From 8c374555ba5fbb78b38ff0c053bed5c72ea4dde4 Mon Sep 17 00:00:00 2001
From: Maxime Dénès <mail@maximedenes.fr>
Date: Tue, 13 Mar 2018 17:40:09 +0100
Subject: [Sphinx] Move chapter 10 to new infrastructure

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-\chapter[Detailed examples of tactics]{Detailed examples of tactics\label{Tactics-examples}}
-%HEVEA\cutname{tactic-examples.html}
-
-This chapter presents detailed examples of certain tactics, to
-illustrate their behavior.
-
-\section[\tt dependent induction]{\tt dependent induction\label{dependent-induction-example}}
-\def\depind{{\tt dependent induction}~}
-\def\depdestr{{\tt dependent destruction}~}
-
-The tactics \depind and \depdestr are another solution for inverting
-inductive predicate instances and potentially doing induction at the
-same time. It is based on the \texttt{BasicElim} tactic of Conor McBride which
-works by abstracting each argument of an inductive instance by a variable
-and constraining it by equalities afterwards. This way, the usual 
-{\tt induction} and {\tt destruct} tactics can be applied to the
-abstracted instance and after simplification of the equalities we get
-the expected goals.
-
-The abstracting tactic is called {\tt generalize\_eqs} and it takes as
-argument an hypothesis to generalize. It uses the {\tt JMeq} datatype
-defined in {\tt Coq.Logic.JMeq}, hence we need to require it before.
-For example, revisiting the first example of the inversion documentation above:
-
-\begin{coq_example*}
-Require Import Coq.Logic.JMeq.
-\end{coq_example*}
-\begin{coq_eval}
-Require Import Coq.Program.Equality.
-\end{coq_eval}
-
-\begin{coq_eval}
-Inductive Le : nat -> nat -> Set :=
-  | LeO : forall n:nat, Le 0 n
-  | LeS : forall n m:nat, Le n m -> Le (S n) (S m).
-Variable P : nat -> nat -> Prop.
-Variable Q : forall n m:nat, Le n m -> Prop.
-\end{coq_eval}
-
-\begin{coq_example*}
-Goal forall n m:nat, Le (S n) m -> P n m.
-intros n m H.
-\end{coq_example*}
-\begin{coq_example}
-generalize_eqs H.
-\end{coq_example}
-
-The index {\tt S n} gets abstracted by a variable here, but a
-corresponding equality is added under the abstract instance so that no
-information is actually lost. The goal is now almost amenable to do induction
-or case analysis. One should indeed first move {\tt n} into the goal to
-strengthen it before doing induction, or {\tt n} will be fixed in
-the inductive hypotheses (this does not matter for case analysis). 
-As a rule of thumb, all the variables that appear inside constructors in
-the indices of the hypothesis should be generalized. This is exactly
-what the \texttt{generalize\_eqs\_vars} variant does:
-
-\begin{coq_eval} 
-Undo 1.
-\end{coq_eval}
-\begin{coq_example}
-generalize_eqs_vars H.
-induction H.
-\end{coq_example}
-
-As the hypothesis itself did not appear in the goal, we did not need to
-use an heterogeneous equality to relate the new hypothesis to the old
-one (which just disappeared here). However, the tactic works just as well
-in this case, e.g.:
-
-\begin{coq_eval}
-Admitted.
-\end{coq_eval}
-
-\begin{coq_example}
-Goal forall n m (p : Le (S n) m), Q (S n) m p.
-intros n m p ; generalize_eqs_vars p.
-\end{coq_example}
-
-One drawback of this approach is that in the branches one will have to
-substitute the equalities back into the instance to get the right
-assumptions. Sometimes injection of constructors will also be needed to
-recover the needed equalities. Also, some subgoals should be directly
-solved because of inconsistent contexts arising from the constraints on 
-indexes. The nice thing is that we can make a tactic based on
-discriminate, injection and variants of substitution to automatically 
-do such simplifications (which may involve the K axiom). 
-This is what the {\tt simplify\_dep\_elim} tactic from
-{\tt Coq.Program.Equality} does. For example, we might simplify the
-previous goals considerably:
-% \begin{coq_eval} 
-% Abort.
-% Goal forall n m:nat, Le (S n) m -> P n m.
-% intros n m H ; generalize_eqs_vars H.
-% \end{coq_eval}
-
-\begin{coq_example}
-induction p ; simplify_dep_elim.
-\end{coq_example}
-
-The higher-order tactic {\tt do\_depind} defined in {\tt
-  Coq.Program.Equality} takes a tactic and combines the
-building blocks we have seen with it: generalizing by equalities
-calling the given tactic with the
-generalized induction hypothesis as argument and cleaning the subgoals
-with respect to equalities. Its most important instantiations are
-\depind and \depdestr that do induction or simply case analysis on the
-generalized hypothesis. For example we can redo what we've done manually
-with \depdestr:
-
-\begin{coq_eval}
-Abort.
-\end{coq_eval}
-\begin{coq_example*}
-Require Import Coq.Program.Equality.
-Lemma ex : forall n m:nat, Le (S n) m -> P n m.
-intros n m H.
-\end{coq_example*}
-\begin{coq_example}
-dependent destruction H.
-\end{coq_example}
-\begin{coq_eval}
-Abort.
-\end{coq_eval}
-
-This gives essentially the same result as inversion. Now if the
-destructed hypothesis actually appeared in the goal, the tactic would
-still be able to invert it, contrary to {\tt dependent
- inversion}. Consider the following example on vectors:
-
-\begin{coq_example*}
-Require Import Coq.Program.Equality.
-Set Implicit Arguments.
-Variable A : Set.
-Inductive vector : nat -> Type := 
-| vnil : vector 0 
-| vcons : A -> forall n, vector n -> vector (S n).
-Goal forall n, forall v : vector (S n), 
-  exists v' : vector n, exists a : A, v = vcons a v'.
-  intros n v.
-\end{coq_example*}
-\begin{coq_example}
-  dependent destruction v.
-\end{coq_example}
-\begin{coq_eval}
-Abort.
-\end{coq_eval}
-
-In this case, the {\tt v} variable can be replaced in the goal by the
-generalized hypothesis only when it has a type of the form {\tt vector
- (S n)}, that is only in the second case of the {\tt destruct}. The
-first one is dismissed because {\tt S n <> 0}.
-
-\subsection{A larger example}
-
-Let's see how the technique works with {\tt induction} on inductive
-predicates on a real example. We will develop an example application to the
-theory of simply-typed lambda-calculus formalized in a dependently-typed style:
-
-\begin{coq_example*}
-Inductive type : Type :=
-| base : type
-| arrow : type -> type -> type.
-Notation " t --> t' " := (arrow t t') (at level 20, t' at next level).
-Inductive ctx : Type :=
-| empty : ctx
-| snoc : ctx -> type -> ctx.
-Notation " G , tau " := (snoc G tau) (at level 20, tau at next level).
-Fixpoint conc (G D : ctx) : ctx :=
-  match D with
-    | empty => G
-    | snoc D' x => snoc (conc G D') x
-  end.
-Notation " G ; D " := (conc G D) (at level 20).
-Inductive term : ctx -> type -> Type :=
-| ax : forall G tau, term (G, tau) tau
-| weak : forall G tau, 
-  term G tau -> forall tau', term (G, tau') tau
-| abs : forall G tau tau', 
-  term (G , tau) tau' -> term G (tau --> tau')
-| app : forall G tau tau', 
-  term G (tau --> tau') -> term G tau -> term G tau'.
-\end{coq_example*}
-
-We have defined types and contexts which are snoc-lists of types. We
-also have a {\tt conc} operation that concatenates two contexts.
-The {\tt term} datatype represents in fact the possible typing
-derivations of the calculus, which are isomorphic to the well-typed
-terms, hence the name. A term is either an application of:
-\begin{itemize}
-\item the axiom rule to type a reference to the first variable in a context,
-\item the weakening rule to type an object in a larger context
-\item the abstraction or lambda rule to type a function
-\item the application to type an application of a function to an argument
-\end{itemize}
-
-Once we have this datatype we want to do proofs on it, like weakening:
-
-\begin{coq_example*}
-Lemma weakening : forall G D tau, term (G ; D) tau -> 
-  forall tau', term (G , tau' ; D) tau.
-\end{coq_example*}
-\begin{coq_eval}
-  Abort.
-\end{coq_eval}
-
-The problem here is that we can't just use {\tt induction} on the typing
-derivation because it will forget about the {\tt G ; D} constraint
-appearing in the instance. A solution would be to rewrite the goal as:
-\begin{coq_example*}
-Lemma weakening' : forall G' tau, term G' tau -> 
-  forall G D, (G ; D) = G' ->
-  forall tau', term (G, tau' ; D) tau.
-\end{coq_example*}
-\begin{coq_eval}
-  Abort.
-\end{coq_eval}
-
-With this proper separation of the index from the instance and the right
-induction loading (putting {\tt G} and {\tt D} after the inducted-on
-hypothesis), the proof will go through, but it is a very tedious
-process. One is also forced to make a wrapper lemma to get back the
-more natural statement. The \depind tactic alleviates this trouble by
-doing all of this plumbing of generalizing and substituting back automatically.
