+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
include "sets/sets.ma".
-ndefinition binary_morph_setoid : setoid → setoid → setoid → setoid.
-#S1; #S2; #T; @ (binary_morphism S1 S2 T); @;
-##[ #f; #g; napply (∀x,y. f x y = g x y);
-##| #f; #x; #y; napply #;
-##| #f; #g; #H; #x; #y; napply ((H x y)^-1);
-##| #f; #g; #h; #H1; #H2; #x; #y; napply (trans … (H1 …) (H2 …)); ##]
-nqed.
-
-ndefinition unary_morph_setoid : setoid → setoid → setoid.
-#S1; #S2; @ (unary_morphism S1 S2); @;
-##[ #f; #g; napply (∀x. f x = g x);
-##| #f; #x; napply #;
-##| #f; #g; #H; #x; napply ((H x)^-1);
-##| #f; #g; #h; #H1; #H2; #x; napply (trans … (H1 …) (H2 …)); ##]
-nqed.
-
nrecord category : Type[2] ≝
{ objs:> Type[1];
arrows: objs → objs → setoid;
id: ∀o:objs. arrows o o;
- comp: ∀o1,o2,o3. binary_morphism (arrows o1 o2) (arrows o2 o3) (arrows o1 o3);
+ comp: ∀o1,o2,o3. unary_morphism (arrows o1 o2) (unary_morph_setoid (arrows o2 o3) (arrows o1 o3));
comp_assoc: ∀o1,o2,o3,o4. ∀a12,a23,a34.
comp o1 o3 o4 (comp o1 o2 o3 a12 a23) a34 = comp o1 o2 o4 a12 (comp o2 o3 o4 a23 a34);
id_neutral_left: ∀o1,o2. ∀a: arrows o1 o2. comp ??? (id o1) a = a;
@;
##[ napply setoid;
##| napply unary_morph_setoid;
-##| #o; @ (λx.x); #a; #b; #H; napply H;
-##| #o1; #o2; #o3; @;
- ##[ #f; #g; @(λx.g (f x)); #a; #b; #H; napply (.= (††H)); napply #;
- ##| #f; #g; #f'; #g'; #H1; #H2; nwhd; #x; napply (.= (H2 (f x)));
- napply (.= (†(H1 x))); napply #; ##]
-##| #o1; #o2; #o3; #o4; #f; #g; #h; nwhd; #x; napply #;
-##|##6,7: #o1; #o2; #f; nwhd; #x; napply #; ##]
+##| #o; @ (λx.x); //
+##| #o1; #o2; #o3; napply mk_binary_morphism [ #f; #g; @(λx.g (f x)) ]
+ nnormalize; /3/
+##| nnormalize; /4/
+##|##6,7: nnormalize; /2/ ]
nqed.
unification hint 0 ≔ ;
(* ----------------------------------- *) ⊢
unary_morphism A B ≡ carr T.
+(*
ndefinition TYPE : setoid1.
@ setoid; @;
interpretation "new I" 'I a = (nI ? a).
ndefinition image ≝ λA:nAx.λa:A.λi. { x | ∃j:𝐃 a i. x = 𝐝 a i j }.
-(*
+
nlemma elim_eq_TYPE : ∀A,B:setoid.∀P:CProp[1]. A=B → ((B ⇒ A) → P) → P.
#A; #B; #P; *; #f; *; #g; #_; #IH; napply IH; napply g;
nqed.
##[ @(f e);
*)
+(*
ndefinition image_is_ext : ∀A:nAx.∀a:A.∀i:𝐈 a.𝛀^A.
#A; #a; #i; @ (image … i); #x; #y; #H; @;
##[ *; #d; #Ex; @ d; napply (.= H^-1); nassumption;
@ (f i); #a; #Ha; napply H1;
ncut (𝐈𝐦[𝐝 y (f i)] = 𝐈𝐦[𝐝 x i]);
- ##[##2: #E; alias symbol "refl" = "refl".
- alias symbol "prop2" = "prop21 mem".
- alias symbol "invert" = "setoid1 symmetry".
- napply (. (#‡E^-1)); napply Ha; ##]
+ ##[##2: #E; napply (. (#‡E^-1)); napply Ha; ##]
@; #w; #Hw; nwhd;
ncut (𝐈𝐦[𝐝 y (f i)] = 𝐈𝐦[𝐝 x i]);
[1]: http://upsilon.cc/~zack/research/publications/notation.pdf
D*)
-*)
\ No newline at end of file
+*)*)
\ No newline at end of file