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

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  14
  15 include "basic_2/relocation/ldrop_leq.ma".
  16 include "basic_2/relocation/lleq_lleq.ma".
  17 include "basic_2/reduction/cpx.ma".
  18
  19 (* CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION FOR TERMS ******************)
  20
  21 (* Properties on lazy equivalence for local environments ********************)
  22
  23 lemma lleq_cpx_trans_leq: ∀h,g,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡[h, g] T2 →
  24                           ∀L1,d. L1 ⋕[d, T1] L2 → L1 ≃[d, ∞] L2 → ⦃G, L1⦄ ⊢ T1 ➡[h, g] T2.
  25 #h #g #G #L2 #T1 #T2 #H elim H -G -L2 -T1 -T2 /2 width=2 by cpx_sort/
  26 [ #I #G #L2 #K2 #V1 #V2 #W2 #i #HLK2 #_ #HVW2 #IHV12 #L1 #d #H #HL12 elim (lleq_inv_lref_dx … H … HLK2) -H *
  27   [ #K1 #HLK1 #HV1 #Hdi elim (ldrop_leq_conf_be … HL12 … HLK1) -HL12 /2 width=1 by yle_inj/
  28     >yminus_Y_inj #J #Y #X #HK12 #H lapply (ldrop_mono … H … HLK2) -L2
  29     #H destruct /3 width=7 by cpx_delta/
  30   | #J #K1 #V #HLK1 #_ #HV1 #Hid elim (ldrop_leq_conf_lt … HL12 … HLK1) -HL12 /2 width=1 by ylt_inj/
  31     <yminus_SO2 >yminus_inj #Y #HK12 #H lapply (ldrop_mono … H … HLK2) -L2
  32     #H destruct /3 width=7 by cpx_delta/
  33   ]
  34 | #a #I #G #L2 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #d #H elim (lleq_inv_bind … H) -H
  35   /4 width=3 by cpx_bind, leq_succ/
  36 | #I #G #L2 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #d #H elim (lleq_inv_flat … H) -H
  37   /3 width=3 by cpx_flat/
  38 | #G #L2 #V2 #T1 #T #T2 #_ #HT2 #IHT1 #L1 #d #H elim (lleq_inv_bind … H) -H
  39   /4 width=3 by cpx_zeta, leq_succ/
  40 | #G #L2 #W1 #T1 #T2 #_ #IHT12 #L1 #d #H elim (lleq_inv_flat … H) -H
  41   /3 width=3 by cpx_tau/
  42 | #G #L2 #W1 #W2 #T1 #_ #IHW12 #L1 #d #H elim (lleq_inv_flat … H) -H
  43   /3 width=3 by cpx_ti/
  44 | #a #G #L1 #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #L1 #d #H elim (lleq_inv_flat … H) -H
  45   #HV1 #H elim (lleq_inv_bind … H) -H /4 width=3 by cpx_beta, leq_succ/
  46 | #a #G #L1 #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #L1 #d #H elim (lleq_inv_flat … H) -H
  47   #HV1 #H elim (lleq_inv_bind … H) -H /4 width=3 by cpx_theta, leq_succ/
  48 ]
  49 qed-.
  50
  51 (* Note: this can be proved directly *)
  52 lemma lleq_cpx_trans: ∀h,g,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡[h, g] T2 →
  53                       ∀L1. L1 ⋕[0, T1] L2 → ⦃G, L1⦄ ⊢ T1 ➡[h, g] T2.
  54 /4 width=6 by lleq_cpx_trans_leq, lleq_fwd_length, leq_O_Y/ qed-.