YES 8.349
H-Termination proof of /home/matraf/haskell/eval_FullyBlown_Fast/empty.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:
↳ HASKELL
↳ IFR
mainModule Main
| ((fromDouble :: Double -> Ratio Int) :: Double -> Ratio Int) |
module Main where
If Reductions:
The following If expression
if primGEqNatS x y then Succ (primDivNatS (primMinusNatS x y) (Succ y)) else Zero
is transformed to
primDivNatS0 | x y True | = Succ (primDivNatS (primMinusNatS x y) (Succ y)) |
primDivNatS0 | x y False | = Zero |
The following If expression
if primGEqNatS x y then primModNatS (primMinusNatS x y) (Succ y) else Succ x
is transformed to
primModNatS0 | x y True | = primModNatS (primMinusNatS x y) (Succ y) |
primModNatS0 | x y False | = Succ x |
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
mainModule Main
| ((fromDouble :: Double -> Ratio Int) :: Double -> Ratio Int) |
module Main where
Replaced joker patterns by fresh variables and removed binding patterns.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
mainModule Main
| ((fromDouble :: Double -> Ratio Int) :: Double -> Ratio Int) |
module Main where
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
is transformed to
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
is transformed to
undefined0 | True | = undefined |
undefined1 | | = undefined0 False |
The following Function with conditions
reduce | x y |
| | y == 0 | |
| | otherwise |
= | x `quot` d :% (y `quot` d) |
|
|
where | |
|
is transformed to
reduce2 | x y | =
reduce1 x y (y == 0) |
where | |
|
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 | |
| | x > 0 | |
| | otherwise | |
|
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 |
signumReal3 | x | = signumReal2 x (x == 0) |
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
mainModule Main
| ((fromDouble :: Double -> Ratio Int) :: Double -> Ratio Int) |
module Main where
Let/Where Reductions:
The bindings of the following Let/Where expression
reduce1 x y (y == 0) |
where | |
|
reduce0 | x y True | = x `quot` d :% (y `quot` d) |
|
|
reduce1 | x y True | = error [] |
reduce1 | x y False | = reduce0 x y otherwise |
|
are unpacked to the following functions on top level
reduce2Reduce1 | vux vuy x y True | = error [] |
reduce2Reduce1 | vux vuy x y False | = reduce2Reduce0 vux vuy x y otherwise |
reduce2D | vux vuy | = gcd vux vuy |
reduce2Reduce0 | vux vuy x y True | = x `quot` reduce2D vux vuy :% (y `quot` reduce2D vux vuy) |
The bindings of the following Let/Where expression
gcd' (abs x) (abs y) |
where |
gcd' | x xz | = gcd'2 x xz |
gcd' | x y | = gcd'0 x y |
|
|
gcd'0 | x y | = gcd' y (x `rem` y) |
|
|
gcd'1 | True x xz | = x |
gcd'1 | yu yv yw | = gcd'0 yv yw |
|
|
gcd'2 | x xz | = gcd'1 (xz == 0) x xz |
gcd'2 | yx yy | = gcd'0 yx yy |
|
are unpacked to the following functions on top level
gcd0Gcd'1 | True x xz | = x |
gcd0Gcd'1 | yu yv yw | = gcd0Gcd'0 yv yw |
gcd0Gcd'2 | x xz | = gcd0Gcd'1 (xz == 0) x xz |
gcd0Gcd'2 | yx yy | = gcd0Gcd'0 yx yy |
gcd0Gcd'0 | x y | = gcd0Gcd' y (x `rem` y) |
gcd0Gcd' | x xz | = gcd0Gcd'2 x xz |
gcd0Gcd' | x y | = gcd0Gcd'0 x y |
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
mainModule Main
| ((fromDouble :: Double -> Ratio Int) :: Double -> Ratio Int) |
module Main where
Num Reduction: All numbers are transformed to thier corresponding representation with Pos, Neg, Succ and Zero.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
mainModule Main
