matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpx_reqg.ma

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  14
  15 include "basic_2/rt_transition/rpx_lpx.ma". (**) (* one dependence *)
  16
  17 (* EXTENDED PARALLEL RT-TRANSITION FOR FULL LOCAL ENVIRONMENTS **************)
  18
  19 (* Properties with generic equivalence for local environments ***************)
  20
  21 lemma reqg_lpx_trans_rpx (S) (G) (L) (T:term):
  22       reflexive … S → symmetric … S →
  23       ∀L1. L1 ≛[S,T] L → ∀L2. ❪G,L❫ ⊢ ⬈ L2 → ❪G,L1❫ ⊢ ⬈[T] L2.
  24 /3 width=6 by lpx_rpx, reqg_rpx_trans/ qed.
  25
  26 (* Basic_2A1: uses: lleq_lpx_trans *)
  27 lemma reqg_lpx_trans (S) (G) (T:term):
  28       reflexive … S → symmetric … S →
  29       ∀L2,K2. ❪G,L2❫ ⊢ ⬈ K2 → ∀L1. L1 ≛[S,T] L2 →
  30       ∃∃K1. ❪G,L1❫ ⊢ ⬈ K1 & K1 ≛[S,T] K2.
  31 #S #G #T #H1S #H2S #L2 #K2 #HLK2 #L1 #HL12
  32 lapply (lpx_rpx … T … HLK2) -HLK2 #HLK2
  33 lapply (reqg_rpx_trans … HL12 … HLK2) -L2 // #H
  34 elim (rpx_fwd_lpx_req … H) -H #K1 #HLK1 #HK12
  35 /3 width=3 by req_fwd_reqg, ex2_intro/
  36 qed-.
  37
  38 (* Inversion lemmas with generic equivalence for local environments *********)
  39
  40 lemma rpx_inv_reqg_lpx (S) (G) (T):
  41       reflexive … S →
  42       ∀L1,L2. ❪G,L1❫ ⊢ ⬈[T] L2 →
  43       ∃∃L. L1 ≛[S,T] L & ❪G,L❫ ⊢ ⬈ L2.
  44 #S #G #T #HS #L1 #L2 #H
  45 elim (rpx_inv_req_lpx … H) -H #L #HL1 #HL2
  46 /3 width=3 by req_fwd_reqg, ex2_intro/
  47 qed-.
  48
  49 (* Forward lemmas with generic equivalence for local environments ***********)
  50
  51 lemma rpx_fwd_lpx_reqg (S) (G) (T):
  52       reflexive … S →
  53       ∀L1,L2. ❪G,L1❫ ⊢ ⬈[T] L2 →
  54       ∃∃L. ❪G,L1❫ ⊢ ⬈ L & L ≛[S,T] L2.
  55 #S #G #T #HS #L1 #L2 #H
  56 elim (rpx_fwd_lpx_req … H) -H #L #HL1 #HL2
  57 /3 width=3 by req_fwd_reqg, ex2_intro/
  58 qed-.