Library Coq.micromega.Tauto


Require Import List.
Require Import Refl.
Require Import Bool.
Require Import Relation_Definitions Setoid.

Set Implicit Arguments.

Formulae are either interpreted over Prop or bool.
Inductive kind : Type :=
|isProp
|isBool.


Section S.
  Context {TA : Type}.   Context {TX : kind -> Type}.   Context {AA : Type}.   Context {AF : Type}.
  Inductive GFormula : kind -> Type :=
  | TT : forall (k: kind), GFormula k
  | FF : forall (k: kind), GFormula k
  | X : forall (k: kind), TX k -> GFormula k
  | A : forall (k: kind), TA -> AA -> GFormula k
  | AND : forall (k: kind), GFormula k -> GFormula k -> GFormula k
  | OR : forall (k: kind), GFormula k -> GFormula k -> GFormula k
  | NOT : forall (k: kind), GFormula k -> GFormula k
  | IMPL : forall (k: kind), GFormula k -> option AF -> GFormula k -> GFormula k
  | IFF : forall (k: kind), GFormula k -> GFormula k -> GFormula k
  | EQ : GFormula isBool -> GFormula isBool -> GFormula isProp.


  Section MAPX.
    Variable F : forall k, TX k -> TX k.

    Fixpoint mapX (k:kind) (f : GFormula k) : GFormula k :=
      match f with
      | TT k => TT k
      | FF k => FF k
      | X x => X (F x)
      | A k a an => A k a an
      | AND f1 f2 => AND (mapX f1) (mapX f2)
      | OR f1 f2 => OR (mapX f1) (mapX f2)
      | NOT f => NOT (mapX f)
      | IMPL f1 o f2 => IMPL (mapX f1) o (mapX f2)
      | IFF f1 f2 => IFF (mapX f1) (mapX f2)
      | EQ f1 f2 => EQ (mapX f1) (mapX f2)
      end.

  End MAPX.

  Section FOLDANNOT.
    Variable ACC : Type.
    Variable F : ACC -> AA -> ACC.

    Fixpoint foldA (k: kind) (f : GFormula k) (acc : ACC) : ACC :=
      match f with
      | TT _ => acc
      | FF _ => acc
      | X x => acc
      | A _ a an => F acc an
      | AND f1 f2
      | OR f1 f2
      | IFF f1 f2
      | IMPL f1 _ f2 | EQ f1 f2 => foldA f1 (foldA f2 acc)
      | NOT f => foldA f acc
      end.

  End FOLDANNOT.

  Definition cons_id (id : option AF) (l : list AF) :=
    match id with
    | None => l
    | Some id => id :: l
    end.

  Fixpoint ids_of_formula (k: kind) (f:GFormula k) :=
    match f with
    | IMPL f id f' => cons_id id (ids_of_formula f')
    | _ => nil
    end.

  Fixpoint collect_annot (k: kind) (f : GFormula k) : list AA :=
    match f with
    | TT _ | FF _ | X _ => nil
    | A _ _ a => a ::nil
    | AND f1 f2
    | OR f1 f2
    | IFF f1 f2 | EQ f1 f2
    | IMPL f1 _ f2 => collect_annot f1 ++ collect_annot f2
    | NOT f => collect_annot f
    end.

  Definition rtyp (k: kind) : Type := if k then Prop else bool.

  Variable ex : forall (k: kind), TX k -> rtyp k.
  Section EVAL.

    Variable ea : forall (k: kind), TA -> rtyp k.

    Definition eTT (k: kind) : rtyp k :=
      if k as k' return rtyp k' then True else true.

    Definition eFF (k: kind) : rtyp k :=
      if k as k' return rtyp k' then False else false.

    Definition eAND (k: kind) : rtyp k -> rtyp k -> rtyp k :=
      if k as k' return rtyp k' -> rtyp k' -> rtyp k'
      then and else andb.

    Definition eOR (k: kind) : rtyp k -> rtyp k -> rtyp k :=
      if k as k' return rtyp k' -> rtyp k' -> rtyp k'
      then or else orb.

    Definition eIMPL (k: kind) : rtyp k -> rtyp k -> rtyp k :=
      if k as k' return rtyp k' -> rtyp k' -> rtyp k'
      then (fun x y => x -> y) else implb.

    Definition eIFF (k: kind) : rtyp k -> rtyp k -> rtyp k :=
      if k as k' return rtyp k' -> rtyp k' -> rtyp k'
      then iff else eqb.

    Definition eNOT (k: kind) : rtyp k -> rtyp k :=
      if k as k' return rtyp k' -> rtyp k'
      then not else negb.

