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

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  14
  15 include "basic_2/notation/relations/sn_4.ma".
  16 include "basic_2/reduction/cnx.ma".
  17
  18 (* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERMS ********************)
  19
  20 definition csn: ∀h. sd h → lenv → predicate term ≝
  21                 λh,g,L. SN … (cpx h g L) (eq …).
  22
  23 interpretation
  24    "context-sensitive extended strong normalization (term)"
  25    'SN h g L T = (csn h g L T).
  26
  27 (* Basic eliminators ********************************************************)
  28
  29 lemma csn_ind: ∀h,g,L. ∀R:predicate term.
  30                (∀T1. ⦃h, L⦄ ⊢ ⬊*[g] T1 →
  31                      (∀T2. ⦃h, L⦄ ⊢ T1 ➡[g] T2 → (T1 = T2 → ⊥) → R T2) →
  32                      R T1
  33                ) →
  34                ∀T. ⦃h, L⦄ ⊢ ⬊*[g] T → R T.
  35 #h #g #L #R #H0 #T1 #H elim H -T1 #T1 #HT1 #IHT1
  36 @H0 -H0 /3 width=1/ -IHT1 /4 width=1/
  37 qed-.
  38
  39 (* Basic properties *********************************************************)
  40
  41 (* Basic_1: was just: sn3_pr2_intro *)
  42 lemma csn_intro: ∀h,g,L,T1.
  43                  (∀T2. ⦃h, L⦄ ⊢ T1 ➡[g] T2 → (T1 = T2 → ⊥) → ⦃h, L⦄ ⊢ ⬊*[g] T2) →
  44                  ⦃h, L⦄ ⊢ ⬊*[g] T1.
  45 /4 width=1/ qed.
  46
  47 lemma csn_cpx_trans: ∀h,g,L,T1. ⦃h, L⦄ ⊢ ⬊*[g] T1 →
  48                      ∀T2. ⦃h, L⦄ ⊢ T1 ➡[g] T2 → ⦃h, L⦄ ⊢ ⬊*[g] T2.
  49 #h #g #L #T1 #H elim H -T1 #T1 #HT1 #IHT1 #T2 #HLT12
  50 @csn_intro #T #HLT2 #HT2
  51 elim (term_eq_dec T1 T2) #HT12
  52 [ -IHT1 -HLT12 destruct /3 width=1/
  53 | -HT1 -HT2 /3 width=4/
  54 qed-.
  55
  56 (* Basic_1: was just: sn3_nf2 *)
  57 lemma cnx_csn: ∀h,g,L,T. ⦃h, L⦄ ⊢ 𝐍[g]⦃T⦄ → ⦃h, L⦄ ⊢ ⬊*[g] T.
  58 /2 width=1/ qed.
  59
  60 lemma cnx_sort: ∀h,g,L,k. ⦃h, L⦄ ⊢ ⬊*[g] ⋆k.
  61 #h #g #L #k elim (deg_total h g k)
  62 #l generalize in match k; -k @(nat_ind_plus … l) -l /3 width=1/
  63 #l #IHl #k #Hkl lapply (deg_next_SO … Hkl) -Hkl
  64 #Hkl @csn_intro #X #H #HX elim (cpx_inv_sort1 … H) -H
  65 [ #H destruct elim HX //
  66 | -HX * #l0 #_ #H destruct -l0 /2 width=1/
  67 ]
  68 qed.
  69
  70 (* Basic_1: was just: sn3_cast *)
  71 lemma csn_cast: ∀h,g,L,W. ⦃h, L⦄ ⊢ ⬊*[g] W →
  72                 ∀T. ⦃h, L⦄ ⊢ ⬊*[g] T → ⦃h, L⦄ ⊢ ⬊*[g] ⓝW.T.
  73 #h #g #L #W #HW @(csn_ind … HW) -W #W #HW #IHW #T #HT @(csn_ind … HT) -T #T #HT #IHT
  74 @csn_intro #X #H1 #H2
  75 elim (cpx_inv_cast1 … H1) -H1
  76 [ * #W0 #T0 #HLW0 #HLT0 #H destruct
  77   elim (eq_false_inv_tpair_sn … H2) -H2
  78   [ /3 width=3 by csn_cpx_trans/
  79   | -HLW0 * #H destruct /3 width=1/
  80   ]
  81 |2,3: /3 width=3 by csn_cpx_trans/
  82 ]
  83 qed.
  84
  85 (* Basic forward lemmas *****************************************************)
  86
  87 fact csn_fwd_pair_sn_aux: ∀h,g,L,U. ⦃h, L⦄ ⊢ ⬊*[g] U →
  88                           ∀I,V,T. U = ②{I}V.T → ⦃h, L⦄ ⊢ ⬊*[g] V.
  89 #h #g #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
  90 @csn_intro #V2 #HLV2 #HV2
  91 @(IH (②{I}V2.T)) -IH // /2 width=1/ -HLV2 #H destruct /2 width=1/
  92 qed-.
  93
  94 (* Basic_1: was just: sn3_gen_head *)
  95 lemma csn_fwd_pair_sn: ∀h,g,I,L,V,T. ⦃h, L⦄ ⊢ ⬊*[g] ②{I}V.T → ⦃h, L⦄ ⊢ ⬊*[g] V.
  96 /2 width=5 by csn_fwd_pair_sn_aux/ qed-.
  97
  98 fact csn_fwd_bind_dx_aux: ∀h,g,L,U. ⦃h, L⦄ ⊢ ⬊*[g] U →
  99                           ∀a,I,V,T. U = ⓑ{a,I}V.T → ⦃h, L.ⓑ{I}V⦄ ⊢ ⬊*[g] T.
 100 #h #g #L #U #H elim H -H #U0 #_ #IH #a #I #V #T #H destruct
 101 @csn_intro #T2 #HLT2 #HT2
 102 @(IH (ⓑ{a,I} V. T2)) -IH // /2 width=1/ -HLT2 #H destruct /2 width=1/
 103 qed-.
 104
 105 (* Basic_1: was just: sn3_gen_bind *)
 106 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.
 107 /2 width=4 by csn_fwd_bind_dx_aux/ qed-.
 108
 109 fact csn_fwd_flat_dx_aux: ∀h,g,L,U. ⦃h, L⦄ ⊢ ⬊*[g] U →
 110                           ∀I,V,T. U = ⓕ{I}V.T → ⦃h, L⦄ ⊢ ⬊*[g] T.
 111 #h #g #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
 112 @csn_intro #T2 #HLT2 #HT2
 113 @(IH (ⓕ{I}V.T2)) -IH // /2 width=1/ -HLT2 #H destruct /2 width=1/
 114 qed-.
 115
 116 (* Basic_1: was just: sn3_gen_flat *)
 117 lemma csn_fwd_flat_dx: ∀h,g,I,L,V,T. ⦃h, L⦄ ⊢ ⬊*[g] ⓕ{I}V.T → ⦃h, L⦄ ⊢ ⬊*[g] T.
 118 /2 width=5 by csn_fwd_flat_dx_aux/ qed-.
 119
 120 (* Basic_1: removed theorems 14:
 121             sn3_cdelta
 122             sn3_gen_cflat sn3_cflat sn3_cpr3_trans sn3_shift sn3_change
 123             sn3_appl_cast sn3_appl_beta sn3_appl_lref sn3_appl_abbr
 124             sn3_appl_appls sn3_bind sn3_appl_bind sn3_appls_bind
 125 *)