summaryrefslogtreecommitdiff
path: root/plugins/micromega/micromega.ml
diff options
context:
space:
mode:
Diffstat (limited to 'plugins/micromega/micromega.ml')
-rw-r--r--plugins/micromega/micromega.ml4625
1 files changed, 3484 insertions, 1141 deletions
diff --git a/plugins/micromega/micromega.ml b/plugins/micromega/micromega.ml
index c350ed0f..564126d2 100644
--- a/plugins/micromega/micromega.ml
+++ b/plugins/micromega/micromega.ml
@@ -1,447 +1,2786 @@
+type __ = Obj.t
+let __ = let rec f _ = Obj.repr f in Obj.repr f
+
(** val negb : bool -> bool **)
let negb = function
- | true -> false
- | false -> true
+| true -> false
+| false -> true
type nat =
- | O
- | S of nat
+| O
+| S of nat
+
+(** val fst : ('a1 * 'a2) -> 'a1 **)
+
+let fst = function
+| x,y -> x
+
+(** val snd : ('a1 * 'a2) -> 'a2 **)
+
+let snd = function
+| x,y -> y
+
+(** val app : 'a1 list -> 'a1 list -> 'a1 list **)
+
+let rec app l m =
+ match l with
+ | [] -> m
+ | a::l1 -> a::(app l1 m)
type comparison =
- | Eq
- | Lt
- | Gt
+| Eq
+| Lt
+| Gt
(** val compOpp : comparison -> comparison **)
let compOpp = function
- | Eq -> Eq
- | Lt -> Gt
- | Gt -> Lt
+| Eq -> Eq
+| Lt -> Gt
+| Gt -> Lt
-(** val plus : nat -> nat -> nat **)
+type compareSpecT =
+| CompEqT
+| CompLtT
+| CompGtT
-let rec plus n0 m =
- match n0 with
- | O -> m
- | S p -> S (plus p m)
+(** val compareSpec2Type : comparison -> compareSpecT **)
-(** val app : 'a1 list -> 'a1 list -> 'a1 list **)
+let compareSpec2Type = function
+| Eq -> CompEqT
+| Lt -> CompLtT
+| Gt -> CompGtT
-let rec app l m =
- match l with
- | [] -> m
- | a :: l1 -> a :: (app l1 m)
+type 'a compSpecT = compareSpecT
-(** val nth : nat -> 'a1 list -> 'a1 -> 'a1 **)
+(** val compSpec2Type : 'a1 -> 'a1 -> comparison -> 'a1 compSpecT **)
-let rec nth n0 l default =
+let compSpec2Type x y c =
+ compareSpec2Type c
+
+type 'a sig0 =
+ 'a
+ (* singleton inductive, whose constructor was exist *)
+
+(** val plus : nat -> nat -> nat **)
+
+let rec plus n0 m =
match n0 with
- | O -> (match l with
- | [] -> default
- | x :: l' -> x)
- | S m -> (match l with
- | [] -> default
- | x :: t0 -> nth m t0 default)
+ | O -> m
+ | S p -> S (plus p m)
-(** val map : ('a1 -> 'a2) -> 'a1 list -> 'a2 list **)
+(** val nat_iter : nat -> ('a1 -> 'a1) -> 'a1 -> 'a1 **)
-let rec map f = function
- | [] -> []
- | a :: t0 -> (f a) :: (map f t0)
+let rec nat_iter n0 f x =
+ match n0 with
+ | O -> x
+ | S n' -> f (nat_iter n' f x)
type positive =
- | XI of positive
- | XO of positive
- | XH
+| XI of positive
+| XO of positive
+| XH
-(** val psucc : positive -> positive **)
+type n =
+| N0
+| Npos of positive
-let rec psucc = function
- | XI p -> XO (psucc p)
+type z =
+| Z0
+| Zpos of positive
+| Zneg of positive
+
+module type TotalOrder' =
+ sig
+ type t
+ end
+
+module MakeOrderTac =
+ functor (O:TotalOrder') ->
+ struct
+
+ end
+
+module MaxLogicalProperties =
+ functor (O:TotalOrder') ->
+ functor (M:sig
+ val max : O.t -> O.t -> O.t
+ end) ->
+ struct
+ module T = MakeOrderTac(O)
+ end
+
+module Pos =
+ struct
+ type t = positive
+
+ (** val succ : positive -> positive **)
+
+ let rec succ = function
+ | XI p -> XO (succ p)
| XO p -> XI p
| XH -> XO XH
-
-(** val pplus : positive -> positive -> positive **)
-
-let rec pplus x y =
- match x with
+
+ (** val add : positive -> positive -> positive **)
+
+ let rec add 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))
+ (match y with
+ | XI q0 -> XO (add_carry p q0)
+ | XO q0 -> XI (add p q0)
+ | XH -> XO (succ p))
| XO p ->
- (match y with
- | XI q0 -> XI (pplus p q0)
- | XO q0 -> XO (pplus p q0)
- | XH -> XI p)
+ (match y with
+ | XI q0 -> XI (add p q0)
+ | XO q0 -> XO (add 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
+ (match y with
+ | XI q0 -> XO (succ q0)
+ | XO q0 -> XI q0
+ | XH -> XO XH)
+
+ (** val add_carry : positive -> positive -> positive **)
+
+ and add_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))
+ (match y with
+ | XI q0 -> XI (add_carry p q0)
+ | XO q0 -> XO (add_carry p q0)
+ | XH -> XI (succ p))
| XO p ->
- (match y with
- | XI q0 -> XO (pplus_carry p q0)
- | XO q0 -> XI (pplus p q0)
- | XH -> XO (psucc p))
+ (match y with
+ | XI q0 -> XO (add_carry p q0)
+ | XO q0 -> XI (add p q0)
+ | XH -> XO (succ 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
+ (match y with
+ | XI q0 -> XI (succ q0)
+ | XO q0 -> XO (succ q0)
+ | XH -> XI XH)
+
+ (** val pred_double : positive -> positive **)
+
+ let rec pred_double = function
| XI p -> XI (XO p)
- | XO p -> XI (pdouble_minus_one p)
+ | XO p -> XI (pred_double p)
| XH -> XH
-
-type positive_mask =
+
+ (** val pred : positive -> positive **)
+
+ let pred = function
+ | XI p -> XO p
+ | XO p -> pred_double p
+ | XH -> XH
+
+ (** val pred_N : positive -> n **)
+
+ let pred_N = function
+ | XI p -> Npos (XO p)
+ | XO p -> Npos (pred_double p)
+ | XH -> N0
+
+ type mask =
| IsNul
| IsPos of positive
| IsNeg
-
-(** val pdouble_plus_one_mask : positive_mask -> positive_mask **)
-
-let pdouble_plus_one_mask = function
+
+ (** val mask_rect : 'a1 -> (positive -> 'a1) -> 'a1 -> mask -> 'a1 **)
+
+ let mask_rect f f0 f1 = function
+ | IsNul -> f
+ | IsPos x -> f0 x
+ | IsNeg -> f1
+
+ (** val mask_rec : 'a1 -> (positive -> 'a1) -> 'a1 -> mask -> 'a1 **)
+
+ let mask_rec f f0 f1 = function
+ | IsNul -> f
+ | IsPos x -> f0 x
+ | IsNeg -> f1
+
+ (** val succ_double_mask : mask -> mask **)
+
+ let succ_double_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
+
+ (** val double_mask : mask -> mask **)
+
+ let double_mask = function
| IsPos p -> IsPos (XO p)
- | IsNeg -> IsNeg
-
-(** val pdouble_minus_two : positive -> positive_mask **)
-
-let pdouble_minus_two = function
+ | x0 -> x0
+
+ (** val double_pred_mask : positive -> mask **)
+
+ let double_pred_mask = function
| XI p -> IsPos (XO (XO p))
- | XO p -> IsPos (XO (pdouble_minus_one p))
+ | XO p -> IsPos (XO (pred_double p))
| XH -> IsNul
-
-(** val pminus_mask : positive -> positive -> positive_mask **)
-
-let rec pminus_mask x y =
- match x with
+
+ (** val pred_mask : mask -> mask **)
+
+ let pred_mask = function
+ | IsPos q0 ->
+ (match q0 with
+ | XH -> IsNul
+ | _ -> IsPos (pred q0))
+ | _ -> IsNeg
+
+ (** val sub_mask : positive -> positive -> mask **)
+
+ let rec sub_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))
+ (match y with
+ | XI q0 -> double_mask (sub_mask p q0)
+ | XO q0 -> succ_double_mask (sub_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
+ (match y with
+ | XI q0 -> succ_double_mask (sub_mask_carry p q0)
+ | XO q0 -> double_mask (sub_mask p q0)
+ | XH -> IsPos (pred_double p))
+ | XH ->
+ (match y with
+ | XH -> IsNul
+ | _ -> IsNeg)
+
+ (** val sub_mask_carry : positive -> positive -> mask **)
+
+ and sub_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))
+ (match y with
+ | XI q0 -> succ_double_mask (sub_mask_carry p q0)
+ | XO q0 -> double_mask (sub_mask p q0)
+ | XH -> IsPos (pred_double 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)
+ (match y with
+ | XI q0 -> double_mask (sub_mask_carry p q0)
+ | XO q0 -> succ_double_mask (sub_mask_carry p q0)
+ | XH -> double_pred_mask p)
| XH -> IsNeg
-
-(** val pminus : positive -> positive -> positive **)
-
-let pminus x y =
- match pminus_mask x y with
+
+ (** val sub : positive -> positive -> positive **)
+
+ let sub x y =
+ match sub_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)
+
+ (** val mul : positive -> positive -> positive **)
+
+ let rec mul x y =
+ match x with
+ | XI p -> add y (XO (mul p y))
+ | XO p -> XO (mul p y)
| XH -> y
-
-(** val pcompare : positive -> positive -> comparison -> comparison **)
-
-let rec pcompare x y r =
- match x with
+
+ (** val iter : positive -> ('a1 -> 'a1) -> 'a1 -> 'a1 **)
+
+ let rec iter n0 f x =
+ match n0 with
+ | XI n' -> f (iter n' f (iter n' f x))
+ | XO n' -> iter n' f (iter n' f x)
+ | XH -> f x
+
+ (** val pow : positive -> positive -> positive **)
+
+ let pow x y =
+ iter y (mul x) XH
+
+ (** val div2 : positive -> positive **)
+
+ let div2 = function
+ | XI p2 -> p2
+ | XO p2 -> p2
+ | XH -> XH
+
+ (** val div2_up : positive -> positive **)
+
+ let div2_up = function
+ | XI p2 -> succ p2
+ | XO p2 -> p2
+ | XH -> XH
+
+ (** val size_nat : positive -> nat **)
+
+ let rec size_nat = function
+ | XI p2 -> S (size_nat p2)
+ | XO p2 -> S (size_nat p2)
+ | XH -> S O
+
+ (** val size : positive -> positive **)
+
+ let rec size = function
+ | XI p2 -> succ (size p2)
+ | XO p2 -> succ (size p2)
+ | XH -> XH
+
+ (** val compare_cont : positive -> positive -> comparison -> comparison **)
+
+ let rec compare_cont x y r =
+ match x with
+ | XI p ->
+ (match y with
+ | XI q0 -> compare_cont p q0 r
+ | XO q0 -> compare_cont p q0 Gt
+ | XH -> Gt)
+ | XO p ->
+ (match y with
+ | XI q0 -> compare_cont p q0 Lt
+ | XO q0 -> compare_cont p q0 r
+ | XH -> Gt)
+ | XH ->
+ (match y with
+ | XH -> r
+ | _ -> Lt)
+
+ (** val compare : positive -> positive -> comparison **)
+
+ let compare x y =
+ compare_cont x y Eq
+
+ (** val min : positive -> positive -> positive **)
+
+ let min p p' =
+ match compare p p' with
+ | Gt -> p'
+ | _ -> p
+
+ (** val max : positive -> positive -> positive **)
+
+ let max p p' =
+ match compare p p' with
+ | Gt -> p
+ | _ -> p'
+
+ (** val eqb : positive -> positive -> bool **)
+
+ let rec eqb p q0 =
+ match p with
+ | XI p2 ->
+ (match q0 with
+ | XI q1 -> eqb p2 q1
+ | _ -> false)
+ | XO p2 ->
+ (match q0 with
+ | XO q1 -> eqb p2 q1
+ | _ -> false)
+ | XH ->
+ (match q0 with
+ | XH -> true
+ | _ -> false)
+
+ (** val leb : positive -> positive -> bool **)
+
+ let leb x y =
+ match compare x y with
+ | Gt -> false
+ | _ -> true
+
+ (** val ltb : positive -> positive -> bool **)
+
+ let ltb x y =
+ match compare x y with
+ | Lt -> true
+ | _ -> false
+
+ (** val sqrtrem_step :
+ (positive -> positive) -> (positive -> positive) -> (positive * mask)
+ -> positive * mask **)
+
+ let sqrtrem_step f g = function
+ | s,y ->
+ (match y with
+ | IsPos r ->
+ let s' = XI (XO s) in
+ let r' = g (f r) in
+ if leb s' r' then (XI s),(sub_mask r' s') else (XO s),(IsPos r')
+ | _ -> (XO s),(sub_mask (g (f XH)) (XO (XO XH))))
+
+ (** val sqrtrem : positive -> positive * mask **)
+
+ let rec sqrtrem = function
+ | XI p2 ->
+ (match p2 with
+ | XI p3 -> sqrtrem_step (fun x -> XI x) (fun x -> XI x) (sqrtrem p3)
+ | XO p3 -> sqrtrem_step (fun x -> XO x) (fun x -> XI x) (sqrtrem p3)
+ | XH -> XH,(IsPos (XO XH)))
+ | XO p2 ->
+ (match p2 with
+ | XI p3 -> sqrtrem_step (fun x -> XI x) (fun x -> XO x) (sqrtrem p3)
+ | XO p3 -> sqrtrem_step (fun x -> XO x) (fun x -> XO x) (sqrtrem p3)
+ | XH -> XH,(IsPos XH))
+ | XH -> XH,IsNul
+
+ (** val sqrt : positive -> positive **)
+
+ let sqrt p =
+ fst (sqrtrem p)
+
+ (** val gcdn : nat -> positive -> positive -> positive **)
+
+ let rec gcdn n0 a b =
+ match n0 with
+ | O -> XH
+ | S n1 ->
+ (match a with
+ | XI a' ->
+ (match b with
+ | XI b' ->
+ (match compare a' b' with
+ | Eq -> a
+ | Lt -> gcdn n1 (sub b' a') a
+ | Gt -> gcdn n1 (sub a' b') b)
+ | XO b0 -> gcdn n1 a b0
+ | XH -> XH)
+ | XO a0 ->
+ (match b with
+ | XI p -> gcdn n1 a0 b
+ | XO b0 -> XO (gcdn n1 a0 b0)
+ | XH -> XH)
+ | XH -> XH)
+
+ (** val gcd : positive -> positive -> positive **)
+
+ let gcd a b =
+ gcdn (plus (size_nat a) (size_nat b)) a b
+
+ (** val ggcdn :
+ nat -> positive -> positive -> positive * (positive * positive) **)
+
+ let rec ggcdn n0 a b =
+ match n0 with
+ | O -> XH,(a,b)
+ | S n1 ->
+ (match a with
+ | XI a' ->
+ (match b with
+ | XI b' ->
+ (match compare a' b' with
+ | Eq -> a,(XH,XH)
+ | Lt ->
+ let g,p = ggcdn n1 (sub b' a') a in
+ let ba,aa = p in g,(aa,(add aa (XO ba)))
+ | Gt ->
+ let g,p = ggcdn n1 (sub a' b') b in
+ let ab,bb = p in g,((add bb (XO ab)),bb))
+ | XO b0 ->
+ let g,p = ggcdn n1 a b0 in let aa,bb = p in g,(aa,(XO bb))
+ | XH -> XH,(a,XH))
+ | XO a0 ->
+ (match b with
+ | XI p ->
+ let g,p2 = ggcdn n1 a0 b in let aa,bb = p2 in g,((XO aa),bb)
+ | XO b0 -> let g,p = ggcdn n1 a0 b0 in (XO g),p
+ | XH -> XH,(a,XH))
+ | XH -> XH,(XH,b))
+
+ (** val ggcd : positive -> positive -> positive * (positive * positive) **)
+
+ let ggcd a b =
+ ggcdn (plus (size_nat a) (size_nat b)) a b
+
+ (** val coq_Nsucc_double : n -> n **)
+
+ let coq_Nsucc_double = function
+ | N0 -> Npos XH
+ | Npos p -> Npos (XI p)
+
+ (** val coq_Ndouble : n -> n **)
+
+ let coq_Ndouble = function
+ | N0 -> N0
+ | Npos p -> Npos (XO p)
+
+ (** val coq_lor : positive -> positive -> positive **)
+
+ let rec coq_lor p q0 =
+ match p with
+ | XI p2 ->
+ (match q0 with
+ | XI q1 -> XI (coq_lor p2 q1)
+ | XO q1 -> XI (coq_lor p2 q1)
+ | XH -> p)
+ | XO p2 ->
+ (match q0 with
+ | XI q1 -> XI (coq_lor p2 q1)
+ | XO q1 -> XO (coq_lor p2 q1)
+ | XH -> XI p2)
+ | XH ->
+ (match q0 with
+ | XO q1 -> XI q1
+ | _ -> q0)
+
+ (** val coq_land : positive -> positive -> n **)
+
+ let rec coq_land p q0 =
+ match p with
+ | XI p2 ->
+ (match q0 with
+ | XI q1 -> coq_Nsucc_double (coq_land p2 q1)
+ | XO q1 -> coq_Ndouble (coq_land p2 q1)
+ | XH -> Npos XH)
+ | XO p2 ->
+ (match q0 with
+ | XI q1 -> coq_Ndouble (coq_land p2 q1)
+ | XO q1 -> coq_Ndouble (coq_land p2 q1)
