-lemma push_comp (M): ∀i. compatible_3 … (push M i) (sq M) (veq M) (veq M).
-#m #i #d1 #d2 #Hd12 #lv1 #lv2 #HLv12 #j
-elim (lt_or_eq_or_gt j i) #Hij destruct
-[ >(push_lt … Hij) >(push_lt … Hij) //
-| >(push_eq …) >(push_eq …) //
-| >(push_gt … Hij) >(push_gt … Hij) //
+theorem vpush_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
+ >vpush_lt // >vpush_lt // >vpush_lt /2 width=1 by lt_S/ >vpush_lt //
+ /2 width=1 by veq_refl/
+| >vpush_eq >vpush_lt /2 width=1 by monotonic_le_plus_l/ >vpush_eq
+ /2 width=1 by mr/
+| >vpush_gt // elim (lt_or_eq_or_gt (↓j) i2) #Hji2 destruct
+ [ >vpush_lt // >vpush_lt /2 width=1 by lt_minus_to_plus/ >vpush_gt //
+ /2 width=1 by veq_refl/
+ | >vpush_eq <(lt_succ_pred … Hji1) >vpush_eq
+ /2 width=1 by mr/
+ | lapply (le_to_lt_to_lt … Hi12 Hji2) #Hi1j
+ >vpush_gt // >vpush_gt /2 width=1 by lt_minus_to_plus_r/ >vpush_gt //
+ /2 width=1 by veq_refl/
+ ]