YES 16.019000000000002
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
↳ LR
mainModule Main
| ((divMod :: Int -> Int -> (Int,Int)) :: Int -> Int -> (Int,Int)) |
module Main where
Lambda Reductions:
The following Lambda expression
\(_,r)→r
is transformed to
The following Lambda expression
\qr→qr
is transformed to
The following Lambda expression
\(q,_)→q
is transformed to
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
mainModule Main
| ((divMod :: Int -> Int -> (Int,Int)) :: Int -> Int -> (Int,Int)) |
module Main where
If Reductions:
The following If expression
if signum r == `negate` signum d then (q - 1,r + d) else qr
is transformed to
divMod0 | d True | = (q - 1,r + d) |
divMod0 | d False | = qr |
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
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
mainModule Main
| ((divMod :: Int -> Int -> (Int,Int)) :: Int -> Int -> (Int,Int)) |
module Main where
Replaced joker patterns by fresh variables and removed binding patterns.
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
mainModule Main
| ((divMod :: Int -> Int -> (Int,Int)) :: Int -> Int -> (Int,Int)) |
module Main where
Cond Reductions:
The following Function with conditions
is transformed to
undefined0 | True | = undefined |
undefined1 | | = undefined0 False |
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
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
mainModule Main
| ((divMod :: Int -> Int -> (Int,Int)) :: Int -> Int -> (Int,Int)) |
module Main where
Let/Where Reductions:
The bindings of the following Let/Where expression
divMod0 d (signum r == `negate` signum d) |
where |
divMod0 | d True | = (q - 1,r + d) |
divMod0 | d False | = qr |
|
| |
| |
| |
| |
| |
| |
| |
are unpacked to the following functions on top level
divModR0 | wz xu (vw,r) | = r |
divModDivMod0 | wz xu d True | = (divModQ wz xu - 1,divModR wz xu + d) |
divModDivMod0 | wz xu d False | = divModQr wz xu |
divModR | wz xu | = divModR0 wz xu (divModVu5 wz xu) |
divModQ1 | wz xu (q,vv) | = q |
divModVu5 | wz xu | = quotRem wz xu |
divModQr | wz xu | = divModQr0 wz xu (divModVu5 wz xu) |
divModQ | wz xu | = divModQ1 wz xu (divModVu5 wz xu) |
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
mainModule Main
| ((divMod :: Int -> Int -> (Int,Int)) :: Int -> Int -> (Int,Int)) |
module Main where
Num Reduction: All numbers are transformed to thier corresponding representation with Pos, Neg, Succ and Zero.
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
mainModule Main
| (divMod :: Int -> Int -> (Int,Int)) |
module Main where
Haskell To QDPs
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primPlusNat(Succ(xv2721000), Succ(xv950)) → new_primPlusNat(xv2721000, xv950)
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_primPlusNat(Succ(xv2721000), Succ(xv950)) → new_primPlusNat(xv2721000, xv950)
The graph contains the following edges 1 > 1, 2 > 2
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primMinusNat(Succ(xv2721000), Succ(xv950)) → new_primMinusNat(xv2721000, xv950)
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_primMinusNat(Succ(xv2721000), Succ(xv950)) → new_primMinusNat(xv2721000, xv950)
The graph contains the following edges 1 > 1, 2 > 2
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primModNatS0(Succ(xv3000), Zero) → new_primModNatS(Succ(xv3000), Zero, Zero)
new_primModNatS00(xv286, xv287, Succ(xv2880), Succ(xv2890)) → new_primModNatS00(xv286, xv287, xv2880, xv2890)
new_primModNatS1(xv169, xv170) → new_primModNatS0(xv169, xv170)
new_primModNatS01(xv286, xv287) → new_primModNatS(Succ(xv286), Succ(xv287), Succ(xv287))
