(* *)
(**************************************************************************)
+include "properties/relations1.ma".
include "sets/setoids.ma".
+include "hints_declaration.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
+nrecord setoid1: Type[2] ≝
+ { carr1:> Type[1];
+ eq1: equivalence_relation1 carr1
}.
-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'))
+ndefinition setoid1_of_setoid: setoid → setoid1.
+ #s; napply mk_setoid1
+ [ napply (carr s)
+ | napply (mk_equivalence_relation1 s)
+ [ napply eq
+ | napply refl
+ | napply sym
+ | napply trans]##]
+nqed.
+
+(*ncoercion setoid1_of_setoid : ∀s:setoid. setoid1 ≝ setoid1_of_setoid
+ on _s: setoid to setoid1.*)
+(*prefer coercion Type_OF_setoid.*)
+
+interpretation "setoid1 eq" 'eq t x y = (eq_rel1 ? (eq1 t) x y).
+interpretation "setoid eq" 'eq t x y = (eq_rel ? (eq t) x y).
+
+notation > "hvbox(a break =_12 b)" non associative with precedence 45
+for @{ eq_rel2 (carr2 (setoid2_of_setoid1 ?)) (eq2 (setoid2_of_setoid1 ?)) $a $b }.
+notation > "hvbox(a break =_0 b)" non associative with precedence 45
+for @{ eq_rel ? (eq ?) $a $b }.
+notation > "hvbox(a break =_1 b)" non associative with precedence 45
+for @{ eq_rel1 ? (eq1 ?) $a $b }.
+
+interpretation "setoid1 symmetry" 'invert r = (sym1 ???? r).
+interpretation "setoid symmetry" 'invert r = (sym ???? r).
+notation ".= r" with precedence 50 for @{'trans $r}.
+interpretation "trans1" 'trans r = (trans1 ????? r).
+interpretation "trans" 'trans r = (trans ????? r).
+
+nrecord unary_morphism1 (A,B: setoid1) : Type[1] ≝
+ { fun11:1> A → B;
+ prop11: ∀a,a'. eq1 ? a a' → eq1 ? (fun11 a) (fun11 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);
- |
- |
- |
-
-*)
+interpretation "prop11" 'prop1 c = (prop11 ????? c).
+interpretation "refl1" 'refl = (refl1 ???).
+
+ndefinition unary_morphism1_setoid1: setoid1 → setoid1 → setoid1.
+ #s; #s1; @ (unary_morphism1 s s1); @
+ [ #f; #g; napply (∀a,a':s. a=a' → f a = g a')
+ | #x; #a; #a'; #Ha; napply (.= †Ha); napply refl1
+ | #x; #y; #H; #a; #a'; #Ha; napply (.= †Ha); napply sym1; /2/
+ | #x; #y; #z; #H1; #H2; #a; #a'; #Ha; napply (.= †Ha); napply trans1; ##[##2: napply H1 | ##skip | napply H2]//;##]
+nqed.
+
+unification hint 0 ≔ S, T ;
+ R ≟ (unary_morphism1_setoid1 S T)
+(* --------------------------------- *) ⊢
+ carr1 R ≡ unary_morphism1 S T.
+
+interpretation "prop21" 'prop2 l r = (prop11 ? (unary_morphism1_setoid1 ??) ? ?? l ?? r).
+
+nlemma unary_morph1_eq1: ∀A,B.∀f,g: unary_morphism1 A B. (∀x. f x = g x) → f=g.
+/3/. nqed.
+
+nlemma mk_binary_morphism1:
+ ∀A,B,C: setoid1. ∀f: A → B → C. (∀a,a',b,b'. a=a' → b=b' → f a b = f a' b') →
+ unary_morphism1 A (unary_morphism1_setoid1 B C).
+ #A; #B; #C; #f; #H; @ [ #x; @ (f x) ] #a; #a'; #Ha [##2: napply unary_morph1_eq1; #y]
+ /2/.
+nqed.
+
+ndefinition composition1 ≝
+ λo1,o2,o3:Type[1].λf:o2 → o3.λg: o1 → o2.λx.f (g x).
+
+interpretation "function composition" 'compose f g = (composition ??? f g).
+interpretation "function composition1" 'compose f g = (composition1 ??? f g).
+
+ndefinition comp1_unary_morphisms:
+ ∀o1,o2,o3:setoid1.
+ unary_morphism1 o2 o3 → unary_morphism1 o1 o2 →
+ unary_morphism1 o1 o3.
+#o1; #o2; #o3; #f; #g; @ (f ∘ g);
+ #a; #a'; #e; nnormalize; napply (.= †(†e)); napply #.
+nqed.
+
+unification hint 0 ≔ o1,o2,o3:setoid1,f:unary_morphism1 o2 o3,g:unary_morphism1 o1 o2;
+ R ≟ (mk_unary_morphism1 ?? (composition1 … f g)
+ (prop11 ?? (comp1_unary_morphisms o1 o2 o3 f g)))
+ (* -------------------------------------------------------------------- *) ⊢
+ fun11 ?? R ≡ (composition1 … f g).
+
+ndefinition comp1_binary_morphisms:
+ ∀o1,o2,o3.
+ unary_morphism1 (unary_morphism1_setoid1 o2 o3)
+ (unary_morphism1_setoid1 (unary_morphism1_setoid1 o1 o2) (unary_morphism1_setoid1 o1 o3)).
+#o1; #o2; #o3; napply mk_binary_morphism1
+ [ #f; #g; napply (comp1_unary_morphisms … f g) (*CSC: why not ∘?*)
+ | #a; #a'; #b; #b'; #ea; #eb; #x; #x'; #Hx; nnormalize; /3/ ]
+nqed.
\ No newline at end of file
projects / helm.git / blobdiff