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15 include "basic_2A/multiple/llpx_sn_alt.ma".
16 include "basic_2A/multiple/lleq.ma".
18 (* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
20 (* Alternative definition (not recursive) ***********************************)
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 →
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 //
32 theorem lleq_inv_alt: ∀L1,L2,T,l. L1 ≡[T, l] 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 →
37 #L1 #L2 #T #l #H elim (llpx_sn_llpx_sn_alt … H) -H
39 #I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2
40 @(IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 //