(** * Abstracting the result of conditions (type [option bool]) *)

Inductive abool :=
  | Bnone               (**r always [None] (undefined) *)
  | Just (b: bool)      (**r always [Some b] (defined and known to be [b]) *)
  | Maybe (b: bool)     (**r either [None] or [Some b] (known to be [b] if defined) *)
  | Btop.(** * Abstracting the result of conditions (type [option bool]) *)

Inductive abool :=
  | Bnone               (**r always [None] (undefined) *)
  | Just (b: bool)      (**r always [Some b] (defined and known to be [b]) *)
  | Maybe (b: bool)     (**r either [None] or [Some b] (known to be [b] if defined) *)
  | Btop.(** * Abstracting the result of conditions (type [option bool]) *)

Inductive abool :=
  | Bnone               (**r always [None] (undefined) *)
  | Just (b: bool)      (**r always [Some b] (defined and known to be [b]) *)
  | Maybe (b: bool)     (**r either [None] or [Some b] (known to be [b] if defined) *)
  | Btop.(** * Abstracting the result of conditions (type [option bool]) *)

Inductive abool :=
  | Bnone               (**r always [None] (undefined) *)
  | Just (b: bool)      (**r always [Some b] (defined and known to be [b]) *)
  | Maybe (b: bool)     (**r either [None] or [Some b] (known to be [b] if defined) *)
  | Btop.(** * Abstracting the result of conditions (type [option bool]) *)

Inductive abool :=
  | Bnone               (**r always [None] (undefined) *)
  | Just (b: bool)      (**r always [Some b] (defined and known to be [b]) *)
  | Maybe (b: bool)     (**r either [None] or [Some b] (known to be [b] if defined) *)
  | Btop.(**r unknown, all results are possible *)

Inductive cmatch: option bool -> abool -> Prop :=
  | cmatch_none: cmatch None Bnone
  | cmatch_just: forall b, cmatch (Some b) (Just b)
  | cmatch_maybe_none: forall b, cmatch None (Maybe b)
  | cmatch_maybe_some: forall b, cmatch (Some b) (Maybe b)
  | cmatch_top: forall ob, cmatch ob Btop.(**r unknown, all results are possible *)

Inductive cmatch: option bool -> abool -> Prop :=
  | cmatch_none: cmatch None Bnone
  | cmatch_just: forall b, cmatch (Some b) (Just b)
  | cmatch_maybe_none: forall b, cmatch None (Maybe b)
  | cmatch_maybe_some: forall b, cmatch (Some b) (Maybe b)
  | cmatch_top: forall ob, cmatch ob Btop.(**r unknown, all results are possible *)

Inductive cmatch: option bool -> abool -> Prop :=
  | cmatch_none: cmatch None Bnone
  | cmatch_just: forall b, cmatch (Some b) (Just b)
  | cmatch_maybe_none: forall b, cmatch None (Maybe b)
  | cmatch_maybe_some: forall b, cmatch (Some b) (Maybe b)
  | cmatch_top: forall ob, cmatch ob Btop.(**r unknown, all results are possible *)

Inductive cmatch: option bool -> abool -> Prop :=
  | cmatch_none: cmatch None Bnone
  | cmatch_just: forall b, cmatch (Some b) (Just b)
  | cmatch_maybe_none: forall b, cmatch None (Maybe b)
  | cmatch_maybe_some: forall b, cmatch (Some b) (Maybe b)
  | cmatch_top: forall ob, cmatch ob Btop.(**r unknown, all results are possible *)

Inductive cmatch: option bool -> abool -> Prop :=
  | cmatch_none: cmatch None Bnone
  | cmatch_just: forall b, cmatch (Some b) (Just b)
  | cmatch_maybe_none: forall b, cmatch None (Maybe b)
  | cmatch_maybe_some: forall b, cmatch (Some b) (Maybe b)
  | cmatch_top: forall ob, cmatch ob Btop.(**r unknown, all results are possible *)

Inductive cmatch: option bool -> abool -> Prop :=
  | cmatch_none: cmatch None Bnone
  | cmatch_just: forall b, cmatch (Some b) (Just b)
  | cmatch_maybe_none: forall b, cmatch None (Maybe b)
  | cmatch_maybe_some: forall b, cmatch (Some b) (Maybe b)
  | cmatch_top: forall ob, cmatch ob Btop.Definition club (x y: abool) : abool :=
  match x, y with
  | Bnone, Bnone => Bnone
  | Bnone, (Just b | Maybe b) => Maybe b
  | (Just b | Maybe b), Bnone => Maybe b
  | Just b1, Just b2 => if eqb b1 b2 then x else Btop
  | Maybe b1, Maybe b2 => if eqb b1 b2 then x else Btop
  | Maybe b1, Just b2 => if eqb b1 b2 then x else Btop
  | Just b1, Maybe b2 => if eqb b1 b2 then y else Btop
  | _, _ => Btop
  end.

Lemma cmatch_lub_r: forall ob x y, cmatch ob y -> cmatch ob (club x y).