YES 8.349 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
  ((fromDouble :: Double  ->  Ratio Int) :: Double  ->  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
  ((fromDouble :: Double  ->  Ratio Int) :: Double  ->  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
  ((fromDouble :: Double  ->  Ratio Int) :: Double  ->  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

signumReal0 x True = -1

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

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

signumReal3 x = signumReal2 x (x == 0)



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

mainModule Main
  ((fromDouble :: Double  ->  Ratio Int) :: Double  ->  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

reduce2D vux vuy = gcd vux vuy

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

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



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

mainModule Main
  ((fromDouble :: Double  ->  Ratio Int) :: Double  ->  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
  (fromDouble :: Double  ->  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_primMulNat(Succ(vuz30000)) → new_primMulNat(vuz30000)

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
                            ↳ DependencyGraphProof
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP

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

new_gcd0Gcd'15(Zero, Succ(vuz1070)) → new_gcd0Gcd'12(vuz1070)
new_gcd0Gcd'14(Succ(vuz1070), Succ(vuz1080)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070)
new_gcd0Gcd'13(Succ(vuz1380), Zero, vuz140) → new_gcd0Gcd'1(vuz1380, vuz140)
new_gcd0Gcd'1(Succ(vuz1080), Succ(vuz1070)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070)
new_gcd0Gcd'1(Succ(vuz1080), Zero) → new_gcd0Gcd'11(vuz1080, Zero)
new_gcd0Gcd'10(vuz113, vuz114, Zero, Succ(vuz1160)) → new_gcd0Gcd'14(Succ(vuz113), vuz114)
new_gcd0Gcd'15(Zero, Zero) → new_gcd0Gcd'13(Zero, Zero, Zero)
new_gcd0Gcd'12(vuz31000) → new_gcd0Gcd'14(Zero, Succ(vuz31000))
new_gcd0Gcd'1(Zero, Zero) → new_gcd0Gcd'13(Zero, Zero, Zero)
new_gcd0Gcd'1(Zero, Succ(vuz1070)) → new_gcd0Gcd'12(vuz1070)
new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Zero) → new_gcd0Gcd'13(Succ(vuz113), vuz114, vuz114)
new_gcd0Gcd'14(Succ(vuz1070), Zero) → new_gcd0Gcd'12(vuz1070)
new_gcd0Gcd'14(Zero, Zero) → new_gcd0Gcd'13(Zero, Zero, Zero)
new_gcd0Gcd'11(vuz113, vuz114) → new_gcd0Gcd'13(Succ(vuz113), vuz114, vuz114)
new_gcd0Gcd'10(vuz113, vuz114, Zero, Zero) → new_gcd0Gcd'11(vuz113, vuz114)
new_gcd0Gcd'13(Succ(vuz1380), Succ(vuz1390), vuz140) → new_gcd0Gcd'13(vuz1380, vuz1390, vuz140)
new_gcd0Gcd'14(Zero, Succ(vuz1080)) → new_gcd0Gcd'11(vuz1080, Zero)
new_gcd0Gcd'15(Succ(vuz1080), Succ(vuz1070)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070)
new_gcd0Gcd'15(Succ(vuz1080), Zero) → new_gcd0Gcd'11(vuz1080, Zero)
new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Succ(vuz1160)) → new_gcd0Gcd'10(vuz113, vuz114, vuz1150, vuz1160)

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 6 less nodes.