    Fixpoint eval_f (k: kind) (f:GFormula k) {struct f}: rtyp k :=
      match f in GFormula k' return rtyp k' with
      | TT tk => eTT tk
      | FF tk => eFF tk
      | A k a _ => ea k a
      | X p => ex p
      | @AND k e1 e2 => eAND k (eval_f e1) (eval_f e2)
      | @OR k e1 e2 => eOR k (eval_f e1) (eval_f e2)
      | @NOT k e => eNOT k (eval_f e)
      | @IMPL k f1 _ f2 => eIMPL k (eval_f f1) (eval_f f2)
      | @IFF k f1 f2 => eIFF k (eval_f f1) (eval_f f2)
      | EQ f1 f2 => (eval_f f1) = (eval_f f2)
      end.

    Lemma eval_f_rew : forall k (f:GFormula k),
        eval_f f =
        match f in GFormula k' return rtyp k' with
        | TT tk => eTT tk
        | FF tk => eFF tk
        | A k a _ => ea k a
        | X p => ex p
        | @AND k e1 e2 => eAND k (eval_f e1) (eval_f e2)
        | @OR k e1 e2 => eOR k (eval_f e1) (eval_f e2)
        | @NOT k e => eNOT k (eval_f e)
        | @IMPL k f1 _ f2 => eIMPL k (eval_f f1) (eval_f f2)
        | @IFF k f1 f2 => eIFF k (eval_f f1) (eval_f f2)
        | EQ f1 f2 => (eval_f f1) = (eval_f f2)
        end.

  End EVAL.

  Definition hold (k: kind) : rtyp k -> Prop :=
    if k as k0 return (rtyp k0 -> Prop) then fun x => x else is_true.

  Definition eiff (k: kind) : rtyp k -> rtyp k -> Prop :=
    if k as k' return rtyp k' -> rtyp k' -> Prop then iff else @eq bool.

  Lemma eiff_refl (k: kind) (x : rtyp k) :
      eiff k x x.

  Lemma eiff_sym k (x y : rtyp k) : eiff k x y -> eiff k y x.

  Lemma eiff_trans k (x y z : rtyp k) : eiff k x y -> eiff k y z -> eiff k x z.

  Lemma hold_eiff (k: kind) (x y : rtyp k) :
      (hold k x <-> hold k y) <-> eiff k x y.

  Instance eiff_eq (k: kind) : Equivalence (eiff k).

  Add Parametric Morphism (k: kind) : (@eAND k) with signature eiff k ==> eiff k ==> eiff k as eAnd_morph.

  Add Parametric Morphism (k: kind) : (@eOR k) with signature eiff k ==> eiff k ==> eiff k as eOR_morph.

  Add Parametric Morphism (k: kind) : (@eIMPL k) with signature eiff k ==> eiff k ==> eiff k as eIMPL_morph.

  Add Parametric Morphism (k: kind) : (@eIFF k) with signature eiff k ==> eiff k ==> eiff k as eIFF_morph.

  Add Parametric Morphism (k: kind) : (@eNOT k) with signature eiff k ==> eiff k as eNOT_morph.

  Lemma eval_f_morph :
    forall (ev ev' : forall (k: kind), TA -> rtyp k),
      (forall k a, eiff k (ev k a) (ev' k a)) ->
      forall (k: kind)(f : GFormula k),
        (eiff k (eval_f ev f) (eval_f ev' f)).

End S.

Typical boolean formulae
Definition eKind (k: kind) := if k then Prop else bool.

Definition BFormula (A : Type) := @GFormula A eKind unit unit.


Section MAPATOMS.
  Context {TA TA':Type}.
  Context {TX : kind -> Type}.
  Context {AA : Type}.
  Context {AF : Type}.

  Fixpoint map_bformula (k: kind)(fct : TA -> TA') (f : @GFormula TA TX AA AF k) : @GFormula TA' TX AA AF k:=
    match f with
    | TT k => TT k
    | FF k => FF k
    | X k p => X k p
    | A k a t => A k (fct a) t
    | AND f1 f2 => AND (map_bformula fct f1) (map_bformula fct f2)
    | OR f1 f2 => OR (map_bformula fct f1) (map_bformula fct f2)
    | NOT f => NOT (map_bformula fct f)
    | IMPL f1 a f2 => IMPL (map_bformula fct f1) a (map_bformula fct f2)
    | IFF f1 f2 => IFF (map_bformula fct f1) (map_bformula fct f2)
    | EQ f1 f2 => EQ (map_bformula fct f1) (map_bformula fct f2)
    end.

End MAPATOMS.

Lemma map_simpl : forall A B f l, @map A B f l = match l with
                                                 | nil => nil
                                                 | a :: l=> (f a) :: (@map A B f l)
                                                 end.

Section S.
A cnf tracking annotations of atoms.
Type parameters
  Variable Env : Type.
  Variable Term : Type.
  Variable Term' : Type.
  Variable Annot : Type.

  Variable unsat : Term' -> bool.   Variable deduce : Term' -> Term' -> option Term'.
  Definition clause := list (Term' * Annot).
  Definition cnf := list clause.

  Variable normalise : Term -> Annot -> cnf.
  Variable negate : Term -> Annot -> cnf.

