matita/matita/contribs/lambdadelta/basic_2A/computation/lprs.ma

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  14
  15 include "basic_2A/notation/relations/predsnstar_3.ma".
  16 include "basic_2A/substitution/lpx_sn_tc.ma".
  17 include "basic_2A/reduction/lpr.ma".
  18
  19 (* SN PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ****************************)
  20
  21 definition lprs: relation3 genv lenv lenv ≝
  22                  λG. TC … (lpr G).
  23
  24 interpretation "parallel computation (local environment, sn variant)"
  25    'PRedSnStar G L1 L2 = (lprs G L1 L2).
  26
  27 (* Basic eliminators ********************************************************)
  28
  29 lemma lprs_ind: ∀G,L1. ∀R:predicate lenv. R L1 →
  30                 (∀L,L2. ⦃G, L1⦄ ⊢ ➡* L → ⦃G, L⦄ ⊢ ➡ L2 → R L → R L2) →
  31                 ∀L2. ⦃G, L1⦄ ⊢ ➡* L2 → R L2.
  32 #G #L1 #R #HL1 #IHL1 #L2 #HL12
  33 @(TC_star_ind … HL1 IHL1 … HL12) //
  34 qed-.
  35
  36 lemma lprs_ind_dx: ∀G,L2. ∀R:predicate lenv. R L2 →
  37                    (∀L1,L. ⦃G, L1⦄ ⊢ ➡ L → ⦃G, L⦄ ⊢ ➡* L2 → R L → R L1) →
  38                    ∀L1. ⦃G, L1⦄ ⊢ ➡* L2 → R L1.
  39 #G #L2 #R #HL2 #IHL2 #L1 #HL12
  40 @(TC_star_ind_dx … HL2 IHL2 … HL12) //
  41 qed-.
  42
  43 (* Basic properties *********************************************************)
  44
  45 lemma lpr_lprs: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L1⦄ ⊢ ➡* L2.
  46 /2 width=1 by inj/ qed.
  47
  48 lemma lprs_refl: ∀G,L. ⦃G, L⦄ ⊢ ➡* L.
  49 /2 width=1 by lpr_lprs/ qed.
  50
  51 lemma lprs_strap1: ∀G,L1,L,L2. ⦃G, L1⦄ ⊢ ➡* L → ⦃G, L⦄ ⊢ ➡ L2 → ⦃G, L1⦄ ⊢ ➡* L2.
  52 /2 width=3 by step/ qed-.
  53
  54 lemma lprs_strap2: ∀G,L1,L,L2. ⦃G, L1⦄ ⊢ ➡ L → ⦃G, L⦄ ⊢ ➡* L2 → ⦃G, L1⦄ ⊢ ➡* L2.
  55 /2 width=3 by TC_strap/ qed-.
  56
  57 lemma lprs_pair_refl: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 → ∀I,V. ⦃G, L1.ⓑ{I}V⦄ ⊢ ➡* L2.ⓑ{I}V.
  58 /2 width=1 by TC_lpx_sn_pair_refl/ qed.
  59
  60 (* Basic inversion lemmas ***************************************************)
  61
  62 lemma lprs_inv_atom1: ∀G,L2. ⦃G, ⋆⦄ ⊢ ➡* L2 → L2 = ⋆.
  63 /2 width=2 by TC_lpx_sn_inv_atom1/ qed-.
  64
  65 lemma lprs_inv_atom2: ∀G,L1. ⦃G, L1⦄ ⊢ ➡* ⋆ → L1 = ⋆.
  66 /2 width=2 by TC_lpx_sn_inv_atom2/ qed-.
  67
  68 (* Basic forward lemmas *****************************************************)
  69
  70 lemma lprs_fwd_length: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 → |L1| = |L2|.
  71 /2 width=2 by TC_lpx_sn_fwd_length/ qed-.