MAYBE 90.843
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
| ((subtract :: Ratio Int -> Ratio Int -> Ratio Int) :: Ratio Int -> Ratio Int -> Ratio Int) |
module Main where
If Reductions:
The following If expression
if primGEqNatS x y then Succ (primDivNatS (primMinusNatS x y) (Succ y)) else Zero
is transformed to
primDivNatS0 | x y True | = Succ (primDivNatS (primMinusNatS x y) (Succ y)) |
primDivNatS0 | x y False | = Zero |
The following If expression
if primGEqNatS x y then primModNatS (primMinusNatS x y) (Succ y) else Succ x
is transformed to
primModNatS0 | x y True | = primModNatS (primMinusNatS x y) (Succ y) |
primModNatS0 | x y False | = Succ x |
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
mainModule Main
| ((subtract :: Ratio Int -> Ratio Int -> Ratio Int) :: Ratio Int -> Ratio Int -> Ratio Int) |
module Main where
Replaced joker patterns by fresh variables and removed binding patterns.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
mainModule Main
| ((subtract :: Ratio Int -> Ratio Int -> Ratio Int) :: Ratio Int -> Ratio Int -> Ratio Int) |
module Main where
Cond Reductions:
The following Function with conditions
gcd' | x 0 | = x |
gcd' | x y | = gcd' y (x `rem` y) |
is transformed to
gcd' | x xz | = gcd'2 x xz |
gcd' | x y | = gcd'0 x y |
gcd'0 | x y | = gcd' y (x `rem` y) |
gcd'1 | True x xz | = x |
gcd'1 | yu yv yw | = gcd'0 yv yw |
gcd'2 | x xz | = gcd'1 (xz == 0) x xz |
gcd'2 | yx yy | = gcd'0 yx yy |
The following Function with conditions
gcd | 0 0 | = error [] |
gcd | x y | =
gcd' (abs x) (abs y) |
where |
gcd' | x 0 | = x |
gcd' | x y | = gcd' y (x `rem` y) |
|
|
is transformed to
gcd | yz zu | = gcd3 yz zu |
gcd | x y | = gcd0 x y |
gcd0 | x y | =
gcd' (abs x) (abs y) |
where |
gcd' | x xz | = gcd'2 x xz |
gcd' | x y | = gcd'0 x y |
|
|
gcd'0 | x y | = gcd' y (x `rem` y) |
|
|
gcd'1 | True x xz | = x |
gcd'1 | yu yv yw | = gcd'0 yv yw |
|
|
gcd'2 | x xz | = gcd'1 (xz == 0) x xz |
gcd'2 | yx yy | = gcd'0 yx yy |
|
|
gcd1 | True yz zu | = error [] |
gcd1 | zv zw zx | = gcd0 zw zx |
gcd2 | True yz zu | = gcd1 (zu == 0) yz zu |
gcd2 | zy zz vuu | = gcd0 zz vuu |
gcd3 | yz zu | = gcd2 (yz == 0) yz zu |
gcd3 | vuv vuw | = gcd0 vuv vuw |
The following Function with conditions
is transformed to
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 |
|
|
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
mainModule Main
| ((subtract :: Ratio Int -> Ratio Int -> Ratio Int) :: Ratio Int -> Ratio Int -> Ratio Int) |
module Main where
Let/Where Reductions:
The bindings of the following Let/Where expression
reduce1 x y (y == 0) |
where | |
|
reduce0 | x y True | = x `quot` d :% (y `quot` d) |
|
|
reduce1 | x y True | = error [] |
reduce1 | x y False | = reduce0 x y otherwise |
|
are unpacked to the following functions on top level
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
| ((subtract :: Ratio Int -> Ratio Int -> Ratio Int) :: Ratio Int -> Ratio Int -> Ratio Int) |
module Main where
Num Reduction: All numbers are transformed to thier corresponding representation with Pos, Neg, Succ and Zero.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ Narrow
mainModule Main
| (subtract :: Ratio Int -> Ratio Int -> Ratio Int) |
module Main where
Haskell To QDPs
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primPlusNat(Succ(vuz6600), Succ(vuz31000)) → new_primPlusNat(vuz6600, vuz31000)
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(vuz6600), Succ(vuz31000)) → new_primPlusNat(vuz6600, vuz31000)
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
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primMulNat(Succ(vuz410000), vuz3100) → new_primMulNat(vuz410000, vuz3100)
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(vuz410000), vuz3100) → new_primMulNat(vuz410000, vuz3100)
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
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primMinusNatS(Succ(vuz3380), Succ(vuz3390)) → new_primMinusNatS(vuz3380, vuz3390)
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(vuz3380), Succ(vuz3390)) → new_primMinusNatS(vuz3380, vuz3390)
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
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primModNatS0(vuz391, vuz392, Succ(vuz3930), Zero) → new_primModNatS(new_primMinusNatS0(Succ(vuz391), Succ(vuz392)), Succ(vuz392))
new_primModNatS00(vuz391, vuz392) → new_primModNatS(new_primMinusNatS0(Succ(vuz391), Succ(vuz392)), Succ(vuz392))
new_primModNatS(Succ(Succ(vuz374000)), Succ(vuz373000)) → new_primModNatS0(vuz374000, vuz373000, vuz374000, vuz373000)
new_primModNatS(Succ(Zero), Zero) → new_primModNatS(new_primMinusNatS0(Zero, Zero), Zero)
new_primModNatS(Succ(Succ(vuz374000)), Zero) → new_primModNatS(new_primMinusNatS0(Succ(vuz374000), Zero), Zero)
new_primModNatS0(vuz391, vuz392, Succ(vuz3930), Succ(vuz3940)) → new_primModNatS0(vuz391, vuz392, vuz3930, vuz3940)
new_primModNatS0(vuz391, vuz392, Zero, Zero) → new_primModNatS00(vuz391, vuz392)
