matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltpss_sn_alt.etc

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  11 (*        v         GNU General Public License Version 2                  *)
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  14
  15 notation "hvbox( T1 break ⊢ ▶ ▶ * [ term 46 d , break term 46 e ] break term 46 T2 )"
  16    non associative with precedence 45
  17    for @{ 'PSubstStarSnAlt $T1 $d $e $T2 }.
  18
  19 include "basic_2/unfold/ltpss_dx_ltpss_dx.ma".
  20 include "basic_2/unfold/ltpss_sn_ltpss_sn.ma".
  21
  22 (* SN PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************)
  23
  24 (* alternative definition of ltpss_sn *)
  25 definition ltpssa: nat → nat → relation lenv ≝
  26                    λd,e. TC … (ltpss_dx d e).
  27
  28 interpretation "parallel unfold (local environment, sn variant) alternative"
  29    'PSubstStarSnAlt L1 d e L2 = (ltpssa d e L1 L2).
  30
  31 (* Basic eliminators ********************************************************)
  32
  33 lemma ltpssa_ind: ∀d,e,L1. ∀R:predicate lenv. R L1 →
  34                   (∀L,L2. L1 ⊢ ▶▶* [d, e] L → L ▶* [d, e] L2 → R L → R L2) →
  35                   ∀L2. L1 ⊢ ▶▶* [d, e] L2 → R L2.
  36 #d #e #L1 #R #HL1 #IHL1 #L2 #HL12 @(TC_star_ind … HL1 IHL1 … HL12) //
  37 qed-.
  38
  39 lemma ltpssa_ind_dx: ∀d,e,L2. ∀R:predicate lenv. R L2 →
  40                      (∀L1,L. L1 ▶* [d, e] L → L ⊢ ▶▶* [d, e] L2 → R L → R L1) →
  41                      ∀L1. L1 ⊢ ▶▶* [d, e] L2 → R L1.
  42 #d #e #L2 #R #HL2 #IHL2 #L1 #HL12 @(TC_star_ind_dx … HL2 IHL2 … HL12) //
  43 qed-.
  44
  45 (* Basic properties *********************************************************)
  46
  47 lemma ltpssa_refl: ∀L,d,e. L ⊢ ▶▶* [d, e] L.
  48 /2 width=1/ qed.
  49
  50 lemma ltpssa_tpss2: ∀I,L1,V1,V2,e. L1 ⊢ V1 ▶*[0, e] V2 →
  51                     ∀L2. L1 ⊢ ▶▶* [0, e] L2 →
  52                     L1.ⓑ{I}V1 ⊢ ▶▶* [O, e + 1] L2.ⓑ{I}V2.
  53 #I #L1 #V1 #V2 #e #HV12 #L2 #H @(ltpssa_ind … H) -L2
  54 [ /3 width=1/ | /3 width=5/ ]
  55 qed.
  56
  57 lemma ltpssa_tpss1: ∀I,L1,V1,V2,d,e. L1 ⊢ V1 ▶*[d, e] V2 →
  58                     ∀L2. L1 ⊢ ▶▶* [d, e] L2 →
  59                     L1.ⓑ{I}V1 ⊢ ▶▶* [d + 1, e] L2.ⓑ{I}V2.
  60 #I #L1 #V1 #V2 #d #e #HV12 #L2 #H @(ltpssa_ind … H) -L2
  61 [ /3 width=1/ | /3 width=5/ ]
  62 qed.
  63
  64 lemma ltpss_sn_ltpssa: ∀L1,L2,d,e. L1 ⊢ ▶* [d, e] L2 → L1 ⊢ ▶▶* [d, e] L2.
  65 #L1 #L2 #d #e #H elim H -L1 -L2 -d -e // /2 width=1/
  66 qed.
  67
  68 lemma ltpss_sn_dx_trans_eq: ∀L1,L,d,e. L1 ⊢ ▶* [d, e] L →
  69                             ∀L2. L ▶* [d, e] L2 → L1 ⊢ ▶* [d, e] L2.
  70 #L1 #L #d #e #H elim H -L1 -L -d -e
  71 [ #d #e #X #H
  72   lapply (ltpss_dx_inv_atom1 … H) -H #H destruct //
  73 | #L #I #V #X #H
  74   lapply (ltpss_dx_inv_refl_O2 … H) -H #H destruct //
  75 | #L1 #L #I #V1 #V #e #_ #HV1 #IHL1 #X #H
  76   elim (ltpss_dx_inv_tpss21 … H ?) -H // <minus_plus_m_m
  77   #L2 #V2 #HL2 #HV2 #H destruct
  78   lapply (IHL1 … HL2) -L #HL12
  79   lapply (ltpss_sn_tpss_trans_eq … HV2 … HL12) -HV2 #HV2
  80   lapply (tpss_trans_eq … HV1 HV2) -V /2 width=1/
  81 | #L1 #L #I #V1 #V #d #e #_ #HV1 #IHL1 #X #H
  82   elim (ltpss_dx_inv_tpss11 … H ?) -H // <minus_plus_m_m
  83   #L2 #V2 #HL2 #HV2 #H destruct
  84   lapply (IHL1 … HL2) -L #HL12
  85   lapply (ltpss_sn_tpss_trans_eq … HV2 … HL12) -HV2 #HV2
  86   lapply (tpss_trans_eq … HV1 HV2) -V /2 width=1/
  87 ]
  88 qed.
  89
  90 lemma ltpss_dx_sn_trans_eq: ∀L1,L,d,e. L1 ▶* [d, e] L →
  91                             ∀L2. L ⊢ ▶* [d, e] L2 → L1 ⊢ ▶* [d, e] L2.
