Merge branch 'master' into sphinx-doc-chapter-21

3 files changed, 0 insertions, 1228 deletions

diff --git a/doc/refman/CanonicalStructures.tex b/doc/refman/CanonicalStructures.tex
deleted file mode 100644
index 8961b00964..0000000000
--- a/doc/refman/CanonicalStructures.tex
+++ /dev/null
@@ -1,383 +0,0 @@
-\achapter{Canonical Structures}
-%HEVEA\cutname{canonical-structures.html}
-\aauthor{Assia Mahboubi and Enrico Tassi}
-
-\label{CS-full}
-\index{Canonical Structures!presentation}
-
-\noindent This chapter explains the basics of Canonical Structure and how they can be used
-to overload notations and build a hierarchy of algebraic structures.
-The examples are taken from~\cite{CSwcu}.  We invite the interested reader
-to refer to this paper for all the details that are omitted here for brevity.
-The interested reader shall also find in~\cite{CSlessadhoc} a detailed
-description of another, complementary, use of Canonical Structures:
-advanced proof search.  This latter papers also presents many techniques one
-can employ to tune the inference of Canonical Structures.
-
-\section{Notation overloading}
-
-We build an infix notation $==$ for a comparison predicate.  Such notation
-will be overloaded, and its meaning will depend on the types of the terms
-that are compared.
-
-\begin{coq_eval}
-Require Import Arith.
-\end{coq_eval}
-
-\begin{coq_example}
-Module EQ.
-  Record class (T : Type) := Class { cmp : T -> T -> Prop }.
-  Structure type := Pack { obj : Type; class_of : class obj }.
-  Definition op (e : type) : obj e -> obj e -> Prop :=
-    let 'Pack _ (Class _ the_cmp) := e in the_cmp.
-  Check op.
-  Arguments op {e} x y : simpl never.
-  Arguments Class {T} cmp.
-  Module theory.
-    Notation "x == y" := (op x y) (at level 70).
-  End theory.
-End EQ.
-\end{coq_example}
-
-We use Coq modules as name spaces.  This allows us to follow the same pattern
-and naming convention for the rest of the chapter.  The base name space
-contains the definitions of the algebraic structure.  To keep the example
-small, the algebraic structure \texttt{EQ.type} we are defining is very simplistic,
-and characterizes terms on which a binary relation is defined, without
-requiring such relation to validate any property.
-The inner \texttt{theory} module contains the overloaded notation \texttt{==} and
-will eventually contain lemmas holding on all the instances of the
-algebraic structure (in this case there are no lemmas).
-
-Note that in practice the user may want to declare \texttt{EQ.obj} as a coercion,
-but we will not do that here.
-
-The following line tests that, when we assume a type \texttt{e} that is in the
-\texttt{EQ} class, then we can relates two of its objects with \texttt{==}.
-
-\begin{coq_example}
-Import EQ.theory.
-Check forall (e : EQ.type) (a b : EQ.obj e), a == b.
-\end{coq_example}
-
-Still, no concrete type is in the \texttt{EQ} class.  We amend that by equipping \texttt{nat}
-with a comparison relation.
-
-\begin{coq_example}
-Fail Check 3 == 3.
-Definition nat_eq (x y : nat) := nat_compare x y = Eq.
-Definition nat_EQcl : EQ.class nat := EQ.Class nat_eq.
-Canonical Structure nat_EQty : EQ.type := EQ.Pack nat nat_EQcl.
-Check 3 == 3.
-Eval compute in 3 == 4.
-\end{coq_example}
-
-This last test shows that Coq is now not only able to typecheck \texttt{3==3}, but
-also that the infix relation was bound to the \texttt{nat\_eq} relation.  This
-relation is selected whenever \texttt{==} is used on terms of type \texttt{nat}.  This
-can be read in the line declaring the canonical structure \texttt{nat\_EQty},
-where the first argument to \texttt{Pack} is the key and its second argument
-a group of canonical values associated to the key.  In this case we associate
-to \texttt{nat} only one canonical value (since its class, \texttt{nat\_EQcl} has just one
-member).  The use of the projection \texttt{op} requires its argument to be in
-the class \texttt{EQ}, and uses such a member (function) to actually compare
-its arguments.
-
-Similarly, we could equip any other type with a comparison relation, and
-use the \texttt{==} notation on terms of this type.
-
-\subsection{Derived Canonical Structures}
-
-We know how to use \texttt{==} on base types, like \texttt{nat}, \texttt{bool}, \texttt{Z}.
-Here we show how to deal with type constructors, i.e. how to make the
-following example work:
-
-\begin{coq_example}
-Fail Check forall (e : EQ.type) (a b : EQ.obj e), (a,b) == (a,b). 
-\end{coq_example}
-
-The error message is telling that Coq has no idea on how to compare
-pairs of objects.  The following construction is telling Coq exactly how to do
-that.
-
-\begin{coq_example}
-Definition pair_eq (e1 e2 : EQ.type) (x y : EQ.obj e1 * EQ.obj e2) :=
-  fst x == fst y /\ snd x == snd y.
-Definition pair_EQcl e1 e2 := EQ.Class (pair_eq e1 e2).
-Canonical Structure pair_EQty (e1 e2 : EQ.type) : EQ.type :=
-  EQ.Pack (EQ.obj e1 * EQ.obj e2) (pair_EQcl e1 e2).
-Check forall (e : EQ.type) (a b : EQ.obj e), (a,b) == (a,b).
-Check forall n m : nat, (3,4) == (n,m).
-\end{coq_example}
-
-Thanks to the \texttt{pair\_EQty} declaration, Coq is able to build a comparison
-relation for pairs whenever it is able to build a comparison relation
-for each component of the pair.  The declaration associates to the key
-\texttt{*} (the type constructor of pairs) the canonical comparison relation
-\texttt{pair\_eq} whenever the type constructor \texttt{*} is applied to two types
-being themselves in the \texttt{EQ} class.
-
-\section{Hierarchy of structures}
-
-To get to an interesting example we need another base class to be available.
-We choose the class of types that are equipped with an order relation,
-to which we associate the infix \texttt{<=} notation.
-
-\begin{coq_example}
-Module LE.
-  Record class T := Class { cmp : T -> T -> Prop }.
-  Structure type := Pack { obj : Type; class_of : class obj }.
