matita/matita/contribs/lambdadelta/basic_2/etc/lfpx/lfpx_lreq.etc

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  14
  15 include "basic_2/multiple/llor_drop.ma".
  16 include "basic_2/multiple/llpx_sn_llor.ma".
  17 include "basic_2/multiple/llpx_sn_lpx_sn.ma".
  18 include "basic_2/multiple/lleq_lreq.ma".
  19 include "basic_2/multiple/lleq_llor.ma".
  20 include "basic_2/reduction/cpx_lreq.ma".
  21 include "basic_2/reduction/cpx_lleq.ma".
  22 include "basic_2/reduction/lpx_frees.ma".
  23
  24 (* SN EXTENDED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ********************)
  25
  26
  27 fact lreq_lpx_trans_lleq_aux: ∀h,o,G,L1,L0,l,k. L1 ⩬[l, k] L0 → k = ∞ →
  28                               ∀L2. ⦃G, L0⦄ ⊢ ➡[h, o] L2 →
  29                               ∃∃L. L ⩬[l, k] L2 & ⦃G, L1⦄ ⊢ ➡[h, o] L &
  30                                    (∀T. L0 ≡[T, l] L2 ↔ L1 ≡[T, l] L).
  31 #h #o #G #L1 #L0 #l #k #H elim H -L1 -L0 -l -k
  32 [ #l #k #_ #L2 #H >(lpx_inv_atom1 … H) -H
  33   /3 width=5 by ex3_intro, conj/
  34 | #I1 #I0 #L1 #L0 #V1 #V0 #_ #_ #Hm destruct
  35 | #I #L1 #L0 #V1 #k #HL10 #IHL10 #Hm #Y #H
  36   elim (lpx_inv_pair1 … H) -H #L2 #V2 #HL02 #HV02 #H destruct
  37   lapply (ysucc_inv_Y_dx … Hm) -Hm #Hm
  38   elim (IHL10 … HL02) // -IHL10 -HL02 #L #HL2 #HL1 #IH
  39   @(ex3_intro … (L.ⓑ{I}V2)) /3 width=3 by lpx_pair, lreq_cpx_trans, lreq_pair/
  40   #T elim (IH T) #HL0dx #HL0sn
  41   @conj #H @(lleq_lreq_repl … H) -H /3 width=1 by lreq_sym, lreq_pair_O_Y/
  42 | #I1 #I0 #L1 #L0 #V1 #V0 #l #k #HL10 #IHL10 #Hm #Y #H
  43   elim (lpx_inv_pair1 … H) -H #L2 #V2 #HL02 #HV02 #H destruct
  44   elim (IHL10 … HL02) // -IHL10 -HL02 #L #HL2 #HL1 #IH
  45   @(ex3_intro … (L.ⓑ{I1}V1)) /3 width=1 by lpx_pair, lreq_succ/
  46   #T elim (IH T) #HL0dx #HL0sn
  47   @conj #H @(lleq_lreq_repl … H) -H /3 width=1 by lreq_sym, lreq_succ/
  48 ]
  49 qed-.
  50
  51 lemma lreq_lpx_trans_lleq: ∀h,o,G,L1,L0,l. L1 ⩬[l, ∞] L0 →
  52                            ∀L2. ⦃G, L0⦄ ⊢ ➡[h, o] L2 →
  53                            ∃∃L. L ⩬[l, ∞] L2 & ⦃G, L1⦄ ⊢ ➡[h, o] L &
  54                                 (∀T. L0 ≡[T, l] L2 ↔ L1 ≡[T, l] L).
  55 /2 width=1 by lreq_lpx_trans_lleq_aux/ qed-.