matita/matita/contribs/lambdadelta/basic_2A/computation/fpbg_fpbs.ma

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  14
  15 include "basic_2A/computation/lpxs_lleq.ma".
  16 include "basic_2A/computation/fpbs_lift.ma".
  17 include "basic_2A/computation/fpbg_fleq.ma".
  18
  19 (* "QRST" PROPER PARALLEL COMPUTATION FOR CLOSURES **************************)
  20
  21 (* Properties on "qrst" parallel reduction on closures **********************)
  22
  23 lemma fpb_fpbg_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2.
  24                       ⦃G1, L1, T1⦄ ≻[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ >≡[h, g] ⦃G2, L2, T2⦄ →
  25                       ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
  26 /3 width=5 by fpbg_fwd_fpbs, ex2_3_intro/ qed-.
  27
  28 lemma fpbq_fpbg_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2.
  29                        ⦃G1, L1, T1⦄ ≽[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ >≡[h, g] ⦃G2, L2, T2⦄ →
  30                        ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
  31 #h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 @(fpbq_ind_alt … H1) -H1
  32 /2 width=5 by fleq_fpbg_trans, fpb_fpbg_trans/
  33 qed-.
  34
  35 (* Properties on "qrst" parallel compuutation on closures *******************)
  36
  37 lemma fpbs_fpbg_trans: ∀h,g,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ →
  38                        ∀G2,L2,T2. ⦃G, L, T⦄ >≡[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
  39 #h #g #G1 #G #L1 #L #T1 #T #H @(fpbs_ind … H) -G -L -T /3 width=5 by fpbq_fpbg_trans/
  40 qed-.
  41
  42 (* Note: this is used in the closure proof *)
  43 lemma fpbg_fpbs_trans: ∀h,g,G,G2,L,L2,T,T2. ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
  44                        ∀G1,L1,T1. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
  45 #h #g #G #G2 #L #L2 #T #T2 #H @(fpbs_ind_dx … H) -G -L -T /3 width=5 by fpbg_fpbq_trans/
  46 qed-.
  47
  48 (* Note: this is used in the closure proof *)
  49 lemma fqup_fpbg: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
  50 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim (fqup_inv_step_sn … H) -H
  51 /3 width=5 by fqus_fpbs, fpb_fqu, ex2_3_intro/
  52 qed.
  53
  54 lemma cpxs_fpbg: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 →
  55                  (T1 = T2 → ⊥) → ⦃G, L, T1⦄ >≡[h, g] ⦃G, L, T2⦄.
  56 #h #g #G #L #T1 #T2 #H #H0 elim (cpxs_neq_inv_step_sn … H … H0) -H -H0
  57 /4 width=5 by cpxs_fpbs, fpb_cpx, ex2_3_intro/
  58 qed.
  59
  60 lemma lstas_fpbg: ∀h,g,G,L,T1,T2,d2. ⦃G, L⦄ ⊢ T1 •*[h, d2] T2 → (T1 = T2 → ⊥) →
  61                   ∀d1. d2 ≤ d1 → ⦃G, L⦄ ⊢ T1 ▪[h, g] d1 → ⦃G, L, T1⦄ >≡[h, g] ⦃G, L, T2⦄.
  62 /3 width=5 by lstas_cpxs, cpxs_fpbg/ qed.
  63
  64 lemma lpxs_fpbg: ∀h,g,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
  65                  (L1 ≡[T, 0] L2 → ⊥) → ⦃G, L1, T⦄ >≡[h, g] ⦃G, L2, T⦄.
  66 #h #g #G #L1 #L2 #T #H #H0 elim (lpxs_nlleq_inv_step_sn … H … H0) -H -H0
  67 /4 width=5 by fpb_lpx, lpxs_lleq_fpbs, ex2_3_intro/
  68 qed.