(* *)
(**************************************************************************)
-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 := Type0.
-definition CProp1 := Type1.
-definition CProp2 := Type2.
-(*
-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;
+ | intros 3; cases i; split; assumption
+ | intros 5; cases i; cases i1; split; intro;
[ apply (f2 (f x1)) | apply (f1 (f3 z1))]]]
qed.
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)
qed.
coercion Type1_OF_SET1.
+definition Type_OF_setoid1_of_carr: ∀U. carr U → Type_OF_setoid1 ?(*(setoid1_of_SET U)*).
+ [ apply setoid1_of_SET; apply U
+ | intros; apply c;]
+qed.
+coercion Type_OF_setoid1_of_carr.
+
interpretation "SET dagger" 'prop1 h = (prop11_SET1 _ _ _ _ _ h).
interpretation "unary morphism1" 'Imply a b = (arrows2 SET1 a b).
interpretation "SET1 eq" 'eq x y = (eq_rel1 _ (eq'' _) x y).