(**************************************************************************)
include "formal_topology/cprop_connectives.ma".
+include "basics/core_notation/compose_2.ma".
+include "basics/core_notation/invert_1.ma".
inductive eq (A:Type[0]) (x:A) : A → CProp[0] ≝
refl: eq A x x.
interpretation "setoid2 symmetry" 'invert r = (sym2 ???? r).
interpretation "setoid1 symmetry" 'invert r = (sym1 ???? r).
interpretation "setoid symmetry" 'invert r = (sym ???? r).
-notation ".= r" with precedence 50 for @{'trans $r}.
+notation ".= r" with precedence 55 for @{'trans $r}.
interpretation "trans3" 'trans r = (trans3 ????? r).
interpretation "trans2" 'trans r = (trans2 ????? r).
interpretation "trans1" 'trans r = (trans1 ????? r).
notation > "B ⇒_2 C" right associative with precedence 72 for @{'arrows2_SET1 $B $C}.
notation > "B ⇒_2. C" right associative with precedence 72 for @{'arrows2_SET1low $B $C}.
-notation "A × term 74 B ⇒ term 19 C" non associative with precedence 72 for @{'binary_morphism0 $A $B $C}.
notation "A × term 74 B ⇒\sub 1 term 19 C" non associative with precedence 72 for @{'binary_morphism1 $A $B $C}.
notation "A × term 74 B ⇒\sub 2 term 19 C" non associative with precedence 72 for @{'binary_morphism2 $A $B $C}.
notation "A × term 74 B ⇒\sub 3 term 19 C" non associative with precedence 72 for @{'binary_morphism3 $A $B $C}.
[ @CProp[0]
| @mk_equivalence_relation1
[ @Iff
- | #H1 @mk_Iff #H3 #H5 assumption
- | #H7 #H8 #i cases i #H14 #H15 @mk_Iff assumption
- | #H17 #H18 #H19 #i1 #i cases i cases i1 #f #f1 #f2 #f3 @mk_Iff
- [ #x1 @(f2 (f x1)) | #z1 @(f1 (f3 z1))]]]
+ | #x @mk_Iff #x1 assumption
+ | #x #y #i cases i #f #f1 @mk_Iff assumption
+ | #x #y #z #i1 #i cases i cases i1 #f #f1 #f2 #f3 @mk_Iff #x1
+ [ @(f2 (f x1)) | @(f1 (f3 x1))]]]
qed.
definition CProp0_of_CPROP: carr1 CPROP → CProp[0] ≝ λx.x.
alias symbol "eq" = "setoid1 eq".
definition fi': ∀A,B:CPROP. A = B → B → A.
- intros; @(fi ?? e); assumption.
+ #A #B #e #b @(fi ?? e) assumption.
qed.
-notation ". r" with precedence 50 for @{'fi $r}.
+notation ". r" with precedence 55 for @{'fi $r}.
interpretation "fi" 'fi r = (fi' ?? r).
definition and_morphism: binary_morphism1 CPROP CPROP CPROP.
- constructor 1;
+ @mk_binary_morphism1
[ @And
- | intros; split; intro; cases a1; split;
+ | #a #a' #b #b' #e #e1 @mk_Iff #a1 cases a1 #a2 #b1 @Conj
[ @(if ?? e a2)
| @(if ?? e1 b1)
| @(fi ?? e a2)
interpretation "and_morphism" 'and a b = (fun21 ??? and_morphism a b).
definition or_morphism: binary_morphism1 CPROP CPROP CPROP.
- constructor 1;
+ @mk_binary_morphism1
[ @Or
- | intros; split; intro; cases o; [1,3:left |2,4: right]
+ | #a #a' #b #b' #e #e1 @mk_Iff #o cases o #a1 [1,3: @Left |2,4: @Right]
[ @(if ?? e a1)
| @(fi ?? e a1)
- | @(if ?? e1 b1)
- | @(fi ?? e1 b1)]]
+ | @(if ?? e1 a1)
+ | @(fi ?? e1 a1)]]
qed.
interpretation "or_morphism" 'or a b = (fun21 ??? or_morphism a b).
definition if_morphism: binary_morphism1 CPROP CPROP CPROP.
- constructor 1;
+ @mk_binary_morphism1
[ @(λA,B. A → B)
- | intros; split; intros;
- [ @(if ?? e1); @f; @(fi ?? e); assumption
- | @(fi ?? e1); @f; @(if ?? e); assumption]]
+ | #a #a' #b #b' #e #e1 @mk_Iff #f #a1
+ [ @(if ?? e1) @f @(fi ?? e) assumption
+ | @(fi ?? e1) @f @(if ?? e) assumption]]
qed.
-
-notation > "hvbox(a break ∘ b)" left associative with precedence 55 for @{ comp ??? $a $b }.
+(*
+notation > "hvbox(a break ∘ b)" left associative with precedence 60 for @{ comp ??? $a $b }.