-Indeed we can simply write:
-
-\begin{coq_example*}
-Require Import Coq.Program.Tactics.
-Lemma weakening : forall G D tau, term (G ; D) tau -> 
-  forall tau', term (G , tau' ; D) tau.
-Proof with simpl in * ; simpl_depind ; auto.
-  intros G D tau H. dependent induction H generalizing G D ; intros.
-\end{coq_example*}
-
-This call to \depind has an additional arguments which is a list of
-variables appearing in the instance that should be generalized in the
-goal, so that they can vary in the induction hypotheses. By default, all
-variables appearing inside constructors (except in a parameter position)
-of the instantiated hypothesis will be generalized automatically but
-one can always give the list explicitly.
-
-\begin{coq_example}
-  Show.
-\end{coq_example}
-
-The {\tt simpl\_depind} tactic includes an automatic tactic that tries
-to simplify equalities appearing at the beginning of induction
-hypotheses, generally using trivial applications of
-reflexivity. In cases where the equality is not between constructor
-forms though, one must help the automation by giving
-some arguments, using the {\tt specialize} tactic for example.
-
-\begin{coq_example*}
-destruct D... apply weak ; apply ax. apply ax.
-destruct D...
-\end{coq_example*}
-\begin{coq_example}
-Show.
-\end{coq_example}
-\begin{coq_example}
-  specialize (IHterm G0 empty eq_refl).
-\end{coq_example}
-
-Once the induction hypothesis has been narrowed to the right equality,
-it can be used directly. 
-
-\begin{coq_example}
-  apply weak, IHterm.
-\end{coq_example}
-
-If there is an easy first-order solution to these equations as in this subgoal, the
-{\tt specialize\_eqs} tactic can be used instead of giving explicit proof
-terms:
-
-\begin{coq_example}
-  specialize_eqs IHterm.
-\end{coq_example}
-This concludes our example.
-\SeeAlso The induction \ref{elim}, case \ref{case} and inversion \ref{inversion} tactics.
-
-\section[\tt autorewrite]{\tt autorewrite\label{autorewrite-example}}
-
-Here are two examples of {\tt autorewrite} use. The first one ({\em Ackermann
-function}) shows actually a quite basic use where there is no conditional
-rewriting. The second one ({\em Mac Carthy function}) involves conditional
-rewritings and shows how to deal with them using the optional tactic of the
-{\tt Hint~Rewrite} command.
-
-\firstexample
-\example{Ackermann function}
-%Here is a basic use of {\tt AutoRewrite} with the Ackermann function:
-
-\begin{coq_example*}
-Reset Initial.
-Require Import Arith.
-Variable Ack : 
-           nat -> nat -> nat.
-Axiom Ack0 : 
-        forall m:nat, Ack 0 m = S m.
-Axiom Ack1 : forall n:nat, Ack (S n) 0 = Ack n 1.
-Axiom Ack2 : forall n m:nat, Ack (S n) (S m) = Ack n (Ack (S n) m).
-\end{coq_example*}
-
-\begin{coq_example}
-Hint Rewrite Ack0 Ack1 Ack2 : base0.
-Lemma ResAck0 : 
- Ack 3 2 = 29.
-autorewrite with base0 using try reflexivity.
-\end{coq_example}
-
-\begin{coq_eval}
-Reset Initial.
-\end{coq_eval}
-
-\example{Mac Carthy function}
-%The Mac Carthy function shows a more complex case:
-
-\begin{coq_example*}
-Require Import Omega.
-Variable g :   
-           nat -> nat -> nat.
-Axiom g0 : 
-        forall m:nat, g 0 m = m.
-Axiom
-  g1 :
-    forall n m:nat,
-      (n > 0) -> (m > 100) -> g n m = g (pred n) (m - 10).
-Axiom
-  g2 :
-    forall n m:nat,
-      (n > 0) -> (m <= 100) -> g n m = g (S n) (m + 11).
-\end{coq_example*}
-
-\begin{coq_example}
-Hint Rewrite g0 g1 g2 using omega : base1.
-Lemma Resg0 : 
- g 1 110 = 100.
-autorewrite with base1 using reflexivity || simpl.
-\end{coq_example}
-
-\begin{coq_eval}
-Abort.
-\end{coq_eval}
-
-\begin{coq_example}
-Lemma Resg1 : g 1 95 = 91.
-autorewrite with base1 using reflexivity || simpl.
-\end{coq_example}
-
-\begin{coq_eval}
-Reset Initial.
-\end{coq_eval}
-
-\section[\tt quote]{\tt quote\tacindex{quote}
-\label{quote-examples}}
-
-The tactic \texttt{quote} allows using Barendregt's so-called
-2-level approach without writing any ML code. Suppose you have a
-language \texttt{L} of 
-'abstract terms' and a type \texttt{A} of 'concrete terms' 
-and a function \texttt{f : L -> A}. If \texttt{L} is a simple
-inductive datatype and \texttt{f} a simple fixpoint, \texttt{quote f}
-will replace the head of current goal by a convertible term of the form 
-\texttt{(f t)}. \texttt{L} must have a constructor of type: \texttt{A
-  -> L}. 
-
-Here is an example:
-
-\begin{coq_example}
-Require Import Quote.
-Parameters A B C : Prop.
-Inductive formula : Type :=
-  | f_and : formula -> formula -> formula (* binary constructor *)
-  | f_or : formula -> formula -> formula
-  | f_not : formula -> formula (* unary constructor *)
-  | f_true : formula (* 0-ary constructor *)
-  | f_const : Prop -> formula (* constructor for constants *).
-Fixpoint interp_f (f:
-                   formula) : Prop :=
-  match f with
-  | f_and f1 f2 => interp_f f1 /\ interp_f f2
-  | f_or f1 f2 => interp_f f1 \/ interp_f f2
-  | f_not f1 => ~ interp_f f1
-  | f_true => True
-  | f_const c => c
-  end.
-Goal A /\ (A \/ True) /\ ~ B /\ (A <-> A).
-quote interp_f.
-\end{coq_example}
-
-The algorithm to perform this inversion is: try to match the
-term with right-hand sides expression of \texttt{f}. If there is a
-match, apply the corresponding left-hand side and call yourself
-recursively on sub-terms. If there is no match, we are at a leaf:
-return the corresponding constructor (here \texttt{f\_const}) applied
-to the term. 
-
-\begin{ErrMsgs}
-\item \errindex{quote: not a simple fixpoint} \\
-  Happens when \texttt{quote} is not able to perform inversion properly.
-\end{ErrMsgs}
-
-\subsection{Introducing variables map}
-
-The normal use of \texttt{quote} is to make proofs by reflection: one
-defines a function \texttt{simplify : formula -> formula} and proves a 
-theorem \texttt{simplify\_ok: (f:formula)(interp\_f (simplify f)) ->
-  (interp\_f f)}. Then, one can simplify formulas by doing:
-\begin{verbatim}
-   quote interp_f.
-   apply simplify_ok.
-   compute.
-\end{verbatim}
-But there is a problem with leafs: in the example above one cannot
-write a function that implements, for example, the logical simplifications 
-$A \land A \ra A$ or $A \land \lnot A \ra \texttt{False}$. This is
-because the \Prop{} is impredicative.
-
-It is better to use that type of formulas:
-
-\begin{coq_eval}
-Reset formula.
-\end{coq_eval}
-\begin{coq_example}
-Inductive formula : Set :=
-  | f_and : formula -> formula -> formula
-  | f_or : formula -> formula -> formula
-  | f_not : formula -> formula
-  | f_true : formula
-  | f_atom : index -> formula.
-\end{coq_example*}
-
-\texttt{index} is defined in module \texttt{quote}. Equality on that
-type is decidable so we are able to simplify $A \land A$ into $A$ at
-the abstract level. 
-
-When there are variables, there are bindings, and \texttt{quote}
-provides also a type \texttt{(varmap A)} of bindings from
-\texttt{index} to any set \texttt{A}, and a function
-\texttt{varmap\_find} to search in such maps. The interpretation
-function has now another argument, a variables map:
-
-\begin{coq_example}
-Fixpoint interp_f (vm:
-                    varmap Prop) (f:formula) {struct f} : Prop :=
-  match f with
-  | f_and f1 f2 => interp_f vm f1 /\ interp_f vm f2
-  | f_or f1 f2 => interp_f vm f1 \/ interp_f vm f2
-  | f_not f1 => ~ interp_f vm f1
-  | f_true => True
-  | f_atom i => varmap_find True i vm
-  end.
-\end{coq_example}
-
-\noindent\texttt{quote} handles this second case properly:
-
-\begin{coq_example}
-Goal A /\ (B \/ A) /\ (A \/ ~ B).
-quote interp_f.
-\end{coq_example}
-
-It builds \texttt{vm} and \texttt{t} such that \texttt{(f vm t)} is
-convertible with the conclusion of current goal.
