propagating the arithmetics library, partial commit

[helm.git] / matita / matita / contribs / lambdadelta / ground / counters / rtc_plus.ma

diff --git a/matita/matita/contribs/lambdadelta/ground/counters/rtc_plus.ma b/matita/matita/contribs/lambdadelta/ground/counters/rtc_plus.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_plus.ma".
+include "ground/counters/rtc.ma".
+
+(* ADDITION FOR RT-TRANSITION COUNTERS **************************************)
+
+definition rtc_plus (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
+  "plus (rtc)"
+  'plus c1 c2 = (rtc_plus c1 c2).
+
+(* Basic constructions ******************************************************)
+
+lemma rtc_plus_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_plus_zz_dx (c): c = c + 𝟘𝟘.
+* #ri #rs #ti #ts <rtc_plus_rew //
+qed.
+
+(* Basic inversions *********************************************************)
+
+lemma rtc_plus_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_plus_rew #H destruct /2 width=14 by ex6_8_intro/
+qed-.
+
+(* Main constructions *******************************************************)
+
+theorem rtc_plus_assoc: associative … rtc_plus.
+* #ri1 #rs1 #ti1 #ts1 * #ri2 #rs2 #ti2 #ts2 * #ri3 #rs3 #ti3 #ts3
+<rtc_plus_rew //
+qed.