Definition disj (s1 s2 : fmap) := 
  all (predC (fun x => x \in supp s2)) (supp s1).CoInductive disj_spec (s1 s2 : fmap) : bool -> Type :=
| disj_true of (forall x, x \in supp s1 -> x \notin supp s2) : 
    disj_spec s1 s2 true
| disj_false x of x \in supp s1 & x \in supp s2 : 
    disj_spec s1 s2 false.CoInductive disj_spec (s1 s2 : fmap) : bool -> Type :=
| disj_true of (forall x, x \in supp s1 -> x \notin supp s2) : 
    disj_spec s1 s2 true
| disj_false x of x \in supp s1 & x \in supp s2 : 
    disj_spec s1 s2 false.CoInductive disj_spec (s1 s2 : fmap) : bool -> Type :=
| disj_true of (forall x, x \in supp s1 -> x \notin supp s2) : 
    disj_spec s1 s2 true
| disj_false x of x \in supp s1 & x \in supp s2 : 
    disj_spec s1 s2 false.

Lemma fcatAC (s1 s2 s3 : fmap) : [&& disj s1 s2, disj s2 s3 & disj s1 s3] -> fcat s1 (fcat s2 s3) = fcat s2 (fcat s1 s3).