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).