matita/matita/contribs/lambdadelta/basic_2A/etc/llpr/llprs.etc

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  14
  15 include "basic_2/notation/relations/lazypredsnstar_5.ma".
  16 include "basic_2/reduction/llpr.ma".
  17
  18 (* LAZY SN PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ***********************)
  19
  20 definition llprs: genv → relation4 ynat term lenv lenv ≝
  21                   λG,d. LTC … (llpr G d).
  22
  23 interpretation "lazy parallel computation (local environment, sn variant)"
  24    'LazyPRedSnStar G L1 L2 T d = (llprs G d T L1 L2).
  25
  26 (* Basic eliminators ********************************************************)
  27
  28 lemma llprs_ind: ∀G,L1,T,d. ∀R:predicate lenv. R L1 →
  29                  (∀L,L2. ⦃G, L1⦄ ⊢ ➡*[T, d] L → ⦃G, L⦄ ⊢ ➡[T, d] L2 → R L → R L2) →
  30                  ∀L2. ⦃G, L1⦄ ⊢ ➡*[T, d] L2 → R L2.
  31 #G #L1 #T #d #R #HL1 #IHL1 #L2 #HL12
  32 @(TC_star_ind … HL1 IHL1 … HL12) //
  33 qed-.
  34
  35 lemma llprs_ind_dx: ∀G,L2,T,d. ∀R:predicate lenv. R L2 →
  36                     (∀L1,L. ⦃G, L1⦄ ⊢ ➡[T, d] L → ⦃G, L⦄ ⊢ ➡*[T, d] L2 → R L → R L1) →
  37                     ∀L1. ⦃G, L1⦄ ⊢ ➡*[T, d] L2 → R L1.
  38 #G #L2 #T #d #R #HL2 #IHL2 #L1 #HL12
  39 @(TC_star_ind_dx … HL2 IHL2 … HL12) //
  40 qed-.
  41
  42 (* Basic properties *********************************************************)
  43
  44 lemma lpr_llprs: ∀G,L1,L2,T,d. ⦃G, L1⦄ ⊢ ➡[T, d] L2 → ⦃G, L1⦄ ⊢ ➡*[T, d] L2.
  45 /2 width=1 by inj/ qed.
  46
  47 lemma llprs_refl: ∀G,L,T,d. ⦃G, L⦄ ⊢ ➡*[T, d] L.
  48 /2 width=1 by lpr_llprs/ qed.
  49
  50 lemma llprs_strap1: ∀G,L1,L,L2,T,d. ⦃G, L1⦄ ⊢ ➡*[T, d] L → ⦃G, L⦄ ⊢ ➡[T, d] L2 → ⦃G, L1⦄ ⊢ ➡*[T, d] L2.
  51 normalize /2 width=3 by step/ qed-.
  52
  53 lemma llprs_strap2: ∀G,L1,L,L2,T,d. ⦃G, L1⦄ ⊢ ➡[T, d] L → ⦃G, L⦄ ⊢ ➡*[T, d] L2 → ⦃G, L1⦄ ⊢ ➡*[T, d] L2.
  54 normalize /2 width=3 by TC_strap/ qed-.
  55
  56 (* Basic forward lemmas *****************************************************)
  57
  58 lemma llprs_fwd_length: ∀G,L1,L2,T,d. ⦃G, L1⦄ ⊢ ➡*[T, d] L2 → |L1| = |L2|.
  59 #G #L1 #L2 #T #d #H @(llprs_ind … H) -L2
  60 /3 width=6 by llpr_fwd_length, trans_eq/
  61 qed-.