Instance set_compat2 (A : Type) x dom :
  Proper (Logic.eq ==> @eq A dom dom ==> full_relation ==> @eq A dom dom) (@set A dom x).

(* Specifications: *)
Theorem add_compat : forall A x y v dom₁ dom₂ (m₁ : t A dom₁) (m₂ : t A dom₂), X.eq x y -> eq m₁ m₂ -> eq (add x v m₁) (add y v m₂).