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15 include "basic_2/relocation/llpx_sn_alt.ma".
16 include "basic_2/relocation/lleq.ma".
18 (* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
20 (* Alternative definition ***************************************************)
22 theorem lleq_intro_alt: ∀L1,L2,T,d. |L1| = |L2| →
23 (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
24 ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
25 ∧∧ I1 = I2 & V1 = V2 & K1 ⋕[V1, 0] K2
27 #L1 #L2 #T #d #HL12 #IH @llpx_sn_intro_alt // -HL12
28 #I1 #I2 #K1 #K2 #HLK1 #HLK2 #i #Hid #HnT #HLK1 #HLK2
29 elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by and3_intro/
32 theorem lleq_fwd_alt: ∀L1,L2,T,d. L1 ⋕[T, d] L2 →
34 ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
35 ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
36 ∧∧ I1 = I2 & V1 = V2 & K1 ⋕[V1, 0] K2.
37 #L1 #L2 #T #d #H elim (llpx_sn_fwd_alt … H) -H
39 #I1 #I2 #K1 #K2 #HLK1 #HLK2 #i #Hid #HnT #HLK1 #HLK2
40 elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by and3_intro/