#s; #s1; @ (unary_morphism1 s s1); @
[ #f; #g; napply (∀a:s. f a = g a)
| #x; #a; napply refl1
- | #x; #y; #H; #a; napply sym1; nauto
+ | #x; #y; #H; #a; napply sym1; //
| #x; #y; #z; #H1; #H2; #a; 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.
\ No newline at end of file
+ carr1 R ≡ unary_morphism1 S T.
+
+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 comp_binary_morphisms:
+ ∀o1,o2,o3.
+ binary_morphism1 (unary_morphism1_setoid1 o2 o3) (unary_morphism1_setoid1 o1 o2)
+ (unary_morphism1_setoid1 o1 o3).
+#o1; #o2; #o3; @
+ [ #f; #g; napply (comp1_unary_morphisms … f g) (*CSC: why not ∘?*)
+ | #a; #a'; #b; #b'; #ea; #eb; #x; nnormalize;
+ napply (.= †(eb x)); napply ea.
+nqed.