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

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.