matita/matita/contribs/lambdadelta/static_2/i_static/rexs_drops.ma

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   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
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  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 include "static_2/relocation/seq_seq.ma".
  16 include "static_2/static/rex_drops.ma".
  17 include "static_2/i_static/rexs.ma".
  18
  19 (* ITERATED EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ***)
  20
  21 definition tc_f_dedropable_sn: predicate (relation3 lenv term term) ≝
  22                                λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 →
  23                                ∀K2,T. K1 ⪤*[R, T] K2 → ∀U. ⬆*[f] T ≘ U →
  24                                ∃∃L2. L1 ⪤*[R, U] L2 & ⬇*[b, f] L2 ≘ K2 & L1 ≡[f] L2.
  25
  26 definition tc_f_dropable_sn: predicate (relation3 lenv term term) ≝
  27                              λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 → 𝐔⦃f⦄ →
  28                              ∀L2,U. L1 ⪤*[R, U] L2 → ∀T. ⬆*[f] T ≘ U →
  29                              ∃∃K2. K1 ⪤*[R, T] K2 & ⬇*[b, f] L2 ≘ K2.
  30
  31 definition tc_f_dropable_dx: predicate (relation3 lenv term term) ≝
  32                              λR. ∀L1,L2,U. L1 ⪤*[R, U] L2 →
  33                              ∀b,f,K2. ⬇*[b, f] L2 ≘ K2 → 𝐔⦃f⦄ → ∀T. ⬆*[f] T ≘ U →
  34                              ∃∃K1. ⬇*[b, f] L1 ≘ K1 & K1 ⪤*[R, T] K2.
  35
  36 (* Properties with generic slicing for local environments *******************)
  37
  38 lemma dedropable_sn_CTC: ∀R. f_dedropable_sn R → tc_f_dedropable_sn R.
  39 #R #HR #b #f #L1 #K1 #HLK1 #K2 #T #H elim H -K2
  40 [ #K2 #HK12 #U #HTU elim (HR … HLK1 … HK12 … HTU) -K1 -T -HR
  41   /3 width=4 by ex3_intro, inj/
  42 | #K #K2 #_ #HK2 #IH #U #HTU -HLK1
  43   elim (IH … HTU) -IH #L #HL1 #HLK
  44   elim (HR … HLK … HK2 … HTU) -K -T -HR
  45   /3 width=6 by seq_trans, rexs_step_dx, ex3_intro/
  46 ]
  47 qed-.
  48
  49 (* Inversion lemmas with generic slicing for local environments *************)
  50
  51 lemma dropable_sn_CTC: ∀R. f_dropable_sn R → tc_f_dropable_sn R.
  52 #R #HR #b #f #L1 #K1 #HLK1 #Hf #L2 #U #H elim H -L2
  53 [ #L2 #HL12 #T #HTU elim (HR … HLK1 … HL12 … HTU) -L1 -U -HR
  54   /3 width=3 by inj, ex2_intro/
  55 | #L #L2 #_ #HL2 #IH #T #HTU -HLK1
  56   elim (IH … HTU) -IH #K #HK1 #HLK
  57   elim (HR … HLK … HL2 … HTU) -L -U -HR
  58   /3 width=3 by rexs_step_dx, ex2_intro/
  59 ]
  60 qed-.
  61
  62 lemma dropable_dx_CTC: ∀R. f_dropable_dx R → tc_f_dropable_dx R.
  63 #R #HR #L1 #L2 #U #H elim H -L2
  64 [ #L2 #HL12 #b #f #K2 #HLK2 #Hf #T #HTU
  65   elim (HR … HL12 … HLK2 … HTU) -L2 -U -HR
  66   /3 width=3 by inj, ex2_intro/
  67 | #L #L2 #_ #HL2 #IH #b #f #K2 #HLK2 #Hf #T #HTU
  68   elim (HR … HL2 … HLK2 … HTU) -L2 -HR // #K #HLK #HK2
  69   elim (IH … HLK … HTU) -IH -L -U
  70   /3 width=5 by rexs_step_dx, ex2_intro/
  71 ]
  72 qed-.