eq: equivalence_relation carr
}.
-definition reflexive1: ∀A:Type1.∀R:A→A→CProp1.CProp1 ≝ λA:Type1.λR:A→A→CProp1.∀x:A.R x x.
-definition symmetric1: ∀A:Type1.∀R:A→A→CProp1.CProp1 ≝ λC:Type1.λlt:C→C→CProp1. ∀x,y:C.lt x y → lt y x.
-definition transitive1: ∀A:Type1.∀R:A→A→CProp1.CProp1 ≝ λA:Type1.λR:A→A→CProp1.∀x,y,z:A.R x y → R y z → R x z.
-
record equivalence_relation1 (A:Type1) : Type2 ≝
{ eq_rel1:2> A → A → CProp1;
refl1: reflexive1 ? eq_rel1;
qed.
coercion setoid1_of_setoid.
-
-definition reflexive2: ∀A:Type2.∀R:A→A→CProp2.CProp2 ≝ λA:Type2.λR:A→A→CProp2.∀x:A.R x x.
-definition symmetric2: ∀A:Type2.∀R:A→A→CProp2.CProp2 ≝ λC:Type2.λlt:C→C→CProp2. ∀x,y:C.lt x y → lt y x.
-definition transitive2: ∀A:Type2.∀R:A→A→CProp2.CProp2 ≝ λA:Type2.λR:A→A→CProp2.∀x,y,z:A.R x y → R y z → R x z.
+prefer coercion Type_OF_setoid.
record equivalence_relation2 (A:Type2) : Type3 ≝
{ eq_rel2:2> A → A → CProp2;
qed.
coercion setoid2_of_setoid1.
+prefer coercion Type_OF_setoid2.
+prefer coercion Type_OF_setoid.
+prefer coercion Type_OF_setoid1.
+(* we prefer 0 < 1 < 2 *)
+
+record equivalence_relation3 (A:Type3) : Type4 ≝
+ { eq_rel3:2> A → A → CProp3;
+ refl3: reflexive3 ? eq_rel3;
+ sym3: symmetric3 ? eq_rel3;
+ trans3: transitive3 ? eq_rel3
+ }.
+
+record setoid3: Type4 ≝
+ { carr3:> Type3;
+ eq3: equivalence_relation3 carr3
+ }.
+
+interpretation "setoid3 eq" 'eq x y = (eq_rel3 _ (eq3 _) x y).
interpretation "setoid2 eq" 'eq x y = (eq_rel2 _ (eq2 _) x y).
interpretation "setoid1 eq" 'eq x y = (eq_rel1 _ (eq1 _) x y).
interpretation "setoid eq" 'eq x y = (eq_rel _ (eq _) x y).
+interpretation "setoid3 symmetry" 'invert r = (sym3 ____ r).
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}.
+interpretation "trans3" 'trans r = (trans3 _____ r).
interpretation "trans2" 'trans r = (trans2 _____ r).
interpretation "trans1" 'trans r = (trans1 _____ r).
interpretation "trans" 'trans r = (trans _____ r).
prop12: ∀a,a'. eq2 ? a a' → eq2 ? (fun12 a) (fun12 a')
}.
+record unary_morphism3 (A,B: setoid3) : Type3 ≝
+ { fun13:1> A → B;
+ prop13: ∀a,a'. eq3 ? a a' → eq3 ? (fun13 a) (fun13 a')
+ }.
+
record binary_morphism (A,B,C:setoid) : Type0 ≝
{ fun2:2> A → B → C;
prop2: ∀a,a',b,b'. eq ? a a' → eq ? b b' → eq ? (fun2 a b) (fun2 a' b')
prop22: ∀a,a',b,b'. eq2 ? a a' → eq2 ? b b' → eq2 ? (fun22 a b) (fun22 a' b')
}.
+record binary_morphism3 (A,B,C:setoid3) : Type3 ≝
+ { fun23:2> A → B → C;
+ prop23: ∀a,a',b,b'. eq3 ? a a' → eq3 ? b b' → eq3 ? (fun23 a b) (fun23 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 "prop11" 'prop1 c = (prop11 _____ c).
interpretation "prop12" 'prop1 c = (prop12 _____ c).
