MAYBE 9.788 H-Termination proof of /home/matraf/haskell/eval_FullyBlown_Fast/empty.hs
H-Termination of the given Haskell-Program with start terms could not be shown:



HASKELL
  ↳ IFR

mainModule Main
  ((toRational :: Float  ->  Ratio Integer) :: Float  ->  Ratio Integer)

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
  ((toRational :: Float  ->  Ratio Integer) :: Float  ->  Ratio Integer)

module Main where
  import qualified Prelude



Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL
  ↳ IFR
    ↳ HASKELL
      ↳ BR
HASKELL
          ↳ COR

mainModule Main
  ((toRational :: Float  ->  Ratio Integer) :: Float  ->  Ratio Integer)

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
  ((toRational :: Float  ->  Ratio Integer) :: Float  ->  Ratio Integer)

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'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

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



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

mainModule Main
  ((toRational :: Float  ->  Ratio Integer) :: Float  ->  Ratio Integer)

module Main where
  import qualified Prelude



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

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

mainModule Main
  (toRational :: Float  ->  Ratio Integer)

module Main where
  import qualified Prelude



Haskell To QDPs


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

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
                            ↳ QDPSizeChangeProof
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

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

new_primMinusNatS(Succ(vuz10200), Succ(vuz1030)) → new_primMinusNatS(vuz10200, vuz1030)

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

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



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

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

new_primDivNatS00(vuz135, vuz136, Succ(vuz1370), Zero) → new_primDivNatS(new_primMinusNatS0(Succ(vuz135), Succ(vuz136)), vuz136)
new_primDivNatS00(vuz135, vuz136, Succ(vuz1370), Succ(vuz1380)) → new_primDivNatS00(vuz135, vuz136, vuz1370, vuz1380)
new_primDivNatS0(vuz115, vuz1170) → new_primDivNatS0(vuz115, vuz1170)
new_primDivNatS01(vuz135, vuz136) → new_primDivNatS(new_primMinusNatS0(Succ(vuz135), Succ(vuz136)), vuz136)
new_primDivNatS00(vuz135, vuz136, Zero, Zero) → new_primDivNatS01(vuz135, vuz136)
new_primDivNatS(Succ(Succ(vuz13900)), vuz136) → new_primDivNatS0(vuz13900, vuz136)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vuz1030)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz10200), Succ(vuz1030)) → new_primMinusNatS0(vuz10200, vuz1030)
new_primMinusNatS0(Succ(vuz10200), Zero) → Succ(vuz10200)

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 4 less nodes.

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

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

new_primDivNatS0(vuz115, vuz1170) → new_primDivNatS0(vuz115, vuz1170)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vuz1030)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz10200), Succ(vuz1030)) → new_primMinusNatS0(vuz10200, vuz1030)
new_primMinusNatS0(Succ(vuz10200), Zero) → Succ(vuz10200)

The set Q consists of the following terms:

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

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
                            ↳ DependencyGraphProof
                              ↳ AND
                                ↳ QDP
                                  ↳ UsableRulesProof
QDP
                                      ↳ QReductionProof
                                ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

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

new_primDivNatS0(vuz115, vuz1170) → new_primDivNatS0(vuz115, vuz1170)

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

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

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



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

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

new_primDivNatS0(vuz115, vuz1170) → new_primDivNatS0(vuz115, vuz1170)

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

The TRS P consists of the following rules:

new_primDivNatS0(vuz115, vuz1170) → new_primDivNatS0(vuz115, vuz1170)

The TRS R consists of the following rules:none


s = new_primDivNatS0(vuz115, vuz1170) evaluates to t =new_primDivNatS0(vuz115, vuz1170)

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




Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from new_primDivNatS0(vuz115, vuz1170) to new_primDivNatS0(vuz115, vuz1170).





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

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

new_primDivNatS00(vuz135, vuz136, Succ(vuz1370), Succ(vuz1380)) → new_primDivNatS00(vuz135, vuz136, vuz1370, vuz1380)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vuz1030)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz10200), Succ(vuz1030)) → new_primMinusNatS0(vuz10200, vuz1030)
new_primMinusNatS0(Succ(vuz10200), Zero) → Succ(vuz10200)

The set Q consists of the following terms:

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

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
                            ↳ DependencyGraphProof
                              ↳ AND
                                ↳ QDP
                                ↳ QDP
                                  ↳ UsableRulesProof
QDP
                                      ↳ QReductionProof
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

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

new_primDivNatS00(vuz135, vuz136, Succ(vuz1370), Succ(vuz1380)) → new_primDivNatS00(vuz135, vuz136, vuz1370, vuz1380)

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

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

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



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

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

new_primDivNatS00(vuz135, vuz136, Succ(vuz1370), Succ(vuz1380)) → new_primDivNatS00(vuz135, vuz136, vuz1370, vuz1380)

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
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

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

new_primDivNatS1(Succ(vuz1070), vuz108) → new_primDivNatS1(vuz1070, vuz108)

