ground_2 released and permanently renamed as ground

[helm.git] / matita / matita / contribs / lambdadelta / ground / lib / arith_2b.ma

diff --git a/matita/matita/contribs/lambdadelta/ground/lib/arith_2b.ma b/matita/matita/contribs/lambdadelta/ground/lib/arith_2b.ma
new file mode 100644 (file)
index 0000000..f9df115
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground/lib/arith_2b.ma
@@ -0,0 +1,44 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/lib/arith.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 >minus_plus
+elim (le_or_ge (m11+m12) m21) #Hm1121
+[ lapply (transitive_le m11 ??? Hm1121) // #Hm121
+  lapply (le_plus_to_minus_l … Hm1121) #Hm12211
+  <plus_minus // @eq_f2 // >(eq_minus_O m11 ?) // >(eq_minus_O m12 ?) //
+| >(eq_minus_O m21 ?) // <plus_O_n <minus_plus <commutative_plus
+  elim (le_or_ge m11 m21) #Hm121
+  [ lapply (le_plus_to_minus_comm … Hm1121) #Hm2112
+    >(eq_minus_O m11 ?) // <plus_minus_associative // <minus_le_minus_minus_comm //
+  | >(eq_minus_O m21 ?) // <minus_le_minus_minus_comm //
+  ]
+]
+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: ∀x. 1 = 1-x+(x-(x-1)).
+#x <arith_l2 //
+qed.