matita/matita/contribs/lambdadelta/basic_2/computation/fpbu_fleq.ma

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  14
  15 include "basic_2/multiple/fleq.ma".
  16 include "basic_2/computation/fpbs_alt.ma".
  17 include "basic_2/computation/fpbu_lleq.ma".
  18
  19 (* UNITARY "BIG TREE" PROPER PARALLEL COMPUTATION FOR CLOSURES **************)
  20
  21 (* Properties on lazy equivalence for closures ******************************)
  22
  23 lemma fleq_fpbu_trans: ∀h,g,F1,F2,K1,K2,T1,T2. ⦃F1, K1, T1⦄ ≡[0] ⦃F2, K2, T2⦄ →
  24                        ∀G2,L2,U2. ⦃F2, K2, T2⦄ ≻[h, g] ⦃G2, L2, U2⦄ →
  25                        ∃∃G1,L1,U1. ⦃F1, K1, T1⦄ ≻[h, g] ⦃G1, L1, U1⦄ & ⦃G1, L1, U1⦄ ≡[0] ⦃G2, L2, U2⦄.
  26 #h #g #F1 #F2 #K1 #K2 #T1 #T2 * -F2 -K2 -T2
  27 #K2 #HK12 #G2 #L2 #U2 #H12 elim (lleq_fpbu_trans … HK12 … H12) -K2
  28 /3 width=5 by fleq_intro, ex2_3_intro/
  29 qed-.
  30
  31 lemma fpb_fpbu: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
  32                 ⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ ∨
  33                 ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄.
  34 #h #g #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
  35 [ #G2 #L2 #T2 #H elim (fquq_inv_gen … H) -H
  36   [ /4 width=1 by fpbu_fqup, fqu_fqup, or_intror/
  37   | * #H1 #H2 #H3 destruct /2 width=1 by or_introl/
  38   ]
  39 | #T2 #HT12 elim (eq_term_dec T1 T2)
  40   #HnT12 destruct /4 width=1 by fpbu_cpxs, cpx_cpxs, or_intror, or_introl/
  41 | #L2 #HL12 elim (lleq_dec … T1 L1 L2 0)
  42   /4 width=3 by fpbu_lpxs, fleq_intro, lpx_lpxs, or_intror, or_introl/
  43 | /3 width=1 by fleq_intro, or_introl/
  44 ]
  45 qed-.
  46
  47 lemma fpbs_fpbu_sn: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
  48                     ⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ ∨
  49                     ∃∃G,L,T. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G, L, T⦄ & ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄.
  50 (* ALTERNATIVE PROOF
  51 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind_dx … H) -G1 -L1 -T1
  52 [ /2 width=1 by or_introl/
  53 | #G1 #G #L1 #L #T1 #T #H1 #_ * [ #H2 | * #G0 #L0 #T0 #H0 #H02 ]
  54   elim (fpb_fpbu … H1) -H1 #H1
  55   [ /3 width=1 by
  56 *)
  57 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim(fpbs_inv_alt … H) -H
  58 #L0 #L #T #HT1 #HT2 #HL0 #HL2 elim (eq_term_dec T1 T) #H destruct
  59 [ -HT1 elim (fqus_inv_gen … HT2) -HT2
  60   [ /4 width=11 by fpbs_intro_alt, fpbu_fqup, ex2_3_intro, or_intror/
  61   | * #HG #HL #HT destruct elim (lleq_dec T2 L0 L 0) #H
  62     [ /4 width=3 by fleq_intro, lleq_trans, or_introl/
  63     | /5 width=5 by fpbu_lpxs, lleq_fpbs, ex2_3_intro, or_intror/
  64     ]
  65   ]
  66 | elim (cpxs_neq_inv_step_sn … HT1 H) -HT1 -H
  67   /5 width=11 by fpbs_intro_alt, fpbu_cpxs, cpx_cpxs, ex2_3_intro, or_intror/
  68 ]
  69 qed-.