| (fromDouble :: Double -> Ratio Int) |
module Main where
Haskell To QDPs
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primMulNat(Succ(vuz30000)) → new_primMulNat(vuz30000)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- new_primMulNat(Succ(vuz30000)) → new_primMulNat(vuz30000)
The graph contains the following edges 1 > 1
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_gcd0Gcd'15(Zero, Succ(vuz1070)) → new_gcd0Gcd'12(vuz1070)
new_gcd0Gcd'14(Succ(vuz1070), Succ(vuz1080)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070)
new_gcd0Gcd'13(Succ(vuz1380), Zero, vuz140) → new_gcd0Gcd'1(vuz1380, vuz140)
new_gcd0Gcd'1(Succ(vuz1080), Succ(vuz1070)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070)
new_gcd0Gcd'1(Succ(vuz1080), Zero) → new_gcd0Gcd'11(vuz1080, Zero)
new_gcd0Gcd'10(vuz113, vuz114, Zero, Succ(vuz1160)) → new_gcd0Gcd'14(Succ(vuz113), vuz114)
new_gcd0Gcd'15(Zero, Zero) → new_gcd0Gcd'13(Zero, Zero, Zero)
new_gcd0Gcd'12(vuz31000) → new_gcd0Gcd'14(Zero, Succ(vuz31000))
new_gcd0Gcd'1(Zero, Zero) → new_gcd0Gcd'13(Zero, Zero, Zero)
new_gcd0Gcd'1(Zero, Succ(vuz1070)) → new_gcd0Gcd'12(vuz1070)
new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Zero) → new_gcd0Gcd'13(Succ(vuz113), vuz114, vuz114)
new_gcd0Gcd'14(Succ(vuz1070), Zero) → new_gcd0Gcd'12(vuz1070)
new_gcd0Gcd'14(Zero, Zero) → new_gcd0Gcd'13(Zero, Zero, Zero)
new_gcd0Gcd'11(vuz113, vuz114) → new_gcd0Gcd'13(Succ(vuz113), vuz114, vuz114)
new_gcd0Gcd'10(vuz113, vuz114, Zero, Zero) → new_gcd0Gcd'11(vuz113, vuz114)
new_gcd0Gcd'13(Succ(vuz1380), Succ(vuz1390), vuz140) → new_gcd0Gcd'13(vuz1380, vuz1390, vuz140)
new_gcd0Gcd'14(Zero, Succ(vuz1080)) → new_gcd0Gcd'11(vuz1080, Zero)
new_gcd0Gcd'15(Succ(vuz1080), Succ(vuz1070)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070)
new_gcd0Gcd'15(Succ(vuz1080), Zero) → new_gcd0Gcd'11(vuz1080, Zero)
new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Succ(vuz1160)) → new_gcd0Gcd'10(vuz113, vuz114, vuz1150, vuz1160)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 6 less nodes.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_gcd0Gcd'14(Succ(vuz1070), Succ(vuz1080)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070)
new_gcd0Gcd'13(Succ(vuz1380), Zero, vuz140) → new_gcd0Gcd'1(vuz1380, vuz140)
new_gcd0Gcd'10(vuz113, vuz114, Zero, Succ(vuz1160)) → new_gcd0Gcd'14(Succ(vuz113), vuz114)
new_gcd0Gcd'1(Succ(vuz1080), Succ(vuz1070)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070)
new_gcd0Gcd'1(Succ(vuz1080), Zero) → new_gcd0Gcd'11(vuz1080, Zero)
new_gcd0Gcd'12(vuz31000) → new_gcd0Gcd'14(Zero, Succ(vuz31000))
new_gcd0Gcd'1(Zero, Succ(vuz1070)) → new_gcd0Gcd'12(vuz1070)
new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Zero) → new_gcd0Gcd'13(Succ(vuz113), vuz114, vuz114)
new_gcd0Gcd'14(Succ(vuz1070), Zero) → new_gcd0Gcd'12(vuz1070)
new_gcd0Gcd'10(vuz113, vuz114, Zero, Zero) → new_gcd0Gcd'11(vuz113, vuz114)
new_gcd0Gcd'11(vuz113, vuz114) → new_gcd0Gcd'13(Succ(vuz113), vuz114, vuz114)
new_gcd0Gcd'13(Succ(vuz1380), Succ(vuz1390), vuz140) → new_gcd0Gcd'13(vuz1380, vuz1390, vuz140)
new_gcd0Gcd'14(Zero, Succ(vuz1080)) → new_gcd0Gcd'11(vuz1080, Zero)
new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Succ(vuz1160)) → new_gcd0Gcd'10(vuz113, vuz114, vuz1150, vuz1160)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
new_gcd0Gcd'14(Succ(vuz1070), Zero) → new_gcd0Gcd'12(vuz1070)
The remaining pairs can at least be oriented weakly.
