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diff --git a/contrib/micromega/micromega.ml b/contrib/micromega/micromega.ml
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+++ b/contrib/micromega/micromega.ml
@@ -0,0 +1,1512 @@
+type __ = Obj.t
+let __ = let rec f _ = Obj.repr f in Obj.repr f
+
+type bool =
+ | True
+ | False
+
+(** val negb : bool -> bool **)
+
+let negb = function
+ | True -> False
+ | False -> True
+
+type nat =
+ | O
+ | S of nat
+
+type 'a option =
+ | Some of 'a
+ | None
+
+type ('a, 'b) prod =
+ | Pair of 'a * 'b
+
+type comparison =
+ | Eq
+ | Lt
+ | Gt
+
+(** val compOpp : comparison -> comparison **)
+
+let compOpp = function
+ | Eq -> Eq
+ | Lt -> Gt
+ | Gt -> Lt
+
+type sumbool =
+ | Left
+ | Right
+
+type 'a sumor =
+ | Inleft of 'a
+ | Inright
+
+type 'a list =
+ | Nil
+ | Cons of 'a * 'a list
+
+(** val app : 'a1 list -> 'a1 list -> 'a1 list **)
+
+let rec app l m =
+ match l with
+ | Nil -> m
+ | Cons (a, l1) -> Cons (a, (app l1 m))
+
+(** val nth : nat -> 'a1 list -> 'a1 -> 'a1 **)
+
+let rec nth n0 l default =
+ match n0 with
+ | O -> (match l with
+ | Nil -> default
+ | Cons (x, l') -> x)
+ | S m ->
+ (match l with
+ | Nil -> default
+ | Cons (x, t0) -> nth m t0 default)
+
+(** val map : ('a1 -> 'a2) -> 'a1 list -> 'a2 list **)
+
+let rec map f = function
+ | Nil -> Nil
+ | Cons (a, t0) -> Cons ((f a), (map f t0))
+
+type positive =
+ | XI of positive
+ | XO of positive
+ | XH
+
+(** val psucc : positive -> positive **)
+
+let rec psucc = function
+ | XI p -> XO (psucc p)
+ | XO p -> XI p
+ | XH -> XO XH
+
+(** val pplus : positive -> positive -> positive **)
+
+let rec pplus x y =
+ match x with
+ | XI p ->
+ (match y with
+ | XI q0 -> XO (pplus_carry p q0)
+ | XO q0 -> XI (pplus p q0)
+ | XH -> XO (psucc p))
+ | XO p ->
+ (match y with
+ | XI q0 -> XI (pplus p q0)
+ | XO q0 -> XO (pplus p q0)
+ | XH -> XI p)
+ | XH ->
+ (match y with
+ | XI q0 -> XO (psucc q0)
+ | XO q0 -> XI q0
+ | XH -> XO XH)
+
+(** val pplus_carry : positive -> positive -> positive **)
+
+and pplus_carry x y =
+ match x with
+ | XI p ->
+ (match y with
+ | XI q0 -> XI (pplus_carry p q0)
+ | XO q0 -> XO (pplus_carry p q0)
+ | XH -> XI (psucc p))
+ | XO p ->
+ (match y with
+ | XI q0 -> XO (pplus_carry p q0)
+ | XO q0 -> XI (pplus p q0)
+ | XH -> XO (psucc p))
+ | XH ->
+ (match y with
+ | XI q0 -> XI (psucc q0)
+ | XO q0 -> XO (psucc q0)
+ | XH -> XI XH)
+
+(** val p_of_succ_nat : nat -> positive **)
+
+let rec p_of_succ_nat = function
+ | O -> XH
+ | S x -> psucc (p_of_succ_nat x)
+
+(** val pdouble_minus_one : positive -> positive **)
+
+let rec pdouble_minus_one = function
+ | XI p -> XI (XO p)
+ | XO p -> XI (pdouble_minus_one p)
+ | XH -> XH
+
+type positive_mask =
+ | IsNul
+ | IsPos of positive
+ | IsNeg
+
+(** val pdouble_plus_one_mask : positive_mask -> positive_mask **)
+
+let pdouble_plus_one_mask = function
+ | IsNul -> IsPos XH
+ | IsPos p -> IsPos (XI p)
+ | IsNeg -> IsNeg
+
+(** val pdouble_mask : positive_mask -> positive_mask **)
+
+let pdouble_mask = function
+ | IsNul -> IsNul
+ | IsPos p -> IsPos (XO p)
+ | IsNeg -> IsNeg
+
+(** val pdouble_minus_two : positive -> positive_mask **)
+
+let pdouble_minus_two = function
+ | XI p -> IsPos (XO (XO p))
+ | XO p -> IsPos (XO (pdouble_minus_one p))
+ | XH -> IsNul
+
+(** val pminus_mask : positive -> positive -> positive_mask **)
+
+let rec pminus_mask x y =
+ match x with
+ | XI p ->
+ (match y with
+ | XI q0 -> pdouble_mask (pminus_mask p q0)
+ | XO q0 -> pdouble_plus_one_mask (pminus_mask p q0)
+ | XH -> IsPos (XO p))
+ | XO p ->
+ (match y with
+ | XI q0 -> pdouble_plus_one_mask (pminus_mask_carry p q0)
+ | XO q0 -> pdouble_mask (pminus_mask p q0)
+ | XH -> IsPos (pdouble_minus_one p))
+ | XH -> (match y with
+ | XH -> IsNul
+ | _ -> IsNeg)
+
+(** val pminus_mask_carry : positive -> positive -> positive_mask **)
+
+and pminus_mask_carry x y =
+ match x with
+ | XI p ->
+ (match y with
+ | XI q0 -> pdouble_plus_one_mask (pminus_mask_carry p q0)
+ | XO q0 -> pdouble_mask (pminus_mask p q0)
+ | XH -> IsPos (pdouble_minus_one p))
+ | XO p ->
+ (match y with
+ | XI q0 -> pdouble_mask (pminus_mask_carry p q0)
+ | XO q0 -> pdouble_plus_one_mask (pminus_mask_carry p q0)
+ | XH -> pdouble_minus_two p)
+ | XH -> IsNeg
+
+(** val pminus : positive -> positive -> positive **)
+
+let pminus x y =
+ match pminus_mask x y with
+ | IsPos z0 -> z0
+ | _ -> XH
+
+(** val pmult : positive -> positive -> positive **)
+
+let rec pmult x y =
+ match x with
+ | XI p -> pplus y (XO (pmult p y))
+ | XO p -> XO (pmult p y)
+ | XH -> y
+
+(** val pcompare : positive -> positive -> comparison -> comparison **)
+
+let rec pcompare x y r =
+ match x with
+ | XI p ->
+ (match y with
+ | XI q0 -> pcompare p q0 r
+ | XO q0 -> pcompare p q0 Gt
+ | XH -> Gt)
+ | XO p ->
+ (match y with
+ | XI q0 -> pcompare p q0 Lt
+ | XO q0 -> pcompare p q0 r
+ | XH -> Gt)
