matita/matita/contribs/lambdadelta/basic_2/equivalence/lfpcs.ma

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