YES 8.521 H-Termination proof of /home/matraf/haskell/eval_FullyBlown_Fast/empty.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:



↳ HASKELL
  ↳ IFR

mainModule Main
  (((/) :: Ratio Int  ->  Ratio Int  ->  Ratio Int) :: Ratio Int  ->  Ratio Int  ->  Ratio Int)

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
  (((/) :: Ratio Int  ->  Ratio Int  ->  Ratio Int) :: Ratio Int  ->  Ratio Int  ->  Ratio Int)

module Main where
  import qualified Prelude


Replaced joker patterns by fresh variables and removed binding patterns.

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

mainModule Main
  (((/) :: Ratio Int  ->  Ratio Int  ->  Ratio Int) :: Ratio Int  ->  Ratio Int  ->  Ratio Int)

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
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
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
signumReal x
 | x == 0
 = 0
 | x > 0
 = 1
 | otherwise
 = -1

is transformed to
signumReal x = signumReal3 x

signumReal2 x True = 0
signumReal2 x False = signumReal1 x (x > 0)

signumReal1 x True = 1
signumReal1 x False = signumReal0 x otherwise

signumReal0 x True = -1

signumReal3 x = signumReal2 x (x == 0)



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

mainModule Main
  (((/) :: Ratio Int  ->  Ratio Int  ->  Ratio Int) :: Ratio Int  ->  Ratio Int  ->  Ratio Int)

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
reduce2Reduce1 vux vuy x y True = error []
reduce2Reduce1 vux vuy x y False = reduce2Reduce0 vux vuy x y otherwise

reduce2Reduce0 vux vuy x y True = x `quot` reduce2D vux vuy :% (y `quot` reduce2D vux vuy)

reduce2D vux vuy = gcd vux vuy

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

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

gcd0Gcd' x xz = gcd0Gcd'2 x xz
gcd0Gcd' x y = gcd0Gcd'0 x y

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



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

mainModule Main
  (((/) :: Ratio Int  ->  Ratio Int  ->  Ratio Int) :: Ratio Int  ->  Ratio Int  ->  Ratio Int)

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

mainModule Main
  ((/) :: Ratio Int  ->  Ratio Int  ->  Ratio Int)

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

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

new_primPlusNat(Succ(vuz4600), Succ(vuz40000)) → new_primPlusNat(vuz4600, vuz40000)

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

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

new_primMulNat(Succ(vuz3100), Succ(vuz4000)) → new_primMulNat(vuz3100, Succ(vuz4000))

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
                            ↳ QDPSizeChangeProof
                          ↳ QDP
                          ↳ QDP

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

new_primMinusNatS(Succ(vuz1180), Succ(vuz1190)) → new_primMinusNatS(vuz1180, vuz1190)

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

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

new_gcd0Gcd'1(Succ(Zero), Succ(vuz1790), vuz183) → new_gcd0Gcd'12(Zero, Succ(vuz1790))
new_gcd0Gcd'10(vuz201, vuz202, Succ(vuz2030), Zero) → new_gcd0Gcd'11(vuz201, vuz202)
new_gcd0Gcd'10(vuz201, vuz202, Zero, Succ(vuz2040)) → new_gcd0Gcd'12(Succ(vuz201), vuz202)
new_gcd0Gcd'10(vuz201, vuz202, Zero, Zero) → new_gcd0Gcd'11(vuz201, vuz202)
new_gcd0Gcd'1(Succ(Succ(vuz18400)), Succ(vuz1790), vuz183) → new_gcd0Gcd'10(vuz18400, Succ(vuz1790), vuz18400, vuz1790)
new_gcd0Gcd'13(vuz900, vuz320) → new_gcd0Gcd'1(Succ(vuz900), vuz320, Succ(vuz900))
new_gcd0Gcd'1(Succ(Zero), Zero, vuz183) → new_gcd0Gcd'1(new_primMinusNatS0(Zero, Zero), Zero, new_primMinusNatS0(Zero, Zero))
new_gcd0Gcd'11(vuz192, vuz193) → new_gcd0Gcd'1(new_primMinusNatS0(Succ(vuz192), vuz193), vuz193, new_primMinusNatS0(Succ(vuz192), vuz193))
new_gcd0Gcd'12(vuz195, vuz196) → new_gcd0Gcd'13(vuz196, vuz195)
new_gcd0Gcd'1(Succ(Succ(vuz18400)), Zero, vuz183) → new_gcd0Gcd'11(vuz18400, Zero)
new_gcd0Gcd'10(vuz201, vuz202, Succ(vuz2030), Succ(vuz2040)) → new_gcd0Gcd'10(vuz201, vuz202, vuz2030, vuz2040)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vuz1190)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz1180), Succ(vuz1190)) → new_primMinusNatS0(vuz1180, vuz1190)
new_primMinusNatS0(Succ(vuz1180), Zero) → Succ(vuz1180)

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Succ(x0), Zero)
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
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ QDPOrderProof
                          ↳ QDP

