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
interpretation "logical iff" 'iff x y = (Iff x y).
interpretation "logical iff type1" 'iff1 x y = (Iff1 x y).
+inductive exT22 (A:Type2) (P:A→CProp2) : CProp2 ≝
+ ex_introT22: ∀w:A. P w → exT22 A P.
+
+interpretation "CProp2 exists" 'exists \eta.x = (exT22 ? x).
+
+definition pi1exT22 ≝ λA,P.λx:exT22 A P.match x with [ex_introT22 x _ ⇒ x].
+definition pi2exT22 ≝
+ λA,P.λx:exT22 A P.match x return λx.P (pi1exT22 ?? x) with [ex_introT22 _ p ⇒ p].
+
+interpretation "exT22 \fst" 'pi1 = (pi1exT22 ? ?).
+interpretation "exT22 \snd" 'pi2 = (pi2exT22 ? ?).
+interpretation "exT22 \fst a" 'pi1a x = (pi1exT22 ? ? x).
+interpretation "exT22 \snd a" 'pi2a x = (pi2exT22 ? ? x).
+interpretation "exT22 \fst b" 'pi1b x y = (pi1exT22 ? ? x y).
+interpretation "exT22 \snd b" 'pi2b x y = (pi2exT22 ? ? x y).
+
inductive exT (A:Type0) (P:A→CProp0) : CProp0 ≝
ex_introT: ∀w:A. P w → exT A P.
-interpretation "CProp exists" 'exists \eta.x = (exT _ x).
+interpretation "CProp exists" 'exists \eta.x = (exT ? x).
notation "\ll term 19 a, break term 19 b \gg"
with precedence 90 for @{'dependent_pair $a $b}.
interpretation "dependent pair" 'dependent_pair a b =
- (ex_introT _ _ a b).
+ (ex_introT ? ? a b).
definition pi1exT ≝ λA,P.λx:exT A P.match x with [ex_introT x _ ⇒ x].
definition pi2exT ≝
λA,P.λx:exT A P.match x return λx.P (pi1exT ?? x) with [ex_introT _ p ⇒ p].
-interpretation "exT \fst" 'pi1 = (pi1exT _ _).
-interpretation "exT \fst" 'pi1a x = (pi1exT _ _ x).
-interpretation "exT \fst" 'pi1b x y = (pi1exT _ _ x y).
-interpretation "exT \snd" 'pi2 = (pi2exT _ _).
-interpretation "exT \snd" 'pi2a x = (pi2exT _ _ x).
-interpretation "exT \snd" 'pi2b x y = (pi2exT _ _ x y).
+interpretation "exT \fst" 'pi1 = (pi1exT ? ?).
+interpretation "exT \fst a" 'pi1a x = (pi1exT ? ? x).
+interpretation "exT \fst b" 'pi1b x y = (pi1exT ? ? x y).
+interpretation "exT \snd" 'pi2 = (pi2exT ? ?).
+interpretation "exT \snd a" 'pi2a x = (pi2exT ? ? x).
+interpretation "exT \snd b" 'pi2b x y = (pi2exT ? ? x y).
inductive exT23 (A:Type0) (P:A→CProp0) (Q:A→CProp0) (R:A→A→CProp0) : CProp0 ≝
ex_introT23: ∀w,p:A. P w → Q p → R w p → exT23 A P Q R.
definition pi2exT23 ≝
λA,P,Q,R.λx:exT23 A P Q R.match x with [ex_introT23 _ x _ _ _ ⇒ x].
-interpretation "exT2 \fst" 'pi1 = (pi1exT23 _ _ _ _).
-interpretation "exT2 \snd" 'pi2 = (pi2exT23 _ _ _ _).
-interpretation "exT2 \fst" 'pi1a x = (pi1exT23 _ _ _ _ x).
-interpretation "exT2 \snd" 'pi2a x = (pi2exT23 _ _ _ _ x).
-interpretation "exT2 \fst" 'pi1b x y = (pi1exT23 _ _ _ _ x y).
-interpretation "exT2 \snd" 'pi2b x y = (pi2exT23 _ _ _ _ x y).
+interpretation "exT2 \fst" 'pi1 = (pi1exT23 ? ? ? ?).
+interpretation "exT2 \snd" 'pi2 = (pi2exT23 ? ? ? ?).
+interpretation "exT2 \fst a" 'pi1a x = (pi1exT23 ? ? ? ? x).
+interpretation "exT2 \snd a" 'pi2a x = (pi2exT23 ? ? ? ? x).
+interpretation "exT2 \fst b" 'pi1b x y = (pi1exT23 ? ? ? ? x y).
+interpretation "exT2 \snd b" 'pi2b x y = (pi2exT23 ? ? ? ? x y).
inductive exT2 (A:Type0) (P,Q:A→CProp0) : CProp0 ≝
ex_introT2: ∀w:A. P w → Q w → exT2 A P Q.
+
definition Not : CProp0 → Prop ≝ λx:CProp.x → False.
interpretation "constructive not" 'not x = (Not x).
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.