller arguments. -The verification that inductive proof arguments are correct is done -only at the time of registering the lemma in the environment. To know -if the use of induction hypotheses is correct at some -time of the interactive development of a proof, use the command {\tt - Guarded} (see Section~\ref{Guarded}). - -\begin{Variants} - \item {\tt fix} {\ident}$_1$ {\num} {\tt with (} {\ident}$_2$ - \nelist{{\binder}$_{2}$}{} \zeroone{{\tt \{ struct {\ident$'_2$} - \}}} {\tt :} {\type}$_2$ {\tt )} {\ldots} {\tt (} {\ident}$_1$ - \nelist{{\binder}$_n$}{} \zeroone{{\tt \{ struct {\ident$'_n$} \}}} - {\tt :} {\type}$_n$ {\tt )} - -This starts a proof by mutual induction. The statements to be -simultaneously proved are respectively {\tt forall} - \nelist{{\binder}$_2$}{}{\tt ,} {\type}$_2$, {\ldots}, {\tt forall} - \nelist{{\binder}$_n$}{}{\tt ,} {\type}$_n$. The identifiers -{\ident}$_1$ {\ldots} {\ident}$_n$ are the names of the induction -hypotheses. The identifiers {\ident}$'_2$ {\ldots} {\ident}$'_n$ are the -respective names of the premises on which the induction is performed -in the statements to be simultaneously proved (if not given, the -system tries to guess itself what they are). - -\end{Variants} - -\subsection{\tt cofix {\ident} -\tacindex{cofix} -\label{tactic:cofix}} - -This tactic starts a proof by coinduction. The identifier {\ident} is -the name given to the coinduction hypothesis. Like in a {\tt cofix} -expression, the use of induction hypotheses have to guarded by a -constructor. The verification that the use of coinductive hypotheses -is correct is done only at the time of registering the lemma in the -environment. To know if the use of coinduction hypotheses is correct -at some time of the interactive development of a proof, use the -command {\tt Guarded} (see Section~\ref{Guarded}). - - -\begin{Variants} - \item {\tt cofix} {\ident}$_1$ {\tt with (} {\ident}$_2$ - \nelist{{\binder}$_2$}{} {\tt :} {\type}$_2$ {\tt )} {\ldots} {\tt - (} {\ident}$_1$ \nelist{{\binder}$_1$}{} {\tt :} {\type}$_n$ - {\tt )} - -This starts a proof by mutual coinduction. The statements to be -simultaneously proved are respectively {\tt forall} -\nelist{{\binder}$_2$}{}{\tt ,} {\type}$_2$, {\ldots}, {\tt forall} - \nelist{{\binder}$_n$}{}{\tt ,} {\type}$_n$. The identifiers - {\ident}$_1$ {\ldots} {\ident}$_n$ are the names of the - coinduction hypotheses. - -\end{Variants} - -\subsection{\tt evar (\ident:\term) -\tacindex{evar} -\label{evar}} - -The {\tt evar} tactic creates a new local definition named \ident\ with -type \term\ in the context. The body of this binding is a fresh -existential variable. - -\subsection{\tt instantiate (\num:= \term) -\tacindex{instantiate} -\label{instantiate}} - -The {\tt instantiate} tactic allows to solve an existential variable -with the term \term. The \num\ argument is the position of the -existential variable from right to left in the conclusion. This cannot be -the number of the existential variable since this number is different -in every session. - -\begin{Variants} - \item {\tt instantiate (\num:=\term) in \ident} - - \item {\tt instantiate (\num:=\term) in (Value of \ident)} - - \item {\tt instantiate (\num:=\term) in (Type of \ident)} - -These allow to refer respectively to existential variables occurring in -a hypothesis or in the body or the type of a local definition. - - \item {\tt instantiate} - - Without argument, the {\tt instantiate} tactic tries to solve as - many existential variables as possible, using information gathered - from other tactics in the same tactical. This is automatically - done after each complete tactic (i.e. after a dot in proof mode), - but not, for example, between each tactic when they are sequenced - by semicolons. - -\end{Variants} - -\subsection{\tt admit -\tacindex{admit} -\label{admit}} - -The {\tt admit} tactic ``solves'' the current subgoal by an -axiom. This typically allows to temporarily skip a subgoal so as to -progress further in the rest of the proof. To know if some proof still -relies on unproved subgoals, one can use the command {\tt Print -Assumptions} (see Section~\ref{PrintAssumptions}). Admitted subgoals -have names of the form {\ident}\texttt{\_admitted} possibly followed -by a number. - -\subsection{\tt constr\_eq \term$_1$ \term$_2$ -\tacindex{constr\_eq} -\label{constreq}} - -This tactic applies to any goal. It checks whether its arguments are -equal modulo alpha conversion and casts. - -\ErrMsg \errindex{Not equal} - -\subsection{\tt is\_evar \term -\tacindex{is\_evar} -\label{isevar}} - -This tactic applies to any goal. It checks whether its argument is an -existential variable. Existential variables are uninstantiated -variables generated by e.g. {\tt eapply} (see Section~\ref{apply}). - -\ErrMsg \errindex{Not an evar} - -\subsection{\tt has\_evar \term -\tacindex{has\_evar} -\label{hasevar}} - -This tactic applies to any goal. It checks whether its argument has an -existential variable as a subterm. Unlike {\tt context} patterns -combined with {\tt is\_evar}, this tactic scans all subterms, -including those under binders. - -\ErrMsg \errindex{No evars} - -\subsection{\tt is\_var \term -\tacindex{is\_var} -\label{isvar}} - -This tactic applies to any goal. It checks whether its argument is a -variable or hypothesis in the current goal context or in the opened sections. - -\ErrMsg \errindex{Not a variable or hypothesis} - -\subsection{Bindings list -\index{Binding list} -\label{Binding-list}} - -Tactics that take a term as argument may also support a bindings list, so -as to instantiate some parameters of the term by name or position. -The general form of a term equipped with a bindings list is {\tt -{\term} with {\bindinglist}} where {\bindinglist} may be of two -different forms: - -\begin{itemize} -\item In a bindings list of the form {\tt (\vref$_1$ := \term$_1$) - \dots\ (\vref$_n$ := \term$_n$)}, {\vref} is either an {\ident} or a - {\num}. The references are determined according to the type of - {\term}. If \vref$_i$ is an identifier, this identifier has to be - bound in the type of {\term} and the binding provides the tactic - with an instance for the parameter of this name. If \vref$_i$ is - some number $n$, this number denotes the $n$-th non dependent - premise of the {\term}, as determined by the type of {\term}. - - \ErrMsg \errindex{No such binder} - -\item A bindings list can also be a simple list of terms {\tt - \term$_1$ \dots\term$_n$}. In that case the references to - which these terms correspond are determined by the tactic. In case - of {\tt induction}, {\tt destruct}, {\tt elim} and {\tt case} (see - Section~\ref{elim}) the terms have to provide instances for all the - dependent products in the type of \term\ while in the case of {\tt - apply}, or of {\tt constructor} and its variants, only instances for - the dependent products which are not bound in the conclusion of the - type are required. - - \ErrMsg \errindex{Not the right number of missing arguments} -\end{itemize} - -\subsection{Occurrences sets and occurrences clauses} -\label{Occurrences clauses} -\index{Occurrences clauses} - -An occurrences clause is a modifier to some tactics that obeys the -following syntax: - -$\!\!\!$\begin{tabular}{lcl} -{\occclause} & ::= & {\tt in} {\occgoalset} \\ -{\occgoalset} & ::= & - \zeroone{{\ident$_1$} \zeroone{\atoccurrences} {\tt ,} \\ -& & {\dots} {\tt ,}\\ -& & {\ident$_m$} \zeroone{\atoccurrences}}\\ -& & \zeroone{{\tt |-} \zeroone{{\tt *} \zeroone{\atoccurrences}}}\\ -& | & - {\tt *} {\tt |-} \zeroone{{\tt *} \zeroone{\atoccurrences}}\\ -& | & - {\tt *}\\ -{\atoccurrences} & ::= & {\tt at} {\occlist}\\ -{\occlist} & ::= & \zeroone{{\tt -}} {\num$_1$} \dots\ {\num$_n$} -\end{tabular} - -The role of an occurrence clause is to select a set of occurrences of -a {\term} in a goal. In the first case, the {{\ident$_i$} -\zeroone{{\tt at} {\num$_1^i$} \dots\ {\num$_{n_i}^i$}}} parts -indicate that occurrences have to be selected in the hypotheses named -{\ident$_i$}. If no numbers are given for hypothesis {\ident$_i$}, -then all occurrences of {\term} in the hypothesis are selected. If -numbers are given, they refer to occurrences of {\term} when the term -is printed using option {\tt Set Printing All} (see -Section~\ref{SetPrintingAll}), counting from left to right. In -particular, occurrences of {\term} in implicit arguments (see -Section~\ref{Implicit Arguments}) or coercions (see -Section~\ref{Coercions}) are counted. - -If a minus sign is given between {\tt at} and the list of occurrences, -it negates the condition so that the clause denotes all the occurrences except -the ones explicitly mentioned after the minus sign. - -As an exception to the left-to-right order, the occurrences in the -{\tt return} subexpression of a {\tt match} are considered {\em -before} the occurrences in the matched term. - -In the second case, the {\tt *} on the left of {\tt |-} means that -all occurrences of {\term} are selected in every hypothesis. - -In the first and second case, if {\tt *} is mentioned on the right of -{\tt |-}, the occurrences of the conclusion of the goal have to be -selected. If some numbers are given, then only the occurrences denoted -by these numbers are selected. In no numbers are given, all -occurrences of {\term} in the goal are selected. - -Finally, the last notation is an abbreviation for {\tt * |- *}. Note -also that {\tt |-} is optional in the first case when no {\tt *} is -given. - -Here are some tactics that understand occurrences clauses: -{\tt set}, {\tt remember}, {\tt induction}, {\tt destruct}. - -\SeeAlso~Sections~\ref{tactic:set}, \ref{Tac-induction}, \ref{SetPrintingAll}. - - -\section{Negation and contradiction} - -\subsection{\tt absurd \term -\tacindex{absurd} -\label{absurd}} - -This tactic applies to any goal. The argument {\term} is any -proposition {\tt P} of type {\tt Prop}. This tactic applies {\tt - False} elimination, that is it deduces the current goal from {\tt - False}, and generates as subgoals {\tt $\sim$P} and {\tt P}. It is -very useful in proofs by cases, where some cases are impossible. In -most cases, \texttt{P} or $\sim$\texttt{P} is one of the hypotheses of -the local context. - -\subsection{\tt contradiction -\label{contradiction} -\tacindex{contradiction}} - -This tactic applies to any goal. The {\tt contradiction} tactic -attempts to find in the current context (after all {\tt intros}) one -hypothesis which is equivalent to {\tt False}. It permits to prune -irrelevant cases. This tactic is a macro for the tactics sequence -{\tt intros; elimtype False; assumption}. - -\begin{ErrMsgs} -\item \errindex{No such assumption} -\end{ErrMsgs} - -\begin{Variants} -\item {\tt contradiction \ident} - -The proof of {\tt False} is searched in the hypothesis named \ident. -\end{Variants} - -\subsection {\tt contradict \ident} -\label{contradict} -\tacindex{contradict} - -This tactic allows to manipulate negated hypothesis and goals. The -name \ident\ should correspond to a hypothesis. With -{\tt contradict H}, the current goal and context is transformed in -the following way: -\begin{itemize} -\item {\tt H:$\neg$A $\vd$ B} \ becomes \ {\tt $\vd$ A} -\item {\tt H:$\neg$A $\vd$ $\neg$B} \ becomes \ {\tt H: B $\vd$ A } -\item {\tt H: A $\vd$ B} \ becomes \ {\tt $\vd$ $\neg$A} -\item {\tt H: A $\vd$ $\neg$B} \ becomes \ {\tt H: B $\vd$ $\neg$A} -\end{itemize} - -\subsection{\tt exfalso} -\label{exfalso} -\tacindex{exfalso} - -This tactic implements the ``ex falso quodlibet'' logical principle: -an elimination of {\tt False} is performed on the current goal, and the -user is then required to prove that {\tt False} is indeed provable in -the current context. This tactic is a macro for {\tt elimtype False}. - -\section{Conversion tactics -\index{Conversion tactics} -\label{Conversion-tactics}} - -This set of tactics implements different specialized usages of the -tactic \texttt{change}. - -All conversion tactics (including \texttt{change}) can be -parameterized by the parts of the goal where the conversion can -occur. This is done using \emph{goal clauses} which consists in a list -of hypotheses and, optionally, of a reference to the conclusion of the -goal. For defined hypothesis it is possible to specify if the -conversion should occur on the type part, the body part or both -(default). - -\index{Clauses} -\index{Goal clauses} -Goal clauses are written after a conversion tactic (tactics -\texttt{set}~\ref{tactic:set}, \texttt{rewrite}~\ref{rewrite}, -\texttt{replace}~\ref{tactic:replace} and -\texttt{autorewrite}~\ref{tactic:autorewrite} also use goal clauses) and -are introduced by the keyword \texttt{in}. If no goal clause is provided, -the default is to perform the conversion only in the conclusion. - -The syntax and description of the various goal clauses is the following: -\begin{description} -\item[]\texttt{in {\ident}$_1$ $\ldots$ {\ident}$_n$ |- } only in hypotheses {\ident}$_1$ - \ldots {\ident}$_n$ -\item[]\texttt{in {\ident}$_1$ $\ldots$ {\ident}$_n$ |- *} in hypotheses {\ident}$_1$ \ldots - {\ident}$_n$ and in the conclusion -\item[]\texttt{in * |-} in every hypothesis -\item[]\texttt{in *} (equivalent to \texttt{in * |- *}) everywhere -\item[]\texttt{in (type of {\ident}$_1$) (value of {\ident}$_2$) $\ldots$ |-} in - type part of {\ident}$_1$, in the value part of {\ident}$_2$, etc. -\end{description} - -For backward compatibility, the notation \texttt{in}~{\ident}$_1$\ldots {\ident}$_n$ -performs the conversion in hypotheses {\ident}$_1$\ldots {\ident}$_n$. - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -%voir reduction__conv_x : histoires d'univers. -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\subsection[{\tt cbv \flag$_1$ \dots\ \flag$_n$}, {\tt lazy \flag$_1$ -\dots\ \flag$_n$} and {\tt compute}] -{{\tt cbv \flag$_1$ \dots\ \flag$_n$}, {\tt lazy \flag$_1$ -\dots\ \flag$_n$} and {\tt compute} -\tacindex{cbv} -\tacindex{lazy} -\tacindex{compute} -\tacindex{vm\_compute}\label{vmcompute}} - -These parameterized reduction tactics apply to any goal and perform -the normalization of the goal according to the specified flags. In -correspondence with the kinds of reduction considered in \Coq\, namely -$\beta$ (reduction of functional application), $\delta$ (unfolding of -transparent constants, see \ref{Transparent}), $\iota$ (reduction of -pattern-matching over a constructed term, and unfolding of {\tt fix} -and {\tt cofix} expressions) and $\zeta$ (contraction of local -definitions), the flag are either {\tt beta}, {\tt delta}, {\tt iota} -or {\tt zeta}. The {\tt delta} flag itself can be refined into {\tt -delta [\qualid$_1$\ldots\qualid$_k$]} or {\tt delta --[\qualid$_1$\ldots\qualid$_k$]}, restricting in the first case the -constants to unfold to the constants listed, and restricting in the -second case the constant to unfold to all but the ones explicitly -mentioned. Notice that the {\tt delta} flag does not apply to -variables bound by a let-in construction inside the term itself (use -here the {\tt zeta} flag). In any cases, opaque constants are not -unfolded (see Section~\ref{Opaque}). - -The goal may be normalized with two strategies: {\em lazy} ({\tt lazy} -tactic), or {\em call-by-value} ({\tt cbv} tactic). The lazy strategy -is a call-by-need strategy, with sharing of reductions: the arguments of a -function call are partially evaluated only when necessary, and if an -argument is used several times then it is computed only once. This -reduction is efficient for reducing expressions with dead code. For -instance, the proofs of a proposition {\tt exists~$x$. $P(x)$} reduce to a -pair of a witness $t$, and a proof that $t$ satisfies the predicate -$P$. Most of the time, $t$ may be computed without computing the proof -of $P(t)$, thanks to the lazy strategy. - -The call-by-value strategy is the one used in ML languages: the -arguments of a function call are evaluated first, using a weak -reduction (no reduction under the $\lambda$-abstractions). Despite the -lazy strategy always performs fewer reductions than the call-by-value -strategy, the latter is generally more efficient for evaluating purely -computational expressions (i.e. with few dead code). - -\begin{Variants} -\item {\tt compute} \tacindex{compute}\\ - {\tt cbv} - - These are synonyms for {\tt cbv beta delta iota zeta}. - -\item {\tt lazy} - - This is a synonym for {\tt lazy beta delta iota zeta}. - -\item {\tt compute [\qualid$_1$\ldots\qualid$_k$]}\\ - {\tt cbv [\qualid$_1$\ldots\qualid$_k$]} - - These are synonyms of {\tt cbv beta delta - [\qualid$_1$\ldots\qualid$_k$] iota zeta}. - -\item {\tt compute -[\qualid$_1$\ldots\qualid$_k$]}\\ - {\tt cbv -[\qualid$_1$\ldots\qualid$_k$]} - - These are synonyms of {\tt cbv beta delta - -[\qualid$_1$\ldots\qualid$_k$] iota zeta}. - -\item {\tt lazy [\qualid$_1$\ldots\qualid$_k$]}\\ - {\tt lazy -[\qualid$_1$\ldots\qualid$_k$]} - - These are respectively synonyms of {\tt cbv beta delta - [\qualid$_1$\ldots\qualid$_k$] iota zeta} and {\tt cbv beta delta - -[\qualid$_1$\ldots\qualid$_k$] iota zeta}. - -\item {\tt vm\_compute} \tacindex{vm\_compute} - - This tactic evaluates the goal using the optimized call-by-value - evaluation bytecode-based virtual machine. This algorithm is - dramatically more efficient than the algorithm used for the {\tt - cbv} tactic, but it cannot be fine-tuned. It is specially - interesting for full evaluation of algebraic objects. This includes - the case of reflexion-based tactics. - -\end{Variants} - -% Obsolete? Anyway not very important message -%\begin{ErrMsgs} -%\item \errindex{Delta must be specified before} -% -% A list of constants appeared before the {\tt delta} flag. -%\end{ErrMsgs} - - -\subsection{{\tt red} -\tacindex{red}} - -This tactic applies to a goal which has the form {\tt - forall (x:T1)\dots(xk:Tk), c t1 \dots\ tn} where {\tt c} is a constant. If -{\tt c} is transparent then it replaces {\tt c} with its definition -(say {\tt t}) and then reduces {\tt (t t1 \dots\ tn)} according to -$\beta\iota\zeta$-reduction rules. - -\begin{ErrMsgs} -\item \errindex{Not reducible} -\end{ErrMsgs} - -\subsection{{\tt hnf} -\tacindex{hnf}} - -This tactic applies to any goal. It replaces the current goal with its -head normal form according to the $\beta\delta\iota\zeta$-reduction -rules, i.e. it reduces the head of the goal until it becomes a -product or an irreducible term. - -\Example -The term \verb+forall n:nat, (plus (S n) (S n))+ is not reduced by {\tt hnf}. - -\Rem The $\delta$ rule only applies to transparent constants -(see Section~\ref{Opaque} on transparency and opacity). - -\subsection{\tt simpl -\tacindex{simpl}} - -This tactic applies to any goal. The tactic {\tt simpl} first applies -$\beta\iota$-reduction rule. Then it expands transparent constants -and tries to reduce {\tt T'} according, once more, to $\beta\iota$ -rules. But when the $\iota$ rule is not applicable then possible -$\delta$-reductions are not applied. For instance trying to use {\tt -simpl} on {\tt (plus n O)=n} changes nothing. Notice that only -transparent constants whose name can be reused as such in the -recursive calls are possibly unfolded. For instance a constant defined -by {\tt plus' := plus} is possibly unfolded and reused in the -recursive calls, but a constant such as {\tt succ := plus (S O)} is -never unfolded. - -\tacindex{simpl \dots\ in} -\begin{Variants} -\item {\tt simpl {\term}} - - This applies {\tt simpl} only to the occurrences of {\term} in the - current goal. - -\item {\tt simpl {\term} at \num$_1$ \dots\ \num$_i$} - - This applies {\tt simpl} only to the \num$_1$, \dots, \num$_i$ - occurrences of {\term} in the current goal. - - \ErrMsg {\tt Too few occurrences} - -\item {\tt simpl {\ident}} - - This applies {\tt simpl} only to the applicative subterms whose head - occurrence is {\ident}. - -\item {\tt simpl {\ident} at \num$_1$ \dots\ \num$_i$} - - This applies {\tt simpl} only to the \num$_1$, \dots, \num$_i$ -applicative subterms whose head occurrence is {\ident}. - -\end{Variants} - -\subsection{\tt unfold \qualid -\tacindex{unfold} -\label{unfold}} - -This tactic applies to any goal. The argument {\qualid} must denote a -defined transparent constant or local definition (see Sections~\ref{Basic-definitions} and~\ref{Transparent}). The tactic {\tt - unfold} applies the $\delta$ rule to each occurrence of the constant -to which {\qualid} refers in the current goal and then replaces it -with its $\beta\iota$-normal form. - -\begin{ErrMsgs} -\item {\qualid} \errindex{does not denote an evaluable constant} - -\end{ErrMsgs} - -\begin{Variants} -\item {\tt unfold {\qualid}$_1$, \dots, \qualid$_n$} - \tacindex{unfold \dots\ in} - - Replaces {\em simultaneously} {\qualid}$_1$, \dots, {\qualid}$_n$ - with their definitions and replaces the current goal with its - $\beta\iota$ normal form. - -\item {\tt unfold {\qualid}$_1$ at \num$_1^1$, \dots, \num$_i^1$, -\dots,\ \qualid$_n$ at \num$_1^n$ \dots\ \num$_j^n$} - - The lists \num$_1^1$, \dots, \num$_i^1$ and \num$_1^n$, \dots, - \num$_j^n$ specify the occurrences of {\qualid}$_1$, \dots, - \qualid$_n$ to be unfolded. Occurrences are located from left to - right. - - \ErrMsg {\tt bad occurrence number of {\qualid}$_i$} - - \ErrMsg {\qualid}$_i$ {\tt does not occur} - -\item {\tt unfold {\qstring}} - - If {\qstring} denotes the discriminating symbol of a notation (e.g. {\tt - "+"}) or an expression defining a notation (e.g. \verb!"_ + _"!), and - this notation refers to an unfoldable constant, then the tactic - unfolds it. - -\item {\tt unfold {\qstring}\%{\delimkey}} - - This is variant of {\tt unfold {\qstring}} where {\qstring} gets its - interpretation from the scope bound to the delimiting key - {\delimkey} instead of its default interpretation (see - Section~\ref{scopechange}). - -\item {\tt unfold \qualidorstring$_1$ at \num$_1^1$, \dots, \num$_i^1$, -\dots,\ \qualidorstring$_n$ at \num$_1^n$ \dots\ \num$_j^n$} - - This is the most general form, where {\qualidorstring} is either a - {\qualid} or a {\qstring} referring to a notation. - -\end{Variants} - -\subsection{{\tt fold} \term -\tacindex{fold}} - -This tactic applies to any goal. The term \term\ is reduced using the {\tt red} -tactic. Every occurrence of the resulting term in the goal is then -replaced by \term. - -\begin{Variants} -\item {\tt fold} \term$_1$ \dots\ \term$_n$ - - Equivalent to {\tt fold} \term$_1${\tt;}\ldots{\tt; fold} \term$_n$. -\end{Variants} - -\subsection{{\tt pattern {\term}} -\tacindex{pattern} -\label{pattern}} - -This command applies to any goal. The argument {\term} must be a free -subterm of the current goal. The command {\tt pattern} performs -$\beta$-expansion (the inverse of $\bt$-reduction) of the current goal -(say \T) by -\begin{enumerate} -\item replacing all occurrences of {\term} in {\T} with a fresh variable -\item abstracting this variable -\item applying the abstracted goal to {\term} -\end{enumerate} - -For instance, if the current goal $T$ is expressible has $\phi(t)$ -where the notation captures all the instances of $t$ in $\phi(t)$, -then {\tt pattern $t$} transforms it into {\tt (fun x:$A$ => $\phi(${\tt -x}$)$) $t$}. This command can be used, for instance, when the tactic -{\tt apply} fails on matching. - -\begin{Variants} -\item {\tt pattern {\term} at {\num$_1$} \dots\ {\num$_n$}} - - Only the occurrences {\num$_1$} \dots\ {\num$_n$} of {\term} are - considered for $\beta$-expansion. Occurrences are located from left - to right. - -\item {\tt pattern {\term} at - {\num$_1$} \dots\ {\num$_n$}} - - All occurrences except the occurrences of indexes {\num$_1$} \dots\ - {\num$_n$} of {\term} are considered for - $\beta$-expansion. Occurrences are located from left to right. - -\item {\tt pattern {\term$_1$}, \dots, {\term$_m$}} - - Starting from a goal $\phi(t_1 \dots\ t_m)$, the tactic - {\tt pattern $t_1$, \dots,\ $t_m$} generates the equivalent goal {\tt - (fun (x$_1$:$A_1$) \dots\ (x$_m$:$A_m$) => $\phi(${\tt x$_1$\dots\ - x$_m$}$)$) $t_1$ \dots\ $t_m$}.\\ If $t_i$ occurs in one of the - generated types $A_j$ these occurrences will also be considered and - possibly abstracted. - -\item {\tt pattern {\term$_1$} at {\num$_1^1$} \dots\ {\num$_{n_1}^1$}, \dots, - {\term$_m$} at {\num$_1^m$} \dots\ {\num$_{n_m}^m$}} - - This behaves as above but processing only the occurrences \num$_1^1$, - \dots, \num$_i^1$ of \term$_1$, \dots, \num$_1^m$, \dots, \num$_j^m$ - of \term$_m$ starting from \term$_m$. - -\item {\tt pattern} {\term$_1$} \zeroone{{\tt at \zeroone{-}} {\num$_1^1$} \dots\ {\num$_{n_1}^1$}} {\tt ,} \dots {\tt ,} - {\term$_m$} \zeroone{{\tt at \zeroone{-}} {\num$_1^m$} \dots\ {\num$_{n_m}^m$}} - - This is the most general syntax that combines the different variants. - -\end{Variants} - -\subsection{Conversion tactics applied to hypotheses} - -{\convtactic} {\tt in} \ident$_1$ \dots\ \ident$_n$ - -Applies the conversion tactic {\convtactic} to the -hypotheses \ident$_1$, \ldots, \ident$_n$. The tactic {\convtactic} is -any of the conversion tactics listed in this section. - -If \ident$_i$ is a local definition, then \ident$_i$ can be replaced -by (Type of \ident$_i$) to address not the body but the type of the -local definition. Example: {\tt unfold not in (Type of H1) (Type of H3).} - -\begin{ErrMsgs} -\item \errindex{No such hypothesis} : {\ident}. -\end{ErrMsgs} - - -\section{Introductions} - -Introduction tactics address goals which are inductive constants. -They are used when one guesses that the goal can be obtained with one -of its constructors' type. - -\subsection{\tt constructor \num -\label{constructor} -\tacindex{constructor}} - -This tactic applies to a goal such that the head of its conclusion is -an inductive constant (say {\tt I}). The argument {\num} must be less -or equal to the numbers of constructor(s) of {\tt I}. Let {\tt ci} be -the {\tt i}-th constructor of {\tt I}, then {\tt constructor i} is -equivalent to {\tt intros; apply ci}. - -\begin{ErrMsgs} -\item \errindex{Not an inductive product} -\item \errindex{Not enough constructors} -\end{ErrMsgs} - -\begin{Variants} -\item \texttt{constructor} - - This tries \texttt{constructor 1} then \texttt{constructor 2}, - \dots\ , then \texttt{constructor} \textit{n} where \textit{n} if - the number of constructors of the head of the goal. - -\item {\tt constructor \num~with} {\bindinglist} - - Let {\tt ci} be the {\tt i}-th constructor of {\tt I}, then {\tt - constructor i with \bindinglist} is equivalent to {\tt intros; - apply ci with \bindinglist}. - - \Warning the terms in the \bindinglist\ are checked - in the context where {\tt constructor} is executed and not in the - context where {\tt apply} is executed (the introductions are not - taken into account). - -% To document? -% \item {\tt constructor {\tactic}} - -\item {\tt split}\tacindex{split} - - Applies if {\tt I} has only one constructor, typically in the case - of conjunction $A\land B$. Then, it is equivalent to {\tt constructor 1}. - -\item {\tt exists {\bindinglist}}\tacindex{exists} - - Applies if {\tt I} has only one constructor, for instance in the - case of existential quantification $\exists x\cdot P(x)$. - Then, it is equivalent to {\tt intros; constructor 1 with \bindinglist}. - -\item {\tt exists \nelist{\bindinglist}{,}} - - This iteratively applies {\tt exists {\bindinglist}}. - -\item {\tt left}\tacindex{left}\\ - {\tt right}\tacindex{right} - - Apply if {\tt I} has two constructors, for instance in the case of - disjunction $A\lor B$. Then, they are respectively equivalent to {\tt - constructor 1} and {\tt constructor 2}. - -\item {\tt left with \bindinglist}\\ - {\tt right with \bindinglist}\\ - {\tt split with \bindinglist} - - As soon as the inductive type has the right number of constructors, - these expressions are equivalent to calling {\tt - constructor $i$ with \bindinglist} for the appropriate $i$. - -\item \texttt{econstructor}\tacindex{econstructor}\\ - \texttt{eexists}\tacindex{eexists}\\ - \texttt{esplit}\tacindex{esplit}\\ - \texttt{eleft}\tacindex{eleft}\\ - \texttt{eright}\tacindex{eright}\\ - - These tactics and their variants behave like \texttt{constructor}, - \texttt{exists}, \texttt{split}, \texttt{left}, \texttt{right} and - their variants but they introduce existential variables instead of - failing when the instantiation of a variable cannot be found (cf - \texttt{eapply} and Section~\ref{eapply-example}). - -\end{Variants} - -\section[Induction and Case Analysis]{Induction and Case Analysis -\label{Tac-induction}} - -The tactics presented in this section implement induction or case -analysis on inductive or coinductive objects (see -Section~\ref{Cic-inductive-definitions}). - -\subsection{\tt induction \term -\tacindex{induction}} - -This tactic applies to any goal. The type of the argument {\term} must -be an inductive constant. Then, the tactic {\tt induction} -generates subgoals, one for each possible form of {\term}, i.e. one -for each constructor of the inductive type. - -The tactic {\tt induction} automatically replaces every occurrences -of {\term} in the conclusion and the hypotheses of the goal. It -automatically adds induction hypotheses (using names of the form {\tt - IHn1}) to the local context. If some hypothesis must not be taken -into account in the induction hypothesis, then it needs to be removed -first (you can also use the tactics {\tt elim} or {\tt simple induction}, -see below). - -There are particular cases: - -\begin{itemize} - -\item If {\term} is an identifier {\ident} denoting a quantified -variable of the conclusion of the goal, then {\tt induction {\ident}} -behaves as {\tt intros until {\ident}; induction {\ident}}. - -\item If {\term} is a {\num}, then {\tt induction {\num}} behaves as -{\tt intros until {\num}} followed by {\tt induction} applied to the -last introduced hypothesis. - -\Rem For simple induction on a numeral, use syntax {\tt induction -({\num})} (not very interesting anyway). - -\end{itemize} - -\Example - -\begin{coq_example} -Lemma induction_test : forall n:nat, n = n -> n <= n. -intros n H. -induction n. -\end{coq_example} - -\begin{ErrMsgs} -\item \errindex{Not an inductive product} -\item \errindex{Unable to find an instance for the variables -{\ident} \ldots {\ident}} - - Use in this case - the variant {\tt elim \dots\ with \dots} below. -\end{ErrMsgs} - -\begin{Variants} -\item{\tt induction {\term} as {\disjconjintropattern}} - - This behaves as {\tt induction {\term}} but uses the names in - {\disjconjintropattern} to name the variables introduced in the context. - The {\disjconjintropattern} must typically be of the form - {\tt [} $p_{11}$ \ldots - $p_{1n_1}$ {\tt |} {\ldots} {\tt |} $p_{m1}$ \ldots $p_{mn_m}$ {\tt - ]} with $m$ being the number of constructors of the type of - {\term}. Each variable introduced by {\tt induction} in the context - of the $i^{th}$ goal gets its name from the list $p_{i1}$ \ldots - $p_{in_i}$ in order. If there are not enough names, {\tt induction} - invents names for the remaining variables to introduce. More - generally, the $p_{ij}$ can be any disjunctive/conjunctive - introduction pattern (see Section~\ref{intros-pattern}). For instance, - for an inductive type with one constructor, the pattern notation - {\tt ($p_{1}$,\ldots,$p_{n}$)} can be used instead of - {\tt [} $p_{1}$ \ldots $p_{n}$ {\tt ]}. - -\item{\tt induction {\term} as {\namingintropattern}} - - This behaves as {\tt induction {\term}} but adds an equation between - {\term} and the value that {\term} takes in each of the induction - case. The name of the equation is built according to - {\namingintropattern} which can be an identifier, a ``?'', etc, as - indicated in Section~\ref{intros-pattern}. - -\item{\tt induction {\term} as {\namingintropattern} {\disjconjintropattern}} - - This combines the two previous forms. - -\item{\tt induction {\term} with \bindinglist} - - This behaves like \texttt{induction {\term}} providing explicit - instances for the premises of the type of {\term} (see the syntax of - bindings in Section~\ref{Binding-list}). - -\item{\tt einduction {\term}\tacindex{einduction}} - - This tactic behaves like \texttt{induction {\term}} excepts that it - does not fail if some dependent premise of the type of {\term} is - not inferable. Instead, the unresolved premises are posed as - existential variables to be inferred later, in the same way as {\tt - eapply} does (see Section~\ref{eapply-example}). - -\item {\tt induction {\term$_1$} using {\term$_2$}} - - This behaves as {\tt induction {\term$_1$}} but using {\term$_2$} as - induction scheme. It does not expect the conclusion of the type of - {\term$_1$} to be inductive. - -\item {\tt induction {\term$_1$} using {\term$_2$} with {\bindinglist}} - - This behaves as {\tt induction {\term$_1$} using {\term$_2$}} but - also providing instances for the premises of the type of {\term$_2$}. - -\item \texttt{induction {\term}$_1$ $\ldots$ {\term}$_n$ using {\qualid}} - - This syntax is used for the case {\qualid} denotes an induction principle - with complex predicates as the induction principles generated by - {\tt Function} or {\tt Functional Scheme} may be. - -\item \texttt{induction {\term} in {\occgoalset}} - - This syntax is used for selecting which occurrences of {\term} the - induction has to be carried on. The {\tt in {\atoccurrences}} clause is an - occurrence clause whose syntax and behavior is described in - Section~\ref{Occurrences clauses}. - - When an occurrence clause is given, an equation between {\term} and - the value it gets in each case of the induction is added to the - context of the subgoals corresponding to the induction cases (even - if no clause {\tt as {\namingintropattern}} is given). - -\item {\tt induction {\term$_1$} with {\bindinglist$_1$} as {\namingintropattern} {\disjconjintropattern} using {\term$_2$} with {\bindinglist$_2$} in {\occgoalset}}\\ - {\tt einduction {\term$_1$} with {\bindinglist$_1$} as {\namingintropattern} {\disjconjintropattern} using {\term$_2$} with {\bindinglist$_2$} in {\occgoalset}} - - These are the most general forms of {\tt induction} and {\tt - einduction}. It combines the effects of the {\tt with}, {\tt as}, - {\tt using}, and {\tt in} clauses. - -\item {\tt elim \term}\label{elim} - - This is a more basic induction tactic. Again, the type of the - argument {\term} must be an inductive type. Then, according to - the type of the goal, the tactic {\tt elim} chooses the appropriate - destructor and applies it as the tactic {\tt apply} - would do. For instance, if the proof context contains {\tt - n:nat} and the current goal is {\tt T} of type {\tt - Prop}, then {\tt elim n} is equivalent to {\tt apply nat\_ind with - (n:=n)}. The tactic {\tt elim} does not modify the context of - the goal, neither introduces the induction loading into the context - of hypotheses. - - More generally, {\tt elim \term} also works when the type of {\term} - is a statement with premises and whose conclusion is inductive. In - that case the tactic performs induction on the conclusion of the - type of {\term} and leaves the non-dependent premises of the type as - subgoals. In the case of dependent products, the tactic tries to - find an instance for which the elimination lemma applies and fails - otherwise. - -\item {\tt elim {\term} with {\bindinglist}} - - Allows to give explicit instances to the premises of the type - of {\term} (see Section~\ref{Binding-list}). - -\item{\tt eelim {\term}\tacindex{eelim}} - - In case the type of {\term} has dependent premises, this turns them into - existential variables to be resolved later on. - -\item{\tt elim {\term$_1$} using {\term$_2$}}\\ - {\tt elim {\term$_1$} using {\term$_2$} with {\bindinglist}\tacindex{elim \dots\ using}} - -Allows the user to give explicitly an elimination predicate -{\term$_2$} which is not the standard one for the underlying inductive -type of {\term$_1$}. The {\bindinglist} clause allows to -instantiate premises of the type of {\term$_2$}. - -\item{\tt elim {\term$_1$} with {\bindinglist$_1$} using {\term$_2$} with {\bindinglist$_2$}}\\ - {\tt eelim {\term$_1$} with {\bindinglist$_1$} using {\term$_2$} with {\bindinglist$_2$}} - - These are the most general forms of {\tt elim} and {\tt eelim}. It - combines the effects of the {\tt using} clause and of the two uses - of the {\tt with} clause. - -\item {\tt elimtype \form}\tacindex{elimtype} - - The argument {\form} must be inductively defined. {\tt elimtype I} - is equivalent to {\tt cut I. intro H{\rm\sl n}; elim H{\rm\sl n}; - clear H{\rm\sl n}}. Therefore the hypothesis {\tt H{\rm\sl n}} will - not appear in the context(s) of the subgoal(s). Conversely, if {\tt - t} is a term of (inductive) type {\tt I} and which does not occur - in the goal then {\tt elim t} is equivalent to {\tt elimtype I; 2: - exact t.} - -\item {\tt simple induction \ident}\tacindex{simple induction} - - This tactic behaves as {\tt intros until - {\ident}; elim {\tt {\ident}}} when {\ident} is a quantified - variable of the goal. - -\item {\tt simple induction {\num}} - - This tactic behaves as {\tt intros until - {\num}; elim {\tt {\ident}}} where {\ident} is the name given by - {\tt intros until {\num}} to the {\num}-th non-dependent premise of - the goal. - -%% \item {\tt simple induction {\term}}\tacindex{simple induction} - -%% If {\term} is an {\ident} corresponding to a quantified variable of -%% the goal then the tactic behaves as {\tt intros until {\ident}; elim -%% {\tt {\ident}}}. If {\term} is a {\num} then the tactic behaves as -%% {\tt intros until {\ident}; elim {\tt {\ident}}}. Otherwise, it is -%% a synonym for {\tt elim {\term}}. - -%% \Rem For simple induction on a numeral, use syntax {\tt simple -%% induction ({\num})}. - -\end{Variants} - -\subsection{\tt destruct \term -\tacindex{destruct}} -\label{destruct} - -The tactic {\tt destruct} is used to perform case analysis without -recursion. Its behavior is similar to {\tt induction} except -that no induction hypothesis is generated. It applies to any goal and -the type of {\term} must be inductively defined. There are particular cases: - -\begin{itemize} - -\item If {\term} is an identifier {\ident} denoting a quantified -variable of the conclusion of the goal, then {\tt destruct {\ident}} -behaves as {\tt intros until {\ident}; destruct {\ident}}. - -\item If {\term} is a {\num}, then {\tt destruct {\num}} behaves as -{\tt intros until {\num}} followed by {\tt destruct} applied to the -last introduced hypothesis. - -\Rem For destruction of a numeral, use syntax {\tt destruct -({\num})} (not very interesting anyway). - -\end{itemize} - -\begin{Variants} -\item{\tt destruct {\term} as {\disjconjintropattern}} - - This behaves as {\tt destruct {\term}} but uses the names in - {\intropattern} to name the variables introduced in the context. - The {\intropattern} must have the form {\tt [} $p_{11}$ \ldots - $p_{1n_1}$ {\tt |} {\ldots} {\tt |} $p_{m1}$ \ldots $p_{mn_m}$ {\tt - ]} with $m$ being the number of constructors of the type of - {\term}. Each variable introduced by {\tt destruct} in the context - of the $i^{th}$ goal gets its name from the list $p_{i1}$ \ldots - $p_{in_i}$ in order. If there are not enough names, {\tt destruct} - invents names for the remaining variables to introduce. More - generally, the $p_{ij}$ can be any disjunctive/conjunctive - introduction pattern (see Section~\ref{intros-pattern}). This - provides a concise notation for nested destruction. - -% It is recommended to use this variant of {\tt destruct} for -% robust proof scripts. - -\item{\tt destruct {\term} as {\disjconjintropattern} \_eqn} - - This behaves as {\tt destruct {\term}} but adds an equation between - {\term} and the value that {\term} takes in each of the possible - cases. The name of the equation is chosen by Coq. If - {\disjconjintropattern} is simply {\tt []}, it is automatically considered - as a disjunctive pattern of the appropriate size. - -\item{\tt destruct {\term} as {\disjconjintropattern} \_eqn: {\namingintropattern}} - - This behaves as {\tt destruct {\term} as - {\disjconjintropattern} \_eqn} but use {\namingintropattern} to - name the equation (see Section~\ref{intros-pattern}). Note that spaces - can generally be removed around {\tt \_eqn}. - -\item{\tt destruct {\term} with \bindinglist} - - This behaves like \texttt{destruct {\term}} providing explicit - instances for the dependent premises of the type of {\term} (see - syntax of bindings in Section~\ref{Binding-list}). - -\item{\tt edestruct {\term}\tacindex{edestruct}} - - This tactic behaves like \texttt{destruct {\term}} excepts that it - does not fail if the instance of a dependent premises of the type of - {\term} is not inferable. Instead, the unresolved instances are left - as existential variables to be inferred later, in the same way as - {\tt eapply} does (see Section~\ref{eapply-example}). - -\item{\tt destruct {\term$_1$} using {\term$_2$}}\\ - {\tt destruct {\term$_1$} using {\term$_2$} with {\bindinglist}} - - These are synonyms of {\tt induction {\term$_1$} using {\term$_2$}} and - {\tt induction {\term$_1$} using {\term$_2$} with {\bindinglist}}. - -\item \texttt{destruct {\term} in {\occgoalset}} - - This syntax is used for selecting which occurrences of {\term} the - case analysis has to be done on. The {\tt in {\occgoalset}} clause is an - occurrence clause whose syntax and behavior is described in - Section~\ref{Occurrences clauses}. - - When an occurrence clause is given, an equation between {\term} and - the value it gets in each case of the analysis is added to the - context of the subgoals corresponding to the cases (even - if no clause {\tt as {\namingintropattern}} is given). - -\item{\tt destruct {\term$_1$} with {\bindinglist$_1$} as {\disjconjintropattern} \_eqn: {\namingintropattern} using {\term$_2$} with {\bindinglist$_2$} in {\occgoalset}}\\ - {\tt edestruct {\term$_1$} with {\bindinglist$_1$} as {\disjconjintropattern} \_eqn: {\namingintropattern} using {\term$_2$} with {\bindinglist$_2$} in {\occgoalset}} - - These are the general forms of {\tt destruct} and {\tt edestruct}. - They combine the effects of the {\tt with}, {\tt as}, {\tt using}, - and {\tt in} clauses. - -\item{\tt case \term}\label{case}\tacindex{case} - - The tactic {\tt case} is a more basic tactic to perform case - analysis without recursion. It behaves as {\tt elim \term} but using - a case-analysis elimination principle and not a recursive one. - -\item{\tt case\_eq \term}\label{case_eq}\tacindex{case\_eq} - - The tactic {\tt case\_eq} is a variant of the {\tt case} tactic that - allow to perform case analysis on a term without completely - forgetting its original form. This is done by generating equalities - between the original form of the term and the outcomes of the case - analysis. The effect of this tactic is similar to the effect of {\tt - destruct {\term} in |- *} with the exception that no new hypotheses - are introduced in the context. - -\item {\tt case {\term} with {\bindinglist}} - - Analogous to {\tt elim {\term} with {\bindinglist}} above. - -\item{\tt ecase {\term}\tacindex{ecase}}\\ - {\tt ecase {\term} with {\bindinglist}} - - In case the type of {\term} has dependent premises, or dependent - premises whose values are not inferable from the {\tt with - {\bindinglist}} clause, {\tt ecase} turns them into existential - variables to be resolved later on. - -\item {\tt simple destruct \ident}\tacindex{simple destruct} - - This tactic behaves as {\tt intros until - {\ident}; case {\tt {\ident}}} when {\ident} is a quantified - variable of the goal. - -\item {\tt simple destruct {\num}} - - This tactic behaves as {\tt intros until - {\num}; case {\tt {\ident}}} where {\ident} is the name given by - {\tt intros until {\num}} to the {\num}-th non-dependent premise of - the goal. - - -\end{Variants} - -\subsection{\tt intros {\intropattern} {\ldots} {\intropattern} -\label{intros-pattern} -\tacindex{intros \intropattern}} -\index{Introduction patterns} -\index{Naming introduction patterns} -\index{Disjunctive/conjunctive introduction patterns} - -This extension of the tactic {\tt intros} combines introduction of -variables or hypotheses and case analysis. An {\em introduction pattern} is -either: -\begin{itemize} -\item A {\em naming introduction pattern}, i.e. either one of: - \begin{itemize} - \item the pattern \texttt{?} - \item the pattern \texttt{?\ident} - \item an identifier - \end{itemize} -\item A {\em disjunctive/conjunctive introduction pattern}, i.e. either one of: - \begin{itemize} - \item a disjunction of lists of patterns: - {\tt [$p_{11}$ {\ldots} $p_{1m_1}$ | {\ldots} | $p_{11}$ {\ldots} $p_{nm_n}$]} - \item a conjunction of patterns: {\tt (} $p_1$ {\tt ,} {\ldots} {\tt ,} $p_n$ {\tt )} - \item a list of patterns {\tt (} $p_1$\ {\tt \&}\ {\ldots}\ {\tt \&}\ $p_n$ {\tt )} - for sequence of right-associative binary constructs - \end{itemize} -\item the wildcard: {\tt \_} -\item the rewriting orientations: {\tt ->} or {\tt <-} -\end{itemize} - -Assuming a goal of type {\tt $Q$ -> $P$} (non dependent product), or -of type {\tt forall $x$:$T$, $P$} (dependent product), the behavior of -{\tt intros $p$} is defined inductively over the structure of the -introduction pattern $p$: -\begin{itemize} -\item introduction on \texttt{?} performs the introduction, and lets {\Coq} - choose a fresh name for the variable; -\item introduction on \texttt{?\ident} performs the introduction, and - lets {\Coq} choose a fresh name for the variable based on {\ident}; -\item introduction on \texttt{\ident} behaves as described in - Section~\ref{intro}; -\item introduction over a disjunction of list of patterns {\tt - [$p_{11}$ {\ldots} $p_{1m_1}$ | {\ldots} | $p_{11}$ {\ldots} - $p_{nm_n}$]} expects the product to be over an inductive type - whose number of constructors is $n$ (or more generally over a type - of conclusion an inductive type built from $n$ constructors, - e.g. {\tt C -> A$\backslash$/B if $n=2$}): it destructs the introduced - hypothesis as {\tt destruct} (see Section~\ref{destruct}) would and - applies on each generated subgoal the corresponding tactic; - \texttt{intros}~$p_{i1}$ {\ldots} $p_{im_i}$; if the disjunctive - pattern is part of a sequence of patterns and is not the last - pattern of the sequence, then {\Coq} completes the pattern so as all - the argument of the constructors of the inductive type are - introduced (for instance, the list of patterns {\tt [$\;$|$\;$] H} - applied on goal {\tt forall x:nat, x=0 -> 0=x} behaves the same as - the list of patterns {\tt [$\,$|$\,$?$\,$] H}); -\item introduction over a conjunction of patterns {\tt ($p_1$, \ldots, - $p_n$)} expects the goal to be a product over an inductive type $I$ with a - single constructor that itself has at least $n$ arguments: it - performs a case analysis over the hypothesis, as {\tt destruct} - would, and applies the patterns $p_1$~\ldots~$p_n$ to the arguments - of the constructor of $I$ (observe that {\tt ($p_1$, {\ldots}, - $p_n$)} is an alternative notation for {\tt [$p_1$ {\ldots} - $p_n$]}); -\item introduction via {\tt ( $p_1$ \& \ldots \& $p_n$ )} - is a shortcut for introduction via - {\tt ($p_1$,(\ldots,(\dots,$p_n$)\ldots))}; it expects the - hypothesis to be a sequence of right-associative binary inductive - constructors such as {\tt conj} or {\tt ex\_intro}; for instance, an - hypothesis with type {\tt A\verb|/\|exists x, B\verb|/\|C\verb|/\|D} can be - introduced via pattern {\tt (a \& x \& b \& c \& d)}; -\item introduction on the wildcard depends on whether the product is - dependent or not: in the non dependent case, it erases the - corresponding hypothesis (i.e. it behaves as an {\tt intro} followed - by a {\tt clear}, cf Section~\ref{clear}) while in the dependent - case, it succeeds and erases the variable only if the wildcard is - part of a more complex list of introduction patterns that also - erases the hypotheses depending on this variable; -\item introduction over {\tt ->} (respectively {\tt <-}) expects the - hypothesis to be an equality and the right-hand-side (respectively - the left-hand-side) is replaced by the left-hand-side (respectively - the right-hand-side) in both the conclusion and the context of the goal; - if moreover the term to substitute is a variable, the hypothesis is - removed. -\end{itemize} - -\Rem {\tt intros $p_1~\ldots~p_n$} is not equivalent to \texttt{intros - $p_1$;\ldots; intros $p_n$} for the following reasons: -\begin{itemize} -\item A wildcard pattern never succeeds when applied isolated on a - dependent product, while it succeeds as part of a list of - introduction patterns if the hypotheses that depends on it are - erased too. -\item A disjunctive or conjunctive pattern followed by an introduction - pattern forces the introduction in the context of all arguments of - the constructors before applying the next pattern while a terminal - disjunctive or conjunctive pattern does not. Here is an example - -\begin{coq_example} -Goal forall n:nat, n = 0 -> n = 0. -intros [ | ] H. -Show 2. -Undo. -intros [ | ]; intros H. -Show 2. -\end{coq_example} - -\end{itemize} - -\begin{coq_example} -Lemma intros_test : forall A B C:Prop, A \/ B /\ C -> (A -> C) -> C. -intros A B C [a| [_ c]] f. -apply (f a). -exact c. -Qed. -\end{coq_example} - -%\subsection[\tt FixPoint \dots]{\tt FixPoint \dots\tacindex{Fixpoint}} -%Not yet documented. - -\subsection{\tt double induction \ident$_1$ \ident$_2$} -%\tacindex{double induction}} -This tactic is deprecated and should be replaced by {\tt induction \ident$_1$; induction \ident$_2$} (or {\tt induction \ident$_1$; destruct \ident$_2$} depending on the exact needs). - -%% This tactic applies to any goal. If the variables {\ident$_1$} and -%% {\ident$_2$} of the goal have an inductive type, then this tactic -%% performs double induction on these variables. For instance, if the -%% current goal is \verb+forall n m:nat, P n m+ then, {\tt double induction n -%% m} yields the four cases with their respective inductive hypotheses. - -%% In particular, for proving \verb+(P (S n) (S m))+, the generated induction -%% hypotheses are \verb+(P (S n) m)+ and \verb+(m:nat)(P n m)+ (of the latter, -%% \verb+(P n m)+ and \verb+(P n (S m))+ are derivable). - -%% \Rem When the induction hypothesis \verb+(P (S n) m)+ is not -%% needed, {\tt induction \ident$_1$; destruct \ident$_2$} produces -%% more concise subgoals. - -\begin{Variant} - -\item {\tt double induction \num$_1$ \num$_2$} - -This tactic is deprecated and should be replaced by {\tt induction - \num$_1$; induction \num$_3$} where \num$_3$ is the result of -\num$_2$-\num$_1$. - -%% This tactic applies to any goal. If the variables {\ident$_1$} and - -%% This applies double induction on the \num$_1^{th}$ and \num$_2^{th}$ {\it -%% non dependent} premises of the goal. More generally, any combination of an -%% {\ident} and a {\num} is valid. - -\end{Variant} - -\subsection{\tt dependent induction \ident - \tacindex{dependent induction} - \label{DepInduction}} - -The \emph{experimental} tactic \texttt{dependent induction} performs -induction-inversion on an instantiated inductive predicate. -One needs to first require the {\tt Coq.Program.Equality} module to use -this tactic. The tactic is based on the BasicElim tactic by Conor -McBride \cite{DBLP:conf/types/McBride00} and the work of Cristina Cornes -around inversion \cite{DBLP:conf/types/CornesT95}. From an instantiated -inductive predicate and a goal it generates an equivalent goal where the -hypothesis has been generalized over its indexes which are then -constrained by equalities to be the right instances. This permits to -state lemmas without resorting to manually adding these equalities and -still get enough information in the proofs. -A simple example is the following: - -\begin{coq_eval} -Reset Initial. -\end{coq_eval} -\begin{coq_example} -Lemma le_minus : forall n:nat, n < 1 -> n = 0. -intros n H ; induction H. -\end{coq_example} - -Here we didn't get any information on the indexes to help fulfill this -proof. The problem is that when we use the \texttt{induction} tactic -we lose information on the hypothesis instance, notably that the second -argument is \texttt{1} here. Dependent induction solves this problem by -adding the corresponding equality to the context. - -\begin{coq_eval} -Reset Initial. -\end{coq_eval} -\begin{coq_example} -Require Import Coq.Program.Equality. -Lemma le_minus : forall n:nat, n < 1 -> n = 0. -intros n H ; dependent induction H. -\end{coq_example} - -The subgoal is cleaned up as the tactic tries to automatically -simplify the subgoals with respect to the generated equalities. -In this enriched context it becomes possible to solve this subgoal. -\begin{coq_example} -reflexivity. -\end{coq_example} - -Now we are in a contradictory context and the proof can be solved. -\begin{coq_example} -inversion H. -\end{coq_example} - -This technique works with any inductive predicate. -In fact, the \texttt{dependent induction} tactic is just a wrapper around -the \texttt{induction} tactic. One can make its own variant by just -writing a new tactic based on the definition found in -\texttt{Coq.Program.Equality}. Common useful variants are the following, -defined in the same file: - -\begin{Variants} -\item {\tt dependent induction {\ident} generalizing {\ident$_1$} \dots - {\ident$_n$}}\tacindex{dependent induction \dots\ generalizing} - - Does dependent induction on the hypothesis {\ident} but first - generalizes the goal by the given variables so that they are - universally quantified in the goal. This is generally what one wants - to do with the variables that are inside some constructors in the - induction hypothesis. The other ones need not be further generalized. - -\item {\tt dependent destruction {\ident}}\tacindex{dependent destruction} - - Does the generalization of the instance {\ident} but uses {\tt destruct} - instead of {\tt induction} on the generalized hypothesis. This gives - results equivalent to {\tt inversion} or {\tt dependent inversion} if - the hypothesis is dependent. -\end{Variants} - -A larger example of dependent induction and an explanation of the -underlying technique are developed in section~\ref{dependent-induction-example}. - -\subsection{\tt decompose [ {\qualid$_1$} \dots\ {\qualid$_n$} ] \term -\label{decompose} -\tacindex{decompose}} - -This tactic allows to recursively decompose a -complex proposition in order to obtain atomic ones. -Example: - -\begin{coq_eval} -Reset Initial. -\end{coq_eval} -\begin{coq_example} -Lemma ex1 : forall A B C:Prop, A /\ B /\ C \/ B /\ C \/ C /\ A -> C. -intros A B C H; decompose [and or] H; assumption. -\end{coq_example} -\begin{coq_example*} -Qed. -\end{coq_example*} - -{\tt decompose} does not work on right-hand sides of implications or products. - -\begin{Variants} - -\item {\tt decompose sum \term}\tacindex{decompose sum} - This decomposes sum types (like \texttt{or}). -\item {\tt decompose record \term}\tacindex{decompose record} - This decomposes record types (inductive types with one constructor, - like \texttt{and} and \texttt{exists} and those defined with the - \texttt{Record} macro, see Section~\ref{Record}). -\end{Variants} - - -\subsection{\tt functional induction (\qualid\ \term$_1$ \dots\ \term$_n$). -\tacindex{functional induction} -\label{FunInduction}} - -The \emph{experimental} tactic \texttt{functional induction} performs -case analysis and induction following the definition of a function. It -makes use of a principle generated by \texttt{Function} -(see Section~\ref{Function}) or \texttt{Functional Scheme} -(see Section~\ref{FunScheme}). - -\begin{coq_eval} -Reset Initial. -\end{coq_eval} -\begin{coq_example} -Functional Scheme minus_ind := Induction for minus Sort Prop. - -Lemma le_minus : forall n m:nat, (n - m <= n). -intros n m. -functional induction (minus n m); simpl; auto. -\end{coq_example} -\begin{coq_example*} -Qed. -\end{coq_example*} - -\Rem \texttt{(\qualid\ \term$_1$ \dots\ \term$_n$)} must be a correct -full application of \qualid. In particular, the rules for implicit -arguments are the same as usual. For example use \texttt{@\qualid} if -you want to write implicit arguments explicitly. - -\Rem Parenthesis over \qualid \dots \term$_n$ are mandatory. - -\Rem \texttt{functional induction (f x1 x2 x3)} is actually a wrapper -for \texttt{induction x1 x2 x3 (f x1 x2 x3) using \qualid} followed by -a cleaning phase, where $\qualid$ is the induction principle -registered for $f$ (by the \texttt{Function} (see Section~\ref{Function}) -or \texttt{Functional Scheme} (see Section~\ref{FunScheme}) command) -corresponding to the sort of the goal. Therefore \texttt{functional - induction} may fail if the induction scheme (\texttt{\qualid}) is -not defined. See also Section~\ref{Function} for the function terms -accepted by \texttt{Function}. - -\Rem There is a difference between obtaining an induction scheme for a -function by using \texttt{Function} (see Section~\ref{Function}) and by -using \texttt{Functional Scheme} after a normal definition using -\texttt{Fixpoint} or \texttt{Definition}. See \ref{Function} for -details. - -\SeeAlso{\ref{Function},\ref{FunScheme},\ref{FunScheme-examples}, - \ref{sec:functional-inversion}} - -\begin{ErrMsgs} -\item \errindex{Cannot find induction information on \qualid} - - ~ - -\item \errindex{Not the right number of induction arguments} -\end{ErrMsgs} - -\begin{Variants} -\item {\tt functional induction (\qualid\ \term$_1$ \dots\ \term$_n$) - using \term$_{m+1}$ with {\term$_{n+1}$} \dots {\term$_m$}} - - Similar to \texttt{Induction} and \texttt{elim} - (see Section~\ref{Tac-induction}), allows to give explicitly the - induction principle and the values of dependent premises of the - elimination scheme, including \emph{predicates} for mutual induction - when {\qualid} is part of a mutually recursive definition. - -\item {\tt functional induction (\qualid\ \term$_1$ \dots\ \term$_n$) - using \term$_{m+1}$ with {\vref$_1$} := {\term$_{n+1}$} \dots\ - {\vref$_m$} := {\term$_n$}} - - Similar to \texttt{induction} and \texttt{elim} - (see Section~\ref{Tac-induction}). - -\item All previous variants can be extended by the usual \texttt{as - \intropattern} construction, similar for example to - \texttt{induction} and \texttt{elim} (see Section~\ref{Tac-induction}). - -\end{Variants} - - - -\section{Equality} - -These tactics use the equality {\tt eq:forall A:Type, A->A->Prop} -defined in file {\tt Logic.v} (see Section~\ref{Equality}). The -notation for {\tt eq}~$T~t~u$ is simply {\tt $t$=$u$} dropping the -implicit type of $t$ and $u$. - -\subsection{\tt rewrite \term -\label{rewrite} -\tacindex{rewrite}} - -This tactic applies to any goal. The type of {\term} -must have the form - -\texttt{forall (x$_1$:A$_1$) \dots\ (x$_n$:A$_n$)}\texttt{eq} \term$_1$ \term$_2$. - -\noindent where \texttt{eq} is the Leibniz equality or a registered -setoid equality. - -\noindent Then {\tt rewrite \term} finds the first subterm matching -\term$_1$ in the goal, resulting in instances \term$_1'$ and \term$_2'$ -and then replaces every occurrence of \term$_1'$ by \term$_2'$. -Hence, some of the variables x$_i$ are -solved by unification, and some of the types \texttt{A}$_1$, \dots, -\texttt{A}$_n$ become new subgoals. - -% \Rem In case the type of -% \term$_1$ contains occurrences of variables bound in the -% type of \term, the tactic tries first to find a subterm of the goal -% which matches this term in order to find a closed instance \term$'_1$ -% of \term$_1$, and then all instances of \term$'_1$ will be replaced. - -\begin{ErrMsgs} -\item \errindex{The term provided does not end with an equation} - -\item \errindex{Tactic generated a subgoal identical to the original goal}\\ -This happens if \term$_1$ does not occur in the goal. -\end{ErrMsgs} - -\begin{Variants} -\item {\tt rewrite -> {\term}}\tacindex{rewrite ->}\\ - Is equivalent to {\tt rewrite \term} - -\item {\tt rewrite <- {\term}}\tacindex{rewrite <-}\\ - Uses the equality \term$_1${\tt=}\term$_2$ from right to left - -\item {\tt rewrite {\term} in \textit{clause}} - \tacindex{rewrite \dots\ in}\\ - Analogous to {\tt rewrite {\term}} but rewriting is done following - \textit{clause} (similarly to \ref{Conversion-tactics}). For - instance: - \begin{itemize} - \item \texttt{rewrite H in H1} will rewrite \texttt{H} in the hypothesis - \texttt{H1} instead of the current goal. - \item \texttt{rewrite H in H1 at 1, H2 at - 2 |- *} means \texttt{rewrite H; rewrite H in H1 at 1; - rewrite H in H2 at - 2}. In particular a failure will happen if any of - these three simpler tactics fails. - \item \texttt{rewrite H in * |- } will do \texttt{rewrite H in - H$_i$} for all hypothesis \texttt{H$_i$ <> H}. A success will happen - as soon as at least one of these simpler tactics succeeds. - \item \texttt{rewrite H in *} is a combination of \texttt{rewrite H} - and \texttt{rewrite H in * |-} that succeeds if at - least one of these two tactics succeeds. - \end{itemize} - Orientation {\tt ->} or {\tt <-} can be - inserted before the term to rewrite. - -\item {\tt rewrite {\term} at {\occlist}} - \tacindex{rewrite \dots\ at} - - Rewrite only the given occurrences of \term$_1'$. Occurrences are - specified from left to right as for \texttt{pattern} (\S - \ref{pattern}). The rewrite is always performed using setoid - rewriting, even for Leibniz's equality, so one has to - \texttt{Import Setoid} to use this variant. - -\item {\tt rewrite {\term} by {\tac}} - \tacindex{rewrite \dots\ by} - - Use {\tac} to completely solve the side-conditions arising from the - rewrite. - -\item {\tt rewrite $\term_1$, \ldots, $\term_n$}\\ - Is equivalent to the $n$ successive tactics {\tt rewrite $\term_1$} - up to {\tt rewrite $\term_n$}, each one working on the first subgoal - generated by the previous one. - Orientation {\tt ->} or {\tt <-} can be - inserted before each term to rewrite. One unique \textit{clause} - can be added at the end after the keyword {\tt in}; it will - then affect all rewrite operations. - -\item In all forms of {\tt rewrite} described above, a term to rewrite - can be immediately prefixed by one of the following modifiers: - \begin{itemize} - \item {\tt ?