new_gcd0Gcd'14(Succ(vuz1070), Succ(vuz1080)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070)
new_gcd0Gcd'13(Succ(vuz1380), Zero, vuz140) → new_gcd0Gcd'1(vuz1380, vuz140)
new_gcd0Gcd'10(vuz113, vuz114, Zero, Succ(vuz1160)) → new_gcd0Gcd'14(Succ(vuz113), vuz114)
new_gcd0Gcd'1(Succ(vuz1080), Succ(vuz1070)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070)
new_gcd0Gcd'1(Succ(vuz1080), Zero) → new_gcd0Gcd'11(vuz1080, Zero)
new_gcd0Gcd'12(vuz31000) → new_gcd0Gcd'14(Zero, Succ(vuz31000))
new_gcd0Gcd'1(Zero, Succ(vuz1070)) → new_gcd0Gcd'12(vuz1070)
new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Zero) → new_gcd0Gcd'13(Succ(vuz113), vuz114, vuz114)
new_gcd0Gcd'10(vuz113, vuz114, Zero, Zero) → new_gcd0Gcd'11(vuz113, vuz114)
new_gcd0Gcd'11(vuz113, vuz114) → new_gcd0Gcd'13(Succ(vuz113), vuz114, vuz114)
new_gcd0Gcd'13(Succ(vuz1380), Succ(vuz1390), vuz140) → new_gcd0Gcd'13(vuz1380, vuz1390, vuz140)
new_gcd0Gcd'14(Zero, Succ(vuz1080)) → new_gcd0Gcd'11(vuz1080, Zero)
new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Succ(vuz1160)) → new_gcd0Gcd'10(vuz113, vuz114, vuz1150, vuz1160)
Used ordering: Polynomial interpretation [25]:
POL(Succ(x1)) = 0
POL(Zero) = 1
POL(new_gcd0Gcd'1(x1, x2)) = 0
POL(new_gcd0Gcd'10(x1, x2, x3, x4)) = x2
POL(new_gcd0Gcd'11(x1, x2)) = 0
POL(new_gcd0Gcd'12(x1)) = 0
POL(new_gcd0Gcd'13(x1, x2, x3)) = 0
POL(new_gcd0Gcd'14(x1, x2)) = x2
The following usable rules [17] were oriented:
none
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_gcd0Gcd'14(Succ(vuz1070), Succ(vuz1080)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070)
new_gcd0Gcd'13(Succ(vuz1380), Zero, vuz140) → new_gcd0Gcd'1(vuz1380, vuz140)
new_gcd0Gcd'10(vuz113, vuz114, Zero, Succ(vuz1160)) → new_gcd0Gcd'14(Succ(vuz113), vuz114)
new_gcd0Gcd'1(Succ(vuz1080), Succ(vuz1070)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070)
new_gcd0Gcd'1(Succ(vuz1080), Zero) → new_gcd0Gcd'11(vuz1080, Zero)
new_gcd0Gcd'12(vuz31000) → new_gcd0Gcd'14(Zero, Succ(vuz31000))
new_gcd0Gcd'1(Zero, Succ(vuz1070)) → new_gcd0Gcd'12(vuz1070)
new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Zero) → new_gcd0Gcd'13(Succ(vuz113), vuz114, vuz114)
new_gcd0Gcd'10(vuz113, vuz114, Zero, Zero) → new_gcd0Gcd'11(vuz113, vuz114)
new_gcd0Gcd'11(vuz113, vuz114) → new_gcd0Gcd'13(Succ(vuz113), vuz114, vuz114)
new_gcd0Gcd'13(Succ(vuz1380), Succ(vuz1390), vuz140) → new_gcd0Gcd'13(vuz1380, vuz1390, vuz140)
new_gcd0Gcd'14(Zero, Succ(vuz1080)) → new_gcd0Gcd'11(vuz1080, Zero)
new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Succ(vuz1160)) → new_gcd0Gcd'10(vuz113, vuz114, vuz1150, vuz1160)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
new_gcd0Gcd'14(Zero, Succ(vuz1080)) → new_gcd0Gcd'11(vuz1080, Zero)
The remaining pairs can at least be oriented weakly.
