1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| A.Asperti, C.Sacerdoti Coen, *)
8 (* ||A|| E.Tassi, S.Zacchiroli *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU Lesser General Public License Version 2.1 *)
13 (**************************************************************************)
15 include "Plogic/equality.ma".
17 ninductive True: Prop ≝
20 default "true" cic:/matita/basics/connectives/True.ind.
22 ninductive False: Prop ≝ .
24 default "false" cic:/matita/basics/connectives/False.ind.
26 ndefinition Not: Prop → Prop ≝
29 interpretation "logical not" 'not x = (Not x).
31 ntheorem absurd : ∀ A,C:Prop. A → ¬A → C.
32 #A; #C; #H; #Hn; nelim (Hn H).
35 ntheorem not_to_not : ∀A,B:Prop. (A → B) → ¬B →¬A.
39 ninductive And (A,B:Prop) : Prop ≝
40 conj : A → B → And A B.
42 interpretation "logical and" 'and x y = (And x y).
44 ntheorem proj1: ∀A,B:Prop. A ∧ B → A.
45 #A; #B; #AB; nelim AB; //.
48 ntheorem proj2: ∀ A,B:Prop. A ∧ B → B.
49 #A; #B; #AB; nelim AB; //.
52 ninductive Or (A,B:Prop) : Prop ≝
53 or_introl : A → (Or A B)
54 | or_intror : B → (Or A B).
56 interpretation "logical or" 'or x y = (Or x y).
58 ndefinition decidable : Prop → Prop ≝
61 ninductive ex (A:Type[0]) (P:A → Prop) : Prop ≝
62 ex_intro: ∀ x:A. P x → ex A P.
64 interpretation "exists" 'exists x = (ex ? x).
66 ninductive ex2 (A:Type[0]) (P,Q:A \to Prop) : Prop ≝
67 ex_intro2: ∀ x:A. P x → Q x → ex2 A P Q.
70 λ A,B. (A → B) ∧ (B → A).