milestone update in ground, partial commit

[helm.git] / matita / matita / contribs / lambdadelta / ground / arith / arith_2b.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/arith/nat_le_minus_plus.ma".
+
+(* ARITHMETICAL PROPERTIES FOR λδ-2B ****************************************)
+
+lemma arith_l4 (m11) (m12) (m21) (m22):
+               m21+m22-(m11+m12) = m21-m11-m12+(m22-(m11-m21)-(m12-(m21-m11))).
+#m11 #m12 #m21 #m22 >nminus_plus_assoc
+elim (nat_split_le_ge (m11+m12) m21) #Hm1121
+[ lapply (nle_trans m11 ??? Hm1121) // #Hm121
+  lapply (nle_minus_dx_dx … Hm1121) #Hm12211
+  <nminus_plus_comm_23 // @eq_f2 // <(nle_inv_eq_zero_minus m11 ?) // <(nle_inv_eq_zero_minus m12 ?) //
+| <(nle_inv_eq_zero_minus m21 ?) // <nplus_zero_sn <nminus_plus_assoc <nplus_comm
+  elim (nat_split_le_ge m11 m21) #Hm121
+  [ lapply (nle_minus_sn_dx … Hm1121) #Hm2112
+    <(nle_inv_eq_zero_minus m11 ?) // >nplus_minus_assoc // >nminus_assoc_comm_23 //
+  | <(nle_inv_eq_zero_minus m21 ?) // >nminus_assoc_comm_23 //
+  ]
+]
+qed.
+
+lemma arith_l3 (m) (n1) (n2): n1+n2-m = n1-m+(n2-(m-n1)).
+// qed.
+
+lemma arith_l2 (n1) (n2): ↑n2-n1 = 𝟏-n1+(n2-(n1-𝟏)).
+#n1 #n2 <arith_l3 //
+qed.
+
+lemma arith_l1 (n): ninj (𝟏) = 𝟏-n+(n-(n-𝟏)).
+#n <arith_l2 //
+qed.