+/2 width=1 by mr/ qed.
+
+lemma veq_repl (M): is_model M →
+ replace_2 … (veq M) (veq M) (veq M).
+/2 width=5 by mq/ qed-.
+
+lemma veq_sym (M): is_model M → symmetric … (veq M).
+/3 width=5 by veq_repl, veq_refl/ qed-.
+
+lemma veq_trans (M): is_model M → Transitive … (veq M).
+/3 width=5 by veq_repl, veq_refl/ qed-.
+
+lemma veq_canc_sn (M): is_model M → left_cancellable … (veq M).
+/3 width=3 by veq_trans, veq_sym/ qed-.
+
+lemma veq_canc_dx (M): is_model M → right_cancellable … (veq M).
+/3 width=3 by veq_trans, veq_sym/ qed-.
+
+(* Properties with evaluation lift ******************************************)
+
+theorem vlift_swap (M): is_model M →
+ ∀i1,i2. i1 ≤ i2 →
+ ∀lv,d1,d2. ⫯[i1←d1] ⫯[i2←d2] lv ≗{M} ⫯[↑i2←d2] ⫯[i1←d1] lv.
+#M #HM #i1 #i2 #Hi12 #lv #d1 #d2 #j
+elim (lt_or_eq_or_gt j i1) #Hji1 destruct
+[ lapply (lt_to_le_to_lt … Hji1 Hi12) #Hji2
+ >vlift_lt // >vlift_lt // >vlift_lt /2 width=1 by lt_S/ >vlift_lt //
+ /2 width=1 by veq_refl/
+| >vlift_eq >vlift_lt /2 width=1 by monotonic_le_plus_l/ >vlift_eq
+ /2 width=1 by mr/
+| >vlift_gt // elim (lt_or_eq_or_gt (↓j) i2) #Hji2 destruct
+ [ >vlift_lt // >vlift_lt /2 width=1 by lt_minus_to_plus/ >vlift_gt //
+ /2 width=1 by veq_refl/
+ | >vlift_eq <(lt_succ_pred … Hji1) >vlift_eq
+ /2 width=1 by mr/
+ | lapply (le_to_lt_to_lt … Hi12 Hji2) #Hi1j
+ >vlift_gt // >vlift_gt /2 width=1 by lt_minus_to_plus_r/ >vlift_gt //
+ /2 width=1 by veq_refl/
+ ]
+]
+qed.
+
+lemma vlift_comp (M): is_model M →
+ ∀i. compatible_3 … (vlift M i) (sq M) (veq M) (veq M).
+#M #HM #i #d1 #d2 #Hd12 #lv1 #lv2 #HLv12 #j
+elim (lt_or_eq_or_gt j i) #Hij destruct
+[ >vlift_lt // >vlift_lt //
+| >vlift_eq >vlift_eq //
+| >vlift_gt // >vlift_gt //
+]
+qed-.
+
+(* Properies with term interpretation ***************************************)
+
+lemma ti_comp (M): is_model M →
+ ∀T,gv1,gv2. gv1 ≗ gv2 → ∀lv1,lv2. lv1 ≗ lv2 →
+ ⟦T⟧[gv1, lv1] ≗{M} ⟦T⟧[gv2, lv2].
+#M #HM #T elim T -T * [||| #p * | * ]
+[ /4 width=5 by seq_trans, seq_sym, ms/
+| /4 width=5 by seq_sym, ml, mq/
+| /4 width=3 by seq_trans, seq_sym, mg/
+| /5 width=5 by vlift_comp, seq_sym, md, mq/
+| /5 width=1 by vlift_comp, mi, mr/
+| /4 width=5 by seq_sym, ma, mp, mq/
+| /4 width=5 by seq_sym, me, mq/
+]
+qed.