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

new_gcd0Gcd'1(Succ(Zero), Succ(vuz1790), vuz183) → new_gcd0Gcd'12(Zero, Succ(vuz1790))
new_gcd0Gcd'10(vuz201, vuz202, Succ(vuz2030), Zero) → new_gcd0Gcd'11(vuz201, vuz202)
new_gcd0Gcd'10(vuz201, vuz202, Zero, Succ(vuz2040)) → new_gcd0Gcd'12(Succ(vuz201), vuz202)
new_gcd0Gcd'10(vuz201, vuz202, Zero, Zero) → new_gcd0Gcd'11(vuz201, vuz202)
new_gcd0Gcd'1(Succ(Succ(vuz18400)), Succ(vuz1790), vuz183) → new_gcd0Gcd'10(vuz18400, Succ(vuz1790), vuz18400, vuz1790)
new_gcd0Gcd'13(vuz900, vuz320) → new_gcd0Gcd'1(Succ(vuz900), vuz320, Succ(vuz900))
new_gcd0Gcd'11(vuz192, vuz193) → new_gcd0Gcd'1(new_primMinusNatS0(Succ(vuz192), vuz193), vuz193, new_primMinusNatS0(Succ(vuz192), vuz193))
new_gcd0Gcd'12(vuz195, vuz196) → new_gcd0Gcd'13(vuz196, vuz195)
new_gcd0Gcd'10(vuz201, vuz202, Succ(vuz2030), Succ(vuz2040)) → new_gcd0Gcd'10(vuz201, vuz202, vuz2030, vuz2040)
new_gcd0Gcd'1(Succ(Succ(vuz18400)), Zero, vuz183) → new_gcd0Gcd'11(vuz18400, Zero)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vuz1190)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz1180), Succ(vuz1190)) → new_primMinusNatS0(vuz1180, vuz1190)
new_primMinusNatS0(Succ(vuz1180), Zero) → Succ(vuz1180)

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Succ(x0), Zero)
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(vuz1790), vuz183) → new_gcd0Gcd'12(Zero, Succ(vuz1790))
The remaining pairs can at least be oriented weakly.

new_gcd0Gcd'10(vuz201, vuz202, Succ(vuz2030), Zero) → new_gcd0Gcd'11(vuz201, vuz202)
new_gcd0Gcd'10(vuz201, vuz202, Zero, Succ(vuz2040)) → new_gcd0Gcd'12(Succ(vuz201), vuz202)
new_gcd0Gcd'10(vuz201, vuz202, Zero, Zero) → new_gcd0Gcd'11(vuz201, vuz202)
new_gcd0Gcd'1(Succ(Succ(vuz18400)), Succ(vuz1790), vuz183) → new_gcd0Gcd'10(vuz18400, Succ(vuz1790), vuz18400, vuz1790)
new_gcd0Gcd'13(vuz900, vuz320) → new_gcd0Gcd'1(Succ(vuz900), vuz320, Succ(vuz900))
new_gcd0Gcd'11(vuz192, vuz193) → new_gcd0Gcd'1(new_primMinusNatS0(Succ(vuz192), vuz193), vuz193, new_primMinusNatS0(Succ(vuz192), vuz193))
new_gcd0Gcd'12(vuz195, vuz196) → new_gcd0Gcd'13(vuz196, vuz195)
new_gcd0Gcd'10(vuz201, vuz202, Succ(vuz2030), Succ(vuz2040)) → new_gcd0Gcd'10(vuz201, vuz202, vuz2030, vuz2040)
new_gcd0Gcd'1(Succ(Succ(vuz18400)), Zero, vuz183) → new_gcd0Gcd'11(vuz18400, Zero)
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( new_primMinusNatS0(x1, x2) ) =
/1\
\0/
+
/00\
\01/
·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(vuz1180), Zero) → Succ(vuz1180)
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz1180), Succ(vuz1190)) → new_primMinusNatS0(vuz1180, vuz1190)
new_primMinusNatS0(Zero, Succ(vuz1190)) → Zero



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

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

new_gcd0Gcd'10(vuz201, vuz202, Succ(vuz2030), Zero) → new_gcd0Gcd'11(vuz201, vuz202)
new_gcd0Gcd'10(vuz201, vuz202, Zero, Succ(vuz2040)) → new_gcd0Gcd'12(Succ(vuz201), vuz202)
new_gcd0Gcd'10(vuz201, vuz202, Zero, Zero) → new_gcd0Gcd'11(vuz201, vuz202)
new_gcd0Gcd'13(vuz900, vuz320) → new_gcd0Gcd'1(Succ(vuz900), vuz320, Succ(vuz900))
new_gcd0Gcd'1(Succ(Succ(vuz18400)), Succ(vuz1790), vuz183) → new_gcd0Gcd'10(vuz18400, Succ(vuz1790), vuz18400, vuz1790)
new_gcd0Gcd'12(vuz195, vuz196) → new_gcd0Gcd'13(vuz196, vuz195)
new_gcd0Gcd'11(vuz192, vuz193) → new_gcd0Gcd'1(new_primMinusNatS0(Succ(vuz192), vuz193), vuz193, new_primMinusNatS0(Succ(vuz192), vuz193))
new_gcd0Gcd'1(Succ(Succ(vuz18400)), Zero, vuz183) → new_gcd0Gcd'11(vuz18400, Zero)
new_gcd0Gcd'10(vuz201, vuz202, Succ(vuz2030), Succ(vuz2040)) → new_gcd0Gcd'10(vuz201, vuz202, vuz2030, vuz2040)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vuz1190)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz1180), Succ(vuz1190)) → new_primMinusNatS0(vuz1180, vuz1190)
new_primMinusNatS0(Succ(vuz1180), Zero) → Succ(vuz1180)

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Succ(x0), Zero)
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(vuz18400)), Zero, vuz183) → new_gcd0Gcd'11(vuz18400, Zero)
The remaining pairs can at least be oriented weakly.