+ | XH -> (match y with
+ | XH -> r
+ | _ -> Lt)
+
+type n =
+ | N0
+ | Npos of positive
+
+type z =
+ | Z0
+ | Zpos of positive
+ | Zneg of positive
+
+(** val zdouble_plus_one : z -> z **)
+
+let zdouble_plus_one = function
+ | Z0 -> Zpos XH
+ | Zpos p -> Zpos (XI p)
+ | Zneg p -> Zneg (pdouble_minus_one p)
+
+(** val zdouble_minus_one : z -> z **)
+
+let zdouble_minus_one = function
+ | Z0 -> Zneg XH
+ | Zpos p -> Zpos (pdouble_minus_one p)
+ | Zneg p -> Zneg (XI p)
+
+(** val zdouble : z -> z **)
+
+let zdouble = function
+ | Z0 -> Z0
+ | Zpos p -> Zpos (XO p)
+ | Zneg p -> Zneg (XO p)
+
+(** val zPminus : positive -> positive -> z **)
+
+let rec zPminus x y =
+ match x with
+ | XI p ->
+ (match y with
+ | XI q0 -> zdouble (zPminus p q0)
+ | XO q0 -> zdouble_plus_one (zPminus p q0)
+ | XH -> Zpos (XO p))
+ | XO p ->
+ (match y with
+ | XI q0 -> zdouble_minus_one (zPminus p q0)
+ | XO q0 -> zdouble (zPminus p q0)
+ | XH -> Zpos (pdouble_minus_one p))
+ | XH ->
+ (match y with
+ | XI q0 -> Zneg (XO q0)
+ | XO q0 -> Zneg (pdouble_minus_one q0)
+ | XH -> Z0)
+
+(** val zplus : z -> z -> z **)
+
+let zplus x y =
+ match x with
+ | Z0 -> y
+ | Zpos x' ->
+ (match y with
+ | Z0 -> Zpos x'
+ | Zpos y' -> Zpos (pplus x' y')
+ | Zneg y' ->
+ (match pcompare x' y' Eq with
+ | Eq -> Z0
+ | Lt -> Zneg (pminus y' x')
+ | Gt -> Zpos (pminus x' y')))
+ | Zneg x' ->
+ (match y with
+ | Z0 -> Zneg x'
+ | Zpos y' ->
+ (match pcompare x' y' Eq with
+ | Eq -> Z0
+ | Lt -> Zpos (pminus y' x')
+ | Gt -> Zneg (pminus x' y'))
+ | Zneg y' -> Zneg (pplus x' y'))
+
+(** val zopp : z -> z **)
+
+let zopp = function
+ | Z0 -> Z0
+ | Zpos x0 -> Zneg x0
+ | Zneg x0 -> Zpos x0
+
+(** val zminus : z -> z -> z **)
+
+let zminus m n0 =
+ zplus m (zopp n0)
+
+(** val zmult : z -> z -> z **)
+
+let zmult x y =
+ match x with
+ | Z0 -> Z0
+ | Zpos x' ->
+ (match y with
+ | Z0 -> Z0
+ | Zpos y' -> Zpos (pmult x' y')
+ | Zneg y' -> Zneg (pmult x' y'))
+ | Zneg x' ->
+ (match y with
+ | Z0 -> Z0
+ | Zpos y' -> Zneg (pmult x' y')
+ | Zneg y' -> Zpos (pmult x' y'))
+
+(** val zcompare : z -> z -> comparison **)
+
+let zcompare x y =
+ match x with
+ | Z0 -> (match y with
+ | Z0 -> Eq
+ | Zpos y' -> Lt
+ | Zneg y' -> Gt)
+ | Zpos x' -> (match y with
+ | Zpos y' -> pcompare x' y' Eq
+ | _ -> Gt)
+ | Zneg x' ->
+ (match y with
+ | Zneg y' -> compOpp (pcompare x' y' Eq)
+ | _ -> Lt)
+
+(** val dcompare_inf : comparison -> sumbool sumor **)
+
+let dcompare_inf = function
+ | Eq -> Inleft Left
+ | Lt -> Inleft Right
+ | Gt -> Inright
+
+(** val zcompare_rec :
+ z -> z -> (__ -> 'a1) -> (__ -> 'a1) -> (__ -> 'a1) -> 'a1 **)
+
+let zcompare_rec x y h1 h2 h3 =
+ match dcompare_inf (zcompare x y) with
+ | Inleft x0 -> (match x0 with
+ | Left -> h1 __
+ | Right -> h2 __)
+ | Inright -> h3 __
+
+(** val z_gt_dec : z -> z -> sumbool **)
+
+let z_gt_dec x y =
+ zcompare_rec x y (fun _ -> Right) (fun _ -> Right) (fun _ -> Left)
+
+(** val zle_bool : z -> z -> bool **)
+
+let zle_bool x y =
+ match zcompare x y with
+ | Gt -> False
+ | _ -> True
+
+(** val zge_bool : z -> z -> bool **)
+
+let zge_bool x y =
+ match zcompare x y with
+ | Lt -> False
+ | _ -> True
+
+(** val zgt_bool : z -> z -> bool **)
+
+let zgt_bool x y =
+ match zcompare x y with
+ | Gt -> True
+ | _ -> False
+
+(** val zeq_bool : z -> z -> bool **)
+
+let zeq_bool x y =
+ match zcompare x y with
+ | Eq -> True
+ | _ -> False
+
+(** val n_of_nat : nat -> n **)
+
+let n_of_nat = function
+ | O -> N0
+ | S n' -> Npos (p_of_succ_nat n')
+
+(** val zdiv_eucl_POS : positive -> z -> (z, z) prod **)
+
+let rec zdiv_eucl_POS a b =
+ match a with
+ | XI a' ->
+ let Pair (q0, r) = zdiv_eucl_POS a' b in
+ let r' = zplus (zmult (Zpos (XO XH)) r) (Zpos XH) in
+ (match zgt_bool b r' with
+ | True -> Pair ((zmult (Zpos (XO XH)) q0), r')
+ | False -> Pair ((zplus (zmult (Zpos (XO XH)) q0) (Zpos XH)),
+ (zminus r' b)))
+ | XO a' ->
+ let Pair (q0, r) = zdiv_eucl_POS a' b in
+ let r' = zmult (Zpos (XO XH)) r in
+ (match zgt_bool b r' with
+ | True -> Pair ((zmult (Zpos (XO XH)) q0), r')
+ | False -> Pair ((zplus (zmult (Zpos (XO XH)) q0) (Zpos XH)),
+ (zminus r' b)))
+ | XH ->
+ (match zge_bool b (Zpos (XO XH)) with
+ | True -> Pair (Z0, (Zpos XH))
+ | False -> Pair ((Zpos XH), Z0))
+
+(** val zdiv_eucl : z -> z -> (z, z) prod **)
+
+let zdiv_eucl a b =
+ match a with
+ | Z0 -> Pair (Z0, Z0)
+ | Zpos a' ->
+ (match b with
+ | Z0 -> Pair (Z0, Z0)
+ | Zpos p -> zdiv_eucl_POS a' b
+ | Zneg b' ->
+ let Pair (q0, r) = zdiv_eucl_POS a' (Zpos b') in
+ (match r with
+ | Z0 -> Pair ((zopp q0), Z0)
+ | _ -> Pair ((zopp (zplus q0 (Zpos XH))), (zplus b r))))
+ | Zneg a' ->
+ (match b with
+ | Z0 -> Pair (Z0, Z0)
+ | Zpos p ->
+ let Pair (q0, r) = zdiv_eucl_POS a' b in
+ (match r with
+ | Z0 -> Pair ((zopp q0), Z0)
+ | _ -> Pair ((zopp (zplus q0 (Zpos XH))), (zminus b r)))
+ | Zneg b' ->
+ let Pair (q0, r) = zdiv_eucl_POS a' (Zpos b') in
