val negb : bool -> bool

type nat =
| O
| S of nat

val app : 'a1 list -> 'a1 list -> 'a1 list

type comparison =
| Eq
| Lt
| Gt

val compOpp : comparison -> comparison

val add : nat -> nat -> nat

type positive =
| XI of positive
| XO of positive
| XH

type n =
| N0
| Npos of positive

type z =
| Z0
| Zpos of positive
| Zneg of positive

module Pos :
 sig
  type mask =
  | IsNul
  | IsPos of positive
  | IsNeg
 end

module Coq_Pos :
 sig
  val succ : positive -> positive

  val add : positive -> positive -> positive

  val add_carry : positive -> positive -> positive

  val pred_double : positive -> positive

  type mask = Pos.mask =
  | IsNul
  | IsPos of positive
  | IsNeg

  val succ_double_mask : mask -> mask

  val double_mask : mask -> mask

  val double_pred_mask : positive -> mask

  val sub_mask : positive -> positive -> mask

  val sub_mask_carry : positive -> positive -> mask

  val sub : positive -> positive -> positive

  val mul : positive -> positive -> positive

  val size_nat : positive -> nat

  val compare_cont : comparison -> positive -> positive -> comparison

  val compare : positive -> positive -> comparison

  val gcdn : nat -> positive -> positive -> positive

  val gcd : positive -> positive -> positive

  val of_succ_nat : nat -> positive
 end

module N :
 sig
  val of_nat : nat -> n
 end

val pow_pos : ('a1 -> 'a1 -> 'a1) -> 'a1 -> positive -> 'a1

val nth : nat -> 'a1 list -> 'a1 -> 'a1

val map : ('a1 -> 'a2) -> 'a1 list -> 'a2 list

val fold_right : ('a2 -> 'a1 -> 'a1) -> 'a1 -> 'a2 list -> 'a1

module Z :
 sig
  val double : z -> z

  val succ_double : z -> z

  val pred_double : z -> z

  val pos_sub : positive -> positive -> z

  val add : z -> z -> z

  val opp : z -> z

  val sub : z -> z -> z

  val mul : z -> z -> z

  val compare : z -> z -> comparison

  val leb : z -> z -> bool

  val ltb : z -> z -> bool

  val gtb : z -> z -> bool

  val max : z -> z -> z

  val abs : z -> z

  val to_N : z -> n

  val pos_div_eucl : positive -> z -> z * z

  val div_eucl : z -> z -> z * z

  val div : z -> z -> z

  val gcd : z -> z -> z
 end

val zeq_bool : z -> z -> bool

type 'c pol =
| Pc of 'c
| Pinj of positive * 'c pol
| PX of 'c pol * positive * 'c pol

val p0 : 'a1 -> 'a1 pol

val p1 : 'a1 -> 'a1 pol

val peq : ('a1 -> 'a1 -> bool) -> 'a1 pol -> 'a1 pol -> bool

val mkPinj : positive -> 'a1 pol -> 'a1 pol

val mkPinj_pred : positive -> 'a1 pol -> 'a1 pol

val mkPX :
  'a1 -> ('a1 -> 'a1 -> bool) -> 'a1 pol -> positive -> 'a1 pol -> 'a1 pol

val mkXi : 'a1 -> 'a1 -> positive -> 'a1 pol

val mkX : 'a1 -> 'a1 -> 'a1 pol

val popp : ('a1 -> 'a1) -> 'a1 pol -> 'a1 pol

val paddC : ('a1 -> 'a1 -> 'a1) -> 'a1 pol -> 'a1 -> 'a1 pol

val psubC : ('a1 -> 'a1 -> 'a1) -> 'a1 pol -> 'a1 -> 'a1 pol

val paddI :
  ('a1 -> 'a1 -> 'a1) -> ('a1 pol -> 'a1 pol -> 'a1 pol) -> 'a1 pol ->
  positive -> 'a1 pol -> 'a1 pol

val psubI :
  ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1) -> ('a1 pol -> 'a1 pol -> 'a1 pol) ->
  'a1 pol -> positive -> 'a1 pol -> 'a1 pol

val paddX :
  'a1 -> ('a1 -> 'a1 -> bool) -> ('a1 pol -> 'a1 pol -> 'a1 pol) -> 'a1 pol
  -> positive -> 'a1 pol -> 'a1 pol

val psubX :
  'a1 -> ('a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> ('a1 pol -> 'a1 pol -> 'a1
  pol) -> 'a1 pol -> positive -> 'a1 pol -> 'a1 pol

val padd :
  'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> 'a1 pol -> 'a1 pol ->
  'a1 pol

val psub :
  'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1) -> ('a1
  -> 'a1 -> bool) -> 'a1 pol -> 'a1 pol -> 'a1 pol

val pmulC_aux :
  'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> 'a1 pol -> 'a1 -> 'a1
  pol

val pmulC :
  'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> 'a1 pol -> 'a1
  -> 'a1 pol

