Section proofs.
Hypothesis RDC_K : RelDec_Correct RD_K.
Hypothesis Refl : Reflexive R.
Hypothesis Sym : Symmetric R.
Hypothesis Trans : Transitive R.
Definition mapsto_alist (m : alist) k v : Prop :=
alist_find k m = Some v.

Theorem mapsto_remove_neq_alist : forall (m : list (K * V)) (k : K),
forall k', ~ R k k' -> forall v', (mapsto_alist m k' v' <-> mapsto_alist (remove k m) k' v').