  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.

  Section REC.
    Context {TX : kind -> Type}.
    Context {AF : Type}.

    Variable REC : forall (pol : bool) (k: kind) (f : TFormula TX AF k), cnf.

    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)).

  End REC.

  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.

  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 _.

  Fixpoint xcnf {TX : kind -> Type} {AF: Type} (pol : bool) (k: kind) (f : TFormula TX AF k) {struct f}: cnf :=
    match f with
    | TT _ => if pol then cnf_tt else cnf_ff
    | FF _ => if pol then cnf_ff else cnf_tt
    | X _ p => if pol then cnf_ff else cnf_ff
    | A _ x t => if pol then normalise x t else negate x t
    | NOT e => xcnf (negb pol) e
    | AND e1 e2 => mk_and xcnf pol e1 e2
    | OR e1 e2 => mk_or xcnf pol e1 e2
    | IMPL e1 _ e2 => mk_impl xcnf pol e1 e2
    | IFF e1 e2 => match is_bool e2 with
                   | Some isb => xcnf (if isb then pol else negb pol) e1
                   | None => mk_iff xcnf pol e1 e2
                   end
    | EQ e1 e2 =>
      match is_bool e2 with
      | Some isb => xcnf (if isb then pol else negb pol) e1
      | None => mk_iff xcnf pol e1 e2
      end
    end.

  Section CNFAnnot.

Records annotations used to optimise the cnf. Those need to be kept when pruning the formula. For efficiency, this is a separate function.

    Fixpoint radd_term (t : Term' * Annot) (cl : clause) : clause + list Annot :=
      match cl with
      | nil =>
        match deduce (fst t) (fst t) with
        | Some u => if unsat u then inr ((snd t)::nil) else inl (t::nil)
        | None => inl (t::nil)
        end
      | t'::cl =>
        match deduce (fst t) (fst t') with
        | Some u => if unsat u then inr ((snd t)::(snd t')::nil)
                    else match radd_term t cl with
                         | inl cl' => inl (t'::cl')
                         | inr l => inr l
                         end
        | None => match radd_term t cl with
                   | inl cl' => inl (t'::cl')
                   | inr l => inr l
                   end
        end
      end.

    Fixpoint ror_clause cl1 cl2 :=
      match cl1 with
      | nil => inl cl2
      | t::cl => match radd_term t cl2 with
                 | inl cl' => ror_clause cl cl'
                 | inr l => inr l
                 end
      end.

    Definition xror_clause_cnf t f :=
      List.fold_left (fun '(acc,tg) e =>
                        match ror_clause t e with
                        | inl cl => (cl :: acc,tg)
                        | inr l => (acc,tg+++l)
                        end) f (nil,nil).

    Definition ror_clause_cnf t f :=
      match t with
      | nil => (f,nil)
      | _ => xror_clause_cnf t f
      end.

    Fixpoint ror_cnf (f f':list clause) :=
      match f with
      | nil => (cnf_tt,nil)
      | e :: rst =>
        let (rst_f',t) := ror_cnf rst f' in
        let (e_f', t') := ror_clause_cnf e f' in
        (rst_f' +++ e_f', t +++ t')
      end.

    Definition annot_of_clause (l : clause) : list Annot :=
      List.map snd l.

    Definition annot_of_cnf (f : cnf) : list Annot :=
      List.fold_left (fun acc e => annot_of_clause e +++ acc ) f nil.

    Definition ror_cnf_opt f1 f2 :=
      if is_cnf_tt f1
      then (cnf_tt , nil)
      else if is_cnf_tt f2
           then (cnf_tt, nil)
           else if is_cnf_ff f2
                then (f1,nil)
                else ror_cnf f1 f2.

    Definition ocons {A : Type} (o : option A) (l : list A) : list A :=
      match o with
      | None => l
      | Some e => e ::l
      end.

    Definition ratom (c : cnf) (a : Annot) : cnf * list Annot :=
      if is_cnf_ff c || is_cnf_tt c
      then (c,a::nil)
      else (c,nil).
    Section REC.
      Context {TX : kind -> Type} {AF : Type}.

      Variable RXCNF : forall (polarity: bool) (k: kind) (f: TFormula TX AF k) , cnf * list Annot.

      Definition rxcnf_and (polarity:bool) (k: kind) (e1 e2 : TFormula TX AF k) :=
        let '(e1,t1) := RXCNF polarity e1 in
        let '(e2,t2) := RXCNF polarity e2 in
        if polarity
        then (and_cnf_opt e1 e2, t1 +++ t2)
        else let (f',t') := ror_cnf_opt e1 e2 in
             (f', t1 +++ t2 +++ t').

      Definition rxcnf_or (polarity:bool) (k: kind) (e1 e2 : TFormula TX AF k) :=
        let '(e1,t1) := RXCNF polarity e1 in
        let '(e2,t2) := RXCNF polarity e2 in
        if polarity
        then let (f',t') := ror_cnf_opt e1 e2 in
             (f', t1 +++ t2 +++ t')
        else (and_cnf_opt e1 e2, t1 +++ t2).

