+(**************************************************************************)
+(* ___ *)
+(* ||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;
- refl: reflexive ? eq;
- sym: symmetric ? eq;
- trans: transitive ? eq
- }.
-
-ndefinition proofs: CProp → setoid.
-#P; napply (mk_setoid ?????);
-##[ napply P;
-##| #x; #y; napply True;
-##|##*: nwhd; nrepeat (#_); napply I;
-##]
-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.
-*)
+notation "hvbox(a break = \sub \ID b)" non associative with precedence 45
+for @{ 'eqID $a $b }.
-(*
-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.
+notation > "hvbox(a break =_\ID b)" non associative with precedence 45
+for @{ cic:/matita/logic/equality/eq.ind#xpointer(1/1) ? $a $b }.
+interpretation "ID eq" 'eqID x y = (cic:/matita/logic/equality/eq.ind#xpointer(1/1) ? x y).
*)
-(************************CSC
-nrecord function_space (A,B: setoid): Type[1] ≝
- { f:1> carr A → carr 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'))
+nrecord setoid : Type[1] ≝
+ { carr:> Type[0];
+ eq: equivalence_relation carr
}.
-*)
-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.
+interpretation "setoid eq" 'eq t x y = (eq_rel ? (eq t) x y).
-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.
+notation > "hvbox(a break =_0 b)" non associative with precedence 45
+for @{ eq_rel ? (eq ?) $a $b }.
-interpretation "function_space_setoid1" 'Imply a b = (function_space_setoid1 a b).
+interpretation "setoid symmetry" 'invert r = (sym ???? r).
+notation ".= r" with precedence 50 for @{'trans $r}.
+interpretation "trans" 'trans r = (trans ????? r).
-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)
+nrecord unary_morphism (A,B: setoid) : Type[0] ≝
+ { fun1:1> A → B;
+ prop1: ∀a,a'. eq ? a a' → eq ? (fun1 a) (fun1 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.
+nrecord binary_morphism (A,B,C:setoid) : Type[0] ≝
+ { fun2:2> A → B → C;
+ prop2: ∀a,a',b,b'. eq ? a a' → eq ? b b' → eq ? (fun2 a b) (fun2 a' b')
+ }.
-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);
- |
- |
- |
-*)
-*******************)
+notation "† c" with precedence 90 for @{'prop1 $c }.
+notation "l ‡ r" with precedence 90 for @{'prop2 $l $r }.
+notation "#" with precedence 90 for @{'refl}.
+interpretation "prop1" 'prop1 c = (prop1 ????? c).
+interpretation "prop2" 'prop2 l r = (prop2 ???????? l r).
+interpretation "refl" 'refl = (refl ???).
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