basic properties of cpr ...

[helm.git] / matita / matita / contribs / lambdadelta / ground_2 / steps / 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_2/steps/rtc_shift.ma".
+
+(* RT-TRANSITION COUNTER ****************************************************)
+
+definition 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 = (max c1 c2).
+
+(* Basic properties *********************************************************)
+
+lemma 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 max_O_dx: ∀c. c = (c ∨ 𝟘𝟘).
+* #ri #rs #ti #ts <max_rew //
+qed.
+
+(* Basic inversion properties ***********************************************)
+
+lemma 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
+<max_rew #H destruct /2 width=14 by ex6_8_intro/
+qed-.
+
+(* Properties with test for constrained rt-transition counter ***************)
+
+lemma isrt_max: ∀n1,n2,c1,c2. 𝐑𝐓⦃n1, c1⦄ → 𝐑𝐓⦃n2, c2⦄ → 𝐑𝐓⦃n1∨n2, c1∨c2⦄.
+#n1 #n2 #c1 #c2 * #ri1 #rs1 #H1 * #ri2 #rs2 #H2 destruct
+/2 width=3 by ex1_2_intro/
+qed.
+
+lemma isrt_max_O1: ∀n,c1,c2. 𝐑𝐓⦃0, c1⦄ → 𝐑𝐓⦃n, c2⦄ → 𝐑𝐓⦃n, c1∨c2⦄.
+/2 width=1 by isrt_max/ qed.
+
+lemma isrt_max_O2: ∀n,c1,c2. 𝐑𝐓⦃n, c1⦄ → 𝐑𝐓⦃0, c2⦄ → 𝐑𝐓⦃n, c1∨c2⦄.
+#n #c1 #c2 #H1 #H2 >(max_O2 n) /2 width=1 by isrt_max/
+qed.
+
+(* Inversion properties with test for constrained rt-transition counter *****)
+
+lemma isrt_inv_max: ∀n,c1,c2. 𝐑𝐓⦃n, c1 ∨ c2⦄ →
+                    ∃∃n1,n2. 𝐑𝐓⦃n1, c1⦄ & 𝐑𝐓⦃n2, c2⦄ & (n1 ∨ n2) = n.
+#n #c1 #c2 * #ri #rs #H
+elim (max_inv_dx … H) -H #ri1 #rs1 #ti1 #ts1 #ri2 #rs2 #ti2 #ts2 #_ #_ #H1 #H2 #H3 #H4
+elim (max_inv_O3 … H1) -H1 /3 width=5 by ex3_2_intro, ex1_2_intro/
+qed-.
+
+lemma isrt_inv_max_O_dx: ∀n,c1,c2. 𝐑𝐓⦃n, c1 ∨ c2⦄ → 𝐑𝐓⦃0, c2⦄ → 𝐑𝐓⦃n, c1⦄.
+#n #c1 #c2 #H #H2
+elim (isrt_inv_max … H) -H #n1 #n2 #Hn1 #Hn2 #H destruct
+lapply (isrt_mono … Hn2 H2) -c2 #H destruct //
+qed-.
+
+(* Properties with shift ****************************************************)
+(*
+lemma max_shift: ∀c1,c2. (↓c1) ∨ (↓c2) = ↓(c1∨c2).
+* #ri1 #rs1 #ti1 #ts1 * #ri2 #rs2 #ti2 #ts2
+<shift_rew <shift_rew <shift_rew <max_rew //
+qed.
+*)