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vuz3390)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz3380), Succ(vuz3390)) → new_primMinusNatS0(vuz3380, vuz3390)
new_primMinusNatS0(Succ(vuz3380), Zero) → Succ(vuz3380)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
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
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primModNatS(Succ(Succ(vuz374000)), Zero) → new_primModNatS(new_primMinusNatS0(Succ(vuz374000), Zero), Zero)
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vuz3390)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz3380), Succ(vuz3390)) → new_primMinusNatS0(vuz3380, vuz3390)
new_primMinusNatS0(Succ(vuz3380), Zero) → Succ(vuz3380)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
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
↳ RuleRemovalProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primModNatS(Succ(Succ(vuz374000)), Zero) → new_primModNatS(new_primMinusNatS0(Succ(vuz374000), Zero), Zero)
The TRS R consists of the following rules:
new_primMinusNatS0(Succ(vuz3380), Zero) → Succ(vuz3380)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(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_primModNatS(Succ(Succ(vuz374000)), Zero) → new_primModNatS(new_primMinusNatS0(Succ(vuz374000), Zero), Zero)
Strictly oriented rules of the TRS R:
new_primMinusNatS0(Succ(vuz3380), Zero) → Succ(vuz3380)
Used ordering: POLO with Polynomial interpretation [25]:
POL(Succ(x1)) = 2 + 2·x1
POL(Zero) = 0
POL(new_primMinusNatS0(x1, x2)) = 1 + 2·x1 + x2
POL(new_primModNatS(x1, x2)) = x1 + x2
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
P is empty.
R is empty.
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primModNatS0(vuz391, vuz392, Succ(vuz3930), Zero) → new_primModNatS(new_primMinusNatS0(Succ(vuz391), Succ(vuz392)), Succ(vuz392))
new_primModNatS00(vuz391, vuz392) → new_primModNatS(new_primMinusNatS0(Succ(vuz391), Succ(vuz392)), Succ(vuz392))
new_primModNatS(Succ(Succ(vuz374000)), Succ(vuz373000)) → new_primModNatS0(vuz374000, vuz373000, vuz374000, vuz373000)
new_primModNatS0(vuz391, vuz392, Succ(vuz3930), Succ(vuz3940)) → new_primModNatS0(vuz391, vuz392, vuz3930, vuz3940)
new_primModNatS0(vuz391, vuz392, Zero, Zero) → new_primModNatS00(vuz391, vuz392)
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vuz3390)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz3380), Succ(vuz3390)) → new_primMinusNatS0(vuz3380, vuz3390)
new_primMinusNatS0(Succ(vuz3380), Zero) → Succ(vuz3380)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_primModNatS0(vuz391, vuz392, Succ(vuz3930), Zero) → new_primModNatS(new_primMinusNatS0(Succ(vuz391), Succ(vuz392)), Succ(vuz392)) at position [0] we obtained the following new rules:
new_primModNatS0(vuz391, vuz392, Succ(vuz3930), Zero) → new_primModNatS(new_primMinusNatS0(vuz391, vuz392), Succ(vuz392))
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primModNatS00(vuz391, vuz392) → new_primModNatS(new_primMinusNatS0(Succ(vuz391), Succ(vuz392)), Succ(vuz392))
new_primModNatS(Succ(Succ(vuz374000)), Succ(vuz373000)) → new_primModNatS0(vuz374000, vuz373000, vuz374000, vuz373000)
new_primModNatS0(vuz391, vuz392, Succ(vuz3930), Zero) → new_primModNatS(new_primMinusNatS0(vuz391, vuz392), Succ(vuz392))
new_primModNatS0(vuz391, vuz392, Zero, Zero) → new_primModNatS00(vuz391, vuz392)
new_primModNatS0(vuz391, vuz392, Succ(vuz3930), Succ(vuz3940)) → new_primModNatS0(vuz391, vuz392, vuz3930, vuz3940)
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vuz3390)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz3380), Succ(vuz3390)) → new_primMinusNatS0(vuz3380, vuz3390)
new_primMinusNatS0(Succ(vuz3380), Zero) → Succ(vuz3380)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_primModNatS00(vuz391, vuz392) → new_primModNatS(new_primMinusNatS0(Succ(vuz391), Succ(vuz392)), Succ(vuz392)) at position [0] we obtained the following new rules:
new_primModNatS00(vuz391, vuz392) → new_primModNatS(new_primMinusNatS0(vuz391, vuz392), Succ(vuz392))
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primModNatS00(vuz391, vuz392) → new_primModNatS(new_primMinusNatS0(vuz391, vuz392), Succ(vuz392))
new_primModNatS(Succ(Succ(vuz374000)), Succ(vuz373000)) → new_primModNatS0(vuz374000, vuz373000, vuz374000, vuz373000)
new_primModNatS0(vuz391, vuz392, Succ(vuz3930), Zero) → new_primModNatS(new_primMinusNatS0(vuz391, vuz392), Succ(vuz392))
new_primModNatS0(vuz391, vuz392, Succ(vuz3930), Succ(vuz3940)) → new_primModNatS0(vuz391, vuz392, vuz3930, vuz3940)
new_primModNatS0(vuz391, vuz392, Zero, Zero) → new_primModNatS00(vuz391, vuz392)
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vuz3390)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz3380), Succ(vuz3390)) → new_primMinusNatS0(vuz3380, vuz3390)
new_primMinusNatS0(Succ(vuz3380), Zero) → Succ(vuz3380)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
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_primModNatS(Succ(Succ(vuz374000)), Succ(vuz373000)) → new_primModNatS0(vuz374000, vuz373000, vuz374000, vuz373000)
The remaining pairs can at least be oriented weakly.