new_primModNatS0(Succ(xv3000), Succ(xv4000)) → new_primModNatS00(xv3000, xv4000, xv3000, xv4000)
new_primModNatS0(Zero, Zero) → new_primModNatS(Zero, Zero, Zero)
new_primModNatS00(xv286, xv287, Zero, Zero) → new_primModNatS01(xv286, xv287)
new_primModNatS(Succ(xv2910), Succ(xv2920), xv293) → new_primModNatS(xv2910, xv2920, xv293)
new_primModNatS(Succ(xv2910), Zero, xv293) → new_primModNatS1(xv2910, xv293)
new_primModNatS00(xv286, xv287, Succ(xv2880), Zero) → new_primModNatS(Succ(xv286), Succ(xv287), Succ(xv287))
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
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primModNatS00(xv286, xv287, Succ(xv2880), Succ(xv2890)) → new_primModNatS00(xv286, xv287, xv2880, xv2890)
new_primModNatS1(xv169, xv170) → new_primModNatS0(xv169, xv170)
new_primModNatS0(Succ(xv3000), Zero) → new_primModNatS(Succ(xv3000), Zero, Zero)
new_primModNatS01(xv286, xv287) → new_primModNatS(Succ(xv286), Succ(xv287), Succ(xv287))
new_primModNatS0(Succ(xv3000), Succ(xv4000)) → new_primModNatS00(xv3000, xv4000, xv3000, xv4000)
new_primModNatS00(xv286, xv287, Zero, Zero) → new_primModNatS01(xv286, xv287)
new_primModNatS(Succ(xv2910), Succ(xv2920), xv293) → new_primModNatS(xv2910, xv2920, xv293)
new_primModNatS(Succ(xv2910), Zero, xv293) → new_primModNatS1(xv2910, xv293)
new_primModNatS00(xv286, xv287, Succ(xv2880), Zero) → new_primModNatS(Succ(xv286), Succ(xv287), Succ(xv287))
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_primModNatS1(xv169, xv170) → new_primModNatS0(xv169, xv170)
new_primModNatS(Succ(xv2910), Succ(xv2920), xv293) → new_primModNatS(xv2910, xv2920, xv293)
The remaining pairs can at least be oriented weakly.
new_primModNatS00(xv286, xv287, Succ(xv2880), Succ(xv2890)) → new_primModNatS00(xv286, xv287, xv2880, xv2890)
new_primModNatS0(Succ(xv3000), Zero) → new_primModNatS(Succ(xv3000), Zero, Zero)
new_primModNatS01(xv286, xv287) → new_primModNatS(Succ(xv286), Succ(xv287), Succ(xv287))
new_primModNatS0(Succ(xv3000), Succ(xv4000)) → new_primModNatS00(xv3000, xv4000, xv3000, xv4000)
new_primModNatS00(xv286, xv287, Zero, Zero) → new_primModNatS01(xv286, xv287)
new_primModNatS(Succ(xv2910), Zero, xv293) → new_primModNatS1(xv2910, xv293)
new_primModNatS00(xv286, xv287, Succ(xv2880), Zero) → new_primModNatS(Succ(xv286), Succ(xv287), Succ(xv287))
Used ordering: Polynomial interpretation [25]:
POL(Succ(x1)) = 1 + x1
POL(Zero) = 0
POL(new_primModNatS(x1, x2, x3)) = x1
POL(new_primModNatS0(x1, x2)) = x1
POL(new_primModNatS00(x1, x2, x3, x4)) = 1 + x1
POL(new_primModNatS01(x1, x2)) = 1 + x1
POL(new_primModNatS1(x1, x2)) = 1 + x1
The following usable rules [17] were oriented:
none
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primModNatS0(Succ(xv3000), Zero) → new_primModNatS(Succ(xv3000), Zero, Zero)
new_primModNatS00(xv286, xv287, Succ(xv2880), Succ(xv2890)) → new_primModNatS00(xv286, xv287, xv2880, xv2890)
new_primModNatS01(xv286, xv287) → new_primModNatS(Succ(xv286), Succ(xv287), Succ(xv287))
new_primModNatS0(Succ(xv3000), Succ(xv4000)) → new_primModNatS00(xv3000, xv4000, xv3000, xv4000)
new_primModNatS00(xv286, xv287, Zero, Zero) → new_primModNatS01(xv286, xv287)
new_primModNatS(Succ(xv2910), Zero, xv293) → new_primModNatS1(xv2910, xv293)
new_primModNatS00(xv286, xv287, Succ(xv2880), Zero) → new_primModNatS(Succ(xv286), Succ(xv287), Succ(xv287))
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
↳ LR
↳ 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
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primModNatS00(xv286, xv287, Succ(xv2880), Succ(xv2890)) → new_primModNatS00(xv286, xv287, xv2880, xv2890)
