MAYBE 8.82
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
| ((realToFrac :: Real a => a -> Float) :: Real a => a -> Float) |
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
| ((realToFrac :: Real a => a -> Float) :: Real a => a -> Float) |
module Main where
Replaced joker patterns by fresh variables and removed binding patterns.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
mainModule Main
| ((realToFrac :: Real a => a -> Float) :: Real a => a -> Float) |
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
absReal0 | x True | = `negate` x |
absReal1 | x True | = x |
absReal1 | x False | = absReal0 x otherwise |
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
| ((realToFrac :: Real a => a -> Float) :: Real a => a -> Float) |
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
reduce2Reduce0 | vux vuy x y True | = x `quot` reduce2D vux vuy :% (y `quot` reduce2D vux vuy) |
reduce2D | vux vuy | = gcd vux vuy |
reduce2Reduce1 | vux vuy x y True | = error [] |
reduce2Reduce1 | vux vuy x y False | = reduce2Reduce0 vux vuy x y otherwise |
The bindings of the following Let/Where expression
gcd' (abs x) (abs y) |
where |
gcd' | x xz | = gcd'2 x xz |
gcd' | x y | = gcd'0 x y |
|
|
gcd'0 | x y | = gcd' y (x `rem` y) |
|
|
gcd'1 | True x xz | = x |
gcd'1 | yu yv yw | = gcd'0 yv yw |
|
|
gcd'2 | x xz | = gcd'1 (xz == 0) x xz |
gcd'2 | yx yy | = gcd'0 yx yy |
|
are unpacked to the following functions on top level
gcd0Gcd'1 | True x xz | = x |
gcd0Gcd'1 | yu yv yw | = gcd0Gcd'0 yv yw |
gcd0Gcd'0 | x y | = gcd0Gcd' y (x `rem` y) |
gcd0Gcd' | x xz | = gcd0Gcd'2 x xz |
gcd0Gcd' | x y | = gcd0Gcd'0 x y |
gcd0Gcd'2 | x xz | = gcd0Gcd'1 (xz == 0) x xz |
gcd0Gcd'2 | yx yy | = gcd0Gcd'0 yx yy |
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
mainModule Main
| ((realToFrac :: Real a => a -> Float) :: Real a => a -> Float) |
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
↳ Narrow
mainModule Main
| (realToFrac :: Real a => a -> Float) |
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
↳ 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:
- 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
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primMinusNatS(Succ(vuz12700), Succ(vuz1280)) → new_primMinusNatS(vuz12700, vuz1280)
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_primMinusNatS(Succ(vuz12700), Succ(vuz1280)) → new_primMinusNatS(vuz12700, vuz1280)
The graph contains the following edges 1 > 1, 2 > 2
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vuz180, vuz181, Zero, Zero) → new_primDivNatS00(vuz180, vuz181)
new_primDivNatS(Succ(Succ(vuz4500)), Succ(vuz31000)) → new_primDivNatS0(vuz4500, vuz31000, vuz4500, vuz31000)
new_primDivNatS0(vuz180, vuz181, Succ(vuz1820), Succ(vuz1830)) → new_primDivNatS0(vuz180, vuz181, vuz1820, vuz1830)
new_primDivNatS00(vuz180, vuz181) → new_primDivNatS(new_primMinusNatS2(Succ(vuz180), Succ(vuz181)), Succ(vuz181))
new_primDivNatS(Succ(Succ(vuz4500)), Zero) → new_primDivNatS(new_primMinusNatS0(vuz4500), Zero)
new_primDivNatS0(vuz180, vuz181, Succ(vuz1820), Zero) → new_primDivNatS(new_primMinusNatS2(Succ(vuz180), Succ(vuz181)), Succ(vuz181))
new_primDivNatS(Succ(Zero), Zero) → new_primDivNatS(new_primMinusNatS1, Zero)
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12700), Succ(vuz1280)) → new_primMinusNatS2(vuz12700, vuz1280)
new_primMinusNatS2(Zero, Succ(vuz1280)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12700), Zero) → Succ(vuz12700)
new_primMinusNatS1 → Zero
new_primMinusNatS0(vuz4500) → Succ(vuz4500)
The set Q consists of the following terms:
new_primMinusNatS0(x0)
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS1
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 1 less node.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS(Succ(Succ(vuz4500)), Zero) → new_primDivNatS(new_primMinusNatS0(vuz4500), Zero)
