From a8839f8646ae0675361483e99c0b937a6b83bfbe Mon Sep 17 00:00:00 2001 From: Maxime Dénès Date: Thu, 15 Mar 2018 09:31:46 +0100 Subject: [Sphinx] Move chapter 3 to new infrastructure --- Makefile.doc | 1 - doc/refman/RefMan-lib.tex | 1112 ----------------------------------- doc/refman/Reference-Manual.tex | 1 - doc/sphinx/language/coq-library.rst | 1112 +++++++++++++++++++++++++++++++++++ 4 files changed, 1112 insertions(+), 1114 deletions(-) delete mode 100644 doc/refman/RefMan-lib.tex create mode 100644 doc/sphinx/language/coq-library.rst diff --git a/Makefile.doc b/Makefile.doc index a6d4584d8..7c68e5d8e 100644 --- a/Makefile.doc +++ b/Makefile.doc @@ -58,7 +58,6 @@ DOCCOMMON:=doc/common/version.tex doc/common/title.tex doc/common/macros.tex REFMANCOQTEXFILES:=$(addprefix doc/refman/, \ RefMan-gal.v.tex \ - RefMan-lib.v.tex \ RefMan-oth.v.tex RefMan-ltac.v.tex \ RefMan-pro.v.tex \ Cases.v.tex Coercion.v.tex CanonicalStructures.v.tex Extraction.v.tex \ diff --git a/doc/refman/RefMan-lib.tex b/doc/refman/RefMan-lib.tex deleted file mode 100644 index 89f5be843..000000000 --- a/doc/refman/RefMan-lib.tex +++ /dev/null @@ -1,1112 +0,0 @@ -\chapter[The {\Coq} library]{The {\Coq} library\index{Theories}\label{Theories}} -%HEVEA\cutname{stdlib.html} - -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 \url{http://coq.inria.fr} (see -Section~\ref{Contributions}). - -The chapter briefly reviews the \Coq\ libraries whose contents can -also be browsed at \url{http://coq.inria.fr/stdlib}. - -\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!_ -> _! & 99 & right \\ -\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{coq_example*} -\begin{coq_eval} -Abort All. -\end{coq_eval} -\begin{coq_example*} -End Projections. -\end{coq_example*} -\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{eq\_refl} - -\begin{coq_example*} -Inductive eq (A:Type) (x:A) : A -> Prop := - eq_refl : 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{eq\_sym} -\ttindex{eq\_trans} -\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 eq_sym : x = y -> y = x. -Theorem eq_trans : x = y -> y = z -> x = z. -Theorem f_equal : x = y -> f x = f y. -Theorem not_eq_sym : 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 eq_sym not_eq_sym : 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 Projections2. -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 Projections2. -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_rect2 : - 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 x : 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}}. They are listed at -\url{http://coq.inria.fr/stdlib} (paragraph {\tt Init}, link {\tt - Tactics}). - -\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 PArith} & Basic positive integer arithmetic \\ - {\bf NArith} & Basic binary natural number 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{\url{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 Z.lt} &&\\ -\verb!x <= y! & {\tt Z.le} &&\\ -\verb!_ > _! & {\tt Z.gt} &&\\ -\verb!x >= y! & {\tt Z.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 Z.compare} & 70 & no\\ -\verb!_ + _! & {\tt Z.add} &&\\ -\verb!_ - _! & {\tt Z.sub} &&\\ -\verb!_ * _! & {\tt Z.mul} &&\\ -\verb!_ / _! & {\tt Z.div} &&\\ -\verb!_ mod _! & {\tt Z.modulo} & 40 & no \\ -\verb!- _! & {\tt Z.opp} &&\\ -\verb!_ ^ _! & {\tt Z.pow} &&\\ -\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. -\end{coq_example*} -\begin{coq_example} -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. -\end{coq_example*} -\begin{coq_example} -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 unfolding the {\tt Rabs} constant and splits -corresponding conjunctions. -\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} splits 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} - -These 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. - -%%% Local Variables: -%%% mode: latex -%%% TeX-master: "Reference-Manual" -%%% End: diff --git a/doc/refman/Reference-Manual.tex b/doc/refman/Reference-Manual.tex index a6a3670bd..5122b2a36 100644 --- a/doc/refman/Reference-Manual.tex +++ b/doc/refman/Reference-Manual.tex @@ -95,7 +95,6 @@ Options A and B of the licence are {\em not} elected.} \defaultheaders %END LATEX \include{RefMan-gal.v}% Gallina -\include{RefMan-lib.v}% The coq library \part{The proof engine} diff --git a/doc/sphinx/language/coq-library.rst b/doc/sphinx/language/coq-library.rst new file mode 100644 index 000000000..89f5be843 --- /dev/null +++ b/doc/sphinx/language/coq-library.rst @@ -0,0 +1,1112 @@ +\chapter[The {\Coq} library]{The {\Coq} library\index{Theories}\label{Theories}} +%HEVEA\cutname{stdlib.html} + +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 \url{http://coq.inria.fr} (see +Section~\ref{Contributions}). + +The chapter briefly reviews the \Coq\ libraries whose contents can +also be browsed at \url{http://coq.inria.fr/stdlib}. + +\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!_ -> _! & 99 & right \\ +\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{coq_example*} +\begin{coq_eval} +Abort All. +\end{coq_eval} +\begin{coq_example*} +End Projections. +\end{coq_example*} +\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{eq\_refl} + +\begin{coq_example*} +Inductive eq (A:Type) (x:A) : A -> Prop := + eq_refl : 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{eq\_sym} +\ttindex{eq\_trans} +\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 eq_sym : x = y -> y = x. +Theorem eq_trans : x = y -> y = z -> x = z. +Theorem f_equal : x = y -> f x = f y. +Theorem not_eq_sym : 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 eq_sym not_eq_sym : 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 Projections2. +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 Projections2. +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_rect2 : + 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 x : 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}}. They are listed at +\url{http://coq.inria.fr/stdlib} (paragraph {\tt Init}, link {\tt + Tactics}). + +\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 PArith} & Basic positive integer arithmetic \\ + {\bf NArith} & Basic binary natural number 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{\url{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 Z.lt} &&\\ +\verb!x <= y! & {\tt Z.le} &&\\ +\verb!_ > _! & {\tt Z.gt} &&\\ +\verb!x >= y! & {\tt Z.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 Z.compare} & 70 & no\\ +\verb!_ + _! & {\tt Z.add} &&\\ +\verb!_ - _! & {\tt Z.sub} &&\\ +\verb!_ * _! & {\tt Z.mul} &&\\ +\verb!_ / _! & {\tt Z.div} &&\\ +\verb!_ mod _! & {\tt Z.modulo} & 40 & no \\ +\verb!- _! & {\tt Z.opp} &&\\ +\verb!_ ^ _! & {\tt Z.pow} &&\\ +\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. +\end{coq_example*} +\begin{coq_example} +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. +\end{coq_example*} +\begin{coq_example} +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 unfolding the {\tt Rabs} constant and splits +corresponding conjunctions. +\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} splits 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} + +These 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. + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "Reference-Manual" +%%% End: -- cgit v1.2.3