matita/matita/contribs/lambdadelta/basic_2/etc_2A1/fpn/fsb_alt.etc

   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/notation/relations/btsnalt_5.ma".
  16 include "basic_2/computation/fpbs_fpbs.ma".
  17 include "basic_2/computation/fsb.ma".
  18
  19 (* "BIG TREE" STRONGLY NORMALIZING TERMS ************************************)
  20
  21 (* Note: alternative definition of fsb *)
  22 inductive fsba (h) (g): relation3 genv lenv term ≝
  23 | fsba_intro: ∀G1,L1,T1. (
  24                  ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
  25                  (⦃G1, L1, T1⦄ ⋕ ⦃G2, L2, T2⦄ → ⊥) → fsba h g G2 L2 T2
  26               ) → fsba h g G1 L1 T1.
  27
  28 interpretation
  29    "'big tree' strong normalization (closure) alternative"
  30    'BTSNAlt h g G L T = (fsba h g G L T).
  31
  32 (* Basic eliminators ********************************************************)
  33
  34 theorem fsba_ind_alt: ∀h,g. ∀R: relation3 …. (
  35                          ∀G1,L1,T1. ⦃G1, L1⦄ ⊢ ⦥⦥[h,g] T1 → (
  36                             ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
  37                             (⦃G1, L1, T1⦄ ⋕ ⦃G2, L2, T2⦄ → ⊥) → R G2 L2 T2
  38                          ) → R G1 L1 T1
  39                       ) →
  40                       ∀G,L,T. ⦃G, L⦄ ⊢ ⦥⦥[h, g] T → R G L T.
  41 #h #g #R #IH #G #L #T #H elim H -G -L -T
  42 /5 width=1 by fsba_intro/
  43 qed-.
  44
  45 (* Basic_properties *********************************************************)
  46
  47 fact fsba_intro_aux: ∀h,g,G1,L1,T1. (
  48                         ∀G,G2,L,L2,T,T2. ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
  49                         ⦃G1, L1, T1⦄ ⋕ ⦃G, L, T⦄ →
  50                         (⦃G1, L1, T1⦄ ⋕ ⦃G2, L2, T2⦄ → ⊥) → fsba h g G2 L2 T2
  51                      ) → fsba h g G1 L1 T1.
  52 /4 width=5 by fsba_intro/ qed-.
  53
  54 lemma fsba_fpbs_trans: ∀h,g,G1,L1,T1. ⦃G1, L1⦄ ⊢ ⦥⦥[h, g] T1 →
  55                        ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ → ⦃G2, L2⦄ ⊢ ⦥⦥[h, g] T2.
  56 #h #g #G1 #L1 #T1 #H @(fsba_ind_alt … H) -G1 -L1 -T1
  57 #G1 #L1 #T1 #H0 #IH0 #G #L #T #H1 @fsba_intro
  58 #G2 #L2 #T2 #H2 #_ lapply (fpbs_trans … H1 … H2) -G -L -T
  59 #H12 elim (bteq_dec G1 G2 L1 L2 T1 T2) /3 width=6 by fpb_fpbs/
  60 -IH0 #H212
  61
  62
  63  -H0 -H #H @(IH0 … H) -IH0 -H // @(fpbs_trans … H1 … H2)
  64
  65 lemma fsba_intro_fpb: ∀h,g,G1,L1,T1. (
  66                          ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
  67                          (⦃G1, L1, T1⦄ ⋕ ⦃G2, L2, T2⦄ → ⊥) → ⦃G2, L2⦄ ⊢ ⦥⦥[h, g] T2
  68                       ) → ⦃G1, L1⦄ ⊢ ⦥⦥[h, g] T1.
  69 #h #g #G1 #L1 #T1 #IH1 @fsba_intro_aux
  70 #G #G2 #L #L2 #T #T2 #H @(fpbs_ind_dx … H) -G -L -T
  71 [ #H1 #H2 -IH1 elim H2 -H2 //
  72 | #G0 #G #L0 #L #T0 #T #H10 #H12 #IH2 #H210 #H212 elim (bteq_dec G1 G L1 L T1 T)
  73   [ -IH1 -H210 -H10 -H12 /3 width=1 by/
  74   | -IH2 -H212 #H21 lapply (IH1 … H21) -IH1 -H21
  75     [
  76     | -H10 -H210 #H
  77 (*
  78 (* Main inversion lemmas ****************************************************)
  79
  80 theorem fsba_inv_fsb: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⦥⦥[h, g] T → ⦃G, L⦄ ⊢ ⦥[h, g] T.
  81 #h #g #G #L #T #H elim H -G -L -T
  82 /5 width=12 by fsb_intro, fpb_fpbs, fpbc_fwd_fpb, fpbc_fwd_gen/
  83 qed-.
  84
  85 (* Main properties **********************************************************)
  86
  87 theorem fsb_fsba: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⦥[h, g] T → ⦃G, L⦄ ⊢ ⦥⦥[h, g] T.
  88 #h #g #G #L #T #H @(fsb_ind_alt … H) -G -L -T
  89 /4 width=1 by fsba_intro_fpb/
  90 qed.
  91 (*
  92 | fsba_intro: ∀G1,L1,T1. (
  93                 ∀G2,L2,T2.  ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ → fsb h g G2 L2 T2
  94               ) → fsb h g G1 L1 T1
  95 .
  96
  97
  98
  99 (****************************************************************************)
 100
 101 include "basic_2/substitution/fqup.ma".
 102
 103 lemma fsb_csx: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → ⦃G, L⦄ ⊢ ⦥[h, g] T.
 104 #h #g #G #L #T #H @(csx_ind … H) -T
 105 *)*)