--- /dev/null
+include "logic/connectives.ma".
+include "properties/relations.ma".
+
+nrecord iff (A,B: CProp) : CProp ≝
+ { if: A → B;
+ fi: B → A
+ }.
+
+notation > "hvbox(a break \liff b)"
+ left associative with precedence 25
+for @{ 'iff $a $b }.
+
+notation "hvbox(a break \leftrightarrow b)"
+ left associative with precedence 25
+for @{ 'iff $a $b }.
+
+interpretation "logical iff" 'iff x y = (iff x y).
+
+nrecord setoid : Type[1] ≝
+ { carr:> Type;
+ eq: carr → carr → CProp;
+ refl: reflexive ? eq;
+ sym: symmetric ? eq;
+ trans: transitive ? eq
+ }.
+
+ndefinition proofs: CProp → setoid.
+#P; napply (mk_setoid ?????);
+##[ napply P;
+##| napply (λ_,_.True);
+ #x; #y; napply True; (* DIVERGE *)
+ intro;
+ constructor 1;
+ [ apply A
+ | intros;
+ apply True
+ | intro;
+ constructor 1
+ | intros 3;
+ constructor 1
+ | intros 5;
+ constructor 1]
+qed.
+
+definition reflexive1 ≝ λA:Type.λR:A→A→CProp.∀x:A.R x x.
+definition symmetric1 ≝ λC:Type.λlt:C→C→CProp. ∀x,y:C.lt x y → lt y x.
+definition transitive1 ≝ λA:Type.λR:A→A→CProp.∀x,y,z:A.R x y → R y z → R x z.
+
+record setoid1 : Type ≝
+ { carr1:> Type;
+ eq1: carr1 → carr1 → CProp;
+ refl1: reflexive1 ? eq1;
+ sym1: symmetric1 ? eq1;
+ trans1: transitive1 ? eq1
+ }.
+
+definition proofs1: CProp → setoid1.
+ intro;
+ constructor 1;
+ [ apply A
+ | intros;
+ apply True
+ | intro;
+ constructor 1
+ | intros 3;
+ constructor 1
+ | intros 5;
+ constructor 1]
+qed.
+
+definition CCProp: setoid1.
+ constructor 1;
+ [ apply CProp
+ | apply iff
+ | intro;
+ split;
+ intro;
+ assumption
+ | intros 3;
+ cases H; clear H;
+ split;
+ assumption
+ | intros 5;
+ cases H; cases H1; clear H H1;
+ split;
+ intros;
+ [ apply (H4 (H2 H))
+ | apply (H3 (H5 H))]]
+qed.
+
+record function_space (A,B: setoid): Type ≝
+ { f:1> A → B;
+ f_ok: ∀a,a':A. proofs (eq ? a a') → proofs (eq ? (f a) (f a'))
+ }.
+
+notation "hbox(a break ⇒ b)" right associative with precedence 20 for @{ 'Imply $a $b }.
+
+record function_space1 (A: setoid1) (B: setoid1): Type ≝
+ { f1:1> A → B;
+ f1_ok: ∀a,a':A. proofs1 (eq1 ? a a') → proofs1 (eq1 ? (f1 a) (f1 a'))
+ }.
+
+definition function_space_setoid: setoid → setoid → setoid.
+ intros (A B);
+ constructor 1;
+ [ apply (function_space A B);
+ | intros;
+ apply (∀a:A. proofs (eq ? (f a) (f1 a)));
+ | simplify;
+ intros;
+ apply (f_ok ? ? x);
+ unfold carr; unfold proofs; simplify;
+ apply (refl A)
+ | simplify;
+ intros;
+ unfold carr; unfold proofs; simplify;
+ apply (sym B);
+ apply (f a)
+ | simplify;
+ intros;
+ unfold carr; unfold proofs; simplify;
+ apply (trans B ? (y a));
+ [ apply (f a)
+ | apply (f1 a)]]
+qed.
+
+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).
+
+record isomorphism (A,B: setoid): Type ≝
+ { map1:> function_space_setoid A B;
+ map2:> function_space_setoid B A;
+ inv1: ∀a:A. proofs (eq ? (map2 (map1 a)) a);
+ inv2: ∀b:B. proofs (eq ? (map1 (map2 b)) b)
+ }.
+
+interpretation "isomorphism" 'iff x y = (isomorphism x y).
+
+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 dependent_product (A:setoid) (B: A ⇒ setoids): Type ≝
+ { dp:> ∀a:A.carr (B a);
+ dp_ok: ∀a,a':A. ∀p:proofs1 (eq1 ? a a'). proofs1 (eq1 ? (dp a) (map2 ?? (f1_ok ?? B ?? p) (dp a')))
+ }.*)
+
+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);
+ |
+ |
+ |
+*)
projects / helm.git / commitdiff
