ground_2 milestone: multiple relocation with lists of booleans

[helm.git] / matita / matita / contribs / lambdadelta / ground_2 / relocation / mr2_minus.ma

diff --git a/matita/matita/contribs/lambdadelta/ground_2/relocation/mr2_minus.ma b/matita/matita/contribs/lambdadelta/ground_2/relocation/mr2_minus.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 "ground_2/notation/relations/rminus_3.ma".
+include "ground_2/relocation/mr2.ma".
+
+(* MULTIPLE RELOCATION WITH PAIRS *******************************************)
+
+inductive minuss: nat → relation mr2 ≝
+| minuss_nil: ∀i. minuss i (◊) (◊)
+| minuss_lt : ∀cs1,cs2,l,m,i. i < l → minuss i cs1 cs2 →
+              minuss i ({l, m} @ cs1) ({l - i, m} @ cs2)
+| minuss_ge : ∀cs1,cs2,l,m,i. l ≤ i → minuss (m + i) cs1 cs2 →
+              minuss i ({l, m} @ cs1) cs2
+.
+
+interpretation "minus (multiple relocation with pairs)"
+   'RMinus cs1 i cs2 = (minuss i cs1 cs2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact minuss_inv_nil1_aux: ∀cs1,cs2,i. cs1 ▭ i ≡ cs2 → cs1 = ◊ → cs2 = ◊.
+#cs1 #cs2 #i * -cs1 -cs2 -i
+[ //
+| #cs1 #cs2 #l #m #i #_ #_ #H destruct
+| #cs1 #cs2 #l #m #i #_ #_ #H destruct
+]
+qed-.
+
+lemma minuss_inv_nil1: ∀cs2,i. ◊ ▭ i ≡ cs2 → cs2 = ◊.
+/2 width=4 by minuss_inv_nil1_aux/ qed-.
+
+fact minuss_inv_cons1_aux: ∀cs1,cs2,i. cs1 ▭ i ≡ cs2 →
+                           ∀l,m,cs. cs1 = {l, m} @ cs →
+                           l ≤ i ∧ cs ▭ m + i ≡ cs2 ∨
+                           ∃∃cs0. i < l & cs ▭ i ≡ cs0 &
+                                   cs2 = {l - i, m} @ cs0.
+#cs1 #cs2 #i * -cs1 -cs2 -i
+[ #i #l #m #cs #H destruct
+| #cs1 #cs #l1 #m1 #i1 #Hil1 #Hcs #l2 #m2 #cs2 #H destruct /3 width=3 by ex3_intro, or_intror/
+| #cs1 #cs #l1 #m1 #i1 #Hli1 #Hcs #l2 #m2 #cs2 #H destruct /3 width=1 by or_introl, conj/
+]
+qed-.
+
+lemma minuss_inv_cons1: ∀cs1,cs2,l,m,i. {l, m} @ cs1 ▭ i ≡ cs2 →
+                        l ≤ i ∧ cs1 ▭ m + i ≡ cs2 ∨
+                        ∃∃cs. i < l & cs1 ▭ i ≡ cs &
+                               cs2 = {l - i, m} @ cs.
+/2 width=3 by minuss_inv_cons1_aux/ qed-.
+
+lemma minuss_inv_cons1_ge: ∀cs1,cs2,l,m,i. {l, m} @ cs1 ▭ i ≡ cs2 →
+                           l ≤ i → cs1 ▭ m + i ≡ cs2.
+#cs1 #cs2 #l #m #i #H
+elim (minuss_inv_cons1 … H) -H * // #cs #Hil #_ #_ #Hli
+elim (lt_le_false … Hil Hli)
+qed-.
+
+lemma minuss_inv_cons1_lt: ∀cs1,cs2,l,m,i. {l, m} @ cs1 ▭ i ≡ cs2 →
+                           i < l →
+                           ∃∃cs. cs1 ▭ i ≡ cs & cs2 = {l - i, m} @ cs.
+#cs1 #cs2 #l #m #i #H elim (minuss_inv_cons1 … H) -H * /2 width=3 by ex2_intro/
+#Hli #_ #Hil elim (lt_le_false … Hil Hli)
+qed-.