new_gcd0Gcd'10(vuz201, vuz202, Succ(vuz2030), Zero) → new_gcd0Gcd'11(vuz201, vuz202)
new_gcd0Gcd'10(vuz201, vuz202, Zero, Succ(vuz2040)) → new_gcd0Gcd'12(Succ(vuz201), vuz202)
new_gcd0Gcd'10(vuz201, vuz202, Zero, Zero) → new_gcd0Gcd'11(vuz201, vuz202)
new_gcd0Gcd'13(vuz900, vuz320) → new_gcd0Gcd'1(Succ(vuz900), vuz320, Succ(vuz900))
new_gcd0Gcd'1(Succ(Succ(vuz18400)), Succ(vuz1790), vuz183) → new_gcd0Gcd'10(vuz18400, Succ(vuz1790), vuz18400, vuz1790)
new_gcd0Gcd'12(vuz195, vuz196) → new_gcd0Gcd'13(vuz196, vuz195)
new_gcd0Gcd'11(vuz192, vuz193) → new_gcd0Gcd'1(new_primMinusNatS0(Succ(vuz192), vuz193), vuz193, new_primMinusNatS0(Succ(vuz192), vuz193))
new_gcd0Gcd'10(vuz201, vuz202, Succ(vuz2030), Succ(vuz2040)) → new_gcd0Gcd'10(vuz201, vuz202, vuz2030, vuz2040)
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( new_primMinusNatS0(x1, x2) ) =
/0\
\0/
+
/10\
\00/
·x1+
/00\
\10/
·x2

M( Succ(x1) ) =
/0\
\1/
+
/11\
\00/
·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+
[1,0]
·x2

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

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

M( new_gcd0Gcd'13(x1, x2) ) = 0+
[1,1]
·x1+
[1,0]
·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(vuz1180), Zero) → Succ(vuz1180)
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz1180), Succ(vuz1190)) → new_primMinusNatS0(vuz1180, vuz1190)
new_primMinusNatS0(Zero, Succ(vuz1190)) → Zero



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

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

new_gcd0Gcd'10(vuz201, vuz202, Succ(vuz2030), Zero) → new_gcd0Gcd'11(vuz201, vuz202)
new_gcd0Gcd'10(vuz201, vuz202, Zero, Succ(vuz2040)) → new_gcd0Gcd'12(Succ(vuz201), vuz202)
new_gcd0Gcd'10(vuz201, vuz202, Zero, Zero) → new_gcd0Gcd'11(vuz201, vuz202)
new_gcd0Gcd'1(Succ(Succ(vuz18400)), Succ(vuz1790), vuz183) → new_gcd0Gcd'10(vuz18400, Succ(vuz1790), vuz18400, vuz1790)
new_gcd0Gcd'13(vuz900, vuz320) → new_gcd0Gcd'1(Succ(vuz900), vuz320, Succ(vuz900))
new_gcd0Gcd'11(vuz192, vuz193) → new_gcd0Gcd'1(new_primMinusNatS0(Succ(vuz192), vuz193), vuz193, new_primMinusNatS0(Succ(vuz192), vuz193))
new_gcd0Gcd'12(vuz195, vuz196) → new_gcd0Gcd'13(vuz196, vuz195)
new_gcd0Gcd'10(vuz201, vuz202, Succ(vuz2030), Succ(vuz2040)) → new_gcd0Gcd'10(vuz201, vuz202, vuz2030, vuz2040)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vuz1190)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz1180), Succ(vuz1190)) → new_primMinusNatS0(vuz1180, vuz1190)
new_primMinusNatS0(Succ(vuz1180), Zero) → Succ(vuz1180)

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Succ(x0), Zero)
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(vuz201, vuz202, Succ(vuz2030), Zero) → new_gcd0Gcd'11(vuz201, vuz202)
new_gcd0Gcd'10(vuz201, vuz202, Zero, Zero) → new_gcd0Gcd'11(vuz201, vuz202)
The remaining pairs can at least be oriented weakly.

new_gcd0Gcd'10(vuz201, vuz202, Zero, Succ(vuz2040)) → new_gcd0Gcd'12(Succ(vuz201), vuz202)
new_gcd0Gcd'1(Succ(Succ(vuz18400)), Succ(vuz1790), vuz183) → new_gcd0Gcd'10(vuz18400, Succ(vuz1790), vuz18400, vuz1790)
new_gcd0Gcd'13(vuz900, vuz320) → new_gcd0Gcd'1(Succ(vuz900), vuz320, Succ(vuz900))
new_gcd0Gcd'11(vuz192, vuz193) → new_gcd0Gcd'1(new_primMinusNatS0(Succ(vuz192), vuz193), vuz193, new_primMinusNatS0(Succ(vuz192), vuz193))
new_gcd0Gcd'12(vuz195, vuz196) → new_gcd0Gcd'13(vuz196, vuz195)
new_gcd0Gcd'10(vuz201, vuz202, Succ(vuz2030), Succ(vuz2040)) → new_gcd0Gcd'10(vuz201, vuz202, vuz2030, vuz2040)
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(vuz1180), Zero) → Succ(vuz1180)
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz1180), Succ(vuz1190)) → new_primMinusNatS0(vuz1180, vuz1190)
new_primMinusNatS0(Zero, Succ(vuz1190)) → Zero



