propagating the arithmetics library, partial commit

[helm.git] / matita / matita / contribs / lambdadelta / ground / steps / rtc_isrt_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_3_2.ma".
-include "ground/steps/rtc_max.ma".
-include "ground/steps/rtc_isrt.ma".
-
-(* RT-TRANSITION COUNTER ****************************************************)
-
-(* 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.
-
-lemma isrt_max_idem1: ∀n,c1,c2. 𝐑𝐓❪n,c1❫ → 𝐑𝐓❪n,c2❫ → 𝐑𝐓❪n,c1∨c2❫.
-#n #c1 #c2 #H1 #H2 >(idempotent_max 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_O_inv_max: ∀c1,c2. 𝐑𝐓❪0,c1 ∨ c2❫ → ∧∧ 𝐑𝐓❪0,c1❫ & 𝐑𝐓❪0,c2❫.
-#c1 #c2 #H
-elim (isrt_inv_max … H) -H #n1 #n2 #Hn1 #Hn2 #H
-elim (max_inv_O3 … H) -H #H1 #H2 destruct
-/2 width=1 by conj/
-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_inj … Hn2 H2) -c2 #H destruct //
-qed-.
-
-lemma isrt_inv_max_eq_t: ∀n,c1,c2. 𝐑𝐓❪n,c1 ∨ c2❫ → rtc_eq_t c1 c2 →
-                         ∧∧ 𝐑𝐓❪n,c1❫ & 𝐑𝐓❪n,c2❫.
-#n #c1 #c2 #H #Hc12
-elim (isrt_inv_max … H) -H #n1 #n2 #Hc1 #Hc2 #H destruct
-lapply (isrt_eq_t_trans … Hc1 … Hc12) -Hc12 #H
-<(isrt_inj … H … Hc2) -Hc2
-<idempotent_max /2 width=1 by conj/
-qed-.