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

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-\newtheorem{cscexample}{Example}
-
-\achapter{\protect{User defined equalities and relations}}
-\aauthor{Matthieu Sozeau}
-\tacindex{setoid\_replace}
-\label{setoid_replace}
-
-This chapter presents the extension of several equality related tactics to
-work over user-defined structures (called setoids) that are equipped with
-ad-hoc equivalence relations meant to behave as equalities.
-Actually, the tactics have also been generalized to relations weaker then
-equivalences (e.g. rewriting systems).
-
-This documentation is adapted from the previous setoid documentation by
-Claudio Sacerdoti Coen (based on previous work by Cl\'ement Renard).
-The new implementation is a drop-in replacement for the old one \footnote{Nicolas
-Tabareau helped with the gluing}, hence most of the documentation still applies.
-
-The work is a complete rewrite of the previous implementation, based on
-the type class infrastructure. It also improves on and generalizes
-the previous implementation in several ways:
-\begin{itemize}
-\item User-extensible algorithm. The algorithm is separated in two
-  parts: generations of the rewriting constraints (done in ML) and
-  solving of these constraints using type class resolution. As type
-  class resolution is extensible using tactics, this allows users to define
-  general ways to solve morphism constraints.
-\item Sub-relations. An example extension to the base algorithm is the
-  ability to define one relation as a subrelation of another so that
-  morphism declarations on one relation can be used automatically for
-  the other. This is done purely using tactics and type class search.
-\item Rewriting under binders. It is possible to rewrite under binders
-  in the new implementation, if one provides the proper
-  morphisms. Again, most of the work is handled in the tactics.
-\item First-class morphisms and signatures. Signatures and morphisms are
-  ordinary Coq terms, hence they can be manipulated inside Coq, put
-  inside structures and lemmas about them can be proved inside the
-  system. Higher-order morphisms are also allowed.
-\item Performance. The implementation is based on a depth-first search for the first
-  solution to a set of constraints which can be as fast as linear in the
-  size of the term, and the size of the proof term is linear
-  in the size of the original term. Besides, the extensibility allows the
-  user to customize the proof-search if necessary.
-\end{itemize}
-
-\asection{Relations and morphisms}
-
-A parametric \emph{relation} \texttt{R} is any term of type
-\texttt{forall ($x_1$:$T_1$) \ldots ($x_n$:$T_n$), relation $A$}. The
-expression $A$, which depends on $x_1$ \ldots $x_n$, is called the
-\emph{carrier} of the relation and \texttt{R} is
-said to be a relation over \texttt{A}; the list $x_1,\ldots,x_n$
-is the (possibly empty) list of parameters of the relation.
-
-\firstexample
-\begin{cscexample}[Parametric relation]
-It is possible to implement finite sets of elements of type \texttt{A}
-as unordered list of elements of type \texttt{A}. The function
-\texttt{set\_eq: forall (A: Type), relation (list A)} satisfied by two lists
-with the same elements is a parametric relation over \texttt{(list A)} with
-one parameter \texttt{A}. The type of \texttt{set\_eq} is convertible with
-\texttt{forall (A: Type), list A -> list A -> Prop}.
-\end{cscexample}
-
-An \emph{instance} of a parametric relation \texttt{R} with $n$ parameters
-is any term \texttt{(R $t_1$ \ldots $t_n$)}.
-
-Let \texttt{R} be a relation over \texttt{A} with $n$ parameters.
-A term is a parametric proof of reflexivity for \texttt{R} if it has type
-\texttt{forall ($x_1$:$T_1$) \ldots ($x_n$:$T_n$),
- reflexive (R $x_1$ \ldots $x_n$)}. Similar definitions are given for
-parametric proofs of symmetry and transitivity.
-
-\begin{cscexample}[Parametric relation (cont.)]
-The \texttt{set\_eq} relation of the previous example can be proved to be
-reflexive, symmetric and transitive.
-\end{cscexample}
-
-A parametric unary function $f$ of type
-\texttt{forall ($x_1$:$T_1$) \ldots ($x_n$:$T_n$), $A_1$ -> $A_2$}
-covariantly respects two parametric relation instances $R_1$ and $R_2$ if,
-whenever $x, y$ satisfy $R_1~x~y$, their images $(f~x)$ and $(f~y)$ 
-satisfy $R_2~(f~x)~(f~y)$ . An $f$ that respects its input and output relations
-will be called a unary covariant \emph{morphism}. We can also say that $f$ is
-a monotone function with respect to $R_1$ and $R_2$. 