-
-\subsection{Combining variables and constants}
-
-One can have both variables and constants in abstracts terms; that is
-the case, for example, for the \texttt{ring} tactic (chapter
-\ref{ring}). Then one must provide to \texttt{quote} a list of
-\emph{constructors of constants}. For example, if the list is
-\texttt{[O S]} then closed natural numbers will be considered as
-constants and other terms as variables. 
-
-Example: 
-
-\begin{coq_eval}
-Reset formula.
-\end{coq_eval}
-\begin{coq_example*}
-Inductive formula : Type :=
-  | f_and : formula -> formula -> formula
-  | f_or : formula -> formula -> formula
-  | f_not : formula -> formula
-  | f_true : formula
-  | f_const : Prop -> formula (* constructor for constants *)
-  | f_atom : index -> formula.
-Fixpoint interp_f
- (vm:            (* constructor for variables *)
-  varmap Prop) (f:formula) {struct f} : Prop :=
-  match f with
-  | f_and f1 f2 => interp_f vm f1 /\ interp_f vm f2
-  | f_or f1 f2 => interp_f vm f1 \/ interp_f vm f2
-  | f_not f1 => ~ interp_f vm f1
-  | f_true => True
-  | f_const c => c
-  | f_atom i => varmap_find True i vm
-  end.
-Goal 
-A /\ (A \/ True) /\ ~ B /\ (C <-> C).
-\end{coq_example*}
-
-\begin{coq_example}
-quote interp_f [ A B ].
-Undo.
-  quote interp_f [ B C iff ].
-\end{coq_example}
-
-\Warning Since function inversion
-is undecidable in general case, don't expect miracles from it!
-
-\begin{Variants}
-
-\item {\tt quote {\ident} in {\term} using {\tac}}
-
-  \tac\ must be a functional tactic (starting with {\tt fun x =>})
-  and will be called with the quoted version of \term\ according to
-  \ident.
-
-\item {\tt quote {\ident} [ \ident$_1$ \dots\ \ident$_n$ ] in {\term} using {\tac}}
-
-  Same as above, but will use \ident$_1$, \dots, \ident$_n$ to
-  chose which subterms are constants (see above).
-
-\end{Variants}
-
-% \SeeAlso file \texttt{theories/DEMOS/DemoQuote.v}
-
-\SeeAlso comments of source file \texttt{plugins/quote/quote.ml}
-
-\SeeAlso the \texttt{ring} tactic (Chapter~\ref{ring})
-
-
-
-\section{Using the tactical language}
-
-\subsection{About the cardinality of the set of natural numbers}
-
-A first example which shows how to use the pattern matching over the proof
-contexts is the proof that natural numbers have more than two elements. The
-proof of such a lemma can be done as %shown on Figure~\ref{cnatltac}.
-follows:
-%\begin{figure}
-%\begin{centerframe}
-\begin{coq_eval}
-Reset Initial.
-Require Import Arith.
-Require Import List.
-\end{coq_eval}
-\begin{coq_example*}
-Lemma card_nat :
- ~ (exists x : nat, exists y : nat, forall z:nat, x = z \/ y = z).
-Proof.
-red; intros (x, (y, Hy)).
-elim (Hy 0); elim (Hy 1); elim (Hy 2); intros;
- match goal with
- | [_:(?a = ?b),_:(?a = ?c) |- _ ] =>
-     cut (b = c); [ discriminate | transitivity a; auto ]
- end.
-Qed.
-\end{coq_example*}
-%\end{centerframe}
-%\caption{A proof on cardinality of natural numbers}
-%\label{cnatltac}
-%\end{figure}
-
-We can notice that all the (very similar) cases coming from the three
-eliminations (with three distinct natural numbers) are successfully solved by
-a {\tt match goal} structure and, in particular, with only one pattern (use
-of non-linear matching).
-
-\subsection{Permutation on closed lists}
-
-Another more complex example is the problem of permutation on closed lists. The
-aim is to show that a closed list is a permutation of another one.
-
-First, we define the permutation predicate as shown in table~\ref{permutpred}.
-
-\begin{figure}
-\begin{centerframe}
-\begin{coq_example*}
-Section Sort.
-Variable A : Set.
-Inductive permut : list A -> list A -> Prop :=
-  | permut_refl   : forall l, permut l l
-  | permut_cons   :
-      forall a l0 l1, permut l0 l1 -> permut (a :: l0) (a :: l1)
-  | permut_append : forall a l, permut (a :: l) (l ++ a :: nil)
-  | permut_trans  :
-      forall l0 l1 l2, permut l0 l1 -> permut l1 l2 -> permut l0 l2.
-End Sort.
-\end{coq_example*}
-\end{centerframe}
-\caption{Definition of the permutation predicate}
-\label{permutpred}
-\end{figure}
-
-A more complex example is the problem of permutation on closed lists.
-The aim is to show that a closed list is a permutation of another one.
-First, we define the permutation predicate as shown on
-Figure~\ref{permutpred}.
-
-\begin{figure}
-\begin{centerframe}
-\begin{coq_example}
-Ltac Permut n :=
-  match goal with
-  | |- (permut _ ?l ?l) => apply permut_refl
-  | |- (permut _ (?a :: ?l1) (?a :: ?l2)) =>
-      let newn := eval compute in (length l1) in
-      (apply permut_cons; Permut newn)
-  | |- (permut ?A (?a :: ?l1) ?l2) =>
-      match eval compute in n with
-      | 1 => fail
-      | _ =>
-          let l1' := constr:(l1 ++ a :: nil) in
-          (apply (permut_trans A (a :: l1) l1' l2);
-            [ apply permut_append | compute; Permut (pred n) ])
-      end
-  end.
-Ltac PermutProve :=
-  match goal with
-  | |- (permut _ ?l1 ?l2) =>
-      match eval compute in (length l1 = length l2) with
-      | (?n = ?n) => Permut n
-      end
-  end.
-\end{coq_example}
-\end{centerframe}
-\caption{Permutation tactic}
-\label{permutltac}
-\end{figure}
-
-Next, we can write naturally the tactic and the result can be seen on
-Figure~\ref{permutltac}. We can notice that we use two toplevel
-definitions {\tt PermutProve} and {\tt Permut}. The function to be
-called is {\tt PermutProve} which computes the lengths of the two
-lists and calls {\tt Permut} with the length if the two lists have the
-same length. {\tt Permut} works as expected.  If the two lists are
-equal, it concludes. Otherwise, if the lists have identical first
-elements, it applies {\tt Permut} on the tail of the lists.  Finally,
-if the lists have different first elements, it puts the first element
-of one of the lists (here the second one which appears in the {\tt
-  permut} predicate) at the end if that is possible, i.e., if the new
-first element has been at this place previously. To verify that all
-rotations have been done for a list, we use the length of the list as
-an argument for {\tt Permut} and this length is decremented for each
-rotation down to, but not including, 1 because for a list of length
-$n$, we can make exactly $n-1$ rotations to generate at most $n$
-distinct lists. Here, it must be noticed that we use the natural
-numbers of {\Coq} for the rotation counter. On Figure~\ref{ltac}, we
-can see that it is possible to use usual natural numbers but they are
-only used as arguments for primitive tactics and they cannot be
-handled, in particular, we cannot make computations with them. So, a
-natural choice is to use {\Coq} data structures so that {\Coq} makes
-the computations (reductions) by {\tt eval compute in} and we can get
-the terms back by {\tt match}.
- 
-With {\tt PermutProve}, we can now prove lemmas as 
-% shown on Figure~\ref{permutlem}.
-follows:
-%\begin{figure}
-%\begin{centerframe}
-
-\begin{coq_example*}
-Lemma permut_ex1 :
-  permut nat (1 :: 2 :: 3 :: nil) (3 :: 2 :: 1 :: nil).
-Proof. PermutProve. Qed.
-Lemma permut_ex2 :
-  permut nat
-    (0 :: 1 :: 2 :: 3 :: 4 :: 5 :: 6 :: 7 :: 8 :: 9 :: nil)
-    (0 :: 2 :: 4 :: 6 :: 8 :: 9 :: 7 :: 5 :: 3 :: 1 :: nil).
-Proof. PermutProve. Qed.
-\end{coq_example*}
-%\end{centerframe}
-%\caption{Examples of {\tt PermutProve} use}
-%\label{permutlem}
-%\end{figure}
-
-
-\subsection{Deciding intuitionistic propositional logic}
-
-\begin{figure}[b]
-\begin{centerframe}
-\begin{coq_example}
-Ltac Axioms :=
-  match goal with
-  | |- True => trivial
-  | _:False |- _  => elimtype False; assumption
-  | _:?A |- ?A  => auto
-  end.
-\end{coq_example}
-\end{centerframe}
-\caption{Deciding intuitionistic propositions (1)}
-\label{tautoltaca}
-\end{figure}
-
-
-\begin{figure}
-\begin{centerframe}
-\begin{coq_example}
-Ltac DSimplif :=
-  repeat
-   (intros;
-    match goal with
-     | id:(~ _) |- _ => red in id
-     | id:(_ /\ _) |- _ =>
-         elim id; do 2 intro; clear id
-     | id:(_ \/ _) |- _ =>
-         elim id; intro; clear id
-     | id:(?A /\ ?B -> ?C) |- _ =>
-         cut (A -> B -> C);
-          [ intro | intros; apply id; split; assumption ]
-     | id:(?A \/ ?B -> ?C) |- _ =>
-         cut (B -> C);
-          [ cut (A -> C);
-             [ intros; clear id
-             | intro; apply id; left; assumption ]
-          | intro; apply id; right; assumption ]
-     | id0:(?A -> ?B),id1:?A |- _ =>
-         cut B; [ intro; clear id0 | apply id0; assumption ]
-     | |- (_ /\ _) => split
-     | |- (~ _) => red
-     end).
