propagating the arithmetics library, partial commit

[helm.git] / matita / matita / contribs / lambdadelta / ground / steps / rtc_isrt.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/notation/relations/isredtype_2.ma".
-include "ground/steps/rtc.ma".
-
-(* RT-TRANSITION COUNTER ****************************************************)
-
-definition isrt: relation2 nat rtc ≝ λts,c.
-                 ∃∃ri,rs. 〈ri,rs,0,ts〉 = c.
-
-interpretation "test for constrained rt-transition counter (rtc)"
-   'IsRedType ts c = (isrt ts c).
-
-(* Basic properties *********************************************************)
-
-lemma isrt_00: 𝐑𝐓❪0,𝟘𝟘❫.
-/2 width=3 by ex1_2_intro/ qed.
-
-lemma isrt_10: 𝐑𝐓❪0,𝟙𝟘❫.
-/2 width=3 by ex1_2_intro/ qed.
-
-lemma isrt_01: 𝐑𝐓❪1,𝟘𝟙❫.
-/2 width=3 by ex1_2_intro/ qed.
-
-lemma isrt_eq_t_trans: ∀n,c1,c2. 𝐑𝐓❪n,c1❫ → rtc_eq_t c1 c2 → 𝐑𝐓❪n,c2❫.
-#n #c1 #c2 * #ri1 #rs1 #H destruct
-#H elim (rtc_eq_t_inv_dx … H) -H /2 width=3 by ex1_2_intro/
-qed-.
-
-(* Basic inversion properties ***********************************************)
-
-lemma isrt_inv_00: ∀n. 𝐑𝐓❪n,𝟘𝟘❫ → 0 = n.
-#n * #ri #rs #H destruct //
-qed-.
-
-lemma isrt_inv_10: ∀n. 𝐑𝐓❪n,𝟙𝟘❫ → 0 = n.
-#n * #ri #rs #H destruct //
-qed-.
-
-lemma isrt_inv_01: ∀n. 𝐑𝐓❪n,𝟘𝟙❫ → 1 = n.
-#n * #ri #rs #H destruct //
-qed-.
-
-(* Main inversion properties ************************************************)
-
-theorem isrt_inj: ∀n1,n2,c. 𝐑𝐓❪n1,c❫ → 𝐑𝐓❪n2,c❫ → n1 = n2.
-#n1 #n2 #c * #ri1 #rs1 #H1 * #ri2 #rs2 #H2 destruct //
-qed-.
-
-theorem isrt_mono: ∀n,c1,c2. 𝐑𝐓❪n,c1❫ → 𝐑𝐓❪n,c2❫ → rtc_eq_t c1 c2.
-#n #c1 #c2 * #ri1 #rs1 #H1 * #ri2 #rs2 #H2 destruct //
-qed-.