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_primModNatS00(xv286, xv287, Succ(xv2880), Succ(xv2890)) → new_primModNatS00(xv286, xv287, xv2880, xv2890)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(xv312, xv313, Zero, Zero) → new_primDivNatS00(xv312, xv313)
new_primDivNatS0(xv312, xv313, Succ(xv3140), Succ(xv3150)) → new_primDivNatS0(xv312, xv313, xv3140, xv3150)
new_primDivNatS(Succ(xv3170), Zero, xv319) → new_primDivNatS1(xv3170, xv319)
new_primDivNatS(Succ(xv3170), Succ(xv3180), xv319) → new_primDivNatS(xv3170, xv3180, xv319)
new_primDivNatS1(Succ(xv1690), Zero) → new_primDivNatS(Succ(xv1690), Zero, Zero)
new_primDivNatS1(Succ(xv1690), Succ(xv1700)) → new_primDivNatS0(xv1690, xv1700, xv1690, xv1700)
new_primDivNatS00(xv312, xv313) → new_primDivNatS(Succ(xv312), Succ(xv313), Succ(xv313))
new_primDivNatS1(Zero, Zero) → new_primDivNatS(Zero, Zero, Zero)
new_primDivNatS0(xv312, xv313, Succ(xv3140), Zero) → new_primDivNatS(Succ(xv312), Succ(xv313), Succ(xv313))
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
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(xv312, xv313, Zero, Zero) → new_primDivNatS00(xv312, xv313)
new_primDivNatS0(xv312, xv313, Succ(xv3140), Succ(xv3150)) → new_primDivNatS0(xv312, xv313, xv3140, xv3150)
new_primDivNatS(Succ(xv3170), Zero, xv319) → new_primDivNatS1(xv3170, xv319)
new_primDivNatS(Succ(xv3170), Succ(xv3180), xv319) → new_primDivNatS(xv3170, xv3180, xv319)
new_primDivNatS1(Succ(xv1690), Zero) → new_primDivNatS(Succ(xv1690), Zero, Zero)
new_primDivNatS1(Succ(xv1690), Succ(xv1700)) → new_primDivNatS0(xv1690, xv1700, xv1690, xv1700)
new_primDivNatS00(xv312, xv313) → new_primDivNatS(Succ(xv312), Succ(xv313), Succ(xv313))
new_primDivNatS0(xv312, xv313, Succ(xv3140), Zero) → new_primDivNatS(Succ(xv312), Succ(xv313), Succ(xv313))
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(xv3170), Succ(xv3180), xv319) → new_primDivNatS(xv3170, xv3180, xv319)
new_primDivNatS1(Succ(xv1690), Zero) → new_primDivNatS(Succ(xv1690), Zero, Zero)
new_primDivNatS1(Succ(xv1690), Succ(xv1700)) → new_primDivNatS0(xv1690, xv1700, xv1690, xv1700)
The remaining pairs can at least be oriented weakly.
new_primDivNatS0(xv312, xv313, Zero, Zero) → new_primDivNatS00(xv312, xv313)
new_primDivNatS0(xv312, xv313, Succ(xv3140), Succ(xv3150)) → new_primDivNatS0(xv312, xv313, xv3140, xv3150)
new_primDivNatS(Succ(xv3170), Zero, xv319) → new_primDivNatS1(xv3170, xv319)
new_primDivNatS00(xv312, xv313) → new_primDivNatS(Succ(xv312), Succ(xv313), Succ(xv313))
new_primDivNatS0(xv312, xv313, Succ(xv3140), Zero) → new_primDivNatS(Succ(xv312), Succ(xv313), Succ(xv313))
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
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(xv312, xv313, Zero, Zero) → new_primDivNatS00(xv312, xv313)
new_primDivNatS(Succ(xv3170), Zero, xv319) → new_primDivNatS1(xv3170, xv319)
new_primDivNatS0(xv312, xv313, Succ(xv3140), Succ(xv3150)) → new_primDivNatS0(xv312, xv313, xv3140, xv3150)
new_primDivNatS00(xv312, xv313) → new_primDivNatS(Succ(xv312), Succ(xv313), Succ(xv313))
new_primDivNatS0(xv312, xv313, Succ(xv3140), Zero) → new_primDivNatS(Succ(xv312), Succ(xv313), Succ(xv313))
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
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(xv312, xv313, Succ(xv3140), Succ(xv3150)) → new_primDivNatS0(xv312, xv313, xv3140, xv3150)
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(xv312, xv313, Succ(xv3140), Succ(xv3150)) → new_primDivNatS0(xv312, xv313, xv3140, xv3150)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_divModDivMod00(xv141, Zero, Succ(Succ(xv14300)), Zero) → new_divModDivMod00(xv141, Zero, Succ(xv14300), Zero)
