MAYBE 28.918 H-Termination proof of /home/matraf/haskell/eval_FullyBlown_Fast/empty.hs
H-Termination of the given Haskell-Program with start terms could not be shown:



HASKELL
  ↳ IFR

mainModule Main
  ((enumFrom :: Enum a => a  ->  [a]) :: Enum a => a  ->  [a])

module Main where
  import qualified Prelude



If Reductions:
The following If expression
if primGEqNatS x y then Succ (primDivNatS (primMinusNatS x y) (Succ y)) else Zero

is transformed to
primDivNatS0 x y True = Succ (primDivNatS (primMinusNatS x y) (Succ y))
primDivNatS0 x y False = Zero

The following If expression
if primGEqNatS x y then primModNatS (primMinusNatS x y) (Succ y) else Succ x

is transformed to
primModNatS0 x y True = primModNatS (primMinusNatS x y) (Succ y)
primModNatS0 x y False = Succ x



↳ HASKELL
  ↳ IFR
HASKELL
      ↳ BR

mainModule Main
  ((enumFrom :: Enum a => a  ->  [a]) :: Enum a => a  ->  [a])

module Main where
  import qualified Prelude



Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
HASKELL
          ↳ COR

mainModule Main
  ((enumFrom :: Enum a => a  ->  [a]) :: Enum a => a  ->  [a])

module Main where
  import qualified Prelude



Cond Reductions:
The following Function with conditions
gcd' x 0 = x
gcd' x y = gcd' y (x `rem` y)

is transformed to
gcd' x xz = gcd'2 x xz
gcd' x y = gcd'0 x y

gcd'0 x y = gcd' y (x `rem` y)

gcd'1 True x xz = x
gcd'1 yu yv yw = gcd'0 yv yw

gcd'2 x xz = gcd'1 (xz == 0) x xz
gcd'2 yx yy = gcd'0 yx yy

The following Function with conditions
gcd 0 0 = error []
gcd x y = 
gcd' (abs x) (abs y)
where 
gcd' x 0 = x
gcd' x y = gcd' y (x `rem` y)

is transformed to
gcd yz zu = gcd3 yz zu
gcd x y = gcd0 x y

gcd0 x y = 
gcd' (abs x) (abs y)
where 
gcd' x xz = gcd'2 x xz
gcd' x y = gcd'0 x y
gcd'0 x y = gcd' y (x `rem` y)
gcd'1 True x xz = x
gcd'1 yu yv yw = gcd'0 yv yw
gcd'2 x xz = gcd'1 (xz == 0) x xz
gcd'2 yx yy = gcd'0 yx yy

gcd1 True yz zu = error []
gcd1 zv zw zx = gcd0 zw zx

gcd2 True yz zu = gcd1 (zu == 0) yz zu
gcd2 zy zz vuu = gcd0 zz vuu

gcd3 yz zu = gcd2 (yz == 0) yz zu
gcd3 vuv vuw = gcd0 vuv vuw

The following Function with conditions
reduce x y
 | y == 0
 = error []
 | otherwise
 = x `quot` d :% (y `quot` d)
where 
d  = gcd x y

is transformed to
reduce x y = reduce2 x y

reduce2 x y = 
reduce1 x y (y == 0)
where 
d  = gcd x y
reduce0 x y True = x `quot` d :% (y `quot` d)
reduce1 x y True = error []
reduce1 x y False = reduce0 x y otherwise

The following Function with conditions
absReal x
 | x >= 0
 = x
 | otherwise
 = `negate` x

is transformed to
absReal x = absReal2 x

absReal1 x True = x
absReal1 x False = absReal0 x otherwise

absReal0 x True = `negate` x

absReal2 x = absReal1 x (x >= 0)

The following Function with conditions
undefined 
 | False
 = undefined

is transformed to
undefined  = undefined1

undefined0 True = undefined

undefined1  = undefined0 False

The following Function with conditions
takeWhile p [] = []
takeWhile p (x : xs)
 | p x
 = x : takeWhile p xs
 | otherwise
 = []

is transformed to
takeWhile p [] = takeWhile3 p []
takeWhile p (x : xs) = takeWhile2 p (x : xs)

takeWhile1 p x xs True = x : takeWhile p xs
takeWhile1 p x xs False = takeWhile0 p x xs otherwise

takeWhile0 p x xs True = []

takeWhile2 p (x : xs) = takeWhile1 p x xs (p x)

takeWhile3 p [] = []
takeWhile3 vuz vvu = takeWhile2 vuz vvu

The following Function with conditions
toEnum 0 = False
toEnum 1 = True

is transformed to
toEnum vvw = toEnum3 vvw
toEnum vvv = toEnum1 vvv

toEnum0 True vvv = True

toEnum1 vvv = toEnum0 (vvv == 1) vvv

toEnum2 True vvw = False
toEnum2 vvx vvy = toEnum1 vvy

toEnum3 vvw = toEnum2 (vvw == 0) vvw
toEnum3 vvz = toEnum1 vvz

The following Function with conditions
toEnum 0 = LT
toEnum 1 = EQ
toEnum 2 = GT

is transformed to
toEnum vwz = toEnum9 vwz
toEnum vwv = toEnum7 vwv
toEnum vwu = toEnum5 vwu

toEnum4 True vwu = GT

toEnum5 vwu = toEnum4 (vwu == 2) vwu

toEnum6 True vwv = EQ
toEnum6 vww vwx = toEnum5 vwx

toEnum7 vwv = toEnum6 (vwv == 1) vwv
toEnum7 vwy = toEnum5 vwy

toEnum8 True vwz = LT
toEnum8 vxu vxv = toEnum7 vxv

toEnum9 vwz = toEnum8 (vwz == 0) vwz
toEnum9 vxw = toEnum7 vxw



↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
HASKELL
              ↳ LetRed

mainModule Main
  ((enumFrom :: Enum a => a  ->  [a]) :: Enum a => a  ->  [a])

module Main where
  import qualified Prelude



Let/Where Reductions:
The bindings of the following Let/Where expression
reduce1 x y (y == 0)
where 
d  = gcd x y
reduce0 x y True = x `quot` d :% (y `quot` d)
reduce1 x y True = error []
reduce1 x y False = reduce0 x y otherwise

are unpacked to the following functions on top level
reduce2D vxx vxy = gcd vxx vxy

reduce2Reduce0 vxx vxy x y True = x `quot` reduce2D vxx vxy :% (y `quot` reduce2D vxx vxy)

reduce2Reduce1 vxx vxy x y True = error []
reduce2Reduce1 vxx vxy x y False = reduce2Reduce0 vxx vxy x y otherwise

The bindings of the following Let/Where expression
gcd' (abs x) (abs y)
where 
gcd' x xz = gcd'2 x xz
gcd' x y = gcd'0 x y
gcd'0 x y = gcd' y (x `rem` y)
gcd'1 True x xz = x
gcd'1 yu yv yw = gcd'0 yv yw
gcd'2 x xz = gcd'1 (xz == 0) x xz
gcd'2 yx yy = gcd'0 yx yy

are unpacked to the following functions on top level
gcd0Gcd' x xz = gcd0Gcd'2 x xz
gcd0Gcd' x y = gcd0Gcd'0 x y

gcd0Gcd'1 True x xz = x
gcd0Gcd'1 yu yv yw = gcd0Gcd'0 yv yw

gcd0Gcd'2 x xz = gcd0Gcd'1 (xz == 0) x xz
gcd0Gcd'2 yx yy = gcd0Gcd'0 yx yy

gcd0Gcd'0 x y = gcd0Gcd' y (x `rem` y)



↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
HASKELL
                  ↳ NumRed

mainModule Main
  ((enumFrom :: Enum a => a  ->  [a]) :: Enum a => a  ->  [a])

module Main where
  import qualified Prelude



Num Reduction: All numbers are transformed to thier corresponding representation with Pos, Neg, Succ and Zero.

↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
HASKELL
                      ↳ Narrow
                      ↳ Narrow

mainModule Main
  (enumFrom :: Enum a => a  ->  [a])

module Main where
  import qualified Prelude



Haskell To QDPs


↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
QDP
                            ↳ QDPSizeChangeProof
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_map0(vxz129, vxz130, :(Neg(Succ(vxz131000)), vxz1311), ba) → new_map(vxz129, vxz131000, vxz1311, Succ(vxz129), Succ(vxz131000), ba)
new_map(vxz129, vxz130, vxz131, Succ(vxz1320), Succ(vxz1330), ba) → new_map(vxz129, vxz130, vxz131, vxz1320, vxz1330, ba)
new_map(vxz129, vxz130, :(Neg(Succ(vxz131000)), vxz1311), Zero, Succ(vxz1330), ba) → new_map(vxz129, vxz131000, vxz1311, Succ(vxz129), Succ(vxz131000), ba)
new_map(vxz129, vxz130, vxz131, Zero, Zero, ba) → new_map0(vxz129, vxz130, vxz131, ba)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:



↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
QDP
                            ↳ QDPSizeChangeProof
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_map2(vxz123, :(Neg(Zero), vxz1251), ba) → new_map2(vxz123, vxz1251, ba)
new_map3(vxz123, vxz124, :(Pos(Zero), vxz1251), ba) → new_map2(vxz123, vxz1251, ba)
new_map2(vxz123, :(Pos(Zero), vxz1251), ba) → new_map2(vxz123, vxz1251, ba)
new_map1(vxz123, vxz124, vxz125, Succ(vxz1260), Succ(vxz1270), ba) → new_map1(vxz123, vxz124, vxz125, vxz1260, vxz1270, ba)
new_map1(vxz123, vxz124, vxz125, Zero, Zero, ba) → new_map3(vxz123, vxz124, vxz125, ba)
new_map1(vxz123, vxz124, :(Pos(Succ(vxz125000)), vxz1251), Zero, Succ(vxz1270), ba) → new_map1(vxz123, vxz125000, vxz1251, Succ(vxz125000), Succ(vxz123), ba)
new_map1(vxz123, vxz124, :(Neg(Succ(vxz125000)), vxz1251), Zero, Succ(vxz1270), ba) → new_map2(vxz123, vxz1251, ba)
new_map1(vxz123, vxz124, :(Pos(Zero), vxz1251), Zero, Succ(vxz1270), ba) → new_map2(vxz123, vxz1251, ba)
new_map2(vxz123, :(Pos(Succ(vxz125000)), vxz1251), ba) → new_map1(vxz123, vxz125000, vxz1251, Succ(vxz125000), Succ(vxz123), ba)
new_map3(vxz123, vxz124, :(Neg(Succ(vxz125000)), vxz1251), ba) → new_map2(vxz123, vxz1251, ba)
new_map2(vxz123, :(Neg(Succ(vxz125000)), vxz1251), ba) → new_map2(vxz123, vxz1251, ba)
new_map3(vxz123, vxz124, :(Pos(Succ(vxz125000)), vxz1251), ba) → new_map1(vxz123, vxz125000, vxz1251, Succ(vxz125000), Succ(vxz123), ba)
new_map1(vxz123, vxz124, :(Neg(Zero), vxz1251), Zero, Succ(vxz1270), ba) → new_map2(vxz123, vxz1251, ba)
new_map3(vxz123, vxz124, :(Neg(Zero), vxz1251), ba) → new_map2(vxz123, vxz1251, ba)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:



↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
QDP
                            ↳ DependencyGraphProof
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_map5(Pos(Zero), Pos(Zero), vxz41) → new_map4(Zero, vxz41)
new_map5(Neg(Zero), Pos(Zero), :(vxz410, vxz411)) → new_map5(Neg(Zero), vxz410, vxz411)
new_map5(Pos(Succ(vxz1500)), Pos(Zero), vxz41) → new_map4(Succ(vxz1500), vxz41)
new_map5(Pos(vxz150), Neg(Succ(vxz4000)), :(vxz410, vxz411)) → new_map5(Pos(vxz150), vxz410, vxz411)
new_map5(Neg(Zero), Neg(Succ(vxz4000)), vxz41) → new_map6(vxz41)
new_map6(:(vxz410, vxz411)) → new_map5(Neg(Zero), vxz410, vxz411)
new_map5(Neg(Zero), Neg(Zero), vxz41) → new_map6(vxz41)
new_map4(vxz150, :(vxz410, vxz411)) → new_map5(Pos(vxz150), vxz410, vxz411)
new_map5(Pos(Succ(vxz1500)), Neg(Zero), vxz41) → new_map4(Succ(vxz1500), vxz41)
new_map5(Pos(Zero), Neg(Zero), vxz41) → new_map4(Zero, vxz41)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs.

↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ AND
QDP
                                  ↳ QDPSizeChangeProof
                                ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_map5(Neg(Zero), Pos(Zero), :(vxz410, vxz411)) → new_map5(Neg(Zero), vxz410, vxz411)
new_map5(Neg(Zero), Neg(Succ(vxz4000)), vxz41) → new_map6(vxz41)
new_map6(:(vxz410, vxz411)) → new_map5(Neg(Zero), vxz410, vxz411)
new_map5(Neg(Zero), Neg(Zero), vxz41) → new_map6(vxz41)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:



↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ AND
                                ↳ QDP
QDP
                                  ↳ QDPSizeChangeProof
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_map5(Pos(Zero), Pos(Zero), vxz41) → new_map4(Zero, vxz41)
new_map5(Pos(Succ(vxz1500)), Pos(Zero), vxz41) → new_map4(Succ(vxz1500), vxz41)
new_map5(Pos(vxz150), Neg(Succ(vxz4000)), :(vxz410, vxz411)) → new_map5(Pos(vxz150), vxz410, vxz411)
new_map4(vxz150, :(vxz410, vxz411)) → new_map5(Pos(vxz150), vxz410, vxz411)
new_map5(Pos(Succ(vxz1500)), Neg(Zero), vxz41) → new_map4(Succ(vxz1500), vxz41)
new_map5(Pos(Zero), Neg(Zero), vxz41) → new_map4(Zero, vxz41)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:



↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
QDP
                            ↳ DependencyGraphProof
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_map7(vxz210, :(vxz910, vxz911)) → new_map8(Pos(vxz210), vxz910, vxz911)
new_map8(Neg(Zero), Neg(Zero), vxz91) → new_map9(vxz91)
new_map8(Pos(Succ(vxz2100)), Pos(Zero), vxz91) → new_map7(Succ(vxz2100), vxz91)
new_map8(Neg(Zero), Neg(Succ(vxz9000)), vxz91) → new_map9(vxz91)
new_map8(Pos(vxz210), Neg(Succ(vxz9000)), :(vxz910, vxz911)) → new_map8(Pos(vxz210), vxz910, vxz911)
new_map8(Neg(Zero), Pos(Zero), :(vxz910, vxz911)) → new_map8(Neg(Zero), vxz910, vxz911)
new_map8(Pos(Succ(vxz2100)), Neg(Zero), vxz91) → new_map7(Succ(vxz2100), vxz91)
new_map8(Pos(Zero), Neg(Zero), vxz91) → new_map7(Zero, vxz91)
new_map9(:(vxz910, vxz911)) → new_map8(Neg(Zero), vxz910, vxz911)
new_map8(Pos(Zero), Pos(Zero), vxz91) → new_map7(Zero, vxz91)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs.

↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ AND
QDP
                                  ↳ QDPSizeChangeProof
                                ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_map8(Neg(Zero), Neg(Zero), vxz91) → new_map9(vxz91)
new_map8(Neg(Zero), Neg(Succ(vxz9000)), vxz91) → new_map9(vxz91)
new_map8(Neg(Zero), Pos(Zero), :(vxz910, vxz911)) → new_map8(Neg(Zero), vxz910, vxz911)
new_map9(:(vxz910, vxz911)) → new_map8(Neg(Zero), vxz910, vxz911)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:



↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ AND
                                ↳ QDP
QDP
                                  ↳ QDPSizeChangeProof
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_map7(vxz210, :(vxz910, vxz911)) → new_map8(Pos(vxz210), vxz910, vxz911)
new_map8(Pos(Succ(vxz2100)), Pos(Zero), vxz91) → new_map7(Succ(vxz2100), vxz91)
new_map8(Pos(vxz210), Neg(Succ(vxz9000)), :(vxz910, vxz911)) → new_map8(Pos(vxz210), vxz910, vxz911)
new_map8(Pos(Succ(vxz2100)), Neg(Zero), vxz91) → new_map7(Succ(vxz2100), vxz91)
new_map8(Pos(Zero), Neg(Zero), vxz91) → new_map7(Zero, vxz91)
new_map8(Pos(Zero), Pos(Zero), vxz91) → new_map7(Zero, vxz91)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:



↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
QDP
                            ↳ DependencyGraphProof
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_map11(Pos(Succ(vxz1900)), Pos(Zero), vxz81) → new_map10(Succ(vxz1900), vxz81)
new_map11(Pos(Succ(vxz1900)), Neg(Zero), vxz81) → new_map10(Succ(vxz1900), vxz81)
new_map11(Neg(Zero), Neg(Succ(vxz8000)), vxz81) → new_map12(vxz81)
new_map11(Neg(Zero), Pos(Zero), :(vxz810, vxz811)) → new_map11(Neg(Zero), vxz810, vxz811)
new_map11(Pos(Zero), Pos(Zero), vxz81) → new_map10(Zero, vxz81)
new_map12(:(vxz810, vxz811)) → new_map11(Neg(Zero), vxz810, vxz811)
new_map11(Neg(Zero), Neg(Zero), vxz81) → new_map12(vxz81)
new_map10(vxz190, :(vxz810, vxz811)) → new_map11(Pos(vxz190), vxz810, vxz811)
new_map11(Pos(Zero), Neg(Zero), vxz81) → new_map10(Zero, vxz81)
new_map11(Pos(vxz190), Neg(Succ(vxz8000)), :(vxz810, vxz811)) → new_map11(Pos(vxz190), vxz810, vxz811)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs.

↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ AND
QDP
                                  ↳ QDPSizeChangeProof
                                ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_map11(Neg(Zero), Neg(Succ(vxz8000)), vxz81) → new_map12(vxz81)
new_map11(Neg(Zero), Pos(Zero), :(vxz810, vxz811)) → new_map11(Neg(Zero), vxz810, vxz811)
new_map12(:(vxz810, vxz811)) → new_map11(Neg(Zero), vxz810, vxz811)
new_map11(Neg(Zero), Neg(Zero), vxz81) → new_map12(vxz81)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:



↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ AND
                                ↳ QDP
QDP
                                  ↳ QDPSizeChangeProof
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_map11(Pos(Succ(vxz1900)), Pos(Zero), vxz81) → new_map10(Succ(vxz1900), vxz81)
new_map11(Pos(Succ(vxz1900)), Neg(Zero), vxz81) → new_map10(Succ(vxz1900), vxz81)
new_map11(Pos(Zero), Pos(Zero), vxz81) → new_map10(Zero, vxz81)
new_map11(Pos(Zero), Neg(Zero), vxz81) → new_map10(Zero, vxz81)
new_map10(vxz190, :(vxz810, vxz811)) → new_map11(Pos(vxz190), vxz810, vxz811)
new_map11(Pos(vxz190), Neg(Succ(vxz8000)), :(vxz810, vxz811)) → new_map11(Pos(vxz190), vxz810, vxz811)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:



↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
QDP
                            ↳ QDPSizeChangeProof
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_primMinusNat(Succ(vxz31000), Succ(vxz28000)) → new_primMinusNat(vxz31000, vxz28000)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:



↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
QDP
                            ↳ RuleRemovalProof
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_numericEnumFrom(vxz3) → new_numericEnumFrom(new_ps(vxz3))

The TRS R consists of the following rules:

new_primMinusNat0(Succ(vxz300)) → new_primMinusNat1(vxz300)
new_primPlusNat0(Succ(vxz300)) → Succ(Succ(new_primPlusNat(vxz300)))
new_primPlusNat(Succ(vxz3000)) → Succ(vxz3000)
new_primMinusNat0(Zero) → Pos(Succ(Zero))
new_primMinusNat1(Zero) → Pos(Zero)
new_primPlusNat(Zero) → Zero
new_primPlusNat0(Zero) → Succ(Zero)
new_primMinusNat1(Succ(vxz3000)) → Neg(Succ(vxz3000))
new_ps(Pos(vxz30)) → Pos(new_primPlusNat0(vxz30))
new_ps(Neg(vxz30)) → new_primMinusNat0(vxz30)

The set Q consists of the following terms:

new_primMinusNat1(Zero)
new_primPlusNat(Zero)
new_ps(Pos(x0))
new_primPlusNat0(Zero)
new_primPlusNat0(Succ(x0))
new_primMinusNat1(Succ(x0))
new_primMinusNat0(Zero)
new_primMinusNat0(Succ(x0))
new_ps(Neg(x0))
new_primPlusNat(Succ(x0))

We have to consider all minimal (P,Q,R)-chains.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.

