matita/matita/contribs/lambdadelta/basic_2/rt_computation/cprs.ma

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  10 (*       \ /        This file is distributed under the terms of the       *)
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  14
  15 include "basic_2/rt_transition/cpr.ma".
  16 include "basic_2/rt_computation/cpms.ma".
  17
  18 (* CONTEXT-SENSITIVE PARALLEL COMPUTATION FOR TERMS *************************)
  19
  20 (* Basic eliminators ********************************************************)
  21
  22 (* Basic_2A1: was: cprs_ind_dx *)
  23 lemma cprs_ind_sn (h) (G) (L) (T2) (R:predicate …):
  24                   R T2 →
  25                   (∀T1,T. ⦃G, L⦄ ⊢ T1 ➡[h] T → ⦃G, L⦄ ⊢ T ➡*[h] T2 → R T → R T1) →
  26                   ∀T1. ⦃G, L⦄ ⊢ T1 ➡*[h] T2 → R T1.
  27 #h #G #L #T2 #R #IH1 #IH2 #T1
  28 @(insert_eq_0 … 0) #n #H
  29 @(cpms_ind_sn … H) -n -T1 //
  30 #n1 #n2 #T1 #T #HT1 #HT2 #IH #H
  31 elim (plus_inv_O3 n1 n2) // -H #H1 #H2 destruct
  32 /3 width=4 by/
  33 qed-.
  34
  35 (* Basic_2A1: was: cprs_ind *)
  36 lemma cprs_ind_dx (h) (G) (L) (T1) (R:predicate …):
  37                   R T1 →
  38                   (∀T,T2. ⦃G, L⦄ ⊢ T1 ➡*[h] T → ⦃G, L⦄ ⊢ T ➡[h] T2 → R T → R T2) →
  39                   ∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h] T2 → R T2.
  40 #h #G #L #T1 #R #IH1 #IH2 #T2
  41 @(insert_eq_0 … 0) #n #H
  42 @(cpms_ind_dx … H) -n -T2 //
  43 #n1 #n2 #T #T2 #HT1 #IH #HT2 #H
  44 elim (plus_inv_O3 n1 n2) // -H #H1 #H2 destruct
  45 /3 width=4 by/
  46 qed-.
  47
  48 (* Basic properties *********************************************************)
  49
  50 (* Basic_1: was: pr3_step *)
  51 (* Basic_2A1: was: cprs_strap2 *)
  52 lemma cprs_step_sn (h) (G) (L):
  53                    ∀T1,T. ⦃G, L⦄ ⊢ T1 ➡[h] T →
  54                    ∀T2. ⦃G, L⦄ ⊢ T ➡*[h] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h] T2.
  55 /2 width=3 by cpms_step_sn/ qed-.
  56
  57 (* Basic_2A1: was: cprs_strap1 *)
  58 lemma cprs_step_dx (h) (G) (L):
  59                    ∀T1,T. ⦃G, L⦄ ⊢ T1 ➡*[h] T →
  60                    ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h] T2.
  61 /2 width=3 by cpms_step_dx/ qed-.
  62
  63 (* Basic_1: was only: pr3_thin_dx *)
  64 lemma cprs_flat_dx (h) (I) (G) (L):
  65                    ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 →
  66                    ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h] T2 →
  67                    ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡*[h] ⓕ{I}V2.T2.
  68 #h #I #G #L #V1 #V2 #HV12 #T1 #T2 #H @(cprs_ind_sn … H) -T1
  69 /3 width=3 by cprs_step_sn, cpm_cpms, cpr_flat/
  70 qed.
  71 (*
  72 lemma cprs_flat_sn: ∀I,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h] T2 → ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h] V2 →
  73                     ⦃G, L⦄ ⊢ ⓕ{I} V1. T1 ➡*[h] ⓕ{I} V2. T2.
  74 #I #G #L #T1 #T2 #HT12 #V1 #V2 #H @(cprs_ind … H) -V2
  75 /3 width=3 by cprs_strap1, cpr_cprs, cpr_pair_sn, cpr_flat/
  76 qed.
