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-rw-r--r--theories/Setoids/Setoid.v116
1 files changed, 105 insertions, 11 deletions
diff --git a/theories/Setoids/Setoid.v b/theories/Setoids/Setoid.v
index d98f94419..91745c98b 100644
--- a/theories/Setoids/Setoid.v
+++ b/theories/Setoids/Setoid.v
@@ -25,6 +25,8 @@ Inductive X_Relation_Class (X: Type) : Type :=
SymmetricReflexive :
forall A Aeq, @is_symmetric A Aeq -> is_reflexive Aeq -> X_Relation_Class X
| AsymmetricReflexive : X -> forall A Aeq, @is_reflexive A Aeq -> X_Relation_Class X
+ | Symmetric : forall A Aeq, @is_symmetric A Aeq -> X_Relation_Class X
+ | Asymmetric : X -> forall A (Aeq : A -> A -> Prop), X_Relation_Class X
| Leibniz : Type -> X_Relation_Class X.
Inductive variance : Set :=
@@ -40,6 +42,8 @@ Definition relation_class_of_argument_class : Argument_Class -> Relation_Class.
destruct 1.
exact (SymmetricReflexive _ i i0).
exact (AsymmetricReflexive tt i).
+ exact (Symmetric _ i).
+ exact (Asymmetric tt Aeq).
exact (Leibniz _ T).
Defined.
@@ -47,6 +51,8 @@ Definition carrier_of_relation_class : forall X, X_Relation_Class X -> Type.
destruct 1.
exact A.
exact A.
+ exact A.
+ exact A.
exact T.
Defined.
@@ -55,6 +61,8 @@ Definition relation_of_relation_class :
destruct R.
exact Aeq.
exact Aeq.
+ exact Aeq.
+ exact Aeq.
exact (@eq T).
Defined.
@@ -90,12 +98,20 @@ Definition make_compatibility_goal_aux:
destruct x.
exact (forall x1 x2, Aeq x1 x2 -> relation_of_relation_class Out (f x1) (g x2)).
exact (forall x1 x2, Aeq x2 x1 -> relation_of_relation_class Out (f x1) (g x2)).
+ exact (forall x1 x2, Aeq x1 x2 -> relation_of_relation_class Out (f x1) (g x2)).
+ destruct x.
+ exact (forall x1 x2, Aeq x1 x2 -> relation_of_relation_class Out (f x1) (g x2)).
+ exact (forall x1 x2, Aeq x2 x1 -> relation_of_relation_class Out (f x1) (g x2)).
exact (forall x, relation_of_relation_class Out (f x) (g x)).
induction a; simpl in f, g.
exact (forall x1 x2, Aeq x1 x2 -> IHIn (f x1) (g x2)).
destruct x.
exact (forall x1 x2, Aeq x1 x2 -> IHIn (f x1) (g x2)).
exact (forall x1 x2, Aeq x2 x1 -> IHIn (f x1) (g x2)).
+ exact (forall x1 x2, Aeq x1 x2 -> IHIn (f x1) (g x2)).
+ destruct x.
+ exact (forall x1 x2, Aeq x1 x2 -> IHIn (f x1) (g x2)).
+ exact (forall x1 x2, Aeq x2 x1 -> IHIn (f x1) (g x2)).
exact (forall x, IHIn (f x) (g x)).
Defined.
@@ -162,6 +178,8 @@ Definition variance_of_argument_class : Argument_Class -> option variance.
exact None.
exact (Some v).
exact None.
+ exact (Some v).
+ exact None.
Defined.
Definition opposite_direction :=
@@ -185,15 +203,55 @@ Inductive check_if_variance_is_respected :
forall dir,
check_if_variance_is_respected (Some Contravariant) dir (opposite_direction dir).
+Inductive Reflexive_Relation_Class : Type :=
+ RSymmetricReflexive :
+ forall A Aeq, @is_symmetric A Aeq -> is_reflexive Aeq -> Reflexive_Relation_Class
+ | RAsymmetricReflexive :
+ forall A Aeq, @is_reflexive A Aeq -> Reflexive_Relation_Class
+ | RLeibniz : Type -> Reflexive_Relation_Class.
+
+Inductive Areflexive_Relation_Class : Type :=
+ | ASymmetric : forall A Aeq, @is_symmetric A Aeq -> Areflexive_Relation_Class
+ | AAsymmetric : forall A (Aeq : A -> A -> Prop), Areflexive_Relation_Class.
+
+Definition relation_class_of_reflexive_relation_class:
+ Reflexive_Relation_Class -> Relation_Class.
+ induction 1.
+ exact (SymmetricReflexive _ i i0).
+ exact (AsymmetricReflexive tt i).
+ exact (Leibniz _ T).
+Defined.
+
+Definition relation_class_of_areflexive_relation_class:
+ Areflexive_Relation_Class -> Relation_Class.
+ induction 1.
+ exact (Symmetric _ i).
+ exact (Asymmetric tt Aeq).
+Defined.
+
+Definition carrier_of_reflexive_relation_class :=
+ fun R => carrier_of_relation_class (relation_class_of_reflexive_relation_class R).
+
+Definition carrier_of_areflexive_relation_class :=
+ fun R => carrier_of_relation_class (relation_class_of_areflexive_relation_class R).
+
+Definition relation_of_areflexive_relation_class :=
+ fun R => relation_of_relation_class (relation_class_of_areflexive_relation_class R).
