Definition set_in_state nextv z s := State (env_set nextv z (stk s)) (hp s).

(* Specifications: *)
Lemma expr_denote_agree: forall e s nextv z, fresh freshmax_expr e nextv -> expr_denote e s = expr_denote e (set_in_state nextv z s).
Lemma pn_atom_denote_agree: forall a s nextv z, fresh freshmax_pn_atom a nextv -> pn_atom_denote a s -> pn_atom_denote a (set_in_state nextv z s).
Lemma space_atom_denote_agree: forall a s nextv z, fresh freshmax_space_atom a nextv -> space_atom_denote a s -> space_atom_denote a (set_in_state nextv z s).
Lemma list_denote__pn_atom_agree: forall pos s nextv z, fresh (freshmax_list freshmax_pn_atom) pos nextv -> list_denote pn_atom_denote (@andp _) TT pos s -> list_denote pn_atom_denote (@andp _) TT pos (set_in_state nextv z s).
Lemma list_denote_space_agree: forall pos s nextv z, fresh (freshmax_list freshmax_space_atom) pos nextv -> list_denote space_atom_denote (@sepcon _ _) emp pos s -> list_denote space_atom_denote (@sepcon _ _) emp pos (set_in_state nextv z s).