-  Definition op (e : type) : obj e -> obj e -> Prop :=
-    let 'Pack _ (Class _ f) := e in f.
-  Arguments op {_} x y : simpl never.
-  Arguments Class {T} cmp.
-  Module theory.
-  Notation "x <= y" := (op x y) (at level 70).
-  End theory.
-End LE.
-\end{coq_example}
-
-As before we register a canonical \texttt{LE} class for \texttt{nat}.
-
-\begin{coq_example}
-Import LE.theory.
-Definition nat_le x y := nat_compare x y <> Gt.
-Definition nat_LEcl : LE.class nat := LE.Class nat_le.
-Canonical Structure nat_LEty : LE.type := LE.Pack nat nat_LEcl.
-\end{coq_example}
-
-And we enable Coq to relate pair of terms with \texttt{<=}.
-
-\begin{coq_example}
-Definition pair_le e1 e2 (x y : LE.obj e1 * LE.obj e2) :=
-  fst x <= fst y /\ snd x <= snd y.
-Definition pair_LEcl e1 e2 := LE.Class (pair_le e1 e2).
-Canonical Structure pair_LEty (e1 e2 : LE.type) : LE.type :=
-  LE.Pack (LE.obj e1 * LE.obj e2) (pair_LEcl e1 e2).
-Check (3,4,5) <= (3,4,5).
-\end{coq_example}
-
-At the current stage we can use \texttt{==} and \texttt{<=} on concrete types,
-like tuples of natural numbers, but we can't develop an algebraic
-theory over the types that are equipped with both relations.
-
-\begin{coq_example}
-Check 2 <= 3 /\ 2 == 2.
-Fail Check forall (e : EQ.type) (x y : EQ.obj e), x <= y -> y <= x -> x == y.
-Fail Check forall (e : LE.type) (x y : LE.obj e), x <= y -> y <= x -> x == y.
-\end{coq_example}
-
-We need to define a new class that inherits from both \texttt{EQ} and \texttt{LE}.
-
-\begin{coq_example}
-Module LEQ.
-  Record mixin (e : EQ.type) (le : EQ.obj e -> EQ.obj e -> Prop) :=
-    Mixin { compat : forall x y : EQ.obj e, le x y /\ le y x <-> x == y }.
-  Record class T := Class {
-    EQ_class    : EQ.class T;
-    LE_class    : LE.class T;
-    extra : mixin (EQ.Pack T EQ_class) (LE.cmp T LE_class) }.
-  Structure type := _Pack { obj : Type; class_of : class obj }.
-  Arguments Mixin {e le} _.
-  Arguments Class {T} _ _ _.
-\end{coq_example}
-
-The \texttt{mixin} component of the \texttt{LEQ} class contains all the extra content
-we are adding to \texttt{EQ} and \texttt{LE}.  In particular it contains the requirement
-that the two relations we are combining are compatible.
-
-Unfortunately there is still an obstacle to developing the algebraic theory
-of this new class.
-
-\begin{coq_example}
-  Module theory.
-  Fail Check forall (le : type) (n m : obj le), n <= m -> n <= m -> n == m.
-\end{coq_example}
-
-The problem is that the two classes \texttt{LE} and \texttt{LEQ} are not yet related by
-a subclass relation.  In other words Coq does not see that an object
-of the \texttt{LEQ} class is also an object of the \texttt{LE} class.
-
-The following two constructions tell Coq how to canonically build
-the \texttt{LE.type} and \texttt{EQ.type} structure given an \texttt{LEQ.type} structure
-on the same type.
-
-\begin{coq_example}
-  Definition to_EQ (e : type) : EQ.type :=
-    EQ.Pack (obj e) (EQ_class _ (class_of e)).
-  Canonical Structure to_EQ.
-  Definition to_LE (e : type) : LE.type :=
-    LE.Pack (obj e) (LE_class _ (class_of e)).
-  Canonical Structure to_LE.
-\end{coq_example}
-We can now formulate out first theorem on the objects of the \texttt{LEQ} structure.
-\begin{coq_example}
-  Lemma lele_eq (e : type) (x y : obj e) : x <= y -> y <= x -> x == y.
-   now intros; apply (compat _ _ (extra _ (class_of e)) x y); split. Qed.
-  Arguments lele_eq {e} x y _ _.
-  End theory.
-End LEQ.
-Import LEQ.theory.
-Check lele_eq.
-\end{coq_example}
-
-Of course one would like to apply results proved in the algebraic
-setting to any concrete instate of the algebraic structure.
-
-\begin{coq_example}
-Example test_algebraic (n m : nat) :  n <= m -> m <= n -> n == m.
- Fail apply (lele_eq n m). Abort.
-Example test_algebraic2 (l1 l2 : LEQ.type) (n m : LEQ.obj l1 * LEQ.obj l2) :
-  n <= m -> m <= n -> n == m.
- Fail apply (lele_eq n m). Abort.
-\end{coq_example}
-
-Again one has to tell Coq that the type \texttt{nat} is in the \texttt{LEQ} class, and how
-the type constructor \texttt{*} interacts with the \texttt{LEQ} class.  In the following
-proofs are omitted for brevity.
-
-\begin{coq_example}
-Lemma nat_LEQ_compat (n m : nat) : n <= m /\ m <= n <-> n == m.
-\end{coq_example}
-\begin{coq_eval}
-
-split.
-  unfold EQ.op; unfold LE.op; simpl; unfold nat_le; unfold nat_eq.
-  case (nat_compare_spec n m); [ reflexivity | | now intros _ [H _]; case H ].
-  now intro H; apply nat_compare_gt in H; rewrite -> H; intros [_ K]; case K.
-unfold EQ.op; unfold LE.op; simpl; unfold nat_le; unfold nat_eq.
-case (nat_compare_spec n m); [ | intros H1 H2; discriminate H2 .. ].
-intro H; rewrite H; intros _; split; [ intro H1; discriminate H1 | ].
-case (nat_compare_eq_iff m m); intros _ H1.
-now rewrite H1; auto; intro H2; discriminate H2.
-Qed.
-\end{coq_eval}
-\begin{coq_example}
-Definition nat_LEQmx := LEQ.Mixin nat_LEQ_compat.
-Lemma pair_LEQ_compat (l1 l2 : LEQ.type) (n m : LEQ.obj l1 * LEQ.obj l2) :
-n <= m /\ m <= n <-> n == m.
