matita/matita/contribs/lambdadelta/basic_2/substitution/lleq_llor.ma

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  14
  15 include "basic_2/substitution/llor.ma".
  16 include "basic_2/substitution/lleq_alt.ma".
  17
  18 (* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
  19
  20 (* Properties on poinwise union for local environments **********************)
  21
  22 lemma llpx_sn_llor_dx: ∀R,L1,L2.
  23                        (∀U,i. L2 ⊢ i ϵ 𝐅*[0]⦃U⦄ → L1 ⊢ i ϵ 𝐅*[0]⦃U⦄) →
  24                        ∀T. llpx_sn R 0 T L1 L2 → ∀L. L1 ⩖[T] L2 ≡ L → L2 ≡[T, 0] L.
  25 #R #L1 #L2 #HR #T #H1 #L #H2
  26 elim (llpx_sn_llpx_sn_alt … H1) -H1 #HL12 #IH1
  27 elim H2 -H2 #_ #HL1 #IH2
  28 @lleq_intro_alt // #I2 #I #K2 #K #V2 #V #i #Hi #HnT #HLK2 #HLK
  29 lapply (ldrop_fwd_length_lt2 … HLK) #HiL
  30 elim (ldrop_O1_lt (Ⓕ) L1 i) // -HiL #I1 #K1 #V1 #HLK1
  31 elim (IH1 … HLK1 HLK2) -IH1 /2 width=1 by/ #H #_ destruct
  32 elim (IH2 … HLK1 HLK2 HLK) -IH2 -HLK1 -HLK2 -HLK * /2 width=1 by conj/ #H
  33 elim H -H /2 width=1 by/
  34 qed.