+ Pair (q0, (zopp r)))
+
+type 'c pol =
+ | Pc of 'c
+ | Pinj of positive * 'c pol
+ | PX of 'c pol * positive * 'c pol
+
+(** val p0 : 'a1 -> 'a1 pol **)
+
+let p0 cO =
+ Pc cO
+
+(** val p1 : 'a1 -> 'a1 pol **)
+
+let p1 cI =
+ Pc cI
+
+(** val peq : ('a1 -> 'a1 -> bool) -> 'a1 pol -> 'a1 pol -> bool **)
+
+let rec peq ceqb p p' =
+ match p with
+ | Pc c -> (match p' with
+ | Pc c' -> ceqb c c'
+ | _ -> False)
+ | Pinj (j, q0) ->
+ (match p' with
+ | Pinj (j', q') ->
+ (match pcompare j j' Eq with
+ | Eq -> peq ceqb q0 q'
+ | _ -> False)
+ | _ -> False)
+ | PX (p2, i, q0) ->
+ (match p' with
+ | PX (p'0, i', q') ->
+ (match pcompare i i' Eq with
+ | Eq ->
+ (match peq ceqb p2 p'0 with
+ | True -> peq ceqb q0 q'
+ | False -> False)
+ | _ -> False)
+ | _ -> False)
+
+(** val mkPinj_pred : positive -> 'a1 pol -> 'a1 pol **)
+
+let mkPinj_pred j p =
+ match j with
+ | XI j0 -> Pinj ((XO j0), p)
+ | XO j0 -> Pinj ((pdouble_minus_one j0), p)
+ | XH -> p
+
+(** val mkPX :
+ 'a1 -> ('a1 -> 'a1 -> bool) -> 'a1 pol -> positive -> 'a1 pol -> 'a1 pol **)
+
+let mkPX cO ceqb p i q0 =
+ match p with
+ | Pc c ->
+ (match ceqb c cO with
+ | True ->
+ (match q0 with
+ | Pc c0 -> q0
+ | Pinj (j', q1) -> Pinj ((pplus XH j'), q1)
+ | PX (p2, p3, p4) -> Pinj (XH, q0))
+ | False -> PX (p, i, q0))
+ | Pinj (p2, p3) -> PX (p, i, q0)
+ | PX (p', i', q') ->
+ (match peq ceqb q' (p0 cO) with
+ | True -> PX (p', (pplus i' i), q0)
+ | False -> PX (p, i, q0))
+
+(** val mkXi : 'a1 -> 'a1 -> positive -> 'a1 pol **)
+
+let mkXi cO cI i =
+ PX ((p1 cI), i, (p0 cO))
+
+(** val mkX : 'a1 -> 'a1 -> 'a1 pol **)
+
+let mkX cO cI =
+ mkXi cO cI XH
+
+(** val popp : ('a1 -> 'a1) -> 'a1 pol -> 'a1 pol **)
+
+let rec popp copp = function
+ | Pc c -> Pc (copp c)
+ | Pinj (j, q0) -> Pinj (j, (popp copp q0))
+ | PX (p2, i, q0) -> PX ((popp copp p2), i, (popp copp q0))
+
+(** val paddC : ('a1 -> 'a1 -> 'a1) -> 'a1 pol -> 'a1 -> 'a1 pol **)
+
+let rec paddC cadd p c =
+ match p with
+ | Pc c1 -> Pc (cadd c1 c)
+ | Pinj (j, q0) -> Pinj (j, (paddC cadd q0 c))
+ | PX (p2, i, q0) -> PX (p2, i, (paddC cadd q0 c))
+
+(** val psubC : ('a1 -> 'a1 -> 'a1) -> 'a1 pol -> 'a1 -> 'a1 pol **)
+
+let rec psubC csub p c =
+ match p with
+ | Pc c1 -> Pc (csub c1 c)
+ | Pinj (j, q0) -> Pinj (j, (psubC csub q0 c))
+ | PX (p2, i, q0) -> PX (p2, i, (psubC csub q0 c))
+
+(** val paddI :
+ ('a1 -> 'a1 -> 'a1) -> ('a1 pol -> 'a1 pol -> 'a1 pol) -> 'a1 pol ->
+ positive -> 'a1 pol -> 'a1 pol **)
+
+let rec paddI cadd pop q0 j = function
+ | Pc c ->
+ let p2 = paddC cadd q0 c in
+ (match p2 with
+ | Pc c0 -> p2
+ | Pinj (j', q1) -> Pinj ((pplus j j'), q1)
+ | PX (p3, p4, p5) -> Pinj (j, p2))
+ | Pinj (j', q') ->
+ (match zPminus j' j with
+ | Z0 ->
+ let p2 = pop q' q0 in
+ (match p2 with
+ | Pc c -> p2
+ | Pinj (j'0, q1) -> Pinj ((pplus j j'0), q1)
+ | PX (p3, p4, p5) -> Pinj (j, p2))
+ | Zpos k ->
+ let p2 = pop (Pinj (k, q')) q0 in
+ (match p2 with
+ | Pc c -> p2
+ | Pinj (j'0, q1) -> Pinj ((pplus j j'0), q1)
+ | PX (p3, p4, p5) -> Pinj (j, p2))
+ | Zneg k ->
+ let p2 = paddI cadd pop q0 k q' in
+ (match p2 with
+ | Pc c -> p2
+ | Pinj (j'0, q1) -> Pinj ((pplus j' j'0), q1)
+ | PX (p3, p4, p5) -> Pinj (j', p2)))
+ | PX (p2, i, q') ->
+ (match j with
+ | XI j0 -> PX (p2, i, (paddI cadd pop q0 (XO j0) q'))
+ | XO j0 -> PX (p2, i, (paddI cadd pop q0 (pdouble_minus_one j0) q'))
+ | XH -> PX (p2, i, (pop q' q0)))
+
+(** val psubI :
+ ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1) -> ('a1 pol -> 'a1 pol -> 'a1 pol) ->
+ 'a1 pol -> positive -> 'a1 pol -> 'a1 pol **)
+
+let rec psubI cadd copp pop q0 j = function
+ | Pc c ->
+ let p2 = paddC cadd (popp copp q0) c in
+ (match p2 with
+ | Pc c0 -> p2
+ | Pinj (j', q1) -> Pinj ((pplus j j'), q1)
+ | PX (p3, p4, p5) -> Pinj (j, p2))
+ | Pinj (j', q') ->
+ (match zPminus j' j with
+ | Z0 ->
+ let p2 = pop q' q0 in
+ (match p2 with
+ | Pc c -> p2
+ | Pinj (j'0, q1) -> Pinj ((pplus j j'0), q1)
+ | PX (p3, p4, p5) -> Pinj (j, p2))
+ | Zpos k ->
+ let p2 = pop (Pinj (k, q')) q0 in
+ (match p2 with
+ | Pc c -> p2
+ | Pinj (j'0, q1) -> Pinj ((pplus j j'0), q1)
+ | PX (p3, p4, p5) -> Pinj (j, p2))
+ | Zneg k ->
+ let p2 = psubI cadd copp pop q0 k q' in
+ (match p2 with
+ | Pc c -> p2
+ | Pinj (j'0, q1) -> Pinj ((pplus j' j'0), q1)
+ | PX (p3, p4, p5) -> Pinj (j', p2)))
+ | PX (p2, i, q') ->
+ (match j with
+ | XI j0 -> PX (p2, i, (psubI cadd copp pop q0 (XO j0) q'))
+ | XO j0 -> PX (p2, i,
+ (psubI cadd copp pop q0 (pdouble_minus_one j0) q'))
+ | XH -> PX (p2, i, (pop q' q0)))
+
+(** val paddX :
+ 'a1 -> ('a1 -> 'a1 -> bool) -> ('a1 pol -> 'a1 pol -> 'a1 pol) -> 'a1 pol
+ -> positive -> 'a1 pol -> 'a1 pol **)
+
+let rec paddX cO ceqb pop p' i' p = match p with
+ | Pc c -> PX (p', i', p)
+ | Pinj (j, q') ->
+ (match j with
+ | XI j0 -> PX (p', i', (Pinj ((XO j0), q')))
+ | XO j0 -> PX (p', i', (Pinj ((pdouble_minus_one j0), q')))
