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 ? (eq 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 ? (eq ?) $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 interpretation "trans1" 'trans r = (trans1 ????? r).
52 interpretation "trans" 'trans r = (trans ????? r).
54 nrecord unary_morphism1 (A,B: setoid1) : Type[1] ≝
56 prop11: ∀a,a'. eq1 ? a a' → eq1 ? (fun11 a) (fun11 a')
59 nrecord binary_morphism1 (A,B,C:setoid1) : Type[1] ≝
61 prop21: ∀a,a',b,b'. eq1 ? a a' → eq1 ? b b' → eq1 ? (fun21 a b) (fun21 a' b')
64 interpretation "prop11" 'prop1 c = (prop11 ????? c).
65 interpretation "prop21" 'prop2 l r = (prop21 ???????? l r).
66 interpretation "refl1" 'refl = (refl1 ???).
68 ndefinition unary_morphism1_setoid1: setoid1 → setoid1 → setoid1.
69 #s; #s1; @ (unary_morphism1 s s1); @
70 [ #f; #g; napply (∀a:s. f a = g a)
71 | #x; #a; napply refl1
72 | #x; #y; #H; #a; napply sym1; nauto
73 | #x; #y; #z; #H1; #H2; #a; napply trans1; ##[##2: napply H1 | ##skip | napply H2]##]
76 unification hint 0 ≔ S, T ;
77 R ≟ (unary_morphism1_setoid1 S T)
78 (* --------------------------------- *) ⊢
79 carr1 R ≡ unary_morphism1 S T.
81 ndefinition composition1 ≝
82 λo1,o2,o3:Type[1].λf:o2 → o3.λg: o1 → o2.λx.f (g x).
84 interpretation "function composition" 'compose f g = (composition ??? f g).
85 interpretation "function composition1" 'compose f g = (composition1 ??? f g).
87 ndefinition comp1_unary_morphisms:
89 unary_morphism1 o2 o3 → unary_morphism1 o1 o2 →
90 unary_morphism1 o1 o3.
91 #o1; #o2; #o3; #f; #g; @ (f ∘ g);
92 #a; #a'; #e; nnormalize; napply (.= †(†e)); napply #.
95 unification hint 0 ≔ o1,o2,o3:setoid1,f:unary_morphism1 o2 o3,g:unary_morphism1 o1 o2;
96 R ≟ (mk_unary_morphism1 ?? (composition1 … f g)
97 (prop11 ?? (comp1_unary_morphisms o1 o2 o3 f g)))
98 (* -------------------------------------------------------------------- *) ⊢
99 fun11 ?? R ≡ (composition1 … f g).
101 ndefinition comp_binary_morphisms:
103 binary_morphism1 (unary_morphism1_setoid1 o2 o3) (unary_morphism1_setoid1 o1 o2)
104 (unary_morphism1_setoid1 o1 o3).
106 [ #f; #g; napply (comp1_unary_morphisms … f g) (*CSC: why not ∘?*)
107 | #a; #a'; #b; #b'; #ea; #eb; #x; nnormalize;
108 napply (.= †(eb x)); napply ea.