From 8f4d4c66134804bbf2d2fe65c893b68387272d31 Mon Sep 17 00:00:00 2001
From: Stephane Glondu <steph@glondu.net>
Date: Sat, 10 Jul 2010 15:57:24 +0100
Subject: Remove non-DFSG contents

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-\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-L�f'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: 
-- 
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