matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv_lstas.ma

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   4 (*      ||A||       A project by Andrea Asperti                           *)
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  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/static/lsubd_da.ma".
  16 include "basic_2/unfold/lstas_alt.ma".
  17 include "basic_2/equivalence/cpes_cpcs.ma".
  18 include "basic_2/dynamic/lsubsv_lsubd.ma".
  19
  20 (* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED NATIVE VALIDITY **************)
  21
  22 (* Properties on nat-iterated static type assignment ************************)
  23
  24 lemma lsubsv_lstas_trans: ∀h,g,G,L2,T,U2,l2. ⦃G, L2⦄ ⊢ T •*[h, l2] U2 →
  25                           ∀l1. l2 ≤ l1 → ⦃G, L2⦄ ⊢ T ▪[h, g] l1 →
  26                           ∀L1. G ⊢ L1 ⫃¡[h, g] L2 →
  27                           ∃∃U1. ⦃G, L1⦄ ⊢ T •*[h, l2] U1 & ⦃G, L1⦄ ⊢ U1 ⬌* U2.
  28 #h #g #G #L2 #T #U #l2 #H @(lstas_ind_alt … H) -G -L2 -T -U -l2
  29 [1,2: /2 width=3 by ex2_intro/
  30 | #G #L2 #K2 #X #Y #U #i #l2 #HLK2 #_ #HYU #IHXY #l1 #Hl21 #Hl1 #L1 #HL12
  31   elim (da_inv_lref … Hl1) -Hl1 * #K0 #V0 [| #l0 ] #HK0 #HV0
  32   lapply (drop_mono … HK0 … HLK2) -HK0 #H destruct
  33   elim (lsubsv_drop_O1_trans … HL12 … HLK2) -L2 #X #H #HLK1
  34   elim (lsubsv_inv_pair2 … H) -H * #K1 [ | -HYU -IHXY -HLK1 ]
  35   [ #HK12 #H destruct
  36     elim (IHXY … Hl21 HV0 … HK12) -K2 -l1 #T #HXT #HTY
  37     lapply (drop_fwd_drop2 … HLK1) #H
  38     elim (lift_total T 0 (i+1))
  39     /3 width=12 by lstas_ldef, cpcs_lift, ex2_intro/
  40   | #V #l0 #_ #_ #_ #_ #_ #H destruct
  41   ]
  42 | #G #L2 #K2 #X #Y #Y0 #U #i #l2 #HLK2 #HXY0 #_ #HYU #IHXY #l1 #Hl21 #Hl1 #L1 #HL12
  43   elim (da_inv_lref … Hl1) -Hl1 * #K0 #V0 [| #l0 ] #HK0 #HV0 [| #H1 ]
  44   lapply (drop_mono … HK0 … HLK2) -HK0 #H2 destruct
  45   lapply (le_plus_to_le_r … Hl21) -Hl21 #Hl21
  46   elim (lsubsv_drop_O1_trans … HL12 … HLK2) -L2 #X #H #HLK1
  47   elim (lsubsv_inv_pair2 … H) -H * #K1
  48   [ #HK12 #H destruct
  49     lapply (lsubsv_fwd_lsubd … HK12) #H
  50     lapply (lsubd_da_trans … HV0 … H) -H #H
  51     elim (da_inv_sta … H) -H
  52     elim (IHXY … Hl21 HV0 … HK12) -K2 -Hl21 #Y1
  53     lapply (drop_fwd_drop2 … HLK1)
  54     elim (lift_total Y1 0 (i+1))
  55     /3 width=12 by lstas_ldec, cpcs_lift, ex2_intro/
  56   | #V #l1 #HXV #_ #HV #HX #HK12 #_ #H destruct
  57     lapply (da_mono … HV0 … HX) -HX #H destruct
  58     elim (hsnv_inv_cast … HXV) -HXV #_ #_ #H
  59     lapply (H … Hl21) -H #HXV
  60     elim (IHXY … Hl21 HV0 … HK12) -K2 -Hl21 #Y0 #HXY0 #HY0
  61     elim (da_inv_sta … HV) -HV #W #HVW
  62     elim (lstas_total … HVW (l2+1)) -W #W #HVW
  63     lapply (cpes_inv_lstas_eq … HXV … HXY0 … HVW) -HXV -HXY0 #HY0W
  64     lapply (cpcs_canc_sn … HY0W … HY0) -Y0 #HYW
  65     elim (lift_total W 0 (i+1))
  66     lapply (drop_fwd_drop2 … HLK1)
  67     /4 width=12 by cpcs_lift, lstas_cast, lstas_ldef, ex2_intro/
  68   ]
  69 | #a #I #G #L2 #V2 #T2 #U2 #l1 #_ #IHTU2 #l2 #Hl12 #Hl2 #L1 #HL12
  70   lapply (da_inv_bind … Hl2) -Hl2 #Hl2
  71   elim (IHTU2 … Hl2 (L1.ⓑ{I}V2) …)
  72   /3 width=3 by lsubsv_pair, lstas_bind, cpcs_bind_dx, ex2_intro/
  73 | #G #L2 #V2 #T2 #U2 #l1 #_ #IHTU2 #l2 #Hl12 #Hl2 #L1 #HL12
  74   lapply (da_inv_flat … Hl2) -Hl2 #Hl2
  75   elim (IHTU2 … Hl2 … HL12) -L2
  76   /3 width=5 by lstas_appl, cpcs_flat, ex2_intro/
  77 | #G #L2 #W2 #T2 #U2 #l1 #_ #IHTU2 #l2 #Hl12 #Hl2 #L1 #HL12
  78   lapply (da_inv_flat … Hl2) -Hl2 #Hl2
  79   elim (IHTU2 … Hl2 … HL12) -L2
  80   /3 width=3 by lstas_cast, ex2_intro/
  81 ]
  82 qed-.
  83
  84 lemma lsubsv_sta_trans: ∀h,g,G,L2,T,U2. ⦃G, L2⦄ ⊢ T •[h] U2 →
  85                         ∀l. ⦃G, L2⦄ ⊢ T ▪[h, g] l+1 →
  86                         ∀L1. G ⊢ L1 ⫃¡[h, g] L2 →
  87                         ∃∃U1. ⦃G, L1⦄ ⊢ T •[h] U1 & ⦃G, L1⦄ ⊢ U1 ⬌* U2.
  88 #h #g #G #L2 #T #U2 #H #l #HTl #L1 #HL12
  89 elim (lsubsv_lstas_trans … U2 1 … HTl … HL12)
  90 /3 width=3 by lstas_inv_SO, sta_lstas, ex2_intro/
  91 qed-.