+interpretation "prop13" 'prop1 c = (prop13 _____ c).
interpretation "prop2" 'prop2 l r = (prop2 ________ l r).
interpretation "prop21" 'prop2 l r = (prop21 ________ l r).
interpretation "prop22" 'prop2 l r = (prop22 ________ l r).
+interpretation "prop23" 'prop2 l r = (prop23 ________ l r).
interpretation "refl" 'refl = (refl ___).
interpretation "refl1" 'refl = (refl1 ___).
interpretation "refl2" 'refl = (refl2 ___).
+interpretation "refl3" 'refl = (refl3 ___).
+
+definition unary_morphism2_of_unary_morphism1: ∀S,T.unary_morphism1 S T → unary_morphism2 S T.
+ intros;
+ constructor 1;
+ [ apply (fun11 ?? u);
+ | apply (prop11 ?? u); ]
+qed.
definition CPROP: setoid1.
constructor 1;
[ apply (f2 (f x1)) | apply (f1 (f3 z1))]]]
qed.
+definition CProp0_of_CPROP: carr1 CPROP → CProp0 ≝ λx.x.
+coercion CProp0_of_CPROP.
+
alias symbol "eq" = "setoid1 eq".
definition fi': ∀A,B:CPROP. A = B → B → A.
intros; apply (fi ?? e); assumption.
| apply (fi ?? e1); apply f; apply (if ?? e); assumption]]
qed.
+
record category : Type1 ≝
{ objs:> Type0;
arrows: objs → objs → setoid;
id: ∀o:objs. arrows o o;
- comp: ∀o1,o2,o3. binary_morphism1 (arrows o1 o2) (arrows o2 o3) (arrows o1 o3);
+ comp: ∀o1,o2,o3. binary_morphism (arrows o1 o2) (arrows o2 o3) (arrows o1 o3);
comp_assoc: ∀o1,o2,o3,o4. ∀a12,a23,a34.
comp o1 o3 o4 (comp o1 o2 o3 a12 a23) a34 = comp o1 o2 o4 a12 (comp o2 o3 o4 a23 a34);
id_neutral_left: ∀o1,o2. ∀a: arrows o1 o2. comp ??? (id o1) a = a;
id_neutral_left2: ∀o1,o2. ∀a: arrows2 o1 o2. comp2 ??? a (id2 o2) = a
}.
+record category3 : Type4 ≝
+ { objs3:> Type3;
+ arrows3: objs3 → objs3 → setoid3;
+ id3: ∀o:objs3. arrows3 o o;
+ comp3: ∀o1,o2,o3. binary_morphism3 (arrows3 o1 o2) (arrows3 o2 o3) (arrows3 o1 o3);
+ comp_assoc3: ∀o1,o2,o3,o4. ∀a12,a23,a34.
+ comp3 o1 o3 o4 (comp3 o1 o2 o3 a12 a23) a34 = comp3 o1 o2 o4 a12 (comp3 o2 o3 o4 a23 a34);
+ id_neutral_right3: ∀o1,o2. ∀a: arrows3 o1 o2. comp3 ??? (id3 o1) a = a;
+ id_neutral_left3: ∀o1,o2. ∀a: arrows3 o1 o2. comp3 ??? a (id3 o2) = a
+ }.
+
notation "'ASSOC'" with precedence 90 for @{'assoc}.
interpretation "category2 composition" 'compose x y = (fun22 ___ (comp2 ____) y x).
interpretation "category composition" 'compose x y = (fun2 ___ (comp ____) y x).
interpretation "category assoc" 'assoc = (comp_assoc ________).
+definition category2_of_category1: category1 → category2.
+ intro;
+ constructor 1;
+ [ apply (objs1 c);
+ | intros; apply (setoid2_of_setoid1 (arrows1 c o o1));
+ | apply (id1 c);
+ | intros;
+ constructor 1;
+ [ intros; apply (comp1 c o1 o2 o3 c1 c2);
+ | intros; whd in e e1 a a' b b'; change with (eq1 ? (b∘a) (b'∘a')); apply (e‡e1); ]
+ | intros; simplify; whd in a12 a23 a34; whd; apply rule (ASSOC);
+ | intros; simplify; whd in a; whd; apply id_neutral_right1;
+ | intros; simplify; whd in a; whd; apply id_neutral_left1; ]
+qed.