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

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

new_gcd0Gcd'14(Succ(vuz1070), Succ(vuz1080)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070)
new_gcd0Gcd'13(Succ(vuz1380), Zero, vuz140) → new_gcd0Gcd'1(vuz1380, vuz140)
new_gcd0Gcd'10(vuz113, vuz114, Zero, Succ(vuz1160)) → new_gcd0Gcd'14(Succ(vuz113), vuz114)
new_gcd0Gcd'1(Succ(vuz1080), Succ(vuz1070)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070)
new_gcd0Gcd'1(Succ(vuz1080), Zero) → new_gcd0Gcd'11(vuz1080, Zero)
new_gcd0Gcd'12(vuz31000) → new_gcd0Gcd'14(Zero, Succ(vuz31000))
new_gcd0Gcd'1(Zero, Succ(vuz1070)) → new_gcd0Gcd'12(vuz1070)
new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Zero) → new_gcd0Gcd'13(Succ(vuz113), vuz114, vuz114)
new_gcd0Gcd'14(Succ(vuz1070), Zero) → new_gcd0Gcd'12(vuz1070)
new_gcd0Gcd'10(vuz113, vuz114, Zero, Zero) → new_gcd0Gcd'11(vuz113, vuz114)
new_gcd0Gcd'11(vuz113, vuz114) → new_gcd0Gcd'13(Succ(vuz113), vuz114, vuz114)
new_gcd0Gcd'13(Succ(vuz1380), Succ(vuz1390), vuz140) → new_gcd0Gcd'13(vuz1380, vuz1390, vuz140)
new_gcd0Gcd'14(Zero, Succ(vuz1080)) → new_gcd0Gcd'11(vuz1080, Zero)
new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Succ(vuz1160)) → new_gcd0Gcd'10(vuz113, vuz114, vuz1150, vuz1160)

R is empty.
Q is empty.
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'14(Succ(vuz1070), Zero) → new_gcd0Gcd'12(vuz1070)
The remaining pairs can at least be oriented weakly.

new_gcd0Gcd'14(Succ(vuz1070), Succ(vuz1080)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070)
new_gcd0Gcd'13(Succ(vuz1380), Zero, vuz140) → new_gcd0Gcd'1(vuz1380, vuz140)
new_gcd0Gcd'10(vuz113, vuz114, Zero, Succ(vuz1160)) → new_gcd0Gcd'14(Succ(vuz113), vuz114)
new_gcd0Gcd'1(Succ(vuz1080), Succ(vuz1070)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070)
new_gcd0Gcd'1(Succ(vuz1080), Zero) → new_gcd0Gcd'11(vuz1080, Zero)
new_gcd0Gcd'12(vuz31000) → new_gcd0Gcd'14(Zero, Succ(vuz31000))
new_gcd0Gcd'1(Zero, Succ(vuz1070)) → new_gcd0Gcd'12(vuz1070)
new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Zero) → new_gcd0Gcd'13(Succ(vuz113), vuz114, vuz114)
new_gcd0Gcd'10(vuz113, vuz114, Zero, Zero) → new_gcd0Gcd'11(vuz113, vuz114)
new_gcd0Gcd'11(vuz113, vuz114) → new_gcd0Gcd'13(Succ(vuz113), vuz114, vuz114)
new_gcd0Gcd'13(Succ(vuz1380), Succ(vuz1390), vuz140) → new_gcd0Gcd'13(vuz1380, vuz1390, vuz140)
new_gcd0Gcd'14(Zero, Succ(vuz1080)) → new_gcd0Gcd'11(vuz1080, Zero)
new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Succ(vuz1160)) → new_gcd0Gcd'10(vuz113, vuz114, vuz1150, vuz1160)
Used ordering: Polynomial interpretation [25]:

POL(Succ(x1)) = 0   
POL(Zero) = 1   
POL(new_gcd0Gcd'1(x1, x2)) = 0   
POL(new_gcd0Gcd'10(x1, x2, x3, x4)) = x2   
POL(new_gcd0Gcd'11(x1, x2)) = 0   
POL(new_gcd0Gcd'12(x1)) = 0   
POL(new_gcd0Gcd'13(x1, x2, x3)) = 0   
POL(new_gcd0Gcd'14(x1, x2)) = x2   

