(* *)
(**************************************************************************)
-include "logic/equality.ma".
-include "datatypes/constructors.ma".
+include "logic/connectives.ma".
inductive Or (A,B:CProp) : CProp ≝
| Left : A → Or A B
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.
-
-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).
+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).
-
+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].
+ 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 \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 \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 \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.