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

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



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

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

new_primDivNatS02(vuz520000) → new_primDivNatS2(new_primMinusNatS1(vuz520000))
new_primDivNatS2(Succ(Succ(vuz520000))) → new_primDivNatS2(new_primMinusNatS1(vuz520000))

The TRS R consists of the following rules:

new_primMinusNatS1(vuz520000) → Succ(vuz520000)

The set Q consists of the following terms:

new_primMinusNatS1(x0)

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

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

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

new_primDivNatS2(Succ(Succ(vuz520000))) → new_primDivNatS2(new_primMinusNatS1(vuz520000))

The TRS R consists of the following rules:

new_primMinusNatS1(vuz520000) → Succ(vuz520000)

The set Q consists of the following terms:

new_primMinusNatS1(x0)

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

new_primDivNatS2(Succ(Succ(vuz520000))) → new_primDivNatS2(new_primMinusNatS1(vuz520000))

Strictly oriented rules of the TRS R:

new_primMinusNatS1(vuz520000) → Succ(vuz520000)

Used ordering: POLO with Polynomial interpretation [25]:

POL(Succ(x1)) = 1 + 2·x1   
POL(new_primDivNatS2(x1)) = x1   
POL(new_primMinusNatS1(x1)) = 2 + 2·x1   



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

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

new_primMinusNatS1(x0)

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

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

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

new_quot(Succ(vuz5000)) → new_quot(vuz5000)

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

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



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

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

new_quot1(vuz115, Succ(Succ(vuz12100)), Zero, vuz120) → new_quot2(vuz115, vuz12100, Zero)
new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Succ(vuz1190)) → new_quot0(vuz115, vuz116, vuz117, vuz1180, vuz1190)
new_quot0(vuz115, vuz116, vuz117, Zero, Succ(vuz1190)) → new_quot1(vuz115, Succ(vuz117), Succ(vuz116), Succ(vuz117))
new_quot2(vuz115, vuz116, vuz117) → new_quot1(vuz115, new_primMinusNatS0(Succ(vuz116), vuz117), vuz117, new_primMinusNatS0(Succ(vuz116), vuz117))
new_quot1(vuz115, Succ(Succ(vuz12100)), Succ(vuz1170), vuz120) → new_quot0(vuz115, vuz12100, Succ(vuz1170), vuz12100, vuz1170)
new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Zero) → new_quot1(vuz115, new_primMinusNatS0(Succ(vuz116), vuz117), vuz117, new_primMinusNatS0(Succ(vuz116), vuz117))
new_quot0(vuz115, vuz116, vuz117, Zero, Zero) → new_quot2(vuz115, vuz116, vuz117)
new_quot1(vuz115, Succ(Zero), Succ(vuz1170), vuz120) → new_quot3(vuz115, vuz1170, Zero)
new_quot3(vuz44, vuz45, vuz14) → new_quot1(vuz44, Succ(Succ(vuz45)), vuz14, Succ(Succ(vuz45)))

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vuz1030)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz10200), Succ(vuz1030)) → new_primMinusNatS0(vuz10200, vuz1030)
new_primMinusNatS0(Succ(vuz10200), Zero) → Succ(vuz10200)

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule new_quot3(vuz44, vuz45, vuz14) → new_quot1(vuz44, Succ(Succ(vuz45)), vuz14, Succ(Succ(vuz45))) we obtained the following new rules:

new_quot3(z0, z1, Zero) → new_quot1(z0, Succ(Succ(z1)), Zero, Succ(Succ(z1)))



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

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

new_quot1(vuz115, Succ(Succ(vuz12100)), Zero, vuz120) → new_quot2(vuz115, vuz12100, Zero)
new_quot3(z0, z1, Zero) → new_quot1(z0, Succ(Succ(z1)), Zero, Succ(Succ(z1)))
new_quot0(vuz115, vuz116, vuz117, Zero, Succ(vuz1190)) → new_quot1(vuz115, Succ(vuz117), Succ(vuz116), Succ(vuz117))
new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Succ(vuz1190)) → new_quot0(vuz115, vuz116, vuz117, vuz1180, vuz1190)
new_quot2(vuz115, vuz116, vuz117) → new_quot1(vuz115, new_primMinusNatS0(Succ(vuz116), vuz117), vuz117, new_primMinusNatS0(Succ(vuz116), vuz117))
new_quot1(vuz115, Succ(Succ(vuz12100)), Succ(vuz1170), vuz120) → new_quot0(vuz115, vuz12100, Succ(vuz1170), vuz12100, vuz1170)
new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Zero) → new_quot1(vuz115, new_primMinusNatS0(Succ(vuz116), vuz117), vuz117, new_primMinusNatS0(Succ(vuz116), vuz117))
new_quot1(vuz115, Succ(Zero), Succ(vuz1170), vuz120) → new_quot3(vuz115, vuz1170, Zero)
new_quot0(vuz115, vuz116, vuz117, Zero, Zero) → new_quot2(vuz115, vuz116, vuz117)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vuz1030)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz10200), Succ(vuz1030)) → new_primMinusNatS0(vuz10200, vuz1030)
new_primMinusNatS0(Succ(vuz10200), Zero) → Succ(vuz10200)

