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

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  14
  15 include "basic_2/conversion/cpc.ma".
  16
  17 (* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************)
  18
  19 definition cpcs: lenv → relation term ≝
  20                  λL. TC … (cpc L).
  21
  22 interpretation "context-sensitive parallel equivalence (term)"
  23    'PConvStar L T1 T2 = (cpcs L T1 T2).
  24
  25 (* Basic eliminators ********************************************************)
  26
  27 lemma cpcs_ind: ∀L,T1. ∀R:predicate term. R T1 →
  28                 (∀T,T2. L ⊢ T1 ⬌* T → L ⊢ T ⬌ T2 → R T → R T2) →
  29                 ∀T2. L ⊢ T1 ⬌* T2 → R T2.
  30 #L #T1 #R #HT1 #IHT1 #T2 #HT12 @(TC_star_ind … HT1 IHT1 … HT12) //
  31 qed-.
  32
  33 lemma cpcs_ind_dx: ∀L,T2. ∀R:predicate term. R T2 →
  34                    (∀T1,T. L ⊢ T1 ⬌ T → L ⊢ T ⬌* T2 → R T → R T1) →
  35                    ∀T1. L ⊢ T1 ⬌* T2 → R T1.
  36 #L #T2 #R #HT2 #IHT2 #T1 #HT12
  37 @(TC_star_ind_dx … HT2 IHT2 … HT12) //
  38 qed-.
  39
  40 (* Basic properties *********************************************************)
  41
  42 (* Basic_1: was: pc3_refl *)
  43 lemma cpcs_refl: ∀L. reflexive … (cpcs L).
  44 /2 width=1/ qed.
  45
  46 (* Basic_1: was: pc3_s *)
  47 lemma cpcs_sym: ∀L. symmetric … (cpcs L).
  48 /3 width=1/ qed.
  49
  50 lemma cpcs_strap1: ∀L,T1,T,T2. L ⊢ T1 ⬌* T → L ⊢ T ⬌ T2 → L ⊢ T1 ⬌* T2.
  51 /2 width=3/ qed.
  52
  53 lemma cpcs_strap2: ∀L,T1,T,T2. L ⊢ T1 ⬌ T → L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2.
  54 /2 width=3/ qed.
  55
  56 (* Basic_1: was: pc3_pr2_r *)
  57 lemma cpcs_cpr_dx: ∀L,T1,T2. L ⊢ T1 ➡ T2 → L ⊢ T1 ⬌* T2.
  58 /3 width=1/ qed.
  59
  60 lemma cpcs_tpr_dx: ∀L,T1,T2. T1 ➡ T2 → L ⊢ T1 ⬌* T2.
  61 /3 width=1/ qed.
  62
  63 (* Basic_1: was: pc3_pr2_x *)
  64 lemma cpcs_cpr_sn: ∀L,T1,T2. L ⊢ T2 ➡ T1 → L ⊢ T1 ⬌* T2.
  65 /3 width=1/ qed.
  66
  67 lemma cpcs_tpr_sn: ∀L,T1,T2. T2 ➡ T1 → L ⊢ T1 ⬌* T2.
  68 /3 width=1/ qed.
  69
  70 lemma cpcs_cpr_strap1: ∀L,T1,T. L ⊢ T1 ⬌* T → ∀T2. L ⊢ T ➡ T2 → L ⊢ T1 ⬌* T2.
  71 /3 width=3/ qed.
  72
  73 (* Basic_1: was: pc3_pr2_u *)
  74 lemma cpcs_cpr_strap2: ∀L,T1,T. L ⊢ T1 ➡ T → ∀T2. L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2.
  75 /3 width=3/ qed.
  76
  77 lemma cpcs_cpr_div: ∀L,T1,T. L ⊢ T1 ⬌* T → ∀T2. L ⊢ T2 ➡ T → L ⊢ T1 ⬌* T2.
  78 /3 width=3/ qed.
  79
  80 lemma cpr_div: ∀L,T1,T. L ⊢ T1 ➡ T → ∀T2. L ⊢ T2 ➡ T → L ⊢ T1 ⬌* T2.
  81 /3 width=3/ qed-.
  82
  83 (* Basic_1: was: pc3_pr2_u2 *)
  84 lemma cpcs_cpr_conf: ∀L,T1,T. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2.
  85 /3 width=3/ qed.
  86
  87 lemma cpcs_tpss_strap1: ∀L,T1,T. L ⊢ T1 ⬌* T →
  88                         ∀T2,d,e. L ⊢ T ▶* [d, e] T2 → L ⊢ T1 ⬌* T2.
  89 #L #T1 #T #HT1 #T2 #d #e #HT2
  90 @(cpcs_cpr_strap1 … HT1) -T1 /2 width=3/
  91 qed-.
  92
  93 lemma cpcs_tpss_conf: ∀L,T,T1,d,e. L ⊢ T ▶* [d, e] T1 →
  94                       ∀T2. L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2.
  95 #L #T #T1 #d #e #HT1 #T2 #HT2
  96 @(cpcs_cpr_conf … HT2) -T2 /2 width=3/
  97 qed-.
  98
  99 (* Basic_1: removed theorems 9:
 100             clear_pc3_trans pc3_ind_left
 101             pc3_head_1 pc3_head_2 pc3_head_12 pc3_head_21
 102             pc3_pr2_fsubst0 pc3_pr2_fsubst0_back pc3_fsubst0
 103 *)
 104 (* Basic_1: removed local theorems 6:
 105             pc3_left_pr3 pc3_left_trans pc3_left_sym pc3_left_pc3 pc3_pc3_left
 106             pc3_wcpr0_t_aux
 107 *)