X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Flibrary%2Fformal_topology%2Fcategories.ma;h=015e245f3f5b142084a4b8a1922e0c07f6b30ade;hb=e880d6eab5e1700f4a625ddcd7d0fa8f0cce2dcc;hp=7adcb279f72287c1a0e5a5e09e30dc2b18a549b2;hpb=be9826d87207e8dcf6eb152bd54417b5a9e80ab9;p=helm.git

diff --git a/helm/software/matita/library/formal_topology/categories.ma b/helm/software/matita/library/formal_topology/categories.ma
index 7adcb279f..015e245f3 100644
--- a/helm/software/matita/library/formal_topology/categories.ma
+++ b/helm/software/matita/library/formal_topology/categories.ma
@@ -12,36 +12,511 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "logic/cprop_connectives.ma".
+include "formal_topology/cprop_connectives.ma".
 
-record equivalence_relation (A:Type) : Type ≝
- { eq_rel:2> A → A → CProp;
+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).
+
+record equivalence_relation (A:Type0) : Type1 ≝
+ { eq_rel:2> A → A → CProp0;
    refl: reflexive ? eq_rel;
    sym: symmetric ? eq_rel;
    trans: transitive ? eq_rel
  }.
 
-record setoid : Type ≝
- { carr:> Type;
+record setoid : Type1 ≝
+ { carr:> Type0;
    eq: equivalence_relation carr
  }.
 
-interpretation "setoid eq" 'eq x y = (eq_rel _ (eq _) x y).
+record equivalence_relation1 (A:Type1) : Type2 ≝
+ { eq_rel1:2> A → A → CProp1;
+   refl1: reflexive1 ? eq_rel1;
+   sym1: symmetric1 ? eq_rel1;
+   trans1: transitive1 ? eq_rel1
+ }.
+
+record setoid1: Type2 ≝
+ { carr1:> Type1;
+   eq1: equivalence_relation1 carr1
+ }.
+
+definition setoid1_of_setoid: setoid → setoid1.
+ intro;
+ constructor 1;
+  [ apply (carr s)
+  | constructor 1;
+    [ apply (eq_rel s);
+      apply (eq s)
+    | apply (refl s)
+    | apply (sym s)
+    | apply (trans s)]]
+qed.
+
+coercion setoid1_of_setoid.
+prefer coercion Type_OF_setoid.
+
+record equivalence_relation2 (A:Type2) : Type3 ≝
+ { eq_rel2:2> A → A → CProp2;
+   refl2: reflexive2 ? eq_rel2;
+   sym2: symmetric2 ? eq_rel2;
+   trans2: transitive2 ? eq_rel2
+ }.
+
+record setoid2: Type3 ≝
+ { carr2:> Type2;
+   eq2: equivalence_relation2 carr2
+ }.
+
+definition setoid2_of_setoid1: setoid1 → setoid2.
+ intro;
+ constructor 1;
+  [ apply (carr1 s)
+  | constructor 1;
+    [ apply (eq_rel1 s);
+      apply (eq1 s)
+    | apply (refl1 s)
+    | apply (sym1 s)
+    | apply (trans1 s)]]
+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 t x y = (eq_rel3 ? (eq3 t) x y).
+interpretation "setoid2 eq" 'eq t x y = (eq_rel2 ? (eq2 t) x y).
+interpretation "setoid1 eq" 'eq t x y = (eq_rel1 ? (eq1 t) x y).
+interpretation "setoid eq" 'eq t x y = (eq_rel ? (eq t) x y).
+
+notation > "hvbox(a break =_12 b)" non associative with precedence 45
+for @{ eq_rel2 (carr2 (setoid2_of_setoid1 ?)) (eq2 (setoid2_of_setoid1 ?)) $a $b }.
+notation > "hvbox(a break =_0 b)" non associative with precedence 45
+for @{ eq_rel ? (eq ?) $a $b }.
+notation > "hvbox(a break =_1 b)" non associative with precedence 45
+for @{ eq_rel1 ? (eq1 ?) $a $b }.
+notation > "hvbox(a break =_2 b)" non associative with precedence 45
+for @{ eq_rel2 ? (eq2 ?) $a $b }.
+notation > "hvbox(a break =_3 b)" non associative with precedence 45
+for @{ eq_rel3 ? (eq3 ?) $a $b }.
+
+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).
 