-\end{coq_example}
-\begin{coq_eval}
-
-case n; case m; unfold EQ.op; unfold LE.op; simpl.
-intros n1 n2 m1 m2; split; [ intros [[Le1 Le2] [Ge1 Ge2]] | intros [H1 H2] ].
-  now split; apply lele_eq.
-case (LEQ.compat _ _ (LEQ.extra _ (LEQ.class_of l1)) m1 n1).
-case (LEQ.compat _ _ (LEQ.extra _ (LEQ.class_of l2)) m2 n2).
-intros _ H3 _ H4; apply H3 in H2; apply H4 in H1; clear H3 H4.
-now case H1; case H2; split; split.
-Qed.
-\end{coq_eval}
-\begin{coq_example}
-Definition pair_LEQmx l1 l2 := LEQ.Mixin (pair_LEQ_compat l1 l2).
-\end{coq_example}
-
-The following script registers an \texttt{LEQ} class for \texttt{nat} and for the
-type constructor \texttt{*}.  It also tests that they work as expected.
-
-Unfortunately, these declarations are very verbose.  In the following
-subsection we show how to make these declaration more compact.
-
-\begin{coq_example}
-Module Add_instance_attempt.
-  Canonical Structure nat_LEQty : LEQ.type :=
-    LEQ._Pack nat (LEQ.Class nat_EQcl nat_LEcl nat_LEQmx).
-  Canonical Structure pair_LEQty (l1 l2 : LEQ.type) : LEQ.type :=
-    LEQ._Pack (LEQ.obj l1 * LEQ.obj l2)
-      (LEQ.Class
-        (EQ.class_of (pair_EQty (to_EQ l1) (to_EQ l2)))
-        (LE.class_of (pair_LEty (to_LE l1) (to_LE l2)))
-        (pair_LEQmx l1 l2)).
-  Example test_algebraic (n m : nat) : n <= m -> m <= n -> n == m.
-   now apply (lele_eq n m). Qed.
-  Example test_algebraic2 (n m : nat * nat) : n <= m -> m <= n -> n == m.
-   now apply (lele_eq n m). Qed.
-End Add_instance_attempt.
-\end{coq_example}
-
-Note that no direct proof of \texttt{n <= m -> m <= n -> n == m} is provided by the
-user for \texttt{n} and \texttt{m} of type \texttt{nat * nat}.  What the user provides is a proof of
-this statement for \texttt{n} and \texttt{m} of type \texttt{nat} and a proof that the pair
-constructor preserves this property.  The combination of these two facts is a
-simple form of proof search that Coq performs automatically while inferring
-canonical structures.
-
-\subsection{Compact declaration of Canonical Structures}
-
-We need some infrastructure for that.
-
-\begin{coq_example*}
-Require Import Strings.String.
-\end{coq_example*}
-\begin{coq_example}
-Module infrastructure.
-  Inductive phantom {T : Type} (t : T) : Type := Phantom.
-  Definition unify {T1 T2} (t1 : T1) (t2 : T2) (s : option string) :=
-    phantom t1 -> phantom t2.
-  Definition id {T} {t : T} (x : phantom t) := x.
-  Notation "[find v | t1 ~ t2 ] p" := (fun v (_ : unify t1 t2 None) => p)
-    (at level 50, v ident, only parsing).
-  Notation "[find v | t1 ~ t2 | s ] p" := (fun v (_ : unify t1 t2 (Some s)) => p)
-    (at level 50, v ident, only parsing).
-  Notation "'Error : t : s" := (unify _ t (Some s))
-    (at level 50, format "''Error'  :  t  :  s").
-  Open Scope string_scope.
-End infrastructure.
-\end{coq_example}
-
-To explain the notation \texttt{[find v | t1 \textasciitilde t2]} let us pick one
-of its instances: \texttt{[find e | EQ.obj e \textasciitilde T | "is not an EQ.type" ]}.
-It should be read as: ``find a class e such that its objects have type T
-or fail with message "T is not an EQ.type"''.
-
-The other utilities are used to ask Coq to solve a specific unification
-problem, that will in turn require the inference of some canonical
-structures.  They are explained in mode details in~\cite{CSwcu}.
-
-We now have all we need to create a compact ``packager'' to declare
-instances of the \texttt{LEQ} class.
-
-\begin{coq_example}
-Import infrastructure.
-Definition packager T e0 le0 (m0 : LEQ.mixin e0 le0) :=
-    [find e  | EQ.obj e ~ T       | "is not an EQ.type" ]
-    [find o  | LE.obj o ~ T       | "is not an LE.type" ]
-    [find ce | EQ.class_of e ~ ce ]
-    [find co | LE.class_of o ~ co ]
-    [find m  | m ~ m0             | "is not the right mixin" ]
-  LEQ._Pack T (LEQ.Class ce co m).
-Notation Pack T m := (packager T _ _ m _ id _ id _ id _ id _ id).
-\end{coq_example}
-
-The object \texttt{Pack} takes a type \texttt{T} (the key) and a mixin \texttt{m}.  It infers all
-the other pieces of the class \texttt{LEQ} and declares them as canonical values
-associated to the \texttt{T} key.  All in all, the only new piece of information
-we add in the \texttt{LEQ} class is the mixin, all the rest is already canonical
-for \texttt{T} and hence can be inferred by Coq.
-
-\texttt{Pack} is a notation, hence it is not type checked at the time of its
-declaration.  It will be type checked when it is used, an in that case
-\texttt{T} is going to be a concrete type.  The odd arguments \texttt{\_} and \texttt{id} we
-pass to the 
-packager represent respectively the classes to be inferred (like \texttt{e}, \texttt{o}, etc) and a token (\texttt{id}) to force their inference.  Again, for all the details the
-reader can refer to~\cite{CSwcu}. 
-
-The declaration of canonical instances can now be way more compact:
-
-\begin{coq_example}
-Canonical Structure nat_LEQty := Eval hnf in Pack nat nat_LEQmx.
-Canonical Structure pair_LEQty (l1 l2 : LEQ.type) :=
-  Eval hnf in Pack (LEQ.obj l1 * LEQ.obj l2) (pair_LEQmx l1 l2).
-\end{coq_example}
-
-Error messages are also quite intelligible (if one skips to the end of
-the message).
