matita/matita/contribs/lambdadelta/basic_2/rt_computation/lfsx_lfsx.ma

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  14
  15 include "basic_2/static/lfdeq_lfdeq.ma".
  16 include "basic_2/rt_transition/lfpx_lfdeq.ma".
  17 include "basic_2/rt_computation/lfsx.ma".
  18
  19 (* STRONGLY NORMALIZING LOCAL ENV.S FOR UNCOUNTED PARALLEL RT-TRANSITION ****)
  20
  21 (* Advanced properties ******************************************************)
  22
  23 (* Basic_2A1: uses: lsx_lleq_trans *)
  24 lemma lfsx_lfdeq_trans: ∀h,o,G,L1,T. G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄ →
  25                         ∀L2. L1 ≛[h, o, T] L2 → G ⊢ ⬈*[h, o, T] 𝐒⦃L2⦄.
  26 #h #o #G #L1 #T #H @(lfsx_ind … H) -L1
  27 #L1 #_ #IHL1 #L2 #HL12 @lfsx_intro
  28 #L #HL2 #HnL2 elim (lfdeq_lfpx_trans … HL2 … HL12) -HL2
  29 /4 width=5 by lfdeq_repl/
  30 qed-.
  31
  32 (* Basic_2A1: uses: lsx_lpx_trans *)
  33 lemma lfsx_lfpx_trans: ∀h,o,G,L1,T. G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄ →
  34                        ∀L2. ⦃G, L1⦄ ⊢ ⬈[h, T] L2 → G ⊢ ⬈*[h, o, T] 𝐒⦃L2⦄.
  35 #h #o #G #L1 #T #H @(lfsx_ind … H) -L1 #L1 #HL1 #IHL1 #L2 #HL12
  36 elim (lfdeq_dec h o L1 L2 T) /3 width=4 by lfsx_lfdeq_trans, lfxs_refl/
  37 qed-.
  38
  39 (* Advanced forward lemmas **************************************************)
  40
  41 (* Basic_2A1: uses: lsx_fwd_pair_sn lsx_fwd_bind_sn lsx_fwd_flat_sn *)
  42 lemma lfsx_fwd_pair_sn: ∀h,o,I,G,L,V,T. G ⊢ ⬈*[h, o, ②{I}V.T] 𝐒⦃L⦄ →
  43                         G ⊢ ⬈*[h, o, V] 𝐒⦃L⦄.
  44 #h #o #I #G #L #V #T #H @(lfsx_ind … H) -L
  45 #L1 #_ #IHL1 @lfsx_intro
  46 #L2 #H #HnL12 elim (lfpx_pair_sn_split … H o I T) -H
  47 /6 width=3 by lfsx_lfdeq_trans, lfdeq_trans, lfdeq_fwd_pair_sn/
  48 qed-.
  49
  50 (* Basic_2A1: uses: lsx_fwd_flat_dx *)
  51 lemma lfsx_fwd_flat_dx: ∀h,o,I,G,L,V,T. G ⊢ ⬈*[h, o, ⓕ{I}V.T] 𝐒⦃L⦄ →
  52                         G ⊢ ⬈*[h, o, T] 𝐒⦃L⦄.
  53 #h #o #I #G #L #V #T #H @(lfsx_ind … H) -L
  54 #L1 #_ #IHL1 @lfsx_intro
  55 #L2 #H #HnL12 elim (lfpx_flat_dx_split … H o I V) -H
  56 /6 width=3 by lfsx_lfdeq_trans, lfdeq_trans, lfdeq_fwd_flat_dx/
  57 qed-.
  58
  59 (* Basic_2A1: uses: lsx_fwd_bind_dx *)
  60 (* Note: the exclusion binder (ⓧ) makes this more elegant and much simpler *)
  61 lemma lfsx_fwd_bind_dx: ∀h,o,p,I,G,L,V,T. G ⊢ ⬈*[h, o, ⓑ{p,I}V.T] 𝐒⦃L⦄ →
  62                         G ⊢ ⬈*[h, o, T] 𝐒⦃L.ⓧ⦄.
  63 #h #o #p #I #G #L #V #T #H @(lfsx_ind … H) -L
  64 #L1 #_ #IH @lfsx_intro
  65 #L2 #H #HT elim (lfpx_bind_dx_split_void … H o p I V) -H
  66 /6 width=5 by lfsx_lfdeq_trans, lfdeq_trans, lfdeq_fwd_bind_dx_void/
  67 qed-.
  68
  69 (* Advanced inversion lemmas ************************************************)
  70
  71 (* Basic_2A1: uses: lsx_inv_flat *)
  72 lemma lfsx_inv_flat: ∀h,o,I,G,L,V,T. G ⊢ ⬈*[h, o, ⓕ{I}V.T] 𝐒⦃L⦄ →
  73                      G ⊢ ⬈*[h, o, V] 𝐒⦃L⦄ ∧ G ⊢ ⬈*[h, o, T] 𝐒⦃L⦄.
  74 /3 width=3 by lfsx_fwd_pair_sn, lfsx_fwd_flat_dx, conj/ qed-.
  75
  76 (* Basic_2A1: uses: lsx_inv_bind *)
  77 lemma lfsx_inv_bind: ∀h,o,p,I,G,L,V,T. G ⊢ ⬈*[h, o, ⓑ{p,I}V.T] 𝐒⦃L⦄ →
  78                      G ⊢ ⬈*[h, o, V] 𝐒⦃L⦄ ∧ G ⊢ ⬈*[h, o, T] 𝐒⦃L.ⓧ⦄.
  79 /3 width=4 by lfsx_fwd_pair_sn, lfsx_fwd_bind_dx, conj/ qed-.