-The sequence $x_1,\ldots x_n$ represents the parameters of the morphism.
-
-Let $R_1$ and $R_2$ be two parametric relations.
-The \emph{signature} of a parametric morphism of type
-\texttt{forall ($x_1$:$T_1$) \ldots ($x_n$:$T_n$), $A_1$ -> $A_2$} that
-covariantly respects two instances $I_{R_1}$ and $I_{R_2}$ of $R_1$ and $R_2$ is written $I_{R_1} \texttt{++>} I_{R_2}$.
-Notice that the special arrow \texttt{++>}, which reminds the reader
-of covariance, is placed between the two relation instances, not
-between the two carriers. The signature relation instances and morphism will
-be typed in a context introducing variables for the parameters.
-
-The previous definitions are extended straightforwardly to $n$-ary morphisms,
-that are required to be simultaneously monotone on every argument.
-
-Morphisms can also be contravariant in one or more of their arguments.
-A morphism is contravariant on an argument associated to the relation instance
-$R$ if it is covariant on the same argument when the inverse relation
-$R^{-1}$ (\texttt{inverse R} in Coq) is considered. 
-The special arrow \texttt{-{}->} is used in signatures
-for contravariant morphisms.
-
-Functions having arguments related by symmetric relations instances are both
-covariant and contravariant in those arguments. The special arrow
-\texttt{==>} is used in signatures for morphisms that are both covariant
-and contravariant.
-
-An instance of a parametric morphism $f$ with $n$ parameters is any term
-\texttt{f $t_1$ \ldots $t_n$}.
-
-\begin{cscexample}[Morphisms]
-Continuing the previous example, let
-\texttt{union: forall (A: Type), list A -> list A -> list A} perform the union
-of two sets by appending one list to the other. \texttt{union} is a binary
-morphism parametric over \texttt{A} that respects the relation instance
-\texttt{(set\_eq A)}. The latter condition is proved by showing
-\texttt{forall (A: Type) (S1 S1' S2 S2': list A), set\_eq A S1 S1' ->
- set\_eq A S2 S2' -> set\_eq A (union A S1 S2) (union A S1' S2')}.
-
-The signature of the function \texttt{union A} is
-\texttt{set\_eq A ==> set\_eq A ==> set\_eq A} for all \texttt{A}.
-\end{cscexample}
-
-\begin{cscexample}[Contravariant morphism]
-The division function \texttt{Rdiv: R -> R -> R} is a morphism of
-signature \texttt{le ++> le -{}-> le} where \texttt{le} is
-the usual order relation over real numbers. Notice that division is
-covariant in its first argument and contravariant in its second
-argument.
-\end{cscexample}
-
-Leibniz equality is a relation and every function is a
-morphism that respects Leibniz equality. Unfortunately, Leibniz equality
-is not always the intended equality for a given structure.
-
-In the next section we will describe the commands to register terms as
-parametric relations and morphisms. Several tactics that deal with equality
-in \Coq\ can also work with the registered relations.
-The exact list of tactic will be given in Sect.~\ref{setoidtactics}.
-For instance, the
-tactic \texttt{reflexivity} can be used to close a goal $R~n~n$ whenever
-$R$ is an instance of a registered reflexive relation. However, the tactics
-that replace in a context $C[]$ one term with another one related by $R$
-must verify that $C[]$ is a morphism that respects the intended relation.
-Currently the verification consists in checking whether $C[]$ is a syntactic
-composition of morphism instances that respects some obvious
-compatibility constraints.
-
-\begin{cscexample}[Rewriting]
-Continuing the previous examples, suppose that the user must prove
-\texttt{set\_eq int (union int (union int S1 S2) S2) (f S1 S2)} under the
-hypothesis \texttt{H: set\_eq int S2 (nil int)}. It is possible to
-use the \texttt{rewrite} tactic to replace the first two occurrences of
-\texttt{S2} with \texttt{nil int} in the goal since the context
-\texttt{set\_eq int (union int (union int S1 nil) nil) (f S1 S2)}, being
-a composition of morphisms instances, is a morphism. However the tactic
-will fail replacing the third occurrence of \texttt{S2} unless \texttt{f}
-has also been declared as a morphism.