new_gcd0Gcd'14(Succ(vuz1070), Succ(vuz1080)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070)
new_gcd0Gcd'13(Succ(vuz1380), Zero, vuz140) → new_gcd0Gcd'1(vuz1380, vuz140)
new_gcd0Gcd'10(vuz113, vuz114, Zero, Succ(vuz1160)) → new_gcd0Gcd'14(Succ(vuz113), vuz114)
new_gcd0Gcd'1(Succ(vuz1080), Succ(vuz1070)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070)
new_gcd0Gcd'1(Succ(vuz1080), Zero) → new_gcd0Gcd'11(vuz1080, Zero)
new_gcd0Gcd'12(vuz31000) → new_gcd0Gcd'14(Zero, Succ(vuz31000))
new_gcd0Gcd'1(Zero, Succ(vuz1070)) → new_gcd0Gcd'12(vuz1070)
new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Zero) → new_gcd0Gcd'13(Succ(vuz113), vuz114, vuz114)
new_gcd0Gcd'10(vuz113, vuz114, Zero, Zero) → new_gcd0Gcd'11(vuz113, vuz114)
new_gcd0Gcd'11(vuz113, vuz114) → new_gcd0Gcd'13(Succ(vuz113), vuz114, vuz114)
new_gcd0Gcd'13(Succ(vuz1380), Succ(vuz1390), vuz140) → new_gcd0Gcd'13(vuz1380, vuz1390, vuz140)
new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Succ(vuz1160)) → new_gcd0Gcd'10(vuz113, vuz114, vuz1150, vuz1160)
Used ordering: Polynomial interpretation [25]:
POL(Succ(x1)) = 1
POL(Zero) = 0
POL(new_gcd0Gcd'1(x1, x2)) = x2
POL(new_gcd0Gcd'10(x1, x2, x3, x4)) = x2
POL(new_gcd0Gcd'11(x1, x2)) = x2
POL(new_gcd0Gcd'12(x1)) = 1
POL(new_gcd0Gcd'13(x1, x2, x3)) = x3
POL(new_gcd0Gcd'14(x1, x2)) = x2
The following usable rules [17] were oriented:
none
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_gcd0Gcd'14(Succ(vuz1070), Succ(vuz1080)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070)
new_gcd0Gcd'13(Succ(vuz1380), Zero, vuz140) → new_gcd0Gcd'1(vuz1380, vuz140)
new_gcd0Gcd'1(Succ(vuz1080), Zero) → new_gcd0Gcd'11(vuz1080, Zero)
new_gcd0Gcd'1(Succ(vuz1080), Succ(vuz1070)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070)
new_gcd0Gcd'10(vuz113, vuz114, Zero, Succ(vuz1160)) → new_gcd0Gcd'14(Succ(vuz113), vuz114)
new_gcd0Gcd'12(vuz31000) → new_gcd0Gcd'14(Zero, Succ(vuz31000))
new_gcd0Gcd'11(vuz113, vuz114) → new_gcd0Gcd'13(Succ(vuz113), vuz114, vuz114)
new_gcd0Gcd'10(vuz113, vuz114, Zero, Zero) → new_gcd0Gcd'11(vuz113, vuz114)
new_gcd0Gcd'13(Succ(vuz1380), Succ(vuz1390), vuz140) → new_gcd0Gcd'13(vuz1380, vuz1390, vuz140)
new_gcd0Gcd'1(Zero, Succ(vuz1070)) → new_gcd0Gcd'12(vuz1070)
new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Zero) → new_gcd0Gcd'13(Succ(vuz113), vuz114, vuz114)
new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Succ(vuz1160)) → new_gcd0Gcd'10(vuz113, vuz114, vuz1150, vuz1160)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_gcd0Gcd'14(Succ(vuz1070), Succ(vuz1080)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070)
new_gcd0Gcd'13(Succ(vuz1380), Zero, vuz140) → new_gcd0Gcd'1(vuz1380, vuz140)
new_gcd0Gcd'1(Succ(vuz1080), Zero) → new_gcd0Gcd'11(vuz1080, Zero)
new_gcd0Gcd'1(Succ(vuz1080), Succ(vuz1070)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070)
new_gcd0Gcd'10(vuz113, vuz114, Zero, Succ(vuz1160)) → new_gcd0Gcd'14(Succ(vuz113), vuz114)
new_gcd0Gcd'11(vuz113, vuz114) → new_gcd0Gcd'13(Succ(vuz113), vuz114, vuz114)
new_gcd0Gcd'10(vuz113, vuz114, Zero, Zero) → new_gcd0Gcd'11(vuz113, vuz114)
new_gcd0Gcd'13(Succ(vuz1380), Succ(vuz1390), vuz140) → new_gcd0Gcd'13(vuz1380, vuz1390, vuz140)
new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Zero) → new_gcd0Gcd'13(Succ(vuz113), vuz114, vuz114)
new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Succ(vuz1160)) → new_gcd0Gcd'10(vuz113, vuz114, vuz1150, vuz1160)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
new_gcd0Gcd'13(Succ(vuz1380), Zero, vuz140) → new_gcd0Gcd'1(vuz1380, vuz140)
new_gcd0Gcd'13(Succ(vuz1380), Succ(vuz1390), vuz140) → new_gcd0Gcd'13(vuz1380, vuz1390, vuz140)
The remaining pairs can at least be oriented weakly.
