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

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  14
  15 include "basic_2/notation/relations/predsnstar_2.ma".
  16 include "basic_2/grammar/lpx_sn_tc.ma".
  17 include "basic_2/reduction/lpr.ma".
  18
  19 (* SN PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ****************************)
  20
  21 definition lprs: relation lenv ≝ TC … lpr.
  22
  23 interpretation "parallel computation (local environment, sn variant)"
  24    'PRedSnStar L1 L2 = (lprs L1 L2).
  25
  26 (* Basic eliminators ********************************************************)
  27
  28 lemma lprs_ind: ∀L1. ∀R:predicate lenv. R L1 →
  29                 (∀L,L2. L1 ⊢ ➡* L → ⦃G, 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 lprs_ind_dx: ∀L2. ∀R:predicate lenv. R L2 →
  36                    (∀L1,L. L1 ⊢ ➡ L → ⦃G, 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 lpr_lprs: ∀L1,L2. L1 ⊢ ➡ L2 → L1 ⊢ ➡* L2.
  45 /2 width=1/ qed.
  46
  47 lemma lprs_refl: ∀L. ⦃G, L⦄ ⊢ ➡* L.
  48 /2 width=1/ qed.
  49
  50 lemma lprs_strap1: ∀L1,L,L2. L1 ⊢ ➡* L → ⦃G, L⦄ ⊢ ➡ L2 → L1 ⊢ ➡* L2.
  51 /2 width=3/ qed.
  52
  53 lemma lprs_strap2: ∀L1,L,L2. L1 ⊢ ➡ L → ⦃G, L⦄ ⊢ ➡* L2 → L1 ⊢ ➡* L2.
  54 /2 width=3/ qed.
  55
  56 lemma lprs_pair_refl: ∀L1,L2. L1 ⊢ ➡* L2 → ∀I,V. L1. ⓑ{I} V ⊢ ➡* L2. ⓑ{I} V.
  57 /2 width=1 by TC_lpx_sn_pair_refl/ qed.
  58
  59 (* Basic inversion lemmas ***************************************************)
  60
  61 lemma lprs_inv_atom1: ∀L2. ⋆ ⊢ ➡* L2 → L2 = ⋆.
  62 /2 width=2 by TC_lpx_sn_inv_atom1/ qed-.
  63
  64 lemma lprs_inv_atom2: ∀L1. L1 ⊢ ➡* ⋆ → L1 = ⋆.
  65 /2 width=2 by TC_lpx_sn_inv_atom2/ qed-.
  66
  67 (* Basic forward lemmas *****************************************************)
  68
  69 lemma lprs_fwd_length: ∀L1,L2. L1 ⊢ ➡* L2 → |L1| = |L2|.
  70 /2 width=2 by TC_lpx_sn_fwd_length/ qed-.