matita/matita/contribs/lambdadelta/basic_2/etc/fpr/fpr_cpr.etc

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  14
  15 include "basic_2/reducibility/cfpr_cpr.ma".
  16
  17 (* CONTEXT-FREE PARALLEL REDUCTION ON CLOSURES ******************************)
  18
  19 (* Properties on context-sensitive parallel reduction for terms *************)
  20
  21 lemma ltpr_tpr_fpr: ∀L1,L2. L1 ➡ L2 → ∀T1,T2. T1 ➡ T2 → ⦃L1, T1⦄ ➡ ⦃L2, T2⦄.
  22 /3 width=4/ qed.
  23
  24 lemma cpr_fpr: ∀L,T1,T2. L ⊢ T1 ➡ T2 → ⦃L, T1⦄ ➡ ⦃L, T2⦄.
  25 /2 width=4/ qed.
  26
  27 lemma fpr_lift: ∀K1,K2,T1,T2. ⦃K1, T1⦄ ➡ ⦃K2, T2⦄ →
  28                 ∀d,e,L1. ⇩[d, e] L1 ≡ K1 →
  29                 ∀U1,U2. ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 →
  30                 ∃∃L2. ⦃L1, U1⦄ ➡ ⦃L2, U2⦄ & ⇩[d, e] L2 ≡ K2.
  31 #K1 #K2 #T1 #T2 #HT12 #d #e #L1 #HLK1 #U1 #U2 #HTU1 #HTU2
  32 elim (fpr_inv_all … HT12) -HT12 #K #HK1 #HT12 #HK2
  33 elim (ldrop_ltpr_trans … HLK1 … HK1) -K1 #L #HL1 #HLK
  34 lapply (cpr_lift … HLK … HTU1 … HTU2 HT12) -T1 -T2 #HU12
  35 elim (le_or_ge (|K|) d) #Hd
  36 [ elim (ldrop_ltpss_sn_trans_ge … HLK … HK2 …)
  37 | elim (ldrop_ltpss_sn_trans_be … HLK … HK2 …)
  38 ] // -Hd #L2 #HL2 #HLK2
  39 lapply (ltpss_sn_weak_full … HL2) -K /3 width=4/
  40 qed-.
  41
  42 (* Advanced properties ******************************************************)
  43
  44 lemma fpr_flat_dx: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → ∀V1,V2. V1 ➡ V2 →
  45                    ∀I. ⦃L1, ⓕ{I}V1.T1⦄ ➡ ⦃L2, ⓕ{I}V2.T2⦄.
  46 #L1 #L2 #T1 #T2 #HT12
  47 elim (fpr_inv_all … HT12) -HT12 /4 width=4/
  48 qed.
  49
  50 lemma fpr_bind_sn: ∀L1,L2,V1,V2. ⦃L1, V1⦄ ➡ ⦃L2, V2⦄ → ∀T1,T2. T1 ➡ T2 →
  51                    ∀a,I. ⦃L1, ⓑ{a,I}V1.T1⦄ ➡ ⦃L2, ⓑ{a,I}V2.T2⦄.
  52 #L1 #L2 #V1 #V2 #H #T1 #T2 #HT12 #a #I
  53 elim (fpr_inv_all … H) /3 width=4/
  54 qed.
  55
  56 lemma fpr_bind2_minus: ∀I,L1,L2,V1,T1,U2. ⦃L1, -ⓑ{I}V1.T1⦄ ➡ ⦃L2, U2⦄ →
  57                        ∃∃V2,T2. ⦃L1.ⓑ{I}V1, T1⦄ ➡ ⦃L2.ⓑ{I}V2, T2⦄ &
  58                                 U2 = -ⓑ{I}V2.T2.
  59 #I1 #L1 #L2 #V1 #T1 #U2 #H
  60 elim (fpr_inv_all … H) -H #L #HL1 #H #HL2
  61 elim (cpr_fwd_bind1_minus … H false) -H /4 width=4/
  62 qed-.
  63
  64 (* Advanced forward lemmas **************************************************)
  65
  66 lemma fpr_fwd_bind2_minus: ∀I,L1,L,V1,T1,T. ⦃L1, -ⓑ{I}V1.T1⦄ ➡ ⦃L, T⦄ → ∀b.
  67                            ∃∃V2,T2. ⦃L1, ⓑ{b,I}V1.T1⦄ ➡ ⦃L, ⓑ{b,I}V2.T2⦄ &
  68                                     T = -ⓑ{I}V2.T2.
  69 #I #L1 #L #V1 #T1 #T #H1 #b
  70 elim (fpr_inv_all … H1) -H1 #L0 #HL10 #HT1 #HL0
  71 elim (cpr_fwd_bind1_minus … HT1 b) -HT1 /3 width=4/
  72 qed-.
