include "properties/relations.ma".
include "hints_declaration.ma".
-(*
-notation "hvbox(a break = \sub \ID b)" non associative with precedence 45
-for @{ 'eqID $a $b }.
-
-notation > "hvbox(a break =_\ID b)" non associative with precedence 45
-for @{ cic:/matita/logic/equality/eq.ind#xpointer(1/1) ? $a $b }.
-
-interpretation "ID eq" 'eqID x y = (cic:/matita/logic/equality/eq.ind#xpointer(1/1) ? x y).
-*)
-
-nrecord setoid : Type[1] ≝
- { carr:> Type[0];
- eq: equivalence_relation carr
- }.
-
-interpretation "setoid eq" 'eq t x y = (eq_rel ? (eq t) x y).
+nrecord setoid : Type[1] ≝ {
+ carr:> Type[0];
+ eq0: equivalence_relation carr
+}.
+
+(* activate non uniform coercions on: Type → setoid *)
+unification hint 0 ≔ R : setoid;
+ MR ≟ carr R,
+ lock ≟ mk_lock1 Type[0] MR setoid R
+(* ---------------------------------------- *) ⊢
+ setoid ≡ force1 ? MR lock.
+
+notation < "[\setoid\ensp\of term 19 x]" non associative with precedence 90 for @{'mk_setoid $x}.
+interpretation "mk_setoid" 'mk_setoid x = (mk_setoid x ?).
+
+interpretation "setoid eq" 'eq t x y = (eq_rel ? (eq0 t) x y).
+(* single = is for the abstract equality of setoids, == is for concrete
+ equalities (that may be lifted to the setoid level when needed *)
+notation < "hvbox(a break mpadded width -50% (=)= b)" non associative with precedence 45 for @{ 'eq_low $a $b }.
+notation > "a == b" non associative with precedence 45 for @{ 'eq_low $a $b }.
notation > "hvbox(a break =_0 b)" non associative with precedence 45
-for @{ eq_rel ? (eq ?) $a $b }.
+for @{ eq_rel ? (eq0 ?) $a $b }.
interpretation "setoid symmetry" 'invert r = (sym ???? r).
notation ".= r" with precedence 50 for @{'trans $r}.
interpretation "trans" 'trans r = (trans ????? r).
+notation > ".=_0 r" with precedence 50 for @{'trans_x0 $r}.
+interpretation "trans_x0" 'trans_x0 r = (trans ????? r).
+
+nrecord unary_morphism (A,B: setoid) : Type[0] ≝ {
+ fun1:1> A → B;
+ prop1: ∀a,a'. a = a' → fun1 a = fun1 a'
+}.
-nrecord unary_morphism (A,B: setoid) : Type[0] ≝
- { fun1:1> A → B;
- prop1: ∀a,a'. eq ? a a' → eq ? (fun1 a) (fun1 a')
- }.
+notation > "B ⇒_0 C" right associative with precedence 72 for @{'umorph0 $B $C}.
+notation "hvbox(B break ⇒\sub 0 C)" right associative with precedence 72 for @{'umorph0 $B $C}.
+interpretation "unary morphism 0" 'umorph0 A B = (unary_morphism A B).
notation "† c" with precedence 90 for @{'prop1 $c }.
notation "l ‡ r" with precedence 90 for @{'prop2 $l $r }.
notation "#" with precedence 90 for @{'refl}.
interpretation "prop1" 'prop1 c = (prop1 ????? c).
interpretation "refl" 'refl = (refl ???).
+notation "┼_0 c" with precedence 89 for @{'prop1_x0 $c }.
+notation "l ╪_0 r" with precedence 89 for @{'prop2_x0 $l $r }.
+interpretation "prop1_x0" 'prop1_x0 c = (prop1 ????? c).
ndefinition unary_morph_setoid : setoid → setoid → setoid.
-#S1; #S2; @ (unary_morphism S1 S2); @;
+#S1; #S2; @ (S1 ⇒_0 S2); @;
##[ #f; #g; napply (∀x,x'. x=x' → f x = g x');
##| #f; #x; #x'; #Hx; napply (.= †Hx); napply #;
##| #f; #g; #H; #x; #x'; #Hx; napply ((H … Hx^-1)^-1);
##| #f; #g; #h; #H1; #H2; #x; #x'; #Hx; napply (trans … (H1 …) (H2 …)); //; ##]
nqed.