new_primModNatS00(vuz391, vuz392) → new_primModNatS(new_primMinusNatS0(vuz391, vuz392), Succ(vuz392))
new_primModNatS0(vuz391, vuz392, Succ(vuz3930), Zero) → new_primModNatS(new_primMinusNatS0(vuz391, vuz392), Succ(vuz392))
new_primModNatS0(vuz391, vuz392, Succ(vuz3930), Succ(vuz3940)) → new_primModNatS0(vuz391, vuz392, vuz3930, vuz3940)
new_primModNatS0(vuz391, vuz392, Zero, Zero) → new_primModNatS00(vuz391, vuz392)
Used ordering: Polynomial interpretation [25]:
POL(Succ(x1)) = 1 + x1
POL(Zero) = 0
POL(new_primMinusNatS0(x1, x2)) = x1
POL(new_primModNatS(x1, x2)) = x1
POL(new_primModNatS0(x1, x2, x3, x4)) = x1
POL(new_primModNatS00(x1, x2)) = x1
The following usable rules [17] were oriented:
new_primMinusNatS0(Succ(vuz3380), Zero) → Succ(vuz3380)
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Zero, Succ(vuz3390)) → Zero
new_primMinusNatS0(Succ(vuz3380), Succ(vuz3390)) → new_primMinusNatS0(vuz3380, vuz3390)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primModNatS00(vuz391, vuz392) → new_primModNatS(new_primMinusNatS0(vuz391, vuz392), Succ(vuz392))
new_primModNatS0(vuz391, vuz392, Succ(vuz3930), Zero) → new_primModNatS(new_primMinusNatS0(vuz391, vuz392), Succ(vuz392))
new_primModNatS0(vuz391, vuz392, Zero, Zero) → new_primModNatS00(vuz391, vuz392)
new_primModNatS0(vuz391, vuz392, Succ(vuz3930), Succ(vuz3940)) → new_primModNatS0(vuz391, vuz392, vuz3930, vuz3940)
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vuz3390)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz3380), Succ(vuz3390)) → new_primMinusNatS0(vuz3380, vuz3390)
new_primMinusNatS0(Succ(vuz3380), Zero) → Succ(vuz3380)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
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
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primModNatS0(vuz391, vuz392, Succ(vuz3930), Succ(vuz3940)) → new_primModNatS0(vuz391, vuz392, vuz3930, vuz3940)
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vuz3390)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz3380), Succ(vuz3390)) → new_primMinusNatS0(vuz3380, vuz3390)
new_primMinusNatS0(Succ(vuz3380), Zero) → Succ(vuz3380)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
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
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primModNatS0(vuz391, vuz392, Succ(vuz3930), Succ(vuz3940)) → new_primModNatS0(vuz391, vuz392, vuz3930, vuz3940)
R is empty.
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primModNatS0(vuz391, vuz392, Succ(vuz3930), Succ(vuz3940)) → new_primModNatS0(vuz391, vuz392, vuz3930, vuz3940)
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_primModNatS0(vuz391, vuz392, Succ(vuz3930), Succ(vuz3940)) → new_primModNatS0(vuz391, vuz392, vuz3930, vuz3940)
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
↳ QDP
↳ Narrowing
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_gcd0Gcd'(vuz374, Neg(Succ(vuz37300))) → new_gcd0Gcd'(Neg(Succ(vuz37300)), new_primRemInt0(vuz374, vuz37300))
new_gcd0Gcd'(vuz374, Pos(Succ(vuz37300))) → new_gcd0Gcd'(Pos(Succ(vuz37300)), new_primRemInt(vuz374, vuz37300))
The TRS R consists of the following rules:
new_primModNatS1(Succ(Zero), Zero) → new_primModNatS1(new_primMinusNatS0(Zero, Zero), Zero)
new_primMinusNatS0(Zero, Succ(vuz3390)) → Zero
new_primModNatS01(vuz391, vuz392) → new_primModNatS1(new_primMinusNatS0(Succ(vuz391), Succ(vuz392)), Succ(vuz392))
new_primModNatS1(Succ(Zero), Succ(vuz373000)) → Succ(Zero)
new_primRemInt0(Pos(vuz3740), vuz37300) → Pos(new_primModNatS1(vuz3740, vuz37300))
new_primRemInt(Neg(vuz3740), vuz37300) → Neg(new_primModNatS1(vuz3740, vuz37300))
