(* *)
(**************************************************************************)
-include "basic_2/relocation/llpx_sn_alt.ma".
+include "basic_2/substitution/llpx_sn_alt1.ma".
include "basic_2/substitution/lleq.ma".
(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
theorem lleq_intro_alt: ∀L1,L2,T,d. |L1| = |L2| →
(∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
- â\88§â\88§ I1 = I2 & V1 = V2 & K1 â\8b\95[V1, 0] K2
- ) â\86\92 L1 â\8b\95[T, d] L2.
-#L1 #L2 #T #d #HL12 #IH @llpx_sn_intro_alt // -HL12
+ â\88§â\88§ I1 = I2 & V1 = V2 & K1 â\89¡[V1, 0] K2
+ ) â\86\92 L1 â\89¡[T, d] L2.
+#L1 #L2 #T #d #HL12 #IH @llpx_sn_intro_alt1 // -HL12
#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by and3_intro/
qed.
(∀L1,L2,T,d. |L1| = |L2| → (
∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
- â\88§â\88§ I1 = I2 & V1 = V2 & K1 â\8b\95[V1, 0] K2 & S 0 V1 K1 K2
+ â\88§â\88§ I1 = I2 & V1 = V2 & K1 â\89¡[V1, 0] K2 & S 0 V1 K1 K2
) → S d T L1 L2) →
- â\88\80L1,L2,T,d. L1 â\8b\95[T, d] L2 → S d T L1 L2.
-#S #IH1 #L1 #L2 #T #d #H @(llpx_sn_ind_alt … H) -L1 -L2 -T -d
+ â\88\80L1,L2,T,d. L1 â\89¡[T, d] L2 → S d T L1 L2.
+#S #IH1 #L1 #L2 #T #d #H @(llpx_sn_ind_alt1 … H) -L1 -L2 -T -d
#L1 #L2 #T #d #HL12 #IH2 @IH1 -IH1 // -HL12
#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
elim (IH2 … HnT HLK1 HLK2) -IH2 -HnT -HLK1 -HLK2 /2 width=1 by and4_intro/
qed-.
-theorem lleq_inv_alt: â\88\80L1,L2,T,d. L1 â\8b\95[T, d] L2 →
+theorem lleq_inv_alt: â\88\80L1,L2,T,d. L1 â\89¡[T, d] L2 →
|L1| = |L2| ∧
∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
- â\88§â\88§ I1 = I2 & V1 = V2 & K1 â\8b\95[V1, 0] K2.
-#L1 #L2 #T #d #H elim (llpx_sn_inv_alt … H) -H
+ â\88§â\88§ I1 = I2 & V1 = V2 & K1 â\89¡[V1, 0] K2.
+#L1 #L2 #T #d #H elim (llpx_sn_inv_alt1 … H) -H
#HL12 #IH @conj //
#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by and3_intro/