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12700), Succ(vuz1280)) → new_primMinusNatS2(vuz12700, vuz1280)
new_primMinusNatS2(Zero, Succ(vuz1280)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12700), Zero) → Succ(vuz12700)
new_primMinusNatS1 → Zero
new_primMinusNatS0(vuz4500) → Succ(vuz4500)
The set Q consists of the following terms:
new_primMinusNatS0(x0)
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS1
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
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS(Succ(Succ(vuz4500)), Zero) → new_primDivNatS(new_primMinusNatS0(vuz4500), Zero)
The TRS R consists of the following rules:
new_primMinusNatS0(vuz4500) → Succ(vuz4500)
The set Q consists of the following terms:
new_primMinusNatS0(x0)
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS1
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS1
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS(Succ(Succ(vuz4500)), Zero) → new_primDivNatS(new_primMinusNatS0(vuz4500), Zero)
The TRS R consists of the following rules:
new_primMinusNatS0(vuz4500) → Succ(vuz4500)
The set Q consists of the following terms:
new_primMinusNatS0(x0)
We have to consider all minimal (P,Q,R)-chains.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:
new_primDivNatS(Succ(Succ(vuz4500)), Zero) → new_primDivNatS(new_primMinusNatS0(vuz4500), Zero)
Strictly oriented rules of the TRS R:
new_primMinusNatS0(vuz4500) → Succ(vuz4500)
Used ordering: POLO with Polynomial interpretation [25]:
POL(Succ(x1)) = 1 + 2·x1
POL(Zero) = 0
POL(new_primDivNatS(x1, x2)) = x1 + x2
POL(new_primMinusNatS0(x1)) = 2 + 2·x1
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
P is empty.
R is empty.
The set Q consists of the following terms:
new_primMinusNatS0(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
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vuz180, vuz181, Zero, Zero) → new_primDivNatS00(vuz180, vuz181)
new_primDivNatS0(vuz180, vuz181, Succ(vuz1820), Succ(vuz1830)) → new_primDivNatS0(vuz180, vuz181, vuz1820, vuz1830)
new_primDivNatS(Succ(Succ(vuz4500)), Succ(vuz31000)) → new_primDivNatS0(vuz4500, vuz31000, vuz4500, vuz31000)
new_primDivNatS00(vuz180, vuz181) → new_primDivNatS(new_primMinusNatS2(Succ(vuz180), Succ(vuz181)), Succ(vuz181))
new_primDivNatS0(vuz180, vuz181, Succ(vuz1820), Zero) → new_primDivNatS(new_primMinusNatS2(Succ(vuz180), Succ(vuz181)), Succ(vuz181))
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12700), Succ(vuz1280)) → new_primMinusNatS2(vuz12700, vuz1280)
new_primMinusNatS2(Zero, Succ(vuz1280)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12700), Zero) → Succ(vuz12700)
new_primMinusNatS1 → Zero
new_primMinusNatS0(vuz4500) → Succ(vuz4500)
The set Q consists of the following terms:
new_primMinusNatS0(x0)
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS1
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
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vuz180, vuz181, Zero, Zero) → new_primDivNatS00(vuz180, vuz181)
new_primDivNatS0(vuz180, vuz181, Succ(vuz1820), Succ(vuz1830)) → new_primDivNatS0(vuz180, vuz181, vuz1820, vuz1830)
new_primDivNatS(Succ(Succ(vuz4500)), Succ(vuz31000)) → new_primDivNatS0(vuz4500, vuz31000, vuz4500, vuz31000)
new_primDivNatS00(vuz180, vuz181) → new_primDivNatS(new_primMinusNatS2(Succ(vuz180), Succ(vuz181)), Succ(vuz181))
new_primDivNatS0(vuz180, vuz181, Succ(vuz1820), Zero) → new_primDivNatS(new_primMinusNatS2(Succ(vuz180), Succ(vuz181)), Succ(vuz181))
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12700), Succ(vuz1280)) → new_primMinusNatS2(vuz12700, vuz1280)
new_primMinusNatS2(Zero, Succ(vuz1280)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12700), Zero) → Succ(vuz12700)
