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15 include "basic_2/multiple/llpx_sn_alt_rec.ma".
16 include "basic_2/multiple/lleq.ma".
18 (* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
20 (* Alternative definition (recursive) ***************************************)
22 theorem lleq_intro_alt_r: ∀L1,L2,T,l. |L1| = |L2| →
23 (∀I1,I2,K1,K2,V1,V2,i. l ≤ 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 #l #HL12 #IH @llpx_sn_intro_alt_r // -HL12
28 #I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2
29 elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by and3_intro/
32 theorem lleq_ind_alt_r: ∀S:relation4 ynat term lenv lenv.
33 (∀L1,L2,T,l. |L1| = |L2| → (
34 ∀I1,I2,K1,K2,V1,V2,i. l ≤ 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 & S 0 V1 K1 K2
38 ∀L1,L2,T,l. L1 ≡[T, l] L2 → S l T L1 L2.
39 #S #IH1 #L1 #L2 #T #l #H @(llpx_sn_ind_alt_r … H) -L1 -L2 -T -l
40 #L1 #L2 #T #l #HL12 #IH2 @IH1 -IH1 // -HL12
41 #I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2
42 elim (IH2 … HnT HLK1 HLK2) -IH2 -HnT -HLK1 -HLK2 /2 width=1 by and4_intro/
45 theorem lleq_inv_alt_r: ∀L1,L2,T,l. L1 ≡[T, l] L2 →
47 ∀I1,I2,K1,K2,V1,V2,i. l ≤ yinj i → (∀U. ⬆[i, 1] U ≡ T → ⊥) →
48 ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 →
49 ∧∧ I1 = I2 & V1 = V2 & K1 ≡[V1, 0] K2.
50 #L1 #L2 #T #l #H elim (llpx_sn_inv_alt_r … H) -H
52 #I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2
53 elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by and3_intro/