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 set "baseuri" "cic:/matita/connectives/".
17 inductive True: Prop \def
20 default "true" cic:/matita/logic/True.ind.
22 inductive False: Prop \def .
24 default "false" cic:/matita/logic/False.ind.
26 definition Not: Prop \to Prop \def
27 \lambda A. (A \to False).
29 theorem absurd : \forall A,C:Prop. A \to Not A \to C.
33 default "absurd" cic:/matita/logic/absurd.ind.
35 inductive And (A,B:Prop) : Prop \def
36 conj : A \to B \to (And A B).
38 theorem proj1: \forall A,B:Prop. (And A B) \to A.
39 intros. elim H. assumption.
42 theorem proj2: \forall A,B:Prop. (And A B) \to B.
43 intros. elim H. assumption.
46 inductive Or (A,B:Prop) : Prop \def
47 or_introl : A \to (Or A B)
48 | or_intror : B \to (Or A B).
50 definition decidable : Prop \to Prop \def \lambda A:Prop. Or A (Not A).
52 inductive ex (A:Type) (P:A \to Prop) : Prop \def
53 ex_intro: \forall x:A. P x \to ex A P.
55 inductive ex2 (A:Type) (P,Q:A \to Prop) : Prop \def
56 ex_intro2: \forall x:A. P x \to Q x \to ex2 A P Q.