matita/matita/contribs/lambdadelta/basic_2/etc/fpr/cfpr_cpr.etc

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  14
  15 include "basic_2/reducibility/cpr_tpss.ma".
  16 include "basic_2/reducibility/cpr_cpr.ma".
  17 include "basic_2/reducibility/cfpr_ltpss.ma".
  18
  19 (* CONTEXT-SENSITIVE PARALLEL REDUCTION ON CLOSURES *************************)
  20
  21 (* Advanced properties ******************************************************)
  22
  23 lemma fpr_all: ∀L1,L. L1 ➡ L → ∀L2,T1,T2. L ⊢ T1 ➡ T2 →
  24                L ⊢ ▶* [0, |L|] L2 → ⦃L1, T1⦄ ➡ ⦃L2, T2⦄.
  25 #L1 #L #H elim H -L1 -L
  26 [ #L2 #T1 #T2 #HT12 #HL2
  27   lapply (ltpss_sn_inv_atom1 … HL2) -HL2 #H destruct
  28   lapply (cpr_inv_atom … HT12) -HT12 /2 width=1/
  29 | #I #L1 #L #V1 #V #_ #HV1 #IH #X #T1 #T2 #HT12 #H
  30   elim (ltpss_sn_inv_tpss21 … H ?) -H // <minus_plus_m_m #L2 #V2 #HL2 #HV2 #H destruct
  31   lapply (cpr_bind_dx false … HV1 HT12) -HV1 -HT12 #HT12
  32   lapply (cpr_tpss_trans … HT12 (-ⓑ{I}V2.T2) 0 (|L|) ?) -HT12 /2 width=1/ -HV2 /3 width=1/
  33 ]
  34 qed.
  35
  36 (* Advanced inversion lemmas ************************************************)
  37
  38 lemma cfpr_inv_all: ∀L1,L2,L0,T1,T2. L0 ⊢ ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ →
  39                     ∃∃L. L0 @@ L1 ➡ L0 @@ L & L0 @@ L ⊢ T1 ➡ T2 &
  40                          L0 @@ L ⊢ ▶* [0, |L0| + |L|] L0 @@ L2.
  41 #L1 @(lenv_ind_dx … L1) -L1
  42 [ #L2 #L0 #T1 #T2 #H
  43   elim (cfpr_inv_atom1 … H) -H #HT12 #H destruct /3 width=4/
  44 | #I #L1 #V1 #IH #X #L0 #T1 #T2 #H
  45   elim (cfpr_inv_pair1 … H) -H #L2 #V #V2 #HV1 #HV2 #HT12 #H destruct
  46   elim (IH … HT12) -IH -HT12 #L #HL1 #HT12 #HL2
  47   elim (ltpr_inv_append1 … HL1) -HL1 #X #Y #HX #HY #H
  48   lapply (ltpr_fwd_length … HX) -HX #HX
  49   elim (append_inj_dx … H ?) -H // -HX #_ #H destruct -X
  50   lapply (ltpss_sn_fwd_length … HL2) >append_length >append_length #H
  51   lapply (injective_plus_r … H) -H #H
  52   @(ex3_intro … (⋆.ⓑ{I}V@@Y)) <append_assoc // -HT12
  53   <append_assoc [ /3 width=1/ ] -HV1 -HY
  54   >append_length <associative_plus
  55   @(ltpss_sn_dx_trans_eq … HL2) -HL2 >H -H >commutative_plus /3 width=1/
  56 ]
  57 qed-.
  58
  59 lemma fpr_inv_all: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ →
  60                    ∃∃L. L1 ➡ L & L ⊢ T1 ➡ T2 & L ⊢ ▶* [0, |L|] L2.
  61 #L1 #L2 #T1 #T2 #H
  62 lapply (fpr_cfpr … H) -H #H
  63 elim (cfpr_inv_all … H) -H /2 width=4/
  64 qed-.