-\end{cscexample}
-
-\asection{Adding new relations and morphisms}
-A parametric relation
-\textit{Aeq}\texttt{: forall ($y_1 : \beta_!$ \ldots $y_m : \beta_m$), relation (A $t_1$ \ldots $t_n$)} over
-\textit{(A : $\alpha_i$ -> \ldots $\alpha_n$ -> }\texttt{Type})
-can be declared with the following command:
-
-\comindex{Add Parametric Relation}
-\begin{quote}
-  \texttt{Add Parametric Relation} ($x_1 : T_1$) \ldots ($x_n : T_k$) :
-  \textit{(A $t_1$ \ldots $t_n$) (Aeq $t'_1$ \ldots $t'_m$)}\\
-  ~\zeroone{\texttt{reflexivity proved by} \textit{refl}}\\
-  ~\zeroone{\texttt{symmetry proved by} \textit{sym}}\\
-  ~\zeroone{\texttt{transitivity proved by} \textit{trans}}\\
-  \texttt{~as} \textit{id}.
-\end{quote}
-after having required the \texttt{Setoid} module with the
-\texttt{Require Setoid} command.
-
-The identifier \textit{id} gives a unique name to the morphism and it is
-used by the command to generate fresh names for automatically provided lemmas
-used internally.
-
-Notice that the carrier and relation parameters may refer to the context 
-of variables introduced at the beginning of the declaration, but the
-instances need not be made only of variables.
-Also notice that \textit{A} is \emph{not} required to be a term
-having the same parameters as \textit{Aeq}, although that is often the
-case in practice (this departs from the previous implementation).
-
-\comindex{Add Relation}
-In case the carrier and relations are not parametric, one can use the
-command \texttt{Add Relation} instead, whose syntax is the same except
-there is no local context.
-
-The proofs of reflexivity, symmetry and transitivity can be omitted if the
-relation is not an equivalence relation. The proofs must be instances of the
-corresponding relation definitions: e.g. the proof of reflexivity must
-have a type convertible to \texttt{reflexive (A $t_1$ \ldots $t_n$) (Aeq $t'_1$ \ldots
-  $t'_n$)}. Each proof may refer to the introduced variables as well. 
-
-\begin{cscexample}[Parametric relation]
-For Leibniz equality, we may declare:
-\texttt{Add Parametric Relation (A : Type) :} \texttt{A (@eq A)}\\
-~\zeroone{\texttt{reflexivity proved by} \texttt{@refl\_equal A}}\\
-\ldots
-\end{cscexample}
-
-Some tactics
-(\texttt{reflexivity}, \texttt{symmetry}, \texttt{transitivity}) work only
-on relations that respect the expected properties. The remaining tactics
-(\texttt{replace}, \texttt{rewrite} and derived tactics such as
-\texttt{autorewrite}) do not require any properties over the relation.
-However, they are able to replace terms with related ones only in contexts
-that are syntactic compositions of parametric morphism instances declared with
-the following command.
-
-\comindex{Add Parametric Morphism}
-\begin{quote}
-  \texttt{Add Parametric Morphism} ($x_1 : \T_!$) \ldots ($x_k : \T_k$)\\
-  (\textit{f $t_1$ \ldots $t_n$})\\
-  \texttt{~with signature} \textit{sig}\\
-  \texttt{~as id}.\\
-  \texttt{Proof}\\
-  ~\ldots\\
-  \texttt{Qed}
-\end{quote}
-
-The command declares \textit{f} as a parametric morphism of signature
-\textit{sig}. The identifier \textit{id} gives a unique name to the morphism
-and it is used as the base name of the type class instance definition 
-and as the name of the lemma that proves the well-definedness of the morphism.
-The parameters of the morphism as well as the signature may refer to the
-context of variables.
-The command asks the user to prove interactively that \textit{f} respects
-the relations identified from the signature.
-
-\begin{cscexample}
-We start the example by assuming a small theory over homogeneous sets and
-we declare set equality as a parametric equivalence relation and
-union of two sets as a parametric morphism.
-\begin{coq_example*}
-Require Export Setoid.
-Require Export Relation_Definitions.
-Set Implicit Arguments.
-Parameter set: Type -> Type.
-Parameter empty: forall A, set A.
-Parameter eq_set: forall A, set A -> set A -> Prop.
-Parameter union: forall A, set A -> set A -> set A.
-Axiom eq_set_refl: forall A, reflexive _ (eq_set (A:=A)).
-Axiom eq_set_sym: forall A, symmetric _ (eq_set (A:=A)).
-Axiom eq_set_trans: forall A, transitive _ (eq_set (A:=A)).
-Axiom empty_neutral: forall A (S: set A), eq_set (union S (empty A)) S.