-record binary_morphism (A,B,C: setoid) : Type ≝
- { fun:2> A → B → C;
-   prop: ∀a,a',b,b'. eq ? a a' → eq ? b b' → eq ? (fun a b) (fun a' b')
+record unary_morphism (A,B: setoid) : Type0 ≝
+ { fun1:1> A → B;
+   prop1: ∀a,a'. eq ? a a' → eq ? (fun1 a) (fun1 a')
  }.
 
-record category : Type ≝
- { objs: Type;
+record unary_morphism1 (A,B: setoid1) : Type1 ≝
+ { fun11:1> A → B;
+   prop11: ∀a,a'. eq1 ? a a' → eq1 ? (fun11 a) (fun11 a')
+ }.
+
+record unary_morphism2 (A,B: setoid2) : Type2 ≝
+ { fun12:1> A → B;
+   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')
+ }.
+
+record binary_morphism1 (A,B,C:setoid1) : Type1 ≝
+ { fun21:2> A → B → C;
+   prop21: ∀a,a',b,b'. eq1 ? a a' → eq1 ? b b' → eq1 ? (fun21 a b) (fun21 a' b')
+ }.
+
+record binary_morphism2 (A,B,C:setoid2) : Type2 ≝
+ { fun22:2> A → B → C;
+   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 ???).
+
+notation > "A × term 74 B ⇒ term 19 C" non associative with precedence 72 for @{'binary_morphism0 $A $B $C}.
+notation > "A × term 74 B ⇒_1 term 19 C" non associative with precedence 72 for @{'binary_morphism1 $A $B $C}.
+notation > "A × term 74 B ⇒_2 term 19 C" non associative with precedence 72 for @{'binary_morphism2 $A $B $C}.
+notation > "A × term 74 B ⇒_3 term 19 C" non associative with precedence 72 for @{'binary_morphism3 $A $B $C}.
+notation > "B ⇒_1 C" right associative with precedence 72 for @{'arrows1_SET $B $C}.
+notation > "B ⇒_1. C" right associative with precedence 72 for @{'arrows1_SETlow $B $C}.
+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}.
+notation "B ⇒\sub 1 C" right associative with precedence 72 for @{'arrows1_SET $B $C}.
+notation "B ⇒\sub 2 C" right associative with precedence 72 for @{'arrows2_SET1 $B $C}.
+notation "B ⇒\sub 1. C" right associative with precedence 72 for @{'arrows1_SETlow $B $C}.
+notation "B ⇒\sub 2. C" right associative with precedence 72 for @{'arrows2_SET1low $B $C}.
+
+interpretation "'binary_morphism0" 'binary_morphism0 A B C = (binary_morphism A B C).
+interpretation "'arrows2_SET1 low" 'arrows2_SET1 A B = (unary_morphism2 A B).
+interpretation "'arrows2_SET1low" 'arrows2_SET1low A B = (unary_morphism2 A B).
+interpretation "'binary_morphism1" 'binary_morphism1 A B C = (binary_morphism1 A B C).
+interpretation "'binary_morphism2" 'binary_morphism2 A B C = (binary_morphism2 A B C).
+interpretation "'binary_morphism3" 'binary_morphism3 A B C = (binary_morphism3 A B C).
+interpretation "'arrows1_SET low" 'arrows1_SET A B = (unary_morphism1 A B).
+interpretation "'arrows1_SETlow" 'arrows1_SETlow A B = (unary_morphism1 A B).
