matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ltpr_tps.etc

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  14
  15 include "basic_2/reducibility/ltpr_ldrop.ma".
  16
  17 (* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
  18
  19 (* Properties concerning parallel substitution on terms *********************)
  20
  21 (* Basic_1: was: pr0_subst1_fwd *)
  22 lemma ltpr_tps_conf: ∀L1,T1,T2,d,e. L1 ⊢ T1 ▶ [d, e] T2 → ∀L2. L1 ➡ L2 →
  23                      ∃∃T. L2 ⊢ T1 ▶ [d, e] T & T2 ➡ T.
  24 #L1 #T1 #T2 #d #e #H elim H -L1 -T1 -T2 -d -e
  25 [ /2 width=3/
  26 | #L1 #K1 #V1 #W1 #i #d #e #Hdi #Hide #HLK1 #HVW1 #L2 #HL12
  27   elim (ltpr_ldrop_conf … HLK1 … HL12) -L1 #X #H #HLK2
  28   elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct -K1
  29   elim (lift_total V2 0 (i+1)) #W2 #HVW2
  30   lapply (tpr_lift … HV12 … HVW1 … HVW2) -V1 /3 width=4/
  31 | #L1 #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L2 #HL12
  32   elim (IHV12 … HL12) -IHV12 #V #HV1 #HV2
  33   elim (IHT12 (L2.ⓑ{I}V) ?) /2 width=1/ -L1 /3 width=5/
  34 | #L1 #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L2 #HL12
  35   elim (IHV12 … HL12) -IHV12
  36   elim (IHT12 … HL12) -L1 /3 width=5/
  37 ]
  38 qed-.
  39
  40 (* Basic_1: was: pr0_subst1_back *)
  41 lemma ltpr_tps_trans: ∀L2,T1,T2,d,e. L2 ⊢ T1 ▶ [d, e] T2 → ∀L1. L1 ➡ L2 →
  42                       ∃∃T. L1 ⊢ T1 ▶ [d, e] T & T ➡ T2.
  43 #L2 #T1 #T2 #d #e #H elim H -L2 -T1 -T2 -d -e
  44 [ /2 width=3/
  45 | #L2 #K2 #V2 #W2 #i #d #e #Hdi #Hide #HLK2 #HVW2 #L1 #HL12
  46   elim (ltpr_ldrop_trans_O1 … HL12 … HLK2) -L2 #X #HLK1 #H
  47   elim (ltpr_inv_pair2 … H) -H #K1 #V1 #HK12 #HV12 #H destruct -K2
  48   elim (lift_total V1 0 (i+1)) #W1 #HVW1
  49   lapply (tpr_lift … HV12 … HVW1 … HVW2) -V2 /3 width=4/
  50 | #L2 #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L1 #HL12
  51   elim (IHV12 … HL12) -IHV12 #V #HV1 #HV2
  52   elim (IHT12 (L1.ⓑ{I}V) ?) /2 width=1/ -L2 /3 width=5/
  53 | #L2 #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L1 #HL12
  54   elim (IHV12 … HL12) -IHV12
  55   elim (IHT12 … HL12) -L2 /3 width=5/
  56 ]
  57 qed-.