-
-\begin{coq_example}
-Fail Canonical Structure err := Eval hnf in Pack bool nat_LEQmx.
-\end{coq_example}
-
-%%% Local Variables: 
-%%% mode: latex
-%%% TeX-master: "Reference-Manual"
-%%% End: 
diff --git a/doc/refman/Cases.tex b/doc/refman/Cases.tex
deleted file mode 100644
index 376ef031db..0000000000
--- a/doc/refman/Cases.tex
+++ /dev/null
@@ -1,843 +0,0 @@
-\achapter{Extended pattern-matching}
-%HEVEA\cutname{cases.html}
-%BEGIN LATEX
-\defaultheaders
-%END LATEX
-\aauthor{Cristina Cornes and Hugo Herbelin}
-
-\label{Mult-match-full}
-\ttindex{Cases}
-\index{ML-like patterns}
-
-This section describes the full form of pattern-matching in {\Coq} terms.
-
-\asection{Patterns}\label{implementation} The full syntax of {\tt
-match} is presented in Figures~\ref{term-syntax}
-and~\ref{term-syntax-aux}.  Identifiers in patterns are either
-constructor names or variables. Any identifier that is not the
-constructor of an inductive or co-inductive type is considered to be a
-variable. A variable name cannot occur more than once in a given
-pattern. It is recommended to start variable names by a lowercase
-letter.
-
-If a pattern has the form $(c~\vec{x})$ where $c$ is a constructor
-symbol and $\vec{x}$ is a linear vector of (distinct) variables, it is
-called {\em simple}: it is the kind of pattern recognized by the basic
-version of {\tt match}. On the opposite, if it is a variable $x$ or
-has the form $(c~\vec{p})$ with $p$ not only made of variables, the
-pattern is called {\em nested}.
-
-A variable pattern matches any value, and the identifier is bound to
-that value. The pattern ``\texttt{\_}'' (called ``don't care'' or
-``wildcard'' symbol) also matches any value, but does not bind
-anything. It may occur an arbitrary number of times in a
-pattern. Alias patterns written \texttt{(}{\sl pattern} \texttt{as}
-{\sl identifier}\texttt{)} are also accepted. This pattern matches the
-same values as {\sl pattern} does and {\sl identifier} is bound to the
-matched value.
-A pattern of the form {\pattern}{\tt |}{\pattern} is called
-disjunctive. A list of patterns separated with commas is also
-considered as a pattern and is called {\em multiple pattern}. However
-multiple patterns can only occur at the root of pattern-matching
-equations. Disjunctions of {\em multiple pattern} are allowed though.
-
-Since extended {\tt match} expressions are compiled into the primitive
-ones, the expressiveness of the theory remains the same. Once the
-stage of parsing has finished only simple patterns remain.  Re-nesting
-of pattern is performed at printing time. An easy way to see the
-result of the expansion is to toggle off the nesting performed at
-printing (use here {\tt Set Printing Matching}), then by printing the term
-with \texttt{Print} if the term is a constant, or using the command
-\texttt{Check}.
-
-The extended \texttt{match} still accepts an optional {\em elimination
-predicate} given after the keyword \texttt{return}.  Given a pattern
-matching expression, if all the right-hand-sides of \texttt{=>} ({\em
-rhs} in short) have the same type, then this type can be sometimes
-synthesized, and so we can omit the \texttt{return} part. Otherwise
-the predicate after \texttt{return} has to be provided, like for the basic
-\texttt{match}.
-
-Let us illustrate through examples the different aspects of extended
-pattern matching. Consider for example the function that computes the
-maximum of two natural numbers. We can write it in primitive syntax
-by:
-
-\begin{coq_example}
-Fixpoint max (n m:nat) {struct m} : nat :=
-  match n with
-  | O => m
-  | S n' => match m with
-            | O => S n'
-            | S m' => S (max n' m')
-            end
-  end.
-\end{coq_example}
-
-\paragraph{Multiple patterns}
-
-Using multiple patterns in the definition of {\tt max} lets us write:
-
-\begin{coq_eval}
-Reset max.
-\end{coq_eval}
-\begin{coq_example}
-Fixpoint max (n m:nat) {struct m} : nat :=
-  match n, m with
-  | O, _ => m
-  | S n', O => S n'
-  | S n', S m' => S (max n' m')
-  end.
-\end{coq_example}
-
-which will be compiled into the previous form.
-
-The pattern-matching compilation strategy examines patterns from left
-to right. A \texttt{match} expression is generated {\bf only} when
-there is at least one constructor in the column of patterns. E.g. the
-following example does not build a \texttt{match} expression.
-
-\begin{coq_example}
-Check (fun x:nat => match x return nat with
-                    | y => y
-                    end).
-\end{coq_example}
-
-\paragraph{Aliasing subpatterns}
-
-We can also use ``\texttt{as} {\ident}'' to associate a name to a
-sub-pattern:
-
-\begin{coq_eval}
-Reset max.
-\end{coq_eval}
-\begin{coq_example}
-Fixpoint max (n m:nat) {struct n} : nat :=
-  match n, m with
-  | O, _ => m
-  | S n' as p, O => p
-  | S n', S m' => S (max n' m')
-  end.
-\end{coq_example}
-
-\paragraph{Nested patterns}
-
-Here is now an example of nested patterns:
-
-\begin{coq_example}
-Fixpoint even (n:nat) : bool :=
-  match n with
-  | O => true
-  | S O => false
-  | S (S n') => even n'
-  end.
-\end{coq_example}
-
-This is compiled into:
-
-\begin{coq_example}
-Unset Printing Matching.
-Print even.
-\end{coq_example}
-\begin{coq_eval}
-Set Printing Matching.
-\end{coq_eval}
-
-In the previous examples patterns do not conflict with, but
-sometimes it is comfortable to write patterns that admit a non
-trivial superposition. Consider
-the boolean function \texttt{lef} that given two natural numbers
-yields \texttt{true} if the first one is less or equal than the second
-one and \texttt{false} otherwise. We can write it as follows:
-
-\begin{coq_example}
-Fixpoint lef (n m:nat) {struct m} : bool :=
-  match n, m with
-  | O, x => true
-  | x, O => false
-  | S n, S m => lef n m
-  end.
