Instance set_compat2 (A : Type) x dom :
Proper (Logic.eq ==> @eq A dom dom ==> full_relation ==> @eq A dom dom) (@set A dom x).
Instance add_compat2 (A : Type) x dom :
Proper (Logic.eq ==> @eq A dom dom ==> @eq A (Dom.add x dom) (Dom.add x dom)) (@add A dom x).
Instance remove_compat2 (A : Type) x dom :
Proper (@eq A dom dom ==> @eq A (Dom.remove x dom) (Dom.remove x dom)) (@remove A dom x).

Theorem add_Some : forall A x y v u dom (m : t A dom),
find y (add x v m) = Some u <-> X.eq y x ∧ u = v ∨ ¬X.eq y x ∧ find y m = Some u.