include "properties/relations1.ma".
include "sets/setoids.ma".
+include "hints_declaration.ma".
nrecord setoid1: Type[2] ≝
{ carr1:> Type[1];
| napply trans]##]
nqed.
-ncoercion setoid1_of_setoid : ∀s:setoid. setoid1 ≝ setoid1_of_setoid
- on _s: setoid to setoid1.
+(*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).
prop11: ∀a,a'. eq1 ? a a' → eq1 ? (fun11 a) (fun11 a')
}.
-nrecord binary_morphism1 (A,B,C:setoid1) : Type[1] ≝
- { fun21:2> A → B → C;
- prop21: ∀a,a',b,b'. eq1 ? a a' → eq1 ? b b' → eq1 ? (fun21 a b) (fun21 a' b')
- }.
-
interpretation "prop11" 'prop1 c = (prop11 ????? c).
-interpretation "prop21" 'prop2 l r = (prop21 ???????? l r).
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