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule new_quot2(vuz115, vuz116, vuz117) → new_quot1(vuz115, new_primMinusNatS0(Succ(vuz116), vuz117), vuz117, new_primMinusNatS0(Succ(vuz116), vuz117)) at position [1] we obtained the following new rules:

new_quot2(y0, x0, Zero) → new_quot1(y0, Succ(x0), Zero, new_primMinusNatS0(Succ(x0), Zero))
new_quot2(y0, x0, Succ(x1)) → new_quot1(y0, new_primMinusNatS0(x0, x1), Succ(x1), new_primMinusNatS0(Succ(x0), Succ(x1)))



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

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

new_quot2(y0, x0, Zero) → new_quot1(y0, Succ(x0), Zero, new_primMinusNatS0(Succ(x0), Zero))
new_quot1(vuz115, Succ(Succ(vuz12100)), Zero, vuz120) → new_quot2(vuz115, vuz12100, Zero)
new_quot3(z0, z1, Zero) → new_quot1(z0, Succ(Succ(z1)), Zero, Succ(Succ(z1)))
new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Succ(vuz1190)) → new_quot0(vuz115, vuz116, vuz117, vuz1180, vuz1190)
new_quot0(vuz115, vuz116, vuz117, Zero, Succ(vuz1190)) → new_quot1(vuz115, Succ(vuz117), Succ(vuz116), Succ(vuz117))
new_quot1(vuz115, Succ(Succ(vuz12100)), Succ(vuz1170), vuz120) → new_quot0(vuz115, vuz12100, Succ(vuz1170), vuz12100, vuz1170)
new_quot2(y0, x0, Succ(x1)) → new_quot1(y0, new_primMinusNatS0(x0, x1), Succ(x1), new_primMinusNatS0(Succ(x0), Succ(x1)))
new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Zero) → new_quot1(vuz115, new_primMinusNatS0(Succ(vuz116), vuz117), vuz117, new_primMinusNatS0(Succ(vuz116), vuz117))
new_quot0(vuz115, vuz116, vuz117, Zero, Zero) → new_quot2(vuz115, vuz116, vuz117)
new_quot1(vuz115, Succ(Zero), Succ(vuz1170), vuz120) → new_quot3(vuz115, vuz1170, Zero)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vuz1030)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz10200), Succ(vuz1030)) → new_primMinusNatS0(vuz10200, vuz1030)
new_primMinusNatS0(Succ(vuz10200), Zero) → Succ(vuz10200)

The set Q consists of the following terms:

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

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

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

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

new_quot1(vuz115, Succ(Succ(vuz12100)), Zero, vuz120) → new_quot2(vuz115, vuz12100, Zero)
new_quot2(y0, x0, Zero) → new_quot1(y0, Succ(x0), Zero, new_primMinusNatS0(Succ(x0), Zero))

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vuz1030)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz10200), Succ(vuz1030)) → new_primMinusNatS0(vuz10200, vuz1030)
new_primMinusNatS0(Succ(vuz10200), Zero) → Succ(vuz10200)

The set Q consists of the following terms:

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

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

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

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

new_quot1(vuz115, Succ(Succ(vuz12100)), Zero, vuz120) → new_quot2(vuz115, vuz12100, Zero)
new_quot2(y0, x0, Zero) → new_quot1(y0, Succ(x0), Zero, new_primMinusNatS0(Succ(x0), Zero))

The TRS R consists of the following rules:

new_primMinusNatS0(Succ(vuz10200), Zero) → Succ(vuz10200)

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_quot2(y0, x0, Zero) → new_quot1(y0, Succ(x0), Zero, new_primMinusNatS0(Succ(x0), Zero)) at position [3] we obtained the following new rules:

new_quot2(y0, x0, Zero) → new_quot1(y0, Succ(x0), Zero, Succ(x0))



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

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

new_quot1(vuz115, Succ(Succ(vuz12100)), Zero, vuz120) → new_quot2(vuz115, vuz12100, Zero)
new_quot2(y0, x0, Zero) → new_quot1(y0, Succ(x0), Zero, Succ(x0))

The TRS R consists of the following rules:

new_primMinusNatS0(Succ(vuz10200), Zero) → Succ(vuz10200)

The set Q consists of the following terms:

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

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

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

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

new_quot1(vuz115, Succ(Succ(vuz12100)), Zero, vuz120) → new_quot2(vuz115, vuz12100, Zero)
new_quot2(y0, x0, Zero) → new_quot1(y0, Succ(x0), Zero, Succ(x0))

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

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

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



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

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

new_quot1(vuz115, Succ(Succ(vuz12100)), Zero, vuz120) → new_quot2(vuz115, vuz12100, Zero)
new_quot2(y0, x0, Zero) → new_quot1(y0, Succ(x0), Zero, Succ(x0))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule new_quot1(vuz115, Succ(Succ(vuz12100)), Zero, vuz120) → new_quot2(vuz115, vuz12100, Zero) we obtained the following new rules:

new_quot1(z0, Succ(Succ(x1)), Zero, Succ(Succ(x1))) → new_quot2(z0, x1, Zero)