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

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

new_gcd0Gcd'10(vuz201, vuz202, Zero, Succ(vuz2040)) → new_gcd0Gcd'12(Succ(vuz201), vuz202)
new_gcd0Gcd'13(vuz900, vuz320) → new_gcd0Gcd'1(Succ(vuz900), vuz320, Succ(vuz900))
new_gcd0Gcd'1(Succ(Succ(vuz18400)), Succ(vuz1790), vuz183) → new_gcd0Gcd'10(vuz18400, Succ(vuz1790), vuz18400, vuz1790)
new_gcd0Gcd'12(vuz195, vuz196) → new_gcd0Gcd'13(vuz196, vuz195)
new_gcd0Gcd'11(vuz192, vuz193) → new_gcd0Gcd'1(new_primMinusNatS0(Succ(vuz192), vuz193), vuz193, new_primMinusNatS0(Succ(vuz192), vuz193))
new_gcd0Gcd'10(vuz201, vuz202, Succ(vuz2030), Succ(vuz2040)) → new_gcd0Gcd'10(vuz201, vuz202, vuz2030, vuz2040)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vuz1190)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz1180), Succ(vuz1190)) → new_primMinusNatS0(vuz1180, vuz1190)
new_primMinusNatS0(Succ(vuz1180), Zero) → Succ(vuz1180)

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Succ(x0), Zero)
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
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ UsableRulesProof
                          ↳ QDP

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

new_gcd0Gcd'10(vuz201, vuz202, Zero, Succ(vuz2040)) → new_gcd0Gcd'12(Succ(vuz201), vuz202)
new_gcd0Gcd'1(Succ(Succ(vuz18400)), Succ(vuz1790), vuz183) → new_gcd0Gcd'10(vuz18400, Succ(vuz1790), vuz18400, vuz1790)
new_gcd0Gcd'13(vuz900, vuz320) → new_gcd0Gcd'1(Succ(vuz900), vuz320, Succ(vuz900))
new_gcd0Gcd'12(vuz195, vuz196) → new_gcd0Gcd'13(vuz196, vuz195)
new_gcd0Gcd'10(vuz201, vuz202, Succ(vuz2030), Succ(vuz2040)) → new_gcd0Gcd'10(vuz201, vuz202, vuz2030, vuz2040)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vuz1190)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz1180), Succ(vuz1190)) → new_primMinusNatS0(vuz1180, vuz1190)
new_primMinusNatS0(Succ(vuz1180), Zero) → Succ(vuz1180)

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Succ(x0), Zero)
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
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ UsableRulesProof
                                                  ↳ QDP
                                                    ↳ QReductionProof
                          ↳ QDP

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

new_gcd0Gcd'10(vuz201, vuz202, Zero, Succ(vuz2040)) → new_gcd0Gcd'12(Succ(vuz201), vuz202)
new_gcd0Gcd'1(Succ(Succ(vuz18400)), Succ(vuz1790), vuz183) → new_gcd0Gcd'10(vuz18400, Succ(vuz1790), vuz18400, vuz1790)
new_gcd0Gcd'13(vuz900, vuz320) → new_gcd0Gcd'1(Succ(vuz900), vuz320, Succ(vuz900))
new_gcd0Gcd'12(vuz195, vuz196) → new_gcd0Gcd'13(vuz196, vuz195)
new_gcd0Gcd'10(vuz201, vuz202, Succ(vuz2030), Succ(vuz2040)) → new_gcd0Gcd'10(vuz201, vuz202, vuz2030, vuz2040)

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

new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Succ(x0), Zero)
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), Succ(x1))
new_primMinusNatS0(Succ(x0), Zero)
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
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ UsableRulesProof
                                                  ↳ QDP
                                                    ↳ QReductionProof
                                                      ↳ QDP
                                                        ↳ Instantiation
                          ↳ QDP

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

new_gcd0Gcd'10(vuz201, vuz202, Zero, Succ(vuz2040)) → new_gcd0Gcd'12(Succ(vuz201), vuz202)
new_gcd0Gcd'13(vuz900, vuz320) → new_gcd0Gcd'1(Succ(vuz900), vuz320, Succ(vuz900))
new_gcd0Gcd'1(Succ(Succ(vuz18400)), Succ(vuz1790), vuz183) → new_gcd0Gcd'10(vuz18400, Succ(vuz1790), vuz18400, vuz1790)
new_gcd0Gcd'12(vuz195, vuz196) → new_gcd0Gcd'13(vuz196, vuz195)
new_gcd0Gcd'10(vuz201, vuz202, Succ(vuz2030), Succ(vuz2040)) → new_gcd0Gcd'10(vuz201, vuz202, vuz2030, vuz2040)

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(vuz18400)), Succ(vuz1790), vuz183) → new_gcd0Gcd'10(vuz18400, Succ(vuz1790), vuz18400, vuz1790) 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
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ UsableRulesProof
                                                  ↳ QDP
                                                    ↳ QReductionProof
                                                      ↳ QDP
                                                        ↳ Instantiation
                                                          ↳ QDP
                                                            ↳ NonInfProof
                          ↳ QDP

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(vuz201, vuz202, Zero, Succ(vuz2040)) → new_gcd0Gcd'12(Succ(vuz201), vuz202)
new_gcd0Gcd'13(vuz900, vuz320) → new_gcd0Gcd'1(Succ(vuz900), vuz320, Succ(vuz900))
new_gcd0Gcd'12(vuz195, vuz196) → new_gcd0Gcd'13(vuz196, vuz195)
new_gcd0Gcd'10(vuz201, vuz202, Succ(vuz2030), Succ(vuz2040)) → new_gcd0Gcd'10(vuz201, vuz202, vuz2030, vuz2040)