  92 /3 width=3/ qed.
  93
  94 lemma ltpssa_strip: ∀L0,L1,d1,e1. L0 ⊢ ▶▶* [d1, e1] L1 →
  95                     ∀L2,d2,e2. L0 ▶* [d2, e2] L2 →
  96                     ∃∃L. L1 ▶* [d2, e2] L & L2 ⊢ ▶▶* [d1, e1] L.
  97 /3 width=3/ qed.
  98
  99 (* Basic inversion lemmas ***************************************************)
 100
 101 lemma ltpssa_ltpss_sn: ∀L1,L2,d,e. L1 ⊢ ▶▶* [d, e] L2 → L1 ⊢ ▶* [d, e] L2.
 102 #L1 #L2 #d #e #H @(ltpssa_ind … H) -L2 // /2 width=3/
 103 qed-.
 104
 105 (* Advanced properties ******************************************************)
 106
 107 lemma ltpss_sn_strip: ∀L0,L1,d1,e1. L0 ⊢ ▶* [d1, e1] L1 →
 108                       ∀L2,d2,e2. L0 ▶* [d2, e2] L2 →
 109                       ∃∃L. L1 ▶* [d2, e2] L & L2 ⊢ ▶* [d1, e1] L.
 110 #L0 #L1 #d1 #e1 #H #L2 #d2 #e2 #HL02
 111 lapply (ltpss_sn_ltpssa … H) -H #HL01
 112 elim (ltpssa_strip … HL01 … HL02) -L0
 113 /3 width=3 by ltpssa_ltpss_sn, ex2_intro/
 114 qed.
 115
 116 (* Note: this should go in ltpss_sn_ltpss_sn.ma *)
 117 lemma ltpss_sn_tpss_conf: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶* [d2, e2] U2 →
 118                           ∀L1,d1,e1. L0 ⊢ ▶* [d1, e1] L1 →
 119                           ∃∃T. L1 ⊢ T2 ▶* [d2, e2] T &
 120                                L0 ⊢ U2 ▶* [d1, e1] T.
 121 #L0 #T2 #U2 #d2 #e2 #HTU2 #L1 #d1 #e1 #H
 122 lapply (ltpss_sn_ltpssa … H) -H #H @(ltpssa_ind … H) -L1 /2 width=3/ -HTU2
 123 #L #L1 #H #HL1 * #T #HT2 #HU2T
 124 lapply (ltpssa_ltpss_sn … H) -H #HL0
 125 lapply (ltpss_sn_dx_trans_eq … HL0 … HL1) -HL0 #HL01
 126 elim (ltpss_dx_tpss_conf … HT2 … HL1) -HT2 -HL1 #T0 #HT20 #HT0
 127 lapply (ltpss_sn_tpss_trans_eq … HT0 … HL01) -HT0 -HL01 #HT0
 128 lapply (tpss_trans_eq … HU2T HT0) -T /2 width=3/
 129 qed.
 130
 131 (* Note: this should go in ltpss_sn_ltpss_sn.ma *)
 132 lemma ltpss_sn_tpss_trans_down: ∀L0,L1,T2,U2,d1,e1,d2,e2. d2 + e2 ≤ d1 →
 133                                 L1 ⊢ ▶* [d1, e1] L0 → L0 ⊢ T2 ▶* [d2, e2] U2 →
 134                                 ∃∃T. L1 ⊢ T2 ▶* [d2, e2] T & L1 ⊢ T ▶* [d1, e1] U2.
 135 #L0 #L1 #T2 #U2 #d1 #e1 #d2 #e2 #Hde2d1 #H #HTU2
 136 lapply (ltpss_sn_ltpssa … H) -H #HL10
 137 @(ltpssa_ind_dx … HL10) -L1 /2 width=3/ -HTU2
 138 #L1 #L #HL1 #_ * #T #HT2 #HTU2
 139 elim (ltpss_dx_tpss_trans_down … HL1 HT2) -HT2 // #T0 #HT20 #HT0 -Hde2d1
 140 lapply (tpss_trans_eq … HT0 HTU2) -T #HT0U2
 141 lapply (ltpss_dx_tpss_trans_eq … HT0U2 … HL1) -HT0U2 -HL1 /2 width=3/
 142 qed.
 143
 144 (* Main properties **********************************************************)
 145
 146 theorem ltpssa_conf: ∀L0,L1,d1,e1. L0 ⊢ ▶▶* [d1, e1] L1 →
 147                      ∀L2,d2,e2. L0 ⊢ ▶▶* [d2, e2] L2 →
 148                      ∃∃L. L1 ⊢ ▶▶* [d2, e2] L & L2 ⊢ ▶▶* [d1, e1] L.
 149 /3 width=3/ qed.
 150
 151 (* Note: this should go in ltpss_sn_ltpss_sn.ma *)
 152 theorem ltpss_sn_conf: ∀L0,L1,d1,e1. L0 ⊢ ▶* [d1, e1] L1 →
 153                        ∀L2,d2,e2. L0 ⊢ ▶* [d2, e2] L2 →
 154                        ∃∃L. L1 ⊢ ▶* [d2, e2] L & L2 ⊢ ▶* [d1, e1] L.
 155 #L0 #L1 #d1 #e1 #H1 #L2 #d2 #e2 #H2
 156 lapply (ltpss_sn_ltpssa … H1) -H1 #HL01
 157 lapply (ltpss_sn_ltpssa … H2) -H2 #HL02
 158 elim (ltpssa_conf … HL01 … HL02) -L0
 159 /3 width=3 by ltpssa_ltpss_sn, ex2_intro/
 160 qed.