} : the tactic {\tt rewrite ?$\term$} performs the - rewrite of $\term$ as many times as possible (perhaps zero time). - This form never fails. - \item {\tt $n$?} : works similarly, except that it will do at most - $n$ rewrites. - \item {\tt !} : works as {\tt ?}, except that at least one rewrite - should succeed, otherwise the tactic fails. - \item {\tt $n$!} (or simply {\tt $n$}) : precisely $n$ rewrites - of $\term$ will be done, leading to failure if these $n$ rewrites are not possible. - \end{itemize} - -\item {\tt erewrite {\term}\tacindex{erewrite}} - -This tactic works as {\tt rewrite {\term}} but turning unresolved -bindings into existential variables, if any, instead of failing. It has -the same variants as {\tt rewrite} has. - -\end{Variants} - - -\subsection{\tt cutrewrite -> \term$_1$ = \term$_2$ -\label{cutrewrite} -\tacindex{cutrewrite}} - -This tactic acts like {\tt replace {\term$_1$} with {\term$_2$}} -(see below). - -\subsection{\tt replace {\term$_1$} with {\term$_2$} -\label{tactic:replace} -\tacindex{replace \dots\ with}} - -This tactic applies to any goal. It replaces all free occurrences of -{\term$_1$} in the current goal with {\term$_2$} and generates the -equality {\term$_2$}{\tt =}{\term$_1$} as a subgoal. This equality is -automatically solved if it occurs amongst the assumption, or if its -symmetric form occurs. It is equivalent to {\tt cut -\term$_2$=\term$_1$; [intro H{\sl n}; rewrite <- H{\sl n}; clear H{\sl -n}| assumption || symmetry; try assumption]}. - -\begin{ErrMsgs} -\item \errindex{terms do not have convertible types} -\end{ErrMsgs} - -\begin{Variants} -\item {\tt replace {\term$_1$} with {\term$_2$} by \tac}\\ This acts - as {\tt replace {\term$_1$} with {\term$_2$}} but applies {\tt \tac} - to solve the generated subgoal {\tt \term$_2$=\term$_1$}. -\item {\tt replace {\term}}\\ Replace {\term} with {\term'} using the - first assumption whose type has the form {\tt \term=\term'} or {\tt - \term'=\term} -\item {\tt replace -> {\term}}\\ Replace {\term} with {\term'} using the - first assumption whose type has the form {\tt \term=\term'} -\item {\tt replace <- {\term}}\\ Replace {\term} with {\term'} using the - first assumption whose type has the form {\tt \term'=\term} -\item {\tt replace {\term$_1$} with {\term$_2$} \textit{clause} }\\ - {\tt replace {\term$_1$} with {\term$_2$} \textit{clause} by \tac }\\ - {\tt replace {\term} \textit{clause}}\\ - {\tt replace -> {\term} \textit{clause}}\\ - {\tt replace <- {\term} \textit{clause}}\\ - Act as before but the replacements take place in - \textit{clause}~(see Section~\ref{Conversion-tactics}) and not only - in the conclusion of the goal.\\ - The \textit{clause} argument must not contain any \texttt{type of} nor \texttt{value of}. -\end{Variants} - -\subsection{\tt reflexivity -\label{reflexivity} -\tacindex{reflexivity}} - -This tactic applies to a goal which has the form {\tt t=u}. It checks -that {\tt t} and {\tt u} are convertible and then solves the goal. -It is equivalent to {\tt apply refl\_equal}. - -\begin{ErrMsgs} -\item \errindex{The conclusion is not a substitutive equation} -\item \errindex{Impossible to unify \dots\ with \dots.} -\end{ErrMsgs} - -\subsection{\tt symmetry -\tacindex{symmetry} -\tacindex{symmetry in}} -This tactic applies to a goal which has the form {\tt t=u} and changes it -into {\tt u=t}. - -\variant {\tt symmetry in {\ident}}\\ -If the statement of the hypothesis {\ident} has the form {\tt t=u}, -the tactic changes it to {\tt u=t}. - -\subsection{\tt transitivity \term -\tacindex{transitivity}} -This tactic applies to a goal which has the form {\tt t=u} -and transforms it into the two subgoals -{\tt t={\term}} and {\tt {\term}=u}. - -\subsection{\tt subst {\ident} -\tacindex{subst}} - -This tactic applies to a goal which has \ident\ in its context and -(at least) one hypothesis, say {\tt H}, of type {\tt - \ident=t} or {\tt t=\ident}. Then it replaces -\ident\ by {\tt t} everywhere in the goal (in the hypotheses -and in the conclusion) and clears \ident\ and {\tt H} from the context. - -\Rem -When several hypotheses have the form {\tt \ident=t} or {\tt - t=\ident}, the first one is used. - -\begin{Variants} - \item {\tt subst \ident$_1$ \dots \ident$_n$} \\ - Is equivalent to {\tt subst \ident$_1$; \dots; subst \ident$_n$}. - \item {\tt subst} \\ - Applies {\tt subst} repeatedly to all identifiers from the context - for which an equality exists. -\end{Variants} - -\subsection[{\tt stepl {\term}}]{{\tt stepl {\term}}\tacindex{stepl}} - -This tactic is for chaining rewriting steps. It assumes a goal of the -form ``$R$ {\term}$_1$ {\term}$_2$'' where $R$ is a binary relation -and relies on a database of lemmas of the form {\tt forall} $x$ $y$ -$z$, $R$ $x$ $y$ {\tt ->} $eq$ $x$ $z$ {\tt ->} $R$ $z$ $y$ where $eq$ -is typically a setoid equality. The application of {\tt stepl {\term}} -then replaces the goal by ``$R$ {\term} {\term}$_2$'' and adds a new -goal stating ``$eq$ {\term} {\term}$_1$''. - -Lemmas are added to the database using the command -\comindex{Declare Left Step} -\begin{quote} -{\tt Declare Left Step {\term}.} -\end{quote} - -The tactic is especially useful for parametric setoids which are not -accepted as regular setoids for {\tt rewrite} and {\tt - setoid\_replace} (see Chapter~\ref{setoid_replace}). - -\tacindex{stepr} -\comindex{Declare Right Step} -\begin{Variants} -\item{\tt stepl {\term} by {\tac}}\\ -This applies {\tt stepl {\term}} then applies {\tac} to the second goal. - -\item{\tt stepr {\term}}\\ - {\tt stepr {\term} by {\tac}}\\ -This behaves as {\tt stepl} but on the right-hand-side of the binary relation. -Lemmas are expected to be of the form -``{\tt forall} $x$ $y$ -$z$, $R$ $x$ $y$ {\tt ->} $eq$ $y$ $z$ {\tt ->} $R$ $x$ $z$'' -and are registered using the command -\begin{quote} -{\tt Declare Right Step {\term}.} -\end{quote} -\end{Variants} - - -\subsection{\tt f\_equal -\label{f-equal} -\tacindex{f\_equal}} - -This tactic applies to a goal of the form $f\ a_1\ \ldots\ a_n = f'\ -a'_1\ \ldots\ a'_n$. Using {\tt f\_equal} on such a goal leads to -subgoals $f=f'$ and $a_1=a'_1$ and so on up to $a_n=a'_n$. Amongst -these subgoals, the simple ones (e.g. provable by -reflexivity or congruence) are automatically solved by {\tt f\_equal}. - - -\section{Equality and inductive sets} - -We describe in this section some special purpose tactics dealing with -equality and inductive sets or types. These tactics use the equality -{\tt eq:forall (A:Type), A->A->Prop}, simply written with the -infix symbol {\tt =}. - -\subsection{\tt decide equality -\label{decideequality} -\tacindex{decide equality}} - -This tactic solves a goal of the form -{\tt forall $x$ $y$:$R$, \{$x$=$y$\}+\{\verb|~|$x$=$y$\}}, where $R$ -is an inductive type such that its constructors do not take proofs or -functions as arguments, nor objects in dependent types. - -\begin{Variants} -\item {\tt decide equality {\term}$_1$ {\term}$_2$ }.\\ - Solves a goal of the form {\tt \{}\term$_1${\tt =}\term$_2${\tt -\}+\{\verb|~|}\term$_1${\tt =}\term$_2${\tt \}}. -\end{Variants} - -\subsection{\tt compare \term$_1$ \term$_2$ -\tacindex{compare}} - -This tactic compares two given objects \term$_1$ and \term$_2$ -of an inductive datatype. If $G$ is the current goal, it leaves the sub-goals -\term$_1${\tt =}\term$_2$ {\tt ->} $G$ and \verb|~|\term$_1${\tt =}\term$_2$ -{\tt ->} $G$. The type -of \term$_1$ and \term$_2$ must satisfy the same restrictions as in the tactic -\texttt{decide equality}. - -\subsection{\tt discriminate {\term} -\label{discriminate} -\tacindex{discriminate} -\tacindex{ediscriminate}} - -This tactic proves any goal from an assumption stating that two -structurally different terms of an inductive set are equal. For -example, from {\tt (S (S O))=(S O)} we can derive by absurdity any -proposition. - -The argument {\term} is assumed to be a proof of a statement -of conclusion {\tt{\term$_1$} = {\term$_2$}} with {\term$_1$} and -{\term$_2$} being elements of an inductive set. To build the proof, -the tactic traverses the normal forms\footnote{Reminder: opaque - constants will not be expanded by $\delta$ reductions} of -{\term$_1$} and {\term$_2$} looking for a couple of subterms {\tt u} -and {\tt w} ({\tt u} subterm of the normal form of {\term$_1$} and -{\tt w} subterm of the normal form of {\term$_2$}), placed at the same -positions and whose head symbols are two different constructors. If -such a couple of subterms exists, then the proof of the current goal -is completed, otherwise the tactic fails. - -\Rem The syntax {\tt discriminate {\ident}} can be used to refer to a -hypothesis quantified in the goal. In this case, the quantified -hypothesis whose name is {\ident} is first introduced in the local -context using \texttt{intros until \ident}. - -\begin{ErrMsgs} -\item \errindex{No primitive equality found} -\item \errindex{Not a discriminable equality} -\end{ErrMsgs} - -\begin{Variants} -\item \texttt{discriminate} \num - - This does the same thing as \texttt{intros until \num} followed by - \texttt{discriminate \ident} where {\ident} is the identifier for - the last introduced hypothesis. - -\item \texttt{discriminate} {\term} {\tt with} {\bindinglist} - - This does the same thing as \texttt{discriminate {\term}} but using -the given bindings to instantiate parameters or hypotheses of {\term}. - -\item \texttt{ediscriminate} \num\\ - \texttt{ediscriminate} {\term} \zeroone{{\tt with} {\bindinglist}} - - This works the same as {\tt discriminate} but if the type of {\term}, - or the type of the hypothesis referred to by {\num}, has uninstantiated - parameters, these parameters are left as existential variables. - -\item \texttt{discriminate} - - This behaves like {\tt discriminate {\ident}} if {\ident} is the - name of an hypothesis to which {\tt discriminate} is applicable; if - the current goal is of the form {\term$_1$} {\tt <>} {\term$_2$}, - this behaves as {\tt intro {\ident}; injection {\ident}}. - - \begin{ErrMsgs} - \item \errindex{No discriminable equalities} \\ - occurs when the goal does not verify the expected preconditions. - \end{ErrMsgs} -\end{Variants} - -\subsection{\tt injection {\term} -\label{injection} -\tacindex{injection} -\tacindex{einjection}} - -The {\tt injection} tactic is based on the fact that constructors of -inductive sets are injections. That means that if $c$ is a constructor -of an inductive set, and if $(c~\vec{t_1})$ and $(c~\vec{t_2})$ are two -terms that are equal then $~\vec{t_1}$ and $~\vec{t_2}$ are equal -too. - -If {\term} is a proof of a statement of conclusion - {\tt {\term$_1$} = {\term$_2$}}, -then {\tt injection} applies injectivity as deep as possible to -derive the equality of all the subterms of {\term$_1$} and {\term$_2$} -placed in the same positions. For example, from {\tt (S - (S n))=(S (S (S m))} we may derive {\tt n=(S m)}. To use this -tactic {\term$_1$} and {\term$_2$} should be elements of an inductive -set and they should be neither explicitly equal, nor structurally -different. We mean by this that, if {\tt n$_1$} and {\tt n$_2$} are -their respective normal forms, then: -\begin{itemize} -\item {\tt n$_1$} and {\tt n$_2$} should not be syntactically equal, -\item there must not exist any pair of subterms {\tt u} and {\tt w}, - {\tt u} subterm of {\tt n$_1$} and {\tt w} subterm of {\tt n$_2$} , - placed in the same positions and having different constructors as - head symbols. -\end{itemize} -If these conditions are satisfied, then, the tactic derives the -equality of all the subterms of {\term$_1$} and {\term$_2$} placed in -the same positions and puts them as antecedents of the current goal. - -\Example Consider the following goal: - -\begin{coq_example*} -Inductive list : Set := - | nil : list - | cons : nat -> list -> list. -Variable P : list -> Prop. -\end{coq_example*} -\begin{coq_eval} -Lemma ex : - forall (l:list) (n:nat), P nil -> cons n l = cons 0 nil -> P l. -intros l n H H0. -\end{coq_eval} -\begin{coq_example} -Show. -injection H0. -\end{coq_example} -\begin{coq_eval} -Abort. -\end{coq_eval} - -Beware that \texttt{injection} yields always an equality in a sigma type -whenever the injected object has a dependent type. - -\Rem There is a special case for dependent pairs. If we have a decidable -equality over the type of the first argument, then it is safe to do -the projection on the second one, and so {\tt injection} will work fine. -To define such an equality, you have to use the {\tt Scheme} command -(see \ref{Scheme}). - -\Rem If some quantified hypothesis of the goal is named {\ident}, then -{\tt injection {\ident}} first introduces the hypothesis in the local -context using \texttt{intros until \ident}. - -\begin{ErrMsgs} -\item \errindex{Not a projectable equality but a discriminable one} -\item \errindex{Nothing to do, it is an equality between convertible terms} -\item \errindex{Not a primitive equality} -\end{ErrMsgs} - -\begin{Variants} -\item \texttt{injection} \num{} - - This does the same thing as \texttt{intros until \num} followed by -\texttt{injection \ident} where {\ident} is the identifier for the last -introduced hypothesis. - -\item \texttt{injection} \term{} {\tt with} {\bindinglist} - - This does the same as \texttt{injection {\term}} but using - the given bindings to instantiate parameters or hypotheses of {\term}. - -\item \texttt{einjection} \num\\ - \texttt{einjection} \term{} \zeroone{{\tt with} {\bindinglist}} - - This works the same as {\tt injection} but if the type of {\term}, - or the type of the hypothesis referred to by {\num}, has uninstantiated - parameters, these parameters are left as existential variables. - -\item{\tt injection} - - If the current goal is of the form {\term$_1$} {\tt <>} {\term$_2$}, - this behaves as {\tt intro {\ident}; injection {\ident}}. - - \ErrMsg \errindex{goal does not satisfy the expected preconditions} - -\item \texttt{injection} \term{} \zeroone{{\tt with} {\bindinglist}} \texttt{as} \nelist{\intropattern}{}\\ -\texttt{injection} \num{} \texttt{as} {\intropattern} {\ldots} {\intropattern}\\ -\texttt{injection} \texttt{as} {\intropattern} {\ldots} {\intropattern}\\ -\texttt{einjection} \term{} \zeroone{{\tt with} {\bindinglist}} \texttt{as} \nelist{\intropattern}{}\\ -\texttt{einjection} \num{} \texttt{as} {\intropattern} {\ldots} {\intropattern}\\ -\texttt{einjection} \texttt{as} {\intropattern} {\ldots} {\intropattern}\\ -\tacindex{injection \ldots{} as} - -These variants apply \texttt{intros} \nelist{\intropattern}{} after -the call to \texttt{injection} or \texttt{einjection}. - -\end{Variants} - -\subsection{\tt simplify\_eq {\term} -\tacindex{simplify\_eq} -\tacindex{esimplify\_eq} -\label{simplify-eq}} - -Let {\term} be the proof of a statement of conclusion {\tt - {\term$_1$}={\term$_2$}}. If {\term$_1$} and -{\term$_2$} are structurally different (in the sense described for the -tactic {\tt discriminate}), then the tactic {\tt simplify\_eq} behaves as {\tt - discriminate {\term}}, otherwise it behaves as {\tt injection - {\term}}. - -\Rem If some quantified hypothesis of the goal is named {\ident}, then -{\tt simplify\_eq {\ident}} first introduces the hypothesis in the local -context using \texttt{intros until \ident}. - -\begin{Variants} -\item \texttt{simplify\_eq} \num - - This does the same thing as \texttt{intros until \num} then -\texttt{simplify\_eq \ident} where {\ident} is the identifier for the last -introduced hypothesis. - -\item \texttt{simplify\_eq} \term{} {\tt with} {\bindinglist} - - This does the same as \texttt{simplify\_eq {\term}} but using - the given bindings to instantiate parameters or hypotheses of {\term}. - -\item \texttt{esimplify\_eq} \num\\ - \texttt{esimplify\_eq} \term{} \zeroone{{\tt with} {\bindinglist}} - - This works the same as {\tt simplify\_eq} but if the type of {\term}, - or the type of the hypothesis referred to by {\num}, has uninstantiated - parameters, these parameters are left as existential variables. - -\item{\tt simplify\_eq} - -If the current goal has form $t_1\verb=<>=t_2$, it behaves as -\texttt{intro {\ident}; simplify\_eq {\ident}}. -\end{Variants} - -\subsection{\tt dependent rewrite -> {\ident} -\tacindex{dependent rewrite ->} -\label{dependent-rewrite}} - -This tactic applies to any goal. If \ident\ has type -\verb+(existT B a b)=(existT B a' b')+ -in the local context (i.e. each term of the -equality has a sigma type $\{ a:A~ \&~(B~a)\}$) this tactic rewrites -\verb+a+ into \verb+a'+ and \verb+b+ into \verb+b'+ in the current -goal. This tactic works even if $B$ is also a sigma type. This kind -of equalities between dependent pairs may be derived by the injection -and inversion tactics. - -\begin{Variants} -\item{\tt dependent rewrite <- {\ident}} -\tacindex{dependent rewrite <-} \\ -Analogous to {\tt dependent rewrite ->} but uses the equality from -right to left. -\end{Variants} - -\section{Inversion -\label{inversion}} - -\subsection{\tt inversion {\ident} -\tacindex{inversion}} - -Let the type of \ident~ in the local context be $(I~\vec{t})$, -where $I$ is a (co)inductive predicate. Then, -\texttt{inversion} applied to \ident~ derives for each possible -constructor $c_i$ of $(I~\vec{t})$, {\bf all} the necessary -conditions that should hold for the instance $(I~\vec{t})$ to be -proved by $c_i$. - -\Rem If {\ident} does not denote a hypothesis in the local context -but refers to a hypothesis quantified in the goal, then the -latter is first introduced in the local context using -\texttt{intros until \ident}. - -\begin{Variants} -\item \texttt{inversion} \num - - This does the same thing as \texttt{intros until \num} then - \texttt{inversion \ident} where {\ident} is the identifier for the - last introduced hypothesis. - -\item \tacindex{inversion\_clear} \texttt{inversion\_clear} \ident - - This behaves as \texttt{inversion} and then erases \ident~ from the - context. - -\item \tacindex{inversion \dots\ as} \texttt{inversion} {\ident} \texttt{as} {\intropattern} - - This behaves as \texttt{inversion} but using names in - {\intropattern} for naming hypotheses. The {\intropattern} must have - the form {\tt [} $p_{11}$ \ldots $p_{1n_1}$ {\tt |} {\ldots} {\tt |} - $p_{m1}$ \ldots $p_{mn_m}$ {\tt ]} with $m$ being the number of - constructors of the type of {\ident}. Be careful that the list must - be of length $m$ even if {\tt inversion} discards some cases (which - is precisely one of its roles): for the discarded cases, just use an - empty list (i.e. $n_i=0$). - - The arguments of the $i^{th}$ constructor and the - equalities that {\tt inversion} introduces in the context of the - goal corresponding to the $i^{th}$ constructor, if it exists, get - their names from the list $p_{i1}$ \ldots $p_{in_i}$ in order. If - there are not enough names, {\tt induction} invents names for the - remaining variables to introduce. In case an equation splits into - several equations (because {\tt inversion} applies {\tt injection} - on the equalities it generates), the corresponding name $p_{ij}$ in - the list must be replaced by a sublist of the form {\tt [$p_{ij1}$ - \ldots $p_{ijq}$]} (or, equivalently, {\tt ($p_{ij1}$, - \ldots, $p_{ijq}$)}) where $q$ is the number of subequalities - obtained from splitting the original equation. Here is an example. - -\begin{coq_eval} -Require Import List. -\end{coq_eval} - -\begin{coq_example} -Inductive contains0 : list nat -> Prop := - | in_hd : forall l, contains0 (0 :: l) - | in_tl : forall l b, contains0 l -> contains0 (b :: l). -Goal forall l:list nat, contains0 (1 :: l) -> contains0 l. -intros l H; inversion H as [ | l' p Hl' [Heqp Heql'] ]. -\end{coq_example} - -\begin{coq_eval} -Abort. -\end{coq_eval} - -\item \texttt{inversion} {\num} {\tt as} {\intropattern} - - This allows to name the hypotheses introduced by - \texttt{inversion} {\num} in the context. - -\item \tacindex{inversion\_cleardots\ as} \texttt{inversion\_clear} - {\ident} {\tt as} {\intropattern} - - This allows to name the hypotheses introduced by - \texttt{inversion\_clear} in the context. - -\item \tacindex{inversion \dots\ in} \texttt{inversion } {\ident} - \texttt{in} \ident$_1$ \dots\ \ident$_n$ - - Let \ident$_1$ \dots\ \ident$_n$, be identifiers in the local context. This - tactic behaves as generalizing \ident$_1$ \dots\ \ident$_n$, and - then performing \texttt{inversion}. - -\item \tacindex{inversion \dots\ as \dots\ in} \texttt{inversion } - {\ident} {\tt as} {\intropattern} \texttt{in} \ident$_1$ \dots\ - \ident$_n$ - - This allows to name the hypotheses introduced in the context by - \texttt{inversion} {\ident} \texttt{in} \ident$_1$ \dots\ - \ident$_n$. - -\item \tacindex{inversion\_clear \dots\ in} \texttt{inversion\_clear} - {\ident} \texttt{in} \ident$_1$ \ldots \ident$_n$ - - Let \ident$_1$ \dots\ \ident$_n$, be identifiers in the local context. This - tactic behaves as generalizing \ident$_1$ \dots\ \ident$_n$, and - then performing {\tt inversion\_clear}. - -\item \tacindex{inversion\_clear \dots\ as \dots\ in} - \texttt{inversion\_clear} {\ident} \texttt{as} {\intropattern} - \texttt{in} \ident$_1$ \ldots \ident$_n$ - - This allows to name the hypotheses introduced in the context by - \texttt{inversion\_clear} {\ident} \texttt{in} \ident$_1$ \ldots - \ident$_n$. - -\item \tacindex{dependent inversion} \texttt{dependent inversion} - {\ident} - - That must be used when \ident\ appears in the current goal. It acts - like \texttt{inversion} and then substitutes \ident\ for the - corresponding term in the goal. - -\item \tacindex{dependent inversion \dots\ as } \texttt{dependent - inversion} {\ident} \texttt{as} {\intropattern} - - This allows to name the hypotheses introduced in the context by - \texttt{dependent inversion} {\ident}. - -\item \tacindex{dependent inversion\_clear} \texttt{dependent - inversion\_clear} {\ident} - - Like \texttt{dependent inversion}, except that {\ident} is cleared - from the local context. - -\item \tacindex{dependent inversion\_clear \dots\ as} - \texttt{dependent inversion\_clear} {\ident}\texttt{as} {\intropattern} - - This allows to name the hypotheses introduced in the context by - \texttt{dependent inversion\_clear} {\ident}. - -\item \tacindex{dependent inversion \dots\ with} \texttt{dependent - inversion } {\ident} \texttt{ with } \term - - This variant allows you to specify the generalization of the goal. It - is useful when the system fails to generalize the goal automatically. If - {\ident} has type $(I~\vec{t})$ and $I$ has type - $forall (\vec{x}:\vec{T}), s$, then \term~ must be of type - $I:forall (\vec{x}:\vec{T}), I~\vec{x}\to s'$ where $s'$ is the - type of the goal. - -\item \tacindex{dependent inversion \dots\ as \dots\ with} - \texttt{dependent inversion } {\ident} \texttt{as} {\intropattern} - \texttt{ with } \term - - This allows to name the hypotheses introduced in the context by - \texttt{dependent inversion } {\ident} \texttt{ with } \term. - -\item \tacindex{dependent inversion\_clear \dots\ with} - \texttt{dependent inversion\_clear } {\ident} \texttt{ with } \term - - Like \texttt{dependent inversion \dots\ with} but clears {\ident} from - the local context. - -\item \tacindex{dependent inversion\_clear \dots\ as \dots\ with} - \texttt{dependent inversion\_clear } {\ident} \texttt{as} - {\intropattern} \texttt{ with } \term - - This allows to name the hypotheses introduced in the context by - \texttt{dependent inversion\_clear } {\ident} \texttt{ with } \term. - -\item \tacindex{simple inversion} \texttt{simple inversion} {\ident} - - It is a very primitive inversion tactic that derives all the necessary - equalities but it does not simplify the constraints as - \texttt{inversion} does. - -\item \tacindex{simple inversion \dots\ as} \texttt{simple inversion} - {\ident} \texttt{as} {\intropattern} - - This allows to name the hypotheses introduced in the context by - \texttt{simple inversion}. - -\item \tacindex{inversion \dots\ using} \texttt{inversion} \ident - \texttt{ using} \ident$'$ - - Let {\ident} have type $(I~\vec{t})$ ($I$ an inductive - predicate) in the local context, and \ident$'$ be a (dependent) inversion - lemma. Then, this tactic refines the current goal with the specified - lemma. - -\item \tacindex{inversion \dots\ using \dots\ in} \texttt{inversion} - {\ident} \texttt{using} \ident$'$ \texttt{in} \ident$_1$\dots\ \ident$_n$ - - This tactic behaves as generalizing \ident$_1$\dots\ \ident$_n$, - then doing \texttt{inversion} {\ident} \texttt{using} \ident$'$. - -\end{Variants} - -\SeeAlso~\ref{inversion-examples} for detailed examples - -\subsection{\tt Derive Inversion {\ident} with - ${\tt forall (}\vec{x}{\tt :}\vec{T}{\tt),} I~\vec{t}$ Sort \sort -\label{Derive-Inversion} -\comindex{Derive Inversion}} - -This command generates an inversion principle for the -\texttt{inversion \dots\ using} tactic. -Let $I$ be an inductive predicate and $\vec{x}$ the variables -occurring in $\vec{t}$. This command generates and stocks the -inversion lemma for the sort \sort~ corresponding to the instance -$forall (\vec{x}:\vec{T}), I~\vec{t}$ with the name {\ident} in the {\bf -global} environment. When applied it is equivalent to have inverted -the instance with the tactic {\tt inversion}. - -\begin{Variants} -\item \texttt{Derive Inversion\_clear} {\ident} \texttt{with} - \comindex{Derive Inversion\_clear} - $forall (\vec{x}:\vec{T}), I~\vec{t}$ \texttt{Sort} \sort~ \\ - \index{Derive Inversion\_clear \dots\ with} - When applied it is equivalent to having - inverted the instance with the tactic \texttt{inversion} - replaced by the tactic \texttt{inversion\_clear}. -\item \texttt{Derive Dependent Inversion} {\ident} \texttt{with} - $forall (\vec{x}:\vec{T}), I~\vec{t}$ \texttt{Sort} \sort~\\ - \comindex{Derive Dependent Inversion} - When applied it is equivalent to having - inverted the instance with the tactic \texttt{dependent inversion}. -\item \texttt{Derive Dependent Inversion\_clear} {\ident} \texttt{with} - $forall (\vec{x}:\vec{T}), I~\vec{t}$ \texttt{Sort} \sort~\\ - \comindex{Derive Dependent Inversion\_clear} - When applied it is equivalent to having - inverted the instance with the tactic \texttt{dependent inversion\_clear}. -\end{Variants} - -\SeeAlso \ref{inversion-examples} for examples - - - -\subsection[\tt functional inversion \ident]{\tt functional inversion \ident\label{sec:functional-inversion}} - -\texttt{functional inversion} is a \emph{highly} experimental tactic -which performs inversion on hypothesis \ident\ of the form -\texttt{\qualid\ \term$_1$\dots\term$_n$\ = \term} or \texttt{\term\ = - \qualid\ \term$_1$\dots\term$_n$} where \qualid\ must have been -defined using \texttt{Function} (see Section~\ref{Function}). - -\begin{ErrMsgs} -\item \errindex{Hypothesis {\ident} must contain at least one Function} -\item \errindex{Cannot find inversion information for hypothesis \ident} - This error may be raised when some inversion lemma failed to be - generated by Function. -\end{ErrMsgs} - -\begin{Variants} -\item {\tt functional inversion \num} - - This does the same thing as \texttt{intros until \num} then - \texttt{functional inversion \ident} where {\ident} is the - identifier for the last introduced hypothesis. -\item {\tt functional inversion \ident\ \qualid}\\ - {\tt functional inversion \num\ \qualid} - - In case the hypothesis {\ident} (or {\num}) has a type of the form - \texttt{\qualid$_1$\ \term$_1$\dots\term$_n$\ =\ \qualid$_2$\ - \term$_{n+1}$\dots\term$_{n+m}$} where \qualid$_1$ and \qualid$_2$ - are valid candidates to functional inversion, this variant allows to - choose which must be inverted. -\end{Variants} - - - -\subsection{\tt quote \ident -\tacindex{quote} -\index{2-level approach}} - -This kind of inversion has nothing to do with the tactic -\texttt{inversion} above. This tactic does \texttt{change (\ident\ - t)}, where \texttt{t} is a term built in order to ensure the -convertibility. In other words, it does inversion of the function -\ident. This function must be a fixpoint on a simple recursive -datatype: see~\ref{quote-examples} for the full details. - -\begin{ErrMsgs} -\item \errindex{quote: not a simple fixpoint}\\ - Happens when \texttt{quote} is not able to perform inversion properly. -\end{ErrMsgs} - -\begin{Variants} -\item \texttt{quote {\ident} [ \ident$_1$ \dots \ident$_n$ ]}\\ - All terms that are built only with \ident$_1$ \dots \ident$_n$ will be - considered by \texttt{quote} as constants rather than variables. -\end{Variants} - -% En attente d'un moyen de valoriser les fichiers de demos -% \SeeAlso file \texttt{theories/DEMOS/DemoQuote.v} in the distribution - -\section[Classical tactics]{Classical tactics\label{ClassicalTactics}} - -In order to ease the proving process, when the {\tt Classical} module is loaded. A few more tactics are available. Make sure to load the module using the \texttt{Require Import} command. - -\subsection{{\tt classical\_left, classical\_right} \tacindex{classical\_left} \tacindex{classical\_right}} - -The tactics \texttt{classical\_left} and \texttt{classical\_right} are the analog of the \texttt{left} and \texttt{right} but using classical logic. They can only be used for disjunctions. -Use \texttt{classical\_left} to prove the left part of the disjunction with the assumption that the negation of right part holds. -Use \texttt{classical\_right} to prove the right part of the disjunction with the assumption that the negation of left part holds. - -\section{Automatizing -\label{Automatizing}} - -\subsection{\tt auto -\label{auto} -\tacindex{auto}} - -This tactic implements a Prolog-like resolution procedure to solve the -current goal. It first tries to solve the goal using the {\tt - assumption} tactic, then it reduces the goal to an atomic one using -{\tt intros} and introducing the newly generated hypotheses as hints. -Then it looks at the list of tactics associated to the head symbol of -the goal and tries to apply one of them (starting from the tactics -with lower cost). This process is recursively applied to the generated -subgoals. - -By default, \texttt{auto} only uses the hypotheses of the current goal and the -hints of the database named {\tt core}. - -\begin{Variants} - -\item {\tt auto \num} - - Forces the search depth to be \num. The maximal search depth is 5 by - default. - -\item {\tt auto with \ident$_1$ \dots\ \ident$_n$} - - Uses the hint databases $\ident_1$ \dots\ $\ident_n$ in addition to - the database {\tt core}. See Section~\ref{Hints-databases} for the - list of pre-defined databases and the way to create or extend a - database. This option can be combined with the previous one. - -\item {\tt auto with *} - - Uses all existing hint databases, minus the special database - {\tt v62}. See Section~\ref{Hints-databases} - -\item \texttt{auto using \nterm{lemma}$_1$ , \ldots , \nterm{lemma}$_n$} - - Uses \nterm{lemma}$_1$, \ldots, \nterm{lemma}$_n$ in addition to - hints (can be combined with the \texttt{with \ident} option). If - $lemma_i$ is an inductive type, it is the collection of its - constructors which is added as hints. - -\item \texttt{auto using \nterm{lemma}$_1$ , \ldots , \nterm{lemma}$_n$ with \ident$_1$ \dots\ \ident$_n$} - - This combines the effects of the {\tt using} and {\tt with} options. - -\item {\tt trivial}\tacindex{trivial} - - This tactic is a restriction of {\tt auto} that is not recursive and - tries only hints which cost 0. Typically it solves trivial - equalities like $X=X$. - -\item \texttt{trivial with \ident$_1$ \dots\ \ident$_n$} - -\item \texttt{trivial with *} - -\end{Variants} - -\Rem {\tt auto} either solves completely the goal or else leaves it -intact. \texttt{auto} and \texttt{trivial} never fail. - -\SeeAlso Section~\ref{Hints-databases} - -\subsection{\tt eauto -\tacindex{eauto} -\label{eauto}} - -This tactic generalizes {\tt auto}. In contrast with -the latter, {\tt eauto} uses unification of the goal -against the hints rather than pattern-matching -(in other words, it uses {\tt eapply} instead of -{\tt apply}). -As a consequence, {\tt eauto} can solve such a goal: - -\begin{coq_example} -Hint Resolve ex_intro. -Goal forall P:nat -> Prop, P 0 -> exists n, P n. -eauto. -\end{coq_example} -\begin{coq_eval} -Abort. -\end{coq_eval} - -Note that {\tt ex\_intro} should be declared as an -hint. - -\SeeAlso Section~\ref{Hints-databases} - -\subsection{\tt autounfold with \ident$_1$ \dots\ \ident$_n$ -\tacindex{autounfold} -\label{autounfold}} - -This tactic unfolds constants that were declared through a {\tt Hint - Unfold} in the given databases. - -\begin{Variants} -\item {\tt autounfold with \ident$_1$ \dots\ \ident$_n$ in \textit{clause}} - - Perform the unfolding in the given clause. - -\item {\tt autounfold with *} - - Uses the unfold hints declared in all the hint databases. -\end{Variants} - - -% EXISTE ENCORE ? -% -% \subsection{\tt Prolog [ \term$_1$ \dots\ \term$_n$ ] \num} -% \tacindex{Prolog}\label{Prolog} -% This tactic, implemented by Chet Murthy, is based upon the concept of -% existential variables of Gilles Dowek, stating that resolution is a -% kind of unification. It tries to solve the current goal using the {\tt -% Assumption} tactic, the {\tt intro} tactic, and applying hypotheses -% of the local context and terms of the given list {\tt [ \term$_1$ -% \dots\ \term$_n$\ ]}. It is more powerful than {\tt auto} since it -% may apply to any theorem, even those of the form {\tt (x:A)(P x) -> Q} -% where {\tt x} does not appear free in {\tt Q}. The maximal search -% depth is {\tt \num}. - -% \begin{ErrMsgs} -% \item \errindex{Prolog failed}\\ -% The Prolog tactic was not able to prove the subgoal. -% \end{ErrMsgs} - -\subsection{\tt tauto -\tacindex{tauto} -\label{tauto}} - -This tactic implements a decision procedure for intuitionistic propositional -calculus based on the contraction-free sequent calculi LJT* of Roy Dyckhoff -\cite{Dyc92}. Note that {\tt tauto} succeeds on any instance of an -intuitionistic tautological proposition. {\tt tauto} unfolds negations -and logical equivalence but does not unfold any other definition. - -The following goal can be proved by {\tt tauto} whereas {\tt auto} -would fail: - -\begin{coq_example} -Goal forall (x:nat) (P:nat -> Prop), x = 0 \/ P x -> x <> 0 -> P x. - intros. - tauto. -\end{coq_example} -\begin{coq_eval} -Abort. -\end{coq_eval} - -Moreover, if it has nothing else to do, {\tt tauto} performs -introductions. Therefore, the use of {\tt intros} in the previous -proof is unnecessary. {\tt tauto} can for instance prove the -following: -\begin{coq_example} -(* auto would fail *) -Goal forall (A:Prop) (P:nat -> Prop), - A \/ (forall x:nat, ~ A -> P x) -> forall x:nat, ~ A -> P x. - - tauto. -\end{coq_example} -\begin{coq_eval} -Abort. -\end{coq_eval} - -\Rem In contrast, {\tt tauto} cannot solve the following goal - -\begin{coq_example*} -Goal forall (A:Prop) (P:nat -> Prop), - A \/ (forall x:nat, ~ A -> P x) -> forall x:nat, ~ ~ (A \/ P x). -\end{coq_example*} -\begin{coq_eval} -Abort. -\end{coq_eval} - -because \verb=(forall x:nat, ~ A -> P x)= cannot be treated as atomic and an -instantiation of \verb=x= is necessary. - -\subsection{\tt intuition {\tac} -\tacindex{intuition} -\label{intuition}} - -The tactic \texttt{intuition} takes advantage of the search-tree built -by the decision procedure involved in the tactic {\tt tauto}. It uses -this information to generate a set of subgoals equivalent to the -original one (but simpler than it) and applies the tactic -{\tac} to them \cite{Mun94}. If this tactic fails on some goals then -{\tt intuition} fails. In fact, {\tt tauto} is simply {\tt intuition - fail}. - -For instance, the tactic {\tt intuition auto} applied to the goal -\begin{verbatim} -(forall (x:nat), P x)/\B -> (forall (y:nat),P y)/\ P O \/B/\ P O -\end{verbatim} -internally replaces it by the equivalent one: -\begin{verbatim} -(forall (x:nat), P x), B |- P O -\end{verbatim} -and then uses {\tt auto} which completes the proof. - -Originally due to C{\'e}sar~Mu{\~n}oz, these tactics ({\tt tauto} and {\tt intuition}) -have been completely re-engineered by David~Delahaye using mainly the tactic -language (see Chapter~\ref{TacticLanguage}). The code is now much shorter and -a significant increase in performance has been noticed. The general behavior -with respect to dependent types, unfolding and introductions has -slightly changed to get clearer semantics. This may lead to some -incompatibilities. - -\begin{Variants} -\item {\tt intuition}\\ - Is equivalent to {\tt intuition auto with *}. -\end{Variants} - -% En attente d'un moyen de valoriser les fichiers de demos -%\SeeAlso file \texttt{contrib/Rocq/DEMOS/Demo\_tauto.v} - - -\subsection{\tt rtauto -\tacindex{rtauto} -\label{rtauto}} - -The {\tt rtauto} tactic solves propositional tautologies similarly to what {\tt tauto} does. The main difference is that the proof term is built using a reflection scheme applied to a sequent calculus proof of the goal. The search procedure is also implemented using a different technique. - -Users should be aware that this difference may result in faster proof-search but slower proof-checking, and {\tt rtauto} might not solve goals that {\tt tauto} would be able to solve (e.g. goals involving universal quantifiers). - -\subsection{{\tt firstorder} -\tacindex{firstorder} -\label{firstorder}} - -The tactic \texttt{firstorder} is an {\it experimental} extension of -\texttt{tauto} to -first-order reasoning, written by Pierre Corbineau. -It is not restricted to usual logical connectives but -instead may reason about any first-order class inductive definition. - -\begin{Variants} - \item {\tt firstorder {\tac}} - \tacindex{firstorder {\tac}} - - Tries to solve the goal with {\tac} when no logical rule may apply. - - \item {\tt firstorder with \ident$_1$ \dots\ \ident$_n$ } - \tacindex{firstorder with} - - Adds lemmas \ident$_1$ \dots\ \ident$_n$ to the proof-search - environment. - - \item {\tt firstorder using {\qualid}$_1$ , \dots\ , {\qualid}$_n$ } - \tacindex{firstorder using} - - Adds lemmas in {\tt auto} hints bases {\qualid}$_1$ \dots\ {\qualid}$_n$ - to the proof-search environment. If {\qualid}$_i$ refers to an inductive - type, it is the collection of its constructors which is added as hints. - -\item \texttt{firstorder using {\qualid}$_1$ , \dots\ , {\qualid}$_n$ with \ident$_1$ \dots\ \ident$_n$} - - This combines the effects of the {\tt using} and {\tt with} options. - -\end{Variants} - -Proof-search is bounded by a depth parameter which can be set by typing the -{\nobreak \tt Set Firstorder Depth $n$} \comindex{Set Firstorder Depth} -vernacular command. - -%% \subsection{{\tt jp} {\em (Jprover)} -%% \tacindex{jp} -%% \label{jprover}} - -%% The tactic \texttt{jp}, due to Huang Guan-Shieng, is an experimental -%% port of the {\em Jprover}\cite{SLKN01} semi-decision procedure for -%% first-order intuitionistic logic implemented in {\em -%% NuPRL}\cite{Kre02}. - -%% The tactic \texttt{jp}, due to Huang Guan-Shieng, is an {\it -%% experimental} port of the {\em Jprover}\cite{SLKN01} semi-decision -%% procedure for first-order intuitionistic logic implemented in {\em -%% NuPRL}\cite{Kre02}. - -%% Search may optionnaly be bounded by a multiplicity parameter -%% indicating how many (at most) copies of a formula may be used in -%% the proof process, its absence may lead to non-termination of the tactic. - -%% %\begin{coq_eval} -%% %Variable S:Set. -%% %Variables P Q:S->Prop. -%% %Variable f:S->S. -%% %\end{coq_eval} - -%% %\begin{coq_example*} -%% %Lemma example: (exists x |P x\/Q x)->(exists x |P x)\/(exists x |Q x). -%% %jp. -%% %Qed. - -%% %Lemma example2: (forall x ,P x->P (f x))->forall x,P x->P (f(f x)). -%% %jp. -%% %Qed. -%% %\end{coq_example*} - -%% \begin{Variants} -%% \item {\tt jp $n$}\\ -%% \tacindex{jp $n$} -%% Tries the {\em Jprover} procedure with multiplicities up to $n$, -%% starting from 1. -%% \item {\tt jp}\\ -%% Tries the {\em Jprover} procedure without multiplicity bound, -%% possibly running forever. -%% \end{Variants} - -%% \begin{ErrMsgs} -%% \item \errindex{multiplicity limit reached}\\ -%% The procedure tried all multiplicities below the limit and -%% failed. Goal might be solved by increasing the multiplicity limit. -%% \item \errindex{formula is not provable}\\ -%% The procedure determined that goal was not provable in -%% intuitionistic first-order logic, no matter how big the -%% multiplicity is. -%% \end{ErrMsgs} - - -% \subsection[\tt Linear]{\tt Linear\tacindex{Linear}\label{Linear}} -% The tactic \texttt{Linear}, due to Jean-Christophe Filli{\^a}atre -% \cite{Fil94}, implements a decision procedure for {\em Direct -% Predicate Calculus}, that is first-order Gentzen's Sequent Calculus -% without contraction rules \cite{KeWe84,BeKe92}. Intuitively, a -% first-order goal is provable in Direct Predicate Calculus if it can be -% proved using each hypothesis at most once. - -% Unlike the previous tactics, the \texttt{Linear} tactic does not belong -% to the initial state of the system, and it must be loaded explicitly -% with the command - -% \begin{coq_example*} -% Require Linear. -% \end{coq_example*} - -% For instance, assuming that \texttt{even} and \texttt{odd} are two -% predicates on natural numbers, and \texttt{a} of type \texttt{nat}, the -% tactic \texttt{Linear} solves the following goal - -% \begin{coq_eval} -% Variables even,odd : nat -> Prop. -% Variable a:nat. -% \end{coq_eval} - -% \begin{coq_example*} -% Lemma example : (even a) -% -> ((x:nat)((even x)->(odd (S x)))) -% -> (EX y | (odd y)). -% \end{coq_example*} - -% You can find examples of the use of \texttt{Linear} in -% \texttt{theories/DEMOS/DemoLinear.v}. -% \begin{coq_eval} -% Abort. -% \end{coq_eval} - -% \begin{Variants} -% \item {\tt Linear with \ident$_1$ \dots\ \ident$_n$}\\ -% \tacindex{Linear with} -% Is equivalent to apply first {\tt generalize \ident$_1$ \dots -% \ident$_n$} (see Section~\ref{generalize}) then the \texttt{Linear} -% tactic. So one can use axioms, lemmas or hypotheses of the local -% context with \texttt{Linear} in this way. -% \end{Variants} - -% \begin{ErrMsgs} -% \item \errindex{Not provable in Direct Predicate Calculus} -% \item \errindex{Found $n$ classical proof(s) but no intuitionistic one}\\ -% The decision procedure looks actually for classical proofs of the -% goals, and then checks that they are intuitionistic. In that case, -% classical proofs have been found, which do not correspond to -% intuitionistic ones. -% \end{ErrMsgs} - -\subsection{\tt congruence -\tacindex{congruence} -\label{congruence}} - -The tactic {\tt congruence}, by Pierre Corbineau, implements the standard Nelson and Oppen -congruence closure algorithm, which is a decision procedure for ground -equalities with uninterpreted symbols. It also include the constructor theory -(see \ref{injection} and \ref{discriminate}). -If the goal is a non-quantified equality, {\tt congruence} tries to -prove it with non-quantified equalities in the context. Otherwise it -tries to infer a discriminable equality from those in the context. Alternatively, congruence tries to prove that a hypothesis is equal to the goal or to the negation of another hypothesis. - -{\tt congruence} is also able to take advantage of hypotheses stating quantified equalities, you have to provide a bound for the number of extra equalities generated that way. Please note that one of the members of the equality must contain all the quantified variables in order for {\tt congruence} to match against it. - -\begin{coq_eval} -Reset Initial. -Variable A:Set. -Variables a b:A. -Variable f:A->A. -Variable g:A->A->A. -\end{coq_eval} - -\begin{coq_example} -Theorem T: - a=(f a) -> (g b (f a))=(f (f a)) -> (g a b)=(f (g b a)) -> (g a b)=a. -intros. -congruence. -\end{coq_example} - -\begin{coq_eval} -Reset Initial. -Variable A:Set. -Variables a c d:A. -Variable f:A->A*A. -\end{coq_eval} - -\begin{coq_example} -Theorem inj : f = pair a -> Some (f c) = Some (f d) -> c=d. -intros. -congruence. -\end{coq_example} - -\begin{Variants} - \item {\tt congruence {\sl n}}\\ - Tries to add at most {\tt \sl n} instances of hypotheses stating quantified equalities to the problem in order to solve it. A bigger value of {\tt \sl n} does not make success slower, only failure. You might consider adding some lemmas as hypotheses using {\tt assert} in order for congruence to use them. - -\end{Variants} - -\begin{Variants} -\item {\tt congruence with \term$_1$ \dots\ \term$_n$}\\ - Adds {\tt \term$_1$ \dots\ \term$_n$} to the pool of terms used by - {\tt congruence}. This helps in case you have partially applied - constructors in your goal. -\end{Variants} - -\begin{ErrMsgs} - \item \errindex{I don't know how to handle dependent equality} \\ - The decision procedure managed to find a proof of the goal or of - a discriminable equality but this proof couldn't be built in {\Coq} - because of dependently-typed functions. - \item \errindex{I couldn't solve goal} \\ - The decision procedure didn't find any way to solve the goal. - \item \errindex{Goal is solvable by congruence but some arguments are missing. Try "congruence with \dots", replacing metavariables by arbitrary terms.} \\ - The decision procedure could solve the goal with the provision - that additional arguments are supplied for some partially applied - constructors. Any term of an appropriate type will allow the - tactic to successfully solve the goal. Those additional arguments - can be given to {\tt congruence} by filling in the holes in the - terms given in the error message, using the {\tt with} variant - described above. -\end{ErrMsgs} - -\subsection{\tt omega -\tacindex{omega} -\label{omega}} - -The tactic \texttt{omega}, due to Pierre Cr{\'e}gut, -is an automatic decision procedure for Presburger -arithmetic. It solves quantifier-free -formulas built with \verb|~|, \verb|\/|, \verb|/\|, -\verb|->| on top of equalities, inequalities and disequalities on -both the type \texttt{nat} of natural numbers and \texttt{Z} of binary -integers. This tactic must be loaded by the command \texttt{Require Import - Omega}. See the additional documentation about \texttt{omega} -(see Chapter~\ref{OmegaChapter}). - -\subsection{{\tt ring} and {\tt ring\_simplify \term$_1$ \dots\ \term$_n$} -\tacindex{ring} -\tacindex{ring\_simplify} -\comindex{Add Ring}} - -The {\tt ring} tactic solves equations upon polynomial expressions of -a ring (or semi-ring) structure. It proceeds by normalizing both hand -sides of the equation (w.r.t. associativity, commutativity and -distributivity, constant propagation) and comparing syntactically the -results. - -{\tt ring\_simplify} applies the normalization procedure described -above to the terms given. The tactic then replaces all occurrences of -the terms given in the conclusion of the goal by their normal -forms. If no term is given, then the conclusion should be an equation -and both hand sides are normalized. - -See Chapter~\ref{ring} for more information on the tactic and how to -declare new ring structures. - -\subsection{{\tt field}, {\tt field\_simplify \term$_1$\dots\ \term$_n$} - and {\tt field\_simplify\_eq} -\tacindex{field} -\tacindex{field\_simplify} -\tacindex{field\_simplify\_eq} -\comindex{Add Field}} - -The {\tt field} tactic is built on the same ideas as {\tt ring}: this -is a reflexive tactic that solves or simplifies equations in a field -structure. The main idea is to reduce a field expression (which is an -extension of ring expressions with the inverse and division -operations) to a fraction made of two polynomial expressions. - -Tactic {\tt field} is used to solve subgoals, whereas {\tt - field\_simplify \term$_1$\dots\term$_n$} replaces the provided terms -by their reduced fraction. {\tt field\_simplify\_eq} applies when the -conclusion is an equation: it simplifies both hand sides and multiplies -so as to cancel denominators. So it produces an equation without -division nor inverse. - -All of these 3 tactics may generate a subgoal in order to prove that -denominators are different from zero. - -See Chapter~\ref{ring} for more information on the tactic and how to -declare new field structures. - -\Example -\begin{coq_example*} -Require Import Reals. -Goal forall x y:R, - (x * y > 0)%R -> - (x * (1 / x + x / (x + y)))%R = - ((- 1 / y) * y * (- x * (x / (x + y)) - 1))%R. -\end{coq_example*} - -\begin{coq_example} -intros; field. -\end{coq_example} - -\begin{coq_eval} -Reset Initial. -\end{coq_eval} - -\SeeAlso file {\tt plugins/setoid\_ring/RealField.v} for an example of instantiation,\\ -\phantom{\SeeAlso}theory {\tt theories/Reals} for many examples of use of {\tt -field}. - -\subsection{\tt fourier -\tacindex{fourier}} - -This tactic written by Lo{\"\i}c Pottier solves linear inequalities on -real numbers using Fourier's method~\cite{Fourier}. This tactic must -be loaded by {\tt Require Import Fourier}. - -\Example -\begin{coq_example*} -Require Import Reals. -Require Import Fourier. -Goal forall x y:R, (x < y)%R -> (y + 1 >= x - 1)%R. -\end{coq_example*} - -\begin{coq_example} -intros; fourier. -\end{coq_example} - -\begin{coq_eval} -Reset Initial. -\end{coq_eval} - -\subsection{\tt autorewrite with \ident$_1$ \dots \ident$_n$. -\label{tactic:autorewrite} -\tacindex{autorewrite}} - -This tactic \footnote{The behavior of this tactic has much changed compared to -the versions available in the previous distributions (V6). This may cause -significant changes in your theories to obtain the same result. As a drawback -of the re-engineering of the code, this tactic has also been completely revised -to get a very compact and readable version.