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

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

new_quot1(z0, Succ(Succ(x1)), Zero, Succ(Succ(x1))) → new_quot2(z0, x1, Zero)
new_quot2(y0, x0, Zero) → new_quot1(y0, Succ(x0), Zero, Succ(x0))

R is empty.
Q is empty.
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_quot1(z0, Succ(Succ(x1)), Zero, Succ(Succ(x1))) → new_quot2(z0, x1, Zero)


Used ordering: POLO with Polynomial interpretation [25]:

POL(Succ(x1)) = 1 + x1   
POL(Zero) = 0   
POL(new_quot1(x1, x2, x3, x4)) = x1 + x2 + 2·x3 + x4   
POL(new_quot2(x1, x2, x3)) = 2 + x1 + 2·x2 + 2·x3   



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

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

new_quot2(y0, x0, Zero) → new_quot1(y0, Succ(x0), Zero, Succ(x0))

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 0 SCCs with 1 less node.

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

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

new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Succ(vuz1190)) → new_quot0(vuz115, vuz116, vuz117, vuz1180, vuz1190)
new_quot0(vuz115, vuz116, vuz117, Zero, Succ(vuz1190)) → new_quot1(vuz115, Succ(vuz117), Succ(vuz116), Succ(vuz117))
new_quot1(vuz115, Succ(Succ(vuz12100)), Succ(vuz1170), vuz120) → new_quot0(vuz115, vuz12100, Succ(vuz1170), vuz12100, vuz1170)
new_quot2(y0, x0, Succ(x1)) → new_quot1(y0, new_primMinusNatS0(x0, x1), Succ(x1), new_primMinusNatS0(Succ(x0), Succ(x1)))
new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Zero) → new_quot1(vuz115, new_primMinusNatS0(Succ(vuz116), vuz117), vuz117, new_primMinusNatS0(Succ(vuz116), vuz117))
new_quot0(vuz115, vuz116, vuz117, Zero, Zero) → new_quot2(vuz115, vuz116, vuz117)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vuz1030)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz10200), Succ(vuz1030)) → new_primMinusNatS0(vuz10200, vuz1030)
new_primMinusNatS0(Succ(vuz10200), Zero) → Succ(vuz10200)

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_quot2(y0, x0, Succ(x1)) → new_quot1(y0, new_primMinusNatS0(x0, x1), Succ(x1), new_primMinusNatS0(Succ(x0), Succ(x1))) at position [3] we obtained the following new rules:

new_quot2(y0, x0, Succ(x1)) → new_quot1(y0, new_primMinusNatS0(x0, x1), Succ(x1), new_primMinusNatS0(x0, x1))



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

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

new_quot0(vuz115, vuz116, vuz117, Zero, Succ(vuz1190)) → new_quot1(vuz115, Succ(vuz117), Succ(vuz116), Succ(vuz117))
new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Succ(vuz1190)) → new_quot0(vuz115, vuz116, vuz117, vuz1180, vuz1190)
new_quot2(y0, x0, Succ(x1)) → new_quot1(y0, new_primMinusNatS0(x0, x1), Succ(x1), new_primMinusNatS0(x0, x1))
new_quot1(vuz115, Succ(Succ(vuz12100)), Succ(vuz1170), vuz120) → new_quot0(vuz115, vuz12100, Succ(vuz1170), vuz12100, vuz1170)
new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Zero) → new_quot1(vuz115, new_primMinusNatS0(Succ(vuz116), vuz117), vuz117, new_primMinusNatS0(Succ(vuz116), vuz117))
new_quot0(vuz115, vuz116, vuz117, Zero, Zero) → new_quot2(vuz115, vuz116, vuz117)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vuz1030)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz10200), Succ(vuz1030)) → new_primMinusNatS0(vuz10200, vuz1030)
new_primMinusNatS0(Succ(vuz10200), Zero) → Succ(vuz10200)

The set Q consists of the following terms:

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

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_quot2(y0, x0, Succ(x1)) → new_quot1(y0, new_primMinusNatS0(x0, x1), Succ(x1), new_primMinusNatS0(x0, x1))
The remaining pairs can at least be oriented weakly.

new_quot0(vuz115, vuz116, vuz117, Zero, Succ(vuz1190)) → new_quot1(vuz115, Succ(vuz117), Succ(vuz116), Succ(vuz117))
new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Succ(vuz1190)) → new_quot0(vuz115, vuz116, vuz117, vuz1180, vuz1190)
new_quot1(vuz115, Succ(Succ(vuz12100)), Succ(vuz1170), vuz120) → new_quot0(vuz115, vuz12100, Succ(vuz1170), vuz12100, vuz1170)
new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Zero) → new_quot1(vuz115, new_primMinusNatS0(Succ(vuz116), vuz117), vuz117, new_primMinusNatS0(Succ(vuz116), vuz117))
new_quot0(vuz115, vuz116, vuz117, Zero, Zero) → new_quot2(vuz115, vuz116, vuz117)
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( new_primMinusNatS0(x1, x2) ) =
/1\
\0/
+
/00\
\01/
·x1+
/00\
\00/
·x2