+ | XH -> N0)
+ | XH ->
+ (match q0 with
+ | XO q1 -> N0
+ | _ -> Npos XH)
+
+ (** val ldiff : positive -> positive -> n **)
+
+ let rec ldiff p q0 =
+ match p with
+ | XI p2 ->
+ (match q0 with
+ | XI q1 -> coq_Ndouble (ldiff p2 q1)
+ | XO q1 -> coq_Nsucc_double (ldiff p2 q1)
+ | XH -> Npos (XO p2))
+ | XO p2 ->
+ (match q0 with
+ | XI q1 -> coq_Ndouble (ldiff p2 q1)
+ | XO q1 -> coq_Ndouble (ldiff p2 q1)
+ | XH -> Npos p)
+ | XH ->
+ (match q0 with
+ | XO q1 -> Npos XH
+ | _ -> N0)
+
+ (** val coq_lxor : positive -> positive -> n **)
+
+ let rec coq_lxor p q0 =
+ match p with
+ | XI p2 ->
+ (match q0 with
+ | XI q1 -> coq_Ndouble (coq_lxor p2 q1)
+ | XO q1 -> coq_Nsucc_double (coq_lxor p2 q1)
+ | XH -> Npos (XO p2))
+ | XO p2 ->
+ (match q0 with
+ | XI q1 -> coq_Nsucc_double (coq_lxor p2 q1)
+ | XO q1 -> coq_Ndouble (coq_lxor p2 q1)
+ | XH -> Npos (XI p2))
+ | XH ->
+ (match q0 with
+ | XI q1 -> Npos (XO q1)
+ | XO q1 -> Npos (XI q1)
+ | XH -> N0)
+
+ (** val shiftl_nat : positive -> nat -> positive **)
+
+ let shiftl_nat p n0 =
+ nat_iter n0 (fun x -> XO x) p
+
+ (** val shiftr_nat : positive -> nat -> positive **)
+
+ let shiftr_nat p n0 =
+ nat_iter n0 div2 p
+
+ (** val shiftl : positive -> n -> positive **)
+
+ let shiftl p = function
+ | N0 -> p
+ | Npos n1 -> iter n1 (fun x -> XO x) p
+
+ (** val shiftr : positive -> n -> positive **)
+
+ let shiftr p = function
+ | N0 -> p
+ | Npos n1 -> iter n1 div2 p
+
+ (** val testbit_nat : positive -> nat -> bool **)
+
+ let rec testbit_nat p n0 =
+ match p with
+ | XI p2 ->
+ (match n0 with
+ | O -> true
+ | S n' -> testbit_nat p2 n')
+ | XO p2 ->
+ (match n0 with
+ | O -> false
+ | S n' -> testbit_nat p2 n')
+ | XH ->
+ (match n0 with
+ | O -> true
+ | S n1 -> false)
+
+ (** val testbit : positive -> n -> bool **)
+
+ let rec testbit p n0 =
+ match p with
+ | XI p2 ->
+ (match n0 with
+ | N0 -> true
+ | Npos n1 -> testbit p2 (pred_N n1))
+ | XO p2 ->
+ (match n0 with
+ | N0 -> false
+ | Npos n1 -> testbit p2 (pred_N n1))
+ | XH ->
+ (match n0 with
+ | N0 -> true
+ | Npos p2 -> false)
+
+ (** val iter_op : ('a1 -> 'a1 -> 'a1) -> positive -> 'a1 -> 'a1 **)
+
+ let rec iter_op op p a =
+ match p with
+ | XI p2 -> op a (iter_op op p2 (op a a))
+ | XO p2 -> iter_op op p2 (op a a)
+ | XH -> a
+
+ (** val to_nat : positive -> nat **)
+
+ let to_nat x =
+ iter_op plus x (S O)
+
+ (** val of_nat : nat -> positive **)
+
+ let rec of_nat = function
+ | O -> XH
+ | S x ->
+ (match x with
+ | O -> XH
+ | S n1 -> succ (of_nat x))
+
+ (** val of_succ_nat : nat -> positive **)
+
+ let rec of_succ_nat = function
+ | O -> XH
+ | S x -> succ (of_succ_nat x)
+ end
+
+module Coq_Pos =
+ struct
+ module Coq__1 = struct
+ type t = positive
+ end
+ type t = Coq__1.t
+
+ (** val succ : positive -> positive **)
+
+ let rec succ = function
+ | XI p -> XO (succ p)
+ | XO p -> XI p
+ | XH -> XO XH
+
+ (** val add : positive -> positive -> positive **)
+
+ let rec add x y =
+ match x with
+ | XI p ->
+ (match y with
+ | XI q0 -> XO (add_carry p q0)
+ | XO q0 -> XI (add p q0)
+ | XH -> XO (succ p))
+ | XO p ->
+ (match y with
+ | XI q0 -> XI (add p q0)
+ | XO q0 -> XO (add p q0)
+ | XH -> XI p)
+ | XH ->
+ (match y with
+ | XI q0 -> XO (succ q0)
+ | XO q0 -> XI q0
+ | XH -> XO XH)
+
+ (** val add_carry : positive -> positive -> positive **)
+
+ and add_carry x y =
+ match x with
+ | XI p ->
+ (match y with
+ | XI q0 -> XI (add_carry p q0)
+ | XO q0 -> XO (add_carry p q0)
+ | XH -> XI (succ p))
+ | XO p ->
+ (match y with
+ | XI q0 -> XO (add_carry p q0)
+ | XO q0 -> XI (add p q0)
+ | XH -> XO (succ p))
+ | XH ->
+ (match y with
+ | XI q0 -> XI (succ q0)
+ | XO q0 -> XO (succ q0)
+ | XH -> XI XH)
+
+ (** val pred_double : positive -> positive **)
+
+ let rec pred_double = function
+ | XI p -> XI (XO p)
+ | XO p -> XI (pred_double p)
+ | XH -> XH
+
+ (** val pred : positive -> positive **)
+
+ let pred = function
+ | XI p -> XO p
+ | XO p -> pred_double p
+ | XH -> XH
+
+ (** val pred_N : positive -> n **)
+
+ let pred_N = function
+ | XI p -> Npos (XO p)
+ | XO p -> Npos (pred_double p)
+ | XH -> N0
+
+ type mask = Pos.mask =
+ | IsNul
+ | IsPos of positive
+ | IsNeg
+
+ (** val mask_rect : 'a1 -> (positive -> 'a1) -> 'a1 -> mask -> 'a1 **)
+
+ let mask_rect f f0 f1 = function
+ | IsNul -> f
+ | IsPos x -> f0 x
+ | IsNeg -> f1
+
+ (** val mask_rec : 'a1 -> (positive -> 'a1) -> 'a1 -> mask -> 'a1 **)
+
+ let mask_rec f f0 f1 = function
+ | IsNul -> f
+ | IsPos x -> f0 x
+ | IsNeg -> f1
+
+ (** val succ_double_mask : mask -> mask **)
+
+ let succ_double_mask = function
+ | IsNul -> IsPos XH
+ | IsPos p -> IsPos (XI p)
+ | IsNeg -> IsNeg
+
+ (** val double_mask : mask -> mask **)
+
+ let double_mask = function
+ | IsPos p -> IsPos (XO p)
+ | x0 -> x0
+
+ (** val double_pred_mask : positive -> mask **)
+
+ let double_pred_mask = function
+ | XI p -> IsPos (XO (XO p))
+ | XO p -> IsPos (XO (pred_double p))
+ | XH -> IsNul
+
+ (** val pred_mask : mask -> mask **)
+
+ let pred_mask = function
+ | IsPos q0 ->
+ (match q0 with
+ | XH -> IsNul
+ | _ -> IsPos (pred q0))
+ | _ -> IsNeg
+
+ (** val sub_mask : positive -> positive -> mask **)
+
+ let rec sub_mask x y =
+ match x with
+ | XI p ->
+ (match y with
+ | XI q0 -> double_mask (sub_mask p q0)
+ | XO q0 -> succ_double_mask (sub_mask p q0)
+ | XH -> IsPos (XO p))
+ | XO p ->
+ (match y with
+ | XI q0 -> succ_double_mask (sub_mask_carry p q0)
+ | XO q0 -> double_mask (sub_mask p q0)
+ | XH -> IsPos (pred_double p))
+ | XH ->
+ (match y with
+ | XH -> IsNul
+ | _ -> IsNeg)
+
+ (** val sub_mask_carry : positive -> positive -> mask **)
+
+ and sub_mask_carry x y =
+ match x with
| XI p ->
- (match y with
- | XI q0 -> pcompare p q0 r
- | XO q0 -> pcompare p q0 Gt
- | XH -> Gt)
+ (match y with
+ | XI q0 -> succ_double_mask (sub_mask_carry p q0)
+ | XO q0 -> double_mask (sub_mask p q0)
+ | XH -> IsPos (pred_double p))
| 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)
-
-(** val psize : positive -> nat **)
-
-let rec psize = function
- | XI p2 -> S (psize p2)
- | XO p2 -> S (psize p2)
+ (match y with
+ | XI q0 -> double_mask (sub_mask_carry p q0)
+ | XO q0 -> succ_double_mask (sub_mask_carry p q0)
+ | XH -> double_pred_mask p)
+ | XH -> IsNeg
+
+ (** val sub : positive -> positive -> positive **)
+
+ let sub x y =
+ match sub_mask x y with
+ | IsPos z0 -> z0
+ | _ -> XH
+
+ (** val mul : positive -> positive -> positive **)
+
+ let rec mul x y =
+ match x with
+ | XI p -> add y (XO (mul p y))
+ | XO p -> XO (mul p y)
+ | XH -> y
+
+ (** val iter : positive -> ('a1 -> 'a1) -> 'a1 -> 'a1 **)
+
+ let rec iter n0 f x =
+ match n0 with
+ | XI n' -> f (iter n' f (iter n' f x))
+ | XO n' -> iter n' f (iter n' f x)
+ | XH -> f x
+
+ (** val pow : positive -> positive -> positive **)
+
+ let pow x y =
+ iter y (mul x) XH
+
+ (** val div2 : positive -> positive **)
+
+ let div2 = function
+ | XI p2 -> p2
+ | XO p2 -> p2
+ | XH -> XH
+
+ (** val div2_up : positive -> positive **)
+
+ let div2_up = function
+ | XI p2 -> succ p2
+ | XO p2 -> p2
+ | XH -> XH
+
+ (** val size_nat : positive -> nat **)
+
+ let rec size_nat = function
+ | XI p2 -> S (size_nat p2)
+ | XO p2 -> S (size_nat p2)
| XH -> S O
-
-type n =
- | N0
- | Npos of positive
+
+ (** val size : positive -> positive **)
+
+ let rec size = function
+ | XI p2 -> succ (size p2)
+ | XO p2 -> succ (size p2)
+ | XH -> XH
+
+ (** val compare_cont : positive -> positive -> comparison -> comparison **)
+
+ let rec compare_cont x y r =
+ match x with
+ | XI p ->
+ (match y with
+ | XI q0 -> compare_cont p q0 r
+ | XO q0 -> compare_cont p q0 Gt
+ | XH -> Gt)
+ | XO p ->
+ (match y with
+ | XI q0 -> compare_cont p q0 Lt
+ | XO q0 -> compare_cont p q0 r
+ | XH -> Gt)
+ | XH ->
+ (match y with
+ | XH -> r
+ | _ -> Lt)
+
+ (** val compare : positive -> positive -> comparison **)
+
+ let compare x y =
+ compare_cont x y Eq
+
+ (** val min : positive -> positive -> positive **)
+
+ let min p p' =
+ match compare p p' with
+ | Gt -> p'
+ | _ -> p
+
+ (** val max : positive -> positive -> positive **)
+
+ let max p p' =
+ match compare p p' with
+ | Gt -> p
+ | _ -> p'
+
+ (** val eqb : positive -> positive -> bool **)
+
+ let rec eqb p q0 =
+ match p with
+ | XI p2 ->
+ (match q0 with
+ | XI q1 -> eqb p2 q1
+ | _ -> false)
+ | XO p2 ->
+ (match q0 with
+ | XO q1 -> eqb p2 q1
+ | _ -> false)
+ | XH ->
+ (match q0 with
+ | XH -> true
+ | _ -> false)
+
+ (** val leb : positive -> positive -> bool **)
+
+ let leb x y =
+ match compare x y with
+ | Gt -> false
+ | _ -> true
+
+ (** val ltb : positive -> positive -> bool **)
+
+ let ltb x y =
+ match compare x y with
+ | Lt -> true
+ | _ -> false
+
+ (** val sqrtrem_step :
+ (positive -> positive) -> (positive -> positive) -> (positive * mask)
+ -> positive * mask **)
+
+ let sqrtrem_step f g = function
+ | s,y ->
+ (match y with
+ | IsPos r ->
+ let s' = XI (XO s) in
+ let r' = g (f r) in
+ if leb s' r' then (XI s),(sub_mask r' s') else (XO s),(IsPos r')
+ | _ -> (XO s),(sub_mask (g (f XH)) (XO (XO XH))))
+
+ (** val sqrtrem : positive -> positive * mask **)
+
+ let rec sqrtrem = function
+ | XI p2 ->
+ (match p2 with
+ | XI p3 -> sqrtrem_step (fun x -> XI x) (fun x -> XI x) (sqrtrem p3)
+ | XO p3 -> sqrtrem_step (fun x -> XO x) (fun x -> XI x) (sqrtrem p3)
+ | XH -> XH,(IsPos (XO XH)))
+ | XO p2 ->
+ (match p2 with
+ | XI p3 -> sqrtrem_step (fun x -> XI x) (fun x -> XO x) (sqrtrem p3)
+ | XO p3 -> sqrtrem_step (fun x -> XO x) (fun x -> XO x) (sqrtrem p3)
+ | XH -> XH,(IsPos XH))
+ | XH -> XH,IsNul
+
+ (** val sqrt : positive -> positive **)
+
+ let sqrt p =
+ fst (sqrtrem p)
+
+ (** val gcdn : nat -> positive -> positive -> positive **)
+
+ let rec gcdn n0 a b =
+ match n0 with
+ | O -> XH
+ | S n1 ->
+ (match a with
+ | XI a' ->
+ (match b with
+ | XI b' ->
+ (match compare a' b' with
+ | Eq -> a
+ | Lt -> gcdn n1 (sub b' a') a
+ | Gt -> gcdn n1 (sub a' b') b)
+ | XO b0 -> gcdn n1 a b0
+ | XH -> XH)
+ | XO a0 ->
+ (match b with
+ | XI p -> gcdn n1 a0 b
+ | XO b0 -> XO (gcdn n1 a0 b0)
+ | XH -> XH)
+ | XH -> XH)
+
+ (** val gcd : positive -> positive -> positive **)
+
+ let gcd a b =
+ gcdn (plus (size_nat a) (size_nat b)) a b
+
+ (** val ggcdn :
+ nat -> positive -> positive -> positive * (positive * positive) **)
+
+ let rec ggcdn n0 a b =
+ match n0 with
+ | O -> XH,(a,b)
+ | S n1 ->
+ (match a with
+ | XI a' ->
+ (match b with
+ | XI b' ->
+ (match compare a' b' with
+ | Eq -> a,(XH,XH)
+ | Lt ->
+ let g,p = ggcdn n1 (sub b' a') a in
+ let ba,aa = p in g,(aa,(add aa (XO ba)))
+ | Gt ->
+ let g,p = ggcdn n1 (sub a' b') b in
+ let ab,bb = p in g,((add bb (XO ab)),bb))
+ | XO b0 ->
+ let g,p = ggcdn n1 a b0 in let aa,bb = p in g,(aa,(XO bb))
+ | XH -> XH,(a,XH))
+ | XO a0 ->
+ (match b with
+ | XI p ->
+ let g,p2 = ggcdn n1 a0 b in let aa,bb = p2 in g,((XO aa),bb)
+ | XO b0 -> let g,p = ggcdn n1 a0 b0 in (XO g),p
+ | XH -> XH,(a,XH))
+ | XH -> XH,(XH,b))
+
+ (** val ggcd : positive -> positive -> positive * (positive * positive) **)
+
+ let ggcd a b =
+ ggcdn (plus (size_nat a) (size_nat b)) a b
+
+ (** val coq_Nsucc_double : n -> n **)
+
+ let coq_Nsucc_double = function
+ | N0 -> Npos XH
+ | Npos p -> Npos (XI p)
+
+ (** val coq_Ndouble : n -> n **)
+
+ let coq_Ndouble = function
+ | N0 -> N0
+ | Npos p -> Npos (XO p)
+
+ (** val coq_lor : positive -> positive -> positive **)
+
+ let rec coq_lor p q0 =
+ match p with
+ | XI p2 ->
+ (match q0 with
+ | XI q1 -> XI (coq_lor p2 q1)
+ | XO q1 -> XI (coq_lor p2 q1)
+ | XH -> p)
+ | XO p2 ->
+ (match q0 with
+ | XI q1 -> XI (coq_lor p2 q1)
+ | XO q1 -> XO (coq_lor p2 q1)
+ | XH -> XI p2)
+ | XH ->
+ (match q0 with
+ | XO q1 -> XI q1
+ | _ -> q0)
+
+ (** val coq_land : positive -> positive -> n **)
+
+ let rec coq_land p q0 =
+ match p with
+ | XI p2 ->
+ (match q0 with
+ | XI q1 -> coq_Nsucc_double (coq_land p2 q1)
+ | XO q1 -> coq_Ndouble (coq_land p2 q1)
+ | XH -> Npos XH)
+ | XO p2 ->
+ (match q0 with
+ | XI q1 -> coq_Ndouble (coq_land p2 q1)
+ | XO q1 -> coq_Ndouble (coq_land p2 q1)
+ | XH -> N0)
+ | XH ->
+ (match q0 with
+ | XO q1 -> N0
+ | _ -> Npos XH)
+
+ (** val ldiff : positive -> positive -> n **)
+
+ let rec ldiff p q0 =
+ match p with
+ | XI p2 ->
+ (match q0 with
+ | XI q1 -> coq_Ndouble (ldiff p2 q1)
+ | XO q1 -> coq_Nsucc_double (ldiff p2 q1)
+ | XH -> Npos (XO p2))
+ | XO p2 ->
+ (match q0 with
+ | XI q1 -> coq_Ndouble (ldiff p2 q1)
+ | XO q1 -> coq_Ndouble (ldiff p2 q1)
+ | XH -> Npos p)
+ | XH ->
+ (match q0 with
+ | XO q1 -> Npos XH
+ | _ -> N0)
+
+ (** val coq_lxor : positive -> positive -> n **)
+
+ let rec coq_lxor p q0 =
+ match p with
+ | XI p2 ->
+ (match q0 with
+ | XI q1 -> coq_Ndouble (coq_lxor p2 q1)
+ | XO q1 -> coq_Nsucc_double (coq_lxor p2 q1)
+ | XH -> Npos (XO p2))
+ | XO p2 ->
+ (match q0 with
+ | XI q1 -> coq_Nsucc_double (coq_lxor p2 q1)
+ | XO q1 -> coq_Ndouble (coq_lxor p2 q1)
+ | XH -> Npos (XI p2))
+ | XH ->
+ (match q0 with
+ | XI q1 -> Npos (XO q1)
+ | XO q1 -> Npos (XI q1)
+ | XH -> N0)
+
+ (** val shiftl_nat : positive -> nat -> positive **)
+
+ let shiftl_nat p n0 =
+ nat_iter n0 (fun x -> XO x) p