-Axiom union_compat:
- forall (A : Type),
-  forall x x' : set A, eq_set x x' ->
-  forall y y' : set A, eq_set y y' ->
-   eq_set (union x y) (union x' y').
-Add Parametric Relation A : (set A) (@eq_set A)
- reflexivity proved by (eq_set_refl (A:=A))
- symmetry proved by (eq_set_sym (A:=A))
- transitivity proved by (eq_set_trans (A:=A))
- as eq_set_rel.
-Add Parametric Morphism A : (@union A) with 
-signature (@eq_set A) ==> (@eq_set A) ==> (@eq_set A) as union_mor.
-Proof. exact (@union_compat A). Qed.
-\end{coq_example*}
-
-\end{cscexample}
-
-Is is possible to reduce the burden of specifying parameters using
-(maximally inserted) implicit arguments. If \texttt{A} is always set as
-maximally implicit in the previous example, one can write:
-
-\begin{coq_eval}
-Reset Initial.
-Require Export Setoid.
-Require Export Relation_Definitions.
-Parameter set: Type -> Type.
-Parameter empty: forall {A}, set A.
-Parameter eq_set: forall {A}, set A -> set A -> Prop.
-Parameter union: forall {A}, set A -> set A -> set A.
-Axiom eq_set_refl: forall {A}, reflexive (set A) eq_set.
-Axiom eq_set_sym: forall {A}, symmetric (set A) eq_set.
-Axiom eq_set_trans: forall {A}, transitive (set A) eq_set.
-Axiom empty_neutral: forall A (S: set A), eq_set (union S empty) S.
-Axiom union_compat:
- forall (A : Type),
-  forall x x' : set A, eq_set x x' ->
-  forall y y' : set A, eq_set y y' ->
-   eq_set (union x y) (union x' y').
-\end{coq_eval}
-
-\begin{coq_example*} 
-Add Parametric Relation A : (set A) eq_set
- reflexivity proved by eq_set_refl
- symmetry proved by eq_set_sym
- transitivity proved by eq_set_trans
- as eq_set_rel.
-Add Parametric Morphism A : (@union A) with
-  signature eq_set ==> eq_set ==> eq_set as union_mor.
-Proof. exact (@union_compat A). Qed.
-\end{coq_example*}
-
-We proceed now by proving a simple lemma performing a rewrite step
-and then applying reflexivity, as we would do working with Leibniz
-equality. Both tactic applications are accepted
-since the required properties over \texttt{eq\_set} and
-\texttt{union} can be established from the two declarations above.
-
-\begin{coq_example*}
-Goal forall (S: set nat),
- eq_set (union (union S empty) S) (union S S).
-Proof. intros. rewrite empty_neutral. reflexivity. Qed.
-\end{coq_example*}
-
-The tables of relations and morphisms are managed by the type class
-instance mechanism. The behavior on section close is to generalize
-the instances by the variables of the section (and possibly hypotheses
-used in the proofs of instance declarations) but not to export them in
-the rest of the development for proof search. One can use the
-\texttt{Existing Instance} command to do so outside the section,
-using the name of the declared morphism suffixed by \texttt{\_Morphism}, 
-or use the \texttt{Global} modifier for the corresponding class instance
-declaration (see \S\ref{setoid:first-class}) at definition time.
-When loading a compiled file or importing a module,
-all the declarations of this module will be loaded.
-
-\asection{Rewriting and non reflexive relations}
-To replace only one argument of an n-ary morphism it is necessary to prove
-that all the other arguments are related to themselves by the respective
-relation instances.
-
-\begin{cscexample}
-To replace \texttt{(union S empty)} with \texttt{S} in
-\texttt{(union (union S empty) S) (union S S)} the rewrite tactic must
-exploit the monotony of \texttt{union} (axiom \texttt{union\_compat} in
-the previous example). Applying \texttt{union\_compat} by hand we are left
-with the goal \texttt{eq\_set (union S S) (union S S)}.
-\end{cscexample}
-
-When the relations associated to some arguments are not reflexive, the tactic
-cannot automatically prove the reflexivity goals, that are left to the user.
-
-Setoids whose relation are partial equivalence relations (PER)
-are useful to deal with partial functions. Let \texttt{R} be a PER. We say
-that an element \texttt{x} is defined if \texttt{R x x}. A partial function
-whose domain comprises all the defined elements only is declared as a
-morphism that respects \texttt{R}. Every time a rewriting step is performed
-the user must prove that the argument of the morphism is defined.