The following usable rules [17] were oriented: none



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

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

new_gcd0Gcd'14(Succ(vuz1070), Succ(vuz1080)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070)
new_gcd0Gcd'13(Succ(vuz1380), Zero, vuz140) → new_gcd0Gcd'1(vuz1380, vuz140)
new_gcd0Gcd'10(vuz113, vuz114, Zero, Succ(vuz1160)) → new_gcd0Gcd'14(Succ(vuz113), vuz114)
new_gcd0Gcd'1(Succ(vuz1080), Succ(vuz1070)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070)
new_gcd0Gcd'1(Succ(vuz1080), Zero) → new_gcd0Gcd'11(vuz1080, Zero)
new_gcd0Gcd'12(vuz31000) → new_gcd0Gcd'14(Zero, Succ(vuz31000))
new_gcd0Gcd'1(Zero, Succ(vuz1070)) → new_gcd0Gcd'12(vuz1070)
new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Zero) → new_gcd0Gcd'13(Succ(vuz113), vuz114, vuz114)
new_gcd0Gcd'10(vuz113, vuz114, Zero, Zero) → new_gcd0Gcd'11(vuz113, vuz114)
new_gcd0Gcd'11(vuz113, vuz114) → new_gcd0Gcd'13(Succ(vuz113), vuz114, vuz114)
new_gcd0Gcd'13(Succ(vuz1380), Succ(vuz1390), vuz140) → new_gcd0Gcd'13(vuz1380, vuz1390, vuz140)
new_gcd0Gcd'14(Zero, Succ(vuz1080)) → new_gcd0Gcd'11(vuz1080, Zero)
new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Succ(vuz1160)) → new_gcd0Gcd'10(vuz113, vuz114, vuz1150, vuz1160)

R is empty.
Q is empty.
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'14(Zero, Succ(vuz1080)) → new_gcd0Gcd'11(vuz1080, Zero)
The remaining pairs can at least be oriented weakly.

new_gcd0Gcd'14(Succ(vuz1070), Succ(vuz1080)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070)
new_gcd0Gcd'13(Succ(vuz1380), Zero, vuz140) → new_gcd0Gcd'1(vuz1380, vuz140)
new_gcd0Gcd'10(vuz113, vuz114, Zero, Succ(vuz1160)) → new_gcd0Gcd'14(Succ(vuz113), vuz114)
new_gcd0Gcd'1(Succ(vuz1080), Succ(vuz1070)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070)
new_gcd0Gcd'1(Succ(vuz1080), Zero) → new_gcd0Gcd'11(vuz1080, Zero)
new_gcd0Gcd'12(vuz31000) → new_gcd0Gcd'14(Zero, Succ(vuz31000))
new_gcd0Gcd'1(Zero, Succ(vuz1070)) → new_gcd0Gcd'12(vuz1070)
new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Zero) → new_gcd0Gcd'13(Succ(vuz113), vuz114, vuz114)
new_gcd0Gcd'10(vuz113, vuz114, Zero, Zero) → new_gcd0Gcd'11(vuz113, vuz114)
new_gcd0Gcd'11(vuz113, vuz114) → new_gcd0Gcd'13(Succ(vuz113), vuz114, vuz114)
new_gcd0Gcd'13(Succ(vuz1380), Succ(vuz1390), vuz140) → new_gcd0Gcd'13(vuz1380, vuz1390, vuz140)
new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Succ(vuz1160)) → new_gcd0Gcd'10(vuz113, vuz114, vuz1150, vuz1160)
Used ordering: Polynomial interpretation [25]:

POL(Succ(x1)) = 1   
POL(Zero) = 0   
POL(new_gcd0Gcd'1(x1, x2)) = x2   
POL(new_gcd0Gcd'10(x1, x2, x3, x4)) = x2   
POL(new_gcd0Gcd'11(x1, x2)) = x2   
POL(new_gcd0Gcd'12(x1)) = 1   
POL(new_gcd0Gcd'13(x1, x2, x3)) = x3   
POL(new_gcd0Gcd'14(x1, x2)) = x2   

The following usable rules [17] were oriented: none



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

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

new_gcd0Gcd'14(Succ(vuz1070), Succ(vuz1080)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070)
new_gcd0Gcd'13(Succ(vuz1380), Zero, vuz140) → new_gcd0Gcd'1(vuz1380, vuz140)
new_gcd0Gcd'1(Succ(vuz1080), Zero) → new_gcd0Gcd'11(vuz1080, Zero)
new_gcd0Gcd'1(Succ(vuz1080), Succ(vuz1070)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070)
new_gcd0Gcd'10(vuz113, vuz114, Zero, Succ(vuz1160)) → new_gcd0Gcd'14(Succ(vuz113), vuz114)
new_gcd0Gcd'12(vuz31000) → new_gcd0Gcd'14(Zero, Succ(vuz31000))
new_gcd0Gcd'11(vuz113, vuz114) → new_gcd0Gcd'13(Succ(vuz113), vuz114, vuz114)
new_gcd0Gcd'10(vuz113, vuz114, Zero, Zero) → new_gcd0Gcd'11(vuz113, vuz114)
new_gcd0Gcd'13(Succ(vuz1380), Succ(vuz1390), vuz140) → new_gcd0Gcd'13(vuz1380, vuz1390, vuz140)
new_gcd0Gcd'1(Zero, Succ(vuz1070)) → new_gcd0Gcd'12(vuz1070)
new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Zero) → new_gcd0Gcd'13(Succ(vuz113), vuz114, vuz114)
new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Succ(vuz1160)) → new_gcd0Gcd'10(vuz113, vuz114, vuz1150, vuz1160)