-\end{coq_example}
-
-Note that the first and the second multiple pattern superpose because
-the couple of values \texttt{O O} matches both. Thus, what is the result
-of the function on those values?  To eliminate ambiguity we use the
-{\em textual priority rule}: we consider patterns ordered from top to
-bottom, then a value is matched by the pattern at the $ith$ row if and
-only if it is not matched by some pattern of a previous row. Thus in the
-example,
-\texttt{O O} is matched by the first pattern, and so \texttt{(lef O O)}
-yields \texttt{true}.
-
-Another way to write  this function is:
-
-\begin{coq_eval}
-Reset lef.
-\end{coq_eval}
-\begin{coq_example}
-Fixpoint lef (n m:nat) {struct m} : bool :=
-  match n, m with
-  | O, x => true
-  | S n, S m => lef n m
-  | _, _ => false
-  end.
-\end{coq_example}
-
-Here the last pattern superposes with the first two. Because
-of the priority rule, the last pattern
-will be used only for values that do not match neither the  first nor
-the second one.
-
-Terms with useless patterns are not accepted by the
-system. Here is an example:
-% Test failure: "This clause is redundant."
-\begin{coq_eval}
-Set Printing Depth 50.
-\end{coq_eval}
-\begin{coq_example}
-Fail Check (fun x:nat =>
-         match x with
-         | O => true
-         | S _ => false
-         | x => true
-         end).
-\end{coq_example}
-
-\paragraph{Disjunctive patterns}
-
-Multiple patterns that share the same right-hand-side can be
-factorized using the notation \nelist{\multpattern}{\tt |}. For instance,
-{\tt max} can be rewritten as follows:
-
-\begin{coq_eval}
-Reset max.
-\end{coq_eval}
-\begin{coq_example}
-Fixpoint max (n m:nat) {struct m} : nat :=
-  match n, m with
-  | S n', S m' => S (max n' m')
-  | 0, p | p, 0 => p
-  end.
-\end{coq_example}
-
-Similarly, factorization of (non necessary multiple) patterns
-that share the same variables is possible by using the notation
-\nelist{\pattern}{\tt |}. Here is an example:
-
-\begin{coq_example}
-Definition filter_2_4 (n:nat) : nat :=
-  match n with
-  | 2 as m | 4 as m => m
-  | _ => 0
-  end.
-\end{coq_example}
-
-Here is another example using disjunctive subpatterns.
-
-\begin{coq_example}
-Definition filter_some_square_corners (p:nat*nat) : nat*nat :=
-  match p with
-  | ((2 as m | 4 as m), (3 as n | 5 as n)) => (m,n)
-  | _ => (0,0)
-  end.
-\end{coq_example}
-
-\asection{About patterns of parametric types}
-\paragraph{Parameters in patterns}
-When matching objects of a parametric type, parameters do not bind in patterns.
-They must be substituted by ``\_''.
-Consider for example the type of polymorphic lists:
-
-\begin{coq_example}
-Inductive List (A:Set) : Set :=
-  | nil : List A
-  | cons : A -> List A -> List A.
-\end{coq_example}
-
-We can check the function {\em tail}:
-
-\begin{coq_example}
-Check
-  (fun l:List nat =>
-     match l with
-     | nil _ => nil nat
-     | cons _ _ l' => l'
-     end).
-\end{coq_example}
-
-
-When we use parameters in patterns there is an error message:
-% Test failure: "The parameters do not bind in patterns."
-\begin{coq_eval}
-Set Printing Depth 50.
-\end{coq_eval}
-\begin{coq_example}
-Fail Check
-  (fun l:List nat =>
-     match l with
-     | nil A => nil nat
-     | cons A _ l' => l'
-     end).
-\end{coq_example}
-
-The option {\tt Set Asymmetric Patterns} \optindex{Asymmetric Patterns}
-(off by default) removes parameters from constructors in patterns:
-\begin{coq_example}
-  Set Asymmetric Patterns.
-  Check (fun l:List nat =>
-    match l with
-    | nil => nil
-    | cons _ l' => l'
-    end)
-  Unset Asymmetric Patterns.
-\end{coq_example}
-
-\paragraph{Implicit arguments in patterns}
-By default, implicit arguments are omitted in patterns. So we write:
-
-\begin{coq_example}
-Arguments nil [A].
-Arguments cons [A] _ _.
-Check
-  (fun l:List nat =>
-     match l with
-     | nil => nil
-     | cons _ l' => l'
-     end).
-\end{coq_example}
-
-But the possibility to use all the arguments is given by ``{\tt @}'' implicit
-explicitations (as for terms~\ref{Implicits-explicitation}).
-
-\begin{coq_example}
-Check
-  (fun l:List nat =>
-     match l with
-     | @nil _ => @nil nat
-     | @cons _ _ l' => l'
-     end).
-\end{coq_example}
-
-\asection{Matching objects of dependent types}
-The previous examples illustrate pattern matching on objects of
-non-dependent types, but we can also
-use the expansion strategy to destructure objects of dependent type.
-Consider the type \texttt{listn} of lists of a certain length:
-\label{listn}
-
-\begin{coq_example}
-Inductive listn : nat -> Set :=
-  | niln : listn 0
-  | consn : forall n:nat, nat -> listn n -> listn (S n).
-\end{coq_example}
-
-\asubsection{Understanding dependencies in patterns}
-We can define the function \texttt{length} over \texttt{listn} by:
-
-\begin{coq_example}
-Definition length (n:nat) (l:listn n) := n.
-\end{coq_example}
-
-Just for illustrating pattern matching,
-we can define it by case analysis:
-
-\begin{coq_eval}
-Reset length.
-\end{coq_eval}
-\begin{coq_example}
-Definition length (n:nat) (l:listn n) :=
-  match l with
-  | niln => 0
-  | consn n _ _ => S n
-  end.
-\end{coq_example}
-
-We can understand the meaning of this definition using the
-same notions of usual pattern matching.
-
-%
-% Constraining of dependencies is not longer valid in V7
-%
-\iffalse
-Now suppose we split the second pattern  of \texttt{length} into two
-cases so to give an
-alternative definition using nested patterns:
-\begin{coq_example}
-Definition length1 (n:nat) (l:listn n) :=
-  match l with
-  | niln => 0
-  | consn n _ niln => S n
-  | consn n _ (consn _ _ _) => S n
-  end.
