include "properties/relations.ma".
include "hints_declaration.ma".
-(*
-notation "hvbox(a break = \sub \ID b)" non associative with precedence 45
-for @{ 'eqID $a $b }.
-
-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
+ eq0: 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}.
nrecord unary_morphism (A,B: setoid) : Type[0] ≝
{ fun1:1> A → B;
- prop1: ∀a,a'. eq ? a a' → eq ? (fun1 a) (fun1 a')
+ prop1: ∀a,a'. a = a' → fun1 a = fun1 a'
}.
notation "† c" with precedence 90 for @{'prop1 $c }.
##| #f; #g; #h; #H1; #H2; #x; #x'; #Hx; napply (trans … (H1 …) (H2 …)); //; ##]
nqed.
-unification hint 0
- (∀o1,o2. (λx,y:Type[0].True) (carr (unary_morph_setoid o1 o2)) (unary_morphism o1 o2)).
+alias symbol "hint_decl" (instance 1) = "hint_decl_Type1".
+unification hint 0 ≔ o1,o2 ;
+ X ≟ unary_morph_setoid o1 o2
+ (* ------------------------------ *) ⊢
+ carr X ≡ unary_morphism o1 o2.
interpretation "prop2" 'prop2 l r = (prop1 ? (unary_morph_setoid ??) ? ?? l ?? r).
nlemma unary_morph_eq: ∀A,B.∀f,g: unary_morphism A B. (∀x. f x = g x) → f=g.
-/3/. nqed.
+#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') →
nqed.
unification hint 0 ≔ o1,o2,o3:setoid,f:unary_morphism o2 o3,g:unary_morphism o1 o2;
- R ≟ (mk_unary_morphism ?? (composition … f g)
- (prop1 ?? (comp_unary_morphisms o1 o2 o3 f g)))
+ R ≟ mk_unary_morphism ?? (composition … f g)
+ (prop1 ?? (comp_unary_morphisms o1 o2 o3 f g))
(* -------------------------------------------------------------------- *) ⊢
fun1 ?? R ≡ (composition … f g).
#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.
\ No newline at end of file
+nqed.