-
-\begin{cscexample}
-Let \texttt{eqO} be \texttt{fun x y => x = y $\land$ ~x$\neq$ 0} (the smaller PER over
-non zero elements). Division can be declared as a morphism of signature
-\texttt{eq ==> eq0 ==> eq}. Replace \texttt{x} with \texttt{y} in
-\texttt{div x n = div y n} opens the additional goal \texttt{eq0 n n} that
-is equivalent to \texttt{n=n $\land$ n$\neq$0}.
-\end{cscexample}
-
-\asection{Rewriting and non symmetric relations}
-When the user works up to relations that are not symmetric, it is no longer
-the case that any covariant morphism argument is also contravariant. As a
-result it is no longer possible to replace a term with a related one in
-every context, since the obtained goal implies the previous one if and
-only if the replacement has been performed in a contravariant position.
-In a similar way, replacement in an hypothesis can be performed only if
-the replaced term occurs in a covariant position.
-
-\begin{cscexample}[Covariance and contravariance]
-Suppose that division over real numbers has been defined as a
-morphism of signature \texttt{Zdiv: Zlt ++> Zlt -{}-> Zlt} (i.e.
-\texttt{Zdiv} is increasing in its first argument, but decreasing on the
-second one). Let \texttt{<} denotes \texttt{Zlt}. 
-Under the hypothesis \texttt{H: x < y} we have
-\texttt{k < x / y -> k < x / x}, but not
-\texttt{k < y / x -> k < x / x}.
-Dually, under the same hypothesis \texttt{k < x / y -> k < y / y} holds,
-but \texttt{k < y / x -> k < y / y} does not.
-Thus, if the current goal is \texttt{k < x / x}, it is possible to replace
-only the second occurrence of \texttt{x} (in contravariant position)
-with \texttt{y} since the obtained goal must imply the current one.
-On the contrary, if \texttt{k < x / x} is
-an hypothesis, it is possible to replace only the first occurrence of
-\texttt{x} (in covariant position) with \texttt{y} since
-the current hypothesis must imply the obtained one.
-\end{cscexample}
-
-Contrary to the previous implementation, no specific error message will
-be raised when trying to replace a term that occurs in the wrong
-position. It will only fail because the rewriting constraints are not
-satisfiable. However it is possible to use the \texttt{at} modifier to
-specify which occurrences should be rewritten.
-
-As expected, composing morphisms together propagates the variance annotations by
-switching the variance every time a contravariant position is traversed.
-\begin{cscexample}
-Let us continue the previous example and let us consider the goal
-\texttt{x / (x / x) < k}. The first and third occurrences of \texttt{x} are
-in a contravariant position, while the second one is in covariant position.
-More in detail, the second occurrence of \texttt{x} occurs
-covariantly in \texttt{(x / x)} (since division is covariant in its first
-argument), and thus contravariantly in \texttt{x / (x / x)} (since division
-is contravariant in its second argument), and finally covariantly in
-\texttt{x / (x / x) < k} (since \texttt{<}, as every transitive relation,
-is contravariant in its first argument with respect to the relation itself).
-\end{cscexample}
-
-\asection{Rewriting in ambiguous setoid contexts}
-One function can respect several different relations and thus it can be
-declared as a morphism having multiple signatures.
-
-\begin{cscexample}
-Union over homogeneous lists can be given all the following signatures:
-\texttt{eq ==> eq ==> eq} (\texttt{eq} being the equality over ordered lists)
-\texttt{set\_eq ==> set\_eq ==> set\_eq} (\texttt{set\_eq} being the equality
-over unordered lists up to duplicates),
-\texttt{multiset\_eq ==> multiset\_eq ==> multiset\_eq} (\texttt{multiset\_eq}
-being the equality over unordered lists).
-\end{cscexample}
-
-To declare multiple signatures for a morphism, repeat the \texttt{Add Morphism}
-command.
-
-When morphisms have multiple signatures it can be the case that a rewrite
-request is ambiguous, since it is unclear what relations should be used to
-perform the rewriting. Contrary to the previous implementation, the
-tactic will always choose the first possible solution to the set of
-constraints generated by a rewrite and will not try to find \emph{all}
-possible solutions to warn the user about.
-
-\asection{First class setoids and morphisms}
-\label{setoid:first-class}
-
-The implementation is based on a first-class representation of
-properties of relations and morphisms as type classes. That is, 
-the various combinations of properties on relations and morphisms 
-are represented as records and instances of theses classes are put
-in a hint database.
