matita/matita/contribs/lambda_delta/basic_2/reducibility/cpr_lift.ma

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  14
  15 include "basic_2/unfold/tpss_lift.ma".
  16 include "basic_2/reducibility/tpr_lift.ma".
  17 include "basic_2/reducibility/cpr.ma".
  18
  19 (* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
  20
  21 (* Advanced properties ******************************************************)
  22
  23 lemma cpr_cdelta: ∀L,K,V1,W1,W2,i.
  24                   ⇩[0, i] L ≡ K. ⓓV1 → K ⊢ V1 ▶* [0, |L| - i - 1] W1 →
  25                   ⇧[0, i + 1] W1 ≡ W2 → L ⊢ #i ➡ W2.
  26 #L #K #V1 #W1 #W2 #i #HLK #HVW1 #HW12
  27 lapply (ldrop_fwd_ldrop2_length … HLK) #Hi
  28 @ex2_1_intro [2: // | skip | @tpss_subst /width=6/ ] (**) (* /3 width=6/ is too slow *)
  29 qed.
  30
  31 lemma cpr_abst: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀V,T1,T2.
  32                 L.ⓛV ⊢ T1 ➡ T2 → L ⊢ ⓛV1. T1 ➡ ⓛV2. T2.
  33 #L #V1 #V2 * #V0 #HV10 #HV02 #V #T1 #T2 * #T0 #HT10 #HT02
  34 lapply (tpss_inv_S2 … HT02 L V ?) -HT02 // #HT02
  35 lapply (tpss_lsubs_trans … HT02 (L.ⓛV2) ?) -HT02 /2 width=1/ #HT02
  36 @(ex2_1_intro … (ⓛV0.T0)) /2 width=1/ (* explicit constructors *)
  37 qed.
  38
  39 lemma cpr_beta: ∀L,V1,V2,W,T1,T2.
  40                 L ⊢ V1 ➡ V2 → L.ⓛW ⊢ T1 ➡ T2 → L ⊢ ⓐV1.ⓛW.T1 ➡ ⓓV2.T2.
  41 #L #V1 #V2 #W #T1 #T2 * #V #HV1 #HV2 * #T #HT1 #HT2
  42 lapply (tpss_inv_S2 … HT2 L W ?) -HT2 // #HT2
  43 lapply (tpss_lsubs_trans … HT2 (L.ⓓV2) ?) -HT2 /2 width=1/ #HT2
  44 @(ex2_1_intro … (ⓓV.T)) /2 width=1/ (**) (* explicit constructor, /3/ is too slow *)
  45 qed.
  46
  47 lemma cpr_beta_dx: ∀L,V1,V2,W,T1,T2.
  48                    V1 ➡ V2 → L.ⓛW ⊢ T1 ➡ T2 → L ⊢ ⓐV1.ⓛW.T1 ➡ ⓓV2.T2.
  49 /3 width=1/ qed.
  50
  51 (* Advanced inversion lemmas ************************************************)
  52
  53 (* Basic_1: was: pr2_gen_lref *)
  54 lemma cpr_inv_lref1: ∀L,T2,i. L ⊢ #i ➡ T2 →
  55                      T2 = #i ∨
  56                      ∃∃K,V1,T1. ⇩[0, i] L ≡ K. ⓓV1 &
  57                                 K ⊢ V1 ▶* [0, |L| - i - 1] T1 &
  58                                 ⇧[0, i + 1] T1 ≡ T2 &
  59                                 i < |L|.
  60 #L #T2 #i * #X #H
  61 >(tpr_inv_atom1 … H) -H #H
  62 elim (tpss_inv_lref1 … H) -H /2 width=1/
  63 * /3 width=6/
  64 qed-.
  65
  66 (* Basic_1: was: pr2_gen_abst *)
  67 lemma cpr_inv_abst1: ∀L,V1,T1,U2. L ⊢ ⓛV1. T1 ➡ U2 → ∀I,W.
  68                      ∃∃V2,T2. L ⊢ V1 ➡ V2 & L. ⓑ{I} W ⊢ T1 ➡ T2 & U2 = ⓛV2. T2.
  69 #L #V1 #T1 #Y * #X #H1 #H2 #I #W
  70 elim (tpr_inv_abst1 … H1) -H1 #V #T #HV1 #HT1 #H destruct
  71 elim (tpss_inv_bind1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct
  72 lapply (tpss_lsubs_trans … HT2 (L. ⓑ{I} W) ?) -HT2 /2 width=1/ /4 width=5/
  73 qed-.