The set Q consists of the following terms:
new_primMinusNatS0(x0)
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS1
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(x0)
new_primMinusNatS1
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vuz180, vuz181, Zero, Zero) → new_primDivNatS00(vuz180, vuz181)
new_primDivNatS(Succ(Succ(vuz4500)), Succ(vuz31000)) → new_primDivNatS0(vuz4500, vuz31000, vuz4500, vuz31000)
new_primDivNatS0(vuz180, vuz181, Succ(vuz1820), Succ(vuz1830)) → new_primDivNatS0(vuz180, vuz181, vuz1820, vuz1830)
new_primDivNatS00(vuz180, vuz181) → new_primDivNatS(new_primMinusNatS2(Succ(vuz180), Succ(vuz181)), Succ(vuz181))
new_primDivNatS0(vuz180, vuz181, Succ(vuz1820), Zero) → new_primDivNatS(new_primMinusNatS2(Succ(vuz180), Succ(vuz181)), Succ(vuz181))
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12700), Succ(vuz1280)) → new_primMinusNatS2(vuz12700, vuz1280)
new_primMinusNatS2(Zero, Succ(vuz1280)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12700), Zero) → Succ(vuz12700)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_primDivNatS00(vuz180, vuz181) → new_primDivNatS(new_primMinusNatS2(Succ(vuz180), Succ(vuz181)), Succ(vuz181)) at position [0] we obtained the following new rules:
new_primDivNatS00(vuz180, vuz181) → new_primDivNatS(new_primMinusNatS2(vuz180, vuz181), Succ(vuz181))
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vuz180, vuz181, Zero, Zero) → new_primDivNatS00(vuz180, vuz181)
new_primDivNatS0(vuz180, vuz181, Succ(vuz1820), Succ(vuz1830)) → new_primDivNatS0(vuz180, vuz181, vuz1820, vuz1830)
new_primDivNatS(Succ(Succ(vuz4500)), Succ(vuz31000)) → new_primDivNatS0(vuz4500, vuz31000, vuz4500, vuz31000)
new_primDivNatS00(vuz180, vuz181) → new_primDivNatS(new_primMinusNatS2(vuz180, vuz181), Succ(vuz181))
new_primDivNatS0(vuz180, vuz181, Succ(vuz1820), Zero) → new_primDivNatS(new_primMinusNatS2(Succ(vuz180), Succ(vuz181)), Succ(vuz181))
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12700), Succ(vuz1280)) → new_primMinusNatS2(vuz12700, vuz1280)
new_primMinusNatS2(Zero, Succ(vuz1280)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12700), Zero) → Succ(vuz12700)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_primDivNatS0(vuz180, vuz181, Succ(vuz1820), Zero) → new_primDivNatS(new_primMinusNatS2(Succ(vuz180), Succ(vuz181)), Succ(vuz181)) at position [0] we obtained the following new rules:
new_primDivNatS0(vuz180, vuz181, Succ(vuz1820), Zero) → new_primDivNatS(new_primMinusNatS2(vuz180, vuz181), Succ(vuz181))
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vuz180, vuz181, Zero, Zero) → new_primDivNatS00(vuz180, vuz181)
new_primDivNatS0(vuz180, vuz181, Succ(vuz1820), Zero) → new_primDivNatS(new_primMinusNatS2(vuz180, vuz181), Succ(vuz181))
new_primDivNatS(Succ(Succ(vuz4500)), Succ(vuz31000)) → new_primDivNatS0(vuz4500, vuz31000, vuz4500, vuz31000)
new_primDivNatS0(vuz180, vuz181, Succ(vuz1820), Succ(vuz1830)) → new_primDivNatS0(vuz180, vuz181, vuz1820, vuz1830)
new_primDivNatS00(vuz180, vuz181) → new_primDivNatS(new_primMinusNatS2(vuz180, vuz181), Succ(vuz181))
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12700), Succ(vuz1280)) → new_primMinusNatS2(vuz12700, vuz1280)
new_primMinusNatS2(Zero, Succ(vuz1280)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12700), Zero) → Succ(vuz12700)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
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_primDivNatS0(vuz180, vuz181, Succ(vuz1820), Zero) → new_primDivNatS(new_primMinusNatS2(vuz180, vuz181), Succ(vuz181))
new_primDivNatS(Succ(Succ(vuz4500)), Succ(vuz31000)) → new_primDivNatS0(vuz4500, vuz31000, vuz4500, vuz31000)
new_primDivNatS00(vuz180, vuz181) → new_primDivNatS(new_primMinusNatS2(vuz180, vuz181), Succ(vuz181))
The remaining pairs can at least be oriented weakly.