-For example, the declaration:
-
-\begin{quote}
-  \texttt{Add Parametric Relation} ($x_1 : T_1$) \ldots ($x_n : T_k$) :
-  \textit{(A $t_1$ \ldots $t_n$) (Aeq $t'_1$ \ldots $t'_m$)}\\
-  ~\zeroone{\texttt{reflexivity proved by} \textit{refl}}\\
-  ~\zeroone{\texttt{symmetry proved by} \textit{sym}}\\
-  ~\zeroone{\texttt{transitivity proved by} \textit{trans}}\\
-  \texttt{~as} \textit{id}.
-\end{quote}
-
-is equivalent to an instance declaration:
-
-\begin{quote}
-  \texttt{Instance} ($x_1 : T_1$) \ldots ($x_n : T_k$) \texttt{=>}
-  \textit{id} : \texttt{@Equivalence} \textit{(A $t_1$ \ldots $t_n$) (Aeq
-    $t'_1$ \ldots $t'_m$)} :=\\
-  ~\zeroone{\texttt{Equivalence\_Reflexive :=} \textit{refl}}\\
-  ~\zeroone{\texttt{Equivalence\_Symmetric :=} \textit{sym}}\\
-  ~\zeroone{\texttt{Equivalence\_Transitive :=} \textit{trans}}.
-\end{quote}
-
-The declaration itself amounts to the definition of an object of the
-record type \texttt{Coq.Classes.RelationClasses.Equivalence} and a
-hint added to the \texttt{typeclass\_instances} hint database. 
-Morphism declarations are also instances of a type class defined in
-\texttt{Classes.Morphisms}.
-See the documentation on type classes \ref{typeclasses} and 
-the theories files in \texttt{Classes} for further explanations. 
-
-One can inform the rewrite tactic about morphisms and relations just by
-using the typeclass mechanism to declare them using \texttt{Instance}
-and \texttt{Context} vernacular commands.
-Any object of type \texttt{Proper} (the type of morphism declarations)
-in the local context will also be automatically used by the rewriting 
-tactic to solve constraints.
-
-Other representations of first class setoids and morphisms can also
-be handled by encoding them as records. In the following example,
-the projections of the setoid relation and of the morphism function 
-can be registered as parametric relations and morphisms.
-\begin{cscexample}[First class setoids]
-
-\begin{coq_example*}
-Require Import Relation_Definitions Setoid.
-Record Setoid: Type :=
-{ car:Type;
-  eq:car->car->Prop;
-  refl: reflexive _ eq;
-  sym: symmetric _ eq;
-  trans: transitive _ eq
-}.
-Add Parametric Relation (s : Setoid) : (@car s) (@eq s)
- reflexivity proved by (refl s)
- symmetry proved by (sym s)
- transitivity proved by (trans s) as eq_rel.
-Record Morphism (S1 S2:Setoid): Type :=
-{ f:car S1 ->car S2;
-  compat: forall (x1 x2: car S1), eq S1 x1 x2 -> eq S2 (f x1) (f x2) }.
-Add Parametric Morphism (S1 S2 : Setoid) (M : Morphism S1 S2) :
- (@f S1 S2 M) with signature (@eq S1 ==> @eq S2) as apply_mor.
-Proof. apply (compat S1 S2 M). Qed.
-Lemma test: forall (S1 S2:Setoid) (m: Morphism S1 S2)
- (x y: car S1), eq S1 x y -> eq S2 (f _ _ m x) (f _ _ m y).
-Proof. intros. rewrite H. reflexivity. Qed.
-\end{coq_example*}
-\end{cscexample}
-
-\asection{Tactics enabled on user provided relations}
-\label{setoidtactics}
-The following tactics, all prefixed by \texttt{setoid\_}, 
-deal with arbitrary
-registered relations and morphisms. Moreover, all the corresponding unprefixed
-tactics (i.e. \texttt{reflexivity, symmetry, transitivity, replace, rewrite})
-have been extended to fall back to their prefixed counterparts when
-the relation involved is not Leibniz equality. Notice, however, that using
-the prefixed tactics it is possible to pass additional arguments such as
-\texttt{using relation}.