-unification hint 0
- (∀o1,o2. (λx,y:Type[0].True) (carr (unary_morph_setoid o1 o2)) (unary_morphism o1 o2)).
+alias symbol "hint_decl" (instance 1) = "hint_decl_Type1".
+unification hint 0 ≔ o1,o2 ;
+ X ≟ unary_morph_setoid o1 o2
+ (* ----------------------------- *) ⊢
+ carr X ≡ o1 ⇒_0 o2.
interpretation "prop2" 'prop2 l r = (prop1 ? (unary_morph_setoid ??) ? ?? l ?? r).
+interpretation "prop2_x0" 'prop2_x0 l r = (prop1 ? (unary_morph_setoid ??) ? ?? l ?? r).
-nlemma unary_morph_eq: ∀A,B.∀f,g: unary_morphism A B. (∀x. f x = g x) → f=g.
-/3/. nqed.
+nlemma unary_morph_eq: ∀A,B.∀f,g:A ⇒_0 B. (∀x. f x = g x) → f = g.
+#A B f g H x1 x2 E; napply (.= †E); napply H; nqed.
nlemma mk_binary_morphism:
∀A,B,C: setoid. ∀f: A → B → C. (∀a,a',b,b'. a=a' → b=b' → f a b = f a' b') →
- unary_morphism A (unary_morph_setoid B C).
- #A; #B; #C; #f; #H; @ [ #x; @ (f x) ] #a; #a'; #Ha [##2: napply unary_morph_eq; #y]
+ A ⇒_0 (unary_morph_setoid B C).
+ #A; #B; #C; #f; #H; @; ##[ #x; @ (f x) ] #a; #a'; #Ha [##2: napply unary_morph_eq; #y]
/2/.
nqed.
interpretation "function composition" 'compose f g = (composition ??? f g).
ndefinition comp_unary_morphisms:
- ∀o1,o2,o3:setoid.
- unary_morphism o2 o3 → unary_morphism o1 o2 →
- unary_morphism o1 o3.
+ ∀o1,o2,o3:setoid.o2 ⇒_0 o3 → o1 ⇒_0 o2 → o1 ⇒_0 o3.
#o1; #o2; #o3; #f; #g; @ (f ∘ g);
#a; #a'; #e; nnormalize; napply (.= †(†e)); napply #.
nqed.
-unification hint 0 ≔ o1,o2,o3:setoid,f:unary_morphism o2 o3,g:unary_morphism o1 o2;
- R ≟ (mk_unary_morphism ?? (composition … f g)
- (prop1 ?? (comp_unary_morphisms o1 o2 o3 f g)))
+unification hint 0 ≔ o1,o2,o3:setoid,f:o2 ⇒_0 o3,g:o1 ⇒_0 o2;
+ R ≟ mk_unary_morphism ?? (composition ??? f g)
+ (prop1 ?? (comp_unary_morphisms o1 o2 o3 f g))
(* -------------------------------------------------------------------- *) ⊢
- fun1 ?? R ≡ (composition … f g).
+ fun1 ?? R ≡ (composition ??? f g).
-ndefinition comp_binary_morphisms:
- ∀o1,o2,o3.
- unary_morphism (unary_morph_setoid o2 o3)
- (unary_morph_setoid (unary_morph_setoid o1 o2) (unary_morph_setoid o1 o3)).
+ndefinition comp_binary_morphisms:
+ ∀o1,o2,o3.(o2 ⇒_0 o3) ⇒_0 ((o1 ⇒_0 o2) ⇒_0 (o1 ⇒_0 o3)).
#o1; #o2; #o3; napply mk_binary_morphism
- [ #f; #g; napply (comp_unary_morphisms … f g) (*CSC: why not ∘?*)
+ [ #f; #g; napply (comp_unary_morphisms ??? f g)
+ (* CSC: why not ∘?
+ GARES: because the coercion to FunClass is not triggered if there
+ are no "extra" arguments. We could fix that in the refiner
+ *)
| #a; #a'; #b; #b'; #ea; #eb; #x; #x'; #Hx; nnormalize; /3/ ]
-nqed.
\ No newline at end of file
+nqed.
projects / helm.git / blobdiff