} carries out rewritings according -the rewriting rule bases {\tt \ident$_1$ \dots \ident$_n$}. - -Each rewriting rule of a base \ident$_i$ is applied to the main subgoal until -it fails. Once all the rules have been processed, if the main subgoal has -progressed (e.g., if it is distinct from the initial main goal) then the rules -of this base are processed again. If the main subgoal has not progressed then -the next base is processed. For the bases, the behavior is exactly similar to -the processing of the rewriting rules. - -The rewriting rule bases are built with the {\tt Hint~Rewrite} vernacular -command. - -\Warning{} This tactic may loop if you build non terminating rewriting systems. - -\begin{Variant} -\item {\tt autorewrite with \ident$_1$ \dots \ident$_n$ using \tac}\\ -Performs, in the same way, all the rewritings of the bases {\tt \ident$_1$ $...$ -\ident$_n$} applying {\tt \tac} to the main subgoal after each rewriting step. - -\item \texttt{autorewrite with {\ident$_1$} \dots \ident$_n$ in {\qualid}} - - Performs all the rewritings in hypothesis {\qualid}. -\item \texttt{autorewrite with {\ident$_1$} \dots \ident$_n$ in {\qualid} using \tac} - - Performs all the rewritings in hypothesis {\qualid} applying {\tt - \tac} to the main subgoal after each rewriting step. - -\item \texttt{autorewrite with {\ident$_1$} \dots \ident$_n$ in \textit{clause}} - Performs all the rewritings in the clause \textit{clause}. \\ - The \textit{clause} argument must not contain any \texttt{type of} nor \texttt{value of}. - -\end{Variant} - -\SeeAlso Section~\ref{HintRewrite} for feeding the database of lemmas used by {\tt autorewrite}. - -\SeeAlso Section~\ref{autorewrite-example} for examples showing the use of -this tactic. - -% En attente d'un moyen de valoriser les fichiers de demos -%\SeeAlso file \texttt{contrib/Rocq/DEMOS/Demo\_AutoRewrite.v} - -\section{Controlling automation} - -\subsection{The hints databases for {\tt auto} and {\tt eauto} -\index{Hints databases} -\label{Hints-databases} -\comindex{Hint}} - -The hints for \texttt{auto} and \texttt{eauto} are stored in -databases. Each database maps head symbols to a list of hints. One can -use the command \texttt{Print Hint \ident} to display the hints -associated to the head symbol \ident{} (see \ref{PrintHint}). Each -hint has a cost that is an nonnegative integer, and an optional pattern. -The hints with lower cost are tried first. A hint is tried by -\texttt{auto} when the conclusion of the current goal -matches its pattern or when it has no pattern. - -\subsubsection*{Creating Hint databases - \label{CreateHintDb}\comindex{CreateHintDb}} - -One can optionally declare a hint database using the command -\texttt{Create HintDb}. If a hint is added to an unknown database, it -will be automatically created. - -\medskip -\texttt{Create HintDb} {\ident} [\texttt{discriminated}] -\medskip - -This command creates a new database named \ident. -The database is implemented by a Discrimination Tree (DT) that serves as -an index of all the lemmas. The DT can use transparency information to decide -if a constant should be indexed or not (c.f. \ref{HintTransparency}), -making the retrieval more efficient. -The legacy implementation (the default one for new databases) uses the -DT only on goals without existentials (i.e., auto goals), for non-Immediate -hints and do not make use of transparency hints, putting more work on the -unification that is run after retrieval (it keeps a list of the lemmas -in case the DT is not used). The new implementation enabled by -the {\tt discriminated} option makes use of DTs in all cases and takes -transparency information into account. However, the order in which hints -are retrieved from the DT may differ from the order in which they were -inserted, making this implementation observationaly different from the -legacy one. - -\begin{Variants} -\item\texttt{Local Hint} \textsl{hint\_definition} \texttt{:} - \ident$_1$ \ldots\ \ident$_n$ - - This is used to declare a hint database that must not be exported to the other - modules that require and import the current module. Inside a - section, the option {\tt Local} is useless since hints do not - survive anyway to the closure of sections. - -\end{Variants} - -The general -command to add a hint to some database \ident$_1$, \dots, \ident$_n$ is: -\begin{tabbing} - \texttt{Hint} \textsl{hint\_definition} \texttt{:} \ident$_1$ \ldots\ \ident$_n$ -\end{tabbing} -where {\sl hint\_definition} is one of the following expressions: - -\begin{itemize} -\item \texttt{Resolve} {\term} - \comindex{Hint Resolve} - - This command adds {\tt apply {\term}} to the hint list - with the head symbol of the type of \term. The cost of that hint is - the number of subgoals generated by {\tt apply {\term}}. - - In case the inferred type of \term\ does not start with a product the - tactic added in the hint list is {\tt exact {\term}}. In case this - type can be reduced to a type starting with a product, the tactic {\tt - apply {\term}} is also stored in the hints list. - - If the inferred type of \term\ contains a dependent - quantification on a predicate, it is added to the hint list of {\tt - eapply} instead of the hint list of {\tt apply}. In this case, a - warning is printed since the hint is only used by the tactic {\tt - eauto} (see \ref{eauto}). A typical example of a hint that is used - only by \texttt{eauto} is a transitivity lemma. - - \begin{ErrMsgs} - \item \errindex{Bound head variable} - - The head symbol of the type of {\term} is a bound variable such - that this tactic cannot be associated to a constant. - - \item \term\ \errindex{cannot be used as a hint} - - The type of \term\ contains products over variables which do not - appear in the conclusion. A typical example is a transitivity axiom. - In that case the {\tt apply} tactic fails, and thus is useless. - - \end{ErrMsgs} - - \begin{Variants} - - \item \texttt{Resolve} {\term$_1$} \dots {\term$_m$} - - Adds each \texttt{Resolve} {\term$_i$}. - - \end{Variants} - -\item \texttt{Immediate {\term}} -\comindex{Hint Immediate} - - This command adds {\tt apply {\term}; trivial} to the hint list - associated with the head symbol of the type of {\ident} in the given - database. This tactic will fail if all the subgoals generated by - {\tt apply {\term}} are not solved immediately by the {\tt trivial} - tactic (which only tries tactics with cost $0$). - - This command is useful for theorems such as the symmetry of equality - or $n+1=m+1 \to n=m$ that we may like to introduce with a - limited use in order to avoid useless proof-search. - - The cost of this tactic (which never generates subgoals) is always 1, - so that it is not used by {\tt trivial} itself. - - \begin{ErrMsgs} - - \item \errindex{Bound head variable} - - \item \term\ \errindex{cannot be used as a hint} - - \end{ErrMsgs} - - \begin{Variants} - - \item \texttt{Immediate} {\term$_1$} \dots {\term$_m$} - - Adds each \texttt{Immediate} {\term$_i$}. - - \end{Variants} - -\item \texttt{Constructors} {\ident} -\comindex{Hint Constructors} - - If {\ident} is an inductive type, this command adds all its - constructors as hints of type \texttt{Resolve}. Then, when the - conclusion of current goal has the form \texttt{({\ident} \dots)}, - \texttt{auto} will try to apply each constructor. - - \begin{ErrMsgs} - - \item {\ident} \errindex{is not an inductive type} - - \item {\ident} \errindex{not declared} - - \end{ErrMsgs} - - \begin{Variants} - - \item \texttt{Constructors} {\ident$_1$} \dots {\ident$_m$} - - Adds each \texttt{Constructors} {\ident$_i$}. - - \end{Variants} - -\item \texttt{Unfold} {\qualid} -\comindex{Hint Unfold} - - This adds the tactic {\tt unfold {\qualid}} to the hint list that - will only be used when the head constant of the goal is \ident. Its - cost is 4. - - \begin{Variants} - - \item \texttt{Unfold} {\ident$_1$} \dots {\ident$_m$} - - Adds each \texttt{Unfold} {\ident$_i$}. - - \end{Variants} - -\item \texttt{Transparent}, \texttt{Opaque} {\qualid} -\label{HintTransparency} -\comindex{Hint Transparent} -\comindex{Hint Opaque} - - This adds a transparency hint to the database, making {\tt {\qualid}} - a transparent or opaque constant during resolution. This information - is used during unification of the goal with any lemma in the database - and inside the discrimination network to relax or constrain it in the - case of \texttt{discriminated} databases. - - \begin{Variants} - - \item \texttt{Transparent}, \texttt{Opaque} {\ident$_1$} \dots {\ident$_m$} - - Declares each {\ident$_i$} as a transparent or opaque constant. - - \end{Variants} - -\item \texttt{Extern \num\ [\pattern]\ => }\textsl{tactic} -\comindex{Hint Extern} - - This hint type is to extend \texttt{auto} with tactics other than - \texttt{apply} and \texttt{unfold}. For that, we must specify a - cost, an optional pattern and a tactic to execute. Here is an example: - -\begin{quotation} -\begin{verbatim} -Hint Extern 4 (~(_ = _)) => discriminate. -\end{verbatim} -\end{quotation} - - Now, when the head of the goal is a disequality, \texttt{auto} will - try \texttt{discriminate} if it does not manage to solve the goal - with hints with a cost less than 4. - - One can even use some sub-patterns of the pattern in the tactic - script. A sub-pattern is a question mark followed by an ident, like - \texttt{?X1} or \texttt{?X2}. Here is an example: - -% Require EqDecide. -\begin{coq_example*} -Require Import List. -\end{coq_example*} -\begin{coq_example} -Hint Extern 5 ({?X1 = ?X2} + {?X1 <> ?X2}) => - generalize X1, X2; decide equality : eqdec. -Goal -forall a b:list (nat * nat), {a = b} + {a <> b}. -info auto with eqdec. -\end{coq_example} -\begin{coq_eval} -Abort. -\end{coq_eval} - -\end{itemize} - -\Rem One can use an \texttt{Extern} hint with no pattern to do -pattern-matching on hypotheses using \texttt{match goal with} inside -the tactic. - -\begin{Variants} -\item \texttt{Hint} \textsl{hint\_definition} - - No database name is given: the hint is registered in the {\tt core} - database. - -\item\texttt{Hint Local} \textsl{hint\_definition} \texttt{:} - \ident$_1$ \ldots\ \ident$_n$ - - This is used to declare hints that must not be exported to the other - modules that require and import the current module. Inside a - section, the option {\tt Local} is useless since hints do not - survive anyway to the closure of sections. - -\item\texttt{Hint Local} \textsl{hint\_definition} - - Idem for the {\tt core} database. - -\end{Variants} - -% There are shortcuts that allow to define several goal at once: - -% \begin{itemize} -% \item \comindex{Hints Resolve}\texttt{Hints Resolve \ident$_1$ \dots\ \ident$_n$ : \ident.}\\ -% This command is a shortcut for the following ones: -% \begin{quotation} -% \noindent\texttt{Hint \ident$_1$ : \ident\ := Resolve \ident$_1$}\\ -% \dots\\ -% \texttt{Hint \ident$_1$ : \ident := Resolve \ident$_1$} -% \end{quotation} -% Notice that the hint name is the same that the theorem given as -% hint. -% \item \comindex{Hints Immediate}\texttt{Hints Immediate \ident$_1$ \dots\ \ident$_n$ : \ident.}\\ -% \item \comindex{Hints Unfold}\texttt{Hints Unfold \qualid$_1$ \dots\ \qualid$_n$ : \ident.}\\ -% \end{itemize} - -%\begin{Warnings} -% \item \texttt{Overriding hint named \dots\ in database \dots} -%\end{Warnings} - - - -\subsection{Hint databases defined in the \Coq\ standard library} - -Several hint databases are defined in the \Coq\ standard library. The -actual content of a database is the collection of the hints declared -to belong to this database in each of the various modules currently -loaded. Especially, requiring new modules potentially extend a -database. At {\Coq} startup, only the {\tt core} and {\tt v62} -databases are non empty and can be used. - -\begin{description} - -\item[\tt core] This special database is automatically used by - \texttt{auto}. It contains only basic lemmas about negation, - conjunction, and so on from. Most of the hints in this database come - from the \texttt{Init} and \texttt{Logic} directories. - -\item[\tt arith] This database contains all lemmas about Peano's - arithmetic proved in the directories \texttt{Init} and - \texttt{Arith} - -\item[\tt zarith] contains lemmas about binary signed integers from - the directories \texttt{theories/ZArith}. When required, the module - {\tt Omega} also extends the database {\tt zarith} with a high-cost - hint that calls {\tt omega} on equations and inequalities in {\tt - nat} or {\tt Z}. - -\item[\tt bool] contains lemmas about booleans, mostly from directory - \texttt{theories/Bool}. - -\item[\tt datatypes] is for lemmas about lists, streams and so on that - are mainly proved in the \texttt{Lists} subdirectory. - -\item[\tt sets] contains lemmas about sets and relations from the - directories \texttt{Sets} and \texttt{Relations}. - -\item[\tt typeclass\_instances] contains all the type class instances - declared in the environment, including those used for \texttt{setoid\_rewrite}, - from the \texttt{Classes} directory. -\end{description} - -There is also a special database called {\tt v62}. It collects all -hints that were declared in the versions of {\Coq} prior to version -6.2.4 when the databases {\tt core}, {\tt arith}, and so on were -introduced. The purpose of the database {\tt v62} is to ensure -compatibility with further versions of {\Coq} for developments done in -versions prior to 6.2.4 ({\tt auto} being replaced by {\tt auto with v62}). -The database {\tt v62} is intended not to be extended (!). It is not -included in the hint databases list used in the {\tt auto with *} tactic. - -Furthermore, you are advised not to put your own hints in the -{\tt core} database, but use one or several databases specific to your -development. - -\subsection{\tt Print Hint -\label{PrintHint} -\comindex{Print Hint}} - -This command displays all hints that apply to the current goal. It -fails if no proof is being edited, while the two variants can be used at -every moment. - -\begin{Variants} - -\item {\tt Print Hint {\ident} } - - This command displays only tactics associated with \ident\ in the - hints list. This is independent of the goal being edited, so this - command will not fail if no goal is being edited. - -\item {\tt Print Hint *} - - This command displays all declared hints. - -\item {\tt Print HintDb {\ident} } -\label{PrintHintDb} -\comindex{Print HintDb} - - This command displays all hints from database \ident. - -\end{Variants} - -\subsection{\tt Hint Rewrite \term$_1$ \dots \term$_n$ : \ident -\label{HintRewrite} -\comindex{Hint Rewrite}} - -This vernacular command adds the terms {\tt \term$_1$ \dots \term$_n$} -(their types must be equalities) in the rewriting base {\tt \ident} -with the default orientation (left to right). Notice that the -rewriting bases are distinct from the {\tt auto} hint bases and that -{\tt auto} does not take them into account. - -This command is synchronous with the section mechanism (see \ref{Section}): -when closing a section, all aliases created by \texttt{Hint Rewrite} in that -section are lost. Conversely, when loading a module, all \texttt{Hint Rewrite} -declarations at the global level of that module are loaded. - -\begin{Variants} -\item {\tt Hint Rewrite -> \term$_1$ \dots \term$_n$ : \ident}\\ -This is strictly equivalent to the command above (we only make explicit the -orientation which otherwise defaults to {\tt ->}). - -\item {\tt Hint Rewrite <- \term$_1$ \dots \term$_n$ : \ident}\\ -Adds the rewriting rules {\tt \term$_1$ \dots \term$_n$} with a right-to-left -orientation in the base {\tt \ident}. - -\item {\tt Hint Rewrite \term$_1$ \dots \term$_n$ using {\tac} : {\ident}}\\ -When the rewriting rules {\tt \term$_1$ \dots \term$_n$} in {\tt \ident} will -be used, the tactic {\tt \tac} will be applied to the generated subgoals, the -main subgoal excluded. - -%% \item -%% {\tt Hint Rewrite [ \term$_1$ \dots \term$_n$ ] in \ident}\\ -%% {\tt Hint Rewrite [ \term$_1$ \dots \term$_n$ ] in {\ident} using {\tac}}\\ -%% These are deprecated syntactic variants for -%% {\tt Hint Rewrite \term$_1$ \dots \term$_n$ : \ident} and -%% {\tt Hint Rewrite \term$_1$ \dots \term$_n$ using {\tac} : {\ident}}. - -\item \texttt{Print Rewrite HintDb {\ident}} - - This command displays all rewrite hints contained in {\ident}. - -\end{Variants} - -\subsection{Hints and sections -\label{Hint-and-Section}} - -Hints provided by the \texttt{Hint} commands are erased when closing a -section. Conversely, all hints of a module \texttt{A} that are not -defined inside a section (and not defined with option {\tt Local}) become -available when the module {\tt A} is imported (using -e.g. \texttt{Require Import A.}). - -\subsection{Setting implicit automation tactics} - -\subsubsection[\tt Proof with {\tac}.]{\tt Proof with {\tac}.\label{ProofWith} -\comindex{Proof with}} - - This command may be used to start a proof. It defines a default - tactic to be used each time a tactic command {\tac$_1$} is ended by - ``\verb#...#''. In this case the tactic command typed by the user is - equivalent to \tac$_1$;{\tac}. - -\SeeAlso {\tt Proof.} in Section~\ref{BeginProof}. - -\subsubsection[\tt Declare Implicit Tactic {\tac}.]{\tt Declare Implicit Tactic {\tac}.\comindex{Declare Implicit Tactic}} - -This command declares a tactic to be used to solve implicit arguments -that {\Coq} does not know how to solve by unification. It is used -every time the term argument of a tactic has one of its holes not -fully resolved. - -Here is an example: - -\begin{coq_example} -Parameter quo : nat -> forall n:nat, n<>0 -> nat. -Notation "x // y" := (quo x y _) (at level 40). - -Declare Implicit Tactic assumption. -Goal forall n m, m<>0 -> { q:nat & { r | q * m + r = n } }. -intros. -exists (n // m). -\end{coq_example} - -The tactic {\tt exists (n // m)} did not fail. The hole was solved by -{\tt assumption} so that it behaved as {\tt exists (quo n m H)}. - -\section{Generation of induction principles with {\tt Scheme} -\label{Scheme} -\index{Schemes} -\comindex{Scheme}} - -The {\tt Scheme} command is a high-level tool for generating -automatically (possibly mutual) induction principles for given types -and sorts. Its syntax follows the schema: -\begin{quote} -{\tt Scheme {\ident$_1$} := Induction for \ident'$_1$ Sort {\sort$_1$} \\ - with\\ - \mbox{}\hspace{0.1cm} \dots\\ - with {\ident$_m$} := Induction for {\ident'$_m$} Sort - {\sort$_m$}} -\end{quote} -where \ident'$_1$ \dots\ \ident'$_m$ are different inductive type -identifiers belonging to the same package of mutual inductive -definitions. This command generates {\ident$_1$}\dots{} {\ident$_m$} -to be mutually recursive definitions. Each term {\ident$_i$} proves a -general principle of mutual induction for objects in type {\term$_i$}. - -\begin{Variants} -\item {\tt Scheme {\ident$_1$} := Minimality for \ident'$_1$ Sort {\sort$_1$} \\ - with\\ - \mbox{}\hspace{0.1cm} \dots\ \\ - with {\ident$_m$} := Minimality for {\ident'$_m$} Sort - {\sort$_m$}} - - Same as before but defines a non-dependent elimination principle more - natural in case of inductively defined relations. - -\item {\tt Scheme Equality for \ident$_1$\comindex{Scheme Equality}} - - Tries to generate a boolean equality and a proof of the - decidability of the usual equality. - -\item {\tt Scheme Induction for \ident$_1$ Sort {\sort$_1$} \\ - with\\ - \mbox{}\hspace{0.1cm} \dots\\ - with Induction for {\ident$_m$} Sort - {\sort$_m$}} - - If you do not provide the name of the schemes, they will be automatically - computed from the sorts involved (works also with Minimality). - -\end{Variants} - -\SeeAlso Section~\ref{Scheme-examples} - -\subsection{Automatic declaration of schemes} -\comindex{Set Equality Schemes} -\comindex{Set Elimination Schemes} -It is possible to deactivate the automatic declaration of the induction - principles when defining a new inductive type with the - {\tt Unset Elimination Schemes} command. It may be -reactivated at any time with {\tt Set Elimination Schemes}. -\\ - -You can also activate the automatic declaration of those boolean equalities -(see the second variant of {\tt Scheme}) with the {\tt Set Equality Schemes} - command. However you have to be careful with this option since -\Coq~ may now reject well-defined inductive types because it cannot compute -a boolean equality for them. - -\subsection{\tt Combined Scheme\label{CombinedScheme} -\comindex{Combined Scheme}} -The {\tt Combined Scheme} command is a tool for combining -induction principles generated by the {\tt Scheme} command. -Its syntax follows the schema : - -\noindent -{\tt Combined Scheme {\ident$_0$} from {\ident$_1$}, .., {\ident$_n$}}\\ -\ident$_1$ \ldots \ident$_n$ are different inductive principles that must belong to -the same package of mutual inductive principle definitions. This command -generates {\ident$_0$} to be the conjunction of the principles: it is -built from the common premises of the principles and concluded by the -conjunction of their conclusions. - -\SeeAlso Section~\ref{CombinedScheme-examples} - -\section{Generation of induction principles with {\tt Functional Scheme} -\label{FunScheme} -\comindex{Functional Scheme}} - -The {\tt Functional Scheme} command is a high-level experimental -tool for generating automatically induction principles -corresponding to (possibly mutually recursive) functions. Its -syntax follows the schema: -\begin{quote} -{\tt Functional Scheme {\ident$_1$} := Induction for \ident'$_1$ Sort {\sort$_1$} \\ - with\\ - \mbox{}\hspace{0.1cm} \dots\ \\ - with {\ident$_m$} := Induction for {\ident'$_m$} Sort - {\sort$_m$}} -\end{quote} -where \ident'$_1$ \dots\ \ident'$_m$ are different mutually defined function -names (they must be in the same order as when they were defined). -This command generates the induction principles -\ident$_1$\dots\ident$_m$, following the recursive structure and case -analyses of the functions \ident'$_1$ \dots\ \ident'$_m$. - - -\paragraph{\texttt{Functional Scheme}} -There is a difference between obtaining an induction scheme by using -\texttt{Functional Scheme} on a function defined by \texttt{Function} -or not. Indeed \texttt{Function} generally produces smaller -principles, closer to the definition written by the user. - - -\SeeAlso Section~\ref{FunScheme-examples} - - -\section{Simple tactic macros -\index{Tactic macros} -\comindex{Tactic Definition} -\label{TacticDefinition}} - -A simple example has more value than a long explanation: - -\begin{coq_example} -Ltac Solve := simpl; intros; auto. -Ltac ElimBoolRewrite b H1 H2 := - elim b; [ intros; rewrite H1; eauto | intros; rewrite H2; eauto ]. -\end{coq_example} - -The tactics macros are synchronous with the \Coq\ section mechanism: -a tactic definition is deleted from the current environment -when you close the section (see also \ref{Section}) -where it was defined. If you want that a -tactic macro defined in a module is usable in the modules that -require it, you should put it outside of any section. - -Chapter~\ref{TacticLanguage} gives examples of more complex -user-defined tactics. - - -% $Id: RefMan-tac.tex 14762 2011-12-04 20:48:36Z herbelin $ - -%%% Local Variables: -%%% mode: latex -%%% TeX-master: "Reference-Manual" -%%% TeX-master: "Reference-Manual" -%%% End: diff --git a/doc/refman/RefMan-tacex.tex b/doc/refman/RefMan-tacex.tex deleted file mode 100644 index 8330a434..00000000 --- a/doc/refman/RefMan-tacex.tex +++ /dev/null @@ -1,1505 +0,0 @@ -\chapter[Detailed examples of tactics]{Detailed examples of tactics\label{Tactics-examples}} - -This chapter presents detailed examples of certain tactics, to -illustrate their behavior. - -\section[\tt refine]{\tt refine\tacindex{refine} -\label{refine-example}} - -This tactic applies to any goal. It behaves like {\tt exact} with a -big difference : the user can leave some holes (denoted by \texttt{\_} or -{\tt (\_:}{\it type}{\tt )}) in the term. -{\tt refine} will generate as many -subgoals as they are holes in the term. The type of holes must be -either synthesized by the system or declared by an -explicit cast like \verb|(\_:nat->Prop)|. This low-level -tactic can be useful to advanced users. - -%\firstexample -\Example - -\begin{coq_example*} -Inductive Option : Set := - | Fail : Option - | Ok : bool -> Option. -\end{coq_example} -\begin{coq_example} -Definition get : forall x:Option, x <> Fail -> bool. -refine - (fun x:Option => - match x return x <> Fail -> bool with - | Fail => _ - | Ok b => fun _ => b - end). -intros; absurd (Fail = Fail); trivial. -\end{coq_example} -\begin{coq_example*} -Defined. -\end{coq_example*} - -% \example{Using Refine to build a poor-man's ``Cases'' tactic} - -% \texttt{Refine} is actually the only way for the user to do -% a proof with the same structure as a {\tt Cases} definition. Actually, -% the tactics \texttt{case} (see \ref{case}) and \texttt{Elim} (see -% \ref{elim}) only allow one step of elementary induction. - -% \begin{coq_example*} -% Require Bool. -% Require Arith. -% \end{coq_example*} -% %\begin{coq_eval} -% %Abort. -% %\end{coq_eval} -% \begin{coq_example} -% Definition one_two_or_five := [x:nat] -% Cases x of -% (1) => true -% | (2) => true -% | (5) => true -% | _ => false -% end. -% Goal (x:nat)(Is_true (one_two_or_five x)) -> x=(1)\/x=(2)\/x=(5). -% \end{coq_example} - -% A traditional script would be the following: - -% \begin{coq_example*} -% Destruct x. -% Tauto. -% Destruct n. -% Auto. -% Destruct n0. -% Auto. -% Destruct n1. -% Tauto. -% Destruct n2. -% Tauto. -% Destruct n3. -% Auto. -% Intros; Inversion H. -% \end{coq_example*} - -% With the tactic \texttt{Refine}, it becomes quite shorter: - -% \begin{coq_example*} -% Restart. -% \end{coq_example*} -% \begin{coq_example} -% Refine [x:nat] -% <[y:nat](Is_true (one_two_or_five y))->(y=(1)\/y=(2)\/y=(5))> -% Cases x of -% (1) => [H]? -% | (2) => [H]? -% | (5) => [H]? -% | n => [H](False_ind ? H) -% end; Auto. -% \end{coq_example} -% \begin{coq_eval} -% Abort. -% \end{coq_eval} - -\section[\tt eapply]{\tt eapply\tacindex{eapply} -\label{eapply-example}} -\Example -Assume we have a relation on {\tt nat} which is transitive: - -\begin{coq_example*} -Variable R : nat -> nat -> Prop. -Hypothesis Rtrans : forall x y z:nat, R x y -> R y z -> R x z. -Variables n m p : nat. -Hypothesis Rnm : R n m. -Hypothesis Rmp : R m p. -\end{coq_example*} - -Consider the goal {\tt (R n p)} provable using the transitivity of -{\tt R}: - -\begin{coq_example*} -Goal R n p. -\end{coq_example*} - -The direct application of {\tt Rtrans} with {\tt apply} fails because -no value for {\tt y} in {\tt Rtrans} is found by {\tt apply}: - -\begin{coq_eval} -Set Printing Depth 50. -(********** The following is not correct and should produce **********) -(**** Error: generated subgoal (R n ?17) has metavariables in it *****) -\end{coq_eval} -\begin{coq_example} -apply Rtrans. -\end{coq_example} - -A solution is to rather apply {\tt (Rtrans n m p)}. - -\begin{coq_example} -apply (Rtrans n m p). -\end{coq_example} - -\begin{coq_eval} -Undo. -\end{coq_eval} - -More elegantly, {\tt apply Rtrans with (y:=m)} allows to only mention -the unknown {\tt m}: - -\begin{coq_example} - - apply Rtrans with (y := m). -\end{coq_example} - -\begin{coq_eval} -Undo. -\end{coq_eval} - -Another solution is to mention the proof of {\tt (R x y)} in {\tt -Rtrans}... - -\begin{coq_example} - - apply Rtrans with (1 := Rnm). -\end{coq_example} - -\begin{coq_eval} -Undo. -\end{coq_eval} - -... or the proof of {\tt (R y z)}: - -\begin{coq_example} - - apply Rtrans with (2 := Rmp). -\end{coq_example} - -\begin{coq_eval} -Undo. -\end{coq_eval} - -On the opposite, one can use {\tt eapply} which postpone the problem -of finding {\tt m}. Then one can apply the hypotheses {\tt Rnm} and {\tt -Rmp}. This instantiates the existential variable and completes the proof. - -\begin{coq_example} -eapply Rtrans. -apply Rnm. -apply Rmp. -\end{coq_example} - -\begin{coq_eval} -Reset R. -\end{coq_eval} - -\section[{\tt Scheme}]{{\tt Scheme}\comindex{Scheme} -\label{Scheme-examples}} - -\firstexample -\example{Induction scheme for \texttt{tree} and \texttt{forest}} - -The definition of principle of mutual induction for {\tt tree} and -{\tt forest} over the sort {\tt Set} is defined by the command: - -\begin{coq_eval} -Reset Initial. -Variables A B : - Set. -\end{coq_eval} - -\begin{coq_example*} -Inductive tree : Set := - node : A -> forest -> tree -with forest : Set := - | leaf : B -> forest - | cons : tree -> forest -> forest. - -Scheme tree_forest_rec := Induction for tree Sort Set - with forest_tree_rec := Induction for forest Sort Set. -\end{coq_example*} - -You may now look at the type of {\tt tree\_forest\_rec}: - -\begin{coq_example} -Check tree_forest_rec. -\end{coq_example} - -This principle involves two different predicates for {\tt trees} and -{\tt forests}; it also has three premises each one corresponding to a -constructor of one of the inductive definitions. - -The principle {\tt forest\_tree\_rec} shares exactly the same -premises, only the conclusion now refers to the property of forests. - -\begin{coq_example} -Check forest_tree_rec. -\end{coq_example} - -\example{Predicates {\tt odd} and {\tt even} on naturals} - -Let {\tt odd} and {\tt even} be inductively defined as: - -% Reset Initial. -\begin{coq_eval} -Open Scope nat_scope. -\end{coq_eval} - -\begin{coq_example*} -Inductive odd : nat -> Prop := - oddS : forall n:nat, even n -> odd (S n) -with even : nat -> Prop := - | evenO : even 0 - | evenS : forall n:nat, odd n -> even (S n). -\end{coq_example*} - -The following command generates a powerful elimination -principle: - -\begin{coq_example} -Scheme odd_even := Minimality for odd Sort Prop - with even_odd := Minimality for even Sort Prop. -\end{coq_example} - -The type of {\tt odd\_even} for instance will be: - -\begin{coq_example} -Check odd_even. -\end{coq_example} - -The type of {\tt even\_odd} shares the same premises but the -conclusion is {\tt (n:nat)(even n)->(Q n)}. - -\subsection[{\tt Combined Scheme}]{{\tt Combined Scheme}\comindex{Combined Scheme} -\label{CombinedScheme-examples}} - -We can define the induction principles for trees and forests using: -\begin{coq_example} -Scheme tree_forest_ind := Induction for tree Sort Prop - with forest_tree_ind := Induction for forest Sort Prop. -\end{coq_example} - -Then we can build the combined induction principle which gives the -conjunction of the conclusions of each individual principle: -\begin{coq_example} -Combined Scheme tree_forest_mutind from tree_forest_ind, forest_tree_ind. -\end{coq_example} - -The type of {\tt tree\_forest\_mutrec} will be: -\begin{coq_example} -Check tree_forest_mutind. -\end{coq_example} - -\section[{\tt Functional Scheme} and {\tt functional induction}]{{\tt Functional Scheme} and {\tt functional induction}\comindex{Functional Scheme}\tacindex{functional induction} -\label{FunScheme-examples}} - -\firstexample -\example{Induction scheme for \texttt{div2}} - -We define the function \texttt{div2} as follows: - -\begin{coq_eval} -Reset Initial. -\end{coq_eval} - -\begin{coq_example*} -Require Import Arith. -Fixpoint div2 (n:nat) : nat := - match n with - | O => 0 - | S O => 0 - | S (S n') => S (div2 n') - end. -\end{coq_example*} - -The definition of a principle of induction corresponding to the -recursive structure of \texttt{div2} is defined by the command: - -\begin{coq_example} -Functional Scheme div2_ind := Induction for div2 Sort Prop. -\end{coq_example} - -You may now look at the type of {\tt div2\_ind}: - -\begin{coq_example} -Check div2_ind. -\end{coq_example} - -We can now prove the following lemma using this principle: - - -\begin{coq_example*} -Lemma div2_le' : forall n:nat, div2 n <= n. -intro n. - pattern n , (div2 n). -\end{coq_example*} - - -\begin{coq_example} -apply div2_ind; intros. -\end{coq_example} - -\begin{coq_example*} -auto with arith. -auto with arith. -simpl; auto with arith. -Qed. -\end{coq_example*} - -We can use directly the \texttt{functional induction} -(\ref{FunInduction}) tactic instead of the pattern/apply trick: - -\begin{coq_example*} -Reset div2_le'. -Lemma div2_le : forall n:nat, div2 n <= n. -intro n. -\end{coq_example*} - -\begin{coq_example} -functional induction (div2 n). -\end{coq_example} - -\begin{coq_example*} -auto with arith. -auto with arith. -auto with arith. -Qed. -\end{coq_example*} - -\Rem There is a difference between obtaining an induction scheme for a -function by using \texttt{Function} (see Section~\ref{Function}) and by -using \texttt{Functional Scheme} after a normal definition using -\texttt{Fixpoint} or \texttt{Definition}. See \ref{Function} for -details. - - -\example{Induction scheme for \texttt{tree\_size}} - -\begin{coq_eval} -Reset Initial. -\end{coq_eval} - -We define trees by the following mutual inductive type: - -\begin{coq_example*} -Variable A : Set. -Inductive tree : Set := - node : A -> forest -> tree -with forest : Set := - | empty : forest - | cons : tree -> forest -> forest. -\end{coq_example*} - -We define the function \texttt{tree\_size} that computes the size -of a tree or a forest. Note that we use \texttt{Function} which -generally produces better principles. - -\begin{coq_example*} -Function tree_size (t:tree) : nat := - match t with - | node A f => S (forest_size f) - end - with forest_size (f:forest) : nat := - match f with - | empty => 0 - | cons t f' => (tree_size t + forest_size f') - end. -\end{coq_example*} - -Remark: \texttt{Function} generates itself non mutual induction -principles {\tt tree\_size\_ind} and {\tt forest\_size\_ind}: - -\begin{coq_example} -Check tree_size_ind. -\end{coq_example} - -The definition of mutual induction principles following the recursive -structure of \texttt{tree\_size} and \texttt{forest\_size} is defined -by the command: - -\begin{coq_example*} -Functional Scheme tree_size_ind2 := Induction for tree_size Sort Prop -with forest_size_ind2 := Induction for forest_size Sort Prop. -\end{coq_example*} - -You may now look at the type of {\tt tree\_size\_ind2}: - -\begin{coq_example} -Check tree_size_ind2. -\end{coq_example} - - - - -\section[{\tt inversion}]{{\tt inversion}\tacindex{inversion} -\label{inversion-examples}} - -\subsection*{Generalities about inversion} - -When working with (co)inductive predicates, we are very often faced to -some of these situations: -\begin{itemize} -\item we have an inconsistent instance of an inductive predicate in the - local context of hypotheses. Thus, the current goal can be trivially - proved by absurdity. -\item we have a hypothesis that is an instance of an inductive - predicate, and the instance has some variables whose constraints we - would like to derive. -\end{itemize} - -The inversion tactics are very useful to simplify the work in these -cases. Inversion tools can be classified in three groups: - -\begin{enumerate} -\item tactics for inverting an instance without stocking the inversion - lemma in the context; this includes the tactics - (\texttt{dependent}) \texttt{inversion} and - (\texttt{dependent}) \texttt{inversion\_clear}. -\item commands for generating and stocking in the context the inversion - lemma corresponding to an instance; this includes \texttt{Derive} - (\texttt{Dependent}) \texttt{Inversion} and \texttt{Derive} - (\texttt{Dependent}) \texttt{Inversion\_clear}. -\item tactics for inverting an instance using an already defined - inversion lemma; this includes the tactic \texttt{inversion \ldots using}. -\end{enumerate} - -As inversion proofs may be large in size, we recommend the user to -stock the lemmas whenever the same instance needs to be inverted -several times. - -\firstexample -\example{Non-dependent inversion} - -Let's consider the relation \texttt{Le} over natural numbers and the -following variables: - -\begin{coq_eval} -Reset Initial. -\end{coq_eval} - -\begin{coq_example*} -Inductive Le : nat -> nat -> Set := - | LeO : forall n:nat, Le 0 n - | LeS : forall n m:nat, Le n m -> Le (S n) (S m). -Variable P : nat -> nat -> Prop. -Variable Q : forall n m:nat, Le n m -> Prop. -\end{coq_example*} - -For example, consider the goal: - -\begin{coq_eval} -Lemma ex : forall n m:nat, Le (S n) m -> P n m. -intros. -\end{coq_eval} - -\begin{coq_example} -Show. -\end{coq_example} - -To prove the goal we may need to reason by cases on \texttt{H} and to - derive that \texttt{m} is necessarily of -the form $(S~m_0)$ for certain $m_0$ and that $(Le~n~m_0)$. -Deriving these conditions corresponds to prove that the -only possible constructor of \texttt{(Le (S n) m)} is -\texttt{LeS} and that we can invert the -\texttt{->} in the type of \texttt{LeS}. -This inversion is possible because \texttt{Le} is the smallest set closed by -the constructors \texttt{LeO} and \texttt{LeS}. - -\begin{coq_example} -inversion_clear H. -\end{coq_example} - -Note that \texttt{m} has been substituted in the goal for \texttt{(S m0)} -and that the hypothesis \texttt{(Le n m0)} has been added to the -context. - -Sometimes it is -interesting to have the equality \texttt{m=(S m0)} in the -context to use it after. In that case we can use \texttt{inversion} that -does not clear the equalities: - -\begin{coq_example*} -Undo. -\end{coq_example*} - -\begin{coq_example} -inversion H. -\end{coq_example} - -\begin{coq_eval} -Undo. -\end{coq_eval} - -\example{Dependent Inversion} - -Let us consider the following goal: - -\begin{coq_eval} -Lemma ex_dep : forall (n m:nat) (H:Le (S n) m), Q (S n) m H. -intros. -\end{coq_eval} - -\begin{coq_example} -Show. -\end{coq_example} - -As \texttt{H} occurs in the goal, we may want to reason by cases on its -structure and so, we would like inversion tactics to -substitute \texttt{H} by the corresponding term in constructor form. -Neither \texttt{Inversion} nor {\tt Inversion\_clear} make such a -substitution. -To have such a behavior we use the dependent inversion tactics: - -\begin{coq_example} -dependent inversion_clear H. -\end{coq_example} - -Note that \texttt{H} has been substituted by \texttt{(LeS n m0 l)} and -\texttt{m} by \texttt{(S m0)}. - -\example{using already defined inversion lemmas} - -\begin{coq_eval} -Abort. -\end{coq_eval} - -For example, to generate the inversion lemma for the instance -\texttt{(Le (S n) m)} and the sort \texttt{Prop} we do: - -\begin{coq_example*} -Derive Inversion_clear leminv with (forall n m:nat, Le (S n) m) Sort - Prop. -\end{coq_example*} - -\begin{coq_example} -Check leminv. -\end{coq_example} - -Then we can use the proven inversion lemma: - -\begin{coq_example} -Show. -\end{coq_example} - -\begin{coq_example} -inversion H using leminv. -\end{coq_example} - -\begin{coq_eval} -Reset Initial. -\end{coq_eval} - -\section[\tt dependent induction]{\tt dependent induction\label{dependent-induction-example}} -\def\depind{{\tt dependent induction}~} -\def\depdestr{{\tt dependent destruction}~} - -The tactics \depind and \depdestr are another solution for inverting -inductive predicate instances and potentially doing induction at the -same time. It is based on the \texttt{BasicElim} tactic of Conor McBride which -works by abstracting each argument of an inductive instance by a variable -and constraining it by equalities afterwards. This way, the usual -{\tt induction} and {\tt destruct} tactics can be applied to the -abstracted instance and after simplification of the equalities we get -the expected goals. - -The abstracting tactic is called {\tt generalize\_eqs} and it takes as -argument an hypothesis to generalize. It uses the {\tt JMeq} datatype -defined in {\tt Coq.Logic.JMeq}, hence we need to require it before. -For example, revisiting the first example of the inversion documentation above: - -\begin{coq_example*} -Require Import Coq.Logic.JMeq. -\end{coq_example*} -\begin{coq_eval} -Require Import Coq.Program.Equality. -\end{coq_eval} - -\begin{coq_eval} -Inductive Le : nat -> nat -> Set := - | LeO : forall n:nat, Le 0 n - | LeS : forall n m:nat, Le n m -> Le (S n) (S m). -Variable P : nat -> nat -> Prop. -Variable Q : forall n m:nat, Le n m -> Prop. -\end{coq_eval} - -\begin{coq_example*} -Goal forall n m:nat, Le (S n) m -> P n m. -intros n m H. -\end{coq_example*} -\begin{coq_example} -generalize_eqs H. -\end{coq_example} - -The index {\tt S n} gets abstracted by a variable here, but a -corresponding equality is added under the abstract instance so that no -information is actually lost. The goal is now almost amenable to do induction -or case analysis. One should indeed first move {\tt n} into the goal to -strengthen it before doing induction, or {\tt n} will be fixed in -the inductive hypotheses (this does not matter for case analysis). -As a rule of thumb, all the variables that appear inside constructors in -the indices of the hypothesis should be generalized. This is exactly -what the \texttt{generalize\_eqs\_vars} variant does: - -\begin{coq_eval} -Undo 1. -\end{coq_eval} -\begin{coq_example} -generalize_eqs_vars H. -induction H. -\end{coq_example} - -As the hypothesis itself did not appear in the goal, we did not need to -use an heterogeneous equality to relate the new hypothesis to the old -one (which just disappeared here). However, the tactic works just a well -in this case, e.g.: - -\begin{coq_eval} -Admitted. -\end{coq_eval} - -\begin{coq_example} -Goal forall n m (p : Le (S n) m), Q (S n) m p. -intros n m p ; generalize_eqs_vars p. -\end{coq_example} - -One drawback of this approach is that in the branches one will have to -substitute the equalities back into the instance to get the right -assumptions. Sometimes injection of constructors will also be needed to -recover the needed equalities. Also, some subgoals should be directly -solved because of inconsistent contexts arising from the constraints on -indexes. The nice thing is that we can make a tactic based on -discriminate, injection and variants of substitution to automatically -do such simplifications (which may involve the K axiom). -This is what the {\tt simplify\_dep\_elim} tactic from -{\tt Coq.Program.Equality} does. For example, we might simplify the -previous goals considerably: -% \begin{coq_eval} -% Abort. -% Goal forall n m:nat, Le (S n) m -> P n m. -% intros n m H ; generalize_eqs_vars H. -% \end{coq_eval} - -\begin{coq_example} -induction p ; simplify_dep_elim. -\end{coq_example} - -The higher-order tactic {\tt do\_depind} defined in {\tt - Coq.Program.Equality} takes a tactic and combines the -building blocks we have seen with it: generalizing by equalities -calling the given tactic with the -generalized induction hypothesis as argument and cleaning the subgoals -with respect to equalities. Its most important instantiations are -\depind and \depdestr that do induction or simply case analysis on the -generalized hypothesis. For example we can redo what we've done manually -with \depdestr: - -\begin{coq_eval} -Abort. -\end{coq_eval} -\begin{coq_example*} -Require Import Coq.Program.Equality. -Lemma ex : forall n m:nat, Le (S n) m -> P n m. -intros n m H. -\end{coq_example*} -\begin{coq_example} -dependent destruction H. -\end{coq_example} -\begin{coq_eval} -Abort. -\end{coq_eval} - -This gives essentially the same result as inversion. Now if the -destructed hypothesis actually appeared in the goal, the tactic would -still be able to invert it, contrary to {\tt dependent - inversion}. Consider the following example on vectors: - -\begin{coq_example*} -Require Import Coq.Program.Equality. -Set Implicit Arguments. -Variable A : Set. -Inductive vector : nat -> Type := -| vnil : vector 0 -| vcons : A -> forall n, vector n -> vector (S n). -Goal forall n, forall v : vector (S n), - exists v' : vector n, exists a : A, v = vcons a v'. - intros n v. -\end{coq_example*} -\begin{coq_example} - dependent destruction v. -\end{coq_example} -\begin{coq_eval} -Abort. -\end{coq_eval} - -In this case, the {\tt v} variable can be replaced in the goal by the -generalized hypothesis only when it has a type of the form {\tt vector - (S n)}, that is only in the second case of the {\tt destruct}. The -first one is dismissed because {\tt S n <> 0}. - -\subsection{A larger example} - -Let's see how the technique works with {\tt induction} on inductive -predicates on a real example. We will develop an example application to the -theory of simply-typed lambda-calculus formalized in a dependently-typed style: - -\begin{coq_example*} -Inductive type : Type := -| base : type -| arrow : type -> type -> type. -Notation " t --> t' " := (arrow t t') (at level 20, t' at next level). -Inductive ctx : Type := -| empty : ctx -| snoc : ctx -> type -> ctx. -Notation " G , tau " := (snoc G tau) (at level 20, t at next level). -Fixpoint conc (G D : ctx) : ctx := - match D with - | empty => G - | snoc D' x => snoc (conc G D') x - end. -Notation " G ; D " := (conc G D) (at level 20). -Inductive term : ctx -> type -> Type := -| ax : forall G tau, term (G, tau) tau -| weak : forall G tau, - term G tau -> forall tau', term (G, tau') tau -| abs : forall G tau tau', - term (G , tau) tau' -> term G (tau --> tau') -| app : forall G tau tau', - term G (tau --> tau') -> term G tau -> term G tau'. -\end{coq_example*} - -We have defined types and contexts which are snoc-lists of types. We -also have a {\tt conc} operation that concatenates two contexts. -The {\tt term} datatype represents in fact the possible typing -derivations of the calculus, which are isomorphic to the well-typed -terms, hence the name. A term is either an application of: -\begin{itemize} -\item the axiom rule to type a reference to the first variable in a context, -\item the weakening rule to type an object in a larger context -\item the abstraction or lambda rule to type a function -\item the application to type an application of a function to an argument -\end{itemize} - -Once we have this datatype we want to do proofs on it, like weakening: - -\begin{coq_example*} -Lemma weakening : forall G D tau, term (G ; D) tau -> - forall tau', term (G , tau' ; D) tau. -\end{coq_example*} -\begin{coq_eval} - Abort. -\end{coq_eval} - -The problem here is that we can't just use {\tt induction} on the typing -derivation because it will forget about the {\tt G ; D} constraint -appearing in the instance. A solution would be to rewrite the goal as: -\begin{coq_example*} -Lemma weakening' : forall G' tau, term G' tau -> - forall G D, (G ; D) = G' -> - forall tau', term (G, tau' ; D) tau. -\end{coq_example*} -\begin{coq_eval} - Abort. -\end{coq_eval} - -With this proper separation of the index from the instance and the right -induction loading (putting {\tt G} and {\tt D} after the inducted-on -hypothesis), the proof will go through, but it is a very tedious -process. One is also forced to make a wrapper lemma to get back the -more natural statement. The \depind tactic alleviates this trouble by -doing all of this plumbing of generalizing and substituting back automatically. -Indeed we can simply write: - -\begin{coq_example*} -Require Import Coq.Program.Tactics. -Lemma weakening : forall G D tau, term (G ; D) tau -> - forall tau', term (G , tau' ; D) tau. -Proof with simpl in * ; simpl_depind ; auto. - intros G D tau H. dependent induction H generalizing G D ; intros. -\end{coq_example*} - -This call to \depind has an additional arguments which is a list of -variables appearing in the instance that should be generalized in the -goal, so that they can vary in the induction hypotheses. By default, all -variables appearing inside constructors (except in a parameter position) -of the instantiated hypothesis will be generalized automatically but -one can always give the list explicitly. - -\begin{coq_example} - Show. -\end{coq_example} - -The {\tt simpl\_depind} tactic includes an automatic tactic that tries -to simplify equalities appearing at the beginning of induction -hypotheses, generally using trivial applications of -reflexivity. In cases where the equality is not between constructor -forms though, one must help the automation by giving -some arguments, using the {\tt specialize} tactic. - -\begin{coq_example*} -destruct D... apply weak ; apply ax. apply ax. -destruct D... -\end{coq_example*} -\begin{coq_example} -Show. -\end{coq_example} -\begin{coq_example} - specialize (IHterm G empty). -\end{coq_example} - -Then the automation can find the needed equality {\tt G = G} to narrow -the induction hypothesis further. This concludes our example. - -\begin{coq_example} - simpl_depind. -\end{coq_example} - -\SeeAlso The induction \ref{elim}, case \ref{case} and inversion \ref{inversion} tactics. - -\section[\tt autorewrite]{\tt autorewrite\label{autorewrite-example}} - -Here are two examples of {\tt autorewrite} use. The first one ({\em Ackermann -function}) shows actually a quite basic use where there is no conditional -rewriting. The second one ({\em Mac Carthy function}) involves conditional -rewritings and shows how to deal with them using the optional tactic of the -{\tt Hint~Rewrite} command. - -\firstexample -\example{Ackermann function} -%Here is a basic use of {\tt AutoRewrite} with the Ackermann function: - -\begin{coq_example*} -Reset Initial. -Require Import Arith. -Variable Ack : - nat -> nat -> nat. -Axiom Ack0 : - forall m:nat, Ack 0 m = S m. -Axiom Ack1 : forall n:nat, Ack (S n) 0 = Ack n 1. -Axiom Ack2 : forall n m:nat, Ack (S n) (S m) = Ack n (Ack (S n) m). -\end{coq_example*} - -\begin{coq_example} -Hint Rewrite Ack0 Ack1 Ack2 : base0. -Lemma ResAck0 : - Ack 3 2 = 29. -autorewrite with base0 using try reflexivity. -\end{coq_example} - -\begin{coq_eval} -Reset Initial. -\end{coq_eval} - -\example{Mac Carthy function} -%The Mac Carthy function shows a more complex case: - -\begin{coq_example*} -Require Import Omega. -Variable g : - nat -> nat -> nat. -Axiom g0 : - forall m:nat, g 0 m = m. -Axiom - g1 : - forall n m:nat, - (n > 0) -> (m > 100) -> g n m = g (pred n) (m - 10). -Axiom - g2 : - forall n m:nat, - (n > 0) -> (m <= 100) -> g n m = g (S n) (m + 11). -\end{coq_example*} - -\begin{coq_example} -Hint Rewrite g0 g1 g2 using omega : base1. -Lemma Resg0 : - g 1 110 = 100. -autorewrite with base1 using reflexivity || simpl. -\end{coq_example} - -\begin{coq_eval} -Abort. -\end{coq_eval} - -\begin{coq_example} -Lemma Resg1 : g 1 95 = 91. -autorewrite with base1 using reflexivity || simpl. -\end{coq_example} - -\begin{coq_eval} -Reset Initial. -\end{coq_eval} - -\section[\tt quote]{\tt quote\tacindex{quote} -\label{quote-examples}} - -The tactic \texttt{quote} allows to use Barendregt's so-called -2-level approach without writing any ML code. Suppose you have a -language \texttt{L} of -'abstract terms' and a type \texttt{A} of 'concrete terms' -and a function \texttt{f : L -> A}. If \texttt{L} is a simple -inductive datatype and \texttt{f} a simple fixpoint, \texttt{quote f} -will replace the head of current goal by a convertible term of the form -\texttt{(f t)}. \texttt{L} must have a constructor of type: \texttt{A - -> L}. - -Here is an example: - -\begin{coq_example} -Require Import Quote. -Parameters A B C : Prop. -Inductive formula : Type := - | f_and : formula -> formula -> formula (* binary constructor *) - | f_or : formula -> formula -> formula - | f_not : formula -> formula (* unary constructor *) - | f_true : formula (* 0-ary constructor *) - | f_const : Prop -> formula (* constructor for constants *). -Fixpoint interp_f (f: - formula) : Prop := - match f with - | f_and f1 f2 => interp_f f1 /\ interp_f f2 - | f_or f1 f2 => interp_f f1 \/ interp_f f2 - | f_not f1 => ~ interp_f f1 - | f_true => True - | f_const c => c - end. -Goal A /\ (A \/ True) /\ ~ B /\ (A <-> A). -quote interp_f. -\end{coq_example} - -The algorithm to perform this inversion is: try to match the -term with right-hand sides expression of \texttt{f}. If there is a -match, apply the corresponding left-hand side and call yourself -recursively on sub-terms. If there is no match, we are at a leaf: -return the corresponding constructor (here \texttt{f\_const}) applied -to the term. - -\begin{ErrMsgs} -\item \errindex{quote: not a simple fixpoint} \\ - Happens when \texttt{quote} is not able to perform inversion properly. -\end{ErrMsgs} - -\subsection{Introducing variables map} - -The normal use of \texttt{quote} is to make proofs by reflection: one -defines a function \texttt{simplify : formula -> formula} and proves a -theorem \texttt{simplify\_ok: (f:formula)(interp\_f (simplify f)) -> - (interp\_f f)}. Then, one can simplify formulas by doing: -\begin{verbatim} - quote interp_f. - apply simplify_ok. - compute. -\end{verbatim} -But there is a problem with leafs: in the example above one cannot -write a function that implements, for example, the logical simplifications -$A \land A \ra A$ or $A \land \lnot A \ra \texttt{False}$. This