new_gcd0Gcd'14(Succ(vuz1070), Succ(vuz1080)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070)
new_gcd0Gcd'1(Succ(vuz1080), Zero) → new_gcd0Gcd'11(vuz1080, Zero)
new_gcd0Gcd'1(Succ(vuz1080), Succ(vuz1070)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070)
new_gcd0Gcd'10(vuz113, vuz114, Zero, Succ(vuz1160)) → new_gcd0Gcd'14(Succ(vuz113), vuz114)
new_gcd0Gcd'11(vuz113, vuz114) → new_gcd0Gcd'13(Succ(vuz113), vuz114, vuz114)
new_gcd0Gcd'10(vuz113, vuz114, Zero, Zero) → new_gcd0Gcd'11(vuz113, vuz114)
new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Zero) → new_gcd0Gcd'13(Succ(vuz113), vuz114, vuz114)
new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Succ(vuz1160)) → new_gcd0Gcd'10(vuz113, vuz114, vuz1150, vuz1160)
Used ordering: Polynomial interpretation [25]:
POL(Succ(x1)) = 1 + x1
POL(Zero) = 0
POL(new_gcd0Gcd'1(x1, x2)) = x1 + x2
POL(new_gcd0Gcd'10(x1, x2, x3, x4)) = 1 + x1 + x2
POL(new_gcd0Gcd'11(x1, x2)) = 1 + x1 + x2
POL(new_gcd0Gcd'13(x1, x2, x3)) = x1 + x3
POL(new_gcd0Gcd'14(x1, x2)) = x1 + x2
The following usable rules [17] were oriented:
none
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_gcd0Gcd'14(Succ(vuz1070), Succ(vuz1080)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070)
new_gcd0Gcd'10(vuz113, vuz114, Zero, Succ(vuz1160)) → new_gcd0Gcd'14(Succ(vuz113), vuz114)
new_gcd0Gcd'1(Succ(vuz1080), Succ(vuz1070)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070)
new_gcd0Gcd'1(Succ(vuz1080), Zero) → new_gcd0Gcd'11(vuz1080, Zero)
new_gcd0Gcd'10(vuz113, vuz114, Zero, Zero) → new_gcd0Gcd'11(vuz113, vuz114)
new_gcd0Gcd'11(vuz113, vuz114) → new_gcd0Gcd'13(Succ(vuz113), vuz114, vuz114)
new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Zero) → new_gcd0Gcd'13(Succ(vuz113), vuz114, vuz114)
new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Succ(vuz1160)) → new_gcd0Gcd'10(vuz113, vuz114, vuz1150, vuz1160)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 5 less nodes.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ NonInfProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_gcd0Gcd'14(Succ(vuz1070), Succ(vuz1080)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070)
new_gcd0Gcd'10(vuz113, vuz114, Zero, Succ(vuz1160)) → new_gcd0Gcd'14(Succ(vuz113), vuz114)
new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Succ(vuz1160)) → new_gcd0Gcd'10(vuz113, vuz114, vuz1150, vuz1160)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The DP Problem is simplified using the Induction Calculus [18] with the following steps:
Note that final constraints are written in bold face.
