matita/matita/contribs/lambdadelta/basic_2/etc/referred/lfsx_lfsx.etc

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  14
  15 include "basic_2/rt_transition/lfpx_lfdeq.ma".
  16 include "basic_2/rt_computation/lfsx.ma".
  17
  18 (* STRONGLY NORMALIZING LOCAL ENV.S FOR UNBOUND PARALLEL RT-TRANSITION ******)
  19
  20 (* Advanced properties ******************************************************)
  21
  22 (* Basic_2A1: uses: lsx_lleq_trans *)
  23 lemma lfsx_lfdeq_trans: ∀h,o,G,L1,T. G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄ →
  24                         ∀L2. L1 ≛[h, o, T] L2 → G ⊢ ⬈*[h, o, T] 𝐒⦃L2⦄.
  25 #h #o #G #L1 #T #H @(lfsx_ind … H) -L1
  26 #L1 #_ #IHL1 #L2 #HL12 @lfsx_intro
  27 #L #HL2 #HnL2 elim (lfdeq_lfpx_trans … HL2 … HL12) -HL2
  28 /4 width=5 by lfdeq_repl/
  29 qed-.
  30
  31 (* Basic_2A1: uses: lsx_lpx_trans *)
  32 lemma lfsx_lfpx_trans: ∀h,o,G,L1,T. G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄ →
  33                        ∀L2. ⦃G, L1⦄ ⊢ ⬈[h, T] L2 → G ⊢ ⬈*[h, o, T] 𝐒⦃L2⦄.
  34 #h #o #G #L1 #T #H @(lfsx_ind … H) -L1 #L1 #HL1 #IHL1 #L2 #HL12
  35 elim (lfdeq_dec h o L1 L2 T) /3 width=4 by lfsx_lfdeq_trans, lfxs_refl/
  36 qed-.
  37
  38 (* Advanced forward lemmas **************************************************)
  39
  40 (* Basic_2A1: uses: lsx_fwd_pair_sn lsx_fwd_bind_sn lsx_fwd_flat_sn *)
  41 lemma lfsx_fwd_pair_sn: ∀h,o,I,G,L,V,T. G ⊢ ⬈*[h, o, ②{I}V.T] 𝐒⦃L⦄ →
  42                         G ⊢ ⬈*[h, o, V] 𝐒⦃L⦄.
  43 #h #o #I #G #L #V #T #H @(lfsx_ind … H) -L
  44 #L1 #_ #IHL1 @lfsx_intro
  45 #L2 #H #HnL12 elim (lfpx_pair_sn_split … H o I T) -H
  46 /6 width=3 by lfsx_lfdeq_trans, lfdeq_trans, lfdeq_fwd_pair_sn/
  47 qed-.
  48
  49 (* Basic_2A1: uses: lsx_fwd_flat_dx *)
  50 lemma lfsx_fwd_flat_dx: ∀h,o,I,G,L,V,T. G ⊢ ⬈*[h, o, ⓕ{I}V.T] 𝐒⦃L⦄ →
  51                         G ⊢ ⬈*[h, o, T] 𝐒⦃L⦄.
  52 #h #o #I #G #L #V #T #H @(lfsx_ind … H) -L
  53 #L1 #_ #IHL1 @lfsx_intro
  54 #L2 #H #HnL12 elim (lfpx_flat_dx_split … H o I V) -H
  55 /6 width=3 by lfsx_lfdeq_trans, lfdeq_trans, lfdeq_fwd_flat_dx/
  56 qed-.
  57
  58 (* Basic_2A1: uses: lsx_fwd_bind_dx *)
  59 (* Note: the exclusion binder (ⓧ) makes this more elegant and much simpler *)
  60 lemma lfsx_fwd_bind_dx: ∀h,o,p,I,G,L,V,T. G ⊢ ⬈*[h, o, ⓑ{p,I}V.T] 𝐒⦃L⦄ →
  61                         G ⊢ ⬈*[h, o, T] 𝐒⦃L.ⓧ⦄.
  62 #h #o #p #I #G #L #V #T #H @(lfsx_ind … H) -L
  63 #L1 #_ #IH @lfsx_intro
  64 #L2 #H #HT elim (lfpx_bind_dx_split_void … H o p I V) -H
  65 /6 width=5 by lfsx_lfdeq_trans, lfdeq_trans, lfdeq_fwd_bind_dx_void/
  66 qed-.
  67
  68 (* Advanced inversion lemmas ************************************************)
  69
  70 (* Basic_2A1: uses: lsx_inv_flat *)
  71 lemma lfsx_inv_flat: ∀h,o,I,G,L,V,T. G ⊢ ⬈*[h, o, ⓕ{I}V.T] 𝐒⦃L⦄ →
  72                      G ⊢ ⬈*[h, o, V] 𝐒⦃L⦄ ∧ G ⊢ ⬈*[h, o, T] 𝐒⦃L⦄.
  73 /3 width=3 by lfsx_fwd_pair_sn, lfsx_fwd_flat_dx, conj/ qed-.
  74
  75 (* Basic_2A1: uses: lsx_inv_bind *)
  76 lemma lfsx_inv_bind: ∀h,o,p,I,G,L,V,T. G ⊢ ⬈*[h, o, ⓑ{p,I}V.T] 𝐒⦃L⦄ →
  77                      G ⊢ ⬈*[h, o, V] 𝐒⦃L⦄ ∧ G ⊢ ⬈*[h, o, T] 𝐒⦃L.ⓧ⦄.
  78 /3 width=4 by lfsx_fwd_pair_sn, lfsx_fwd_bind_dx, conj/ qed-.