+
+
+definition unary_morphism2_of_unary_morphism1: 
+  ∀S,T:setoid1.unary_morphism1 S T → unary_morphism2 (setoid2_of_setoid1 S) T.
+ intros;
+ constructor 1;
+  [ apply (fun11 ?? u);
+  | apply (prop11 ?? u); ]
+qed.
+
+definition CPROP: setoid1.
+ constructor 1;
+  [ apply CProp0
+  | constructor 1;
+     [ apply Iff
+     | intros 1; split; intro; assumption
+     | intros 3; cases i; split; assumption
+     | intros 5; cases i; cases i1; split; intro;
+        [ 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.
+qed.
+
+notation ". r" with precedence 50 for @{'fi $r}.
+interpretation "fi" 'fi r = (fi' ?? r).
+
+definition and_morphism: binary_morphism1 CPROP CPROP CPROP.
+ constructor 1;
+  [ apply And
+  | intros; split; intro; cases a1; split;
+     [ apply (if ?? e a2)
+     | apply (if ?? e1 b1)
+     | apply (fi ?? e a2)
+     | apply (fi ?? e1 b1)]]
+qed.
+
+interpretation "and_morphism" 'and a b = (fun21 ??? and_morphism a b).
+
+definition or_morphism: binary_morphism1 CPROP CPROP CPROP.
+ constructor 1;
+  [ apply Or
+  | intros; split; intro; cases o; [1,3:left |2,4: right]
+     [ apply (if ?? e a1)
+     | apply (fi ?? e a1)
+     | apply (if ?? e1 b1)
+     | apply (fi ?? e1 b1)]]
+qed.
+
+interpretation "or_morphism" 'or a b = (fun21 ??? or_morphism a b).
+
+definition if_morphism: binary_morphism1 CPROP CPROP CPROP.
+ constructor 1;
+  [ apply (λA,B. A → B)
+  | intros; split; intros;
+     [ apply (if ?? e1); apply f; apply (fi ?? e); assumption
+     | apply (fi ?? e1); apply f; apply (if ?? e); assumption]]
+qed.
+
+notation > "hvbox(a break ∘ b)" left associative with precedence 55 for @{ comp ??? $a $b }.
+record category : Type1 ≝ { 
+   objs:> Type0;
    arrows: objs → objs → setoid;
    id: ∀o:objs. arrows o o;
-   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_right: ∀o1,o2. ∀a: arrows o1 o2. comp ??? a (id o2) = a
+   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
+}.
+notation "hvbox(a break ∘ b)" left associative with precedence 55 for @{ 'compose $a $b }.
+
+record category1 : Type2 ≝
+ { objs1:> Type1;
+   arrows1: objs1 → objs1 → setoid1;
+   id1: ∀o:objs1. arrows1 o o;
+   comp1: ∀o1,o2,o3. binary_morphism1 (arrows1 o1 o2) (arrows1 o2 o3) (arrows1 o1 o3);
+   comp_assoc1: ∀o1,o2,o3,o4. ∀a12,a23,a34.
+    comp1 o1 o3 o4 (comp1 o1 o2 o3 a12 a23) a34 =_1 comp1 o1 o2 o4 a12 (comp1 o2 o3 o4 a23 a34);
+   id_neutral_right1: ∀o1,o2. ∀a: arrows1 o1 o2. comp1 ??? (id1 o1) a =_1 a;
+   id_neutral_left1: ∀o1,o2. ∀a: arrows1 o1 o2. comp1 ??? a (id1 o2) =_1 a
  }.
 