new_primModNatS02(vuz391, vuz392, Zero, Succ(vuz3940)) → Succ(Succ(vuz391))
new_primRemInt0(Neg(vuz3740), vuz37300) → Neg(new_primModNatS1(vuz3740, vuz37300))
new_primModNatS1(Zero, vuz37300) → Zero
new_primModNatS1(Succ(Succ(vuz374000)), Succ(vuz373000)) → new_primModNatS02(vuz374000, vuz373000, vuz374000, vuz373000)
new_primModNatS02(vuz391, vuz392, Succ(vuz3930), Zero) → new_primModNatS01(vuz391, vuz392)
new_primMinusNatS0(Zero, Zero) → Zero
new_primModNatS02(vuz391, vuz392, Zero, Zero) → new_primModNatS01(vuz391, vuz392)
new_primModNatS02(vuz391, vuz392, Succ(vuz3930), Succ(vuz3940)) → new_primModNatS02(vuz391, vuz392, vuz3930, vuz3940)
new_primRemInt(Pos(vuz3740), vuz37300) → Pos(new_primModNatS1(vuz3740, vuz37300))
new_primMinusNatS0(Succ(vuz3380), Succ(vuz3390)) → new_primMinusNatS0(vuz3380, vuz3390)
new_primMinusNatS0(Succ(vuz3380), Zero) → Succ(vuz3380)
new_primModNatS1(Succ(Succ(vuz374000)), Zero) → new_primModNatS1(new_primMinusNatS0(Succ(vuz374000), Zero), Zero)
The set Q consists of the following terms:
new_primModNatS02(x0, x1, Zero, Succ(x2))
new_primModNatS1(Succ(Zero), Succ(x0))
new_primRemInt0(Neg(x0), x1)
new_primModNatS1(Zero, x0)
new_primModNatS1(Succ(Succ(x0)), Zero)
new_primModNatS1(Succ(Succ(x0)), Succ(x1))
new_primModNatS01(x0, x1)
new_primModNatS02(x0, x1, Succ(x2), Zero)
new_primMinusNatS0(Zero, Succ(x0))
new_primModNatS02(x0, x1, Zero, Zero)
new_primRemInt(Neg(x0), x1)
new_primMinusNatS0(Succ(x0), Zero)
new_primRemInt(Pos(x0), x1)
new_primRemInt0(Pos(x0), x1)
new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
new_primModNatS1(Succ(Zero), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule new_gcd0Gcd'(vuz374, Neg(Succ(vuz37300))) → new_gcd0Gcd'(Neg(Succ(vuz37300)), new_primRemInt0(vuz374, vuz37300)) at position [1] we obtained the following new rules:
new_gcd0Gcd'(Neg(x0), Neg(Succ(x1))) → new_gcd0Gcd'(Neg(Succ(x1)), Neg(new_primModNatS1(x0, x1)))
new_gcd0Gcd'(Pos(x0), Neg(Succ(x1))) → new_gcd0Gcd'(Neg(Succ(x1)), Pos(new_primModNatS1(x0, x1)))
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_gcd0Gcd'(vuz374, Pos(Succ(vuz37300))) → new_gcd0Gcd'(Pos(Succ(vuz37300)), new_primRemInt(vuz374, vuz37300))
new_gcd0Gcd'(Pos(x0), Neg(Succ(x1))) → new_gcd0Gcd'(Neg(Succ(x1)), Pos(new_primModNatS1(x0, x1)))
new_gcd0Gcd'(Neg(x0), Neg(Succ(x1))) → new_gcd0Gcd'(Neg(Succ(x1)), Neg(new_primModNatS1(x0, x1)))
The TRS R consists of the following rules:
new_primModNatS1(Succ(Zero), Zero) → new_primModNatS1(new_primMinusNatS0(Zero, Zero), Zero)
new_primMinusNatS0(Zero, Succ(vuz3390)) → Zero
new_primModNatS01(vuz391, vuz392) → new_primModNatS1(new_primMinusNatS0(Succ(vuz391), Succ(vuz392)), Succ(vuz392))
new_primModNatS1(Succ(Zero), Succ(vuz373000)) → Succ(Zero)
new_primRemInt0(Pos(vuz3740), vuz37300) → Pos(new_primModNatS1(vuz3740, vuz37300))
new_primRemInt(Neg(vuz3740), vuz37300) → Neg(new_primModNatS1(vuz3740, vuz37300))
new_primModNatS02(vuz391, vuz392, Zero, Succ(vuz3940)) → Succ(Succ(vuz391))
new_primRemInt0(Neg(vuz3740), vuz37300) → Neg(new_primModNatS1(vuz3740, vuz37300))
new_primModNatS1(Zero, vuz37300) → Zero
new_primModNatS1(Succ(Succ(vuz374000)), Succ(vuz373000)) → new_primModNatS02(vuz374000, vuz373000, vuz374000, vuz373000)
new_primModNatS02(vuz391, vuz392, Succ(vuz3930), Zero) → new_primModNatS01(vuz391, vuz392)
new_primMinusNatS0(Zero, Zero) → Zero
new_primModNatS02(vuz391, vuz392, Zero, Zero) → new_primModNatS01(vuz391, vuz392)
new_primModNatS02(vuz391, vuz392, Succ(vuz3930), Succ(vuz3940)) → new_primModNatS02(vuz391, vuz392, vuz3930, vuz3940)
new_primRemInt(Pos(vuz3740), vuz37300) → Pos(new_primModNatS1(vuz3740, vuz37300))
