matita/matita/contribs/lambda_delta/basic_2/etc/ltpsss/ltpsss.etc

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  14
  15 include "basic_2/unfold/ltpss.ma".
  16
  17 (* ITERATED PARTIAL UNFOLD ON LOCAL ENVIRONMENTS ****************************)
  18
  19 definition ltpsss: nat → nat → relation lenv ≝
  20                    λd,e. TC … (ltpss d e).
  21
  22 interpretation "repeated partial unfold (local environment)"
  23    'PSubstStars L1 d e L2 = (ltpsss d e L1 L2).
  24
  25 (* Basic eliminators ********************************************************)
  26
  27 lemma ltpsss_ind: ∀d,e,L1. ∀R:predicate lenv. R L1 →
  28                   (∀L,L2. L1 [d, e] ▶** L → L [d, e] ▶* L2 → R L → R L2) →
  29                   ∀L2. L1 [d, e] ▶** L2 → R L2.
  30 #d #e #L1 #R #HL1 #IHL1 #L2 #HL12 @(TC_star_ind … HL1 IHL1 … HL12) //
  31 qed-.
  32
  33 lemma ltpsss_ind_dx: ∀d,e,L2. ∀R:predicate lenv. R L2 →
  34                      (∀L1,L. L1 [d, e] ▶* L → L [d, e] ▶** L2 → R L → R L1) →
  35                      ∀L1. L1 [d, e] ▶** L2 → R L1.
  36 #d #e #L2 #R #HL2 #IHL2 #L1 #HL12 @(TC_star_ind_dx … HL2 IHL2 … HL12) //
  37 qed-.
  38
  39 (* Basic properties *********************************************************)
  40
  41 lemma ltpsss_strap1: ∀L1,L,L2,d,e.
  42                      L1 [d, e] ▶** L → L [d, e] ▶* L2 → L1 [d, e] ▶** L2.
  43 /2 width=3/ qed.
  44
  45 lemma ltpsss_strap2: ∀L1,L,L2,d,e.
  46                      L1 [d, e] ▶* L → L [d, e] ▶** L2 → L1 [d, e] ▶** L2.
  47 /2 width=3/ qed.
  48
  49 lemma ltpsss_refl: ∀L,d,e. L [d, e] ▶** L.
  50 /2 width=1/ qed.
  51
  52 lemma ltpsss_weak_all: ∀L1,L2,d,e. L1 [d, e] ▶** L2 → L1 [0, |L2|] ▶** L2.
  53 #L1 #L2 #d #e #H @(ltpsss_ind … H) -L2 //
  54 #L #L2 #_ #HL2
  55 >(ltpss_fwd_length … HL2) /3 width=5/
  56 qed.
  57
  58 (* Basic forward lemmas *****************************************************)
  59
  60 lemma ltpsss_fwd_length: ∀L1,L2,d,e. L1 [d, e] ▶** L2 → |L1| = |L2|.
  61 #L1 #L2 #d #e #H @(ltpsss_ind … H) -L2 //
  62 #L #L2 #_ #HL2 #IHL12 >IHL12 -IHL12
  63 /2 width=3 by ltpss_fwd_length/
  64 qed-.
  65
  66 (* Basic inversion lemmas ***************************************************)
  67
  68 lemma ltpsss_inv_refl_O2: ∀d,L1,L2. L1 [d, 0] ▶** L2 → L1 = L2.
  69 #d #L1 #L2 #H @(ltpsss_ind … H) -L2 //
  70 #L #L2 #_ #HL2 #IHL <(ltpss_inv_refl_O2 … HL2) -HL2 //
  71 qed-.
  72
  73 lemma ltpsss_inv_atom1: ∀d,e,L2. ⋆ [d, e] ▶** L2 → L2 = ⋆.
  74 #d #e #L2 #H @(ltpsss_ind … H) -L2 //
  75 #L #L2 #_ #HL2 #IHL destruct
  76 >(ltpss_inv_atom1 … HL2) -HL2 //
  77 qed-.
  78
  79 lemma ltpsss_inv_atom2: ∀d,e,L1. L1 [d, e] ▶** ⋆ → L1 = ⋆.
  80 #d #e #L1 #H @(ltpsss_ind_dx … H) -L1 //
  81 #L1 #L #HL1 #_ #IHL2 destruct
  82 >(ltpss_inv_atom2 … HL1) -HL1 //
  83 qed.