+
+ (** val shiftr_nat : positive -> nat -> positive **)
+
+ let shiftr_nat p n0 =
+ nat_iter n0 div2 p
+
+ (** val shiftl : positive -> n -> positive **)
+
+ let shiftl p = function
+ | N0 -> p
+ | Npos n1 -> iter n1 (fun x -> XO x) p
+
+ (** val shiftr : positive -> n -> positive **)
+
+ let shiftr p = function
+ | N0 -> p
+ | Npos n1 -> iter n1 div2 p
+
+ (** val testbit_nat : positive -> nat -> bool **)
+
+ let rec testbit_nat p n0 =
+ match p with
+ | XI p2 ->
+ (match n0 with
+ | O -> true
+ | S n' -> testbit_nat p2 n')
+ | XO p2 ->
+ (match n0 with
+ | O -> false
+ | S n' -> testbit_nat p2 n')
+ | XH ->
+ (match n0 with
+ | O -> true
+ | S n1 -> false)
+
+ (** val testbit : positive -> n -> bool **)
+
+ let rec testbit p n0 =
+ match p with
+ | XI p2 ->
+ (match n0 with
+ | N0 -> true
+ | Npos n1 -> testbit p2 (pred_N n1))
+ | XO p2 ->
+ (match n0 with
+ | N0 -> false
+ | Npos n1 -> testbit p2 (pred_N n1))
+ | XH ->
+ (match n0 with
+ | N0 -> true
+ | Npos p2 -> false)
+
+ (** val iter_op : ('a1 -> 'a1 -> 'a1) -> positive -> 'a1 -> 'a1 **)
+
+ let rec iter_op op p a =
+ match p with
+ | XI p2 -> op a (iter_op op p2 (op a a))
+ | XO p2 -> iter_op op p2 (op a a)
+ | XH -> a
+
+ (** val to_nat : positive -> nat **)
+
+ let to_nat x =
+ iter_op plus x (S O)
+
+ (** val of_nat : nat -> positive **)
+
+ let rec of_nat = function
+ | O -> XH
+ | S x ->
+ (match x with
+ | O -> XH
+ | S n1 -> succ (of_nat x))
+
+ (** val of_succ_nat : nat -> positive **)
+
+ let rec of_succ_nat = function
+ | O -> XH
+ | S x -> succ (of_succ_nat x)
+
+ (** val eq_dec : positive -> positive -> bool **)
+
+ let rec eq_dec p y0 =
+ match p with
+ | XI p2 ->
+ (match y0 with
+ | XI p3 -> eq_dec p2 p3
+ | _ -> false)
+ | XO p2 ->
+ (match y0 with
+ | XO p3 -> eq_dec p2 p3
+ | _ -> false)
+ | XH ->
+ (match y0 with
+ | XH -> true
+ | _ -> false)
+
+ (** val peano_rect : 'a1 -> (positive -> 'a1 -> 'a1) -> positive -> 'a1 **)
+
+ let rec peano_rect a f p =
+ let f2 = peano_rect (f XH a) (fun p2 x -> f (succ (XO p2)) (f (XO p2) x))
+ in
+ (match p with
+ | XI q0 -> f (XO q0) (f2 q0)
+ | XO q0 -> f2 q0
+ | XH -> a)
+
+ (** val peano_rec : 'a1 -> (positive -> 'a1 -> 'a1) -> positive -> 'a1 **)
+
+ let peano_rec =
+ peano_rect
+
+ type coq_PeanoView =
+ | PeanoOne
+ | PeanoSucc of positive * coq_PeanoView
+
+ (** val coq_PeanoView_rect :
+ 'a1 -> (positive -> coq_PeanoView -> 'a1 -> 'a1) -> positive ->
+ coq_PeanoView -> 'a1 **)
+
+ let rec coq_PeanoView_rect f f0 p = function
+ | PeanoOne -> f
+ | PeanoSucc (p3, p4) -> f0 p3 p4 (coq_PeanoView_rect f f0 p3 p4)
+
+ (** val coq_PeanoView_rec :
+ 'a1 -> (positive -> coq_PeanoView -> 'a1 -> 'a1) -> positive ->
+ coq_PeanoView -> 'a1 **)
+
+ let rec coq_PeanoView_rec f f0 p = function
+ | PeanoOne -> f
+ | PeanoSucc (p3, p4) -> f0 p3 p4 (coq_PeanoView_rec f f0 p3 p4)
+
+ (** val peanoView_xO : positive -> coq_PeanoView -> coq_PeanoView **)
+
+ let rec peanoView_xO p = function
+ | PeanoOne -> PeanoSucc (XH, PeanoOne)
+ | PeanoSucc (p2, q1) ->
+ PeanoSucc ((succ (XO p2)), (PeanoSucc ((XO p2), (peanoView_xO p2 q1))))
+
+ (** val peanoView_xI : positive -> coq_PeanoView -> coq_PeanoView **)
+
+ let rec peanoView_xI p = function
+ | PeanoOne -> PeanoSucc ((succ XH), (PeanoSucc (XH, PeanoOne)))
+ | PeanoSucc (p2, q1) ->
+ PeanoSucc ((succ (XI p2)), (PeanoSucc ((XI p2), (peanoView_xI p2 q1))))
+
+ (** val peanoView : positive -> coq_PeanoView **)
+
+ let rec peanoView = function
+ | XI p2 -> peanoView_xI p2 (peanoView p2)
+ | XO p2 -> peanoView_xO p2 (peanoView p2)
+ | XH -> PeanoOne
+
+ (** val coq_PeanoView_iter :
+ 'a1 -> (positive -> 'a1 -> 'a1) -> positive -> coq_PeanoView -> 'a1 **)
+
+ let rec coq_PeanoView_iter a f p = function
+ | PeanoOne -> a
+ | PeanoSucc (p2, q1) -> f p2 (coq_PeanoView_iter a f p2 q1)
+
+ (** val switch_Eq : comparison -> comparison -> comparison **)
+
+ let switch_Eq c = function
+ | Eq -> c
+ | x -> x
+
+ (** val mask2cmp : mask -> comparison **)
+
+ let mask2cmp = function
+ | IsNul -> Eq
+ | IsPos p2 -> Gt
+ | IsNeg -> Lt
+
+ module T =
+ struct
+
+ end
+
+ module ORev =
+ struct
+ type t = Coq__1.t
+ end
+
+ module MRev =
+ struct
+ (** val max : t -> t -> t **)
+
+ let max x y =
+ min y x
+ end
+
+ module MPRev = MaxLogicalProperties(ORev)(MRev)
+
+ module P =
+ struct
+ (** val max_case_strong :
+ t -> t -> (t -> t -> __ -> 'a1 -> 'a1) -> (__ -> 'a1) -> (__ -> 'a1)
+ -> 'a1 **)
+
+ let max_case_strong n0 m compat hl hr =
+ let c = compSpec2Type n0 m (compare n0 m) in
+ (match c with
+ | CompGtT -> compat n0 (max n0 m) __ (hl __)
+ | _ -> compat m (max n0 m) __ (hr __))
+
+ (** val max_case :
+ t -> t -> (t -> t -> __ -> 'a1 -> 'a1) -> 'a1 -> 'a1 -> 'a1 **)
+
+ let max_case n0 m x x0 x1 =
+ max_case_strong n0 m x (fun _ -> x0) (fun _ -> x1)
+
+ (** val max_dec : t -> t -> bool **)
+
+ let max_dec n0 m =
+ max_case n0 m (fun x y _ h0 -> h0) true false
+
+ (** val min_case_strong :
+ t -> t -> (t -> t -> __ -> 'a1 -> 'a1) -> (__ -> 'a1) -> (__ -> 'a1)
+ -> 'a1 **)
+
+ let min_case_strong n0 m compat hl hr =
+ let c = compSpec2Type n0 m (compare n0 m) in
+ (match c with
+ | CompGtT -> compat m (min n0 m) __ (hr __)
+ | _ -> compat n0 (min n0 m) __ (hl __))
+
+ (** val min_case :
+ t -> t -> (t -> t -> __ -> 'a1 -> 'a1) -> 'a1 -> 'a1 -> 'a1 **)
+
+ let min_case n0 m x x0 x1 =
+ min_case_strong n0 m x (fun _ -> x0) (fun _ -> x1)
+
+ (** val min_dec : t -> t -> bool **)
+
+ let min_dec n0 m =
+ min_case n0 m (fun x y _ h0 -> h0) true false
+ end
+
+ (** val max_case_strong : t -> t -> (__ -> 'a1) -> (__ -> 'a1) -> 'a1 **)
+
+ let max_case_strong n0 m x x0 =
+ P.max_case_strong n0 m (fun x1 y _ x2 -> x2) x x0
+
+ (** val max_case : t -> t -> 'a1 -> 'a1 -> 'a1 **)
+
+ let max_case n0 m x x0 =
+ max_case_strong n0 m (fun _ -> x) (fun _ -> x0)
+
+ (** val max_dec : t -> t -> bool **)
+
+ let max_dec =
+ P.max_dec
+
+ (** val min_case_strong : t -> t -> (__ -> 'a1) -> (__ -> 'a1) -> 'a1 **)
+
+ let min_case_strong n0 m x x0 =
+ P.min_case_strong n0 m (fun x1 y _ x2 -> x2) x x0
+
+ (** val min_case : t -> t -> 'a1 -> 'a1 -> 'a1 **)
+
+ let min_case n0 m x x0 =
+ min_case_strong n0 m (fun _ -> x) (fun _ -> x0)
+
+ (** val min_dec : t -> t -> bool **)
+
+ let min_dec =
+ P.min_dec
+ end
+
+module N =
+ struct
+ type t = n
+
+ (** val zero : n **)
+
+ let zero =
+ N0
+
+ (** val one : n **)
+
+ let one =
+ Npos XH
+
+ (** val two : n **)
+
+ let two =
+ Npos (XO XH)
+
+ (** val succ_double : n -> n **)
+
+ let succ_double = function
+ | N0 -> Npos XH
+ | Npos p -> Npos (XI p)
+
+ (** val double : n -> n **)
+
+ let double = function
+ | N0 -> N0
+ | Npos p -> Npos (XO p)
+
+ (** val succ : n -> n **)
+
+ let succ = function
+ | N0 -> Npos XH
+ | Npos p -> Npos (Coq_Pos.succ p)
+
+ (** val pred : n -> n **)
+
+ let pred = function
+ | N0 -> N0
+ | Npos p -> Coq_Pos.pred_N p
+
+ (** val succ_pos : n -> positive **)
+
+ let succ_pos = function
+ | N0 -> XH
+ | Npos p -> Coq_Pos.succ p
+
+ (** val add : n -> n -> n **)
+
+ let add n0 m =
+ match n0 with
+ | N0 -> m
+ | Npos p ->
+ (match m with
+ | N0 -> n0
+ | Npos q0 -> Npos (Coq_Pos.add p q0))
+
+ (** val sub : n -> n -> n **)
+
+ let sub n0 m =
+ match n0 with
+ | N0 -> N0
+ | Npos n' ->
+ (match m with
+ | N0 -> n0
+ | Npos m' ->
+ (match Coq_Pos.sub_mask n' m' with
+ | Coq_Pos.IsPos p -> Npos p
+ | _ -> N0))
+
+ (** val mul : n -> n -> n **)
+
+ let mul n0 m =
+ match n0 with
+ | N0 -> N0
+ | Npos p ->
+ (match m with
+ | N0 -> N0
+ | Npos q0 -> Npos (Coq_Pos.mul p q0))
+
+ (** val compare : n -> n -> comparison **)
+
+ let compare n0 m =
+ match n0 with
+ | N0 ->
+ (match m with
+ | N0 -> Eq
+ | Npos m' -> Lt)
+ | Npos n' ->
+ (match m with
+ | N0 -> Gt
+ | Npos m' -> Coq_Pos.compare n' m')
+
+ (** val eqb : n -> n -> bool **)
+
+ let rec eqb n0 m =
+ match n0 with
+ | N0 ->
+ (match m with
+ | N0 -> true
+ | Npos p -> false)
+ | Npos p ->
+ (match m with
+ | N0 -> false
+ | Npos q0 -> Coq_Pos.eqb p q0)
+
+ (** val leb : n -> n -> bool **)
+
+ let leb x y =
+ match compare x y with
+ | Gt -> false
+ | _ -> true
+
+ (** val ltb : n -> n -> bool **)
+
+ let ltb x y =
+ match compare x y with
+ | Lt -> true
+ | _ -> false
+
+ (** val min : n -> n -> n **)
+
+ let min n0 n' =
+ match compare n0 n' with
+ | Gt -> n'
+ | _ -> n0
+
+ (** val max : n -> n -> n **)
+
+ let max n0 n' =
+ match compare n0 n' with
+ | Gt -> n0
+ | _ -> n'
+
+ (** val div2 : n -> n **)
+
+ let div2 = function
+ | N0 -> N0
+ | Npos p2 ->
+ (match p2 with
+ | XI p -> Npos p
+ | XO p -> Npos p
+ | XH -> N0)
+
+ (** val even : n -> bool **)
+
+ let even = function
+ | N0 -> true
+ | Npos p ->
+ (match p with
+ | XO p2 -> true
+ | _ -> false)
+
+ (** val odd : n -> bool **)
+
+ let odd n0 =
+ negb (even n0)
+
+ (** val pow : n -> n -> n **)
+
+ let pow n0 = function
+ | N0 -> Npos XH
+ | Npos p2 ->
+ (match n0 with
+ | N0 -> N0
+ | Npos q0 -> Npos (Coq_Pos.pow q0 p2))
+
+ (** val log2 : n -> n **)
+
+ let log2 = function
+ | N0 -> N0
+ | Npos p2 ->
+ (match p2 with
+ | XI p -> Npos (Coq_Pos.size p)
+ | XO p -> Npos (Coq_Pos.size p)
+ | XH -> N0)
+
+ (** val size : n -> n **)
+
+ let size = function
+ | N0 -> N0
+ | Npos p -> Npos (Coq_Pos.size p)
+
+ (** val size_nat : n -> nat **)
+
+ let size_nat = function
+ | N0 -> O
+ | Npos p -> Coq_Pos.size_nat p
+
+ (** val pos_div_eucl : positive -> n -> n * n **)
+
+ let rec pos_div_eucl a b =
+ match a with
+ | XI a' ->
+ let q0,r = pos_div_eucl a' b in
+ let r' = succ_double r in
+ if leb b r' then (succ_double q0),(sub r' b) else (double q0),r'
+ | XO a' ->
+ let q0,r = pos_div_eucl a' b in
+ let r' = double r in
+ if leb b r' then (succ_double q0),(sub r' b) else (double q0),r'
+ | XH ->
+ (match b with
+ | N0 -> N0,(Npos XH)
+ | Npos p ->
+ (match p with
+ | XH -> (Npos XH),N0
+ | _ -> N0,(Npos XH)))
+
+ (** val div_eucl : n -> n -> n * n **)
+
+ let div_eucl a b =
+ match a with
+ | N0 -> N0,N0
+ | Npos na ->
+ (match b with
+ | N0 -> N0,a
+ | Npos p -> pos_div_eucl na b)
+
+ (** val div : n -> n -> n **)
+
+ let div a b =
+ fst (div_eucl a b)
+
+ (** val modulo : n -> n -> n **)
+
+ let modulo a b =
+ snd (div_eucl a b)
+
+ (** val gcd : n -> n -> n **)
+
+ let gcd a b =
+ match a with
+ | N0 -> b
+ | Npos p ->
+ (match b with
+ | N0 -> a
+ | Npos q0 -> Npos (Coq_Pos.gcd p q0))
+
+ (** val ggcd : n -> n -> n * (n * n) **)
+
+ let ggcd a b =
+ match a with
+ | N0 -> b,(N0,(Npos XH))
+ | Npos p ->
+ (match b with
+ | N0 -> a,((Npos XH),N0)
+ | Npos q0 ->
+ let g,p2 = Coq_Pos.ggcd p q0 in
+ let aa,bb = p2 in (Npos g),((Npos aa),(Npos bb)))
+
+ (** val sqrtrem : n -> n * n **)
+
+ let sqrtrem = function
+ | N0 -> N0,N0
+ | Npos p ->
+ let s,m = Coq_Pos.sqrtrem p in
+ (match m with
+ | Coq_Pos.IsPos r -> (Npos s),(Npos r)
+ | _ -> (Npos s),N0)
+
+ (** val sqrt : n -> n **)
+
+ let sqrt = function
+ | N0 -> N0
+ | Npos p -> Npos (Coq_Pos.sqrt p)
+
+ (** val coq_lor : n -> n -> n **)
+
+ let coq_lor n0 m =
+ match n0 with
+ | N0 -> m
+ | Npos p ->
+ (match m with
+ | N0 -> n0
+ | Npos q0 -> Npos (Coq_Pos.coq_lor p q0))
+
+ (** val coq_land : n -> n -> n **)
+
+ let coq_land n0 m =
+ match n0 with
+ | N0 -> N0
+ | Npos p ->
+ (match m with
+ | N0 -> N0
+ | Npos q0 -> Coq_Pos.coq_land p q0)
+
+ (** val ldiff : n -> n -> n **)
+
+ let rec ldiff n0 m =
+ match n0 with
+ | N0 -> N0
+ | Npos p ->
+ (match m with
+ | N0 -> n0
+ | Npos q0 -> Coq_Pos.ldiff p q0)
+
+ (** val coq_lxor : n -> n -> n **)
+
+ let coq_lxor n0 m =
+ match n0 with
+ | N0 -> m
+ | Npos p ->
+ (match m with
+ | N0 -> n0
+ | Npos q0 -> Coq_Pos.coq_lxor p q0)
+
+ (** val shiftl_nat : n -> nat -> n **)
+
+ let shiftl_nat a n0 =
+ nat_iter n0 double a
+
+ (** val shiftr_nat : n -> nat -> n **)
+
+ let shiftr_nat a n0 =
+ nat_iter n0 div2 a
+
+ (** val shiftl : n -> n -> n **)
+
+ let shiftl a n0 =
+ match a with
+ | N0 -> N0
+ | Npos a0 -> Npos (Coq_Pos.shiftl a0 n0)
+
+ (** val shiftr : n -> n -> n **)
+
+ let shiftr a = function
+ | N0 -> a
+ | Npos p -> Coq_Pos.iter p div2 a
+
+ (** val testbit_nat : n -> nat -> bool **)
+
+ let testbit_nat = function
+ | N0 -> (fun x -> false)
+ | Npos p -> Coq_Pos.testbit_nat p
+
+ (** val testbit : n -> n -> bool **)
+
+ let testbit a n0 =
+ match a with
+ | N0 -> false
+ | Npos p -> Coq_Pos.testbit p n0
+
+ (** val to_nat : n -> nat **)
+
+ let to_nat = function
+ | N0 -> O
+ | Npos p -> Coq_Pos.to_nat p
+
+ (** val of_nat : nat -> n **)
+
+ let of_nat = function
+ | O -> N0
+ | S n' -> Npos (Coq_Pos.of_succ_nat n')
+
+ (** val iter : n -> ('a1 -> 'a1) -> 'a1 -> 'a1 **)
+
+ let iter n0 f x =
+ match n0 with
+ | N0 -> x
+ | Npos p -> Coq_Pos.iter p f x
+
+ (** val eq_dec : n -> n -> bool **)
+
+ let eq_dec n0 m =
+ match n0 with
+ | N0 ->
+ (match m with
+ | N0 -> true
+ | Npos p -> false)
+ | Npos x ->
+ (match m with
+ | N0 -> false
+ | Npos p2 -> Coq_Pos.eq_dec x p2)
+
+ (** val discr : n -> positive option **)
+
+ let discr = function
+ | N0 -> None
+ | Npos p -> Some p
+
+ (** val binary_rect :
+ 'a1 -> (n -> 'a1 -> 'a1) -> (n -> 'a1 -> 'a1) -> n -> 'a1 **)
+
+ let binary_rect f0 f2 fS2 n0 =
+ let f2' = fun p -> f2 (Npos p) in