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(vuz201, vuz202, Zero, Succ(vuz2040)) → new_gcd0Gcd'12(Succ(vuz201), vuz202) the following chains were created:




For Pair new_gcd0Gcd'13(vuz900, vuz320) → new_gcd0Gcd'1(Succ(vuz900), vuz320, Succ(vuz900)) the following chains were created:




For Pair new_gcd0Gcd'12(vuz195, vuz196) → new_gcd0Gcd'13(vuz196, vuz195) the following chains were created:




For Pair new_gcd0Gcd'10(vuz201, vuz202, Succ(vuz2030), Succ(vuz2040)) → new_gcd0Gcd'10(vuz201, vuz202, vuz2030, vuz2040) 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(vuz201, vuz202, Zero, Succ(vuz2040)) → new_gcd0Gcd'12(Succ(vuz201), vuz202)
new_gcd0Gcd'13(vuz900, vuz320) → new_gcd0Gcd'1(Succ(vuz900), vuz320, Succ(vuz900))
new_gcd0Gcd'12(vuz195, vuz196) → new_gcd0Gcd'13(vuz196, vuz195)
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
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ UsableRulesProof
                                                  ↳ QDP
                                                    ↳ QReductionProof
                                                      ↳ QDP
                                                        ↳ Instantiation
                                                          ↳ QDP
                                                            ↳ NonInfProof
                                                              ↳ QDP
                                                                ↳ DependencyGraphProof
                          ↳ QDP

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

new_gcd0Gcd'10(vuz201, vuz202, Zero, Succ(vuz2040)) → new_gcd0Gcd'12(Succ(vuz201), vuz202)
new_gcd0Gcd'13(vuz900, vuz320) → new_gcd0Gcd'1(Succ(vuz900), vuz320, Succ(vuz900))
new_gcd0Gcd'12(vuz195, vuz196) → new_gcd0Gcd'13(vuz196, vuz195)
new_gcd0Gcd'10(vuz201, vuz202, Succ(vuz2030), Succ(vuz2040)) → new_gcd0Gcd'10(vuz201, vuz202, vuz2030, vuz2040)

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
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ UsableRulesProof
                                                  ↳ QDP
                                                    ↳ QReductionProof
                                                      ↳ QDP
                                                        ↳ Instantiation
                                                          ↳ QDP
                                                            ↳ NonInfProof
                                                              ↳ QDP
                                                                ↳ DependencyGraphProof
                                                                  ↳ QDP
                                                                    ↳ QDPSizeChangeProof
                          ↳ QDP

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

new_gcd0Gcd'10(vuz201, vuz202, Succ(vuz2030), Succ(vuz2040)) → new_gcd0Gcd'10(vuz201, vuz202, vuz2030, vuz2040)

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

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

new_primDivNatS0(vuz118, vuz119, Zero, Zero) → new_primDivNatS00(vuz118, vuz119)
new_primDivNatS0(vuz118, vuz119, Succ(vuz1200), Succ(vuz1210)) → new_primDivNatS0(vuz118, vuz119, vuz1200, vuz1210)
new_primDivNatS(Succ(Succ(vuz8800)), Succ(vuz89000)) → new_primDivNatS0(vuz8800, vuz89000, vuz8800, vuz89000)
new_primDivNatS0(vuz118, vuz119, Succ(vuz1200), Zero) → new_primDivNatS(new_primMinusNatS1(vuz118, vuz119), Succ(vuz119))
new_primDivNatS(Succ(Zero), Zero) → new_primDivNatS(new_primMinusNatS3, Zero)
new_primDivNatS00(vuz118, vuz119) → new_primDivNatS(new_primMinusNatS1(vuz118, vuz119), Succ(vuz119))
new_primDivNatS(Succ(Succ(vuz8800)), Zero) → new_primDivNatS(new_primMinusNatS2(vuz8800), Zero)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vuz1190)) → Zero
new_primMinusNatS3 → Zero
new_primMinusNatS1(vuz118, vuz119) → new_primMinusNatS0(vuz118, vuz119)
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz1180), Succ(vuz1190)) → new_primMinusNatS0(vuz1180, vuz1190)
new_primMinusNatS0(Succ(vuz1180), Zero) → Succ(vuz1180)
new_primMinusNatS2(vuz8800) → Succ(vuz8800)

The set Q consists of the following terms:

new_primMinusNatS1(x0, x1)
new_primMinusNatS2(x0)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS3
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
                            ↳ DependencyGraphProof
                              ↳ AND
                                ↳ QDP
                                  ↳ UsableRulesProof
                                ↳ QDP

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

new_primDivNatS(Succ(Succ(vuz8800)), Zero) → new_primDivNatS(new_primMinusNatS2(vuz8800), Zero)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vuz1190)) → Zero
new_primMinusNatS3 → Zero
new_primMinusNatS1(vuz118, vuz119) → new_primMinusNatS0(vuz118, vuz119)
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz1180), Succ(vuz1190)) → new_primMinusNatS0(vuz1180, vuz1190)
new_primMinusNatS0(Succ(vuz1180), Zero) → Succ(vuz1180)
new_primMinusNatS2(vuz8800) → Succ(vuz8800)

The set Q consists of the following terms:

new_primMinusNatS1(x0, x1)
new_primMinusNatS2(x0)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS3
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
                            ↳ DependencyGraphProof
                              ↳ AND
                                ↳ QDP
                                  ↳ UsableRulesProof
                                    ↳ QDP
                                      ↳ QReductionProof
                                ↳ QDP