+ | XH -> PX (p', i', q'))
+ | PX (p2, i, q') ->
+ (match zPminus i i' with
+ | Z0 -> mkPX cO ceqb (pop p2 p') i q'
+ | Zpos k -> mkPX cO ceqb (pop (PX (p2, k, (p0 cO))) p') i' q'
+ | Zneg k -> mkPX cO ceqb (paddX cO ceqb pop p' k p2) i q')
+
+(** val psubX :
+ 'a1 -> ('a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> ('a1 pol -> 'a1 pol -> 'a1
+ pol) -> 'a1 pol -> positive -> 'a1 pol -> 'a1 pol **)
+
+let rec psubX cO copp ceqb pop p' i' p = match p with
+ | Pc c -> PX ((popp copp p'), i', p)
+ | Pinj (j, q') ->
+ (match j with
+ | XI j0 -> PX ((popp copp p'), i', (Pinj ((XO j0), q')))
+ | XO j0 -> PX ((popp copp p'), i', (Pinj (
+ (pdouble_minus_one j0), q')))
+ | XH -> PX ((popp copp p'), i', q'))
+ | PX (p2, i, q') ->
+ (match zPminus i i' with
+ | Z0 -> mkPX cO ceqb (pop p2 p') i q'
+ | Zpos k -> mkPX cO ceqb (pop (PX (p2, k, (p0 cO))) p') i' q'
+ | Zneg k -> mkPX cO ceqb (psubX cO copp ceqb pop p' k p2) i q')
+
+(** val padd :
+ 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> 'a1 pol -> 'a1 pol
+ -> 'a1 pol **)
+
+let rec padd cO cadd ceqb p = function
+ | Pc c' -> paddC cadd p c'
+ | Pinj (j', q') -> paddI cadd (fun x x0 -> padd cO cadd ceqb x x0) q' j' p
+ | PX (p'0, i', q') ->
+ (match p with
+ | Pc c -> PX (p'0, i', (paddC cadd q' c))
+ | Pinj (j, q0) ->
+ (match j with
+ | XI j0 -> PX (p'0, i',
+ (padd cO cadd ceqb (Pinj ((XO j0), q0)) q'))
+ | XO j0 -> PX (p'0, i',
+ (padd cO cadd ceqb (Pinj ((pdouble_minus_one j0), q0))
+ q'))
+ | XH -> PX (p'0, i', (padd cO cadd ceqb q0 q')))
+ | PX (p2, i, q0) ->
+ (match zPminus i i' with
+ | Z0 ->
+ mkPX cO ceqb (padd cO cadd ceqb p2 p'0) i
+ (padd cO cadd ceqb q0 q')
+ | Zpos k ->
+ mkPX cO ceqb
+ (padd cO cadd ceqb (PX (p2, k, (p0 cO))) p'0) i'
+ (padd cO cadd ceqb q0 q')
+ | Zneg k ->
+ mkPX cO ceqb
+ (paddX cO ceqb (fun x x0 -> padd cO cadd ceqb x x0) p'0
+ k p2) i (padd cO cadd ceqb q0 q')))
+
+(** val psub :
+ 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1) -> ('a1
+ -> 'a1 -> bool) -> 'a1 pol -> 'a1 pol -> 'a1 pol **)
+
+let rec psub cO cadd csub copp ceqb p = function
+ | Pc c' -> psubC csub p c'
+ | Pinj (j', q') ->
+ psubI cadd copp (fun x x0 -> psub cO cadd csub copp ceqb x x0) q' j' p
+ | PX (p'0, i', q') ->
+ (match p with
+ | Pc c -> PX ((popp copp p'0), i', (paddC cadd (popp copp q') c))
+ | Pinj (j, q0) ->
+ (match j with
+ | XI j0 -> PX ((popp copp p'0), i',
+ (psub cO cadd csub copp ceqb (Pinj ((XO j0), q0)) q'))
+ | XO j0 -> PX ((popp copp p'0), i',
+ (psub cO cadd csub copp ceqb (Pinj
+ ((pdouble_minus_one j0), q0)) q'))
+ | XH -> PX ((popp copp p'0), i',
+ (psub cO cadd csub copp ceqb q0 q')))
+ | PX (p2, i, q0) ->
+ (match zPminus i i' with
+ | Z0 ->
+ mkPX cO ceqb (psub cO cadd csub copp ceqb p2 p'0) i
+ (psub cO cadd csub copp ceqb q0 q')
+ | Zpos k ->
+ mkPX cO ceqb
+ (psub cO cadd csub copp ceqb (PX (p2, k, (p0 cO))) p'0)
+ i' (psub cO cadd csub copp ceqb q0 q')
+ | Zneg k ->
+ mkPX cO ceqb
+ (psubX cO copp ceqb (fun x x0 ->
+ psub cO cadd csub copp ceqb x x0) p'0 k p2) i
+ (psub cO cadd csub copp ceqb q0 q')))
+
+(** val pmulC_aux :
+ 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> 'a1 pol -> 'a1 ->
+ 'a1 pol **)
+
+let rec pmulC_aux cO cmul ceqb p c =
+ match p with
+ | Pc c' -> Pc (cmul c' c)
+ | Pinj (j, q0) ->
+ let p2 = pmulC_aux cO cmul ceqb q0 c in
+ (match p2 with
+ | Pc c0 -> p2
+ | Pinj (j', q1) -> Pinj ((pplus j j'), q1)
+ | PX (p3, p4, p5) -> Pinj (j, p2))
+ | PX (p2, i, q0) ->
+ mkPX cO ceqb (pmulC_aux cO cmul ceqb p2 c) i
+ (pmulC_aux cO cmul ceqb q0 c)
+
+(** val pmulC :
+ 'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> 'a1 pol ->
+ 'a1 -> 'a1 pol **)
+
+let pmulC cO cI cmul ceqb p c =
+ match ceqb c cO with
+ | True -> p0 cO
+ | False ->
+ (match ceqb c cI with
+ | True -> p
+ | False -> pmulC_aux cO cmul ceqb p c)
+
+(** val pmulI :
+ 'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> ('a1 pol ->
+ 'a1 pol -> 'a1 pol) -> 'a1 pol -> positive -> 'a1 pol -> 'a1 pol **)
+
+let rec pmulI cO cI cmul ceqb pmul0 q0 j = function
+ | Pc c ->
+ let p2 = pmulC cO cI cmul ceqb q0 c in
+ (match p2 with
+ | Pc c0 -> p2
+ | Pinj (j', q1) -> Pinj ((pplus j j'), q1)
+ | PX (p3, p4, p5) -> Pinj (j, p2))
+ | Pinj (j', q') ->
+ (match zPminus j' j with
+ | Z0 ->
+ let p2 = pmul0 q' q0 in
+ (match p2 with
+ | Pc c -> p2
+ | Pinj (j'0, q1) -> Pinj ((pplus j j'0), q1)
+ | PX (p3, p4, p5) -> Pinj (j, p2))
+ | Zpos k ->
+ let p2 = pmul0 (Pinj (k, q')) q0 in
+ (match p2 with
+ | Pc c -> p2
+ | Pinj (j'0, q1) -> Pinj ((pplus j j'0), q1)
+ | PX (p3, p4, p5) -> Pinj (j, p2))
+ | Zneg k ->
+ let p2 = pmulI cO cI cmul ceqb pmul0 q0 k q' in
+ (match p2 with
+ | Pc c -> p2
+ | Pinj (j'0, q1) -> Pinj ((pplus j' j'0), q1)
+ | PX (p3, p4, p5) -> Pinj (j', p2)))
+ | PX (p', i', q') ->
+ (match j with
+ | XI j' ->
+ mkPX cO ceqb (pmulI cO cI cmul ceqb pmul0 q0 j p') i'
+ (pmulI cO cI cmul ceqb pmul0 q0 (XO j') q')
+ | XO j' ->
