Definition agree_env (fr : positive) (env env' : positive -> Z) : Prop :=
forall x, Pos.le x fr -> env x = env' x.

(* Specifications: *)
Lemma agree_env_subset : forall v1 v2 env env',
agree_env v1 env env' ->
Pos.le v2 v1 ->
agree_env v2 env env'.
Lemma agree_env_jump : forall fr j env env',
agree_env (fr + j) env env' ->
agree_env fr (Env.jump j env) (Env.jump j env').
Lemma agree_env_tail : forall fr env env',
agree_env (Pos.succ fr) env env' ->
agree_env fr (Env.tail env) (Env.tail env').
Lemma max_var_acc : forall p i j,
(max_var (i + j) p = max_var i p + j)%positive.
Lemma agree_env_eval_nformula :
forall env env' e
(AGREE : agree_env (max_var xH (fst e))  env env'),
eval_nformula env e  <->  eval_nformula env' e.
Lemma agree_env_eval_nformulae :
forall env env' l
(AGREE : agree_env (max_var_nformulae l) env env'),
make_conj (eval_nformula env) l <->
make_conj (eval_nformula env') l.
Lemma eq_true_iff_eq :
forall b1 b2 : bool, (b1 = true <-> b2 = true) <-> b1 = b2.