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

new_primDivNatS(Succ(Succ(vuz8800)), Zero) → new_primDivNatS(new_primMinusNatS2(vuz8800), Zero)

The TRS R consists of the following rules:

new_primMinusNatS2(vuz8800) → Succ(vuz8800)

The set Q consists of the following terms:

new_primMinusNatS1(x0, x1)
new_primMinusNatS2(x0)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS3
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_primMinusNatS1(x0, x1)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS3
new_primMinusNatS0(Zero, Succ(x0))



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

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

new_primDivNatS(Succ(Succ(vuz8800)), Zero) → new_primDivNatS(new_primMinusNatS2(vuz8800), Zero)

The TRS R consists of the following rules:

new_primMinusNatS2(vuz8800) → Succ(vuz8800)

The set Q consists of the following terms:

new_primMinusNatS2(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(vuz8800)), Zero) → new_primDivNatS(new_primMinusNatS2(vuz8800), Zero)

Strictly oriented rules of the TRS R:

new_primMinusNatS2(vuz8800) → Succ(vuz8800)

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_primMinusNatS2(x1)) = 2 + 2·x1   



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

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

new_primMinusNatS2(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
                            ↳ DependencyGraphProof
                              ↳ AND
                                ↳ QDP
                                ↳ QDP
                                  ↳ UsableRulesProof

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

new_primDivNatS0(vuz118, vuz119, Zero, Zero) → new_primDivNatS00(vuz118, vuz119)
new_primDivNatS0(vuz118, vuz119, Succ(vuz1200), Succ(vuz1210)) → new_primDivNatS0(vuz118, vuz119, vuz1200, vuz1210)
new_primDivNatS(Succ(Succ(vuz8800)), Succ(vuz89000)) → new_primDivNatS0(vuz8800, vuz89000, vuz8800, vuz89000)
new_primDivNatS0(vuz118, vuz119, Succ(vuz1200), Zero) → new_primDivNatS(new_primMinusNatS1(vuz118, vuz119), Succ(vuz119))
new_primDivNatS00(vuz118, vuz119) → new_primDivNatS(new_primMinusNatS1(vuz118, vuz119), Succ(vuz119))

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vuz1190)) → Zero
new_primMinusNatS3 → Zero
new_primMinusNatS1(vuz118, vuz119) → new_primMinusNatS0(vuz118, vuz119)
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz1180), Succ(vuz1190)) → new_primMinusNatS0(vuz1180, vuz1190)
new_primMinusNatS0(Succ(vuz1180), Zero) → Succ(vuz1180)
new_primMinusNatS2(vuz8800) → Succ(vuz8800)

The set Q consists of the following terms:

new_primMinusNatS1(x0, x1)
new_primMinusNatS2(x0)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS3
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
                            ↳ DependencyGraphProof
                              ↳ AND
                                ↳ QDP
                                ↳ QDP
                                  ↳ UsableRulesProof
                                    ↳ QDP
                                      ↳ QReductionProof

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

new_primDivNatS0(vuz118, vuz119, Zero, Zero) → new_primDivNatS00(vuz118, vuz119)
new_primDivNatS0(vuz118, vuz119, Succ(vuz1200), Succ(vuz1210)) → new_primDivNatS0(vuz118, vuz119, vuz1200, vuz1210)
new_primDivNatS(Succ(Succ(vuz8800)), Succ(vuz89000)) → new_primDivNatS0(vuz8800, vuz89000, vuz8800, vuz89000)
new_primDivNatS0(vuz118, vuz119, Succ(vuz1200), Zero) → new_primDivNatS(new_primMinusNatS1(vuz118, vuz119), Succ(vuz119))
new_primDivNatS00(vuz118, vuz119) → new_primDivNatS(new_primMinusNatS1(vuz118, vuz119), Succ(vuz119))

The TRS R consists of the following rules:

new_primMinusNatS1(vuz118, vuz119) → new_primMinusNatS0(vuz118, vuz119)
new_primMinusNatS0(Zero, Succ(vuz1190)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz1180), Succ(vuz1190)) → new_primMinusNatS0(vuz1180, vuz1190)
new_primMinusNatS0(Succ(vuz1180), Zero) → Succ(vuz1180)

The set Q consists of the following terms:

new_primMinusNatS1(x0, x1)
new_primMinusNatS2(x0)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS3
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(x0)
new_primMinusNatS3



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

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

new_primDivNatS0(vuz118, vuz119, Zero, Zero) → new_primDivNatS00(vuz118, vuz119)
new_primDivNatS(Succ(Succ(vuz8800)), Succ(vuz89000)) → new_primDivNatS0(vuz8800, vuz89000, vuz8800, vuz89000)
new_primDivNatS0(vuz118, vuz119, Succ(vuz1200), Succ(vuz1210)) → new_primDivNatS0(vuz118, vuz119, vuz1200, vuz1210)
new_primDivNatS0(vuz118, vuz119, Succ(vuz1200), Zero) → new_primDivNatS(new_primMinusNatS1(vuz118, vuz119), Succ(vuz119))
new_primDivNatS00(vuz118, vuz119) → new_primDivNatS(new_primMinusNatS1(vuz118, vuz119), Succ(vuz119))

The TRS R consists of the following rules:

new_primMinusNatS1(vuz118, vuz119) → new_primMinusNatS0(vuz118, vuz119)
new_primMinusNatS0(Zero, Succ(vuz1190)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz1180), Succ(vuz1190)) → new_primMinusNatS0(vuz1180, vuz1190)
new_primMinusNatS0(Succ(vuz1180), Zero) → Succ(vuz1180)