  74
  75 (* Basic_1: was pr2_gen_appl *)
  76 lemma cpr_inv_appl1: ∀L,V1,U0,U2. L ⊢ ⓐV1. U0 ➡ U2 →
  77                      ∨∨ ∃∃V2,T2.            L ⊢ V1 ➡ V2 & L ⊢ U0 ➡ T2 &
  78                                             U2 = ⓐV2. T2
  79                       | ∃∃V2,W,T1,T2.       L ⊢ V1 ➡ V2 & L. ⓓV2 ⊢ T1 ➡ T2 &
  80                                             U0 = ⓛW. T1 &
  81                                             U2 = ⓓV2. T2
  82                       | ∃∃V2,V,W1,W2,T1,T2. L ⊢ V1 ➡ V2 & L ⊢ W1 ➡ W2 & L. ⓓW2 ⊢ T1 ➡ T2 &
  83                                             ⇧[0,1] V2 ≡ V &
  84                                             U0 = ⓓW1. T1 &
  85                                             U2 = ⓓW2. ⓐV. T2.
  86 #L #V1 #U0 #Y * #X #H1 #H2
  87 elim (tpr_inv_appl1 … H1) -H1 *
  88 [ #V #U #HV1 #HU0 #H destruct
  89   elim (tpss_inv_flat1 … H2) -H2 #V2 #U2 #HV2 #HU2 #H destruct /4 width=5/
  90 | #V #W #T0 #T #HV1 #HT0 #H #H1 destruct
  91   elim (tpss_inv_bind1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct
  92   lapply (tpss_weak … HT2 0 (|L|+1) ? ?) -HT2 // /4 width=8/
  93 | #V0 #V #W #W0 #T #T0 #HV10 #HW0 #HT0 #HV0 #H #H1 destruct
  94   elim (tpss_inv_bind1 … H2) -H2 #W2 #X #HW02 #HX #HY destruct
  95   elim (tpss_inv_flat1 … HX) -HX #V2 #T2 #HV2 #HT2 #H destruct
  96   elim (tpss_inv_lift1_ge … HV2 … HV0 ?) -V // [3: /2 width=1/ |2: skip ] #V <minus_plus_m_m
  97   lapply (tpss_weak … HT2 0 (|L|+1) ? ?) -HT2 // /4 width=12/
  98 ]
  99 qed-.
 100
 101 (* Note: the main property of simple terms *)
 102 lemma cpr_inv_appl1_simple: ∀L,V1,T1,U. L ⊢ ⓐV1. T1 ➡ U → 𝐒⦃T1⦄ →
 103                             ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ T1 ➡ T2 &
 104                                      U = ⓐV2. T2.
 105 #L #V1 #T1 #U #H #HT1
 106 elim (cpr_inv_appl1 … H) -H *
 107 [ /2 width=5/
 108 | #V2 #W #W1 #W2 #_ #_ #H #_ destruct
 109   elim (simple_inv_bind … HT1)
 110 | #V2 #V #W1 #W2 #U1 #U2 #_ #_ #_ #_ #H #_ destruct
 111   elim (simple_inv_bind … HT1)
 112 ]
 113 qed-.
 114
 115 (* Relocation properties ****************************************************)
 116
 117 (* Basic_1: was: pr2_lift *)
 118 lemma cpr_lift: ∀L,K,d,e. ⇩[d, e] L ≡ K →
 119                 ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀T2,U2. ⇧[d, e] T2 ≡ U2 →
 120                 K ⊢ T1 ➡ T2 → L ⊢ U1 ➡ U2.
 121 #L #K #d #e #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 * #T #HT1 #HT2
 122 elim (lift_total T d e) #U #HTU
 123 lapply (tpr_lift … HT1 … HTU1 … HTU) -T1 #HU1
 124 elim (lt_or_ge (|K|) d) #HKd
 125 [ lapply (tpss_lift_le … HT2 … HLK HTU … HTU2) -T2 -T -HLK [ /2 width=2/ | /3 width=4/ ]
 126 | lapply (tpss_lift_be … HT2 … HLK HTU … HTU2) -T2 -T -HLK // /3 width=4/
 127 ]
 128 qed.
 129
 130 (* Basic_1: was: pr2_gen_lift *)
 131 lemma cpr_inv_lift: ∀L,K,d,e. ⇩[d, e] L ≡ K →
 132                     ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀U2. L ⊢ U1 ➡ U2 →
 133                     ∃∃T2. ⇧[d, e] T2 ≡ U2 & K ⊢ T1 ➡ T2.
 134 #L #K #d #e #HLK #T1 #U1 #HTU1 #U2 * #U #HU1 #HU2
 135 elim (tpr_inv_lift … HU1 … HTU1) -U1 #T #HTU #T1
 136 elim (lt_or_ge (|L|) d) #HLd
 137 [ elim (tpss_inv_lift1_le … HU2 … HLK … HTU ?) -U -HLK [ /5 width=4/ | /2 width=2/ ]
 138 | elim (lt_or_ge (|L|) (d + e)) #HLde
 139   [ elim (tpss_inv_lift1_be_up … HU2 … HLK … HTU ? ?) -U -HLK // [ /5 width=4/ | /2 width=2/ ]
 140   | elim (tpss_inv_lift1_be … HU2 … HLK … HTU ? ?) -U -HLK // /5 width=4/
 141   ]
 142 ]
 143 qed.