new_primMinusNatS0(Succ(vuz3380), Succ(vuz3390)) → new_primMinusNatS0(vuz3380, vuz3390)
new_primMinusNatS0(Succ(vuz3380), Zero) → Succ(vuz3380)
new_primModNatS1(Succ(Succ(vuz374000)), Zero) → new_primModNatS1(new_primMinusNatS0(Succ(vuz374000), Zero), Zero)
The set Q consists of the following terms:
new_primModNatS02(x0, x1, Zero, Succ(x2))
new_primModNatS1(Succ(Zero), Succ(x0))
new_primRemInt0(Neg(x0), x1)
new_primModNatS1(Zero, x0)
new_primModNatS1(Succ(Succ(x0)), Zero)
new_primModNatS1(Succ(Succ(x0)), Succ(x1))
new_primModNatS01(x0, x1)
new_primModNatS02(x0, x1, Succ(x2), Zero)
new_primMinusNatS0(Zero, Succ(x0))
new_primModNatS02(x0, x1, Zero, Zero)
new_primRemInt(Neg(x0), x1)
new_primMinusNatS0(Succ(x0), Zero)
new_primRemInt(Pos(x0), x1)
new_primRemInt0(Pos(x0), x1)
new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
new_primModNatS1(Succ(Zero), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_gcd0Gcd'(Neg(x0), Neg(Succ(x1))) → new_gcd0Gcd'(Neg(Succ(x1)), Neg(new_primModNatS1(x0, x1)))
The TRS R consists of the following rules:
new_primModNatS1(Succ(Zero), Zero) → new_primModNatS1(new_primMinusNatS0(Zero, Zero), Zero)
new_primMinusNatS0(Zero, Succ(vuz3390)) → Zero
new_primModNatS01(vuz391, vuz392) → new_primModNatS1(new_primMinusNatS0(Succ(vuz391), Succ(vuz392)), Succ(vuz392))
new_primModNatS1(Succ(Zero), Succ(vuz373000)) → Succ(Zero)
new_primRemInt0(Pos(vuz3740), vuz37300) → Pos(new_primModNatS1(vuz3740, vuz37300))
new_primRemInt(Neg(vuz3740), vuz37300) → Neg(new_primModNatS1(vuz3740, vuz37300))
new_primModNatS02(vuz391, vuz392, Zero, Succ(vuz3940)) → Succ(Succ(vuz391))
new_primRemInt0(Neg(vuz3740), vuz37300) → Neg(new_primModNatS1(vuz3740, vuz37300))
new_primModNatS1(Zero, vuz37300) → Zero
new_primModNatS1(Succ(Succ(vuz374000)), Succ(vuz373000)) → new_primModNatS02(vuz374000, vuz373000, vuz374000, vuz373000)
new_primModNatS02(vuz391, vuz392, Succ(vuz3930), Zero) → new_primModNatS01(vuz391, vuz392)
new_primMinusNatS0(Zero, Zero) → Zero
new_primModNatS02(vuz391, vuz392, Zero, Zero) → new_primModNatS01(vuz391, vuz392)
new_primModNatS02(vuz391, vuz392, Succ(vuz3930), Succ(vuz3940)) → new_primModNatS02(vuz391, vuz392, vuz3930, vuz3940)
new_primRemInt(Pos(vuz3740), vuz37300) → Pos(new_primModNatS1(vuz3740, vuz37300))
new_primMinusNatS0(Succ(vuz3380), Succ(vuz3390)) → new_primMinusNatS0(vuz3380, vuz3390)
new_primMinusNatS0(Succ(vuz3380), Zero) → Succ(vuz3380)
new_primModNatS1(Succ(Succ(vuz374000)), Zero) → new_primModNatS1(new_primMinusNatS0(Succ(vuz374000), Zero), Zero)
The set Q consists of the following terms:
new_primModNatS02(x0, x1, Zero, Succ(x2))
new_primModNatS1(Succ(Zero), Succ(x0))
new_primRemInt0(Neg(x0), x1)
new_primModNatS1(Zero, x0)
new_primModNatS1(Succ(Succ(x0)), Zero)
new_primModNatS1(Succ(Succ(x0)), Succ(x1))
new_primModNatS01(x0, x1)
new_primModNatS02(x0, x1, Succ(x2), Zero)
new_primMinusNatS0(Zero, Succ(x0))
new_primModNatS02(x0, x1, Zero, Zero)
new_primRemInt(Neg(x0), x1)
new_primMinusNatS0(Succ(x0), Zero)
new_primRemInt(Pos(x0), x1)
new_primRemInt0(Pos(x0), x1)
new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
new_primModNatS1(Succ(Zero), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_gcd0Gcd'(Neg(x0), Neg(Succ(x1))) → new_gcd0Gcd'(Neg(Succ(x1)), Neg(new_primModNatS1(x0, x1)))
The TRS R consists of the following rules:
new_primModNatS1(Succ(Zero), Zero) → new_primModNatS1(new_primMinusNatS0(Zero, Zero), Zero)
new_primModNatS1(Succ(Succ(vuz374000)), Zero) → new_primModNatS1(new_primMinusNatS0(Succ(vuz374000), Zero), Zero)
new_primModNatS1(Succ(Zero), Succ(vuz373000)) → Succ(Zero)
