--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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".
+
+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).
+
+record Iff (A,B:CProp) : CProp ≝
+ { if: A → B;
+ fi: B → A
+ }.
+
+record Iff1 (A,B:CProp) : CProp ≝
+ { 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:Type) (P:A→CProp) : CProp ≝
+ ex_introT: ∀w:A. P w → exT A P.
+
+interpretation "CProp exists" 'exists 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].
+
+interpretation "exT \fst" 'pi1 = (pi1exT ? ?).
+interpretation "exT \fst" 'pi1a x = (pi1exT ? ? x).
+interpretation "exT \fst" 'pi1b x y = (pi1exT ? ? x y).
+
+definition pi2exT ≝
+ λA,P.λx:exT A P.match x return λx.P (pi1exT ? ? x) with [ex_introT _ p ⇒ p].
+
+interpretation "exT \snd" 'pi2 = (pi2exT ? ?).
+interpretation "exT \snd" 'pi2a x = (pi2exT ? ? x).
+interpretation "exT \snd" 'pi2b x y = (pi2exT ? ? x y).
+
+inductive exP (A:Type) (P:A→Prop) : CProp ≝
+ ex_introP: ∀w:A. P w → exP A P.
+
+interpretation "dependent pair for Prop" 'dependent_pair a b =
+ (ex_introP ?? a b).
+
+interpretation "CProp exists for Prop" 'exists x = (exP ? x).
+
+definition pi1exP ≝ λA,P.λx:exP A P.match x with [ex_introP x _ ⇒ x].
+
+interpretation "exP \fst" 'pi1 = (pi1exP ? ?).
+interpretation "exP \fst" 'pi1a x = (pi1exP ? ? x).
+interpretation "exP \fst" 'pi1b x y = (pi1exP ? ? x y).
+
+definition pi2exP ≝
+ λA,P.λx:exP A P.match x return λx.P (pi1exP ?? x) with [ex_introP _ p ⇒ p].
+
+interpretation "exP \snd" 'pi2 = (pi2exP ? ?).
+interpretation "exP \snd" 'pi2a x = (pi2exP ? ? x).
+interpretation "exP \snd" 'pi2b x y = (pi2exP ? ? 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].
+
+interpretation "exT2 \fst" 'pi1 = (pi1exT23 ? ? ? ?).
+interpretation "exT2 \fst" 'pi1a x = (pi1exT23 ? ? ? ? x).
+interpretation "exT2 \fst" 'pi1b x y = (pi1exT23 ? ? ? ? x y).
+
+definition pi2exT23 ≝
+ λA,P,Q,R.λx:exT23 A P Q R.match x with [ex_introT23 _ x _ _ _ ⇒ x].
+
+interpretation "exT2 \snd" 'pi2 = (pi2exT23 ? ? ? ?).
+interpretation "exT2 \snd" 'pi2a x = (pi2exT23 ? ? ? ? x).
+interpretation "exT2 \snd" 'pi2b x y = (pi2exT23 ? ? ? ? x y).
+
+inductive exT2 (A:Type) (P,Q:A→CProp) : CProp ≝
+ ex_introT2: ∀w:A. P w → Q w → exT2 A P Q.
+
+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.
+
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