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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/equality.ma".
-include "datatypes/constructors.ma".
-
-inductive Or (A,B:CProp) : CProp ≝
- | Left : A → Or A B
- | Right : B → Or A B.
-
-interpretation "constructive or" 'or x y = (Or x y).
-
-inductive Or3 (A,B,C:CProp) : CProp ≝
- | 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:CProp) : CProp ≝
- | 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:CProp) : CProp ≝
- | Conj : A → B → And A B.
- 
-interpretation "constructive and" 'and x y = (And x y).
-
-inductive And3 (A,B,C:CProp) : CProp ≝
- | 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 And4 (A,B,C,D:CProp) : CProp ≝
- | Conj4 : A → B → C → D → And4 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 and" 'and4 x y z t = (And4 x y z t).
-
-inductive exT (A:Type) (P:A→CProp) : CProp ≝
-  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:Type) (P:A→CProp) (Q:A→CProp) (R:A→A→CProp) : CProp ≝
-  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).
-
-definition Not : CProp → Prop ≝ λx:CProp.x → False.
-
-interpretation "constructive not" 'not x = (Not x).
-  
-definition cotransitive ≝
- λC:Type.λlt:C→C→CProp.∀x,y,z:C. lt x y → lt x z ∨ lt z y. 
-
-definition coreflexive ≝ λC:Type.λlt:C→C→CProp. ∀x:C. ¬ (lt x x).
-
-definition symmetric ≝ λC:Type.λlt:C→C→CProp. ∀x,y:C.lt x y → lt y x.
-
-definition antisymmetric ≝ λA:Type.λR:A→A→CProp.λeq:A→A→Prop.∀x:A.∀y:A.R x y→R y x→eq x y.
-
-definition reflexive ≝ λA:Type.λR:A→A→CProp.∀x:A.R x x.
-
-definition transitive ≝ λA:Type.λR:A→A→CProp.∀x,y,z:A.R x y → R y z → R x z.
-