new_primModNatS1(Zero, vuz37300) → Zero
new_primModNatS02(vuz391, vuz392, Succ(vuz3930), Succ(vuz3940)) → new_primModNatS02(vuz391, vuz392, vuz3930, vuz3940)
new_primModNatS1(Succ(Succ(vuz374000)), Succ(vuz373000)) → new_primModNatS02(vuz374000, vuz373000, vuz374000, vuz373000)
new_primModNatS02(vuz391, vuz392, Succ(vuz3930), Zero) → new_primModNatS01(vuz391, vuz392)
new_primModNatS01(vuz391, vuz392) → new_primModNatS1(new_primMinusNatS0(Succ(vuz391), Succ(vuz392)), Succ(vuz392))
new_primModNatS02(vuz391, vuz392, Zero, Zero) → new_primModNatS01(vuz391, vuz392)
new_primModNatS02(vuz391, vuz392, Zero, Succ(vuz3940)) → Succ(Succ(vuz391))
new_primMinusNatS0(Succ(vuz3380), Succ(vuz3390)) → new_primMinusNatS0(vuz3380, vuz3390)
new_primMinusNatS0(Zero, Succ(vuz3390)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz3380), Zero) → Succ(vuz3380)
The set Q consists of the following terms:
new_primModNatS02(x0, x1, Zero, Succ(x2))
new_primModNatS1(Succ(Zero), Succ(x0))
new_primRemInt0(Neg(x0), x1)
new_primModNatS1(Zero, x0)
new_primModNatS1(Succ(Succ(x0)), Zero)
new_primModNatS1(Succ(Succ(x0)), Succ(x1))
new_primModNatS01(x0, x1)
new_primModNatS02(x0, x1, Succ(x2), Zero)
new_primMinusNatS0(Zero, Succ(x0))
new_primModNatS02(x0, x1, Zero, Zero)
new_primRemInt(Neg(x0), x1)
new_primMinusNatS0(Succ(x0), Zero)
new_primRemInt(Pos(x0), x1)
new_primRemInt0(Pos(x0), x1)
new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
new_primModNatS1(Succ(Zero), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
new_primRemInt0(Neg(x0), x1)
new_primRemInt(Neg(x0), x1)
new_primRemInt(Pos(x0), x1)
new_primRemInt0(Pos(x0), x1)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_gcd0Gcd'(Neg(x0), Neg(Succ(x1))) → new_gcd0Gcd'(Neg(Succ(x1)), Neg(new_primModNatS1(x0, x1)))
The TRS R consists of the following rules:
new_primModNatS1(Succ(Zero), Zero) → new_primModNatS1(new_primMinusNatS0(Zero, Zero), Zero)
new_primModNatS1(Succ(Succ(vuz374000)), Zero) → new_primModNatS1(new_primMinusNatS0(Succ(vuz374000), Zero), Zero)
new_primModNatS1(Succ(Zero), Succ(vuz373000)) → Succ(Zero)
new_primModNatS1(Zero, vuz37300) → Zero
new_primModNatS02(vuz391, vuz392, Succ(vuz3930), Succ(vuz3940)) → new_primModNatS02(vuz391, vuz392, vuz3930, vuz3940)
new_primModNatS1(Succ(Succ(vuz374000)), Succ(vuz373000)) → new_primModNatS02(vuz374000, vuz373000, vuz374000, vuz373000)
new_primModNatS02(vuz391, vuz392, Succ(vuz3930), Zero) → new_primModNatS01(vuz391, vuz392)
new_primModNatS01(vuz391, vuz392) → new_primModNatS1(new_primMinusNatS0(Succ(vuz391), Succ(vuz392)), Succ(vuz392))
new_primModNatS02(vuz391, vuz392, Zero, Zero) → new_primModNatS01(vuz391, vuz392)
new_primModNatS02(vuz391, vuz392, Zero, Succ(vuz3940)) → Succ(Succ(vuz391))
new_primMinusNatS0(Succ(vuz3380), Succ(vuz3390)) → new_primMinusNatS0(vuz3380, vuz3390)
new_primMinusNatS0(Zero, Succ(vuz3390)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz3380), Zero) → Succ(vuz3380)
The set Q consists of the following terms:
new_primModNatS02(x0, x1, Zero, Succ(x2))
new_primModNatS1(Succ(Zero), Succ(x0))
new_primModNatS1(Zero, x0)
new_primModNatS1(Succ(Succ(x0)), Zero)
new_primModNatS1(Succ(Succ(x0)), Succ(x1))
new_primModNatS01(x0, x1)
new_primModNatS02(x0, x1, Succ(x2), Zero)
new_primMinusNatS0(Zero, Succ(x0))
new_primModNatS02(x0, x1, Zero, Zero)
new_primMinusNatS0(Succ(x0), Zero)
new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
new_primModNatS1(Succ(Zero), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule new_gcd0Gcd'(Neg(x0), Neg(Succ(x1))) → new_gcd0Gcd'(Neg(Succ(x1)), Neg(new_primModNatS1(x0, x1))) we obtained the following new rules:
new_gcd0Gcd'(Neg(Succ(z1)), Neg(Succ(x1))) → new_gcd0Gcd'(Neg(Succ(x1)), Neg(new_primModNatS1(Succ(z1), x1)))
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ MNOCProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_gcd0Gcd'(Neg(Succ(z1)), Neg(Succ(x1))) → new_gcd0Gcd'(Neg(Succ(x1)), Neg(new_primModNatS1(Succ(z1), x1)))
The TRS R consists of the following rules:
new_primModNatS1(Succ(Zero), Zero) → new_primModNatS1(new_primMinusNatS0(Zero, Zero), Zero)
new_primModNatS1(Succ(Succ(vuz374000)), Zero) → new_primModNatS1(new_primMinusNatS0(Succ(vuz374000), Zero), Zero)
new_primModNatS1(Succ(Zero), Succ(vuz373000)) → Succ(Zero)
new_primModNatS1(Zero, vuz37300) → Zero
new_primModNatS02(vuz391, vuz392, Succ(vuz3930), Succ(vuz3940)) → new_primModNatS02(vuz391, vuz392, vuz3930, vuz3940)
new_primModNatS1(Succ(Succ(vuz374000)), Succ(vuz373000)) → new_primModNatS02(vuz374000, vuz373000, vuz374000, vuz373000)
new_primModNatS02(vuz391, vuz392, Succ(vuz3930), Zero) → new_primModNatS01(vuz391, vuz392)
new_primModNatS01(vuz391, vuz392) → new_primModNatS1(new_primMinusNatS0(Succ(vuz391), Succ(vuz392)), Succ(vuz392))
new_primModNatS02(vuz391, vuz392, Zero, Zero) → new_primModNatS01(vuz391, vuz392)
new_primModNatS02(vuz391, vuz392, Zero, Succ(vuz3940)) → Succ(Succ(vuz391))
new_primMinusNatS0(Succ(vuz3380), Succ(vuz3390)) → new_primMinusNatS0(vuz3380, vuz3390)
new_primMinusNatS0(Zero, Succ(vuz3390)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz3380), Zero) → Succ(vuz3380)
The set Q consists of the following terms:
new_primModNatS02(x0, x1, Zero, Succ(x2))
new_primModNatS1(Succ(Zero), Succ(x0))
new_primModNatS1(Zero, x0)
new_primModNatS1(Succ(Succ(x0)), Zero)
new_primModNatS1(Succ(Succ(x0)), Succ(x1))
new_primModNatS01(x0, x1)
new_primModNatS02(x0, x1, Succ(x2), Zero)
new_primMinusNatS0(Zero, Succ(x0))
new_primModNatS02(x0, x1, Zero, Zero)
new_primMinusNatS0(Succ(x0), Zero)
new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
new_primModNatS1(Succ(Zero), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
We have to consider all minimal (P,Q,R)-chains.
We use the modular non-overlap check [17] to decrease Q to the empty set.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ MNOCProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_gcd0Gcd'(Neg(Succ(z1)), Neg(Succ(x1))) → new_gcd0Gcd'(Neg(Succ(x1)), Neg(new_primModNatS1(Succ(z1), x1)))
The TRS R consists of the following rules:
new_primModNatS1(Succ(Zero), Zero) → new_primModNatS1(new_primMinusNatS0(Zero, Zero), Zero)
new_primModNatS1(Succ(Succ(vuz374000)), Zero) → new_primModNatS1(new_primMinusNatS0(Succ(vuz374000), Zero), Zero)
new_primModNatS1(Succ(Zero), Succ(vuz373000)) → Succ(Zero)
new_primModNatS1(Zero, vuz37300) → Zero
new_primModNatS02(vuz391, vuz392, Succ(vuz3930), Succ(vuz3940)) → new_primModNatS02(vuz391, vuz392, vuz3930, vuz3940)
new_primModNatS1(Succ(Succ(vuz374000)), Succ(vuz373000)) → new_primModNatS02(vuz374000, vuz373000, vuz374000, vuz373000)
new_primModNatS02(vuz391, vuz392, Succ(vuz3930), Zero) → new_primModNatS01(vuz391, vuz392)
new_primModNatS01(vuz391, vuz392) → new_primModNatS1(new_primMinusNatS0(Succ(vuz391), Succ(vuz392)), Succ(vuz392))
new_primModNatS02(vuz391, vuz392, Zero, Zero) → new_primModNatS01(vuz391, vuz392)
new_primModNatS02(vuz391, vuz392, Zero, Succ(vuz3940)) → Succ(Succ(vuz391))
new_primMinusNatS0(Succ(vuz3380), Succ(vuz3390)) → new_primMinusNatS0(vuz3380, vuz3390)
new_primMinusNatS0(Zero, Succ(vuz3390)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz3380), Zero) → Succ(vuz3380)
Q is empty.
