matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lpc/lfpcs.etc

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  14
  15 notation "hvbox( ⦃ term 46 L1 ⦄ ⬌ ⬌ * break ⦃ term 46 L2 ⦄ )"
  16    non associative with precedence 45
  17    for @{ 'FocalizedPConvStarAlt $L1 $L2 }.
  18
  19 include "basic_2/conversion/lfpc.ma".
  20
  21 (* FOCALIZED PARALLEL EQUIVALENCE ON LOCAL ENVIRONMENTS *********************)
  22
  23 definition lfpcs: relation lenv ≝ TC … lfpc.
  24
  25 interpretation "focalized parallel equivalence (local environment)"
  26    'FocalizedPConvStar L1 L2 = (lfpcs L1 L2).
  27
  28 (* Basic eliminators ********************************************************)
  29
  30 lemma lfpcs_ind: ∀L1. ∀R:predicate lenv. R L1 →
  31                  (∀L,L2. ⦃L1⦄ ⬌* ⦃L⦄ → ⦃L⦄ ⬌ ⦃L2⦄ → R L → R L2) →
  32                  ∀L2. ⦃L1⦄ ⬌* ⦃L2⦄ → R L2.
  33 #L1 #R #HL1 #IHL1 #L2 #HL12 @(TC_star_ind … HL1 IHL1 … HL12) //
  34 qed-.
  35
  36 lemma lfpcs_ind_dx: ∀L2. ∀R:predicate lenv. R L2 →
  37                     (∀L1,L. ⦃L1⦄ ⬌ ⦃L⦄ → ⦃L⦄ ⬌* ⦃L2⦄ → R L → R L1) →
  38                     ∀L1. ⦃L1⦄ ⬌* ⦃L2⦄ → R L1.
  39 #L2 #R #HL2 #IHL2 #L1 #HL12
  40 @(TC_star_ind_dx … HL2 IHL2 … HL12) //
  41 qed-.
  42
  43 (* Basic properties *********************************************************)
  44
  45 lemma lfpcs_refl: reflexive … lfpcs.
  46 /2 width=1/ qed.
  47
  48 lemma lfpcs_sym: symmetric … lfpcs.
  49 /3 width=1/ qed.
  50
  51 lemma lfpc_lfpcs: ∀L1,L2. ⦃L1⦄ ⬌ ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
  52 /2 width=1/ qed.
  53
  54 lemma lfpcs_strap1: ∀L1,L,L2. ⦃L1⦄ ⬌* ⦃L⦄ → ⦃L⦄ ⬌ ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
  55 /2 width=3/ qed.
  56
  57 lemma lfpcs_strap2: ∀L1,L,L2. ⦃L1⦄ ⬌ ⦃L⦄ → ⦃L⦄ ⬌* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
  58 /2 width=3/ qed.
  59
  60 lemma lfpcs_lfpr_dx: ∀L1,L2. ⦃L1⦄ ➡ ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
  61 /3 width=1/ qed.
  62
  63 lemma lfpcs_lfpr_sn: ∀L1,L2. ⦃L2⦄ ➡ ⦃L1⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
  64 /3 width=1/ qed.
  65
  66 lemma lfpcs_lfpr_strap1: ∀L1,L. ⦃L1⦄ ⬌* ⦃L⦄ → ∀L2. ⦃L⦄ ➡ ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
  67 /3 width=3/ qed.
  68
  69 lemma lfpcs_lfpr_strap2: ∀L1,L. ⦃L1⦄ ➡ ⦃L⦄ → ∀L2. ⦃L⦄ ⬌* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
  70 /3 width=3/ qed.
  71
  72 lemma lfpcs_lfpr_div: ∀L1,L. ⦃L1⦄ ⬌* ⦃L⦄ → ∀L2. ⦃L2⦄ ➡ ⦃L⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
  73 /3 width=3/ qed.
  74
  75 lemma lfpcs_lfpr_conf: ∀L1,L. ⦃L⦄ ➡ ⦃L1⦄ → ∀L2. ⦃L⦄ ⬌* ⦃L2⦄ → ⦃L1⦄ ⬌* ⦃L2⦄.
  76 /3 width=3/ qed.