val pmulI :
  'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> ('a1 pol ->
  'a1 pol -> 'a1 pol) -> 'a1 pol -> positive -> 'a1 pol -> 'a1 pol

val pmul :
  'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 ->
  bool) -> 'a1 pol -> 'a1 pol -> 'a1 pol

val psquare :
  'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 ->
  bool) -> 'a1 pol -> 'a1 pol

type 'c pExpr =
| PEc of 'c
| PEX of positive
| PEadd of 'c pExpr * 'c pExpr
| PEsub of 'c pExpr * 'c pExpr
| PEmul of 'c pExpr * 'c pExpr
| PEopp of 'c pExpr
| PEpow of 'c pExpr * n

val mk_X : 'a1 -> 'a1 -> positive -> 'a1 pol

val ppow_pos :
  'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 ->
  bool) -> ('a1 pol -> 'a1 pol) -> 'a1 pol -> 'a1 pol -> positive -> 'a1 pol

val ppow_N :
  'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 ->
  bool) -> ('a1 pol -> 'a1 pol) -> 'a1 pol -> n -> 'a1 pol

val norm_aux :
  'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 ->
  'a1) -> ('a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> 'a1 pExpr -> 'a1 pol

type 'a bFormula =
| TT
| FF
| X
| A of 'a
| Cj of 'a bFormula * 'a bFormula
| D of 'a bFormula * 'a bFormula
| N of 'a bFormula
| I of 'a bFormula * 'a bFormula

val map_bformula : ('a1 -> 'a2) -> 'a1 bFormula -> 'a2 bFormula

type 'x clause = 'x list

type 'x cnf = 'x clause list

val tt : 'a1 cnf

val ff : 'a1 cnf

val add_term :
  ('a1 -> bool) -> ('a1 -> 'a1 -> 'a1 option) -> 'a1 -> 'a1 clause -> 'a1
  clause option

val or_clause :
  ('a1 -> bool) -> ('a1 -> 'a1 -> 'a1 option) -> 'a1 clause -> 'a1 clause ->
  'a1 clause option

val or_clause_cnf :
  ('a1 -> bool) -> ('a1 -> 'a1 -> 'a1 option) -> 'a1 clause -> 'a1 cnf -> 'a1
  cnf

val or_cnf :
  ('a1 -> bool) -> ('a1 -> 'a1 -> 'a1 option) -> 'a1 cnf -> 'a1 cnf -> 'a1 cnf

val and_cnf : 'a1 cnf -> 'a1 cnf -> 'a1 cnf

val xcnf :
  ('a2 -> bool) -> ('a2 -> 'a2 -> 'a2 option) -> ('a1 -> 'a2 cnf) -> ('a1 ->
  'a2 cnf) -> bool -> 'a1 bFormula -> 'a2 cnf

val cnf_checker : ('a1 list -> 'a2 -> bool) -> 'a1 cnf -> 'a2 list -> bool

val tauto_checker :
  ('a2 -> bool) -> ('a2 -> 'a2 -> 'a2 option) -> ('a1 -> 'a2 cnf) -> ('a1 ->
  'a2 cnf) -> ('a2 list -> 'a3 -> bool) -> 'a1 bFormula -> 'a3 list -> bool

val cneqb : ('a1 -> 'a1 -> bool) -> 'a1 -> 'a1 -> bool

val cltb : ('a1 -> 'a1 -> bool) -> ('a1 -> 'a1 -> bool) -> 'a1 -> 'a1 -> bool

type 'c polC = 'c pol

type op1 =
| Equal
| NonEqual
| Strict
| NonStrict

type 'c nFormula = 'c polC * op1

val opMult : op1 -> op1 -> op1 option

val opAdd : op1 -> op1 -> op1 option

type 'c psatz =
| PsatzIn of nat
| PsatzSquare of 'c polC
| PsatzMulC of 'c polC * 'c psatz
| PsatzMulE of 'c psatz * 'c psatz
| PsatzAdd of 'c psatz * 'c psatz
| PsatzC of 'c
| PsatzZ

val map_option : ('a1 -> 'a2 option) -> 'a1 option -> 'a2 option

val map_option2 :
  ('a1 -> 'a2 -> 'a3 option) -> 'a1 option -> 'a2 option -> 'a3 option

val pexpr_times_nformula :
  'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 ->
  bool) -> 'a1 polC -> 'a1 nFormula -> 'a1 nFormula option

val nformula_times_nformula :
  'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 ->
  bool) -> 'a1 nFormula -> 'a1 nFormula -> 'a1 nFormula option

val nformula_plus_nformula :
  'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> 'a1 nFormula -> 'a1
  nFormula -> 'a1 nFormula option

val eval_Psatz :
  'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 ->
  bool) -> ('a1 -> 'a1 -> bool) -> 'a1 nFormula list -> 'a1 psatz -> 'a1
  nFormula option

