arithmetics for λδ

[helm.git] / matita / matita / contribs / lambdadelta / ground / arith / pnat_tri.ma

diff --git a/matita/matita/contribs/lambdadelta/ground/arith/pnat_tri.ma b/matita/matita/contribs/lambdadelta/ground/arith/pnat_tri.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/pnat.ma".
+
+(* TRICHOTOMY OPERATOR FOR POSITIVE INTEGERS ********************************)
+
+rec definition ptri (A:Type[0]) p1 p2 a1 a2 a3 on p1 : A ≝
+  match p1 with
+  [ punit    ⇒ match p2 with [ punit ⇒ a2 | psucc p2 ⇒ a1 ]
+  | psucc p1 ⇒ match p2 with [ punit ⇒ a3 | psucc p2 ⇒ ptri A p1 p2 a1 a2 a3 ]
+  ].
+
+(* Basic rewrites ***********************************************************)
+
+lemma ptri_unit_bi (A) (a1) (a2) (a3):
+      a2 = ptri A (𝟏) (𝟏) a1 a2 a3.
+// qed.
+
+lemma ptri_unit_succ (A) (a1) (a2) (a3) (p):
+      a1 = ptri A (𝟏) (↑p) a1 a2 a3.
+// qed.
+
+lemma ptri_succ_unit (A) (a1) (a2) (a3) (p):
+      a3 = ptri A (↑p) (𝟏) a1 a2 a3.
+// qed.
+
+lemma ptri_succ_bi (A) (a1) (a2) (a3) (p1) (p2):
+      ptri A (p1) (p2) a1 a2 a3 = ptri A (↑p1) (↑p2) a1 a2 a3.
+// qed.
+
+(* Advanced rewrites ********************************************************)
+
+lemma ptri_eq (A) (a1) (a2) (a3) (p): a2 = ptri A p p a1 a2 a3.
+#A #a1 #a2 #a3 #p elim p -p //
+qed.
+
+lemma ptri_f_tri (A) (B) (f) (a1) (a2) (a3) (p1) (p2):
+      f (ptri A p1 p2 a1 a2 a3) = ptri B p1 p2 (f a1) (f a2) (f a3).
+#A #B #f #a1 #a2 #a3 #p1
+elim p1 -p1 [| #p1 #IH ] * //
+qed.