Definition lt_eq_lt_dec : forall a b : A, {lt a b} + {a = b} + {lt b a}.Definition key := Key.A.Definition data := Val.data.Definition dict := key -> option data.Definition empty (k : key) := None (A:=data).Definition find (k : key) (d : dict) := d k.Definition add (k : key) (v : data) (d : dict) : dict :=
   fun k' : key =>
   match Key.eqdec k' k with
   | left _ => Some v
   | right _ => d k'
   end.

Theorem empty_def : forall k : key, find k empty = None.