Stuff moved from old Matita.

[helm.git] / matita / matita / contribs / RELATIONAL / NPlus / inv.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 "NPlus/defs.ma".
+
+(* Inversion lemmas *********************************************************)
+
+theorem nplus_inv_zero_1: ∀q,r. zero ⊕ q ≍ r → q = r.
+ intros. elim H; clear H q r; autobatch.
+qed.
+
+theorem nplus_inv_succ_1: ∀p,q,r. succ p ⊕ q ≍ r → 
+                          ∃s. r = succ s ∧ p ⊕ q ≍ s.
+ intros. elim H; clear H q r; intros;
+ [ autobatch depth = 3
+ | clear H1; decompose; destruct; autobatch depth = 4
+ ]
+qed.
+
+theorem nplus_inv_zero_2: ∀p,r. p ⊕ zero ≍ r → p = r.
+ intros; inversion H; clear H; intros; destruct; autobatch.
+qed.
+
+theorem nplus_inv_succ_2: ∀p,q,r. p ⊕ succ q ≍ r → 
+                          ∃s. r = succ s ∧ p ⊕ q ≍ s.
+ intros; inversion H; clear H; intros; destruct.
+ autobatch depth = 3.
+qed.
+
+theorem nplus_inv_zero_3: ∀p,q. p ⊕ q ≍ zero → 
+                          p = zero ∧ q = zero.
+ intros; inversion H; clear H; intros; destruct; autobatch.
+qed.
+
+theorem nplus_inv_succ_3: ∀p,q,r. p ⊕ q ≍ succ r →
+                             ∃s. p = succ s ∧ s ⊕ q ≍ r ∨
+                               q = succ s ∧ p ⊕ s ≍ r.
+ intros; inversion H; clear H; intros; destruct;
+ autobatch depth = 4.
+qed.
+
+(* Corollaries to inversion lemmas ******************************************)
+
+theorem nplus_inv_succ_2_3: ∀p,q,r.
+                            p ⊕ succ q ≍ succ r → p ⊕ q ≍ r.
+ intros;
+ lapply linear nplus_inv_succ_2 to H; decompose; destruct; autobatch.
+qed.
+
+theorem nplus_inv_succ_1_3: ∀p,q,r.
+                            succ p ⊕ q ≍ succ r → p ⊕ q ≍ r.
+ intros;
+ lapply linear nplus_inv_succ_1 to H; decompose; destruct; autobatch.
+qed.
+
+theorem nplus_inv_eq_2_3: ∀p,q. p ⊕ q ≍ q → 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: ∀p,q. p ⊕ q ≍ p → 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.