-Ltac TautoProp :=
-  DSimplif;
-   Axioms ||
-     match goal with
-     | id:((?A -> ?B) -> ?C) |- _ =>
-          cut (B -> C);
-          [ intro; cut (A -> B);
-             [ intro; cut C;
-                [ intro; clear id | apply id; assumption ]
-             | clear id ]
-          | intro; apply id; intro; assumption ]; TautoProp
-     | id:(~ ?A -> ?B) |- _ =>
-         cut (False -> B);
-          [ intro; cut (A -> False);
-             [ intro; cut B;
-                [ intro; clear id | apply id; assumption ]
-             | clear id ]
-          | intro; apply id; red; intro; assumption ]; TautoProp
-     | |- (_ \/ _) => (left; TautoProp) || (right; TautoProp)
-     end.
-\end{coq_example}
-\end{centerframe}
-\caption{Deciding intuitionistic propositions (2)}
-\label{tautoltacb}
-\end{figure}
-
-The pattern matching on goals allows a complete and so a powerful
-backtracking when returning tactic values. An interesting application
-is the problem of deciding intuitionistic propositional logic.
-Considering the contraction-free sequent calculi {\tt LJT*} of
-Roy~Dyckhoff (\cite{Dyc92}), it is quite natural to code such a tactic
-using the tactic language as shown on Figures~\ref{tautoltaca}
-and~\ref{tautoltacb}. The tactic {\tt Axioms} tries to conclude using
-usual axioms. The tactic {\tt DSimplif} applies all the reversible
-rules of Dyckhoff's system. Finally, the tactic {\tt TautoProp} (the
-main tactic to be called) simplifies with {\tt DSimplif}, tries to
-conclude with {\tt Axioms} and tries several paths using the
-backtracking rules (one of the four Dyckhoff's rules for the left
-implication to get rid of the contraction and the right or).
-
-For example, with {\tt TautoProp}, we can prove tautologies like
- those:
-% on Figure~\ref{tautolem}.
-%\begin{figure}[tbp]
-%\begin{centerframe}
-\begin{coq_example*}
-Lemma tauto_ex1 : forall A B:Prop, A /\ B -> A \/ B.
-Proof. TautoProp. Qed.
-Lemma tauto_ex2 :
-   forall A B:Prop, (~ ~ B -> B) -> (A -> B) -> ~ ~ A -> B.
-Proof. TautoProp. Qed.
-\end{coq_example*}
-%\end{centerframe}
-%\caption{Proofs of tautologies with {\tt TautoProp}}
-%\label{tautolem}
-%\end{figure}
-
-\subsection{Deciding type isomorphisms}
-
-A more tricky problem is to decide equalities between types and modulo
-isomorphisms. Here, we choose to use the isomorphisms of the simply typed
-$\lb{}$-calculus with Cartesian product and $unit$ type (see, for example,
-\cite{RC95}). The axioms of this $\lb{}$-calculus are given by
-table~\ref{isosax}.
-
-\begin{figure}
-\begin{centerframe}
-\begin{coq_eval}
-Reset Initial.
-\end{coq_eval}
-\begin{coq_example*}
-Open Scope type_scope.
-Section Iso_axioms.
-Variables A B C : Set.
-Axiom Com : A * B = B * A.
-Axiom Ass : A * (B * C) = A * B * C.
-Axiom Cur : (A * B -> C) = (A -> B -> C).
-Axiom Dis : (A -> B * C) = (A -> B) * (A -> C).
-Axiom P_unit : A * unit = A.
-Axiom AR_unit : (A -> unit) = unit.
-Axiom AL_unit : (unit -> A) = A.
-Lemma Cons : B = C -> A * B = A * C.
-Proof.
-intro Heq; rewrite Heq; reflexivity.
-Qed.
-End Iso_axioms.
-\end{coq_example*}
-\end{centerframe}
-\caption{Type isomorphism axioms}
-\label{isosax}
-\end{figure}
-
-A more tricky problem is to decide equalities between types and modulo
-isomorphisms. Here, we choose to use the isomorphisms of the simply typed
-$\lb{}$-calculus with Cartesian product and $unit$ type (see, for example,
-\cite{RC95}). The axioms of this $\lb{}$-calculus are given on
-Figure~\ref{isosax}.
-
-\begin{figure}[ht]
-\begin{centerframe}
-\begin{coq_example}
-Ltac DSimplif trm :=
-  match trm with
-  | (?A * ?B * ?C) =>
-      rewrite <- (Ass A B C); try MainSimplif
-  | (?A * ?B -> ?C) =>
-      rewrite (Cur A B C); try MainSimplif
-  | (?A -> ?B * ?C) =>
-      rewrite (Dis A B C); try MainSimplif
-  | (?A * unit) =>
-      rewrite (P_unit A); try MainSimplif
-  | (unit * ?B) =>
-      rewrite (Com unit B); try MainSimplif
-  | (?A -> unit) =>
-      rewrite (AR_unit A); try MainSimplif
-  | (unit -> ?B) =>
-      rewrite (AL_unit B); try MainSimplif
-  | (?A * ?B) =>
-      (DSimplif A; try MainSimplif) || (DSimplif B; try MainSimplif)
-  | (?A -> ?B) =>
-      (DSimplif A; try MainSimplif) || (DSimplif B; try MainSimplif)
-  end
- with MainSimplif :=
-  match goal with
-  | |- (?A = ?B) => try DSimplif A; try DSimplif B
-  end.
-Ltac Length trm :=
-  match trm with
-  | (_ * ?B) => let succ := Length B in constr:(S succ)
-  | _ => constr:(1)
-  end.
-Ltac assoc := repeat rewrite <- Ass.
-\end{coq_example}
-\end{centerframe}
-\caption{Type isomorphism tactic (1)}
-\label{isosltac1}
-\end{figure}
-
-\begin{figure}[ht]
-\begin{centerframe}
-\begin{coq_example}
-Ltac DoCompare n :=
-  match goal with
-  | [ |- (?A = ?A) ] => reflexivity
-  | [ |- (?A * ?B = ?A * ?C) ] =>
-      apply Cons; let newn := Length B in
-                  DoCompare newn
-  | [ |- (?A * ?B = ?C) ] =>
-      match eval compute in n with
-      | 1 => fail
-      | _ =>
-          pattern (A * B) at 1; rewrite Com; assoc; DoCompare (pred n)
-      end
-  end.
-Ltac CompareStruct :=
-  match goal with
-  | [ |- (?A = ?B) ] =>
-      let l1 := Length A
-      with l2 := Length B in
-      match eval compute in (l1 = l2) with
-      | (?n = ?n) => DoCompare n
-      end
-  end.
-Ltac IsoProve := MainSimplif; CompareStruct.
-\end{coq_example}
-\end{centerframe}
-\caption{Type isomorphism tactic (2)}
-\label{isosltac2}
-\end{figure}
-
-The tactic to judge equalities modulo this axiomatization can be written as
-shown on Figures~\ref{isosltac1} and~\ref{isosltac2}. The algorithm is quite
-simple. Types are reduced using axioms that can be oriented (this done by {\tt
-MainSimplif}). The normal forms are sequences of Cartesian
-products without Cartesian product in the left component. These normal forms
-are then compared modulo permutation of the components (this is done by {\tt
-CompareStruct}). The main tactic to be called and realizing this algorithm is
-{\tt IsoProve}.
-
-% Figure~\ref{isoslem} gives 
-Here are examples of what can be solved by {\tt IsoProve}.
-%\begin{figure}[ht]
-%\begin{centerframe}
-\begin{coq_example*}
-Lemma isos_ex1 : 
-  forall A B:Set, A * unit * B = B * (unit * A).
-Proof.
-intros; IsoProve.
-Qed.
-
-Lemma isos_ex2 :
-  forall A B C:Set,
-    (A * unit -> B * (C * unit)) =
-    (A * unit -> (C -> unit) * C) * (unit -> A -> B).
-Proof.
-intros; IsoProve.
-Qed.
