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[helm.git] / matita / tests / demodulation_coq.ma

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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+
+
+include "coq.ma".
+
+alias num = "Coq natural number".
+alias symbol "times" = "Coq's natural times".
+alias symbol "plus" = "Coq's natural plus".
+alias symbol "eq" = "Coq's leibnitz's equality".
+alias id "nat" = "cic:/Coq/Init/Datatypes/nat.ind#xpointer(1/1)".
+alias id "S" = "cic:/Coq/Init/Datatypes/nat.ind#xpointer(1/1/2)".
+alias id "O" = "cic:/Coq/Init/Datatypes/nat.ind#xpointer(1/1/1)".
+
+theorem p0 : \forall m:nat. m+O = m.
+intro. demodulate.reflexivity.
+qed.
+
+theorem p: \forall m.1*m = m.
+intros.demodulate.reflexivity.
+qed.
+
+theorem p2: \forall x,y:nat.(S x)*y = (y+x*y).
+intros.demodulate.reflexivity.
+qed.
+
+theorem p1: \forall x,y:nat.(S ((S x)*y+x))=(S x)+(y*x+y).
+intros.demodulate.reflexivity.
+qed.
+
+theorem p3: \forall x,y:nat. (x+y)*(x+y) = x*x + 2*(x*y) + (y*y).
+intros.demodulate.reflexivity.
+qed.
+
+theorem p4: \forall x:nat. (x+1)*(x-1)=x*x - 1.
+intro.
+apply (nat_case x)
+[simplify.reflexivity
+|intro.demodulate.reflexivity]
+qed.
+