branch for universe

[helm.git] / matita / contribs / RELATIONAL / NPlus / inv.ma

diff --git a/matita/contribs/RELATIONAL/NPlus/inv.ma b/matita/contribs/RELATIONAL/NPlus/inv.ma
new file mode 100644 (file)
index 0000000..e3ec168
--- /dev/null
+++ b/matita/contribs/RELATIONAL/NPlus/inv.ma
@@ -0,0 +1,81 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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 "NPlus/defs.ma".
+
+(* Inversion lemmas *********************************************************)
+
+theorem nplus_inv_zero_1: \forall q,r. (zero + q == r) \to q = r.
+ intros. elim H; clear H q r; autobatch.
+qed.
+
+theorem nplus_inv_succ_1: \forall p,q,r. ((succ p) + q == r) \to 
+                          \exists s. r = (succ s) \land p + q == s.
+ intros. elim H; clear H q r; intros;
+ [ autobatch depth = 4
+ | clear H1. decompose. destruct. autobatch depth = 4
+ ]
+qed.
+
+theorem nplus_inv_zero_2: \forall p,r. (p + zero == r) \to p = r.
+ intros. inversion H; clear H; intros; destruct. autobatch.
+qed.
+
+theorem nplus_inv_succ_2: \forall p,q,r. (p + (succ q) == r) \to 
+                          \exists s. r = (succ s) \land p + q == s.
+ intros. inversion H; clear H; intros; destruct.
+ autobatch depth = 4.
+qed.
+
+theorem nplus_inv_zero_3: \forall p,q. (p + q == zero) \to 
+                          p = zero \land q = zero.
+ intros. inversion H; clear H; intros; destruct. autobatch.
+qed.
+
+theorem nplus_inv_succ_3: \forall p,q,r. (p + q == (succ r)) \to
+                          \exists s. p = succ s \land (s + q == r) \lor
+                                     q = succ s \land p + s == r.
+ intros. inversion H; clear H; intros; destruct;
+ autobatch depth = 4.
+qed.
+
+(* Corollaries to inversion lemmas ******************************************)
+
+theorem nplus_inv_succ_2_3: \forall p,q,r.
+                            (p + (succ q) == (succ r)) \to p + q == r.
+ intros. 
+ lapply linear nplus_inv_succ_2 to H. decompose. destruct. autobatch.
+qed.
+
+theorem nplus_inv_succ_1_3: \forall p,q,r.
+                            ((succ p) + q == (succ r)) \to p + q == r.
+ intros. 
+ lapply linear nplus_inv_succ_1 to H. decompose. destruct. autobatch.
+qed.
+
+theorem nplus_inv_eq_2_3: \forall p,q. (p + q == q) \to p = zero.
+ intros 2. elim q; clear q;
+ [ lapply linear nplus_inv_zero_2 to H
+ | lapply linear nplus_inv_succ_2_3 to H1
+ ]; autobatch.
+qed.
+
+theorem nplus_inv_eq_1_3: \forall p,q. (p + q == p) \to q = zero.
+ intros 1. elim p; clear p;
+ [ lapply linear nplus_inv_zero_1 to H
+ | lapply linear nplus_inv_succ_1_3 to H1.
+ ]; autobatch.
+qed.