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_primDivNatS0(vuz118, vuz119, Succ(vuz1200), Zero) → new_primDivNatS(new_primMinusNatS1(vuz118, vuz119), Succ(vuz119)) at position [0] we obtained the following new rules:

new_primDivNatS0(vuz118, vuz119, Succ(vuz1200), Zero) → new_primDivNatS(new_primMinusNatS0(vuz118, vuz119), Succ(vuz119))



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

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

new_primDivNatS0(vuz118, vuz119, Zero, Zero) → new_primDivNatS00(vuz118, vuz119)
new_primDivNatS0(vuz118, vuz119, Succ(vuz1200), Succ(vuz1210)) → new_primDivNatS0(vuz118, vuz119, vuz1200, vuz1210)
new_primDivNatS(Succ(Succ(vuz8800)), Succ(vuz89000)) → new_primDivNatS0(vuz8800, vuz89000, vuz8800, vuz89000)
new_primDivNatS0(vuz118, vuz119, Succ(vuz1200), Zero) → new_primDivNatS(new_primMinusNatS0(vuz118, vuz119), Succ(vuz119))
new_primDivNatS00(vuz118, vuz119) → new_primDivNatS(new_primMinusNatS1(vuz118, vuz119), Succ(vuz119))

The TRS R consists of the following rules:

new_primMinusNatS1(vuz118, vuz119) → new_primMinusNatS0(vuz118, vuz119)
new_primMinusNatS0(Zero, Succ(vuz1190)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz1180), Succ(vuz1190)) → new_primMinusNatS0(vuz1180, vuz1190)
new_primMinusNatS0(Succ(vuz1180), Zero) → Succ(vuz1180)

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_primDivNatS00(vuz118, vuz119) → new_primDivNatS(new_primMinusNatS1(vuz118, vuz119), Succ(vuz119)) at position [0] we obtained the following new rules:

new_primDivNatS00(vuz118, vuz119) → new_primDivNatS(new_primMinusNatS0(vuz118, vuz119), Succ(vuz119))



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

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

new_primDivNatS0(vuz118, vuz119, Zero, Zero) → new_primDivNatS00(vuz118, vuz119)
new_primDivNatS00(vuz118, vuz119) → new_primDivNatS(new_primMinusNatS0(vuz118, vuz119), Succ(vuz119))
new_primDivNatS(Succ(Succ(vuz8800)), Succ(vuz89000)) → new_primDivNatS0(vuz8800, vuz89000, vuz8800, vuz89000)
new_primDivNatS0(vuz118, vuz119, Succ(vuz1200), Succ(vuz1210)) → new_primDivNatS0(vuz118, vuz119, vuz1200, vuz1210)
new_primDivNatS0(vuz118, vuz119, Succ(vuz1200), Zero) → new_primDivNatS(new_primMinusNatS0(vuz118, vuz119), Succ(vuz119))

The TRS R consists of the following rules:

new_primMinusNatS1(vuz118, vuz119) → new_primMinusNatS0(vuz118, vuz119)
new_primMinusNatS0(Zero, Succ(vuz1190)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz1180), Succ(vuz1190)) → new_primMinusNatS0(vuz1180, vuz1190)
new_primMinusNatS0(Succ(vuz1180), Zero) → Succ(vuz1180)

The set Q consists of the following terms:

new_primMinusNatS1(x0, x1)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Succ(x0), Zero)
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
                            ↳ DependencyGraphProof
                              ↳ AND
                                ↳ QDP
                                ↳ QDP
                                  ↳ UsableRulesProof
                                    ↳ QDP
                                      ↳ QReductionProof
                                        ↳ QDP
                                          ↳ Rewriting
                                            ↳ QDP
                                              ↳ Rewriting
                                                ↳ QDP
                                                  ↳ UsableRulesProof
                                                    ↳ QDP
                                                      ↳ QReductionProof

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

new_primDivNatS0(vuz118, vuz119, Zero, Zero) → new_primDivNatS00(vuz118, vuz119)
new_primDivNatS00(vuz118, vuz119) → new_primDivNatS(new_primMinusNatS0(vuz118, vuz119), Succ(vuz119))
new_primDivNatS(Succ(Succ(vuz8800)), Succ(vuz89000)) → new_primDivNatS0(vuz8800, vuz89000, vuz8800, vuz89000)
new_primDivNatS0(vuz118, vuz119, Succ(vuz1200), Succ(vuz1210)) → new_primDivNatS0(vuz118, vuz119, vuz1200, vuz1210)
new_primDivNatS0(vuz118, vuz119, Succ(vuz1200), Zero) → new_primDivNatS(new_primMinusNatS0(vuz118, vuz119), Succ(vuz119))

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vuz1190)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz1180), Succ(vuz1190)) → new_primMinusNatS0(vuz1180, vuz1190)
new_primMinusNatS0(Succ(vuz1180), Zero) → Succ(vuz1180)

The set Q consists of the following terms:

new_primMinusNatS1(x0, x1)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Succ(x0), Zero)
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_primMinusNatS1(x0, x1)



