matita/matita/contribs/lambda_delta/basic_2/unfold/ltpss.ma

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  14
  15 include "basic_2/substitution/ltps.ma".
  16 include "basic_2/unfold/tpss.ma".
  17
  18 (* PARTIAL UNFOLD ON LOCAL ENVIRONMENTS *************************************)
  19
  20 definition ltpss: nat → nat → relation lenv ≝
  21                   λd,e. TC … (ltps d e).
  22
  23 interpretation "partial unfold (local environment)"
  24    'PSubstStar L1 d e L2 = (ltpss d e L1 L2).
  25
  26 (* Basic eliminators ********************************************************)
  27
  28 lemma ltpss_ind: ∀d,e,L1. ∀R:predicate lenv. R L1 →
  29                  (∀L,L2. L1 [d, e] ▶* L → L [d, e] ▶ L2 → R L → R L2) →
  30                  ∀L2. L1 [d, e] ▶* L2 → R L2.
  31 #d #e #L1 #R #HL1 #IHL1 #L2 #HL12 @(TC_star_ind … HL1 IHL1 … HL12) //
  32 qed-.
  33
  34 lemma ltpss_ind_dx: ∀d,e,L2. ∀R:predicate lenv. R L2 →
  35                     (∀L1,L. L1 [d, e] ▶ L → L [d, e] ▶* L2 → R L → R L1) →
  36                     ∀L1. L1 [d, e] ▶* L2 → R L1.
  37 #d #e #L2 #R #HL2 #IHL2 #L1 #HL12 @(TC_star_ind_dx … HL2 IHL2 … HL12) //
  38 qed-.
  39
  40 (* Basic properties *********************************************************)
  41
  42 lemma ltpss_strap: ∀L1,L,L2,d,e.
  43                    L1 [d, e] ▶ L → L [d, e] ▶* L2 → L1 [d, e] ▶* L2.
  44 /2 width=3/ qed.
  45
  46 lemma ltpss_refl: ∀L,d,e. L [d, e] ▶* L.
  47 /2 width=1/ qed.
  48
  49 lemma ltpss_weak_all: ∀L1,L2,d,e. L1 [d, e] ▶* L2 → L1 [0, |L2|] ▶* L2.
  50 #L1 #L2 #d #e #H @(ltpss_ind … H) -L2 //
  51 #L #L2 #_ #HL2
  52 >(ltps_fwd_length … HL2) /3 width=5/
  53 qed.
  54
  55 (* Basic forward lemmas *****************************************************)
  56
  57 lemma ltpss_fwd_length: ∀L1,L2,d,e. L1 [d, e] ▶* L2 → |L1| = |L2|.
  58 #L1 #L2 #d #e #H @(ltpss_ind … H) -L2 //
  59 #L #L2 #_ #HL2 #IHL12 >IHL12 -IHL12
  60 /2 width=3 by ltps_fwd_length/
  61 qed-.
  62
  63 (* Basic inversion lemmas ***************************************************)
  64
  65 lemma ltpss_inv_refl_O2: ∀d,L1,L2. L1 [d, 0] ▶* L2 → L1 = L2.
  66 #d #L1 #L2 #H @(ltpss_ind … H) -L2 //
  67 #L #L2 #_ #HL2 #IHL <(ltps_inv_refl_O2 … HL2) -HL2 //
  68 qed-.
  69
  70 lemma ltpss_inv_atom1: ∀d,e,L2. ⋆ [d, e] ▶* L2 → L2 = ⋆.
  71 #d #e #L2 #H @(ltpss_ind … H) -L2 //
  72 #L #L2 #_ #HL2 #IHL destruct
  73 >(ltps_inv_atom1 … HL2) -HL2 //
  74 qed-.
  75
  76 fact ltpss_inv_atom2_aux: ∀d,e,L1,L2. L1 [d, e] ▶* L2 → L2 = ⋆ → L1 = ⋆.
  77 #d #e #L1 #L2 #H @(ltpss_ind … H) -L2 //
  78 #L2 #L #_ #HL2 #IHL2 #H destruct
  79 lapply (ltps_inv_atom2 … HL2) -HL2 /2 width=1/
  80 qed.
  81
  82 lemma ltpss_inv_atom2: ∀d,e,L1. L1 [d, e] ▶* ⋆ → L1 = ⋆.
  83 /2 width=5/ qed-.
  84 (*
  85 fact ltps_inv_tps22_aux: ∀d,e,L1,L2. L1 [d, e] ▶ L2 → d = 0 → 0 < e →
  86                          ∀K2,I,V2. L2 = K2. ⓑ{I} V2 →
  87                          ∃∃K1,V1. K1 [0, e - 1] ▶ K2 &
  88                                   K2 ⊢ V1 [0, e - 1] ▶ V2 &
  89                                   L1 = K1. ⓑ{I} V1.
  90 #d #e #L1 #L2 * -d e L1 L2
  91 [ #d #e #_ #_ #K1 #I #V1 #H destruct
  92 | #L1 #I #V #_ #H elim (lt_refl_false … H)
  93 | #L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #_ #_ #K2 #J #W2 #H destruct /2 width=5/
  94 | #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H elim (plus_S_eq_O_false … H)
  95 ]
  96 qed.
  97
  98 lemma ltps_inv_tps22: ∀e,L1,K2,I,V2. L1 [0, e] ▶ K2. ⓑ{I} V2 → 0 < e →
  99                       ∃∃K1,V1. K1 [0, e - 1] ▶ K2 & K2 ⊢ V1 [0, e - 1] ▶ V2 &
 100                                L1 = K1. ⓑ{I} V1.
 101 /2 width=5/ qed.
 102
 103 fact ltps_inv_tps12_aux: ∀d,e,L1,L2. L1 [d, e] ▶ L2 → 0 < d →
 104                          ∀I,K2,V2. L2 = K2. ⓑ{I} V2 →
 105                          ∃∃K1,V1. K1 [d - 1, e] ▶ K2 &
 106                                   K2 ⊢ V1 [d - 1, e] ▶ V2 &
 107                                   L1 = K1. ⓑ{I} V1.
 108 #d #e #L1 #L2 * -d e L1 L2
 109 [ #d #e #_ #I #K2 #V2 #H destruct
 110 | #L #I #V #H elim (lt_refl_false … H)
 111 | #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false … H)
 112 | #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K2 #W2 #H destruct /2 width=5/
 113 ]
 114 qed.
 115
 116 lemma ltps_inv_tps12: ∀L1,K2,I,V2,d,e. L1 [d, e] ▶ K2. ⓑ{I} V2 → 0 < d →
 117                       ∃∃K1,V1. K1 [d - 1, e] ▶ K2 &
 118                                   K2 ⊢ V1 [d - 1, e] ▶ V2 &
 119                                   L1 = K1. ⓑ{I} V1.
 120 /2 width=1/ qed.
 121 *)