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 "properties/relations1.ma".
16 include "sets/setoids.ma".
17 include "hints_declaration.ma".
19 record setoid1: Type[2] ≝ {
21 eq1: equivalence_relation1 carr1
24 unification hint 0 ≔ R : setoid1;
26 lock ≟ mk_lock2 Type[1] MR setoid1 R
27 (* ---------------------------------------- *) ⊢
28 setoid1 ≡ force2 ? MR lock.
30 notation < "[\setoid1\ensp\of term 19 x]" non associative with precedence 90 for @{'mk_setoid1 $x}.
31 interpretation "mk_setoid1" 'mk_setoid1 x = (mk_setoid1 x ?).
33 (* da capire se mettere come coercion *)
34 definition setoid1_of_setoid: setoid → setoid1.
35 #s % [@(carr s)] % [@(eq0…)|@(refl…)|@(sym…)|@(trans…)]
38 alias symbol "hint_decl" = "hint_decl_CProp2".
39 alias symbol "hint_decl" (instance 1) = "hint_decl_Type2".
40 unification hint 0 ≔ A,x,y;
42 R ≟ setoid1_of_setoid A,
44 (*-----------------------------------------------*) ⊢
45 eq_rel T (eq0 A) x y ≡ eq_rel1 T1 (eq1 R) x y.
47 unification hint 0 ≔ A;
48 R ≟ setoid1_of_setoid A
49 (*-----------------------------------------------*) ⊢
52 interpretation "setoid1 eq" 'eq t x y = (eq_rel1 ? (eq1 t) x y).
53 interpretation "setoid eq" 'eq t x y = (eq_rel ? (eq0 t) x y).
55 notation > "hvbox(a break =_12 b)" non associative with precedence 45
56 for @{ eq_rel2 (carr2 (setoid2_of_setoid1 ?)) (eq2 (setoid2_of_setoid1 ?)) $a $b }.
57 notation > "hvbox(a break =_0 b)" non associative with precedence 45
58 for @{ eq_rel ? (eq0 ?) $a $b }.
59 notation > "hvbox(a break =_1 b)" non associative with precedence 45
60 for @{ eq_rel1 ? (eq1 ?) $a $b }.
62 interpretation "setoid1 symmetry" 'invert r = (sym1 ???? r).
63 interpretation "setoid symmetry" 'invert r = (sym ???? r).
64 notation ".=_1 r" with precedence 50 for @{'trans_x1 $r}.
65 interpretation "trans1" 'trans r = (trans1 ????? r).
66 interpretation "trans" 'trans r = (trans ????? r).
67 interpretation "trans1_x1" 'trans_x1 r = (trans1 ????? r).
69 record unary_morphism1 (A,B: setoid1) : Type[1] ≝ {
71 prop11: ∀a,a'. eq1 ? a a' → eq1 ? (fun11 a) (fun11 a')
74 notation > "B ⇒_1 C" right associative with precedence 72 for @{'umorph1 $B $C}.
75 notation "hvbox(B break ⇒\sub 1 C)" right associative with precedence 72 for @{'umorph1 $B $C}.
76 interpretation "unary morphism 1" 'umorph1 A B = (unary_morphism1 A B).
78 notation "┼_1 c" with precedence 89 for @{'prop1_x1 $c }.
79 interpretation "prop11" 'prop1 c = (prop11 ????? c).
80 interpretation "prop11_x1" 'prop1_x1 c = (prop11 ????? c).
81 interpretation "refl1" 'refl = (refl1 ???).
83 definition unary_morphism1_setoid1: setoid1 → setoid1 → setoid1.
84 #s #s1 % [@(s ⇒_1 s1)] %
85 [ #f #g @(∀a,a':s. a=a' → f a = g a')
86 | #x #a #a' #Ha @(.= †Ha) @refl1
87 | #x #y #H #a #a' #Ha @(.= †Ha) @sym1 /2/
88 | #x #y #z #H1 #H2 #a #a' #Ha @(.= †Ha) @trans1
89 [2: @H1 | skip | @H2] // ]
92 unification hint 0 ≔ S, T ;
93 R ≟ (unary_morphism1_setoid1 S T)
94 (* --------------------------------- *) ⊢
95 carr1 R ≡ unary_morphism1 S T.
97 notation "l ╪_1 r" with precedence 89 for @{'prop2_x1 $l $r }.
98 interpretation "prop21" 'prop2 l r = (prop11 ? (unary_morphism1_setoid1 ??) ? ?? l ?? r).
99 interpretation "prop21_x1" 'prop2_x1 l r = (prop11 ? (unary_morphism1_setoid1 ??) ? ?? l ?? r).
101 lemma unary_morph1_eq1: ∀A,B.∀f,g: A ⇒_1 B. (∀x. f x = g x) → f = g.
105 (* DISAMBIGUATION XXX: this takes some time to disambiguate *)
106 lemma mk_binary_morphism1:
107 ∀A,B,C: setoid1. ∀f: A → B → C. (∀a,a',b,b'. a=a' → b=b' → f a b = f a' b') →
108 A ⇒_1 (unary_morphism1_setoid1 B C).
109 #A #B #C #f #H % [ #x % [@(f x)]] #a #a' #Ha [2: @unary_morph1_eq1 #y]
113 definition composition1 ≝
114 λo1,o2,o3:Type[1].λf:o2 → o3.λg: o1 → o2.λx.f (g x).
116 interpretation "function composition" 'compose f g = (composition ??? f g).
117 interpretation "function composition1" 'compose f g = (composition1 ??? f g).
119 definition comp1_unary_morphisms:
120 ∀o1,o2,o3:setoid1.o2 ⇒_1 o3 → o1 ⇒_1 o2 → o1 ⇒_1 o3.
121 #o1 #o2 #o3 #f #g % [@ (f ∘ g)]
122 #a #a' #e normalize @(.= †(†e)) @#
125 unification hint 0 ≔ o1,o2,o3:setoid1,f:o2 ⇒_1 o3,g:o1 ⇒_1 o2;
126 R ≟ (mk_unary_morphism1 ?? (composition1 ??? (fun11 ?? f) (fun11 ?? g))
127 (prop11 ?? (comp1_unary_morphisms o1 o2 o3 f g)))
128 (* -------------------------------------------------------------------- *) ⊢
129 fun11 o1 o3 R ≡ composition1 ??? (fun11 ?? f) (fun11 ?? g).
131 definition comp1_binary_morphisms:
132 ∀o1,o2,o3. (o2 ⇒_1 o3) ⇒_1 ((o1 ⇒_1 o2) ⇒_1 (o1 ⇒_1 o3)).
133 #o1 #o2 #o3 @mk_binary_morphism1
134 [ #f #g @(comp1_unary_morphisms … f g)
135 | #a #a' #b #b' #ea #eb #x #x' #Hx normalize /3/ ]