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 (* activate non uniform coercions on: Type → setoid *)
25 unification hint 0 ≔ R : setoid;
27 lock ≟ mk_lock1 Type[0] MR setoid R
28 (* ---------------------------------------- *) ⊢
29 setoid ≡ force1 ? MR lock.
31 notation < "[\setoid\ensp\of term 19 x]" non associative with precedence 90 for @{'mk_setoid $x}.
32 interpretation "mk_setoid" 'mk_setoid x = (mk_setoid x ?).
34 interpretation "setoid eq" 'eq t x y = (eq_rel ? (eq0 t) x y).
36 notation > "hvbox(a break =_0 b)" non associative with precedence 45
37 for @{ eq_rel ? (eq0 ?) $a $b }.
39 interpretation "setoid symmetry" 'invert r = (sym ???? r).
40 notation ".= r" with precedence 50 for @{'trans $r}.
41 interpretation "trans" 'trans r = (trans ????? r).
42 notation > ".=_0 r" with precedence 50 for @{'trans_x0 $r}.
43 interpretation "trans_x0" 'trans_x0 r = (trans ????? r).
45 nrecord unary_morphism (A,B: setoid) : Type[0] ≝ {
47 prop1: ∀a,a'. a = a' → fun1 a = fun1 a'
50 notation > "B ⇒_0 C" right associative with precedence 72 for @{'umorph0 $B $C}.
51 notation "hvbox(B break ⇒\sub 0 C)" right associative with precedence 72 for @{'umorph0 $B $C}.
52 interpretation "unary morphism 0" 'umorph0 A B = (unary_morphism A B).
54 notation "† c" with precedence 90 for @{'prop1 $c }.
55 notation "l ‡ r" with precedence 90 for @{'prop2 $l $r }.
56 notation "#" with precedence 90 for @{'refl}.
57 interpretation "prop1" 'prop1 c = (prop1 ????? c).
58 interpretation "refl" 'refl = (refl ???).
59 notation "┼_0 c" with precedence 89 for @{'prop1_x0 $c }.
60 notation "l ╪_0 r" with precedence 89 for @{'prop2_x0 $l $r }.
61 interpretation "prop1_x0" 'prop1_x0 c = (prop1 ????? c).
63 ndefinition unary_morph_setoid : setoid → setoid → setoid.
64 #S1; #S2; @ (S1 ⇒_0 S2); @;
65 ##[ #f; #g; napply (∀x,x'. x=x' → f x = g x');
66 ##| #f; #x; #x'; #Hx; napply (.= †Hx); napply #;
67 ##| #f; #g; #H; #x; #x'; #Hx; napply ((H … Hx^-1)^-1);
68 ##| #f; #g; #h; #H1; #H2; #x; #x'; #Hx; napply (trans … (H1 …) (H2 …)); //; ##]
71 alias symbol "hint_decl" (instance 1) = "hint_decl_Type1".
72 unification hint 0 ≔ o1,o2 ;
73 X ≟ unary_morph_setoid o1 o2
74 (* ----------------------------- *) ⊢
77 interpretation "prop2" 'prop2 l r = (prop1 ? (unary_morph_setoid ??) ? ?? l ?? r).
78 interpretation "prop2_x0" 'prop2_x0 l r = (prop1 ? (unary_morph_setoid ??) ? ?? l ?? r).
80 nlemma unary_morph_eq: ∀A,B.∀f,g:A ⇒_0 B. (∀x. f x = g x) → f = g.
81 #A B f g H x1 x2 E; napply (.= †E); napply H; nqed.
83 nlemma mk_binary_morphism:
84 ∀A,B,C: setoid. ∀f: A → B → C. (∀a,a',b,b'. a=a' → b=b' → f a b = f a' b') →
85 A ⇒_0 (unary_morph_setoid B C).
86 #A; #B; #C; #f; #H; @; ##[ #x; @ (f x) ] #a; #a'; #Ha [##2: napply unary_morph_eq; #y]
90 ndefinition composition ≝
91 λo1,o2,o3:Type[0].λf:o2 → o3.λg: o1 → o2.λx.f (g x).
93 interpretation "function composition" 'compose f g = (composition ??? f g).
95 ndefinition comp_unary_morphisms:
96 ∀o1,o2,o3:setoid.o2 ⇒_0 o3 → o1 ⇒_0 o2 → o1 ⇒_0 o3.
97 #o1; #o2; #o3; #f; #g; @ (f ∘ g);
98 #a; #a'; #e; nnormalize; napply (.= †(†e)); napply #.
101 unification hint 0 ≔ o1,o2,o3:setoid,f:o2 ⇒_0 o3,g:o1 ⇒_0 o2;
102 R ≟ mk_unary_morphism ?? (composition ??? f g)
103 (prop1 ?? (comp_unary_morphisms o1 o2 o3 f g))
104 (* -------------------------------------------------------------------- *) ⊢
105 fun1 ?? R ≡ (composition ??? f g).
107 ndefinition comp_binary_morphisms:
108 ∀o1,o2,o3.(o2 ⇒_0 o3) ⇒_0 ((o1 ⇒_0 o2) ⇒_0 (o1 ⇒_0 o3)).
109 #o1; #o2; #o3; napply mk_binary_morphism
110 [ #f; #g; napply (comp_unary_morphisms ??? f g)
112 GARES: because the coercion to FunClass is not triggered if there
113 are no "extra" arguments. We could fix that in the refiner
115 | #a; #a'; #b; #b'; #ea; #eb; #x; #x'; #Hx; nnormalize; /3/ ]