Definition valid_cache_extension (ext base: cache) : Prop :=
  forall t: term, assoc_cache t ext <> None -> assoc_cache t base = None.

(* Specifications: *)
Lemma cache_extension_app (ext' ext base: cache) : valid_cache_extension ext' (ext ++ base) -> valid_cache_extension ext base -> valid_cache_extension (ext' ++ ext) base.