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
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ DependencyGraphProof
QDP
                                            ↳ QDPOrderProof
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP

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

new_gcd0Gcd'14(Succ(vuz1070), Succ(vuz1080)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070)
new_gcd0Gcd'13(Succ(vuz1380), Zero, vuz140) → new_gcd0Gcd'1(vuz1380, vuz140)
new_gcd0Gcd'1(Succ(vuz1080), Zero) → new_gcd0Gcd'11(vuz1080, Zero)
new_gcd0Gcd'1(Succ(vuz1080), Succ(vuz1070)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070)
new_gcd0Gcd'10(vuz113, vuz114, Zero, Succ(vuz1160)) → new_gcd0Gcd'14(Succ(vuz113), vuz114)
new_gcd0Gcd'11(vuz113, vuz114) → new_gcd0Gcd'13(Succ(vuz113), vuz114, vuz114)
new_gcd0Gcd'10(vuz113, vuz114, Zero, Zero) → new_gcd0Gcd'11(vuz113, vuz114)
new_gcd0Gcd'13(Succ(vuz1380), Succ(vuz1390), vuz140) → new_gcd0Gcd'13(vuz1380, vuz1390, vuz140)
new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Zero) → new_gcd0Gcd'13(Succ(vuz113), vuz114, vuz114)
new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Succ(vuz1160)) → new_gcd0Gcd'10(vuz113, vuz114, vuz1150, vuz1160)

R is empty.
Q is empty.
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'13(Succ(vuz1380), Zero, vuz140) → new_gcd0Gcd'1(vuz1380, vuz140)
new_gcd0Gcd'13(Succ(vuz1380), Succ(vuz1390), vuz140) → new_gcd0Gcd'13(vuz1380, vuz1390, vuz140)
The remaining pairs can at least be oriented weakly.

new_gcd0Gcd'14(Succ(vuz1070), Succ(vuz1080)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070)
new_gcd0Gcd'1(Succ(vuz1080), Zero) → new_gcd0Gcd'11(vuz1080, Zero)
new_gcd0Gcd'1(Succ(vuz1080), Succ(vuz1070)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070)
new_gcd0Gcd'10(vuz113, vuz114, Zero, Succ(vuz1160)) → new_gcd0Gcd'14(Succ(vuz113), vuz114)
new_gcd0Gcd'11(vuz113, vuz114) → new_gcd0Gcd'13(Succ(vuz113), vuz114, vuz114)
new_gcd0Gcd'10(vuz113, vuz114, Zero, Zero) → new_gcd0Gcd'11(vuz113, vuz114)
new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Zero) → new_gcd0Gcd'13(Succ(vuz113), vuz114, vuz114)
new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Succ(vuz1160)) → new_gcd0Gcd'10(vuz113, vuz114, vuz1150, vuz1160)
Used ordering: Polynomial interpretation [25]:

POL(Succ(x1)) = 1 + x1   
POL(Zero) = 0   
POL(new_gcd0Gcd'1(x1, x2)) = x1 + x2   
POL(new_gcd0Gcd'10(x1, x2, x3, x4)) = 1 + x1 + x2   
POL(new_gcd0Gcd'11(x1, x2)) = 1 + x1 + x2   
POL(new_gcd0Gcd'13(x1, x2, x3)) = x1 + x3   
POL(new_gcd0Gcd'14(x1, x2)) = x1 + x2   