-\end{coq_example}
-
-It is obvious that \texttt{length1} is  another version of
-\texttt{length}. We can also give the following definition:
-\begin{coq_example}
-Definition length2 (n:nat) (l:listn n) :=
-  match l with
-  | niln => 0
-  | consn n _ niln => 1
-  | consn n _ (consn m _ _) => S (S m)
-  end.
-\end{coq_example}
-
-If we forget that \texttt{listn} is a dependent type and we read these
-definitions using the usual semantics of pattern matching,  we can conclude
-that \texttt{length1}
-and \texttt{length2} are different functions.
-In fact, they are equivalent
-because the pattern \texttt{niln} implies that \texttt{n} can only match
-the value $0$ and analogously the pattern \texttt{consn} determines that \texttt{n} can
-only match  values of the form  $(S~v)$ where $v$ is the value matched by
-\texttt{m}.
-
-The converse is also true. If
-we destructure the  length  value with the pattern \texttt{O} then the list
-value should be $niln$.
-Thus, the following term \texttt{length3} corresponds to the function
-\texttt{length} but this time defined by case analysis on the dependencies instead of on the list:
-
-\begin{coq_example}
-Definition length3 (n:nat) (l:listn n) :=
-  match l with
-  | niln => 0
-  | consn O _ _ => 1
-  | consn (S n) _ _ => S (S n)
-  end.
-\end{coq_example}
-
-When we have nested patterns of dependent types, the semantics of
-pattern matching becomes a little more difficult because
-the set of values that are matched by a sub-pattern may be conditioned by the
-values matched by another sub-pattern. Dependent nested patterns are
-somehow constrained patterns.
-In the examples, the expansion of
-\texttt{length1} and \texttt{length2} yields exactly the same term
- but the
-expansion of \texttt{length3} is completely different. \texttt{length1} and
-\texttt{length2} are expanded into two nested case analysis on
-\texttt{listn} while \texttt{length3} is expanded into a case analysis on
-\texttt{listn} containing a case analysis on natural numbers inside.
-
-
-In practice the user can think about the patterns as independent and
-it is the expansion algorithm that cares to relate them. \\
-\fi
-%
-%
-%
-
-\asubsection{When the elimination predicate must be provided}
-\paragraph{Dependent pattern matching}
-The examples  given so far do not need an explicit elimination predicate
- because all the rhs have the same type and the
-strategy succeeds to synthesize it.
-Unfortunately when dealing with dependent patterns it often happens
-that we need to write cases where the type of the rhs are
-different  instances of the elimination  predicate.
-The function  \texttt{concat} for \texttt{listn}
-is an example where the branches have different type
-and we need to provide the elimination predicate:
-
-\begin{coq_example}
-Fixpoint concat (n:nat) (l:listn n) (m:nat) (l':listn m) {struct l} :
- listn (n + m) :=
-  match l in listn n return listn (n + m) with
-  | niln => l'
-  | consn n' a y => consn (n' + m) a (concat n' y m l')
-  end.
-\end{coq_example}
-The elimination predicate is {\tt fun (n:nat) (l:listn n) => listn~(n+m)}.
-In general if $m$ has type {\tt (}$I$ $q_1$ {\ldots} $q_r$ $t_1$ {\ldots} $t_s${\tt )} where
-$q_1$, {\ldots}, $q_r$ are parameters, the elimination predicate should be of
-the form~:
-{\tt fun} $y_1$ {\ldots} $y_s$ $x${\tt :}($I$~$q_1$ {\ldots} $q_r$ $y_1$ {\ldots}
-  $y_s${\tt ) =>} $Q$.
-
-In the concrete syntax, it should be written~:
-\[ \kw{match}~m~\kw{as}~x~\kw{in}~(I~\_~\mbox{\ldots}~\_~y_1~\mbox{\ldots}~y_s)~\kw{return}~Q~\kw{with}~\mbox{\ldots}~\kw{end}\]
-
-The variables which appear in the \kw{in} and \kw{as} clause are new
-and bounded in the property $Q$ in the \kw{return} clause. The
-parameters of the inductive definitions should not be mentioned and
-are replaced by \kw{\_}.
-
-\paragraph{Multiple dependent pattern matching}
-Recall that a list of patterns is also a pattern. So, when we destructure several
-terms at the same time and the branches have different types we need to provide the
-elimination predicate for this multiple pattern. It is done using the same
-scheme, each term may be associated to an \kw{as} and \kw{in} clause in order to
-introduce a dependent product.
-
-For example, an equivalent definition for \texttt{concat} (even though the
-matching on the second term is trivial) would have been:
-
-\begin{coq_eval}
-Reset concat.
-\end{coq_eval}
-\begin{coq_example}
-Fixpoint concat (n:nat) (l:listn n) (m:nat) (l':listn m) {struct l} :
- listn (n + m) :=
-  match l in listn n, l' return listn (n + m) with
-  | niln, x => x
-  | consn n' a y, x => consn (n' + m) a (concat n' y m x)
-  end.
-\end{coq_example}
-
-Even without real matching over the second term, this construction can be used to
-keep types linked.  If {\tt a} and {\tt b} are two {\tt listn} of the same length,
-by writing
-\begin{coq_eval}
-  Unset Printing Matching.
-\end{coq_eval}
-\begin{coq_example}
-Check (fun n (a b: listn n) => match a,b with
- |niln,b0 => tt
- |consn n' a y, bS => tt
-end).
-\end{coq_example}
-\begin{coq_eval}
-  Set Printing Matching.
-\end{coq_eval}
-
-I have a copy of {\tt b} in type {\tt listn 0} resp {\tt listn (S n')}.
-
-% Notice that this time, the predicate \texttt{[n,\_:nat](listn (plus n
-%   m))}  is binary because we
-% destructure both \texttt{l} and \texttt{l'} whose types have arity one.
-% In general, if we destructure the terms $e_1\ldots e_n$
-% the predicate will be of arity $m$ where $m$ is the sum of the
-% number of dependencies of the type of $e_1, e_2,\ldots e_n$
-% (the $\lambda$-abstractions
-% should correspond from left to right to each dependent argument of the
-% type of $e_1\ldots e_n$).
-% When the arity of the predicate (i.e. number of abstractions) is not
-% correct Coq raises an error message. For example:
-
-% % Test failure
-% \begin{coq_eval}
-% Reset concat.
-% Set Printing Depth 50.
