matita/matita/contribs/lambdadelta/basic_2/computation/lpxs.ma

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  14
  15 include "basic_2/notation/relations/predsnstar_4.ma".
  16 include "basic_2/reduction/lpx.ma".
  17 include "basic_2/computation/lprs.ma".
  18
  19 (* SN EXTENDED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *******************)
  20
  21 definition lpxs: ∀h. sd h → relation lenv ≝ λh,g. TC … (lpx h g).
  22
  23 interpretation "extended parallel computation (local environment, sn variant)"
  24    'PRedSnStar h g L1 L2 = (lpxs h g L1 L2).
  25
  26 (* Basic eliminators ********************************************************)
  27
  28 lemma lpxs_ind: ∀h,g,L1. ∀R:predicate lenv. R L1 →
  29                 (∀L,L2. ⦃h, L1⦄ ⊢ ➡*[h, g] L → ⦃G, L⦄ ⊢ ➡[h, g] L2 → R L → R L2) →
  30                 ∀L2. ⦃h, L1⦄ ⊢ ➡*[h, g] L2 → R L2.
  31 #h #g #L1 #R #HL1 #IHL1 #L2 #HL12
  32 @(TC_star_ind … HL1 IHL1 … HL12) //
  33 qed-.
  34
  35 lemma lpxs_ind_dx: ∀h,g,L2. ∀R:predicate lenv. R L2 →
  36                    (∀L1,L. ⦃h, L1⦄ ⊢ ➡[h, g] L → ⦃G, L⦄ ⊢ ➡*[h, g] L2 → R L → R L1) →
  37                    ∀L1. ⦃h, L1⦄ ⊢ ➡*[h, g] L2 → R L1.
  38 #h #g #L2 #R #HL2 #IHL2 #L1 #HL12
  39 @(TC_star_ind_dx … HL2 IHL2 … HL12) //
  40 qed-.
  41
  42 (* Basic properties *********************************************************)
  43
  44 lemma lprs_lpxs: ∀h,g,L1,L2. L1 ⊢ ➡* L2 → ⦃h, L1⦄ ⊢ ➡*[h, g] L2.
  45 /3 width=3/ qed.
  46
  47 lemma lpx_lpxs: ∀h,g,L1,L2. ⦃h, L1⦄ ⊢ ➡[h, g] L2 → ⦃h, L1⦄ ⊢ ➡*[h, g] L2.
  48 /2 width=1/ qed.
  49
  50 lemma lpxs_refl: ∀h,g,L. ⦃G, L⦄ ⊢ ➡*[h, g] L.
  51 /2 width=1/ qed.
  52
  53 lemma lpxs_strap1: ∀h,g,L1,L,L2. ⦃h, L1⦄ ⊢ ➡*[h, g] L → ⦃G, L⦄ ⊢ ➡[h, g] L2 → ⦃h, L1⦄ ⊢ ➡*[h, g] L2.
  54 /2 width=3/ qed.
  55
  56 lemma lpxs_strap2: ∀h,g,L1,L,L2. ⦃h, L1⦄ ⊢ ➡[h, g] L → ⦃G, L⦄ ⊢ ➡*[h, g] L2 → ⦃h, L1⦄ ⊢ ➡*[h, g] L2.
  57 /2 width=3/ qed.
  58
  59 lemma lpxs_pair_refl: ∀h,g,L1,L2. ⦃h, L1⦄ ⊢ ➡*[h, g] L2 → ∀I,V. ⦃h, L1. ⓑ{I}V⦄ ⊢ ➡*[h, g] L2. ⓑ{I}V.
  60 /2 width=1 by TC_lpx_sn_pair_refl/ qed.
  61
  62 (* Basic inversion lemmas ***************************************************)
  63
  64 lemma lpxs_inv_atom1: ∀h,g,L2. ⦃h, ⋆⦄ ⊢ ➡*[h, g] L2 → L2 = ⋆.
  65 /2 width=2 by TC_lpx_sn_inv_atom1/ qed-.
  66
  67 lemma lpxs_inv_atom2: ∀h,g,L1. ⦃h, L1⦄ ⊢ ➡*[h, g] ⋆ → L1 = ⋆.
  68 /2 width=2 by TC_lpx_sn_inv_atom2/ qed-.
  69
  70 (* Basic forward lemmas *****************************************************)
  71
  72 lemma lpxs_fwd_length: ∀h,g,L1,L2. ⦃h, L1⦄ ⊢ ➡*[h, g] L2 → |L1| = |L2|.
  73 /2 width=2 by TC_lpx_sn_fwd_length/ qed-.