matita/matita/contribs/lambdadelta/basic_2/unfold/ltpss_sn_ltpss_sn.ma

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
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   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/unfold/tpss_tpss.ma".
  16 include "basic_2/unfold/tpss_alt.ma".
  17 include "basic_2/unfold/ltpss_sn_tpss.ma".
  18
  19 (* PARTIAL UNFOLD ON LOCAL ENVIRONMENTS *************************************)
  20
  21 (* Advanced properties ******************************************************)
  22
  23 fact ltpss_sn_tpss_trans_eq_aux: ∀Y1,X2,L1,T2,U2,d,e.
  24                                  L1 ⊢ T2 ▶* [d, e] U2 → ∀L0. L0 ⊢ ▶* [d, e] L1 →
  25                                  Y1 = L1 → X2 = T2 → L0 ⊢ T2 ▶* [d, e] U2.
  26 #Y1 #X2 @(fw_ind … Y1 X2) -Y1 -X2 #Y1 #X2 #IH
  27 #L1 #T2 #U2 #d #e #H @(tpss_ind_alt … H) -L1 -T2 -U2 -d -e
  28 [ //
  29 | #L1 #K1 #V1 #V2 #W2 #i #d #e #Hdi #Hide #HLK1 #HV12 #HVW2 #_ #L0 #HL01 #H1 #H2 destruct
  30   lapply (ldrop_fwd_lw … HLK1) #H1 normalize in H1;
  31   elim (ltpss_sn_ldrop_trans_be … HL01 … HLK1 ? ?) -HL01 -HLK1 // /2 width=2/ #X #H #HLK0
  32   elim (ltpss_sn_inv_tpss22 … H ?) -H /2 width=1/ #K0 #V0 #HK01 #HV01 #H destruct
  33   lapply (IH … HV12 … HK01 ? ?) -IH -HV12 -HK01 [1,3: // |2,4: skip | normalize /2 width=1/ ] #HV12
  34   lapply (tpss_trans_eq … HV01 HV12) -V1 /2 width=6/
  35 | #L #a #I #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #_ #_ #L0 #HL0 #H1 #H2 destruct
  36   lapply (tpss_lsubs_trans … HT12 (L. ⓑ{I} V1) ?) -HT12 /2 width=1/ #HT12
  37   lapply (IH … HV12 … HL0 ? ?) -HV12 [1,3: // |2,4: skip |5: /2 width=2/ ] #HV12
  38   lapply (IH … HT12 (L0. ⓑ{I} V1) ? ? ?) -IH -HT12 [1,3,5: /2 width=2/ |2,4: skip | normalize // ] -HL0 #HT12
  39   lapply (tpss_lsubs_trans … HT12 (L0. ⓑ{I} V2) ?) -HT12 /2 width=1/
  40 | #L #I #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #_ #_ #L0 #HL0 #H1 #H2 destruct
  41   lapply (IH … HV12 … HL0 ? ?) -HV12 [1,3: // |2,4: skip |5: /2 width=3/ ]
  42   lapply (IH … HT12 … HL0 ? ?) -IH -HT12 [1,3,5: normalize // |2,4: skip ] -HL0 /2 width=1/
  43 ]
  44 qed.
  45
  46 lemma ltpss_sn_tpss_trans_eq: ∀L1,T2,U2,d,e. L1 ⊢ T2 ▶* [d, e] U2 →
  47                               ∀L0. L0 ⊢ ▶* [d, e] L1 → L0 ⊢ T2 ▶* [d, e] U2.
  48 /2 width=5/ qed.
  49
  50 lemma ltpss_sn_tps_trans_eq: ∀L0,L1,T2,U2,d,e. L0 ⊢ ▶* [d, e] L1 →
  51                              L1 ⊢ T2 ▶ [d, e] U2 → L0 ⊢ T2 ▶* [d, e] U2.
  52 /3 width=3/ qed.
  53
  54 (* Main properties **********************************************************)
  55
  56 theorem ltpss_sn_trans_eq: ∀L1,L,d,e. L1 ⊢ ▶* [d, e] L →
  57                            ∀L2. L ⊢ ▶* [d, e] L2 → L1 ⊢ ▶* [d, e] L2.
  58 #L1 #L #d #e #H elim H -L1 -L -d -e //
  59 [ #L1 #L #I #V1 #V #e #HL1 #HV1 #IHL1 #X #H
  60   elim (ltpss_sn_inv_tpss21 … H ?) -H // <minus_plus_m_m #L2 #V2 #HL2 #HV2 #H destruct
  61   lapply (ltpss_sn_tpss_trans_eq … HV2 … HL1) -HV2 -HL1 #HV2
  62   lapply (tpss_trans_eq … HV1 … HV2) -V /3 width=1/
  63 | #L1 #L #I #V1 #V #d #e #HL1 #HV1 #IHL1 #X #H
  64   elim (ltpss_sn_inv_tpss11 … H ?) -H // <minus_plus_m_m #L2 #V2 #HL2 #HV2 #H destruct
  65   lapply (ltpss_sn_tpss_trans_eq … HV2 … HL1) -HV2 -HL1 #HV2
  66   lapply (tpss_trans_eq … HV1 … HV2) -V /3 width=1/
  67 ]
  68 qed.
  69
  70 (* Advanced forward lemmas **************************************************)
  71
  72 lemma tps_fwd_shift1: ∀L1,L,T1,T,d,e. L ⊢ L1 @@ T1 ▶ [d, e] T →
  73                       ∃∃L2,T2. L @@ L1 ⊢ ▶* [d + |L1|, e] L @@ L2 &
  74                                L @@ L2 ⊢ T1 ▶ [d + |L1|, e] T2 &
  75                                T = L2 @@ T2.
  76 #L1 @(lenv_ind_dx … L1) -L1
  77 [ #L #T1 #T #d #e #HT1
  78   @ex3_2_intro [3: // |5: // |4: normalize /2 width=1/ |1,2: skip ] (**) (* /2 width=4/ does not work *)
  79 | #I #L1 #V1 #IH #L #T1 #T #d #e >shift_append_assoc #H
  80   elim (tps_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
  81   elim (IH … HT12) -IH -HT12 #L2 #T #HL12 #HT1 #H destruct
  82   <append_assoc >append_length <associative_plus
  83   lapply (ltpss_sn_trans_eq (L.ⓑ{I}V1@@L1) … HL12) -HL12 /3 width=1/ #HL12
  84   @(ex3_2_intro … (⋆.ⓑ{I}V2@@L2)) [4: /2 width=3/ | skip ] <append_assoc // (**) (* explicit constructor *)
  85 ]
  86 qed-.