-% (********** The following is not correct and should produce ***********)
-% (** Error: the term l' has type listn m while it is expected to have **)
-% (** type listn (?31 + ?32)                                           **)
-% \end{coq_eval}
-% \begin{coq_example}
-% Fixpoint concat
-%  (n:nat) (l:listn n) (m:nat)
-%  (l':listn m) {struct l} : listn (n + m) :=
-%   match l, l' with
-%   | niln, x => x
-%   | consn n' a y, x => consn (n' + m) a (concat n' y m x)
-%   end.
-% \end{coq_example}
-
-\paragraph{Patterns in {\tt in}}
-\label{match-in-patterns}
-
-If the type of the matched term is more precise than an inductive applied to
-variables, arguments of the inductive in the {\tt in} branch can be more
-complicated patterns than a variable.
-
-Moreover, constructors whose type do not follow the same pattern will
-become impossible branches. In an impossible branch, you can answer
-anything but {\tt False\_rect unit} has the advantage to be subterm of
-anything. % ???
-
-To be concrete: the {\tt tail} function can be written:
-\begin{coq_example}
-Definition tail n (v: listn (S n)) :=
-  match v in listn (S m) return listn m with
-  | niln => False_rect unit
-  | consn n' a y => y
-  end.
-\end{coq_example}
-and {\tt tail n v} will be subterm of {\tt v}.
-
-\asection{Using pattern matching to write proofs}
-In all the previous examples the elimination predicate does not depend
-on the object(s) matched. But it may depend and the typical case
-is when we write a proof by induction or a function that yields an
-object of dependent type. An example of proof using \texttt{match} in
-given in Section~\ref{refine-example}.
-
-For example, we can write
-the function \texttt{buildlist} that given a natural number
-$n$ builds a list of length $n$ containing zeros as follows:
-
-\begin{coq_example}
-Fixpoint buildlist (n:nat) : listn n :=
-  match n return listn n with
-  | O => niln
-  | S n => consn n 0 (buildlist n)
-  end.
-\end{coq_example}
-
-We can also use multiple patterns.
-Consider the following definition of the predicate less-equal
-\texttt{Le}:
-
-\begin{coq_example}
-Inductive LE : nat -> nat -> Prop :=
-  | LEO : forall n:nat, LE 0 n
-  | LES : forall n m:nat, LE n m -> LE (S n) (S m).
-\end{coq_example}
-
-We can use multiple patterns to write  the proof of the lemma
- \texttt{forall (n m:nat), (LE n m)}\verb=\/=\texttt{(LE m n)}:
-
-\begin{coq_example}
-Fixpoint dec (n m:nat) {struct n} : LE n m \/ LE m n :=
-  match n, m return LE n m \/ LE m n with
-  | O, x => or_introl (LE x 0) (LEO x)
-  | x, O => or_intror (LE x 0) (LEO x)
-  | S n as n', S m as m' =>
-      match dec n m with
-      | or_introl h => or_introl (LE m' n') (LES n m h)
-      | or_intror h => or_intror (LE n' m') (LES m n h)
-      end
-  end.
-\end{coq_example}
-In the example of \texttt{dec},
-the first \texttt{match} is dependent while
-the second is not.
-
-% In general, consider the terms $e_1\ldots e_n$,
-% where  the type of $e_i$ is an instance of a family type
-% $\lb (\vec{d_i}:\vec{D_i}) \mto T_i$  ($1\leq i
-% \leq n$). Then, in expression \texttt{match}  $e_1,\ldots,
-% e_n$ \texttt{of} \ldots \texttt{end}, the
-% elimination predicate ${\cal P}$ should be of the form:
-% $[\vec{d_1}:\vec{D_1}][x_1:T_1]\ldots [\vec{d_n}:\vec{D_n}][x_n:T_n]Q.$
-
-The user can also use \texttt{match} in combination with the tactic
-\texttt{refine} (see Section~\ref{refine}) to build incomplete proofs
-beginning with a \texttt{match} construction.
-
-\asection{Pattern-matching on inductive objects involving local
-definitions}
-
-If local definitions occur in the type of a constructor, then there are two ways
-to match on this constructor. Either the local definitions are skipped and
-matching is done only on the true arguments of the constructors, or the bindings
-for local definitions can also be caught in the matching.
-
-Example.
-
-\begin{coq_eval}
-Reset Initial.
-Require Import Arith.
-\end{coq_eval}
-
-\begin{coq_example*}
-Inductive list : nat -> Set :=
-  | nil : list 0
-  | cons : forall n:nat, let m := (2 * n) in list m -> list (S (S m)).
-\end{coq_example*}
-
-In the next example, the local definition is not caught.
-
-\begin{coq_example}
-Fixpoint length n (l:list n) {struct l} : nat :=
-  match l with
-  | nil => 0
-  | cons n l0 => S (length (2 * n) l0)
-  end.
-\end{coq_example}
-
-But in this example, it is.
-
-\begin{coq_example}
-Fixpoint length' n (l:list n) {struct l} : nat :=
-  match l with
-  | nil => 0
-  | @cons _ m l0 => S (length' m l0)
-  end.
-\end{coq_example}
-
-\Rem for a given matching clause, either none of the local definitions or all of
-them can be caught.
-
-\Rem you can only catch {\tt let} bindings in mode where you bind all variables and so you
-have to use @ syntax.
-
-\Rem this feature is incoherent with the fact that parameters cannot be caught and
-consequently is somehow hidden. For example, there is no mention of it in error messages.
-
-\asection{Pattern-matching and coercions}
-
-If a mismatch occurs between the expected type of a pattern and its
-actual type, a coercion made from constructors is sought. If such a
-coercion can be found, it is automatically inserted around the
-pattern.
-
-Example:
-
-\begin{coq_example}
-Inductive I : Set :=
-  | C1 : nat -> I
-  | C2 : I -> I.
-Coercion C1 : nat >-> I.
-Check (fun x => match x with
-                | C2 O => 0
-                | _ => 0
-                end).
-\end{coq_example}
-
-
-\asection{When does the expansion strategy fail ?}\label{limitations}
-The strategy works very like in ML languages when treating
-patterns of non-dependent type.
-But there are new cases of failure that are due to the presence of
-dependencies.