+
Inductive Morphism_Context Hole dir : Relation_Class -> rewrite_direction -> Type :=
App :
forall In Out dir',
Morphism_Theory In Out -> Morphism_Context_List Hole dir dir' In ->
Morphism_Context Hole dir Out dir'
- | Toreplace : Morphism_Context Hole dir Hole dir
- | Tokeep :
+ | ToReplace : Morphism_Context Hole dir Hole dir
+ | ToKeep :
forall S dir',
- carrier_of_relation_class S -> Morphism_Context Hole dir S dir'
+ carrier_of_reflexive_relation_class S ->
+ Morphism_Context Hole dir (relation_class_of_reflexive_relation_class S) dir'
+ | ProperElementToKeep :
+ forall S dir' (x: carrier_of_areflexive_relation_class S),
+ relation_of_areflexive_relation_class S x x ->
+ Morphism_Context Hole dir (relation_class_of_areflexive_relation_class S) dir'
with Morphism_Context_List Hole dir :
rewrite_direction -> Arguments -> Type
:=
@@ -287,9 +345,11 @@ Theorem apply_morphism_compatibility_Right2Left:
destruct a; simpl in H; hnf in H0.
apply H; exact H0.
destruct v; simpl in H0; apply H; exact H0.
+ apply H; exact H0.
+ destruct v; simpl in H0; apply H; exact H0.
rewrite H0; apply H; exact H0.
- simpl in m1, m2, args1, args2, H0 |- *.
+ simpl in m1, m2, args1, args2, H0 |- *.
destruct args1; destruct args2; simpl.
destruct H0.
simpl in H.
@@ -304,6 +364,16 @@ Theorem apply_morphism_compatibility_Right2Left:
apply IHIn.
apply H; exact H0.
exact H1.
+ apply IHIn.
+ apply H; exact H0.
+ exact H1.
+ destruct v.
+ apply IHIn.
+ apply H; exact H0.
+ exact H1.
+ apply IHIn.
+ apply H; exact H0.
+ exact H1.
rewrite H0; apply IHIn.
apply H.
exact H1.
@@ -322,6 +392,8 @@ Theorem apply_morphism_compatibility_Left2Right:
destruct a; simpl in H; hnf in H0.
apply H; exact H0.
destruct v; simpl in H0; apply H; exact H0.
+ apply H; exact H0.
+ destruct v; simpl in H0; apply H; exact H0.
rewrite H0; apply H; exact H0.
simpl in m1, m2, args1, args2, H0 |- *.
@@ -339,6 +411,12 @@ Theorem apply_morphism_compatibility_Left2Right:
apply IHIn.
apply H; exact H0.
exact H1.
+ apply IHIn.
+ apply H; exact H0.
+ exact H1.
+ apply IHIn.
+ destruct v; simpl in H, H0; apply H; exact H0.
+ exact H1.
rewrite H0; apply IHIn.
apply H.
exact H1.
@@ -355,6 +433,7 @@ Definition interp :
exact (apply_morphism _ _ (Function m) X).
exact H.
exact c.
+ exact x.
simpl;
rewrite <-
(about_carrier_of_relation_class_and_relation_class_of_argument_class S);
@@ -380,6 +459,7 @@ Definition interp_relation_class_list :
exact (apply_morphism _ _ (Function m) X).
exact H.
exact c.
+ exact x.
simpl;
rewrite <-
(about_carrier_of_relation_class_and_relation_class_of_argument_class S);
@@ -427,17 +507,27 @@ Theorem setoid_rewrite:
reflexivity.
reflexivity.
+ destruct dir'0; exact r.
+
destruct S; unfold directed_relation_of_argument_class; simpl in H0 |- *;
unfold get_rewrite_direction; simpl.
destruct dir'0; destruct dir'';
(exact H0 ||
unfold directed_relation_of_argument_class; simpl; apply i; exact H0).
- (* the following mess with generalize/clear/intros is to help Coq resolving *)
- (* second order unification problems. *)
- generalize m c H0; clear H0 m c; inversion c;
- generalize m c; clear m c; rewrite <- H1; rewrite <- H2; intros;
- (exact H3 || rewrite (opposite_direction_idempotent dir'0); apply H3).
- destruct dir'0; destruct dir''; (exact H0 || hnf; symmetry; exact H0).
+ (* the following mess with generalize/clear/intros is to help Coq resolving *)
+ (* second order unification problems. *)
+ generalize m c H0; clear H0 m c; inversion c;
+ generalize m c; clear m c; rewrite <- H1; rewrite <- H2; intros;
+ (exact H3 || rewrite (opposite_direction_idempotent dir'0); apply H3).
+ destruct dir'0; destruct dir'';
+ (exact H0 ||
+ unfold directed_relation_of_argument_class; simpl; apply i; exact H0).
+(* the following mess with generalize/clear/intros is to help Coq resolving *)
+ (* second order unification problems. *)
+ generalize m c H0; clear H0 m c; inversion c;
+ generalize m c; clear m c; rewrite <- H1; rewrite <- H2; intros;
+ (exact H3 || rewrite (opposite_direction_idempotent dir'0); apply H3).
+ destruct dir'0; destruct dir''; (exact H0 || hnf; symmetry; exact H0).
change
(directed_relation_of_argument_class (get_rewrite_direction dir'' S) S
@@ -453,7 +543,11 @@ Theorem setoid_rewrite:
inversion c.
rewrite <- H3; exact H0.
rewrite (opposite_direction_idempotent dir'0); exact H0.
- destruct dir''; destruct dir'0; (exact H0 || hnf; symmetry; exact H0).
+ destruct dir''; destruct dir'0; (exact H0 || hnf; apply i; exact H0).
+ inversion c.
+ rewrite <- H3; exact H0.
+ rewrite (opposite_direction_idempotent dir'0); exact H0.
+ destruct dir''; destruct dir'0; (exact H0 || hnf; symmetry; exact H0).
exact H1.
Qed.
generated by cgit v1.2.3 (git 2.39.1) at 2024-11-26 18:32:44 +0000