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

M( Zero ) =
/0\
\0/

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

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

M( new_quot0(x1, ..., x5) ) = 1+
[0,0]
·x1+
[1,1]
·x2+
[1,1]
·x3+
[0,0]
·x4+
[0,0]
·x5


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(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz10200), Succ(vuz1030)) → new_primMinusNatS0(vuz10200, vuz1030)
new_primMinusNatS0(Zero, Succ(vuz1030)) → Zero
new_primMinusNatS0(Succ(vuz10200), Zero) → Succ(vuz10200)



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

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

new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Succ(vuz1190)) → new_quot0(vuz115, vuz116, vuz117, vuz1180, vuz1190)
new_quot0(vuz115, vuz116, vuz117, Zero, Succ(vuz1190)) → new_quot1(vuz115, Succ(vuz117), Succ(vuz116), Succ(vuz117))
new_quot1(vuz115, Succ(Succ(vuz12100)), Succ(vuz1170), vuz120) → new_quot0(vuz115, vuz12100, Succ(vuz1170), vuz12100, vuz1170)
new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Zero) → new_quot1(vuz115, new_primMinusNatS0(Succ(vuz116), vuz117), vuz117, new_primMinusNatS0(Succ(vuz116), vuz117))
new_quot0(vuz115, vuz116, vuz117, Zero, Zero) → new_quot2(vuz115, vuz116, vuz117)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vuz1030)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz10200), Succ(vuz1030)) → new_primMinusNatS0(vuz10200, vuz1030)
new_primMinusNatS0(Succ(vuz10200), Zero) → Succ(vuz10200)

The set Q consists of the following terms:

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

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

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

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

new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Succ(vuz1190)) → new_quot0(vuz115, vuz116, vuz117, vuz1180, vuz1190)
new_quot0(vuz115, vuz116, vuz117, Zero, Succ(vuz1190)) → new_quot1(vuz115, Succ(vuz117), Succ(vuz116), Succ(vuz117))
new_quot1(vuz115, Succ(Succ(vuz12100)), Succ(vuz1170), vuz120) → new_quot0(vuz115, vuz12100, Succ(vuz1170), vuz12100, vuz1170)
new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Zero) → new_quot1(vuz115, new_primMinusNatS0(Succ(vuz116), vuz117), vuz117, new_primMinusNatS0(Succ(vuz116), vuz117))

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vuz1030)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz10200), Succ(vuz1030)) → new_primMinusNatS0(vuz10200, vuz1030)
new_primMinusNatS0(Succ(vuz10200), Zero) → Succ(vuz10200)

The set Q consists of the following terms:

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

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_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Zero) → new_quot1(vuz115, new_primMinusNatS0(Succ(vuz116), vuz117), vuz117, new_primMinusNatS0(Succ(vuz116), vuz117))
The remaining pairs can at least be oriented weakly.

new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Succ(vuz1190)) → new_quot0(vuz115, vuz116, vuz117, vuz1180, vuz1190)
new_quot0(vuz115, vuz116, vuz117, Zero, Succ(vuz1190)) → new_quot1(vuz115, Succ(vuz117), Succ(vuz116), Succ(vuz117))
new_quot1(vuz115, Succ(Succ(vuz12100)), Succ(vuz1170), vuz120) → new_quot0(vuz115, vuz12100, Succ(vuz1170), vuz12100, vuz1170)
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( new_primMinusNatS0(x1, x2) ) =
/0\
\0/
+
/10\
\00/
·x1+
/00\
\11/
·x2

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

M( Zero ) =
/0\
\1/

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

M( new_quot0(x1, ..., x5) ) = 1+
[0,0]
·x1+
[1,1]
·x2+
[1,1]
·x3+
[0,0]
·x4+
[0,0]
·x5


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(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz10200), Succ(vuz1030)) → new_primMinusNatS0(vuz10200, vuz1030)
new_primMinusNatS0(Zero, Succ(vuz1030)) → Zero
new_primMinusNatS0(Succ(vuz10200), Zero) → Succ(vuz10200)



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

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

new_quot0(vuz115, vuz116, vuz117, Zero, Succ(vuz1190)) → new_quot1(vuz115, Succ(vuz117), Succ(vuz116), Succ(vuz117))
new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Succ(vuz1190)) → new_quot0(vuz115, vuz116, vuz117, vuz1180, vuz1190)
new_quot1(vuz115, Succ(Succ(vuz12100)), Succ(vuz1170), vuz120) → new_quot0(vuz115, vuz12100, Succ(vuz1170), vuz12100, vuz1170)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vuz1030)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz10200), Succ(vuz1030)) → new_primMinusNatS0(vuz10200, vuz1030)
new_primMinusNatS0(Succ(vuz10200), Zero) → Succ(vuz10200)