+ mkPX cO ceqb (pmulI cO cI cmul ceqb pmul0 q0 j p') i'
+ (pmulI cO cI cmul ceqb pmul0 q0 (pdouble_minus_one j') q')
+ | XH ->
+ mkPX cO ceqb (pmulI cO cI cmul ceqb pmul0 q0 XH p') i'
+ (pmul0 q' q0))
+
+(** val pmul :
+ 'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1
+ -> bool) -> 'a1 pol -> 'a1 pol -> 'a1 pol **)
+
+let rec pmul cO cI cadd cmul ceqb p p'' = match p'' with
+ | Pc c -> pmulC cO cI cmul ceqb p c
+ | Pinj (j', q') ->
+ pmulI cO cI cmul ceqb (fun x x0 -> pmul cO cI cadd cmul ceqb x x0) q'
+ j' p
+ | PX (p', i', q') ->
+ (match p with
+ | Pc c -> pmulC cO cI cmul ceqb p'' c
+ | Pinj (j, q0) ->
+ mkPX cO ceqb (pmul cO cI cadd cmul ceqb p p') i'
+ (match j with
+ | XI j0 ->
+ pmul cO cI cadd cmul ceqb (Pinj ((XO j0), q0)) q'
+ | XO j0 ->
+ pmul cO cI cadd cmul ceqb (Pinj
+ ((pdouble_minus_one j0), q0)) q'
+ | XH -> pmul cO cI cadd cmul ceqb q0 q')
+ | PX (p2, i, q0) ->
+ padd cO cadd ceqb
+ (mkPX cO ceqb
+ (padd cO cadd ceqb
+ (mkPX cO ceqb (pmul cO cI cadd cmul ceqb p2 p') i (p0 cO))
+ (pmul cO cI cadd cmul ceqb
+ (match q0 with
+ | Pc c -> q0
+ | Pinj (j', q1) -> Pinj ((pplus XH j'), q1)
+ | PX (p3, p4, p5) -> Pinj (XH, q0)) p')) i'
+ (p0 cO))
+ (mkPX cO ceqb
+ (pmulI cO cI cmul ceqb (fun x x0 ->
+ pmul cO cI cadd cmul ceqb x x0) q' XH p2) i
+ (pmul cO cI cadd cmul ceqb q0 q')))
+
+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 **)
+
+let mk_X cO cI j =
+ mkPinj_pred j (mkX cO cI)
+
+(** 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 **)
+
+let rec ppow_pos cO cI cadd cmul ceqb subst_l res p = function
+ | XI p3 ->
+ subst_l
+ (pmul cO cI cadd cmul ceqb
+ (ppow_pos cO cI cadd cmul ceqb subst_l
+ (ppow_pos cO cI cadd cmul ceqb subst_l res p p3) p p3) p)
+ | XO p3 ->
+ ppow_pos cO cI cadd cmul ceqb subst_l
+ (ppow_pos cO cI cadd cmul ceqb subst_l res p p3) p p3
+ | XH -> subst_l (pmul cO cI cadd cmul ceqb res p)
+
+(** val ppow_N :
+ 'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1
+ -> bool) -> ('a1 pol -> 'a1 pol) -> 'a1 pol -> n -> 'a1 pol **)
+
+let ppow_N cO cI cadd cmul ceqb subst_l p = function
+ | N0 -> p1 cI
+ | Npos p2 -> ppow_pos cO cI cadd cmul ceqb subst_l (p1 cI) p p2
+
+(** val norm_aux :
+ 'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1
+ -> 'a1) -> ('a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> 'a1 pExpr -> 'a1 pol **)
+
+let rec norm_aux cO cI cadd cmul csub copp ceqb = function
+ | PEc c -> Pc c
+ | PEX j -> mk_X cO cI j
+ | PEadd (pe1, pe2) ->
+ (match pe1 with
+ | PEopp pe3 ->
+ psub cO cadd csub copp ceqb
+ (norm_aux cO cI cadd cmul csub copp ceqb pe2)
+ (norm_aux cO cI cadd cmul csub copp ceqb pe3)
+ | _ ->
+ (match pe2 with
+ | PEopp pe3 ->
+ psub cO cadd csub copp ceqb
+ (norm_aux cO cI cadd cmul csub copp ceqb pe1)
+ (norm_aux cO cI cadd cmul csub copp ceqb pe3)
+ | _ ->
+ padd cO cadd ceqb
+ (norm_aux cO cI cadd cmul csub copp ceqb pe1)
+ (norm_aux cO cI cadd cmul csub copp ceqb pe2)))
+ | PEsub (pe1, pe2) ->
+ psub cO cadd csub copp ceqb
+ (norm_aux cO cI cadd cmul csub copp ceqb pe1)
+ (norm_aux cO cI cadd cmul csub copp ceqb pe2)
+ | PEmul (pe1, pe2) ->
+ pmul cO cI cadd cmul ceqb (norm_aux cO cI cadd cmul csub copp ceqb pe1)
+ (norm_aux cO cI cadd cmul csub copp ceqb pe2)
+ | PEopp pe1 -> popp copp (norm_aux cO cI cadd cmul csub copp ceqb pe1)
+ | PEpow (pe1, n0) ->
+ ppow_N cO cI cadd cmul ceqb (fun p -> p)
+ (norm_aux cO cI cadd cmul csub copp ceqb pe1) n0
+
+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
+
+type 'term' clause = 'term' list
+
+type 'term' cnf = 'term' clause list
+
+(** val tt : 'a1 cnf **)
+
+let tt =
+ Nil
+
+(** val ff : 'a1 cnf **)
+
+let ff =
+ Cons (Nil, Nil)
+
+(** val or_clause_cnf : 'a1 clause -> 'a1 cnf -> 'a1 cnf **)
+
+let or_clause_cnf t0 f =
+ map (fun x -> app t0 x) f
+
+(** val or_cnf : 'a1 cnf -> 'a1 cnf -> 'a1 cnf **)
+
+let rec or_cnf f f' =
+ match f with
+ | Nil -> tt
+ | Cons (e, rst) -> app (or_cnf rst f') (or_clause_cnf e f')
+
+(** val and_cnf : 'a1 cnf -> 'a1 cnf -> 'a1 cnf **)
+
+let and_cnf f1 f2 =
+ app f1 f2
+
+(** val xcnf :
+ ('a1 -> 'a2 cnf) -> ('a1 -> 'a2 cnf) -> bool -> 'a1 bFormula -> 'a2 cnf **)
+
+let rec xcnf normalise0 negate0 pol0 = function
+ | TT -> (match pol0 with
+ | True -> tt
+ | False -> ff)
+ | FF -> (match pol0 with
+ | True -> ff
+ | False -> tt)
+ | X -> ff
+ | A x -> (match pol0 with
+ | True -> normalise0 x
+ | False -> negate0 x)
+ | Cj (e1, e2) ->
+ (match pol0 with
+ | True ->
+ and_cnf (xcnf normalise0 negate0 pol0 e1)
+ (xcnf normalise0 negate0 pol0 e2)
+ | False ->
+ or_cnf (xcnf normalise0 negate0 pol0 e1)
+ (xcnf normalise0 negate0 pol0 e2))
+ | D (e1, e2) ->
+ (match pol0 with
+ | True ->
+ or_cnf (xcnf normalise0 negate0 pol0 e1)
+ (xcnf normalise0 negate0 pol0 e2)
+ | False ->
+ and_cnf (xcnf normalise0 negate0 pol0 e1)
+ (xcnf normalise0 negate0 pol0 e2))
+ | N e -> xcnf normalise0 negate0 (negb pol0) e
+ | I (e1, e2) ->
+ (match pol0 with
