Inductive pre_ad : Set :=
  | pre_ad_empty : pre_ad
  | pre_ad_O : pre_ad -> pre_ad
  | pre_ad_I : pre_ad -> pre_ad.Inductive pre_ad : Set :=
  | pre_ad_empty : pre_ad
  | pre_ad_O : pre_ad -> pre_ad
  | pre_ad_I : pre_ad -> pre_ad.Inductive pre_ad : Set :=
  | pre_ad_empty : pre_ad
  | pre_ad_O : pre_ad -> pre_ad
  | pre_ad_I : pre_ad -> pre_ad.Inductive pre_ad : Set :=
  | pre_ad_empty : pre_ad
  | pre_ad_O : pre_ad -> pre_ad
  | pre_ad_I : pre_ad -> pre_ad.Fixpoint pre_ad_concat (pa : pre_ad) : ad -> ad :=
  fun a : ad =>
  match pa with
  | pre_ad_empty => a
  | pre_ad_O pa' => pre_ad_concat pa' (Ndouble a)
  | pre_ad_I pa' => pre_ad_concat pa' (Ndouble_plus_one a)
  end.Fixpoint st_compat_check_0 (pa : pre_ad) (sigma : signature) 
 (s : state) {struct s} : bool :=
  match s with
  | M0 => true
  | M1 a p =>
      match pl_compat_check p, MapGet nat sigma (pre_ad_concat pa a) with
      | None, _ => false
      | _, None => false
      | Some n, Some m => beq_nat n m
      end
  | M2 x y =>
      st_compat_check_0 (pre_ad_O pa) sigma x &&
      st_compat_check_0 (pre_ad_I pa) sigma y
  end.Definition st_compat_check (s : state) (sigma : signature) : bool :=
  st_compat_check_0 pre_ad_empty sigma s.Fixpoint predta_compat_check (d : preDTA) : signature -> bool :=
  fun sigma : signature =>
  match d with
  | M0 => true
  | M1 a s => st_compat_check s sigma
  | M2 x y => predta_compat_check x sigma && predta_compat_check y sigma
  end.Definition dta_compat_check (d : DTA) (sigma : signature) : bool :=
  match d with
  | dta p a => predta_compat_check p sigma
  end.Definition state_correct_wrt_sign_with_offset (s : state) 
  (sigma : signature) (pa : pre_ad) : Prop :=
  forall (a : ad) (p : prec_list),
  MapGet prec_list s a = Some p ->
  exists n : nat,
    MapGet nat sigma (pre_ad_concat pa a) = Some n /\ pl_tl_length p n.

Lemma state_correct_wrt_sign_with_offset_M2 : forall (s0 s1 : state) (sigma : signature) (pa : pre_ad), state_correct_wrt_sign_with_offset (M2 prec_list s0 s1) sigma pa -> state_correct_wrt_sign_with_offset s0 sigma (pre_ad_O pa) /\ state_correct_wrt_sign_with_offset s1 sigma (pre_ad_I pa).