The set Q consists of the following terms:

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

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

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

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

new_quot0(vuz115, vuz116, vuz117, Zero, Succ(vuz1190)) → new_quot1(vuz115, Succ(vuz117), Succ(vuz116), Succ(vuz117))
new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Succ(vuz1190)) → new_quot0(vuz115, vuz116, vuz117, vuz1180, vuz1190)
new_quot1(vuz115, Succ(Succ(vuz12100)), Succ(vuz1170), vuz120) → new_quot0(vuz115, vuz12100, Succ(vuz1170), vuz12100, vuz1170)

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

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

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



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

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

new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Succ(vuz1190)) → new_quot0(vuz115, vuz116, vuz117, vuz1180, vuz1190)
new_quot0(vuz115, vuz116, vuz117, Zero, Succ(vuz1190)) → new_quot1(vuz115, Succ(vuz117), Succ(vuz116), Succ(vuz117))
new_quot1(vuz115, Succ(Succ(vuz12100)), Succ(vuz1170), vuz120) → new_quot0(vuz115, vuz12100, Succ(vuz1170), vuz12100, vuz1170)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule new_quot1(vuz115, Succ(Succ(vuz12100)), Succ(vuz1170), vuz120) → new_quot0(vuz115, vuz12100, Succ(vuz1170), vuz12100, vuz1170) we obtained the following new rules:

new_quot1(z0, Succ(Succ(x1)), Succ(z1), Succ(Succ(x1))) → new_quot0(z0, x1, Succ(z1), x1, z1)



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

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

new_quot1(z0, Succ(Succ(x1)), Succ(z1), Succ(Succ(x1))) → new_quot0(z0, x1, Succ(z1), x1, z1)
new_quot0(vuz115, vuz116, vuz117, Zero, Succ(vuz1190)) → new_quot1(vuz115, Succ(vuz117), Succ(vuz116), Succ(vuz117))
new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Succ(vuz1190)) → new_quot0(vuz115, vuz116, vuz117, vuz1180, vuz1190)

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_quot0(vuz115, vuz116, vuz117, Zero, Succ(vuz1190)) → new_quot1(vuz115, Succ(vuz117), Succ(vuz116), Succ(vuz117)) the following chains were created:




For Pair new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Succ(vuz1190)) → new_quot0(vuz115, vuz116, vuz117, vuz1180, vuz1190) the following chains were created:




For Pair new_quot1(z0, Succ(Succ(x1)), Succ(z1), Succ(Succ(x1))) → new_quot0(z0, x1, Succ(z1), x1, z1) the following chains were created:




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



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

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

The following pairs are in P>:

new_quot1(z0, Succ(Succ(x1)), Succ(z1), Succ(Succ(x1))) → new_quot0(z0, x1, Succ(z1), x1, z1)
The following pairs are in Pbound:

new_quot0(vuz115, vuz116, vuz117, Zero, Succ(vuz1190)) → new_quot1(vuz115, Succ(vuz117), Succ(vuz116), Succ(vuz117))
new_quot1(z0, Succ(Succ(x1)), Succ(z1), Succ(Succ(x1))) → new_quot0(z0, x1, Succ(z1), x1, z1)
There are no usable rules

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

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

new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Succ(vuz1190)) → new_quot0(vuz115, vuz116, vuz117, vuz1180, vuz1190)
new_quot0(vuz115, vuz116, vuz117, Zero, Succ(vuz1190)) → new_quot1(vuz115, Succ(vuz117), Succ(vuz116), Succ(vuz117))

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
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                            ↳ Instantiation
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ AND
                                        ↳ QDP
                                        ↳ QDP
                                          ↳ Rewriting
                                            ↳ QDP
                                              ↳ QDPOrderProof
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ QDPOrderProof
                                                        ↳ QDP
                                                          ↳ UsableRulesProof
                                                            ↳ QDP
                                                              ↳ QReductionProof
                                                                ↳ QDP
                                                                  ↳ Instantiation
                                                                    ↳ QDP
                                                                      ↳ NonInfProof
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
QDP
                                                                              ↳ QDPSizeChangeProof
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

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

new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Succ(vuz1190)) → new_quot0(vuz115, vuz116, vuz117, 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
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
QDP
                            ↳ DependencyGraphProof
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

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

new_quot4(vuz46, Succ(vuz470)) → new_quot4(vuz46, vuz470)
new_quot5(vuz46, Succ(vuz470)) → new_quot4(vuz46, vuz470)

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
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                          ↳ QDP
                            ↳ DependencyGraphProof
QDP
                                ↳ QDPSizeChangeProof
                          ↳ QDP
                          ↳ QDP
                      ↳ Narrow

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

new_quot4(vuz46, Succ(vuz470)) → new_quot4(vuz46, vuz470)

