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[helm.git] / matita / tests / apply.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                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* test _with_ the WHD on the apply argument *)
+
+include "coq.ma".
+
+alias id "not" = "cic:/Coq/Init/Logic/not.con".
+alias id "False" = "cic:/Coq/Init/Logic/False.ind#xpointer(1/1)".
+
+theorem b:
+  \forall x:Prop.
+  (not x) \to x \to False.
+intros.
+apply H.
+assumption.
+qed.
+
+(* test _without_ the WHD on the apply argument *)
+
+alias symbol "eq" (instance 0) = "Coq's leibnitz's equality".
+
+theorem a:
+  \forall A:Set.
+  \forall x: A.
+  not (x=x) \to not (x=x).
+intros.
+apply H.
+qed.
+
+
+(* this test shows what happens when a term of type A -> ? is applied to
+   a goal of type A' -> B: if A unifies with A' the unifier becomes ? := B
+   and no goal is opened; otherwise the unifier becomes ? := A' -> B and a
+   new goal of type A is created. *)
+theorem c:
+ \forall A,B:Prop.
+   A \to (\forall P: Prop. A \to P) \to (A \to B) \land (B \to B).
+ intros 4; split; [ apply H1 | apply H1; exact H ].
+qed.
+
+(* this test requires the delta-expansion of not in the type of the applied
+   term (to reveal a product) *)
+theorem d: \forall A: Prop. \lnot A \to A \to False.
+ intros. apply H. assumption.
+qed.