new_primDivNatS0(vuz180, vuz181, Zero, Zero) → new_primDivNatS00(vuz180, vuz181)
new_primDivNatS0(vuz180, vuz181, Succ(vuz1820), Succ(vuz1830)) → new_primDivNatS0(vuz180, vuz181, vuz1820, vuz1830)
Used ordering: Polynomial interpretation [25]:
POL(Succ(x1)) = 1 + x1
POL(Zero) = 0
POL(new_primDivNatS(x1, x2)) = x1
POL(new_primDivNatS0(x1, x2, x3, x4)) = 1 + x1
POL(new_primDivNatS00(x1, x2)) = 1 + x1
POL(new_primMinusNatS2(x1, x2)) = x1
The following usable rules [17] were oriented:
new_primMinusNatS2(Succ(vuz12700), Succ(vuz1280)) → new_primMinusNatS2(vuz12700, vuz1280)
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Zero, Succ(vuz1280)) → Zero
new_primMinusNatS2(Succ(vuz12700), Zero) → Succ(vuz12700)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vuz180, vuz181, Zero, Zero) → new_primDivNatS00(vuz180, vuz181)
new_primDivNatS0(vuz180, vuz181, Succ(vuz1820), Succ(vuz1830)) → new_primDivNatS0(vuz180, vuz181, vuz1820, vuz1830)
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12700), Succ(vuz1280)) → new_primMinusNatS2(vuz12700, vuz1280)
new_primMinusNatS2(Zero, Succ(vuz1280)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12700), Zero) → Succ(vuz12700)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
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
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vuz180, vuz181, Succ(vuz1820), Succ(vuz1830)) → new_primDivNatS0(vuz180, vuz181, vuz1820, vuz1830)
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12700), Succ(vuz1280)) → new_primMinusNatS2(vuz12700, vuz1280)
new_primMinusNatS2(Zero, Succ(vuz1280)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12700), Zero) → Succ(vuz12700)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
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
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vuz180, vuz181, Succ(vuz1820), Succ(vuz1830)) → new_primDivNatS0(vuz180, vuz181, vuz1820, vuz1830)
R is empty.
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vuz180, vuz181, Succ(vuz1820), Succ(vuz1830)) → new_primDivNatS0(vuz180, vuz181, vuz1820, vuz1830)
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(vuz180, vuz181, Succ(vuz1820), Succ(vuz1830)) → new_primDivNatS0(vuz180, vuz181, vuz1820, vuz1830)
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
↳ DependencyGraphProof
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot0(vuz160, Succ(Succ(vuz17000)), Succ(vuz1620), vuz169) → new_quot(vuz160, vuz17000, Succ(vuz1620), vuz17000, vuz1620)
new_quot0(vuz160, Succ(Succ(vuz17000)), Zero, vuz169) → new_quot0(vuz160, new_primMinusNatS2(Succ(vuz17000), Zero), Zero, new_primMinusNatS2(Succ(vuz17000), Zero))
new_quot0(vuz160, Succ(Zero), Zero, vuz169) → new_quot0(vuz160, new_primMinusNatS2(Zero, Zero), Zero, new_primMinusNatS2(Zero, Zero))
new_quot(vuz213, vuz214, vuz215, Zero, Succ(vuz2170)) → new_quot1(vuz213, vuz215, Succ(vuz214))
new_quot2(vuz213, vuz214, vuz215) → new_quot0(vuz213, new_primMinusNatS2(Succ(vuz214), vuz215), vuz215, new_primMinusNatS2(Succ(vuz214), vuz215))
new_quot(vuz213, vuz214, vuz215, Succ(vuz2160), Succ(vuz2170)) → new_quot(vuz213, vuz214, vuz215, vuz2160, vuz2170)
new_quot(vuz213, vuz214, vuz215, Succ(vuz2160), Zero) → new_quot0(vuz213, new_primMinusNatS2(Succ(vuz214), vuz215), vuz215, new_primMinusNatS2(Succ(vuz214), vuz215))
new_quot1(vuz63, vuz640, vuz3100) → new_quot3(vuz63, vuz640, vuz3100)
new_quot0(vuz160, Succ(Zero), Succ(vuz1620), vuz169) → new_quot1(vuz160, Succ(vuz1620), Zero)
new_quot3(vuz63, vuz640, vuz3100) → new_quot3(vuz63, vuz640, vuz3100)
new_quot(vuz213, vuz214, vuz215, Zero, Zero) → new_quot2(vuz213, vuz214, vuz215)
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12700), Succ(vuz1280)) → new_primMinusNatS2(vuz12700, vuz1280)
new_primMinusNatS2(Zero, Succ(vuz1280)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12700), Zero) → Succ(vuz12700)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 3 SCCs with 4 less nodes.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot3(vuz63, vuz640, vuz3100) → new_quot3(vuz63, vuz640, vuz3100)
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12700), Succ(vuz1280)) → new_primMinusNatS2(vuz12700, vuz1280)
new_primMinusNatS2(Zero, Succ(vuz1280)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12700), Zero) → Succ(vuz12700)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot3(vuz63, vuz640, vuz3100) → new_quot3(vuz63, vuz640, vuz3100)
R is empty.