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

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



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

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

new_quot6(vuz89, Succ(Succ(vuz9000)), Succ(Zero), Succ(vuz920), Zero) → new_quot6(vuz89, vuz9000, Succ(Zero), vuz9000, Zero)
new_quot8(vuz89, Zero, Succ(Succ(vuz9100))) → new_quot7(vuz89, Zero, vuz9100, Zero)
new_quot6(vuz89, vuz90, Succ(vuz910), Zero, Succ(vuz930)) → new_quot6(vuz89, vuz910, Succ(vuz90), vuz910, vuz90)
new_quot6(vuz89, Succ(vuz900), Succ(Succ(vuz9100)), Succ(vuz920), Zero) → new_quot7(vuz89, new_primMinusNatS0(vuz900, vuz9100), vuz9100, new_primMinusNatS0(vuz900, vuz9100))
new_quot8(vuz89, Succ(vuz900), Succ(Succ(vuz9100))) → new_quot7(vuz89, new_primMinusNatS0(vuz900, vuz9100), vuz9100, new_primMinusNatS0(vuz900, vuz9100))
new_quot8(vuz89, Succ(Succ(vuz9000)), Succ(Zero)) → new_quot6(vuz89, vuz9000, Succ(Zero), vuz9000, Zero)
new_quot6(vuz89, Zero, Succ(Succ(vuz9100)), Succ(vuz920), Zero) → new_quot7(vuz89, Zero, vuz9100, Zero)
new_quot6(vuz89, vuz90, vuz91, Zero, Zero) → new_quot8(vuz89, vuz90, vuz91)
new_quot7(vuz89, Succ(Succ(vuz11300)), vuz9100, vuz112) → new_quot6(vuz89, vuz11300, Succ(Succ(vuz9100)), vuz11300, Succ(vuz9100))
new_quot6(vuz89, vuz90, vuz91, Succ(vuz920), Succ(vuz930)) → new_quot6(vuz89, vuz90, vuz91, vuz920, vuz930)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vuz1030)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz10200), Succ(vuz1030)) → new_primMinusNatS0(vuz10200, vuz1030)
new_primMinusNatS0(Succ(vuz10200), Zero) → Succ(vuz10200)

The set Q consists of the following terms:

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

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

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

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

new_quot6(vuz89, Succ(Succ(vuz9000)), Succ(Zero), Succ(vuz920), Zero) → new_quot6(vuz89, vuz9000, Succ(Zero), vuz9000, Zero)
new_quot6(vuz89, vuz90, Succ(vuz910), Zero, Succ(vuz930)) → new_quot6(vuz89, vuz910, Succ(vuz90), vuz910, vuz90)
new_quot6(vuz89, Succ(vuz900), Succ(Succ(vuz9100)), Succ(vuz920), Zero) → new_quot7(vuz89, new_primMinusNatS0(vuz900, vuz9100), vuz9100, new_primMinusNatS0(vuz900, vuz9100))
new_quot8(vuz89, Succ(vuz900), Succ(Succ(vuz9100))) → new_quot7(vuz89, new_primMinusNatS0(vuz900, vuz9100), vuz9100, new_primMinusNatS0(vuz900, vuz9100))
new_quot8(vuz89, Succ(Succ(vuz9000)), Succ(Zero)) → new_quot6(vuz89, vuz9000, Succ(Zero), vuz9000, Zero)
new_quot6(vuz89, vuz90, vuz91, Zero, Zero) → new_quot8(vuz89, vuz90, vuz91)
new_quot7(vuz89, Succ(Succ(vuz11300)), vuz9100, vuz112) → new_quot6(vuz89, vuz11300, Succ(Succ(vuz9100)), vuz11300, Succ(vuz9100))
new_quot6(vuz89, vuz90, vuz91, Succ(vuz920), Succ(vuz930)) → new_quot6(vuz89, vuz90, vuz91, vuz920, vuz930)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vuz1030)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz10200), Succ(vuz1030)) → new_primMinusNatS0(vuz10200, vuz1030)
new_primMinusNatS0(Succ(vuz10200), Zero) → Succ(vuz10200)

The set Q consists of the following terms:

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

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_quot6(vuz89, Succ(Succ(vuz9000)), Succ(Zero), Succ(vuz920), Zero) → new_quot6(vuz89, vuz9000, Succ(Zero), vuz9000, Zero)
new_quot6(vuz89, Succ(vuz900), Succ(Succ(vuz9100)), Succ(vuz920), Zero) → new_quot7(vuz89, new_primMinusNatS0(vuz900, vuz9100), vuz9100, new_primMinusNatS0(vuz900, vuz9100))
new_quot8(vuz89, Succ(vuz900), Succ(Succ(vuz9100))) → new_quot7(vuz89, new_primMinusNatS0(vuz900, vuz9100), vuz9100, new_primMinusNatS0(vuz900, vuz9100))
new_quot8(vuz89, Succ(Succ(vuz9000)), Succ(Zero)) → new_quot6(vuz89, vuz9000, Succ(Zero), vuz9000, Zero)
The remaining pairs can at least be oriented weakly.