-\end{coq_example*}
-%\end{centerframe}
-%\caption{Type equalities solved by {\tt IsoProve}}
-%\label{isoslem}
-%\end{figure}
-
-%%% Local Variables: 
-%%% mode: latex
-%%% TeX-master: "Reference-Manual"
-%%% End: 
diff --git a/doc/refman/Reference-Manual.tex b/doc/refman/Reference-Manual.tex
index 85364870b..e63e6731e 100644
--- a/doc/refman/Reference-Manual.tex
+++ b/doc/refman/Reference-Manual.tex
@@ -102,7 +102,6 @@ Options A and B of the licence are {\em not} elected.}
 \include{RefMan-oth.v}% Vernacular commands
 \include{RefMan-pro.v}% Proof handling
 \include{RefMan-ltac.v}% Writing tactics
-\include{RefMan-tacex.v}% Detailed Examples of tactics
 
 \lstset{language=SSR}
 \lstset{moredelim=[is][]{|*}{*|}}
diff --git a/doc/sphinx/proof-engine/detailed-tactic-examples.rst b/doc/sphinx/proof-engine/detailed-tactic-examples.rst
new file mode 100644
index 000000000..7cdb1a527
--- /dev/null
+++ b/doc/sphinx/proof-engine/detailed-tactic-examples.rst
@@ -0,0 +1,930 @@
+\chapter[Detailed examples of tactics]{Detailed examples of tactics\label{Tactics-examples}}
+%HEVEA\cutname{tactic-examples.html}
+
+This chapter presents detailed examples of certain tactics, to
+illustrate their behavior.
+
+\section[\tt dependent induction]{\tt dependent induction\label{dependent-induction-example}}
+\def\depind{{\tt dependent induction}~}
+\def\depdestr{{\tt dependent destruction}~}
+
+The tactics \depind and \depdestr are another solution for inverting
+inductive predicate instances and potentially doing induction at the
+same time. It is based on the \texttt{BasicElim} tactic of Conor McBride which
+works by abstracting each argument of an inductive instance by a variable
+and constraining it by equalities afterwards. This way, the usual 
+{\tt induction} and {\tt destruct} tactics can be applied to the
+abstracted instance and after simplification of the equalities we get
+the expected goals.
+
+The abstracting tactic is called {\tt generalize\_eqs} and it takes as
+argument an hypothesis to generalize. It uses the {\tt JMeq} datatype
+defined in {\tt Coq.Logic.JMeq}, hence we need to require it before.
+For example, revisiting the first example of the inversion documentation above:
+
+\begin{coq_example*}
+Require Import Coq.Logic.JMeq.
+\end{coq_example*}
+\begin{coq_eval}
+Require Import Coq.Program.Equality.
+\end{coq_eval}
+
+\begin{coq_eval}
+Inductive Le : nat -> nat -> Set :=
+  | LeO : forall n:nat, Le 0 n
+  | LeS : forall n m:nat, Le n m -> Le (S n) (S m).
+Variable P : nat -> nat -> Prop.
+Variable Q : forall n m:nat, Le n m -> Prop.
+\end{coq_eval}
+
+\begin{coq_example*}
+Goal forall n m:nat, Le (S n) m -> P n m.
+intros n m H.
+\end{coq_example*}
+\begin{coq_example}
+generalize_eqs H.
+\end{coq_example}
+
+The index {\tt S n} gets abstracted by a variable here, but a
+corresponding equality is added under the abstract instance so that no
+information is actually lost. The goal is now almost amenable to do induction
+or case analysis. One should indeed first move {\tt n} into the goal to
+strengthen it before doing induction, or {\tt n} will be fixed in
+the inductive hypotheses (this does not matter for case analysis). 
+As a rule of thumb, all the variables that appear inside constructors in
+the indices of the hypothesis should be generalized. This is exactly
+what the \texttt{generalize\_eqs\_vars} variant does:
+
+\begin{coq_eval} 
+Undo 1.
+\end{coq_eval}
+\begin{coq_example}
+generalize_eqs_vars H.
+induction H.
+\end{coq_example}
+
+As the hypothesis itself did not appear in the goal, we did not need to
+use an heterogeneous equality to relate the new hypothesis to the old
+one (which just disappeared here). However, the tactic works just as well
+in this case, e.g.:
+
+\begin{coq_eval}
+Admitted.
+\end{coq_eval}
+
+\begin{coq_example}
+Goal forall n m (p : Le (S n) m), Q (S n) m p.
+intros n m p ; generalize_eqs_vars p.
+\end{coq_example}
+
+One drawback of this approach is that in the branches one will have to
+substitute the equalities back into the instance to get the right
+assumptions. Sometimes injection of constructors will also be needed to
+recover the needed equalities. Also, some subgoals should be directly
+solved because of inconsistent contexts arising from the constraints on 
+indexes. The nice thing is that we can make a tactic based on
+discriminate, injection and variants of substitution to automatically 
+do such simplifications (which may involve the K axiom). 
+This is what the {\tt simplify\_dep\_elim} tactic from
+{\tt Coq.Program.Equality} does. For example, we might simplify the
+previous goals considerably:
+% \begin{coq_eval} 
+% Abort.
+% Goal forall n m:nat, Le (S n) m -> P n m.
+% intros n m H ; generalize_eqs_vars H.
+% \end{coq_eval}
+
+\begin{coq_example}
+induction p ; simplify_dep_elim.
+\end{coq_example}
+
+The higher-order tactic {\tt do\_depind} defined in {\tt
+  Coq.Program.Equality} takes a tactic and combines the
+building blocks we have seen with it: generalizing by equalities
+calling the given tactic with the
+generalized induction hypothesis as argument and cleaning the subgoals
+with respect to equalities. Its most important instantiations are
+\depind and \depdestr that do induction or simply case analysis on the
+generalized hypothesis. For example we can redo what we've done manually
+with \depdestr:
+
+\begin{coq_eval}
+Abort.
+\end{coq_eval}
+\begin{coq_example*}
+Require Import Coq.Program.Equality.
+Lemma ex : forall n m:nat, Le (S n) m -> P n m.
+intros n m H.
+\end{coq_example*}
+\begin{coq_example}
+dependent destruction H.
+\end{coq_example}
+\begin{coq_eval}
+Abort.
+\end{coq_eval}
+
+This gives essentially the same result as inversion. Now if the
+destructed hypothesis actually appeared in the goal, the tactic would
+still be able to invert it, contrary to {\tt dependent
+ inversion}. Consider the following example on vectors:
+
+\begin{coq_example*}
+Require Import Coq.Program.Equality.
+Set Implicit Arguments.
+Variable A : Set.
+Inductive vector : nat -> Type := 
+| vnil : vector 0 
+| vcons : A -> forall n, vector n -> vector (S n).
+Goal forall n, forall v : vector (S n), 
+  exists v' : vector n, exists a : A, v = vcons a v'.
+  intros n v.
+\end{coq_example*}
+\begin{coq_example}
+  dependent destruction v.
+\end{coq_example}
+\begin{coq_eval}
+Abort.
+\end{coq_eval}
+
+In this case, the {\tt v} variable can be replaced in the goal by the
+generalized hypothesis only when it has a type of the form {\tt vector
+ (S n)}, that is only in the second case of the {\tt destruct}. The
+first one is dismissed because {\tt S n <> 0}.
+
+\subsection{A larger example}
+
+Let's see how the technique works with {\tt induction} on inductive
+predicates on a real example. We will develop an example application to the
+theory of simply-typed lambda-calculus formalized in a dependently-typed style:
+
+\begin{coq_example*}
+Inductive type : Type :=
+| base : type
+| arrow : type -> type -> type.
+Notation " t --> t' " := (arrow t t') (at level 20, t' at next level).
+Inductive ctx : Type :=
+| empty : ctx
+| snoc : ctx -> type -> ctx.
+Notation " G , tau " := (snoc G tau) (at level 20, tau at next level).
+Fixpoint conc (G D : ctx) : ctx :=
+  match D with
+    | empty => G
+    | snoc D' x => snoc (conc G D') x
+  end.
+Notation " G ; D " := (conc G D) (at level 20).
+Inductive term : ctx -> type -> Type :=
+| ax : forall G tau, term (G, tau) tau
+| weak : forall G tau, 
+  term G tau -> forall tau', term (G, tau') tau
+| abs : forall G tau tau', 
+  term (G , tau) tau' -> term G (tau --> tau')
+| app : forall G tau tau', 
+  term G (tau --> tau') -> term G tau -> term G tau'.
+\end{coq_example*}
+
+We have defined types and contexts which are snoc-lists of types. We
+also have a {\tt conc} operation that concatenates two contexts.
+The {\tt term} datatype represents in fact the possible typing
+derivations of the calculus, which are isomorphic to the well-typed
+terms, hence the name. A term is either an application of:
+\begin{itemize}
+\item the axiom rule to type a reference to the first variable in a context,
+\item the weakening rule to type an object in a larger context
+\item the abstraction or lambda rule to type a function
+\item the application to type an application of a function to an argument
+\end{itemize}
+
+Once we have this datatype we want to do proofs on it, like weakening:
+
+\begin{coq_example*}
+Lemma weakening : forall G D tau, term (G ; D) tau -> 
+  forall tau', term (G , tau' ; D) tau.
+\end{coq_example*}
+\begin{coq_eval}
+  Abort.
+\end{coq_eval}
+
+The problem here is that we can't just use {\tt induction} on the typing
+derivation because it will forget about the {\tt G ; D} constraint
+appearing in the instance. A solution would be to rewrite the goal as:
+\begin{coq_example*}
+Lemma weakening' : forall G' tau, term G' tau -> 
+  forall G D, (G ; D) = G' ->
+  forall tau', term (G, tau' ; D) tau.