For Pair new_gcd0Gcd'10(vuz113, vuz114, Zero, Succ(vuz1160)) → new_gcd0Gcd'14(Succ(vuz113), vuz114) the following chains were created:
- We consider the chain new_gcd0Gcd'14(Succ(vuz1070), Succ(vuz1080)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070), new_gcd0Gcd'10(vuz113, vuz114, Zero, Succ(vuz1160)) → new_gcd0Gcd'14(Succ(vuz113), vuz114) which results in the following constraint:
(1) (new_gcd0Gcd'10(x4, Succ(x3), x4, x3)=new_gcd0Gcd'10(x5, x6, Zero, Succ(x7)) ⇒ new_gcd0Gcd'10(x5, x6, Zero, Succ(x7))≥new_gcd0Gcd'14(Succ(x5), x6))
We simplified constraint (1) using rules (I), (II), (III) which results in the following new constraint:
(2) (new_gcd0Gcd'10(Zero, Succ(Succ(x7)), Zero, Succ(x7))≥new_gcd0Gcd'14(Succ(Zero), Succ(Succ(x7))))
- We consider the chain new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Succ(vuz1160)) → new_gcd0Gcd'10(vuz113, vuz114, vuz1150, vuz1160), new_gcd0Gcd'10(vuz113, vuz114, Zero, Succ(vuz1160)) → new_gcd0Gcd'14(Succ(vuz113), vuz114) which results in the following constraint:
(3) (new_gcd0Gcd'10(x8, x9, x10, x11)=new_gcd0Gcd'10(x12, x13, Zero, Succ(x14)) ⇒ new_gcd0Gcd'10(x12, x13, Zero, Succ(x14))≥new_gcd0Gcd'14(Succ(x12), x13))
We simplified constraint (3) using rules (I), (II), (III) which results in the following new constraint:
(4) (new_gcd0Gcd'10(x8, x9, Zero, Succ(x14))≥new_gcd0Gcd'14(Succ(x8), x9))
For Pair new_gcd0Gcd'14(Succ(vuz1070), Succ(vuz1080)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070) the following chains were created:
- We consider the chain new_gcd0Gcd'10(vuz113, vuz114, Zero, Succ(vuz1160)) → new_gcd0Gcd'14(Succ(vuz113), vuz114), new_gcd0Gcd'14(Succ(vuz1070), Succ(vuz1080)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070) which results in the following constraint:
(5) (new_gcd0Gcd'14(Succ(x15), x16)=new_gcd0Gcd'14(Succ(x18), Succ(x19)) ⇒ new_gcd0Gcd'14(Succ(x18), Succ(x19))≥new_gcd0Gcd'10(x19, Succ(x18), x19, x18))
We simplified constraint (5) using rules (I), (II), (III) which results in the following new constraint:
(6) (new_gcd0Gcd'14(Succ(x15), Succ(x19))≥new_gcd0Gcd'10(x19, Succ(x15), x19, x15))
For Pair new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Succ(vuz1160)) → new_gcd0Gcd'10(vuz113, vuz114, vuz1150, vuz1160) the following chains were created:
- We consider the chain new_gcd0Gcd'14(Succ(vuz1070), Succ(vuz1080)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070), new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Succ(vuz1160)) → new_gcd0Gcd'10(vuz113, vuz114, vuz1150, vuz1160) which results in the following constraint:
(7) (new_gcd0Gcd'10(x30, Succ(x29), x30, x29)=new_gcd0Gcd'10(x31, x32, Succ(x33), Succ(x34)) ⇒ new_gcd0Gcd'10(x31, x32, Succ(x33), Succ(x34))≥new_gcd0Gcd'10(x31, x32, x33, x34))
We simplified constraint (7) using rules (I), (II), (III) which results in the following new constraint:
(8) (new_gcd0Gcd'10(Succ(x33), Succ(Succ(x34)), Succ(x33), Succ(x34))≥new_gcd0Gcd'10(Succ(x33), Succ(Succ(x34)), x33, x34))
- We consider the chain new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Succ(vuz1160)) → new_gcd0Gcd'10(vuz113, vuz114, vuz1150, vuz1160), new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Succ(vuz1160)) → new_gcd0Gcd'10(vuz113, vuz114, vuz1150, vuz1160) which results in the following constraint:
(9) (new_gcd0Gcd'10(x35, x36, x37, x38)=new_gcd0Gcd'10(x39, x40, Succ(x41), Succ(x42)) ⇒ new_gcd0Gcd'10(x39, x40, Succ(x41), Succ(x42))≥new_gcd0Gcd'10(x39, x40, x41, x42))
We simplified constraint (9) using rules (I), (II), (III) which results in the following new constraint:
(10) (new_gcd0Gcd'10(x35, x36, Succ(x41), Succ(x42))≥new_gcd0Gcd'10(x35, x36, x41, x42))
To summarize, we get the following constraints P≥ for the following pairs.
