+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| A.Asperti, C.Sacerdoti Coen, *)
+(* ||A|| E.Tassi, S.Zacchiroli *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU Lesser General Public License Version 2.1 *)
+(* *)
+(**************************************************************************)
+
+include "Plogic/equality.ma".
+
+inductive True: Prop ≝
+I : True.
+
+inductive False: Prop ≝ .
+
+(*
+ndefinition Not: Prop → Prop ≝
+λA. A → False. *)
+
+inductive Not (A:Prop): Prop ≝
+nmk: (A → False) → Not A.
+
+interpretation "logical not" 'not x = (Not x).
+
+theorem absurd : ∀ A:Prop. A → ¬A → False.
+#A #H #Hn elim Hn /2/ qed.
+
+(*
+ntheorem absurd : ∀ A,C:Prop. A → ¬A → C.
+#A #C #H #Hn nelim (Hn H).
+nqed. *)
+
+theorem not_to_not : ∀A,B:Prop. (A → B) → ¬B →¬A.
+/4/ qed.
+
+inductive And (A,B:Prop) : Prop ≝
+ conj : A → B → And A B.
+
+interpretation "logical and" 'and x y = (And x y).
+
+theorem proj1: ∀A,B:Prop. A ∧ B → A.
+#A #B #AB elim AB //.
+qed.
+
+theorem proj2: ∀ A,B:Prop. A ∧ B → B.
+#A #B #AB elim AB //.
+qed.
+
+inductive Or (A,B:Prop) : Prop ≝
+ or_introl : A → (Or A B)
+ | or_intror : B → (Or A B).
+
+interpretation "logical or" 'or x y = (Or x y).
+
+definition decidable : Prop → Prop ≝
+λ A:Prop. A ∨ ¬ A.
+
+inductive ex (A:Type[0]) (P:A → Prop) : Prop ≝
+ ex_intro: ∀ x:A. P x → ex A P.
+
+interpretation "exists" 'exists x = (ex ? x).
+
+inductive ex2 (A:Type[0]) (P,Q:A \to Prop) : Prop ≝
+ ex_intro2: ∀ x:A. P x → Q x → ex2 A P Q.
+
+definition iff :=
+ λ A,B. (A → B) ∧ (B → A).
+
+interpretation "iff" 'iff a b = (iff a b).