+\end{coq_example*}
+\begin{coq_eval}
+  Abort.
+\end{coq_eval}
+
+With this proper separation of the index from the instance and the right
+induction loading (putting {\tt G} and {\tt D} after the inducted-on
+hypothesis), the proof will go through, but it is a very tedious
+process. One is also forced to make a wrapper lemma to get back the
+more natural statement. The \depind tactic alleviates this trouble by
+doing all of this plumbing of generalizing and substituting back automatically.
+Indeed we can simply write:
+
+\begin{coq_example*}
+Require Import Coq.Program.Tactics.
+Lemma weakening : forall G D tau, term (G ; D) tau -> 
+  forall tau', term (G , tau' ; D) tau.
+Proof with simpl in * ; simpl_depind ; auto.
+  intros G D tau H. dependent induction H generalizing G D ; intros.
+\end{coq_example*}
+
+This call to \depind has an additional arguments which is a list of
+variables appearing in the instance that should be generalized in the
+goal, so that they can vary in the induction hypotheses. By default, all
+variables appearing inside constructors (except in a parameter position)
+of the instantiated hypothesis will be generalized automatically but
+one can always give the list explicitly.
+
+\begin{coq_example}
+  Show.
+\end{coq_example}
+
+The {\tt simpl\_depind} tactic includes an automatic tactic that tries
+to simplify equalities appearing at the beginning of induction
+hypotheses, generally using trivial applications of
+reflexivity. In cases where the equality is not between constructor
+forms though, one must help the automation by giving
+some arguments, using the {\tt specialize} tactic for example.
+
+\begin{coq_example*}
+destruct D... apply weak ; apply ax. apply ax.
+destruct D...
+\end{coq_example*}
+\begin{coq_example}
+Show.
+\end{coq_example}
+\begin{coq_example}
+  specialize (IHterm G0 empty eq_refl).
+\end{coq_example}
+
+Once the induction hypothesis has been narrowed to the right equality,
+it can be used directly. 
+
+\begin{coq_example}
+  apply weak, IHterm.
+\end{coq_example}
+
+If there is an easy first-order solution to these equations as in this subgoal, the
+{\tt specialize\_eqs} tactic can be used instead of giving explicit proof
+terms:
+
+\begin{coq_example}
+  specialize_eqs IHterm.
+\end{coq_example}
+This concludes our example.
+\SeeAlso The induction \ref{elim}, case \ref{case} and inversion \ref{inversion} tactics.
+
+\section[\tt autorewrite]{\tt autorewrite\label{autorewrite-example}}
+
+Here are two examples of {\tt autorewrite} use. The first one ({\em Ackermann
+function}) shows actually a quite basic use where there is no conditional
+rewriting. The second one ({\em Mac Carthy function}) involves conditional
+rewritings and shows how to deal with them using the optional tactic of the
+{\tt Hint~Rewrite} command.
+
+\firstexample
+\example{Ackermann function}
+%Here is a basic use of {\tt AutoRewrite} with the Ackermann function:
+
+\begin{coq_example*}
+Reset Initial.
+Require Import Arith.
+Variable Ack : 
+           nat -> nat -> nat.
+Axiom Ack0 : 
+        forall m:nat, Ack 0 m = S m.
+Axiom Ack1 : forall n:nat, Ack (S n) 0 = Ack n 1.
+Axiom Ack2 : forall n m:nat, Ack (S n) (S m) = Ack n (Ack (S n) m).
+\end{coq_example*}
+
+\begin{coq_example}
+Hint Rewrite Ack0 Ack1 Ack2 : base0.
+Lemma ResAck0 : 
+ Ack 3 2 = 29.
+autorewrite with base0 using try reflexivity.
+\end{coq_example}
+
+\begin{coq_eval}
+Reset Initial.
+\end{coq_eval}
+
+\example{Mac Carthy function}
+%The Mac Carthy function shows a more complex case:
+
+\begin{coq_example*}
+Require Import Omega.
+Variable g :   
+           nat -> nat -> nat.
+Axiom g0 : 
+        forall m:nat, g 0 m = m.
+Axiom
+  g1 :
+    forall n m:nat,
+      (n > 0) -> (m > 100) -> g n m = g (pred n) (m - 10).
+Axiom
+  g2 :
+    forall n m:nat,
+      (n > 0) -> (m <= 100) -> g n m = g (S n) (m + 11).
+\end{coq_example*}
+
+\begin{coq_example}
+Hint Rewrite g0 g1 g2 using omega : base1.
+Lemma Resg0 : 
+ g 1 110 = 100.
+autorewrite with base1 using reflexivity || simpl.
+\end{coq_example}
+
+\begin{coq_eval}
+Abort.
+\end{coq_eval}
+
+\begin{coq_example}
+Lemma Resg1 : g 1 95 = 91.
+autorewrite with base1 using reflexivity || simpl.
+\end{coq_example}
+
+\begin{coq_eval}
+Reset Initial.
+\end{coq_eval}
+
+\section[\tt quote]{\tt quote\tacindex{quote}
+\label{quote-examples}}
+
+The tactic \texttt{quote} allows using Barendregt's so-called
+2-level approach without writing any ML code. Suppose you have a
+language \texttt{L} of 
+'abstract terms' and a type \texttt{A} of 'concrete terms' 
+and a function \texttt{f : L -> A}. If \texttt{L} is a simple
+inductive datatype and \texttt{f} a simple fixpoint, \texttt{quote f}
+will replace the head of current goal by a convertible term of the form 
+\texttt{(f t)}. \texttt{L} must have a constructor of type: \texttt{A
+  -> L}. 
+
+Here is an example:
+
+\begin{coq_example}
+Require Import Quote.
+Parameters A B C : Prop.
+Inductive formula : Type :=
+  | f_and : formula -> formula -> formula (* binary constructor *)
+  | f_or : formula -> formula -> formula
+  | f_not : formula -> formula (* unary constructor *)
+  | f_true : formula (* 0-ary constructor *)
+  | f_const : Prop -> formula (* constructor for constants *).
+Fixpoint interp_f (f:
+                   formula) : Prop :=
+  match f with
+  | f_and f1 f2 => interp_f f1 /\ interp_f f2
+  | f_or f1 f2 => interp_f f1 \/ interp_f f2
+  | f_not f1 => ~ interp_f f1
+  | f_true => True
+  | f_const c => c
+  end.
+Goal A /\ (A \/ True) /\ ~ B /\ (A <-> A).
+quote interp_f.
+\end{coq_example}
+
+The algorithm to perform this inversion is: try to match the
+term with right-hand sides expression of \texttt{f}. If there is a
+match, apply the corresponding left-hand side and call yourself
+recursively on sub-terms. If there is no match, we are at a leaf:
+return the corresponding constructor (here \texttt{f\_const}) applied
+to the term. 
+
+\begin{ErrMsgs}
+\item \errindex{quote: not a simple fixpoint} \\
+  Happens when \texttt{quote} is not able to perform inversion properly.
+\end{ErrMsgs}
+
+\subsection{Introducing variables map}
+
+The normal use of \texttt{quote} is to make proofs by reflection: one
+defines a function \texttt{simplify : formula -> formula} and proves a 
+theorem \texttt{simplify\_ok: (f:formula)(interp\_f (simplify f)) ->
+  (interp\_f f)}. Then, one can simplify formulas by doing:
+\begin{verbatim}
+   quote interp_f.
+   apply simplify_ok.
+   compute.
+\end{verbatim}
+But there is a problem with leafs: in the example above one cannot
+write a function that implements, for example, the logical simplifications 
+$A \land A \ra A$ or $A \land \lnot A \ra \texttt{False}$. This is
+because the \Prop{} is impredicative.
+
+It is better to use that type of formulas:
+
+\begin{coq_eval}
+Reset formula.
+\end{coq_eval}
+\begin{coq_example}
+Inductive formula : Set :=
+  | f_and : formula -> formula -> formula
+  | f_or : formula -> formula -> formula
+  | f_not : formula -> formula
+  | f_true : formula
+  | f_atom : index -> formula.
+\end{coq_example*}
+
+\texttt{index} is defined in module \texttt{quote}. Equality on that
+type is decidable so we are able to simplify $A \land A$ into $A$ at
+the abstract level. 
+
+When there are variables, there are bindings, and \texttt{quote}
+provides also a type \texttt{(varmap A)} of bindings from
+\texttt{index} to any set \texttt{A}, and a function
+\texttt{varmap\_find} to search in such maps. The interpretation
+function has now another argument, a variables map:
+
+\begin{coq_example}
+Fixpoint interp_f (vm:
+                    varmap Prop) (f:formula) {struct f} : Prop :=
+  match f with
+  | f_and f1 f2 => interp_f vm f1 /\ interp_f vm f2
+  | f_or f1 f2 => interp_f vm f1 \/ interp_f vm f2
+  | f_not f1 => ~ interp_f vm f1
+  | f_true => True
+  | f_atom i => varmap_find True i vm
+  end.
+\end{coq_example}
+
+\noindent\texttt{quote} handles this second case properly:
+
+\begin{coq_example}
+Goal A /\ (B \/ A) /\ (A \/ ~ B).
+quote interp_f.
