matita/matita/contribs/lambdadelta/basic_2/computation/csn.ma

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  14
  15 include "basic_2/reduction/cnx.ma".
  16
  17 (* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERMS ********************)
  18
  19 definition csn: ∀h. sd h → lenv → predicate term ≝
  20                 λh,g,L. SN … (cpx h g L) (eq …).
  21
  22 interpretation
  23    "context-sensitive extended strong normalization (term)"
  24    'SN h g L T = (csn h g L T).
  25
  26 (* Basic eliminators ********************************************************)
  27
  28 lemma csn_ind: ∀h,g,L. ∀R:predicate term.
  29                (∀T1. ⦃h, L⦄ ⊢ ⬊*[g] T1 →
  30                      (∀T2. ⦃h, L⦄ ⊢ T1 ➡[g] T2 → (T1 = T2 → ⊥) → R T2) →
  31                      R T1
  32                ) →
  33                ∀T. ⦃h, L⦄ ⊢ ⬊*[g] T → R T.
  34 #h #g #L #R #H0 #T1 #H elim H -T1 #T1 #HT1 #IHT1
  35 @H0 -H0 /3 width=1/ -IHT1 /4 width=1/
  36 qed-.
  37
  38 (* Basic properties *********************************************************)
  39
  40 (* Basic_1: was just: sn3_pr2_intro *)
  41 lemma csn_intro: ∀h,g,L,T1.
  42                  (∀T2. ⦃h, L⦄ ⊢ T1 ➡[g] T2 → (T1 = T2 → ⊥) → ⦃h, L⦄ ⊢ ⬊*[g] T2) →
  43                  ⦃h, L⦄ ⊢ ⬊*[g] T1.
  44 /4 width=1/ qed.
  45
  46 (* Basic_1: was just: sn3_nf2 *)
  47 lemma cnx_csn: ∀h,g,L,T. ⦃h, L⦄ ⊢ 𝐍[g]⦃T⦄ → ⦃h, L⦄ ⊢ ⬊*[g] T.
  48 /2 width=1/ qed.
  49
  50 lemma csn_cpx_trans: ∀h,g,L,T1. ⦃h, L⦄ ⊢ ⬊*[g] T1 →
  51                      ∀T2. ⦃h, L⦄ ⊢ T1 ➡[g] T2 → ⦃h, L⦄ ⊢ ⬊*[g] T2.
  52 #h #g #L #T1 #H elim H -T1 #T1 #HT1 #IHT1 #T2 #HLT12
  53 @csn_intro #T #HLT2 #HT2
  54 elim (term_eq_dec T1 T2) #HT12
  55 [ -IHT1 -HLT12 destruct /3 width=1/
  56 | -HT1 -HT2 /3 width=4/
  57 qed-.
  58
  59 (* Basic_1: was just: sn3_cast *)
  60 lemma csn_cast: ∀h,g,L,W. ⦃h, L⦄ ⊢ ⬊*[g] W →
  61                 ∀T. ⦃h, L⦄ ⊢ ⬊*[g] T → ⦃h, L⦄ ⊢ ⬊*[g] ⓝW.T.
  62 #h #g #L #W #HW elim HW -W #W #_ #IHW #T #HT @(csn_ind … HT) -T #T #HT #IHT
  63 @csn_intro #X #H1 #H2
  64 elim (cpx_inv_cast1 … H1) -H1
  65 [ * #W0 #T0 #HLW0 #HLT0 #H destruct
  66   elim (eq_false_inv_tpair_sn … H2) -H2
  67   [ /3 width=3 by csn_cpx_trans/
  68   | -HLW0 * #H destruct /3 width=1/
  69   ]
  70 | /3 width=3 by csn_cpx_trans/
  71 ]
  72 qed.
  73
  74 (* Basic forward lemmas *****************************************************)
  75
  76 fact csn_fwd_pair_sn_aux: ∀h,g,L,U. ⦃h, L⦄ ⊢ ⬊*[g] U →
  77                           ∀I,V,T. U = ②{I}V.T → ⦃h, L⦄ ⊢ ⬊*[g] V.
  78 #h #g #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
  79 @csn_intro #V2 #HLV2 #HV2
  80 @(IH (②{I}V2.T)) -IH // /2 width=1/ -HLV2 #H destruct /2 width=1/
  81 qed-.
  82
  83 (* Basic_1: was just: sn3_gen_head *)
  84 lemma csn_fwd_pair_sn: ∀h,g,I,L,V,T. ⦃h, L⦄ ⊢ ⬊*[g] ②{I}V.T → ⦃h, L⦄ ⊢ ⬊*[g] V.
  85 /2 width=5 by csn_fwd_pair_sn_aux/ qed-.
  86
  87 fact csn_fwd_bind_dx_aux: ∀h,g,L,U. ⦃h, L⦄ ⊢ ⬊*[g] U →
  88                           ∀a,I,V,T. U = ⓑ{a,I}V.T → ⦃h, L.ⓑ{I}V⦄ ⊢ ⬊*[g] T.
  89 #h #g #L #U #H elim H -H #U0 #_ #IH #a #I #V #T #H destruct
  90 @csn_intro #T2 #HLT2 #HT2
  91 @(IH (ⓑ{a,I} V. T2)) -IH // /2 width=1/ -HLT2 #H destruct /2 width=1/
  92 qed-.
  93
  94 (* Basic_1: was just: sn3_gen_bind *)
  95 lemma csn_fwd_bind_dx: ∀h,g,a,I,L,V,T. ⦃h, L⦄ ⊢ ⬊*[g] ⓑ{a,I}V.T → ⦃h, L.ⓑ{I}V⦄ ⊢ ⬊*[g] T.
  96 /2 width=4 by csn_fwd_bind_dx_aux/ qed-.
  97
  98 fact csn_fwd_flat_dx_aux: ∀h,g,L,U. ⦃h, L⦄ ⊢ ⬊*[g] U →
  99                           ∀I,V,T. U = ⓕ{I}V.T → ⦃h, L⦄ ⊢ ⬊*[g] T.
 100 #h #g #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
 101 @csn_intro #T2 #HLT2 #HT2
 102 @(IH (ⓕ{I}V.T2)) -IH // /2 width=1/ -HLT2 #H destruct /2 width=1/
 103 qed-.
 104
 105 (* Basic_1: was just: sn3_gen_flat *)
 106 lemma csn_fwd_flat_dx: ∀h,g,I,L,V,T. ⦃h, L⦄ ⊢ ⬊*[g] ⓕ{I}V.T → ⦃h, L⦄ ⊢ ⬊*[g] T.
 107 /2 width=5 by csn_fwd_flat_dx_aux/ qed-.
 108
 109 (* Basic_1: removed theorems 14:
 110             sn3_cdelta
 111             sn3_gen_cflat sn3_cflat sn3_cpr3_trans sn3_shift sn3_change
 112             sn3_appl_cast sn3_appl_beta sn3_appl_lref sn3_appl_abbr
 113             sn3_appl_appls sn3_bind sn3_appl_bind sn3_appls_bind
 114 *)