include "logic/connectives.ma".
-definition Type3 : Type := Type.
+definition Type4 : Type := Type.
+definition Type3 : Type4 := Type.
definition Type2 : Type3 := Type.
definition Type1 : Type2 := Type.
definition Type0 : Type1 := Type.
definition Type_of_Type1: Type1 → Type := λx.x.
definition Type_of_Type2: Type2 → Type := λx.x.
definition Type_of_Type3: Type3 → Type := λx.x.
+definition Type_of_Type4: Type4 → Type := λx.x.
coercion Type_of_Type0.
coercion Type_of_Type1.
coercion Type_of_Type2.
coercion Type_of_Type3.
+coercion Type_of_Type4.
definition CProp0 : Type1 := Type0.
definition CProp1 : Type2 := Type1.
definition CProp2 : Type3 := Type2.
+definition CProp3 : Type4 := Type3.
definition CProp_of_CProp0: CProp0 → CProp ≝ λx.x.
definition CProp_of_CProp1: CProp1 → CProp ≝ λx.x.
definition CProp_of_CProp2: CProp2 → CProp ≝ λx.x.
+definition CProp_of_CProp3: CProp3 → CProp ≝ λx.x.
coercion CProp_of_CProp0.
coercion CProp_of_CProp1.
coercion CProp_of_CProp2.
+coercion CProp_of_CProp3.
inductive Or (A,B:CProp0) : CProp0 ≝
| Left : A → Or A B
definition reflexive: ∀C:Type0. ∀lt:C→C→CProp0.CProp0 ≝ λA:Type0.λR:A→A→CProp0.∀x:A.R x x.
definition transitive: ∀C:Type0. ∀lt:C→C→CProp0.CProp0 ≝ λA:Type0.λR:A→A→CProp0.∀x,y,z:A.R x y → R y z → R x z.
+
+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.
+
+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.
+
+definition reflexive3: ∀A:Type3.∀R:A→A→CProp3.CProp3 ≝ λA:Type3.λR:A→A→CProp3.∀x:A.R x x.
+definition symmetric3: ∀A:Type3.∀R:A→A→CProp3.CProp3 ≝ λC:Type3.λlt:C→C→CProp3. ∀x,y:C.lt x y → lt y x.
+definition transitive3: ∀A:Type3.∀R:A→A→CProp3.CProp3 ≝ λA:Type3.λR:A→A→CProp3.∀x,y,z:A.R x y → R y z → R x z.