is -because the \Prop{} is impredicative. - -It is better to use that type of formulas: - -\begin{coq_eval} -Reset formula. -\end{coq_eval} -\begin{coq_example} -Inductive formula : Set := - | f_and : formula -> formula -> formula - | f_or : formula -> formula -> formula - | f_not : formula -> formula - | f_true : formula - | f_atom : index -> formula. -\end{coq_example*} - -\texttt{index} is defined in module \texttt{quote}. Equality on that -type is decidable so we are able to simplify $A \land A$ into $A$ at -the abstract level. - -When there are variables, there are bindings, and \texttt{quote} -provides also a type \texttt{(varmap A)} of bindings from -\texttt{index} to any set \texttt{A}, and a function -\texttt{varmap\_find} to search in such maps. The interpretation -function has now another argument, a variables map: - -\begin{coq_example} -Fixpoint interp_f (vm: - varmap Prop) (f:formula) {struct f} : Prop := - match f with - | f_and f1 f2 => interp_f vm f1 /\ interp_f vm f2 - | f_or f1 f2 => interp_f vm f1 \/ interp_f vm f2 - | f_not f1 => ~ interp_f vm f1 - | f_true => True - | f_atom i => varmap_find True i vm - end. -\end{coq_example} - -\noindent\texttt{quote} handles this second case properly: - -\begin{coq_example} -Goal A /\ (B \/ A) /\ (A \/ ~ B). -quote interp_f. -\end{coq_example} - -It builds \texttt{vm} and \texttt{t} such that \texttt{(f vm t)} is -convertible with the conclusion of current goal. - -\subsection{Combining variables and constants} - -One can have both variables and constants in abstracts terms; that is -the case, for example, for the \texttt{ring} tactic (chapter -\ref{ring}). Then one must provide to \texttt{quote} a list of -\emph{constructors of constants}. For example, if the list is -\texttt{[O S]} then closed natural numbers will be considered as -constants and other terms as variables. - -Example: - -\begin{coq_eval} -Reset formula. -\end{coq_eval} -\begin{coq_example*} -Inductive formula : Type := - | f_and : formula -> formula -> formula - | f_or : formula -> formula -> formula - | f_not : formula -> formula - | f_true : formula - | f_const : Prop -> formula (* constructor for constants *) - | f_atom : index -> formula. -Fixpoint interp_f - (vm: (* constructor for variables *) - varmap Prop) (f:formula) {struct f} : Prop := - match f with - | f_and f1 f2 => interp_f vm f1 /\ interp_f vm f2 - | f_or f1 f2 => interp_f vm f1 \/ interp_f vm f2 - | f_not f1 => ~ interp_f vm f1 - | f_true => True - | f_const c => c - | f_atom i => varmap_find True i vm - end. -Goal -A /\ (A \/ True) /\ ~ B /\ (C <-> C). -\end{coq_example*} - -\begin{coq_example} -quote interp_f [ A B ]. -Undo. - quote interp_f [ B C iff ]. -\end{coq_example} - -\Warning Since function inversion -is undecidable in general case, don't expect miracles from it! - -\begin{Variants} - -\item {\tt quote {\ident} in {\term} using {\tac}} - - \tac\ must be a functional tactic (starting with {\tt fun x =>}) - and will be called with the quoted version of \term\ according to - \ident. - -\item {\tt quote {\ident} [ \ident$_1$ \dots\ \ident$_n$ ] in {\term} using {\tac}} - - Same as above, but will use \ident$_1$, \dots, \ident$_n$ to - chose which subterms are constants (see above). - -\end{Variants} - -% \SeeAlso file \texttt{theories/DEMOS/DemoQuote.v} - -\SeeAlso comments of source file \texttt{plugins/quote/quote.ml} - -\SeeAlso the \texttt{ring} tactic (Chapter~\ref{ring}) - - - -\section{Using the tactical language} - -\subsection{About the cardinality of the set of natural numbers} - -A first example which shows how to use the pattern matching over the proof -contexts is the proof that natural numbers have more than two elements. The -proof of such a lemma can be done as %shown on Figure~\ref{cnatltac}. -follows: -%\begin{figure} -%\begin{centerframe} -\begin{coq_eval} -Reset Initial. -Require Import Arith. -Require Import List. -\end{coq_eval} -\begin{coq_example*} -Lemma card_nat : - ~ (exists x : nat, exists y : nat, forall z:nat, x = z \/ y = z). -Proof. -red; intros (x, (y, Hy)). -elim (Hy 0); elim (Hy 1); elim (Hy 2); intros; - match goal with - | [_:(?a = ?b),_:(?a = ?c) |- _ ] => - cut (b = c); [ discriminate | apply trans_equal with a; auto ] - end. -Qed. -\end{coq_example*} -%\end{centerframe} -%\caption{A proof on cardinality of natural numbers} -%\label{cnatltac} -%\end{figure} - -We can notice that all the (very similar) cases coming from the three -eliminations (with three distinct natural numbers) are successfully solved by -a {\tt match goal} structure and, in particular, with only one pattern (use -of non-linear matching). - -\subsection{Permutation on closed lists} - -Another more complex example is the problem of permutation on closed lists. The -aim is to show that a closed list is a permutation of another one. - -First, we define the permutation predicate as shown in table~\ref{permutpred}. - -\begin{figure} -\begin{centerframe} -\begin{coq_example*} -Section Sort. -Variable A : Set. -Inductive permut : list A -> list A -> Prop := - | permut_refl : forall l, permut l l - | permut_cons : - forall a l0 l1, permut l0 l1 -> permut (a :: l0) (a :: l1) - | permut_append : forall a l, permut (a :: l) (l ++ a :: nil) - | permut_trans : - forall l0 l1 l2, permut l0 l1 -> permut l1 l2 -> permut l0 l2. -End Sort. -\end{coq_example*} -\end{centerframe} -\caption{Definition of the permutation predicate} -\label{permutpred} -\end{figure} - -A more complex example is the problem of permutation on closed lists. -The aim is to show that a closed list is a permutation of another one. -First, we define the permutation predicate as shown on -Figure~\ref{permutpred}. - -\begin{figure} -\begin{centerframe} -\begin{coq_example} -Ltac Permut n := - match goal with - | |- (permut _ ?l ?l) => apply permut_refl - | |- (permut _ (?a :: ?l1) (?a :: ?l2)) => - let newn := eval compute in (length l1) in - (apply permut_cons; Permut newn) - | |- (permut ?A (?a :: ?l1) ?l2) => - match eval compute in n with - | 1 => fail - | _ => - let l1' := constr:(l1 ++ a :: nil) in - (apply (permut_trans A (a :: l1) l1' l2); - [ apply permut_append | compute; Permut (pred n) ]) - end - end. -Ltac PermutProve := - match goal with - | |- (permut _ ?l1 ?l2) => - match eval compute in (length l1 = length l2) with - | (?n = ?n) => Permut n - end - end. -\end{coq_example} -\end{centerframe} -\caption{Permutation tactic} -\label{permutltac} -\end{figure} - -Next, we can write naturally the tactic and the result can be seen on -Figure~\ref{permutltac}. We can notice that we use two toplevel -definitions {\tt PermutProve} and {\tt Permut}. The function to be -called is {\tt PermutProve} which computes the lengths of the two -lists and calls {\tt Permut} with the length if the two lists have the -same length. {\tt Permut} works as expected. If the two lists are -equal, it concludes. Otherwise, if the lists have identical first -elements, it applies {\tt Permut} on the tail of the lists. Finally, -if the lists have different first elements, it puts the first element -of one of the lists (here the second one which appears in the {\tt - permut} predicate) at the end if that is possible, i.e., if the new -first element has been at this place previously. To verify that all -rotations have been done for a list, we use the length of the list as -an argument for {\tt Permut} and this length is decremented for each -rotation down to, but not including, 1 because for a list of length -$n$, we can make exactly $n-1$ rotations to generate at most $n$ -distinct lists. Here, it must be noticed that we use the natural -numbers of {\Coq} for the rotation counter. On Figure~\ref{ltac}, we -can see that it is possible to use usual natural numbers but they are -only used as arguments for primitive tactics and they cannot be -handled, in particular, we cannot make computations with them. So, a -natural choice is to use {\Coq} data structures so that {\Coq} makes -the computations (reductions) by {\tt eval compute in} and we can get -the terms back by {\tt match}. - -With {\tt PermutProve}, we can now prove lemmas as -% shown on Figure~\ref{permutlem}. -follows: -%\begin{figure} -%\begin{centerframe} - -\begin{coq_example*} -Lemma permut_ex1 : - permut nat (1 :: 2 :: 3 :: nil) (3 :: 2 :: 1 :: nil). -Proof. PermutProve. Qed. -Lemma permut_ex2 : - permut nat - (0 :: 1 :: 2 :: 3 :: 4 :: 5 :: 6 :: 7 :: 8 :: 9 :: nil) - (0 :: 2 :: 4 :: 6 :: 8 :: 9 :: 7 :: 5 :: 3 :: 1 :: nil). -Proof. PermutProve. Qed. -\end{coq_example*} -%\end{centerframe} -%\caption{Examples of {\tt PermutProve} use} -%\label{permutlem} -%\end{figure} - - -\subsection{Deciding intuitionistic propositional logic} - -\begin{figure}[b] -\begin{centerframe} -\begin{coq_example} -Ltac Axioms := - match goal with - | |- True => trivial - | _:False |- _ => elimtype False; assumption - | _:?A |- ?A => auto - end. -\end{coq_example} -\end{centerframe} -\caption{Deciding intuitionistic propositions (1)} -\label{tautoltaca} -\end{figure} - - -\begin{figure} -\begin{centerframe} -\begin{coq_example} -Ltac DSimplif := - repeat - (intros; - match goal with - | id:(~ _) |- _ => red in id - | id:(_ /\ _) |- _ => - elim id; do 2 intro; clear id - | id:(_ \/ _) |- _ => - elim id; intro; clear id - | id:(?A /\ ?B -> ?C) |- _ => - cut (A -> B -> C); - [ intro | intros; apply id; split; assumption ] - | id:(?A \/ ?B -> ?C) |- _ => - cut (B -> C); - [ cut (A -> C); - [ intros; clear id - | intro; apply id; left; assumption ] - | intro; apply id; right; assumption ] - | id0:(?A -> ?B),id1:?A |- _ => - cut B; [ intro; clear id0 | apply id0; assumption ] - | |- (_ /\ _) => split - | |- (~ _) => red - end). -Ltac TautoProp := - DSimplif; - Axioms || - match goal with - | id:((?A -> ?B) -> ?C) |- _ => - cut (B -> C); - [ intro; cut (A -> B); - [ intro; cut C; - [ intro; clear id | apply id; assumption ] - | clear id ] - | intro; apply id; intro; assumption ]; TautoProp - | id:(~ ?A -> ?B) |- _ => - cut (False -> B); - [ intro; cut (A -> False); - [ intro; cut B; - [ intro; clear id | apply id; assumption ] - | clear id ] - | intro; apply id; red; intro; assumption ]; TautoProp - | |- (_ \/ _) => (left; TautoProp) || (right; TautoProp) - end. -\end{coq_example} -\end{centerframe} -\caption{Deciding intuitionistic propositions (2)} -\label{tautoltacb} -\end{figure} - -The pattern matching on goals allows a complete and so a powerful -backtracking when returning tactic values. An interesting application -is the problem of deciding intuitionistic propositional logic. -Considering the contraction-free sequent calculi {\tt LJT*} of -Roy~Dyckhoff (\cite{Dyc92}), it is quite natural to code such a tactic -using the tactic language as shown on Figures~\ref{tautoltaca} -and~\ref{tautoltacb}. The tactic {\tt Axioms} tries to conclude using -usual axioms. The tactic {\tt DSimplif} applies all the reversible -rules of Dyckhoff's system. Finally, the tactic {\tt TautoProp} (the -main tactic to be called) simplifies with {\tt DSimplif}, tries to -conclude with {\tt Axioms} and tries several paths using the -backtracking rules (one of the four Dyckhoff's rules for the left -implication to get rid of the contraction and the right or). - -For example, with {\tt TautoProp}, we can prove tautologies like - those: -% on Figure~\ref{tautolem}. -%\begin{figure}[tbp] -%\begin{centerframe} -\begin{coq_example*} -Lemma tauto_ex1 : forall A B:Prop, A /\ B -> A \/ B. -Proof. TautoProp. Qed. -Lemma tauto_ex2 : - forall A B:Prop, (~ ~ B -> B) -> (A -> B) -> ~ ~ A -> B. -Proof. TautoProp. Qed. -\end{coq_example*} -%\end{centerframe} -%\caption{Proofs of tautologies with {\tt TautoProp}} -%\label{tautolem} -%\end{figure} - -\subsection{Deciding type isomorphisms} - -A more tricky problem is to decide equalities between types and modulo -isomorphisms. Here, we choose to use the isomorphisms of the simply typed -$\lb{}$-calculus with Cartesian product and $unit$ type (see, for example, -\cite{RC95}). The axioms of this $\lb{}$-calculus are given by -table~\ref{isosax}. - -\begin{figure} -\begin{centerframe} -\begin{coq_eval} -Reset Initial. -\end{coq_eval} -\begin{coq_example*} -Open Scope type_scope. -Section Iso_axioms. -Variables A B C : Set. -Axiom Com : A * B = B * A. -Axiom Ass : A * (B * C) = A * B * C. -Axiom Cur : (A * B -> C) = (A -> B -> C). -Axiom Dis : (A -> B * C) = (A -> B) * (A -> C). -Axiom P_unit : A * unit = A. -Axiom AR_unit : (A -> unit) = unit. -Axiom AL_unit : (unit -> A) = A. -Lemma Cons : B = C -> A * B = A * C. -Proof. -intro Heq; rewrite Heq; apply refl_equal. -Qed. -End Iso_axioms. -\end{coq_example*} -\end{centerframe} -\caption{Type isomorphism axioms} -\label{isosax} -\end{figure} - -A more tricky problem is to decide equalities between types and modulo -isomorphisms. Here, we choose to use the isomorphisms of the simply typed -$\lb{}$-calculus with Cartesian product and $unit$ type (see, for example, -\cite{RC95}). The axioms of this $\lb{}$-calculus are given on -Figure~\ref{isosax}. - -\begin{figure}[ht] -\begin{centerframe} -\begin{coq_example} -Ltac DSimplif trm := - match trm with - | (?A * ?B * ?C) => - rewrite <- (Ass A B C); try MainSimplif - | (?A * ?B -> ?C) => - rewrite (Cur A B C); try MainSimplif - | (?A -> ?B * ?C) => - rewrite (Dis A B C); try MainSimplif - | (?A * unit) => - rewrite (P_unit A); try MainSimplif - | (unit * ?B) => - rewrite (Com unit B); try MainSimplif - | (?A -> unit) => - rewrite (AR_unit A); try MainSimplif - | (unit -> ?B) => - rewrite (AL_unit B); try MainSimplif - | (?A * ?B) => - (DSimplif A; try MainSimplif) || (DSimplif B; try MainSimplif) - | (?A -> ?B) => - (DSimplif A; try MainSimplif) || (DSimplif B; try MainSimplif) - end - with MainSimplif := - match goal with - | |- (?A = ?B) => try DSimplif A; try DSimplif B - end. -Ltac Length trm := - match trm with - | (_ * ?B) => let succ := Length B in constr:(S succ) - | _ => constr:1 - end. -Ltac assoc := repeat rewrite <- Ass. -\end{coq_example} -\end{centerframe} -\caption{Type isomorphism tactic (1)} -\label{isosltac1} -\end{figure} - -\begin{figure}[ht] -\begin{centerframe} -\begin{coq_example} -Ltac DoCompare n := - match goal with - | [ |- (?A = ?A) ] => apply refl_equal - | [ |- (?A * ?B = ?A * ?C) ] => - apply Cons; let newn := Length B in - DoCompare newn - | [ |- (?A * ?B = ?C) ] => - match eval compute in n with - | 1 => fail - | _ => - pattern (A * B) at 1; rewrite Com; assoc; DoCompare (pred n) - end - end. -Ltac CompareStruct := - match goal with - | [ |- (?A = ?B) ] => - let l1 := Length A - with l2 := Length B in - match eval compute in (l1 = l2) with - | (?n = ?n) => DoCompare n - end - end. -Ltac IsoProve := MainSimplif; CompareStruct. -\end{coq_example} -\end{centerframe} -\caption{Type isomorphism tactic (2)} -\label{isosltac2} -\end{figure} - -The tactic to judge equalities modulo this axiomatization can be written as -shown on Figures~\ref{isosltac1} and~\ref{isosltac2}. The algorithm is quite -simple. Types are reduced using axioms that can be oriented (this done by {\tt -MainSimplif}). The normal forms are sequences of Cartesian -products without Cartesian product in the left component. These normal forms -are then compared modulo permutation of the components (this is done by {\tt -CompareStruct}). The main tactic to be called and realizing this algorithm is -{\tt IsoProve}. - -% Figure~\ref{isoslem} gives -Here are examples of what can be solved by {\tt IsoProve}. -%\begin{figure}[ht] -%\begin{centerframe} -\begin{coq_example*} -Lemma isos_ex1 : - forall A B:Set, A * unit * B = B * (unit * A). -Proof. -intros; IsoProve. -Qed. - -Lemma isos_ex2 : - forall A B C:Set, - (A * unit -> B * (C * unit)) = - (A * unit -> (C -> unit) * C) * (unit -> A -> B). -Proof. -intros; IsoProve. -Qed. -\end{coq_example*} -%\end{centerframe} -%\caption{Type equalities solved by {\tt IsoProve}} -%\label{isoslem} -%\end{figure} - -%%% Local Variables: -%%% mode: latex -%%% TeX-master: "Reference-Manual" -%%% End: diff --git a/doc/refman/RefMan-tus.tex b/doc/refman/RefMan-tus.tex deleted file mode 100644 index 3e298867..00000000 --- a/doc/refman/RefMan-tus.tex +++ /dev/null @@ -1,2001 +0,0 @@ -%\documentclass[11pt]{article} -%\usepackage{fullpage,euler} -%\usepackage[latin1]{inputenc} -%\begin{document} -%\title{Writing ad-hoc Tactics in Coq} -%\author{} -%\date{} -%\maketitle -%\tableofcontents -%\clearpage - -\chapter[Writing ad-hoc Tactics in Coq]{Writing ad-hoc Tactics in Coq\label{WritingTactics}} - -\section{Introduction} - -\Coq\ is an open proof environment, in the sense that the collection of -proof strategies offered by the system can be extended by the user. -This feature has two important advantages. First, the user can develop -his/her own ad-hoc proof procedures, customizing the system for a -particular domain of application. Second, the repetitive and tedious -aspects of the proofs can be abstracted away implementing new tactics -for dealing with them. For example, this may be useful when a theorem -needs several lemmas which are all proven in a similar but not exactly -the same way. Let us illustrate this with an example. - -Consider the problem of deciding the equality of two booleans. The -theorem establishing that this is always possible is state by -the following theorem: - -\begin{coq_example*} -Theorem decideBool : (x,y:bool){x=y}+{~x=y}. -\end{coq_example*} - -The proof proceeds by case analysis on both $x$ and $y$. This yields -four cases to solve. The cases $x=y=\textsl{true}$ and -$x=y=\textsl{false}$ are immediate by the reflexivity of equality. - -The other two cases follow by discrimination. The following script -describes the proof: - -\begin{coq_example*} -Destruct x. - Destruct y. - Left ; Reflexivity. - Right; Discriminate. - Destruct y. - Right; Discriminate. - Left ; Reflexivity. -\end{coq_example*} -\begin{coq_eval} -Abort. -\end{coq_eval} - -Now, consider the theorem stating the same property but for the -following enumerated type: - -\begin{coq_example*} -Inductive Set Color := Blue:Color | White:Color | Red:Color. -Theorem decideColor : (c1,c2:Color){c1=c2}+{~c1=c2}. -\end{coq_example*} - -This theorem can be proven in a very similar way, reasoning by case -analysis on $c_1$ and $c_2$. Once more, each of the (now six) cases is -solved either by reflexivity or by discrimination: - -\begin{coq_example*} -Destruct c1. - Destruct c2. - Left ; Reflexivity. - Right ; Discriminate. - Right ; Discriminate. - Destruct c2. - Right ; Discriminate. - Left ; Reflexivity. - Right ; Discriminate. - Destruct c2. - Right ; Discriminate. - Right ; Discriminate. - Left ; Reflexivity. -\end{coq_example*} -\begin{coq_eval} -Abort. -\end{coq_eval} - -If we face the same theorem for an enumerated datatype corresponding -to the days of the week, it would still follow a similar pattern. In -general, the general pattern for proving the property -$(x,y:R)\{x=y\}+\{\neg x =y\}$ for an enumerated type $R$ proceeds as -follow: -\begin{enumerate} -\item Analyze the cases for $x$. -\item For each of the sub-goals generated by the first step, analyze -the cases for $y$. -\item The remaining subgoals follow either by reflexivity or -by discrimination. -\end{enumerate} - -Let us describe how this general proof procedure can be introduced in -\Coq. - -\section{Tactic Macros} - -The simplest way to introduce it is to define it as new a -\textsl{tactic macro}, as follows: - -\begin{coq_example*} -Tactic Definition DecideEq [$a $b] := - [<:tactic:<Destruct $a; - Destruct $b; - (Left;Reflexivity) Orelse (Right;Discriminate)>>]. -\end{coq_example*} - -The general pattern of the proof is abstracted away using the -tacticals ``\texttt{;}'' and \texttt{Orelse}, and introducing two -parameters for the names of the arguments to be analyzed. - -Once defined, this tactic can be called like any other tactic, just -supplying the list of terms corresponding to its real arguments. Let us -revisit the proof of the former theorems using the new tactic -\texttt{DecideEq}: - -\begin{coq_example*} -Theorem decideBool : (x,y:bool){x=y}+{~x=y}. -DecideEq x y. -Defined. -\end{coq_example*} -\begin{coq_example*} -Theorem decideColor : (c1,c2:Color){c1=c2}+{~c1=c2}. -DecideEq c1 c2. -Defined. -\end{coq_example*} - -In general, the command \texttt{Tactic Definition} associates a name -to a parameterized tactic expression, built up from the tactics and -tacticals that are already available. The general syntax rule for this -command is the following: - -\begin{tabbing} -\texttt{Tactic Definition} \textit{tactic-name} \= -\texttt{[}\$$id_1\ldots \$id_n$\texttt{]}\\ -\> := \texttt{[<:tactic:<} \textit{tactic-expression} \verb+>>]+ -\end{tabbing} - -This command provides a quick but also very primitive mechanism for -introducing new tactics. It does not support recursive definitions, -and the arguments of a tactic macro are restricted to term -expressions. Moreover, there is no static checking of the definition -other than the syntactical one. Any error in the definition of the -tactic ---for instance, a call to an undefined tactic--- will not be -noticed until the tactic is called. - -%This command provides a very primitive mechanism for introducing new -%tactics. The arguments of a tactic macro are restricted to term -%expressions. Hence, it is not possible to define higher order tactics -%with this command. Also, there is no static checking of the definition -%other than syntactical. If the tactic contain errors in its definition -%--for instance, a call to an undefined tactic-- this will be noticed -%during the tactic call. - -Let us illustrate the weakness of this way of introducing new tactics -trying to extend our proof procedure to work on a larger class of -inductive types. Consider for example the decidability of equality -for pairs of booleans and colors: - -\begin{coq_example*} -Theorem decideBoolXColor : (p1,p2:bool*Color){p1=p2}+{~p1=p2}. -\end{coq_example*} - -The proof still proceeds by a double case analysis, but now the -constructors of the type take two arguments. Therefore, the sub-goals -that can not be solved by discrimination need further considerations -about the equality of such arguments: - -\begin{coq_example} - Destruct p1; - Destruct p2; Try (Right;Discriminate);Intros. -\end{coq_example} - -The half of the disjunction to be chosen depends on whether or not -$b=b_0$ and $c=c_0$. These equalities can be decided automatically -using the previous lemmas about booleans and colors. If both -equalities are satisfied, then it is sufficient to rewrite $b$ into -$b_0$ and $c$ into $c_0$, so that the left half of the goal follows by -reflexivity. Otherwise, the right half follows by first contraposing -the disequality, and then applying the invectiveness of the pairing -constructor. - -As the cases associated to each argument of the pair are very similar, -a tactic macro can be introduced to abstract this part of the proof: - -\begin{coq_example*} -Hints Resolve decideBool decideColor. -Tactic Definition SolveArg [$t1 $t2] := - [<:tactic:< - ElimType {$t1=$t2}+{~$t1=$t2}; - [(Intro equality;Rewrite equality;Clear equality) | - (Intro diseq; Right; Red; Intro absurd; - Apply diseq;Injection absurd;Trivial) | - Auto]>>]. -\end{coq_example*} - -This tactic is applied to each corresponding pair of arguments of the -arguments, until the goal can be solved by reflexivity: - -\begin{coq_example*} -SolveArg b b0; - SolveArg c c0; - Left; Reflexivity. -Defined. -\end{coq_example*} - -Therefore, a more general strategy for deciding the property -$(x,y:R)\{x=y\}+\{\neg x =y\}$ on $R$ can be sketched as follows: -\begin{enumerate} -\item Eliminate $x$ and then $y$. -\item Try discrimination to solve those goals where $x$ and $y$ has -been introduced by different constructors. -\item If $x$ and $y$ have been introduced by the same constructor, -then iterate the tactic \textsl{SolveArg} for each pair of -arguments. -\item Finally, solve the left half of the goal by reflexivity. -\end{enumerate} - -The implementation of this stronger proof strategy needs to perform a -term decomposition, in order to extract the list of arguments of each -constructor. It also requires the introduction of recursively defined -tactics, so that the \textsl{SolveArg} can be iterated on the lists of -arguments. These features are not supported by the \texttt{Tactic -Definition} command. One possibility could be extended this command in -order to introduce recursion, general parameter passing, -pattern-matching, etc, but this would quickly lead us to introduce the -whole \ocaml{} into \Coq\footnote{This is historically true. In fact, -\ocaml{} is a direct descendent of ML, a functional programming language -conceived language for programming the tactics of the theorem prover -LCF.}. Instead of doing this, we prefer to give to the user the -possibility of writing his/her own tactics directly in \ocaml{}, and then -to link them dynamically with \Coq's code. This requires a minimal -knowledge about \Coq's implementation. The next section provides an -overview of \Coq's architecture. - -%It is important to point out that the introduction of a new tactic -%never endangers the correction of the theorems proven in the extended -%system. In order to understand why, let us introduce briefly the system -%architecture. - -\section{An Overview of \Coq's Architecture} - -The implementation of \Coq\ is based on eight \textsl{logical -modules}. By ``module'' we mean here a logical piece of code having a -conceptual unity, that may concern several \ocaml{} files. By the sake of -organization, all the \ocaml{} files concerning a logical module are -grouped altogether into the same sub-directory. The eight modules -are: - -\begin{tabular}{lll} -1. & The logical framework & (directory \texttt{src/generic})\\ -2. & The language of constructions & (directory \texttt{src/constr})\\ -3. & The type-checker & (directory \texttt{src/typing})\\ -4. & The proof engine & (directory \texttt{src/proofs})\\ -5. & The language of basic tactics & (directory \texttt{src/tactics})\\ -6. & The vernacular interpreter & (directory \texttt{src/env})\\ -7. & The parser and the pretty-printer & (directory \texttt{src/parsing})\\ -8. & The standard library & (directory \texttt{src/lib}) -\end{tabular} - -\vspace{1em} - -The following sections briefly present each of the modules above. -This presentation is not intended to be a complete description of \Coq's -implementation, but rather a guideline to be read before taking a look -at the sources. For each of the modules, we also present some of its -most important functions, which are sufficient to implement a large -class of tactics. - - -\subsection[The Logical Framework]{The Logical Framework\label{LogicalFramework}} - -At the very heart of \Coq there is a generic untyped language for -expressing abstractions, applications and global constants. This -language is used as a meta-language for expressing the terms of the -Calculus of Inductive Constructions. General operations on terms like -collecting the free variables of an expression, substituting a term for -a free variable, etc, are expressed in this language. - -The meta-language \texttt{'op term} of terms has seven main -constructors: -\begin{itemize} -\item $(\texttt{VAR}\;id)$, a reference to a global identifier called $id$; -\item $(\texttt{Rel}\;n)$, a bound variable, whose binder is the $nth$ - binder up in the term; -\item $\texttt{DLAM}\;(x,t)$, a deBruijn's binder on the term $t$; -\item $\texttt{DLAMV}\;(x,vt)$, a deBruijn's binder on all the terms of - the vector $vt$; -\item $(\texttt{DOP0}\;op)$, a unary operator $op$; -\item $\texttt{DOP2}\;(op,t_1,t_2)$, the application of a binary -operator $op$ to the terms $t_1$ and $t_2$; -\item $\texttt{DOPN} (op,vt)$, the application of an n-ary operator $op$ to the -vector of terms $vt$. -\end{itemize} - -In this meta-language, bound variables are represented using the -so-called deBrujin's indexes. In this representation, an occurrence of -a bound variable is denoted by an integer, meaning the number of -binders that must be traversed to reach its own -binder\footnote{Actually, $(\texttt{Rel}\;n)$ means that $(n-1)$ binders -have to be traversed, since indexes are represented by strictly -positive integers.}. On the other hand, constants are referred by its -name, as usual. For example, if $A$ is a variable of the current -section, then the lambda abstraction $[x:A]x$ of the Calculus of -Constructions is represented in the meta-language by the term: - -\begin{displaymath} -(DOP2 (Lambda,(Var\;A),DLAM (x,(Rel\;1))) -\end{displaymath} - -In this term, $Lambda$ is a binary operator. Its first argument -correspond to the type $A$ of the bound variable, while the second is -a body of the abstraction, where $x$ is bound. The name $x$ is just kept -to pretty-print the occurrences of the bound variable. - -%Similarly, the product -%$(A:Prop)A$ of the Calculus of Constructions is represented by the -%term: -%\begin{displaumath} -%DOP2 (Prod, DOP0 (Sort (Prop Null)), DLAM (Name \#A, Rel 1)) -%\end{displaymath} - -The following functions perform some of the most frequent operations -on the terms of the meta-language: -\begin{description} -\fun{val Generic.subst1 : 'op term -> 'op term -> 'op term} - {$(\texttt{subst1}\;t_1\;t_2)$ substitutes $t_1$ for - $\texttt{(Rel}\;1)$ in $t_2$.} -\fun{val Generic.occur\_var : identifier -> 'op term -> bool} - {Returns true when the given identifier appears in the term, - and false otherwise.} -\fun{val Generic.eq\_term : 'op term -> 'op term -> bool} - {Implements $\alpha$-equality for terms.} -\fun{val Generic.dependent : 'op term -> 'op term -> bool} - {Returns true if the first term is a sub-term of the second.} -%\fun{val Generic.subst\_var : identifier -> 'op term -> 'op term} -% { $(\texttt{subst\_var}\;id\;t)$ substitutes the deBruijn's index -% associated to $id$ to every occurrence of the term -% $(\texttt{VAR}\;id)$ in $t$.} -\end{description} - -\subsubsection{Identifiers, names and sections paths.} - -Three different kinds of names are used in the meta-language. They are -all defined in the \ocaml{} file \texttt{Names}. - -\paragraph{Identifiers.} The simplest kind of names are -\textsl{identifiers}. An identifier is a string possibly indexed by an -integer. They are used to represent names that are not unique, like -for example the name of a variable in the scope of a section. The -following operations can be used for handling identifiers: - -\begin{description} -\fun{val Names.make\_ident : string -> int -> identifier} - {The value $(\texttt{make\_ident}\;x\;i)$ creates the - identifier $x_i$. If $i=-1$, then the identifier has - is created with no index at all.} -\fun{val Names.repr\_ident : identifier -> string * int} - {The inverse operation of \texttt{make\_ident}: - it yields the string and the index of the identifier.} -\fun{val Names.lift\_ident : identifier -> identifier} - {Increases the index of the identifier by one.} -\fun{val Names.next\_ident\_away : \\ -\qquad identifier -> identifier list -> identifier} - {\\ Generates a new identifier with the same root string than the - given one, but with a new index, different from all the indexes of - a given list of identifiers.} -\fun{val Names.id\_of\_string : string -> - identifier} - {Creates an identifier from a string.} -\fun{val Names.string\_of\_id : identifier -> string} - {The inverse operation: transforms an identifier into a string} -\end{description} - -\paragraph{Names.} A \textsl{name} is either an identifier or the -special name \texttt{Anonymous}. Names are used as arguments of -binders, in order to pretty print bound variables. -The following operations can be used for handling names: - -\begin{description} -\fun{val Names.Name: identifier -> Name} - {Constructs a name from an identifier.} -\fun{val Names.Anonymous : Name} - {Constructs a special, anonymous identifier, like the variable abstracted - in the term $[\_:A]0$.} -\fun{val - Names.next\_name\_away\_with\_default : \\ \qquad - string->name->identifier list->identifier} -{\\ If the name is not anonymous, then this function generates a new - identifier different from all the ones in a given list. Otherwise, it - generates an identifier from the given string.} -\end{description} - -\paragraph[Section paths.]{Section paths.\label{SectionPaths}} -A \textsl{section-path} is a global name to refer to an object without -ambiguity. It can be seen as a sort of filename, where open sections -play the role of directories. Each section path is formed by three -components: a \textsl{directory} (the list of open sections); a -\textsl{basename} (the identifier for the object); and a \textsl{kind} -(either CCI for the terms of the Calculus of Constructions, FW for the -the terms of $F_\omega$, or OBJ for other objects). For example, the -name of the following constant: -\begin{verbatim} - Section A. - Section B. - Section C. - Definition zero := O. -\end{verbatim} - -is internally represented by the section path: - -$$\underbrace{\mathtt{\#A\#B\#C}}_{\mbox{dirpath}} -\underbrace{\mathtt{\tt \#zero}}_{\mbox{basename}} -\underbrace{\mathtt{\tt .cci}_{\;}}_{\mbox{kind}}$$ - -When one of the sections is closed, a new constant is created with an -updated section-path,a nd the old one is no longer reachable. In our -example, after closing the section \texttt{C}, the new section-path -for the constant {\tt zero} becomes: -\begin{center} -\texttt{ \#A\#B\#zero.cci} -\end{center} - -The following operations can be used to handle section paths: - -\begin{description} -\fun{val Names.string\_of\_path : section\_path -> string} - {Transforms the section path into a string.} -\fun{val Names.path\_of\_string : string -> section\_path} - {Parses a string an returns the corresponding section path.} -\fun{val Names.basename : section\_path -> identifier} - {Provides the basename of a section path} -\fun{val Names.dirpath : section\_path -> string list} - {Provides the directory of a section path} -\fun{val Names.kind\_of\_path : section\_path -> path\_kind} - {Provides the kind of a section path} -\end{description} - -\subsubsection{Signatures} - -A \textsl{signature} is a mapping associating different informations -to identifiers (for example, its type, its definition, etc). The -following operations could be useful for working with signatures: - -\begin{description} -\fun{val Names.ids\_of\_sign : 'a signature -> identifier list} - {Gets the list of identifiers of the signature.} -\fun{val Names.vals\_of\_sign : 'a signature -> 'a list} - {Gets the list of values associated to the identifiers of the signature.} -\fun{val Names.lookup\_glob1 : \\ \qquad -identifier -> 'a signature -> (identifier * - 'a)} - {\\ Gets the value associated to a given identifier of the signature.} -\end{description} - - -\subsection{The Terms of the Calculus of Constructions} - -The language of the Calculus of Inductive Constructions described in -Chapter \ref{Cic} is implemented on the top of the logical framework, -instantiating the parameter $op$ of the meta-language with a -particular set of operators. In the implementation this language is -called \texttt{constr}, the language of constructions. - -% The only difference -%with respect to the one described in Section \ref{} is that the terms -%of \texttt{constr} may contain \textsl{existential variables}. An -%existential variable is a place holder representing a part of the term -%that is still to be constructed. Such ``open terms'' are necessary -%when building proofs interactively. - -\subsubsection{Building Constructions} - -The user does not need to know the choices made to represent -\texttt{constr} in the meta-language. They are abstracted away by the -following constructor functions: - -\begin{description} -\fun{val Term.mkRel : int -> constr} - {$(\texttt{mkRel}\;n)$ represents deBrujin's index $n$.} - -\fun{val Term.mkVar : identifier -> constr} - {$(\texttt{mkVar}\;id)$ - represents a global identifier named $id$, like a variable - inside the scope of a section, or a hypothesis in a proof}. - -\fun{val Term.mkExistential : constr} - {\texttt{mkExistential} represents an implicit sub-term, like the question - marks in the term \texttt{(pair ? ? O true)}.} - -%\fun{val Term.mkMeta : int -> constr} -% {$(\texttt{mkMeta}\;n)$ represents an existential variable, whose -% name is the integer $n$.} - -\fun{val Term.mkProp : constr} - {$\texttt{mkProp}$ represents the sort \textsl{Prop}.} - -\fun{val Term.mkSet : constr} - {$\texttt{mkSet}$ represents the sort \textsl{Set}.} - -\fun{val Term.mkType : Impuniv.universe -> constr} - {$(\texttt{mkType}\;u)$ represents the term - $\textsl{Type}(u)$. The universe $u$ is represented as a - section path indexed by an integer. } - -\fun{val Term.mkConst : section\_path -> constr array -> constr} - {$(\texttt{mkConst}\;c\;v)$ represents a constant whose name is - $c$. The body of the constant is stored in a global table, - accessible through the name of the constant. The array of terms - $v$ corresponds to the variables of the environment appearing in - the body of the constant when it was defined. For instance, a - constant defined in the section \textsl{Foo} containing the - variable $A$, and whose body is $[x:Prop\ra Prop](x\;A)$ is - represented inside the scope of the section by - $(\texttt{mkConst}\;\texttt{\#foo\#f.cci}\;[| \texttt{mkVAR}\;A - |])$. Once the section is closed, the constant is represented by - the term $(\texttt{mkConst}\;\#f.cci\;[| |])$, and its body - becomes $[A:Prop][x:Prop\ra Prop](x\;A)$}. - -\fun{val Term.mkMutInd : section\_path -> int -> constr array ->constr} - {$(\texttt{mkMutInd}\;c\;i)$ represents the $ith$ type - (starting from zero) of the block of mutually dependent - (co)inductive types, whose first type is $c$. Similarly to the - case of constants, the array of terms represents the current - environment of the (co)inductive type. The definition of the type - (its arity, its constructors, whether it is inductive or co-inductive, etc.) - is stored in a global hash table, accessible through the name of - the type.} - -\fun{val Term.mkMutConstruct : \\ \qquad section\_path -> int -> int -> constr array - ->constr} {\\ $(\texttt{mkMutConstruct}\;c\;i\;j)$ represents the - $jth$ constructor of the $ith$ type of the block of mutually - dependent (co)inductive types whose first type is $c$. The array - of terms represents the current environment of the (co)inductive - type.} - -\fun{val Term.mkCast : constr -> constr -> constr} - {$(\texttt{mkCast}\;t\;T)$ represents the annotated term $t::T$ in - \Coq's syntax.} - -\fun{val Term.mkProd : name ->constr ->constr -> constr} - {$(\texttt{mkProd}\;x\;A\;B)$ represents the product $(x:A)B$. - The free ocurrences of $x$ in $B$ are represented by deBrujin's - indexes.} - -\fun{val Term.mkNamedProd : identifier -> constr -> constr -> constr} - {$(\texttt{produit}\;x\;A\;B)$ represents the product $(x:A)B$, - but the bound occurrences of $x$ in $B$ are denoted by - the identifier $(\texttt{mkVar}\;x)$. The function automatically - changes each occurrences of this identifier into the corresponding - deBrujin's index.} - -\fun{val Term.mkArrow : constr -> constr -> constr} - {$(\texttt{arrow}\;A\;B)$ represents the type $(A\rightarrow B)$.} - -\fun{val Term.mkLambda : name -> constr -> constr -> constr} - {$(\texttt{mkLambda}\;x\;A\;b)$ represents the lambda abstraction - $[x:A]b$. The free ocurrences of $x$ in $B$ are represented by deBrujin's - indexes.} - -\fun{val Term.mkNamedLambda : identifier -> constr -> constr -> constr} - {$(\texttt{lambda}\;x\;A\;b)$ represents the lambda abstraction - $[x:A]b$, but the bound occurrences of $x$ in $B$ are denoted by - the identifier $(\texttt{mkVar}\;x)$. } - -\fun{val Term.mkAppLA : constr array -> constr} - {$(\texttt{mkAppLA}\;t\;[|t_1\ldots t_n|])$ represents the application - $(t\;t_1\;\ldots t_n)$.} - -\fun{val Term.mkMutCaseA : \\ \qquad - case\_info -> constr ->constr - ->constr array -> constr} - {\\ $(\texttt{mkMutCaseA}\;r\;P\;m\;[|f_1\ldots f_n|])$ - represents the term \Case{P}{m}{f_1\ldots f_n}. The first argument - $r$ is either \texttt{None} or $\texttt{Some}\;(c,i)$, where the - pair $(c,i)$ refers to the inductive type that $m$ belongs to.} - -\fun{val Term.mkFix : \\ \qquad -int array->int->constr array->name - list->constr array->constr} - {\\ $(\texttt{mkFix}\;[|k_1\ldots k_n |]\;i\;[|A_1\ldots - A_n|]\;[|f_1\ldots f_n|]\;[|t_1\ldots t_n|])$ represents the term - $\Fix{f_i}{f_1/k_1:A_1:=t_1 \ldots f_n/k_n:A_n:=t_n}$} - -\fun{val Term.mkCoFix : \\ \qquad - int -> constr array -> name list -> - constr array -> constr} - {\\ $(\texttt{mkCoFix}\;i\;[|A_1\ldots - A_n|]\;[|f_1\ldots f_n|]\;[|t_1\ldots t_n|])$ represents the term - $\CoFix{f_i}{f_1:A_1:=t_1 \ldots f_n:A_n:=t_n}$. There are no - decreasing indexes in this case.} -\end{description} - -\subsubsection{Decomposing Constructions} - -Each of the construction functions above has its corresponding -(partial) destruction function, whose name is obtained changing the -prefix \texttt{mk} by \texttt{dest}. In addition to these functions, a -concrete datatype \texttt{kindOfTerm} can be used to do pattern -matching on terms without dealing with their internal representation -in the meta-language. This concrete datatype is described in the \ocaml{} -file \texttt{term.mli}. The following function transforms a construction -into an element of type \texttt{kindOfTerm}: - -\begin{description} -\fun{val Term.kind\_of\_term : constr -> kindOfTerm} - {Destructs a term of the language \texttt{constr}, -yielding the direct components of the term. Hence, in order to do -pattern matching on an object $c$ of \texttt{constr}, it is sufficient -to do pattern matching on the value $(\texttt{kind\_of\_term}\;c)$.} -\end{description} - -Part of the information associated to the constants is stored in -global tables. The following functions give access to such -information: - -\begin{description} -\fun{val Termenv.constant\_value : constr -> constr} - {If the term denotes a constant, projects the body of a constant} -\fun{Termenv.constant\_type : constr -> constr} - {If the term denotes a constant, projects the type of the constant} -\fun{val mind\_arity : constr -> constr} - {If the term denotes an inductive type, projects its arity (i.e., - the type of the inductive type).} -\fun{val Termenv.mis\_is\_finite : mind\_specif -> bool} - {Determines whether a recursive type is inductive or co-inductive.} -\fun{val Termenv.mind\_nparams : constr -> int} - {If the term denotes an inductive type, projects the number of - its general parameters.} -\fun{val Termenv.mind\_is\_recursive : constr -> bool} - {If the term denotes an inductive type, - determines if the type has at least one recursive constructor. } -\fun{val Termenv.mind\_recargs : constr -> recarg list array array} - {If the term denotes an inductive type, returns an array $v$ such - that the nth element of $v.(i).(j)$ is - \texttt{Mrec} if the $nth$ argument of the $jth$ constructor of - the $ith$ type is recursive, and \texttt{Norec} if it is not.}. -\end{description} - -\subsection[The Type Checker]{The Type Checker\label{TypeChecker}} - -The third logical module is the type checker. It concentrates two main -tasks concerning the language of constructions. - -On one hand, it contains the type inference and type-checking -functions. The type inference function takes a term -$a$ and a signature $\Gamma$, and yields a term $A$ such that -$\Gamma \vdash a:A$. The type-checking function takes two terms $a$ -and $A$ and a signature $\Gamma$, and determines whether or not -$\Gamma \vdash a:A$. - -On the other hand, this module is in charge of the compilation of -\Coq's abstract syntax trees into the language \texttt{constr} of -constructions. This compilation seeks to eliminate all the ambiguities -contained in \Coq's abstract syntax, restoring the information -necessary to type-check it. It concerns at least the following steps: -\begin{enumerate} -\item Compiling the pattern-matching expressions containing -constructor patterns, wild-cards, etc, into terms that only -use the primitive \textsl{Case} described in Chapter \ref{Cic} -\item Restoring type coercions and synthesizing the implicit arguments -(the one denoted by question marks in -{\Coq} syntax: see Section~\ref{Coercions}). -\item Transforming the named bound variables into deBrujin's indexes. -\item Classifying the global names into the different classes of -constants (defined constants, constructors, inductive types, etc). -\end{enumerate} - -\subsection{The Proof Engine} - -The fourth stage of \Coq's implementation is the \textsl{proof engine}: -the interactive machine for constructing proofs. The aim of the proof -engine is to construct a top-down derivation or \textsl{proof tree}, -by the application of \textsl{tactics}. A proof tree has the following -general structure:\\ - -\begin{displaymath} -\frac{\Gamma \vdash ? = t(?_1,\ldots?_n) : G} - {\hspace{3ex}\frac{\displaystyle \Gamma_1 \vdash ?_1 = t_1(\ldots) : G_1} - {\stackrel{\vdots}{\displaystyle {\Gamma_{i_1} \vdash ?_{i_1} - : G_{i_1}}}}(tac_1) - \;\;\;\;\;\;\;\;\; - \frac{\displaystyle \Gamma_n \vdash ?_n = t_n(\ldots) : G_n} - {\displaystyle \stackrel{\vdots}{\displaystyle {\Gamma_{i_m} \vdash ?_{i_m} : - G_{i_m}}}}(tac_n)} (tac) -\end{displaymath} - - -\noindent Each node of the tree is called a \textsl{goal}. A goal -is a record type containing the following three fields: -\begin{enumerate} -\item the conclusion $G$ to be proven; -\item a typing signature $\Gamma$ for the free variables in $G$; -\item if the goal is an internal node of the proof tree, the -definition $t(?_1,\ldots?_n)$ of an \textsl{existential variable} -(i.e. a possible undefined constant) $?$ of type $G$ in terms of the -existential variables of the children sub-goals. If the node is a -leaf, the existential variable maybe still undefined. -\end{enumerate} - -Once all the existential variables have been defined the derivation is -completed, and a construction can be generated from the proof tree, -replacing each of the existential variables by its definition. This -is exactly what happens when one of the commands -\texttt{Qed}, \texttt{Save} or \texttt{Defined} is invoked -(see Section~\ref{Qed}). The saved theorem becomes a defined constant, -whose body is the proof object generated. - -\paragraph{Important:} Before being added to the -context, the proof object is type-checked, in order to verify that it is -actually an object of the expected type $G$. Hence, the correctness -of the proof actually does not depend on the tactics applied to -generate it or the machinery of the proof engine, but only on the -type-checker. In other words, extending the system with a potentially -bugged new tactic never endangers the consistency of the system. - -\subsubsection[What is a Tactic?]{What is a Tactic?\label{WhatIsATactic}} -%Let us now explain what is a tactic, and how the user can introduce -%new ones. - -From an operational point of view, the current state of the proof -engine is given by the mapping $emap$ from existential variables into -goals, plus a pointer to one of the leaf goals $g$. Such a pointer -indicates where the proof tree will be refined by the application of a -\textsl{tactic}. A tactic is a function from the current state -$(g,emap)$ of the proof engine into a pair $(l,val)$. The first -component of this pair is the list of children sub-goals $g_1,\ldots -g_n$ of $g$ to be yielded by the tactic. The second one is a -\textsl{validation function}. Once the proof trees $\pi_1,\ldots -\pi_n$ for $g_1,\ldots g_n$ have been completed, this validation -function must yield a proof tree $(val\;\pi_1,\ldots \pi_n)$ deriving -$g$. - -Tactics can be classified into \textsl{primitive} ones and -\textsl{defined} ones. Primitive tactics correspond to the five basic -operations of the proof engine: - -\begin{enumerate} -\item Introducing a universally quantified variable into the local -context of the goal. -\item Defining an undefined existential variable -\item Changing the conclusion of the goal for another ---definitionally equal-- term. -\item Changing the type of a variable in the local context for another -definitionally equal term. -\item Erasing a variable from the local context. -\end{enumerate} - -\textsl{Defined} tactics are tactics constructed by combining these -primitive operations. Defined tactics are registered in a hash table, -so that they can be introduced dynamically. In order to define such a -tactic table, it is necessary to fix what a \textsl{possible argument} -of a tactic may be. The type \texttt{tactic\_arg} of the possible -arguments for tactics is a union type including: -\begin{itemize} -\item quoted strings; -\item integers; -\item identifiers; -\item lists of identifiers; -\item plain terms, represented by its abstract syntax tree; -\item well-typed terms, represented by a construction; -\item a substitution for bound variables, like the -substitution in the tactic \\$\texttt{Apply}\;t\;\texttt{with}\;x:=t_1\ldots -x_n:=t_n$, (see Section~\ref{apply}); -\item a reduction expression, denoting the reduction strategy to be -followed. -\end{itemize} -Therefore, for each function $tac:a \rightarrow tactic$ implementing a -defined tactic, an associated dynamic tactic $tacargs\_tac: -\texttt{tactic\_arg}\;list \rightarrow tactic$ calling $tac$ must be -written. The aim of the auxiliary function $tacargs\_tac$ is to inject -the arguments of the tactic $tac$ into the type of possible arguments -for a tactic. - -The following function can be used for registering and calling a -defined tactic: - -\begin{description} -\fun{val Tacmach.add\_tactic : \\ \qquad -string -> (tactic\_arg list ->tactic) -> unit} - {\\ Registers a dynamic tactic with the given string as access index.} -\fun{val Tacinterp.vernac\_tactic : string*tactic\_arg list -> tactic} - {Interprets a defined tactic given by its entry in the - tactics table with a particular list of possible arguments.} -\fun{val Tacinterp.vernac\_interp : CoqAst.t -> tactic} - {Interprets a tactic expression formed combining \Coq's tactics and - tacticals, and described by its abstract syntax tree.} -\end{description} - -When programming a new tactic that calls an already defined tactic -$tac$, we have the choice between using the \ocaml{} function -implementing $tac$, or calling the tactic interpreter with the name -and arguments for interpreting $tac$. In the first case, a tactic call -will left the trace of the whole implementation of $tac$ in the proof -tree. In the second, the implementation of $tac$ will be hidden, and -only an invocation of $tac$ will be recalled (cf. the example of -Section \ref{ACompleteExample}. The following combinators can be used -to hide the implementation of a tactic: - -\begin{verbatim} -type 'a hiding_combinator = string -> ('a -> tactic) -> ('a -> tactic) -val Tacmach.hide_atomic_tactic : string -> tactic -> tactic -val Tacmach.hide_constr_tactic : constr hiding_combinator -val Tacmach.hide_constrl_tactic : (constr list) hiding_combinator -val Tacmach.hide_numarg_tactic : int hiding_combinator -val Tacmach.hide_ident_tactic : identifier hiding_combinator -val Tacmach.hide_identl_tactic : identifier hiding_combinator -val Tacmach.hide_string_tactic : string hiding_combinator -val Tacmach.hide_bindl_tactic : substitution hiding_combinator -val Tacmach.hide_cbindl_tactic : - (constr * substitution) hiding_combinator -\end{verbatim} - -These functions first register the tactic by a side effect, and then -yield a function calling the interpreter with the registered name and -the right injection into the type of possible arguments. - -\subsection{Tactics and Tacticals Provided by \Coq} - -The fifth logical module is the library of tacticals and basic tactics -provided by \Coq. This library is distributed into the directories -\texttt{tactics} and \texttt{src/tactics}. The former contains those -basic tactics that make use of the types contained in the basic state -of \Coq. For example, inversion or rewriting tactics are in the -directory \texttt{tactics}, since they make use of the propositional -equality type. Those tactics which are independent from the context ---like for example \texttt{Cut}, \texttt{Intros}, etc-- are defined in -the directory \texttt{src/tactics}. This latter directory also -contains some useful tools for programming new tactics, referred in -Section \ref{SomeUsefulToolsforWrittingTactics}. - -In practice, it is very unusual that the list of sub-goals and the -validation function of the tactic must be explicitly constructed by -the user. In most of the cases, the implementation of a new tactic -consists in supplying the appropriate arguments to the basic tactics -and tacticals. - -\subsubsection{Basic Tactics} - -The file \texttt{Tactics} contain the implementation of the basic -tactics provided by \Coq. The following tactics are some of the most -used ones: - -\begin{verbatim} -val Tactics.intro : tactic -val Tactics.assumption : tactic -val Tactics.clear : identifier list -> tactic -val Tactics.apply : constr -> constr substitution -> tactic -val Tactics.one_constructor : int -> constr substitution -> tactic -val Tactics.simplest_elim : constr -> tactic -val Tactics.elimType : constr -> tactic -val Tactics.simplest_case : constr -> tactic -val Tactics.caseType : constr -> tactic -val Tactics.cut : constr -> tactic -val Tactics.reduce : redexpr -> tactic -val Tactics.exact : constr -> tactic -val Auto.auto : int option -> tactic -val Auto.trivial : tactic -\end{verbatim} - -The functions hiding the implementation of these tactics are defined -in the module \texttt{Hiddentac}. Their names are prefixed by ``h\_''. - -\subsubsection[Tacticals]{Tacticals\label{OcamlTacticals}} - -The following tacticals can be used to combine already existing -tactics: - -\begin{description} -\fun{val Tacticals.tclIDTAC : tactic} - {The identity tactic: it leaves the goal as it is.} - -\fun{val Tacticals.tclORELSE : tactic -> tactic -> tactic} - {Tries the first tactic and in case of failure applies the second one.} - -\fun{val Tacticals.tclTHEN : tactic -> tactic -> tactic} - {Applies the first tactic and then the second one to each generated subgoal.} - -\fun{val Tacticals.tclTHENS : tactic -> tactic list -> tactic} - {Applies a tactic, and then applies each tactic of the tactic list to the - corresponding generated subgoal.} - -\fun{val Tacticals.tclTHENL : tactic -> tactic -> tactic} - {Applies the first tactic, and then applies the second one to the last - generated subgoal.} - -\fun{val Tacticals.tclREPEAT : tactic -> tactic} - {If the given tactic succeeds in producing a subgoal, then it - is recursively applied to each generated subgoal, - and so on until it fails. } - -\fun{val Tacticals.tclFIRST : tactic list -> tactic} - {Tries the tactics of the given list one by one, until one of them - succeeds.