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ NonTerminationProof
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot3(vuz63, vuz640, vuz3100) → new_quot3(vuz63, vuz640, vuz3100)
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_quot3(vuz63, vuz640, vuz3100) → new_quot3(vuz63, vuz640, vuz3100)
The TRS R consists of the following rules:none
s = new_quot3(vuz63, vuz640, vuz3100) evaluates to t =new_quot3(vuz63, vuz640, vuz3100)
Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
- Semiunifier: [ ]
- Matcher: [ ]
Rewriting sequence
The DP semiunifies directly so there is only one rewrite step from new_quot3(vuz63, vuz640, vuz3100) to new_quot3(vuz63, vuz640, vuz3100).
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot0(vuz160, Succ(Succ(vuz17000)), Zero, vuz169) → new_quot0(vuz160, new_primMinusNatS2(Succ(vuz17000), Zero), Zero, new_primMinusNatS2(Succ(vuz17000), Zero))
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12700), Succ(vuz1280)) → new_primMinusNatS2(vuz12700, vuz1280)
new_primMinusNatS2(Zero, Succ(vuz1280)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12700), Zero) → Succ(vuz12700)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot0(vuz160, Succ(Succ(vuz17000)), Zero, vuz169) → new_quot0(vuz160, new_primMinusNatS2(Succ(vuz17000), Zero), Zero, new_primMinusNatS2(Succ(vuz17000), Zero))
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12700), Zero) → Succ(vuz12700)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
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_quot0(vuz160, Succ(Succ(vuz17000)), Zero, vuz169) → new_quot0(vuz160, new_primMinusNatS2(Succ(vuz17000), Zero), Zero, new_primMinusNatS2(Succ(vuz17000), Zero))
Used ordering: POLO with Polynomial interpretation [25]:
POL(Succ(x1)) = 1 + 2·x1
POL(Zero) = 0
POL(new_primMinusNatS2(x1, x2)) = x1 + 2·x2
POL(new_quot0(x1, x2, x3, x4)) = x1 + x2 + x3 + x4
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
P is empty.
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12700), Zero) → Succ(vuz12700)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
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
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot0(vuz160, Succ(Succ(vuz17000)), Succ(vuz1620), vuz169) → new_quot(vuz160, vuz17000, Succ(vuz1620), vuz17000, vuz1620)
new_quot2(vuz213, vuz214, vuz215) → new_quot0(vuz213, new_primMinusNatS2(Succ(vuz214), vuz215), vuz215, new_primMinusNatS2(Succ(vuz214), vuz215))
new_quot(vuz213, vuz214, vuz215, Succ(vuz2160), Succ(vuz2170)) → new_quot(vuz213, vuz214, vuz215, vuz2160, vuz2170)
new_quot(vuz213, vuz214, vuz215, Succ(vuz2160), Zero) → new_quot0(vuz213, new_primMinusNatS2(Succ(vuz214), vuz215), vuz215, new_primMinusNatS2(Succ(vuz214), vuz215))
new_quot(vuz213, vuz214, vuz215, Zero, Zero) → new_quot2(vuz213, vuz214, vuz215)
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12700), Succ(vuz1280)) → new_primMinusNatS2(vuz12700, vuz1280)
new_primMinusNatS2(Zero, Succ(vuz1280)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12700), Zero) → Succ(vuz12700)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
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(vuz160, Succ(Succ(vuz17000)), Succ(vuz1620), vuz169) → new_quot(vuz160, vuz17000, Succ(vuz1620), vuz17000, vuz1620)
The remaining pairs can at least be oriented weakly.