-
-The error messages of the current implementation may be sometimes
-confusing.  When the tactic fails because patterns are somehow
-incorrect then error messages refer to the initial expression. But the
-strategy may succeed to build an expression whose sub-expressions are
-well typed when the whole expression is not. In this situation the
-message makes reference to the expanded expression.  We encourage
-users, when they have patterns with the same outer constructor in
-different equations, to name the variable patterns in the same
-positions with the same name.
-E.g. to write {\small\texttt{(cons n O x) => e1}}
-and {\small\texttt{(cons n \_ x) => e2}} instead of
-{\small\texttt{(cons n O x) => e1}} and
-{\small\texttt{(cons n' \_ x') => e2}}.
-This helps to maintain certain name correspondence between the
-generated expression and the original.
-
-Here is a summary of the error messages corresponding to each situation:
-
-\begin{ErrMsgs}
-\item \sverb{The constructor } {\sl
-    ident} \sverb{ expects } {\sl num} \sverb{ arguments}
-
- \sverb{The variable } {\sl ident} \sverb{ is bound several times
-    in pattern } {\sl term}
-
- \sverb{Found a constructor of inductive type } {\term}
- \sverb{ while a constructor of } {\term} \sverb{ is expected}
-
- Patterns are incorrect (because constructors are not applied to
-  the correct number of the arguments, because they are not linear or
-  they are wrongly typed).
-
-\item \errindex{Non exhaustive pattern-matching}
-
-The pattern matching is not exhaustive.
-
-\item \sverb{The elimination predicate } {\sl term} \sverb{ should be
-    of arity } {\sl num} \sverb{ (for non dependent case) or } {\sl
-    num} \sverb{ (for dependent case)}
-
-The elimination predicate provided to \texttt{match} has not the
-  expected arity.
-
-
-%\item the whole expression is wrongly typed
-
-% CADUC ?
-% , or the synthesis of
-%   implicit arguments fails (for example to find the elimination
-%   predicate or to resolve implicit arguments in the rhs).
-
-%   There are {\em nested patterns of dependent type}, the elimination
-%   predicate corresponds to non-dependent case and has the form
-%   $[x_1:T_1]...[x_n:T_n]T$ and {\bf some} $x_i$ occurs {\bf free} in
-%   $T$.  Then, the strategy may fail to find out a correct elimination
-%   predicate during some step of compilation.  In this situation we
-%   recommend the user to rewrite the nested dependent patterns into
-%   several \texttt{match} with {\em simple patterns}.
-
-\item {\tt Unable to infer a match predicate\\
-    Either there is a type incompatibility or the problem involves\\
-    dependencies}
-
-  There is a type mismatch between the different branches.
-  The user should provide an elimination predicate.
-
-% Obsolete ?
-% \item because of nested patterns, it may happen that even though all
-%   the rhs have the same type, the strategy needs dependent elimination
-%   and so an elimination predicate must be provided. The system warns
-%   about this situation, trying to compile anyway with the
-%   non-dependent strategy. The risen message is:
-
-% \begin{itemize}
-% \item {\tt Warning: This pattern matching may need dependent
-%     elimination to be compiled.  I will try, but if fails try again
-%     giving dependent elimination predicate.}
-% \end{itemize}
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% % LA PROPAGATION DES CONTRAINTES ARRIERE N'EST PAS FAITE DANS LA V7
-% TODO
-% \item there are {\em nested patterns of dependent type} and the
-%   strategy builds a term that is well typed but recursive calls in fix
-%   point are reported as illegal:
-% \begin{itemize}
-% \item {\tt Error: Recursive call applied to an illegal term ...}
-% \end{itemize}
-
-% This is because the strategy generates a term that is correct w.r.t.
-% the initial term but which does not pass the guard condition.  In
-% this situation we recommend the user to transform the nested dependent
-% patterns into {\em several \texttt{match} of simple patterns}.  Let us
-% explain this with an example.  Consider the following definition of a
-% function that yields the last element of a list and \texttt{O} if it is
-% empty:
-
-% \begin{coq_example}
-%   Fixpoint last [n:nat; l:(listn n)] : nat :=
-%    match l of
-%      (consn _ a niln) => a
-%    | (consn m _ x) => (last m x) | niln => O
-%    end.
-% \end{coq_example}
-
-% It fails because of the priority between patterns, we know that this
-% definition is equivalent to the following more explicit one (which
-% fails too):
-
-% \begin{coq_example*}
-%   Fixpoint last [n:nat; l:(listn n)] : nat :=
-%    match l of
-%      (consn _ a niln) => a
-%    | (consn n _ (consn m b x)) => (last n (consn m b x))
-%    | niln => O
-%    end.
-% \end{coq_example*}
-
-% Note that the recursive call {\tt (last n (consn m b x))} is not
-% guarded. When treating with patterns of dependent types the strategy
-% interprets the first definition of \texttt{last} as the second
-% one\footnote{In languages of the ML family the first definition would
-%   be translated into a term where the variable \texttt{x} is shared in
-%   the expression.  When patterns are of non-dependent types, Coq
-%   compiles as in ML languages using sharing. When patterns are of
-%   dependent types the compilation reconstructs the term as in the
-%   second definition of \texttt{last} so to ensure the result of
-%   expansion is well typed.}.  Thus it generates a term where the
-% recursive call is rejected by the guard condition.
-
-% You can get rid of this problem by writing the definition with
-% \emph{simple patterns}:
-
-% \begin{coq_example}
-%   Fixpoint last [n:nat; l:(listn n)] : nat :=
-%   <[_:nat]nat>match l of
-%     (consn m a x) => Cases x of niln => a | _ => (last m x) end
-%   | niln => O
-%   end.
-% \end{coq_example}
-
-\end{ErrMsgs}
-
-
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: "Reference-Manual"
-%%% End:
diff --git a/doc/refman/Reference-Manual.tex b/doc/refman/Reference-Manual.tex
index afc1b9c57f..a49637bb2a 100644
--- a/doc/refman/Reference-Manual.tex
+++ b/doc/refman/Reference-Manual.tex
@@ -117,9 +117,7 @@ Options A and B of the licence are {\em not} elected.}
 %END LATEX
 \part{Addendum to the Reference Manual}
 \include{AddRefMan-pre}%
-\include{Cases.v}%
 \include{Coercion.v}%
-\include{CanonicalStructures.v}%
 \include{Classes.v}%
 \include{Micromega.v}
 \include{Extraction.v}%