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

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

new_primDivNatS0(vuz118, vuz119, Zero, Zero) → new_primDivNatS00(vuz118, vuz119)
new_primDivNatS0(vuz118, vuz119, Succ(vuz1200), Succ(vuz1210)) → new_primDivNatS0(vuz118, vuz119, vuz1200, vuz1210)
new_primDivNatS(Succ(Succ(vuz8800)), Succ(vuz89000)) → new_primDivNatS0(vuz8800, vuz89000, vuz8800, vuz89000)
new_primDivNatS00(vuz118, vuz119) → new_primDivNatS(new_primMinusNatS0(vuz118, vuz119), Succ(vuz119))
new_primDivNatS0(vuz118, vuz119, Succ(vuz1200), Zero) → new_primDivNatS(new_primMinusNatS0(vuz118, vuz119), Succ(vuz119))

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vuz1190)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz1180), Succ(vuz1190)) → new_primMinusNatS0(vuz1180, vuz1190)
new_primMinusNatS0(Succ(vuz1180), Zero) → Succ(vuz1180)

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Succ(x0), Zero)
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_primDivNatS(Succ(Succ(vuz8800)), Succ(vuz89000)) → new_primDivNatS0(vuz8800, vuz89000, vuz8800, vuz89000)
The remaining pairs can at least be oriented weakly.

new_primDivNatS0(vuz118, vuz119, Zero, Zero) → new_primDivNatS00(vuz118, vuz119)
new_primDivNatS0(vuz118, vuz119, Succ(vuz1200), Succ(vuz1210)) → new_primDivNatS0(vuz118, vuz119, vuz1200, vuz1210)
new_primDivNatS00(vuz118, vuz119) → new_primDivNatS(new_primMinusNatS0(vuz118, vuz119), Succ(vuz119))
new_primDivNatS0(vuz118, vuz119, Succ(vuz1200), Zero) → new_primDivNatS(new_primMinusNatS0(vuz118, vuz119), Succ(vuz119))
Used ordering: Polynomial interpretation [25]:

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

The following usable rules [17] were oriented:

new_primMinusNatS0(Succ(vuz1180), Succ(vuz1190)) → new_primMinusNatS0(vuz1180, vuz1190)
new_primMinusNatS0(Zero, Succ(vuz1190)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz1180), Zero) → Succ(vuz1180)



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

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

new_primDivNatS0(vuz118, vuz119, Zero, Zero) → new_primDivNatS00(vuz118, vuz119)
new_primDivNatS00(vuz118, vuz119) → new_primDivNatS(new_primMinusNatS0(vuz118, vuz119), Succ(vuz119))
new_primDivNatS0(vuz118, vuz119, Succ(vuz1200), Succ(vuz1210)) → new_primDivNatS0(vuz118, vuz119, vuz1200, vuz1210)
new_primDivNatS0(vuz118, vuz119, Succ(vuz1200), Zero) → new_primDivNatS(new_primMinusNatS0(vuz118, vuz119), Succ(vuz119))

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vuz1190)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz1180), Succ(vuz1190)) → new_primMinusNatS0(vuz1180, vuz1190)
new_primMinusNatS0(Succ(vuz1180), Zero) → Succ(vuz1180)

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Succ(x0), Zero)
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 3 less nodes.

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

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

new_primDivNatS0(vuz118, vuz119, Succ(vuz1200), Succ(vuz1210)) → new_primDivNatS0(vuz118, vuz119, vuz1200, vuz1210)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vuz1190)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz1180), Succ(vuz1190)) → new_primMinusNatS0(vuz1180, vuz1190)
new_primMinusNatS0(Succ(vuz1180), Zero) → Succ(vuz1180)

The set Q consists of the following terms:

new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Succ(x0), Zero)
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
                            ↳ DependencyGraphProof
                              ↳ AND
                                ↳ QDP
                                ↳ QDP
                                  ↳ UsableRulesProof
                                    ↳ QDP
                                      ↳ QReductionProof
                                        ↳ QDP
                                          ↳ Rewriting
                                            ↳ QDP
                                              ↳ Rewriting
                                                ↳ QDP
                                                  ↳ UsableRulesProof
                                                    ↳ QDP
                                                      ↳ QReductionProof
                                                        ↳ QDP
                                                          ↳ QDPOrderProof
                                                            ↳ QDP
                                                              ↳ DependencyGraphProof
                                                                ↳ QDP
                                                                  ↳ UsableRulesProof
                                                                    ↳ QDP
                                                                      ↳ QReductionProof

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

new_primDivNatS0(vuz118, vuz119, Succ(vuz1200), Succ(vuz1210)) → new_primDivNatS0(vuz118, vuz119, vuz1200, vuz1210)

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

new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Succ(x0), Zero)
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), Succ(x1))
new_primMinusNatS0(Succ(x0), Zero)
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
                            ↳ DependencyGraphProof
                              ↳ AND
                                ↳ QDP
                                ↳ QDP
                                  ↳ UsableRulesProof
                                    ↳ QDP
                                      ↳ QReductionProof
                                        ↳ QDP
                                          ↳ Rewriting
                                            ↳ QDP
                                              ↳ Rewriting
                                                ↳ QDP
                                                  ↳ UsableRulesProof
                                                    ↳ QDP
                                                      ↳ QReductionProof
                                                        ↳ QDP
                                                          ↳ QDPOrderProof
                                                            ↳ QDP
                                                              ↳ DependencyGraphProof
                                                                ↳ QDP
                                                                  ↳ UsableRulesProof
                                                                    ↳ QDP
                                                                      ↳ QReductionProof
                                                                        ↳ QDP
                                                                          ↳ QDPSizeChangeProof

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

new_primDivNatS0(vuz118, vuz119, Succ(vuz1200), Succ(vuz1210)) → new_primDivNatS0(vuz118, vuz119, vuz1200, vuz1210)

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: