+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
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;
##]
nqed.
-(*
-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.
-*)
-
-(*
-ndefinition 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.
-
-*)
-
-(************************CSC
-nrecord function_space (A,B: setoid): Type[1] ≝
- { f:1> carr A → carr B}.;
+nrecord 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);
+ndefinition function_space_setoid: setoid → setoid → setoid.
+ #A; #B; napply (mk_setoid ?????);
+##[ napply (function_space A B);
+##| #f; #f1; napply (∀a:A. proofs (eq ? (f a) (f1 a)));
+##| nwhd; #x; #a;
+ napply (f_ok ? ? x ? ? ?); (* QUI!! *)
unfold carr; unfold proofs; simplify;
apply (refl A)
| simplify;
| 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 ≝
+nrecord 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);
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);
- |
- |
- |
-*)
-*******************)
+
+ *)
\ No newline at end of file
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "sets/setoids.ma".
+
+(*
+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.
+
+ndefinition 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_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_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);
+ |
+ |
+ |
+
+*)
projects / helm.git / commitdiff