Strictly oriented rules of the TRS R:

new_primMinusNat0(Zero) → Pos(Succ(Zero))
new_primMinusNat1(Zero) → Pos(Zero)

Used ordering: POLO with Polynomial interpretation [25]:

POL(Neg(x1)) = 2 + 2·x1   
POL(Pos(x1)) = 2·x1   
POL(Succ(x1)) = x1   
POL(Zero) = 0   
POL(new_numericEnumFrom(x1)) = x1   
POL(new_primMinusNat0(x1)) = 2 + 2·x1   
POL(new_primMinusNat1(x1)) = 2 + 2·x1   
POL(new_primPlusNat(x1)) = x1   
POL(new_primPlusNat0(x1)) = x1   
POL(new_ps(x1)) = x1   



↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                            ↳ RuleRemovalProof
QDP
                                ↳ NonTerminationProof
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_numericEnumFrom(vxz3) → new_numericEnumFrom(new_ps(vxz3))

The TRS R consists of the following rules:

new_primMinusNat0(Succ(vxz300)) → new_primMinusNat1(vxz300)
new_primPlusNat0(Succ(vxz300)) → Succ(Succ(new_primPlusNat(vxz300)))
new_primPlusNat(Succ(vxz3000)) → Succ(vxz3000)
new_primPlusNat(Zero) → Zero
new_primPlusNat0(Zero) → Succ(Zero)
new_primMinusNat1(Succ(vxz3000)) → Neg(Succ(vxz3000))
new_ps(Pos(vxz30)) → Pos(new_primPlusNat0(vxz30))
new_ps(Neg(vxz30)) → new_primMinusNat0(vxz30)

The set Q consists of the following terms:

new_primMinusNat1(Zero)
new_primPlusNat(Zero)
new_ps(Pos(x0))
new_primPlusNat0(Zero)
new_primPlusNat0(Succ(x0))
new_primMinusNat1(Succ(x0))
new_primMinusNat0(Zero)
new_primMinusNat0(Succ(x0))
new_ps(Neg(x0))
new_primPlusNat(Succ(x0))

We have to consider all minimal (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

new_numericEnumFrom(vxz3) → new_numericEnumFrom(new_ps(vxz3))

The TRS R consists of the following rules:

new_primMinusNat0(Succ(vxz300)) → new_primMinusNat1(vxz300)
new_primPlusNat0(Succ(vxz300)) → Succ(Succ(new_primPlusNat(vxz300)))
new_primPlusNat(Succ(vxz3000)) → Succ(vxz3000)
new_primPlusNat(Zero) → Zero
new_primPlusNat0(Zero) → Succ(Zero)
new_primMinusNat1(Succ(vxz3000)) → Neg(Succ(vxz3000))
new_ps(Pos(vxz30)) → Pos(new_primPlusNat0(vxz30))
new_ps(Neg(vxz30)) → new_primMinusNat0(vxz30)


s = new_numericEnumFrom(vxz3) evaluates to t =new_numericEnumFrom(new_ps(vxz3))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:




Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from new_numericEnumFrom(vxz3) to new_numericEnumFrom(new_ps(vxz3)).





↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
QDP
                            ↳ RuleRemovalProof
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_numericEnumFrom0(vxz3) → new_numericEnumFrom0(new_ps0(vxz3))

The TRS R consists of the following rules:

new_primMinusNat0(Succ(vxz300)) → new_primMinusNat1(vxz300)
new_primPlusInt(Neg(vxz300)) → new_primMinusNat0(vxz300)
new_primPlusNat0(Succ(vxz300)) → Succ(Succ(new_primPlusNat(vxz300)))
new_primPlusInt(Pos(vxz300)) → Pos(new_primPlusNat0(vxz300))
new_primPlusNat(Succ(vxz3000)) → Succ(vxz3000)
new_ps0(Integer(vxz30)) → Integer(new_primPlusInt(vxz30))
new_primMinusNat0(Zero) → Pos(Succ(Zero))
new_primMinusNat1(Zero) → Pos(Zero)
new_primPlusNat(Zero) → Zero
new_primPlusNat0(Zero) → Succ(Zero)
new_primMinusNat1(Succ(vxz3000)) → Neg(Succ(vxz3000))

The set Q consists of the following terms:

new_primMinusNat1(Zero)
new_primPlusInt(Pos(x0))
new_primPlusNat(Zero)
new_primPlusInt(Neg(x0))
new_primPlusNat0(Zero)
new_primPlusNat0(Succ(x0))
new_primMinusNat1(Succ(x0))
new_primMinusNat0(Zero)
new_ps0(Integer(x0))
new_primMinusNat0(Succ(x0))
new_primPlusNat(Succ(x0))

We have to consider all minimal (P,Q,R)-chains.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.

Strictly oriented rules of the TRS R:

new_primMinusNat0(Zero) → Pos(Succ(Zero))
new_primMinusNat1(Zero) → Pos(Zero)

Used ordering: POLO with Polynomial interpretation [25]:

POL(Integer(x1)) = x1   
POL(Neg(x1)) = 2·x1   
POL(Pos(x1)) = x1   
POL(Succ(x1)) = x1   
POL(Zero) = 1   
POL(new_numericEnumFrom0(x1)) = 2·x1   
POL(new_primMinusNat0(x1)) = 2·x1   
POL(new_primMinusNat1(x1)) = 2·x1   
POL(new_primPlusInt(x1)) = x1   
POL(new_primPlusNat(x1)) = x1   
POL(new_primPlusNat0(x1)) = x1   
POL(new_ps0(x1)) = x1   



↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                            ↳ RuleRemovalProof
QDP
                                ↳ NonTerminationProof
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_numericEnumFrom0(vxz3) → new_numericEnumFrom0(new_ps0(vxz3))

The TRS R consists of the following rules:

new_primMinusNat0(Succ(vxz300)) → new_primMinusNat1(vxz300)
new_primPlusInt(Neg(vxz300)) → new_primMinusNat0(vxz300)
new_primPlusNat0(Succ(vxz300)) → Succ(Succ(new_primPlusNat(vxz300)))
new_primPlusInt(Pos(vxz300)) → Pos(new_primPlusNat0(vxz300))
new_primPlusNat(Succ(vxz3000)) → Succ(vxz3000)
new_ps0(Integer(vxz30)) → Integer(new_primPlusInt(vxz30))
new_primPlusNat(Zero) → Zero
new_primPlusNat0(Zero) → Succ(Zero)
new_primMinusNat1(Succ(vxz3000)) → Neg(Succ(vxz3000))

The set Q consists of the following terms:

new_primMinusNat1(Zero)
new_primPlusInt(Pos(x0))
new_primPlusNat(Zero)
new_primPlusInt(Neg(x0))
new_primPlusNat0(Zero)
new_primPlusNat0(Succ(x0))
new_primMinusNat1(Succ(x0))
new_primMinusNat0(Zero)
new_ps0(Integer(x0))
new_primMinusNat0(Succ(x0))
new_primPlusNat(Succ(x0))

We have to consider all minimal (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

new_numericEnumFrom0(vxz3) → new_numericEnumFrom0(new_ps0(vxz3))

The TRS R consists of the following rules:

new_primMinusNat0(Succ(vxz300)) → new_primMinusNat1(vxz300)
new_primPlusInt(Neg(vxz300)) → new_primMinusNat0(vxz300)
new_primPlusNat0(Succ(vxz300)) → Succ(Succ(new_primPlusNat(vxz300)))
new_primPlusInt(Pos(vxz300)) → Pos(new_primPlusNat0(vxz300))
new_primPlusNat(Succ(vxz3000)) → Succ(vxz3000)
new_ps0(Integer(vxz30)) → Integer(new_primPlusInt(vxz30))
new_primPlusNat(Zero) → Zero
new_primPlusNat0(Zero) → Succ(Zero)
new_primMinusNat1(Succ(vxz3000)) → Neg(Succ(vxz3000))


s = new_numericEnumFrom0(vxz3) evaluates to t =new_numericEnumFrom0(new_ps0(vxz3))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:




Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from new_numericEnumFrom0(vxz3) to new_numericEnumFrom0(new_ps0(vxz3)).





↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
QDP
                            ↳ QDPSizeChangeProof
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_primMulNat(Succ(vxz3100)) → new_primMulNat(vxz3100)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:



↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
QDP
                            ↳ RuleRemovalProof
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_numericEnumFrom1(vxz3) → new_numericEnumFrom1(new_ps1(vxz3))

The TRS R consists of the following rules:

new_primPlusInt(Neg(vxz300)) → new_primMinusNat0(vxz300)
new_primMinusNat0(Succ(vxz300)) → new_primMinusNat1(vxz300)
new_primPlusNat0(Succ(vxz300)) → Succ(Succ(new_primPlusNat(vxz300)))
new_primPlusInt(Pos(vxz300)) → Pos(new_primPlusNat0(vxz300))
new_ps1(Float(vxz30, vxz31)) → Float(new_ps2(vxz30), new_sr(vxz31))
new_ps2(vxz30) → new_primPlusInt(vxz30)
new_primPlusNat0(Zero) → Succ(Zero)
new_primMinusNat1(Succ(vxz3000)) → Neg(Succ(vxz3000))
new_primMulNat0(Succ(vxz3100)) → new_primPlusNat0(new_primMulNat0(vxz3100))
new_primMulNat0(Zero) → Zero
new_primPlusNat(Succ(vxz3000)) → Succ(vxz3000)
new_primMinusNat0(Zero) → Pos(Succ(Zero))
new_primMulInt(Pos(vxz310)) → Pos(new_primMulNat0(vxz310))
new_primMinusNat1(Zero) → Pos(Zero)
new_primPlusNat(Zero) → Zero
new_primMulInt(Neg(vxz310)) → Neg(new_primMulNat0(vxz310))
new_sr(vxz31) → new_primMulInt(vxz31)

The set Q consists of the following terms:

new_ps1(Float(x0, x1))
new_primMulInt(Neg(x0))
new_primPlusInt(Neg(x0))
new_primPlusNat0(Succ(x0))
new_ps2(x0)
new_primMinusNat1(Succ(x0))
new_primMinusNat0(Succ(x0))
new_primPlusNat(Succ(x0))
new_primMinusNat1(Zero)
new_primPlusInt(Pos(x0))
new_sr(x0)
new_primPlusNat(Zero)
new_primMulNat0(Succ(x0))
new_primPlusNat0(Zero)
new_primMulNat0(Zero)
new_primMinusNat0(Zero)
new_primMulInt(Pos(x0))

We have to consider all minimal (P,Q,R)-chains.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.

Strictly oriented rules of the TRS R:

new_primMinusNat0(Zero) → Pos(Succ(Zero))
new_primMinusNat1(Zero) → Pos(Zero)

Used ordering: POLO with Polynomial interpretation [25]:

POL(Float(x1, x2)) = 2·x1 + 2·x2   
POL(Neg(x1)) = 1 + x1   
POL(Pos(x1)) = x1   
POL(Succ(x1)) = x1   
POL(Zero) = 0   
POL(new_numericEnumFrom1(x1)) = x1   
POL(new_primMinusNat0(x1)) = 1 + x1   
POL(new_primMinusNat1(x1)) = 1 + x1   
POL(new_primMulInt(x1)) = x1   
POL(new_primMulNat0(x1)) = x1   
POL(new_primPlusInt(x1)) = x1   
POL(new_primPlusNat(x1)) = x1   
POL(new_primPlusNat0(x1)) = x1   
POL(new_ps1(x1)) = x1   
POL(new_ps2(x1)) = x1   
POL(new_sr(x1)) = x1   



↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                            ↳ RuleRemovalProof
QDP
                                ↳ NonTerminationProof
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_numericEnumFrom1(vxz3) → new_numericEnumFrom1(new_ps1(vxz3))

The TRS R consists of the following rules:

new_primPlusInt(Neg(vxz300)) → new_primMinusNat0(vxz300)
new_primMinusNat0(Succ(vxz300)) → new_primMinusNat1(vxz300)
new_primPlusNat0(Succ(vxz300)) → Succ(Succ(new_primPlusNat(vxz300)))
new_primPlusInt(Pos(vxz300)) → Pos(new_primPlusNat0(vxz300))
new_ps1(Float(vxz30, vxz31)) → Float(new_ps2(vxz30), new_sr(vxz31))
new_ps2(vxz30) → new_primPlusInt(vxz30)
new_primPlusNat0(Zero) → Succ(Zero)
new_primMinusNat1(Succ(vxz3000)) → Neg(Succ(vxz3000))
new_primMulNat0(Succ(vxz3100)) → new_primPlusNat0(new_primMulNat0(vxz3100))
new_primMulNat0(Zero) → Zero
new_primPlusNat(Succ(vxz3000)) → Succ(vxz3000)
new_primMulInt(Pos(vxz310)) → Pos(new_primMulNat0(vxz310))
new_primPlusNat(Zero) → Zero
new_primMulInt(Neg(vxz310)) → Neg(new_primMulNat0(vxz310))
new_sr(vxz31) → new_primMulInt(vxz31)

The set Q consists of the following terms:

new_ps1(Float(x0, x1))
new_primMulInt(Neg(x0))
new_primPlusInt(Neg(x0))
new_primPlusNat0(Succ(x0))
new_ps2(x0)
new_primMinusNat1(Succ(x0))
new_primMinusNat0(Succ(x0))
new_primPlusNat(Succ(x0))
new_primMinusNat1(Zero)
new_primPlusInt(Pos(x0))
new_sr(x0)
new_primPlusNat(Zero)
new_primMulNat0(Succ(x0))
new_primPlusNat0(Zero)
new_primMulNat0(Zero)
new_primMinusNat0(Zero)
new_primMulInt(Pos(x0))

We have to consider all minimal (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

new_numericEnumFrom1(vxz3) → new_numericEnumFrom1(new_ps1(vxz3))

The TRS R consists of the following rules:

new_primPlusInt(Neg(vxz300)) → new_primMinusNat0(vxz300)
new_primMinusNat0(Succ(vxz300)) → new_primMinusNat1(vxz300)
new_primPlusNat0(Succ(vxz300)) → Succ(Succ(new_primPlusNat(vxz300)))
new_primPlusInt(Pos(vxz300)) → Pos(new_primPlusNat0(vxz300))
new_ps1(Float(vxz30, vxz31)) → Float(new_ps2(vxz30), new_sr(vxz31))
new_ps2(vxz30) → new_primPlusInt(vxz30)
new_primPlusNat0(Zero) → Succ(Zero)
new_primMinusNat1(Succ(vxz3000)) → Neg(Succ(vxz3000))
new_primMulNat0(Succ(vxz3100)) → new_primPlusNat0(new_primMulNat0(vxz3100))
new_primMulNat0(Zero) → Zero
new_primPlusNat(Succ(vxz3000)) → Succ(vxz3000)
new_primMulInt(Pos(vxz310)) → Pos(new_primMulNat0(vxz310))
new_primPlusNat(Zero) → Zero
new_primMulInt(Neg(vxz310)) → Neg(new_primMulNat0(vxz310))
new_sr(vxz31) → new_primMulInt(vxz31)


s = new_numericEnumFrom1(vxz3) evaluates to t =new_numericEnumFrom1(new_ps1(vxz3))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:




Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from new_numericEnumFrom1(vxz3) to new_numericEnumFrom1(new_ps1(vxz3)).





↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
QDP
                            ↳ RuleRemovalProof
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_numericEnumFrom2(vxz3) → new_numericEnumFrom2(new_ps3(vxz3))

The TRS R consists of the following rules:

new_primPlusInt(Neg(vxz300)) → new_primMinusNat0(vxz300)
new_primMinusNat0(Succ(vxz300)) → new_primMinusNat1(vxz300)
new_primPlusNat0(Succ(vxz300)) → Succ(Succ(new_primPlusNat(vxz300)))
new_primPlusInt(Pos(vxz300)) → Pos(new_primPlusNat0(vxz300))
new_ps2(vxz30) → new_primPlusInt(vxz30)
new_ps3(Double(vxz30, vxz31)) → Double(new_ps2(vxz30), new_sr(vxz31))
new_primPlusNat0(Zero) → Succ(Zero)
new_primMinusNat1(Succ(vxz3000)) → Neg(Succ(vxz3000))
new_primMulNat0(Succ(vxz3100)) → new_primPlusNat0(new_primMulNat0(vxz3100))
new_primMulNat0(Zero) → Zero
new_primPlusNat(Succ(vxz3000)) → Succ(vxz3000)
new_primMinusNat0(Zero) → Pos(Succ(Zero))
new_primMulInt(Pos(vxz310)) → Pos(new_primMulNat0(vxz310))
new_primMinusNat1(Zero) → Pos(Zero)
new_primPlusNat(Zero) → Zero
new_primMulInt(Neg(vxz310)) → Neg(new_primMulNat0(vxz310))
new_sr(vxz31) → new_primMulInt(vxz31)

The set Q consists of the following terms:

new_primMulInt(Neg(x0))
new_primPlusInt(Neg(x0))
new_primPlusNat0(Succ(x0))
new_ps2(x0)
new_primMinusNat1(Succ(x0))
new_primMinusNat0(Succ(x0))
new_primPlusNat(Succ(x0))
new_primMinusNat1(Zero)
new_primPlusInt(Pos(x0))
new_sr(x0)
new_primPlusNat(Zero)
new_primMulNat0(Succ(x0))
new_primPlusNat0(Zero)
new_primMulNat0(Zero)
new_primMinusNat0(Zero)
new_ps3(Double(x0, x1))
new_primMulInt(Pos(x0))

We have to consider all minimal (P,Q,R)-chains.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.

Strictly oriented rules of the TRS R:

new_primMinusNat0(Zero) → Pos(Succ(Zero))
new_primMinusNat1(Zero) → Pos(Zero)

Used ordering: POLO with Polynomial interpretation [25]:

POL(Double(x1, x2)) = 2·x1 + 2·x2   
POL(Neg(x1)) = 2 + 2·x1   
POL(Pos(x1)) = 2·x1   
POL(Succ(x1)) = x1   
POL(Zero) = 1   
POL(new_numericEnumFrom2(x1)) = x1   
POL(new_primMinusNat0(x1)) = 2 + 2·x1   
POL(new_primMinusNat1(x1)) = 2 + 2·x1   
POL(new_primMulInt(x1)) = x1   
POL(new_primMulNat0(x1)) = x1   
POL(new_primPlusInt(x1)) = x1   
POL(new_primPlusNat(x1)) = x1   
POL(new_primPlusNat0(x1)) = x1   
POL(new_ps2(x1)) = x1   
POL(new_ps3(x1)) = x1   
POL(new_sr(x1)) = x1   



↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                            ↳ RuleRemovalProof
QDP
                                ↳ NonTerminationProof
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_numericEnumFrom2(vxz3) → new_numericEnumFrom2(new_ps3(vxz3))

The TRS R consists of the following rules:

new_primPlusInt(Neg(vxz300)) → new_primMinusNat0(vxz300)
new_primMinusNat0(Succ(vxz300)) → new_primMinusNat1(vxz300)
new_primPlusNat0(Succ(vxz300)) → Succ(Succ(new_primPlusNat(vxz300)))
new_primPlusInt(Pos(vxz300)) → Pos(new_primPlusNat0(vxz300))
new_ps2(vxz30) → new_primPlusInt(vxz30)
new_ps3(Double(vxz30, vxz31)) → Double(new_ps2(vxz30), new_sr(vxz31))
new_primPlusNat0(Zero) → Succ(Zero)
new_primMinusNat1(Succ(vxz3000)) → Neg(Succ(vxz3000))
new_primMulNat0(Succ(vxz3100)) → new_primPlusNat0(new_primMulNat0(vxz3100))
new_primMulNat0(Zero) → Zero
new_primPlusNat(Succ(vxz3000)) → Succ(vxz3000)
new_primMulInt(Pos(vxz310)) → Pos(new_primMulNat0(vxz310))
new_primPlusNat(Zero) → Zero
new_primMulInt(Neg(vxz310)) → Neg(new_primMulNat0(vxz310))
new_sr(vxz31) → new_primMulInt(vxz31)

The set Q consists of the following terms:

new_primMulInt(Neg(x0))
new_primPlusInt(Neg(x0))
new_primPlusNat0(Succ(x0))
new_ps2(x0)
new_primMinusNat1(Succ(x0))
new_primMinusNat0(Succ(x0))
new_primPlusNat(Succ(x0))
new_primMinusNat1(Zero)
new_primPlusInt(Pos(x0))
new_sr(x0)
new_primPlusNat(Zero)
new_primMulNat0(Succ(x0))
new_primPlusNat0(Zero)
new_primMulNat0(Zero)
new_primMinusNat0(Zero)
new_ps3(Double(x0, x1))
new_primMulInt(Pos(x0))

We have to consider all minimal (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

new_numericEnumFrom2(vxz3) → new_numericEnumFrom2(new_ps3(vxz3))

The TRS R consists of the following rules:

new_primPlusInt(Neg(vxz300)) → new_primMinusNat0(vxz300)
new_primMinusNat0(Succ(vxz300)) → new_primMinusNat1(vxz300)
new_primPlusNat0(Succ(vxz300)) → Succ(Succ(new_primPlusNat(vxz300)))
new_primPlusInt(Pos(vxz300)) → Pos(new_primPlusNat0(vxz300))
new_ps2(vxz30) → new_primPlusInt(vxz30)
new_ps3(Double(vxz30, vxz31)) → Double(new_ps2(vxz30), new_sr(vxz31))
new_primPlusNat0(Zero) → Succ(Zero)
new_primMinusNat1(Succ(vxz3000)) → Neg(Succ(vxz3000))
new_primMulNat0(Succ(vxz3100)) → new_primPlusNat0(new_primMulNat0(vxz3100))
new_primMulNat0(Zero) → Zero
new_primPlusNat(Succ(vxz3000)) → Succ(vxz3000)
new_primMulInt(Pos(vxz310)) → Pos(new_primMulNat0(vxz310))
new_primPlusNat(Zero) → Zero
new_primMulInt(Neg(vxz310)) → Neg(new_primMulNat0(vxz310))
new_sr(vxz31) → new_primMulInt(vxz31)


s = new_numericEnumFrom2(vxz3) evaluates to t =new_numericEnumFrom2(new_ps3(vxz3))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:




Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from new_numericEnumFrom2(vxz3) to new_numericEnumFrom2(new_ps3(vxz3)).





↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
QDP
                            ↳ QDPSizeChangeProof
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_primPlusNat1(Succ(vxz28000), Succ(vxz31000)) → new_primPlusNat1(vxz28000, vxz31000)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:



↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
QDP
                            ↳ QDPSizeChangeProof
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_primMinusNatS(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS(vxz1680, vxz1690)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:



↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
QDP
                            ↳ DependencyGraphProof
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_gcd0Gcd'1(Succ(Zero), Succ(vxz2320), vxz236) → new_gcd0Gcd'12(Zero, Succ(vxz2320))
new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Zero) → new_gcd0Gcd'11(vxz272, vxz273)
new_gcd0Gcd'10(vxz272, vxz273, Zero, Succ(vxz2750)) → new_gcd0Gcd'12(Succ(vxz272), vxz273)
new_gcd0Gcd'10(vxz272, vxz273, Zero, Zero) → new_gcd0Gcd'11(vxz272, vxz273)
new_gcd0Gcd'1(Succ(Succ(vxz23700)), Succ(vxz2320), vxz236) → new_gcd0Gcd'10(vxz23700, Succ(vxz2320), vxz23700, vxz2320)
new_gcd0Gcd'13(vxz5200, vxz2700) → new_gcd0Gcd'1(Succ(vxz5200), vxz2700, Succ(vxz5200))
new_gcd0Gcd'1(Succ(Zero), Zero, vxz236) → new_gcd0Gcd'1(new_primMinusNatS0(Zero, Zero), Zero, new_primMinusNatS0(Zero, Zero))
new_gcd0Gcd'11(vxz263, vxz264) → new_gcd0Gcd'1(new_primMinusNatS0(Succ(vxz263), vxz264), vxz264, new_primMinusNatS0(Succ(vxz263), vxz264))
new_gcd0Gcd'12(vxz266, vxz267) → new_gcd0Gcd'13(vxz267, vxz266)
new_gcd0Gcd'1(Succ(Succ(vxz23700)), Zero, vxz236) → new_gcd0Gcd'11(vxz23700, Zero)
new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Succ(vxz2750)) → new_gcd0Gcd'10(vxz272, vxz273, vxz2740, vxz2750)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                            ↳ DependencyGraphProof
QDP
                                ↳ QDPOrderProof
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_gcd0Gcd'1(Succ(Zero), Succ(vxz2320), vxz236) → new_gcd0Gcd'12(Zero, Succ(vxz2320))
new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Zero) → new_gcd0Gcd'11(vxz272, vxz273)
new_gcd0Gcd'10(vxz272, vxz273, Zero, Succ(vxz2750)) → new_gcd0Gcd'12(Succ(vxz272), vxz273)
new_gcd0Gcd'10(vxz272, vxz273, Zero, Zero) → new_gcd0Gcd'11(vxz272, vxz273)
new_gcd0Gcd'1(Succ(Succ(vxz23700)), Succ(vxz2320), vxz236) → new_gcd0Gcd'10(vxz23700, Succ(vxz2320), vxz23700, vxz2320)
new_gcd0Gcd'13(vxz5200, vxz2700) → new_gcd0Gcd'1(Succ(vxz5200), vxz2700, Succ(vxz5200))
new_gcd0Gcd'11(vxz263, vxz264) → new_gcd0Gcd'1(new_primMinusNatS0(Succ(vxz263), vxz264), vxz264, new_primMinusNatS0(Succ(vxz263), vxz264))
new_gcd0Gcd'12(vxz266, vxz267) → new_gcd0Gcd'13(vxz267, vxz266)
new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Succ(vxz2750)) → new_gcd0Gcd'10(vxz272, vxz273, vxz2740, vxz2750)
new_gcd0Gcd'1(Succ(Succ(vxz23700)), Zero, vxz236) → new_gcd0Gcd'11(vxz23700, Zero)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))

We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


new_gcd0Gcd'1(Succ(Zero), Succ(vxz2320), vxz236) → new_gcd0Gcd'12(Zero, Succ(vxz2320))
The remaining pairs can at least be oriented weakly.

new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Zero) → new_gcd0Gcd'11(vxz272, vxz273)
new_gcd0Gcd'10(vxz272, vxz273, Zero, Succ(vxz2750)) → new_gcd0Gcd'12(Succ(vxz272), vxz273)
new_gcd0Gcd'10(vxz272, vxz273, Zero, Zero) → new_gcd0Gcd'11(vxz272, vxz273)
new_gcd0Gcd'1(Succ(Succ(vxz23700)), Succ(vxz2320), vxz236) → new_gcd0Gcd'10(vxz23700, Succ(vxz2320), vxz23700, vxz2320)
new_gcd0Gcd'13(vxz5200, vxz2700) → new_gcd0Gcd'1(Succ(vxz5200), vxz2700, Succ(vxz5200))
new_gcd0Gcd'11(vxz263, vxz264) → new_gcd0Gcd'1(new_primMinusNatS0(Succ(vxz263), vxz264), vxz264, new_primMinusNatS0(Succ(vxz263), vxz264))
new_gcd0Gcd'12(vxz266, vxz267) → new_gcd0Gcd'13(vxz267, vxz266)
new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Succ(vxz2750)) → new_gcd0Gcd'10(vxz272, vxz273, vxz2740, vxz2750)
new_gcd0Gcd'1(Succ(Succ(vxz23700)), Zero, vxz236) → new_gcd0Gcd'11(vxz23700, Zero)
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( new_primMinusNatS0(x1, x2) ) =
/1\
\0/
+
/00\
\11/
·x1+
/00\
\00/
·x2

M( Succ(x1) ) =
/0\
\0/
+
/00\
\11/
·x1

M( Zero ) =
/1\
\0/

Tuple symbols:
M( new_gcd0Gcd'10(x1, ..., x4) ) = 0+
[1,1]
·x1+
[1,1]
·x2+
[0,0]
·x3+
[0,0]
·x4

M( new_gcd0Gcd'11(x1, x2) ) = 0+
[1,1]
·x1+
[0,1]
·x2

M( new_gcd0Gcd'1(x1, ..., x3) ) = 0+
[0,1]
·x1+
[0,1]
·x2+
[0,0]
·x3

M( new_gcd0Gcd'12(x1, x2) ) = 0+
[0,1]
·x1+
[1,1]
·x2

M( new_gcd0Gcd'13(x1, x2) ) = 0+
[1,1]
·x1+
[0,1]
·x2


Matrix type:
We used a basic matrix type which is not further parametrizeable.


As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
The following usable rules [17] were oriented:

new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)



↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ QDPOrderProof
QDP
                                    ↳ QDPOrderProof
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Zero) → new_gcd0Gcd'11(vxz272, vxz273)
new_gcd0Gcd'10(vxz272, vxz273, Zero, Succ(vxz2750)) → new_gcd0Gcd'12(Succ(vxz272), vxz273)
new_gcd0Gcd'10(vxz272, vxz273, Zero, Zero) → new_gcd0Gcd'11(vxz272, vxz273)
new_gcd0Gcd'13(vxz5200, vxz2700) → new_gcd0Gcd'1(Succ(vxz5200), vxz2700, Succ(vxz5200))
new_gcd0Gcd'1(Succ(Succ(vxz23700)), Succ(vxz2320), vxz236) → new_gcd0Gcd'10(vxz23700, Succ(vxz2320), vxz23700, vxz2320)
new_gcd0Gcd'12(vxz266, vxz267) → new_gcd0Gcd'13(vxz267, vxz266)
new_gcd0Gcd'11(vxz263, vxz264) → new_gcd0Gcd'1(new_primMinusNatS0(Succ(vxz263), vxz264), vxz264, new_primMinusNatS0(Succ(vxz263), vxz264))
new_gcd0Gcd'1(Succ(Succ(vxz23700)), Zero, vxz236) → new_gcd0Gcd'11(vxz23700, Zero)
new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Succ(vxz2750)) → new_gcd0Gcd'10(vxz272, vxz273, vxz2740, vxz2750)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))

We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


new_gcd0Gcd'1(Succ(Succ(vxz23700)), Zero, vxz236) → new_gcd0Gcd'11(vxz23700, Zero)
The remaining pairs can at least be oriented weakly.

new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Zero) → new_gcd0Gcd'11(vxz272, vxz273)
new_gcd0Gcd'10(vxz272, vxz273, Zero, Succ(vxz2750)) → new_gcd0Gcd'12(Succ(vxz272), vxz273)
new_gcd0Gcd'10(vxz272, vxz273, Zero, Zero) → new_gcd0Gcd'11(vxz272, vxz273)
new_gcd0Gcd'13(vxz5200, vxz2700) → new_gcd0Gcd'1(Succ(vxz5200), vxz2700, Succ(vxz5200))
new_gcd0Gcd'1(Succ(Succ(vxz23700)), Succ(vxz2320), vxz236) → new_gcd0Gcd'10(vxz23700, Succ(vxz2320), vxz23700, vxz2320)
new_gcd0Gcd'12(vxz266, vxz267) → new_gcd0Gcd'13(vxz267, vxz266)
new_gcd0Gcd'11(vxz263, vxz264) → new_gcd0Gcd'1(new_primMinusNatS0(Succ(vxz263), vxz264), vxz264, new_primMinusNatS0(Succ(vxz263), vxz264))
new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Succ(vxz2750)) → new_gcd0Gcd'10(vxz272, vxz273, vxz2740, vxz2750)
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( new_primMinusNatS0(x1, x2) ) =
/1\
\0/
+
/01\
\01/
·x1+
/00\
\00/
·x2

M( Succ(x1) ) =
/1\
\0/
+
/00\
\11/
·x1

M( Zero ) =
/0\
\0/

Tuple symbols:
M( new_gcd0Gcd'10(x1, ..., x4) ) = 0+
[1,1]
·x1+
[1,1]
·x2+
[0,0]
·x3+
[0,0]
·x4

M( new_gcd0Gcd'11(x1, x2) ) = 0+
[1,1]
·x1+
[0,1]
·x2

M( new_gcd0Gcd'1(x1, ..., x3) ) = 0+
[0,1]
·x1+
[0,1]
·x2+
[0,0]
·x3

M( new_gcd0Gcd'12(x1, x2) ) = 0+
[0,1]
·x1+
[1,1]
·x2

M( new_gcd0Gcd'13(x1, x2) ) = 0+
[1,1]
·x1+
[0,1]
·x2


Matrix type:
We used a basic matrix type which is not further parametrizeable.


As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
The following usable rules [17] were oriented:

new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)



↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
QDP
                                        ↳ QDPOrderProof
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Zero) → new_gcd0Gcd'11(vxz272, vxz273)
new_gcd0Gcd'10(vxz272, vxz273, Zero, Succ(vxz2750)) → new_gcd0Gcd'12(Succ(vxz272), vxz273)
new_gcd0Gcd'10(vxz272, vxz273, Zero, Zero) → new_gcd0Gcd'11(vxz272, vxz273)
new_gcd0Gcd'1(Succ(Succ(vxz23700)), Succ(vxz2320), vxz236) → new_gcd0Gcd'10(vxz23700, Succ(vxz2320), vxz23700, vxz2320)
new_gcd0Gcd'13(vxz5200, vxz2700) → new_gcd0Gcd'1(Succ(vxz5200), vxz2700, Succ(vxz5200))
new_gcd0Gcd'11(vxz263, vxz264) → new_gcd0Gcd'1(new_primMinusNatS0(Succ(vxz263), vxz264), vxz264, new_primMinusNatS0(Succ(vxz263), vxz264))
new_gcd0Gcd'12(vxz266, vxz267) → new_gcd0Gcd'13(vxz267, vxz266)
new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Succ(vxz2750)) → new_gcd0Gcd'10(vxz272, vxz273, vxz2740, vxz2750)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))

We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Zero) → new_gcd0Gcd'11(vxz272, vxz273)
new_gcd0Gcd'10(vxz272, vxz273, Zero, Zero) → new_gcd0Gcd'11(vxz272, vxz273)
The remaining pairs can at least be oriented weakly.

new_gcd0Gcd'10(vxz272, vxz273, Zero, Succ(vxz2750)) → new_gcd0Gcd'12(Succ(vxz272), vxz273)
new_gcd0Gcd'1(Succ(Succ(vxz23700)), Succ(vxz2320), vxz236) → new_gcd0Gcd'10(vxz23700, Succ(vxz2320), vxz23700, vxz2320)
new_gcd0Gcd'13(vxz5200, vxz2700) → new_gcd0Gcd'1(Succ(vxz5200), vxz2700, Succ(vxz5200))
new_gcd0Gcd'11(vxz263, vxz264) → new_gcd0Gcd'1(new_primMinusNatS0(Succ(vxz263), vxz264), vxz264, new_primMinusNatS0(Succ(vxz263), vxz264))
new_gcd0Gcd'12(vxz266, vxz267) → new_gcd0Gcd'13(vxz267, vxz266)
new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Succ(vxz2750)) → new_gcd0Gcd'10(vxz272, vxz273, vxz2740, vxz2750)
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( new_primMinusNatS0(x1, x2) ) =
/0\
\1/
+
/10\
\00/
·x1+
/00\
\00/
·x2

M( Succ(x1) ) =
/0\
\1/
+
/11\
\00/
·x1

M( Zero ) =
/0\
\0/

Tuple symbols:
M( new_gcd0Gcd'10(x1, ..., x4) ) = 1+
[1,1]
·x1+
[1,1]
·x2+
[0,0]
·x3+
[0,0]
·x4

M( new_gcd0Gcd'11(x1, x2) ) = 0+
[1,1]
·x1+
[1,1]
·x2

M( new_gcd0Gcd'1(x1, ..., x3) ) = 0+
[1,0]
·x1+
[1,1]
·x2+
[0,0]
·x3

M( new_gcd0Gcd'12(x1, x2) ) = 0+
[1,1]
·x1+
[1,1]
·x2

M( new_gcd0Gcd'13(x1, x2) ) = 0+
[1,1]
·x1+
[1,1]
·x2


Matrix type:
We used a basic matrix type which is not further parametrizeable.


As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
The following usable rules [17] were oriented:

new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)



↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
QDP
                                            ↳ DependencyGraphProof
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_gcd0Gcd'10(vxz272, vxz273, Zero, Succ(vxz2750)) → new_gcd0Gcd'12(Succ(vxz272), vxz273)
new_gcd0Gcd'13(vxz5200, vxz2700) → new_gcd0Gcd'1(Succ(vxz5200), vxz2700, Succ(vxz5200))
new_gcd0Gcd'1(Succ(Succ(vxz23700)), Succ(vxz2320), vxz236) → new_gcd0Gcd'10(vxz23700, Succ(vxz2320), vxz23700, vxz2320)
new_gcd0Gcd'12(vxz266, vxz267) → new_gcd0Gcd'13(vxz267, vxz266)
new_gcd0Gcd'11(vxz263, vxz264) → new_gcd0Gcd'1(new_primMinusNatS0(Succ(vxz263), vxz264), vxz264, new_primMinusNatS0(Succ(vxz263), vxz264))
new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Succ(vxz2750)) → new_gcd0Gcd'10(vxz272, vxz273, vxz2740, vxz2750)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ DependencyGraphProof
QDP
                                                ↳ UsableRulesProof
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_gcd0Gcd'10(vxz272, vxz273, Zero, Succ(vxz2750)) → new_gcd0Gcd'12(Succ(vxz272), vxz273)
new_gcd0Gcd'1(Succ(Succ(vxz23700)), Succ(vxz2320), vxz236) → new_gcd0Gcd'10(vxz23700, Succ(vxz2320), vxz23700, vxz2320)
new_gcd0Gcd'13(vxz5200, vxz2700) → new_gcd0Gcd'1(Succ(vxz5200), vxz2700, Succ(vxz5200))
new_gcd0Gcd'12(vxz266, vxz267) → new_gcd0Gcd'13(vxz267, vxz266)
new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Succ(vxz2750)) → new_gcd0Gcd'10(vxz272, vxz273, vxz2740, vxz2750)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))

We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.

↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ UsableRulesProof
QDP
                                                    ↳ QReductionProof
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_gcd0Gcd'10(vxz272, vxz273, Zero, Succ(vxz2750)) → new_gcd0Gcd'12(Succ(vxz272), vxz273)
new_gcd0Gcd'1(Succ(Succ(vxz23700)), Succ(vxz2320), vxz236) → new_gcd0Gcd'10(vxz23700, Succ(vxz2320), vxz23700, vxz2320)
new_gcd0Gcd'13(vxz5200, vxz2700) → new_gcd0Gcd'1(Succ(vxz5200), vxz2700, Succ(vxz5200))
new_gcd0Gcd'12(vxz266, vxz267) → new_gcd0Gcd'13(vxz267, vxz266)
new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Succ(vxz2750)) → new_gcd0Gcd'10(vxz272, vxz273, vxz2740, vxz2750)

R is empty.
The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))



↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ UsableRulesProof
                                                  ↳ QDP
                                                    ↳ QReductionProof
QDP
                                                        ↳ Instantiation
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_gcd0Gcd'10(vxz272, vxz273, Zero, Succ(vxz2750)) → new_gcd0Gcd'12(Succ(vxz272), vxz273)
new_gcd0Gcd'13(vxz5200, vxz2700) → new_gcd0Gcd'1(Succ(vxz5200), vxz2700, Succ(vxz5200))
new_gcd0Gcd'1(Succ(Succ(vxz23700)), Succ(vxz2320), vxz236) → new_gcd0Gcd'10(vxz23700, Succ(vxz2320), vxz23700, vxz2320)
new_gcd0Gcd'12(vxz266, vxz267) → new_gcd0Gcd'13(vxz267, vxz266)
new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Succ(vxz2750)) → new_gcd0Gcd'10(vxz272, vxz273, vxz2740, vxz2750)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule new_gcd0Gcd'1(Succ(Succ(vxz23700)), Succ(vxz2320), vxz236) → new_gcd0Gcd'10(vxz23700, Succ(vxz2320), vxz23700, vxz2320) we obtained the following new rules:

new_gcd0Gcd'1(Succ(Succ(x0)), Succ(x1), Succ(Succ(x0))) → new_gcd0Gcd'10(x0, Succ(x1), x0, x1)



↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ UsableRulesProof
                                                  ↳ QDP
                                                    ↳ QReductionProof
                                                      ↳ QDP
                                                        ↳ Instantiation
QDP
                                                            ↳ NonInfProof
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_gcd0Gcd'1(Succ(Succ(x0)), Succ(x1), Succ(Succ(x0))) → new_gcd0Gcd'10(x0, Succ(x1), x0, x1)
new_gcd0Gcd'10(vxz272, vxz273, Zero, Succ(vxz2750)) → new_gcd0Gcd'12(Succ(vxz272), vxz273)
new_gcd0Gcd'13(vxz5200, vxz2700) → new_gcd0Gcd'1(Succ(vxz5200), vxz2700, Succ(vxz5200))
new_gcd0Gcd'12(vxz266, vxz267) → new_gcd0Gcd'13(vxz267, vxz266)
new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Succ(vxz2750)) → new_gcd0Gcd'10(vxz272, vxz273, vxz2740, vxz2750)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The DP Problem is simplified using the Induction Calculus [18] with the following steps:
Note that final constraints are written in bold face.


