matita/matita/contribs/lambdadelta/basic_2/etc_2A1/cpr/tprs.etc

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  14
  15 notation "hvbox( T1 ➡ * break term 46 T2 )"
  16    non associative with precedence 45
  17    for @{ 'PRedStar $T1 $T2 }.
  18
  19 include "basic_2/reducibility/tpr.ma".
  20
  21 (* CONTEXT-FREE PARALLEL COMPUTATION ON TERMS *******************************)
  22
  23 (* Basic_1: includes: pr1_pr0 *)
  24 definition tprs: relation term ≝ TC … tpr.
  25
  26 interpretation "context-free parallel computation (term)"
  27    'PRedStar T1 T2 = (tprs T1 T2).
  28
  29 (* Basic eliminators ********************************************************)
  30
  31 lemma tprs_ind: ∀T1. ∀R:predicate term. R T1 →
  32                 (∀T,T2. T1 ➡* T → T ➡ T2 → R T → R T2) →
  33                 ∀T2. T1 ➡* T2 → R T2.
  34 #T1 #R #HT1 #IHT1 #T2 #HT12
  35 @(TC_star_ind … HT1 IHT1 … HT12) //
  36 qed-.
  37
  38 lemma tprs_ind_dx: ∀T2. ∀R:predicate term. R T2 →
  39                    (∀T1,T. T1 ➡ T → T ➡* T2 → R T → R T1) →
  40                    ∀T1. T1 ➡* T2 → R T1.
  41 #T2 #R #HT2 #IHT2 #T1 #HT12
  42 @(TC_star_ind_dx … HT2 IHT2 … HT12) //
  43 qed-.
  44
  45 (* Basic properties *********************************************************)
  46
  47 lemma tprs_refl: reflexive … tprs.
  48 /2 width=1/ qed.
  49
  50 lemma tpr_tprs: ∀T1,T2. T1 ➡ T2 → T2 ➡* T2.
  51 /2 width=1/ qed.
  52
  53 lemma tprs_strap1: ∀T1,T,T2. T1 ➡* T → T ➡ T2 → T1 ➡* T2.
  54 /2 width=3/ qed.
  55
  56 lemma tprs_strap2: ∀T1,T,T2. T1 ➡ T → T ➡* T2 → T1 ➡* T2.
  57 /2 width=3/ qed.
  58
  59 (* Basic_1: was only: pr1_head_1 *)
  60 lemma tprs_pair_sn: ∀I,T1,T2. T1 ➡ T2 → ∀V1,V2. V1 ➡* V2 →
  61                     ②{I} V1. T1 ➡* ②{I} V2. T2.
  62 * [ #a ] #I #T1 #T2 #HT12 #V1 #V2 #H @(tprs_ind … H) -V2
  63 [1,3: /3 width=1/
  64 |2,4: #V #V2 #_ #HV2 #IHV1
  65       @(tprs_strap1 … IHV1) -IHV1 /2 width=1/
  66 ]
  67 qed.
  68
  69 (* Basic_1: was only: pr1_head_2 *)
  70 lemma tprs_pair_dx: ∀I,V1,V2. V1 ➡ V2 → ∀T1,T2. T1 ➡* T2 →
  71                     ②{I} V1. T1 ➡* ②{I} V2. T2.
  72 * [ #a ] #I #V1 #V2 #HV12 #T1 #T2 #H @(tprs_ind … H) -T2
  73 [1,3: /3 width=1/
  74 |2,4: #T #T2 #_ #HT2 #IHT1
  75       @(tprs_strap1 … IHT1) -IHT1 /2 width=1/
  76 ]
  77 qed.
  78
  79 (* Basic inversion lemmas ***************************************************)
  80
  81 lemma tprs_inv_atom1: ∀U2,k. ⋆k ➡* U2 → U2 = ⋆k.
  82 #U2 #k #H @(tprs_ind … H) -U2 //
  83 #U #U2 #_ #HU2 #IHU1 destruct
  84 >(tpr_inv_atom1 … HU2) -HU2 //
  85 qed-.
  86
  87 lemma tprs_inv_cast1: ∀W1,T1,U2. ⓝW1.T1 ➡* U2 → T1 ➡* U2 ∨
  88                       ∃∃W2,T2. W1 ➡* W2 & T1 ➡* T2 & U2 = ⓝW2.T2.
  89 #W1 #T1 #U2 #H @(tprs_ind … H) -U2 /3 width=5/
  90 #U #U2 #_ #HU2 * /3 width=3/ *
  91 #W #T #HW1 #HT1 #H destruct
  92 elim (tpr_inv_cast1 … HU2) -HU2 /3 width=3/ *
  93 #W2 #T2 #HW2 #HT2 #H destruct /4 width=5/
  94 qed-.