matita/matita/contribs/lambdadelta/basic_2/reduction/cpx_lleq.ma

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  14
  15 include "basic_2/substitution/lleq_ext.ma".
  16 include "basic_2/reduction/cpx.ma".
  17
  18 (* CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION FOR TERMS ******************)
  19
  20 (* Properties on lazy equivalence for local environments ********************)
  21
  22 lemma lleq_cpx_trans: ∀h,g,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡[h, g] T2 →
  23                       ∀L1. L1 ⋕[T1, 0] L2 → ⦃G, L1⦄ ⊢ T1 ➡[h, g] T2.
  24 #h #g #G #L2 #T1 #T2 #H elim H -G -L2 -T1 -T2 /2 width=2 by cpx_sort/
  25 [ #I #G #L2 #K2 #V1 #V2 #W2 #i #HLK2 #_ #HVW2 #IHV12 #L1 #H elim (lleq_fwd_lref_dx … H … HLK2) -L2
  26   [ #H elim (ylt_yle_false … H) //
  27   | * /3 width=7 by cpx_delta/
  28   ]
  29 | #a #I #G #L2 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #H elim (lleq_inv_bind_O … H) -H
  30   /3 width=1 by cpx_bind/
  31 | #I #G #L2 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #H elim (lleq_inv_flat … H) -H
  32   /3 width=1 by cpx_flat/
  33 | #G #L2 #V2 #T1 #T #T2 #_ #HT2 #IHT1 #L1 #H elim (lleq_inv_bind_O … H) -H
  34   /3 width=3 by cpx_zeta/
  35 | #G #L2 #W1 #T1 #T2 #_ #IHT12 #L1 #H elim (lleq_inv_flat … H) -H
  36   /3 width=1 by cpx_tau/
  37 | #G #L2 #W1 #W2 #T1 #_ #IHW12 #L1 #H elim (lleq_inv_flat … H) -H
  38   /3 width=1 by cpx_ti/
  39 | #a #G #L1 #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #L1 #H elim (lleq_inv_flat … H) -H
  40   #HV1 #H elim (lleq_inv_bind_O … H) -H /3 width=1 by cpx_beta/
  41 | #a #G #L1 #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #L1 #H elim (lleq_inv_flat … H) -H
  42   #HV1 #H elim (lleq_inv_bind_O … H) -H /3 width=3 by cpx_theta/
  43 ]
  44 qed-.
  45
  46 (* Note: lemma 1000 *)
  47 lemma lleq_cpx_conf_dx: ∀h,g,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡[h, g] T2 →
  48                         ∀L1. L1 ⋕[T1, 0] L2 → L1 ⋕[T2, 0] L2.
  49 #h #g #G #L2 #T1 #T2 #H elim H -G -L2 -T1 -T2
  50 [ //
  51 | /3 width=3 by lleq_fwd_length, lleq_sort/
  52 | #I2 #G #L2 #K2 #V1 #V2 #W2 #i #HLK2 #_ #HVW2 #IHV12 #L1 #H elim (lleq_inv_lref_ge_dx … H … HLK2) // -H
  53   #I1 #K1 #HLK1 #HV1
  54   lapply (ldrop_fwd_drop2 … HLK1) -HLK1 #HLK1
  55   lapply (ldrop_fwd_drop2 … HLK2) -HLK2 #HLK2
  56   @(lleq_lift_le … HLK1 HLK2 … HVW2) -HLK1 -HLK2 -HVW2 /2 width=1 by/ (**) (* full auto too slow *)
  57 | #a #I #G #L2 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #H elim (lleq_inv_bind_O … H) -H
  58   /4 width=5 by lleq_bind_repl_SO, lleq_bind/
  59 | #I #G #L2 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #H elim (lleq_inv_flat … H) -H
  60   /3 width=1 by lleq_flat/
  61 | #G #L2 #V #T1 #T2 #T #_ #HT2 #IHT12 #L1 #H elim (lleq_inv_bind_O … H) -H
  62   /3 width=10 by lleq_inv_lift_le, ldrop_drop/
  63 | #G #L2 #V #T1 #T2 #_ #IHT12 #L1 #H elim (lleq_inv_flat … H) -H /2 width=1 by/
  64 | #G #L2 #V1 #V2 #T #_ #IHV12 #L1 #H elim (lleq_inv_flat … H) -H /2 width=1 by/
  65 | #a #G #L2 #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #L1 #H elim (lleq_inv_flat … H) -H
  66   #HV1 #H elim (lleq_inv_bind_O … H) -H
  67   /4 width=5 by lleq_bind_repl_SO, lleq_flat, lleq_bind/
  68 | #a #G #L2 #V1 #V2 #V #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #L1 #H elim (lleq_inv_flat … H) -H
  69   #HV1 #H elim (lleq_inv_bind_O … H) -H
  70   #HW1 #HT1 @lleq_bind_O /2 width=1 by/ @lleq_flat (**) (* full auto too slow *)
  71   [ /3 width=10 by lleq_lift_le, ldrop_drop/
  72   | /3 width=3 by lleq_bind_repl_O/
  73 ]
  74 qed-.