      Definition rxcnf_impl (polarity:bool) (k: kind) (e1 e2 : TFormula TX AF k) :=
        let '(e1 , t1) := (RXCNF (negb polarity) e1) in
        if polarity
        then
          if is_cnf_tt e1
          then (e1,t1)
          else if is_cnf_ff e1
            then
              RXCNF polarity e2
            else
              let '(e2 , t2) := (RXCNF polarity e2) in
              let (f',t') := ror_cnf_opt e1 e2 in
              (f', t1 +++ t2 +++ t')
        else
          let '(e2 , t2) := (RXCNF polarity e2) in
          (and_cnf_opt e1 e2, t1 +++ t2).

      Definition rxcnf_iff (polarity:bool) (k: kind) (e1 e2 : TFormula TX AF k) :=
        let '(c1,t1) := RXCNF (negb polarity) e1 in
        let '(c2,t2) := RXCNF false e2 in
        let '(c3,t3) := RXCNF polarity e1 in
        let '(c4,t4) := RXCNF true e2 in
        let (f',t') := ror_cnf_opt (and_cnf_opt c1 c2) (and_cnf_opt c3 c4) in
        (f', t1+++ t2+++t3 +++ t4 +++ t')
      .

    End REC.

    Fixpoint rxcnf {TX : kind -> Type} {AF: Type}(polarity : bool) (k: kind) (f : TFormula TX AF k) : cnf * list Annot :=

      match f with
      | TT _ => if polarity then (cnf_tt,nil) else (cnf_ff,nil)
      | FF _ => if polarity then (cnf_ff,nil) else (cnf_tt,nil)
      | X b p => if polarity then (cnf_ff,nil) else (cnf_ff,nil)
      | A _ x t => ratom (if polarity then normalise x t else negate x t) t
      | NOT e => rxcnf (negb polarity) e
      | AND e1 e2 => rxcnf_and rxcnf polarity e1 e2
      | OR e1 e2 => rxcnf_or rxcnf polarity e1 e2
      | IMPL e1 a e2 => rxcnf_impl rxcnf polarity e1 e2
      | IFF e1 e2 => rxcnf_iff rxcnf polarity e1 e2
      | EQ e1 e2 => rxcnf_iff rxcnf polarity e1 e2
      end.

    Section Abstraction.
      Variable TX : kind -> Type.
      Variable AF : Type.

      Class to_constrT : Type :=
        {
          mkTT : forall (k: kind), TX k;
          mkFF : forall (k: kind), TX k;
          mkA : forall (k: kind), Term -> Annot -> TX k;
          mkAND : forall (k: kind), TX k -> TX k -> TX k;
          mkOR : forall (k: kind), TX k -> TX k -> TX k;
          mkIMPL : forall (k: kind), TX k -> TX k -> TX k;
          mkIFF : forall (k: kind), TX k -> TX k -> TX k;
          mkNOT : forall (k: kind), TX k -> TX k;
          mkEQ : TX isBool -> TX isBool -> TX isProp

        }.

      Context {to_constr : to_constrT}.

      Fixpoint aformula (k: kind) (f : TFormula TX AF k) : TX k :=
        match f with
        | TT b => mkTT b
        | FF b => mkFF b
        | X b p => p
        | A b x t => mkA b x t
        | AND f1 f2 => mkAND (aformula f1) (aformula f2)
        | OR f1 f2 => mkOR (aformula f1) (aformula f2)
        | IMPL f1 o f2 => mkIMPL (aformula f1) (aformula f2)
        | IFF f1 f2 => mkIFF (aformula f1) (aformula f2)
        | NOT f => mkNOT (aformula f)
        | EQ f1 f2 => mkEQ (aformula f1) (aformula f2)
        end.

      Definition is_X (k: kind) (f : TFormula TX AF k) : option (TX k) :=
        match f with
        | X _ p => Some p
        | _ => None
        end.

      Definition is_X_inv : forall (k: kind) (f: TFormula TX AF k) x,
          is_X f = Some x -> f = X k x.

      Variable needA : Annot -> bool.

      Definition abs_and (k: kind) (f1 f2 : TFormula TX AF k)
                 (c : forall (k: kind), TFormula TX AF k -> TFormula TX AF k -> TFormula TX AF k) :=
        match is_X f1 , is_X f2 with
        | Some _ , _ | _ , Some _ => X k (aformula (c k f1 f2))
        | _ , _ => c k f1 f2
        end.

      Definition abs_or (k: kind) (f1 f2 : TFormula TX AF k)
                 (c : forall (k: kind), TFormula TX AF k -> TFormula TX AF k -> TFormula TX AF k) :=
        match is_X f1 , is_X f2 with
        | Some _ , Some _ => X k (aformula (c k f1 f2))
        | _ , _ => c k f1 f2
        end.

