+
+ninductive Ex1 (A:Type[1]) (P:A → CProp[0]) : CProp[1] ≝
+ ex_intro1: ∀x:A. P x → Ex1 A P.
+
+interpretation "exists1" 'exists x = (Ex1 ? x).
+interpretation "exists" 'exists x = (Ex ? x).
+
+ninductive sigma (A : Type[0]) (P : A → CProp[0]) : Type[0] ≝
+ sig_intro : ∀x:A.P x → sigma A P.
+
+interpretation "sigma" 'sigma \eta.p = (sigma ? p).
+
+nrecord iff (A,B: CProp[0]) : CProp[0] ≝
+ { if: A → B;
+ fi: B → A
+ }.
+
+notation > "hvbox(a break \liff b)"
+ left associative with precedence 25
+for @{ 'iff $a $b }.
+
+notation "hvbox(a break \leftrightarrow b)"
+ left associative with precedence 25
+for @{ 'iff $a $b }.
+
+interpretation "logical iff" 'iff x y = (iff x y).