definition symmetric1 ≝ λC:Type1.λlt:C→C→CProp1. ∀x,y:C.lt x y → lt y x.
definition transitive1 ≝ λA:Type1.λR:A→A→CProp1.∀x,y,z:A.R x y → R y z → R x z.
-record equivalence_relation1 (A:Type1) : Type1 ≝
+record equivalence_relation1 (A:Type1) : Type2 ≝
{ eq_rel1:2> A → A → CProp1;
refl1: reflexive1 ? eq_rel1;
sym1: symmetric1 ? eq_rel1;
definition symmetric2 ≝ λC:Type2.λlt:C→C→CProp2. ∀x,y:C.lt x y → lt y x.
definition transitive2 ≝ λA:Type2.λR:A→A→CProp2.∀x,y,z:A.R x y → R y z → R x z.
-record equivalence_relation2 (A:Type2) : Type2 ≝
+record equivalence_relation2 (A:Type2) : Type3 ≝
{ eq_rel2:2> A → A → CProp2;
refl2: reflexive2 ? eq_rel2;
sym2: symmetric2 ? eq_rel2;
| apply (trans1 s)]]
qed.
-coercion setoid2_of_setoid1.
+(*coercion setoid2_of_setoid1.*)
(*
definition Leibniz: Type → setoid.
interpretation "prop12" 'prop1 c = (prop12 _____ 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 "refl" 'refl = (refl ___).
interpretation "refl1" 'refl = (refl1 ___).
interpretation "refl2" 'refl = (refl2 ___).
notation "'ASSOC1'" with precedence 90 for @{'assoc1}.
notation "'ASSOC2'" with precedence 90 for @{'assoc2}.
-interpretation "category1 composition" 'compose x y = (fun22 ___ (comp2 ____) y x).
-interpretation "category1 assoc" 'assoc1 = (comp_assoc2 ________).
+interpretation "category2 composition" 'compose x y = (fun22 ___ (comp2 ____) y x).
+interpretation "category2 assoc" 'assoc1 = (comp_assoc2 ________).
interpretation "category1 composition" 'compose x y = (fun21 ___ (comp1 ____) y x).
interpretation "category1 assoc" 'assoc1 = (comp_assoc1 ________).
interpretation "category composition" 'compose x y = (fun2 ___ (comp ____) y x).
(* bug grande come una casa?
Ma come fa a passare la quantificazione larga??? *)
-definition unary_morphism_setoid: setoid → setoid → setoid.
+definition unary_morphism_setoid: setoid → setoid → setoid1.
intros;
constructor 1;
[ apply (unary_morphism s s1);
intros; apply (prop11 A B w a b e);
qed.
-definition hint: Type_OF_category2 SET1 → Type1.
+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.
+
+definition Type1_OF_SET1: Type_OF_category2 SET1 → Type1.
intro; whd in t; apply (carr1 t);
qed.
-coercion hint.
+coercion Type1_OF_SET1.
interpretation "SET dagger" 'prop1 h = (prop11_SET1 _ _ _ _ _ h).
-notation "hbox(a break ⇒ b)" right associative with precedence 20 for @{ 'Imply $a $b }.
interpretation "unary morphism1" 'Imply a b = (arrows2 SET1 a b).
interpretation "SET1 eq" 'eq x y = (eq_rel1 _ (eq'' _) x y).