+(*coercion category2_of_category1.*)
+
+record functor2 (C1: category2) (C2: category2) : Type3 ≝
+ { map_objs2:1> C1 → C2;
+ map_arrows2: ∀S,T. unary_morphism2 (arrows2 ? S T) (arrows2 ? (map_objs2 S) (map_objs2 T));
+ respects_id2: ∀o:C1. map_arrows2 ?? (id2 ? o) = id2 ? (map_objs2 o);
+ respects_comp2:
+ ∀o1,o2,o3.∀f1:arrows2 ? o1 o2.∀f2:arrows2 ? o2 o3.
+ map_arrows2 ?? (f2 ∘ f1) = map_arrows2 ?? f2 ∘ map_arrows2 ?? f1}.
+
+definition functor2_setoid: category2 → category2 → setoid3.
+ intros (C1 C2);
+ constructor 1;
+ [ apply (functor2 C1 C2);
+ | constructor 1;
+ [ intros (f g);
+ apply (∀c:C1. cic:/matita/logic/equality/eq.ind#xpointer(1/1) ? (f c) (g c));
+ | simplify; intros; apply cic:/matita/logic/equality/eq.ind#xpointer(1/1/1);
+ | simplify; intros; apply cic:/matita/logic/equality/sym_eq.con; apply H;
+ | simplify; intros; apply cic:/matita/logic/equality/trans_eq.con;
+ [2: apply H; | skip | apply H1;]]]
+qed.
+
+definition functor2_of_functor2_setoid: ∀S,T. functor2_setoid S T → functor2 S T ≝ λS,T,x.x.
+coercion functor2_of_functor2_setoid.
+
+definition CAT2: category3.
+ constructor 1;
+ [ apply category2;
+ | apply functor2_setoid;
+ | intros; constructor 1;
+ [ apply (λx.x);
+ | intros; constructor 1;
+ [ apply (λx.x);
+ | intros; assumption;]
+ | intros; apply rule #;
+ | intros; apply rule #; ]
+ | intros; constructor 1;
+ [ intros; constructor 1;
+ [ intros; apply (c1 (c o));
+ | intros; constructor 1;
+ [ intro; apply (map_arrows2 ?? c1 ?? (map_arrows2 ?? c ?? c2));
+ | intros; apply (††e); ]
+ | intros; simplify;
+ apply (.= †(respects_id2 : ?));
+ apply (respects_id2 : ?);
+ | intros; simplify;
+ apply (.= †(respects_comp2 : ?));
+ apply (respects_comp2 : ?); ]
+ | intros; intro; simplify;
+ apply (cic:/matita/logic/equality/eq_ind.con ????? (e ?));
+ apply (cic:/matita/logic/equality/eq_ind.con ????? (e1 ?));
+ constructor 1; ]
+ | intros; intro; simplify; constructor 1;
+ | intros; intro; simplify; constructor 1;
+ | intros; intro; simplify; constructor 1; ]
+qed.
+
+definition category2_of_objs3_CAT2: objs3 CAT2 → category2 ≝ λx.x.
+coercion category2_of_objs3_CAT2.
+
+definition functor2_setoid_of_arrows3_CAT2: ∀S,T. arrows3 CAT2 S T → functor2_setoid S T ≝ λS,T,x.x.
+coercion functor2_setoid_of_arrows3_CAT2.
+
definition unary_morphism_setoid: setoid → setoid → setoid.
intros;
constructor 1;
definition SET: category1.
constructor 1;
[ apply setoid;
- | apply rule (λS,T:setoid.unary_morphism_setoid S T);
+ | apply rule (λS,T:setoid.setoid1_of_setoid (unary_morphism_setoid S T));
| intros; constructor 1; [ apply (λx:carr o.x); | intros; assumption ]
- | intros; constructor 1; [ intros; constructor 1; [ apply (λx. t1 (t x)); | intros;
+ | intros; constructor 1; [ intros; constructor 1; [ apply (λx. c1 (c x)); | intros;
apply († (†e));]
| intros; whd; intros; simplify; whd in H1; whd in H;
apply trans; [ apply (b (a' a1)); | lapply (prop1 ?? b (a a1) (a' a1));
]
qed.