+\end{coq_example}
+
+It builds \texttt{vm} and \texttt{t} such that \texttt{(f vm t)} is
+convertible with the conclusion of current goal.
+
+\subsection{Combining variables and constants}
+
+One can have both variables and constants in abstracts terms; that is
+the case, for example, for the \texttt{ring} tactic (chapter
+\ref{ring}). Then one must provide to \texttt{quote} a list of
+\emph{constructors of constants}. For example, if the list is
+\texttt{[O S]} then closed natural numbers will be considered as
+constants and other terms as variables. 
+
+Example: 
+
+\begin{coq_eval}
+Reset formula.
+\end{coq_eval}
+\begin{coq_example*}
+Inductive formula : Type :=
+  | f_and : formula -> formula -> formula
+  | f_or : formula -> formula -> formula
+  | f_not : formula -> formula
+  | f_true : formula
+  | f_const : Prop -> formula (* constructor for constants *)
+  | f_atom : index -> formula.
+Fixpoint interp_f
+ (vm:            (* constructor for variables *)
+  varmap Prop) (f:formula) {struct f} : Prop :=
+  match f with
+  | f_and f1 f2 => interp_f vm f1 /\ interp_f vm f2
+  | f_or f1 f2 => interp_f vm f1 \/ interp_f vm f2
+  | f_not f1 => ~ interp_f vm f1
+  | f_true => True
+  | f_const c => c
+  | f_atom i => varmap_find True i vm
+  end.
+Goal 
+A /\ (A \/ True) /\ ~ B /\ (C <-> C).
+\end{coq_example*}
+
+\begin{coq_example}
+quote interp_f [ A B ].
+Undo.
+  quote interp_f [ B C iff ].
+\end{coq_example}
+
+\Warning Since function inversion
+is undecidable in general case, don't expect miracles from it!
+
+\begin{Variants}
+
+\item {\tt quote {\ident} in {\term} using {\tac}}
+
+  \tac\ must be a functional tactic (starting with {\tt fun x =>})
+  and will be called with the quoted version of \term\ according to
+  \ident.
+
+\item {\tt quote {\ident} [ \ident$_1$ \dots\ \ident$_n$ ] in {\term} using {\tac}}
+
+  Same as above, but will use \ident$_1$, \dots, \ident$_n$ to
+  chose which subterms are constants (see above).
+
+\end{Variants}
+
+% \SeeAlso file \texttt{theories/DEMOS/DemoQuote.v}
+
+\SeeAlso comments of source file \texttt{plugins/quote/quote.ml}
+
+\SeeAlso the \texttt{ring} tactic (Chapter~\ref{ring})
+
+
+
+\section{Using the tactical language}
+
+\subsection{About the cardinality of the set of natural numbers}
+
+A first example which shows how to use the pattern matching over the proof
+contexts is the proof that natural numbers have more than two elements. The
+proof of such a lemma can be done as %shown on Figure~\ref{cnatltac}.
+follows:
+%\begin{figure}
+%\begin{centerframe}
+\begin{coq_eval}
+Reset Initial.
+Require Import Arith.
+Require Import List.
+\end{coq_eval}
+\begin{coq_example*}
+Lemma card_nat :
+ ~ (exists x : nat, exists y : nat, forall z:nat, x = z \/ y = z).
+Proof.
+red; intros (x, (y, Hy)).
+elim (Hy 0); elim (Hy 1); elim (Hy 2); intros;
+ match goal with
+ | [_:(?a = ?b),_:(?a = ?c) |- _ ] =>
+     cut (b = c); [ discriminate | transitivity a; auto ]
+ end.
+Qed.
+\end{coq_example*}
+%\end{centerframe}
+%\caption{A proof on cardinality of natural numbers}
+%\label{cnatltac}
+%\end{figure}
+
+We can notice that all the (very similar) cases coming from the three
+eliminations (with three distinct natural numbers) are successfully solved by
+a {\tt match goal} structure and, in particular, with only one pattern (use
+of non-linear matching).
+
+\subsection{Permutation on closed lists}
+
+Another more complex example is the problem of permutation on closed lists. The
+aim is to show that a closed list is a permutation of another one.
+
+First, we define the permutation predicate as shown in table~\ref{permutpred}.
+
+\begin{figure}
+\begin{centerframe}
+\begin{coq_example*}
+Section Sort.
+Variable A : Set.
+Inductive permut : list A -> list A -> Prop :=
+  | permut_refl   : forall l, permut l l
+  | permut_cons   :
+      forall a l0 l1, permut l0 l1 -> permut (a :: l0) (a :: l1)
+  | permut_append : forall a l, permut (a :: l) (l ++ a :: nil)
+  | permut_trans  :
+      forall l0 l1 l2, permut l0 l1 -> permut l1 l2 -> permut l0 l2.
+End Sort.
+\end{coq_example*}
+\end{centerframe}
+\caption{Definition of the permutation predicate}
+\label{permutpred}
+\end{figure}
+
+A more complex example is the problem of permutation on closed lists.
+The aim is to show that a closed list is a permutation of another one.
+First, we define the permutation predicate as shown on
+Figure~\ref{permutpred}.
+
+\begin{figure}
+\begin{centerframe}
+\begin{coq_example}
+Ltac Permut n :=
+  match goal with
+  | |- (permut _ ?l ?l) => apply permut_refl
+  | |- (permut _ (?a :: ?l1) (?a :: ?l2)) =>
+      let newn := eval compute in (length l1) in
+      (apply permut_cons; Permut newn)
+  | |- (permut ?A (?a :: ?l1) ?l2) =>
+      match eval compute in n with
+      | 1 => fail
+      | _ =>
+          let l1' := constr:(l1 ++ a :: nil) in
+          (apply (permut_trans A (a :: l1) l1' l2);
+            [ apply permut_append | compute; Permut (pred n) ])
+      end
+  end.
+Ltac PermutProve :=
+  match goal with
+  | |- (permut _ ?l1 ?l2) =>
+      match eval compute in (length l1 = length l2) with
+      | (?n = ?n) => Permut n
+      end
+  end.
+\end{coq_example}
+\end{centerframe}
+\caption{Permutation tactic}
+\label{permutltac}
+\end{figure}
+
+Next, we can write naturally the tactic and the result can be seen on
+Figure~\ref{permutltac}. We can notice that we use two toplevel
+definitions {\tt PermutProve} and {\tt Permut}. The function to be
+called is {\tt PermutProve} which computes the lengths of the two
+lists and calls {\tt Permut} with the length if the two lists have the
+same length. {\tt Permut} works as expected.  If the two lists are
+equal, it concludes. Otherwise, if the lists have identical first
+elements, it applies {\tt Permut} on the tail of the lists.  Finally,
+if the lists have different first elements, it puts the first element
+of one of the lists (here the second one which appears in the {\tt
+  permut} predicate) at the end if that is possible, i.e., if the new
+first element has been at this place previously. To verify that all
+rotations have been done for a list, we use the length of the list as
+an argument for {\tt Permut} and this length is decremented for each
+rotation down to, but not including, 1 because for a list of length
+$n$, we can make exactly $n-1$ rotations to generate at most $n$
+distinct lists. Here, it must be noticed that we use the natural
+numbers of {\Coq} for the rotation counter. On Figure~\ref{ltac}, we
+can see that it is possible to use usual natural numbers but they are
+only used as arguments for primitive tactics and they cannot be
+handled, in particular, we cannot make computations with them. So, a
+natural choice is to use {\Coq} data structures so that {\Coq} makes
+the computations (reductions) by {\tt eval compute in} and we can get
+the terms back by {\tt match}.
+ 
+With {\tt PermutProve}, we can now prove lemmas as 
+% shown on Figure~\ref{permutlem}.
+follows:
+%\begin{figure}
+%\begin{centerframe}
+
+\begin{coq_example*}
+Lemma permut_ex1 :
+  permut nat (1 :: 2 :: 3 :: nil) (3 :: 2 :: 1 :: nil).
+Proof. PermutProve. Qed.
+Lemma permut_ex2 :
+  permut nat
+    (0 :: 1 :: 2 :: 3 :: 4 :: 5 :: 6 :: 7 :: 8 :: 9 :: nil)
+    (0 :: 2 :: 4 :: 6 :: 8 :: 9 :: 7 :: 5 :: 3 :: 1 :: nil).
+Proof. PermutProve. Qed.
+\end{coq_example*}
+%\end{centerframe}
+%\caption{Examples of {\tt PermutProve} use}
+%\label{permutlem}
+%\end{figure}
+
+
+\subsection{Deciding intuitionistic propositional logic}
+
+\begin{figure}[b]
+\begin{centerframe}
+\begin{coq_example}
+Ltac Axioms :=
+  match goal with
+  | |- True => trivial
+  | _:False |- _  => elimtype False; assumption
+  | _:?A |- ?A  => auto
+  end.
