Proof with simpl; auto.
intros H1 H2 H3. revert sigma0 H2. induction l... intros sigma0 H2.
inversion H1; apply IHl...
Qed.

(* Specifications: *)
Lemma space_denote_map_id sigma0 st :
space_denote sigma0 st ->
space_denote (map (fun atm : space_atom => atm) sigma0) st.
Lemma simplify_atoms_sound l sigma st :
listd clause_denote inter TT l st -> space_denote sigma st ->
space_denote (simplify_atoms l sigma) st.