new_quot2(vuz213, vuz214, vuz215) → new_quot0(vuz213, new_primMinusNatS2(Succ(vuz214), vuz215), vuz215, new_primMinusNatS2(Succ(vuz214), vuz215))
new_quot(vuz213, vuz214, vuz215, Succ(vuz2160), Succ(vuz2170)) → new_quot(vuz213, vuz214, vuz215, vuz2160, vuz2170)
new_quot(vuz213, vuz214, vuz215, Succ(vuz2160), Zero) → new_quot0(vuz213, new_primMinusNatS2(Succ(vuz214), vuz215), vuz215, new_primMinusNatS2(Succ(vuz214), vuz215))
new_quot(vuz213, vuz214, vuz215, Zero, Zero) → new_quot2(vuz213, vuz214, vuz215)
Used ordering: Polynomial interpretation [25]:
POL(Succ(x1)) = 1 + x1
POL(Zero) = 0
POL(new_primMinusNatS2(x1, x2)) = x1
POL(new_quot(x1, x2, x3, x4, x5)) = 1 + x2 + x3
POL(new_quot0(x1, x2, x3, x4)) = x2 + x3
POL(new_quot2(x1, x2, x3)) = 1 + x2 + x3
The following usable rules [17] were oriented:
new_primMinusNatS2(Succ(vuz12700), Zero) → Succ(vuz12700)
new_primMinusNatS2(Succ(vuz12700), Succ(vuz1280)) → new_primMinusNatS2(vuz12700, vuz1280)
new_primMinusNatS2(Zero, Succ(vuz1280)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot2(vuz213, vuz214, vuz215) → new_quot0(vuz213, new_primMinusNatS2(Succ(vuz214), vuz215), vuz215, new_primMinusNatS2(Succ(vuz214), vuz215))
new_quot(vuz213, vuz214, vuz215, Succ(vuz2160), Zero) → new_quot0(vuz213, new_primMinusNatS2(Succ(vuz214), vuz215), vuz215, new_primMinusNatS2(Succ(vuz214), vuz215))
new_quot(vuz213, vuz214, vuz215, Succ(vuz2160), Succ(vuz2170)) → new_quot(vuz213, vuz214, vuz215, vuz2160, vuz2170)
new_quot(vuz213, vuz214, vuz215, Zero, Zero) → new_quot2(vuz213, vuz214, vuz215)
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12700), Succ(vuz1280)) → new_primMinusNatS2(vuz12700, vuz1280)
new_primMinusNatS2(Zero, Succ(vuz1280)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12700), Zero) → Succ(vuz12700)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 3 less nodes.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot(vuz213, vuz214, vuz215, Succ(vuz2160), Succ(vuz2170)) → new_quot(vuz213, vuz214, vuz215, vuz2160, vuz2170)
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12700), Succ(vuz1280)) → new_primMinusNatS2(vuz12700, vuz1280)
new_primMinusNatS2(Zero, Succ(vuz1280)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12700), Zero) → Succ(vuz12700)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot(vuz213, vuz214, vuz215, Succ(vuz2160), Succ(vuz2170)) → new_quot(vuz213, vuz214, vuz215, vuz2160, vuz2170)
R is empty.
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot(vuz213, vuz214, vuz215, Succ(vuz2160), Succ(vuz2170)) → new_quot(vuz213, vuz214, vuz215, vuz2160, vuz2170)
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_quot(vuz213, vuz214, vuz215, Succ(vuz2160), Succ(vuz2170)) → new_quot(vuz213, vuz214, vuz215, vuz2160, vuz2170)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4, 5 > 5
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot6(vuz146, Succ(Succ(vuz15800)), Succ(vuz1480), vuz157) → new_quot7(vuz146, vuz15800, Succ(vuz1480), vuz15800, vuz1480)
new_quot6(vuz146, Succ(Zero), Zero, vuz157) → new_quot6(vuz146, new_primMinusNatS2(Zero, Zero), Zero, new_primMinusNatS2(Zero, Zero))
new_quot7(vuz199, vuz200, vuz201, Succ(vuz2020), Succ(vuz2030)) → new_quot7(vuz199, vuz200, vuz201, vuz2020, vuz2030)
new_quot6(vuz146, Succ(Succ(vuz15800)), Zero, vuz157) → new_quot6(vuz146, new_primMinusNatS2(Succ(vuz15800), Zero), Zero, new_primMinusNatS2(Succ(vuz15800), Zero))
new_quot8(vuz45, vuz460, vuz3100) → new_quot5(vuz45, vuz460, vuz3100)
