-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.