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

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