      Definition abs_not (k: kind) (f1 : TFormula TX AF k)
                 (c : forall (k: kind), TFormula TX AF k -> TFormula TX AF k) :=
        match is_X f1 with
        | Some _ => X k (aformula (c k f1 ))
        | _ => c k f1
        end.

      Definition mk_arrow (o : option AF) (k: kind) (f1 f2: TFormula TX AF k) :=
        match o with
        | None => IMPL f1 None f2
        | Some _ => if is_X f1 then f2 else IMPL f1 o f2
        end.

      Fixpoint abst_simpl (k: kind) (f : TFormula TX AF k) : TFormula TX AF k:=
        match f with
        | TT k => TT k
        | FF k => FF k
        | X k p => X k p
        | A k x t => if needA t then A k x t else X k (mkA k x t)
        | AND f1 f2 => AND (abst_simpl f1) (abst_simpl f2)
        | OR f1 f2 => OR (abst_simpl f1) (abst_simpl f2)
        | IMPL f1 o f2 => IMPL (abst_simpl f1) o (abst_simpl f2)
        | NOT f => NOT (abst_simpl f)
        | IFF f1 f2 => IFF (abst_simpl f1) (abst_simpl f2)
        | EQ f1 f2 => EQ (abst_simpl f1) (abst_simpl f2)
        end.

      Section REC.
        Variable REC : forall (pol : bool) (k: kind) (f : TFormula TX AF k), TFormula TX AF k.

        Definition abst_and (pol : bool) (k: kind) (f1 f2:TFormula TX AF k) : TFormula TX AF k:=
          (if pol then abs_and else abs_or) k (REC pol f1) (REC pol f2) AND.

        Definition abst_or (pol : bool) (k: kind) (f1 f2:TFormula TX AF k) : TFormula TX AF k:=
          (if pol then abs_or else abs_and) k (REC pol f1) (REC pol f2) OR.

        Definition abst_impl (pol : bool) (o :option AF) (k: kind) (f1 f2:TFormula TX AF k) : TFormula TX AF k:=
          (if pol then abs_or else abs_and) k (REC (negb pol) f1) (REC pol f2) (mk_arrow o).

        Definition or_is_X (k: kind) (f1 f2: TFormula TX AF k) : bool :=
          match is_X f1 , is_X f2 with
          | Some _ , _
          | _ , Some _ => true
          | _ , _ => false
          end.

        Definition abs_iff (k: kind) (nf1 ff2 f1 tf2 : TFormula TX AF k) (r: kind) (def : TFormula TX AF r) : TFormula TX AF r :=
          if andb (or_is_X nf1 ff2) (or_is_X f1 tf2)
          then X r (aformula def)
          else def.

        Definition abst_iff (pol : bool) (k: kind) (f1 f2: TFormula TX AF k) : TFormula TX AF k :=
          abs_iff (REC (negb pol) f1) (REC false f2) (REC pol f1) (REC true f2) (IFF (abst_simpl f1) (abst_simpl f2)).

        Definition abst_eq (pol : bool) (f1 f2: TFormula TX AF isBool) : TFormula TX AF isProp :=
          abs_iff (REC (negb pol) f1) (REC false f2) (REC pol f1) (REC true f2) (EQ (abst_simpl f1) (abst_simpl f2)).

      End REC.

      Fixpoint abst_form (pol : bool) (k: kind) (f : TFormula TX AF k) : TFormula TX AF k:=
        match f with
        | TT k => if pol then TT k else X k (mkTT k)
        | FF k => if pol then X k (mkFF k) else FF k
        | X k p => X k p
        | A k x t => if needA t then A k x t else X k (mkA k x t)
        | AND f1 f2 => abst_and abst_form pol f1 f2
        | OR f1 f2 => abst_or abst_form pol f1 f2
        | IMPL f1 o f2 => abst_impl abst_form pol o f1 f2
        | NOT f => abs_not (abst_form (negb pol) f) NOT
        | IFF f1 f2 => abst_iff abst_form pol f1 f2
        | EQ f1 f2 => abst_eq abst_form pol f1 f2
        end.

      Lemma if_same : forall {A: Type} (b: bool) (t:A),
          (if b then t else t) = t.

      Lemma is_cnf_tt_cnf_ff :
        is_cnf_tt cnf_ff = false.

      Lemma is_cnf_ff_cnf_ff :
        is_cnf_ff cnf_ff = true.

      Lemma is_cnf_tt_inv : forall f1,
          is_cnf_tt f1 = true -> f1 = cnf_tt.

      Lemma is_cnf_ff_inv : forall f1,
          is_cnf_ff f1 = true -> f1 = cnf_ff.

      Lemma if_cnf_tt : forall f, (if is_cnf_tt f then cnf_tt else f) = f.

      Lemma or_cnf_opt_cnf_ff : forall f,
          or_cnf_opt cnf_ff f = f.