+ let fS2' = fun p -> fS2 (Npos p) in
+ (match n0 with
+ | N0 -> f0
+ | Npos p ->
+ let rec f = function
+ | XI p3 -> fS2' p3 (f p3)
+ | XO p3 -> f2' p3 (f p3)
+ | XH -> fS2 N0 f0
+ in f p)
+
+ (** val binary_rec :
+ 'a1 -> (n -> 'a1 -> 'a1) -> (n -> 'a1 -> 'a1) -> n -> 'a1 **)
+
+ let binary_rec =
+ binary_rect
+
+ (** val peano_rect : 'a1 -> (n -> 'a1 -> 'a1) -> n -> 'a1 **)
+
+ let peano_rect f0 f n0 =
+ let f' = fun p -> f (Npos p) in
+ (match n0 with
+ | N0 -> f0
+ | Npos p -> Coq_Pos.peano_rect (f N0 f0) f' p)
+
+ (** val peano_rec : 'a1 -> (n -> 'a1 -> 'a1) -> n -> 'a1 **)
+
+ let peano_rec =
+ peano_rect
+
+ module BootStrap =
+ struct
+
+ end
+
+ (** val recursion : 'a1 -> (n -> 'a1 -> 'a1) -> n -> 'a1 **)
+
+ let recursion x =
+ peano_rect x
+
+ module OrderElts =
+ struct
+ type t = n
+ end
+
+ module OrderTac = MakeOrderTac(OrderElts)
+
+ module NZPowP =
+ struct
+
+ end
+
+ module NZSqrtP =
+ struct
+
+ end
+
+ (** val sqrt_up : n -> n **)
+
+ let sqrt_up a =
+ match compare N0 a with
+ | Lt -> succ (sqrt (pred a))
+ | _ -> N0
+
+ (** val log2_up : n -> n **)
+
+ let log2_up a =
+ match compare (Npos XH) a with
+ | Lt -> succ (log2 (pred a))
+ | _ -> N0
+
+ module NZDivP =
+ struct
+
+ end
+
+ (** val lcm : n -> n -> n **)
+
+ let lcm a b =
+ mul a (div b (gcd a b))
+
+ (** val b2n : bool -> n **)
+
+ let b2n = function
+ | true -> Npos XH
+ | false -> N0
+
+ (** val setbit : n -> n -> n **)
+
+ let setbit a n0 =
+ coq_lor a (shiftl (Npos XH) n0)
+
+ (** val clearbit : n -> n -> n **)
+
+ let clearbit a n0 =
+ ldiff a (shiftl (Npos XH) n0)
+
+ (** val ones : n -> n **)
+
+ let ones n0 =
+ pred (shiftl (Npos XH) n0)
+
+ (** val lnot : n -> n -> n **)
+
+ let lnot a n0 =
+ coq_lxor a (ones n0)
+
+ module T =
+ struct
+
+ end
+
+ module ORev =
+ struct
+ type t = n
+ end
+
+ module MRev =
+ struct
+ (** val max : n -> n -> n **)
+
+ let max x y =
+ min y x
+ end
+
+ module MPRev = MaxLogicalProperties(ORev)(MRev)
+
+ module P =
+ struct
+ (** val max_case_strong :
+ n -> n -> (n -> n -> __ -> 'a1 -> 'a1) -> (__ -> 'a1) -> (__ -> 'a1)
+ -> 'a1 **)
+
+ let max_case_strong n0 m compat hl hr =
+ let c = compSpec2Type n0 m (compare n0 m) in
+ (match c with
+ | CompGtT -> compat n0 (max n0 m) __ (hl __)
+ | _ -> compat m (max n0 m) __ (hr __))
+
+ (** val max_case :
+ n -> n -> (n -> n -> __ -> 'a1 -> 'a1) -> 'a1 -> 'a1 -> 'a1 **)
+
+ let max_case n0 m x x0 x1 =
+ max_case_strong n0 m x (fun _ -> x0) (fun _ -> x1)
+
+ (** val max_dec : n -> n -> bool **)
+
+ let max_dec n0 m =
+ max_case n0 m (fun x y _ h0 -> h0) true false
+
+ (** val min_case_strong :
+ n -> n -> (n -> n -> __ -> 'a1 -> 'a1) -> (__ -> 'a1) -> (__ -> 'a1)
+ -> 'a1 **)
+
+ let min_case_strong n0 m compat hl hr =
+ let c = compSpec2Type n0 m (compare n0 m) in
+ (match c with
+ | CompGtT -> compat m (min n0 m) __ (hr __)
+ | _ -> compat n0 (min n0 m) __ (hl __))
+
+ (** val min_case :
+ n -> n -> (n -> n -> __ -> 'a1 -> 'a1) -> 'a1 -> 'a1 -> 'a1 **)
+
+ let min_case n0 m x x0 x1 =
+ min_case_strong n0 m x (fun _ -> x0) (fun _ -> x1)
+
+ (** val min_dec : n -> n -> bool **)
+
+ let min_dec n0 m =
+ min_case n0 m (fun x y _ h0 -> h0) true false
+ end
+
+ (** val max_case_strong : n -> n -> (__ -> 'a1) -> (__ -> 'a1) -> 'a1 **)
+
+ let max_case_strong n0 m x x0 =
+ P.max_case_strong n0 m (fun x1 y _ x2 -> x2) x x0
+
+ (** val max_case : n -> n -> 'a1 -> 'a1 -> 'a1 **)
+
+ let max_case n0 m x x0 =
+ max_case_strong n0 m (fun _ -> x) (fun _ -> x0)
+
+ (** val max_dec : n -> n -> bool **)
+
+ let max_dec =
+ P.max_dec
+
+ (** val min_case_strong : n -> n -> (__ -> 'a1) -> (__ -> 'a1) -> 'a1 **)
+
+ let min_case_strong n0 m x x0 =
+ P.min_case_strong n0 m (fun x1 y _ x2 -> x2) x x0
+
+ (** val min_case : n -> n -> 'a1 -> 'a1 -> 'a1 **)
+
+ let min_case n0 m x x0 =
+ min_case_strong n0 m (fun _ -> x) (fun _ -> x0)
+
+ (** val min_dec : n -> n -> bool **)
+
+ let min_dec =
+ P.min_dec
+ end
(** val pow_pos : ('a1 -> 'a1 -> 'a1) -> 'a1 -> positive -> 'a1 **)
let rec pow_pos rmul x = function
- | XI i0 -> let p = pow_pos rmul x i0 in rmul x (rmul p p)
- | XO i0 -> let p = pow_pos rmul x i0 in rmul p p
- | XH -> x
+| XI i0 -> let p = pow_pos rmul x i0 in rmul x (rmul p p)
+| XO i0 -> let p = pow_pos rmul x i0 in rmul p p
+| XH -> x
-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 **)
+(** val nth : nat -> 'a1 list -> 'a1 -> 'a1 **)
-let zdouble_minus_one = function
- | Z0 -> Zneg XH
- | Zpos p -> Zpos (pdouble_minus_one p)
- | Zneg p -> Zneg (XI p)
+let rec nth n0 l default =
+ match n0 with
+ | O ->
+ (match l with
+ | [] -> default
+ | x::l' -> x)
+ | S m ->
+ (match l with
+ | [] -> default
+ | x::t1 -> nth m t1 default)
-(** val zdouble : z -> z **)
+(** val map : ('a1 -> 'a2) -> 'a1 list -> 'a2 list **)
-let zdouble = function
+let rec map f = function
+| [] -> []
+| a::t1 -> (f a)::(map f t1)
+
+(** val fold_right : ('a2 -> 'a1 -> 'a1) -> 'a1 -> 'a2 list -> 'a1 **)
+
+let rec fold_right f a0 = function
+| [] -> a0
+| b::t1 -> f b (fold_right f a0 t1)
+
+module Z =
+ struct
+ type t = z
+
+ (** val zero : z **)
+
+ let zero =
+ Z0
+
+ (** val one : z **)
+
+ let one =
+ Zpos XH
+
+ (** val two : z **)
+
+ let two =
+ Zpos (XO XH)
+
+ (** val double : z -> z **)
+
+ let double = 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
+
+ (** val succ_double : z -> z **)
+
+ let succ_double = function
+ | Z0 -> Zpos XH
+ | Zpos p -> Zpos (XI p)
+ | Zneg p -> Zneg (Coq_Pos.pred_double p)
+
+ (** val pred_double : z -> z **)
+
+ let pred_double = function
+ | Z0 -> Zneg XH
+ | Zpos p -> Zpos (Coq_Pos.pred_double p)
+ | Zneg p -> Zneg (XI p)
+
+ (** val pos_sub : positive -> positive -> z **)
+
+ let rec pos_sub 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))
+ (match y with
+ | XI q0 -> double (pos_sub p q0)
+ | XO q0 -> succ_double (pos_sub 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))
+ (match y with
+ | XI q0 -> pred_double (pos_sub p q0)
+ | XO q0 -> double (pos_sub p q0)
+ | XH -> Zpos (Coq_Pos.pred_double 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
+ (match y with
+ | XI q0 -> Zneg (XO q0)
+ | XO q0 -> Zneg (Coq_Pos.pred_double q0)
+ | XH -> Z0)
+
+ (** val add : z -> z -> z **)
+
+ let add 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')))
+ (match y with
+ | Z0 -> x
+ | Zpos y' -> Zpos (Coq_Pos.add x' y')
+ | Zneg y' -> pos_sub 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
+ (match y with
+ | Z0 -> x
+ | Zpos y' -> pos_sub y' x'
+ | Zneg y' -> Zneg (Coq_Pos.add x' y'))
+
+ (** val opp : z -> z **)
+
+ let opp = 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
+
+ (** val succ : z -> z **)
+
+ let succ x =
+ add x (Zpos XH)
+
+ (** val pred : z -> z **)
+
+ let pred x =
+ add x (Zneg XH)
+
+ (** val sub : z -> z -> z **)
+
+ let sub m n0 =
+ add m (opp n0)
+
+ (** val mul : z -> z -> z **)
+
+ let mul 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'))
+ (match y with
+ | Z0 -> Z0
+ | Zpos y' -> Zpos (Coq_Pos.mul x' y')
+ | Zneg y' -> Zneg (Coq_Pos.mul 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)
+ (match y with
+ | Z0 -> Z0
+ | Zpos y' -> Zneg (Coq_Pos.mul x' y')
+ | Zneg y' -> Zpos (Coq_Pos.mul x' y'))
+
+ (** val pow_pos : z -> positive -> z **)
+
+ let pow_pos z0 n0 =
+ Coq_Pos.iter n0 (mul z0) (Zpos XH)
+
+ (** val pow : z -> z -> z **)
+
+ let pow x = function
+ | Z0 -> Zpos XH
+ | Zpos p -> pow_pos x p
+ | Zneg p -> Z0
+
+ (** val compare : z -> z -> comparison **)
+
+ let compare x y =
+ match x with
+ | Z0 ->
+ (match y with
+ | Z0 -> Eq
+ | Zpos y' -> Lt
+ | Zneg y' -> Gt)
+ | Zpos x' ->
+ (match y with
+ | Zpos y' -> Coq_Pos.compare x' y'
+ | _ -> Gt)
| Zneg x' ->
- (match y with
- | Zneg y' -> compOpp (pcompare x' y' Eq)
- | _ -> Lt)
-
-(** val zabs : z -> z **)
-
-let zabs = function
+ (match y with
+ | Zneg y' -> compOpp (Coq_Pos.compare x' y')
+ | _ -> Lt)
+
+ (** val sgn : z -> z **)
+
+ let sgn = function
| Z0 -> Z0
- | Zpos p -> Zpos p
- | Zneg p -> Zpos p
-
-(** val zmax : z -> z -> z **)
-
-let zmax m n0 =
- match zcompare m n0 with
- | Lt -> n0
- | _ -> m
-
-(** val zle_bool : z -> z -> bool **)
-
-let zle_bool x y =
- match zcompare x y with
+ | Zpos p -> Zpos XH
+ | Zneg p -> Zneg XH
+
+ (** val leb : z -> z -> bool **)
+
+ let leb x y =
+ match compare x y with
| Gt -> false
| _ -> true
-
-(** val zge_bool : z -> z -> bool **)
-
-let zge_bool x y =
- match zcompare x y with
+
+ (** val geb : z -> z -> bool **)
+
+ let geb x y =
+ match compare x y with
| Lt -> false
| _ -> true
-
-(** val zgt_bool : z -> z -> bool **)
-
-let zgt_bool x y =
- match zcompare x y with
- | Gt -> true
+
+ (** val ltb : z -> z -> bool **)
+
+ let ltb x y =
+ match compare x y with
+ | Lt -> true
| _ -> false
-
-(** val zeq_bool : z -> z -> bool **)
-
-let zeq_bool x y =
- match zcompare x y with
- | Eq -> true
+
+ (** val gtb : z -> z -> bool **)
+
+ let gtb x y =
+ match compare x y with
+ | Gt -> 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 **)
-
-let rec zdiv_eucl_POS a b =
- match a with
+
+ (** val eqb : z -> z -> bool **)
+
+ let rec eqb x y =
+ match x with
+ | Z0 ->
+ (match y with
+ | Z0 -> true
+ | _ -> false)
+ | Zpos p ->
+ (match y with
+ | Zpos q0 -> Coq_Pos.eqb p q0
+ | _ -> false)
+ | Zneg p ->
+ (match y with
+ | Zneg q0 -> Coq_Pos.eqb p q0
+ | _ -> false)
+
+ (** val max : z -> z -> z **)
+
+ let max n0 m =
+ match compare n0 m with
+ | Lt -> m
+ | _ -> n0
+
+ (** val min : z -> z -> z **)
+
+ let min n0 m =
+ match compare n0 m with
+ | Gt -> m
+ | _ -> n0
+
+ (** val abs : z -> z **)
+
+ let abs = function
+ | Zneg p -> Zpos p
+ | x -> x
+
+ (** val abs_nat : z -> nat **)
+
+ let abs_nat = function
+ | Z0 -> O
+ | Zpos p -> Coq_Pos.to_nat p
+ | Zneg p -> Coq_Pos.to_nat p
+
+ (** val abs_N : z -> n **)
+
+ let abs_N = function
+ | Z0 -> N0
+ | Zpos p -> Npos p
+ | Zneg p -> Npos p
+
+ (** val to_nat : z -> nat **)
+
+ let to_nat = function
+ | Zpos p -> Coq_Pos.to_nat p
+ | _ -> O
+
+ (** val to_N : z -> n **)
+
+ let to_N = function
+ | Zpos p -> Npos p
+ | _ -> N0
+
+ (** val of_nat : nat -> z **)
+
+ let of_nat = function
+ | O -> Z0
+ | S n1 -> Zpos (Coq_Pos.of_succ_nat n1)
+
+ (** val of_N : n -> z **)
+
+ let of_N = function
+ | N0 -> Z0
+ | Npos p -> Zpos p
+
+ (** val iter : z -> ('a1 -> 'a1) -> 'a1 -> 'a1 **)
+
+ let iter n0 f x =
+ match n0 with
+ | Zpos p -> Coq_Pos.iter p f x
+ | _ -> x
+
+ (** val pos_div_eucl : positive -> z -> z * z **)
+
+ let rec pos_div_eucl a b =
+ match a with
| XI a' ->
- let q0 , r = zdiv_eucl_POS a' b in
- let r' = zplus (zmult (Zpos (XO XH)) r) (Zpos XH) in
- if zgt_bool b r'
- then (zmult (Zpos (XO XH)) q0) , r'
- else (zplus (zmult (Zpos (XO XH)) q0) (Zpos XH)) , (zminus r' b)
+ let q0,r = pos_div_eucl a' b in
+ let r' = add (mul (Zpos (XO XH)) r) (Zpos XH) in
+ if gtb b r'
+ then (mul (Zpos (XO XH)) q0),r'
+ else (add (mul (Zpos (XO XH)) q0) (Zpos XH)),(sub r' b)
| XO a' ->
- let q0 , r = zdiv_eucl_POS a' b in
- let r' = zmult (Zpos (XO XH)) r in
- if zgt_bool b r'
- then (zmult (Zpos (XO XH)) q0) , r'
- else (zplus (zmult (Zpos (XO XH)) q0) (Zpos XH)) , (zminus r' b)
- | XH ->
- if zge_bool b (Zpos (XO XH)) then Z0 , (Zpos XH) else (Zpos XH) , Z0
-
-(** val zdiv_eucl : z -> z -> z * z **)
-
-let zdiv_eucl a b =
- match a with
- | Z0 -> Z0 , Z0
+ let q0,r = pos_div_eucl a' b in
+ let r' = mul (Zpos (XO XH)) r in
+ if gtb b r'
+ then (mul (Zpos (XO XH)) q0),r'
+ else (add (mul (Zpos (XO XH)) q0) (Zpos XH)),(sub r' b)
+ | XH -> if geb b (Zpos (XO XH)) then Z0,(Zpos XH) else (Zpos XH),Z0
+
+ (** val div_eucl : z -> z -> z * z **)
+
+ let div_eucl a b =
+ match a with
+ | Z0 -> Z0,Z0
| Zpos a' ->
- (match b with
- | Z0 -> Z0 , Z0
- | Zpos p -> zdiv_eucl_POS a' b
- | Zneg b' ->
- let q0 , r = zdiv_eucl_POS a' (Zpos b') in
- (match r with
- | Z0 -> (zopp q0) , Z0
- | _ -> (zopp (zplus q0 (Zpos XH))) , (zplus b r)))
+ (match b with
+ | Z0 -> Z0,Z0
+ | Zpos p -> pos_div_eucl a' b
+ | Zneg b' ->
+ let q0,r = pos_div_eucl a' (Zpos b') in
+ (match r with
+ | Z0 -> (opp q0),Z0
+ | _ -> (opp (add q0 (Zpos XH))),(add b r)))
| Zneg a' ->
- (match b with
- | Z0 -> Z0 , Z0
- | Zpos p ->
- let q0 , r = zdiv_eucl_POS a' b in
- (match r with
- | Z0 -> (zopp q0) , Z0
- | _ -> (zopp (zplus q0 (Zpos XH))) , (zminus b r))
- | Zneg b' ->
- let q0 , r = zdiv_eucl_POS a' (Zpos b') in q0 , (zopp r))
+ (match b with
+ | Z0 -> Z0,Z0
+ | Zpos p ->
+ let q0,r = pos_div_eucl a' b in
+ (match r with
+ | Z0 -> (opp q0),Z0
+ | _ -> (opp (add q0 (Zpos XH))),(sub b r))
+ | Zneg b' -> let q0,r = pos_div_eucl a' (Zpos b') in q0,(opp r))
+
+ (** val div : z -> z -> z **)
+
+ let div a b =
+ let q0,x = div_eucl a b in q0
+
+ (** val modulo : z -> z -> z **)
+
+ let modulo a b =
+ let x,r = div_eucl a b in r
+
+ (** val quotrem : z -> z -> z * z **)
+
+ let quotrem a b =
+ match a with
+ | Z0 -> Z0,Z0
+ | Zpos a0 ->
+ (match b with
+ | Z0 -> Z0,a
+ | Zpos b0 ->
+ let q0,r = N.pos_div_eucl a0 (Npos b0) in (of_N q0),(of_N r)
+ | Zneg b0 ->
+ let q0,r = N.pos_div_eucl a0 (Npos b0) in (opp (of_N q0)),(of_N r))
+ | Zneg a0 ->
+ (match b with
+ | Z0 -> Z0,a
+ | Zpos b0 ->
+ let q0,r = N.pos_div_eucl a0 (Npos b0) in
+ (opp (of_N q0)),(opp (of_N r))
+ | Zneg b0 ->
+ let q0,r = N.pos_div_eucl a0 (Npos b0) in (of_N q0),(opp (of_N r)))
+
+ (** val quot : z -> z -> z **)
+
+ let quot a b =
+ fst (quotrem a b)
+
+ (** val rem : z -> z -> z **)
+
+ let rem a b =
+ snd (quotrem a b)
+
+ (** val even : z -> bool **)
+
+ let even = function
+ | Z0 -> true
+ | Zpos p ->
+ (match p with
+ | XO p2 -> true
+ | _ -> false)
+ | Zneg p ->
+ (match p with
+ | XO p2 -> true
+ | _ -> false)
+
+ (** val odd : z -> bool **)
+
+ let odd = function
+ | Z0 -> false
+ | Zpos p ->
+ (match p with
+ | XO p2 -> false
+ | _ -> true)
+ | Zneg p ->
+ (match p with
+ | XO p2 -> false
+ | _ -> true)
+
+ (** val div2 : z -> z **)
+
+ let div2 = function
+ | Z0 -> Z0
+ | Zpos p ->
+ (match p with
+ | XH -> Z0
+ | _ -> Zpos (Coq_Pos.div2 p))
+ | Zneg p -> Zneg (Coq_Pos.div2_up p)
+
+ (** val quot2 : z -> z **)
+
+ let quot2 = function
+ | Z0 -> Z0
+ | Zpos p ->
+ (match p with
+ | XH -> Z0
+ | _ -> Zpos (Coq_Pos.div2 p))
+ | Zneg p ->
+ (match p with
+ | XH -> Z0
+ | _ -> Zneg (Coq_Pos.div2 p))
+
+ (** val log2 : z -> z **)
+
+ let log2 = function
+ | Zpos p2 ->
+ (match p2 with
+ | XI p -> Zpos (Coq_Pos.size p)
+ | XO p -> Zpos (Coq_Pos.size p)
+ | XH -> Z0)
+ | _ -> Z0
+
+ (** val sqrtrem : z -> z * z **)
+
+ let sqrtrem = function
+ | Zpos p ->
+ let s,m = Coq_Pos.sqrtrem p in
+ (match m with
+ | Coq_Pos.IsPos r -> (Zpos s),(Zpos r)
+ | _ -> (Zpos s),Z0)
+ | _ -> Z0,Z0
+
+ (** val sqrt : z -> z **)
+
+ let sqrt = function
+ | Zpos p -> Zpos (Coq_Pos.sqrt p)
+ | _ -> Z0
+
+ (** val gcd : z -> z -> z **)
+
+ let gcd a b =
+ match a with
+ | Z0 -> abs b
+ | Zpos a0 ->
+ (match b with
+ | Z0 -> abs a
+ | Zpos b0 -> Zpos (Coq_Pos.gcd a0 b0)
+ | Zneg b0 -> Zpos (Coq_Pos.gcd a0 b0))
+ | Zneg a0 ->
+ (match b with
+ | Z0 -> abs a
+ | Zpos b0 -> Zpos (Coq_Pos.gcd a0 b0)
+ | Zneg b0 -> Zpos (Coq_Pos.gcd a0 b0))
+
+ (** val ggcd : z -> z -> z * (z * z) **)
+
+ let ggcd a b =
+ match a with
+ | Z0 -> (abs b),(Z0,(sgn b))
+ | Zpos a0 ->
+ (match b with
+ | Z0 -> (abs a),((sgn a),Z0)
+ | Zpos b0 ->
+ let g,p = Coq_Pos.ggcd a0 b0 in
+ let aa,bb = p in (Zpos g),((Zpos aa),(Zpos bb))
+ | Zneg b0 ->
+ let g,p = Coq_Pos.ggcd a0 b0 in
+ let aa,bb = p in (Zpos g),((Zpos aa),(Zneg bb)))
+ | Zneg a0 ->
+ (match b with
+ | Z0 -> (abs a),((sgn a),Z0)
+ | Zpos b0 ->
+ let g,p = Coq_Pos.ggcd a0 b0 in
+ let aa,bb = p in (Zpos g),((Zneg aa),(Zpos bb))
+ | Zneg b0 ->
+ let g,p = Coq_Pos.ggcd a0 b0 in
+ let aa,bb = p in (Zpos g),((Zneg aa),(Zneg bb)))
+
+ (** val testbit : z -> z -> bool **)
+
+ let testbit a = function
+ | Z0 -> odd a
+ | Zpos p ->
+ (match a with
+ | Z0 -> false
+ | Zpos a0 -> Coq_Pos.testbit a0 (Npos p)
+ | Zneg a0 -> negb (N.testbit (Coq_Pos.pred_N a0) (Npos p)))
+ | Zneg p -> false
+
+ (** val shiftl : z -> z -> z **)
+
+ let shiftl a = function
+ | Z0 -> a
+ | Zpos p -> Coq_Pos.iter p (mul (Zpos (XO XH))) a
+ | Zneg p -> Coq_Pos.iter p div2 a
+
+ (** val shiftr : z -> z -> z **)
+
+ let shiftr a n0 =
+ shiftl a (opp n0)
+
+ (** val coq_lor : z -> z -> z **)
+
+ let coq_lor a b =
+ match a with
+ | Z0 -> b
+ | Zpos a0 ->
+ (match b with
+ | Z0 -> a
+ | Zpos b0 -> Zpos (Coq_Pos.coq_lor a0 b0)
+ | Zneg b0 -> Zneg (N.succ_pos (N.ldiff (Coq_Pos.pred_N b0) (Npos a0))))
+ | Zneg a0 ->
+ (match b with
+ | Z0 -> a
+ | Zpos b0 -> Zneg (N.succ_pos (N.ldiff (Coq_Pos.pred_N a0) (Npos b0)))
+ | Zneg b0 ->
+ Zneg
+ (N.succ_pos (N.coq_land (Coq_Pos.pred_N a0) (Coq_Pos.pred_N b0))))
+
+ (** val coq_land : z -> z -> z **)
+
+ let coq_land a b =
+ match a with
+ | Z0 -> Z0
+ | Zpos a0 ->
+ (match b with
+ | Z0 -> Z0
+ | Zpos b0 -> of_N (Coq_Pos.coq_land a0 b0)
+ | Zneg b0 -> of_N (N.ldiff (Npos a0) (Coq_Pos.pred_N b0)))
+ | Zneg a0 ->
+ (match b with
+ | Z0 -> Z0
+ | Zpos b0 -> of_N (N.ldiff (Npos b0) (Coq_Pos.pred_N a0))
+ | Zneg b0 ->
+ Zneg
+ (N.succ_pos (N.coq_lor (Coq_Pos.pred_N a0) (Coq_Pos.pred_N b0))))
+
+ (** val ldiff : z -> z -> z **)
+
+ let ldiff a b =
+ match a with
+ | Z0 -> Z0
+ | Zpos a0 ->
+ (match b with
+ | Z0 -> a
+ | Zpos b0 -> of_N (Coq_Pos.ldiff a0 b0)
+ | Zneg b0 -> of_N (N.coq_land (Npos a0) (Coq_Pos.pred_N b0)))
+ | Zneg a0 ->
+ (match b with
+ | Z0 -> a
+ | Zpos b0 ->
+ Zneg (N.succ_pos (N.coq_lor (Coq_Pos.pred_N a0) (Npos b0)))
+ | Zneg b0 -> of_N (N.ldiff (Coq_Pos.pred_N b0) (Coq_Pos.pred_N a0)))
+
+ (** val coq_lxor : z -> z -> z **)
+
+ let coq_lxor a b =
+ match a with
+ | Z0 -> b
+ | Zpos a0 ->
+ (match b with
+ | Z0 -> a
+ | Zpos b0 -> of_N (Coq_Pos.coq_lxor a0 b0)
+ | Zneg b0 ->
+ Zneg (N.succ_pos (N.coq_lxor (Npos a0) (Coq_Pos.pred_N b0))))
+ | Zneg a0 ->
+ (match b with
+ | Z0 -> a
+ | Zpos b0 ->
+ Zneg (N.succ_pos (N.coq_lxor (Coq_Pos.pred_N a0) (Npos b0)))
+ | Zneg b0 -> of_N (N.coq_lxor (Coq_Pos.pred_N a0) (Coq_Pos.pred_N b0)))
+
+ (** val eq_dec : z -> z -> bool **)
+
+ let eq_dec x y =
+ match x with
+ | Z0 ->
+ (match y with
+ | Z0 -> true
+ | _ -> false)
+ | Zpos x0 ->
+ (match y with
+ | Zpos p2 -> Coq_Pos.eq_dec x0 p2
+ | _ -> false)
+ | Zneg x0 ->
+ (match y with
+ | Zneg p2 -> Coq_Pos.eq_dec x0 p2
+ | _ -> false)
+
+ module BootStrap =
+ struct
+
+ end
+
+ module OrderElts =
+ struct
+ type t = z
+ end
+
+ module OrderTac = MakeOrderTac(OrderElts)
+
+ (** val sqrt_up : z -> z **)
+
+ let sqrt_up a =
+ match compare Z0 a with
+ | Lt -> succ (sqrt (pred a))
+ | _ -> Z0
+
+ (** val log2_up : z -> z **)
+
+ let log2_up a =
+ match compare (Zpos XH) a with
+ | Lt -> succ (log2 (pred a))
+ | _ -> Z0
+
+ module NZDivP =
+ struct
+
+ end
+
+ module Quot2Div =
+ struct
+ (** val div : z -> z -> z **)
+
+ let div =
+ quot
+
+ (** val modulo : z -> z -> z **)
+
+ let modulo =
+ rem
+ end
+
+ module NZQuot =
+ struct
+
+ end
+
+ (** val lcm : z -> z -> z **)
+
+ let lcm a b =
+ abs (mul a (div b (gcd a b)))
+
+ (** val b2z : bool -> z **)
+
+ let b2z = function
+ | true -> Zpos XH
+ | false -> Z0
+
+ (** val setbit : z -> z -> z **)
+
+ let setbit a n0 =
+ coq_lor a (shiftl (Zpos XH) n0)
+
+ (** val clearbit : z -> z -> z **)
+
+ let clearbit a n0 =
+ ldiff a (shiftl (Zpos XH) n0)
+
+ (** val lnot : z -> z **)
+
+ let lnot a =
+ pred (opp a)
+
+ (** val ones : z -> z **)
+
+ let ones n0 =
+ pred (shiftl (Zpos XH) n0)
+
+ module T =
+ struct
+
+ end
+
+ module ORev =
+ struct
+ type t = z
+ end
+
+ module MRev =
+ struct
+ (** val max : z -> z -> z **)
+
+ let max x y =
+ min y x
+ end
+
+ module MPRev = MaxLogicalProperties(ORev)(MRev)
+
+ module P =
+ struct
+ (** val max_case_strong :
+ z -> z -> (z -> z -> __ -> 'a1 -> 'a1) -> (__ -> 'a1) -> (__ -> 'a1)
+ -> 'a1 **)
+
+ let max_case_strong n0 m compat hl hr =
+ let c = compSpec2Type n0 m (compare n0 m) in
+ (match c with
+ | CompGtT -> compat n0 (max n0 m) __ (hl __)
+ | _ -> compat m (max n0 m) __ (hr __))
+
+ (** val max_case :
+ z -> z -> (z -> z -> __ -> 'a1 -> 'a1) -> 'a1 -> 'a1 -> 'a1 **)
+
+ let max_case n0 m x x0 x1 =
+ max_case_strong n0 m x (fun _ -> x0) (fun _ -> x1)
+
+ (** val max_dec : z -> z -> bool **)
+
+ let max_dec n0 m =
+ max_case n0 m (fun x y _ h0 -> h0) true false
+
+ (** val min_case_strong :
+ z -> z -> (z -> z -> __ -> 'a1 -> 'a1) -> (__ -> 'a1) -> (__ -> 'a1)
+ -> 'a1 **)
+
+ let min_case_strong n0 m compat hl hr =
+ let c = compSpec2Type n0 m (compare n0 m) in
+ (match c with
+ | CompGtT -> compat m (min n0 m) __ (hr __)
+ | _ -> compat n0 (min n0 m) __ (hl __))
+
+ (** val min_case :
+ z -> z -> (z -> z -> __ -> 'a1 -> 'a1) -> 'a1 -> 'a1 -> 'a1 **)
+
+ let min_case n0 m x x0 x1 =
+ min_case_strong n0 m x (fun _ -> x0) (fun _ -> x1)
+
+ (** val min_dec : z -> z -> bool **)
+
+ let min_dec n0 m =
+ min_case n0 m (fun x y _ h0 -> h0) true false
+ end
+
+ (** val max_case_strong : z -> z -> (__ -> 'a1) -> (__ -> 'a1) -> 'a1 **)
+
+ let max_case_strong n0 m x x0 =
+ P.max_case_strong n0 m (fun x1 y _ x2 -> x2) x x0
+
+ (** val max_case : z -> z -> 'a1 -> 'a1 -> 'a1 **)
+
+ let max_case n0 m x x0 =
+ max_case_strong n0 m (fun _ -> x) (fun _ -> x0)
+
+ (** val max_dec : z -> z -> bool **)
+
+ let max_dec =
+ P.max_dec
+
+ (** val min_case_strong : z -> z -> (__ -> 'a1) -> (__ -> 'a1) -> 'a1 **)
+
+ let min_case_strong n0 m x x0 =
+ P.min_case_strong n0 m (fun x1 y _ x2 -> x2) x x0
+
+ (** val min_case : z -> z -> 'a1 -> 'a1 -> 'a1 **)
+
+ let min_case n0 m x x0 =
+ min_case_strong n0 m (fun _ -> x) (fun _ -> x0)
+
+ (** val min_dec : z -> z -> bool **)
+
+ let min_dec =
+ P.min_dec
+ end
-(** val zdiv : z -> z -> z **)
+(** val zeq_bool : z -> z -> bool **)
-let zdiv a b =
- let q0 , x = zdiv_eucl a b in q0
+let zeq_bool x y =
+ match Z.compare x y with
+ | Eq -> true
+ | _ -> false
type 'c pol =
- | Pc of 'c
- | Pinj of positive * 'c pol
- | PX of 'c pol * positive * 'c pol
+| Pc of 'c
+| Pinj of positive * 'c pol
+| PX of 'c pol * positive * 'c pol
(** val p0 : 'a1 -> 'a1 pol **)
@@ -457,49 +2796,51 @@ let p1 cI =
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 -> if peq ceqb p2 p'0 then peq ceqb q0 q' else false
- | _ -> false)
- | _ -> false)
+ | Pc c ->
+ (match p' with
+ | Pc c' -> ceqb c c'
+ | _ -> false)
+ | Pinj (j, q0) ->
+ (match p' with
+ | Pinj (j', q') ->
+ (match Coq_Pos.compare j j' with
+ | Eq -> peq ceqb q0 q'
+ | _ -> false)
+ | _ -> false)
+ | PX (p2, i, q0) ->
+ (match p' with
+ | PX (p'0, i', q') ->
+ (match Coq_Pos.compare i i' with
+ | Eq -> if peq ceqb p2 p'0 then peq ceqb q0 q' else false
+ | _ -> false)
+ | _ -> false)
+
+(** val mkPinj : positive -> 'a1 pol -> 'a1 pol **)
+
+let mkPinj j p = match p with
+| Pc c -> p
+| Pinj (j', q0) -> Pinj ((Coq_Pos.add j j'), q0)
+| PX (p2, p3, p4) -> Pinj (j, p)
(** 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
+ | XI j0 -> Pinj ((XO j0), p)
+ | XO j0 -> Pinj ((Coq_Pos.pred_double 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 ->
- if ceqb c cO
- then (match q0 with
- | Pc c0 -> q0
- | Pinj (j', q1) -> Pinj ((pplus XH j'), q1)
- | PX (p2, p3, p4) -> Pinj (XH, q0))
- else PX (p, i, q0)
- | Pinj (p2, p3) -> PX (p, i, q0)
- | PX (p', i', q') ->
- if peq ceqb q' (p0 cO)
- then PX (p', (pplus i' i), q0)
- else PX (p, i, q0)
+ | Pc c -> if ceqb c cO then mkPinj XH q0 else PX (p, i, q0)
+ | Pinj (p2, p3) -> PX (p, i, q0)
+ | PX (p', i', q') ->
+ if peq ceqb q' (p0 cO)
+ then PX (p', (Coq_Pos.add i' i), q0)
+ else PX (p, i, q0)
(** val mkXi : 'a1 -> 'a1 -> positive -> 'a1 pol **)
@@ -514,202 +2855,155 @@ let mkX cO cI =
(** 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))
+| 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))
+ | 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))
+ | 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)))
+| Pc c -> mkPinj j (paddC cadd q0 c)
+| Pinj (j', q') ->
+ (match Z.pos_sub j' j with
+ | Z0 -> mkPinj j (pop q' q0)
+ | Zpos k -> mkPinj j (pop (Pinj (k, q')) q0)
+ | Zneg k -> mkPinj j' (paddI cadd pop q0 k q'))
+| 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 (Coq_Pos.pred_double 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)))
+| Pc c -> mkPinj j (paddC cadd (popp copp q0) c)
+| Pinj (j', q') ->
+ (match Z.pos_sub j' j with
+ | Z0 -> mkPinj j (pop q' q0)
+ | Zpos k -> mkPinj j (pop (Pinj (k, q')) q0)
+ | Zneg k -> mkPinj j' (psubI cadd copp pop q0 k q'))
+| 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 (Coq_Pos.pred_double 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')
+| 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 ((Coq_Pos.pred_double j0), q')))
+ | XH -> PX (p', i', q'))
+| PX (p2, i, q') ->
+ (match Z.pos_sub 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')
+| 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 ((Coq_Pos.pred_double j0), q')))
+ | XH -> PX ((popp copp p'), i', q'))
+| PX (p2, i, q') ->
+ (match Z.pos_sub 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')))
+| Pc c' -> paddC cadd p c'
+| Pinj (j', q') -> paddI cadd (padd cO cadd ceqb) 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 ((Coq_Pos.pred_double j0), q0)) q'))
+ | XH -> PX (p'0, i', (padd cO cadd ceqb q0 q')))
+ | PX (p2, i, q0) ->
+ (match Z.pos_sub 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 (padd cO cadd ceqb) 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')))
+| Pc c' -> psubC csub p c'
+| Pinj (j', q') -> psubI cadd copp (psub cO cadd csub copp ceqb) 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 ((Coq_Pos.pred_double j0), q0))
+ q'))
+ | XH -> PX ((popp copp p'0), i', (psub cO cadd csub copp ceqb q0 q')))
+ | PX (p2, i, q0) ->
+ (match Z.pos_sub 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 (psub cO cadd csub copp ceqb) 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 ->
@@ -717,16 +3011,11 @@ let rec psub cO cadd csub copp ceqb p = function
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)
+ | Pc c' -> Pc (cmul c' c)
+ | Pinj (j, q0) -> mkPinj j (pmulC_aux cO cmul ceqb q0 c)
+ | 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 ->
@@ -742,108 +3031,75 @@ let pmulC cO cI cmul ceqb p c =
'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))
+| Pc c -> mkPinj j (pmulC cO cI cmul ceqb q0 c)
+| Pinj (j', q') ->
+ (match Z.pos_sub j' j with
+ | Z0 -> mkPinj j (pmul0 q' q0)
+ | Zpos k -> mkPinj j (pmul0 (Pinj (k, q')) q0)
+ | Zneg k -> mkPinj j' (pmulI cO cI cmul ceqb pmul0 q0 k q'))
+| 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 (Coq_Pos.pred_double 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')))
+| Pc c -> pmulC cO cI cmul ceqb p c
+| Pinj (j', q') -> pmulI cO cI cmul ceqb (pmul cO cI cadd cmul ceqb) q' j' p
+| PX (p', i', q') ->
+ (match p with
+ | Pc c -> pmulC cO cI cmul ceqb p'' c
+ | Pinj (j, q0) ->
+ let qQ' =
+ 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 ((Coq_Pos.pred_double j0), q0)) q'
+ | XH -> pmul cO cI cadd cmul ceqb q0 q'
+ in
+ mkPX cO ceqb (pmul cO cI cadd cmul ceqb p p') i' qQ'
+ | PX (p2, i, q0) ->
+ let qQ' = pmul cO cI cadd cmul ceqb q0 q' in
+ let pQ' = pmulI cO cI cmul ceqb (pmul cO cI cadd cmul ceqb) q' XH p2 in
+ let qP' = pmul cO cI cadd cmul ceqb (mkPinj XH q0) p' in
+ let pP' = pmul cO cI cadd cmul ceqb p2 p' in
+ padd cO cadd ceqb
+ (mkPX cO ceqb (padd cO cadd ceqb (mkPX cO ceqb pP' i (p0 cO)) qP') i'
+ (p0 cO)) (mkPX cO ceqb pQ' i qQ'))
(** val psquare :
'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1
-> bool) -> 'a1 pol -> 'a1 pol **)
let rec psquare cO cI cadd cmul ceqb = function
- | Pc c -> Pc (cmul c c)
- | Pinj (j, q0) -> Pinj (j, (psquare cO cI cadd cmul ceqb q0))
- | PX (p2, i, q0) ->
- mkPX cO ceqb
- (padd cO cadd ceqb
- (mkPX cO ceqb (psquare cO cI cadd cmul ceqb p2) i (p0 cO))
- (pmul cO cI cadd cmul ceqb p2
- (let p3 = pmulC cO cI cmul ceqb q0 (cadd cI cI) in
- match p3 with
- | Pc c -> p3
- | Pinj (j', q1) -> Pinj ((pplus XH j'), q1)
- | PX (p4, p5, p6) -> Pinj (XH, p3)))) i
- (psquare cO cI cadd cmul ceqb q0)
+| Pc c -> Pc (cmul c c)
+| Pinj (j, q0) -> Pinj (j, (psquare cO cI cadd cmul ceqb q0))
+| PX (p2, i, q0) ->
+ let twoPQ =
+ pmul cO cI cadd cmul ceqb p2
+ (mkPinj XH (pmulC cO cI cmul ceqb q0 (cadd cI cI)))
+ in
+ let q2 = psquare cO cI cadd cmul ceqb q0 in
+ let p3 = psquare cO cI cadd cmul ceqb p2 in
+ mkPX cO ceqb (padd cO cadd ceqb (mkPX cO ceqb p3 i (p0 cO)) twoPQ) i q2
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
+| 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 **)
@@ -856,68 +3112,78 @@ let mk_X cO cI j =
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)
+| 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
+| 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
+| 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
+| 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 **)
+
+let rec map_bformula fct = function
+| TT -> TT
+| FF -> FF
+| X -> X
+| A a -> A (fct a)
+| Cj (f1, f2) -> Cj ((map_bformula fct f1), (map_bformula fct f2))
+| D (f1, f2) -> D ((map_bformula fct f1), (map_bformula fct f2))
+| N f0 -> N (map_bformula fct f0)
+| I (f1, f2) -> I ((map_bformula fct f1), (map_bformula fct f2))
type 'term' clause = 'term' list
@@ -931,19 +3197,61 @@ let tt =
(** val ff : 'a1 cnf **)
let ff =
- [] :: []
+ []::[]
+
+(** val add_term :
+ ('a1 -> bool) -> ('a1 -> 'a1 -> 'a1 option) -> 'a1 -> 'a1 clause -> 'a1
+ clause option **)
+
+let rec add_term unsat deduce t1 = function
+| [] ->
+ (match deduce t1 t1 with
+ | Some u -> if unsat u then None else Some (t1::[])
+ | None -> Some (t1::[]))
+| t'::cl0 ->
+ (match deduce t1 t' with
+ | Some u ->
+ if unsat u
+ then None
+ else (match add_term unsat deduce t1 cl0 with
+ | Some cl' -> Some (t'::cl')
+ | None -> None)
+ | None ->
+ (match add_term unsat deduce t1 cl0 with
+ | Some cl' -> Some (t'::cl')
+ | None -> None))
+
+(** val or_clause :
+ ('a1 -> bool) -> ('a1 -> 'a1 -> 'a1 option) -> 'a1 clause -> 'a1 clause
+ -> 'a1 clause option **)
+
+let rec or_clause unsat deduce cl1 cl2 =
+ match cl1 with
+ | [] -> Some cl2
+ | t1::cl ->
+ (match add_term unsat deduce t1 cl2 with
+ | Some cl' -> or_clause unsat deduce cl cl'
+ | None -> None)
-(** val or_clause_cnf : 'a1 clause -> 'a1 cnf -> 'a1 cnf **)
+(** val or_clause_cnf :
+ ('a1 -> bool) -> ('a1 -> 'a1 -> 'a1 option) -> 'a1 clause -> 'a1 cnf ->
+ 'a1 cnf **)
-let or_clause_cnf t0 f =
- map (fun x -> app t0 x) f
+let or_clause_cnf unsat deduce t1 f =
+ fold_right (fun e acc ->
+ match or_clause unsat deduce t1 e with
+ | Some cl -> cl::acc
+ | None -> acc) [] f
-(** val or_cnf : 'a1 cnf -> 'a1 cnf -> 'a1 cnf **)
+(** val or_cnf :
+ ('a1 -> bool) -> ('a1 -> 'a1 -> 'a1 option) -> 'a1 cnf -> 'a1 cnf -> 'a1
+ cnf **)
-let rec or_cnf f f' =
+let rec or_cnf unsat deduce f f' =
match f with
- | [] -> tt
- | e :: rst -> app (or_cnf rst f') (or_clause_cnf e f')
+ | [] -> tt
+ | e::rst ->
+ app (or_cnf unsat deduce rst f') (or_clause_cnf unsat deduce e f')
(** val and_cnf : 'a1 cnf -> 'a1 cnf -> 'a1 cnf **)
@@ -951,133 +3259,168 @@ 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 -> if pol0 then tt else ff
- | FF -> if pol0 then ff else tt
- | X -> ff
- | A x -> if pol0 then normalise0 x else negate0 x
- | Cj (e1, e2) ->
- if pol0
- then and_cnf (xcnf normalise0 negate0 pol0 e1)
- (xcnf normalise0 negate0 pol0 e2)
- else or_cnf (xcnf normalise0 negate0 pol0 e1)
- (xcnf normalise0 negate0 pol0 e2)
- | D (e1, e2) ->
- if pol0
- then or_cnf (xcnf normalise0 negate0 pol0 e1)
- (xcnf normalise0 negate0 pol0 e2)
- else and_cnf (xcnf normalise0 negate0 pol0 e1)
- (xcnf normalise0 negate0 pol0 e2)
- | N e -> xcnf normalise0 negate0 (negb pol0) e
- | I (e1, e2) ->
- if pol0
- then or_cnf (xcnf normalise0 negate0 (negb pol0) e1)
- (xcnf normalise0 negate0 pol0 e2)
- else and_cnf (xcnf normalise0 negate0 (negb pol0) e1)
- (xcnf normalise0 negate0 pol0 e2)
+ ('a2 -> bool) -> ('a2 -> 'a2 -> 'a2 option) -> ('a1 -> 'a2 cnf) -> ('a1
+ -> 'a2 cnf) -> bool -> 'a1 bFormula -> 'a2 cnf **)
+
+let rec xcnf unsat deduce normalise0 negate0 pol0 = function
+| TT -> if pol0 then tt else ff
+| FF -> if pol0 then ff else tt
+| X -> ff
+| A x -> if pol0 then normalise0 x else negate0 x
+| Cj (e1, e2) ->
+ if pol0
+ then and_cnf (xcnf unsat deduce normalise0 negate0 pol0 e1)
+ (xcnf unsat deduce normalise0 negate0 pol0 e2)
+ else or_cnf unsat deduce (xcnf unsat deduce normalise0 negate0 pol0 e1)
+ (xcnf unsat deduce normalise0 negate0 pol0 e2)
+| D (e1, e2) ->
+ if pol0
+ then or_cnf unsat deduce (xcnf unsat deduce normalise0 negate0 pol0 e1)
+ (xcnf unsat deduce normalise0 negate0 pol0 e2)
+ else and_cnf (xcnf unsat deduce normalise0 negate0 pol0 e1)
+ (xcnf unsat deduce normalise0 negate0 pol0 e2)
+| N e -> xcnf unsat deduce normalise0 negate0 (negb pol0) e
+| I (e1, e2) ->
+ if pol0
+ then or_cnf unsat deduce
+ (xcnf unsat deduce normalise0 negate0 (negb pol0) e1)
+ (xcnf unsat deduce normalise0 negate0 pol0 e2)
+ else and_cnf (xcnf unsat deduce normalise0 negate0 (negb pol0) e1)
+ (xcnf unsat deduce 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
- | [] -> true
- | e :: f0 ->
- (match l with
- | [] -> false
- | c :: l0 ->
- if checker e c then cnf_checker checker f0 l0 else false)
+ | [] -> true
+ | e::f0 ->
+ (match l with
+ | [] -> false
+ | c::l0 -> if checker e c then cnf_checker checker f0 l0 else false)
(** val tauto_checker :
- ('a1 -> 'a2 cnf) -> ('a1 -> 'a2 cnf) -> ('a2 list -> 'a3 -> bool) -> 'a1
- bFormula -> 'a3 list -> bool **)
+ ('a2 -> bool) -> ('a2 -> 'a2 -> 'a2 option) -> ('a1 -> 'a2 cnf) -> ('a1
+ -> 'a2 cnf) -> ('a2 list -> 'a3 -> bool) -> 'a1 bFormula -> 'a3 list ->
+ bool **)
+
+let tauto_checker unsat deduce normalise0 negate0 checker f w =
+ cnf_checker checker (xcnf unsat deduce normalise0 negate0 true f) w
+
+(** val cneqb : ('a1 -> 'a1 -> bool) -> 'a1 -> 'a1 -> bool **)
+
+let cneqb ceqb x y =
+ negb (ceqb x y)
-let tauto_checker normalise0 negate0 checker f w =
- cnf_checker checker (xcnf normalise0 negate0 true f) w
+(** val cltb :
+ ('a1 -> 'a1 -> bool) -> ('a1 -> 'a1 -> bool) -> 'a1 -> 'a1 -> bool **)
+
+let cltb ceqb cleb x y =
+ (&&) (cleb x y) (cneqb ceqb x y)
type 'c polC = 'c pol
type op1 =
- | Equal
- | NonEqual
- | Strict
- | NonStrict
+| Equal
+| NonEqual
+| Strict
+| NonStrict
+
+type 'c nFormula = 'c polC * op1
-type 'c nFormula = 'c polC * op1
+(** val opMult : op1 -> op1 -> op1 option **)
+
+let opMult o o' =
+ match o with
+ | Equal -> Some Equal
+ | NonEqual ->
+ (match o' with
+ | Strict -> None
+ | NonStrict -> None
+ | x -> Some x)
+ | Strict ->
+ (match o' with
+ | NonEqual -> None
+ | _ -> Some o')
+ | NonStrict ->
+ (match o' with
+ | NonEqual -> None
+ | Strict -> Some NonStrict
+ | x -> Some x)
(** val opAdd : op1 -> op1 -> op1 option **)
let opAdd o o' =
match o with
- | Equal -> Some o'
- | NonEqual -> (match o' with
- | Equal -> Some NonEqual
- | _ -> None)
- | Strict -> (match o' with
- | NonEqual -> None
- | _ -> Some Strict)
- | NonStrict ->
- (match o' with
- | NonEqual -> None
- | Strict -> Some Strict
- | _ -> Some NonStrict)
+ | Equal -> Some o'
+ | NonEqual ->
+ (match o' with
+ | Equal -> Some NonEqual
+ | _ -> None)
+ | Strict ->
+ (match o' with
+ | NonEqual -> None
+ | _ -> Some Strict)
+ | NonStrict ->
+ (match o' with
+ | Equal -> Some NonStrict
+ | NonEqual -> None
+ | x -> Some x)
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
+| 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 **)
+
+let map_option f = function
+| Some x -> f x
+| None -> None
+
+(** val map_option2 :
+ ('a1 -> 'a2 -> 'a3 option) -> 'a1 option -> 'a2 option -> 'a3 option **)
+
+let map_option2 f o o' =
+ match o with
+ | Some x ->
+ (match o' with
+ | Some x' -> f x x'
+ | None -> None)
+ | None -> None
(** val pexpr_times_nformula :
'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1
-> bool) -> 'a1 polC -> 'a1 nFormula -> 'a1 nFormula option **)
let pexpr_times_nformula cO cI cplus ctimes ceqb e = function
- | ef , o ->
- (match o with
- | Equal -> Some ((pmul cO cI cplus ctimes ceqb e ef) , Equal)
- | _ -> None)
+| ef,o ->
+ (match o with
+ | Equal -> Some ((pmul cO cI cplus ctimes ceqb e ef),Equal)
+ | _ -> None)
(** val nformula_times_nformula :
'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1
-> bool) -> 'a1 nFormula -> 'a1 nFormula -> 'a1 nFormula option **)
let nformula_times_nformula cO cI cplus ctimes ceqb f1 f2 =
- let e1 , o1 = f1 in
- let e2 , o2 = f2 in
- (match o1 with
- | Equal -> Some ((pmul cO cI cplus ctimes ceqb e1 e2) , Equal)
- | NonEqual ->
- (match o2 with
- | Equal -> Some ((pmul cO cI cplus ctimes ceqb e1 e2) , Equal)
- | NonEqual -> Some ((pmul cO cI cplus ctimes ceqb e1 e2) ,
- NonEqual)
- | _ -> None)
- | Strict ->
- (match o2 with
- | NonEqual -> None
- | _ -> Some ((pmul cO cI cplus ctimes ceqb e1 e2) , o2))
- | NonStrict ->
- (match o2 with
- | Equal -> Some ((pmul cO cI cplus ctimes ceqb e1 e2) , Equal)
- | NonEqual -> None
- | _ -> Some ((pmul cO cI cplus ctimes ceqb e1 e2) , NonStrict)))
+ let e1,o1 = f1 in
+ let e2,o2 = f2 in
+ map_option (fun x -> Some ((pmul cO cI cplus ctimes ceqb e1 e2),x))
+ (opMult o1 o2)
(** val nformula_plus_nformula :
'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> 'a1 nFormula -> 'a1
nFormula -> 'a1 nFormula option **)
let nformula_plus_nformula cO cplus ceqb f1 f2 =
- let e1 , o1 = f1 in
- let e2 , o2 = f2 in
- (match opAdd o1 o2 with
- | Some x -> Some ((padd cO cplus ceqb e1 e2) , x)
- | None -> None)
+ let e1,o1 = f1 in
+ let e2,o2 = f2 in
+ map_option (fun x -> Some ((padd cO cplus ceqb e1 e2),x)) (opAdd o1 o2)
(** val eval_Psatz :
'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1
@@ -1085,47 +3428,36 @@ let nformula_plus_nformula cO cplus ceqb f1 f2 =
nFormula option **)
let rec eval_Psatz cO cI cplus ctimes ceqb cleb l = function
- | PsatzIn n0 -> Some (nth n0 l ((Pc cO) , Equal))
- | PsatzSquare e0 -> Some ((psquare cO cI cplus ctimes ceqb e0) , NonStrict)
- | PsatzMulC (re, e0) ->
- (match eval_Psatz cO cI cplus ctimes ceqb cleb l e0 with
- | Some x -> pexpr_times_nformula cO cI cplus ctimes ceqb re x
- | None -> None)
- | PsatzMulE (f1, f2) ->
- (match eval_Psatz cO cI cplus ctimes ceqb cleb l f1 with
- | Some x ->
- (match eval_Psatz cO cI cplus ctimes ceqb cleb l f2 with
- | Some x' ->
- nformula_times_nformula cO cI cplus ctimes ceqb x x'
- | None -> None)
- | None -> None)
- | PsatzAdd (f1, f2) ->
- (match eval_Psatz cO cI cplus ctimes ceqb cleb l f1 with
- | Some x ->
- (match eval_Psatz cO cI cplus ctimes ceqb cleb l f2 with
- | Some x' -> nformula_plus_nformula cO cplus ceqb x x'
- | None -> None)
- | None -> None)
- | PsatzC c ->
- if (&&) (cleb cO c) (negb (ceqb cO c))
- then Some ((Pc c) , Strict)
- else None
- | PsatzZ -> Some ((Pc cO) , Equal)
+| PsatzIn n0 -> Some (nth n0 l ((Pc cO),Equal))
+| PsatzSquare e0 -> Some ((psquare cO cI cplus ctimes ceqb e0),NonStrict)
+| PsatzMulC (re, e0) ->
+ map_option (pexpr_times_nformula cO cI cplus ctimes ceqb re)
+ (eval_Psatz cO cI cplus ctimes ceqb cleb l e0)
+| PsatzMulE (f1, f2) ->
+ map_option2 (nformula_times_nformula cO cI cplus ctimes ceqb)
+ (eval_Psatz cO cI cplus ctimes ceqb cleb l f1)
+ (eval_Psatz cO cI cplus ctimes ceqb cleb l f2)
+| PsatzAdd (f1, f2) ->
+ map_option2 (nformula_plus_nformula cO cplus ceqb)
+ (eval_Psatz cO cI cplus ctimes ceqb cleb l f1)
+ (eval_Psatz cO cI cplus ctimes ceqb cleb l f2)
+| PsatzC c -> if cltb ceqb cleb cO c then Some ((Pc c),Strict) else None