new_quot6(vuz89, vuz90, Succ(vuz910), Zero, Succ(vuz930)) → new_quot6(vuz89, vuz910, Succ(vuz90), vuz910, vuz90)
new_quot6(vuz89, vuz90, vuz91, Zero, Zero) → new_quot8(vuz89, vuz90, vuz91)
new_quot7(vuz89, Succ(Succ(vuz11300)), vuz9100, vuz112) → new_quot6(vuz89, vuz11300, Succ(Succ(vuz9100)), vuz11300, Succ(vuz9100))
new_quot6(vuz89, vuz90, vuz91, Succ(vuz920), Succ(vuz930)) → new_quot6(vuz89, vuz90, vuz91, vuz920, vuz930)
Used ordering: Polynomial interpretation [25]:

POL(Succ(x1)) = 1 + x1   
POL(Zero) = 1   
POL(new_primMinusNatS0(x1, x2)) = x1   
POL(new_quot6(x1, x2, x3, x4, x5)) = x2 + x3   
POL(new_quot7(x1, x2, x3, x4)) = x2 + x3   
POL(new_quot8(x1, x2, x3)) = x2 + x3   

The following usable rules [17] were oriented:

new_primMinusNatS0(Succ(vuz10200), Zero) → Succ(vuz10200)
new_primMinusNatS0(Zero, Succ(vuz1030)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz10200), Succ(vuz1030)) → new_primMinusNatS0(vuz10200, vuz1030)



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

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

new_quot6(vuz89, vuz90, Succ(vuz910), Zero, Succ(vuz930)) → new_quot6(vuz89, vuz910, Succ(vuz90), vuz910, vuz90)
new_quot6(vuz89, vuz90, vuz91, Zero, Zero) → new_quot8(vuz89, vuz90, vuz91)
new_quot7(vuz89, Succ(Succ(vuz11300)), vuz9100, vuz112) → new_quot6(vuz89, vuz11300, Succ(Succ(vuz9100)), vuz11300, Succ(vuz9100))
new_quot6(vuz89, vuz90, vuz91, Succ(vuz920), Succ(vuz930)) → new_quot6(vuz89, vuz90, vuz91, vuz920, vuz930)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vuz1030)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz10200), Succ(vuz1030)) → new_primMinusNatS0(vuz10200, vuz1030)
new_primMinusNatS0(Succ(vuz10200), Zero) → Succ(vuz10200)

The set Q consists of the following terms:

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

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

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

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

new_quot6(vuz89, vuz90, Succ(vuz910), Zero, Succ(vuz930)) → new_quot6(vuz89, vuz910, Succ(vuz90), vuz910, vuz90)
new_quot6(vuz89, vuz90, vuz91, Succ(vuz920), Succ(vuz930)) → new_quot6(vuz89, vuz90, vuz91, vuz920, vuz930)

The TRS R consists of the following rules:

new_primMinusNatS0(Zero, Succ(vuz1030)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz10200), Succ(vuz1030)) → new_primMinusNatS0(vuz10200, vuz1030)
new_primMinusNatS0(Succ(vuz10200), Zero) → Succ(vuz10200)

The set Q consists of the following terms:

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

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

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

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

new_quot6(vuz89, vuz90, Succ(vuz910), Zero, Succ(vuz930)) → new_quot6(vuz89, vuz910, Succ(vuz90), vuz910, vuz90)
new_quot6(vuz89, vuz90, vuz91, Succ(vuz920), Succ(vuz930)) → new_quot6(vuz89, vuz90, vuz91, vuz920, vuz930)

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

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

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



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

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

new_quot6(vuz89, vuz90, Succ(vuz910), Zero, Succ(vuz930)) → new_quot6(vuz89, vuz910, Succ(vuz90), vuz910, vuz90)
new_quot6(vuz89, vuz90, vuz91, Succ(vuz920), Succ(vuz930)) → new_quot6(vuz89, vuz90, vuz91, vuz920, vuz930)

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_quot6(vuz89, vuz90, Succ(vuz910), Zero, Succ(vuz930)) → new_quot6(vuz89, vuz910, Succ(vuz90), vuz910, vuz90) the following chains were created:




For Pair new_quot6(vuz89, vuz90, vuz91, Succ(vuz920), Succ(vuz930)) → new_quot6(vuz89, vuz90, vuz91, vuz920, vuz930) 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_quot6(x1, x2, x3, x4, x5)) = -1 + x1 + x2 - x4 + x5   

The following pairs are in P>:

new_quot6(vuz89, vuz90, Succ(vuz910), Zero, Succ(vuz930)) → new_quot6(vuz89, vuz910, Succ(vuz90), vuz910, vuz90)
The following pairs are in Pbound:

new_quot6(vuz89, vuz90, Succ(vuz910), Zero, Succ(vuz930)) → new_quot6(vuz89, vuz910, Succ(vuz90), vuz910, vuz90)
There are no usable rules

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

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

new_quot6(vuz89, vuz90, vuz91, Succ(vuz920), Succ(vuz930)) → new_quot6(vuz89, vuz90, vuz91, vuz920, vuz930)

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

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



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

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

new_quot9(Zero, Succ(vuz1600)) → new_quot9(Zero, vuz1600)

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 To QDPs