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15 include "basic_2A/multiple/frees.ma".
16 include "basic_2A/multiple/llpx_sn_alt_rec.ma".
18 (* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
20 (* Properties on context-sensitive free variables ***************************)
22 fact llpx_sn_frees_trans_aux: ∀R. (s_r_confluent1 … R (llpx_sn R 0)) → (frees_trans R) →
23 ∀L2,U,l,i. L2 ⊢ i ϵ 𝐅*[l]⦃U⦄ →
24 ∀L1. llpx_sn R l U L1 L2 → L1 ⊢ i ϵ 𝐅*[l]⦃U⦄.
25 #R #H1R #H2R #L2 #U #l #i #H elim H -L2 -U -l -i /3 width=2 by frees_eq/
26 #I2 #L2 #K2 #U #W2 #l #i #j #Hlj #Hji #HnU #HLK2 #_ #IHW2 #L1 #HL12
27 elim (llpx_sn_inv_alt_r … HL12) -HL12 #HL12 #IH
28 lapply (drop_fwd_length_lt2 … HLK2) #Hj
29 elim (drop_O1_lt (Ⓕ) L1 j) // -Hj -HL12 #I1 #K1 #W1 #HLK1
30 elim (IH … HnU HLK1 HLK2) // -IH -HLK2 /5 width=11 by frees_be/
33 lemma llpx_sn_frees_trans: ∀R. (s_r_confluent1 … R (llpx_sn R 0)) → (frees_trans R) →
34 ∀L1,L2,U,l. llpx_sn R l U L1 L2 →
35 ∀i. L2 ⊢ i ϵ 𝐅*[l]⦃U⦄ → L1 ⊢ i ϵ 𝐅*[l]⦃U⦄.
36 /2 width=6 by llpx_sn_frees_trans_aux/ qed-.