include "logic/connectives.ma".
include "properties/relations.ma".
-include "hints_declaration.ma".
-nrecord setoid : Type[1] ≝ {
- carr:> Type[0];
- eq0: equivalence_relation carr
-}.
+(*
+notation "hvbox(a break = \sub \ID b)" non associative with precedence 45
+for @{ 'eqID $a $b }.
-(* activate non uniform coercions on: Type → setoid *)
-unification hint 0 ≔ R : setoid;
- MR ≟ carr R,
- lock ≟ mk_lock1 Type[0] MR setoid R
-(* ---------------------------------------- *) ⊢
- setoid ≡ force1 ? MR lock.
+notation > "hvbox(a break =_\ID b)" non associative with precedence 45
+for @{ cic:/matita/logic/equality/eq.ind#xpointer(1/1) ? $a $b }.
-notation < "[\setoid\ensp\of term 19 x]" non associative with precedence 90 for @{'mk_setoid $x}.
-interpretation "mk_setoid" 'mk_setoid x = (mk_setoid x ?).
+interpretation "ID eq" 'eqID x y = (cic:/matita/logic/equality/eq.ind#xpointer(1/1) ? x y).
+*)
-interpretation "setoid eq" 'eq t x y = (eq_rel ? (eq0 t) x y).
-(* single = is for the abstract equality of setoids, == is for concrete
- equalities (that may be lifted to the setoid level when needed *)
-notation < "hvbox(a break mpadded width -50% (=)= b)" non associative with precedence 45 for @{ 'eq_low $a $b }.
-notation > "a == b" non associative with precedence 45 for @{ 'eq_low $a $b }.
+nrecord setoid : Type[1] ≝
+ { carr:> Type[0];
+ eq: equivalence_relation carr
+ }.
+
+interpretation "setoid eq" 'eq t x y = (eq_rel ? (eq t) x y).
notation > "hvbox(a break =_0 b)" non associative with precedence 45
-for @{ eq_rel ? (eq0 ?) $a $b }.
+for @{ eq_rel ? (eq ?) $a $b }.
interpretation "setoid symmetry" 'invert r = (sym ???? r).
notation ".= r" with precedence 50 for @{'trans $r}.
interpretation "trans" 'trans r = (trans ????? r).
-notation > ".=_0 r" with precedence 50 for @{'trans_x0 $r}.
-interpretation "trans_x0" 'trans_x0 r = (trans ????? r).
-nrecord unary_morphism (A,B: setoid) : Type[0] ≝ {
- fun1:1> A → B;
- prop1: ∀a,a'. a = a' → fun1 a = fun1 a'
-}.
+nrecord unary_morphism (A,B: setoid) : Type[0] ≝
+ { fun1:1> A → B;
+ prop1: ∀a,a'. eq ? a a' → eq ? (fun1 a) (fun1 a')
+ }.
-notation > "B ⇒_0 C" right associative with precedence 72 for @{'umorph0 $B $C}.
-notation "hvbox(B break ⇒\sub 0 C)" right associative with precedence 72 for @{'umorph0 $B $C}.
-interpretation "unary morphism 0" 'umorph0 A B = (unary_morphism A B).
+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')
+ }.
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 ???).
-notation "┼_0 c" with precedence 89 for @{'prop1_x0 $c }.
-notation "l ╪_0 r" with precedence 89 for @{'prop2_x0 $l $r }.
-interpretation "prop1_x0" 'prop1_x0 c = (prop1 ????? c).
-
-ndefinition unary_morph_setoid : setoid → setoid → setoid.
-#S1; #S2; @ (S1 ⇒_0 S2); @;
-##[ #f; #g; napply (∀x,x'. x=x' → f x = g x');
-##| #f; #x; #x'; #Hx; napply (.= †Hx); napply #;
-##| #f; #g; #H; #x; #x'; #Hx; napply ((H … Hx^-1)^-1);
-##| #f; #g; #h; #H1; #H2; #x; #x'; #Hx; napply (trans … (H1 …) (H2 …)); //; ##]
-nqed.
-
-alias symbol "hint_decl" (instance 1) = "hint_decl_Type1".
-unification hint 0 ≔ o1,o2 ;
- X ≟ unary_morph_setoid o1 o2
- (* ----------------------------- *) ⊢
- carr X ≡ o1 ⇒_0 o2.
-
-interpretation "prop2" 'prop2 l r = (prop1 ? (unary_morph_setoid ??) ? ?? l ?? r).
-interpretation "prop2_x0" 'prop2_x0 l r = (prop1 ? (unary_morph_setoid ??) ? ?? l ?? r).
-
-nlemma unary_morph_eq: ∀A,B.∀f,g:A ⇒_0 B. (∀x. f x = g x) → f = g.
-#A B f g H x1 x2 E; napply (.= †E); napply H; nqed.
-
-nlemma mk_binary_morphism:
- ∀A,B,C: setoid. ∀f: A → B → C. (∀a,a',b,b'. a=a' → b=b' → f a b = f a' b') →
- A ⇒_0 (unary_morph_setoid B C).
- #A; #B; #C; #f; #H; @; ##[ #x; @ (f x) ] #a; #a'; #Ha [##2: napply unary_morph_eq; #y]
- /2/.
-nqed.
-
-ndefinition composition ≝
- λo1,o2,o3:Type[0].λf:o2 → o3.λg: o1 → o2.λx.f (g x).
-
-interpretation "function composition" 'compose f g = (composition ??? f g).
-
-ndefinition comp_unary_morphisms:
- ∀o1,o2,o3:setoid.o2 ⇒_0 o3 → o1 ⇒_0 o2 → o1 ⇒_0 o3.
-#o1; #o2; #o3; #f; #g; @ (f ∘ g);
- #a; #a'; #e; nnormalize; napply (.= †(†e)); napply #.
-nqed.
-
-unification hint 0 ≔ o1,o2,o3:setoid,f:o2 ⇒_0 o3,g:o1 ⇒_0 o2;
- R ≟ mk_unary_morphism o1 o3
- (composition ??? (fun1 o2 o3 f) (fun1 o1 o2 g))
- (prop1 o1 o3 (comp_unary_morphisms o1 o2 o3 f g))
- (* -------------------------------------------------------------------- *) ⊢
- fun1 o1 o3 R ≡ composition ??? (fun1 o2 o3 f) (fun1 o1 o2 g).
-
-ndefinition comp_binary_morphisms:
- ∀o1,o2,o3.(o2 ⇒_0 o3) ⇒_0 ((o1 ⇒_0 o2) ⇒_0 (o1 ⇒_0 o3)).
-#o1; #o2; #o3; napply mk_binary_morphism
- [ #f; #g; napply (comp_unary_morphisms ??? f g)
- (* CSC: why not ∘?
- GARES: because the coercion to FunClass is not triggered if there
- are no "extra" arguments. We could fix that in the refiner
- *)
- | #a; #a'; #b; #b'; #ea; #eb; #x; #x'; #Hx; nnormalize; /3/ ]
-nqed.
projects / helm.git / blobdiff