-definition setoid_of_SET: objs1 SET → setoid.
- intros; apply o; qed.
+definition setoid_of_SET: objs1 SET → setoid ≝ λx.x.
coercion setoid_of_SET.
-definition setoid1_of_SET: SET → setoid1.
- intro; whd in t; apply setoid1_of_setoid; apply t.
-qed.
-coercion setoid1_of_SET.
+definition unary_morphism_setoid_of_arrows1_SET:
+ ∀P,Q.arrows1 SET P Q → unary_morphism_setoid P Q ≝ λP,Q,x.x.
+coercion unary_morphism_setoid_of_arrows1_SET.
notation "hbox(a break ⇒ b)" right associative with precedence 20 for @{ 'Imply $a $b }.
interpretation "unary morphism" 'Imply a b = (arrows1 SET a b).
| simplify; intros; apply trans1; [2: apply f; | skip | apply f1]]]
qed.
-definition SET1: category2.
+definition unary_morphism1_of_unary_morphism1_setoid1 :
+ ∀S,T. unary_morphism1_setoid1 S T → unary_morphism1 S T ≝ λP,Q,x.x.
+coercion unary_morphism1_of_unary_morphism1_setoid1.
+
+definition SET1: objs3 CAT2.
constructor 1;
[ apply setoid1;
- | apply rule (λS,T.unary_morphism1_setoid1 S T);
+ | apply rule (λS,T.setoid2_of_setoid1 (unary_morphism1_setoid1 S T));
| intros; constructor 1; [ apply (λx.x); | intros; assumption ]
- | intros; constructor 1; [ intros; constructor 1; [ apply (λx. t1 (t x)); | intros;
+ | intros; constructor 1; [ intros; constructor 1; [ apply (λx. c1 (c x)); | intros;
apply († (†e));]
| intros; whd; intros; simplify; whd in H1; whd in H;
apply trans1; [ apply (b (a' a1)); | lapply (prop11 ?? b (a a1) (a' a1));
]
qed.
-definition setoid1_OF_SET1: objs2 SET1 → setoid1.
- intros; apply o; qed.
+definition setoid1_of_SET1: objs2 SET1 → setoid1 ≝ λx.x.
+coercion setoid1_of_SET1.
-coercion setoid1_OF_SET1.
+definition unary_morphism1_setoid1_of_arrows2_SET1:
+ ∀P,Q.arrows2 SET1 P Q → unary_morphism1_setoid1 P Q ≝ λP,Q,x.x.
+coercion unary_morphism1_setoid1_of_arrows2_SET1.
-definition setoid2_OF_category2: Type_OF_category2 SET1 → setoid2.
- intro; apply (setoid2_of_setoid1 t); qed.
-coercion setoid2_OF_category2.
-
-definition objs2_OF_category1: Type_OF_category1 SET → objs2 SET1.
- intro; apply (setoid1_of_setoid t); qed.
-coercion objs2_OF_category1.
+variant objs2_of_category1: objs1 SET → objs2 SET1 ≝ setoid1_of_setoid.
+coercion objs2_of_category1.
-definition Type1_OF_SET1: Type_OF_category2 SET1 → Type1.
- intro; whd in t; apply (carr1 t);
-qed.
-coercion Type1_OF_SET1.
-
-definition Type_OF_setoid1_of_carr: ∀U. carr U → Type_OF_setoid1 ?(*(setoid1_of_SET U)*).
- [ apply setoid1_of_SET; apply U
- | intros; apply c;]
-qed.
-coercion Type_OF_setoid1_of_carr.
+prefer coercion Type_OF_setoid. (* we prefer the lower carrier projection *)
+prefer coercion Type_OF_objs1.
-interpretation "unary morphism1" 'Imply a b = (arrows2 SET1 a b).
-
-lemma unary_morphism1_of_arrows1_SET1: ∀S,T. (S ⇒ T) → unary_morphism1 S T.
- intros; apply t;
-qed.
-coercion unary_morphism1_of_arrows1_SET1.
+interpretation "unary morphism1" 'Imply a b = (arrows2 SET1 a b).
\ No newline at end of file
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