+ | True ->
+ or_cnf (xcnf normalise0 negate0 (negb pol0) e1)
+ (xcnf normalise0 negate0 pol0 e2)
+ | False ->
+ and_cnf (xcnf normalise0 negate0 (negb pol0) e1)
+ (xcnf normalise0 negate0 pol0 e2))
+
+(** val cnf_checker :
+ ('a1 list -> 'a2 -> bool) -> 'a1 cnf -> 'a2 list -> bool **)
+
+let rec cnf_checker checker f l =
+ match f with
+ | Nil -> True
+ | Cons (e, f0) ->
+ (match l with
+ | Nil -> False
+ | Cons (c, l0) ->
+ (match checker e c with
+ | True -> cnf_checker checker f0 l0
+ | False -> False))
+
+(** val tauto_checker :
+ ('a1 -> 'a2 cnf) -> ('a1 -> 'a2 cnf) -> ('a2 list -> 'a3 -> bool) -> 'a1
+ bFormula -> 'a3 list -> bool **)
+
+let tauto_checker normalise0 negate0 checker f w =
+ cnf_checker checker (xcnf normalise0 negate0 True f) w
+
+type 'c pExprC = 'c pExpr
+
+type 'c polC = 'c pol
+
+type op1 =
+ | Equal
+ | NonEqual
+ | Strict
+ | NonStrict
+
+type 'c nFormula = ('c pExprC, op1) prod
+
+type monoidMember = nat list
+
+type 'c coneMember =
+ | S_In of nat
+ | S_Ideal of 'c pExprC * 'c coneMember
+ | S_Square of 'c pExprC
+ | S_Monoid of monoidMember
+ | S_Mult of 'c coneMember * 'c coneMember
+ | S_Add of 'c coneMember * 'c coneMember
+ | S_Pos of 'c
+ | S_Z
+
+(** val nformula_times : 'a1 nFormula -> 'a1 nFormula -> 'a1 nFormula **)
+
+let nformula_times f f' =
+ let Pair (p, op) = f in
+ let Pair (p', op') = f' in
+ Pair ((PEmul (p, p')),
+ (match op with
+ | Equal -> Equal
+ | NonEqual -> NonEqual
+ | Strict -> op'
+ | NonStrict -> NonStrict))
+
+(** val nformula_plus : 'a1 nFormula -> 'a1 nFormula -> 'a1 nFormula **)
+
+let nformula_plus f f' =
+ let Pair (p, op) = f in
+ let Pair (p', op') = f' in
+ Pair ((PEadd (p, p')),
+ (match op with
+ | Equal -> op'
+ | NonEqual -> NonEqual
+ | Strict -> Strict
+ | NonStrict -> (match op' with
+ | Strict -> Strict
+ | _ -> NonStrict)))
+
+(** val eval_monoid :
+ 'a1 -> 'a1 nFormula list -> monoidMember -> 'a1 pExprC **)
+
+let rec eval_monoid cI l = function
+ | Nil -> PEc cI
+ | Cons (n0, ns0) -> PEmul
+ ((let Pair (q0, o) = nth n0 l (Pair ((PEc cI), NonEqual)) in
+ (match o with
+ | NonEqual -> q0
+ | _ -> PEc cI)), (eval_monoid cI l ns0))
+
+(** val eval_cone :
+ 'a1 -> 'a1 -> ('a1 -> 'a1 -> bool) -> ('a1 -> 'a1 -> bool) -> 'a1
+ nFormula list -> 'a1 coneMember -> 'a1 nFormula **)
+
+let rec eval_cone cO cI ceqb cleb l = function
+ | S_In n0 ->
+ let Pair (p, o) = nth n0 l (Pair ((PEc cO), Equal)) in
+ (match o with
+ | NonEqual -> Pair ((PEc cO), Equal)
+ | _ -> nth n0 l (Pair ((PEc cO), Equal)))
+ | S_Ideal (p, cm') ->
+ let f = eval_cone cO cI ceqb cleb l cm' in
+ let Pair (q0, op) = f in
+ (match op with
+ | Equal -> Pair ((PEmul (q0, p)), Equal)
+ | _ -> f)
+ | S_Square p -> Pair ((PEmul (p, p)), NonStrict)
+ | S_Monoid m -> let p = eval_monoid cI l m in Pair ((PEmul (p, p)), Strict)
+ | S_Mult (p, q0) ->
+ nformula_times (eval_cone cO cI ceqb cleb l p)
+ (eval_cone cO cI ceqb cleb l q0)
+ | S_Add (p, q0) ->
+ nformula_plus (eval_cone cO cI ceqb cleb l p)
+ (eval_cone cO cI ceqb cleb l q0)
+ | S_Pos c ->
+ (match match cleb cO c with
+ | True -> negb (ceqb cO c)
+ | False -> False with
+ | True -> Pair ((PEc c), Strict)
+ | False -> Pair ((PEc cO), Equal))
+ | S_Z -> Pair ((PEc cO), Equal)
+
+(** val normalise_pexpr :
+ 'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1
+ -> 'a1) -> ('a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> 'a1 pExprC -> 'a1 polC **)
+
+let normalise_pexpr cO cI cplus ctimes cminus copp ceqb x =
+ norm_aux cO cI cplus ctimes cminus copp ceqb x
+
+(** val check_inconsistent :
+ 'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1
+ -> 'a1) -> ('a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> ('a1 -> 'a1 -> bool)
+ -> 'a1 nFormula -> bool **)
+
+let check_inconsistent cO cI cplus ctimes cminus copp ceqb cleb = function
+ | Pair (e, op) ->
+ (match normalise_pexpr cO cI cplus ctimes cminus copp ceqb e with
+ | Pc c ->
+ (match op with
+ | Equal -> negb (ceqb c cO)
+ | NonEqual -> False
+ | Strict -> cleb c cO
+ | NonStrict ->
+ (match cleb c cO with
+ | True -> negb (ceqb c cO)
+ | False -> False))
+ | _ -> False)
+
+(** val check_normalised_formulas :
+ 'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1
+ -> 'a1) -> ('a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> ('a1 -> 'a1 -> bool)
+ -> 'a1 nFormula list -> 'a1 coneMember -> bool **)
+
+let check_normalised_formulas cO cI cplus ctimes cminus copp ceqb cleb l cm =
+ check_inconsistent cO cI cplus ctimes cminus copp ceqb cleb
+ (eval_cone cO cI ceqb cleb l cm)
+
+type op2 =
+ | OpEq
+ | OpNEq
+ | OpLe
+ | OpGe
+ | OpLt
+ | OpGt
+
+type 'c formula = { flhs : 'c pExprC; fop : op2; frhs : 'c pExprC }
+
+(** val flhs : 'a1 formula -> 'a1 pExprC **)
+
+let flhs x = x.flhs
+
+(** val fop : 'a1 formula -> op2 **)
+
+let fop x = x.fop
+
+(** val frhs : 'a1 formula -> 'a1 pExprC **)
+
+let frhs x = x.frhs
+
