matita/matita/contribs/lambda_delta/basic_2/equivalence/lcpcs.ma

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  14
  15 include "basic_2/conversion/lcpc.ma".
  16
  17 (* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON LOCAL EBVIRONMENTS *************)
  18
  19 definition lcpcs: relation lenv ≝ TC … lcpc.
  20
  21 interpretation "context-sensitive parallel equivalence (local environment)"
  22    'CPConvStar L1 L2 = (lcpcs L1 L2).
  23
  24 (* Basic eliminators ********************************************************)
  25
  26 lemma lcpcs_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 lcpcs_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 lcpcs_refl: ∀L. L ⊢ ⬌* L.
  42 /2 width=1/ qed.
  43
  44 lemma lcpcs_strap1: ∀L1,L,L2. L1 ⊢ ⬌* L → L ⊢ ⬌ L2 → L1 ⊢ ⬌* L2.
  45 /2 width=3/ qed.
  46
  47 lemma lcpcs_strap2: ∀L1,L,L2. L1 ⊢ ⬌ L → L ⊢ ⬌* L2 → L1 ⊢ ⬌* L2.
  48 /2 width=3/ qed.
  49
  50 lemma lcpcs_lcpr_dx: ∀L1,L2. L1 ⊢ ➡ L2 → L1 ⊢ ⬌* L2.
  51 /3 width=1/ qed.
  52
  53 lemma lcpcs_lcpr_sn: ∀L1,L2. L2 ⊢ ➡ L1 → L1 ⊢ ⬌* L2.
  54 /3 width=1/ qed.
  55
  56 lemma lcpcs_lcpr_strap1: ∀L1,L. L1 ⊢ ⬌* L → ∀L2. L ⊢ ➡ L2 → L1 ⊢ ⬌* L2.
  57 /3 width=3/ qed.
  58
  59 lemma lcpcs_lcpr_strap2: ∀L1,L. L1 ⊢ ➡ L → ∀L2. L ⊢ ⬌* L2 → L1 ⊢ ⬌* L2.
  60 /3 width=3/ qed.
  61
  62 lemma lcpcs_lcpr_div: ∀L1,L. L1 ⊢ ⬌* L → ∀L2. L2 ⊢ ➡ L → L1 ⊢ ⬌* L2.
  63 /3 width=3/ qed.
  64
  65 lemma lcpcs_lcpr_conf: ∀L1,L. L ⊢ ➡ L1 → ∀L2. L ⊢ ⬌* L2 → L1 ⊢ ⬌* L2.
  66 /3 width=3/ qed.
  67
  68 lemma lcprs_comm: ∀L1,L2. L1 ⊢ ⬌* L2 → L2 ⊢ ⬌* L1.
  69 #L1 #L2 #H @(lcpcs_ind … H) -L2 // /3 width=3/
  70 qed.