include "logic/connectives.ma".
include "properties/relations.ma".
+include "hints_declaration.ma".
-(*
-notation "hvbox(a break = \sub \ID b)" non associative with precedence 45
-for @{ 'eqID $a $b }.
+nrecord setoid : Type[1] ≝ {
+ carr:> Type[0];
+ eq0: equivalence_relation carr
+}.
-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).
-*)
-
-nrecord setoid : Type[1] ≝
- { carr:> Type[0];
- eq: equivalence_relation carr
- }.
-
-interpretation "setoid eq" 'eq t x y = (eq_rel ? (eq t) x y).
+interpretation "setoid eq" 'eq t x y = (eq_rel ? (eq0 t) x y).
notation > "hvbox(a break =_0 b)" non associative with precedence 45
-for @{ eq_rel ? (eq ?) $a $b }.
+for @{ eq_rel ? (eq0 ?) $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'. eq ? a a' → eq ? (fun1 a) (fun1 a')
- }.
+nrecord unary_morphism (A,B: setoid) : Type[0] ≝ {
+ fun1:1> A → B;
+ prop1: ∀a,a'. a = a' → fun1 a = fun1 a'
+}.
-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 > "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).
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 ?? (composition … f g)
+ (prop1 ?? (comp_unary_morphisms o1 o2 o3 f g))
+ (* -------------------------------------------------------------------- *) ⊢
+ fun1 ?? R ≡ (composition … f g).
+
+ndefinition comp_binary_morphisms:
+ ∀o1,o2,o3.
+ (unary_morph_setoid o2 o3) ⇒_0
+ (unary_morph_setoid (unary_morph_setoid o1 o2) (unary_morph_setoid o1 o3)).
+#o1; #o2; #o3; napply mk_binary_morphism
+ [ #f; #g; napply (comp_unary_morphisms … f g) (*CSC: why not ∘?*)
+ | #a; #a'; #b; #b'; #ea; #eb; #x; #x'; #Hx; nnormalize; /3/ ]
+nqed.