matita/matita/contribs/lambdadelta/basic_2A/multiple/lleq_alt.ma

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  14
  15 include "basic_2A/multiple/llpx_sn_alt.ma".
  16 include "basic_2A/multiple/lleq.ma".
  17
  18 (* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
  19
  20 (* Alternative definition (not recursive) ***********************************)
  21
  22 theorem lleq_intro_alt: ∀L1,L2,T,l. |L1| = |L2| →
  23                         (∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → L1 ⊢ i ϵ 𝐅*[l]⦃T⦄ →
  24                            ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 →
  25                            I1 = I2 ∧ V1 = V2
  26                         ) → L1 ≡[T, l] L2.
  27 #L1 #L2 #T #l #HL12 #IH @llpx_sn_alt_inv_llpx_sn @conj // -HL12
  28 #I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2
  29 @(IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 //
  30 qed.
  31
  32 theorem lleq_inv_alt: ∀L1,L2,T,l. L1 ≡[T, l] L2 →
  33                       |L1| = |L2| ∧
  34                       ∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → L1 ⊢ i ϵ 𝐅*[l]⦃T⦄ →
  35                       ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 →
  36                       I1 = I2 ∧ V1 = V2.
  37 #L1 #L2 #T #l #H elim (llpx_sn_llpx_sn_alt … H) -H
  38 #HL12 #IH @conj //
  39 #I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2
  40 @(IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 //
  41 qed-.