} - -\fun{val Tacticals.tclTRY : tactic -> tactic} - {Tries the given tactic and in case of failure applies the {\tt - tclIDTAC} tactical to the original goal.} - -\fun{val Tacticals.tclDO : int -> tactic -> tactic} - {Applies the tactic a given number of times.} - -\fun{val Tacticals.tclFAIL : tactic} - {The always failing tactic: it raises a {\tt UserError} exception.} - -\fun{val Tacticals.tclPROGRESS : tactic -> tactic} - {Applies the given tactic to the current goal and fails if the - tactic leaves the goal unchanged} - -\fun{val Tacticals.tclNTH\_HYP : int -> (constr -> tactic) -> tactic} - {Applies a tactic to the nth hypothesis of the local context. - The last hypothesis introduced correspond to the integer 1.} - -\fun{val Tacticals.tclLAST\_HYP : (constr -> tactic) -> tactic} - {Applies a tactic to the last hypothesis introduced.} - -\fun{val Tacticals.tclCOMPLETE : tactic -> tactic} - {Applies a tactic and fails if the tactic did not solve completely the - goal} - -\fun{val Tacticals.tclMAP : ('a -> tactic) -> 'a list -> tactic} - {Applied to the function \texttt{f} and the list \texttt{[x\_1; - ... ; x\_n]}, this tactical applies the tactic - \texttt{tclTHEN (f x1) (tclTHEN (f x2) ... ))))}} - -\fun{val Tacicals.tclIF : (goal sigma -> bool) -> tactic -> tactic -> tactic} - {If the condition holds, apply the first tactic; otherwise, - apply the second one} - -\end{description} - - -\subsection{The Vernacular Interpreter} - -The sixth logical module of the implementation corresponds to the -interpreter of the vernacular phrases of \Coq. These phrases may be -expressions from the \gallina{} language (definitions), general -directives (setting commands) or tactics to be applied by the proof -engine. - -\subsection[The Parser and the Pretty-Printer]{The Parser and the Pretty-Printer\label{PrettyPrinter}} - -The last logical module is the parser and pretty printer of \Coq, -which is the interface between the vernacular interpreter and the -user. They translate the chains of characters entered at the input -into abstract syntax trees, and vice versa. Abstract syntax trees are -represented by labeled n-ary trees, and its type is called -\texttt{CoqAst.t}. For instance, the abstract syntax tree associated -to the term $[x:A]x$ is: - -\begin{displaymath} -\texttt{Node} - ((0,6), "LAMBDA", - [\texttt{Nvar}~((3, 4),"A");~\texttt{Slam}~((0,6),~Some~"x",~\texttt{Nvar}~((5,6),"x"))]) -\end{displaymath} - -The numbers correspond to \textsl{locations}, used to point to some -input line and character positions in the error messages. As it was -already explained in Section \ref{TypeChecker}, this term is then -translated into a construction term in order to be typed. - -The parser of \Coq\ is implemented using \camlpppp. The lexer and the data -used by \camlpppp\ to generate the parser lay in the directory -\texttt{src/parsing}. This directory also contains \Coq's -pretty-printer. The printing rules lay in the directory -\texttt{src/syntax}. The different entries of the grammar are -described in the module \texttt{Pcoq.Entry}. Let us present here two -important functions of this logical module: - -\begin{description} -\fun{val Pcoq.parse\_string : 'a Grammar.Entry.e -> string -> 'a} - {Parses a given string, trying to recognize a phrase - corresponding to some entry in the grammar. If it succeeds, - it yields a value associated to the grammar entry. For example, - applied to the entry \texttt{Pcoq.Command.command}, this function - parses a term of \Coq's language, and yields a value of type - \texttt{CoqAst.t}. When applied to the entry - \texttt{Pcoq.Vernac.vernac}, it parses a vernacular command and - returns the corresponding Ast.} -\fun{val gentermpr : \\ \qquad -path\_kind -> constr assumptions -> constr -> std\_ppcmds} - {\\ Pretty-prints a well-typed term of certain kind (cf. Section - \ref{SectionPaths}) under its context of typing assumption.} -\fun{val gentacpr : CoqAst.t -> std\_ppcmds} - {Pretty-prints a given abstract syntax tree representing a tactic - expression.} -\end{description} - -\subsection{The General Library} - -In addition to the ones laying in the standard library of \ocaml{}, -several useful modules about lists, arrays, sets, mappings, balanced -trees, and other frequently used data structures can be found in the -directory \texttt{lib}. Before writing a new one, check if it is not -already there! - -\subsubsection{The module \texttt{Std}} -This module in the directory \texttt{src/lib/util} is opened by almost -all modules of \Coq{}. Among other things, it contains a definition of -the different kinds of errors used in \Coq{} : - -\begin{description} -\fun{exception UserError of string * std\_ppcmds} - {This is the class of ``users exceptions''. Such errors arise when - the user attempts to do something illegal, for example \texttt{Intro} - when the current goal conclusion is not a product.} - -\fun{val Std.error : string -> 'a} - {For simple error messages} -\fun{val Std.errorlabstrm : string -> std\_ppcmds -> 'a} - {See Section~\ref{PrettyPrinter} : this can be used if the user - want to display a term or build a complex error message} - -\fun{exception Anomaly of string * std\_ppcmds} - {This for reporting bugs or things that should not - happen. The tacticals \texttt{tclTRY} and - \texttt{tclTRY} described in Section~\ref{OcamlTacticals} catch the - exceptions of type \texttt{UserError}, but they don't catch the - anomalies. So, in your code, don't raise any anomaly, unless you - know what you are doing. We also recommend to avoid constructs - such as \texttt{try ... with \_ -> ...} : such constructs can trap - an anomaly and make the debugging process harder.} - -\fun{val Std.anomaly : string -> 'a}{} -\fun{val Std.anomalylabstrm : string -> std\_ppcmds -> 'a}{} -\end{description} - -\section{The tactic writer mini-HOWTO} - -\subsection{How to add a vernacular command} - -The command to register a vernacular command can be found -in module \texttt{Vernacinterp}: - -\begin{verbatim} -val vinterp_add : string * (vernac_arg list -> unit -> unit) -> unit;; -\end{verbatim} - -The first argument is the name, the second argument is a function that -parses the arguments and returns a function of type -\texttt{unit}$\rightarrow$\texttt{unit} that do the job. - -In this section we will show how to add a vernacular command -\texttt{CheckCheck} that print a type of a term and the type of its -type. - -File \texttt{dcheck.ml}: - -\begin{verbatim} -open Vernacinterp;; -open Trad;; -let _ = - vinterp_add - ("DblCheck", - function [VARG_COMMAND com] -> - (fun () -> - let evmap = Evd.mt_evd () - and sign = Termenv.initial_sign () in - let {vAL=c;tYP=t;kIND=k} = - fconstruct_with_univ evmap sign com in - Pp.mSGNL [< Printer.prterm c; 'sTR ":"; - Printer.prterm t; 'sTR ":"; - Printer.prterm k >] ) - | _ -> bad_vernac_args "DblCheck") -;; -\end{verbatim} - -Like for a new tactic, a new syntax entry must be created. - -File \texttt{DCheck.v}: - -\begin{verbatim} -Declare ML Module "dcheck.ml". - -Grammar vernac vernac := - dblcheck [ "CheckCheck" comarg($c) ] -> [(DblCheck $c)]. -\end{verbatim} - -We are now able to test our new command: - -\begin{verbatim} -Coq < Require DCheck. -Coq < CheckCheck O. -O:nat:Set -\end{verbatim} - -Most Coq vernacular commands are registered in the module - \verb+src/env/vernacentries.ml+. One can see more examples here. - -\subsection{How to keep a hashtable synchronous with the reset mechanism} - -This is far more tricky. Some vernacular commands modify some -sort of state (for example by adding something in a hashtable). One -wants that \texttt{Reset} has the expected behavior with this -commands. - -\Coq{} provides a general mechanism to do that. \Coq{} environments -contains objects of three kinds: CCI, FW and OBJ. CCI and FW are for -constants of the calculus. OBJ is a dynamically extensible datatype -that contains sections, tactic definitions, hints for auto, and so -on. - -The simplest example of use of such a mechanism is in file -\verb+src/proofs/macros.ml+ (which implements the \texttt{Tactic - Definition} command). Tactic macros are stored in the imperative -hashtable \texttt{mactab}. There are two functions freeze and unfreeze -to make a copy of the table and to restore the state of table from the -copy. Then this table is declared using \texttt{Library.declare\_summary}. - -What does \Coq{} with that ? \Coq{} defines synchronization points. -At each synchronisation point, the declared tables are frozen (that -is, a copy of this tables is stored). - -When \texttt{Reset }$i$ is called, \Coq{} goes back to the first -synchronisation point that is above $i$ and ``replays'' all objects -between that point -and $i$. It will re-declare constants, re-open section, etc. - -So we need to declare a new type of objects, TACTIC-MACRO-DATA. To -``replay'' on object of that type is to add the corresponding tactic -macro to \texttt{mactab} - -So, now, we can say that \texttt{mactab} is synchronous with the Reset -mechanism$^{\mathrm{TM}}$. - -Notice that this works for hash tables but also for a single integer -(the Undo stack size, modified by the \texttt{Set Undo} command, for -example). - -\subsection{The right way to access to Coq constants from your ML code} - -With their long names, Coq constants are stored using: - -\begin{itemize} -\item a section path -\item an identifier -\end{itemize} - -The identifier is exactly the identifier that is used in \Coq{} to -denote the constant; the section path can be known using the -\texttt{Locate} command: - -\begin{coq_example} - Locate S. - Locate nat. - Locate eq. -\end{coq_example} - -Now it is easy to get a constant by its name and section path: - - -\begin{verbatim} -let constant sp id = - Machops.global_reference (Names.gLOB (Termenv.initial_sign ())) - (Names.path_of_string sp) (Names.id_of_string id);; -\end{verbatim} - - -The only issue is that if one cannot put: - - -\begin{verbatim} -let coq_S = constant "#Datatypes#nat.cci" "S";; -\end{verbatim} - - -in his tactic's code. That is because this sentence is evaluated -\emph{before} the module \texttt{Datatypes} is loaded. The solution is -to use the lazy evaluation of \ocaml{}: - - -\begin{verbatim} -let coq_S = lazy (constant "#Datatypes#nat.cci" "S");; - -... (Lazy.force coq_S) ... -\end{verbatim} - - -Be sure to call always Lazy.force behind a closure -- i.e. inside a -function body or behind the \texttt{lazy} keyword. - -One can see examples of that technique in the source code of \Coq{}, -for example -\verb+plugins/omega/coq_omega.ml+. - -\section[Some Useful Tools for Writing Tactics]{Some Useful Tools for Writing Tactics\label{SomeUsefulToolsforWrittingTactics}} -When the implementation of a tactic is not a straightforward -combination of tactics and tacticals, the module \texttt{Tacmach} -provides several useful functions for handling goals, calling the -type-checker, parsing terms, etc. This module is intended to be -the interface of the proof engine for the user. - -\begin{description} -\fun{val Tacmach.pf\_hyps : goal sigma -> constr signature} - {Projects the local typing context $\Gamma$ from a given goal $\Gamma\vdash ?:G$.} -\fun{val pf\_concl : goal sigma -> constr} - {Projects the conclusion $G$ from a given goal $\Gamma\vdash ?:G$.} -\fun{val Tacmach.pf\_nth\_hyp : goal sigma -> int -> identifier * - constr} - {Projects the $ith$ typing constraint $x_i:A_i$ from the local - context of the given goal.} -\fun{val Tacmach.pf\_fexecute : goal sigma -> constr -> judgement} - {Given a goal whose local context is $\Gamma$ and a term $a$, this - function infers a type $A$ and a kind $K$ such that the judgement - $a:A:K$ is valid under $\Gamma$, or raises an exception if there - is no such judgement. A judgement is just a record type containing - the three terms $a$, $A$ and $K$.} -\fun{val Tacmach.pf\_infexecute : \\ - \qquad -goal sigma -> constr -> judgement * information} - {\\ In addition to the typing judgement, this function also extracts - the $F_{\omega}$ program underlying the term.} -\fun{val Tacmach.pf\_type\_of : goal sigma -> constr -> constr} - {Infers a term $A$ such that $\Gamma\vdash a:A$ for a given term - $a$, where $\Gamma$ is the local typing context of the goal.} -\fun{val Tacmach.pf\_check\_type : goal sigma -> constr -> constr -> bool} - {This function yields a type $A$ if the two given terms $a$ and $A$ verify $\Gamma\vdash - a:A$ in the local typing context $\Gamma$ of the goal. Otherwise, - it raises an exception.} -\fun{val Tacmach.pf\_constr\_of\_com : goal sigma -> CoqAst.t -> constr} - {Transforms an abstract syntax tree into a well-typed term of the - language of constructions. Raises an exception if the term cannot - be typed.} -\fun{val Tacmach.pf\_constr\_of\_com\_sort : goal sigma -> CoqAst.t -> constr} - {Transforms an abstract syntax tree representing a type into - a well-typed term of the language of constructions. Raises an - exception if the term cannot be typed.} -\fun{val Tacmach.pf\_parse\_const : goal sigma -> string -> constr} - {Constructs the constant whose name is the given string.} -\fun{val -Tacmach.pf\_reduction\_of\_redexp : \\ - \qquad goal sigma -> red\_expr -> constr -> constr} - {\\ Applies a certain kind of reduction function, specified by an - element of the type red\_expr.} -\fun{val Tacmach.pf\_conv\_x : goal sigma -> constr -> constr -> bool} - {Test whether two given terms are definitionally equal.} -\end{description} - -\subsection[Patterns]{Patterns\label{Patterns}} - -The \ocaml{} file \texttt{Pattern} provides a quick way for describing a -term pattern and performing second-order, binding-preserving, matching -on it. Patterns are described using an extension of \Coq's concrete -syntax, where the second-order meta-variables of the pattern are -denoted by indexed question marks. - -Patterns may depend on constants, and therefore only to make have -sense when certain theories have been loaded. For this reason, they -are stored with a \textsl{module-marker}, telling us which modules -have to be open in order to use the pattern. The following functions -can be used to store and retrieve patterns form the pattern table: - -\begin{description} -\fun{val Pattern.make\_module\_marker : string list -> module\_mark} - {Constructs a module marker from a list of module names.} -\fun{val Pattern.put\_pat : module\_mark -> string -> marked\_term} - {Constructs a pattern from a parseable string containing holes - and a module marker.} -\fun{val Pattern.somatches : constr -> marked\_term-> bool} - {Tests if a term matches a pattern.} -\fun{val dest\_somatch : constr -> marked\_term -> constr list} - {If the term matches the pattern, yields the list of sub-terms - matching the occurrences of the pattern variables (ordered from - left to right). Raises a \texttt{UserError} exception if the term - does not match the pattern.} -\fun{val Pattern.soinstance : marked\_term -> constr list -> constr} - {Substitutes each hole in the pattern - by the corresponding term of the given the list.} -\end{description} - -\paragraph{Warning:} Sometimes, a \Coq\ term may have invisible -sub-terms that the matching functions are nevertheless sensible to. -For example, the \Coq\ term $(?_1,?_2)$ is actually a shorthand for -the expression $(\texttt{pair}\;?\;?\;?_1\;?_2)$. -Hence, matching this term pattern -with the term $(\texttt{true},\texttt{O})$ actually yields the list -$[?;?;\texttt{true};\texttt{O}]$ as result (and \textbf{not} -$[\texttt{true};\texttt{O}]$, as could be expected). - -\subsection{Patterns on Inductive Definitions} - -The module \texttt{Pattern} also includes some functions for testing -if the definition of an inductive type satisfies certain -properties. Such functions may be used to perform pattern matching -independently from the name given to the inductive type and the -universe it inhabits. They yield the value $(\texttt{Some}\;r::l)$ if -the input term reduces into an application of an inductive type $r$ to -a list of terms $l$, and the definition of $r$ satisfies certain -conditions. Otherwise, they yield the value \texttt{None}. - -\begin{description} -\fun{val Pattern.match\_with\_non\_recursive\_type : constr list option} - {Tests if the inductive type $r$ has no recursive constructors} -\fun{val Pattern.match\_with\_disjunction : constr list option} - {Tests if the inductive type $r$ is a non-recursive type - such that all its constructors have a single argument.} -\fun{val Pattern.match\_with\_conjunction : constr list option} - {Tests if the inductive type $r$ is a non-recursive type - with a unique constructor.} -\fun{val Pattern.match\_with\_empty\_type : constr list option} - {Tests if the inductive type $r$ has no constructors at all} -\fun{val Pattern.match\_with\_equation : constr list option} - {Tests if the inductive type $r$ has a single constructor - expressing the property of reflexivity for some type. For - example, the types $a=b$, $A\mbox{==}B$ and $A\mbox{===}B$ satisfy - this predicate.} -\end{description} - -\subsection{Elimination Tacticals} - -It is frequently the case that the subgoals generated by an -elimination can all be solved in a similar way, possibly parametrized -on some information about each case, like for example: -\begin{itemize} -\item the inductive type of the object being eliminated; -\item its arguments (if it is an inductive predicate); -\item the branch number; -\item the predicate to be proven; -\item the number of assumptions to be introduced by the case -\item the signature of the branch, i.e., for each argument of -the branch whether it is recursive or not. -\end{itemize} - -The following tacticals can be useful to deal with such situations. -They - -\begin{description} -\fun{val Elim.simple\_elimination\_then : \\ \qquad -(branch\_args -> tactic) -> constr -> tactic} - {\\ Performs the default elimination on the last argument, and then - tries to solve the generated subgoals using a given parametrized - tactic. The type branch\_args is a record type containing all - information mentioned above.} -\fun{val Elim.simple\_case\_then : \\ \qquad -(branch\_args -> tactic) -> constr -> tactic} - {\\ Similarly, but it performs case analysis instead of induction.} -\end{description} - -\section[A Complete Example]{A Complete Example\label{ACompleteExample}} - -In order to illustrate the implementation of a new tactic, let us come -back to the problem of deciding the equality of two elements of an -inductive type. - -\subsection{Preliminaries} - -Let us call \texttt{newtactic} the directory that will contain the -implementation of the new tactic. In this directory will lay two -files: a file \texttt{eqdecide.ml}, containing the \ocaml{} sources that -implements the tactic, and a \Coq\ file \texttt{Eqdecide.v}, containing -its associated grammar rules and the commands to generate a module -that can be loaded dynamically from \Coq's toplevel. - -To compile our project, we will create a \texttt{Makefile} with the -command \texttt{do\_Makefile} (see Section~\ref{Makefile}) : - -\begin{quotation} - \texttt{do\_Makefile eqdecide.ml EqDecide.v > Makefile}\\ - \texttt{touch .depend}\\ - \texttt{make depend} -\end{quotation} - -We must have kept the sources of \Coq{} somewhere and to set an -environment variable \texttt{COQTOP} that points to that directory. - -\subsection{Implementing the Tactic} - -The file \texttt{eqdecide.ml} contains the implementation of the -tactic in \ocaml{}. Let us recall the main steps of the proof strategy -for deciding the proposition $(x,y:R)\{x=y\}+\{\neg x=y\}$ on the -inductive type $R$: -\begin{enumerate} -\item Eliminate $x$ and then $y$. -\item Try discrimination to solve those goals where $x$ and $y$ has -been introduced by different constructors. -\item If $x$ and $y$ have been introduced by the same constructor, - then analyze one by one the corresponding pairs of arguments. - If they are equal, rewrite one into the other. If they are - not, derive a contradiction from the invectiveness of the - constructor. -\item Once all the arguments have been rewritten, solve the left half -of the goal by reflexivity. -\end{enumerate} - -In the sequel we implement these steps one by one. We start opening -the modules necessary for the implementation of the tactic: - -\begin{verbatim} -open Names -open Term -open Tactics -open Tacticals -open Hiddentac -open Equality -open Auto -open Pattern -open Names -open Termenv -open Std -open Proof_trees -open Tacmach -\end{verbatim} - -The first step of the procedure can be straightforwardly implemented as -follows: - -\begin{verbatim} -let clear_last = (tclLAST_HYP (fun c -> (clear_one (destVar c))));; -\end{verbatim} - -\begin{verbatim} -let mkBranches = - (tclTHEN intro - (tclTHEN (tclLAST_HYP h_simplest_elim) - (tclTHEN clear_last - (tclTHEN intros - (tclTHEN (tclLAST_HYP h_simplest_case) - (tclTHEN clear_last - intros))))));; -\end{verbatim} - -Notice the use of the tactical \texttt{tclLAST\_HYP}, which avoids to -give a (potentially clashing) name to the quantified variables of the -goal when they are introduced. - -The second step of the procedure is implemented by the following -tactic: - -\begin{verbatim} -let solveRightBranch = (tclTHEN simplest_right discrConcl);; -\end{verbatim} - -In order to illustrate how the implementation of a tactic can be -hidden, let us do it with the tactic above: - -\begin{verbatim} -let h_solveRightBranch = - hide_atomic_tactic "solveRightBranch" solveRightBranch -;; -\end{verbatim} - -As it was already mentioned in Section \ref{WhatIsATactic}, the -combinator \texttt{hide\_atomic\_tactic} first registers the tactic -\texttt{solveRightBranch} in the table, and returns a tactic which -calls the interpreter with the used to register it. Hence, when the -tactical \texttt{Info} is used, our tactic will just inform that -\texttt{solveRightBranch} was applied, omitting all the details -corresponding to \texttt{simplest\_right} and \texttt{discrConcl}. - - - -The third step requires some auxiliary functions for constructing the -type $\{c_1=c_2\}+\{\neg c_1=c_2\}$ for a given inductive type $R$ and -two constructions $c_1$ and $c_2$, and for generalizing this type over -$c_1$ and $c_2$: - -\begin{verbatim} -let mmk = make_module_marker ["#Logic.obj";"#Specif.obj"];; -let eqpat = put_pat mmk "eq";; -let sumboolpat = put_pat mmk "sumbool";; -let notpat = put_pat mmk "not";; -let eq = get_pat eqpat;; -let sumbool = get_pat sumboolpat;; -let not = get_pat notpat;; - -let mkDecideEqGoal rectype c1 c2 g = - let equality = mkAppL [eq;rectype;c1;c2] in - let disequality = mkAppL [not;equality] - in mkAppL [sumbool;equality;disequality] -;; -let mkGenDecideEqGoal rectype g = - let hypnames = ids_of_sign (pf_hyps g) in - let xname = next_ident_away (id_of_string "x") hypnames - and yname = next_ident_away (id_of_string "y") hypnames - in (mkNamedProd xname rectype - (mkNamedProd yname rectype - (mkDecideEqGoal rectype (mkVar xname) (mkVar yname) g))) -;; -\end{verbatim} - -The tactic will depend on the \Coq modules \texttt{Logic} and -\texttt{Specif}, since we use the constants corresponding to -propositional equality (\texttt{eq}), computational disjunction -(\texttt{sumbool}), and logical negation (\texttt{not}), defined in -that modules. This is specified creating the module maker -\texttt{mmk} (see Section~\ref{Patterns}). - -The third step of the procedure can be divided into three sub-steps. -Assume that both $x$ and $y$ have been introduced by the same -constructor. For each corresponding pair of arguments of that -constructor, we have to consider whether they are equal or not. If -they are equal, the following tactic is applied to rewrite one into -the other: - -\begin{verbatim} -let eqCase tac = - (tclTHEN intro - (tclTHEN (tclLAST_HYP h_rewriteLR) - (tclTHEN clear_last - tac))) -;; -\end{verbatim} - - -If they are not equal, then the goal is contraposed and a -contradiction is reached form the invectiveness of the constructor: - -\begin{verbatim} -let diseqCase = - let diseq = (id_of_string "diseq") in - let absurd = (id_of_string "absurd") - in (tclTHEN (intro_using diseq) - (tclTHEN h_simplest_right - (tclTHEN red_in_concl - (tclTHEN (intro_using absurd) - (tclTHEN (h_simplest_apply (mkVar diseq)) - (tclTHEN (h_injHyp absurd) - trivial )))))) -;; -\end{verbatim} - -In the tactic above we have chosen to name the hypotheses because -they have to be applied later on. This introduces a potential risk -of name clashing if the context already contains other hypotheses -also named ``diseq'' or ``absurd''. - -We are now ready to implement the tactic \textsl{SolveArg}. Given the -two arguments $a_1$ and $a_2$ of the constructor, this tactic cuts the -goal with the proposition $\{a_1=a_2\}+\{\neg a_1=a_2\}$, and then -applies the tactics above to each of the generated cases. If the -disjunction cannot be solved automatically, it remains as a sub-goal -to be proven. - -\begin{verbatim} -let solveArg a1 a2 tac g = - let rectype = pf_type_of g a1 in - let decide = mkDecideEqGoal rectype a1 a2 g - in (tclTHENS (h_elimType decide) - [(eqCase tac);diseqCase;default_auto]) g -;; -\end{verbatim} - -The following tactic implements the third and fourth steps of the -proof procedure: - -\begin{verbatim} -let conclpatt = put_pat mmk "{<?1>?2=?3}+{?4}" -;; -let solveLeftBranch rectype g = - let (_::(lhs::(rhs::_))) = - try (dest_somatch (pf_concl g) conclpatt) - with UserError ("somatch",_)-> error "Unexpected conclusion!" in - let nparams = mind_nparams rectype in - let getargs l = snd (chop_list nparams (snd (decomp_app l))) in - let rargs = getargs rhs - and largs = getargs lhs - in List.fold_right2 - solveArg largs rargs (tclTHEN h_simplest_left h_reflexivity) g -;; -\end{verbatim} - -Notice the use of a pattern to decompose the goal and obtain the -inductive type and the left and right hand sides of the equality. A -certain number of arguments correspond to the general parameters of -the type, and must be skipped over. Once the corresponding list of -arguments \texttt{rargs} and \texttt{largs} have been obtained, the -tactic \texttt{solveArg} is iterated on them, leaving a disjunction -whose left half can be solved by reflexivity. - -The following tactic joints together the three steps of the -proof procedure: - -\begin{verbatim} -let initialpatt = put_pat mmk "(x,y:?1){<?1>x=y}+{~(<?1>x=y)}" -;; -let decideGralEquality g = - let (typ::_) = try (dest_somatch (pf_concl g) initialpatt) - with UserError ("somatch",_) -> - error "The goal does not have the expected form" in - let headtyp = hd_app (pf_compute g typ) in - let rectype = match (kind_of_term headtyp) with - IsMutInd _ -> headtyp - | _ -> error ("This decision procedure only" - " works for inductive objects") - in (tclTHEN mkBranches - (tclORELSE h_solveRightBranch (solveLeftBranch rectype))) g -;; -;; -\end{verbatim} - -The tactic above can be specialized in two different ways: either to -decide a particular instance $\{c_1=c_2\}+\{\neg c_1=c_2\}$ of the -universal quantification; or to eliminate this property and obtain two -subgoals containing the hypotheses $c_1=c_2$ and $\neg c_1=c_2$ -respectively. - -\begin{verbatim} -let decideGralEquality = - (tclTHEN mkBranches (tclORELSE h_solveRightBranch solveLeftBranch)) -;; -let decideEquality c1 c2 g = - let rectype = pf_type_of g c1 in - let decide = mkGenDecideEqGoal rectype g - in (tclTHENS (cut decide) [default_auto;decideGralEquality]) g -;; -let compare c1 c2 g = - let rectype = pf_type_of g c1 in - let decide = mkDecideEqGoal rectype c1 c2 g - in (tclTHENS (cut decide) - [(tclTHEN intro - (tclTHEN (tclLAST_HYP simplest_case) - clear_last)); - decideEquality c1 c2]) g -;; -\end{verbatim} - -Next, for each of the tactics that will have an entry in the grammar -we construct the associated dynamic one to be registered in the table -of tactics. This function can be used to overload a tactic name with -several similar tactics. For example, the tactic proving the general -decidability property and the one proving a particular instance for -two terms can be grouped together with the following convention: if -the user provides two terms as arguments, then the specialized tactic -is used; if no argument is provided then the general tactic is invoked. - -\begin{verbatim} -let dyn_decideEquality args g = - match args with - [(COMMAND com1);(COMMAND com2)] -> - let c1 = pf_constr_of_com g com1 - and c2 = pf_constr_of_com g com2 - in decideEquality c1 c2 g - | [] -> decideGralEquality g - | _ -> error "Invalid arguments for dynamic tactic" -;; -add_tactic "DecideEquality" dyn_decideEquality -;; - -let dyn_compare args g = - match args with - [(COMMAND com1);(COMMAND com2)] -> - let c1 = pf_constr_of_com g com1 - and c2 = pf_constr_of_com g com2 - in compare c1 c2 g - | _ -> error "Invalid arguments for dynamic tactic" -;; -add_tactic "Compare" tacargs_compare -;; -\end{verbatim} - -This completes the implementation of the tactic. We turn now to the -\Coq file \texttt{Eqdecide.v}. - - -\subsection{The Grammar Rules} - -Associated to the implementation of the tactic there is a \Coq\ file -containing the grammar and pretty-printing rules for the new tactic, -and the commands to generate an object module that can be then loaded -dynamically during a \Coq\ session. In order to generate an ML module, -the \Coq\ file must contain a -\texttt{Declare ML module} command for all the \ocaml{} files concerning -the implementation of the tactic --in our case there is only one file, -the file \texttt{eqdecide.ml}: - -\begin{verbatim} -Declare ML Module "eqdecide". -\end{verbatim} - -The following grammar and pretty-printing rules are -self-explanatory. We refer the reader to the Section \ref{Grammar} for -the details: - -\begin{verbatim} -Grammar tactic simple_tactic := - EqDecideRuleG1 - [ "Decide" "Equality" comarg($com1) comarg($com2)] -> - [(DecideEquality $com1 $com2)] -| EqDecideRuleG2 - [ "Decide" "Equality" ] -> - [(DecideEquality)] -| CompareRule - [ "Compare" comarg($com1) comarg($com2)] -> - [(Compare $com1 $com2)]. - -Syntax tactic level 0: - EqDecideRulePP1 - [(DecideEquality)] -> - ["Decide" "Equality"] -| EqDecideRulePP2 - [(DecideEquality $com1 $com2)] -> - ["Decide" "Equality" $com1 $com2] -| ComparePP - [(Compare $com1 $com2)] -> - ["Compare" $com1 $com2]. -\end{verbatim} - - -\paragraph{Important:} The names used to label the abstract syntax tree -in the grammar rules ---in this case ``DecideEquality'' and -``Compare''--- must be the same as the name used to register the -tactic in the tactics table. This is what makes the links between the -input entered by the user and the tactic executed by the interpreter. - -\subsection{Loading the Tactic} - -Once the module \texttt{EqDecide.v} has been compiled, the tactic can -be dynamically loaded using the \texttt{Require} command. - -\begin{coq_example} -Require EqDecide. -Goal (x,y:nat){x=y}+{~x=y}. -Decide Equality. -\end{coq_example} - -The implementation of the tactic can be accessed through the -tactical \texttt{Info}: -\begin{coq_example} -Undo. -Info Decide Equality. -\end{coq_example} -\begin{coq_eval} -Abort. -\end{coq_eval} - -Remark that the task performed by the tactic \texttt{solveRightBranch} -is not displayed, since we have chosen to hide its implementation. - -\section[Testing and Debugging your Tactic]{Testing and Debugging your Tactic\label{test-and-debug}} - -When your tactic does not behave as expected, it is possible to trace -it dynamically from \Coq. In order to do this, you have first to leave -the toplevel of \Coq, and come back to the \ocaml{} interpreter. This can -be done using the command \texttt{Drop} (see Section~\ref{Drop}). Once -in the \ocaml{} toplevel, load the file \texttt{tactics/include.ml}. -This file installs several pretty printers for proof trees, goals, -terms, abstract syntax trees, names, etc. It also contains the -function \texttt{go:unit -> unit} that enables to go back to \Coq's -toplevel. - -The modules \texttt{Tacmach} and \texttt{Pfedit} contain some basic -functions for extracting information from the state of the proof -engine. Such functions can be used to debug your tactic if -necessary. Let us mention here some of them: - -\begin{description} -\fun{val get\_pftreestate : unit -> pftreestate} - {Projects the current state of the proof engine.} -\fun{val proof\_of\_pftreestate : pftreestate -> proof} - {Projects the current state of the proof tree. A pretty-printer - displays it in a readable form. } -\fun{val top\_goal\_of\_pftreestate : pftreestate -> goal sigma} - {Projects the goal and the existential variables mapping from - the current state of the proof engine.} -\fun{val nth\_goal\_of\_pftreestate : int -> pftreestate -> goal sigma} - {Projects the goal and mapping corresponding to the $nth$ subgoal - that remains to be proven} -\fun{val traverse : int -> pftreestate -> pftreestate} - {Yields the children of the node that the current state of the - proof engine points to.} -\fun{val solve\_nth\_pftreestate : \\ \qquad -int -> tactic -> pftreestate -> pftreestate} - {\\ Provides the new state of the proof engine obtained applying - a given tactic to some unproven sub-goal.} -\end{description} - -Finally, the traditional \ocaml{} debugging tools like the directives -\texttt{trace} and \texttt{untrace} can be used to follow the -execution of your functions. Frequently, a better solution is to use -the \ocaml{} debugger, see Chapter \ref{Utilities}. - -\section[Concrete syntax for ML tactic and vernacular command]{Concrete syntax for ML tactic and vernacular command\label{Notations-for-ML-command}} - -\subsection{The general case} - -The standard way to bind an ML-written tactic or vernacular command to -a concrete {\Coq} syntax is to use the -\verb=TACTIC EXTEND= and \verb=VERNAC COMMAND EXTEND= macros. - -These macros can be used in any {\ocaml} file defining a (new) ML tactic -or vernacular command. They are expanded into pure {\ocaml} code by -the {\camlpppp} preprocessor of {\ocaml}. Concretely, files that use -these macros need to be compiled by giving to {\tt ocamlc} the option - -\verb=-pp "camlp4o -I $(COQTOP)/parsing grammar.cma pa_extend.cmo"= - -\noindent which is the default for every file compiled by means of a Makefile -generated by {\tt coq\_makefile} (see Chapter~\ref{Addoc-coqc}). So, -just do \verb=make= in this latter case. - -The syntax of the macros is given on figure -\ref{EXTEND-syntax}. They can be used at any place of an {\ocaml} -files where an ML sentence (called \verb=str_item= in the {\tt ocamlc} -parser) is expected. For each rule, the left-hand-side describes the -grammar production and the right-hand-side its interpretation which -must be an {\ocaml} expression. Each grammar production starts with -the concrete name of the tactic or command in {\Coq} and is followed -by arguments, possibly separated by terminal symbols or words. -Here is an example: - -\begin{verbatim} -TACTIC EXTEND Replace - [ "replace" constr(c1) "with" constr(c2) ] -> [ replace c1 c2 ] -END -\end{verbatim} - -\newcommand{\grule}{\textrm{\textsl{rule}}} -\newcommand{\stritem}{\textrm{\textsl{ocaml\_str\_item}}} -\newcommand{\camlexpr}{\textrm{\textsl{ocaml\_expr}}} -\newcommand{\arginfo}{\textrm{\textsl{argument\_infos}}} -\newcommand{\lident}{\textrm{\textsl{lower\_ident}}} -\newcommand{\argument}{\textrm{\textsl{argument}}} -\newcommand{\entry}{\textrm{\textsl{entry}}} -\newcommand{\argtype}{\textrm{\textsl{argtype}}} - -\begin{figure} -\begin{tabular}{|lcll|} -\hline -{\stritem} - & ::= & -\multicolumn{2}{l|}{{\tt TACTIC EXTEND} {\ident} \nelist{\grule}{$|$} {\tt END}}\\ - & $|$ & \multicolumn{2}{l|}{{\tt VERNAC COMMAND EXTEND} {\ident} \nelist{\grule}{$|$} {\tt END}}\\ -&&\multicolumn{2}{l|}{}\\ -{\grule} & ::= & -\multicolumn{2}{l|}{{\tt [} {\str} \sequence{\argument}{} {\tt ] -> [} {\camlexpr} {\tt ]}}\\ -&&\multicolumn{2}{l|}{}\\ -{\argument} & ::= & {\str} &\mbox{(terminal)}\\ - & $|$ & {\entry} {\tt (} {\lident} {\tt )} &\mbox{(non-terminal)}\\ -&&\multicolumn{2}{l|}{}\\ -{\entry} - & ::= & {\tt string} & (a string)\\ - & $|$ & {\tt preident} & (an identifier typed as a {\tt string})\\ - & $|$ & {\tt ident} & (an identifier of type {\tt identifier})\\ - & $|$ & {\tt global} & (a qualified identifier)\\ - & $|$ & {\tt constr} & (a {\Coq} term)\\ - & $|$ & {\tt openconstr} & (a {\Coq} term with holes)\\ - & $|$ & {\tt sort} & (a {\Coq} sort)\\ - & $|$ & {\tt tactic} & (an ${\cal L}_{tac}$ expression)\\ - & $|$ & {\tt constr\_with\_bindings} & (a {\Coq} term with a list of bindings\footnote{as for the tactics {\tt apply} and {\tt elim}})\\ - & $|$ & {\tt int\_or\_var} & (an integer or an identifier denoting an integer)\\ - & $|$ & {\tt quantified\_hypothesis} & (a quantified hypothesis\footnote{as for the tactics {\tt intros until}})\\ - & $|$ & {\tt {\entry}\_opt} & (an optional {\entry} )\\ - & $|$ & {\tt ne\_{\entry}\_list} & (a non empty list of {\entry})\\ - & $|$ & {\tt {\entry}\_list} & (a list of {\entry})\\ - & $|$ & {\tt bool} & (a boolean: no grammar rule, just for typing)\\ - & $|$ & {\lident} & (a user-defined entry)\\ -\hline -\end{tabular} -\caption{Syntax of the macros binding {\ocaml} tactics or commands to a {\Coq} syntax} -\label{EXTEND-syntax} -\end{figure} - -There is a set of predefined non-terminal entries which are -automatically translated into an {\ocaml} object of a given type. The -type is not the same for tactics and for vernacular commands. It is -given in the following table: - -\begin{small} -\noindent \begin{tabular}{|l|l|l|} -\hline -{\entry} & {\it type for tactics} & {\it type for commands} \\ -{\tt string} & {\tt string} & {\tt string}\\ -{\tt preident} & {\tt string} & {\tt string}\\ -{\tt ident} & {\tt identifier} & {\tt identifier}\\ -{\tt global} & {\tt global\_reference} & {\tt qualid}\\ -{\tt constr} & {\tt constr} & {\tt constr\_expr}\\ -{\tt openconstr} & {\tt open\_constr} & {\tt constr\_expr}\\ -{\tt sort} & {\tt sorts} & {\tt rawsort}\\ -{\tt tactic} & {\tt glob\_tactic\_expr * tactic} & {\tt raw\_tactic\_expr}\\ -{\tt constr\_with\_bindings} & {\tt constr with\_bindings} & {\tt constr\_expr with\_bindings}\\\\ -{\tt int\_or\_var} & {\tt int or\_var} & {\tt int or\_var}\\ -{\tt quantified\_hypothesis} & {\tt quantified\_hypothesis} & {\tt quantified\_hypothesis}\\ -{\tt {\entry}\_opt} & {\it the type of entry} {\tt option} & {\it the type of entry} {\tt option}\\ -{\tt ne\_{\entry}\_list} & {\it the type of entry} {\tt list} & {\it the type of entry} {\tt list}\\ -{\tt {\entry}\_list} & {\it the type of entry} {\tt list} & {\it the type of entry} {\tt list}\\ -{\tt bool} & {\tt bool} & {\tt bool}\\ -{\lident} & {user-provided, cf next section} & {user-provided, cf next section}\\ -\hline -\end{tabular} -\end{small} - -\bigskip - -Notice that {\entry} consists in a single identifier and that the {\tt -\_opt}, {\tt \_list}, ... modifiers are part of the identifier. -Here is now another example of a tactic which takes either a non empty -list of identifiers and executes the {\ocaml} function {\tt subst} or -takes no arguments and executes the{\ocaml} function {\tt subst\_all}. - -\begin{verbatim} -TACTIC EXTEND Subst -| [ "subst" ne_ident_list(l) ] -> [ subst l ] -| [ "subst" ] -> [ subst_all ] -END -\end{verbatim} - -\subsection{Adding grammar entries for tactic or command arguments} - -In case parsing the arguments of the tactic or the vernacular command -involves grammar entries other than the predefined entries listed -above, you have to declare a new entry using the macros -\verb=ARGUMENT EXTEND= or \verb=VERNAC ARGUMENT EXTEND=. The syntax is -given on Figure~\ref{ARGUMENT-EXTEND-syntax}. Notice that arguments -declared by \verb=ARGUMENT EXTEND= can be used for arguments of both -tactics and vernacular commands while arguments declared by -\verb=VERNAC ARGUMENT EXTEND= can only be used by vernacular commands. - -For \verb=VERNAC ARGUMENT EXTEND=, the identifier is the name of the -entry and it must be a valid {\ocaml} identifier (especially it must -be lowercase). The grammar rules works as before except that they do -not have to start by a terminal symbol or word. As an example, here -is how the {\Coq} {\tt Extraction Language {\it language}} parses its -argument: - -\begin{verbatim} -VERNAC ARGUMENT EXTEND language -| [ "Ocaml" ] -> [ Ocaml ] -| [ "Haskell" ] -> [ Haskell ] -| [ "Scheme" ] -> [ Scheme ] -END -\end{verbatim} - -For tactic arguments, and especially for \verb=ARGUMENT EXTEND=, the -procedure is more subtle because tactics are objects of the {\Coq} -environment which can be printed and interpreted. Then the syntax -requires extra information providing a printer and a type telling how -the argument behaves. Here is an example of entry parsing a pair of -optional {\Coq} terms. - -\begin{verbatim} -let pp_minus_div_arg pr_constr pr_tactic (omin,odiv) = - if omin=None && odiv=None then mt() else - spc() ++ str "with" ++ - pr_opt (fun c -> str "minus := " ++ pr_constr c) omin ++ - pr_opt (fun c -> str "div := " ++ pr_constr c) odiv - -ARGUMENT EXTEND minus_div_arg - TYPED AS constr_opt * constr_opt - PRINTED BY pp_minus_div_arg -| [ "with" minusarg(m) divarg_opt(d) ] -> [ Some m, d ] -| [ "with" divarg(d) minusarg_opt(m) ] -> [ m, Some d ] -| [ ] -> [ None, None ] -END -\end{verbatim} - -Notice that the type {\tt constr\_opt * constr\_opt} tells that the -object behaves as a pair of optional {\Coq} terms, i.e. as an object -of {\ocaml} type {\tt constr option * constr option} if in a -\verb=TACTIC EXTEND= macro and of type {\tt constr\_expr option * -constr\_expr option} if in a \verb=VERNAC COMMAND EXTEND= macro. - -As for the printer, it must be a function expecting a printer for -terms, a printer for tactics and returning a printer for the created -argument. Especially, each sub-{\term} and each sub-{\tac} in the -argument must be typed by the corresponding printers. Otherwise, the -{\ocaml} code will not be well-typed. - -\Rem The entry {\tt bool} is bound to no syntax but it can be used to -give the type of an argument as in the following example: - -\begin{verbatim} -let pr_orient _prc _prt = function - | true -> mt () - | false -> str " <-" - -ARGUMENT EXTEND orient TYPED AS bool PRINTED BY pr_orient -| [ "->" ] -> [ true ] -| [ "<-" ] -> [ false ] -| [ ] -> [ true ] -END -\end{verbatim} - -\begin{figure} -\begin{tabular}{|lcl|} -\hline -{\stritem} & ::= & - {\tt ARGUMENT EXTEND} {\ident} {\arginfo} {\nelist{\grule}{$|$}} {\tt END}\\ -& $|$ & {\tt VERNAC ARGUMENT EXTEND} {\ident} {\nelist{\grule}{$|$}} {\tt END}\\ -\\ -{\arginfo} & ::= & {\tt TYPED AS} {\argtype} \\ -&& {\tt PRINTED BY} {\lident} \\ -%&& \zeroone{{\tt INTERPRETED BY} {\lident}}\\ -%&& \zeroone{{\tt GLOBALIZED BY} {\lident}}\\ -%&& \zeroone{{\tt SUBSTITUTED BY} {\lident}}\\ -%&& \zeroone{{\tt RAW\_TYPED AS} {\lident} {\tt RAW\_PRINTED BY} {\lident}}\\ -%&& \zeroone{{\tt GLOB\_TYPED AS} {\lident} {\tt GLOB\_PRINTED BY} {\lident}}\\ -\\ -{\argtype} & ::= & {\argtype} {\tt *} {\argtype} \\ -& $|$ & {\entry} \\ -\hline -\end{tabular} -\caption{Syntax of the macros binding {\ocaml} tactics or commands to a {\Coq} syntax} -\label{ARGUMENT-EXTEND-syntax} -\end{figure} - -%\end{document} diff --git a/doc/refman/RefMan-uti.tex b/doc/refman/RefMan-uti.tex deleted file mode 100644 index 5f201b67..00000000 --- a/doc/refman/RefMan-uti.tex +++ /dev/null @@ -1,272 +0,0 @@ -\chapter[Utilities]{Utilities\label{Utilities}} - -The distribution provides utilities to simplify some tedious works -beside proof development, tactics writing or documentation. - -\section[Building a toplevel extended with user tactics]{Building a toplevel extended with user tactics\label{Coqmktop}\index{Coqmktop@{\tt coqmktop}}} - -The native-code version of \Coq\ cannot dynamically load user tactics -using Objective Caml code. It is possible to build a toplevel of \Coq, -with Objective Caml code statically linked, with the tool {\tt - coqmktop}. - -For example, one can build a native-code \Coq\ toplevel extended with a tactic -which source is in {\tt tactic.ml} with the command -\begin{verbatim} - % coqmktop -opt -o mytop.out tactic.cmx -\end{verbatim} -where {\tt tactic.ml} has been compiled with the native-code -compiler {\tt ocamlopt}. This command generates an executable -called {\tt mytop.out}. To use this executable to compile your \Coq\ -files, use {\tt coqc -image mytop.out}. - -A basic example is the native-code version of \Coq\ ({\tt coqtop.opt}), -which can be generated by {\tt coqmktop -opt -o coqopt.opt}. - - -\paragraph[Application: how to use the Objective Caml debugger with Coq.]{Application: how to use the Objective Caml debugger with Coq.\index{Debugger}} - -One useful application of \texttt{coqmktop} is to build a \Coq\ toplevel in -order to debug your tactics with the Objective Caml debugger. -You need to have configured and compiled \Coq\ for debugging -(see the file \texttt{INSTALL} included in the distribution). -Then, you must compile the Caml modules of your tactic with the -option \texttt{-g} (with the bytecode compiler) and build a stand-alone -bytecode toplevel with the following command: - -\begin{quotation} -\texttt{\% coqmktop -g -o coq-debug}~\emph{<your \texttt{.cmo} files>} -\end{quotation} - - -To launch the \ocaml\ debugger with the image you need to execute it in -an environment which correctly sets the \texttt{COQLIB} variable. -Moreover, you have to indicate the directories in which -\texttt{ocamldebug} should search for Caml modules. - -A possible solution is to use a wrapper around \texttt{ocamldebug} -which detects the executables containing the word \texttt{coq}. In -this case, the debugger is called with the required additional -arguments. In other cases, the debugger is simply called without additional -arguments. Such a wrapper can be found in the \texttt{dev/} -subdirectory of the sources. - -\section[Modules dependencies]{Modules dependencies\label{Dependencies}\index{Dependencies} - \index{Coqdep@{\tt coqdep}}} - -In order to compute modules dependencies (so to use {\tt make}), -\Coq\ comes with an appropriate tool, {\tt coqdep}. - -{\tt coqdep} computes inter-module dependencies for \Coq\ and -\ocaml\ programs, and prints the dependencies on the standard -output in a format readable by make. When a directory is given as -argument, it is recursively looked at. - -Dependencies of \Coq\ modules are computed by looking at {\tt Require} -commands ({\tt Require}, {\tt Requi\-re Export}, {\tt Require Import}, -but also at the command {\tt Declare ML Module}. - -Dependencies of \ocaml\ modules are computed by looking at -\verb!open! commands and the dot notation {\em module.value}. However, -this is done approximatively and you are advised to use {\tt ocamldep} -instead for the \ocaml\ modules dependencies. - -See the man page of {\tt coqdep} for more details and options. - - -\section[Creating a {\tt Makefile} for \Coq\ modules]{Creating a {\tt Makefile} for \Coq\ modules\label{Makefile} -\index{Makefile@{\tt Makefile}} -\index{CoqMakefile@{\tt coq\_Makefile}}} - -When a proof development becomes large and is split into several files, -it becomes crucial to use a tool like {\tt make} to compile \Coq\ -modules. - -The writing of a generic and complete {\tt Makefile} may be a tedious work -and that's why \Coq\ provides a tool to automate its creation, -{\tt coq\_makefile}. Given the files to compile, the command {\tt -coq\_makefile} prints a -{\tt Makefile} on the standard output. So one has just to run the -command: - -\begin{quotation} -\texttt{\% coq\_makefile} {\em file$_1$.v \dots\ file$_n$.v} \texttt{> Makefile} -\end{quotation} - -The resulted {\tt Makefile} has a target {\tt depend} which computes the -dependencies and puts them in a separate file {\tt .depend}, which is -included by the {\tt Makefile}. -Therefore, you should create such a file before the first invocation -of make. You can for instance use the command - -\begin{quotation} -\texttt{\% touch .depend} -\end{quotation} - -Then, to initialize or update the modules dependencies, type in: - -\begin{quotation} -\texttt{\% make depend} -\end{quotation} - -There is a target {\tt all} to compile all the files {\em file$_1$ -\dots\ file$_n$}, and a generic target to produce a {\tt .vo} file from -the corresponding {\tt .v} file (so you can do {\tt make} {\em file}{\tt.vo} -to compile the file {\em file}{\tt.v}). - -{\tt coq\_makefile} can also handle the case of ML files and -subdirectories. For more options type - -\begin{quotation} -\texttt{\% coq\_makefile --help} -\end{quotation} - -\Warning To compile a project containing \ocaml{} files you must keep -the sources of \Coq{} somewhere and have an environment variable named -\texttt{COQTOP} that points to that directory. - -% \section{{\sf Coq\_SearchIsos}: information retrieval in a \Coq\ proofs -% library} -% \label{coqsearchisos} -% \index{Coq\_SearchIsos@{\sf Coq\_SearchIsos}} - -% In the \Coq\ distribution, there is also a separated and independent tool, -% called {\sf Coq\_SearchIsos}, which allows the search in accordance with {\tt -% SearchIsos}\index{SearchIsos@{\tt SearchIsos}} (see Section~\ref{searchisos}) -% in a \Coq\ proofs library. More precisely, this program begins, once launched -% by {\tt coqtop -searchisos}\index{coqtopsearchisos@{\tt -% coqtop -searchisos}}, loading lightly (by using specifications functions) -% all the \Coq\ objects files ({\tt .vo}) accessible by the {\tt LoadPath} (see -% Section~\ref{loadpath}). Next, a prompt appears and four commands are then -% available: - -% \begin{description} -% \item [{\tt SearchIsos}]\ \\ -% Scans the fixed context. -% \item [{\tt Time}]\index{Time@{\tt Time}}\ \\ -% Turns on the Time Search Display mode (see Section~\ref{time}). -% \item [{\tt Untime}]\index{Untime@{\tt Untime}}\ \\ -% Turns off the Time Search Display mode (see Section~\ref{time}). -% \item [{\tt Quit}]\index{Quit@{\tt Quit}}\ \\ -% Ends the {\tt coqtop -searchisos} session. -% \end{description} - -% When running {\tt coqtop -searchisos} you can use the two options: - -% \begin{description} -% \item[{\tt -opt}]\ \\ -% Runs the native-code version of {\sf Coq\_SearchIsos}. - -% \item[{\tt -image} {\em file}]\ \\ -% This option sets the binary image to be used to be {\em file} -% instead of the standard one. Not of general use. -% \end{description} - - -\section[Documenting \Coq\ files with coqdoc]{Documenting \Coq\ files with coqdoc\label{coqdoc} -\index{Coqdoc@{\sf coqdoc}}} - -\input{./coqdoc} - -\section{Exporting \Coq\ theories to XML} - -\input{./Helm} - -\section[Embedded \Coq\ phrases inside \LaTeX\ documents]{Embedded \Coq\ phrases inside \LaTeX\ documents\label{Latex} - \index{Coqtex@{\tt coq-tex}}\index{Latex@{\LaTeX}}} - -When writing a documentation about a proof development, one may want -to insert \Coq\ phrases inside a \LaTeX\ document, possibly together with -the corresponding answers of the system. We provide a -mechanical way to process such \Coq\ phrases embedded in \LaTeX\ files: the -{\tt coq-tex} filter. This filter extracts Coq phrases embedded in -LaTeX files, evaluates them, and insert the outcome of the evaluation -after each phrase. - -Starting with a file {\em file}{\tt.tex} containing \Coq\ phrases, -the {\tt coq-tex} filter produces a file named {\em file}{\tt.v.tex} with -the \Coq\ outcome. - -There are options to produce the \Coq\ parts in smaller font, italic, -between horizontal rules, etc. -See the man page of {\tt coq-tex} for more details. - -\medskip\noindent {\bf Remark.} This Reference Manual and the Tutorial -have been completely produced with {\tt coq-tex}. - - -\section[\Coq\ and \emacs]{\Coq\ and \emacs\label{Emacs}\index{Emacs}} - -\subsection{The \Coq\ Emacs mode} - -\Coq\ comes with a Major mode for \emacs, {\tt coq.el}. This mode provides -syntax highlighting (assuming your \emacs\ library provides -{\tt hilit19.el}) and also a rudimentary indentation facility -in the style of the Caml \emacs\ mode. - -Add the following lines to your \verb!.emacs! file: - -\begin{verbatim} - (setq auto-mode-alist (cons '("\\.v$" . coq-mode) auto-mode-alist)) - (autoload 'coq-mode "coq" "Major mode for editing Coq vernacular." t) -\end{verbatim} - -The \Coq\ major mode is triggered by visiting a file with extension {\tt .v}, -or manually with the command \verb!M-x coq-mode!. -It gives you the correct syntax table for -the \Coq\ language, and also a rudimentary indentation facility: -\begin{itemize} - \item pressing {\sc Tab} at the beginning of a line indents the line like - the line above; - - \item extra {\sc Tab}s increase the indentation level - (by 2 spaces by default); - - \item M-{\sc Tab} decreases the indentation level. -\end{itemize} - -An inferior mode to run \Coq\ under Emacs, by Marco Maggesi, is also -included in the distribution, in file \texttt{coq-inferior.el}. -Instructions to use it are contained in this file. - -\subsection[Proof General]{Proof General\index{Proof General}} - -Proof General is a generic interface for proof assistants based on -Emacs (or XEmacs). The main idea is that the \Coq\ commands you are -editing are sent to a \Coq\ toplevel running behind Emacs and the -answers of the system automatically inserted into other Emacs buffers. -Thus you don't need to copy-paste the \Coq\ material from your files -to the \Coq\ toplevel or conversely from the \Coq\ toplevel to some -files. - -Proof General is developped and distributed independently of the -system \Coq. It is freely available at \verb!proofgeneral.inf.ed.ac.uk!. - - -\section[Module specification]{Module specification\label{gallina}\index{Gallina@{\tt gallina}}} - -Given a \Coq\ vernacular file, the {\tt gallina} filter extracts its -specification (inductive types declarations, definitions, type of -lemmas and theorems), removing the proofs parts of the file. The \Coq\ -file {\em file}{\tt.v} gives birth to the specification file -{\em file}{\tt.g} (where the suffix {\tt.g} stands for \gallina). - -See the man page of {\tt gallina} for more details and options. - - -\section[Man pages]{Man pages\label{ManPages}\index{Man pages}} - -There are man pages for the commands {\tt coqdep}, {\tt gallina} and -{\tt coq-tex}. Man pages are installed at installation time -(see installation instructions in file {\tt INSTALL}, step 6). - -%BEGIN LATEX -\RefManCutCommand{ENDREFMAN=\thepage} -%END LATEX - -% $Id: RefMan-uti.tex 11975 2009-03-14 11:29:36Z letouzey $ - -%%% Local Variables: -%%% mode: latex -%%% TeX-master: t -%%% End: diff --git a/doc/refman/Reference-Manual.tex b/doc/refman/Reference-Manual.tex deleted file mode 100644 index b0ada528..00000000 --- a/doc/refman/Reference-Manual.tex +++ /dev/null @@ -1,151 +0,0 @@ -%\RequirePackage{ifpdf} -%\ifpdf -% \documentclass[11pt,a4paper,pdftex]{book} -%\else - \documentclass[11pt,a4paper]{book} -%\fi - -\usepackage[latin1]{inputenc} -\usepackage[T1]{fontenc} -\usepackage{times} -\usepackage{url} -\usepackage{verbatim} -\usepackage{amsmath} -\usepackage{amssymb} -\usepackage{alltt} -\usepackage{hevea} -\usepackage{ifpdf} -\usepackage[headings]{fullpage} -\usepackage{headers} % in this directory -\usepackage{multicol} -\usepackage{xspace} - -% for coqide -\ifpdf % si on est pas en pdflatex - \usepackage[pdftex]{graphicx} -\else - \usepackage[dvips]{graphicx} -\fi - - -%\includeonly{Setoid} - -\input{../common/version.tex} -\input{../common/macros.tex}% extension .tex pour htmlgen -\input{../common/title.tex}% extension .tex pour htmlgen -%\input{headers} - -\usepackage[linktocpage,colorlinks,bookmarks=false]{hyperref} -% The manual advises to load hyperref package last to be able to redefine -% necessary commands. -% The above should work for both latex and pdflatex. Even if PDF is produced -% through DVI and PS using dvips and ps2pdf, hyperlinks should still work. -% linktocpage option makes page numbers, not section names, to be links in -% the table of contents. -% colorlinks option colors the links instead of using boxes. - -% The command \tocnumber was added to HEVEA in version 1.06-6. -% It instructs HEVEA to put chapter numbers into the table of -% content entries. The table of content is produced by HACHA using -% the options -tocbis -o toc.html. HEVEA produces a warning when -% a command is not recognized, so versions earlier than 1.06-6 can -% still be used. -%HEVEA\tocnumber - -\begin{document} -%BEGIN LATEX -\sloppy\hbadness=5000 -%END LATEX - -%BEGIN LATEX -\coverpage{Reference Manual} -{The Coq Development Team} -{This material may be distributed only subject to the terms and -conditions set forth in the Open Publication License, v1.0 or later -(the latest version is presently available at -\url{http://www.opencontent.org/openpub}). -Options A and B of the licence are {\em not} elected.