include "properties/relations.ma".
include "hints_declaration.ma".
-nrecord setoid : Type[1] ≝ {
+record setoid : Type[1] ≝ {
carr:> Type[0];
eq0: equivalence_relation carr
}.
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] ≝ {
+record unary_morphism (A,B: setoid) : Type[0] ≝ {
fun1:1> A → B;
prop1: ∀a,a'. a = a' → fun1 a = fun1 a'
}.
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.
+definition unary_morph_setoid : setoid → setoid → setoid.
+#S1 #S2 @(mk_setoid … (S1 ⇒_0 S2)) %
+[ #f #g @(∀x,x'. x=x' → f x = g x')
+| #f #x #x' #Hx @(.= †Hx) @#
+| #f #g #H #x #x' #Hx @((H … Hx^-1)^-1)
+| #f #g #h #H1 #H2 #x #x' #Hx @(trans … (H1 …) (H2 …)) // ]
+qed.
alias symbol "hint_decl" (instance 1) = "hint_decl_Type1".
unification hint 0 ≔ o1,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.
+lemma 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 @(.= †E) @H
+qed.
-nlemma mk_binary_morphism:
+lemma 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.
+ #A #B #C #f #H % [ #x % [@(f x)|/2/] |#a #a' #Ha @unary_morph_eq #y]
+ /2/
+qed.
-ndefinition composition ≝
+definition 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:
+definition 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.
+#o1 #o2 #o3 #f #g % [@(f ∘ g)]
+ #a #a' #e normalize @(.= †(†e)) @#
+qed.
unification hint 0 ≔ o1,o2,o3:setoid,f:o2 ⇒_0 o3,g:o1 ⇒_0 o2;
R ≟ mk_unary_morphism o1 o3
(* -------------------------------------------------------------------- *) ⊢
fun1 o1 o3 R ≡ composition ??? (fun1 o2 o3 f) (fun1 o1 o2 g).
-ndefinition comp_binary_morphisms:
+definition 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)
+#o1 #o2 #o3 @mk_binary_morphism
+ [ #f #g @(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.
+ | #a #a' #b #b' #ea #eb #x #x' #Hx normalize /3/ ]
+qed.