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[helm.git] / matita / library / logic / connectives.ma

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+(**************************************************************************)
+(*       ___                                                               *)
+(*      ||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         *)
+(*                                                                        *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/logic/connectives/".
+
+inductive True: Prop \def
+I : True.
+
+default "true" cic:/matita/logic/connectives/True.ind.
+
+inductive False: Prop \def .
+
+default "false" cic:/matita/logic/connectives/False.ind.
+
+definition Not: Prop \to Prop \def
+\lambda A. (A \to False).
+
+(*CSC: the URI must disappear: there is a bug now *)
+interpretation "logical not" 'not x = (cic:/matita/logic/connectives/Not.con x).
+
+theorem absurd : \forall A,C:Prop. A \to \lnot A \to C.
+intros. elim (H1 H).
+qed.
+
+default "absurd" cic:/matita/logic/connectives/absurd.con.
+
+inductive And (A,B:Prop) : Prop \def
+    conj : A \to B \to (And A B).
+
+(*CSC: the URI must disappear: there is a bug now *)
+interpretation "logical and" 'and x y = (cic:/matita/logic/connectives/And.ind#xpointer(1/1) x y).
+
+theorem proj1: \forall A,B:Prop. A \land B \to A.
+intros. elim H. assumption.
+qed.
+
+theorem proj2: \forall A,B:Prop. A \land B \to B.
+intros. elim H. assumption.
+qed.
+
+inductive Or (A,B:Prop) : Prop \def
+     or_introl : A \to (Or A B)
+   | or_intror : B \to (Or A B).
+
+(*CSC: the URI must disappear: there is a bug now *)
+interpretation "logical or" 'or x y =
+  (cic:/matita/logic/connectives/Or.ind#xpointer(1/1) x y).
+
+theorem Or_ind':
+ \forall A,B:Prop.
+  \forall P: A \lor B \to Prop.
+   (\forall p:A. P (or_introl ? ? p)) \to
+   (\forall q:B. P (or_intror ? ? q)) \to
+    \forall p:A \lor B. P p.
+ intros.
+ apply
+  (match p return \lambda p.P p with
+    [(or_introl p) \Rightarrow H p
+    |(or_intror q) \Rightarrow H1 q]).
+qed.
+
+definition decidable : Prop \to Prop \def \lambda A:Prop. A \lor \lnot A.
+
+inductive ex (A:Type) (P:A \to Prop) : Prop \def
+    ex_intro: \forall x:A. P x \to ex A P.
+
+(*CSC: the URI must disappear: there is a bug now *)
+interpretation "exists" 'exists \eta.x =
+  (cic:/matita/logic/connectives/ex.ind#xpointer(1/1) _ x).
+
+inductive ex2 (A:Type) (P,Q:A \to Prop) : Prop \def
+    ex_intro2: \forall x:A. P x \to Q x \to ex2 A P Q.
+
+definition iff :=
+ \lambda A,B. (A -> B) \land (B -> A).
\ No newline at end of file