include "logic/cprop_connectives.ma".
+axiom daemon: False.
+
record iff (A,B: CProp) : CProp ≝
{ if: A → B;
fi: B → A
interpretation "logical iff" 'iff x y = (iff x y).
+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 setoid : Type ≝
{ carr:> Type;
eq: carr → carr → CProp;
constructor 1;
[ apply A
| intros;
- alias id "True" = "cic:/Coq/Init/Logic/True.ind#xpointer(1/1)".
apply True
| intro;
constructor 1
record setoid1 : Type ≝
{ carr1:> Type;
eq1: carr1 → carr1 → CProp;
- refl1: reflexive ? eq1;
- sym1: symmetric ? eq1;
- trans1: transitive ? eq1
+ 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
record function_space (A,B: setoid): Type ≝
{ f:1> A → B;
- f_ok: ∀a,a':A. eq ? a a' → eq ? (f a) (f a')
+ 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 }.
-interpretation "function_space" 'Imply a b = (function_space a b).
-record function_space1 (A: setoid) (B: setoid1): Type ≝
+record function_space1 (A: setoid1) (B: setoid1): Type ≝
{ f1:1> A → B;
- f1_ok: ∀a,a':A. eq ? a a' → eq1 ? (f1 a) (f1 a')
+ 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 (A ⇒ B);
+ [ apply (function_space A B);
| intros;
- apply (∀a:A. eq ? (f a) (f1 a));
+ 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 (H a)
+ apply (f a)
| simplify;
intros;
+ unfold carr; unfold proofs; simplify;
apply (trans B ? (y a));
- [ apply (H a)
- | apply (H1 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_setoid" 'Imply a b = (function_space_setoid a b).
+
+interpretation "function_space_setoid1" 'Imply a b = (function_space_setoid1 a b).
record isomorphism (A,B: setoid): Type ≝
- { map1:> A ⇒ B;
- map2:> B ⇒ A;
- inv1: ∀a:A. eq ? (map2 (map1 a)) a;
- inv2: ∀b:B. eq ? (map1 (map2 b)) b
+ { 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).
-axiom daemon: False.
-
definition setoids: setoid1.
constructor 1;
[ apply setoid;
|3,4:
intros;
simplify;
+ unfold proofs; simplify;
apply refl;]
- |*: elim daemon]
+ |*: cases daemon]
qed.
-record dependent_product (A:setoid) (B: function_space1 A setoids): Type ≝
+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:eq ? a a'. eq ? (dp a) (map2 ?? (f1_ok ?? B ?? p) (dp 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: function_space1 A CCProp): Type ≝
- { fo:> ∀a:A.proofs (B a)
- }.
+record forall (A:setoid) (B: A ⇒ CCProp): CProp ≝
+ { fo:> ∀a:A.proofs (B a) }.
-record subset (A: setoid) : Type ≝
- { mem: function_space1 A CCProp
- }.
-
-definition subset_eq ≝ λA:setoid.λU,V: subset A. ∀a:A. mem ? U a \liff mem ? V a.
+record subset (A: setoid) : CProp ≝
+ { mem: A ⇒ CCProp }.
-lemma mem_ok:
- ∀A:setoid.∀a,a':A.∀U,V: subset A.
- eq ? a a' → subset_eq ? U V → mem ? U a \liff mem ? V a'.
- intros;
- cases (H1 a);
- split; intro H4;
- [ lapply (H2 H4); clear H2 H3 H4;
- apply (if ?? (f1_ok ?? (mem ? V) ?? H));
+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
- | apply H3; clear H2 H3;
- apply (fi ?? (f1_ok ?? (mem ? V) ?? H));
- apply H4;]
+ | 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 / blobdiff