-\medskip
-
-\comindex{setoid\_reflexivity}
-\texttt{setoid\_reflexivity}
-
-\comindex{setoid\_symmetry}
-\texttt{setoid\_symmetry} \zeroone{\texttt{in} \textit{ident}}
-
-\comindex{setoid\_transitivity}
-\texttt{setoid\_transitivity}
-
-\comindex{setoid\_rewrite}
-\texttt{setoid\_rewrite} \zeroone{\textit{orientation}} \textit{term}
-~\zeroone{\texttt{at} \textit{occs}} ~\zeroone{\texttt{in} \textit{ident}}
-
-\comindex{setoid\_replace}
-\texttt{setoid\_replace} \textit{term} \texttt{with} \textit{term}
-~\zeroone{\texttt{in} \textit{ident}}
-~\zeroone{\texttt{using relation} \textit{term}}
-~\zeroone{\texttt{by} \textit{tactic}}
-\medskip
-
-The \texttt{using relation}
-arguments cannot be passed to the unprefixed form. The latter argument
-tells the tactic what parametric relation should be used to replace
-the first tactic argument with the second one. If omitted, it defaults
-to the \texttt{DefaultRelation} instance on the type of the objects.
-By default, it means the most recent \texttt{Equivalence} instance in
-the environment, but it can be customized by declaring new
-\texttt{DefaultRelation} instances. As Leibniz equality is a declared
-equivalence, it will fall back to it if no other relation is declared on
-a given type.
-
-Every derived tactic that is based on the unprefixed forms of the tactics
-considered above will also work up to user defined relations. For instance,
-it is possible to register hints for \texttt{autorewrite} that are
-not proof of Leibniz equalities. In particular it is possible to exploit
-\texttt{autorewrite} to simulate normalization in a term rewriting system
-up to user defined equalities.
-
-\asection{Printing relations and morphisms}
-The \texttt{Print Instances} command can be used to show the list of
-currently registered \texttt{Reflexive} (using \texttt{Print Instances Reflexive}),
-\texttt{Symmetric} or \texttt{Transitive} relations,
-\texttt{Equivalence}s, \texttt{PreOrder}s, \texttt{PER}s, and
-Morphisms (implemented as \texttt{Proper} instances). When
- the rewriting tactics refuse to replace a term in a context
-because the latter is not a composition of morphisms, the \texttt{Print Instances}
-commands can be useful to understand what additional morphisms should be
-registered.
-
-\asection{Deprecated syntax and backward incompatibilities}
-Due to backward compatibility reasons, the following syntax for the
-declaration of setoids and morphisms is also accepted.
-
-\comindex{Add Setoid}
-\begin{quote}
-  \texttt{Add Setoid} \textit{A Aeq ST} \texttt{as} \textit{ident}
-\end{quote}
-where \textit{Aeq} is a congruence relation without parameters,
-\textit{A} is its carrier and \textit{ST} is an object of type
-\texttt{(Setoid\_Theory A Aeq)} (i.e. a record packing together the reflexivity,
-symmetry and transitivity lemmas). Notice that the syntax is not completely
-backward compatible since the identifier was not required.
-
-\comindex{Add Morphism}
-\begin{quote}
-  \texttt{Add Morphism} \textit{f}:\textit{ident}.\\
-  Proof.\\
-  \ldots\\
-  Qed.
-\end{quote}
-
-The latter command also is restricted to the declaration of morphisms without
-parameters. It is not fully backward compatible since the property the user
-is asked to prove is slightly different: for $n$-ary morphisms the hypotheses
-of the property are permuted; moreover, when the morphism returns a
-proposition, the property is now stated using a bi-implication in place of
-a simple implication. In practice, porting an old development to the new
-semantics is usually quite simple.
-
-Notice that several limitations of the old implementation have been lifted.
-In particular, it is now possible to declare several relations with the
-same carrier and several signatures for the same morphism. Moreover, it is
-now also possible to declare several morphisms having the same signature.
-Finally, the replace and rewrite tactics can be used to replace terms in
-contexts that were refused by the old implementation. As discussed in
-the next section, the semantics of the new \texttt{setoid\_rewrite}
-command differs slightly from the old one and \texttt{rewrite}.
-
-\asection{Rewriting under binders}
-
-\textbf{Warning}: Due to compatibility issues, this feature is enabled only when calling 
-the \texttt{setoid\_rewrite} tactics directly and not \texttt{rewrite}.
-
-To be able to rewrite under binding constructs, one must declare
-morphisms with respect to pointwise (setoid) equivalence of functions. 
-Example of such morphisms are the standard \texttt{all} and \texttt{ex}
-combinators for universal and existential quantification respectively. 
-They are declared as morphisms in the \texttt{Classes.Morphisms\_Prop}
-module. For example, to declare that universal quantification is a
-morphism for logical equivalence:
-
-\begin{coq_eval}
-Reset Initial.