new_divModDivMod0(xv255, xv256, xv257, Succ(xv2580), Zero) → new_divModDivMod00(xv255, xv256, Succ(xv257), xv256)
new_divModDivMod01(xv255, xv256, xv257) → new_divModDivMod00(xv255, xv256, Succ(xv257), xv256)
new_divModDivMod0(xv255, xv256, xv257, Zero, Zero) → new_divModDivMod01(xv255, xv256, xv257)
new_divModDivMod00(xv141, xv142, Succ(xv1430), Succ(xv1440)) → new_divModDivMod00(xv141, xv142, xv1430, xv1440)
new_divModDivMod00(xv141, Succ(xv1420), Succ(Succ(xv14300)), Zero) → new_divModDivMod0(xv141, Succ(xv1420), xv14300, xv14300, xv1420)
new_divModDivMod00(xv141, Zero, Succ(Zero), Zero) → new_divModDivMod00(xv141, Zero, Zero, Zero)
new_divModDivMod0(xv255, xv256, xv257, Succ(xv2580), Succ(xv2590)) → new_divModDivMod0(xv255, xv256, xv257, xv2580, xv2590)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 1 less node.
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_divModDivMod00(xv141, Zero, Succ(Succ(xv14300)), Zero) → new_divModDivMod00(xv141, Zero, Succ(xv14300), Zero)
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_divModDivMod00(xv141, Zero, Succ(Succ(xv14300)), Zero) → new_divModDivMod00(xv141, Zero, Succ(xv14300), Zero)
The graph contains the following edges 1 >= 1, 2 >= 2, 4 >= 2, 3 > 3, 2 >= 4, 4 >= 4
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_divModDivMod01(xv255, xv256, xv257) → new_divModDivMod00(xv255, xv256, Succ(xv257), xv256)
new_divModDivMod0(xv255, xv256, xv257, Succ(xv2580), Zero) → new_divModDivMod00(xv255, xv256, Succ(xv257), xv256)
new_divModDivMod0(xv255, xv256, xv257, Zero, Zero) → new_divModDivMod01(xv255, xv256, xv257)
new_divModDivMod00(xv141, xv142, Succ(xv1430), Succ(xv1440)) → new_divModDivMod00(xv141, xv142, xv1430, xv1440)
new_divModDivMod00(xv141, Succ(xv1420), Succ(Succ(xv14300)), Zero) → new_divModDivMod0(xv141, Succ(xv1420), xv14300, xv14300, xv1420)
new_divModDivMod0(xv255, xv256, xv257, Succ(xv2580), Succ(xv2590)) → new_divModDivMod0(xv255, xv256, xv257, xv2580, xv2590)
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_divModDivMod00(xv141, xv142, Succ(xv1430), Succ(xv1440)) → new_divModDivMod00(xv141, xv142, xv1430, xv1440)
new_divModDivMod00(xv141, Succ(xv1420), Succ(Succ(xv14300)), Zero) → new_divModDivMod0(xv141, Succ(xv1420), xv14300, xv14300, xv1420)
The remaining pairs can at least be oriented weakly.
new_divModDivMod01(xv255, xv256, xv257) → new_divModDivMod00(xv255, xv256, Succ(xv257), xv256)
new_divModDivMod0(xv255, xv256, xv257, Succ(xv2580), Zero) → new_divModDivMod00(xv255, xv256, Succ(xv257), xv256)
new_divModDivMod0(xv255, xv256, xv257, Zero, Zero) → new_divModDivMod01(xv255, xv256, xv257)
new_divModDivMod0(xv255, xv256, xv257, Succ(xv2580), Succ(xv2590)) → new_divModDivMod0(xv255, xv256, xv257, xv2580, xv2590)
Used ordering: Polynomial interpretation [25]:
POL(Succ(x1)) = 1 + x1
POL(Zero) = 0
POL(new_divModDivMod0(x1, x2, x3, x4, x5)) = 1 + x3
POL(new_divModDivMod00(x1, x2, x3, x4)) = x3
POL(new_divModDivMod01(x1, x2, x3)) = 1 + x3
The following usable rules [17] were oriented:
none
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_divModDivMod0(xv255, xv256, xv257, Succ(xv2580), Zero) → new_divModDivMod00(xv255, xv256, Succ(xv257), xv256)
new_divModDivMod01(xv255, xv256, xv257) → new_divModDivMod00(xv255, xv256, Succ(xv257), xv256)
new_divModDivMod0(xv255, xv256, xv257, Zero, Zero) → new_divModDivMod01(xv255, xv256, xv257)
new_divModDivMod0(xv255, xv256, xv257, Succ(xv2580), Succ(xv2590)) → new_divModDivMod0(xv255, xv256, xv257, xv2580, xv2590)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 3 less nodes.
