--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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.
+