} -%END LATEX - -%\defaultheaders -\include{RefMan-int}% Introduction -\include{RefMan-pre}% Credits - -%BEGIN LATEX -\tableofcontents -%END LATEX - -\part{The language} -%BEGIN LATEX -\defaultheaders -%END LATEX -\include{RefMan-gal.v}% Gallina -\include{RefMan-ext.v}% Gallina extensions -\include{RefMan-lib.v}% The coq library -\include{RefMan-cic.v}% The Calculus of Constructions -\include{RefMan-modr}% The module system - - -\part{The proof engine} -\include{RefMan-oth.v}% Vernacular commands -\include{RefMan-pro}% Proof handling -\include{RefMan-tac.v}% Tactics and tacticals -\include{RefMan-ltac.v}% Writing tactics -\include{RefMan-tacex.v}% Detailed Examples of tactics -\include{RefMan-decl.v}% The mathematical proof language - -\part{User extensions} -\include{RefMan-syn.v}% The Syntax and the Grammad commands -%%SUPPRIME \include{RefMan-tus.v}% Writing tactics - -\part{Practical tools} -\include{RefMan-com}% The coq commands (coqc coqtop) -\include{RefMan-uti}% utilities (gallina, do_Makefile, etc) -\include{RefMan-ide}% Coq IDE - -%BEGIN LATEX -\RefManCutCommand{BEGINADDENDUM=\thepage} -%END LATEX -\part{Addendum to the Reference Manual} -\include{AddRefMan-pre}% -\include{Cases.v}% -\include{Coercion.v}% -\include{Classes.v}% -%%SUPPRIME \include{Natural.v}% -\include{Omega.v}% -\include{Micromega.v} -%%SUPPRIME \include{Correctness.v}% = preuve de pgms imperatifs -\include{Extraction.v}% -\include{Program.v}% -\include{Polynom.v}% = Ring -\include{Nsatz.v}% -\include{Setoid.v}% Tactique pour les setoides -%BEGIN LATEX -\RefManCutCommand{ENDADDENDUM=\thepage} -%END LATEX -\nocite{*} -\bibliographystyle{plain} -\bibliography{biblio} -\cutname{biblio.html} - -\printindex -\cutname{general-index.html} - -\printindex[tactic] -\cutname{tactic-index.html} - -\printindex[command] -\cutname{command-index.html} - -\printindex[error] -\cutname{error-index.html} - -%BEGIN LATEX -\listoffigures -\addcontentsline{toc}{chapter}{\listfigurename} -%END LATEX - -\end{document} - - -% $Id: Reference-Manual.tex 14090 2011-05-03 13:34:16Z pboutill $ diff --git a/doc/refman/Setoid.tex b/doc/refman/Setoid.tex deleted file mode 100644 index 20c8c02b..00000000 --- a/doc/refman/Setoid.tex +++ /dev/null @@ -1,714 +0,0 @@ -\newtheorem{cscexample}{Example} - -\achapter{\protect{User defined equalities and relations}} -\aauthor{Matthieu Sozeau} -\tacindex{setoid\_replace} -\label{setoid_replace} - -This chapter presents the extension of several equality related tactics to -work over user-defined structures (called setoids) that are equipped with -ad-hoc equivalence relations meant to behave as equalities. -Actually, the tactics have also been generalized to relations weaker then -equivalences (e.g. rewriting systems). - -This documentation is adapted from the previous setoid documentation by -Claudio Sacerdoti Coen (based on previous work by Cl\'ement Renard). -The new implementation is a drop-in replacement for the old one \footnote{Nicolas -Tabareau helped with the gluing}, hence most of the documentation still applies. - -The work is a complete rewrite of the previous implementation, based on -the type class infrastructure. It also improves on and generalizes -the previous implementation in several ways: -\begin{itemize} -\item User-extensible algorithm. The algorithm is separated in two - parts: generations of the rewriting constraints (done in ML) and - solving of these constraints using type class resolution. As type - class resolution is extensible using tactics, this allows users to define - general ways to solve morphism constraints. -\item Sub-relations. An example extension to the base algorithm is the - ability to define one relation as a subrelation of another so that - morphism declarations on one relation can be used automatically for - the other. This is done purely using tactics and type class search. -\item Rewriting under binders. It is possible to rewrite under binders - in the new implementation, if one provides the proper - morphisms. Again, most of the work is handled in the tactics. -\item First-class morphisms and signatures. Signatures and morphisms are - ordinary Coq terms, hence they can be manipulated inside Coq, put - inside structures and lemmas about them can be proved inside the - system. Higher-order morphisms are also allowed. -\item Performance. The implementation is based on a depth-first search for the first - solution to a set of constraints which can be as fast as linear in the - size of the term, and the size of the proof term is linear - in the size of the original term. Besides, the extensibility allows the - user to customize the proof-search if necessary. -\end{itemize} - -\asection{Relations and morphisms} - -A parametric \emph{relation} \texttt{R} is any term of type -\texttt{forall ($x_1$:$T_1$) \ldots ($x_n$:$T_n$), relation $A$}. The -expression $A$, which depends on $x_1$ \ldots $x_n$, is called the -\emph{carrier} of the relation and \texttt{R} is -said to be a relation over \texttt{A}; the list $x_1,\ldots,x_n$ -is the (possibly empty) list of parameters of the relation. - -\firstexample -\begin{cscexample}[Parametric relation] -It is possible to implement finite sets of elements of type \texttt{A} -as unordered list of elements of type \texttt{A}. The function -\texttt{set\_eq: forall (A: Type), relation (list A)} satisfied by two lists -with the same elements is a parametric relation over \texttt{(list A)} with -one parameter \texttt{A}. The type of \texttt{set\_eq} is convertible with -\texttt{forall (A: Type), list A -> list A -> Prop}. -\end{cscexample} - -An \emph{instance} of a parametric relation \texttt{R} with $n$ parameters -is any term \texttt{(R $t_1$ \ldots $t_n$)}. - -Let \texttt{R} be a relation over \texttt{A} with $n$ parameters. -A term is a parametric proof of reflexivity for \texttt{R} if it has type -\texttt{forall ($x_1$:$T_1$) \ldots ($x_n$:$T_n$), - reflexive (R $x_1$ \ldots $x_n$)}. Similar definitions are given for -parametric proofs of symmetry and transitivity. - -\begin{cscexample}[Parametric relation (cont.)] -The \texttt{set\_eq} relation of the previous example can be proved to be -reflexive, symmetric and transitive. -\end{cscexample} - -A parametric unary function $f$ of type -\texttt{forall ($x_1$:$T_1$) \ldots ($x_n$:$T_n$), $A_1$ -> $A_2$} -covariantly respects two parametric relation instances $R_1$ and $R_2$ if, -whenever $x, y$ satisfy $R_1~x~y$, their images $(f~x)$ and $(f~y)$ -satisfy $R_2~(f~x)~(f~y)$ . An $f$ that respects its input and output relations -will be called a unary covariant \emph{morphism}. We can also say that $f$ is -a monotone function with respect to $R_1$ and $R_2$. -The sequence $x_1,\ldots x_n$ represents the parameters of the morphism. - -Let $R_1$ and $R_2$ be two parametric relations. -The \emph{signature} of a parametric morphism of type -\texttt{forall ($x_1$:$T_1$) \ldots ($x_n$:$T_n$), $A_1$ -> $A_2$} that -covariantly respects two instances $I_{R_1}$ and $I_{R_2}$ of $R_1$ and $R_2$ is written $I_{R_1} \texttt{++>} I_{R_2}$. -Notice that the special arrow \texttt{++>}, which reminds the reader -of covariance, is placed between the two relation instances, not -between the two carriers. The signature relation instances and morphism will -be typed in a context introducing variables for the parameters. - -The previous definitions are extended straightforwardly to $n$-ary morphisms, -that are required to be simultaneously monotone on every argument. - -Morphisms can also be contravariant in one or more of their arguments. -A morphism is contravariant on an argument associated to the relation instance -$R$ if it is covariant on the same argument when the inverse relation -$R^{-1}$ (\texttt{inverse R} in Coq) is considered. -The special arrow \texttt{-{}->} is used in signatures -for contravariant morphisms. - -Functions having arguments related by symmetric relations instances are both -covariant and contravariant in those arguments. The special arrow -\texttt{==>} is used in signatures for morphisms that are both covariant -and contravariant. - -An instance of a parametric morphism $f$ with $n$ parameters is any term -\texttt{f $t_1$ \ldots $t_n$}. - -\begin{cscexample}[Morphisms] -Continuing the previous example, let -\texttt{union: forall (A: Type), list A -> list A -> list A} perform the union -of two sets by appending one list to the other. \texttt{union} is a binary -morphism parametric over \texttt{A} that respects the relation instance -\texttt{(set\_eq A)}. The latter condition is proved by showing -\texttt{forall (A: Type) (S1 S1' S2 S2': list A), set\_eq A S1 S1' -> - set\_eq A S2 S2' -> set\_eq A (union A S1 S2) (union A S1' S2')}. - -The signature of the function \texttt{union A} is -\texttt{set\_eq A ==> set\_eq A ==> set\_eq A} for all \texttt{A}. -\end{cscexample} - -\begin{cscexample}[Contravariant morphism] -The division function \texttt{Rdiv: R -> R -> R} is a morphism of -signature \texttt{le ++> le -{}-> le} where \texttt{le} is -the usual order relation over real numbers. Notice that division is -covariant in its first argument and contravariant in its second -argument. -\end{cscexample} - -Leibniz equality is a relation and every function is a -morphism that respects Leibniz equality. Unfortunately, Leibniz equality -is not always the intended equality for a given structure. - -In the next section we will describe the commands to register terms as -parametric relations and morphisms. Several tactics that deal with equality -in \Coq\ can also work with the registered relations. -The exact list of tactic will be given in Sect.~\ref{setoidtactics}. -For instance, the -tactic \texttt{reflexivity} can be used to close a goal $R~n~n$ whenever -$R$ is an instance of a registered reflexive relation. However, the tactics -that replace in a context $C[]$ one term with another one related by $R$ -must verify that $C[]$ is a morphism that respects the intended relation. -Currently the verification consists in checking whether $C[]$ is a syntactic -composition of morphism instances that respects some obvious -compatibility constraints. - -\begin{cscexample}[Rewriting] -Continuing the previous examples, suppose that the user must prove -\texttt{set\_eq int (union int (union int S1 S2) S2) (f S1 S2)} under the -hypothesis \texttt{H: set\_eq int S2 (nil int)}. It is possible to -use the \texttt{rewrite} tactic to replace the first two occurrences of -\texttt{S2} with \texttt{nil int} in the goal since the context -\texttt{set\_eq int (union int (union int S1 nil) nil) (f S1 S2)}, being -a composition of morphisms instances, is a morphism. However the tactic -will fail replacing the third occurrence of \texttt{S2} unless \texttt{f} -has also been declared as a morphism. -\end{cscexample} - -\asection{Adding new relations and morphisms} -A parametric relation -\textit{Aeq}\texttt{: forall ($y_1 : \beta_!$ \ldots $y_m : \beta_m$), relation (A $t_1$ \ldots $t_n$)} over -\textit{(A : $\alpha_i$ -> \ldots $\alpha_n$ -> }\texttt{Type}) -can be declared with the following command: - -\comindex{Add Parametric Relation} -\begin{quote} - \texttt{Add Parametric Relation} ($x_1 : T_1$) \ldots ($x_n : T_k$) : - \textit{(A $t_1$ \ldots $t_n$) (Aeq $t'_1$ \ldots $t'_m$)}\\ - ~\zeroone{\texttt{reflexivity proved by} \textit{refl}}\\ - ~\zeroone{\texttt{symmetry proved by} \textit{sym}}\\ - ~\zeroone{\texttt{transitivity proved by} \textit{trans}}\\ - \texttt{~as} \textit{id}. -\end{quote} -after having required the \texttt{Setoid} module with the -\texttt{Require Setoid} command. - -The identifier \textit{id} gives a unique name to the morphism and it is -used by the command to generate fresh names for automatically provided lemmas -used internally. - -Notice that the carrier and relation parameters may refer to the context -of variables introduced at the beginning of the declaration, but the -instances need not be made only of variables. -Also notice that \textit{A} is \emph{not} required to be a term -having the same parameters as \textit{Aeq}, although that is often the -case in practice (this departs from the previous implementation). - -\comindex{Add Relation} -In case the carrier and relations are not parametric, one can use the -command \texttt{Add Relation} instead, whose syntax is the same except -there is no local context. - -The proofs of reflexivity, symmetry and transitivity can be omitted if the -relation is not an equivalence relation. The proofs must be instances of the -corresponding relation definitions: e.g. the proof of reflexivity must -have a type convertible to \texttt{reflexive (A $t_1$ \ldots $t_n$) (Aeq $t'_1$ \ldots - $t'_n$)}. Each proof may refer to the introduced variables as well. - -\begin{cscexample}[Parametric relation] -For Leibniz equality, we may declare: -\texttt{Add Parametric Relation (A : Type) :} \texttt{A (@eq A)}\\ -~\zeroone{\texttt{reflexivity proved by} \texttt{@refl\_equal A}}\\ -\ldots -\end{cscexample} - -Some tactics -(\texttt{reflexivity}, \texttt{symmetry}, \texttt{transitivity}) work only -on relations that respect the expected properties. The remaining tactics -(\texttt{replace}, \texttt{rewrite} and derived tactics such as -\texttt{autorewrite}) do not require any properties over the relation. -However, they are able to replace terms with related ones only in contexts -that are syntactic compositions of parametric morphism instances declared with -the following command. - -\comindex{Add Parametric Morphism} -\begin{quote} - \texttt{Add Parametric Morphism} ($x_1 : \T_!$) \ldots ($x_k : \T_k$)\\ - (\textit{f $t_1$ \ldots $t_n$})\\ - \texttt{~with signature} \textit{sig}\\ - \texttt{~as id}.\\ - \texttt{Proof}\\ - ~\ldots\\ - \texttt{Qed} -\end{quote} - -The command declares \textit{f} as a parametric morphism of signature -\textit{sig}. The identifier \textit{id} gives a unique name to the morphism -and it is used as the base name of the type class instance definition -and as the name of the lemma that proves the well-definedness of the morphism. -The parameters of the morphism as well as the signature may refer to the -context of variables. -The command asks the user to prove interactively that \textit{f} respects -the relations identified from the signature. - -\begin{cscexample} -We start the example by assuming a small theory over homogeneous sets and -we declare set equality as a parametric equivalence relation and -union of two sets as a parametric morphism. -\begin{coq_example*} -Require Export Setoid. -Require Export Relation_Definitions. -Set Implicit Arguments. -Parameter set: Type -> Type. -Parameter empty: forall A, set A. -Parameter eq_set: forall A, set A -> set A -> Prop. -Parameter union: forall A, set A -> set A -> set A. -Axiom eq_set_refl: forall A, reflexive _ (eq_set (A:=A)). -Axiom eq_set_sym: forall A, symmetric _ (eq_set (A:=A)). -Axiom eq_set_trans: forall A, transitive _ (eq_set (A:=A)). -Axiom empty_neutral: forall A (S: set A), eq_set (union S (empty A)) S. -Axiom union_compat: - forall (A : Type), - forall x x' : set A, eq_set x x' -> - forall y y' : set A, eq_set y y' -> - eq_set (union x y) (union x' y'). -Add Parametric Relation A : (set A) (@eq_set A) - reflexivity proved by (eq_set_refl (A:=A)) - symmetry proved by (eq_set_sym (A:=A)) - transitivity proved by (eq_set_trans (A:=A)) - as eq_set_rel. -Add Parametric Morphism A : (@union A) with -signature (@eq_set A) ==> (@eq_set A) ==> (@eq_set A) as union_mor. -Proof. exact (@union_compat A). Qed. -\end{coq_example*} - -\end{cscexample} - -Is is possible to reduce the burden of specifying parameters using -(maximally inserted) implicit arguments. If \texttt{A} is always set as -maximally implicit in the previous example, one can write: - -\begin{coq_eval} -Reset Initial. -Require Export Setoid. -Require Export Relation_Definitions. -Parameter set: Type -> Type. -Parameter empty: forall {A}, set A. -Parameter eq_set: forall {A}, set A -> set A -> Prop. -Parameter union: forall {A}, set A -> set A -> set A. -Axiom eq_set_refl: forall {A}, reflexive (set A) eq_set. -Axiom eq_set_sym: forall {A}, symmetric (set A) eq_set. -Axiom eq_set_trans: forall {A}, transitive (set A) eq_set. -Axiom empty_neutral: forall A (S: set A), eq_set (union S empty) S. -Axiom union_compat: - forall (A : Type), - forall x x' : set A, eq_set x x' -> - forall y y' : set A, eq_set y y' -> - eq_set (union x y) (union x' y'). -\end{coq_eval} - -\begin{coq_example*} -Add Parametric Relation A : (set A) eq_set - reflexivity proved by eq_set_refl - symmetry proved by eq_set_sym - transitivity proved by eq_set_trans - as eq_set_rel. -Add Parametric Morphism A : (@union A) with - signature eq_set ==> eq_set ==> eq_set as union_mor. -Proof. exact (@union_compat A). Qed. -\end{coq_example*} - -We proceed now by proving a simple lemma performing a rewrite step -and then applying reflexivity, as we would do working with Leibniz -equality. Both tactic applications are accepted -since the required properties over \texttt{eq\_set} and -\texttt{union} can be established from the two declarations above. - -\begin{coq_example*} -Goal forall (S: set nat), - eq_set (union (union S empty) S) (union S S). -Proof. intros. rewrite empty_neutral. reflexivity. Qed. -\end{coq_example*} - -The tables of relations and morphisms are managed by the type class -instance mechanism. The behavior on section close is to generalize -the instances by the variables of the section (and possibly hypotheses -used in the proofs of instance declarations) but not to export them in -the rest of the development for proof search. One can use the -\texttt{Existing Instance} command to do so outside the section, -using the name of the declared morphism suffixed by \texttt{\_Morphism}, -or use the \texttt{Global} modifier for the corresponding class instance -declaration (see \S\ref{setoid:first-class}) at definition time. -When loading a compiled file or importing a module, -all the declarations of this module will be loaded. - -\asection{Rewriting and non reflexive relations} -To replace only one argument of an n-ary morphism it is necessary to prove -that all the other arguments are related to themselves by the respective -relation instances. - -\begin{cscexample} -To replace \texttt{(union S empty)} with \texttt{S} in -\texttt{(union (union S empty) S) (union S S)} the rewrite tactic must -exploit the monotony of \texttt{union} (axiom \texttt{union\_compat} in -the previous example). Applying \texttt{union\_compat} by hand we are left -with the goal \texttt{eq\_set (union S S) (union S S)}. -\end{cscexample} - -When the relations associated to some arguments are not reflexive, the tactic -cannot automatically prove the reflexivity goals, that are left to the user. - -Setoids whose relation are partial equivalence relations (PER) -are useful to deal with partial functions. Let \texttt{R} be a PER. We say -that an element \texttt{x} is defined if \texttt{R x x}. A partial function -whose domain comprises all the defined elements only is declared as a -morphism that respects \texttt{R}. Every time a rewriting step is performed -the user must prove that the argument of the morphism is defined. - -\begin{cscexample} -Let \texttt{eqO} be \texttt{fun x y => x = y $\land$ ~x$\neq$ 0} (the smaller PER over -non zero elements). Division can be declared as a morphism of signature -\texttt{eq ==> eq0 ==> eq}. Replace \texttt{x} with \texttt{y} in -\texttt{div x n = div y n} opens the additional goal \texttt{eq0 n n} that -is equivalent to \texttt{n=n $\land$ n$\neq$0}. -\end{cscexample} - -\asection{Rewriting and non symmetric relations} -When the user works up to relations that are not symmetric, it is no longer -the case that any covariant morphism argument is also contravariant. As a -result it is no longer possible to replace a term with a related one in -every context, since the obtained goal implies the previous one if and -only if the replacement has been performed in a contravariant position. -In a similar way, replacement in an hypothesis can be performed only if -the replaced term occurs in a covariant position. - -\begin{cscexample}[Covariance and contravariance] -Suppose that division over real numbers has been defined as a -morphism of signature \texttt{Zdiv: Zlt ++> Zlt -{}-> Zlt} (i.e. -\texttt{Zdiv} is increasing in its first argument, but decreasing on the -second one). Let \texttt{<} denotes \texttt{Zlt}. -Under the hypothesis \texttt{H: x < y} we have -\texttt{k < x / y -> k < x / x}, but not -\texttt{k < y / x -> k < x / x}. -Dually, under the same hypothesis \texttt{k < x / y -> k < y / y} holds, -but \texttt{k < y / x -> k < y / y} does not. -Thus, if the current goal is \texttt{k < x / x}, it is possible to replace -only the second occurrence of \texttt{x} (in contravariant position) -with \texttt{y} since the obtained goal must imply the current one. -On the contrary, if \texttt{k < x / x} is -an hypothesis, it is possible to replace only the first occurrence of -\texttt{x} (in covariant position) with \texttt{y} since -the current hypothesis must imply the obtained one. -\end{cscexample} - -Contrary to the previous implementation, no specific error message will -be raised when trying to replace a term that occurs in the wrong -position. It will only fail because the rewriting constraints are not -satisfiable. However it is possible to use the \texttt{at} modifier to -specify which occurrences should be rewritten. - -As expected, composing morphisms together propagates the variance annotations by -switching the variance every time a contravariant position is traversed. -\begin{cscexample} -Let us continue the previous example and let us consider the goal -\texttt{x / (x / x) < k}. The first and third occurrences of \texttt{x} are -in a contravariant position, while the second one is in covariant position. -More in detail, the second occurrence of \texttt{x} occurs -covariantly in \texttt{(x / x)} (since division is covariant in its first -argument), and thus contravariantly in \texttt{x / (x / x)} (since division -is contravariant in its second argument), and finally covariantly in -\texttt{x / (x / x) < k} (since \texttt{<}, as every transitive relation, -is contravariant in its first argument with respect to the relation itself). -\end{cscexample} - -\asection{Rewriting in ambiguous setoid contexts} -One function can respect several different relations and thus it can be -declared as a morphism having multiple signatures. - -\begin{cscexample} -Union over homogeneous lists can be given all the following signatures: -\texttt{eq ==> eq ==> eq} (\texttt{eq} being the equality over ordered lists) -\texttt{set\_eq ==> set\_eq ==> set\_eq} (\texttt{set\_eq} being the equality -over unordered lists up to duplicates), -\texttt{multiset\_eq ==> multiset\_eq ==> multiset\_eq} (\texttt{multiset\_eq} -being the equality over unordered lists). -\end{cscexample} - -To declare multiple signatures for a morphism, repeat the \texttt{Add Morphism} -command. - -When morphisms have multiple signatures it can be the case that a rewrite -request is ambiguous, since it is unclear what relations should be used to -perform the rewriting. Contrary to the previous implementation, the -tactic will always choose the first possible solution to the set of -constraints generated by a rewrite and will not try to find \emph{all} -possible solutions to warn the user about. - -\asection{First class setoids and morphisms} -\label{setoid:first-class} - -The implementation is based on a first-class representation of -properties of relations and morphisms as type classes. That is, -the various combinations of properties on relations and morphisms -are represented as records and instances of theses classes are put -in a hint database. -For example, the declaration: - -\begin{quote} - \texttt{Add Parametric Relation} ($x_1 : T_1$) \ldots ($x_n : T_k$) : - \textit{(A $t_1$ \ldots $t_n$) (Aeq $t'_1$ \ldots $t'_m$)}\\ - ~\zeroone{\texttt{reflexivity proved by} \textit{refl}}\\ - ~\zeroone{\texttt{symmetry proved by} \textit{sym}}\\ - ~\zeroone{\texttt{transitivity proved by} \textit{trans}}\\ - \texttt{~as} \textit{id}. -\end{quote} - -is equivalent to an instance declaration: - -\begin{quote} - \texttt{Instance} ($x_1 : T_1$) \ldots ($x_n : T_k$) \texttt{=>} - \textit{id} : \texttt{@Equivalence} \textit{(A $t_1$ \ldots $t_n$) (Aeq - $t'_1$ \ldots $t'_m$)} :=\\ - ~\zeroone{\texttt{Equivalence\_Reflexive :=} \textit{refl}}\\ - ~\zeroone{\texttt{Equivalence\_Symmetric :=} \textit{sym}}\\ - ~\zeroone{\texttt{Equivalence\_Transitive :=} \textit{trans}}. -\end{quote} - -The declaration itself amounts to the definition of an object of the -record type \texttt{Coq.Classes.RelationClasses.Equivalence} and a -hint added to the \texttt{typeclass\_instances} hint database. -Morphism declarations are also instances of a type class defined in -\texttt{Classes.Morphisms}. -See the documentation on type classes \ref{typeclasses} and -the theories files in \texttt{Classes} for further explanations. - -One can inform the rewrite tactic about morphisms and relations just by -using the typeclass mechanism to declare them using \texttt{Instance} -and \texttt{Context} vernacular commands. -Any object of type \texttt{Proper} (the type of morphism declarations) -in the local context will also be automatically used by the rewriting -tactic to solve constraints. - -Other representations of first class setoids and morphisms can also -be handled by encoding them as records. In the following example, -the projections of the setoid relation and of the morphism function -can be registered as parametric relations and morphisms. -\begin{cscexample}[First class setoids] - -\begin{coq_example*} -Require Import Relation_Definitions Setoid. -Record Setoid: Type := -{ car:Type; - eq:car->car->Prop; - refl: reflexive _ eq; - sym: symmetric _ eq; - trans: transitive _ eq -}. -Add Parametric Relation (s : Setoid) : (@car s) (@eq s) - reflexivity proved by (refl s) - symmetry proved by (sym s) - transitivity proved by (trans s) as eq_rel. -Record Morphism (S1 S2:Setoid): Type := -{ f:car S1 ->car S2; - compat: forall (x1 x2: car S1), eq S1 x1 x2 -> eq S2 (f x1) (f x2) }. -Add Parametric Morphism (S1 S2 : Setoid) (M : Morphism S1 S2) : - (@f S1 S2 M) with signature (@eq S1 ==> @eq S2) as apply_mor. -Proof. apply (compat S1 S2 M). Qed. -Lemma test: forall (S1 S2:Setoid) (m: Morphism S1 S2) - (x y: car S1), eq S1 x y -> eq S2 (f _ _ m x) (f _ _ m y). -Proof. intros. rewrite H. reflexivity. Qed. -\end{coq_example*} -\end{cscexample} - -\asection{Tactics enabled on user provided relations} -\label{setoidtactics} -The following tactics, all prefixed by \texttt{setoid\_}, -deal with arbitrary -registered relations and morphisms. Moreover, all the corresponding unprefixed -tactics (i.e. \texttt{reflexivity, symmetry, transitivity, replace, rewrite}) -have been extended to fall back to their prefixed counterparts when -the relation involved is not Leibniz equality. Notice, however, that using -the prefixed tactics it is possible to pass additional arguments such as -\texttt{using relation}. -\medskip - -\comindex{setoid\_reflexivity} -\texttt{setoid\_reflexivity} - -\comindex{setoid\_symmetry} -\texttt{setoid\_symmetry} \zeroone{\texttt{in} \textit{ident}} - -\comindex{setoid\_transitivity} -\texttt{setoid\_transitivity} - -\comindex{setoid\_rewrite} -\texttt{setoid\_rewrite} \zeroone{\textit{orientation}} \textit{term} -~\zeroone{\texttt{at} \textit{occs}} ~\zeroone{\texttt{in} \textit{ident}} - -\comindex{setoid\_replace} -\texttt{setoid\_replace} \textit{term} \texttt{with} \textit{term} -~\zeroone{\texttt{in} \textit{ident}} -~\zeroone{\texttt{using relation} \textit{term}} -~\zeroone{\texttt{by} \textit{tactic}} -\medskip - -The \texttt{using relation} -arguments cannot be passed to the unprefixed form. The latter argument -tells the tactic what parametric relation should be used to replace -the first tactic argument with the second one. If omitted, it defaults -to the \texttt{DefaultRelation} instance on the type of the objects. -By default, it means the most recent \texttt{Equivalence} instance in -the environment, but it can be customized by declaring new -\texttt{DefaultRelation} instances. As Leibniz equality is a declared -equivalence, it will fall back to it if no other relation is declared on -a given type. - -Every derived tactic that is based on the unprefixed forms of the tactics -considered above will also work up to user defined relations. For instance, -it is possible to register hints for \texttt{autorewrite} that are -not proof of Leibniz equalities. In particular it is possible to exploit -\texttt{autorewrite} to simulate normalization in a term rewriting system -up to user defined equalities. - -\asection{Printing relations and morphisms} -The \texttt{Print Instances} command can be used to show the list of -currently registered \texttt{Reflexive} (using \texttt{Print Instances Reflexive}), -\texttt{Symmetric} or \texttt{Transitive} relations, -\texttt{Equivalence}s, \texttt{PreOrder}s, \texttt{PER}s, and -Morphisms (implemented as \texttt{Proper} instances). When - the rewriting tactics refuse to replace a term in a context -because the latter is not a composition of morphisms, the \texttt{Print Instances} -commands can be useful to understand what additional morphisms should be -registered. - -\asection{Deprecated syntax and backward incompatibilities} -Due to backward compatibility reasons, the following syntax for the -declaration of setoids and morphisms is also accepted. - -\comindex{Add Setoid} -\begin{quote} - \texttt{Add Setoid} \textit{A Aeq ST} \texttt{as} \textit{ident} -\end{quote} -where \textit{Aeq} is a congruence relation without parameters, -\textit{A} is its carrier and \textit{ST} is an object of type -\texttt{(Setoid\_Theory A Aeq)} (i.e. a record packing together the reflexivity, -symmetry and transitivity lemmas). Notice that the syntax is not completely -backward compatible since the identifier was not required. - -\comindex{Add Morphism} -\begin{quote} - \texttt{Add Morphism} \textit{f}:\textit{ident}.\\ - Proof.\\ - \ldots\\ - Qed. -\end{quote} - -The latter command also is restricted to the declaration of morphisms without -parameters. It is not fully backward compatible since the property the user -is asked to prove is slightly different: for $n$-ary morphisms the hypotheses -of the property are permuted; moreover, when the morphism returns a -proposition, the property is now stated using a bi-implication in place of -a simple implication. In practice, porting an old development to the new -semantics is usually quite simple. - -Notice that several limitations of the old implementation have been lifted. -In particular, it is now possible to declare several relations with the -same carrier and several signatures for the same morphism. Moreover, it is -now also possible to declare several morphisms having the same signature. -Finally, the replace and rewrite tactics can be used to replace terms in -contexts that were refused by the old implementation. As discussed in -the next section, the semantics of the new \texttt{setoid\_rewrite} -command differs slightly from the old one and \texttt{rewrite}. - -\asection{Rewriting under binders} - -\textbf{Warning}: Due to compatibility issues, this feature is enabled only when calling -the \texttt{setoid\_rewrite} tactics directly and not \texttt{rewrite}. - -To be able to rewrite under binding constructs, one must declare -morphisms with respect to pointwise (setoid) equivalence of functions. -Example of such morphisms are the standard \texttt{all} and \texttt{ex} -combinators for universal and existential quantification respectively. -They are declared as morphisms in the \texttt{Classes.Morphisms\_Prop} -module. For example, to declare that universal quantification is a -morphism for logical equivalence: - -\begin{coq_eval} -Reset Initial. -Require Import Setoid Morphisms. -\end{coq_eval} -\begin{coq_example} -Instance all_iff_morphism (A : Type) : - Proper (pointwise_relation A iff ==> iff) (@all A). -Proof. simpl_relation. -\end{coq_example} -\begin{coq_eval} -Admitted. -\end{coq_eval} - -One then has to show that if two predicates are equivalent at every -point, their universal quantifications are equivalent. Once we have -declared such a morphism, it will be used by the setoid rewriting tactic -each time we try to rewrite under an \texttt{all} application (products -in \Prop{} are implicitly translated to such applications). - -Indeed, when rewriting under a lambda, binding variable $x$, say from -$P~x$ to $Q~x$ using the relation \texttt{iff}, the tactic will generate -a proof of \texttt{pointwise\_relation A iff (fun x => P x) (fun x => Q -x)} from the proof of \texttt{iff (P x) (Q x)} and a constraint of the -form \texttt{Proper (pointwise\_relation A iff ==> ?) m} will be -generated for the surrounding morphism \texttt{m}. - -Hence, one can add higher-order combinators as morphisms by providing -signatures using pointwise extension for the relations on the functional -arguments (or whatever subrelation of the pointwise extension). -For example, one could declare the \texttt{map} combinator on lists as -a morphism: -\begin{coq_eval} -Require Import List. -Set Implicit Arguments. -Inductive list_equiv {A:Type} (eqA : relation A) : relation (list A) := -| eq_nil : list_equiv eqA nil nil -| eq_cons : forall x y, eqA x y -> - forall l l', list_equiv eqA l l' -> list_equiv eqA (x :: l) (y :: l'). -\end{coq_eval} -\begin{coq_example*} -Instance map_morphism `{Equivalence A eqA, Equivalence B eqB} : - Proper ((eqA ==> eqB) ==> list_equiv eqA ==> list_equiv eqB) - (@map A B). -\end{coq_example*} - -where \texttt{list\_equiv} implements an equivalence on lists -parameterized by an equivalence on the elements. - -Note that when one does rewriting with a lemma under a binder -using \texttt{setoid\_rewrite}, the application of the lemma may capture -the bound variable, as the semantics are different from rewrite where -the lemma is first matched on the whole term. With the new -\texttt{setoid\_rewrite}, matching is done on each subterm separately -and in its local environment, and all matches are rewritten -\emph{simultaneously} by default. The semantics of the previous -\texttt{setoid\_rewrite} implementation can almost be recovered using -the \texttt{at 1} modifier. - -\asection{Sub-relations} - -Sub-relations can be used to specify that one relation is included in -another, so that morphisms signatures for one can be used for the other. -If a signature mentions a relation $R$ on the left of an arrow -\texttt{==>}, then the signature also applies for any relation $S$ that -is smaller than $R$, and the inverse applies on the right of an arrow. -One can then declare only a few morphisms instances that generate the complete set -of signatures for a particular constant. By default, the only declared -subrelation is \texttt{iff}, which is a subrelation of \texttt{impl} -and \texttt{inverse impl} (the dual of implication). That's why we can -declare only two morphisms for conjunction: -\texttt{Proper (impl ==> impl ==> impl) and} and -\texttt{Proper (iff ==> iff ==> iff) and}. This is sufficient to satisfy -any rewriting constraints arising from a rewrite using \texttt{iff}, -\texttt{impl} or \texttt{inverse impl} through \texttt{and}. - -Sub-relations are implemented in \texttt{Classes.Morphisms} and are a -prime example of a mostly user-space extension of the algorithm. - -\asection{Constant unfolding} - -The resolution tactic is based on type classes and hence regards user-defined -constants as transparent by default. This may slow down the resolution -due to a lot of unifications (all the declared \texttt{Proper} -instances are tried at each node of the search tree). -To speed it up, declare your constant as rigid for proof search -using the command \texttt{Typeclasses Opaque} (see \S \ref{TypeclassesTransparency}). - -%%% Local Variables: -%%% mode: latex -%%% TeX-master: "Reference-Manual" -%%% compile-command: "make -C ../.. -f Makefile.stage3 doc/refman/Reference-Manual.pdf" -%%% End: diff --git a/doc/refman/biblio.bib b/doc/refman/biblio.bib deleted file mode 100644 index f93b66f9..00000000 --- a/doc/refman/biblio.bib +++ /dev/null @@ -1,1273 +0,0 @@ -@String{jfp = "Journal of Functional Programming"} -@String{lncs = "Lecture Notes in Computer Science"} -@String{lnai = "Lecture Notes in Artificial Intelligence"} -@String{SV = "{Springer-Verlag}"} - -@InProceedings{Aud91, - author = {Ph. Audebaud}, - booktitle = {Proceedings of the sixth Conf. on Logic in Computer Science.}, - publisher = {IEEE}, - title = {Partial {Objects} in the {Calculus of Constructions}}, - year = {1991} -} - -@PhDThesis{Aud92, - author = {Ph. Audebaud}, - school = {{Universit\'e} Bordeaux I}, - title = {Extension du Calcul des Constructions par Points fixes}, - year = {1992} -} - -@InProceedings{Audebaud92b, - author = {Ph. Audebaud}, - booktitle = {{Proceedings of the 1992 Workshop on Types for Proofs and Programs}}, - editor = {{B. Nordstr\"om and K. Petersson and G. Plotkin}}, - note = {Also Research Report LIP-ENS-Lyon}, - pages = {21--34}, - title = {{CC+ : an extension of the Calculus of Constructions with fixpoints}}, - year = {1992} -} - -@InProceedings{Augustsson85, - author = {L. Augustsson}, - title = {{Compiling Pattern Matching}}, - booktitle = {Conference Functional Programming and -Computer Architecture}, - year = {1985} -} - -@Article{BaCo85, - author = {J.L. Bates and R.L. 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Smith}, - booktitle = {Information Processing 83}, - publisher = {Oxford Science Publications}, - series = {International Series of Monographs on Computer Science}, - title = {Programming in {Martin-L\"of's} Type Theory}, - year = {1990} -} - -@Article{Nor88, - author = {B. {Nordstr\"om}}, - journal = {BIT}, - title = {Terminating General Recursion}, - volume = {28}, - year = {1988} -} - -@Book{Odi90, - editor = {P. Odifreddi}, - publisher = {Academic Press}, - title = {Logic and Computer Science}, - year = {1990} -} - -@InProceedings{PaMS92, - author = {M. Parigot and P. Manoury and M. Simonot}, - address = {St. Petersburg, Russia}, - booktitle = {Logic Programming and automated reasoning}, - editor = {A. Voronkov}, - month = jul, - number = {624}, - publisher = SV, - series = {LNCS}, - title = {{ProPre : A Programming language with proofs}}, - year = {1992} -} - -@Article{PaWe92, - author = {C. Paulin-Mohring and B. Werner}, - journal = {Journal of Symbolic Computation}, - pages = {607--640}, - title = {{Synthesis of ML programs in the system Coq}}, - volume = {15}, - year = {1993} -} - -@Article{Par92, - author = {M. Parigot}, - journal = {Theoretical Computer Science}, - number = {2}, - pages = {335--356}, - title = {{Recursive Programming with Proofs}}, - volume = {94}, - year = {1992} -} - -@InProceedings{Parent95b, - author = {C. Parent}, - booktitle = {{Mathematics of Program Construction'95}}, - publisher = SV, - series = {LNCS}, - title = {{Synthesizing proofs from programs in -the Calculus of Inductive Constructions}}, - volume = {947}, - year = {1995} -} - -@InProceedings{Prasad93, - author = {K.V. Prasad}, - booktitle = {{Proceedings of CONCUR'93}}, - publisher = SV, - series = {LNCS}, - title = {{Programming with broadcasts}}, - volume = {715}, - year = {1993} -} - -@Book{RC95, - author = {di~Cosmo, R.}, - title = {Isomorphisms of Types: from $\lambda$-calculus to information - retrieval and language design}, - series = {Progress in Theoretical Computer Science}, - publisher = {Birkhauser}, - year = {1995}, - note = {ISBN-0-8176-3763-X} -} - -@TechReport{Rou92, - author = {J. Rouyer}, - institution = {INRIA}, - month = nov, - number = {1795}, - title = {{Développement de l'Algorithme d'Unification dans le Calcul des Constructions}}, - year = {1992} -} - -@Article{Rushby98, - title = {Subtypes for Specifications: Predicate Subtyping in - {PVS}}, - author = {John Rushby and Sam Owre and N. Shankar}, - journal = {IEEE Transactions on Software Engineering}, - pages = {709--720}, - volume = 24, - number = 9, - month = sep, - year = 1998 -} - -@TechReport{Saibi94, - author = {A. Sa\"{\i}bi}, - institution = {INRIA}, - month = dec, - number = {2345}, - title = {{Axiomatization of a lambda-calculus with explicit-substitutions in the Coq System}}, - year = {1994} -} - - -@MastersThesis{Ter92, - author = {D. Terrasse}, - month = sep, - school = {IARFA}, - title = {{Traduction de TYPOL en COQ. Application \`a Mini ML}}, - year = {1992} -} - -@TechReport{ThBeKa92, - author = {L. Th\'ery and Y. Bertot and G. Kahn}, - institution = {INRIA Sophia}, - month = may, - number = {1684}, - title = {Real theorem provers deserve real user-interfaces}, - type = {Research Report}, - year = {1992} -} - -@Book{TrDa89, - author = {A.S. Troelstra and D. van Dalen}, - publisher = {North-Holland}, - series = {Studies in Logic and the foundations of Mathematics, volumes 121 and 123}, - title = {Constructivism in Mathematics, an introduction}, - year = {1988} -} - -@PhDThesis{Wer94, - author = {B. Werner}, - school = {Universit\'e Paris 7}, - title = {Une th\'eorie des constructions inductives}, - type = {Th\`ese de Doctorat}, - year = {1994} -} - -@PhDThesis{Bar99, - author = {B. Barras}, - school = {Universit\'e Paris 7}, - title = {Auto-validation d'un système de preuves avec familles inductives}, - type = {Th\`ese de Doctorat}, - year = {1999} -} - -@Unpublished{ddr98, - author = {D. de Rauglaudre}, - title = {Camlp4 version 1.07.2}, - year = {1998}, - note = {In Camlp4 distribution} -} - -@Article{dowek93, - author = {G. Dowek}, - title = {{A Complete Proof Synthesis Method for the Cube of Type Systems}}, - journal = {Journal Logic Computation}, - volume = {3}, - number = {3}, - pages = {287--315}, - month = {June}, - year = {1993} -} - -@InProceedings{manoury94, - author = {P. Manoury}, - title = {{A User's Friendly Syntax to Define -Recursive Functions as Typed $\lambda-$Terms}}, - booktitle = {{Types for Proofs and Programs, TYPES'94}}, - series = {LNCS}, - volume = {996}, - month = jun, - year = {1994} -} - -@TechReport{maranget94, - author = {L. Maranget}, - institution = {INRIA}, - number = {2385}, - title = {{Two Techniques for Compiling Lazy Pattern Matching}}, - year = {1994} -} - -@InProceedings{puel-suarez90, - author = {L.Puel and A. Su\'arez}, - booktitle = {{Conference Lisp and Functional Programming}}, - series = {ACM}, - publisher = SV, - title = {{Compiling Pattern Matching by Term -Decomposition}}, - year = {1990} -} - -@MastersThesis{saidi94, - author = {H. Saidi}, - month = sep, - school = {DEA d'Informatique Fondamentale, Universit\'e Paris 7}, - title = {R\'esolution d'\'equations dans le syst\`eme T - de G\"odel}, - year = {1994} -} - -@inproceedings{sozeau06, - author = {Matthieu Sozeau}, - title = {Subset Coercions in {C}oq}, - year = {2007}, - booktitle = {TYPES'06}, - pages = {237-252}, - volume = {4502}, - publisher = "Springer", - series = {LNCS} -} - -@inproceedings{sozeau08, - Author = {Matthieu Sozeau and Nicolas Oury}, - booktitle = {TPHOLs'08}, - Pdf = {http://www.lri.fr/~sozeau/research/publications/drafts/classes.pdf}, - Title = {{F}irst-{C}lass {T}ype {C}lasses}, - Year = {2008}, -} - -@Misc{streicher93semantical, - author = {T. Streicher}, - title = {Semantical Investigations into Intensional Type Theory}, - note = {Habilitationsschrift, LMU Munchen.}, - year = {1993} -} - -@Misc{Pcoq, - author = {Lemme Team}, - title = {Pcoq a graphical user-interface for {Coq}}, - note = {\url{http://www-sop.inria.fr/lemme/pcoq/}} -} - -@Misc{ProofGeneral, - author = {David Aspinall}, - title = {Proof General}, - note = {\url{http://proofgeneral.inf.ed.ac.uk/}} -} - -@Book{CoqArt, - title = {Interactive Theorem Proving and Program Development. - Coq'Art: The Calculus of Inductive Constructions}, - author = {Yves Bertot and Pierre Castéran}, - publisher = {Springer Verlag}, - series = {Texts in Theoretical Computer Science. An EATCS series}, - year = 2004 -} - -@InCollection{wadler87, - author = {P. Wadler}, - title = {Efficient Compilation of Pattern Matching}, - booktitle = {The Implementation of Functional Programming -Languages}, - editor = {S.L. Peyton Jones}, - publisher = {Prentice-Hall}, - year = {1987} -} - -@inproceedings{DBLP:conf/types/CornesT95, - author = {Cristina Cornes and - Delphine Terrasse}, - title = {Automating Inversion of Inductive Predicates in Coq}, - booktitle = {TYPES}, - year = {1995}, - pages = {85-104}, - crossref = {DBLP:conf/types/1995}, - bibsource = {DBLP, http://dblp.uni-trier.de} -} -@proceedings{DBLP:conf/types/1995, - editor = {Stefano Berardi and - Mario Coppo}, - title = {Types for Proofs and Programs, International Workshop TYPES'95, - Torino, Italy, June 5-8, 1995, Selected Papers}, - booktitle = {TYPES}, - publisher = {Springer}, - series = {Lecture Notes in Computer Science}, - volume = {1158}, - year = {1996}, - isbn = {3-540-61780-9}, - bibsource = {DBLP, http://dblp.uni-trier.de} -} - -@inproceedings{DBLP:conf/types/McBride00, - author = {Conor McBride}, - title = {Elimination with a Motive}, - booktitle = {TYPES}, - year = {2000}, - pages = {197-216}, - ee = {http://link.springer.de/link/service/series/0558/bibs/2277/22770197.htm}, - crossref = {DBLP:conf/types/2000}, - bibsource = {DBLP, http://dblp.uni-trier.de} -} - -@proceedings{DBLP:conf/types/2000, - editor = {Paul Callaghan and - Zhaohui Luo and - James McKinna and - Robert Pollack}, - title = {Types for Proofs and Programs, International Workshop, TYPES - 2000, Durham, UK, December 8-12, 2000, Selected Papers}, - booktitle = {TYPES}, - publisher = {Springer}, - series = {Lecture Notes in Computer Science}, - volume = {2277}, - year = {2002}, - isbn = {3-540-43287-6}, - bibsource = {DBLP, http://dblp.uni-trier.de} -} - -@INPROCEEDINGS{sugar, - author = {Alessandro Giovini and Teo Mora and Gianfranco Niesi and Lorenzo Robbiano and Carlo Traverso}, - title = {"One sugar cube, please" or Selection strategies in the Buchberger algorithm}, - booktitle = { Proceedings of the ISSAC'91, ACM Press}, - year = {1991}, - pages = {5--4}, - publisher = {} -} - -@Comment{cross-references, must be at end} - -@Book{Bastad92, - editor = {B. Nordstr\"om and K. Petersson and G. Plotkin}, - publisher = {Available by ftp at site ftp.inria.fr}, - title = {Proceedings of the 1992 Workshop on Types for Proofs and Programs}, - year = {1992} -} - -@Book{Nijmegen93, - editor = {H. Barendregt and T. Nipkow}, - publisher = SV, - series = LNCS, - title = {Types for Proofs and Programs}, - volume = {806}, - year = {1994} -} - -@article{ TheOmegaPaper, - author = "W. Pugh", - title = "The Omega test: a fast and practical integer programming algorithm for dependence analysis", - journal = "Communication of the ACM", - pages = "102--114", - year = "1992", -} diff --git a/doc/refman/coqdoc.tex b/doc/refman/coqdoc.tex deleted file mode 100644 index 271a13f7..00000000 --- a/doc/refman/coqdoc.tex +++ /dev/null @@ -1,561 +0,0 @@ - -%\newcommand{\Coq}{\textsf{Coq}} -\newcommand{\javadoc}{\textsf{javadoc}} -\newcommand{\ocamldoc}{\textsf{ocamldoc}} -\newcommand{\coqdoc}{\textsf{coqdoc}} -\newcommand{\texmacs}{\TeX{}macs} -\newcommand{\monurl}[1]{#1} -%HEVEA\renewcommand{\monurl}[1]{\ahref{#1}{#1}} -%\newcommand{\lnot}{not} % Hevea handles these symbols nicely -%\newcommand{\lor}{or} -%\newcommand{\land}{\&} -%%% attention : -- dans un argument de \texttt est affiché comme un -%%% seul - d'où l'utilisation de la macro suivante -\newcommand{\mm}{\symbol{45}\symbol{45}} - - -\coqdoc\ is a documentation tool for the proof assistant -\Coq, similar to \javadoc\ or \ocamldoc. -The task of \coqdoc\ is -\begin{enumerate} -\item to produce a nice \LaTeX\ and/or HTML document from the \Coq\ - sources, readable for a human and not only for the proof assistant; -\item to help the user navigating in his own (or third-party) sources. -\end{enumerate} - - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\subsection{Principles} - -Documentation is inserted into \Coq\ files as \emph{special comments}. -Thus your files will compile as usual, whether you use \coqdoc\ or not. -\coqdoc\ presupposes that the given \Coq\ files are well-formed (at -least lexically). Documentation starts with -\texttt{(**}, followed by a space, and ends with the pending \texttt{*)}. -The documentation format is inspired - by Todd~A.~Coram's \emph{Almost Free Text (AFT)} tool: it is mainly -ASCII text with some syntax-light controls, described below. -\coqdoc\ is robust: it shouldn't fail, whatever the input is. But -remember: ``garbage in, garbage out''. - -\paragraph{\Coq\ material inside documentation.} -\Coq\ material is quoted between the -delimiters \texttt{[} and \texttt{]}. Square brackets may be nested, -the inner ones being understood as being part of the quoted code (thus -you can quote a term like $[x:T]u$ by writing -\texttt{[[x:T]u]}). Inside quotations, the code is pretty-printed in -the same way as it is in code parts. - -Pre-formatted vernacular is enclosed by \texttt{[[} and -\texttt{]]}. The former must be followed by a newline and the latter -must follow a newline. - -\paragraph{Pretty-printing.} -\coqdoc\ uses different faces for identifiers and keywords. -The pretty-printing of \Coq\ tokens (identifiers or symbols) can be -controlled using one of the following commands: -\begin{alltt} -(** printing \emph{token} %...\LaTeX...% #...HTML...# *) -\end{alltt} -or -\begin{alltt} -(** printing \emph{token} $...\LaTeX\ math...$ #...HTML...# *) -\end{alltt} -It gives the \LaTeX\ and HTML texts to be produced for the given \Coq\ -token. One of the \LaTeX\ or HTML text may be ommitted, causing the -default pretty-printing to be used for this token. - -The printing for one token can be removed with -\begin{alltt} -(** remove printing \emph{token} *) -\end{alltt} - -Initially, the pretty-printing table contains the following mapping: -\begin{center} - \begin{tabular}{ll@{\qquad\qquad}ll@{\qquad\qquad}ll@{\qquad\qquad}} - \verb!->! & $\rightarrow$ & - \verb!<-! & $\leftarrow$ & - \verb|*| & $\times$ \\ - \verb|<=| & $\le$ & - \verb|>=| & $\ge$ & - \verb|=>| & $\Rightarrow$ \\ - \verb|<>| & $\not=$ & - \verb|<->| & $\leftrightarrow$ & - \verb!|-! & $\vdash$ \\ - \verb|\/| & $\lor$ & - \verb|/\| & $\land$ & - \verb|~| & $\lnot$ - \end{tabular} -\end{center} -Any of these can be overwritten or suppressed using the -\texttt{printing} commands. - -Important note: the recognition of tokens is done by a (ocaml)lex -automaton and thus applies the longest-match rule. For instance, -\verb!->~! is recognized as a single token, where \Coq\ sees two -tokens. It is the responsability of the user to insert space between -tokens \emph{or} to give pretty-printing rules for the possible -combinations, e.g. -\begin{verbatim} -(** printing ->~ %\ensuremath{\rightarrow\lnot}% *) -\end{verbatim} - - -\paragraph{Sections.} -Sections are introduced by 1 to 4 leading stars (i.e. at the beginning of the -line) followed by a space. One star is a section, two stars a sub-section, etc. -The section title is given on the remaining of the line. -Example: -\begin{verbatim} - (** * Well-founded relations - - In this section, we introduce... *) -\end{verbatim} - - -%TODO \paragraph{Fonts.} - - -\paragraph{Lists.} -List items are introduced by a leading dash. \coqdoc\ uses whitespace -to determine the depth of a new list item and which text belongs in -which list items. A list ends when a line of text starts at or before -the level of indenting of the list's dash. A list item's dash must -always be the first non-space character on its line (so, in -particular, a list can not begin on the first line of a comment - -start it on the second line instead). - -Example: -\begin{verbatim} - We go by induction on [n]: - - If [n] is 0... - - If [n] is [S n'] we require... - - two paragraphs of reasoning, and two subcases: - - - In the first case... - - In the second case... - - So the theorem holds. -\end{verbatim} - -\paragraph{Rules.} -More than 4 leading dashes produce an horizontal rule. - -\paragraph{Emphasis.} -Text can be italicized by placing it in underscores. A non-identifier -character must precede the leading underscore and follow the trailing -underscore, so that uses of underscores in names aren't mistaken for -emphasis. Usually, these are spaces or punctuation. - -\begin{verbatim} - This sentence contains some _emphasized text_. -\end{verbatim} - -\paragraph{Escapings to \LaTeX\ and HTML.} -Pure \LaTeX\ or HTML material can be inserted using the following -escape sequences: -\begin{itemize} -\item \verb+$...LaTeX stuff...$+ inserts some \LaTeX\ material in math mode. - Simply discarded in HTML output. - -\item \verb+%...LaTeX stuff...%+ inserts some \LaTeX\ material. - Simply discarded in HTML output. - -\item \verb+#...HTML stuff...