Stuff moved from old Matita.

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

diff --git a/matita/matita/tests/fguidi.ma b/matita/matita/tests/fguidi.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 "logic/connectives.ma".
+include "nat/nat.ma".
+
+theorem eq_S_S: \forall m,n. m = n \to (S m) = (S n).
+intros; destruct; autobatch.
+qed.
+
+inductive le: nat \to nat \to Prop \def
+     le_zero: \forall n. (le O n)
+   | le_succ: \forall m, n. (le m n) \to (le (S m) (S n)).
+
+theorem le_refl: \forall x. (le x x).
+intros; elim x; clear x; autobatch.
+qed.
+
+theorem le_gen_x_O_aux: \forall x, y. (le x y) \to (y =O) \to (x = O).
+intros 3; elim H; clear H x y;
+[ autobatch
+| destruct
+]
+qed.
+
+theorem le_gen_x_O: \forall x. (le x O) \to (x = O).
+intros; lapply linear le_gen_x_O_aux to H;
+[ destruct; autobatch
+| autobatch
+].
+qed.
+
+theorem le_x_O: \forall x. (x = O) \to (le x O).
+intros; destruct; autobatch.
+qed.
+
+theorem le_gen_S_x_aux: \forall m,x,y. (le y x) \to (y = S m) \to 
+                        (\exists n. x = (S n) \land (le m n)).
+intros 4; elim H; clear H x y;
+[ destruct
+| destruct; autobatch
+].
+qed.
+
+theorem le_gen_S_x: \forall m,x. (le (S m) x) \to 
+                    (\exists n. x = (S n) \land (le m n)).
+intros; lapply le_gen_S_x_aux to H; autobatch.
+qed.
+
+theorem le_S_x: \forall m,x. (\exists n. x = (S n) \land (le m n)) \to
+                (le (S m) x).
+intros; decompose; destruct; autobatch.
+qed.
+
+theorem le_gen_S_S: \forall m,n. (le (S m) (S n)) \to (le m n).
+intros.
+lapply linear le_gen_S_x to H as H0; decompose; destruct; autobatch.
+qed.
+
+theorem le_S_S: \forall m,n. (le m n) \to (le (S m) (S n)).
+intros; autobatch.
+qed.
+
+theorem le_trans: \forall x,y. (le x y) \to \forall z. (le y z) \to (le x z).
+intros 3. elim H; clear H x y;
+[ autobatch
+| lapply linear le_gen_S_x to H3; decompose; destruct; autobatch.
+].
+qed.