-interpretation "category composition" 'compose x y = (comp ____ x y).
\ No newline at end of file
+record category2 : Type3 ≝
+ { objs2:> Type2;
+   arrows2: objs2 → objs2 → setoid2;
+   id2: ∀o:objs2. arrows2 o o;
+   comp2: ∀o1,o2,o3. binary_morphism2 (arrows2 o1 o2) (arrows2 o2 o3) (arrows2 o1 o3);
+   comp_assoc2: ∀o1,o2,o3,o4. ∀a12,a23,a34.
+    comp2 o1 o3 o4 (comp2 o1 o2 o3 a12 a23) a34 =_2 comp2 o1 o2 o4 a12 (comp2 o2 o3 o4 a23 a34);
+   id_neutral_right2: ∀o1,o2. ∀a: arrows2 o1 o2. comp2 ??? (id2 o1) a =_2 a;
+   id_neutral_left2: ∀o1,o2. ∀a: arrows2 o1 o2. comp2 ??? a (id2 o2) =_2 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 =_3 comp3 o1 o2 o4 a12 (comp3 o2 o3 o4 a23 a34);
+   id_neutral_right3: ∀o1,o2. ∀a: arrows3 o1 o2. comp3 ??? (id3 o1) a =_3 a;
+   id_neutral_left3: ∀o1,o2. ∀a: arrows3 o1 o2. comp3 ??? a (id3 o2) =_3 a
+ }.
+
+notation "'ASSOC'" with precedence 90 for @{'assoc}.
+
+interpretation "category2 composition" 'compose x y = (fun22 ??? (comp2 ????) y x).
+interpretation "category2 assoc" 'assoc = (comp_assoc2 ????????).
+interpretation "category1 composition" 'compose x y = (fun21 ??? (comp1 ????) y x).
+interpretation "category1 assoc" 'assoc = (comp_assoc1 ????????).
+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; 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')); 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}.
+
+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}.
+interpretation "map_arrows2" 'map_arrows2 F x = (fun12 ?? (map_arrows2 ?? F ??) x).
+
+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.
+
+notation > "B ⇒_\c3 C" right associative with precedence 72 for @{'arrows3_CAT $B $C}.
+notation "B ⇒\sub (\c 3) C" right associative with precedence 72 for @{'arrows3_CAT $B $C}.
+interpretation "'arrows3_CAT" 'arrows3_CAT A B = (arrows3 CAT2 A B).
+
+definition unary_morphism_setoid: setoid → setoid → setoid.
+ intros;
+ constructor 1;
+  [ apply (unary_morphism s s1);
+  | constructor 1;
+     [ intros (f g); apply (∀a:s. eq ? (f a) (g a));
+     | intros 1; simplify; intros; apply refl;
+     | simplify; intros; apply sym; apply f;
+     | simplify; intros; apply trans; [2: apply f; | skip | apply f1]]]
+qed.
+
+definition SET: category1.
+ constructor 1;
+  [ apply setoid;
+  | 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. 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));
+     [ apply Hletin | apply (e a1); ]  | apply e1; ]]
+  | intros; whd; intros; simplify; apply refl;
+  | intros; simplify; whd; intros; simplify; apply refl;
+  | intros; simplify; whd; intros; simplify; apply refl;
+  ]
+qed.
+
+definition setoid_of_SET: objs1 SET → setoid ≝ λx.x.
+coercion setoid_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.
+
+interpretation "'arrows1_SET" 'arrows1_SET A B = (arrows1 SET A B).
+
+definition unary_morphism1_setoid1: setoid1 → setoid1 → setoid1.
+ intros;
+ constructor 1;
+  [ apply (unary_morphism1 s s1);
+  | constructor 1;
+     [ intros (f g);
+       alias symbol "eq" = "setoid1 eq".
+       apply (∀a: carr1 s. f a = g a);
+     | intros 1; simplify; intros; apply refl1;
+     | simplify; intros; apply sym1; apply f;
+     | simplify; intros; apply trans1; [2: apply f; | skip | apply f1]]]
+qed.
+
+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.setoid2_of_setoid1 (unary_morphism1_setoid1 S T));
+  | intros; constructor 1; [ apply (λx.x); | intros; assumption ]
+  | 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));
+     [ apply Hletin | apply (e a1); ]  | apply e1; ]]
+  | intros; whd; intros; simplify; apply refl1;
+  | intros; simplify; whd; intros; simplify; apply refl1;
+  | intros; simplify; whd; intros; simplify; apply refl1;
+  ]
+qed.
+
+interpretation "'arrows2_SET1" 'arrows2_SET1 A B = (arrows2 SET1 A B).
+
+definition setoid1_of_SET1: objs2 SET1 → setoid1 ≝ λx.x.
+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.
+ 
+variant objs2_of_category1: objs1 SET → objs2 SET1 ≝ setoid1_of_setoid.
+coercion objs2_of_category1.
+
+prefer coercion Type_OF_setoid. (* we prefer the lower carrier projection *)
+prefer coercion Type_OF_objs1.
+
+alias symbol "exists" (instance 1) = "CProp2 exists".
+definition full2 ≝  
+  λA,B:CAT2.λF:carr3 (arrows3 CAT2 A B).
+    ∀o1,o2:A.∀f.∃g:arrows2 A o1 o2.F⎽⇒ g =_2 f.
+alias symbol "exists" (instance 1) = "CProp exists".
+    
+definition faithful2 ≝  
+  λA,B:CAT2.λF:carr3 (arrows3 CAT2 A B).
+    ∀o1,o2:A.∀f,g:arrows2 A o1 o2.F⎽⇒ f =_2 F⎽⇒ g → f =_2 g.
+    
+
+notation "r \sup *" non associative with precedence 90 for @{'OR_f_star $r}.
+notation > "r *" non associative with precedence 90 for @{'OR_f_star $r}.
+
+notation "r \sup (⎻* )" non associative with precedence 90 for @{'OR_f_minus_star $r}.
+notation > "r⎻*" non associative with precedence 90 for @{'OR_f_minus_star $r}.
+
+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}.