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

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  14
  15 include "basic_2/reducibility/ltpr.ma".
  16 include "basic_2/computation/tprs.ma".
  17
  18 (* CONTEXT-FREE PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ******************)
  19
  20 definition ltprs: relation lenv ≝ TC … ltpr.
  21
  22 interpretation
  23   "context-free parallel computation (environment)"
  24   'PRedStar L1 L2 = (ltprs L1 L2).
  25
  26 (* Basic eliminators ********************************************************)
  27
  28 lemma ltprs_ind: ∀L1. ∀R:predicate lenv. R L1 →
  29                  (∀L,L2. L1 ➡* L → L ➡ L2 → R L → R L2) →
  30                  ∀L2. L1 ➡* L2 → R L2.
  31 #L1 #R #HL1 #IHL1 #L2 #HL12
  32 @(TC_star_ind … HL1 IHL1 … HL12) //
  33 qed-.
  34
  35 lemma ltprs_ind_dx: ∀L2. ∀R:predicate lenv. R L2 →
  36                     (∀L1,L. L1 ➡ L → L ➡* L2 → R L → R L1) →
  37                     ∀L1. L1 ➡* L2 → R L1.
  38 #L2 #R #HL2 #IHL2 #L1 #HL12
  39 @(TC_star_ind_dx … HL2 IHL2 … HL12) //
  40 qed-.
  41
  42 (* Basic properties *********************************************************)
  43
  44 lemma ltprs_refl: reflexive … ltprs.
  45 /2 width=1/ qed.
  46
  47 lemma ltpr_ltprs: ∀L1,L2. L1 ➡ L2 → L1 ➡* L2.
  48 /2 width=1/ qed.
  49
  50 lemma ltprs_strap1: ∀L1,L,L2. L1 ➡* L → L ➡ L2 → L1 ➡* L2.
  51 /2 width=3/ qed.
  52
  53 lemma ltprs_strap2: ∀L1,L,L2. L1 ➡ L → L ➡* L2 → L1 ➡* L2.
  54 /2 width=3/ qed.
  55
  56 (* Basic inversion lemmas ***************************************************)
  57
  58 lemma ltprs_inv_atom1: ∀L2. ⋆ ➡* L2 → L2 = ⋆.
  59 #L2 #H @(ltprs_ind … H) -L2 //
  60 #L #L2 #_ #HL2 #IHL1 destruct
  61 >(ltpr_inv_atom1 … HL2) -L2 //
  62 qed-.
  63
  64 lemma ltprs_inv_pair1: ∀I,K1,L2,V1. K1. ⓑ{I} V1 ➡* L2 →
  65                        ∃∃K2,V2. K1 ➡* K2 & V1 ➡* V2 & L2 = K2. ⓑ{I} V2.
  66 #I #K1 #L2 #V1 #H @(ltprs_ind … H) -L2 /2 width=5/
  67 #L #L2 #_ #HL2 * #K #V #HK1 #HV1 #H destruct
  68 elim (ltpr_inv_pair1 … HL2) -HL2 #K2 #V2 #HK2 #HV2 #H destruct /3 width=5/
  69 qed-.
  70
  71 lemma ltprs_inv_atom2: ∀L1. L1 ➡* ⋆ → L1 = ⋆.
  72 #L1 #H @(ltprs_ind_dx … H) -L1 //
  73 #L1 #L #HL1 #_ #IHL2 destruct
  74 >(ltpr_inv_atom2 … HL1) -L1 //
  75 qed-.
  76
  77 lemma ltprs_inv_pair2: ∀I,L1,K2,V2. L1 ➡* K2. ⓑ{I} V2 →
  78                        ∃∃K1,V1. K1 ➡* K2 & V1 ➡* V2 & L1 = K1. ⓑ{I} V1.
  79 #I #L1 #K2 #V2 #H @(ltprs_ind_dx … H) -L1 /2 width=5/
  80 #L1 #L #HL1 #_ * #K #V #HK2 #HV2 #H destruct
  81 elim (ltpr_inv_pair2 … HL1) -HL1 #K1 #V1 #HK1 #HV1 #H destruct /3 width=5/
  82 qed-.
  83
  84 (* Basic forward lemmas *****************************************************)
  85
  86 lemma ltprs_fwd_length: ∀L1,L2. L1 ➡* L2 → |L1| = |L2|.
  87 #L1 #L2 #H @(ltprs_ind … H) -L2 //
  88 #L #L2 #_ #HL2 #IHL1
  89 >IHL1 -L1 >(ltpr_fwd_length … HL2) -HL2 //
  90 qed-.