The following usable rules [17] were oriented: none



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

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

new_gcd0Gcd'14(Succ(vuz1070), Succ(vuz1080)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070)
new_gcd0Gcd'10(vuz113, vuz114, Zero, Succ(vuz1160)) → new_gcd0Gcd'14(Succ(vuz113), vuz114)
new_gcd0Gcd'1(Succ(vuz1080), Succ(vuz1070)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070)
new_gcd0Gcd'1(Succ(vuz1080), Zero) → new_gcd0Gcd'11(vuz1080, Zero)
new_gcd0Gcd'10(vuz113, vuz114, Zero, Zero) → new_gcd0Gcd'11(vuz113, vuz114)
new_gcd0Gcd'11(vuz113, vuz114) → new_gcd0Gcd'13(Succ(vuz113), vuz114, vuz114)
new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Zero) → new_gcd0Gcd'13(Succ(vuz113), vuz114, vuz114)
new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Succ(vuz1160)) → new_gcd0Gcd'10(vuz113, vuz114, vuz1150, vuz1160)

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 5 less nodes.

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

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

new_gcd0Gcd'14(Succ(vuz1070), Succ(vuz1080)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070)
new_gcd0Gcd'10(vuz113, vuz114, Zero, Succ(vuz1160)) → new_gcd0Gcd'14(Succ(vuz113), vuz114)
new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Succ(vuz1160)) → new_gcd0Gcd'10(vuz113, vuz114, vuz1150, vuz1160)

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(vuz113, vuz114, Zero, Succ(vuz1160)) → new_gcd0Gcd'14(Succ(vuz113), vuz114) the following chains were created:




For Pair new_gcd0Gcd'14(Succ(vuz1070), Succ(vuz1080)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070) the following chains were created:




For Pair new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Succ(vuz1160)) → new_gcd0Gcd'10(vuz113, vuz114, vuz1150, vuz1160) 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'10(x1, x2, x3, x4)) = -1 + x1 - x3 + x4   
POL(new_gcd0Gcd'14(x1, x2)) = -1 + x1   

The following pairs are in P>:

new_gcd0Gcd'14(Succ(vuz1070), Succ(vuz1080)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070)
The following pairs are in Pbound:

new_gcd0Gcd'10(vuz113, vuz114, Zero, Succ(vuz1160)) → new_gcd0Gcd'14(Succ(vuz113), vuz114)
new_gcd0Gcd'14(Succ(vuz1070), Succ(vuz1080)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070)
There are no usable rules

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

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

new_gcd0Gcd'10(vuz113, vuz114, Zero, Succ(vuz1160)) → new_gcd0Gcd'14(Succ(vuz113), vuz114)
new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Succ(vuz1160)) → new_gcd0Gcd'10(vuz113, vuz114, vuz1150, vuz1160)

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 1 less node.

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

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

new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Succ(vuz1160)) → new_gcd0Gcd'10(vuz113, vuz114, vuz1150, vuz1160)

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

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

new_primDivNatS0(vuz133, vuz134, Zero, Zero) → new_primDivNatS00(vuz133, vuz134)
new_primDivNatS0(vuz133, vuz134, Succ(vuz1350), Succ(vuz1360)) → new_primDivNatS0(vuz133, vuz134, vuz1350, vuz1360)
new_primDivNatS(Succ(vuz1420), Zero, vuz144) → new_primDivNatS1(vuz1420, vuz144)
new_primDivNatS(Succ(vuz1420), Succ(vuz1430), vuz144) → new_primDivNatS(vuz1420, vuz1430, vuz144)
new_primDivNatS1(Succ(vuz990), Zero) → new_primDivNatS(Succ(vuz990), Zero, Zero)
new_primDivNatS1(Succ(vuz990), Succ(vuz1000)) → new_primDivNatS0(vuz990, vuz1000, vuz990, vuz1000)
new_primDivNatS00(vuz133, vuz134) → new_primDivNatS(Succ(vuz133), Succ(vuz134), Succ(vuz134))
new_primDivNatS1(Zero, Zero) → new_primDivNatS(Zero, Zero, Zero)
new_primDivNatS0(vuz133, vuz134, Succ(vuz1350), Zero) → new_primDivNatS(Succ(vuz133), Succ(vuz134), Succ(vuz134))

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 1 less node.