+| PsatzZ -> Some ((Pc cO),Equal)
(** val check_inconsistent :
'a1 -> ('a1 -> 'a1 -> bool) -> ('a1 -> 'a1 -> bool) -> 'a1 nFormula ->
bool **)
let check_inconsistent cO ceqb cleb = function
- | e , op ->
- (match e with
- | Pc c ->
- (match op with
- | Equal -> negb (ceqb c cO)
- | NonEqual -> ceqb c cO
- | Strict -> cleb c cO
- | NonStrict -> (&&) (cleb c cO) (negb (ceqb c cO)))
- | _ -> false)
+| e,op ->
+ (match e with
+ | Pc c ->
+ (match op with
+ | Equal -> cneqb ceqb c cO
+ | NonEqual -> ceqb c cO
+ | Strict -> cleb c cO
+ | NonStrict -> cltb ceqb cleb c cO)
+ | _ -> false)
(** val check_normalised_formulas :
'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1
@@ -1134,18 +3466,18 @@ let check_inconsistent cO ceqb cleb = function
let check_normalised_formulas cO cI cplus ctimes ceqb cleb l cm =
match eval_Psatz cO cI cplus ctimes ceqb cleb l cm with
- | Some f -> check_inconsistent cO ceqb cleb f
- | None -> false
+ | Some f -> check_inconsistent cO ceqb cleb f
+ | None -> false
type op2 =
- | OpEq
- | OpNEq
- | OpLe
- | OpGe
- | OpLt
- | OpGt
+| OpEq
+| OpNEq
+| OpLe
+| OpGe
+| OpLt
+| OpGt
-type 'c formula = { flhs : 'c pExpr; fop : op2; frhs : 'c pExpr }
+type 't formula = { flhs : 't pExpr; fop : op2; frhs : 't pExpr }
(** val flhs : 'a1 formula -> 'a1 pExpr **)
@@ -1163,157 +3495,170 @@ let frhs x = x.frhs
'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1
-> 'a1) -> ('a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> 'a1 pExpr -> 'a1 pol **)
-let norm cO cI cplus ctimes cminus copp ceqb pe =
- norm_aux cO cI cplus ctimes cminus copp ceqb pe
+let norm cO cI cplus ctimes cminus copp ceqb =
+ norm_aux cO cI cplus ctimes cminus copp ceqb
(** val psub0 :
'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1) -> ('a1
-> 'a1 -> bool) -> 'a1 pol -> 'a1 pol -> 'a1 pol **)
-let psub0 cO cplus cminus copp ceqb p p' =
- psub cO cplus cminus copp ceqb p p'
+let psub0 cO cplus cminus copp ceqb =
+ psub cO cplus cminus copp ceqb
(** val padd0 :
'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> 'a1 pol -> 'a1 pol
-> 'a1 pol **)
-let padd0 cO cplus ceqb p p' =
- padd cO cplus ceqb p p'
+let padd0 cO cplus ceqb =
+ padd cO cplus ceqb
(** val xnormalise :
'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1
-> 'a1) -> ('a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> 'a1 formula -> 'a1
nFormula list **)
-let xnormalise cO cI cplus ctimes cminus copp ceqb t0 =
- let { flhs = lhs; fop = o; frhs = rhs } = t0 in
+let xnormalise cO cI cplus ctimes cminus copp ceqb t1 =
+ let { flhs = lhs; fop = o; frhs = rhs } = t1 in
let lhs0 = norm cO cI cplus ctimes cminus copp ceqb lhs in
let rhs0 = norm cO cI cplus ctimes cminus copp ceqb rhs in
(match o with
- | OpEq -> ((psub0 cO cplus cminus copp ceqb lhs0 rhs0) , Strict) ::
- (((psub0 cO cplus cminus copp ceqb rhs0 lhs0) , Strict) :: [])
- | OpNEq -> ((psub0 cO cplus cminus copp ceqb lhs0 rhs0) , Equal) :: []
- | OpLe -> ((psub0 cO cplus cminus copp ceqb lhs0 rhs0) , Strict) :: []
- | OpGe -> ((psub0 cO cplus cminus copp ceqb rhs0 lhs0) , Strict) :: []
- | OpLt -> ((psub0 cO cplus cminus copp ceqb lhs0 rhs0) , NonStrict) ::
- []
- | OpGt -> ((psub0 cO cplus cminus copp ceqb rhs0 lhs0) , NonStrict) ::
- [])
+ | OpEq ->
+ ((psub0 cO cplus cminus copp ceqb lhs0 rhs0),Strict)::(((psub0 cO cplus
+ cminus copp
+ ceqb rhs0
+ lhs0),Strict)::[])
+ | OpNEq -> ((psub0 cO cplus cminus copp ceqb lhs0 rhs0),Equal)::[]
+ | OpLe -> ((psub0 cO cplus cminus copp ceqb lhs0 rhs0),Strict)::[]
+ | OpGe -> ((psub0 cO cplus cminus copp ceqb rhs0 lhs0),Strict)::[]
+ | OpLt -> ((psub0 cO cplus cminus copp ceqb lhs0 rhs0),NonStrict)::[]
+ | OpGt -> ((psub0 cO cplus cminus copp ceqb rhs0 lhs0),NonStrict)::[])
(** val cnf_normalise :
'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1
-> 'a1) -> ('a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> 'a1 formula -> 'a1
nFormula cnf **)
-let cnf_normalise cO cI cplus ctimes cminus copp ceqb t0 =
- map (fun x -> x :: []) (xnormalise cO cI cplus ctimes cminus copp ceqb t0)
+let cnf_normalise cO cI cplus ctimes cminus copp ceqb t1 =
+ map (fun x -> x::[]) (xnormalise cO cI cplus ctimes cminus copp ceqb t1)
(** val xnegate :
'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1
-> 'a1) -> ('a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> 'a1 formula -> 'a1
nFormula list **)
-let xnegate cO cI cplus ctimes cminus copp ceqb t0 =
- let { flhs = lhs; fop = o; frhs = rhs } = t0 in
+let xnegate cO cI cplus ctimes cminus copp ceqb t1 =
+ let { flhs = lhs; fop = o; frhs = rhs } = t1 in
let lhs0 = norm cO cI cplus ctimes cminus copp ceqb lhs in
let rhs0 = norm cO cI cplus ctimes cminus copp ceqb rhs in
(match o with
- | OpEq -> ((psub0 cO cplus cminus copp ceqb lhs0 rhs0) , Equal) :: []
- | OpNEq -> ((psub0 cO cplus cminus copp ceqb lhs0 rhs0) , Strict) ::
- (((psub0 cO cplus cminus copp ceqb rhs0 lhs0) , Strict) :: [])
- | OpLe -> ((psub0 cO cplus cminus copp ceqb rhs0 lhs0) , NonStrict) ::
- []
- | OpGe -> ((psub0 cO cplus cminus copp ceqb lhs0 rhs0) , NonStrict) ::
- []
- | OpLt -> ((psub0 cO cplus cminus copp ceqb rhs0 lhs0) , Strict) :: []
- | OpGt -> ((psub0 cO cplus cminus copp ceqb lhs0 rhs0) , Strict) :: [])
+ | OpEq -> ((psub0 cO cplus cminus copp ceqb lhs0 rhs0),Equal)::[]
+ | OpNEq ->
+ ((psub0 cO cplus cminus copp ceqb lhs0 rhs0),Strict)::(((psub0 cO cplus
+ cminus copp
+ ceqb rhs0
+ lhs0),Strict)::[])
+ | OpLe -> ((psub0 cO cplus cminus copp ceqb rhs0 lhs0),NonStrict)::[]
+ | OpGe -> ((psub0 cO cplus cminus copp ceqb lhs0 rhs0),NonStrict)::[]
+ | OpLt -> ((psub0 cO cplus cminus copp ceqb rhs0 lhs0),Strict)::[]
+ | OpGt -> ((psub0 cO cplus cminus copp ceqb lhs0 rhs0),Strict)::[])
(** val cnf_negate :
'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1
-> 'a1) -> ('a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> 'a1 formula -> 'a1
nFormula cnf **)
-let cnf_negate cO cI cplus ctimes cminus copp ceqb t0 =
- map (fun x -> x :: []) (xnegate cO cI cplus ctimes cminus copp ceqb t0)
+let cnf_negate cO cI cplus ctimes cminus copp ceqb t1 =
+ map (fun x -> x::[]) (xnegate cO cI cplus ctimes cminus copp ceqb t1)
(** val xdenorm : positive -> 'a1 pol -> 'a1 pExpr **)
let rec xdenorm jmp = function
- | Pc c -> PEc c
- | Pinj (j, p2) -> xdenorm (pplus j jmp) p2
- | PX (p2, j, q0) -> PEadd ((PEmul ((xdenorm jmp p2), (PEpow ((PEX jmp),
- (Npos j))))), (xdenorm (psucc jmp) q0))
+| Pc c -> PEc c
+| Pinj (j, p2) -> xdenorm (Coq_Pos.add j jmp) p2
+| PX (p2, j, q0) ->
+ PEadd ((PEmul ((xdenorm jmp p2), (PEpow ((PEX jmp), (Npos j))))),
+ (xdenorm (Coq_Pos.succ jmp) q0))
(** val denorm : 'a1 pol -> 'a1 pExpr **)
let denorm p =
xdenorm XH p
+(** val map_PExpr : ('a2 -> 'a1) -> 'a2 pExpr -> 'a1 pExpr **)
+
+let rec map_PExpr c_of_S = function
+| PEc c -> PEc (c_of_S c)
+| PEX p -> PEX p
+| PEadd (e1, e2) -> PEadd ((map_PExpr c_of_S e1), (map_PExpr c_of_S e2))
+| PEsub (e1, e2) -> PEsub ((map_PExpr c_of_S e1), (map_PExpr c_of_S e2))
+| PEmul (e1, e2) -> PEmul ((map_PExpr c_of_S e1), (map_PExpr c_of_S e2))
+| PEopp e0 -> PEopp (map_PExpr c_of_S e0)
+| PEpow (e0, n0) -> PEpow ((map_PExpr c_of_S e0), n0)
+
+(** val map_Formula : ('a2 -> 'a1) -> 'a2 formula -> 'a1 formula **)
+
+let map_Formula c_of_S f =
+ let { flhs = l; fop = o; frhs = r } = f in
+ { flhs = (map_PExpr c_of_S l); fop = o; frhs = (map_PExpr c_of_S r) }
+
(** val simpl_cone :
'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> 'a1 psatz ->
'a1 psatz **)
let simpl_cone cO cI ctimes ceqb e = match e with
- | PsatzSquare t0 ->
- (match t0 with
- | Pc c -> if ceqb cO c then PsatzZ else PsatzC (ctimes c c)
- | _ -> PsatzSquare t0)
- | PsatzMulE (t1, t2) ->
- (match t1 with
- | PsatzMulE (x, x0) ->
- (match x with
- | PsatzC p2 ->
- (match t2 with
- | PsatzC c -> PsatzMulE ((PsatzC (ctimes c p2)), x0)
- | PsatzZ -> PsatzZ
- | _ -> e)
- | _ ->
- (match x0 with
- | PsatzC p2 ->
- (match t2 with
- | PsatzC c -> PsatzMulE ((PsatzC
- (ctimes c p2)), x)
- | PsatzZ -> PsatzZ
- | _ -> e)
- | _ ->
- (match t2 with
- | PsatzC c ->
- if ceqb cI c
- then t1
- else PsatzMulE (t1, t2)
- | PsatzZ -> PsatzZ
- | _ -> e)))
- | PsatzC c ->
- (match t2 with
- | PsatzMulE (x, x0) ->
- (match x with
- | PsatzC p2 -> PsatzMulE ((PsatzC (ctimes c p2)), x0)
- | _ ->
- (match x0 with
- | PsatzC p2 -> PsatzMulE ((PsatzC
- (ctimes c p2)), x)
- | _ ->
- if ceqb cI c
- then t2
- else PsatzMulE (t1, t2)))
- | PsatzAdd (y, z0) -> PsatzAdd ((PsatzMulE ((PsatzC c), y)),
- (PsatzMulE ((PsatzC c), z0)))
- | PsatzC c0 -> PsatzC (ctimes c c0)
- | PsatzZ -> PsatzZ
- | _ -> if ceqb cI c then t2 else PsatzMulE (t1, t2))
+| PsatzSquare t1 ->
+ (match t1 with
+ | Pc c -> if ceqb cO c then PsatzZ else PsatzC (ctimes c c)
+ | _ -> PsatzSquare t1)
+| PsatzMulE (t1, t2) ->
+ (match t1 with
+ | PsatzMulE (x, x0) ->
+ (match x with
+ | PsatzC p2 ->
+ (match t2 with
+ | PsatzC c -> PsatzMulE ((PsatzC (ctimes c p2)), x0)
| PsatzZ -> PsatzZ
+ | _ -> e)
+ | _ ->
+ (match x0 with
+ | PsatzC p2 ->
+ (match t2 with
+ | PsatzC c -> PsatzMulE ((PsatzC (ctimes c p2)), x)
+ | PsatzZ -> PsatzZ
+ | _ -> e)
+ | _ ->
+ (match t2 with
+ | PsatzC c -> if ceqb cI c then t1 else PsatzMulE (t1, t2)
+ | PsatzZ -> PsatzZ
+ | _ -> e)))
+ | PsatzC c ->
+ (match t2 with
+ | PsatzMulE (x, x0) ->
+ (match x with
+ | PsatzC p2 -> PsatzMulE ((PsatzC (ctimes c p2)), x0)
| _ ->
- (match t2 with
- | PsatzC c -> if ceqb cI c then t1 else PsatzMulE (t1, t2)
- | PsatzZ -> PsatzZ
- | _ -> e))
- | PsatzAdd (t1, t2) ->
- (match t1 with
- | PsatzZ -> t2
- | _ -> (match t2 with
- | PsatzZ -> t1
- | _ -> PsatzAdd (t1, t2)))
- | _ -> e
+ (match x0 with
+ | PsatzC p2 -> PsatzMulE ((PsatzC (ctimes c p2)), x)
+ | _ -> if ceqb cI c then t2 else PsatzMulE (t1, t2)))
+ | PsatzAdd (y, z0) ->
+ PsatzAdd ((PsatzMulE ((PsatzC c), y)), (PsatzMulE ((PsatzC c), z0)))
+ | PsatzC c0 -> PsatzC (ctimes c c0)
+ | PsatzZ -> PsatzZ
+ | _ -> if ceqb cI c then t2 else PsatzMulE (t1, t2))
+ | PsatzZ -> PsatzZ
+ | _ ->
+ (match t2 with
+ | PsatzC c -> if ceqb cI c then t1 else PsatzMulE (t1, t2)
+ | PsatzZ -> PsatzZ
+ | _ -> e))
+| PsatzAdd (t1, t2) ->
+ (match t1 with
+ | PsatzZ -> t2
+ | _ ->
+ (match t2 with
+ | PsatzZ -> t1
+ | _ -> PsatzAdd (t1, t2)))
+| _ -> e
type q = { qnum : z; qden : positive }
@@ -1328,28 +3673,28 @@ let qden x = x.qden
(** val qeq_bool : q -> q -> bool **)
let qeq_bool x y =
- zeq_bool (zmult x.qnum (Zpos y.qden)) (zmult y.qnum (Zpos x.qden))
+ zeq_bool (Z.mul x.qnum (Zpos y.qden)) (Z.mul y.qnum (Zpos x.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))
+ Z.leb (Z.mul x.qnum (Zpos y.qden)) (Z.mul y.qnum (Zpos 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) }
+ { qnum = (Z.add (Z.mul x.qnum (Zpos y.qden)) (Z.mul y.qnum (Zpos x.qden)));
+ qden = (Coq_Pos.mul 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) }
+ { qnum = (Z.mul x.qnum y.qnum); qden = (Coq_Pos.mul x.qden y.qden) }
(** val qopp : q -> q **)
let qopp x =
- { qnum = (zopp x.qnum); qden = x.qden }
+ { qnum = (Z.opp x.qnum); qden = x.qden }
(** val qminus : q -> q -> q **)
@@ -1360,9 +3705,9 @@ let qminus x y =
let qinv x =
match x.qnum with
- | Z0 -> { qnum = Z0; qden = XH }
- | Zpos p -> { qnum = (Zpos x.qden); qden = p }
- | Zneg p -> { qnum = (Zneg x.qden); qden = p }
+ | Z0 -> { qnum = Z0; qden = XH }
+ | Zpos p -> { qnum = (Zpos x.qden); qden = p }
+ | Zneg p -> { qnum = (Zneg x.qden); qden = p }
(** val qpower_positive : q -> positive -> q **)
@@ -1372,332 +3717,330 @@ let qpower_positive q0 p =
(** val qpower : q -> z -> q **)
let qpower q0 = function
- | Z0 -> { qnum = (Zpos XH); qden = XH }
- | Zpos p -> qpower_positive q0 p
- | Zneg p -> qinv (qpower_positive q0 p)
+| Z0 -> { qnum = (Zpos XH); qden = XH }
+| Zpos p -> qpower_positive q0 p
+| Zneg p -> qinv (qpower_positive q0 p)
-(** val pgcdn : nat -> positive -> positive -> positive **)
+type 'a t0 =
+| Empty
+| Leaf of 'a
+| Node of 'a t0 * 'a * 'a t0
-let rec pgcdn n0 a b =
- match n0 with
- | O -> XH
- | S n1 ->
- (match a with
- | XI a' ->
- (match b with
- | XI b' ->
- (match pcompare a' b' Eq with
- | Eq -> a
- | Lt -> pgcdn n1 (pminus b' a') a
- | Gt -> pgcdn n1 (pminus a' b') b)
- | XO b0 -> pgcdn n1 a b0
- | XH -> XH)
- | XO a0 ->
- (match b with
- | XI p -> pgcdn n1 a0 b
- | XO b0 -> XO (pgcdn n1 a0 b0)
- | XH -> XH)
- | XH -> XH)
-
-(** val pgcd : positive -> positive -> positive **)
-
-let pgcd a b =
- pgcdn (plus (psize a) (psize b)) a b
-
-(** val zgcd : z -> z -> z **)
-
-let zgcd a b =
- match a with
- | Z0 -> zabs b
- | Zpos a0 ->
- (match b with
- | Z0 -> zabs a
- | Zpos b0 -> Zpos (pgcd a0 b0)
- | Zneg b0 -> Zpos (pgcd a0 b0))
- | Zneg a0 ->
- (match b with
- | Z0 -> zabs a
- | Zpos b0 -> Zpos (pgcd a0 b0)
- | Zneg b0 -> Zpos (pgcd a0 b0))
-
-type 'a t =
- | Empty
- | Leaf of 'a
- | Node of 'a t * 'a * 'a t
-
-(** val find : 'a1 -> 'a1 t -> positive -> 'a1 **)
+(** val find : 'a1 -> 'a1 t0 -> 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)
+ | 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 psatz
(** val zWeakChecker : z nFormula list -> z psatz -> bool **)
-let zWeakChecker x x0 =
