(* Properties with length for local environments ****************************)
lemma fsle_sort_bi: ∀L1,L2,s1,s2. |L1| = |L2| → ❪L1,⋆s1❫ ⊆ ❪L2,⋆s2❫.
-/3 width=8 by lveq_length_eq, frees_sort, sle_refl, ex4_4_intro/ qed.
+/3 width=8 by lveq_length_eq, frees_sort, pr_sle_refl, ex4_4_intro/ qed.
lemma fsle_gref_bi: ∀L1,L2,l1,l2. |L1| = |L2| → ❪L1,§l1❫ ⊆ ❪L2,§l2❫.
-/3 width=8 by lveq_length_eq, frees_gref, sle_refl, ex4_4_intro/ qed.
+/3 width=8 by lveq_length_eq, frees_gref, pr_sle_refl, ex4_4_intro/ qed.
lemma fsle_pair_bi: ∀K1,K2. |K1| = |K2| → ∀V1,V2. ❪K1,V1❫ ⊆ ❪K2,V2❫ →
∀I1,I2. ❪K1.ⓑ[I1]V1,#O❫ ⊆ ❪K2.ⓑ[I2]V2,#O❫.
* #n1 #n2 #f1 #f2 #Hf1 #Hf2 #HK12 #Hf12
#I1 #I2
elim (lveq_inj_length … HK12) // -HK #H1 #H2 destruct
-/3 width=12 by frees_pair, lveq_bind, sle_next, ex4_4_intro/
+/3 width=12 by frees_pair, lveq_bind, pr_sle_next, ex4_4_intro/
qed.
lemma fsle_unit_bi: ∀K1,K2. |K1| = |K2| →
∀I1,I2. ❪K1.ⓤ[I1],#O❫ ⊆ ❪K2.ⓤ[I2],#O❫.
-/3 width=8 by frees_unit, lveq_length_eq, sle_refl, ex4_4_intro/
+/3 width=8 by frees_unit, lveq_length_eq, pr_sle_refl, ex4_4_intro/
qed.