      Lemma abs_and_pol : forall (k: kind) (f1 f2: TFormula TX AF k) pol,
          and_cnf_opt (xcnf pol f1) (xcnf pol f2) =
          xcnf pol (abs_and f1 f2 (if pol then AND else OR)).

      Lemma abs_or_pol : forall (k: kind) (f1 f2:TFormula TX AF k) pol,
          or_cnf_opt (xcnf pol f1) (xcnf pol f2) =
          xcnf pol (abs_or f1 f2 (if pol then OR else AND)).

      Variable needA_all : forall a, needA a = true.

      Lemma xcnf_true_mk_arrow_l : forall b o t (f:TFormula TX AF b),
          xcnf true (mk_arrow o (X b t) f) = xcnf true f.

      Lemma or_cnf_opt_cnf_ff_r : forall f,
          or_cnf_opt f cnf_ff = f.

      Lemma xcnf_true_mk_arrow_r : forall b o t (f:TFormula TX AF b),
          xcnf true (mk_arrow o f (X b t)) = xcnf false f.

      Lemma and_cnf_opt_cnf_ff_r : forall f,
          and_cnf_opt f cnf_ff = cnf_ff.

      Lemma and_cnf_opt_cnf_ff : forall f,
          and_cnf_opt cnf_ff f = cnf_ff.

      Lemma and_cnf_opt_cnf_tt : forall f,
          and_cnf_opt f cnf_tt = f.

      Lemma is_bool_abst_simpl : forall b (f:TFormula TX AF b),
          is_bool (abst_simpl f) = is_bool f.

      Lemma abst_simpl_correct : forall b (f:TFormula TX AF b) pol,
          xcnf pol f = xcnf pol (abst_simpl f).

      Ltac is_X t :=
        match goal with
        | |-context[is_X ?X] =>
          let f := fresh "EQ" in
          destruct (is_X X) as [t|] eqn:f ;
          [apply is_X_inv in f|]
        end.

      Ltac cnf_simpl :=
        repeat match goal with
               | |- context[and_cnf_opt cnf_ff _ ] => rewrite and_cnf_opt_cnf_ff
               | |- context[and_cnf_opt _ cnf_ff] => rewrite and_cnf_opt_cnf_ff_r
               | |- context[and_cnf_opt _ cnf_tt] => rewrite and_cnf_opt_cnf_tt
               | |- context[or_cnf_opt cnf_ff _] => rewrite or_cnf_opt_cnf_ff
               | |- context[or_cnf_opt _ cnf_ff] => rewrite or_cnf_opt_cnf_ff_r
               end.

      Lemma or_is_X_inv : forall (k: kind) (f1 f2 : TFormula TX AF k),
          or_is_X f1 f2 = true ->
          exists k1, is_X f1 = Some k1 \/ is_X f2 = Some k1.

      Lemma mk_iff_is_bool : forall (k: kind) (f1 f2:TFormula TX AF k) pol,
          match is_bool f2 with
          | Some isb => xcnf (if isb then pol else negb pol) f1
          | None => mk_iff xcnf pol f1 f2
          end = mk_iff xcnf pol f1 f2.

      Lemma abst_iff_correct : forall
          (k: kind)
          (f1 f2 : GFormula k)
          (IHf1 : forall pol : bool, xcnf pol f1 = xcnf pol (abst_form pol f1))
          (IHf2 : forall pol : bool, xcnf pol f2 = xcnf pol (abst_form pol f2))
          (pol : bool),
          xcnf pol (IFF f1 f2) = xcnf pol (abst_iff abst_form pol f1 f2).

      Lemma abst_eq_correct : forall
          (f1 f2 : GFormula isBool)
          (IHf1 : forall pol : bool, xcnf pol f1 = xcnf pol (abst_form pol f1))
          (IHf2 : forall pol : bool, xcnf pol f2 = xcnf pol (abst_form pol f2))
          (pol : bool),
          xcnf pol (EQ f1 f2) = xcnf pol (abst_form pol (EQ f1 f2)).

      Lemma abst_form_correct : forall b (f:TFormula TX AF b) pol,
          xcnf pol f = xcnf pol (abst_form pol f).

    End Abstraction.

  End CNFAnnot.

  Lemma radd_term_term : forall a' a cl, radd_term a a' = inl cl -> add_term a a' = Some cl.

  Lemma radd_term_term' : forall a' a cl, add_term a a' = Some cl -> radd_term a a' = inl cl.

  Lemma xror_clause_clause : forall a f,
      fst (xror_clause_cnf a f) = xor_clause_cnf a f.

  Lemma ror_clause_clause : forall a f,
      fst (ror_clause_cnf a f) = or_clause_cnf a f.

  Lemma ror_cnf_cnf : forall f1 f2, fst (ror_cnf f1 f2) = or_cnf f1 f2.

  Lemma ror_opt_cnf_cnf : forall f1 f2, fst (ror_cnf_opt f1 f2) = or_cnf_opt f1 f2.

