(* *)
(**************************************************************************)
-include "logic/cprop_connectives.ma".
-
-definition Type0 := Type.
-definition Type1 := Type.
-definition Type2 := Type.
-definition Type3 := Type.
-definition Type0_lt_Type1 := (Type0 : Type1).
-definition Type1_lt_Type2 := (Type1 : Type2).
-definition Type2_lt_Type3 := (Type2 : Type3).
-
-definition Type_OF_Type0: Type0 → Type := λx.x.
-definition Type_OF_Type1: Type1 → Type := λx.x.
-definition Type_OF_Type2: Type2 → Type := λx.x.
-definition Type_OF_Type3: Type3 → Type := λx.x.
-coercion Type_OF_Type0.
-coercion Type_OF_Type1.
-coercion Type_OF_Type2.
-coercion Type_OF_Type3.
-
-definition CProp0 := CProp.
-definition CProp1 := CProp.
-definition CProp2 := CProp.
-definition CProp0_lt_CProp1 := (CProp0 : CProp1).
-definition CProp1_lt_CProp2 := (CProp1 : CProp2).
-
-definition CProp_OF_CProp0: CProp0 → CProp := λx.x.
-definition CProp_OF_CProp1: CProp1 → CProp := λx.x.
-definition CProp_OF_CProp2: CProp2 → CProp := λx.x.
+include "cprop_connectives.ma".
record equivalence_relation (A:Type0) : Type1 ≝
{ eq_rel:2> A → A → CProp0;
| constructor 1;
[ apply Iff
| intros 1; split; intro; assumption
- | intros 3; cases H; split; assumption
- | intros 5; cases H; cases H1; split; intro;
- [ apply (H4 (H2 x1)) | apply (H3 (H5 z1))]]]
+ | intros 3; cases i; split; assumption
+ | intros 5; cases i; cases i1; split; intro;
+ [ apply (f2 (f x1)) | apply (f1 (f3 z1))]]]
qed.
alias symbol "eq" = "setoid1 eq".
definition and_morphism: binary_morphism1 CPROP CPROP CPROP.
constructor 1;
[ apply And
- | intros; split; intro; cases H; split;
- [ apply (if ?? e a1)
+ | intros; split; intro; cases a1; split;
+ [ apply (if ?? e a2)
| apply (if ?? e1 b1)
- | apply (fi ?? e a1)
+ | apply (fi ?? e a2)
| apply (fi ?? e1 b1)]]
qed.
definition or_morphism: binary_morphism1 CPROP CPROP CPROP.
constructor 1;
[ apply Or
- | intros; split; intro; cases H; [1,3:left |2,4: right]
+ | intros; split; intro; cases o; [1,3:left |2,4: right]
[ apply (if ?? e a1)
| apply (fi ?? e a1)
| apply (if ?? e1 b1)
constructor 1;
[ apply (λA,B. A → B)
| intros; split; intros;
- [ apply (if ?? e1); apply H; apply (fi ?? e); assumption
- | apply (fi ?? e1); apply H; apply (if ?? e); assumption]]
+ [ apply (if ?? e1); apply f; apply (fi ?? e); assumption
+ | apply (fi ?? e1); apply f; apply (if ?? e); assumption]]
qed.
(*
| constructor 1;
[ intros (f g); apply (∀a:s. eq ? (f a) (g a));
| intros 1; simplify; intros; apply refl;
- | simplify; intros; apply sym; apply H;
- | simplify; intros; apply trans; [2: apply H; | skip | apply H1]]]
+ | simplify; intros; apply sym; apply f;
+ | simplify; intros; apply trans; [2: apply f; | skip | apply f1]]]
qed.
definition SET: category1.
alias symbol "eq" = "setoid1 eq".
apply (∀a: carr1 s. f a = g a);
| intros 1; simplify; intros; apply refl1;
- | simplify; intros; apply sym1; apply H;
- | simplify; intros; apply trans1; [2: apply H; | skip | apply H1]]]
+ | simplify; intros; apply sym1; apply f;
+ | simplify; intros; apply trans1; [2: apply f; | skip | apply f1]]]
qed.
definition SET1: category2.