-Require Import Setoid Morphisms.
-\end{coq_eval}
-\begin{coq_example}
-Instance all_iff_morphism (A : Type) :
-  Proper (pointwise_relation A iff ==> iff) (@all A).
-Proof. simpl_relation. 
-\end{coq_example}
-\begin{coq_eval}
-Admitted.
-\end{coq_eval}
-
-One then has to show that if two predicates are equivalent at every
-point, their universal quantifications are equivalent. Once we have
-declared such a morphism, it will be used by the setoid rewriting tactic
-each time we try to rewrite under an \texttt{all} application (products
-in \Prop{} are implicitly translated to such applications).
-
-Indeed, when rewriting under a lambda, binding variable $x$, say from
-$P~x$ to $Q~x$ using the relation \texttt{iff}, the tactic will generate
-a proof of \texttt{pointwise\_relation A iff (fun x => P x) (fun x => Q
-x)} from the proof of \texttt{iff (P x) (Q x)} and a constraint of the
-form \texttt{Proper (pointwise\_relation A iff ==> ?) m} will be
-generated for the surrounding morphism \texttt{m}.
-
-Hence, one can add higher-order combinators as morphisms by providing
-signatures using pointwise extension for the relations on the functional
-arguments (or whatever subrelation of the pointwise extension).
-For example, one could declare the \texttt{map} combinator on lists as 
-a morphism:
-\begin{coq_eval}
-Require Import List.
-Set Implicit Arguments.
-Inductive list_equiv {A:Type} (eqA : relation A) : relation (list A) :=
-| eq_nil : list_equiv eqA nil nil
-| eq_cons : forall x y, eqA x y -> 
-  forall l l', list_equiv eqA l l' -> list_equiv eqA (x :: l) (y :: l').
-\end{coq_eval}
-\begin{coq_example*}
-Instance map_morphism `{Equivalence A eqA, Equivalence B eqB} :
-  Proper ((eqA ==> eqB) ==> list_equiv eqA ==> list_equiv eqB) 
-     (@map A B).
-\end{coq_example*}
-
-where \texttt{list\_equiv} implements an equivalence on lists
-parameterized by an equivalence on the elements.
-
-Note that when one does rewriting with a lemma under a binder
-using \texttt{setoid\_rewrite}, the application of the lemma may capture
-the bound variable, as the semantics are different from rewrite where
-the lemma is first matched on the whole term. With the new
-\texttt{setoid\_rewrite}, matching is done on each subterm separately
-and in its local environment, and all matches are rewritten
-\emph{simultaneously} by default. The semantics of the previous
-\texttt{setoid\_rewrite} implementation can almost be recovered using
-the \texttt{at 1} modifier.
-
-\asection{Sub-relations}
-
-Sub-relations can be used to specify that one relation is included in
-another, so that morphisms signatures for one can be used for the other.
-If a signature mentions a relation $R$ on the left of an arrow
-\texttt{==>}, then the signature also applies for any relation $S$ that
-is smaller than $R$, and the inverse applies on the right of an arrow. 
-One can then declare only a few morphisms instances that generate the complete set
-of signatures for a particular constant. By default, the only declared
-subrelation is \texttt{iff}, which is a subrelation of \texttt{impl}
-and \texttt{inverse impl} (the dual of implication). That's why we can
-declare only two morphisms for conjunction:
-\texttt{Proper (impl ==> impl ==> impl) and} and 
-\texttt{Proper (iff ==> iff ==> iff) and}. This is sufficient to satisfy
-any rewriting constraints arising from a rewrite using \texttt{iff},
-\texttt{impl} or \texttt{inverse impl} through \texttt{and}.
-
-Sub-relations are implemented in \texttt{Classes.Morphisms} and are a 
-prime example of a mostly user-space extension of the algorithm.
-
-\asection{Constant unfolding}
-
-The resolution tactic is based on type classes and hence regards user-defined 
-constants as transparent by default. This may slow down the resolution
-due to a lot of unifications (all the declared \texttt{Proper}
-instances are tried at each node of the search tree).
-To speed it up, declare your constant as rigid for proof search
-using the command \texttt{Typeclasses Opaque} (see \S \ref{TypeclassesTransparency}).
-
-%%% Local Variables: 
-%%% mode: latex
-%%% TeX-master: "Reference-Manual"
-%%% compile-command: "make -C ../.. -f Makefile.stage3 doc/refman/Reference-Manual.pdf"
-%%% End: 
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