  77
  78 lemma cprs_zeta: ∀G,L,V,T1,T,T2. ⬆[0, 1] T2 ≘ T →
  79                  ⦃G, L.ⓓV⦄ ⊢ T1 ➡*[h] T → ⦃G, L⦄ ⊢ +ⓓV.T1 ➡*[h] T2.
  80 #G #L #V #T1 #T #T2 #HT2 #H @(cprs_ind_dx … H) -T1
  81 /3 width=3 by cprs_strap2, cpr_cprs, cpr_bind, cpr_zeta/
  82 qed.
  83
  84 lemma cprs_eps: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h] T2 → ∀V. ⦃G, L⦄ ⊢ ⓝV.T1 ➡*[h] T2.
  85 #G #L #T1 #T2 #H @(cprs_ind … H) -T2
  86 /3 width=3 by cprs_strap1, cpr_cprs, cpr_eps/
  87 qed.
  88
  89 lemma cprs_beta_dx: ∀a,G,L,V1,V2,W1,W2,T1,T2.
  90                     ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L⦄ ⊢ W1 ➡[h] W2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡*[h] T2 →
  91                     ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡*[h] ⓓ{a}ⓝW2.V2.T2.
  92 #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 * -T2
  93 /4 width=7 by cprs_strap1, cpr_cprs, cprs_bind_dx, cprs_flat_dx, cpr_beta/
  94 qed.
  95
  96 lemma cprs_theta_dx: ∀a,G,L,V1,V,V2,W1,W2,T1,T2.
  97                      ⦃G, L⦄ ⊢ V1 ➡[h] V → ⬆[0, 1] V ≘ V2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡*[h] T2 →
  98                      ⦃G, L⦄ ⊢ W1 ➡[h] W2 → ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡*[h] ⓓ{a}W2.ⓐV2.T2.
  99 #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 * -T2
 100 /4 width=9 by cprs_strap1, cpr_cprs, cprs_bind_dx, cprs_flat_dx, cpr_theta/
 101 qed.
 102
 103 (* Basic inversion lemmas ***************************************************)
 104
 105 (* Basic_1: was: pr3_gen_sort *)
 106 lemma cprs_inv_sort1: ∀G,L,U2,s. ⦃G, L⦄ ⊢ ⋆s ➡*[h] U2 → U2 = ⋆s.
 107 #G #L #U2 #s #H @(cprs_ind … H) -U2 //
 108 #U2 #U #_ #HU2 #IHU2 destruct
 109 >(cpr_inv_sort1 … HU2) -HU2 //
 110 qed-.
 111
 112 (* Basic_1: was: pr3_gen_cast *)
 113 lemma cprs_inv_cast1: ∀G,L,W1,T1,U2. ⦃G, L⦄ ⊢ ⓝW1.T1 ➡*[h] U2 → ⦃G, L⦄ ⊢ T1 ➡*[h] U2 ∨
 114                       ∃∃W2,T2. ⦃G, L⦄ ⊢ W1 ➡*[h] W2 & ⦃G, L⦄ ⊢ T1 ➡*[h] T2 & U2 = ⓝW2.T2.
 115 #G #L #W1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5 by ex3_2_intro, or_intror/
 116 #U2 #U #_ #HU2 * /3 width=3 by cprs_strap1, or_introl/ *
 117 #W #T #HW1 #HT1 #H destruct
 118 elim (cpr_inv_cast1 … HU2) -HU2 /3 width=3 by cprs_strap1, or_introl/ *
 119 #W2 #T2 #HW2 #HT2 #H destruct /4 width=5 by cprs_strap1, ex3_2_intro, or_intror/
 120 qed-.
 121 *)
 122 (* Basic_1: removed theorems 13:
 123    pr1_head_1 pr1_head_2 pr1_comp
 124    clear_pr3_trans pr3_cflat pr3_gen_bind
 125    pr3_head_1 pr3_head_2 pr3_head_21 pr3_head_12
 126    pr3_iso_appl_bind pr3_iso_appls_appl_bind pr3_iso_appls_bind
 127 *)