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 nrecord setoid1: Type[2] ≝
21 eq1: equivalence_relation1 carr1
24 ndefinition setoid1_of_setoid: setoid → setoid1.
27 | napply (mk_equivalence_relation1 s)
34 (*ncoercion setoid1_of_setoid : ∀s:setoid. setoid1 ≝ setoid1_of_setoid
35 on _s: setoid to setoid1.*)
36 (*prefer coercion Type_OF_setoid.*)
38 interpretation "setoid1 eq" 'eq t x y = (eq_rel1 ? (eq1 t) x y).
39 interpretation "setoid eq" 'eq t x y = (eq_rel ? (eq0 t) x y).
41 notation > "hvbox(a break =_12 b)" non associative with precedence 45
42 for @{ eq_rel2 (carr2 (setoid2_of_setoid1 ?)) (eq2 (setoid2_of_setoid1 ?)) $a $b }.
43 notation > "hvbox(a break =_0 b)" non associative with precedence 45
44 for @{ eq_rel ? (eq0 ?) $a $b }.
45 notation > "hvbox(a break =_1 b)" non associative with precedence 45
46 for @{ eq_rel1 ? (eq1 ?) $a $b }.
48 interpretation "setoid1 symmetry" 'invert r = (sym1 ???? r).
49 interpretation "setoid symmetry" 'invert r = (sym ???? r).
50 notation ".= r" with precedence 50 for @{'trans $r}.
51 notation ".=_1 r" with precedence 50 for @{'trans_x1 $r}.
52 interpretation "trans1" 'trans r = (trans1 ????? r).
53 interpretation "trans" 'trans r = (trans ????? r).
54 interpretation "trans1_x1" 'trans_x1 r = (trans1 ????? r).
56 nrecord unary_morphism1 (A,B: setoid1) : Type[1] ≝
58 prop11: ∀a,a'. eq1 ? a a' → eq1 ? (fun11 a) (fun11 a')
61 notation "┼_1 c" with precedence 89 for @{'prop1_x1 $c }.
62 interpretation "prop11" 'prop1 c = (prop11 ????? c).
63 interpretation "prop11_x1" 'prop1_x1 c = (prop11 ????? c).
64 interpretation "refl1" 'refl = (refl1 ???).
66 ndefinition unary_morphism1_setoid1: setoid1 → setoid1 → setoid1.
67 #s; #s1; @ (unary_morphism1 s s1); @
68 [ #f; #g; napply (∀a,a':s. a=a' → f a = g a')
69 | #x; #a; #a'; #Ha; napply (.= †Ha); napply refl1
70 | #x; #y; #H; #a; #a'; #Ha; napply (.= †Ha); napply sym1; /2/
71 | #x; #y; #z; #H1; #H2; #a; #a'; #Ha; napply (.= †Ha); napply trans1; ##[##2: napply H1 | ##skip | napply H2]//;##]
74 unification hint 0 ≔ S, T ;
75 R ≟ (unary_morphism1_setoid1 S T)
76 (* --------------------------------- *) ⊢
77 carr1 R ≡ unary_morphism1 S T.
79 notation "l ╪_1 r" with precedence 89 for @{'prop2_x1 $l $r }.
80 interpretation "prop21" 'prop2 l r = (prop11 ? (unary_morphism1_setoid1 ??) ? ?? l ?? r).
81 interpretation "prop21_x1" 'prop2_x1 l r = (prop11 ? (unary_morphism1_setoid1 ??) ? ?? l ?? r).
83 nlemma unary_morph1_eq1: ∀A,B.∀f,g: unary_morphism1 A B. (∀x. f x = g x) → f=g.
86 nlemma mk_binary_morphism1:
87 ∀A,B,C: setoid1. ∀f: A → B → C. (∀a,a',b,b'. a=a' → b=b' → f a b = f a' b') →
88 unary_morphism1 A (unary_morphism1_setoid1 B C).
89 #A; #B; #C; #f; #H; @ [ #x; @ (f x) ] #a; #a'; #Ha [##2: napply unary_morph1_eq1; #y]
93 ndefinition composition1 ≝
94 λo1,o2,o3:Type[1].λf:o2 → o3.λg: o1 → o2.λx.f (g x).
96 interpretation "function composition" 'compose f g = (composition ??? f g).
97 interpretation "function composition1" 'compose f g = (composition1 ??? f g).
99 ndefinition comp1_unary_morphisms:
101 unary_morphism1 o2 o3 → unary_morphism1 o1 o2 →
102 unary_morphism1 o1 o3.
103 #o1; #o2; #o3; #f; #g; @ (f ∘ g);
104 #a; #a'; #e; nnormalize; napply (.= †(†e)); napply #.
107 unification hint 0 ≔ o1,o2,o3:setoid1,f:unary_morphism1 o2 o3,g:unary_morphism1 o1 o2;
108 R ≟ (mk_unary_morphism1 ?? (composition1 … f g)
109 (prop11 ?? (comp1_unary_morphisms o1 o2 o3 f g)))
110 (* -------------------------------------------------------------------- *) ⊢
111 fun11 ?? R ≡ (composition1 … f g).
113 ndefinition comp1_binary_morphisms:
115 unary_morphism1 (unary_morphism1_setoid1 o2 o3)
116 (unary_morphism1_setoid1 (unary_morphism1_setoid1 o1 o2) (unary_morphism1_setoid1 o1 o3)).
117 #o1; #o2; #o3; napply mk_binary_morphism1
118 [ #f; #g; napply (comp1_unary_morphisms … f g) (*CSC: why not ∘?*)
119 | #a; #a'; #b; #b'; #ea; #eb; #x; #x'; #Hx; nnormalize; /3/ ]