matita/matita/contribs/lambdadelta/basic_2/static/rex_rex.ma

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
   7 (*      ||T||         The HELM team.                                      *)
   8 (*      ||A||         http://helm.cs.unibo.it                             *)
   9 (*      \   /                                                             *)
  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 include "basic_2/relocation/sex_sex.ma".
  16 include "basic_2/static/frees_fqup.ma".
  17 include "basic_2/static/rex.ma".
  18
  19 (* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
  20
  21 (* Advanced inversion lemmas ************************************************)
  22
  23 lemma rex_inv_frees: ∀R,L1,L2,T. L1 ⪤[R, T] L2 →
  24                      ∀f. L1 ⊢ 𝐅*⦃T⦄ ≘ f → L1 ⪤[cext2 R, cfull, f] L2.
  25 #R #L1 #L2 #T * /3 width=6 by frees_mono, sex_eq_repl_back/
  26 qed-.
  27
  28 (* Advanced properties ******************************************************)
  29
  30 (* Basic_2A1: uses: llpx_sn_dec *)
  31 lemma rex_dec: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
  32                ∀L1,L2,T. Decidable (L1 ⪤[R, T] L2).
  33 #R #HR #L1 #L2 #T
  34 elim (frees_total L1 T) #f #Hf
  35 elim (sex_dec (cext2 R) cfull … L1 L2 f)
  36 /4 width=3 by rex_inv_frees, cfull_dec, ext2_dec, ex2_intro, or_intror, or_introl/
  37 qed-.
  38
  39 (* Main properties **********************************************************)
  40
  41 (* Basic_2A1: uses: llpx_sn_bind llpx_sn_bind_O *)
  42 theorem rex_bind: ∀R,p,I,L1,L2,V1,V2,T.
  43                   L1 ⪤[R, V1] L2 → L1.ⓑ{I}V1 ⪤[R, T] L2.ⓑ{I}V2 →
  44                   L1 ⪤[R, ⓑ{p,I}V1.T] L2.
  45 #R #p #I #L1 #L2 #V1 #V2 #T * #f1 #HV #Hf1 * #f2 #HT #Hf2
  46 lapply (sex_fwd_bind … Hf2) -Hf2 #Hf2 elim (sor_isfin_ex f1 (⫱f2))
  47 /3 width=7 by frees_fwd_isfin, frees_bind, sex_join, isfin_tl, ex2_intro/
  48 qed.
  49
  50 (* Basic_2A1: llpx_sn_flat *)
  51 theorem rex_flat: ∀R,I,L1,L2,V,T.
  52                   L1 ⪤[R, V] L2 → L1 ⪤[R, T] L2 →
  53                   L1 ⪤[R, ⓕ{I}V.T] L2.
  54 #R #I #L1 #L2 #V #T * #f1 #HV #Hf1 * #f2 #HT #Hf2 elim (sor_isfin_ex f1 f2)
  55 /3 width=7 by frees_fwd_isfin, frees_flat, sex_join, ex2_intro/
  56 qed.
  57
  58 theorem rex_bind_void: ∀R,p,I,L1,L2,V,T.
  59                        L1 ⪤[R, V] L2 → L1.ⓧ ⪤[R, T] L2.ⓧ →
  60                        L1 ⪤[R, ⓑ{p,I}V.T] L2.
  61 #R #p #I #L1 #L2 #V #T * #f1 #HV #Hf1 * #f2 #HT #Hf2
  62 lapply (sex_fwd_bind … Hf2) -Hf2 #Hf2 elim (sor_isfin_ex f1 (⫱f2))
  63 /3 width=7 by frees_fwd_isfin, frees_bind_void, sex_join, isfin_tl, ex2_intro/
  64 qed.
  65
  66 (* Negated inversion lemmas *************************************************)
  67
  68 (* Basic_2A1: uses: nllpx_sn_inv_bind nllpx_sn_inv_bind_O *)
  69 lemma rnex_inv_bind: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
  70                      ∀p,I,L1,L2,V,T. (L1 ⪤[R, ⓑ{p,I}V.T] L2 → ⊥) →
  71                      (L1 ⪤[R, V] L2 → ⊥) ∨ (L1.ⓑ{I}V ⪤[R, T] L2.ⓑ{I}V → ⊥).
  72 #R #HR #p #I #L1 #L2 #V #T #H elim (rex_dec … HR L1 L2 V)
  73 /4 width=2 by rex_bind, or_intror, or_introl/
  74 qed-.
  75
  76 (* Basic_2A1: uses: nllpx_sn_inv_flat *)
  77 lemma rnex_inv_flat: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
  78                      ∀I,L1,L2,V,T. (L1 ⪤[R, ⓕ{I}V.T] L2 → ⊥) →
  79                      (L1 ⪤[R, V] L2 → ⊥) ∨ (L1 ⪤[R, T] L2 → ⊥).
  80 #R #HR #I #L1 #L2 #V #T #H elim (rex_dec … HR L1 L2 V)
  81 /4 width=1 by rex_flat, or_intror, or_introl/
  82 qed-.
  83
  84 lemma rnex_inv_bind_void: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
  85                           ∀p,I,L1,L2,V,T. (L1 ⪤[R, ⓑ{p,I}V.T] L2 → ⊥) →
  86                           (L1 ⪤[R, V] L2 → ⊥) ∨ (L1.ⓧ ⪤[R, T] L2.ⓧ → ⊥).
  87 #R #HR #p #I #L1 #L2 #V #T #H elim (rex_dec … HR L1 L2 V)
  88 /4 width=2 by rex_bind_void, or_intror, or_introl/
  89 qed-.