arithmetics for λδ

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

diff --git a/matita/matita/contribs/lambdadelta/ground/arith/nat_tri.ma b/matita/matita/contribs/lambdadelta/ground/arith/nat_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_tri.ma".
+include "ground/arith/nat.ma".
+
+(* TRICHOTOMY OPERATOR FOR NON-NEGATIVE INTEGERS ****************************)
+
+(* Note: this is "if eqb n1 n2 then a2 else if leb n1 n2 then a1 else a3" *)
+(*** tri *)
+definition ntri (A:Type[0]) (n1) (n2) (a1) (a2) (a3): A ≝
+  match n1 with
+  [ nzero    ⇒ match n2 with [ nzero ⇒ a2 | ninj p2 ⇒ a1 ]
+  | ninj  p1 ⇒ match n2 with [ nzero ⇒ a3 | ninj p2 ⇒ ptri A p1 p2 a1 a2 a3 ]
+  ].
+
+(* Basic rewrites ***********************************************************)
+
+lemma ntri_zero_bi (A) (a1) (a2) (a3):
+      a2 = ntri A (𝟎) (𝟎) a1 a2 a3.
+// qed.
+
+lemma ntri_zero_inj (A) (a1) (a2) (a3) (p):
+      a1 = ntri A (𝟎) (ninj p) a1 a2 a3.
+// qed.
+
+lemma ntri_inj_zero (A) (a1) (a2) (a3) (p):
+      a3 = ntri A (ninj p) (𝟎) a1 a2 a3.
+// qed.
+
+lemma ntri_inj_bi (A) (a1) (a2) (a3) (p1) (p2):
+      ptri A (p1) (p2) a1 a2 a3 = ntri A (p1) (p2) a1 a2 a3.
+// qed.
+
+(* Advanced rewrites ********************************************************)
+
+(*** tri_eq *)
+lemma ntri_eq (A) (a1) (a2) (a3) (n): a2 = ntri A n n a1 a2 a3.
+#A #a1 #a2 #a3 * //
+qed.
+
+lemma ntri_f_tri (A) (B) (f) (a1) (a2) (a3) (n1) (n2):
+      f (ntri A n1 n2 a1 a2 a3) = ntri B n1 n2 (f a1) (f a2) (f a3).
+#A #B #f #a1 #a2 #a3
+* [| #p1 ] * // #p2
+<ntri_inj_bi <ntri_inj_bi //
+qed.