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

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  14
  15 include "basic_2/notation/relations/pconvstar_3.ma".
  16 include "basic_2/conversion/cpc.ma".
  17
  18 (* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************)
  19
  20 definition cpcs: lenv → relation term ≝ LTC … cpc.
  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 #L @TC_symmetric // qed.
  49
  50 lemma cpc_cpcs: ∀L,T1,T2. L ⊢ T1 ⬌ T2 → L ⊢ T2 ⬌* T2.
  51 /2 width=1/ qed.
  52
  53 lemma cpcs_strap1: ∀L,T1,T,T2. L ⊢ T1 ⬌* T → L ⊢ T ⬌ T2 → L ⊢ T1 ⬌* T2.
  54 #L @step qed.
  55
  56 lemma cpcs_strap2: ∀L,T1,T,T2. L ⊢ T1 ⬌ T → L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2.
  57 #L @TC_strap qed.
  58
  59 (* Basic_1: was: pc3_pr2_r *)
  60 lemma cpr_cpcs_dx: ∀L,T1,T2. L ⊢ T1 ➡ T2 → L ⊢ T1 ⬌* T2.
  61 /3 width=1/ qed.
  62
  63 (* Basic_1: was: pc3_pr2_x *)
  64 lemma cpr_cpcs_sn: ∀L,T1,T2. L ⊢ T2 ➡ T1 → L ⊢ T1 ⬌* T2.
  65 /3 width=1/ qed.
  66
  67 lemma cpcs_cpr_strap1: ∀L,T1,T. L ⊢ T1 ⬌* T → ∀T2. L ⊢ T ➡ T2 → L ⊢ T1 ⬌* T2.
  68 /3 width=3/ qed.
  69
  70 (* Basic_1: was: pc3_pr2_u *)
  71 lemma cpcs_cpr_strap2: ∀L,T1,T. L ⊢ T1 ➡ T → ∀T2. L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2.
  72 /3 width=3/ qed.
  73
  74 lemma cpcs_cpr_div: ∀L,T1,T. L ⊢ T1 ⬌* T → ∀T2. L ⊢ T2 ➡ T → L ⊢ T1 ⬌* T2.
  75 /3 width=3/ qed.
  76
  77 lemma cpr_div: ∀L,T1,T. L ⊢ T1 ➡ T → ∀T2. L ⊢ T2 ➡ T → L ⊢ T1 ⬌* T2.
  78 /3 width=3/ qed-.
  79
  80 (* Basic_1: was: pc3_pr2_u2 *)
  81 lemma cpcs_cpr_conf: ∀L,T1,T. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2.
  82 /3 width=3/ qed.
  83
  84 (* Basic_1: removed theorems 9:
  85             clear_pc3_trans pc3_ind_left
  86             pc3_head_1 pc3_head_2 pc3_head_12 pc3_head_21
  87             pc3_pr2_fsubst0 pc3_pr2_fsubst0_back pc3_fsubst0
  88             pc3_gen_abst pc3_gen_abst_shift
  89 *)
  90 (* Basic_1: removed local theorems 6:
  91             pc3_left_pr3 pc3_left_trans pc3_left_sym pc3_left_pc3 pc3_pc3_left
  92             pc3_wcpr0_t_aux
  93 *)