Definition cnf_normalise {T: Type} (t: Formula C) (tg: T) : cnf NFormula T :=
  let f := normalise t in
  if check_inconsistent f then cnf_ff _ _
  else cnf_of_list (xnormalise f) tg.Definition cnf_negate {T: Type} (t: Formula C) (tg: T) : cnf NFormula T :=
  let f := normalise t in
  if check_inconsistent f then cnf_tt _ _
  else cnf_of_list (xnegate f) tg.

(* Specifications: *)
Lemma eq0_cnf : forall x, (0 < x -> False) /\ (0 < - x -> False) <-> x == 0.
Lemma xnormalise_correct : forall env f, (make_conj (fun x => eval_nformula env x -> False) (xnormalise f)) <-> eval_nformula env f.
Lemma xnegate_correct : forall env f, (make_conj (fun x => eval_nformula env x -> False) (xnegate f)) <-> ~ eval_nformula env f.
Lemma cnf_normalise_correct : forall (T : Type) env t tg, eval_cnf (Annot:=T) eval_nformula env (cnf_normalise t tg) <-> eval_formula env t.
Lemma cnf_negate_correct : forall (T : Type) env t (tg:T), eval_cnf eval_nformula env (cnf_negate t tg) <-> ~ eval_formula env t.
Lemma eval_nformula_dec : forall env d, (eval_nformula env d) \/ ~ (eval_nformula env d).