For Pair new_gcd0Gcd'10(vxz272, vxz273, Zero, Succ(vxz2750)) → new_gcd0Gcd'12(Succ(vxz272), vxz273) the following chains were created:




For Pair new_gcd0Gcd'13(vxz5200, vxz2700) → new_gcd0Gcd'1(Succ(vxz5200), vxz2700, Succ(vxz5200)) the following chains were created:




For Pair new_gcd0Gcd'12(vxz266, vxz267) → new_gcd0Gcd'13(vxz267, vxz266) the following chains were created:




For Pair new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Succ(vxz2750)) → new_gcd0Gcd'10(vxz272, vxz273, vxz2740, vxz2750) the following chains were created:




For Pair new_gcd0Gcd'1(Succ(Succ(x0)), Succ(x1), Succ(Succ(x0))) → new_gcd0Gcd'10(x0, Succ(x1), x0, x1) the following chains were created:




To summarize, we get the following constraints P for the following pairs.



The constraints for P> respective Pbound are constructed from P where we just replace every occurence of "t ≥ s" in P by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation [18]:

POL(Succ(x1)) = 1 + x1   
POL(Zero) = 0   
POL(c) = -1   
POL(new_gcd0Gcd'1(x1, x2, x3)) = -1 - x1 + x2 + x3   
POL(new_gcd0Gcd'10(x1, x2, x3, x4)) = -1 + x1 - x3 + x4   
POL(new_gcd0Gcd'12(x1, x2)) = -1 + x1   
POL(new_gcd0Gcd'13(x1, x2)) = -1 + x2   

The following pairs are in P>:

new_gcd0Gcd'1(Succ(Succ(x0)), Succ(x1), Succ(Succ(x0))) → new_gcd0Gcd'10(x0, Succ(x1), x0, x1)
The following pairs are in Pbound:

new_gcd0Gcd'10(vxz272, vxz273, Zero, Succ(vxz2750)) → new_gcd0Gcd'12(Succ(vxz272), vxz273)
new_gcd0Gcd'13(vxz5200, vxz2700) → new_gcd0Gcd'1(Succ(vxz5200), vxz2700, Succ(vxz5200))
new_gcd0Gcd'12(vxz266, vxz267) → new_gcd0Gcd'13(vxz267, vxz266)
new_gcd0Gcd'1(Succ(Succ(x0)), Succ(x1), Succ(Succ(x0))) → new_gcd0Gcd'10(x0, Succ(x1), x0, x1)
There are no usable rules

↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ UsableRulesProof
                                                  ↳ QDP
                                                    ↳ QReductionProof
                                                      ↳ QDP
                                                        ↳ Instantiation
                                                          ↳ QDP
                                                            ↳ NonInfProof
QDP
                                                                ↳ DependencyGraphProof
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_gcd0Gcd'10(vxz272, vxz273, Zero, Succ(vxz2750)) → new_gcd0Gcd'12(Succ(vxz272), vxz273)
new_gcd0Gcd'13(vxz5200, vxz2700) → new_gcd0Gcd'1(Succ(vxz5200), vxz2700, Succ(vxz5200))
new_gcd0Gcd'12(vxz266, vxz267) → new_gcd0Gcd'13(vxz267, vxz266)
new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Succ(vxz2750)) → new_gcd0Gcd'10(vxz272, vxz273, vxz2740, vxz2750)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 3 less nodes.

↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ UsableRulesProof
                                                  ↳ QDP
                                                    ↳ QReductionProof
                                                      ↳ QDP
                                                        ↳ Instantiation
                                                          ↳ QDP
                                                            ↳ NonInfProof
                                                              ↳ QDP
                                                                ↳ DependencyGraphProof
QDP
                                                                    ↳ QDPSizeChangeProof
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Succ(vxz2750)) → new_gcd0Gcd'10(vxz272, vxz273, vxz2740, vxz2750)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:



↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
QDP
                            ↳ DependencyGraphProof
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_primDivNatS0(vxz168, vxz169, Zero, Zero) → new_primDivNatS00(vxz168, vxz169)
new_primDivNatS(Succ(Succ(vxz13500)), Succ(vxz136000)) → new_primDivNatS0(vxz13500, vxz136000, vxz13500, vxz136000)
new_primDivNatS0(vxz168, vxz169, Succ(vxz1700), Succ(vxz1710)) → new_primDivNatS0(vxz168, vxz169, vxz1700, vxz1710)
new_primDivNatS00(vxz168, vxz169) → new_primDivNatS(new_primMinusNatS0(vxz168, vxz169), Succ(vxz169))
new_primDivNatS0(vxz168, vxz169, Succ(vxz1700), Zero) → new_primDivNatS(new_primMinusNatS0(vxz168, vxz169), Succ(vxz169))
new_primDivNatS(Succ(Zero), Zero) → new_primDivNatS(new_primMinusNatS2, Zero)
new_primDivNatS(Succ(Succ(vxz13500)), Zero) → new_primDivNatS(new_primMinusNatS1(vxz13500), Zero)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS1(vxz13500) → Succ(vxz13500)
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)
new_primMinusNatS2Zero

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS2
new_primMinusNatS1(x0)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 1 less node.

↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ AND
QDP
                                  ↳ UsableRulesProof
                                ↳ QDP
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_primDivNatS(Succ(Succ(vxz13500)), Zero) → new_primDivNatS(new_primMinusNatS1(vxz13500), Zero)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS1(vxz13500) → Succ(vxz13500)
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)
new_primMinusNatS2Zero

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS2
new_primMinusNatS1(x0)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))

We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.

↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ AND
                                ↳ QDP
                                  ↳ UsableRulesProof
QDP
                                      ↳ QReductionProof
                                ↳ QDP
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_primDivNatS(Succ(Succ(vxz13500)), Zero) → new_primDivNatS(new_primMinusNatS1(vxz13500), Zero)

The TRS R consists of the following rules:

new_primMinusNatS1(vxz13500) → Succ(vxz13500)

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS2
new_primMinusNatS1(x0)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS2
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))



↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ AND
                                ↳ QDP
                                  ↳ UsableRulesProof
                                    ↳ QDP
                                      ↳ QReductionProof
QDP
                                          ↳ RuleRemovalProof
                                ↳ QDP
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_primDivNatS(Succ(Succ(vxz13500)), Zero) → new_primDivNatS(new_primMinusNatS1(vxz13500), Zero)

The TRS R consists of the following rules:

new_primMinusNatS1(vxz13500) → Succ(vxz13500)

The set Q consists of the following terms:

new_primMinusNatS1(x0)

We have to consider all minimal (P,Q,R)-chains.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

new_primDivNatS(Succ(Succ(vxz13500)), Zero) → new_primDivNatS(new_primMinusNatS1(vxz13500), Zero)

Strictly oriented rules of the TRS R:

new_primMinusNatS1(vxz13500) → Succ(vxz13500)

Used ordering: POLO with Polynomial interpretation [25]:

POL(Succ(x1)) = 1 + 2·x1   
POL(Zero) = 0   
POL(new_primDivNatS(x1, x2)) = x1 + x2   
POL(new_primMinusNatS1(x1)) = 2 + 2·x1   



↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ AND
                                ↳ QDP
                                  ↳ UsableRulesProof
                                    ↳ QDP
                                      ↳ QReductionProof
                                        ↳ QDP
                                          ↳ RuleRemovalProof
QDP
                                              ↳ PisEmptyProof
                                ↳ QDP
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
P is empty.
R is empty.
The set Q consists of the following terms:

new_primMinusNatS1(x0)

We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ AND
                                ↳ QDP
QDP
                                  ↳ UsableRulesProof
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_primDivNatS0(vxz168, vxz169, Zero, Zero) → new_primDivNatS00(vxz168, vxz169)
new_primDivNatS0(vxz168, vxz169, Succ(vxz1700), Succ(vxz1710)) → new_primDivNatS0(vxz168, vxz169, vxz1700, vxz1710)
new_primDivNatS(Succ(Succ(vxz13500)), Succ(vxz136000)) → new_primDivNatS0(vxz13500, vxz136000, vxz13500, vxz136000)
new_primDivNatS00(vxz168, vxz169) → new_primDivNatS(new_primMinusNatS0(vxz168, vxz169), Succ(vxz169))
new_primDivNatS0(vxz168, vxz169, Succ(vxz1700), Zero) → new_primDivNatS(new_primMinusNatS0(vxz168, vxz169), Succ(vxz169))

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS1(vxz13500) → Succ(vxz13500)
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)
new_primMinusNatS2Zero

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS2
new_primMinusNatS1(x0)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))

We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.

↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ AND
                                ↳ QDP
                                ↳ QDP
                                  ↳ UsableRulesProof
QDP
                                      ↳ QReductionProof
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_primDivNatS0(vxz168, vxz169, Zero, Zero) → new_primDivNatS00(vxz168, vxz169)
new_primDivNatS0(vxz168, vxz169, Succ(vxz1700), Succ(vxz1710)) → new_primDivNatS0(vxz168, vxz169, vxz1700, vxz1710)
new_primDivNatS(Succ(Succ(vxz13500)), Succ(vxz136000)) → new_primDivNatS0(vxz13500, vxz136000, vxz13500, vxz136000)
new_primDivNatS00(vxz168, vxz169) → new_primDivNatS(new_primMinusNatS0(vxz168, vxz169), Succ(vxz169))
new_primDivNatS0(vxz168, vxz169, Succ(vxz1700), Zero) → new_primDivNatS(new_primMinusNatS0(vxz168, vxz169), Succ(vxz169))

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS2
new_primMinusNatS1(x0)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

new_primMinusNatS2
new_primMinusNatS1(x0)



↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ AND
                                ↳ QDP
                                ↳ QDP
                                  ↳ UsableRulesProof
                                    ↳ QDP
                                      ↳ QReductionProof
QDP
                                          ↳ QDPOrderProof
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_primDivNatS0(vxz168, vxz169, Zero, Zero) → new_primDivNatS00(vxz168, vxz169)
new_primDivNatS00(vxz168, vxz169) → new_primDivNatS(new_primMinusNatS0(vxz168, vxz169), Succ(vxz169))
new_primDivNatS(Succ(Succ(vxz13500)), Succ(vxz136000)) → new_primDivNatS0(vxz13500, vxz136000, vxz13500, vxz136000)
new_primDivNatS0(vxz168, vxz169, Succ(vxz1700), Succ(vxz1710)) → new_primDivNatS0(vxz168, vxz169, vxz1700, vxz1710)
new_primDivNatS0(vxz168, vxz169, Succ(vxz1700), Zero) → new_primDivNatS(new_primMinusNatS0(vxz168, vxz169), Succ(vxz169))

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))

We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


new_primDivNatS00(vxz168, vxz169) → new_primDivNatS(new_primMinusNatS0(vxz168, vxz169), Succ(vxz169))
new_primDivNatS(Succ(Succ(vxz13500)), Succ(vxz136000)) → new_primDivNatS0(vxz13500, vxz136000, vxz13500, vxz136000)
new_primDivNatS0(vxz168, vxz169, Succ(vxz1700), Zero) → new_primDivNatS(new_primMinusNatS0(vxz168, vxz169), Succ(vxz169))
The remaining pairs can at least be oriented weakly.

new_primDivNatS0(vxz168, vxz169, Zero, Zero) → new_primDivNatS00(vxz168, vxz169)
new_primDivNatS0(vxz168, vxz169, Succ(vxz1700), Succ(vxz1710)) → new_primDivNatS0(vxz168, vxz169, vxz1700, vxz1710)
Used ordering: Polynomial interpretation [25]:

POL(Succ(x1)) = 1 + x1   
POL(Zero) = 0   
POL(new_primDivNatS(x1, x2)) = x1   
POL(new_primDivNatS0(x1, x2, x3, x4)) = 1 + x1   
POL(new_primDivNatS00(x1, x2)) = 1 + x1   
POL(new_primMinusNatS0(x1, x2)) = x1   

The following usable rules [17] were oriented:

new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)



↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ AND
                                ↳ QDP
                                ↳ QDP
                                  ↳ UsableRulesProof
                                    ↳ QDP
                                      ↳ QReductionProof
                                        ↳ QDP
                                          ↳ QDPOrderProof
QDP
                                              ↳ DependencyGraphProof
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_primDivNatS0(vxz168, vxz169, Zero, Zero) → new_primDivNatS00(vxz168, vxz169)
new_primDivNatS0(vxz168, vxz169, Succ(vxz1700), Succ(vxz1710)) → new_primDivNatS0(vxz168, vxz169, vxz1700, vxz1710)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ AND
                                ↳ QDP
                                ↳ QDP
                                  ↳ UsableRulesProof
                                    ↳ QDP
                                      ↳ QReductionProof
                                        ↳ QDP
                                          ↳ QDPOrderProof
                                            ↳ QDP
                                              ↳ DependencyGraphProof
QDP
                                                  ↳ UsableRulesProof
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_primDivNatS0(vxz168, vxz169, Succ(vxz1700), Succ(vxz1710)) → new_primDivNatS0(vxz168, vxz169, vxz1700, vxz1710)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))

We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.

↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ AND
                                ↳ QDP
                                ↳ QDP
                                  ↳ UsableRulesProof
                                    ↳ QDP
                                      ↳ QReductionProof
                                        ↳ QDP
                                          ↳ QDPOrderProof
                                            ↳ QDP
                                              ↳ DependencyGraphProof
                                                ↳ QDP
                                                  ↳ UsableRulesProof
QDP
                                                      ↳ QReductionProof
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_primDivNatS0(vxz168, vxz169, Succ(vxz1700), Succ(vxz1710)) → new_primDivNatS0(vxz168, vxz169, vxz1700, vxz1710)

R is empty.
The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))



↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ AND
                                ↳ QDP
                                ↳ QDP
                                  ↳ UsableRulesProof
                                    ↳ QDP
                                      ↳ QReductionProof
                                        ↳ QDP
                                          ↳ QDPOrderProof
                                            ↳ QDP
                                              ↳ DependencyGraphProof
                                                ↳ QDP
                                                  ↳ UsableRulesProof
                                                    ↳ QDP
                                                      ↳ QReductionProof
QDP
                                                          ↳ QDPSizeChangeProof
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_primDivNatS0(vxz168, vxz169, Succ(vxz1700), Succ(vxz1710)) → new_primDivNatS0(vxz168, vxz169, vxz1700, vxz1710)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:



↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
QDP
                            ↳ DependencyGraphProof
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_quot3(vxz285, vxz286, vxz287) → new_quot0(vxz285, new_primMinusNatS0(Succ(vxz286), vxz287), vxz287, new_primMinusNatS0(Succ(vxz286), vxz287))
new_quot1(vxz285, vxz286, vxz287, Succ(vxz2880), Zero) → new_quot0(vxz285, new_primMinusNatS0(Succ(vxz286), vxz287), vxz287, new_primMinusNatS0(Succ(vxz286), vxz287))
new_quot0(vxz249, Succ(Succ(vxz25500)), Succ(vxz2510), vxz254) → new_quot1(vxz249, vxz25500, Succ(vxz2510), vxz25500, vxz2510)
new_quot0(vxz249, Succ(Zero), Succ(vxz2510), vxz254) → new_quot2(vxz249, Succ(vxz2510), Zero)
new_quot1(vxz285, vxz286, vxz287, Zero, Succ(vxz2890)) → new_quot2(vxz285, vxz287, Succ(vxz286))
new_quot0(vxz249, Succ(Zero), Zero, vxz254) → new_quot0(vxz249, new_primMinusNatS0(Zero, Zero), Zero, new_primMinusNatS0(Zero, Zero))
new_quot1(vxz285, vxz286, vxz287, Succ(vxz2880), Succ(vxz2890)) → new_quot1(vxz285, vxz286, vxz287, vxz2880, vxz2890)
new_quot(vxz79, vxz8000, vxz3000) → new_quot0(vxz79, Succ(vxz8000), vxz3000, Succ(vxz8000))
new_quot2(vxz79, vxz8000, vxz3000) → new_quot0(vxz79, Succ(vxz8000), vxz3000, Succ(vxz8000))
new_quot0(vxz249, Succ(Succ(vxz25500)), Zero, vxz254) → new_quot0(vxz249, new_primMinusNatS0(Succ(vxz25500), Zero), Zero, new_primMinusNatS0(Succ(vxz25500), Zero))
new_quot1(vxz285, vxz286, vxz287, Zero, Zero) → new_quot3(vxz285, vxz286, vxz287)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 2 less nodes.

↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ AND
QDP
                                  ↳ UsableRulesProof
                                ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_quot0(vxz249, Succ(Succ(vxz25500)), Zero, vxz254) → new_quot0(vxz249, new_primMinusNatS0(Succ(vxz25500), Zero), Zero, new_primMinusNatS0(Succ(vxz25500), Zero))

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))

We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.

↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ AND
                                ↳ QDP
                                  ↳ UsableRulesProof
QDP
                                      ↳ RuleRemovalProof
                                ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_quot0(vxz249, Succ(Succ(vxz25500)), Zero, vxz254) → new_quot0(vxz249, new_primMinusNatS0(Succ(vxz25500), Zero), Zero, new_primMinusNatS0(Succ(vxz25500), Zero))

The TRS R consists of the following rules:

new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))

We have to consider all minimal (P,Q,R)-chains.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

new_quot0(vxz249, Succ(Succ(vxz25500)), Zero, vxz254) → new_quot0(vxz249, new_primMinusNatS0(Succ(vxz25500), Zero), Zero, new_primMinusNatS0(Succ(vxz25500), Zero))


Used ordering: POLO with Polynomial interpretation [25]:

POL(Succ(x1)) = 1 + 2·x1   
POL(Zero) = 0   
POL(new_primMinusNatS0(x1, x2)) = x1 + 2·x2   
POL(new_quot0(x1, x2, x3, x4)) = x1 + x2 + x3 + x4   



↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ AND
                                ↳ QDP
                                  ↳ UsableRulesProof
                                    ↳ QDP
                                      ↳ RuleRemovalProof
QDP
                                          ↳ PisEmptyProof
                                ↳ QDP
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
P is empty.
The TRS R consists of the following rules:

new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))

We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ AND
                                ↳ QDP
QDP
                                  ↳ Instantiation
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_quot3(vxz285, vxz286, vxz287) → new_quot0(vxz285, new_primMinusNatS0(Succ(vxz286), vxz287), vxz287, new_primMinusNatS0(Succ(vxz286), vxz287))
new_quot1(vxz285, vxz286, vxz287, Succ(vxz2880), Zero) → new_quot0(vxz285, new_primMinusNatS0(Succ(vxz286), vxz287), vxz287, new_primMinusNatS0(Succ(vxz286), vxz287))
new_quot1(vxz285, vxz286, vxz287, Zero, Succ(vxz2890)) → new_quot2(vxz285, vxz287, Succ(vxz286))
new_quot0(vxz249, Succ(Succ(vxz25500)), Succ(vxz2510), vxz254) → new_quot1(vxz249, vxz25500, Succ(vxz2510), vxz25500, vxz2510)
new_quot0(vxz249, Succ(Zero), Succ(vxz2510), vxz254) → new_quot2(vxz249, Succ(vxz2510), Zero)
new_quot1(vxz285, vxz286, vxz287, Succ(vxz2880), Succ(vxz2890)) → new_quot1(vxz285, vxz286, vxz287, vxz2880, vxz2890)
new_quot2(vxz79, vxz8000, vxz3000) → new_quot0(vxz79, Succ(vxz8000), vxz3000, Succ(vxz8000))
new_quot1(vxz285, vxz286, vxz287, Zero, Zero) → new_quot3(vxz285, vxz286, vxz287)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule new_quot2(vxz79, vxz8000, vxz3000) → new_quot0(vxz79, Succ(vxz8000), vxz3000, Succ(vxz8000)) we obtained the following new rules:

new_quot2(z0, z2, Succ(z1)) → new_quot0(z0, Succ(z2), Succ(z1), Succ(z2))
new_quot2(z0, Succ(z1), Zero) → new_quot0(z0, Succ(Succ(z1)), Zero, Succ(Succ(z1)))



↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ AND
                                ↳ QDP
                                ↳ QDP
                                  ↳ Instantiation
QDP
                                      ↳ DependencyGraphProof
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_quot2(z0, z2, Succ(z1)) → new_quot0(z0, Succ(z2), Succ(z1), Succ(z2))
new_quot3(vxz285, vxz286, vxz287) → new_quot0(vxz285, new_primMinusNatS0(Succ(vxz286), vxz287), vxz287, new_primMinusNatS0(Succ(vxz286), vxz287))
new_quot2(z0, Succ(z1), Zero) → new_quot0(z0, Succ(Succ(z1)), Zero, Succ(Succ(z1)))
new_quot1(vxz285, vxz286, vxz287, Succ(vxz2880), Zero) → new_quot0(vxz285, new_primMinusNatS0(Succ(vxz286), vxz287), vxz287, new_primMinusNatS0(Succ(vxz286), vxz287))
new_quot0(vxz249, Succ(Zero), Succ(vxz2510), vxz254) → new_quot2(vxz249, Succ(vxz2510), Zero)
new_quot0(vxz249, Succ(Succ(vxz25500)), Succ(vxz2510), vxz254) → new_quot1(vxz249, vxz25500, Succ(vxz2510), vxz25500, vxz2510)
new_quot1(vxz285, vxz286, vxz287, Zero, Succ(vxz2890)) → new_quot2(vxz285, vxz287, Succ(vxz286))
new_quot1(vxz285, vxz286, vxz287, Succ(vxz2880), Succ(vxz2890)) → new_quot1(vxz285, vxz286, vxz287, vxz2880, vxz2890)
new_quot1(vxz285, vxz286, vxz287, Zero, Zero) → new_quot3(vxz285, vxz286, vxz287)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.

↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ AND
                                ↳ QDP
                                ↳ QDP
                                  ↳ Instantiation
                                    ↳ QDP
                                      ↳ DependencyGraphProof
QDP
                                          ↳ QDPOrderProof
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_quot3(vxz285, vxz286, vxz287) → new_quot0(vxz285, new_primMinusNatS0(Succ(vxz286), vxz287), vxz287, new_primMinusNatS0(Succ(vxz286), vxz287))
new_quot2(z0, z2, Succ(z1)) → new_quot0(z0, Succ(z2), Succ(z1), Succ(z2))
new_quot1(vxz285, vxz286, vxz287, Succ(vxz2880), Zero) → new_quot0(vxz285, new_primMinusNatS0(Succ(vxz286), vxz287), vxz287, new_primMinusNatS0(Succ(vxz286), vxz287))
new_quot1(vxz285, vxz286, vxz287, Zero, Succ(vxz2890)) → new_quot2(vxz285, vxz287, Succ(vxz286))
new_quot0(vxz249, Succ(Succ(vxz25500)), Succ(vxz2510), vxz254) → new_quot1(vxz249, vxz25500, Succ(vxz2510), vxz25500, vxz2510)
new_quot1(vxz285, vxz286, vxz287, Succ(vxz2880), Succ(vxz2890)) → new_quot1(vxz285, vxz286, vxz287, vxz2880, vxz2890)
new_quot1(vxz285, vxz286, vxz287, Zero, Zero) → new_quot3(vxz285, vxz286, vxz287)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))

We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


new_quot3(vxz285, vxz286, vxz287) → new_quot0(vxz285, new_primMinusNatS0(Succ(vxz286), vxz287), vxz287, new_primMinusNatS0(Succ(vxz286), vxz287))
new_quot1(vxz285, vxz286, vxz287, Succ(vxz2880), Zero) → new_quot0(vxz285, new_primMinusNatS0(Succ(vxz286), vxz287), vxz287, new_primMinusNatS0(Succ(vxz286), vxz287))
The remaining pairs can at least be oriented weakly.

new_quot2(z0, z2, Succ(z1)) → new_quot0(z0, Succ(z2), Succ(z1), Succ(z2))
new_quot1(vxz285, vxz286, vxz287, Zero, Succ(vxz2890)) → new_quot2(vxz285, vxz287, Succ(vxz286))
new_quot0(vxz249, Succ(Succ(vxz25500)), Succ(vxz2510), vxz254) → new_quot1(vxz249, vxz25500, Succ(vxz2510), vxz25500, vxz2510)
new_quot1(vxz285, vxz286, vxz287, Succ(vxz2880), Succ(vxz2890)) → new_quot1(vxz285, vxz286, vxz287, vxz2880, vxz2890)
new_quot1(vxz285, vxz286, vxz287, Zero, Zero) → new_quot3(vxz285, vxz286, vxz287)
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( new_primMinusNatS0(x1, x2) ) =
/0\
\0/
+
/00\
\01/
·x1+
/01\
\00/
·x2

M( Succ(x1) ) =
/1\
\0/
+
/00\
\11/
·x1

M( Zero ) =
/0\
\1/

Tuple symbols:
M( new_quot2(x1, ..., x3) ) = 1+
[0,0]
·x1+
[1,1]
·x2+
[0,1]
·x3

M( new_quot1(x1, ..., x5) ) = 1+
[0,0]
·x1+
[1,1]
·x2+
[1,1]
·x3+
[0,0]
·x4+
[0,0]
·x5

M( new_quot0(x1, ..., x4) ) = 0+
[0,0]
·x1+
[0,1]
·x2+
[1,1]
·x3+
[0,0]
·x4

M( new_quot3(x1, ..., x3) ) = 1+
[0,0]
·x1+
[1,1]
·x2+
[1,1]
·x3


Matrix type:
We used a basic matrix type which is not further parametrizeable.


As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
The following usable rules [17] were oriented:

new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero



↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ AND
                                ↳ QDP
                                ↳ QDP
                                  ↳ Instantiation
                                    ↳ QDP
                                      ↳ DependencyGraphProof
                                        ↳ QDP
                                          ↳ QDPOrderProof
QDP
                                              ↳ DependencyGraphProof
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_quot2(z0, z2, Succ(z1)) → new_quot0(z0, Succ(z2), Succ(z1), Succ(z2))
new_quot0(vxz249, Succ(Succ(vxz25500)), Succ(vxz2510), vxz254) → new_quot1(vxz249, vxz25500, Succ(vxz2510), vxz25500, vxz2510)
new_quot1(vxz285, vxz286, vxz287, Zero, Succ(vxz2890)) → new_quot2(vxz285, vxz287, Succ(vxz286))
new_quot1(vxz285, vxz286, vxz287, Succ(vxz2880), Succ(vxz2890)) → new_quot1(vxz285, vxz286, vxz287, vxz2880, vxz2890)
new_quot1(vxz285, vxz286, vxz287, Zero, Zero) → new_quot3(vxz285, vxz286, vxz287)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ AND
                                ↳ QDP
                                ↳ QDP
                                  ↳ Instantiation
                                    ↳ QDP
                                      ↳ DependencyGraphProof
                                        ↳ QDP
                                          ↳ QDPOrderProof
                                            ↳ QDP
                                              ↳ DependencyGraphProof
QDP
                                                  ↳ UsableRulesProof
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_quot2(z0, z2, Succ(z1)) → new_quot0(z0, Succ(z2), Succ(z1), Succ(z2))
new_quot1(vxz285, vxz286, vxz287, Zero, Succ(vxz2890)) → new_quot2(vxz285, vxz287, Succ(vxz286))
new_quot0(vxz249, Succ(Succ(vxz25500)), Succ(vxz2510), vxz254) → new_quot1(vxz249, vxz25500, Succ(vxz2510), vxz25500, vxz2510)
new_quot1(vxz285, vxz286, vxz287, Succ(vxz2880), Succ(vxz2890)) → new_quot1(vxz285, vxz286, vxz287, vxz2880, vxz2890)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))

We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.

↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ AND
                                ↳ QDP
                                ↳ QDP
                                  ↳ Instantiation
                                    ↳ QDP
                                      ↳ DependencyGraphProof
                                        ↳ QDP
                                          ↳ QDPOrderProof
                                            ↳ QDP
                                              ↳ DependencyGraphProof
                                                ↳ QDP
                                                  ↳ UsableRulesProof
QDP
                                                      ↳ QReductionProof
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_quot2(z0, z2, Succ(z1)) → new_quot0(z0, Succ(z2), Succ(z1), Succ(z2))
new_quot1(vxz285, vxz286, vxz287, Zero, Succ(vxz2890)) → new_quot2(vxz285, vxz287, Succ(vxz286))
new_quot0(vxz249, Succ(Succ(vxz25500)), Succ(vxz2510), vxz254) → new_quot1(vxz249, vxz25500, Succ(vxz2510), vxz25500, vxz2510)
new_quot1(vxz285, vxz286, vxz287, Succ(vxz2880), Succ(vxz2890)) → new_quot1(vxz285, vxz286, vxz287, vxz2880, vxz2890)

R is empty.
The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))



↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ AND
                                ↳ QDP
                                ↳ QDP
                                  ↳ Instantiation
                                    ↳ QDP
                                      ↳ DependencyGraphProof
                                        ↳ QDP
                                          ↳ QDPOrderProof
                                            ↳ QDP
                                              ↳ DependencyGraphProof
                                                ↳ QDP
                                                  ↳ UsableRulesProof
                                                    ↳ QDP
                                                      ↳ QReductionProof
QDP
                                                          ↳ Instantiation
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_quot2(z0, z2, Succ(z1)) → new_quot0(z0, Succ(z2), Succ(z1), Succ(z2))
new_quot0(vxz249, Succ(Succ(vxz25500)), Succ(vxz2510), vxz254) → new_quot1(vxz249, vxz25500, Succ(vxz2510), vxz25500, vxz2510)
new_quot1(vxz285, vxz286, vxz287, Zero, Succ(vxz2890)) → new_quot2(vxz285, vxz287, Succ(vxz286))
new_quot1(vxz285, vxz286, vxz287, Succ(vxz2880), Succ(vxz2890)) → new_quot1(vxz285, vxz286, vxz287, vxz2880, vxz2890)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule new_quot0(vxz249, Succ(Succ(vxz25500)), Succ(vxz2510), vxz254) → new_quot1(vxz249, vxz25500, Succ(vxz2510), vxz25500, vxz2510) we obtained the following new rules:

new_quot0(z0, Succ(Succ(x1)), Succ(z2), Succ(Succ(x1))) → new_quot1(z0, x1, Succ(z2), x1, z2)



↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ AND
                                ↳ QDP
                                ↳ QDP
                                  ↳ Instantiation
                                    ↳ QDP
                                      ↳ DependencyGraphProof
                                        ↳ QDP
                                          ↳ QDPOrderProof
                                            ↳ QDP
                                              ↳ DependencyGraphProof
                                                ↳ QDP
                                                  ↳ UsableRulesProof
                                                    ↳ QDP
                                                      ↳ QReductionProof
                                                        ↳ QDP
                                                          ↳ Instantiation
QDP
                                                              ↳ NonInfProof
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_quot2(z0, z2, Succ(z1)) → new_quot0(z0, Succ(z2), Succ(z1), Succ(z2))
new_quot1(vxz285, vxz286, vxz287, Zero, Succ(vxz2890)) → new_quot2(vxz285, vxz287, Succ(vxz286))
new_quot1(vxz285, vxz286, vxz287, Succ(vxz2880), Succ(vxz2890)) → new_quot1(vxz285, vxz286, vxz287, vxz2880, vxz2890)
new_quot0(z0, Succ(Succ(x1)), Succ(z2), Succ(Succ(x1))) → new_quot1(z0, x1, Succ(z2), x1, z2)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The DP Problem is simplified using the Induction Calculus [18] with the following steps:
Note that final constraints are written in bold face.


For Pair new_quot2(z0, z2, Succ(z1)) → new_quot0(z0, Succ(z2), Succ(z1), Succ(z2)) the following chains were created:




For Pair new_quot1(vxz285, vxz286, vxz287, Zero, Succ(vxz2890)) → new_quot2(vxz285, vxz287, Succ(vxz286)) the following chains were created:




For Pair new_quot1(vxz285, vxz286, vxz287, Succ(vxz2880), Succ(vxz2890)) → new_quot1(vxz285, vxz286, vxz287, vxz2880, vxz2890) the following chains were created:




For Pair new_quot0(z0, Succ(Succ(x1)), Succ(z2), Succ(Succ(x1))) → new_quot1(z0, x1, Succ(z2), x1, z2) the following chains were created:




To summarize, we get the following constraints P for the following pairs.



The constraints for P> respective Pbound are constructed from P where we just replace every occurence of "t ≥ s" in P by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation [18]:

POL(Succ(x1)) = 1 + x1   
POL(Zero) = 0   
POL(c) = -1   
POL(new_quot0(x1, x2, x3, x4)) = -1 + x2 + x3 - x4   
POL(new_quot1(x1, x2, x3, x4, x5)) = -1 + x2 - x4 + x5   
POL(new_quot2(x1, x2, x3)) = -1 + x3   

The following pairs are in P>:

new_quot0(z0, Succ(Succ(x1)), Succ(z2), Succ(Succ(x1))) → new_quot1(z0, x1, Succ(z2), x1, z2)
The following pairs are in Pbound:

new_quot2(z0, z2, Succ(z1)) → new_quot0(z0, Succ(z2), Succ(z1), Succ(z2))
new_quot1(vxz285, vxz286, vxz287, Zero, Succ(vxz2890)) → new_quot2(vxz285, vxz287, Succ(vxz286))
new_quot0(z0, Succ(Succ(x1)), Succ(z2), Succ(Succ(x1))) → new_quot1(z0, x1, Succ(z2), x1, z2)
There are no usable rules

↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ AND
                                ↳ QDP
                                ↳ QDP
                                  ↳ Instantiation
                                    ↳ QDP
                                      ↳ DependencyGraphProof
                                        ↳ QDP
                                          ↳ QDPOrderProof
                                            ↳ QDP
                                              ↳ DependencyGraphProof
                                                ↳ QDP
                                                  ↳ UsableRulesProof
                                                    ↳ QDP
                                                      ↳ QReductionProof
                                                        ↳ QDP
                                                          ↳ Instantiation
                                                            ↳ QDP
                                                              ↳ NonInfProof
QDP
                                                                  ↳ DependencyGraphProof
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_quot2(z0, z2, Succ(z1)) → new_quot0(z0, Succ(z2), Succ(z1), Succ(z2))
new_quot1(vxz285, vxz286, vxz287, Zero, Succ(vxz2890)) → new_quot2(vxz285, vxz287, Succ(vxz286))
new_quot1(vxz285, vxz286, vxz287, Succ(vxz2880), Succ(vxz2890)) → new_quot1(vxz285, vxz286, vxz287, vxz2880, vxz2890)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.

↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ AND
                                ↳ QDP
                                ↳ QDP
                                  ↳ Instantiation
                                    ↳ QDP
                                      ↳ DependencyGraphProof
                                        ↳ QDP
                                          ↳ QDPOrderProof
                                            ↳ QDP
                                              ↳ DependencyGraphProof
                                                ↳ QDP
                                                  ↳ UsableRulesProof
                                                    ↳ QDP
                                                      ↳ QReductionProof
                                                        ↳ QDP
                                                          ↳ Instantiation
                                                            ↳ QDP
                                                              ↳ NonInfProof
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
QDP
                                                                      ↳ QDPSizeChangeProof
                          ↳ QDP
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_quot1(vxz285, vxz286, vxz287, Succ(vxz2880), Succ(vxz2890)) → new_quot1(vxz285, vxz286, vxz287, vxz2880, vxz2890)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:



↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
QDP
                            ↳ MNOCProof
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_numericEnumFrom3(vxz3, ba) → new_numericEnumFrom3(new_ps4(vxz3, ba), ba)

The TRS R consists of the following rules:

new_primDivNatS1(Succ(Zero), Zero) → Succ(new_primDivNatS1(new_primMinusNatS2, Zero))
new_quot14(vxz79, Pos(Zero), vxz80, vxz30) → new_quot19(vxz79, vxz80, vxz30)
new_quot19(vxz79, vxz80, Pos(Zero)) → new_quot15(vxz79)
new_gcd0Gcd'19(vxz263, vxz264) → new_gcd0Gcd'17(new_primMinusNatS0(Succ(vxz263), vxz264), vxz264, new_primMinusNatS0(Succ(vxz263), vxz264))
new_quot18(vxz79, Pos(Succ(vxz8000))) → new_quot8(vxz79, vxz8000)
new_primQuotInt0(Pos(vxz510), vxz52, vxz2700) → new_primQuotInt1(vxz510, new_gcd0(vxz52, vxz2700))
new_quot18(vxz79, Pos(Zero)) → Integer(new_primQuotInt4(vxz79))
new_reduce2Reduce10(vxz30, vxz310, vxz30, vxz29, Neg(Succ(vxz3100))) → new_reduce2Reduce1(vxz30, vxz310, vxz30, vxz29)
new_quot13(vxz249, Succ(Succ(vxz25500)), Succ(vxz2510), vxz254) → new_quot9(vxz249, vxz25500, Succ(vxz2510), vxz25500, vxz2510)
new_quot15(vxz79) → error([])
new_primQuotInt5(Neg(vxz790), vxz8000) → new_primQuotInt3(vxz790, Pos(Succ(vxz8000)))
new_ps5(Neg(vxz280), Pos(vxz310)) → new_primPlusInt2(vxz280, vxz310)
new_primQuotInt2(Pos(vxz530), vxz54, vxz2700) → new_primQuotInt1(vxz530, new_gcd(vxz54, vxz2700))
new_gcd(Neg(Succ(vxz5400)), vxz2700) → new_gcd0Gcd'14(vxz2700, vxz5400)
new_primMinusNatS0(Zero, Zero) → Zero
new_gcd0Gcd'18(vxz272, vxz273, Zero, Zero) → new_gcd0Gcd'19(vxz272, vxz273)
new_quot17(vxz79, vxz80, Neg(Succ(vxz3000))) → new_quot16(vxz79, vxz3000, vxz80)
new_reduce2D(vxz137, vxz2700) → new_gcd0(vxz137, vxz2700)
new_primMulInt(Neg(vxz310)) → Neg(new_primMulNat0(vxz310))
new_primPlusInt4(Neg(vxz390), Neg(vxz3100)) → new_primPlusInt3(vxz390, vxz3100)
new_primQuotInt1(vxz135, Pos(Succ(vxz13600))) → Pos(new_primDivNatS1(vxz135, vxz13600))
new_primPlusInt1(Succ(vxz2800), Succ(vxz3100)) → new_primMinusNat2(vxz2800, Succ(vxz3100))
new_reduce(vxz28, vxz31, Pos(Succ(vxz2700))) → :%(new_primQuotInt0(new_ps5(vxz28, vxz31), new_ps5(vxz28, vxz31), vxz2700), new_primQuotInt1(Succ(vxz2700), new_reduce2D(new_ps5(vxz28, vxz31), vxz2700)))
new_quot11(vxz79, vxz8000, vxz3000) → new_quot12(vxz79, vxz8000, vxz3000)
new_primQuotInt5(Pos(vxz790), vxz8000) → new_primQuotInt(Pos(vxz790), vxz8000)
new_primPlusNat4(Zero, Zero) → Zero
new_primDivNatS1(Succ(Succ(vxz13500)), Zero) → Succ(new_primDivNatS1(new_primMinusNatS1(vxz13500), Zero))
new_primQuotInt3(vxz207, Pos(Zero)) → new_error0
new_ps4(:%(vxz30, vxz31), ty_Int) → new_reduce(new_sr(vxz30), vxz31, new_sr(vxz31))
new_primMinusNat3(Zero, Succ(vxz28000)) → Neg(Succ(vxz28000))
new_primPlusInt4(Pos(vxz390), Pos(vxz3100)) → new_primPlusInt0(vxz390, vxz3100)
new_gcd0Gcd'17(Succ(Zero), Zero, vxz236) → new_gcd0Gcd'17(new_primMinusNatS0(Zero, Zero), Zero, new_primMinusNatS0(Zero, Zero))
new_ps4(:%(vxz30, Integer(vxz310)), ty_Integer) → new_reduce2Reduce10(vxz30, vxz310, new_primMulInt(vxz310), new_primMulInt(vxz310), new_primMulInt(vxz310))
new_primPlusInt1(Succ(vxz2800), Zero) → Pos(Succ(vxz2800))
new_gcd0Gcd'112(vxz2700) → new_gcd0Gcd'17(Zero, vxz2700, Zero)
new_primMinusNat2(vxz3100, Zero) → Pos(Succ(vxz3100))
new_quot18(vxz79, Neg(Succ(vxz8000))) → Integer(new_primQuotInt5(vxz79, vxz8000))
new_primDivNatS01(vxz168, vxz169, Zero, Succ(vxz1710)) → Zero
new_primPlusNat(Zero) → Zero
new_primMinusNatS1(vxz13500) → Succ(vxz13500)
new_reduce2Reduce11(vxz30, vxz310, vxz30, vxz29) → error([])
new_sr(vxz31) → new_primMulInt(vxz31)
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)
new_primQuotInt6(Neg(vxz790)) → new_error0
new_ps5(Neg(vxz280), Neg(vxz310)) → new_primPlusInt3(vxz280, vxz310)
new_primQuotInt0(Neg(vxz510), vxz52, vxz2700) → new_primQuotInt3(vxz510, new_gcd0(vxz52, vxz2700))
new_gcd0Gcd'111(vxz2700, vxz5200) → new_gcd0Gcd'15(vxz5200, vxz2700)
new_quot9(vxz285, vxz286, vxz287, Zero, Zero) → new_quot10(vxz285, vxz286, vxz287)
new_primPlusNat2(vxz3100) → Succ(vxz3100)
new_quot16(vxz79, vxz3000, Pos(Zero)) → new_quot13(vxz79, Zero, vxz3000, Zero)
new_quot6(vxz39, vxz310, vxz40, vxz30) → new_quot14(new_primPlusInt4(vxz39, vxz310), new_primPlusInt4(vxz39, vxz310), new_primPlusInt4(vxz39, vxz310), vxz30)
new_primPlusNat4(Succ(vxz28000), Succ(vxz31000)) → Succ(Succ(new_primPlusNat4(vxz28000, vxz31000)))
new_error0error([])
new_primPlusInt4(Pos(vxz390), Neg(vxz3100)) → new_primPlusInt1(vxz390, vxz3100)
new_quot14(vxz79, Neg(Succ(vxz8100)), vxz80, vxz30) → new_quot20(vxz79, vxz80, vxz30)
new_quot14(vxz79, Neg(Zero), vxz80, vxz30) → new_quot19(vxz79, vxz80, vxz30)
new_quot9(vxz285, vxz286, vxz287, Succ(vxz2880), Zero) → new_quot10(vxz285, vxz286, vxz287)
new_gcd0(Neg(Succ(vxz5200)), vxz2700) → new_gcd0Gcd'14(vxz2700, vxz5200)
new_primPlusInt3(vxz280, vxz310) → Neg(new_primPlusNat3(vxz280, vxz310))
new_primQuotInt1(vxz135, Neg(Zero)) → new_error0
new_primPlusNat3(Zero, Succ(vxz3100)) → new_primPlusNat2(vxz3100)
new_quot19(vxz79, vxz80, Neg(Zero)) → new_quot15(vxz79)
new_primPlusInt2(vxz280, Succ(vxz3100)) → new_primMinusNat2(vxz3100, vxz280)
new_primPlusInt4(Neg(vxz390), Pos(vxz3100)) → new_primPlusInt2(vxz390, vxz3100)
new_quot19(vxz79, vxz80, Neg(Succ(vxz3000))) → new_quot17(vxz79, vxz80, Neg(Succ(vxz3000)))
new_primPlusNat3(Succ(vxz2800), Zero) → new_primPlusNat(Succ(vxz2800))
new_reduce(vxz28, vxz31, Pos(Zero)) → new_error
new_reduce2Reduce1(vxz30, vxz310, vxz30, vxz29) → :%(new_quot4(vxz30, vxz310, vxz30), new_quot5(vxz29, vxz30, vxz310, vxz30))
new_quot16(vxz79, vxz3000, Neg(Succ(vxz8000))) → new_quot12(vxz79, vxz8000, vxz3000)
new_quot4(Integer(vxz300), vxz310, vxz30) → new_quot6(new_primMulInt(vxz300), vxz310, new_primMulInt(vxz300), vxz30)
new_quot17(vxz79, vxz80, Pos(Succ(vxz3000))) → new_quot16(vxz79, vxz3000, vxz80)
new_gcd0Gcd'16(vxz2700) → Pos(Succ(vxz2700))
new_primQuotInt1(vxz135, Pos(Zero)) → new_error0
new_gcd0Gcd'15(vxz5200, vxz2700) → new_gcd0Gcd'17(Succ(vxz5200), vxz2700, Succ(vxz5200))
new_primPlusInt1(Zero, Succ(vxz3100)) → new_primMinusNat1(new_primPlusNat2(vxz3100))
new_gcd0Gcd'14(vxz2700, vxz5200) → new_gcd0Gcd'15(vxz5200, vxz2700)
new_primMulNat0(Succ(vxz3100)) → new_primPlusNat0(new_primMulNat0(vxz3100))
new_quot13(vxz249, Succ(Zero), Zero, vxz254) → new_quot13(vxz249, new_primMinusNatS0(Zero, Zero), Zero, new_primMinusNatS0(Zero, Zero))
new_primMinusNat3(Succ(vxz31000), Succ(vxz28000)) → new_primMinusNat3(vxz31000, vxz28000)
new_primQuotInt1(vxz135, Neg(Succ(vxz13600))) → Neg(new_primDivNatS1(vxz135, vxz13600))
new_reduce2Reduce10(vxz30, vxz310, vxz30, vxz29, Neg(Zero)) → new_reduce2Reduce11(vxz30, vxz310, vxz30, vxz29)
new_quot18(vxz79, Neg(Zero)) → Integer(new_primQuotInt6(vxz79))
new_gcd0(Neg(Zero), vxz2700) → new_gcd0Gcd'16(vxz2700)
new_primMulInt(Pos(vxz310)) → Pos(new_primMulNat0(vxz310))
new_reduce(vxz28, vxz31, Neg(Succ(vxz2700))) → :%(new_primQuotInt2(new_ps5(vxz28, vxz31), new_ps5(vxz28, vxz31), vxz2700), new_primQuotInt3(Succ(vxz2700), new_reduce2D0(new_ps5(vxz28, vxz31), vxz2700)))
new_ps5(Pos(vxz280), Pos(vxz310)) → new_primPlusInt0(vxz280, vxz310)
new_primPlusNat4(Zero, Succ(vxz31000)) → Succ(vxz31000)
new_primPlusNat4(Succ(vxz28000), Zero) → Succ(vxz28000)
new_primQuotInt2(Neg(vxz530), vxz54, vxz2700) → new_primQuotInt3(vxz530, new_gcd(vxz54, vxz2700))
new_quot14(vxz79, Pos(Succ(vxz8100)), vxz80, vxz30) → new_quot20(vxz79, vxz80, vxz30)
new_quot13(vxz249, Succ(Zero), Succ(vxz2510), vxz254) → new_quot11(vxz249, Succ(vxz2510), Zero)
new_reduce2Reduce10(vxz30, vxz310, vxz30, vxz29, Pos(Zero)) → new_reduce2Reduce11(vxz30, vxz310, vxz30, vxz29)
new_quot17(vxz79, vxz80, Pos(Zero)) → new_quot18(vxz79, vxz80)
new_primQuotInt3(vxz207, Neg(Succ(vxz22600))) → Pos(new_primDivNatS1(vxz207, vxz22600))
new_gcd0Gcd'18(vxz272, vxz273, Zero, Succ(vxz2750)) → new_gcd0Gcd'110(Succ(vxz272), vxz273)
new_primQuotInt3(vxz207, Neg(Zero)) → new_error0
new_gcd0Gcd'18(vxz272, vxz273, Succ(vxz2740), Succ(vxz2750)) → new_gcd0Gcd'18(vxz272, vxz273, vxz2740, vxz2750)
new_primDivNatS01(vxz168, vxz169, Succ(vxz1700), Succ(vxz1710)) → new_primDivNatS01(vxz168, vxz169, vxz1700, vxz1710)
new_primPlusNat(Succ(vxz3000)) → Succ(vxz3000)
new_primPlusNat3(Succ(vxz2800), Succ(vxz3100)) → Succ(Succ(new_primPlusNat4(vxz2800, vxz3100)))
new_primDivNatS01(vxz168, vxz169, Zero, Zero) → new_primDivNatS02(vxz168, vxz169)
new_quot17(vxz79, vxz80, Neg(Zero)) → new_quot18(vxz79, vxz80)
new_reduce2D0(vxz227, vxz2700) → new_gcd(vxz227, vxz2700)
new_primPlusInt2(vxz280, Zero) → new_primMinusNat1(vxz280)
new_primQuotInt4(Neg(vxz790)) → new_error0
new_quot7(vxz29, vxz48, vxz310, vxz47, vxz30) → new_quot14(vxz29, new_primPlusInt4(vxz48, vxz310), new_primPlusInt4(vxz48, vxz310), vxz30)
new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_reduce(vxz28, vxz31, Neg(Zero)) → new_error
new_primQuotInt4(Pos(vxz790)) → new_error0
new_gcd0Gcd'17(Succ(Zero), Succ(vxz2320), vxz236) → new_gcd0Gcd'110(Zero, Succ(vxz2320))
new_quot9(vxz285, vxz286, vxz287, Succ(vxz2880), Succ(vxz2890)) → new_quot9(vxz285, vxz286, vxz287, vxz2880, vxz2890)
new_quot13(vxz249, Zero, vxz251, vxz254) → new_quot8(vxz249, vxz251)
new_quot16(vxz79, vxz3000, Pos(Succ(vxz8000))) → new_quot11(vxz79, vxz8000, vxz3000)
new_quot13(vxz249, Succ(Succ(vxz25500)), Zero, vxz254) → new_quot13(vxz249, new_primMinusNatS0(Succ(vxz25500), Zero), Zero, new_primMinusNatS0(Succ(vxz25500), Zero))
new_primQuotInt(Pos(vxz790), vxz8000) → Pos(new_primDivNatS1(vxz790, vxz8000))
new_primQuotInt6(Pos(vxz790)) → new_error0
new_primPlusInt1(Zero, Zero) → new_primMinusNat1(Zero)
new_gcd(Pos(Succ(vxz5400)), vxz2700) → new_gcd0Gcd'111(vxz2700, vxz5400)
new_primDivNatS01(vxz168, vxz169, Succ(vxz1700), Zero) → new_primDivNatS02(vxz168, vxz169)
new_quot12(vxz79, vxz8000, vxz3000) → new_quot13(vxz79, Succ(vxz8000), vxz3000, Succ(vxz8000))
new_primMulNat0(Zero) → Zero
new_primDivNatS02(vxz168, vxz169) → Succ(new_primDivNatS1(new_primMinusNatS0(vxz168, vxz169), Succ(vxz169)))
new_quot19(vxz79, vxz80, Pos(Succ(vxz3000))) → new_quot17(vxz79, vxz80, Pos(Succ(vxz3000)))
new_gcd0(Pos(Zero), vxz2700) → new_gcd0Gcd'112(vxz2700)
new_gcd(Neg(Zero), vxz2700) → new_gcd0Gcd'16(vxz2700)
new_primDivNatS1(Succ(Succ(vxz13500)), Succ(vxz136000)) → new_primDivNatS01(vxz13500, vxz136000, vxz13500, vxz136000)
new_quot8(vxz79, vxz8000) → Integer(new_primQuotInt(vxz79, vxz8000))
new_primDivNatS1(Zero, vxz13600) → Zero
new_gcd0Gcd'17(Succ(Succ(vxz23700)), Zero, vxz236) → new_gcd0Gcd'19(vxz23700, Zero)
new_primPlusNat0(Succ(vxz300)) → Succ(Succ(new_primPlusNat(vxz300)))
new_primPlusInt0(vxz280, vxz310) → Pos(new_primPlusNat3(vxz280, vxz310))
new_primPlusNat0(Zero) → Succ(Zero)
new_gcd0Gcd'110(vxz266, vxz267) → new_gcd0Gcd'15(vxz267, vxz266)
new_ps5(Pos(vxz280), Neg(vxz310)) → new_primPlusInt1(vxz280, vxz310)
new_errorerror([])
new_reduce2Reduce10(vxz30, vxz310, vxz30, vxz29, Pos(Succ(vxz3100))) → new_reduce2Reduce1(vxz30, vxz310, vxz30, vxz29)
new_gcd(Pos(Zero), vxz2700) → new_gcd0Gcd'112(vxz2700)
new_gcd0Gcd'17(Succ(Succ(vxz23700)), Succ(vxz2320), vxz236) → new_gcd0Gcd'18(vxz23700, Succ(vxz2320), vxz23700, vxz2320)
new_quot5(vxz29, Integer(vxz300), vxz310, vxz30) → new_quot7(vxz29, new_primMulInt(vxz300), vxz310, new_primMulInt(vxz300), vxz30)
new_primMinusNat3(Succ(vxz31000), Zero) → Pos(Succ(vxz31000))
new_primMinusNat1(Zero) → Pos(Zero)
new_quot16(vxz79, vxz3000, Neg(Zero)) → new_quot8(vxz79, vxz3000)
new_primQuotInt(Neg(vxz790), vxz8000) → Neg(new_primDivNatS1(vxz790, vxz8000))
new_primMinusNat3(Zero, Zero) → Pos(Zero)
new_gcd0Gcd'18(vxz272, vxz273, Succ(vxz2740), Zero) → new_gcd0Gcd'19(vxz272, vxz273)
new_primMinusNat1(Succ(vxz3000)) → Neg(Succ(vxz3000))
new_primMinusNatS2Zero
new_gcd0(Pos(Succ(vxz5200)), vxz2700) → new_gcd0Gcd'111(vxz2700, vxz5200)
new_primPlusNat3(Zero, Zero) → new_primPlusNat(Zero)
new_quot9(vxz285, vxz286, vxz287, Zero, Succ(vxz2890)) → new_quot11(vxz285, vxz287, Succ(vxz286))
new_quot10(vxz285, vxz286, vxz287) → new_quot13(vxz285, new_primMinusNatS0(Succ(vxz286), vxz287), vxz287, new_primMinusNatS0(Succ(vxz286), vxz287))
new_gcd0Gcd'17(Zero, vxz232, vxz236) → Pos(Succ(vxz232))
new_quot20(vxz79, vxz80, vxz30) → new_quot17(vxz79, vxz80, vxz30)
new_primDivNatS1(Succ(Zero), Succ(vxz136000)) → Zero
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primQuotInt3(vxz207, Pos(Succ(vxz22600))) → Neg(new_primDivNatS1(vxz207, vxz22600))
new_primMinusNat2(vxz3100, Succ(vxz2800)) → new_primMinusNat3(vxz3100, vxz2800)

The set Q consists of the following terms:

new_gcd0Gcd'16(x0)
new_gcd0Gcd'17(Zero, x0, x1)
new_ps5(Neg(x0), Neg(x1))
new_primMinusNatS2
new_reduce2D0(x0, x1)
new_gcd0Gcd'111(x0, x1)
new_gcd0Gcd'14(x0, x1)
new_primQuotInt2(Pos(x0), x1, x2)
new_quot19(x0, x1, Neg(Succ(x2)))
new_primPlusNat3(Succ(x0), Succ(x1))
new_reduce(x0, x1, Pos(Zero))
new_primPlusInt4(Neg(x0), Neg(x1))
new_primPlusNat3(Zero, Zero)
new_quot16(x0, x1, Pos(Succ(x2)))
new_primQuotInt4(Pos(x0))
new_gcd(Pos(Succ(x0)), x1)
new_quot4(Integer(x0), x1, x2)
new_quot10(x0, x1, x2)
new_primPlusInt2(x0, Zero)
new_primMinusNatS0(Zero, Succ(x0))
new_primPlusNat4(Succ(x0), Succ(x1))
new_reduce(x0, x1, Neg(Succ(x2)))
new_primDivNatS02(x0, x1)
new_reduce(x0, x1, Pos(Succ(x2)))
new_primPlusNat2(x0)
new_primMulInt(Neg(x0))
new_primQuotInt5(Neg(x0), x1)
new_gcd0(Pos(Succ(x0)), x1)
new_error
new_quot13(x0, Zero, x1, x2)
new_primPlusNat(Succ(x0))
new_primQuotInt3(x0, Pos(Succ(x1)))
new_quot16(x0, x1, Neg(Zero))
new_primDivNatS01(x0, x1, Succ(x2), Zero)
new_quot17(x0, x1, Neg(Zero))
new_ps4(:%(x0, x1), ty_Int)
new_quot9(x0, x1, x2, Succ(x3), Succ(x4))
new_quot11(x0, x1, x2)
new_quot16(x0, x1, Pos(Zero))
new_quot14(x0, Neg(Succ(x1)), x2, x3)
new_quot17(x0, x1, Pos(Succ(x2)))
new_primDivNatS01(x0, x1, Zero, Zero)
new_quot13(x0, Succ(Succ(x1)), Succ(x2), x3)
new_primPlusNat4(Zero, Zero)
new_primQuotInt6(Neg(x0))
new_primQuotInt(Neg(x0), x1)
new_primQuotInt4(Neg(x0))
new_primQuotInt1(x0, Pos(Succ(x1)))
new_primMinusNat1(Zero)
new_reduce2Reduce10(x0, x1, x0, x2, Neg(Succ(x3)))
new_reduce(x0, x1, Neg(Zero))
new_quot13(x0, Succ(Zero), Succ(x1), x2)
new_primQuotInt6(Pos(x0))
new_primPlusInt2(x0, Succ(x1))
new_quot15(x0)
new_gcd0Gcd'17(Succ(Succ(x0)), Zero, x1)
new_quot17(x0, x1, Pos(Zero))
new_primMinusNatS1(x0)
new_gcd0(Neg(Succ(x0)), x1)
new_quot18(x0, Neg(Zero))
new_ps5(Pos(x0), Pos(x1))
new_primQuotInt3(x0, Neg(Zero))
new_gcd0Gcd'18(x0, x1, Succ(x2), Succ(x3))
new_primQuotInt3(x0, Pos(Zero))
new_gcd0Gcd'18(x0, x1, Zero, Zero)
new_primPlusNat3(Zero, Succ(x0))
new_primMinusNat2(x0, Succ(x1))
new_sr(x0)
new_gcd0Gcd'15(x0, x1)
new_reduce2Reduce10(x0, x1, x0, x2, Pos(Zero))
new_gcd0Gcd'17(Succ(Zero), Succ(x0), x1)
new_primQuotInt1(x0, Neg(Zero))
new_primDivNatS1(Zero, x0)
new_quot9(x0, x1, x2, Zero, Succ(x3))
new_quot12(x0, x1, x2)
new_primPlusInt0(x0, x1)
new_primPlusInt1(Succ(x0), Zero)
new_primDivNatS1(Succ(Succ(x0)), Zero)
new_gcd0(Pos(Zero), x0)
new_primQuotInt2(Neg(x0), x1, x2)
new_primPlusNat0(Succ(x0))
new_gcd0Gcd'18(x0, x1, Succ(x2), Zero)
new_primMinusNat3(Zero, Succ(x0))
new_primPlusInt3(x0, x1)
new_primPlusNat4(Succ(x0), Zero)
new_primPlusInt1(Succ(x0), Succ(x1))
new_reduce2Reduce10(x0, x1, x0, x2, Pos(Succ(x3)))
new_quot19(x0, x1, Neg(Zero))
new_gcd0Gcd'112(x0)
new_quot18(x0, Pos(Zero))
new_primMinusNat2(x0, Zero)
new_gcd0Gcd'18(x0, x1, Zero, Succ(x2))
new_reduce2Reduce1(x0, x1, x0, x2)
new_primPlusInt4(Pos(x0), Neg(x1))
new_primPlusInt4(Neg(x0), Pos(x1))
new_primMulNat0(Zero)
new_primMinusNatS0(Zero, Zero)
new_primQuotInt0(Neg(x0), x1, x2)
new_quot19(x0, x1, Pos(Succ(x2)))
new_quot18(x0, Neg(Succ(x1)))
new_quot16(x0, x1, Neg(Succ(x2)))
new_quot17(x0, x1, Neg(Succ(x2)))
new_primDivNatS1(Succ(Zero), Zero)
new_gcd(Neg(Zero), x0)
new_gcd0Gcd'19(x0, x1)
new_primDivNatS01(x0, x1, Succ(x2), Succ(x3))
new_primMinusNat3(Succ(x0), Succ(x1))
new_error0
new_quot9(x0, x1, x2, Zero, Zero)
new_primQuotInt5(Pos(x0), x1)
new_quot7(x0, x1, x2, x3, x4)
new_reduce2Reduce11(x0, x1, x0, x2)
new_primMulNat0(Succ(x0))
new_primPlusNat0(Zero)
new_gcd0Gcd'17(Succ(Zero), Zero, x0)
new_primMinusNat3(Zero, Zero)
new_gcd0Gcd'17(Succ(Succ(x0)), Succ(x1), x2)
new_reduce2Reduce10(x0, x1, x0, x2, Neg(Zero))
new_primPlusNat4(Zero, Succ(x0))
new_primQuotInt(Pos(x0), x1)
new_ps4(:%(x0, Integer(x1)), ty_Integer)
new_primQuotInt3(x0, Neg(Succ(x1)))
new_quot8(x0, x1)
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNat3(Succ(x0), Zero)
new_primPlusNat3(Succ(x0), Zero)
new_primPlusInt4(Pos(x0), Pos(x1))
new_quot5(x0, Integer(x1), x2, x3)
new_primPlusInt1(Zero, Zero)
new_primPlusNat(Zero)
new_quot6(x0, x1, x2, x3)
new_quot14(x0, Neg(Zero), x1, x2)
new_primQuotInt1(x0, Pos(Zero))
new_gcd0(Neg(Zero), x0)
new_reduce2D(x0, x1)
new_primMulInt(Pos(x0))
new_primDivNatS1(Succ(Zero), Succ(x0))
new_gcd0Gcd'110(x0, x1)
new_quot18(x0, Pos(Succ(x1)))
new_primMinusNat1(Succ(x0))
new_primDivNatS1(Succ(Succ(x0)), Succ(x1))
new_quot9(x0, x1, x2, Succ(x3), Zero)
new_quot14(x0, Pos(Succ(x1)), x2, x3)
new_gcd(Neg(Succ(x0)), x1)
new_primDivNatS01(x0, x1, Zero, Succ(x2))
new_primQuotInt0(Pos(x0), x1, x2)
new_ps5(Pos(x0), Neg(x1))
new_ps5(Neg(x0), Pos(x1))
new_primPlusInt1(Zero, Succ(x0))
new_gcd(Pos(Zero), x0)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_quot14(x0, Pos(Zero), x1, x2)
new_quot13(x0, Succ(Zero), Zero, x1)
new_primQuotInt1(x0, Neg(Succ(x1)))
new_quot19(x0, x1, Pos(Zero))
new_quot13(x0, Succ(Succ(x1)), Zero, x2)
new_quot20(x0, x1, x2)

We have to consider all minimal (P,Q,R)-chains.
We use the modular non-overlap check [17] to decrease Q to the empty set.

↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
        ↳ HASKELL
          ↳ COR
            ↳ HASKELL
              ↳ LetRed
                ↳ HASKELL
                  ↳ NumRed
                    ↳ HASKELL
                      ↳ Narrow
                        ↳ AND
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                            ↳ MNOCProof
QDP
                                ↳ NonTerminationProof
                      ↳ Narrow

Q DP problem:
The TRS P consists of the following rules:

new_numericEnumFrom3(vxz3, ba) → new_numericEnumFrom3(new_ps4(vxz3, ba), ba)

The TRS R consists of the following rules:

new_primDivNatS1(Succ(Zero), Zero) → Succ(new_primDivNatS1(new_primMinusNatS2, Zero))
new_quot14(vxz79, Pos(Zero), vxz80, vxz30) → new_quot19(vxz79, vxz80, vxz30)
new_quot19(vxz79, vxz80, Pos(Zero)) → new_quot15(vxz79)
new_gcd0Gcd'19(vxz263, vxz264) → new_gcd0Gcd'17(new_primMinusNatS0(Succ(vxz263), vxz264), vxz264, new_primMinusNatS0(Succ(vxz263), vxz264))
new_quot18(vxz79, Pos(Succ(vxz8000))) → new_quot8(vxz79, vxz8000)
new_primQuotInt0(Pos(vxz510), vxz52, vxz2700) → new_primQuotInt1(vxz510, new_gcd0(vxz52, vxz2700))
new_quot18(vxz79, Pos(Zero)) → Integer(new_primQuotInt4(vxz79))
new_reduce2Reduce10(vxz30, vxz310, vxz30, vxz29, Neg(Succ(vxz3100))) → new_reduce2Reduce1(vxz30, vxz310, vxz30, vxz29)
new_quot13(vxz249, Succ(Succ(vxz25500)), Succ(vxz2510), vxz254) → new_quot9(vxz249, vxz25500, Succ(vxz2510), vxz25500, vxz2510)
new_quot15(vxz79) → error([])
new_primQuotInt5(Neg(vxz790), vxz8000) → new_primQuotInt3(vxz790, Pos(Succ(vxz8000)))
new_ps5(Neg(vxz280), Pos(vxz310)) → new_primPlusInt2(vxz280, vxz310)
new_primQuotInt2(Pos(vxz530), vxz54, vxz2700) → new_primQuotInt1(vxz530, new_gcd(vxz54, vxz2700))
new_gcd(Neg(Succ(vxz5400)), vxz2700) → new_gcd0Gcd'14(vxz2700, vxz5400)
new_primMinusNatS0(Zero, Zero) → Zero
new_gcd0Gcd'18(vxz272, vxz273, Zero, Zero) → new_gcd0Gcd'19(vxz272, vxz273)
new_quot17(vxz79, vxz80, Neg(Succ(vxz3000))) → new_quot16(vxz79, vxz3000, vxz80)
new_reduce2D(vxz137, vxz2700) → new_gcd0(vxz137, vxz2700)
new_primMulInt(Neg(vxz310)) → Neg(new_primMulNat0(vxz310))
new_primPlusInt4(Neg(vxz390), Neg(vxz3100)) → new_primPlusInt3(vxz390, vxz3100)
new_primQuotInt1(vxz135, Pos(Succ(vxz13600))) → Pos(new_primDivNatS1(vxz135, vxz13600))
new_primPlusInt1(Succ(vxz2800), Succ(vxz3100)) → new_primMinusNat2(vxz2800, Succ(vxz3100))
new_reduce(vxz28, vxz31, Pos(Succ(vxz2700))) → :%(new_primQuotInt0(new_ps5(vxz28, vxz31), new_ps5(vxz28, vxz31), vxz2700), new_primQuotInt1(Succ(vxz2700), new_reduce2D(new_ps5(vxz28, vxz31), vxz2700)))
new_quot11(vxz79, vxz8000, vxz3000) → new_quot12(vxz79, vxz8000, vxz3000)
new_primQuotInt5(Pos(vxz790), vxz8000) → new_primQuotInt(Pos(vxz790), vxz8000)
new_primPlusNat4(Zero, Zero) → Zero
new_primDivNatS1(Succ(Succ(vxz13500)), Zero) → Succ(new_primDivNatS1(new_primMinusNatS1(vxz13500), Zero))
new_primQuotInt3(vxz207, Pos(Zero)) → new_error0
new_ps4(:%(vxz30, vxz31), ty_Int) → new_reduce(new_sr(vxz30), vxz31, new_sr(vxz31))
new_primMinusNat3(Zero, Succ(vxz28000)) → Neg(Succ(vxz28000))
new_primPlusInt4(Pos(vxz390), Pos(vxz3100)) → new_primPlusInt0(vxz390, vxz3100)
new_gcd0Gcd'17(Succ(Zero), Zero, vxz236) → new_gcd0Gcd'17(new_primMinusNatS0(Zero, Zero), Zero, new_primMinusNatS0(Zero, Zero))
new_ps4(:%(vxz30, Integer(vxz310)), ty_Integer) → new_reduce2Reduce10(vxz30, vxz310, new_primMulInt(vxz310), new_primMulInt(vxz310), new_primMulInt(vxz310))
new_primPlusInt1(Succ(vxz2800), Zero) → Pos(Succ(vxz2800))
new_gcd0Gcd'112(vxz2700) → new_gcd0Gcd'17(Zero, vxz2700, Zero)
new_primMinusNat2(vxz3100, Zero) → Pos(Succ(vxz3100))
new_quot18(vxz79, Neg(Succ(vxz8000))) → Integer(new_primQuotInt5(vxz79, vxz8000))
new_primDivNatS01(vxz168, vxz169, Zero, Succ(vxz1710)) → Zero
new_primPlusNat(Zero) → Zero
new_primMinusNatS1(vxz13500) → Succ(vxz13500)
new_reduce2Reduce11(vxz30, vxz310, vxz30, vxz29) → error([])
new_sr(vxz31) → new_primMulInt(vxz31)
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)
new_primQuotInt6(Neg(vxz790)) → new_error0
new_ps5(Neg(vxz280), Neg(vxz310)) → new_primPlusInt3(vxz280, vxz310)
new_primQuotInt0(Neg(vxz510), vxz52, vxz2700) → new_primQuotInt3(vxz510, new_gcd0(vxz52, vxz2700))
new_gcd0Gcd'111(vxz2700, vxz5200) → new_gcd0Gcd'15(vxz5200, vxz2700)
new_quot9(vxz285, vxz286, vxz287, Zero, Zero) → new_quot10(vxz285, vxz286, vxz287)
new_primPlusNat2(vxz3100) → Succ(vxz3100)
new_quot16(vxz79, vxz3000, Pos(Zero)) → new_quot13(vxz79, Zero, vxz3000, Zero)
new_quot6(vxz39, vxz310, vxz40, vxz30) → new_quot14(new_primPlusInt4(vxz39, vxz310), new_primPlusInt4(vxz39, vxz310), new_primPlusInt4(vxz39, vxz310), vxz30)
new_primPlusNat4(Succ(vxz28000), Succ(vxz31000)) → Succ(Succ(new_primPlusNat4(vxz28000, vxz31000)))
new_error0error([])
new_primPlusInt4(Pos(vxz390), Neg(vxz3100)) → new_primPlusInt1(vxz390, vxz3100)
new_quot14(vxz79, Neg(Succ(vxz8100)), vxz80, vxz30) → new_quot20(vxz79, vxz80, vxz30)
new_quot14(vxz79, Neg(Zero), vxz80, vxz30) → new_quot19(vxz79, vxz80, vxz30)
new_quot9(vxz285, vxz286, vxz287, Succ(vxz2880), Zero) → new_quot10(vxz285, vxz286, vxz287)
new_gcd0(Neg(Succ(vxz5200)), vxz2700) → new_gcd0Gcd'14(vxz2700, vxz5200)
new_primPlusInt3(vxz280, vxz310) → Neg(new_primPlusNat3(vxz280, vxz310))
new_primQuotInt1(vxz135, Neg(Zero)) → new_error0
new_primPlusNat3(Zero, Succ(vxz3100)) → new_primPlusNat2(vxz3100)
new_quot19(vxz79, vxz80, Neg(Zero)) → new_quot15(vxz79)
new_primPlusInt2(vxz280, Succ(vxz3100)) → new_primMinusNat2(vxz3100, vxz280)
new_primPlusInt4(Neg(vxz390), Pos(vxz3100)) → new_primPlusInt2(vxz390, vxz3100)
new_quot19(vxz79, vxz80, Neg(Succ(vxz3000))) → new_quot17(vxz79, vxz80, Neg(Succ(vxz3000)))
new_primPlusNat3(Succ(vxz2800), Zero) → new_primPlusNat(Succ(vxz2800))
new_reduce(vxz28, vxz31, Pos(Zero)) → new_error
new_reduce2Reduce1(vxz30, vxz310, vxz30, vxz29) → :%(new_quot4(vxz30, vxz310, vxz30), new_quot5(vxz29, vxz30, vxz310, vxz30))
new_quot16(vxz79, vxz3000, Neg(Succ(vxz8000))) → new_quot12(vxz79, vxz8000, vxz3000)
new_quot4(Integer(vxz300), vxz310, vxz30) → new_quot6(new_primMulInt(vxz300), vxz310, new_primMulInt(vxz300), vxz30)
new_quot17(vxz79, vxz80, Pos(Succ(vxz3000))) → new_quot16(vxz79, vxz3000, vxz80)
new_gcd0Gcd'16(vxz2700) → Pos(Succ(vxz2700))
new_primQuotInt1(vxz135, Pos(Zero)) → new_error0
new_gcd0Gcd'15(vxz5200, vxz2700) → new_gcd0Gcd'17(Succ(vxz5200), vxz2700, Succ(vxz5200))
new_primPlusInt1(Zero, Succ(vxz3100)) → new_primMinusNat1(new_primPlusNat2(vxz3100))
new_gcd0Gcd'14(vxz2700, vxz5200) → new_gcd0Gcd'15(vxz5200, vxz2700)
new_primMulNat0(Succ(vxz3100)) → new_primPlusNat0(new_primMulNat0(vxz3100))
new_quot13(vxz249, Succ(Zero), Zero, vxz254) → new_quot13(vxz249, new_primMinusNatS0(Zero, Zero), Zero, new_primMinusNatS0(Zero, Zero))
new_primMinusNat3(Succ(vxz31000), Succ(vxz28000)) → new_primMinusNat3(vxz31000, vxz28000)
new_primQuotInt1(vxz135, Neg(Succ(vxz13600))) → Neg(new_primDivNatS1(vxz135, vxz13600))
new_reduce2Reduce10(vxz30, vxz310, vxz30, vxz29, Neg(Zero)) → new_reduce2Reduce11(vxz30, vxz310, vxz30, vxz29)
new_quot18(vxz79, Neg(Zero)) → Integer(new_primQuotInt6(vxz79))
new_gcd0(Neg(Zero), vxz2700) → new_gcd0Gcd'16(vxz2700)
new_primMulInt(Pos(vxz310)) → Pos(new_primMulNat0(vxz310))
new_reduce(vxz28, vxz31, Neg(Succ(vxz2700))) → :%(new_primQuotInt2(new_ps5(vxz28, vxz31), new_ps5(vxz28, vxz31), vxz2700), new_primQuotInt3(Succ(vxz2700), new_reduce2D0(new_ps5(vxz28, vxz31), vxz2700)))
new_ps5(Pos(vxz280), Pos(vxz310)) → new_primPlusInt0(vxz280, vxz310)
new_primPlusNat4(Zero, Succ(vxz31000)) → Succ(vxz31000)
new_primPlusNat4(Succ(vxz28000), Zero) → Succ(vxz28000)
new_primQuotInt2(Neg(vxz530), vxz54, vxz2700) → new_primQuotInt3(vxz530, new_gcd(vxz54, vxz2700))
new_quot14(vxz79, Pos(Succ(vxz8100)), vxz80, vxz30) → new_quot20(vxz79, vxz80, vxz30)
new_quot13(vxz249, Succ(Zero), Succ(vxz2510), vxz254) → new_quot11(vxz249, Succ(vxz2510), Zero)
new_reduce2Reduce10(vxz30, vxz310, vxz30, vxz29, Pos(Zero)) → new_reduce2Reduce11(vxz30, vxz310, vxz30, vxz29)
new_quot17(vxz79, vxz80, Pos(Zero)) → new_quot18(vxz79, vxz80)
new_primQuotInt3(vxz207, Neg(Succ(vxz22600))) → Pos(new_primDivNatS1(vxz207, vxz22600))
new_gcd0Gcd'18(vxz272, vxz273, Zero, Succ(vxz2750)) → new_gcd0Gcd'110(Succ(vxz272), vxz273)
new_primQuotInt3(vxz207, Neg(Zero)) → new_error0
new_gcd0Gcd'18(vxz272, vxz273, Succ(vxz2740), Succ(vxz2750)) → new_gcd0Gcd'18(vxz272, vxz273, vxz2740, vxz2750)
new_primDivNatS01(vxz168, vxz169, Succ(vxz1700), Succ(vxz1710)) → new_primDivNatS01(vxz168, vxz169, vxz1700, vxz1710)
new_primPlusNat(Succ(vxz3000)) → Succ(vxz3000)
new_primPlusNat3(Succ(vxz2800), Succ(vxz3100)) → Succ(Succ(new_primPlusNat4(vxz2800, vxz3100)))
new_primDivNatS01(vxz168, vxz169, Zero, Zero) → new_primDivNatS02(vxz168, vxz169)
new_quot17(vxz79, vxz80, Neg(Zero)) → new_quot18(vxz79, vxz80)
new_reduce2D0(vxz227, vxz2700) → new_gcd(vxz227, vxz2700)
new_primPlusInt2(vxz280, Zero) → new_primMinusNat1(vxz280)
new_primQuotInt4(Neg(vxz790)) → new_error0
new_quot7(vxz29, vxz48, vxz310, vxz47, vxz30) → new_quot14(vxz29, new_primPlusInt4(vxz48, vxz310), new_primPlusInt4(vxz48, vxz310), vxz30)
new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_reduce(vxz28, vxz31, Neg(Zero)) → new_error
new_primQuotInt4(Pos(vxz790)) → new_error0
new_gcd0Gcd'17(Succ(Zero), Succ(vxz2320), vxz236) → new_gcd0Gcd'110(Zero, Succ(vxz2320))
new_quot9(vxz285, vxz286, vxz287, Succ(vxz2880), Succ(vxz2890)) → new_quot9(vxz285, vxz286, vxz287, vxz2880, vxz2890)
new_quot13(vxz249, Zero, vxz251, vxz254) → new_quot8(vxz249, vxz251)
new_quot16(vxz79, vxz3000, Pos(Succ(vxz8000))) → new_quot11(vxz79, vxz8000, vxz3000)
new_quot13(vxz249, Succ(Succ(vxz25500)), Zero, vxz254) → new_quot13(vxz249, new_primMinusNatS0(Succ(vxz25500), Zero), Zero, new_primMinusNatS0(Succ(vxz25500), Zero))
new_primQuotInt(Pos(vxz790), vxz8000) → Pos(new_primDivNatS1(vxz790, vxz8000))
new_primQuotInt6(Pos(vxz790)) → new_error0
new_primPlusInt1(Zero, Zero) → new_primMinusNat1(Zero)
new_gcd(Pos(Succ(vxz5400)), vxz2700) → new_gcd0Gcd'111(vxz2700, vxz5400)
new_primDivNatS01(vxz168, vxz169, Succ(vxz1700), Zero) → new_primDivNatS02(vxz168, vxz169)
new_quot12(vxz79, vxz8000, vxz3000) → new_quot13(vxz79, Succ(vxz8000), vxz3000, Succ(vxz8000))
new_primMulNat0(Zero) → Zero
new_primDivNatS02(vxz168, vxz169) → Succ(new_primDivNatS1(new_primMinusNatS0(vxz168, vxz169), Succ(vxz169)))
new_quot19(vxz79, vxz80, Pos(Succ(vxz3000))) → new_quot17(vxz79, vxz80, Pos(Succ(vxz3000)))
new_gcd0(Pos(Zero), vxz2700) → new_gcd0Gcd'112(vxz2700)
new_gcd(Neg(Zero), vxz2700) → new_gcd0Gcd'16(vxz2700)
new_primDivNatS1(Succ(Succ(vxz13500)), Succ(vxz136000)) → new_primDivNatS01(vxz13500, vxz136000, vxz13500, vxz136000)
new_quot8(vxz79, vxz8000) → Integer(new_primQuotInt(vxz79, vxz8000))
new_primDivNatS1(Zero, vxz13600) → Zero
new_gcd0Gcd'17(Succ(Succ(vxz23700)), Zero, vxz236) → new_gcd0Gcd'19(vxz23700, Zero)
new_primPlusNat0(Succ(vxz300)) → Succ(Succ(new_primPlusNat(vxz300)))
new_primPlusInt0(vxz280, vxz310) → Pos(new_primPlusNat3(vxz280, vxz310))
new_primPlusNat0(Zero) → Succ(Zero)
new_gcd0Gcd'110(vxz266, vxz267) → new_gcd0Gcd'15(vxz267, vxz266)
new_ps5(Pos(vxz280), Neg(vxz310)) → new_primPlusInt1(vxz280, vxz310)
new_errorerror([])
new_reduce2Reduce10(vxz30, vxz310, vxz30, vxz29, Pos(Succ(vxz3100))) → new_reduce2Reduce1(vxz30, vxz310, vxz30, vxz29)
new_gcd(Pos(Zero), vxz2700) → new_gcd0Gcd'112(vxz2700)
new_gcd0Gcd'17(Succ(Succ(vxz23700)), Succ(vxz2320), vxz236) → new_gcd0Gcd'18(vxz23700, Succ(vxz2320), vxz23700, vxz2320)
new_quot5(vxz29, Integer(vxz300), vxz310, vxz30) → new_quot7(vxz29, new_primMulInt(vxz300), vxz310, new_primMulInt(vxz300), vxz30)
new_primMinusNat3(Succ(vxz31000), Zero) → Pos(Succ(vxz31000))
new_primMinusNat1(Zero) → Pos(Zero)
new_quot16(vxz79, vxz3000, Neg(Zero)) → new_quot8(vxz79, vxz3000)
new_primQuotInt(Neg(vxz790), vxz8000) → Neg(new_primDivNatS1(vxz790, vxz8000))
new_primMinusNat3(Zero, Zero) → Pos(Zero)
new_gcd0Gcd'18(vxz272, vxz273, Succ(vxz2740), Zero) → new_gcd0Gcd'19(vxz272, vxz273)
new_primMinusNat1(Succ(vxz3000)) → Neg(Succ(vxz3000))
new_primMinusNatS2Zero
new_gcd0(Pos(Succ(vxz5200)), vxz2700) → new_gcd0Gcd'111(vxz2700, vxz5200)
new_primPlusNat3(Zero, Zero) → new_primPlusNat(Zero)
new_quot9(vxz285, vxz286, vxz287, Zero, Succ(vxz2890)) → new_quot11(vxz285, vxz287, Succ(vxz286))
new_quot10(vxz285, vxz286, vxz287) → new_quot13(vxz285, new_primMinusNatS0(Succ(vxz286), vxz287), vxz287, new_primMinusNatS0(Succ(vxz286), vxz287))
new_gcd0Gcd'17(Zero, vxz232, vxz236) → Pos(Succ(vxz232))
new_quot20(vxz79, vxz80, vxz30) → new_quot17(vxz79, vxz80, vxz30)
new_primDivNatS1(Succ(Zero), Succ(vxz136000)) → Zero
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primQuotInt3(vxz207, Pos(Succ(vxz22600))) → Neg(new_primDivNatS1(vxz207, vxz22600))
new_primMinusNat2(vxz3100, Succ(vxz2800)) → new_primMinusNat3(vxz3100, vxz2800)

