1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 include "logic/connectives.ma".
16 include "properties/relations.ma".
17 include "hints_declaration.ma".
19 nrecord setoid : Type[1] ≝
21 eq0: equivalence_relation carr
24 interpretation "setoid eq" 'eq t x y = (eq_rel ? (eq0 t) x y).
26 notation > "hvbox(a break =_0 b)" non associative with precedence 45
27 for @{ eq_rel ? (eq0 ?) $a $b }.
29 interpretation "setoid symmetry" 'invert r = (sym ???? r).
30 notation ".= r" with precedence 50 for @{'trans $r}.
31 interpretation "trans" 'trans r = (trans ????? r).
33 nrecord unary_morphism (A,B: setoid) : Type[0] ≝
35 prop1: ∀a,a'. a = a' → fun1 a = fun1 a'
38 notation "† c" with precedence 90 for @{'prop1 $c }.
39 notation "l ‡ r" with precedence 90 for @{'prop2 $l $r }.
40 notation "#" with precedence 90 for @{'refl}.
41 interpretation "prop1" 'prop1 c = (prop1 ????? c).
42 interpretation "refl" 'refl = (refl ???).
44 ndefinition unary_morph_setoid : setoid → setoid → setoid.
45 #S1; #S2; @ (unary_morphism S1 S2); @;
46 ##[ #f; #g; napply (∀x,x'. x=x' → f x = g x');
47 ##| #f; #x; #x'; #Hx; napply (.= †Hx); napply #;
48 ##| #f; #g; #H; #x; #x'; #Hx; napply ((H … Hx^-1)^-1);
49 ##| #f; #g; #h; #H1; #H2; #x; #x'; #Hx; napply (trans … (H1 …) (H2 …)); //; ##]
52 alias symbol "hint_decl" (instance 1) = "hint_decl_Type1".
53 unification hint 0 ≔ o1,o2 ;
54 X ≟ unary_morph_setoid o1 o2
55 (* ------------------------------ *) ⊢
56 carr X ≡ unary_morphism o1 o2.
58 interpretation "prop2" 'prop2 l r = (prop1 ? (unary_morph_setoid ??) ? ?? l ?? r).
60 nlemma unary_morph_eq: ∀A,B.∀f,g: unary_morphism A B. (∀x. f x = g x) → f=g.
61 #A B f g H x1 x2 E; napply (.= †E); napply H; nqed.
63 nlemma mk_binary_morphism:
64 ∀A,B,C: setoid. ∀f: A → B → C. (∀a,a',b,b'. a=a' → b=b' → f a b = f a' b') →
65 unary_morphism A (unary_morph_setoid B C).
66 #A; #B; #C; #f; #H; @ [ #x; @ (f x) ] #a; #a'; #Ha [##2: napply unary_morph_eq; #y]
70 ndefinition composition ≝
71 λo1,o2,o3:Type[0].λf:o2 → o3.λg: o1 → o2.λx.f (g x).
73 interpretation "function composition" 'compose f g = (composition ??? f g).
75 ndefinition comp_unary_morphisms:
77 unary_morphism o2 o3 → unary_morphism o1 o2 →
79 #o1; #o2; #o3; #f; #g; @ (f ∘ g);
80 #a; #a'; #e; nnormalize; napply (.= †(†e)); napply #.
83 unification hint 0 ≔ o1,o2,o3:setoid,f:unary_morphism o2 o3,g:unary_morphism o1 o2;
84 R ≟ mk_unary_morphism ?? (composition … f g)
85 (prop1 ?? (comp_unary_morphisms o1 o2 o3 f g))
86 (* -------------------------------------------------------------------- *) ⊢
87 fun1 ?? R ≡ (composition … f g).
89 ndefinition comp_binary_morphisms:
91 unary_morphism (unary_morph_setoid o2 o3)
92 (unary_morph_setoid (unary_morph_setoid o1 o2) (unary_morph_setoid o1 o3)).
93 #o1; #o2; #o3; napply mk_binary_morphism
94 [ #f; #g; napply (comp_unary_morphisms … f g) (*CSC: why not ∘?*)
95 | #a; #a'; #b; #b'; #ea; #eb; #x; #x'; #Hx; nnormalize; /3/ ]