#+ inserts some HTML material. Simply - discarded in \LaTeX\ output. -\end{itemize} - -Note: to simply output the characters \verb+$+, \verb+%+ and \verb+#+ -and escaping their escaping role, these characters must be doubled. - -\paragraph{Verbatim.} -Verbatim material is introduced by a leading \verb+<<+ and closed by -\verb+>>+ at the beginning of a line. Example: -\begin{verbatim} -Here is the corresponding caml code: -<< - let rec fact n = - if n <= 1 then 1 else n * fact (n-1) ->> -\end{verbatim} - - -\paragraph{Hyperlinks.} -Hyperlinks can be inserted into the HTML output, so that any -identifier is linked to the place of its definition. - -\texttt{coqc \emph{file}.v} automatically dumps localization information -in \texttt{\emph{file}.glob} or appends it to a file specified using option -\texttt{\mm{}dump-glob \emph{file}}. Take care of erasing this global file, if -any, when starting the whole compilation process. - -Then invoke \texttt{coqdoc} or \texttt{coqdoc \mm{}glob-from \emph{file}} to tell -\coqdoc\ to look for name resolutions into the file \texttt{\emph{file}} -(it will look in \texttt{\emph{file}.glob} by default). - -Identifiers from the \Coq\ standard library are linked to the \Coq\ -web site at \url{http://coq.inria.fr/library/}. This behavior can be -changed using command line options \url{--no-externals} and -\url{--coqlib}; see below. - - -\paragraph{Hiding / Showing parts of the source.} -Some parts of the source can be hidden using command line options -\texttt{-g} and \texttt{-l} (see below), or using such comments: -\begin{alltt} -(* begin hide *) -\emph{some Coq material} -(* end hide *) -\end{alltt} -Conversely, some parts of the source which would be hidden can be -shown using such comments: -\begin{alltt} -(* begin show *) -\emph{some Coq material} -(* end show *) -\end{alltt} -The latter cannot be used around some inner parts of a proof, but can -be used around a whole proof. - - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\subsection{Usage} - -\coqdoc\ is invoked on a shell command line as follows: -\begin{displaymath} - \texttt{coqdoc }<\textit{options and files}> -\end{displaymath} -Any command line argument which is not an option is considered to be a -file (even if it starts with a \verb!-!). \Coq\ files are identified -by the suffixes \verb!.v! and \verb!.g! and \LaTeX\ files by the -suffix \verb!.tex!. - -\begin{description} -\item[HTML output] ~\par - This is the default output. - One HTML file is created for each \Coq\ file given on the command line, - together with a file \texttt{index.html} (unless option - \texttt{-no-index} is passed). The HTML pages use a style sheet - named \texttt{style.css}. Such a file is distributed with \coqdoc. - -\item[\LaTeX\ output] ~\par - A single \LaTeX\ file is created, on standard output. It can be - redirected to a file with option \texttt{-o}. - The order of files on the command line is kept in the final - document. \LaTeX\ files given on the command line are copied `as is' - in the final document . - DVI and PostScript can be produced directly with the options - \texttt{-dvi} and \texttt{-ps} respectively. - -\item[\texmacs\ output] ~\par - To translate the input files to \texmacs\ format, to be used by - the \texmacs\ Coq interface - (see \url{http://www-sop.inria.fr/lemme/Philippe.Audebaud/tmcoq/}). -\end{description} - - -\subsubsection*{Command line options} - - -\paragraph{Overall options} - -\begin{description} - -\item[\texttt{\mm{}html}] ~\par - - Select a HTML output. - -\item[\texttt{\mm{}latex}] ~\par - - Select a \LaTeX\ output. - -\item[\texttt{\mm{}dvi}] ~\par - - Select a DVI output. - -\item[\texttt{\mm{}ps}] ~\par - - Select a PostScript output. - -\item[\texttt{\mm{}texmacs}] ~\par - - Select a \texmacs\ output. - -\item[\texttt{--stdout}] ~\par - - Write output to stdout. - -\item[\texttt{-o }\textit{file}, \texttt{\mm{}output }\textit{file}] ~\par - - Redirect the output into the file `\textit{file}' (meaningless with - \texttt{-html}). - -\item[\texttt{-d }\textit{dir}, \texttt{\mm{}directory }\textit{dir}] ~\par - - Output files into directory `\textit{dir}' instead of current - directory (option \texttt{-d} does not change the filename specified - with option \texttt{-o}, if any). - -\item[\texttt{\mm{}body-only}] ~\par - - Suppress the header and trailer of the final document. Thus, you can - insert the resulting document into a larger one. - -\item[\texttt{-p} \textit{string}, \texttt{\mm{}preamble} \textit{string}]~\par - - Insert some material in the \LaTeX\ preamble, right before - \verb!\begin{document}! (meaningless with \texttt{-html}). - -\item[\texttt{\mm{}vernac-file }\textit{file}, - \texttt{\mm{}tex-file }\textit{file}] ~\par - - Considers the file `\textit{file}' respectively as a \verb!.v! - (or \verb!.g!) file or a \verb!.tex! file. - -\item[\texttt{\mm{}files-from }\textit{file}] ~\par - - Read file names to process in file `\textit{file}' as if they were - given on the command line. Useful for program sources splitted in - several directories. - -\item[\texttt{-q}, \texttt{\mm{}quiet}] ~\par - - Be quiet. Do not print anything except errors. - -\item[\texttt{-h}, \texttt{\mm{}help}] ~\par - - Give a short summary of the options and exit. - -\item[\texttt{-v}, \texttt{\mm{}version}] ~\par - - Print the version and exit. - -\end{description} - -\paragraph{Index options} - -Default behavior is to build an index, for the HTML output only, into -\texttt{index.html}. - -\begin{description} - -\item[\texttt{\mm{}no-index}] ~\par - - Do not output the index. - -\item[\texttt{\mm{}multi-index}] ~\par - - Generate one page for each category and each letter in the index, - together with a top page \texttt{index.html}. - -\item[\texttt{\mm{}index }\textit{string}] ~\par - - Make the filename of the index \textit{string} instead of ``index''. - Useful since ``index.html'' is special. - -\end{description} - -\paragraph{Table of contents option} - -\begin{description} - -\item[\texttt{-toc}, \texttt{\mm{}table-of-contents}] ~\par - - Insert a table of contents. - For a \LaTeX\ output, it inserts a \verb!\tableofcontents! at the - beginning of the document. For a HTML output, it builds a table of - contents into \texttt{toc.html}. - -\item[\texttt{\mm{}toc-depth }\textit{int}] ~\par - - Only include headers up to depth \textit{int} in the table of - contents. - -\end{description} - -\paragraph{Hyperlinks options} -\begin{description} - -\item[\texttt{\mm{}glob-from }\textit{file}] ~\par - - Make references using \Coq\ globalizations from file \textit{file}. - (Such globalizations are obtained with \Coq\ option \texttt{-dump-glob}). - -\item[\texttt{\mm{}no-externals}] ~\par - - Do not insert links to the \Coq\ standard library. - -\item[\texttt{\mm{}external }\textit{url}~\textit{coqdir}] ~\par - - Use given URL for linking references whose name starts with prefix - \textit{coqdir}. - -\item[\texttt{\mm{}coqlib }\textit{url}] ~\par - - Set base URL for the \Coq\ standard library (default is - \url{http://coq.inria.fr/library/}). This is equivalent to - \texttt{\mm{}external }\textit{url}~\texttt{Coq}. - -\item[\texttt{-R }\textit{dir }\textit{coqdir}] ~\par - - Map physical directory \textit{dir} to \Coq\ logical directory - \textit{coqdir} (similarly to \Coq\ option \texttt{-R}). - - Note: option \texttt{-R} only has effect on the files - \emph{following} it on the command line, so you will probably need - to put this option first. - -\end{description} - -\paragraph{Title options} -\begin{description} -\item[\texttt{-s }, \texttt{\mm{}short}] ~\par - - Do not insert titles for the files. The default behavior is to - insert a title like ``Library Foo'' for each file. - -\item[\texttt{\mm{}lib-name }\textit{string}] ~\par - - Print ``\textit{string} Foo'' instead of ``Library Foo'' in titles. - For example ``Chapter'' and ``Module'' are reasonable choices. - -\item[\texttt{\mm{}no-lib-name}] ~\par - - Print just ``Foo'' instead of ``Library Foo'' in titles. - -\item[\texttt{\mm{}lib-subtitles}] ~\par - - Look for library subtitles. When enabled, the beginning of each - file is checked for a comment of the form: -\begin{alltt} -(** * ModuleName : text *) -\end{alltt} - where \texttt{ModuleName} must be the name of the file. If it is - present, the \texttt{text} is used as a subtitle for the module in - appropriate places. - -\item[\texttt{-t }\textit{string}, - \texttt{\mm{}title }\textit{string}] ~\par - - Set the document title. - -\end{description} - -\paragraph{Contents options} -\begin{description} - -\item[\texttt{-g}, \texttt{\mm{}gallina}] ~\par - - Do not print proofs. - -\item[\texttt{-l}, \texttt{\mm{}light}] ~\par - - Light mode. Suppress proofs (as with \texttt{-g}) and the following commands: - \begin{itemize} - \item {}[\texttt{Recursive}] \texttt{Tactic Definition} - \item \texttt{Hint / Hints} - \item \texttt{Require} - \item \texttt{Transparent / Opaque} - \item \texttt{Implicit Argument / Implicits} - \item \texttt{Section / Variable / Hypothesis / End} - \end{itemize} - -\end{description} -The behavior of options \texttt{-g} and \texttt{-l} can be locally -overridden using the \texttt{(* begin show *)} \dots\ \texttt{(* end - show *)} environment (see above). - -There are a few options to drive the parsing of comments: -\begin{description} -\item[\texttt{\mm{}parse-comments}] ~\par - - Parses regular comments delimited by \texttt{(*} and \texttt{*)} as - well. They are typeset inline. - -\item[\texttt{\mm{}plain-comments}] ~\par - - Do not interpret comments, simply copy them as plain-text. - -\item[\texttt{\mm{}interpolate}] ~\par - - Use the globalization information to typeset identifiers appearing in - \Coq{} escapings inside comments. -\end{description} - - -\paragraph{Language options} - -Default behavior is to assume ASCII 7 bits input files. - -\begin{description} - -\item[\texttt{-latin1}, \texttt{\mm{}latin1}] ~\par - - Select ISO-8859-1 input files. It is equivalent to - \texttt{--inputenc latin1 --charset iso-8859-1}. - -\item[\texttt{-utf8}, \texttt{\mm{}utf8}] ~\par - - Select UTF-8 (Unicode) input files. It is equivalent to - \texttt{--inputenc utf8 --charset utf-8}. - \LaTeX\ UTF-8 support can be found at - \url{http://www.ctan.org/tex-archive/macros/latex/contrib/supported/unicode/}. - -\item[\texttt{\mm{}inputenc} \textit{string}] ~\par - - Give a \LaTeX\ input encoding, as an option to \LaTeX\ package - \texttt{inputenc}. - -\item[\texttt{\mm{}charset} \textit{string}] ~\par - - Specify the HTML character set, to be inserted in the HTML header. - -\end{description} - - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\subsection[The coqdoc \LaTeX{} style file]{The coqdoc \LaTeX{} style file\label{section:coqdoc.sty}} - -In case you choose to produce a document without the default \LaTeX{} -preamble (by using option \verb|--no-preamble|), then you must insert -into your own preamble the command -\begin{quote} - \verb|\usepackage{coqdoc}| -\end{quote} - -The package optionally takes the argument \verb|[color]| to typeset -identifiers with colors (this requires the \verb|xcolor| package). - -Then you may alter the rendering of the document by -redefining some macros: -\begin{description} - -\item[\texttt{coqdockw}, \texttt{coqdocid}, \ldots] ~ - - The one-argument macros for typesetting keywords and identifiers. - Defaults are sans-serif for keywords and italic for identifiers. - - For example, if you would like a slanted font for keywords, you - may insert -\begin{verbatim} - \renewcommand{\coqdockw}[1]{\textsl{#1}} -\end{verbatim} - anywhere between \verb|\usepackage{coqdoc}| and - \verb|\begin{document}|. - -\item[\texttt{coqdocmodule}] ~ - - One-argument macro for typesetting the title of a \verb|.v| file. - Default is -\begin{verbatim} -\newcommand{\coqdocmodule}[1]{\section*{Module #1}} -\end{verbatim} - and you may redefine it using \verb|\renewcommand|. - -\end{description} - - diff --git a/doc/refman/coqide-queries.eps b/doc/refman/coqide-queries.eps deleted file mode 100644 index 6345bb29..00000000 --- a/doc/refman/coqide-queries.eps 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\ No newline at end of file diff --git a/doc/refman/menu.html b/doc/refman/menu.html deleted file mode 100644 index db19678f..00000000 --- a/doc/refman/menu.html +++ /dev/null @@ -1,29 +0,0 @@ -<HTML> - -<BODY> - -<CENTER> - -<TABLE BORDER="0" CELLPADDING=10> -<TR> -<TD><CENTER><A HREF="cover.html" TARGET="UP"><FONT SIZE=2>Cover page</FONT></A></CENTER></TD> -<TD><CENTER><A HREF="toc.html" TARGET="UP"><FONT SIZE=2>Table of contents</FONT></A></CENTER></TD> -<TD><CENTER><A HREF="biblio.html" TARGET="UP"><FONT SIZE=2> -Bibliography</FONT></A></CENTER></TD> -<TD><CENTER><A HREF="general-index.html" TARGET="UP"><FONT SIZE=2> -Global Index -</FONT></A></CENTER></TD> -<TD><CENTER><A HREF="tactic-index.html" TARGET="UP"><FONT SIZE=2> -Tactics Index -</FONT></A></CENTER></TD> -<TD><CENTER><A HREF="command-index.html" TARGET="UP"><FONT SIZE=2> -Vernacular Commands Index -</FONT></A></CENTER></TD> -<TD><CENTER><A HREF="error-index.html" TARGET="UP"><FONT SIZE=2> -Index of Error Messages -</FONT></A></CENTER></TD> -</TABLE> - -</CENTER> - -</BODY></HTML>
\ No newline at end of file diff --git a/doc/rt/RefMan-cover.tex b/doc/rt/RefMan-cover.tex deleted file mode 100644 index d881329a..00000000 --- a/doc/rt/RefMan-cover.tex +++ /dev/null @@ -1,46 +0,0 @@ -\documentstyle[RRcover]{book} - % L'utilisation du style `french' force le résumé français à - % apparaître en premier. - -\RRtitle{Manuel de r\'ef\'erence du syst\`eme Coq \\ version V7.1} -\RRetitle{The Coq Proof Assistant \\ Reference Manual \\ Version 7.1 -\thanks -{This research was partly supported by ESPRIT Basic Research -Action ``Types'' and by the GDR ``Programmation'' co-financed by MRE-PRC and CNRS.} -} -\RRauthor{Bruno Barras, Samuel Boutin, Cristina Cornes, -Judica\"el Courant, Jean-Christophe Filli\^atre, Eduardo Gim\'enez, -Hugo Herbelin, G\'erard Huet, C\'esar Mu\~noz, Chetan Murthy, -Catherine Parent, Christine Paulin-Mohring, -Amokrane Sa{\"\i}bi, Benjamin Werner} -\authorhead{} -\titlehead{Coq V7.1 Reference Manual} -\RRtheme{2} -\RRprojet{Coq} -\RRNo{0123456789} -\RRdate{May 1997} -%\RRpages{} -\URRocq - -\RRresume{Coq est un syst\`eme permettant le d\'eveloppement et la -v\'erification de preuves formelles dans une logique d'ordre -sup\'erieure incluant un riche langage de d\'efinitions de fonctions. -Ce document constitue le manuel de r\'ef\'erence de la version V7.1 -qui est distribu\'ee par ftp anonyme à l'adresse -\url{ftp://ftp.inria.fr/INRIA/coq/}} - -\RRmotcle{Coq, Syst\`eme d'aide \`a la preuve, Preuves formelles, -Calcul des Constructions Inductives} - - -\RRabstract{Coq is a proof assistant based on a higher-order logic -allowing powerful definitions of functions. -Coq V7.1 is available by anonymous -ftp at \url{ftp://ftp.inria.fr/INRIA/coq/}} - -\RRkeyword{Coq, Proof Assistant, Formal Proofs, Calculus of Inductives -Constructions} - -\begin{document} -\makeRT -\end{document} diff --git a/doc/rt/Tutorial-cover.tex b/doc/rt/Tutorial-cover.tex deleted file mode 100644 index b747b812..00000000 --- a/doc/rt/Tutorial-cover.tex +++ /dev/null @@ -1,48 +0,0 @@ -\documentstyle[RRcover]{book} - % L'utilisation du style `french' force le résumé français à - % apparaître en premier. -\RRetitle{ -The Coq Proof Assistant \\ A Tutorial \\ Version 7.1 -\thanks{This research was partly supported by ESPRIT Basic Research -Action ``Types'' and by the GDR ``Programmation'' co-financed by MRE-PRC and CNRS.} -} -\RRtitle{Coq \\ Une introduction \\ V7.1 } -\RRauthor{G\'erard Huet, Gilles Kahn and Christine Paulin-Mohring} -\RRtheme{2} -\RRprojet{{Coq -\\[15pt] -{INRIA Rocquencourt} -{\hskip -5.25pt} -~~{\bf ---}~~ - \def\thefootnote{\arabic{footnote}\hss} -{CNRS - ENS Lyon} -\footnote[1]{LIP URA 1398 du CNRS, -46 All\'ee d'Italie, 69364 Lyon CEDEX 07, France.} -{\hskip -14pt}}} - -%\RRNo{0123456789} -\RRNo{0204} -\RRdate{Ao\^ut 1997} - -\URRocq -\RRresume{Coq est un syst\`eme permettant le d\'eveloppement et la -v\'erification de preuves formelles dans une logique d'ordre -sup\'erieure incluant un riche langage de d\'efinitions de fonctions. -Ce document constitue une introduction pratique \`a l'utilisation de -la version V7.1 qui est distribu\'ee par ftp anonyme à l'adresse -\url{ftp://ftp.inria.fr/INRIA/coq/}} - -\RRmotcle{Coq, Syst\`eme d'aide \`a la preuve, Preuves formelles, Calcul -des Constructions Inductives} - -\RRabstract{Coq is a proof assistant based on a higher-order logic -allowing powerful definitions of functions. This document is a -tutorial for the version V7.1 of Coq. This version is available by -anonymous ftp at \url{ftp://ftp.inria.fr/INRIA/coq/}} - -\RRkeyword{Coq, Proof Assistant, Formal Proofs, Calculus of Inductives -Constructions} - -\begin{document} -\makeRT -\end{document} diff --git a/doc/tools/Translator.tex b/doc/tools/Translator.tex deleted file mode 100644 index 005ca9c0..00000000 --- a/doc/tools/Translator.tex +++ /dev/null @@ -1,898 +0,0 @@ -\ifx\pdfoutput\undefined % si on est pas en pdflatex -\documentclass[11pt,a4paper]{article} -\else -\documentclass[11pt,a4paper,pdftex]{article} -\fi -\usepackage[latin1]{inputenc} -\usepackage[T1]{fontenc} -\usepackage{pslatex} -\usepackage{url} -\usepackage{verbatim} -\usepackage{amsmath} -\usepackage{amssymb} -\usepackage{array} -\usepackage{fullpage} - -\title{Translation from Coq V7 to V8} -\author{The Coq Development Team} - -%% Macros etc. -\catcode`\_=13 -\let\subscr=_ -\def_{\ifmmode\sb\else\subscr\fi} - -\def\NT#1{\langle\textit{#1}\rangle} -\def\NTL#1#2{\langle\textit{#1}\rangle_{#2}} -%\def\TERM#1{\textsf{\bf #1}} -\def\TERM#1{\texttt{#1}} -\newenvironment{transbox} - {\begin{center}\tt\begin{tabular}{l|ll} \hfil\textrm{V7} & \hfil\textrm{V8} \\ \hline} - {\end{tabular}\end{center}} -\def\TRANS#1#2 - {\begin{tabular}[t]{@{}l@{}}#1\end{tabular} & - \begin{tabular}[t]{@{}l@{}}#2\end{tabular} \\} -\def\TRANSCOM#1#2#3 - {\begin{tabular}[t]{@{}l@{}}#1\end{tabular} & - \begin{tabular}[t]{@{}l@{}}#2\end{tabular} & #3 \\} - -%% -%% -%% -\begin{document} -\maketitle - -\section{Introduction} - -Coq version 8.0 is a major version and carries major changes: the -concrete syntax was redesigned almost from scratch, and many notions -of the libraries were renamed for uniformisation purposes. We felt -that these changes could discourage users with large theories from -switching to the new version. - -The goal of this document is to introduce these changes on simple -examples (mainly the syntactic changes), and describe the automated -tools to help moving to V8.0. Essentially, it consists of a translator -that takes as input a Coq source file in old syntax and produces a -file in new syntax and adapted to the new standard library. The main -extra features of this translator is that it keeps comments, even -those within expressions\footnote{The position of those comment might -differ slightly since there is no exact matching of positions between -old and new syntax.}. - -The document is organised as follows: first section describes the new -syntax on simple examples. It is very translation-oriented. This -should give users of older versions the flavour of the new syntax, and -allow them to make translation manually on small -examples. Section~\ref{Translation} explains how the translation -process can be automatised for the most part (the boring one: applying -similar changes over thousands of lines of code). We strongly advise -users to follow these indications, in order to avoid many potential -complications of the translation process. - - -\section{The new syntax on examples} - -The goal of this section is to introduce to the new syntax of Coq on -simple examples, rather than just giving the new grammar. It is -strongly recommended to read first the definition of the new syntax -(in the reference manual), but this document should also be useful for -the eager user who wants to start with the new syntax quickly. - -The toplevel has an option {\tt -translate} which allows to -interactively translate commands. This toplevel translator accepts a -command, prints the translation on standard output (after a % -\verb+New syntax:+ balise), executes the command, and waits for another -command. The only requirements is that they should be syntactically -correct, but they do not have to be well-typed. - -This interactive translator proved to be useful in two main -usages. First as a ``debugger'' of the translation. Before the -translation, it may help in spotting possible conflicts between the -new syntax and user notations. Or when the translation fails for some -reason, it makes it easy to find the exact reason why it failed and -make attempts in fixing the problem. - -The second usage of the translator is when trying to make the first -proofs in new syntax. Well trained users will automatically think -their scripts in old syntax and might waste much time (and the -intuition of the proof) if they have to search the translation in a -document. Running a translator in the background will allow the user -to instantly have the answer. - -The rest of this section is a description of all the aspects of the -syntax that changed and how they were translated. All the examples -below can be tested by entering the V7 commands in the toplevel -translator. - - -%% - -\subsection{Changes in lexical conventions w.r.t. V7} - -\subsubsection{Identifiers} - -The lexical conventions changed: \TERM{_} is not a regular identifier -anymore. It is used in terms as a placeholder for subterms to be inferred -at type-checking, and in patterns as a non-binding variable. - -Furthermore, only letters (Unicode letters), digits, single quotes and -_ are allowed after the first character. - -\subsubsection{Quoted string} - -Quoted strings are used typically to give a filename (which may not -be a regular identifier). As before they are written between double -quotes ("). Unlike for V7, there is no escape character: characters -are written normally except the double quote which is doubled. - -\begin{transbox} -\TRANS{"abcd$\backslash\backslash$efg"}{"abcd$\backslash$efg"} -\TRANS{"abcd$\backslash$"efg"}{"abcd""efg"} -\end{transbox} - - -\subsection{Main changes in terms w.r.t. V7} - - -\subsubsection{Precedence of application} - -In the new syntax, parentheses are not really part of the syntax of -application. The precedence of application (10) is tighter than all -prefix and infix notations. It makes it possible to remove parentheses -in many contexts. - -\begin{transbox} -\TRANS{(A x)->(f x)=(g y)}{A x -> f x = g y} -\TRANS{(f [x]x)}{f (fun x => x)} -\end{transbox} - - -\subsubsection{Arithmetics and scopes} - -The specialized notation for \TERM{Z} and \TERM{R} (introduced by -symbols \TERM{`} and \TERM{``}) have disappeared. They have been -replaced by the general notion of scope. - -\begin{center} -\begin{tabular}{l|l|l} -type & scope name & delimiter \\ -\hline -types & type_scope & \TERM{type} \\ -\TERM{bool} & bool_scope & \\ -\TERM{nat} & nat_scope & \TERM{nat} \\ -\TERM{Z} & Z_scope & \TERM{Z} \\ -\TERM{R} & R_scope & \TERM{R} \\ -\TERM{positive} & positive_scope & \TERM{P} -\end{tabular} -\end{center} - -In order to use notations of arithmetics on \TERM{Z}, its scope must -be opened with command \verb+Open Scope Z_scope.+ Another possibility -is using the scope change notation (\TERM{\%}). The latter notation is -to be used when notations of several scopes appear in the same -expression. - -In examples below, scope changes are not needed if the appropriate scope -has been opened. Scope \verb|nat_scope| is opened in the initial state of Coq. -\begin{transbox} -\TRANSCOM{`0+x=x+0`}{0+x=x+0}{\textrm{Z_scope}} -\TRANSCOM{``0 + [if b then ``1`` else ``2``]``}{0 + if b then 1 else 2}{\textrm{R_scope}} -\TRANSCOM{(0)}{0}{\textrm{nat_scope}} -\end{transbox} - -Below is a table that tells which notation is available in which -scope. The relative precedences and associativity of operators is the -same as in usual mathematics. See the reference manual for more -details. However, it is important to remember that unlike V7, the type -operators for product and sum are left-associative, in order not to -clash with arithmetic operators. - -\begin{center} -\begin{tabular}{l|l} -scope & notations \\ -\hline -nat_scope & \texttt{+ - * < <= > >=} \\ -Z_scope & \texttt{+ - * / mod < <= > >= ?=} \\ -R_scope & \texttt{+ - * / < <= > >=} \\ -type_scope & \texttt{* +} \\ -bool_scope & \texttt{\&\& || -} \\ -list_scope & \texttt{:: ++} -\end{tabular} -\end{center} - - - -\subsubsection{Notation for implicit arguments} - -The explicitation of arguments is closer to the \emph{bindings} -notation in tactics. Argument positions follow the argument names of -the head constant. The example below assumes \verb+f+ is a function -with two implicit dependent arguments named \verb+x+ and \verb+y+. -\begin{transbox} -\TRANS{f 1!t1 2!t2 t3}{f (x:=t1) (y:=t2) t3} -\TRANS{!f t1 t2}{@f t1 t2} -\end{transbox} - - -\subsubsection{Inferred subterms} - -Subterms that can be automatically inferred by the type-checker is now -written {\tt _} - -\begin{transbox} -\TRANS{?}{_} -\end{transbox} - -\subsubsection{Universal quantification} - -The universal quantification and dependent product types are now -introduced by the \texttt{forall} keyword before the binders and a -comma after the binders. - -The syntax of binders also changed significantly. A binder can simply be -a name when its type can be inferred. In other cases, the name and the type -of the variable are put between parentheses. When several consecutive -variables have the same type, they can be grouped. Finally, if all variables -have the same type, parentheses can be omitted. - -\begin{transbox} -\TRANS{(x:A)B}{forall (x:~A), B ~~\textrm{or}~~ forall x:~A, B} -\TRANS{(x,y:nat)P}{forall (x y :~nat), P ~~\textrm{or}~~ forall x y :~nat, P} -\TRANS{(x,y:nat;z:A)P}{forall (x y :~nat) (z:A), P} -\TRANS{(x,y,z,t:?)P}{forall x y z t, P} -\TRANS{(x,y:nat;z:?)P}{forall (x y :~nat) z, P} -\end{transbox} - -\subsubsection{Abstraction} - -The notation for $\lambda$-abstraction follows that of universal -quantification. The binders are surrounded by keyword \texttt{fun} -and \verb+=>+. - -\begin{transbox} -\TRANS{[x,y:nat; z](f a b c)}{fun (x y:nat) z => f a b c} -\end{transbox} - - -\subsubsection{Pattern-matching} - -Beside the usage of the keyword pair \TERM{match}/\TERM{with} instead of -\TERM{Cases}/\TERM{of}, the main change is the notation for the type of -branches and return type. It is no longer written between \TERM{$<$ $>$} before -the \TERM{Cases} keyword, but interleaved with the destructured objects. - -The idea is that for each destructured object, one may specify a -variable name (after the \TERM{as} keyword) to tell how the branches -types depend on this destructured objects (case of a dependent -elimination), and also how they depend on the value of the arguments -of the inductive type of the destructured objects (after the \TERM{in} -keyword). The type of branches is then given after the keyword -\TERM{return}, unless it can be inferred. - -Moreover, when the destructured object is a variable, one may use this -variable in the return type. - -\begin{transbox} -\TRANS{Cases n of\\~~ O => O \\| (S k) => (1) end}{match n with\\~~ 0 => 0 \\| S k => 1 end} -\TRANS{Cases m n of \\~~0 0 => t \\| ... end}{match m, n with \\~~0, 0 => t \\| ... end} -\TRANS{<[n:nat](P n)>Cases T of ... end}{match T as n return P n with ... end} -\TRANS{<[n:nat][p:(even n)]\~{}(odd n)>Cases p of\\~~ ... \\end}{match p in even n return \~{} odd n with\\~~ ...\\end} -\end{transbox} - -The annotations of the special pattern-matching operators -(\TERM{if}/\TERM{then}/\TERM{else}) and \TERM{let()} also changed. The -only restriction is that the destructuring \TERM{let} does not allow -dependent case analysis. - -\begin{transbox} -\TRANS{ - \begin{tabular}{@{}l} - <[n:nat;x:(I n)](P n x)>if t then t1 \\ - else t2 - \end{tabular}}% -{\begin{tabular}{@{}l} - if t as x in I n return P n x then t1 \\ - else t2 - \end{tabular}} -\TRANS{<[n:nat](P n)>let (p,q) = t1 in t2}% -{let (p,q) in I n return P n := t1 in t2} -\end{transbox} - - -\subsubsection{Fixpoints and cofixpoints} - -An simpler syntax for non-mutual fixpoints is provided, making it very close -to the usual notation for non-recursive functions. The decreasing argument -is now indicated by an annotation between curly braces, regardless of the -binders grouping. The annotation can be omitted if the binders introduce only -one variable. The type of the result can be omitted if inferable. - -\begin{transbox} -\TRANS{Fix plus\{plus [n:nat] : nat -> nat :=\\~~ [m]...\}}{fix plus (n m:nat) \{struct n\}: nat := ...} -\TRANS{Fix fact\{fact [n:nat]: nat :=\\ -~~Cases n of\\~~~~ O => (1) \\~~| (S k) => (mult n (fact k)) end\}}{fix fact - (n:nat) :=\\ -~~match n with \\~~~~0 => 1 \\~~| (S k) => n * fact k end} -\end{transbox} - -There is a syntactic sugar for single fixpoints (defining one -variable) associated to a local definition: - -\begin{transbox} -\TRANS{let f := Fix f \{f [x:A] : T := M\} in\\(g (f y))}{let fix f (x:A) : T := M in\\g (f x)} -\end{transbox} - -The same applies to cofixpoints, annotations are not allowed in that case. - -\subsubsection{Notation for type cast} - -\begin{transbox} -\TRANS{O :: nat}{0 : nat} -\end{transbox} - -\subsection{Main changes in tactics w.r.t. V7} - -The main change is that all tactic names are lowercase. This also holds for -Ltac keywords. - -\subsubsection{Renaming of induction tactics} - -\begin{transbox} -\TRANS{NewDestruct}{destruct} -\TRANS{NewInduction}{induction} -\TRANS{Induction}{simple induction} -\TRANS{Destruct}{simple destruct} -\end{transbox} - -\subsubsection{Ltac} - -Definitions of macros are introduced by \TERM{Ltac} instead of -\TERM{Tactic Definition}, \TERM{Meta Definition} or \TERM{Recursive -Definition}. They are considered recursive by default. - -\begin{transbox} -\TRANS{Meta Definition my_tac t1 t2 := t1; t2.}% -{Ltac my_tac t1 t2 := t1; t2.} -\end{transbox} - -Rules of a match command are not between square brackets anymore. - -Context (understand a term with a placeholder) instantiation \TERM{inst} -became \TERM{context}. Syntax is unified with subterm matching. - -\begin{transbox} -\TRANS{Match t With [C[x=y]] -> Inst C[y=x]}% -{match t with context C[x=y] => context C[y=x] end} -\end{transbox} - -Arguments of macros use the term syntax. If a general Ltac expression -is to be passed, it must be prefixed with ``{\tt ltac :}''. In other -cases, when a \'{} was necessary, it is replaced by ``{\tt constr :}'' - -\begin{transbox} -\TRANS{my_tac '(S x)}{my_tac (S x)} -\TRANS{my_tac (Let x=tac In x)}{my_tac ltac:(let x:=tac in x)} -\TRANS{Let x = '[x](S (S x)) In Apply x}% -{let x := constr:(fun x => S (S x)) in apply x} -\end{transbox} - -{\tt Match Context With} is now called {\tt match goal with}. Its -argument is an Ltac expression by default. - - -\subsubsection{Named arguments of theorems ({\em bindings})} - -\begin{transbox} -\TRANS{Apply thm with x:=t 1:=u}{apply thm with (x:=t) (1:=u)} -\end{transbox} - - -\subsubsection{Occurrences} - -To avoid ambiguity between a numeric literal and the optional -occurrence numbers of this term, the occurrence numbers are put after -the term itself and after keyword \TERM{as}. -\begin{transbox} -\TRANS{Pattern 1 2 (f x) 3 4 d y z}{pattern f x at 1 2, d at 3 4, y, z} -\end{transbox} - - -\subsubsection{{\tt LetTac} and {\tt Pose}} - -Tactic {\tt LetTac} was renamed into {\tt set}, and tactic {\tt Pose} -was a particular case of {\tt LetTac} where the abbreviation is folded -in the conclusion\footnote{There is a tactic called {\tt pose} in V8, -but its behaviour is not to fold the abbreviation at all.}. - -\begin{transbox} -\TRANS{LetTac x = t in H}{set (x := t) in H} -\TRANS{Pose x := t}{set (x := t)} -\end{transbox} - -{\tt LetTac} could be followed by a specification (called a clause) of -the places where the abbreviation had to be folded (hypothese and/or -conclusion). Clauses are the syntactic notion to denote in which parts -of a goal a given transformation shold occur. Its basic notation is -either \TERM{*} (meaning everywhere), or {\tt\textrm{\em hyps} |- -\textrm{\em concl}} where {\em hyps} is either \TERM{*} (to denote all -the hypotheses), or a comma-separated list of either hypothesis name, -or {\tt (value of $H$)} or {\tt (type of $H$)}. Moreover, occurrences -can be specified after every hypothesis after the {\TERM{at}} -keyword. {\em concl} is either empty or \TERM{*}, and can be followed -by occurences. - -\begin{transbox} -\TRANS{in Goal}{in |- *} -\TRANS{in H H1}{in H1, H2 |-} -\TRANS{in H H1 ...}{in * |-} -\TRANS{in H H1 Goal}{in H1, H2 |- *} -\TRANS{in H H1 H2 ... Goal}{in *} -\TRANS{in 1 2 H 3 4 H0 1 3 Goal}{in H at 1 2, H0 at 3 4 |- * at 1 3} -\end{transbox} - -\subsection{Main changes in vernacular commands w.r.t. V7} - - -\subsubsection{Require} - -The default behaviour of {\tt Require} is not to open the loaded -module. - -\begin{transbox} -\TRANS{Require Arith}{Require Import Arith} -\end{transbox} - -\subsubsection{Binders} - -The binders of vernacular commands changed in the same way as those of -fixpoints. This also holds for parameters of inductive definitions. - - -\begin{transbox} -\TRANS{Definition x [a:A] : T := M}{Definition x (a:A) : T := M} -\TRANS{Inductive and [A,B:Prop]: Prop := \\~~conj : A->B->(and A B)}% - {Inductive and (A B:Prop): Prop := \\~~conj : A -> B -> and A B} -\end{transbox} - -\subsubsection{Hints} - -Both {\tt Hints} and {\tt Hint} commands are beginning with {\tt Hint}. - -Command {\tt HintDestruct} has disappeared. - - -The syntax of \emph{Extern} hints changed: the pattern and the tactic -to be applied are separated by a {\tt =>}. -\begin{transbox} -\TRANS{Hint name := Resolve (f ? x)}% -{Hint Resolve (f _ x)} -\TRANS{Hint name := Extern 4 (toto ?) Apply lemma}% -{Hint Extern 4 (toto _) => apply lemma} -\TRANS{Hints Resolve x y z}{Hint Resolve x y z} -\TRANS{Hints Resolve f : db1 db2}{Hint Resolve f : db1 db2} -\TRANS{Hints Immediate x y z}{Hint Immediate x y z} -\TRANS{Hints Unfold x y z}{Hint Unfold x y z} -%% \TRANS{\begin{tabular}{@{}l} -%% HintDestruct Local Conclusion \\ -%% ~~name (f ? ?) 3 [Apply thm] -%% \end{tabular}}% -%% {\begin{tabular}{@{}l} -%% Hint Local Destuct name := \\ -%% ~~3 Conclusion (f _ _) => apply thm -%% \end{tabular}} -\end{transbox} - - -\subsubsection{Implicit arguments} - - -{\tt Set Implicit Arguments} changed its meaning in V8: the default is -to turn implicit only the arguments that are {\em strictly} implicit -(or rigid), i.e. that remains inferable whatever the other arguments -are. For instance {\tt x} inferable from {\tt P x} is not strictly -inferable since it can disappears if {\tt P} is instanciated by a term -which erases {\tt x}. - -\begin{transbox} -\TRANS{Set Implicit Arguments}% -{\begin{tabular}{l} - Set Implicit Arguments. \\ - Unset Strict Implicits. - \end{tabular}} -\end{transbox} - -However, you may wish to adopt the new semantics of {\tt Set Implicit -Arguments} (for instance because you think that the choice of -arguments it sets implicit is more ``natural'' for you). - - -\subsection{Changes in standard library} - -Many lemmas had their named changed to improve uniformity. The user -generally do not have to care since the translators performs the -renaming. - - Type {\tt entier} from fast_integer.v is renamed into {\tt N} by the -translator. As a consequence, user-defined objects of same name {\tt N} -are systematically qualified even tough it may not be necessary. The -following table lists the main names with which the same problem -arises: -\begin{transbox} -\TRANS{IF}{IF_then_else} -\TRANS{ZERO}{Z0} -\TRANS{POS}{Zpos} -\TRANS{NEG}{Zneg} -\TRANS{SUPERIEUR}{Gt} -\TRANS{EGAL}{Eq} -\TRANS{INFERIEUR}{Lt} -\TRANS{add}{Pplus} -\TRANS{true_sub}{Pminus} -\TRANS{entier}{N} -\TRANS{Un_suivi_de}{Ndouble_plus_one} -\TRANS{Zero_suivi_de}{Ndouble} -\TRANS{Nul}{N0} -\TRANS{Pos}{Npos} -\end{transbox} - - -\subsubsection{Implicit arguments} - -%% Hugo: -Main definitions of standard library have now implicit -arguments. These arguments are dropped in the translated files. This -can exceptionally be a source of incompatibilities which has to be -solved by hand (it typically happens for polymorphic functions applied -to {\tt nil} or {\tt None}). -%% preciser: avant ou apres trad ? - -\subsubsection{Logic about {\tt Type}} - -Many notations that applied to {\tt Set} have been extended to {\tt -Type}, so several definitions in {\tt Type} are superseded by them. - -\begin{transbox} -\TRANS{x==y}{x=y} -\TRANS{(EXT x:Prop | Q)}{exists x:Prop, Q} -\TRANS{identityT}{identity} -\end{transbox} - - - -%% Doc of the translator -\section{A guide to translation} -\label{Translation} - -%%\subsection{Overview of the translation process} - -Here is a short description of the tools involved in the translation process: -\begin{description} -\item{\tt coqc -translate} -is the automatic translator. It is a parser/pretty-printer. This means -that the translation is made by parsing every command using a parser -of old syntax, which is printed using the new syntax. Many efforts -were made to preserve as much as possible of the quality of the -presentation: it avoids expansion of syntax extensions, comments are -not discarded and placed at the same place. -\item{\tt translate-v8} (in the translation package) is a small -shell-script that will help translate developments that compile with a -Makefile with minimum requirements. -\end{description} - -\subsection{Preparation to translation} - -This step is very important because most of work shall be done before -translation. If a problem occurs during translation, it often means -that you will have to modify the original source and restart the -translation process. This also means that it is recommended not to -edit the output of the translator since it would be overwritten if -the translation has to be restarted. - -\subsubsection{Compilation with {\tt coqc -v7}} - -First of all, it is mandatory that files compile with the current -version of Coq (8.0) with option {\tt -v7}. Translation is a -complicated task that involves the full compilation of the -development. If your development was compiled with older versions, -first upgrade to Coq V8.0 with option {\tt -v7}. If you use a Makefile -similar to those produced by {\tt coq\_makefile}, you probably just -have to do - -{\tt make OPT="-opt -v7"} ~~~or~~~ {\tt make OPT="-byte -v7"} - -When the development compiles successfully, there are several changes -that might be necessary for the translation. Essentially, this is -about syntax extensions (see section below dedicated to porting syntax -extensions). If you do not use such features, then you are ready to -try and make the translation. - -\subsection{Translation} - -\subsubsection{The general case} - -The preferred way is to use script {\tt translate-v8} if your development -is compiled by a Makefile with the following constraints: -\begin{itemize} -\item compilation is achieved by invoking make without specifying a target -\item options are passed to Coq with make variable COQFLAGS that - includes variables OPT, COQLIBS, OTHERFLAGS and COQ_XML. -\end{itemize} -These constraints are met by the makefiles produced by {\tt coq\_makefile} - -Otherwise, modify your build program so as to pass option {\tt --translate} to program {\tt coqc}. The effect of this option is to -ouptut the translated source of any {\tt .v} file in a file with -extension {\tt .v8} located in the same directory than the original -file. - -\subsubsection{What may happen during the translation} - -This section describes events that may happen during the -translation and measures to adopt. - -These are the warnings that may arise during the translation, but they -generally do not require any modification for the user: -Warnings: -\begin{itemize} -\item {\tt Unable to detect if $id$ denotes a local definition}\\ -This is due to a semantic change in clauses. In a command such as {\tt -simpl in H}, the old semantics were to perform simplification in the -type of {\tt H}, or in its body if it is defined. With the new -semantics, it is performed both in the type and the body (if any). It -might lead to incompatibilities - -\item {\tt Forgetting obsolete module}\\ -Some modules have disappeared in V8.0 (new syntax). The user does not -need to worry about it, since the translator deals with it. - -\item {\tt Replacing obsolete module}\\ -Same as before but with the module that were renamed. Here again, the -translator deals with it. -\end{itemize} - -\subsection{Verification of the translation} - -The shell-script {\tt translate-v8} also renames {\tt .v8} files into -{\tt .v} files (older {\tt .v} files are put in a subdirectory called -{\tt v7}) and tries to recompile them. To do so it invokes {\tt make} -without option (which should cause the compilation using {\tt coqc} -without particular option). - -If compilation fails at this stage, you should refrain from repairing -errors manually on the new syntax, but rather modify the old syntax -script and restart the translation. We insist on that because the -problem encountered can show up in many instances (especially if the -problem comes from a syntactic extension), and fixing the original -sources (for instance the {\tt V8only} parts of notations) once will -solve all occurrences of the problem. - -%%\subsubsection{Errors occurring after translation} -%%Equality in {\tt Z} or {\tt R}... - -\subsection{Particular cases} - -\subsubsection{Lexical conventions} - -The definition of identifiers changed. Most of those changes are -handled by the translator. They include: -\begin{itemize} -\item {\tt \_} is not an identifier anymore: it is tranlated to {\tt -x\_} -\item avoid clash with new keywords by adding a trailing {\tt \_} -\end{itemize} - -If the choices made by translation is not satisfactory -or in the following cases: -\begin{itemize} -\item use of latin letters -\item use of iso-latin characters in notations -\end{itemize} -the user should change his development prior to translation. - -\subsubsection{{\tt Case} and {\tt Match}} - -These very low-level case analysis are no longer supported. The -translator tries hard to translate them into a user-friendly one, but -it might lack type information to do so\footnote{The translator tries -to typecheck terms before printing them, but it is not always possible -to determine the context in which terms appearing in tactics -live.}. If this happens, it is preferable to transform it manually -before translation. - -\subsubsection{Syntax extensions with {\tt Grammar} and {\tt Syntax}} - - -{\tt Grammar} and {\tt Syntax} are no longer supported. They -should be replaced by an equivalent {\tt Notation} command and be -processed as described above. Before attempting translation, users -should verify that compilation with option {\tt -v7} succeeds. - -In the cases where {\tt Grammar} and {\tt Syntax} cannot be emulated -by {\tt Notation}, users have to change manually they development as -they wish to avoid the use of {\tt Grammar}. If this is not done, the -translator will simply expand the notations and the output of the -translator will use the regular Coq syntax. - -\subsubsection{Syntax extensions with {\tt Notation} and {\tt Infix}} - -These commands do not necessarily need to be changed. - -Some work will have to be done manually if the notation conflicts with -the new syntax (for instance, using keywords like {\tt fun} or {\tt -exists}, overloading of symbols of the old syntax, etc.) or if the -precedences are not right. - - Precedence levels are now from 0 to 200. In V8, the precedence and -associativity of an operator cannot be redefined. Typical level are -(refer to the chapter on notations in the Reference Manual for the -full list): - -\begin{center} -\begin{tabular}{|cll|} -\hline -Notation & Precedence & Associativity \\ -\hline -\verb!_ <-> _! & 95 & no \\ -\verb!_ \/ _! & 85 & right \\ -\verb!_ /\ _! & 80 & right \\ -\verb!~ _! & 75 & right \\ -\verb!_ = _!, \verb!_ <> _!, \verb!_ < _!, \verb!_ > _!, - \verb!_ <= _!, \verb!_ >= _! & 70 & no \\ -\verb!_ + _!, \verb!_ - _! & 50 & left \\ -\verb!_ * _!, \verb!_ / _! & 40 & left \\ -\verb!- _! & 35 & right \\ -\verb!_ ^ _! & 30 & left \\ -\hline -\end{tabular} -\end{center} - - - By default, the translator keeps the associativity given in V7 while -the levels are mapped according to the following table: - -\begin{center} -\begin{tabular}{l|l|l} -V7 level & mapped to & associativity \\ -\hline -0 & 0 & no \\ -1 & 20 & left \\ -2 & 30 & right \\ -3 & 40 & left \\ -4 & 50 & left \\ -5 & 70 & no \\ -6 & 80 & right \\ -7 & 85 & right \\ -8 & 90 & right \\ -9 & 95 & no \\ -10 & 100 & left -\end{tabular} -\end{center} - -If this is OK, just simply apply the translator. - - -\paragraph{Associativity conflict} - - Since the associativity of the levels obtained by translating a V7 -level (as shown on table above) cannot be changed, you have to choose -another level with a compatible associativity. - - You can choose any level between 0 and 200, knowing that the -standard operators are already set at the levels shown on the list -above. - -Assume you have a notation -\begin{verbatim} -Infix NONA 2 "=_S" my_setoid_eq. -\end{verbatim} -By default, the translator moves it to level 30 which is right -associative, hence a conflict with the expected no associativity. - -To solve the problem, just add the "V8only" modifier to reset the -level and enforce the associativity as follows: -\begin{verbatim} -Infix NONA 2 "=_S" my_setoid_eq V8only (at level 70, no associativity). -\end{verbatim} -The translator now knows that it has to translate "=_S" at level 70 -with no associativity. - -Remark: 70 is the "natural" level for relations, hence the choice of 70 -here, but any other level accepting a no-associativity would have been -OK. - -Second example: assume you have a notation -\begin{verbatim} -Infix RIGHTA 1 "o" my_comp. -\end{verbatim} -By default, the translator moves it to level 20 which is left -associative, hence a conflict with the expected right associativity. - -To solve the problem, just add the "V8only" modifier to reset the -level and enforce the associativity as follows: -\begin{verbatim} -Infix RIGHTA 1 "o" my_comp V8only (at level 20, right associativity). -\end{verbatim} -The translator now knows that it has to translate "o" at level 20 -which has the correct "right associativity". - -Remark: we assumed here that the user wants a strong precedence for -composition, in such a way, say, that "f o g + h" is parsed as -"(f o g) + h". To get "o" binding less than the arithmetical operators, -an appropriated level would have been close of 70, and below, e.g. 65. - - -\paragraph{Conflict: notation hides another notation} - -Remark: use {\tt Print Grammar constr} in V8 to diagnose the overlap -and see the section on factorization in the chapter on notations of -the Reference Manual for hints on how to factorize. - -Example: -\begin{verbatim} -Notation "{ x }" := (my_embedding x) (at level 1). -\end{verbatim} -overlaps in V8 with notation \verb#{ x : A & P }# at level 0 and with -x at level 99. The conflicts can be solved by left-factorizing the -notation as follows: -\begin{verbatim} -Notation "{ x }" := (my_embedding x) (at level 1) - V8only (at level 0, x at level 99). -\end{verbatim} - -\paragraph{Conflict: a notation conflicts with the V8 grammar} - -Again, use the {\tt V8only} modifier to tell the translator to -automatically take in charge the new syntax. - -Example: -\begin{verbatim} -Infix 3 "@" app. -\end{verbatim} -Since {\tt @} is used in the new syntax for deactivating the implicit -arguments, another symbol has to be used, e.g. {\tt @@}. This is done via -the {\tt V8only} option as follows: -\begin{verbatim} -Infix 3 "@" app V8only "@@" (at level 40, left associativity). -\end{verbatim} -or, alternatively by -\begin{verbatim} -Notation "x @ y" := (app x y) (at level 3, left associativity) - V8only "x @@ y" (at level 40, left associativity). -\end{verbatim} - -\paragraph{Conflict: my notation is already defined at another level - (or with another associativity)} - -In V8, the level and associativity of a given notation can no longer -be changed. Then, either you adopt the standard reserved levels and -associativity for this notation (as given on the list above) or you -change your notation. -\begin{itemize} -\item To change the notation, follow the directions in the previous -paragraph -\item To adopt the standard level, just use {\tt V8only} without any -argument. -\end{itemize} - -Example: -\begin{verbatim} -Infix 6 "*" my_mult. -\end{verbatim} -is not accepted as such in V8. Write -\begin{verbatim} -Infix 6 "*" my_mult V8only. -\end{verbatim} -to tell the translator to use {\tt *} at the reserved level (i.e. 40 -with left associativity). Even better, use interpretation scopes (look -at the Reference Manual). - - -\subsubsection{Strict implicit arguments} - -In the case you want to adopt the new semantics of {\tt Set Implicit - Arguments} (only setting rigid arguments as implicit), add the option -{\tt -strict-implicit} to the translator. - -Warning: changing the number of implicit arguments can break the -notations. Then use the {\tt V8only} modifier of {\tt Notation}. - -\end{document} diff --git a/doc/tools/latex_filter b/doc/tools/latex_filter deleted file mode 100755 index 044a8642..00000000 --- a/doc/tools/latex_filter +++ /dev/null @@ -1,43 +0,0 @@ -#!/bin/sh - -# First argument is the number of lines to treat -# Second argument is optional and, if it is "no", overfull are not displayed - -i=$1 -nooverfull=$2 -error=0 -verbose=0 -chapter="" -file="" -while : ; do - read -r line; - case $line in - "! "*) - echo $line $file; - error=1 - verbose=1 - ;; - "LaTeX Font Info"*|"LaTeX Info"*|"Underfull "*) - verbose=0 - ;; - "Overfull "*) - verbose=0 - if [ "$nooverfull" != "no" ]; then echo $line $file; fi - ;; - "LaTeX "*) - verbose=0 - echo $line $chapter - ;; - "["*|"Chapter "*) - verbose=0 - ;; - "(./"*) - file="(file `echo $line | cut -b 4- | cut -d' ' -f 1`)" - verbose=0 - ;; - *) - if [ $verbose = 1 ]; then echo $line; fi - esac; - if [ "$i" = "0" ]; then break; else i=`expr $i - 1`; fi; -done -exit $error diff --git a/doc/tools/show_latex_messages b/doc/tools/show_latex_messages deleted file mode 100755 index 8f1470ec..00000000 --- a/doc/tools/show_latex_messages +++ /dev/null @@ -1,8 +0,0 @@ -#!/bin/sh - -if [ "$1" = "-no-overfull" ]; then - cat $2 | ../tools/latex_filter `cat $2 | wc -l` no -else - cat $1 | ../tools/latex_filter `cat $1 | wc -l` yes -fi - diff --git a/doc/tutorial/Tutorial.tex b/doc/tutorial/Tutorial.tex deleted file mode 100755 index 723295bf..00000000 --- a/doc/tutorial/Tutorial.tex +++ /dev/null @@ -1,1577 +0,0 @@ -\documentclass[11pt,a4paper]{book} -\usepackage[T1]{fontenc} -\usepackage[latin1]{inputenc} -\usepackage{pslatex} - -\input{../common/version.tex} -\input{../common/macros.tex} -\input{../common/title.tex} - -%\makeindex - -\begin{document} -\coverpage{A Tutorial}{Gérard Huet, Gilles Kahn and Christine Paulin-Mohring}{} - -%\tableofcontents - -\chapter*{Getting started} - -\Coq\ is a Proof Assistant for a Logical Framework known as the Calculus -of Inductive Constructions. It allows the interactive construction of -formal proofs, and also the manipulation of functional programs -consistently with their specifications. It runs as a computer program -on many architectures. -%, and mainly on Unix machines. -It is available with a variety of user interfaces. The present -document does not attempt to present a comprehensive view of all the -possibilities of \Coq, but rather to present in the most elementary -manner a tutorial on the basic specification language, called Gallina, -in which formal axiomatisations may be developed, and on the main -proof tools. For more advanced information, the reader could refer to -the \Coq{} Reference Manual or the \textit{Coq'Art}, a new book by Y. -Bertot and P. Castéran on practical uses of the \Coq{} system. - -Coq can be used from a standard teletype-like shell window but -preferably through the graphical user interface -CoqIde\footnote{Alternative graphical interfaces exist: Proof General -and Pcoq.