Q is empty.
We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

new_numericEnumFrom3(vxz3, ba) → new_numericEnumFrom3(new_ps4(vxz3, ba), ba)

The TRS R consists of the following rules:

new_primDivNatS1(Succ(Zero), Zero) → Succ(new_primDivNatS1(new_primMinusNatS2, Zero))
new_quot14(vxz79, Pos(Zero), vxz80, vxz30) → new_quot19(vxz79, vxz80, vxz30)
new_quot19(vxz79, vxz80, Pos(Zero)) → new_quot15(vxz79)
new_gcd0Gcd'19(vxz263, vxz264) → new_gcd0Gcd'17(new_primMinusNatS0(Succ(vxz263), vxz264), vxz264, new_primMinusNatS0(Succ(vxz263), vxz264))
new_quot18(vxz79, Pos(Succ(vxz8000))) → new_quot8(vxz79, vxz8000)
new_primQuotInt0(Pos(vxz510), vxz52, vxz2700) → new_primQuotInt1(vxz510, new_gcd0(vxz52, vxz2700))
new_quot18(vxz79, Pos(Zero)) → Integer(new_primQuotInt4(vxz79))
new_reduce2Reduce10(vxz30, vxz310, vxz30, vxz29, Neg(Succ(vxz3100))) → new_reduce2Reduce1(vxz30, vxz310, vxz30, vxz29)
new_quot13(vxz249, Succ(Succ(vxz25500)), Succ(vxz2510), vxz254) → new_quot9(vxz249, vxz25500, Succ(vxz2510), vxz25500, vxz2510)
new_quot15(vxz79) → error([])
new_primQuotInt5(Neg(vxz790), vxz8000) → new_primQuotInt3(vxz790, Pos(Succ(vxz8000)))
new_ps5(Neg(vxz280), Pos(vxz310)) → new_primPlusInt2(vxz280, vxz310)
new_primQuotInt2(Pos(vxz530), vxz54, vxz2700) → new_primQuotInt1(vxz530, new_gcd(vxz54, vxz2700))
new_gcd(Neg(Succ(vxz5400)), vxz2700) → new_gcd0Gcd'14(vxz2700, vxz5400)
new_primMinusNatS0(Zero, Zero) → Zero
new_gcd0Gcd'18(vxz272, vxz273, Zero, Zero) → new_gcd0Gcd'19(vxz272, vxz273)
new_quot17(vxz79, vxz80, Neg(Succ(vxz3000))) → new_quot16(vxz79, vxz3000, vxz80)
new_reduce2D(vxz137, vxz2700) → new_gcd0(vxz137, vxz2700)
new_primMulInt(Neg(vxz310)) → Neg(new_primMulNat0(vxz310))
new_primPlusInt4(Neg(vxz390), Neg(vxz3100)) → new_primPlusInt3(vxz390, vxz3100)
new_primQuotInt1(vxz135, Pos(Succ(vxz13600))) → Pos(new_primDivNatS1(vxz135, vxz13600))
new_primPlusInt1(Succ(vxz2800), Succ(vxz3100)) → new_primMinusNat2(vxz2800, Succ(vxz3100))
new_reduce(vxz28, vxz31, Pos(Succ(vxz2700))) → :%(new_primQuotInt0(new_ps5(vxz28, vxz31), new_ps5(vxz28, vxz31), vxz2700), new_primQuotInt1(Succ(vxz2700), new_reduce2D(new_ps5(vxz28, vxz31), vxz2700)))
new_quot11(vxz79, vxz8000, vxz3000) → new_quot12(vxz79, vxz8000, vxz3000)
new_primQuotInt5(Pos(vxz790), vxz8000) → new_primQuotInt(Pos(vxz790), vxz8000)
new_primPlusNat4(Zero, Zero) → Zero
new_primDivNatS1(Succ(Succ(vxz13500)), Zero) → Succ(new_primDivNatS1(new_primMinusNatS1(vxz13500), Zero))
new_primQuotInt3(vxz207, Pos(Zero)) → new_error0
new_ps4(:%(vxz30, vxz31), ty_Int) → new_reduce(new_sr(vxz30), vxz31, new_sr(vxz31))
new_primMinusNat3(Zero, Succ(vxz28000)) → Neg(Succ(vxz28000))
new_primPlusInt4(Pos(vxz390), Pos(vxz3100)) → new_primPlusInt0(vxz390, vxz3100)
new_gcd0Gcd'17(Succ(Zero), Zero, vxz236) → new_gcd0Gcd'17(new_primMinusNatS0(Zero, Zero), Zero, new_primMinusNatS0(Zero, Zero))
new_ps4(:%(vxz30, Integer(vxz310)), ty_Integer) → new_reduce2Reduce10(vxz30, vxz310, new_primMulInt(vxz310), new_primMulInt(vxz310), new_primMulInt(vxz310))
new_primPlusInt1(Succ(vxz2800), Zero) → Pos(Succ(vxz2800))
new_gcd0Gcd'112(vxz2700) → new_gcd0Gcd'17(Zero, vxz2700, Zero)
new_primMinusNat2(vxz3100, Zero) → Pos(Succ(vxz3100))
new_quot18(vxz79, Neg(Succ(vxz8000))) → Integer(new_primQuotInt5(vxz79, vxz8000))
new_primDivNatS01(vxz168, vxz169, Zero, Succ(vxz1710)) → Zero
new_primPlusNat(Zero) → Zero
new_primMinusNatS1(vxz13500) → Succ(vxz13500)
new_reduce2Reduce11(vxz30, vxz310, vxz30, vxz29) → error([])
new_sr(vxz31) → new_primMulInt(vxz31)
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)
new_primQuotInt6(Neg(vxz790)) → new_error0
new_ps5(Neg(vxz280), Neg(vxz310)) → new_primPlusInt3(vxz280, vxz310)
new_primQuotInt0(Neg(vxz510), vxz52, vxz2700) → new_primQuotInt3(vxz510, new_gcd0(vxz52, vxz2700))
new_gcd0Gcd'111(vxz2700, vxz5200) → new_gcd0Gcd'15(vxz5200, vxz2700)
new_quot9(vxz285, vxz286, vxz287, Zero, Zero) → new_quot10(vxz285, vxz286, vxz287)
new_primPlusNat2(vxz3100) → Succ(vxz3100)
new_quot16(vxz79, vxz3000, Pos(Zero)) → new_quot13(vxz79, Zero, vxz3000, Zero)
new_quot6(vxz39, vxz310, vxz40, vxz30) → new_quot14(new_primPlusInt4(vxz39, vxz310), new_primPlusInt4(vxz39, vxz310), new_primPlusInt4(vxz39, vxz310), vxz30)
new_primPlusNat4(Succ(vxz28000), Succ(vxz31000)) → Succ(Succ(new_primPlusNat4(vxz28000, vxz31000)))
new_error0error([])
new_primPlusInt4(Pos(vxz390), Neg(vxz3100)) → new_primPlusInt1(vxz390, vxz3100)
new_quot14(vxz79, Neg(Succ(vxz8100)), vxz80, vxz30) → new_quot20(vxz79, vxz80, vxz30)
new_quot14(vxz79, Neg(Zero), vxz80, vxz30) → new_quot19(vxz79, vxz80, vxz30)
new_quot9(vxz285, vxz286, vxz287, Succ(vxz2880), Zero) → new_quot10(vxz285, vxz286, vxz287)
new_gcd0(Neg(Succ(vxz5200)), vxz2700) → new_gcd0Gcd'14(vxz2700, vxz5200)
new_primPlusInt3(vxz280, vxz310) → Neg(new_primPlusNat3(vxz280, vxz310))
new_primQuotInt1(vxz135, Neg(Zero)) → new_error0
new_primPlusNat3(Zero, Succ(vxz3100)) → new_primPlusNat2(vxz3100)
new_quot19(vxz79, vxz80, Neg(Zero)) → new_quot15(vxz79)
new_primPlusInt2(vxz280, Succ(vxz3100)) → new_primMinusNat2(vxz3100, vxz280)
new_primPlusInt4(Neg(vxz390), Pos(vxz3100)) → new_primPlusInt2(vxz390, vxz3100)
new_quot19(vxz79, vxz80, Neg(Succ(vxz3000))) → new_quot17(vxz79, vxz80, Neg(Succ(vxz3000)))
new_primPlusNat3(Succ(vxz2800), Zero) → new_primPlusNat(Succ(vxz2800))
new_reduce(vxz28, vxz31, Pos(Zero)) → new_error
new_reduce2Reduce1(vxz30, vxz310, vxz30, vxz29) → :%(new_quot4(vxz30, vxz310, vxz30), new_quot5(vxz29, vxz30, vxz310, vxz30))
new_quot16(vxz79, vxz3000, Neg(Succ(vxz8000))) → new_quot12(vxz79, vxz8000, vxz3000)
new_quot4(Integer(vxz300), vxz310, vxz30) → new_quot6(new_primMulInt(vxz300), vxz310, new_primMulInt(vxz300), vxz30)
new_quot17(vxz79, vxz80, Pos(Succ(vxz3000))) → new_quot16(vxz79, vxz3000, vxz80)
new_gcd0Gcd'16(vxz2700) → Pos(Succ(vxz2700))
new_primQuotInt1(vxz135, Pos(Zero)) → new_error0
new_gcd0Gcd'15(vxz5200, vxz2700) → new_gcd0Gcd'17(Succ(vxz5200), vxz2700, Succ(vxz5200))
new_primPlusInt1(Zero, Succ(vxz3100)) → new_primMinusNat1(new_primPlusNat2(vxz3100))
new_gcd0Gcd'14(vxz2700, vxz5200) → new_gcd0Gcd'15(vxz5200, vxz2700)
new_primMulNat0(Succ(vxz3100)) → new_primPlusNat0(new_primMulNat0(vxz3100))
new_quot13(vxz249, Succ(Zero), Zero, vxz254) → new_quot13(vxz249, new_primMinusNatS0(Zero, Zero), Zero, new_primMinusNatS0(Zero, Zero))
new_primMinusNat3(Succ(vxz31000), Succ(vxz28000)) → new_primMinusNat3(vxz31000, vxz28000)
new_primQuotInt1(vxz135, Neg(Succ(vxz13600))) → Neg(new_primDivNatS1(vxz135, vxz13600))
new_reduce2Reduce10(vxz30, vxz310, vxz30, vxz29, Neg(Zero)) → new_reduce2Reduce11(vxz30, vxz310, vxz30, vxz29)
new_quot18(vxz79, Neg(Zero)) → Integer(new_primQuotInt6(vxz79))
new_gcd0(Neg(Zero), vxz2700) → new_gcd0Gcd'16(vxz2700)
new_primMulInt(Pos(vxz310)) → Pos(new_primMulNat0(vxz310))
new_reduce(vxz28, vxz31, Neg(Succ(vxz2700))) → :%(new_primQuotInt2(new_ps5(vxz28, vxz31), new_ps5(vxz28, vxz31), vxz2700), new_primQuotInt3(Succ(vxz2700), new_reduce2D0(new_ps5(vxz28, vxz31), vxz2700)))
new_ps5(Pos(vxz280), Pos(vxz310)) → new_primPlusInt0(vxz280, vxz310)
new_primPlusNat4(Zero, Succ(vxz31000)) → Succ(vxz31000)
new_primPlusNat4(Succ(vxz28000), Zero) → Succ(vxz28000)
new_primQuotInt2(Neg(vxz530), vxz54, vxz2700) → new_primQuotInt3(vxz530, new_gcd(vxz54, vxz2700))
new_quot14(vxz79, Pos(Succ(vxz8100)), vxz80, vxz30) → new_quot20(vxz79, vxz80, vxz30)
new_quot13(vxz249, Succ(Zero), Succ(vxz2510), vxz254) → new_quot11(vxz249, Succ(vxz2510), Zero)
new_reduce2Reduce10(vxz30, vxz310, vxz30, vxz29, Pos(Zero)) → new_reduce2Reduce11(vxz30, vxz310, vxz30, vxz29)
new_quot17(vxz79, vxz80, Pos(Zero)) → new_quot18(vxz79, vxz80)
new_primQuotInt3(vxz207, Neg(Succ(vxz22600))) → Pos(new_primDivNatS1(vxz207, vxz22600))
new_gcd0Gcd'18(vxz272, vxz273, Zero, Succ(vxz2750)) → new_gcd0Gcd'110(Succ(vxz272), vxz273)
new_primQuotInt3(vxz207, Neg(Zero)) → new_error0
new_gcd0Gcd'18(vxz272, vxz273, Succ(vxz2740), Succ(vxz2750)) → new_gcd0Gcd'18(vxz272, vxz273, vxz2740, vxz2750)
new_primDivNatS01(vxz168, vxz169, Succ(vxz1700), Succ(vxz1710)) → new_primDivNatS01(vxz168, vxz169, vxz1700, vxz1710)
new_primPlusNat(Succ(vxz3000)) → Succ(vxz3000)
new_primPlusNat3(Succ(vxz2800), Succ(vxz3100)) → Succ(Succ(new_primPlusNat4(vxz2800, vxz3100)))
new_primDivNatS01(vxz168, vxz169, Zero, Zero) → new_primDivNatS02(vxz168, vxz169)
new_quot17(vxz79, vxz80, Neg(Zero)) → new_quot18(vxz79, vxz80)
new_reduce2D0(vxz227, vxz2700) → new_gcd(vxz227, vxz2700)
new_primPlusInt2(vxz280, Zero) → new_primMinusNat1(vxz280)
new_primQuotInt4(Neg(vxz790)) → new_error0
new_quot7(vxz29, vxz48, vxz310, vxz47, vxz30) → new_quot14(vxz29, new_primPlusInt4(vxz48, vxz310), new_primPlusInt4(vxz48, vxz310), vxz30)
new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_reduce(vxz28, vxz31, Neg(Zero)) → new_error
new_primQuotInt4(Pos(vxz790)) → new_error0
new_gcd0Gcd'17(Succ(Zero), Succ(vxz2320), vxz236) → new_gcd0Gcd'110(Zero, Succ(vxz2320))
new_quot9(vxz285, vxz286, vxz287, Succ(vxz2880), Succ(vxz2890)) → new_quot9(vxz285, vxz286, vxz287, vxz2880, vxz2890)
new_quot13(vxz249, Zero, vxz251, vxz254) → new_quot8(vxz249, vxz251)
new_quot16(vxz79, vxz3000, Pos(Succ(vxz8000))) → new_quot11(vxz79, vxz8000, vxz3000)
new_quot13(vxz249, Succ(Succ(vxz25500)), Zero, vxz254) → new_quot13(vxz249, new_primMinusNatS0(Succ(vxz25500), Zero), Zero, new_primMinusNatS0(Succ(vxz25500), Zero))
new_primQuotInt(Pos(vxz790), vxz8000) → Pos(new_primDivNatS1(vxz790, vxz8000))
new_primQuotInt6(Pos(vxz790)) → new_error0
new_primPlusInt1(Zero, Zero) → new_primMinusNat1(Zero)
new_gcd(Pos(Succ(vxz5400)), vxz2700) → new_gcd0Gcd'111(vxz2700, vxz5400)
new_primDivNatS01(vxz168, vxz169, Succ(vxz1700), Zero) → new_primDivNatS02(vxz168, vxz169)
new_quot12(vxz79, vxz8000, vxz3000) → new_quot13(vxz79, Succ(vxz8000), vxz3000, Succ(vxz8000))
new_primMulNat0(Zero) → Zero
new_primDivNatS02(vxz168, vxz169) → Succ(new_primDivNatS1(new_primMinusNatS0(vxz168, vxz169), Succ(vxz169)))
new_quot19(vxz79, vxz80, Pos(Succ(vxz3000))) → new_quot17(vxz79, vxz80, Pos(Succ(vxz3000)))
new_gcd0(Pos(Zero), vxz2700) → new_gcd0Gcd'112(vxz2700)
new_gcd(Neg(Zero), vxz2700) → new_gcd0Gcd'16(vxz2700)
new_primDivNatS1(Succ(Succ(vxz13500)), Succ(vxz136000)) → new_primDivNatS01(vxz13500, vxz136000, vxz13500, vxz136000)
new_quot8(vxz79, vxz8000) → Integer(new_primQuotInt(vxz79, vxz8000))
new_primDivNatS1(Zero, vxz13600) → Zero
new_gcd0Gcd'17(Succ(Succ(vxz23700)), Zero, vxz236) → new_gcd0Gcd'19(vxz23700, Zero)
new_primPlusNat0(Succ(vxz300)) → Succ(Succ(new_primPlusNat(vxz300)))
new_primPlusInt0(vxz280, vxz310) → Pos(new_primPlusNat3(vxz280, vxz310))
new_primPlusNat0(Zero) → Succ(Zero)
new_gcd0Gcd'110(vxz266, vxz267) → new_gcd0Gcd'15(vxz267, vxz266)
new_ps5(Pos(vxz280), Neg(vxz310)) → new_primPlusInt1(vxz280, vxz310)
new_errorerror([])
new_reduce2Reduce10(vxz30, vxz310, vxz30, vxz29, Pos(Succ(vxz3100))) → new_reduce2Reduce1(vxz30, vxz310, vxz30, vxz29)
new_gcd(Pos(Zero), vxz2700) → new_gcd0Gcd'112(vxz2700)
new_gcd0Gcd'17(Succ(Succ(vxz23700)), Succ(vxz2320), vxz236) → new_gcd0Gcd'18(vxz23700, Succ(vxz2320), vxz23700, vxz2320)
new_quot5(vxz29, Integer(vxz300), vxz310, vxz30) → new_quot7(vxz29, new_primMulInt(vxz300), vxz310, new_primMulInt(vxz300), vxz30)
new_primMinusNat3(Succ(vxz31000), Zero) → Pos(Succ(vxz31000))
new_primMinusNat1(Zero) → Pos(Zero)
new_quot16(vxz79, vxz3000, Neg(Zero)) → new_quot8(vxz79, vxz3000)
new_primQuotInt(Neg(vxz790), vxz8000) → Neg(new_primDivNatS1(vxz790, vxz8000))
new_primMinusNat3(Zero, Zero) → Pos(Zero)
new_gcd0Gcd'18(vxz272, vxz273, Succ(vxz2740), Zero) → new_gcd0Gcd'19(vxz272, vxz273)
new_primMinusNat1(Succ(vxz3000)) → Neg(Succ(vxz3000))
new_primMinusNatS2Zero
new_gcd0(Pos(Succ(vxz5200)), vxz2700) → new_gcd0Gcd'111(vxz2700, vxz5200)
new_primPlusNat3(Zero, Zero) → new_primPlusNat(Zero)
new_quot9(vxz285, vxz286, vxz287, Zero, Succ(vxz2890)) → new_quot11(vxz285, vxz287, Succ(vxz286))
new_quot10(vxz285, vxz286, vxz287) → new_quot13(vxz285, new_primMinusNatS0(Succ(vxz286), vxz287), vxz287, new_primMinusNatS0(Succ(vxz286), vxz287))
new_gcd0Gcd'17(Zero, vxz232, vxz236) → Pos(Succ(vxz232))
new_quot20(vxz79, vxz80, vxz30) → new_quot17(vxz79, vxz80, vxz30)
new_primDivNatS1(Succ(Zero), Succ(vxz136000)) → Zero
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primQuotInt3(vxz207, Pos(Succ(vxz22600))) → Neg(new_primDivNatS1(vxz207, vxz22600))
new_primMinusNat2(vxz3100, Succ(vxz2800)) → new_primMinusNat3(vxz3100, vxz2800)


s = new_numericEnumFrom3(vxz3, ba) evaluates to t =new_numericEnumFrom3(new_ps4(vxz3, ba), ba)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:




Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from new_numericEnumFrom3(vxz3, ba) to new_numericEnumFrom3(new_ps4(vxz3, ba), ba).




Haskell To QDPs