+*)
record category : Type[1] ≝ {
objs:> Type[0];
arrows: objs → objs → setoid;
id: ∀o:objs. arrows o o;
comp: ∀o1,o2,o3. (arrows o1 o2) × (arrows o2 o3) ⇒ (arrows o1 o3);
comp_assoc: ∀o1,o2,o3,o4.∀a12:arrows o1 ?.∀a23:arrows o2 ?.∀a34:arrows o3 o4.
- (a12 ∘ a23) ∘ a34 =_0 a12 ∘ (a23 ∘ a34);
- id_neutral_left : ∀o1,o2. ∀a: arrows o1 o2. (id o1) ∘ a =_0 a;
- id_neutral_right: ∀o1,o2. ∀a: arrows o1 o2. a ∘ (id o2) =_0 a
+ comp o1 o3 o4 (comp o1 o2 o3 a12 a23) a34 =_0 comp o1 o2 o4 a12 (comp o2 o3 o4 a23 a34);
+ id_neutral_left : ∀o1,o2. ∀a: arrows o1 o2. comp ??? (id o1) a =_0 a;
+ id_neutral_right: ∀o1,o2. ∀a: arrows o1 o2. comp ??? a (id o2) =_0 a
}.
-notation "hvbox(a break ∘ b)" left associative with precedence 55 for @{ 'compose $a $b }.
-
+(*
+notation "hvbox(a break ∘ b)" left associative with precedence 60 for @{ 'compose $a $b }.
+*)
record category1 : Type[2] ≝
{ objs1:> Type[1];
arrows1: objs1 → objs1 → setoid1;
interpretation "category assoc" 'assoc = (comp_assoc ????????).
definition category2_of_category1: category1 → category2.
- intro;
- constructor 1;
- [ @(objs1 c);
- | intros; @(setoid2_of_setoid1 (arrows1 c o o1));
- | @(id1 c);
- | intros;
- constructor 1;
- [ intros; @(comp1 c o1 o2 o3 c1 c2);
- | intros; unfold setoid2_of_setoid1 in e e1 a a' b b'; simplify in e e1 a a' b b';
- change with ((b∘a) =_1 (b'∘a')); @(e‡e1); ]
- | intros; simplify; whd in a12 a23 a34; whd; @rule (ASSOC);
- | intros; simplify; whd in a; whd; @id_neutral_right1;
- | intros; simplify; whd in a; whd; @id_neutral_left1; ]
+ #c
+ @mk_category2
+ [ @(objs1 c)
+ | #o #o1 @(setoid2_of_setoid1 (arrows1 c o o1))
+ | @(id1 c)
+ | #o1 #o2 #o3
+ @mk_binary_morphism2
+ [ #c1 #c2 @(comp1 c o1 o2 o3 c1 c2)
+ | #a #a' #b #b' #e #e1 @(e‡e1) ]
+ | #o1 #o2 #o3 #o4 #a12 #a23 #a34 @comp_assoc1
+ | #o1 #o2 #a @id_neutral_right1
+ | #o1 #o2 #a @id_neutral_left1 ]
qed.
(*coercion category2_of_category1.*)
∀o1,o2,o3.∀f1:arrows2 ? o1 o2.∀f2:arrows2 ? o2 o3.
map_arrows2 ?? (f2 ∘ f1) = map_arrows2 ?? f2 ∘ map_arrows2 ?? f1}.
-notation > "F⎽⇒ x" left associative with precedence 60 for @{'map_arrows2 $F $x}.
-notation "F\sub⇒ x" left associative with precedence 60 for @{'map_arrows2 $F $x}.
+notation > "F⎽⇒ x" left associative with precedence 65 for @{'map_arrows2 $F $x}.
+notation "F\sub⇒ x" left associative with precedence 65 for @{'map_arrows2 $F $x}.
interpretation "map_arrows2" 'map_arrows2 F x = (fun12 ?? (map_arrows2 ?? F ??) x).
-
+(* BEGIN HERE
definition functor2_setoid: category2 → category2 → setoid3.
- intros (C1 C2);
- constructor 1;
- [ @(functor2 C1 C2);
- | constructor 1;
- [ intros (f g);
+ #C1 #C2
+ @mk_setoid3
+ [ @(functor2 C1 C2)
+ | @mk_equivalence_relation3
+ [ #f #g
@(∀c:C1. cic:/matita/logic/equality/eq.ind#xpointer(1/1) ? (f c) (g c));
| simplify; intros; @cic:/matita/logic/equality/eq.ind#xpointer(1/1/1);
| simplify; intros; @cic:/matita/logic/equality/sym_eq.con; @H;
notation "r \sup ⎻" non associative with precedence 90 for @{'OR_f_minus $r}.
notation > "r⎻" non associative with precedence 90 for @{'OR_f_minus $r}.
+*)
projects / helm.git / blobdiff