  73
  74 lemma fpr_fwd_shift_bind_minus: ∀I1,I2,L1,L2,V1,V2,T1,T2.
  75                                 ⦃L1, -ⓑ{I1}V1.T1⦄ ➡ ⦃L2, -ⓑ{I2}V2.T2⦄ →
  76                                 ⦃L1, V1⦄ ➡ ⦃L2, V2⦄ ∧ I1 = I2.
  77 * #I2 #L1 #L2 #V1 #V2 #T1 #T2 #H
  78 elim (fpr_inv_all … H) -H #L #HL1 #H #HL2
  79 [ elim (cpr_inv_abbr1 … H) -H *
  80   [ #V #V0 #T #HV1 #HV0 #_ #H destruct /4 width=4/
  81   | #T #_ #_ #H destruct
  82   ]
  83 | elim (cpr_inv_abst1 … H Abst V2) -H
  84   #V #T #HV1 #_ #H destruct /3 width=4/
  85 ]
  86 qed-.
  87
  88 lemma fpr_fwd_abst2: ∀a,L1,L2,V1,T1,U2. ⦃L1, ⓛ{a}V1.T1⦄ ➡ ⦃L2, U2⦄ → ∀b,I,W.
  89                      ∃∃V2,T2. ⦃L1, ⓑ{b,I}W.T1⦄ ➡ ⦃L2, ⓑ{b,I}W.T2⦄ &
  90                               U2 = ⓛ{a}V2.T2.
  91 #a #L1 #L2 #V1 #T1 #U2 #H
  92 elim (fpr_inv_all … H) #L #HL1 #H #HL2 #b #I #W
  93 elim (cpr_fwd_abst1 … H b I W) -H /3 width=4/
  94 qed-.
  95
  96 (* Advanced inversion lemmas ************************************************)
  97
  98 lemma fpr_inv_pair1: ∀I,K1,L2,V1,T1,T2. ⦃K1.ⓑ{I}V1, T1⦄ ➡ ⦃L2, T2⦄ →
  99                      ∃∃K2,V2. ⦃K1, V1⦄  ➡ ⦃K2, V2⦄ &
 100                               ⦃K1, -ⓑ{I}V1.T1⦄ ➡ ⦃K2, -ⓑ{I}V2.T2⦄ &
 101                               L2 = K2.ⓑ{I}V2.
 102 #I1 #K1 #X #V1 #T1 #T2 #H
 103 elim (fpr_fwd_pair1 … H) -H #I2 #K2 #V2 #HT12 #H destruct
 104 elim (fpr_fwd_shift_bind_minus … HT12) #HV12 #H destruct /2 width=5/
 105 qed-.
 106
 107 lemma fpr_inv_pair3: ∀I,L1,K2,V2,T1,T2. ⦃L1, T1⦄ ➡ ⦃K2.ⓑ{I}V2, T2⦄ →
 108                      ∃∃K1,V1. ⦃K1, V1⦄  ➡ ⦃K2, V2⦄ &
 109                               ⦃K1, -ⓑ{I}V1.T1⦄ ➡ ⦃K2, -ⓑ{I}V2.T2⦄ &
 110                               L1 = K1.ⓑ{I}V1.
 111 #I2 #X #K2 #V2 #T1 #T2 #H
 112 elim (fpr_fwd_pair3 … H) -H #I1 #K1 #V1 #HT12 #H destruct
 113 elim (fpr_fwd_shift_bind_minus … HT12) #HV12 #H destruct /2 width=5/
 114 qed-.
 115
 116 (* More advanced forward lemmas *********************************************)
 117
 118 lemma fpr_fwd_pair1_full: ∀I,K1,L2,V1,T1,T2. ⦃K1.ⓑ{I}V1, T1⦄ ➡ ⦃L2, T2⦄ →
 119                           ∀b. ∃∃K2,V2. ⦃K1, V1⦄  ➡ ⦃K2, V2⦄ &
 120                                        ⦃K1, ⓑ{b,I}V1.T1⦄ ➡ ⦃K2, ⓑ{b,I}V2.T2⦄ &
 121                                        L2 = K2.ⓑ{I}V2.
 122 #I #K1 #L2 #V1 #T1 #T2 #H #b
 123 elim (fpr_inv_pair1 … H) -H #K2 #V2 #HV12 #HT12 #H destruct
 124 elim (fpr_fwd_bind2_minus … HT12 b) -HT12 #W1 #U1 #HTU1 #H destruct /2 width=5/
 125 qed-.