}. - -Instructions on installation procedures, as well as more comprehensive -documentation, may be found in the standard distribution of \Coq, -which may be obtained from \Coq{} web site \texttt{http://coq.inria.fr}. - -In the following, we assume that \Coq~ is called from a standard -teletype-like shell window. All examples preceded by the prompting -sequence \verb:Coq < : represent user input, terminated by a -period. - -The following lines usually show \Coq's answer as it appears on the -users screen. When used from a graphical user interface such as -CoqIde, the prompt is not displayed: user input is given in one window -and \Coq's answers are displayed in a different window. - -The sequence of such examples is a valid \Coq~ -session, unless otherwise specified. This version of the tutorial has -been prepared on a PC workstation running Linux. The standard -invocation of \Coq\ delivers a message such as: - -\begin{small} -\begin{flushleft} -\begin{verbatim} -unix:~> coqtop -Welcome to Coq 8.3 (October 2010) - -Coq < -\end{verbatim} -\end{flushleft} -\end{small} - -The first line gives a banner stating the precise version of \Coq~ -used. You should always return this banner when you report an anomaly -to our bug-tracking system -\verb|http://coq.inria.fr/bugs| - -\chapter{Basic Predicate Calculus} - -\section{An overview of the specification language Gallina} - -A formal development in Gallina consists in a sequence of {\sl declarations} -and {\sl definitions}. You may also send \Coq~ {\sl commands} which are -not really part of the formal development, but correspond to information -requests, or service routine invocations. For instance, the command: -\begin{verbatim} -Coq < Quit. -\end{verbatim} -terminates the current session. - -\subsection{Declarations} - -A declaration associates a {\sl name} with -a {\sl specification}. -A name corresponds roughly to an identifier in a programming -language, i.e. to a string of letters, digits, and a few ASCII symbols like -underscore (\verb"_") and prime (\verb"'"), starting with a letter. -We use case distinction, so that the names \verb"A" and \verb"a" are distinct. -Certain strings are reserved as key-words of \Coq, and thus are forbidden -as user identifiers. - -A specification is a formal expression which classifies the notion which is -being declared. There are basically three kinds of specifications: -{\sl logical propositions}, {\sl mathematical collections}, and -{\sl abstract types}. They are classified by the three basic sorts -of the system, called respectively \verb:Prop:, \verb:Set:, and -\verb:Type:, which are themselves atomic abstract types. - -Every valid expression $e$ in Gallina is associated with a specification, -itself a valid expression, called its {\sl type} $\tau(E)$. We write -$e:\tau(E)$ for the judgment that $e$ is of type $E$. -You may request \Coq~ to return to you the type of a valid expression by using -the command \verb:Check:: - -\begin{coq_eval} -Set Printing Width 60. -\end{coq_eval} - -\begin{coq_example} -Check O. -\end{coq_example} - -Thus we know that the identifier \verb:O: (the name `O', not to be -confused with the numeral `0' which is not a proper identifier!) is -known in the current context, and that its type is the specification -\verb:nat:. This specification is itself classified as a mathematical -collection, as we may readily check: - -\begin{coq_example} -Check nat. -\end{coq_example} - -The specification \verb:Set: is an abstract type, one of the basic -sorts of the Gallina language, whereas the notions $nat$ and $O$ are -notions which are defined in the arithmetic prelude, -automatically loaded when running the \Coq\ system. - -We start by introducing a so-called section name. The role of sections -is to structure the modelisation by limiting the scope of parameters, -hypotheses and definitions. It will also give a convenient way to -reset part of the development. - -\begin{coq_example} -Section Declaration. -\end{coq_example} -With what we already know, we may now enter in the system a declaration, -corresponding to the informal mathematics {\sl let n be a natural - number}. - -\begin{coq_example} -Variable n : nat. -\end{coq_example} - -If we want to translate a more precise statement, such as -{\sl let n be a positive natural number}, -we have to add another declaration, which will declare explicitly the -hypothesis \verb:Pos_n:, with specification the proper logical -proposition: -\begin{coq_example} -Hypothesis Pos_n : (gt n 0). -\end{coq_example} - -Indeed we may check that the relation \verb:gt: is known with the right type -in the current context: - -\begin{coq_example} -Check gt. -\end{coq_example} - -which tells us that \verb:gt: is a function expecting two arguments of -type \verb:nat: in order to build a logical proposition. -What happens here is similar to what we are used to in a functional -programming language: we may compose the (specification) type \verb:nat: -with the (abstract) type \verb:Prop: of logical propositions through the -arrow function constructor, in order to get a functional type -\verb:nat->Prop:: -\begin{coq_example} -Check (nat -> Prop). -\end{coq_example} -which may be composed one more times with \verb:nat: in order to obtain the -type \verb:nat->nat->Prop: of binary relations over natural numbers. -Actually the type \verb:nat->nat->Prop: is an abbreviation for -\verb:nat->(nat->Prop):. - -Functional notions may be composed in the usual way. An expression $f$ -of type $A\ra B$ may be applied to an expression $e$ of type $A$ in order -to form the expression $(f~e)$ of type $B$. Here we get that -the expression \verb:(gt n): is well-formed of type \verb:nat->Prop:, -and thus that the expression \verb:(gt n O):, which abbreviates -\verb:((gt n) O):, is a well-formed proposition. -\begin{coq_example} -Check gt n O. -\end{coq_example} - -\subsection{Definitions} - -The initial prelude contains a few arithmetic definitions: -\verb:nat: is defined as a mathematical collection (type \verb:Set:), constants -\verb:O:, \verb:S:, \verb:plus:, are defined as objects of types -respectively \verb:nat:, \verb:nat->nat:, and \verb:nat->nat->nat:. -You may introduce new definitions, which link a name to a well-typed value. -For instance, we may introduce the constant \verb:one: as being defined -to be equal to the successor of zero: -\begin{coq_example} -Definition one := (S O). -\end{coq_example} -We may optionally indicate the required type: -\begin{coq_example} -Definition two : nat := S one. -\end{coq_example} - -Actually \Coq~ allows several possible syntaxes: -\begin{coq_example} -Definition three : nat := S two. -\end{coq_example} - -Here is a way to define the doubling function, which expects an -argument \verb:m: of type \verb:nat: in order to build its result as -\verb:(plus m m):: - -\begin{coq_example} -Definition double (m:nat) := plus m m. -\end{coq_example} -This introduces the constant \texttt{double} defined as the -expression \texttt{fun m:nat => plus m m}. -The abstraction introduced by \texttt{fun} is explained as follows. The expression -\verb+fun x:A => e+ is well formed of type \verb+A->B+ in a context -whenever the expression \verb+e+ is well-formed of type \verb+B+ in -the given context to which we add the declaration that \verb+x+ -is of type \verb+A+. Here \verb+x+ is a bound, or dummy variable in -the expression \verb+fun x:A => e+. For instance we could as well have -defined \verb:double: as \verb+fun n:nat => (plus n n)+. - -Bound (local) variables and free (global) variables may be mixed. -For instance, we may define the function which adds the constant \verb:n: -to its argument as -\begin{coq_example} -Definition add_n (m:nat) := plus m n. -\end{coq_example} -However, note that here we may not rename the formal argument $m$ into $n$ -without capturing the free occurrence of $n$, and thus changing the meaning -of the defined notion. - -Binding operations are well known for instance in logic, where they -are called quantifiers. Thus we may universally quantify a -proposition such as $m>0$ in order to get a universal proposition -$\forall m\cdot m>0$. Indeed this operator is available in \Coq, with -the following syntax: \verb+forall m:nat, gt m O+. Similarly to the -case of the functional abstraction binding, we are obliged to declare -explicitly the type of the quantified variable. We check: -\begin{coq_example} -Check (forall m:nat, gt m 0). -\end{coq_example} -We may clean-up the development by removing the contents of the -current section: -\begin{coq_example} -Reset Declaration. -\end{coq_example} - -\section{Introduction to the proof engine: Minimal Logic} - -In the following, we are going to consider various propositions, built -from atomic propositions $A, B, C$. This may be done easily, by -introducing these atoms as global variables declared of type \verb:Prop:. -It is easy to declare several names with the same specification: -\begin{coq_example} -Section Minimal_Logic. -Variables A B C : Prop. -\end{coq_example} - -We shall consider simple implications, such as $A\ra B$, read as -``$A$ implies $B$''. Remark that we overload the arrow symbol, which -has been used above as the functionality type constructor, and which -may be used as well as propositional connective: -\begin{coq_example} -Check (A -> B). -\end{coq_example} - -Let us now embark on a simple proof. We want to prove the easy tautology -$((A\ra (B\ra C))\ra (A\ra B)\ra (A\ra C)$. -We enter the proof engine by the command -\verb:Goal:, followed by the conjecture we want to verify: -\begin{coq_example} -Goal (A -> B -> C) -> (A -> B) -> A -> C. -\end{coq_example} - -The system displays the current goal below a double line, local hypotheses -(there are none initially) being displayed above the line. We call -the combination of local hypotheses with a goal a {\sl judgment}. -We are now in an inner -loop of the system, in proof mode. -New commands are available in this -mode, such as {\sl tactics}, which are proof combining primitives. -A tactic operates on the current goal by attempting to construct a proof -of the corresponding judgment, possibly from proofs of some -hypothetical judgments, which are then added to the current -list of conjectured judgments. -For instance, the \verb:intro: tactic is applicable to any judgment -whose goal is an implication, by moving the proposition to the left -of the application to the list of local hypotheses: -\begin{coq_example} -intro H. -\end{coq_example} - -Several introductions may be done in one step: -\begin{coq_example} -intros H' HA. -\end{coq_example} - -We notice that $C$, the current goal, may be obtained from hypothesis -\verb:H:, provided the truth of $A$ and $B$ are established. -The tactic \verb:apply: implements this piece of reasoning: -\begin{coq_example} -apply H. -\end{coq_example} - -We are now in the situation where we have two judgments as conjectures -that remain to be proved. Only the first is listed in full, for the -others the system displays only the corresponding subgoal, without its -local hypotheses list. Remark that \verb:apply: has kept the local -hypotheses of its father judgment, which are still available for -the judgments it generated. - -In order to solve the current goal, we just have to notice that it is -exactly available as hypothesis $HA$: -\begin{coq_example} -exact HA. -\end{coq_example} - -Now $H'$ applies: -\begin{coq_example} -apply H'. -\end{coq_example} - -And we may now conclude the proof as before, with \verb:exact HA.: -Actually, we may not bother with the name \verb:HA:, and just state that -the current goal is solvable from the current local assumptions: -\begin{coq_example} -assumption. -\end{coq_example} - -The proof is now finished. We may either discard it, by using the -command \verb:Abort: which returns to the standard \Coq~ toplevel loop -without further ado, or else save it as a lemma in the current context, -under name say \verb:trivial_lemma:: -\begin{coq_example} -Save trivial_lemma. -\end{coq_example} - -As a comment, the system shows the proof script listing all tactic -commands used in the proof. - -Let us redo the same proof with a few variations. First of all we may name -the initial goal as a conjectured lemma: -\begin{coq_example} -Lemma distr_impl : (A -> B -> C) -> (A -> B) -> A -> C. -\end{coq_example} - -Next, we may omit the names of local assumptions created by the introduction -tactics, they can be automatically created by the proof engine as new -non-clashing names. -\begin{coq_example} -intros. -\end{coq_example} - -The \verb:intros: tactic, with no arguments, effects as many individual -applications of \verb:intro: as is legal. - -Then, we may compose several tactics together in sequence, or in parallel, -through {\sl tacticals}, that is tactic combinators. The main constructions -are the following: -\begin{itemize} -\item $T_1 ; T_2$ (read $T_1$ then $T_2$) applies tactic $T_1$ to the current -goal, and then tactic $T_2$ to all the subgoals generated by $T_1$. -\item $T; [T_1 | T_2 | ... | T_n]$ applies tactic $T$ to the current -goal, and then tactic $T_1$ to the first newly generated subgoal, -..., $T_n$ to the nth. -\end{itemize} - -We may thus complete the proof of \verb:distr_impl: with one composite tactic: -\begin{coq_example} -apply H; [ assumption | apply H0; assumption ]. -\end{coq_example} - -Let us now save lemma \verb:distr_impl:: -\begin{coq_example} -Save. -\end{coq_example} - -Here \verb:Save: needs no argument, since we gave the name \verb:distr_impl: -in advance; -it is however possible to override the given name by giving a different -argument to command \verb:Save:. - -Actually, such an easy combination of tactics \verb:intro:, \verb:apply: -and \verb:assumption: may be found completely automatically by an automatic -tactic, called \verb:auto:, without user guidance: -\begin{coq_example} -Lemma distr_imp : (A -> B -> C) -> (A -> B) -> A -> C. -auto. -\end{coq_example} - -This time, we do not save the proof, we just discard it with the \verb:Abort: -command: - -\begin{coq_example} -Abort. -\end{coq_example} - -At any point during a proof, we may use \verb:Abort: to exit the proof mode -and go back to Coq's main loop. We may also use \verb:Restart: to restart -from scratch the proof of the same lemma. We may also use \verb:Undo: to -backtrack one step, and more generally \verb:Undo n: to -backtrack n steps. - -We end this section by showing a useful command, \verb:Inspect n.:, -which inspects the global \Coq~ environment, showing the last \verb:n: declared -notions: -\begin{coq_example} -Inspect 3. -\end{coq_example} - -The declarations, whether global parameters or axioms, are shown preceded by -\verb:***:; definitions and lemmas are stated with their specification, but -their value (or proof-term) is omitted. - -\section{Propositional Calculus} - -\subsection{Conjunction} - -We have seen how \verb:intro: and \verb:apply: tactics could be combined -in order to prove implicational statements. More generally, \Coq~ favors a style -of reasoning, called {\sl Natural Deduction}, which decomposes reasoning into -so called {\sl introduction rules}, which tell how to prove a goal whose main -operator is a given propositional connective, and {\sl elimination rules}, -which tell how to use an hypothesis whose main operator is the propositional -connective. Let us show how to use these ideas for the propositional connectives -\verb:/\: and \verb:\/:. - -\begin{coq_example} -Lemma and_commutative : A /\ B -> B /\ A. -intro. -\end{coq_example} - -We make use of the conjunctive hypothesis \verb:H: with the \verb:elim: tactic, -which breaks it into its components: -\begin{coq_example} -elim H. -\end{coq_example} - -We now use the conjunction introduction tactic \verb:split:, which splits the -conjunctive goal into the two subgoals: -\begin{coq_example} -split. -\end{coq_example} -and the proof is now trivial. Indeed, the whole proof is obtainable as follows: -\begin{coq_example} -Restart. -intro H; elim H; auto. -Qed. -\end{coq_example} - -The tactic \verb:auto: succeeded here because it knows as a hint the -conjunction introduction operator \verb+conj+ -\begin{coq_example} -Check conj. -\end{coq_example} - -Actually, the tactic \verb+Split+ is just an abbreviation for \verb+apply conj.+ - -What we have just seen is that the \verb:auto: tactic is more powerful than -just a simple application of local hypotheses; it tries to apply as well -lemmas which have been specified as hints. A -\verb:Hint Resolve: command registers a -lemma as a hint to be used from now on by the \verb:auto: tactic, whose power -may thus be incrementally augmented. - -\subsection{Disjunction} - -In a similar fashion, let us consider disjunction: - -\begin{coq_example} -Lemma or_commutative : A \/ B -> B \/ A. -intro H; elim H. -\end{coq_example} - -Let us prove the first subgoal in detail. We use \verb:intro: in order to -be left to prove \verb:B\/A: from \verb:A:: - -\begin{coq_example} -intro HA. -\end{coq_example} - -Here the hypothesis \verb:H: is not needed anymore. We could choose to -actually erase it with the tactic \verb:clear:; in this simple proof it -does not really matter, but in bigger proof developments it is useful to -clear away unnecessary hypotheses which may clutter your screen. -\begin{coq_example} -clear H. -\end{coq_example} - -The disjunction connective has two introduction rules, since \verb:P\/Q: -may be obtained from \verb:P: or from \verb:Q:; the two corresponding -proof constructors are called respectively \verb:or_introl: and -\verb:or_intror:; they are applied to the current goal by tactics -\verb:left: and \verb:right: respectively. For instance: -\begin{coq_example} -right. -trivial. -\end{coq_example} -The tactic \verb:trivial: works like \verb:auto: with the hints -database, but it only tries those tactics that can solve the goal in one -step. - -As before, all these tedious elementary steps may be performed automatically, -as shown for the second symmetric case: - -\begin{coq_example} -auto. -\end{coq_example} - -However, \verb:auto: alone does not succeed in proving the full lemma, because -it does not try any elimination step. -It is a bit disappointing that \verb:auto: is not able to prove automatically -such a simple tautology. The reason is that we want to keep -\verb:auto: efficient, so that it is always effective to use. - -\subsection{Tauto} - -A complete tactic for propositional -tautologies is indeed available in \Coq~ as the \verb:tauto: tactic. -\begin{coq_example} -Restart. -tauto. -Qed. -\end{coq_example} - -It is possible to inspect the actual proof tree constructed by \verb:tauto:, -using a standard command of the system, which prints the value of any notion -currently defined in the context: -\begin{coq_example} -Print or_commutative. -\end{coq_example} - -It is not easy to understand the notation for proof terms without a few -explanations. The \texttt{fun} prefix, such as \verb+fun H:A\/B =>+, -corresponds -to \verb:intro H:, whereas a subterm such as -\verb:(or_intror: \verb:B H0): -corresponds to the sequence of tactics \verb:apply or_intror; exact H0:. -The generic combinator \verb:or_intror: needs to be instantiated by -the two properties \verb:B: and \verb:A:. Because \verb:A: can be -deduced from the type of \verb:H0:, only \verb:B: is printed. -The two instantiations are effected automatically by the tactic -\verb:apply: when pattern-matching a goal. The specialist will of course -recognize our proof term as a $\lambda$-term, used as notation for the -natural deduction proof term through the Curry-Howard isomorphism. The -naive user of \Coq~ may safely ignore these formal details. - -Let us exercise the \verb:tauto: tactic on a more complex example: -\begin{coq_example} -Lemma distr_and : A -> B /\ C -> (A -> B) /\ (A -> C). -tauto. -Qed. -\end{coq_example} - -\subsection{Classical reasoning} - -The tactic \verb:tauto: always comes back with an answer. Here is an example where it -fails: -\begin{coq_example} -Lemma Peirce : ((A -> B) -> A) -> A. -try tauto. -\end{coq_example} - -Note the use of the \verb:Try: tactical, which does nothing if its tactic -argument fails. - -This may come as a surprise to someone familiar with classical reasoning. -Peirce's lemma is true in Boolean logic, i.e. it evaluates to \verb:true: for -every truth-assignment to \verb:A: and \verb:B:. Indeed the double negation -of Peirce's law may be proved in \Coq~ using \verb:tauto:: -\begin{coq_example} -Abort. -Lemma NNPeirce : ~ ~ (((A -> B) -> A) -> A). -tauto. -Qed. -\end{coq_example} - -In classical logic, the double negation of a proposition is equivalent to this -proposition, but in the constructive logic of \Coq~ this is not so. If you -want to use classical logic in \Coq, you have to import explicitly the -\verb:Classical: module, which will declare the axiom \verb:classic: -of excluded middle, and classical tautologies such as de Morgan's laws. -The \verb:Require: command is used to import a module from \Coq's library: -\begin{coq_example} -Require Import Classical. -Check NNPP. -\end{coq_example} - -and it is now easy (although admittedly not the most direct way) to prove -a classical law such as Peirce's: -\begin{coq_example} -Lemma Peirce : ((A -> B) -> A) -> A. -apply NNPP; tauto. -Qed. -\end{coq_example} - -Here is one more example of propositional reasoning, in the shape of -a Scottish puzzle. A private club has the following rules: -\begin{enumerate} -\item Every non-scottish member wears red socks -\item Every member wears a kilt or doesn't wear red socks -\item The married members don't go out on Sunday -\item A member goes out on Sunday if and only if he is Scottish -\item Every member who wears a kilt is Scottish and married -\item Every scottish member wears a kilt -\end{enumerate} -Now, we show that these rules are so strict that no one can be accepted. -\begin{coq_example} -Section club. -Variables Scottish RedSocks WearKilt Married GoOutSunday : Prop. -Hypothesis rule1 : ~ Scottish -> RedSocks. -Hypothesis rule2 : WearKilt \/ ~ RedSocks. -Hypothesis rule3 : Married -> ~ GoOutSunday. -Hypothesis rule4 : GoOutSunday <-> Scottish. -Hypothesis rule5 : WearKilt -> Scottish /\ Married. -Hypothesis rule6 : Scottish -> WearKilt. -Lemma NoMember : False. -tauto. -Qed. -\end{coq_example} -At that point \verb:NoMember: is a proof of the absurdity depending on -hypotheses. -We may end the section, in that case, the variables and hypotheses -will be discharged, and the type of \verb:NoMember: will be -generalised. - -\begin{coq_example} -End club. -Check NoMember. -\end{coq_example} - -\section{Predicate Calculus} - -Let us now move into predicate logic, and first of all into first-order -predicate calculus. The essence of predicate calculus is that to try to prove -theorems in the most abstract possible way, without using the definitions of -the mathematical notions, but by formal manipulations of uninterpreted -function and predicate symbols. - -\subsection{Sections and signatures} - -Usually one works in some domain of discourse, over which range the individual -variables and function symbols. In \Coq~ we speak in a language with a rich -variety of types, so me may mix several domains of discourse, in our -multi-sorted language. For the moment, we just do a few exercises, over a -domain of discourse \verb:D: axiomatised as a \verb:Set:, and we consider two -predicate symbols \verb:P: and \verb:R: over \verb:D:, of arities -respectively 1 and 2. Such abstract entities may be entered in the context -as global variables. But we must be careful about the pollution of our -global environment by such declarations. For instance, we have already -polluted our \Coq~ session by declaring the variables -\verb:n:, \verb:Pos_n:, \verb:A:, \verb:B:, and \verb:C:. If we want to revert to the clean state of -our initial session, we may use the \Coq~ \verb:Reset: command, which returns -to the state just prior the given global notion as we did before to -remove a section, or we may return to the initial state using~: -\begin{coq_example} -Reset Initial. -\end{coq_example} -\begin{coq_eval} -Set Printing Width 60. -\end{coq_eval} - -We shall now declare a new \verb:Section:, which will allow us to define -notions local to a well-delimited scope. We start by assuming a domain of -discourse \verb:D:, and a binary relation \verb:R: over \verb:D:: -\begin{coq_example} -Section Predicate_calculus. -Variable D : Set. -Variable R : D -> D -> Prop. -\end{coq_example} - -As a simple example of predicate calculus reasoning, let us assume -that relation \verb:R: is symmetric and transitive, and let us show that -\verb:R: is reflexive in any point \verb:x: which has an \verb:R: successor. -Since we do not want to make the assumptions about \verb:R: global axioms of -a theory, but rather local hypotheses to a theorem, we open a specific -section to this effect. -\begin{coq_example} -Section R_sym_trans. -Hypothesis R_symmetric : forall x y:D, R x y -> R y x. -Hypothesis R_transitive : forall x y z:D, R x y -> R y z -> R x z. -\end{coq_example} - -Remark the syntax \verb+forall x:D,+ which stands for universal quantification -$\forall x : D$. - -\subsection{Existential quantification} - -We now state our lemma, and enter proof mode. -\begin{coq_example} -Lemma refl_if : forall x:D, (exists y, R x y) -> R x x. -\end{coq_example} - -Remark that the hypotheses which are local to the currently opened sections -are listed as local hypotheses to the current goals. -The rationale is that these hypotheses are going to be discharged, as we -shall see, when we shall close the corresponding sections. - -Note the functional syntax for existential quantification. The existential -quantifier is built from the operator \verb:ex:, which expects a -predicate as argument: -\begin{coq_example} -Check ex. -\end{coq_example} -and the notation \verb+(exists x:D, P x)+ is just concrete syntax for -the expression \verb+(ex D (fun x:D => P x))+. -Existential quantification is handled in \Coq~ in a similar -fashion to the connectives \verb:/\: and \verb:\/: : it is introduced by -the proof combinator \verb:ex_intro:, which is invoked by the specific -tactic \verb:Exists:, and its elimination provides a witness \verb+a:D+ to -\verb:P:, together with an assumption \verb+h:(P a)+ that indeed \verb+a+ -verifies \verb:P:. Let us see how this works on this simple example. -\begin{coq_example} -intros x x_Rlinked. -\end{coq_example} - -Remark that \verb:intros: treats universal quantification in the same way -as the premises of implications. Renaming of bound variables occurs -when it is needed; for instance, had we started with \verb:intro y:, -we would have obtained the goal: -\begin{coq_eval} -Undo. -\end{coq_eval} -\begin{coq_example} -intro y. -\end{coq_example} -\begin{coq_eval} -Undo. -intros x x_Rlinked. -\end{coq_eval} - -Let us now use the existential hypothesis \verb:x_Rlinked: to -exhibit an R-successor y of x. This is done in two steps, first with -\verb:elim:, then with \verb:intros: - -\begin{coq_example} -elim x_Rlinked. -intros y Rxy. -\end{coq_example} - -Now we want to use \verb:R_transitive:. The \verb:apply: tactic will know -how to match \verb:x: with \verb:x:, and \verb:z: with \verb:x:, but needs -help on how to instantiate \verb:y:, which appear in the hypotheses of -\verb:R_transitive:, but not in its conclusion. We give the proper hint -to \verb:apply: in a \verb:with: clause, as follows: -\begin{coq_example} -apply R_transitive with y. -\end{coq_example} - -The rest of the proof is routine: -\begin{coq_example} -assumption. -apply R_symmetric; assumption. -\end{coq_example} -\begin{coq_example*} -Qed. -\end{coq_example*} - -Let us now close the current section. -\begin{coq_example} -End R_sym_trans. -\end{coq_example} - -Here \Coq's printout is a warning that all local hypotheses have been -discharged in the statement of \verb:refl_if:, which now becomes a general -theorem in the first-order language declared in section -\verb:Predicate_calculus:. In this particular example, the use of section -\verb:R_sym_trans: has not been really significant, since we could have -instead stated theorem \verb:refl_if: in its general form, and done -basically the same proof, obtaining \verb:R_symmetric: and -\verb:R_transitive: as local hypotheses by initial \verb:intros: rather -than as global hypotheses in the context. But if we had pursued the -theory by proving more theorems about relation \verb:R:, -we would have obtained all general statements at the closing of the section, -with minimal dependencies on the hypotheses of symmetry and transitivity. - -\subsection{Paradoxes of classical predicate calculus} - -Let us illustrate this feature by pursuing our \verb:Predicate_calculus: -section with an enrichment of our language: we declare a unary predicate -\verb:P: and a constant \verb:d:: -\begin{coq_example} -Variable P : D -> Prop. -Variable d : D. -\end{coq_example} - -We shall now prove a well-known fact from first-order logic: a universal -predicate is non-empty, or in other terms existential quantification -follows from universal quantification. -\begin{coq_example} -Lemma weird : (forall x:D, P x) -> exists a, P a. - intro UnivP. -\end{coq_example} - -First of all, notice the pair of parentheses around -\verb+forall x:D, P x+ in -the statement of lemma \verb:weird:. -If we had omitted them, \Coq's parser would have interpreted the -statement as a truly trivial fact, since we would -postulate an \verb:x: verifying \verb:(P x):. Here the situation is indeed -more problematic. If we have some element in \verb:Set: \verb:D:, we may -apply \verb:UnivP: to it and conclude, otherwise we are stuck. Indeed -such an element \verb:d: exists, but this is just by virtue of our -new signature. This points out a subtle difference between standard -predicate calculus and \Coq. In standard first-order logic, -the equivalent of lemma \verb:weird: always holds, -because such a rule is wired in the inference rules for quantifiers, the -semantic justification being that the interpretation domain is assumed to -be non-empty. Whereas in \Coq, where types are not assumed to be -systematically inhabited, lemma \verb:weird: only holds in signatures -which allow the explicit construction of an element in the domain of -the predicate. - -Let us conclude the proof, in order to show the use of the \verb:Exists: -tactic: -\begin{coq_example} -exists d; trivial. -Qed. -\end{coq_example} - -Another fact which illustrates the sometimes disconcerting rules of -classical -predicate calculus is Smullyan's drinkers' paradox: ``In any non-empty -bar, there is a person such that if she drinks, then everyone drinks''. -We modelize the bar by Set \verb:D:, drinking by predicate \verb:P:. -We shall need classical reasoning. Instead of loading the \verb:Classical: -module as we did above, we just state the law of excluded middle as a -local hypothesis schema at this point: -\begin{coq_example} -Hypothesis EM : forall A:Prop, A \/ ~ A. -Lemma drinker : exists x:D, P x -> forall x:D, P x. -\end{coq_example} -The proof goes by cases on whether or not -there is someone who does not drink. Such reasoning by cases proceeds -by invoking the excluded middle principle, via \verb:elim: of the -proper instance of \verb:EM:: -\begin{coq_example} -elim (EM (exists x, ~ P x)). -\end{coq_example} - -We first look at the first case. Let Tom be the non-drinker: -\begin{coq_example} -intro Non_drinker; elim Non_drinker; - intros Tom Tom_does_not_drink. -\end{coq_example} - -We conclude in that case by considering Tom, since his drinking leads to -a contradiction: -\begin{coq_example} -exists Tom; intro Tom_drinks. -\end{coq_example} - -There are several ways in which we may eliminate a contradictory case; -a simple one is to use the \verb:absurd: tactic as follows: -\begin{coq_example} -absurd (P Tom); trivial. -\end{coq_example} - -We now proceed with the second case, in which actually any person will do; -such a John Doe is given by the non-emptiness witness \verb:d:: -\begin{coq_example} -intro No_nondrinker; exists d; intro d_drinks. -\end{coq_example} - -Now we consider any Dick in the bar, and reason by cases according to its -drinking or not: -\begin{coq_example} -intro Dick; elim (EM (P Dick)); trivial. -\end{coq_example} - -The only non-trivial case is again treated by contradiction: -\begin{coq_example} -intro Dick_does_not_drink; absurd (exists x, ~ P x); trivial. -exists Dick; trivial. -Qed. -\end{coq_example} - -Now, let us close the main section and look at the complete statements -we proved: -\begin{coq_example} -End Predicate_calculus. -Check refl_if. -Check weird. -Check drinker. -\end{coq_example} - -Remark how the three theorems are completely generic in the most general -fashion; -the domain \verb:D: is discharged in all of them, \verb:R: is discharged in -\verb:refl_if: only, \verb:P: is discharged only in \verb:weird: and -\verb:drinker:, along with the hypothesis that \verb:D: is inhabited. -Finally, the excluded middle hypothesis is discharged only in -\verb:drinker:. - -Note that the name \verb:d: has vanished as well from -the statements of \verb:weird: and \verb:drinker:, -since \Coq's pretty-printer replaces -systematically a quantification such as \verb+forall d:D, E+, where \verb:d: -does not occur in \verb:E:, by the functional notation \verb:D->E:. -Similarly the name \verb:EM: does not appear in \verb:drinker:. - -Actually, universal quantification, implication, -as well as function formation, are -all special cases of one general construct of type theory called -{\sl dependent product}. This is the mathematical construction -corresponding to an indexed family of functions. A function -$f\in \Pi x:D\cdot Cx$ maps an element $x$ of its domain $D$ to its -(indexed) codomain $Cx$. Thus a proof of $\forall x:D\cdot Px$ is -a function mapping an element $x$ of $D$ to a proof of proposition $Px$. - - -\subsection{Flexible use of local assumptions} - -Very often during the course of a proof we want to retrieve a local -assumption and reintroduce it explicitly in the goal, for instance -in order to get a more general induction hypothesis. The tactic -\verb:generalize: is what is needed here: - -\begin{coq_example} -Section Predicate_Calculus. -Variables P Q : nat -> Prop. -Variable R : nat -> nat -> Prop. -Lemma PQR : - forall x y:nat, (R x x -> P x -> Q x) -> P x -> R x y -> Q x. -intros. -generalize H0. -\end{coq_example} - -Sometimes it may be convenient to use a lemma, although we do not have -a direct way to appeal to such an already proven fact. The tactic \verb:cut: -permits to use the lemma at this point, keeping the corresponding proof -obligation as a new subgoal: -\begin{coq_example} -cut (R x x); trivial. -\end{coq_example} -We clean the goal by doing an \verb:Abort: command. -\begin{coq_example*} -Abort. -\end{coq_example*} - - -\subsection{Equality} - -The basic equality provided in \Coq~ is Leibniz equality, noted infix like -\verb+x=y+, when \verb:x: and \verb:y: are two expressions of -type the same Set. The replacement of \verb:x: by \verb:y: in any -term is effected by a variety of tactics, such as \verb:rewrite: -and \verb:replace:. - -Let us give a few examples of equality replacement. Let us assume that -some arithmetic function \verb:f: is null in zero: -\begin{coq_example} -Variable f : nat -> nat. -Hypothesis foo : f 0 = 0. -\end{coq_example} - -We want to prove the following conditional equality: -\begin{coq_example*} -Lemma L1 : forall k:nat, k = 0 -> f k = k. -\end{coq_example*} - -As usual, we first get rid of local assumptions with \verb:intro:: -\begin{coq_example} -intros k E. -\end{coq_example} - -Let us now use equation \verb:E: as a left-to-right rewriting: -\begin{coq_example} -rewrite E. -\end{coq_example} -This replaced both occurrences of \verb:k: by \verb:O:. - -Now \verb:apply foo: will finish the proof: - -\begin{coq_example} -apply foo. -Qed. -\end{coq_example} - -When one wants to rewrite an equality in a right to left fashion, we should -use \verb:rewrite <- E: rather than \verb:rewrite E: or the equivalent -\verb:rewrite -> E:. -Let us now illustrate the tactic \verb:replace:. -\begin{coq_example} -Hypothesis f10 : f 1 = f 0. -Lemma L2 : f (f 1) = 0. -replace (f 1) with 0. -\end{coq_example} -What happened here is that the replacement left the first subgoal to be -proved, but another proof obligation was generated by the \verb:replace: -tactic, as the second subgoal. The first subgoal is solved immediately -by applying lemma \verb:foo:; the second one transitivity and then -symmetry of equality, for instance with tactics \verb:transitivity: and -\verb:symmetry:: -\begin{coq_example} -apply foo. -transitivity (f 0); symmetry; trivial. -\end{coq_example} -In case the equality $t=u$ generated by \verb:replace: $u$ \verb:with: -$t$ is an assumption -(possibly modulo symmetry), it will be automatically proved and the -corresponding goal will not appear. For instance: -\begin{coq_example} -Restart. -replace (f 0) with 0. -rewrite f10; rewrite foo; trivial. -Qed. -\end{coq_example} - -\section{Using definitions} - -The development of mathematics does not simply proceed by logical -argumentation from first principles: definitions are used in an essential way. -A formal development proceeds by a dual process of abstraction, where one -proves abstract statements in predicate calculus, and use of definitions, -which in the contrary one instantiates general statements with particular -notions in order to use the structure of mathematical values for the proof of -more specialised properties. - -\subsection{Unfolding definitions} - -Assume that we want to develop the theory of sets represented as characteristic -predicates over some universe \verb:U:. For instance: -\begin{coq_example} -Variable U : Type. -Definition set := U -> Prop. -Definition element (x:U) (S:set) := S x. -Definition subset (A B:set) := - forall x:U, element x A -> element x B. -\end{coq_example} - -Now, assume that we have loaded a module of general properties about -relations over some abstract type \verb:T:, such as transitivity: - -\begin{coq_example} -Definition transitive (T:Type) (R:T -> T -> Prop) := - forall x y z:T, R x y -> R y z -> R x z. -\end{coq_example} - -Now, assume that we want to prove that \verb:subset: is a \verb:transitive: -relation. -\begin{coq_example} -Lemma subset_transitive : transitive set subset. -\end{coq_example} - -In order to make any progress, one needs to use the definition of -\verb:transitive:. The \verb:unfold: tactic, which replaces all -occurrences of a defined notion by its definition in the current goal, -may be used here. -\begin{coq_example} -unfold transitive. -\end{coq_example} - -Now, we must unfold \verb:subset:: -\begin{coq_example} -unfold subset. -\end{coq_example} -Now, unfolding \verb:element: would be a mistake, because indeed a simple proof -can be found by \verb:auto:, keeping \verb:element: an abstract predicate: -\begin{coq_example} -auto. -\end{coq_example} - -Many variations on \verb:unfold: are provided in \Coq. For instance, -we may selectively unfold one designated occurrence: -\begin{coq_example} -Undo 2. -unfold subset at 2. -\end{coq_example} - -One may also unfold a definition in a given local hypothesis, using the -\verb:in: notation: -\begin{coq_example} -intros. -unfold subset in H. -\end{coq_example} - -Finally, the tactic \verb:red: does only unfolding of the head occurrence -of the current goal: -\begin{coq_example} -red. -auto. -Qed. -\end{coq_example} - - -\subsection{Principle of proof irrelevance} - -Even though in principle the proof term associated with a verified lemma -corresponds to a defined value of the corresponding specification, such -definitions cannot be unfolded in \Coq: a lemma is considered an {\sl opaque} -definition. This conforms to the mathematical tradition of {\sl proof -irrelevance}: the proof of a logical proposition does not matter, and the -mathematical justification of a logical development relies only on -{\sl provability} of the lemmas used in the formal proof. - -Conversely, ordinary mathematical definitions can be unfolded at will, they -are {\sl transparent}. -\chapter{Induction} - -\section{Data Types as Inductively Defined Mathematical Collections} - -All the notions which were studied until now pertain to traditional -mathematical logic. Specifications of objects were abstract properties -used in reasoning more or less constructively; we are now entering -the realm of inductive types, which specify the existence of concrete -mathematical constructions. - -\subsection{Booleans} - -Let us start with the collection of booleans, as they are specified -in the \Coq's \verb:Prelude: module: -\begin{coq_example} -Inductive bool : Set := true | false. -\end{coq_example} - -Such a declaration defines several objects at once. First, a new -\verb:Set: is declared, with name \verb:bool:. Then the {\sl constructors} -of this \verb:Set: are declared, called \verb:true: and \verb:false:. -Those are analogous to introduction rules of the new Set \verb:bool:. -Finally, a specific elimination rule for \verb:bool: is now available, which -permits to reason by cases on \verb:bool: values. Three instances are -indeed defined as new combinators in the global context: \verb:bool_ind:, -a proof combinator corresponding to reasoning by cases, -\verb:bool_rec:, an if-then-else programming construct, -and \verb:bool_rect:, a similar combinator at the level of types. -Indeed: -\begin{coq_example} -Check bool_ind. -Check bool_rec. -Check bool_rect. -\end{coq_example} - -Let us for instance prove that every Boolean is true or false. -\begin{coq_example} -Lemma duality : forall b:bool, b = true \/ b = false. -intro b. -\end{coq_example} - -We use the knowledge that \verb:b: is a \verb:bool: by calling tactic -\verb:elim:, which is this case will appeal to combinator \verb:bool_ind: -in order to split the proof according to the two cases: -\begin{coq_example} -elim b. -\end{coq_example} - -It is easy to conclude in each case: -\begin{coq_example} -left; trivial. -right; trivial. -\end{coq_example} - -Indeed, the whole proof can be done with the combination of the - \verb:simple: \verb:induction:, which combines \verb:intro: and \verb:elim:, -with good old \verb:auto:: -\begin{coq_example} -Restart. -simple induction b; auto. -Qed. -\end{coq_example} - -\subsection{Natural numbers} - -Similarly to Booleans, natural numbers are defined in the \verb:Prelude: -module with constructors \verb:S: and \verb:O:: -\begin{coq_example} -Inductive nat : Set := - | O : nat - | S : nat -> nat. -\end{coq_example} - -The elimination principles which are automatically generated are Peano's -induction principle, and a recursion operator: -\begin{coq_example} -Check nat_ind. -Check nat_rec. -\end{coq_example} - -Let us start by showing how to program the standard primitive recursion -operator \verb:prim_rec: from the more general \verb:nat_rec:: -\begin{coq_example} -Definition prim_rec := nat_rec (fun i:nat => nat). -\end{coq_example} - -That is, instead of computing for natural \verb:i: an element of the indexed -\verb:Set: \verb:(P i):, \verb:prim_rec: computes uniformly an element of -\verb:nat:. Let us check the type of \verb:prim_rec:: -\begin{coq_example} -Check prim_rec. -\end{coq_example} - -Oops! Instead of the expected type \verb+nat->(nat->nat->nat)->nat->nat+ we -get an apparently more complicated expression. Indeed the type of -\verb:prim_rec: is equivalent by rule $\beta$ to its expected type; this may -be checked in \Coq~ by command \verb:Eval Cbv Beta:, which $\beta$-reduces -an expression to its {\sl normal form}: -\begin{coq_example} -Eval cbv beta in - ((fun _:nat => nat) O -> - (forall y:nat, - (fun _:nat => nat) y -> (fun _:nat => nat) (S y)) -> - forall n:nat, (fun _:nat => nat) n). -\end{coq_example} - -Let us now show how to program addition with primitive recursion: -\begin{coq_example} -Definition addition (n m:nat) := - prim_rec m (fun p rec:nat => S rec) n. -\end{coq_example} - -That is, we specify that \verb+(addition n m)+ computes by cases on \verb:n: -according to its main constructor; when \verb:n = O:, we get \verb:m:; - when \verb:n = S p:, we get \verb:(S rec):, where \verb:rec: is the result -of the recursive computation \verb+(addition p m)+. Let us verify it by -asking \Coq~to compute for us say $2+3$: -\begin{coq_example} -Eval compute in (addition (S (S O)) (S (S (S O)))). -\end{coq_example} - -Actually, we do not have to do all explicitly. {\Coq} provides a -special syntax {\tt Fixpoint/match} for generic primitive recursion, -and we could thus have defined directly addition as: - -\begin{coq_example} -Fixpoint plus (n m:nat) {struct n} : nat := - match n with - | O => m - | S p => S (plus p m) - end. -\end{coq_example} - -For the rest of the session, we shall clean up what we did so far with -types \verb:bool: and \verb:nat:, in order to use the initial definitions -given in \Coq's \verb:Prelude: module, and not to get confusing error -messages due to our redefinitions. We thus revert to the state before -our definition of \verb:bool: with the \verb:Reset: command: -\begin{coq_example} -Reset bool. -\end{coq_example} - - -\subsection{Simple proofs by induction} - -\begin{coq_eval} -Reset Initial. -\end{coq_eval} -\begin{coq_eval} -Set Printing Width 60. -\end{coq_eval} - -Let us now show how to do proofs by structural induction. We start with easy -properties of the \verb:plus: function we just defined. Let us first -show that $n=n+0$. -\begin{coq_example} -Lemma plus_n_O : forall n:nat, n = n + 0. -intro n; elim n. -\end{coq_example} - -What happened was that \verb:elim n:, in order to construct a \verb:Prop: -(the initial goal) from a \verb:nat: (i.e. \verb:n:), appealed to the -corresponding induction principle \verb:nat_ind: which we saw was indeed -exactly Peano's induction scheme. Pattern-matching instantiated the -corresponding predicate \verb:P: to \verb+fun n:nat => n = n + 0+, and we get -as subgoals the corresponding instantiations of the base case \verb:(P O): , -and of the inductive step \verb+forall y:nat, P y -> P (S y)+. -In each case we get an instance of function \verb:plus: in which its second -argument starts with a constructor, and is thus amenable to simplification -by primitive recursion. The \Coq~tactic \verb:simpl: can be used for -this purpose: -\begin{coq_example} -simpl. -auto. -\end{coq_example} - -We proceed in the same way for the base step: -\begin{coq_example} -simpl; auto. -Qed. -\end{coq_example} - -Here \verb:auto: succeeded, because it used as a hint lemma \verb:eq_S:, -which say that successor preserves equality: -\begin{coq_example} -Check eq_S. -\end{coq_example} - -Actually, let us see how to declare our lemma \verb:plus_n_O: as a hint -to be used by \verb:auto:: -\begin{coq_example} -Hint Resolve plus_n_O . -\end{coq_example} - -We now proceed to the similar property concerning the other constructor -\verb:S:: -\begin{coq_example} -Lemma plus_n_S : forall n m:nat, S (n + m) = n + S m. -\end{coq_example} - -We now go faster, remembering that tactic \verb:simple induction: does the -necessary \verb:intros: before applying \verb:elim:. Factoring simplification -and automation in both cases thanks to tactic composition, we prove this -lemma in one line: -\begin{coq_example} -simple induction n; simpl; auto. -Qed. -Hint Resolve plus_n_S . -\end{coq_example} - -Let us end this exercise with the commutativity of \verb:plus:: - -\begin{coq_example} -Lemma plus_com : forall n m:nat, n + m = m + n. -\end{coq_example} - -Here we have a choice on doing an induction on \verb:n: or on \verb:m:, the -situation being symmetric. For instance: -\begin{coq_example} -simple induction m; simpl; auto. -\end{coq_example} - -Here \verb:auto: succeeded on the base case, thanks to our hint -\verb:plus_n_O:, but the induction step requires rewriting, which -\verb:auto: does not handle: - -\begin{coq_example} -intros m' E; rewrite <- E; auto. -Qed. -\end{coq_example} - -\subsection{Discriminate} - -It is also possible to define new propositions by primitive recursion. -Let us for instance define the predicate which discriminates between -the constructors \verb:O: and \verb:S:: it computes to \verb:False: -when its argument is \verb:O:, and to \verb:True: when its argument is -of the form \verb:(S n):: -\begin{coq_example} -Definition Is_S (n:nat) := match n with - | O => False - | S p => True - end. -\end{coq_example} - -Now we may use the computational power of \verb:Is_S: in order to prove -trivially that \verb:(Is_S (S n)):: -\begin{coq_example} -Lemma S_Is_S : forall n:nat, Is_S (S n). -simpl; trivial. -Qed. -\end{coq_example} - -But we may also use it to transform a \verb:False: goal into -\verb:(Is_S O):. Let us show a particularly important use of this feature; -we want to prove that \verb:O: and \verb:S: construct different values, one -of Peano's axioms: -\begin{coq_example} -Lemma no_confusion : forall n:nat, 0 <> S n. -\end{coq_example} - -First of all, we replace negation by its definition, by reducing the -goal with tactic \verb:red:; then we get contradiction by successive -\verb:intros:: -\begin{coq_example} -red; intros n H. -\end{coq_example} - -Now we use our trick: -\begin{coq_example} -change (Is_S 0). -\end{coq_example} - -Now we use equality in order to get a subgoal which computes out to -\verb:True:, which finishes the proof: -\begin{coq_example} -rewrite H; trivial. -simpl; trivial. -\end{coq_example} - -Actually, a specific tactic \verb:discriminate: is provided -to produce mechanically such proofs, without the need for the user to define -explicitly the relevant discrimination predicates: - -\begin{coq_example} -Restart. -intro n; discriminate. -Qed. -\end{coq_example} - - -\section{Logic programming} - -In the same way as we defined standard data-types above, we -may define inductive families, and for instance inductive predicates. -Here is the definition of predicate $\le$ over type \verb:nat:, as -given in \Coq's \verb:Prelude: module: -\begin{coq_example*} -Inductive le (n:nat) : nat -> Prop := - | le_n : le n n - | le_S : forall m:nat, le n m -> le n (S m). -\end{coq_example*} - -This definition introduces a new predicate \verb+le:nat->nat->Prop+, -and the two constructors \verb:le_n: and \verb:le_S:, which are the -defining clauses of \verb:le:. That is, we get not only the ``axioms'' -\verb:le_n: and \verb:le_S:, but also the converse property, that -\verb:(le n m): if and only if this statement can be obtained as a -consequence of these defining clauses; that is, \verb:le: is the -minimal predicate verifying clauses \verb:le_n: and \verb:le_S:. This is -insured, as in the case of inductive data types, by an elimination principle, -which here amounts to an induction principle \verb:le_ind:, stating this -minimality property: -\begin{coq_example} -Check le. -Check le_ind. -\end{coq_example} - -Let us show how proofs may be conducted with this principle. -First we show that $n\le m \Rightarrow n+1\le m+1$: -\begin{coq_example} -Lemma le_n_S : forall n m:nat, le n m -> le (S n) (S m). -intros n m n_le_m. -elim n_le_m. -\end{coq_example} - -What happens here is similar to the behaviour of \verb:elim: on natural -numbers: it appeals to the relevant induction principle, here \verb:le_ind:, -which generates the two subgoals, which may then be solved easily -with the help of the defining clauses of \verb:le:. -\begin{coq_example} -apply le_n; trivial. -intros; apply le_S; trivial. -\end{coq_example} - -Now we know that it is a good idea to give the defining clauses as hints, -so that the proof may proceed with a simple combination of -\verb:induction: and \verb:auto:. -\begin{coq_example} -Restart. -Hint Resolve le_n le_S . -\end{coq_example} - -We have a slight problem however. We want to say ``Do an induction on -hypothesis \verb:(le n m):'', but we have no explicit name for it. What we -do in this case is to say ``Do an induction on the first unnamed hypothesis'', -as follows. -\begin{coq_example} -simple induction 1; auto. -Qed. -\end{coq_example} - -Here is a more tricky problem. Assume we want to show that -$n\le 0 \Rightarrow n=0$. This reasoning ought to follow simply from the -fact that only the first defining clause of \verb:le: applies. -\begin{coq_example} -Lemma tricky : forall n:nat, le n 0 -> n = 0. -\end{coq_example} - -However, here trying something like \verb:induction 1: would lead -nowhere (try it and see what happens). -An induction on \verb:n: would not be convenient either. -What we must do here is analyse the definition of \verb"le" in order -to match hypothesis \verb:(le n O): with the defining clauses, to find -that only \verb:le_n: applies, whence the result. -This analysis may be performed by the ``inversion'' tactic -\verb:inversion_clear: as follows: -\begin{coq_example} -intros n H; inversion_clear H. -trivial. -Qed. -\end{coq_example} - -\chapter{Modules} - -\section{Opening library modules} - -When you start \Coq~ without further requirements in the command line, -you get a bare system with few libraries loaded. As we saw, a standard -prelude module provides the standard logic connectives, and a few -arithmetic notions. If you want to load and open other modules from -the library, you have to use the \verb"Require" command, as we saw for -classical logic above. For instance, if you want more arithmetic -constructions, you should request: -\begin{coq_example*} -Require Import Arith. -\end{coq_example*} - -Such a command looks for a (compiled) module file \verb:Arith.vo: in -the libraries registered by \Coq. Libraries inherit the structure of -the file system of the operating system and are registered with the -command \verb:Add LoadPath:. Physical directories are mapped to -logical directories. Especially the standard library of \Coq~ is -pre-registered as a library of name \verb=Coq=. Modules have absolute -unique names denoting their place in \Coq~ libraries. An absolute -name is a sequence of single identifiers separated by dots. E.g. the -module \verb=Arith= has full name \verb=Coq.Arith.Arith= and because -it resides in eponym subdirectory \verb=Arith= of the standard -library, it can be as well required by the command - -\begin{coq_example*} -Require Import Coq.Arith.Arith. -\end{coq_example*} - -This may be useful to avoid ambiguities if somewhere, in another branch -of the libraries known by Coq, another module is also called -\verb=Arith=. Notice that by default, when a library is registered, -all its contents, and all the contents of its subdirectories recursively are -visible and accessible by a short (relative) name as \verb=Arith=. -Notice also that modules or definitions not explicitly registered in -a library are put in a default library called \verb=Top=. - -The loading of a compiled file is quick, because the corresponding -development is not type-checked again. - -\section{Creating your own modules} - -You may create your own module files, by writing {\Coq} commands in a file, -say \verb:my_module.v:. Such a module may be simply loaded in the current -context, with command \verb:Load my_module:. It may also be compiled, -in ``batch'' mode, using the UNIX command -\verb:coqc:. Compiling the module \verb:my_module.v: creates a -file \verb:my_module.vo:{} that can be reloaded with command -\verb:Require: \verb:Import: \verb:my_module:. - -If a required module depends on other modules then the latters are -automatically required beforehand. However their contents is not -automatically visible. If you want a module \verb=M= required in a -module \verb=N= to be automatically visible when \verb=N= is required, -you should use \verb:Require Export M: in your module \verb:N:. - -\section{Managing the context} - -It is often difficult to remember the names of all lemmas and -definitions available in the current context, especially if large -libraries have been loaded. A convenient \verb:SearchAbout: command -is available to lookup all known facts -concerning a given predicate. For instance, if you want to know all the -known lemmas about the less or equal relation, just ask: -\begin{coq_example} -SearchAbout le. -\end{coq_example} -Another command \verb:Search: displays only lemmas where the searched -predicate appears at the head position in the conclusion. -\begin{coq_example} -Search le. -\end{coq_example} - -A new and more convenient search tool is \textsf{SearchPattern} -developed by Yves Bertot. It allows to find the theorems with a -conclusion matching a given pattern, where \verb:\_: can be used in -place of an arbitrary term. We remark in this example, that \Coq{} -provides usual infix notations for arithmetic operators. - -\begin{coq_example} -SearchPattern (_ + _ = _). -\end{coq_example} - -\section{Now you are on your own} - -This tutorial is necessarily incomplete. If you wish to pursue serious -proving in \Coq, you should now get your hands on \Coq's Reference Manual, -which contains a complete description of all the tactics we saw, -plus many more. -You also should look in the library of developed theories which is distributed -with \Coq, in order to acquaint yourself with various proof techniques. - - -\end{document} - -% $Id: Tutorial.tex 13547 2010-10-14 12:35:00Z notin $ |
