matita/matita/contribs/lambdadelta/basic_2A/etc/fpr/fpcs_cpcs.etc

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  14
  15 include "basic_2/computation/fprs_cprs.ma".
  16 include "basic_2/equivalence/cpcs_cpcs.ma".
  17 include "basic_2/equivalence/fpcs_fpcs.ma".
  18
  19 (* CONTEXT-FREE PARALLEL EQUIVALENCE ON CLOSURES ****************************)
  20
  21 (* Advanced properties ******************************************************)
  22
  23 lemma fpcs_flat_dx_tpr: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⬌* ⦃L2, T2⦄ → ∀V1,V2. V1 ➡ V2 →
  24                         ∀I. ⦃L1, ⓕ{I}V1.T1⦄ ⬌* ⦃L2, ⓕ{I}V2.T2⦄.
  25 #L1 #L2 #T1 #T2 #HT12
  26 elim (fpcs_inv_fprs … HT12) -HT12
  27 /3 width=6 by fprs_flat_dx_tpr, fprs_div/ (**) (* auto too slow without trace *)
  28 qed.
  29
  30 lemma fpcs_shift: ∀I,L1,L2,V1,V2,T1,T2. ⦃L1, -ⓑ{I}V1.T1⦄ ⬌* ⦃L2, -ⓑ{I}V2.T2⦄ →
  31                   ⦃L1.ⓑ{I}V1, T1⦄ ⬌* ⦃L2.ⓑ{I}V2, T2⦄.
  32 #I #L1 #L2 #V1 #V2 #T1 #T2 #H12
  33 elim (fpcs_inv_fprs … H12) -H12 #L #T #H1 #H2
  34 elim (fprs_bind2_minus … H1) -H1 #W1 #U1 #HTU1 #H destruct
  35 elim (fprs_bind2_minus … H2) -H2 #W2 #U2 #HTU2 #H destruct /2 width=4/
  36 qed.
  37
  38 (* Advanced inversion lemmas ************************************************)
  39
  40 lemma fpcs_inv_shift: ∀I,L1,L2,V1,V2,T1,T2. ⦃L1.ⓑ{I}V1, T1⦄ ⬌* ⦃L2.ⓑ{I}V2, T2⦄ →
  41                       ⦃L1, -ⓑ{I}V1.T1⦄ ⬌* ⦃L2, -ⓑ{I}V2.T2⦄.
  42 #I #L1 #L2 #V1 #V2 #T1 #T2 #H12
  43 elim (fpcs_inv_fprs … H12) -H12 #L #T #H1 #H2
  44 elim (fprs_inv_pair1 … H1) -H1 #K1 #U1 #_ #HTU1 #H destruct
  45 elim (fprs_inv_pair1 … H2) -H2 #K2 #U2 #_ #HTU2 #H destruct /2 width=4/
  46 qed-.
  47
  48 (* Advanced forward lemmas **************************************************)
  49
  50 lemma fpcs_fwd_bind_minus: ∀I,L1,L2,V1,V2,T1,T2. ⦃L1, -ⓑ{I}V1.T1⦄ ⬌* ⦃L2, -ⓑ{I}V2.T2⦄ →
  51                            ∀b. ⦃L1, ⓑ{b,I}V1.T1⦄ ⬌* ⦃L2, ⓑ{b,I}V2.T2⦄.
  52 #I #L1 #L2 #V1 #V2 #T1 #T2 #H12 #b
  53 elim (fpcs_inv_fprs … H12) -H12 #L #T #H1 #H2
  54 elim (fprs_fwd_bind2_minus … H1 b) -H1 #W1 #U1 #HTU1 #H destruct
  55 elim (fprs_fwd_bind2_minus … H2 b) -H2 #W2 #U2 #HTU2 #H destruct /2 width=4/
  56 qed-.
  57
  58 lemma fpcs_fwd_shift: ∀I,L1,L2,V1,V2,T1,T2. ⦃L1.ⓑ{I}V1, T1⦄ ⬌* ⦃L2.ⓑ{I}V2, T2⦄ →
  59                       ∀b. ⦃L1, ⓑ{b,I}V1.T1⦄ ⬌* ⦃L2, ⓑ{b,I}V2.T2⦄.
  60 /3 width=1 by fpcs_inv_shift, fpcs_fwd_bind_minus/ qed-.
  61
  62 lemma fpcs_fwd_abst24: ∀a,L1,L2,V1,V2,T1,T2. ⦃L1, ⓛ{a}V1.T1⦄ ⬌* ⦃L2, ⓛ{a}V2.T2⦄ →
  63                        ∀b,I,W. ⦃L1, ⓑ{b,I}W.T1⦄ ⬌* ⦃L2, ⓑ{b,I}W.T2⦄.
  64 #a #L1 #L2 #V1 #V2 #T1 #T2 #H12 #b #I #W
  65 elim (fpcs_inv_fprs … H12) -H12 #L #U #H1 #H2
  66 elim (fprs_fwd_abst2 … H1 b I W) -H1 #W1 #U1 #HTU1 #H destruct
  67 elim (fprs_fwd_abst2 … H2 b I W) -H2 #W2 #U2 #HTU2 #H destruct /2 width=4/
  68 qed-.
  69
  70 lemma fpcs_fwd_abst13: ∀L1,L2,V1,V2,T1,T2. ⦃L1.ⓛV1, T1⦄ ⬌* ⦃L2.ⓛV2, T2⦄ →
  71                        ∀I,W. ⦃L1.ⓑ{I}W, T1⦄ ⬌* ⦃L2.ⓑ{I}W, T2⦄.
  72 /4 width=4 by fpcs_fwd_shift, fpcs_fwd_abst24, fpcs_shift/ qed-.
  73
  74 (* Properties on context-sensitive parallel equivalence for terms ***********)
  75
  76 lemma cpcs_fpcs: ∀L,T1,T2. L ⊢ T1 ⬌* T2 → ⦃L, T1⦄ ⬌* ⦃L, T2⦄.
  77 #L #T1 #T2 #H
  78 elim (cpcs_inv_cprs … H) -H /3 width=4 by fprs_div, cprs_fprs/ (**) (* too slow without trace *)
  79 qed.
  80
  81 (* Inversion lemmas on context-sensitive parallel equivalence for terms *****)
  82
  83 lemma fpcs_inv_cpcs: ∀L,T1,T2. ⦃L, T1⦄ ⬌* ⦃L, T2⦄ → L ⊢ T1 ⬌* T2.
  84 #L #T1 #T2 #H
  85 elim (fpcs_inv_fprs … H) -H /3 width=4 by cprs_div, fprs_fwd_cprs/
  86 qed-.