  Lemma ratom_cnf : forall f a,
      fst (ratom f a) = f.

  Lemma rxcnf_and_xcnf : forall {TX : kind -> Type} {AF:Type} (k: kind) (f1 f2:TFormula TX AF k)
                                (IHf1 : forall pol : bool, fst (rxcnf pol f1) = xcnf pol f1)
                                (IHf2 : forall pol : bool, fst (rxcnf pol f2) = xcnf pol f2),
      forall pol : bool, fst (rxcnf_and rxcnf pol f1 f2) = mk_and xcnf pol f1 f2.

  Lemma rxcnf_or_xcnf :
    forall {TX : kind -> Type} {AF:Type} (k: kind) (f1 f2:TFormula TX AF k)
           (IHf1 : forall pol : bool, fst (rxcnf pol f1) = xcnf pol f1)
           (IHf2 : forall pol : bool, fst (rxcnf pol f2) = xcnf pol f2),
    forall pol : bool, fst (rxcnf_or rxcnf pol f1 f2) = mk_or xcnf pol f1 f2.

  Lemma rxcnf_impl_xcnf :
    forall {TX : kind -> Type} {AF:Type} (k: kind) (f1 f2:TFormula TX AF k)
           (IHf1 : forall pol : bool, fst (rxcnf pol f1) = xcnf pol f1)
           (IHf2 : forall pol : bool, fst (rxcnf pol f2) = xcnf pol f2),
    forall pol : bool, fst (rxcnf_impl rxcnf pol f1 f2) = mk_impl xcnf pol f1 f2.

  Lemma rxcnf_iff_xcnf :
    forall {TX : kind -> Type} {AF:Type} (k: kind) (f1 f2:TFormula TX AF k)
           (IHf1 : forall pol : bool, fst (rxcnf pol f1) = xcnf pol f1)
           (IHf2 : forall pol : bool, fst (rxcnf pol f2) = xcnf pol f2),
    forall pol : bool, fst (rxcnf_iff rxcnf pol f1 f2) = mk_iff xcnf pol f1 f2.

  Lemma rxcnf_xcnf : forall {TX : kind -> Type} {AF:Type} (k: kind) (f:TFormula TX AF k) pol,
      fst (rxcnf pol f) = xcnf pol f.

  Variable eval' : Env -> Term' -> Prop.

  Variable no_middle_eval' : forall env d, (eval' env d) \/ ~ (eval' env d).

  Variable unsat_prop : forall t, unsat t = true ->
                                  forall env, eval' env t -> False.

  Variable deduce_prop : forall t t' u,
      deduce t t' = Some u -> forall env,
        eval' env t -> eval' env t' -> eval' env u.

  Definition eval_tt (env : Env) (tt : Term' * Annot) := eval' env (fst tt).

  Definition eval_clause (env : Env) (cl : clause) := ~ make_conj (eval_tt env) cl.

  Definition eval_cnf (env : Env) (f:cnf) := make_conj (eval_clause env) f.

  Lemma eval_cnf_app : forall env x y, eval_cnf env (x+++y) <-> eval_cnf env x /\ eval_cnf env y.

  Lemma eval_cnf_ff : forall env, eval_cnf env cnf_ff <-> False.

  Lemma eval_cnf_tt : forall env, eval_cnf env cnf_tt <-> True.

  Lemma eval_cnf_and_opt : forall env x y, eval_cnf env (and_cnf_opt x y) <-> eval_cnf env (and_cnf x y).

  Definition eval_opt_clause (env : Env) (cl: option clause) :=
    match cl with
    | None => True
    | Some cl => eval_clause env cl
    end.

  Lemma add_term_correct : forall env t cl , eval_opt_clause env (add_term t cl) <-> eval_clause env (t::cl).

  Lemma no_middle_eval_tt : forall env a,
      eval_tt env a \/ ~ eval_tt env a.

  #[local]
  Hint Resolve no_middle_eval_tt : tauto.

  Lemma or_clause_correct : forall cl cl' env, eval_opt_clause env (or_clause cl cl') <-> eval_clause env cl \/ eval_clause env cl'.

  Lemma or_clause_cnf_correct : forall env t f, eval_cnf env (or_clause_cnf t f) <-> (eval_clause env t) \/ (eval_cnf env f).

  Lemma eval_cnf_cons : forall env a f, (~ make_conj (eval_tt env) a /\ eval_cnf env f) <-> eval_cnf env (a::f).

  Lemma eval_cnf_cons_iff : forall env a f, ((~ make_conj (eval_tt env) a) /\ eval_cnf env f) <-> eval_cnf env (a::f).

  Lemma or_cnf_correct : forall env f f', eval_cnf env (or_cnf f f') <-> (eval_cnf env f) \/ (eval_cnf env f').

  Lemma or_cnf_opt_correct : forall env f f', eval_cnf env (or_cnf_opt f f') <-> eval_cnf env (or_cnf f f').

