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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "logic/connectives.ma".
+
+definition Type3 : Type := Type.
+definition Type2 : Type3 := Type.
+definition Type1 : Type2 := Type.
+definition Type0 : Type1 := Type.
+
+definition Type_OF_Type0: Type0 → Type := λx.x.
+definition Type_OF_Type1: Type1 → Type := λx.x.
+definition Type_OF_Type2: Type2 → Type := λx.x.
+definition Type_OF_Type3: Type3 → Type := λx.x.
+coercion Type_OF_Type0.
+coercion Type_OF_Type1.
+coercion Type_OF_Type2.
+coercion Type_OF_Type3.
+
+definition CProp0 : Type1 := Type0.
+definition CProp1 : Type2 := Type1.
+definition CProp2 : Type3 := Type2.
+
+inductive Or (A,B:CProp0) : CProp0 ≝
+ | Left : A → Or A B
+ | Right : B → Or A B.
+
+interpretation "constructive or" 'or x y = (Or x y).
+
+inductive Or3 (A,B,C:CProp0) : CProp0 ≝
+ | Left3 : A → Or3 A B C
+ | Middle3 : B → Or3 A B C
+ | Right3 : C → Or3 A B C.
+
+interpretation "constructive ternary or" 'or3 x y z= (Or3 x y z).
+
+notation < "hvbox(a break ∨ b break ∨ c)" with precedence 35 for @{'or3 $a $b $c}.
+
+inductive Or4 (A,B,C,D:CProp0) : CProp0 ≝
+ | Left3 : A → Or4 A B C D
+ | Middle3 : B → Or4 A B C D
+ | Right3 : C → Or4 A B C D
+ | Extra3: D → Or4 A B C D.
+
+interpretation "constructive ternary or" 'or4 x y z t = (Or4 x y z t).
+
+notation < "hvbox(a break ∨ b break ∨ c break ∨ d)" with precedence 35 for @{'or4 $a $b $c $d}.
+
+inductive And (A,B:CProp0) : CProp0 ≝
+ | Conj : A → B → And A B.
+ 
+interpretation "constructive and" 'and x y = (And x y).
+
+inductive And3 (A,B,C:CProp0) : CProp0 ≝
+ | Conj3 : A → B → C → And3 A B C.
+
+notation < "hvbox(a break ∧ b break ∧ c)" with precedence 35 for @{'and3 $a $b $c}.
+ 
+interpretation "constructive ternary and" 'and3 x y z = (And3 x y z).
+
+inductive And42 (A,B,C,D:CProp2) : CProp2 ≝
+ | Conj42 : A → B → C → D → And42 A B C D.
+
+notation < "hvbox(a break ∧ b break ∧ c break ∧ d)" with precedence 35 for @{'and4 $a $b $c $d}.
+ 
+interpretation "constructive quaternary and2" 'and4 x y z t = (And42 x y z t).
+
+record Iff (A,B:CProp0) : CProp0 ≝
+ { if: A → B;
+   fi: B → A
+ }.
+ 
+record Iff1 (A,B:CProp1) : CProp1 ≝
+ { if1: A → B;
+   fi1: B → A
+ }.
+ 
+interpretation "logical iff" 'iff x y = (Iff x y).
+
+notation "hvbox(a break ⇔ b)" right associative with precedence 25 for @{'iff1 $a $b}.
+interpretation "logical iff type1" 'iff1 x y = (Iff1 x y).
+
+inductive exT (A:Type0) (P:A→CProp0) : CProp0 ≝
+  ex_introT: ∀w:A. P w → exT A P.
+  
+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).
+
+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).
+
+
+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).
+
+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 pi1exT23 ≝
+  λA,P,Q,R.λx:exT23 A P Q R.match x with [ex_introT23 x _ _ _ _ ⇒ x].
+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).
+
+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 cotransitive ≝
+ λC:Type0.λlt:C→C→CProp0.∀x,y,z:C. lt x y → lt x z ∨ lt z y. 
+
+definition coreflexive ≝ λC:Type0.λlt:C→C→CProp0. ∀x:C. ¬ (lt x x).
+
+definition symmetric ≝ λC:Type0.λlt:C→C→CProp0. ∀x,y:C.lt x y → lt y x.
+
+definition antisymmetric ≝ λA:Type0.λR:A→A→CProp0.λeq:A→A→Prop.∀x:A.∀y:A.R x y→R y x→eq x y.
+
+definition reflexive ≝ λA:Type0.λR:A→A→CProp0.∀x:A.R x x.
+
+definition transitive ≝ λA:Type0.λR:A→A→CProp0.∀x,y,z:A.R x y → R y z → R x z.
+