propagating the arithmetics library, partial commit

[helm.git] / matita / matita / contribs / lambdadelta / ground / counters / rtc_max.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/xoa/ex_6_8.ma".
+include "ground/arith/nat_max.ma".
+include "ground/counters/rtc.ma".
+
+(* MAXIMUM FOR RT-TRANSITION COUNTERS ***************************************)
+
+definition rtc_max (c1:rtc) (c2:rtc): rtc ≝
+match c1 with
+[ mk_rtc ri1 rs1 ti1 ts1 ⇒
+  match c2 with
+  [ mk_rtc ri2 rs2 ti2 ts2 ⇒ 〈ri1∨ri2,rs1∨rs2,ti1∨ti2,ts1∨ts2〉
+  ]
+].
+
+interpretation
+  "maximum (rtc)"
+  'or c1 c2 = (rtc_max c1 c2).
+
+(* Basic constructions ******************************************************)
+
+lemma rtc_max_rew (ri1) (ri2) (rs1) (rs2) (ti1) (ti2) (ts1) (ts2):
+      〈ri1∨ri2,rs1∨rs2,ti1∨ti2,ts1∨ts2〉 =
+      (〈ri1,rs1,ti1,ts1〉 ∨ 〈ri2,rs2,ti2,ts2〉).
+// qed.
+
+lemma rtc_max_zz_dx (c): c = (c ∨ 𝟘𝟘).
+* #ri #rs #ti #ts <rtc_max_rew //
+qed.
+
+lemma rtc_max_idem (c): c = (c ∨ c).
+* #ri #rs #ti #ts <rtc_max_rew //
+qed.
+
+(* Basic inversions *********************************************************)
+
+lemma rtc_max_inv_dx (ri) (rs) (ti) (ts) (c1) (c2):
+      〈ri,rs,ti,ts〉 = (c1 ∨ c2) →
+      ∃∃ri1,rs1,ti1,ts1,ri2,rs2,ti2,ts2.
+      (ri1∨ri2) = ri & (rs1∨rs2) = rs & (ti1∨ti2) = ti & (ts1∨ts2) = ts &
+      〈ri1,rs1,ti1,ts1〉 = c1 & 〈ri2,rs2,ti2,ts2〉 = c2.
+#ri #rs #ti #ts * #ri1 #rs1 #ti1 #ts1 * #ri2 #rs2 #ti2 #ts2
+<rtc_max_rew #H destruct /2 width=14 by ex6_8_intro/
+qed-.
+
+(* Main constructions *******************************************************)
+
+theorem rtc_max_assoc: associative … rtc_max.
+* #ri1 #rs1 #ti1 #ts1 * #ri2 #rs2 #ti2 #ts2 * #ri3 #rs3 #ti3 #ts3
+<rtc_max_rew <rtc_max_rew //
+qed.