matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lsubr/cprs_lpss.etc

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  14
  15 include "basic_2/substitution/lpss_lpss.ma".
  16 include "basic_2/reduction/lpr_lpss.ma".
  17 include "basic_2/computation/cprs.ma".
  18
  19 (* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************)
  20
  21 (* Properties on parallel substitution for terms ****************************)
  22
  23 (* Basic_1: was: pr3_subst1 *)
  24 lemma cprs_cpss_conf: ∀L,T0,T1. L ⊢ T0 ➡* T1 → ∀T2. L ⊢ T0 ▶* T2 →
  25                       ∃∃T. L ⊢ T1 ▶* T & L ⊢ T2 ➡* T.
  26 #L @TC_strip1 /2 width=3 by cpr_cpss_conf/ qed-. (**) (* auto /3 width=3/ fails because a δ-expansion gets in the way *)
  27
  28 (* Properties on sn parallel substitution for local environments ************)
  29
  30 lemma cprs_lpss_conf_dx: ∀L0,T0,T1. L0 ⊢ T0 ➡* T1 → ∀L1. L0 ⊢ ▶* L1 →
  31                          ∃∃T. L1 ⊢ T1 ▶* T & L1 ⊢ T0 ➡* T.
  32 #L0 #T0 #T1 #H elim H -T1
  33 [ #T1 #HT01 #L1 #HL01
  34   elim (cpr_lpss_conf_dx … HT01 … HL01) -L0 /3 width=3/
  35 | #T #T1 #_ #HT1 #IHT0 #L1 #HL01
  36   elim (IHT0 … HL01) #T2 #HT2 #HT02
  37   elim (cpr_lpss_conf_dx … HT1 … HL01) -L0 #T3 #HT13 #HT3
  38   elim (cpr_cpss_conf … HT3 … HT2) -T #T #HT3 #HT2
  39   lapply (cpss_trans … HT13 … HT3) -T3
  40   lapply (cprs_strap1 … HT02 … HT2) -T2 /2 width=3/
  41 ]
  42 qed-.
  43
  44 lemma cprs_lpss_conf_sn: ∀L0,T0,T1. L0 ⊢ T0 ➡* T1 → ∀L1. L0 ⊢ ▶* L1 →
  45                          ∃∃T. L0 ⊢ T1 ▶* T & L1 ⊢ T0 ➡* T.
  46 #L0 #T0 #T1 #HT01 #L1 #HL01
  47 elim (cprs_lpss_conf_dx … HT01 … HL01) -HT01 #T #HT1
  48 lapply (lpss_cpss_trans … HL01 … HT1) -HT1 /2 width=3/
  49 qed-.
  50
  51 lemma cprs_cpss_lpss_conf_sn: ∀L1,T1,U1. L1 ⊢ T1 ➡* U1 →
  52                               ∀T2. L1 ⊢ T1 ▶* T2 → ∀L2. L1 ⊢ ▶* L2 →
  53                               ∃∃U2. L2 ⊢ T2 ➡* U2 & L1 ⊢ U1 ▶* U2.
  54 #L1 #T1 #U1 #HTU1 #T2 #HT12 #L2 #HL12
  55 elim (cprs_cpss_conf … HTU1 … HT12) -T1 #U #HU1 #HT2U
  56 elim (cprs_lpss_conf_sn … HT2U … HL12) -HT2U -HL12 #U2 #HU2 #HTU2
  57 lapply (cpss_trans … HU1 … HU2) -U /2 width=3/
  58 qed-.
  59
  60 lemma cprs_cpss_lpss_conf_dx: ∀L1,T1,U1. L1 ⊢ T1 ➡* U1 →
  61                               ∀T2. L1 ⊢ T1 ▶* T2 → ∀L2. L1 ⊢ ▶* L2 →
  62                               ∃∃U2. L2 ⊢ T2 ➡* U2 & L2 ⊢ U1 ▶* U2.
  63 #L1 #T1 #U1 #HTU1 #T2 #HT12 #L2 #HL12
  64 elim (cprs_lpss_conf_dx … HTU1 … HL12) -HTU1 #U2 #HU12 #HT1U2
  65 elim (lpss_cpss_conf_dx … HT12 … HL12) -L1 #T #HT1 #HT2
  66 elim (cprs_cpss_conf … HT1U2 … HT1) -T1 #U #HU2 #HTU
  67 lapply (cpss_trans … HU12 … HU2) -U2
  68 lapply (cpss_cprs_trans … HT2 … HTU) -T /2 width=3/
  69 qed-.
  70
  71
  72 lemma cprs_cpss2_lpss_conf_sn: ∀L1,T1,U1. L1 ⊢ T1 ➡* U1 → ∀T2. L1 ⊢ T1 ▶* T2 →
  73                                ∀U2. L1 ⊢ U1 ▶* U2 → ∀L2. L1 ⊢ ▶* L2 →
  74                                ∃∃U. L2 ⊢ T2 ➡* U & L1 ⊢ U2 ▶* U.
  75 #L1 #T1 #U1 #HTU1 #T2 #HT12 #U2 #HU12 #L2 #HL12
  76 elim (cprs_cpss_lpss_conf_sn … HTU1 … HT12 … HL12) -T1 #T1 #HT21 #HUT1
  77 elim (cpss_conf … HU12 … HUT1) -U1 #U1 #HU21 #HTU1
  78 elim (lpss_cpss_conf_sn … HTU1 … HL12) -HTU1 -HL12 #U2 #HT1U2 #HU12
  79 lapply (cpss_trans … HU21 … HU12) -U1
  80 lapply (cprs_cpss_trans … HT21 … HT1U2) -T1 /2 width=3/
  81 qed-.
  82
  83 lemma cprs_cpss2_lpss_conf_dx: ∀L1,T1,U1. L1 ⊢ T1 ➡* U1 → ∀T2. L1 ⊢ T1 ▶* T2 →
  84                                ∀U2. L1 ⊢ U1 ▶* U2 → ∀L2. L1 ⊢ ▶* L2 →
  85                                ∃∃U. L2 ⊢ T2 ➡* U & L2 ⊢ U2 ▶* U.
  86 #L1 #T1 #U1 #HTU1 #T2 #HT12 #U2 #HU12 #L2 #HL12
  87 elim (cprs_cpss_lpss_conf_dx … HTU1 … HT12 … HL12) -T1 #T1 #HT21 #HUT1
  88 elim (lpss_cpss_conf_dx … HU12 … HL12) -L1 #U #HU1 #HU2
  89 elim (cpss_conf … HU1 … HUT1) -U1 #U1 #HU1 #HTU1
  90 lapply (cpss_trans … HU2 … HU1) -U
  91 lapply (cprs_cpss_trans … HT21 … HTU1) -T1 /2 width=3/
  92 qed-.