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_divModDivMod0(xv255, xv256, xv257, Succ(xv2580), Succ(xv2590)) → new_divModDivMod0(xv255, xv256, xv257, xv2580, xv2590)
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_divModDivMod0(xv255, xv256, xv257, Succ(xv2580), Succ(xv2590)) → new_divModDivMod0(xv255, xv256, xv257, xv2580, xv2590)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4, 5 > 5
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_divModDivMod02(xv62, xv63, Succ(xv640), Succ(xv650)) → new_divModDivMod02(xv62, xv63, xv640, xv650)
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_divModDivMod02(xv62, xv63, Succ(xv640), Succ(xv650)) → new_divModDivMod02(xv62, xv63, xv640, xv650)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_divModDivMod04(xv174, Zero, Succ(Succ(xv17600)), Zero) → new_divModDivMod04(xv174, Zero, Succ(xv17600), Zero)
new_divModDivMod04(xv174, Zero, Succ(Zero), Zero) → new_divModDivMod04(xv174, Zero, Zero, Zero)
new_divModDivMod03(xv249, xv250, xv251, Succ(xv2520), Succ(xv2530)) → new_divModDivMod03(xv249, xv250, xv251, xv2520, xv2530)
new_divModDivMod05(xv249, xv250, xv251) → new_divModDivMod04(xv249, xv250, Succ(xv251), xv250)
new_divModDivMod04(xv174, Succ(xv1750), Succ(Succ(xv17600)), Zero) → new_divModDivMod03(xv174, Succ(xv1750), xv17600, xv17600, xv1750)
new_divModDivMod04(xv174, xv175, Succ(xv1760), Succ(xv1770)) → new_divModDivMod04(xv174, xv175, xv1760, xv1770)
new_divModDivMod03(xv249, xv250, xv251, Succ(xv2520), Zero) → new_divModDivMod04(xv249, xv250, Succ(xv251), xv250)
new_divModDivMod03(xv249, xv250, xv251, Zero, Zero) → new_divModDivMod05(xv249, xv250, xv251)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 1 less node.
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_divModDivMod04(xv174, Zero, Succ(Succ(xv17600)), Zero) → new_divModDivMod04(xv174, Zero, Succ(xv17600), Zero)
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_divModDivMod04(xv174, Zero, Succ(Succ(xv17600)), Zero) → new_divModDivMod04(xv174, Zero, Succ(xv17600), Zero)
The graph contains the following edges 1 >= 1, 2 >= 2, 4 >= 2, 3 > 3, 2 >= 4, 4 >= 4
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_divModDivMod05(xv249, xv250, xv251) → new_divModDivMod04(xv249, xv250, Succ(xv251), xv250)
new_divModDivMod03(xv249, xv250, xv251, Succ(xv2520), Succ(xv2530)) → new_divModDivMod03(xv249, xv250, xv251, xv2520, xv2530)
new_divModDivMod04(xv174, Succ(xv1750), Succ(Succ(xv17600)), Zero) → new_divModDivMod03(xv174, Succ(xv1750), xv17600, xv17600, xv1750)
new_divModDivMod04(xv174, xv175, Succ(xv1760), Succ(xv1770)) → new_divModDivMod04(xv174, xv175, xv1760, xv1770)
new_divModDivMod03(xv249, xv250, xv251, Succ(xv2520), Zero) → new_divModDivMod04(xv249, xv250, Succ(xv251), xv250)
new_divModDivMod03(xv249, xv250, xv251, Zero, Zero) → new_divModDivMod05(xv249, xv250, xv251)
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_divModDivMod04(xv174, Succ(xv1750), Succ(Succ(xv17600)), Zero) → new_divModDivMod03(xv174, Succ(xv1750), xv17600, xv17600, xv1750)
new_divModDivMod04(xv174, xv175, Succ(xv1760), Succ(xv1770)) → new_divModDivMod04(xv174, xv175, xv1760, xv1770)
The remaining pairs can at least be oriented weakly.
