-definition function_space_setoid1: setoid1 → setoid1 → setoid1.
- intros (A B);
- constructor 1;
- [ apply (function_space1 A B);
- | intros;
- apply (∀a:A. proofs1 (eq1 ? (f a) (f1 a)));
- |*: cases daemon] (* simplify;
- intros;
- apply (f1_ok ? ? x);
- unfold proofs; simplify;
- apply (refl1 A)
- | simplify;
- intros;
- unfold proofs; simplify;
- apply (sym1 B);
- apply (f a)
- | simplify;
- intros;
- unfold carr; unfold proofs; simplify;
- apply (trans1 B ? (y a));
- [ apply (f a)
- | apply (f1 a)]] *)
-qed.
-
-interpretation "function_space_setoid1" 'Imply a b = (function_space_setoid1 a b).
-
-definition setoids: setoid1.
- constructor 1;
- [ apply setoid;
- | apply isomorphism;
- | intro;
- split;
- [1,2: constructor 1;
- [1,3: intro; assumption;
- |*: intros; assumption]
- |3,4:
- intros;
- simplify;
- unfold proofs; simplify;
- apply refl;]
- |*: cases daemon]
-qed.
-
-definition setoid1_of_setoid: setoid → setoid1.
- intro;
- constructor 1;
- [ apply (carr s)
- | apply (eq s)
- | apply (refl s)
- | apply (sym s)
- | apply (trans s)]
-qed.
-
-coercion setoid1_of_setoid.
-
-record forall (A:setoid) (B: A ⇒ CCProp): CProp ≝
- { fo:> ∀a:A.proofs (B a) }.
-
-record subset (A: setoid) : CProp ≝
- { mem: A ⇒ CCProp }.
-
-definition ssubset: setoid → setoid1.
- intro;
- constructor 1;
- [ apply (subset s);
- | apply (λU,V:subset s. ∀a. mem ? U a \liff mem ? V a)
- | simplify;
- intros;
- split;
- intro;
- assumption
- | simplify;
- cases daemon
- | cases daemon]
-qed.
-
-definition mmem: ∀A:setoid. (ssubset A) ⇒ A ⇒ CCProp.
- intros;
- constructor 1;
- [ apply mem;
- | unfold function_space_setoid1; simplify;
- intros (b b');
- change in ⊢ (? (? (?→? (? %)))) with (mem ? b a \liff mem ? b' a);
- unfold proofs1; simplify; intros;
- unfold proofs1 in c; simplify in c;
- unfold ssubset in c; simplify in c;
- cases (c a); clear c;
- split;
- assumption]
-qed.
-
-definition sand: CCProp ⇒ CCProp.
-
-definition intersection: ∀A. ssubset A ⇒ ssubset A ⇒ ssubset A.
- intro;
- constructor 1;
- [ intro;
- constructor 1;
- [ intro;
- constructor 1;
- constructor 1;
- intro;
- apply (mem ? c c2 ∧ mem ? c1 c2);
- |
- |
- |