- new_gcd0Gcd'10(vuz113, vuz114, Zero, Succ(vuz1160)) → new_gcd0Gcd'14(Succ(vuz113), vuz114)
- (new_gcd0Gcd'10(Zero, Succ(Succ(x7)), Zero, Succ(x7))≥new_gcd0Gcd'14(Succ(Zero), Succ(Succ(x7))))
- (new_gcd0Gcd'10(x8, x9, Zero, Succ(x14))≥new_gcd0Gcd'14(Succ(x8), x9))
- new_gcd0Gcd'14(Succ(vuz1070), Succ(vuz1080)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070)
- (new_gcd0Gcd'14(Succ(x15), Succ(x19))≥new_gcd0Gcd'10(x19, Succ(x15), x19, x15))
- new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Succ(vuz1160)) → new_gcd0Gcd'10(vuz113, vuz114, vuz1150, vuz1160)
- (new_gcd0Gcd'10(Succ(x33), Succ(Succ(x34)), Succ(x33), Succ(x34))≥new_gcd0Gcd'10(Succ(x33), Succ(Succ(x34)), x33, x34))
- (new_gcd0Gcd'10(x35, x36, Succ(x41), Succ(x42))≥new_gcd0Gcd'10(x35, x36, x41, x42))
The constraints for P> respective Pbound are constructed from P≥ where we just replace every occurence of "t ≥ s" in P≥ by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation [18]:
POL(Succ(x1)) = 1 + x1
POL(Zero) = 0
POL(c) = -1
POL(new_gcd0Gcd'10(x1, x2, x3, x4)) = -1 + x1 - x3 + x4
POL(new_gcd0Gcd'14(x1, x2)) = -1 + x1
The following pairs are in P>:
new_gcd0Gcd'14(Succ(vuz1070), Succ(vuz1080)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070)
The following pairs are in Pbound:
new_gcd0Gcd'10(vuz113, vuz114, Zero, Succ(vuz1160)) → new_gcd0Gcd'14(Succ(vuz113), vuz114)
new_gcd0Gcd'14(Succ(vuz1070), Succ(vuz1080)) → new_gcd0Gcd'10(vuz1080, Succ(vuz1070), vuz1080, vuz1070)
There are no usable rules
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ NonInfProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_gcd0Gcd'10(vuz113, vuz114, Zero, Succ(vuz1160)) → new_gcd0Gcd'14(Succ(vuz113), vuz114)
new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Succ(vuz1160)) → new_gcd0Gcd'10(vuz113, vuz114, vuz1150, vuz1160)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ NonInfProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Succ(vuz1160)) → new_gcd0Gcd'10(vuz113, vuz114, vuz1150, vuz1160)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- new_gcd0Gcd'10(vuz113, vuz114, Succ(vuz1150), Succ(vuz1160)) → new_gcd0Gcd'10(vuz113, vuz114, vuz1150, vuz1160)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vuz133, vuz134, Zero, Zero) → new_primDivNatS00(vuz133, vuz134)
new_primDivNatS0(vuz133, vuz134, Succ(vuz1350), Succ(vuz1360)) → new_primDivNatS0(vuz133, vuz134, vuz1350, vuz1360)
new_primDivNatS(Succ(vuz1420), Zero, vuz144) → new_primDivNatS1(vuz1420, vuz144)
new_primDivNatS(Succ(vuz1420), Succ(vuz1430), vuz144) → new_primDivNatS(vuz1420, vuz1430, vuz144)
new_primDivNatS1(Succ(vuz990), Zero) → new_primDivNatS(Succ(vuz990), Zero, Zero)
new_primDivNatS1(Succ(vuz990), Succ(vuz1000)) → new_primDivNatS0(vuz990, vuz1000, vuz990, vuz1000)
new_primDivNatS00(vuz133, vuz134) → new_primDivNatS(Succ(vuz133), Succ(vuz134), Succ(vuz134))
new_primDivNatS1(Zero, Zero) → new_primDivNatS(Zero, Zero, Zero)
new_primDivNatS0(vuz133, vuz134, Succ(vuz1350), Zero) → new_primDivNatS(Succ(vuz133), Succ(vuz134), Succ(vuz134))
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vuz133, vuz134, Zero, Zero) → new_primDivNatS00(vuz133, vuz134)
new_primDivNatS0(vuz133, vuz134, Succ(vuz1350), Succ(vuz1360)) → new_primDivNatS0(vuz133, vuz134, vuz1350, vuz1360)