val check_inconsistent :
  'a1 -> ('a1 -> 'a1 -> bool) -> ('a1 -> 'a1 -> bool) -> 'a1 nFormula -> bool

val check_normalised_formulas :
  'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 ->
  bool) -> ('a1 -> 'a1 -> bool) -> 'a1 nFormula list -> 'a1 psatz -> bool

type op2 =
| OpEq
| OpNEq
| OpLe
| OpGe
| OpLt
| OpGt

type 't formula = { flhs : 't pExpr; fop : op2; frhs : 't pExpr }

val norm :
  'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 ->
  'a1) -> ('a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> 'a1 pExpr -> 'a1 pol

val psub0 :
  'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1) -> ('a1
  -> 'a1 -> bool) -> 'a1 pol -> 'a1 pol -> 'a1 pol

val padd0 :
  'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> 'a1 pol -> 'a1 pol ->
  'a1 pol

val xnormalise :
  'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 ->
  'a1) -> ('a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> 'a1 formula -> 'a1 nFormula
  list

val cnf_normalise :
  'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 ->
  'a1) -> ('a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> 'a1 formula -> 'a1 nFormula
  cnf

val xnegate :
  'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 ->
  'a1) -> ('a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> 'a1 formula -> 'a1 nFormula
  list

val cnf_negate :
  'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 ->
  'a1) -> ('a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> 'a1 formula -> 'a1 nFormula
  cnf

val xdenorm : positive -> 'a1 pol -> 'a1 pExpr

val denorm : 'a1 pol -> 'a1 pExpr

val map_PExpr : ('a2 -> 'a1) -> 'a2 pExpr -> 'a1 pExpr

val map_Formula : ('a2 -> 'a1) -> 'a2 formula -> 'a1 formula

val simpl_cone :
  'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> 'a1 psatz ->
  'a1 psatz

type q = { qnum : z; qden : positive }

val qnum : q -> z

val qden : q -> positive

val qeq_bool : q -> q -> bool

val qle_bool : q -> q -> bool

val qplus : q -> q -> q

val qmult : q -> q -> q

val qopp : q -> q

val qminus : q -> q -> q

val qinv : q -> q

val qpower_positive : q -> positive -> q

val qpower : q -> z -> q

type 'a t =
| Empty
| Leaf of 'a
| Node of 'a t * 'a * 'a t

val find : 'a1 -> 'a1 t -> positive -> 'a1

val singleton : 'a1 -> positive -> 'a1 -> 'a1 t

val vm_add : 'a1 -> positive -> 'a1 -> 'a1 t -> 'a1 t

type zWitness = z psatz

val zWeakChecker : z nFormula list -> z psatz -> bool

val psub1 : z pol -> z pol -> z pol

val padd1 : z pol -> z pol -> z pol

val norm0 : z pExpr -> z pol

val xnormalise0 : z formula -> z nFormula list

val normalise : z formula -> z nFormula cnf

val xnegate0 : z formula -> z nFormula list

val negate : z formula -> z nFormula cnf

val zunsat : z nFormula -> bool

val zdeduce : z nFormula -> z nFormula -> z nFormula option

val ceiling : z -> z -> z

type zArithProof =
| DoneProof
| RatProof of zWitness * zArithProof
| CutProof of zWitness * zArithProof
| EnumProof of zWitness * zWitness * zArithProof list

val zgcdM : z -> z -> z

val zgcd_pol : z polC -> z * z

val zdiv_pol : z polC -> z -> z polC

val makeCuttingPlane : z polC -> z polC * z

val genCuttingPlane : z nFormula -> ((z polC * z) * op1) option

val nformula_of_cutting_plane : ((z polC * z) * op1) -> z nFormula

val is_pol_Z0 : z polC -> bool

val eval_Psatz0 : z nFormula list -> zWitness -> z nFormula option

val valid_cut_sign : op1 -> bool

val zChecker : z nFormula list -> zArithProof -> bool

val zTautoChecker : z formula bFormula -> zArithProof list -> bool

type qWitness = q psatz

val qWeakChecker : q nFormula list -> q psatz -> bool

val qnormalise : q formula -> q nFormula cnf

val qnegate : q formula -> q nFormula cnf

val qunsat : q nFormula -> bool

val qdeduce : q nFormula -> q nFormula -> q nFormula option

val qTautoChecker : q formula bFormula -> qWitness list -> bool

type rcst =
| C0
| C1
| CQ of q
| CZ of z
| CPlus of rcst * rcst
| CMinus of rcst * rcst
| CMult of rcst * rcst
| CInv of rcst
| COpp of rcst

val q_of_Rcst : rcst -> q

type rWitness = q psatz

val rWeakChecker : q nFormula list -> q psatz -> bool

val rnormalise : q formula -> q nFormula cnf

val rnegate : q formula -> q nFormula cnf

val runsat : q nFormula -> bool

val rdeduce : q nFormula -> q nFormula -> q nFormula option

val rTautoChecker : rcst formula bFormula -> rWitness list -> bool