new_quot4(vuz45, vuz460, vuz3100) → new_quot5(vuz45, vuz460, vuz3100)
new_quot7(vuz199, vuz200, vuz201, Zero, Succ(vuz2030)) → new_quot8(vuz199, vuz201, Succ(vuz200))
new_quot7(vuz199, vuz200, vuz201, Zero, Zero) → new_quot9(vuz199, vuz200, vuz201)
new_quot6(vuz146, Succ(Zero), Succ(vuz1480), vuz157) → new_quot8(vuz146, Succ(vuz1480), Zero)
new_quot9(vuz199, vuz200, vuz201) → new_quot6(vuz199, new_primMinusNatS2(Succ(vuz200), vuz201), vuz201, new_primMinusNatS2(Succ(vuz200), vuz201))
new_quot5(vuz45, vuz460, vuz3100) → new_quot5(vuz45, vuz460, vuz3100)
new_quot7(vuz199, vuz200, vuz201, Succ(vuz2020), Zero) → new_quot6(vuz199, new_primMinusNatS2(Succ(vuz200), vuz201), vuz201, new_primMinusNatS2(Succ(vuz200), vuz201))
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12700), Succ(vuz1280)) → new_primMinusNatS2(vuz12700, vuz1280)
new_primMinusNatS2(Zero, Succ(vuz1280)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12700), Zero) → Succ(vuz12700)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 3 SCCs with 5 less nodes.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot5(vuz45, vuz460, vuz3100) → new_quot5(vuz45, vuz460, vuz3100)
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12700), Succ(vuz1280)) → new_primMinusNatS2(vuz12700, vuz1280)
new_primMinusNatS2(Zero, Succ(vuz1280)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12700), Zero) → Succ(vuz12700)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot5(vuz45, vuz460, vuz3100) → new_quot5(vuz45, vuz460, vuz3100)
R is empty.
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ NonTerminationProof
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot5(vuz45, vuz460, vuz3100) → new_quot5(vuz45, vuz460, vuz3100)
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_quot5(vuz45, vuz460, vuz3100) → new_quot5(vuz45, vuz460, vuz3100)
The TRS R consists of the following rules:none
s = new_quot5(vuz45, vuz460, vuz3100) evaluates to t =new_quot5(vuz45, vuz460, vuz3100)
Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
- Semiunifier: [ ]
- Matcher: [ ]
Rewriting sequence
The DP semiunifies directly so there is only one rewrite step from new_quot5(vuz45, vuz460, vuz3100) to new_quot5(vuz45, vuz460, vuz3100).
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot6(vuz146, Succ(Succ(vuz15800)), Zero, vuz157) → new_quot6(vuz146, new_primMinusNatS2(Succ(vuz15800), Zero), Zero, new_primMinusNatS2(Succ(vuz15800), Zero))
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12700), Succ(vuz1280)) → new_primMinusNatS2(vuz12700, vuz1280)
new_primMinusNatS2(Zero, Succ(vuz1280)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12700), Zero) → Succ(vuz12700)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot6(vuz146, Succ(Succ(vuz15800)), Zero, vuz157) → new_quot6(vuz146, new_primMinusNatS2(Succ(vuz15800), Zero), Zero, new_primMinusNatS2(Succ(vuz15800), Zero))
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12700), Zero) → Succ(vuz12700)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
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_quot6(vuz146, Succ(Succ(vuz15800)), Zero, vuz157) → new_quot6(vuz146, new_primMinusNatS2(Succ(vuz15800), Zero), Zero, new_primMinusNatS2(Succ(vuz15800), Zero))
Used ordering: POLO with Polynomial interpretation [25]:
POL(Succ(x1)) = 1 + 2·x1
POL(Zero) = 0
POL(new_primMinusNatS2(x1, x2)) = x1 + 2·x2
POL(new_quot6(x1, x2, x3, x4)) = x1 + x2 + x3 + x4
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ Narrow
Q DP problem:
P is empty.
