matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/fpcs.etc

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  14
  15 notation "hvbox( ⦃ term 46 L1 , break term 46 T1 ⦄ ⬌ * break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
  16    non associative with precedence 45
  17    for @{ 'FocalizedPConvStar $L1 $T1 $L2 $T2 }.
  18
  19 notation "hvbox( ⦃ term 46 L1 , break term 46 T1 ⦄ ⬌ ⬌ * break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
  20    non associative with precedence 45
  21    for @{ 'FocalizedPConvStarAlt $L1 $T1 $L2 $T2 }.
  22
  23 include "basic_2/conversion/fpc.ma".
  24
  25 (* CONTEXT-FREE PARALLEL EQUIVALENCE ON CLOSURES ****************************)
  26
  27 definition fpcs: bi_relation lenv term ≝ bi_TC … fpc.
  28
  29 interpretation "context-free parallel equivalence (closure)"
  30    'FocalizedPConvStar L1 T1 L2 T2 = (fpcs L1 T1 L2 T2).
  31
  32 (* Basic eliminators ********************************************************)
  33
  34 lemma fpcs_ind: ∀L1,T1. ∀R:relation2 lenv term. R L1 T1 →
  35                 (∀L,L2,T,T2. ⦃L1, T1⦄ ⬌* ⦃L, T⦄ → ⦃L, T⦄ ⬌ ⦃L2, T2⦄ → R L T → R L2 T2) →
  36                 ∀L2,T2. ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄ → R L2 T2.
  37 /3 width=7 by bi_TC_star_ind/ qed-.
  38
  39 lemma fpcs_ind_dx: ∀L2,T2. ∀R:relation2 lenv term. R L2 T2 →
  40                    (∀L1,L,T1,T. ⦃L1, T1⦄ ⬌ ⦃L, T⦄ → ⦃L, T⦄ ⬌* ⦃L2, T2⦄ → R L T → R L1 T1) →
  41                    ∀L1,T1. ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄ → R L1 T1.
  42 /3 width=7 by bi_TC_star_ind_dx/ qed-.
  43
  44 (* Basic properties *********************************************************)
  45
  46 lemma fpcs_refl: bi_reflexive … fpcs.
  47 /2 width=1/ qed.
  48
  49 lemma fpcs_sym: bi_symmetric … fpcs.
  50 /3 width=1/ qed.
  51
  52 lemma fpc_fpcs: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⬌ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
  53 /2 width=1/ qed.
  54
  55 lemma fpcs_strap1: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⬌* ⦃L, T⦄ → ⦃L, T⦄ ⬌ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
  56 /2 width=4/ qed.
  57
  58 lemma fpcs_strap2: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⬌ ⦃L, T⦄ → ⦃L, T⦄ ⬌* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
  59 /2 width=4/ qed.
  60
  61 lemma fpcs_fpr_dx: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
  62 /3 width=1/ qed.
  63
  64 lemma fpcs_fpr_sn: ∀L1,L2,T1,T2. ⦃L2, T2⦄ ➡ ⦃L1, T1⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
  65 /3 width=1/ qed.
  66
  67 lemma fpcs_fpr_strap1: ∀L1,L,T1,T. ⦃L1, T1⦄ ⬌* ⦃L, T⦄ →
  68                        ∀L2,T2. ⦃L, T⦄ ➡ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
  69 /3 width=4/ qed.
  70
  71 lemma fpcs_fpr_strap2: ∀L1,L,T1,T. ⦃L1, T1⦄ ➡ ⦃L, T⦄ →
  72                        ∀L2,T2. ⦃L, T⦄ ⬌* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
  73 /3 width=4/ qed.
  74
  75 lemma fpcs_fpr_div: ∀L1,L,T1,T. ⦃L1, T1⦄ ⬌* ⦃L, T⦄ →
  76                     ∀L2,T2. ⦃L2, T2⦄ ➡ ⦃L, T⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
  77 /3 width=4/ qed.
  78
  79 lemma fpr_div: ∀L1,L,T1,T. ⦃L1, T1⦄ ➡ ⦃L, T⦄ → ∀L2,T2. ⦃L2, T2⦄ ➡ ⦃L, T⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
  80 /3 width=4/ qed-.
  81
  82 lemma fpcs_fpr_conf: ∀L1,L,T1,T. ⦃L, T⦄ ➡ ⦃L1, T1⦄ →
  83                      ∀L2,T2. ⦃L, T⦄ ⬌* ⦃L2, T2⦄ → ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄.
  84 /3 width=4/ qed.