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

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  14
  15 include "basic_2/notation/relations/pconvstar_4.ma".
  16 include "basic_2/conversion/cpc.ma".
  17
  18 (* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************)
  19
  20 definition cpcs: relation4 genv lenv term term ≝
  21            λG. LTC … (cpc G).
  22
  23 interpretation "context-sensitive parallel equivalence (term)"
  24    'PConvStar G L T1 T2 = (cpcs G L T1 T2).
  25
  26 (* Basic eliminators ********************************************************)
  27
  28 lemma cpcs_ind: ∀G,L,T1. ∀R:predicate term. R T1 →
  29                 (∀T,T2. ⦃G, L⦄ ⊢ T1 ⬌* T → ⦃G, L⦄ ⊢ T ⬌ T2 → R T → R T2) →
  30                 ∀T2. ⦃G, L⦄ ⊢ T1 ⬌* T2 → R T2.
  31 #G #L #T1 #R #HT1 #IHT1 #T2 #HT12 @(TC_star_ind … HT1 IHT1 … HT12) //
  32 qed-.
  33
  34 lemma cpcs_ind_dx: ∀G,L,T2. ∀R:predicate term. R T2 →
  35                    (∀T1,T. ⦃G, L⦄ ⊢ T1 ⬌ T → ⦃G, L⦄ ⊢ T ⬌* T2 → R T → R T1) →
  36                    ∀T1. ⦃G, L⦄ ⊢ T1 ⬌* T2 → R T1.
  37 #G #L #T2 #R #HT2 #IHT2 #T1 #HT12
  38 @(TC_star_ind_dx … HT2 IHT2 … HT12) //
  39 qed-.
  40
  41 (* Basic properties *********************************************************)
  42
  43 (* Basic_1: was: pc3_refl *)
  44 lemma cpcs_refl: ∀G,L. reflexive … (cpcs G L).
  45 /2 width=1/ qed.
  46
  47 (* Basic_1: was: pc3_s *)
  48 lemma cpcs_sym: ∀G,L. symmetric … (cpcs G L).
  49 #G #L @TC_symmetric // qed.
  50
  51 lemma cpc_cpcs: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬌ T2 → ⦃G, L⦄ ⊢ T2 ⬌* T2.
  52 /2 width=1/ qed.
  53
  54 lemma cpcs_strap1: ∀G,L,T1,T,T2. ⦃G, L⦄ ⊢ T1 ⬌* T → ⦃G, L⦄ ⊢ T ⬌ T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
  55 #G #L @step qed.
  56
  57 lemma cpcs_strap2: ∀G,L,T1,T,T2. ⦃G, L⦄ ⊢ T1 ⬌ T → ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
  58 #G #L @TC_strap qed.
  59
  60 (* Basic_1: was: pc3_pr2_r *)
  61 lemma cpr_cpcs_dx: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
  62 /3 width=1/ qed.
  63
  64 (* Basic_1: was: pc3_pr2_x *)
  65 lemma cpr_cpcs_sn: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T2 ➡ T1 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
  66 /3 width=1/ qed.
  67
  68 lemma cpcs_cpr_strap1: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ⬌* T → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
  69 /3 width=3/ qed.
  70
  71 (* Basic_1: was: pc3_pr2_u *)
  72 lemma cpcs_cpr_strap2: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡ T → ∀T2. ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
  73 /3 width=3/ qed.
  74
  75 lemma cpcs_cpr_div: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ⬌* T → ∀T2. ⦃G, L⦄ ⊢ T2 ➡ T → ⦃G, L⦄ ⊢ T1 ⬌* T2.
  76 /3 width=3/ qed.
  77
  78 lemma cpr_div: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡ T → ∀T2. ⦃G, L⦄ ⊢ T2 ➡ T → ⦃G, L⦄ ⊢ T1 ⬌* T2.
  79 /3 width=3/ qed-.
  80
  81 (* Basic_1: was: pc3_pr2_u2 *)
  82 lemma cpcs_cpr_conf: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
  83 /3 width=3/ qed.
  84
  85 (* Basic_1: removed theorems 9:
  86             clear_pc3_trans pc3_ind_left
  87             pc3_head_1 pc3_head_2 pc3_head_12 pc3_head_21
  88             pc3_pr2_fqubst0 pc3_pr2_fqubst0_back pc3_fqubst0
  89             pc3_gen_abst pc3_gen_abst_shift
  90 *)
  91 (* Basic_1: removed local theorems 6:
  92             pc3_left_pr3 pc3_left_trans pc3_left_sym pc3_left_pc3 pc3_pc3_left
  93             pc3_wcpr0_t_aux
  94 *)