+(** val xnormalise : 'a1 formula -> 'a1 nFormula list **)
+
+let xnormalise t0 =
+ let { flhs = lhs; fop = o; frhs = rhs } = t0 in
+ (match o with
+ | OpEq -> Cons ((Pair ((PEsub (lhs, rhs)), Strict)), (Cons ((Pair
+ ((PEsub (rhs, lhs)), Strict)), Nil)))
+ | OpNEq -> Cons ((Pair ((PEsub (lhs, rhs)), Equal)), Nil)
+ | OpLe -> Cons ((Pair ((PEsub (lhs, rhs)), Strict)), Nil)
+ | OpGe -> Cons ((Pair ((PEsub (rhs, lhs)), Strict)), Nil)
+ | OpLt -> Cons ((Pair ((PEsub (lhs, rhs)), NonStrict)), Nil)
+ | OpGt -> Cons ((Pair ((PEsub (rhs, lhs)), NonStrict)), Nil))
+
+(** val cnf_normalise : 'a1 formula -> 'a1 nFormula cnf **)
+
+let cnf_normalise t0 =
+ map (fun x -> Cons (x, Nil)) (xnormalise t0)
+
+(** val xnegate : 'a1 formula -> 'a1 nFormula list **)
+
+let xnegate t0 =
+ let { flhs = lhs; fop = o; frhs = rhs } = t0 in
+ (match o with
+ | OpEq -> Cons ((Pair ((PEsub (lhs, rhs)), Equal)), Nil)
+ | OpNEq -> Cons ((Pair ((PEsub (lhs, rhs)), Strict)), (Cons ((Pair
+ ((PEsub (rhs, lhs)), Strict)), Nil)))
+ | OpLe -> Cons ((Pair ((PEsub (rhs, lhs)), NonStrict)), Nil)
+ | OpGe -> Cons ((Pair ((PEsub (lhs, rhs)), NonStrict)), Nil)
+ | OpLt -> Cons ((Pair ((PEsub (rhs, lhs)), Strict)), Nil)
+ | OpGt -> Cons ((Pair ((PEsub (lhs, rhs)), Strict)), Nil))
+
+(** val cnf_negate : 'a1 formula -> 'a1 nFormula cnf **)
+
+let cnf_negate t0 =
+ map (fun x -> Cons (x, Nil)) (xnegate t0)
+
+(** val simpl_expr :
+ 'a1 -> ('a1 -> 'a1 -> bool) -> 'a1 pExprC -> 'a1 pExprC **)
+
+let rec simpl_expr cI ceqb e = match e with
+ | PEadd (x, y) -> PEadd ((simpl_expr cI ceqb x), (simpl_expr cI ceqb y))
+ | PEmul (y, z0) ->
+ let y' = simpl_expr cI ceqb y in
+ (match y' with
+ | PEc c ->
+ (match ceqb c cI with
+ | True -> simpl_expr cI ceqb z0
+ | False -> PEmul (y', (simpl_expr cI ceqb z0)))
+ | _ -> PEmul (y', (simpl_expr cI ceqb z0)))
+ | _ -> e
+
+(** val simpl_cone :
+ 'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> 'a1
+ coneMember -> 'a1 coneMember **)
+
+let simpl_cone cO cI ctimes ceqb e = match e with
+ | S_Square t0 ->
+ (match simpl_expr cI ceqb t0 with
+ | PEc c ->
+ (match ceqb cO c with
+ | True -> S_Z
+ | False -> S_Pos (ctimes c c))
+ | _ -> S_Square (simpl_expr cI ceqb t0))
+ | S_Mult (t1, t2) ->
+ (match t1 with
+ | S_Mult (x, x0) ->
+ (match x with
+ | S_Pos p2 ->
+ (match t2 with
+ | S_Pos c -> S_Mult ((S_Pos (ctimes c p2)), x0)
+ | S_Z -> S_Z
+ | _ -> e)
+ | _ ->
+ (match x0 with
+ | S_Pos p2 ->
+ (match t2 with
+ | S_Pos c -> S_Mult ((S_Pos (ctimes c p2)), x)
+ | S_Z -> S_Z
+ | _ -> e)
+ | _ ->
+ (match t2 with
+ | S_Pos c ->
+ (match ceqb cI c with
+ | True -> t1
+ | False -> S_Mult (t1, t2))
+ | S_Z -> S_Z
+ | _ -> e)))
+ | S_Pos c ->
+ (match t2 with
+ | S_Mult (x, x0) ->
+ (match x with
+ | S_Pos p2 -> S_Mult ((S_Pos (ctimes c p2)), x0)
+ | _ ->
+ (match x0 with
+ | S_Pos p2 -> S_Mult ((S_Pos (ctimes c p2)), x)
+ | _ ->
+ (match ceqb cI c with
+ | True -> t2
+ | False -> S_Mult (t1, t2))))
+ | S_Add (y, z0) -> S_Add ((S_Mult ((S_Pos c), y)), (S_Mult
+ ((S_Pos c), z0)))
+ | S_Pos c0 -> S_Pos (ctimes c c0)
+ | S_Z -> S_Z
+ | _ ->
+ (match ceqb cI c with
+ | True -> t2
+ | False -> S_Mult (t1, t2)))
+ | S_Z -> S_Z
+ | _ ->
+ (match t2 with
+ | S_Pos c ->
+ (match ceqb cI c with
+ | True -> t1
+ | False -> S_Mult (t1, t2))
+ | S_Z -> S_Z
+ | _ -> e))
+ | S_Add (t1, t2) ->
+ (match t1 with
+ | S_Z -> t2
+ | _ -> (match t2 with
+ | S_Z -> t1
+ | _ -> S_Add (t1, t2)))
+ | _ -> e
+
+type q = { qnum : z; qden : positive }
+
+(** val qnum : q -> z **)
+
+let qnum x = x.qnum
+
+(** val qden : q -> positive **)
+
+let qden x = x.qden
+
+(** val qplus : q -> q -> q **)
+
+let qplus x y =
+ { qnum = (zplus (zmult x.qnum (Zpos y.qden)) (zmult y.qnum (Zpos x.qden)));
+ qden = (pmult x.qden y.qden) }
+
+(** val qmult : q -> q -> q **)
+
+let qmult x y =
+ { qnum = (zmult x.qnum y.qnum); qden = (pmult x.qden y.qden) }
+
+(** val qopp : q -> q **)
+
+let qopp x =
+ { qnum = (zopp x.qnum); qden = x.qden }
+
+(** val qminus : q -> q -> q **)
+
+let qminus x y =
+ qplus x (qopp y)
+
+type 'a t =
+ | Empty
+ | Leaf of 'a
+ | Node of 'a t * 'a * 'a t
+
+(** val find : 'a1 -> 'a1 t -> positive -> 'a1 **)
+
+let rec find default vm p =
+ match vm with
+ | Empty -> default
+ | Leaf i -> i
+ | Node (l, e, r) ->
+ (match p with
+ | XI p2 -> find default r p2
+ | XO p2 -> find default l p2
+ | XH -> e)
+
+type zWitness = z coneMember
+
+(** val zWeakChecker : z nFormula list -> z coneMember -> bool **)
+
+let zWeakChecker x x0 =
+ check_normalised_formulas Z0 (Zpos XH) zplus zmult zminus zopp zeq_bool
+ zle_bool x x0
+
+(** val xnormalise0 : z formula -> z nFormula list **)
+
+let xnormalise0 t0 =
+ let { flhs = lhs; fop = o; frhs = rhs } = t0 in
+ (match o with
+ | OpEq -> Cons ((Pair ((PEsub (lhs, (PEadd (rhs, (PEc (Zpos XH)))))),
+ NonStrict)), (Cons ((Pair ((PEsub (rhs, (PEadd (lhs, (PEc (Zpos
+ XH)))))), NonStrict)), Nil)))
+ | OpNEq -> Cons ((Pair ((PEsub (lhs, rhs)), Equal)), Nil)