- check_normalised_formulas Z0 (Zpos XH) zplus zmult zeq_bool zle_bool x x0
+let zWeakChecker =
+ check_normalised_formulas Z0 (Zpos XH) Z.add Z.mul zeq_bool Z.leb
(** val psub1 : z pol -> z pol -> z pol **)
-let psub1 p p' =
- psub0 Z0 zplus zminus zopp zeq_bool p p'
+let psub1 =
+ psub0 Z0 Z.add Z.sub Z.opp zeq_bool
(** val padd1 : z pol -> z pol -> z pol **)
-let padd1 p p' =
- padd0 Z0 zplus zeq_bool p p'
+let padd1 =
+ padd0 Z0 Z.add zeq_bool
(** val norm0 : z pExpr -> z pol **)
-let norm0 pe =
- norm Z0 (Zpos XH) zplus zmult zminus zopp zeq_bool pe
+let norm0 =
+ norm Z0 (Zpos XH) Z.add Z.mul Z.sub Z.opp zeq_bool
(** val xnormalise0 : z formula -> z nFormula list **)
-let xnormalise0 t0 =
- let { flhs = lhs; fop = o; frhs = rhs } = t0 in
+let xnormalise0 t1 =
+ let { flhs = lhs; fop = o; frhs = rhs } = t1 in
let lhs0 = norm0 lhs in
let rhs0 = norm0 rhs in
(match o with
- | OpEq -> ((psub1 lhs0 (padd1 rhs0 (Pc (Zpos XH)))) , NonStrict) ::
- (((psub1 rhs0 (padd1 lhs0 (Pc (Zpos XH)))) , NonStrict) :: [])
- | OpNEq -> ((psub1 lhs0 rhs0) , Equal) :: []
- | OpLe -> ((psub1 lhs0 (padd1 rhs0 (Pc (Zpos XH)))) , NonStrict) :: []
- | OpGe -> ((psub1 rhs0 (padd1 lhs0 (Pc (Zpos XH)))) , NonStrict) :: []
- | OpLt -> ((psub1 lhs0 rhs0) , NonStrict) :: []
- | OpGt -> ((psub1 rhs0 lhs0) , NonStrict) :: [])
+ | OpEq ->
+ ((psub1 lhs0 (padd1 rhs0 (Pc (Zpos XH)))),NonStrict)::(((psub1 rhs0
+ (padd1 lhs0
+ (Pc (Zpos
+ XH)))),NonStrict)::[])
+ | OpNEq -> ((psub1 lhs0 rhs0),Equal)::[]
+ | OpLe -> ((psub1 lhs0 (padd1 rhs0 (Pc (Zpos XH)))),NonStrict)::[]
+ | OpGe -> ((psub1 rhs0 (padd1 lhs0 (Pc (Zpos XH)))),NonStrict)::[]
+ | OpLt -> ((psub1 lhs0 rhs0),NonStrict)::[]
+ | OpGt -> ((psub1 rhs0 lhs0),NonStrict)::[])
(** val normalise : z formula -> z nFormula cnf **)
-let normalise t0 =
- map (fun x -> x :: []) (xnormalise0 t0)
+let normalise t1 =
+ map (fun x -> x::[]) (xnormalise0 t1)
(** val xnegate0 : z formula -> z nFormula list **)
-let xnegate0 t0 =
- let { flhs = lhs; fop = o; frhs = rhs } = t0 in
+let xnegate0 t1 =
+ let { flhs = lhs; fop = o; frhs = rhs } = t1 in
let lhs0 = norm0 lhs in
let rhs0 = norm0 rhs in
(match o with
- | OpEq -> ((psub1 lhs0 rhs0) , Equal) :: []
- | OpNEq -> ((psub1 lhs0 (padd1 rhs0 (Pc (Zpos XH)))) , NonStrict) ::
- (((psub1 rhs0 (padd1 lhs0 (Pc (Zpos XH)))) , NonStrict) :: [])
- | OpLe -> ((psub1 rhs0 lhs0) , NonStrict) :: []
- | OpGe -> ((psub1 lhs0 rhs0) , NonStrict) :: []
- | OpLt -> ((psub1 rhs0 (padd1 lhs0 (Pc (Zpos XH)))) , NonStrict) :: []
- | OpGt -> ((psub1 lhs0 (padd1 rhs0 (Pc (Zpos XH)))) , NonStrict) :: [])
+ | OpEq -> ((psub1 lhs0 rhs0),Equal)::[]
+ | OpNEq ->
+ ((psub1 lhs0 (padd1 rhs0 (Pc (Zpos XH)))),NonStrict)::(((psub1 rhs0
+ (padd1 lhs0
+ (Pc (Zpos
+ XH)))),NonStrict)::[])
+ | OpLe -> ((psub1 rhs0 lhs0),NonStrict)::[]
+ | OpGe -> ((psub1 lhs0 rhs0),NonStrict)::[]
+ | OpLt -> ((psub1 rhs0 (padd1 lhs0 (Pc (Zpos XH)))),NonStrict)::[]
+ | OpGt -> ((psub1 lhs0 (padd1 rhs0 (Pc (Zpos XH)))),NonStrict)::[])
(** val negate : z formula -> z nFormula cnf **)
-let negate t0 =
- map (fun x -> x :: []) (xnegate0 t0)
+let negate t1 =
+ map (fun x -> x::[]) (xnegate0 t1)
+
+(** val zunsat : z nFormula -> bool **)
+
+let zunsat =
+ check_inconsistent Z0 zeq_bool Z.leb
+
+(** val zdeduce : z nFormula -> z nFormula -> z nFormula option **)
+
+let zdeduce =
+ nformula_plus_nformula Z0 Z.add zeq_bool
(** val ceiling : z -> z -> z **)
let ceiling a b =
- let q0 , r = zdiv_eucl a b in
+ let q0,r = Z.div_eucl a b in
(match r with
- | Z0 -> q0
- | _ -> zplus q0 (Zpos XH))
+ | Z0 -> q0
+ | _ -> Z.add q0 (Zpos XH))
type zArithProof =
- | DoneProof
- | RatProof of zWitness * zArithProof
- | CutProof of zWitness * zArithProof
- | EnumProof of zWitness * zWitness * zArithProof list
+| DoneProof
+| RatProof of zWitness * zArithProof
+| CutProof of zWitness * zArithProof
+| EnumProof of zWitness * zWitness * zArithProof list
(** val zgcdM : z -> z -> z **)
let zgcdM x y =
- zmax (zgcd x y) (Zpos XH)
+ Z.max (Z.gcd x y) (Zpos XH)
-(** val zgcd_pol : z polC -> z * z **)
+(** val zgcd_pol : z polC -> z * z **)
let rec zgcd_pol = function
- | Pc c -> Z0 , c
- | Pinj (p2, p3) -> zgcd_pol p3
- | PX (p2, p3, q0) ->
- let g1 , c1 = zgcd_pol p2 in
- let g2 , c2 = zgcd_pol q0 in (zgcdM (zgcdM g1 c1) g2) , c2
+| Pc c -> Z0,c
+| Pinj (p2, p3) -> zgcd_pol p3
+| PX (p2, p3, q0) ->
+ let g1,c1 = zgcd_pol p2 in
+ let g2,c2 = zgcd_pol q0 in (zgcdM (zgcdM g1 c1) g2),c2
(** val zdiv_pol : z polC -> z -> z polC **)
let rec zdiv_pol p x =
match p with
- | Pc c -> Pc (zdiv c x)
- | Pinj (j, p2) -> Pinj (j, (zdiv_pol p2 x))
- | PX (p2, j, q0) -> PX ((zdiv_pol p2 x), j, (zdiv_pol q0 x))
+ | Pc c -> Pc (Z.div c x)
+ | Pinj (j, p2) -> Pinj (j, (zdiv_pol p2 x))
+ | PX (p2, j, q0) -> PX ((zdiv_pol p2 x), j, (zdiv_pol q0 x))
-(** val makeCuttingPlane : z polC -> z polC * z **)
+(** val makeCuttingPlane : z polC -> z polC * z **)
let makeCuttingPlane p =
- let g , c = zgcd_pol p in
- if zgt_bool g Z0
- then (zdiv_pol (psubC zminus p c) g) , (zopp (ceiling (zopp c) g))
- else p , Z0
+ let g,c = zgcd_pol p in
+ if Z.gtb g Z0
+ then (zdiv_pol (psubC Z.sub p c) g),(Z.opp (ceiling (Z.opp c) g))
+ else p,Z0
-(** val genCuttingPlane : z nFormula -> ((z polC * z) * op1) option **)
+(** val genCuttingPlane : z nFormula -> ((z polC * z) * op1) option **)
let genCuttingPlane = function
- | e , op ->
- (match op with
- | Equal ->
- let g , c = zgcd_pol e in
- if (&&) (zgt_bool g Z0)
- ((&&) (zgt_bool c Z0) (negb (zeq_bool (zgcd g c) g)))
- then None
- else Some ((e , Z0) , op)
- | NonEqual -> Some ((e , Z0) , op)
- | Strict ->
- let p , c = makeCuttingPlane (psubC zminus e (Zpos XH)) in
- Some ((p , c) , NonStrict)
- | NonStrict ->
- let p , c = makeCuttingPlane e in Some ((p , c) , NonStrict))
-
-(** val nformula_of_cutting_plane :
- ((z polC * z) * op1) -> z nFormula **)
+| e,op ->
+ (match op with
+ | Equal ->
+ let g,c = zgcd_pol e in
+ if (&&) (Z.gtb g Z0)
+ ((&&) (negb (zeq_bool c Z0)) (negb (zeq_bool (Z.gcd g c) g)))
+ then None
+ else Some ((makeCuttingPlane e),Equal)
+ | NonEqual -> Some ((e,Z0),op)
+ | Strict -> Some ((makeCuttingPlane (psubC Z.sub e (Zpos XH))),NonStrict)
+ | NonStrict -> Some ((makeCuttingPlane e),NonStrict))
+
+(** val nformula_of_cutting_plane : ((z polC * z) * op1) -> z nFormula **)
let nformula_of_cutting_plane = function
- | e_z , o -> let e , z0 = e_z in (padd1 e (Pc z0)) , o
+| e_z,o -> let e,z0 = e_z in (padd1 e (Pc z0)),o
(** val is_pol_Z0 : z polC -> bool **)
let is_pol_Z0 = function
- | Pc z0 -> (match z0 with
- | Z0 -> true
- | _ -> false)
- | _ -> false
+| Pc z0 ->
+ (match z0 with
+ | Z0 -> true
+ | _ -> false)
+| _ -> false
(** val eval_Psatz0 : z nFormula list -> zWitness -> z nFormula option **)
-let eval_Psatz0 x x0 =
- eval_Psatz Z0 (Zpos XH) zplus zmult zeq_bool zle_bool x x0
+let eval_Psatz0 =
+ eval_Psatz Z0 (Zpos XH) Z.add Z.mul zeq_bool Z.leb
-(** val check_inconsistent0 : z nFormula -> bool **)
+(** val valid_cut_sign : op1 -> bool **)
-let check_inconsistent0 f =
- check_inconsistent Z0 zeq_bool zle_bool f
+let valid_cut_sign = function
+| Equal -> true
+| NonStrict -> true
+| _ -> false
(** val zChecker : z nFormula list -> zArithProof -> bool **)
let rec zChecker l = function
- | DoneProof -> false
- | RatProof (w, pf0) ->
- (match eval_Psatz0 l w with
- | Some f ->
- if check_inconsistent0 f then true else zChecker (f :: l) pf0
- | None -> false)
- | CutProof (w, pf0) ->
- (match eval_Psatz0 l w with
- | Some f ->
- (match genCuttingPlane f with
- | Some cp ->
- zChecker ((nformula_of_cutting_plane cp) :: l) pf0
- | None -> true)
- | None -> false)
- | EnumProof (w1, w2, pf0) ->
- (match eval_Psatz0 l w1 with
- | Some f1 ->
- (match eval_Psatz0 l w2 with
- | Some f2 ->
- (match genCuttingPlane f1 with
- | Some p ->
- let p2 , op3 = p in
- let e1 , z1 = p2 in
- (match genCuttingPlane f2 with
- | Some p3 ->
- let p4 , op4 = p3 in
- let e2 , z2 = p4 in
- (match op3 with
- | NonStrict ->
- (match op4 with
- | NonStrict ->
- if is_pol_Z0 (padd1 e1 e2)
- then
- let rec label pfs lb ub =
-
- match pfs with
- |
- [] -> zgt_bool lb ub
- |
- pf1 :: rsr ->
- (&&)
- (zChecker
- (((psub1 e1 (Pc lb)) ,
- Equal) :: l) pf1)
- (label rsr
- (zplus lb (Zpos XH)) ub)
- in label pf0 (zopp z1) z2
- else false
- | _ -> false)
- | _ -> false)
- | None -> false)
- | None -> false)
- | None -> false)
- | None -> false)
+| DoneProof -> false
+| RatProof (w, pf0) ->
+ (match eval_Psatz0 l w with
+ | Some f -> if zunsat f then true else zChecker (f::l) pf0
+ | None -> false)
+| CutProof (w, pf0) ->
+ (match eval_Psatz0 l w with
+ | Some f ->
+ (match genCuttingPlane f with
+ | Some cp -> zChecker ((nformula_of_cutting_plane cp)::l) pf0
+ | None -> true)
+ | None -> false)
+| EnumProof (w1, w2, pf0) ->
+ (match eval_Psatz0 l w1 with
+ | Some f1 ->
+ (match eval_Psatz0 l w2 with
+ | Some f2 ->
+ (match genCuttingPlane f1 with
+ | Some p ->
+ let p2,op3 = p in
+ let e1,z1 = p2 in
+ (match genCuttingPlane f2 with
+ | Some p3 ->
+ let p4,op4 = p3 in
+ let e2,z2 = p4 in
+ if (&&) ((&&) (valid_cut_sign op3) (valid_cut_sign op4))
+ (is_pol_Z0 (padd1 e1 e2))
+ then let rec label pfs lb ub =
+ match pfs with
+ | [] -> Z.gtb lb ub
+ | pf1::rsr ->
+ (&&) (zChecker (((psub1 e1 (Pc lb)),Equal)::l) pf1)
+ (label rsr (Z.add lb (Zpos XH)) ub)
+ in label pf0 (Z.opp z1) z2
+ else false
+ | None -> true)
+ | None -> true)
+ | None -> false)
+ | None -> false)
(** val zTautoChecker : z formula bFormula -> zArithProof list -> bool **)
let zTautoChecker f w =
- tauto_checker normalise negate zChecker f w
-
-(** val n_of_Z : z -> n **)
-
-let n_of_Z = function
- | Zpos p -> Npos p
- | _ -> N0
+ tauto_checker zunsat zdeduce normalise negate zChecker f w
type qWitness = q psatz
(** val qWeakChecker : q nFormula list -> q psatz -> bool **)
-let qWeakChecker x x0 =
+let qWeakChecker =
check_normalised_formulas { qnum = Z0; qden = XH } { qnum = (Zpos XH);
- qden = XH } qplus qmult qeq_bool qle_bool x x0
+ qden = XH } qplus qmult qeq_bool qle_bool
(** val qnormalise : q formula -> q nFormula cnf **)
-let qnormalise t0 =
+let qnormalise =
cnf_normalise { qnum = Z0; qden = XH } { qnum = (Zpos XH); qden = XH }
- qplus qmult qminus qopp qeq_bool t0
+ qplus qmult qminus qopp qeq_bool
(** val qnegate : q formula -> q nFormula cnf **)
-let qnegate t0 =
+let qnegate =
cnf_negate { qnum = Z0; qden = XH } { qnum = (Zpos XH); qden = XH } qplus
- qmult qminus qopp qeq_bool t0
+ qmult qminus qopp qeq_bool
+
+(** val qunsat : q nFormula -> bool **)
+
+let qunsat =
+ check_inconsistent { qnum = Z0; qden = XH } qeq_bool qle_bool
+
+(** val qdeduce : q nFormula -> q nFormula -> q nFormula option **)
+
+let qdeduce =
+ nformula_plus_nformula { qnum = Z0; qden = XH } qplus qeq_bool
(** val qTautoChecker : q formula bFormula -> qWitness list -> bool **)
let qTautoChecker f w =
- tauto_checker qnormalise qnegate qWeakChecker f w
+ tauto_checker qunsat qdeduce qnormalise qnegate qWeakChecker f w
+
+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 **)
+
+let rec q_of_Rcst = function
+| C0 -> { qnum = Z0; qden = XH }
+| C1 -> { qnum = (Zpos XH); qden = XH }
+| CQ q0 -> q0
+| CZ z0 -> { qnum = z0; qden = XH }
+| CPlus (r1, r2) -> qplus (q_of_Rcst r1) (q_of_Rcst r2)
+| CMinus (r1, r2) -> qminus (q_of_Rcst r1) (q_of_Rcst r2)
+| CMult (r1, r2) -> qmult (q_of_Rcst r1) (q_of_Rcst r2)
+| CInv r0 -> qinv (q_of_Rcst r0)
+| COpp r0 -> qopp (q_of_Rcst r0)
+
+type rWitness = q psatz
+
+(** val rWeakChecker : q nFormula list -> q psatz -> bool **)
+
+let rWeakChecker =
+ check_normalised_formulas { qnum = Z0; qden = XH } { qnum = (Zpos XH);
+ qden = XH } qplus qmult qeq_bool qle_bool
+
+(** val rnormalise : q formula -> q nFormula cnf **)
-type rWitness = z psatz
+let rnormalise =
+ cnf_normalise { qnum = Z0; qden = XH } { qnum = (Zpos XH); qden = XH }
+ qplus qmult qminus qopp qeq_bool
-(** val rWeakChecker : z nFormula list -> z psatz -> bool **)
+(** val rnegate : q formula -> q nFormula cnf **)
-let rWeakChecker x x0 =
- check_normalised_formulas Z0 (Zpos XH) zplus zmult zeq_bool zle_bool x x0
+let rnegate =
+ cnf_negate { qnum = Z0; qden = XH } { qnum = (Zpos XH); qden = XH } qplus
+ qmult qminus qopp qeq_bool
-(** val rnormalise : z formula -> z nFormula cnf **)
+(** val runsat : q nFormula -> bool **)
-let rnormalise t0 =
- cnf_normalise Z0 (Zpos XH) zplus zmult zminus zopp zeq_bool t0
+let runsat =
+ check_inconsistent { qnum = Z0; qden = XH } qeq_bool qle_bool
-(** val rnegate : z formula -> z nFormula cnf **)
+(** val rdeduce : q nFormula -> q nFormula -> q nFormula option **)
-let rnegate t0 =
- cnf_negate Z0 (Zpos XH) zplus zmult zminus zopp zeq_bool t0
+let rdeduce =
+ nformula_plus_nformula { qnum = Z0; qden = XH } qplus qeq_bool
-(** val rTautoChecker : z formula bFormula -> rWitness list -> bool **)
+(** val rTautoChecker : rcst formula bFormula -> rWitness list -> bool **)
let rTautoChecker f w =
- tauto_checker rnormalise rnegate rWeakChecker f w
+ tauto_checker runsat rdeduce rnormalise rnegate rWeakChecker
+ (map_bformula (map_Formula q_of_Rcst) f) w
generated by cgit v1.2.3 (git 2.39.1) at 2024-11-15 23:07:58 +0000