  Variable eval : Env -> forall (k: kind), Term -> rtyp k.

  Variable normalise_correct : forall env b t tg, eval_cnf env (normalise t tg) -> hold b (eval env b t).

  Variable negate_correct : forall env b t tg, eval_cnf env (negate t tg) -> hold b (eNOT b (eval env b t)).

  Definition e_rtyp (k: kind) (x : rtyp k) : rtyp k := x.

  Lemma hold_eTT : forall k, hold k (eTT k).

  #[local]
  Hint Resolve hold_eTT : tauto.

  Lemma hold_eFF : forall k,
      hold k (eNOT k (eFF k)).

  #[local]
  Hint Resolve hold_eFF : tauto.

  Lemma hold_eAND : forall k r1 r2,
      hold k (eAND k r1 r2) <-> (hold k r1 /\ hold k r2).

  Lemma hold_eOR : forall k r1 r2,
      hold k (eOR k r1 r2) <-> (hold k r1 \/ hold k r2).

  Lemma hold_eNOT : forall k e,
      hold k (eNOT k e) <-> not (hold k e).

  Lemma hold_eIMPL : forall k e1 e2,
      hold k (eIMPL k e1 e2) <-> (hold k e1 -> hold k e2).

  Lemma hold_eIFF : forall k e1 e2,
      hold k (eIFF k e1 e2) <-> (hold k e1 <-> hold k e2).

  Lemma xcnf_impl :
    forall
      (k: kind)
      (f1 : GFormula k)
      (o : option unit)
      (f2 : GFormula k)
      (IHf1 : forall (pol : bool) (env : Env),
          eval_cnf env (xcnf pol f1) ->
          hold k (eval_f e_rtyp (eval env) (if pol then f1 else NOT f1)))
      (IHf2 : forall (pol : bool) (env : Env),
          eval_cnf env (xcnf pol f2) ->
          hold k (eval_f e_rtyp (eval env) (if pol then f2 else NOT f2))),
    forall (pol : bool) (env : Env),
      eval_cnf env (xcnf pol (IMPL f1 o f2)) ->
      hold k (eval_f e_rtyp (eval env) (if pol then IMPL f1 o f2 else NOT (IMPL f1 o f2))).

  Lemma hold_eIFF_IMPL : forall k e1 e2,
      hold k (eIFF k e1 e2) <-> (hold k (eAND k (eIMPL k e1 e2) (eIMPL k e2 e1))).

  Lemma hold_eEQ : forall e1 e2,
      hold isBool (eIFF isBool e1 e2) <-> e1 = e2.

  Lemma xcnf_iff : forall
      (k : kind)
      (f1 f2 : @GFormula Term rtyp Annot unit k)
      (IHf1 : forall (pol : bool) (env : Env),
          eval_cnf env (xcnf pol f1) ->
          hold k (eval_f e_rtyp (eval env) (if pol then f1 else NOT f1)))
      (IHf2 : forall (pol : bool) (env : Env),
          eval_cnf env (xcnf pol f2) ->
          hold k (eval_f e_rtyp (eval env) (if pol then f2 else NOT f2))),
      forall (pol : bool) (env : Env),
        eval_cnf env (xcnf pol (IFF f1 f2)) ->
        hold k (eval_f e_rtyp (eval env) (if pol then IFF f1 f2 else NOT (IFF f1 f2))).

  Lemma xcnf_correct : forall (k: kind) (f : @GFormula Term rtyp Annot unit k) pol env,
      eval_cnf env (xcnf pol f) -> hold k (eval_f e_rtyp (eval env) (if pol then f else NOT f)).

  Variable Witness : Type.
  Variable checker : list (Term'*Annot) -> Witness -> bool.

  Variable checker_sound : forall t w, checker t w = true -> forall env, make_impl (eval_tt env) t False.

  Fixpoint cnf_checker (f : cnf) (l : list Witness) {struct f}: bool :=
    match f with
    | nil => true
    | e::f => match l with
              | nil => false
              | c::l => match checker e c with
                        | true => cnf_checker f l
                        | _ => false
                        end
              end
    end.

  Lemma cnf_checker_sound : forall t w, cnf_checker t w = true -> forall env, eval_cnf env t.

  Definition tauto_checker (f:@GFormula Term rtyp Annot unit isProp) (w:list Witness) : bool :=
    cnf_checker (xcnf true f) w.

  Lemma tauto_checker_sound : forall t w, tauto_checker t w = true -> forall env, eval_f e_rtyp (eval env) t.

  Definition eval_bf {A : Type} (ea : forall (k: kind), A -> rtyp k) (k: kind) (f: BFormula A k) := eval_f e_rtyp ea f.

  Lemma eval_bf_map : forall T U (fct: T-> U) env (k: kind) (f:BFormula T k) ,
      eval_bf env (map_bformula fct f) = eval_bf (fun b x => env b (fct x)) f.

End S.