+ | OpLe -> Cons ((Pair ((PEsub (lhs, (PEadd (rhs, (PEc (Zpos XH)))))),
+ NonStrict)), Nil)
+ | OpGe -> Cons ((Pair ((PEsub (rhs, (PEadd (lhs, (PEc (Zpos XH)))))),
+ NonStrict)), Nil)
+ | OpLt -> Cons ((Pair ((PEsub (lhs, rhs)), NonStrict)), Nil)
+ | OpGt -> Cons ((Pair ((PEsub (rhs, lhs)), NonStrict)), Nil))
+
+(** val normalise : z formula -> z nFormula cnf **)
+
+let normalise t0 =
+ map (fun x -> Cons (x, Nil)) (xnormalise0 t0)
+
+(** val xnegate0 : z formula -> z nFormula list **)
+
+let xnegate0 t0 =
+ let { flhs = lhs; fop = o; frhs = rhs } = t0 in
+ (match o with
+ | OpEq -> Cons ((Pair ((PEsub (lhs, rhs)), Equal)), Nil)
+ | OpNEq -> Cons ((Pair ((PEsub (lhs, (PEadd (rhs, (PEc (Zpos XH)))))),
+ NonStrict)), (Cons ((Pair ((PEsub (rhs, (PEadd (lhs, (PEc (Zpos
+ XH)))))), NonStrict)), Nil)))
+ | OpLe -> Cons ((Pair ((PEsub (rhs, lhs)), NonStrict)), Nil)
+ | OpGe -> Cons ((Pair ((PEsub (lhs, rhs)), NonStrict)), Nil)
+ | OpLt -> Cons ((Pair ((PEsub (rhs, (PEadd (lhs, (PEc (Zpos XH)))))),
+ NonStrict)), Nil)
+ | OpGt -> Cons ((Pair ((PEsub (lhs, (PEadd (rhs, (PEc (Zpos XH)))))),
+ NonStrict)), Nil))
+
+(** val negate : z formula -> z nFormula cnf **)
+
+let negate t0 =
+ map (fun x -> Cons (x, Nil)) (xnegate0 t0)
+
+(** val ceiling : z -> z -> z **)
+
+let ceiling a b =
+ let Pair (q0, r) = zdiv_eucl a b in
+ (match r with
+ | Z0 -> q0
+ | _ -> zplus q0 (Zpos XH))
+
+type proofTerm =
+ | RatProof of zWitness
+ | CutProof of z pExprC * q * zWitness * proofTerm
+ | EnumProof of q * z pExprC * q * zWitness * zWitness * proofTerm list
+
+(** val makeLb : z pExpr -> q -> z nFormula **)
+
+let makeLb v q0 =
+ let { qnum = n0; qden = d } = q0 in
+ Pair ((PEsub ((PEmul ((PEc (Zpos d)), v)), (PEc n0))), NonStrict)
+
+(** val qceiling : q -> z **)
+
+let qceiling q0 =
+ let { qnum = n0; qden = d } = q0 in ceiling n0 (Zpos d)
+
+(** val makeLbCut : z pExprC -> q -> z nFormula **)
+
+let makeLbCut v q0 =
+ Pair ((PEsub (v, (PEc (qceiling q0)))), NonStrict)
+
+(** val neg_nformula : z nFormula -> (z pExpr, op1) prod **)
+
+let neg_nformula = function
+ | Pair (e, o) -> Pair ((PEopp (PEadd (e, (PEc (Zpos XH))))), o)
+
+(** val cutChecker :
+ z nFormula list -> z pExpr -> q -> zWitness -> z nFormula option **)
+
+let cutChecker l e lb pf =
+ match zWeakChecker (Cons ((neg_nformula (makeLb e lb)), l)) pf with
+ | True -> Some (makeLbCut e lb)
+ | False -> None
+
+(** val zChecker : z nFormula list -> proofTerm -> bool **)
+
+let rec zChecker l = function
+ | RatProof pf0 -> zWeakChecker l pf0
+ | CutProof (e, q0, pf0, rst) ->
+ (match cutChecker l e q0 pf0 with
+ | Some c -> zChecker (Cons (c, l)) rst
+ | None -> False)
+ | EnumProof (lb, e, ub, pf1, pf2, rst) ->
+ (match cutChecker l e lb pf1 with
+ | Some n0 ->
+ (match cutChecker l (PEopp e) (qopp ub) pf2 with
+ | Some n1 ->
+ let rec label pfs lb0 ub0 =
+ match pfs with
+ | Nil ->
+ (match z_gt_dec lb0 ub0 with
+ | Left -> True
+ | Right -> False)
+ | Cons (pf0, rsr) ->
+ (match zChecker (Cons ((Pair ((PEsub (e, (PEc
+ lb0))), Equal)), l)) pf0 with
+ | True -> label rsr (zplus lb0 (Zpos XH)) ub0
+ | False -> False)
+ in label rst (qceiling lb) (zopp (qceiling (qopp ub)))
+ | None -> False)
+ | None -> False)
+
+(** val zTautoChecker : z formula bFormula -> proofTerm list -> bool **)
+
+let zTautoChecker f w =
+ tauto_checker normalise negate zChecker f w
+
+(** val map_cone : (nat -> nat) -> zWitness -> zWitness **)
+
+let rec map_cone f e = match e with
+ | S_In n0 -> S_In (f n0)
+ | S_Ideal (e0, cm) -> S_Ideal (e0, (map_cone f cm))
+ | S_Monoid l -> S_Monoid (map f l)
+ | S_Mult (cm1, cm2) -> S_Mult ((map_cone f cm1), (map_cone f cm2))
+ | S_Add (cm1, cm2) -> S_Add ((map_cone f cm1), (map_cone f cm2))
+ | _ -> e
+
+(** val indexes : zWitness -> nat list **)
+
+let rec indexes = function
+ | S_In n0 -> Cons (n0, Nil)
+ | S_Ideal (e0, cm) -> indexes cm
+ | S_Monoid l -> l
+ | S_Mult (cm1, cm2) -> app (indexes cm1) (indexes cm2)
+ | S_Add (cm1, cm2) -> app (indexes cm1) (indexes cm2)
+ | _ -> Nil
+
+(** val n_of_Z : z -> n **)
+
+let n_of_Z = function
+ | Zpos p -> Npos p
+ | _ -> N0
+
+(** val qeq_bool : q -> q -> bool **)
+
+let qeq_bool p q0 =
+ zeq_bool (zmult p.qnum (Zpos q0.qden)) (zmult q0.qnum (Zpos p.qden))
+
+(** val qle_bool : q -> q -> bool **)
+
+let qle_bool x y =
+ zle_bool (zmult x.qnum (Zpos y.qden)) (zmult y.qnum (Zpos x.qden))
+
+type qWitness = q coneMember
+
+(** val qWeakChecker : q nFormula list -> q coneMember -> bool **)
+
+let qWeakChecker x x0 =
+ check_normalised_formulas { qnum = Z0; qden = XH } { qnum = (Zpos XH);
+ qden = XH } qplus qmult qminus qopp qeq_bool qle_bool x x0
+
+(** val qTautoChecker : q formula bFormula -> qWitness list -> bool **)
+
+let qTautoChecker f w =
+ tauto_checker (fun x -> cnf_normalise x) (fun x ->
+ cnf_negate x) qWeakChecker f w
+
generated by cgit v1.2.3 (git 2.39.1) at 2024-11-12 18:32:14 +0000