new_primDivNatS(Succ(vuz1420), Zero, vuz144) → new_primDivNatS1(vuz1420, vuz144)
new_primDivNatS(Succ(vuz1420), Succ(vuz1430), vuz144) → new_primDivNatS(vuz1420, vuz1430, vuz144)
new_primDivNatS1(Succ(vuz990), Zero) → new_primDivNatS(Succ(vuz990), Zero, Zero)
new_primDivNatS1(Succ(vuz990), Succ(vuz1000)) → new_primDivNatS0(vuz990, vuz1000, vuz990, vuz1000)
new_primDivNatS00(vuz133, vuz134) → new_primDivNatS(Succ(vuz133), Succ(vuz134), Succ(vuz134))
new_primDivNatS0(vuz133, vuz134, Succ(vuz1350), Zero) → new_primDivNatS(Succ(vuz133), Succ(vuz134), Succ(vuz134))
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
new_primDivNatS(Succ(vuz1420), Succ(vuz1430), vuz144) → new_primDivNatS(vuz1420, vuz1430, vuz144)
new_primDivNatS1(Succ(vuz990), Zero) → new_primDivNatS(Succ(vuz990), Zero, Zero)
new_primDivNatS1(Succ(vuz990), Succ(vuz1000)) → new_primDivNatS0(vuz990, vuz1000, vuz990, vuz1000)
The remaining pairs can at least be oriented weakly.
new_primDivNatS0(vuz133, vuz134, Zero, Zero) → new_primDivNatS00(vuz133, vuz134)
new_primDivNatS0(vuz133, vuz134, Succ(vuz1350), Succ(vuz1360)) → new_primDivNatS0(vuz133, vuz134, vuz1350, vuz1360)
new_primDivNatS(Succ(vuz1420), Zero, vuz144) → new_primDivNatS1(vuz1420, vuz144)
new_primDivNatS00(vuz133, vuz134) → new_primDivNatS(Succ(vuz133), Succ(vuz134), Succ(vuz134))
new_primDivNatS0(vuz133, vuz134, Succ(vuz1350), Zero) → new_primDivNatS(Succ(vuz133), Succ(vuz134), Succ(vuz134))
Used ordering: Polynomial interpretation [25]:
POL(Succ(x1)) = 1 + x1
POL(Zero) = 0
POL(new_primDivNatS(x1, x2, x3)) = x1
POL(new_primDivNatS0(x1, x2, x3, x4)) = 1 + x1
POL(new_primDivNatS00(x1, x2)) = 1 + x1
POL(new_primDivNatS1(x1, x2)) = 1 + x1
The following usable rules [17] were oriented:
none
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vuz133, vuz134, Zero, Zero) → new_primDivNatS00(vuz133, vuz134)
new_primDivNatS(Succ(vuz1420), Zero, vuz144) → new_primDivNatS1(vuz1420, vuz144)
new_primDivNatS0(vuz133, vuz134, Succ(vuz1350), Succ(vuz1360)) → new_primDivNatS0(vuz133, vuz134, vuz1350, vuz1360)
new_primDivNatS00(vuz133, vuz134) → new_primDivNatS(Succ(vuz133), Succ(vuz134), Succ(vuz134))
new_primDivNatS0(vuz133, vuz134, Succ(vuz1350), Zero) → new_primDivNatS(Succ(vuz133), Succ(vuz134), Succ(vuz134))
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 4 less nodes.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vuz133, vuz134, Succ(vuz1350), Succ(vuz1360)) → new_primDivNatS0(vuz133, vuz134, vuz1350, vuz1360)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- new_primDivNatS0(vuz133, vuz134, Succ(vuz1350), Succ(vuz1360)) → new_primDivNatS0(vuz133, vuz134, vuz1350, vuz1360)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primQuotInt(Succ(vuz1200)) → new_primQuotInt(vuz1200)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- new_primQuotInt(Succ(vuz1200)) → new_primQuotInt(vuz1200)
The graph contains the following edges 1 > 1
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
Q DP problem:
The TRS P consists of the following rules:
new_primQuotInt0(Zero, Succ(vuz1400)) → new_primQuotInt0(Zero, vuz1400)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- new_primQuotInt0(Zero, Succ(vuz1400)) → new_primQuotInt0(Zero, vuz1400)
The graph contains the following edges 1 >= 1, 2 > 2