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

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

new_primDivNatS0(vuz133, vuz134, Zero, Zero) → new_primDivNatS00(vuz133, vuz134)
new_primDivNatS0(vuz133, vuz134, Succ(vuz1350), Succ(vuz1360)) → new_primDivNatS0(vuz133, vuz134, vuz1350, vuz1360)
new_primDivNatS(Succ(vuz1420), Zero, vuz144) → new_primDivNatS1(vuz1420, vuz144)
new_primDivNatS(Succ(vuz1420), Succ(vuz1430), vuz144) → new_primDivNatS(vuz1420, vuz1430, vuz144)
new_primDivNatS1(Succ(vuz990), Zero) → new_primDivNatS(Succ(vuz990), Zero, Zero)
new_primDivNatS1(Succ(vuz990), Succ(vuz1000)) → new_primDivNatS0(vuz990, vuz1000, vuz990, vuz1000)
new_primDivNatS00(vuz133, vuz134) → new_primDivNatS(Succ(vuz133), Succ(vuz134), Succ(vuz134))
new_primDivNatS0(vuz133, vuz134, Succ(vuz1350), Zero) → new_primDivNatS(Succ(vuz133), Succ(vuz134), Succ(vuz134))

R is empty.
Q is empty.
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(vuz1420), Succ(vuz1430), vuz144) → new_primDivNatS(vuz1420, vuz1430, vuz144)
new_primDivNatS1(Succ(vuz990), Zero) → new_primDivNatS(Succ(vuz990), Zero, Zero)
new_primDivNatS1(Succ(vuz990), Succ(vuz1000)) → new_primDivNatS0(vuz990, vuz1000, vuz990, vuz1000)
The remaining pairs can at least be oriented weakly.

new_primDivNatS0(vuz133, vuz134, Zero, Zero) → new_primDivNatS00(vuz133, vuz134)
new_primDivNatS0(vuz133, vuz134, Succ(vuz1350), Succ(vuz1360)) → new_primDivNatS0(vuz133, vuz134, vuz1350, vuz1360)
new_primDivNatS(Succ(vuz1420), Zero, vuz144) → new_primDivNatS1(vuz1420, vuz144)
new_primDivNatS00(vuz133, vuz134) → new_primDivNatS(Succ(vuz133), Succ(vuz134), Succ(vuz134))
new_primDivNatS0(vuz133, vuz134, Succ(vuz1350), Zero) → new_primDivNatS(Succ(vuz133), Succ(vuz134), Succ(vuz134))
Used ordering: Polynomial interpretation [25]:

POL(Succ(x1)) = 1 + x1   
POL(Zero) = 0   
POL(new_primDivNatS(x1, x2, x3)) = x1   
POL(new_primDivNatS0(x1, x2, x3, x4)) = 1 + x1   
POL(new_primDivNatS00(x1, x2)) = 1 + x1   
POL(new_primDivNatS1(x1, x2)) = 1 + x1   

The following usable rules [17] were oriented: none



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

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

new_primDivNatS0(vuz133, vuz134, Zero, Zero) → new_primDivNatS00(vuz133, vuz134)
new_primDivNatS(Succ(vuz1420), Zero, vuz144) → new_primDivNatS1(vuz1420, vuz144)
new_primDivNatS0(vuz133, vuz134, Succ(vuz1350), Succ(vuz1360)) → new_primDivNatS0(vuz133, vuz134, vuz1350, vuz1360)
new_primDivNatS00(vuz133, vuz134) → new_primDivNatS(Succ(vuz133), Succ(vuz134), Succ(vuz134))
new_primDivNatS0(vuz133, vuz134, Succ(vuz1350), Zero) → new_primDivNatS(Succ(vuz133), Succ(vuz134), Succ(vuz134))

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 4 less nodes.

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

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

new_primDivNatS0(vuz133, vuz134, Succ(vuz1350), Succ(vuz1360)) → new_primDivNatS0(vuz133, vuz134, vuz1350, vuz1360)

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
                            ↳ QDPSizeChangeProof
                          ↳ QDP

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

new_primQuotInt(Succ(vuz1200)) → new_primQuotInt(vuz1200)

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
                            ↳ QDPSizeChangeProof

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

new_primQuotInt0(Zero, Succ(vuz1400)) → new_primQuotInt0(Zero, vuz1400)

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: