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(* -*- coq-prog-args: ("-emacs-U" "-nois") -*- *)
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(* Certified Haskell Prelude.
* Author: Matthieu Sozeau
* Institution: LRI, CNRS UMR 8623 - Universit�copyright Paris Sud
* 91405 Orsay, France *)
(* $Id: FSetAVL_prog.v 616 2007-08-08 12:28:10Z msozeau $ *)
Require Import Coq.Program.Program.
Require Import Coq.Classes.Init.
Set Implicit Arguments.
Unset Strict Implicit.
(** We first define setoids on a carrier, it amounts to an equivalence relation.
Now [equiv] is overloaded for every [Setoid].
*)
Require Export Coq.Relations.Relations.
Class Setoid (carrier : Type) (equiv : relation carrier) :=
equiv_prf : equivalence carrier equiv.
(** Overloaded notations for setoid equivalence and inequivalence. Not to be confused with [eq] and [=]. *)
Definition equiv [ Setoid A R ] : _ := R.
Infix "==" := equiv (at level 70, no associativity) : type_scope.
Notation " x =!= y " := (~ x == y) (at level 70, no associativity).
Definition equiv_refl [ s : Setoid A R ] : forall x : A, R x x := equiv_refl _ _ equiv_prf.
Definition equiv_sym [ s : Setoid A R ] : forall x y : A, R x y -> R y x := equiv_sym _ _ equiv_prf.
Definition equiv_trans [ s : Setoid A R ] : forall x y z : A, R x y -> R y z -> R x z := equiv_trans _ _ equiv_prf.
Lemma nequiv_sym : forall [ s : Setoid A R ] (x y : A), x =!= y -> y =!= x.
Proof.
intros ; red ; intros.
apply equiv_sym in H0...
apply (H H0).
Qed.
(** Tactics to do some progress on the goal, called by the user. *)
Ltac do_setoid_refl :=
match goal with
| [ |- @equiv ?A ?R ?s ?X _ ] => apply (equiv_refl (A:=A) (R:=R) (s:=s) X)
| [ |- ?R ?X _ ] => apply (equiv_refl (R:=R) X)
| [ |- ?R ?A ?X _ ] => apply (equiv_refl (R:=R A) X)
| [ |- ?R ?A ?B ?X _ ] => apply (equiv_refl (R:=R A B) X)
| [ |- ?R ?A ?B ?C ?X _ ] => apply (equiv_refl (R:=R A B C) X)
end.
Ltac refl := do_setoid_refl.
Ltac do_setoid_sym :=
match goal with
| [ |- @equiv ?A ?R ?s ?X ?Y ] => apply (equiv_sym (A:=A) (R:=R) (s:=s) (x:=X) (y:=Y))
| [ |- not (@equiv ?A ?R ?s ?X ?Y) ] => apply (nequiv_sym (A:=A) (R:=R) (s:=s) (x:=X) (y:=Y))
| [ |- ?R ?X ?Y ] => apply (equiv_sym (R:=R) (x:=Y) (y:=X))
| [ |- ?R ?A ?X ?Y ] => apply (equiv_sym (R:=R A) (x:=Y) (y:=X))
| [ |- ?R ?A ?B ?X ?Y ] => apply (equiv_sym (R:=R A B) (x:=Y) (y:=X))
| [ |- ?R ?A ?B ?C ?X ?Y ] => apply (equiv_sym (R:=R A B C) (x:=Y) (y:=X))
end.
Ltac sym := do_setoid_sym.
Ltac do_setoid_trans Y :=
match goal with
| [ |- @equiv ?A ?R ?s ?X ?Z ] => apply (equiv_trans (A:=A) (R:=R) (s:=s) (x:=X) (y:=Y) (z:=Z))
| [ |- ?R ?X ?Z ] => apply (equiv_trans (R:=R) (x:=X) (y:=Y) (z:=Z))
| [ |- ?R ?A ?X ?Z ] => apply (equiv_trans (R:=R A) (x:=X) (y:=Y) (z:=Z))
| [ |- ?R ?A ?B ?X ?Z ] => apply (equiv_trans (R:=R A B) (x:=X) (y:=Y) (z:=Z))
| [ |- ?R ?A ?B ?C ?X ?Z ] => apply (equiv_trans (R:=R A B C) (x:=X) (y:=Y) (z:=Z))
end.
Ltac trans y := do_setoid_trans y.
(** Tactics to immediately solve the goal without user help. *)
Ltac setoid_refl :=
match goal with
| [ |- @equiv ?A ?R ?s ?X _ ] => apply (equiv_refl (A:=A) (R:=R) (s:=s) X)
| [ |- ?R ?X _ ] => apply (equiv_refl (R:=R) X)
| [ |- ?R ?A ?X _ ] => apply (equiv_refl (R:=R A) X)
| [ |- ?R ?A ?B ?X _ ] => apply (equiv_refl (R:=R A B) X)
| [ |- ?R ?A ?B ?C ?X _ ] => apply (equiv_refl (R:=R A B C) X)
end.
Ltac setoid_sym :=
match goal with
| [ H : ?X == ?Y |- ?Y == ?X ] => apply (equiv_sym (x:=X) (y:=Y) H)
| [ H : ?X =!= ?Y |- ?Y =!= ?X ] => apply (nequiv_sym (x:=X) (y:=Y) H)
end.
Ltac setoid_trans :=
match goal with
| [ H : ?X == ?Y, H' : ?Y == ?Z |- @equiv ?A ?R ?s ?X ?Z ] => apply (equiv_trans (A:=A) (R:=R) (s:=s) (x:=X) (y:=Y) (z:=Z) H H')
end.
(** To immediatly solve a goal on setoid equality. *)
Ltac setoid_tac := setoid_refl || setoid_sym || setoid_trans.
Lemma nequiv_equiv : forall [ Setoid A ] (x y z : A), x =!= y -> y == z -> x =!= z.
Proof.
intros; intro.
assert(z == y) by setoid_sym.
assert(x == y) by setoid_trans.
contradiction.
Qed.
Lemma equiv_nequiv : forall [ Setoid A ] (x y z : A), x == y -> y =!= z -> x =!= z.
Proof.
intros; intro.
assert(y == x) by setoid_sym.
assert(y == z) by setoid_trans.
contradiction.
Qed.
Open Scope type_scope.
Ltac setoid_sat :=
let add H t := let name := fresh H in add_hypothesis name t in
match goal with
| [ H : ?x == ?y |- _ ] => let name:=fresh "Heq" y x in add name (equiv_sym H)
| [ H : ?x =!= ?y |- _ ] => let name:=fresh "Hneq" y x in add name (nequiv_sym H)
| [ H : ?x == ?y, H' : ?y == ?z |- _ ] => let name:=fresh "Heq" x z in add name (equiv_trans H H')
| [ H : ?x == ?y, H' : ?y =!= ?z |- _ ] => let name:=fresh "Hneq" x z in add name (equiv_nequiv H H')
| [ H : ?x =!= ?y, H' : ?y == ?z |- _ ] => let name:=fresh "Hneq" x z in add name (nequiv_equiv H H')
end.
Ltac setoid_saturate := repeat setoid_sat.
Ltac setoidify_tac :=
match goal with
| [ s : Setoid ?A ?R, H : ?R ?x ?y |- _ ] => change R with (@equiv A R s) in H
| [ s : Setoid ?A ?R |- ?R ?x ?y ] => change R with (@equiv A R s)
end.
Ltac setoidify := repeat setoidify_tac.
Definition respectful [ sa : Setoid a eqa, sb : Setoid b eqb ] (m : a -> b) : Prop :=
forall x y, eqa x y -> eqb (m x) (m y).
Class [ Setoid a eqa, Setoid b eqb ] => Morphism (m : a -> b) :=
respect : respectful m.
(** Here we build a setoid instance for functions which relates respectful ones only. *)
Definition respecting [ Setoid a R, Setoid b R' ] : Type := { morph : a -> b | respectful morph }.
Obligations Tactic := try red ; program_simpl ; unfold equiv in * ; try tauto.
Program Instance [ sa : Setoid a R, sb : Setoid b R' ] => arrow_setoid :
Setoid ({ morph : a -> b | respectful morph })
(fun (f g : respecting) => forall (x y : a), R x y -> R' (`f x) (`g y)) :=
equiv_prf := Build_equivalence _ _ _ _ _.
Next Obligation.
Proof.
trans (y x0) ; eauto.
apply H.
refl.
Qed.
Next Obligation.
Proof.
sym ; apply H.
sym ; auto.
Qed.
(** We redefine respect for binary and ternary morphims because we cannot get a satisfying instance of [Setoid (a -> b)] from
some arbitrary domain and codomain setoids. We can define it on respectful Coq functions though, see [arrow_setoid] above. *)
Definition binary_respectful [ sa : Setoid a eqa, sb : Setoid b eqb, Setoid c eqc ] (m : a -> b -> c) : Prop :=
forall x y, eqa x y ->
forall z w, eqb z w -> eqc (m x z) (m y w).
Class [ sa : Setoid a eqa, sb : Setoid b eqb, sc : Setoid c eqc ] => BinaryMorphism (m : a -> b -> c) :=
respect2 : binary_respectful m.
Definition ternary_respectful [ sa : Setoid a eqa, sb : Setoid b eqb, sc : Setoid c eqc, Setoid d eqd ] (m : a -> b -> c -> d) : Prop :=
forall x y, eqa x y -> forall z w, eqb z w -> forall u v, eqc u v -> eqd (m x z u) (m y w v).
Class [ sa : Setoid a eqa, sb : Setoid b eqb, sc : Setoid c eqc, sd : Setoid d eqd ] => TernaryMorphism (m : a -> b -> c -> d) :=
respect3 : ternary_respectful m.
(** Definition of the usual morphisms in [Prop]. *)
Program Instance iff_setoid : Setoid Prop iff :=
equiv_prf := @Build_equivalence _ _ iff_refl iff_trans iff_sym.
Program Instance not_morphism : Morphism Prop iff Prop iff not.
Program Instance and_morphism : ? BinaryMorphism iff_setoid iff_setoid iff_setoid and.
(* We make the setoids implicit, they're always [iff] *)
Implicit Arguments Enriching BinaryMorphism [[!sa] [!sb] [!sc]].
Program Instance or_morphism : ? BinaryMorphism or.
Definition impl (A B : Prop) := A -> B.
Program Instance impl_morphism : ? BinaryMorphism impl.
Next Obligation.
Proof.
unfold impl. tauto.
Qed.
(** Every setoid relation gives rise to a morphism, in fact every partial setoid does. *)
Program Instance [ Setoid a R ] => setoid_morphism : ? BinaryMorphism R.
Next Obligation.
Proof with auto.
split ; intros.
trans x. sym... trans z...
trans y... trans w... sym...
Qed.
Definition iff_morphism : BinaryMorphism iff := setoid_morphism.
Existing Instance iff_morphism.
Implicit Arguments eq [[A]].
Program Instance eq_setoid : Setoid A eq :=
equiv_prf := Build_equivalence _ _ _ _ _.
Program Instance eq_morphism : BinaryMorphism A eq A eq Prop iff eq.
Program Instance arrow_morphism : BinaryMorphism A eq B eq C eq m.
Implicit Arguments arrow_morphism [[A] [B] [C]].
Program Instance type_setoid : Setoid Type (fun x y => x = y) :=
equiv_prf := Build_equivalence _ _ _ _ _.
Lemma setoid_subst : forall (x y : Type), x == y -> x -> y.
Proof.
intros.
rewrite <- H.
apply X.
Qed.
Lemma prop_setoid_subst : forall (x y : Prop), x == y -> x -> y.
Proof.
intros.
clrewrite <- H.
apply H0.
Qed.
Program Instance [ sa : Setoid a eqa, sb : Setoid b eqb, sc : Setoid c eqc,
? Morphism sb sc g, ? Morphism sa sb f ] =>
compose_morphism : ? Morphism sa sc (fun x => g (f x)).
Next Obligation.
Proof.
apply (respect (m0:=m)).
apply (respect (m0:=m0)).
assumption.
Qed.
(** Partial setoids don't require reflexivity so we can build a partial setoid on the function space. *)
Class PartialSetoid (carrier : Type) (equiv : relation carrier) :=
pequiv_prf : PER carrier equiv.
(** Overloaded notation for partial setoid equivalence. *)
Definition pequiv [ PartialSetoid A R ] : _ := R.
Infix "=~=" := pequiv (at level 70, no associativity) : type_scope.
Definition pequiv_sym [ s : PartialSetoid A R ] : forall x y : A, R x y -> R y x := per_sym _ _ pequiv_prf.
Definition pequiv_trans [ s : PartialSetoid A R ] : forall x y z : A, R x y -> R y z -> R x z := per_trans _ _ pequiv_prf.
Program Instance [ sa : Setoid a R, sb : Setoid b R' ] => arrow_partial_setoid :
PartialSetoid (a -> b)
(fun f g => forall (x y : a), R x y -> R' (f x) (g y)) :=
pequiv_prf := Build_PER _ _ _ _.
Next Obligation.
Proof.
sym.
apply H.
sym ; assumption.
Qed.
Next Obligation.
Proof.
trans (y x0).
apply H ; auto.
trans y0 ; auto.
sym ; auto.
apply H0 ; auto.
Qed.
(** The following is not definable. *)
(*
Program Instance [ sa : Setoid a R, sb : Setoid b R' ] => arrow_setoid :
Setoid (a -> b)
(fun f g => forall (x y : a), R x y -> R' (f x) (g y)) :=
equiv_prf := Build_equivalence _ _ _ _ _.
*)
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