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12700), Zero) → Succ(vuz12700)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot7(vuz199, vuz200, vuz201, Succ(vuz2020), Succ(vuz2030)) → new_quot7(vuz199, vuz200, vuz201, vuz2020, vuz2030)
new_quot6(vuz146, Succ(Succ(vuz15800)), Succ(vuz1480), vuz157) → new_quot7(vuz146, vuz15800, Succ(vuz1480), vuz15800, vuz1480)
new_quot7(vuz199, vuz200, vuz201, Zero, Zero) → new_quot9(vuz199, vuz200, vuz201)
new_quot9(vuz199, vuz200, vuz201) → new_quot6(vuz199, new_primMinusNatS2(Succ(vuz200), vuz201), vuz201, new_primMinusNatS2(Succ(vuz200), vuz201))
new_quot7(vuz199, vuz200, vuz201, Succ(vuz2020), Zero) → new_quot6(vuz199, new_primMinusNatS2(Succ(vuz200), vuz201), vuz201, new_primMinusNatS2(Succ(vuz200), vuz201))
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12700), Succ(vuz1280)) → new_primMinusNatS2(vuz12700, vuz1280)
new_primMinusNatS2(Zero, Succ(vuz1280)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12700), Zero) → Succ(vuz12700)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
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(vuz146, Succ(Succ(vuz15800)), Succ(vuz1480), vuz157) → new_quot7(vuz146, vuz15800, Succ(vuz1480), vuz15800, vuz1480)
The remaining pairs can at least be oriented weakly.
new_quot7(vuz199, vuz200, vuz201, Succ(vuz2020), Succ(vuz2030)) → new_quot7(vuz199, vuz200, vuz201, vuz2020, vuz2030)
new_quot7(vuz199, vuz200, vuz201, Zero, Zero) → new_quot9(vuz199, vuz200, vuz201)
new_quot9(vuz199, vuz200, vuz201) → new_quot6(vuz199, new_primMinusNatS2(Succ(vuz200), vuz201), vuz201, new_primMinusNatS2(Succ(vuz200), vuz201))
new_quot7(vuz199, vuz200, vuz201, Succ(vuz2020), Zero) → new_quot6(vuz199, new_primMinusNatS2(Succ(vuz200), vuz201), vuz201, new_primMinusNatS2(Succ(vuz200), vuz201))
Used ordering: Polynomial interpretation [25]:
POL(Succ(x1)) = 1 + x1
POL(Zero) = 0
POL(new_primMinusNatS2(x1, x2)) = x1
POL(new_quot6(x1, x2, x3, x4)) = x2
POL(new_quot7(x1, x2, x3, x4, x5)) = 1 + x2
POL(new_quot9(x1, x2, x3)) = 1 + x2
The following usable rules [17] were oriented:
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Zero, Succ(vuz1280)) → Zero
new_primMinusNatS2(Succ(vuz12700), Zero) → Succ(vuz12700)
new_primMinusNatS2(Succ(vuz12700), Succ(vuz1280)) → new_primMinusNatS2(vuz12700, vuz1280)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot7(vuz199, vuz200, vuz201, Succ(vuz2020), Succ(vuz2030)) → new_quot7(vuz199, vuz200, vuz201, vuz2020, vuz2030)
new_quot7(vuz199, vuz200, vuz201, Zero, Zero) → new_quot9(vuz199, vuz200, vuz201)
new_quot9(vuz199, vuz200, vuz201) → new_quot6(vuz199, new_primMinusNatS2(Succ(vuz200), vuz201), vuz201, new_primMinusNatS2(Succ(vuz200), vuz201))
new_quot7(vuz199, vuz200, vuz201, Succ(vuz2020), Zero) → new_quot6(vuz199, new_primMinusNatS2(Succ(vuz200), vuz201), vuz201, new_primMinusNatS2(Succ(vuz200), vuz201))
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12700), Succ(vuz1280)) → new_primMinusNatS2(vuz12700, vuz1280)
new_primMinusNatS2(Zero, Succ(vuz1280)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12700), Zero) → Succ(vuz12700)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 3 less nodes.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot7(vuz199, vuz200, vuz201, Succ(vuz2020), Succ(vuz2030)) → new_quot7(vuz199, vuz200, vuz201, vuz2020, vuz2030)
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12700), Succ(vuz1280)) → new_primMinusNatS2(vuz12700, vuz1280)
new_primMinusNatS2(Zero, Succ(vuz1280)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12700), Zero) → Succ(vuz12700)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot7(vuz199, vuz200, vuz201, Succ(vuz2020), Succ(vuz2030)) → new_quot7(vuz199, vuz200, vuz201, vuz2020, vuz2030)
R is empty.
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Zero)
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Succ(x1))
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPSizeChangeProof
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot7(vuz199, vuz200, vuz201, Succ(vuz2020), Succ(vuz2030)) → new_quot7(vuz199, vuz200, vuz201, vuz2020, vuz2030)
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_quot7(vuz199, vuz200, vuz201, Succ(vuz2020), Succ(vuz2030)) → new_quot7(vuz199, vuz200, vuz201, vuz2020, vuz2030)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4, 5 > 5
Haskell To QDPs