new_divModDivMod05(xv249, xv250, xv251) → new_divModDivMod04(xv249, xv250, Succ(xv251), xv250)
new_divModDivMod03(xv249, xv250, xv251, Succ(xv2520), Succ(xv2530)) → new_divModDivMod03(xv249, xv250, xv251, xv2520, xv2530)
new_divModDivMod03(xv249, xv250, xv251, Succ(xv2520), Zero) → new_divModDivMod04(xv249, xv250, Succ(xv251), xv250)
new_divModDivMod03(xv249, xv250, xv251, Zero, Zero) → new_divModDivMod05(xv249, xv250, xv251)
Used ordering: Polynomial interpretation [25]:
POL(Succ(x1)) = 1 + x1
POL(Zero) = 0
POL(new_divModDivMod03(x1, x2, x3, x4, x5)) = 1 + x3
POL(new_divModDivMod04(x1, x2, x3, x4)) = x3
POL(new_divModDivMod05(x1, x2, x3)) = 1 + x3
The following usable rules [17] were oriented:
none
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_divModDivMod03(xv249, xv250, xv251, Succ(xv2520), Succ(xv2530)) → new_divModDivMod03(xv249, xv250, xv251, xv2520, xv2530)
new_divModDivMod05(xv249, xv250, xv251) → new_divModDivMod04(xv249, xv250, Succ(xv251), xv250)
new_divModDivMod03(xv249, xv250, xv251, Zero, Zero) → new_divModDivMod05(xv249, xv250, xv251)
new_divModDivMod03(xv249, xv250, xv251, Succ(xv2520), Zero) → new_divModDivMod04(xv249, xv250, Succ(xv251), xv250)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 3 less nodes.
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_divModDivMod03(xv249, xv250, xv251, Succ(xv2520), Succ(xv2530)) → new_divModDivMod03(xv249, xv250, xv251, xv2520, xv2530)
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_divModDivMod03(xv249, xv250, xv251, Succ(xv2520), Succ(xv2530)) → new_divModDivMod03(xv249, xv250, xv251, xv2520, xv2530)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4, 5 > 5
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_divModDivMod06(xv94, xv95, Succ(xv960), Succ(xv970)) → new_divModDivMod06(xv94, xv95, xv960, xv970)
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_divModDivMod06(xv94, xv95, Succ(xv960), Succ(xv970)) → new_divModDivMod06(xv94, xv95, xv960, xv970)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_divModDivMod08(xv243, xv244, xv245, Succ(xv2460), Succ(xv2470)) → new_divModDivMod08(xv243, xv244, xv245, xv2460, xv2470)
new_divModDivMod07(xv169, xv170, Succ(xv1710), Succ(xv1720)) → new_divModDivMod07(xv169, xv170, xv1710, xv1720)
new_divModDivMod07(xv169, Zero, Succ(Succ(xv17100)), Zero) → new_divModDivMod07(xv169, Zero, Succ(xv17100), Zero)
new_divModDivMod08(xv243, xv244, xv245, Succ(xv2460), Zero) → new_divModDivMod07(xv243, xv244, Succ(xv245), xv244)
new_divModDivMod08(xv243, xv244, xv245, Zero, Zero) → new_divModDivMod09(xv243, xv244, xv245)
new_divModDivMod07(xv169, Succ(xv1700), Succ(Succ(xv17100)), Zero) → new_divModDivMod08(xv169, Succ(xv1700), xv17100, xv17100, xv1700)
new_divModDivMod09(xv243, xv244, xv245) → new_divModDivMod07(xv243, xv244, Succ(xv245), xv244)
new_divModDivMod07(xv169, Zero, Succ(Zero), Zero) → new_divModDivMod07(xv169, Zero, Zero, Zero)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 1 less node.
