Definition clause := list  (Term' * Annot).Definition cnf := list clause.Definition cnf_tt : cnf := @nil clause.Definition cnf_ff : cnf :=  cons (@nil (Term' * Annot)) nil.(** Our cnf is optimised and detects contradictions on the fly. *)

  Fixpoint add_term (t: Term' * Annot) (cl : clause) : option clause :=
    match cl with
    | nil =>
      match deduce (fst t) (fst t) with
      | None =>  Some (t ::nil)
      | Some u => if unsat u then None else Some (t::nil)
      end
    | t'::cl =>
      match deduce (fst t) (fst t') with
      | None =>
        match add_term t cl with
        | None => None
        | Some cl' => Some (t' :: cl')
        end
      | Some u =>
        if unsat u then None else
          match add_term t cl with
          | None => None
          | Some cl' => Some (t' :: cl')
          end
      end
    end.Fixpoint or_clause (cl1 cl2 : clause) : option clause :=
    match cl1 with
    | nil => Some cl2
    | t::cl => match add_term t cl2 with
               | None => None
               | Some cl' => or_clause cl cl'
               end
    end.Definition xor_clause_cnf (t:clause) (f:cnf) : cnf :=
    List.fold_left (fun acc e =>
                      match or_clause t e with
                      | None => acc
                      | Some cl => cl :: acc
                      end) f nil .Definition or_clause_cnf (t: clause) (f:cnf) : cnf :=
    match t with
    | nil => f
    | _   => xor_clause_cnf t f
    end.Fixpoint or_cnf (f : cnf) (f' : cnf) {struct f}: cnf :=
    match f with
    | nil => cnf_tt
    | e :: rst => (or_cnf rst f') +++ (or_clause_cnf e f')
    end.Definition and_cnf (f1 : cnf) (f2 : cnf) : cnf :=
    f1 +++ f2.(** TX is Prop in Coq and EConstr.constr in Ocaml.
      AF is unit in Coq and Names.Id.t in Ocaml
   *)
  Definition TFormula (TX: kind -> Type) (AF: Type) := @GFormula Term TX Annot AF.Definition is_cnf_tt (c : cnf) : bool :=
    match c with
    | nil => true
    | _  => false
    end.Definition is_cnf_ff (c : cnf) : bool :=
    match c with
    | nil::nil => true
    | _        => false
    end.Definition and_cnf_opt (f1 : cnf) (f2 : cnf) : cnf :=
    if is_cnf_ff f1 || is_cnf_ff f2
    then cnf_ff
    else
      if is_cnf_tt f2
      then f1
      else and_cnf f1 f2.Definition or_cnf_opt (f1 : cnf) (f2 : cnf) : cnf :=
    if is_cnf_tt f1 || is_cnf_tt f2
    then cnf_tt
    else if is_cnf_ff f2
         then f1 else or_cnf f1 f2.Definition mk_and (k: kind) (pol:bool) (f1 f2 : TFormula TX AF k):=
      (if pol then and_cnf_opt else or_cnf_opt) (REC pol f1) (REC pol f2).Definition mk_or (k: kind) (pol:bool) (f1 f2 : TFormula TX AF k):=
      (if pol then or_cnf_opt else and_cnf_opt) (REC pol f1) (REC pol f2).Definition mk_impl (k: kind) (pol:bool) (f1 f2 : TFormula TX AF k):=
      (if pol then or_cnf_opt else and_cnf_opt) (REC (negb pol) f1) (REC pol f2).Definition mk_iff (k: kind) (pol:bool) (f1 f2: TFormula TX AF k):=
      or_cnf_opt (and_cnf_opt (REC (negb pol) f1) (REC false f2))
                 (and_cnf_opt (REC pol f1) (REC true f2)).Definition is_bool {TX : kind -> Type} {AF: Type} (k: kind) (f : TFormula TX AF k) :=
    match f with
    | TT _ => Some true
    | FF _ => Some false
    | _    => None
    end.

(* Specifications: *)
Lemma is_bool_inv : forall {TX : kind -> Type} {AF: Type} (k: kind) (f : TFormula TX AF k) res, is_bool f = Some res -> f = if res then TT _ else FF _.