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.