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_divModDivMod07(xv169, Zero, Succ(Succ(xv17100)), Zero) → new_divModDivMod07(xv169, Zero, Succ(xv17100), Zero)
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_divModDivMod07(xv169, Zero, Succ(Succ(xv17100)), Zero) → new_divModDivMod07(xv169, Zero, Succ(xv17100), Zero)
The graph contains the following edges 1 >= 1, 2 >= 2, 4 >= 2, 3 > 3, 2 >= 4, 4 >= 4
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_divModDivMod08(xv243, xv244, xv245, Succ(xv2460), Succ(xv2470)) → new_divModDivMod08(xv243, xv244, xv245, xv2460, xv2470)
new_divModDivMod08(xv243, xv244, xv245, Succ(xv2460), Zero) → new_divModDivMod07(xv243, xv244, Succ(xv245), xv244)
new_divModDivMod07(xv169, xv170, Succ(xv1710), Succ(xv1720)) → new_divModDivMod07(xv169, xv170, xv1710, xv1720)
new_divModDivMod08(xv243, xv244, xv245, Zero, Zero) → new_divModDivMod09(xv243, xv244, xv245)
new_divModDivMod07(xv169, Succ(xv1700), Succ(Succ(xv17100)), Zero) → new_divModDivMod08(xv169, Succ(xv1700), xv17100, xv17100, xv1700)
new_divModDivMod09(xv243, xv244, xv245) → new_divModDivMod07(xv243, xv244, Succ(xv245), xv244)
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_divModDivMod07(xv169, xv170, Succ(xv1710), Succ(xv1720)) → new_divModDivMod07(xv169, xv170, xv1710, xv1720)
new_divModDivMod07(xv169, Succ(xv1700), Succ(Succ(xv17100)), Zero) → new_divModDivMod08(xv169, Succ(xv1700), xv17100, xv17100, xv1700)
The remaining pairs can at least be oriented weakly.
new_divModDivMod08(xv243, xv244, xv245, Succ(xv2460), Succ(xv2470)) → new_divModDivMod08(xv243, xv244, xv245, xv2460, xv2470)
new_divModDivMod08(xv243, xv244, xv245, Succ(xv2460), Zero) → new_divModDivMod07(xv243, xv244, Succ(xv245), xv244)
new_divModDivMod08(xv243, xv244, xv245, Zero, Zero) → new_divModDivMod09(xv243, xv244, xv245)
new_divModDivMod09(xv243, xv244, xv245) → new_divModDivMod07(xv243, xv244, Succ(xv245), xv244)
Used ordering: Polynomial interpretation [25]:
POL(Succ(x1)) = 1 + x1
POL(Zero) = 0
POL(new_divModDivMod07(x1, x2, x3, x4)) = x3
POL(new_divModDivMod08(x1, x2, x3, x4, x5)) = 1 + x3
POL(new_divModDivMod09(x1, x2, x3)) = 1 + x3
The following usable rules [17] were oriented:
none
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_divModDivMod08(xv243, xv244, xv245, Succ(xv2460), Succ(xv2470)) → new_divModDivMod08(xv243, xv244, xv245, xv2460, xv2470)
new_divModDivMod08(xv243, xv244, xv245, Zero, Zero) → new_divModDivMod09(xv243, xv244, xv245)
new_divModDivMod08(xv243, xv244, xv245, Succ(xv2460), Zero) → new_divModDivMod07(xv243, xv244, Succ(xv245), xv244)
new_divModDivMod09(xv243, xv244, xv245) → new_divModDivMod07(xv243, xv244, Succ(xv245), xv244)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 3 less nodes.
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_divModDivMod08(xv243, xv244, xv245, Succ(xv2460), Succ(xv2470)) → new_divModDivMod08(xv243, xv244, xv245, xv2460, xv2470)
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_divModDivMod08(xv243, xv244, xv245, Succ(xv2460), Succ(xv2470)) → new_divModDivMod08(xv243, xv244, xv245, xv2460, xv2470)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4, 5 > 5
↳ HASKELL
↳ LR
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
Q DP problem:
The TRS P consists of the following rules:
new_divModDivMod010(xv89, xv90, Succ(xv910), Succ(xv920)) → new_divModDivMod010(xv89, xv90, xv910, xv920)
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_divModDivMod010(xv89, xv90, Succ(xv910), Succ(xv920)) → new_divModDivMod010(xv89, xv90, xv910, xv920)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4