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[helm.git] / matita / tests / rewrite.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 id "nat" = "cic:/Coq/Init/Datatypes/nat.ind#xpointer(1/1)".
+alias num (instance 0) = "Coq natural number".
+alias symbol "eq" (instance 0) = "Coq's leibnitz's equality". 
+alias symbol "plus" (instance 0) = "Coq's natural plus".
+alias id "plus_n_O" = "cic:/Coq/Init/Peano/plus_n_O.con".
+
+theorem a:
+  \forall a,b:nat.
+  a = b \to b + a + b + a= (\lambda j.((\lambda w.((\lambda x.x + b + w + j) a)) b)) a.
+intros.
+rewrite < H in \vdash (? ? ? ((\lambda j.((\lambda w.%) ?)) ?)).
+
+rewrite < H in \vdash (? ? % ?).
+
+simplify in \vdash (? ? ? ((\lambda _.((\lambda _.%) ?)) ?)).
+
+rewrite < H in \vdash (? ? ? (% ?)).
+simplify.
+reflexivity.
+qed.
+ 
+theorem t: \forall n. 0=0 \to n = n + 0.
+ intros.
+ apply plus_n_O.
+qed.
+
+(* In this test "rewrite < t" should open a new goal 0=0 and put it in *)
+(* the goallist so that the THEN tactical closes it using reflexivity. *)
+theorem foo: \forall n. n = n + 0.
+ intros.
+ rewrite < t; reflexivity.
+qed.
+
+theorem test_rewrite_in_hyp:
+          \forall n,m. n + 0 = m \to m = n + 0 \to n=m \land m+0=n+0.
+ intros.
+ rewrite < plus_n_O in H.
+ rewrite > plus_n_O in H1.
+ split; [ exact H | exact H1].
+qed.
+
+theorem test_rewrite_in_hyp2:
+          \forall n,m. n + 0 = m \to n + 0 = m \to n=m \land n+0=m.
+ intros.
+ rewrite < plus_n_O in H H1 \vdash (? ? %).
+ split; [ exact H | exact H1].
+qed.
+
+alias id "O" = "cic:/Coq/Init/Datatypes/nat.ind#xpointer(1/1/1)".
+theorem test_rewrite_under_pi: \forall x,y. x = O \to y = O \to x = x \to O = x.
+intros 3.
+rewrite > H in \vdash (? \to ? ? % % \to ? ? ? %).
+intros. reflexivity.
+qed.