+\end{coq_example}
+\end{centerframe}
+\caption{Deciding intuitionistic propositions (1)}
+\label{tautoltaca}
+\end{figure}
+
+
+\begin{figure}
+\begin{centerframe}
+\begin{coq_example}
+Ltac DSimplif :=
+  repeat
+   (intros;
+    match goal with
+     | id:(~ _) |- _ => red in id
+     | id:(_ /\ _) |- _ =>
+         elim id; do 2 intro; clear id
+     | id:(_ \/ _) |- _ =>
+         elim id; intro; clear id
+     | id:(?A /\ ?B -> ?C) |- _ =>
+         cut (A -> B -> C);
+          [ intro | intros; apply id; split; assumption ]
+     | id:(?A \/ ?B -> ?C) |- _ =>
+         cut (B -> C);
+          [ cut (A -> C);
+             [ intros; clear id
+             | intro; apply id; left; assumption ]
+          | intro; apply id; right; assumption ]
+     | id0:(?A -> ?B),id1:?A |- _ =>
+         cut B; [ intro; clear id0 | apply id0; assumption ]
+     | |- (_ /\ _) => split
+     | |- (~ _) => red
+     end).
+Ltac TautoProp :=
+  DSimplif;
+   Axioms ||
+     match goal with
+     | id:((?A -> ?B) -> ?C) |- _ =>
+          cut (B -> C);
+          [ intro; cut (A -> B);
+             [ intro; cut C;
+                [ intro; clear id | apply id; assumption ]
+             | clear id ]
+          | intro; apply id; intro; assumption ]; TautoProp
+     | id:(~ ?A -> ?B) |- _ =>
+         cut (False -> B);
+          [ intro; cut (A -> False);
+             [ intro; cut B;
+                [ intro; clear id | apply id; assumption ]
+             | clear id ]
+          | intro; apply id; red; intro; assumption ]; TautoProp
+     | |- (_ \/ _) => (left; TautoProp) || (right; TautoProp)
+     end.
+\end{coq_example}
+\end{centerframe}
+\caption{Deciding intuitionistic propositions (2)}
+\label{tautoltacb}
+\end{figure}
+
+The pattern matching on goals allows a complete and so a powerful
+backtracking when returning tactic values. An interesting application
+is the problem of deciding intuitionistic propositional logic.
+Considering the contraction-free sequent calculi {\tt LJT*} of
+Roy~Dyckhoff (\cite{Dyc92}), it is quite natural to code such a tactic
+using the tactic language as shown on Figures~\ref{tautoltaca}
+and~\ref{tautoltacb}. The tactic {\tt Axioms} tries to conclude using
+usual axioms. The tactic {\tt DSimplif} applies all the reversible
+rules of Dyckhoff's system. Finally, the tactic {\tt TautoProp} (the
+main tactic to be called) simplifies with {\tt DSimplif}, tries to
+conclude with {\tt Axioms} and tries several paths using the
+backtracking rules (one of the four Dyckhoff's rules for the left
+implication to get rid of the contraction and the right or).
+
+For example, with {\tt TautoProp}, we can prove tautologies like
+ those:
+% on Figure~\ref{tautolem}.
+%\begin{figure}[tbp]
+%\begin{centerframe}
+\begin{coq_example*}
+Lemma tauto_ex1 : forall A B:Prop, A /\ B -> A \/ B.
+Proof. TautoProp. Qed.
+Lemma tauto_ex2 :
+   forall A B:Prop, (~ ~ B -> B) -> (A -> B) -> ~ ~ A -> B.
+Proof. TautoProp. Qed.
+\end{coq_example*}
+%\end{centerframe}
+%\caption{Proofs of tautologies with {\tt TautoProp}}
+%\label{tautolem}
+%\end{figure}
+
+\subsection{Deciding type isomorphisms}
+
+A more tricky problem is to decide equalities between types and modulo
+isomorphisms. Here, we choose to use the isomorphisms of the simply typed
+$\lb{}$-calculus with Cartesian product and $unit$ type (see, for example,
+\cite{RC95}). The axioms of this $\lb{}$-calculus are given by
+table~\ref{isosax}.
+
+\begin{figure}
+\begin{centerframe}
+\begin{coq_eval}
+Reset Initial.
+\end{coq_eval}
+\begin{coq_example*}
+Open Scope type_scope.
+Section Iso_axioms.
+Variables A B C : Set.
+Axiom Com : A * B = B * A.
+Axiom Ass : A * (B * C) = A * B * C.
+Axiom Cur : (A * B -> C) = (A -> B -> C).
+Axiom Dis : (A -> B * C) = (A -> B) * (A -> C).
+Axiom P_unit : A * unit = A.
+Axiom AR_unit : (A -> unit) = unit.
+Axiom AL_unit : (unit -> A) = A.
+Lemma Cons : B = C -> A * B = A * C.
+Proof.
+intro Heq; rewrite Heq; reflexivity.
+Qed.
+End Iso_axioms.
+\end{coq_example*}
+\end{centerframe}
+\caption{Type isomorphism axioms}
+\label{isosax}
+\end{figure}
+
+A more tricky problem is to decide equalities between types and modulo
+isomorphisms. Here, we choose to use the isomorphisms of the simply typed
+$\lb{}$-calculus with Cartesian product and $unit$ type (see, for example,
+\cite{RC95}). The axioms of this $\lb{}$-calculus are given on
+Figure~\ref{isosax}.
+
+\begin{figure}[ht]
+\begin{centerframe}
+\begin{coq_example}
+Ltac DSimplif trm :=
+  match trm with
+  | (?A * ?B * ?C) =>
+      rewrite <- (Ass A B C); try MainSimplif
+  | (?A * ?B -> ?C) =>
+      rewrite (Cur A B C); try MainSimplif
+  | (?A -> ?B * ?C) =>
+      rewrite (Dis A B C); try MainSimplif
+  | (?A * unit) =>
+      rewrite (P_unit A); try MainSimplif
+  | (unit * ?B) =>
+      rewrite (Com unit B); try MainSimplif
+  | (?A -> unit) =>
+      rewrite (AR_unit A); try MainSimplif
+  | (unit -> ?B) =>
+      rewrite (AL_unit B); try MainSimplif
+  | (?A * ?B) =>
+      (DSimplif A; try MainSimplif) || (DSimplif B; try MainSimplif)
+  | (?A -> ?B) =>
+      (DSimplif A; try MainSimplif) || (DSimplif B; try MainSimplif)
+  end
+ with MainSimplif :=
+  match goal with
+  | |- (?A = ?B) => try DSimplif A; try DSimplif B
+  end.
+Ltac Length trm :=
+  match trm with
+  | (_ * ?B) => let succ := Length B in constr:(S succ)
+  | _ => constr:(1)
+  end.
+Ltac assoc := repeat rewrite <- Ass.
+\end{coq_example}
+\end{centerframe}
+\caption{Type isomorphism tactic (1)}
+\label{isosltac1}
+\end{figure}
+
+\begin{figure}[ht]
+\begin{centerframe}
+\begin{coq_example}
+Ltac DoCompare n :=
+  match goal with
+  | [ |- (?A = ?A) ] => reflexivity
+  | [ |- (?A * ?B = ?A * ?C) ] =>
+      apply Cons; let newn := Length B in
+                  DoCompare newn
+  | [ |- (?A * ?B = ?C) ] =>
+      match eval compute in n with
+      | 1 => fail
+      | _ =>
+          pattern (A * B) at 1; rewrite Com; assoc; DoCompare (pred n)
+      end
+  end.
+Ltac CompareStruct :=
+  match goal with
+  | [ |- (?A = ?B) ] =>
+      let l1 := Length A
+      with l2 := Length B in
+      match eval compute in (l1 = l2) with
+      | (?n = ?n) => DoCompare n
+      end
+  end.
+Ltac IsoProve := MainSimplif; CompareStruct.
+\end{coq_example}
+\end{centerframe}
+\caption{Type isomorphism tactic (2)}
+\label{isosltac2}
+\end{figure}
+
+The tactic to judge equalities modulo this axiomatization can be written as
+shown on Figures~\ref{isosltac1} and~\ref{isosltac2}. The algorithm is quite
+simple. Types are reduced using axioms that can be oriented (this done by {\tt
+MainSimplif}). The normal forms are sequences of Cartesian
+products without Cartesian product in the left component. These normal forms
+are then compared modulo permutation of the components (this is done by {\tt
+CompareStruct}). The main tactic to be called and realizing this algorithm is
+{\tt IsoProve}.
+
+% Figure~\ref{isoslem} gives 
+Here are examples of what can be solved by {\tt IsoProve}.
+%\begin{figure}[ht]
+%\begin{centerframe}
+\begin{coq_example*}
+Lemma isos_ex1 : 
+  forall A B:Set, A * unit * B = B * (unit * A).
+Proof.
+intros; IsoProve.
+Qed.
+
+Lemma isos_ex2 :
+  forall A B C:Set,
+    (A * unit -> B * (C * unit)) =
+    (A * unit -> (C -> unit) * C) * (unit -> A -> B).
+Proof.
+intros; IsoProve.
+Qed.
+\end{coq_example*}
+%\end{centerframe}
+%\caption{Type equalities solved by {\tt IsoProve}}
+%\label{isoslem}
+%\end{figure}
+
+%%% Local Variables: 
+%%% mode: latex
+%%% TeX-master: "Reference-Manual"
+%%% End: 
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