(* Properties with length for local environments ****************************)
-lemma fsle_sort_bi: ∀L1,L2,s1,s2. |L1| = |L2| → ⦃L1, ⋆s1⦄ ⊆ ⦃L2, ⋆s2⦄.
+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.
-lemma fsle_gref_bi: ∀L1,L2,l1,l2. |L1| = |L2| → ⦃L1, §l1⦄ ⊆ ⦃L2, §l2⦄.
+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.
-lemma fsle_pair_bi: ∀K1,K2. |K1| = |K2| → ∀V1,V2. ⦃K1, V1⦄ ⊆ ⦃K2, V2⦄ →
- ∀I1,I2. ⦃K1.ⓑ{I1}V1, #O⦄ ⊆ ⦃K2.ⓑ{I2}V2, #O⦄.
+lemma fsle_pair_bi: ∀K1,K2. |K1| = |K2| → ∀V1,V2. ⦃K1,V1⦄ ⊆ ⦃K2,V2⦄ →
+ ∀I1,I2. ⦃K1.ⓑ{I1}V1,#O⦄ ⊆ ⦃K2.ⓑ{I2}V2,#O⦄.
#K1 #K2 #HK #V1 #V2
* #n1 #n2 #f1 #f2 #Hf1 #Hf2 #HK12 #Hf12
#I1 #I2
qed.
lemma fsle_unit_bi: ∀K1,K2. |K1| = |K2| →
- ∀I1,I2. ⦃K1.ⓤ{I1}, #O⦄ ⊆ ⦃K2.ⓤ{I2}, #O⦄.
+ ∀I1,I2. ⦃K1.ⓤ{I1},#O⦄ ⊆ ⦃K2.ⓤ{I2},#O⦄.
/3 width=8 by frees_unit, lveq_length_eq, sle_refl, ex4_4_intro/
qed.