We have to consider all (P,Q,R)-chains.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_gcd0Gcd'(vuz374, Pos(Succ(vuz37300))) → new_gcd0Gcd'(Pos(Succ(vuz37300)), new_primRemInt(vuz374, vuz37300))
new_gcd0Gcd'(Pos(x0), Neg(Succ(x1))) → new_gcd0Gcd'(Neg(Succ(x1)), Pos(new_primModNatS1(x0, x1)))
The TRS R consists of the following rules:
new_primModNatS1(Succ(Zero), Zero) → new_primModNatS1(new_primMinusNatS0(Zero, Zero), Zero)
new_primMinusNatS0(Zero, Succ(vuz3390)) → Zero
new_primModNatS01(vuz391, vuz392) → new_primModNatS1(new_primMinusNatS0(Succ(vuz391), Succ(vuz392)), Succ(vuz392))
new_primModNatS1(Succ(Zero), Succ(vuz373000)) → Succ(Zero)
new_primRemInt0(Pos(vuz3740), vuz37300) → Pos(new_primModNatS1(vuz3740, vuz37300))
new_primRemInt(Neg(vuz3740), vuz37300) → Neg(new_primModNatS1(vuz3740, vuz37300))
new_primModNatS02(vuz391, vuz392, Zero, Succ(vuz3940)) → Succ(Succ(vuz391))
new_primRemInt0(Neg(vuz3740), vuz37300) → Neg(new_primModNatS1(vuz3740, vuz37300))
new_primModNatS1(Zero, vuz37300) → Zero
new_primModNatS1(Succ(Succ(vuz374000)), Succ(vuz373000)) → new_primModNatS02(vuz374000, vuz373000, vuz374000, vuz373000)
new_primModNatS02(vuz391, vuz392, Succ(vuz3930), Zero) → new_primModNatS01(vuz391, vuz392)
new_primMinusNatS0(Zero, Zero) → Zero
new_primModNatS02(vuz391, vuz392, Zero, Zero) → new_primModNatS01(vuz391, vuz392)
new_primModNatS02(vuz391, vuz392, Succ(vuz3930), Succ(vuz3940)) → new_primModNatS02(vuz391, vuz392, vuz3930, vuz3940)
new_primRemInt(Pos(vuz3740), vuz37300) → Pos(new_primModNatS1(vuz3740, vuz37300))
new_primMinusNatS0(Succ(vuz3380), Succ(vuz3390)) → new_primMinusNatS0(vuz3380, vuz3390)
new_primMinusNatS0(Succ(vuz3380), Zero) → Succ(vuz3380)
new_primModNatS1(Succ(Succ(vuz374000)), Zero) → new_primModNatS1(new_primMinusNatS0(Succ(vuz374000), Zero), Zero)
The set Q consists of the following terms:
new_primModNatS02(x0, x1, Zero, Succ(x2))
new_primModNatS1(Succ(Zero), Succ(x0))
new_primRemInt0(Neg(x0), x1)
new_primModNatS1(Zero, x0)
new_primModNatS1(Succ(Succ(x0)), Zero)
new_primModNatS1(Succ(Succ(x0)), Succ(x1))
new_primModNatS01(x0, x1)
new_primModNatS02(x0, x1, Succ(x2), Zero)
new_primMinusNatS0(Zero, Succ(x0))
new_primModNatS02(x0, x1, Zero, Zero)
new_primRemInt(Neg(x0), x1)
new_primMinusNatS0(Succ(x0), Zero)
new_primRemInt(Pos(x0), x1)
new_primRemInt0(Pos(x0), x1)
new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
new_primModNatS1(Succ(Zero), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_gcd0Gcd'(vuz374, Pos(Succ(vuz37300))) → new_gcd0Gcd'(Pos(Succ(vuz37300)), new_primRemInt(vuz374, vuz37300))
new_gcd0Gcd'(Pos(x0), Neg(Succ(x1))) → new_gcd0Gcd'(Neg(Succ(x1)), Pos(new_primModNatS1(x0, x1)))
The TRS R consists of the following rules:
new_primModNatS1(Succ(Zero), Zero) → new_primModNatS1(new_primMinusNatS0(Zero, Zero), Zero)
new_primModNatS1(Succ(Succ(vuz374000)), Zero) → new_primModNatS1(new_primMinusNatS0(Succ(vuz374000), Zero), Zero)
new_primModNatS1(Succ(Zero), Succ(vuz373000)) → Succ(Zero)
new_primModNatS1(Zero, vuz37300) → Zero
new_primModNatS02(vuz391, vuz392, Succ(vuz3930), Succ(vuz3940)) → new_primModNatS02(vuz391, vuz392, vuz3930, vuz3940)
new_primModNatS1(Succ(Succ(vuz374000)), Succ(vuz373000)) → new_primModNatS02(vuz374000, vuz373000, vuz374000, vuz373000)
new_primModNatS02(vuz391, vuz392, Succ(vuz3930), Zero) → new_primModNatS01(vuz391, vuz392)
new_primModNatS01(vuz391, vuz392) → new_primModNatS1(new_primMinusNatS0(Succ(vuz391), Succ(vuz392)), Succ(vuz392))
new_primModNatS02(vuz391, vuz392, Zero, Zero) → new_primModNatS01(vuz391, vuz392)
new_primModNatS02(vuz391, vuz392, Zero, Succ(vuz3940)) → Succ(Succ(vuz391))
new_primMinusNatS0(Succ(vuz3380), Succ(vuz3390)) → new_primMinusNatS0(vuz3380, vuz3390)
new_primMinusNatS0(Zero, Succ(vuz3390)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz3380), Zero) → Succ(vuz3380)
new_primRemInt(Neg(vuz3740), vuz37300) → Neg(new_primModNatS1(vuz3740, vuz37300))
new_primRemInt(Pos(vuz3740), vuz37300) → Pos(new_primModNatS1(vuz3740, vuz37300))
The set Q consists of the following terms:
new_primModNatS02(x0, x1, Zero, Succ(x2))
new_primModNatS1(Succ(Zero), Succ(x0))
new_primRemInt0(Neg(x0), x1)
new_primModNatS1(Zero, x0)
new_primModNatS1(Succ(Succ(x0)), Zero)
new_primModNatS1(Succ(Succ(x0)), Succ(x1))
new_primModNatS01(x0, x1)
new_primModNatS02(x0, x1, Succ(x2), Zero)
new_primMinusNatS0(Zero, Succ(x0))
new_primModNatS02(x0, x1, Zero, Zero)
new_primRemInt(Neg(x0), x1)
new_primMinusNatS0(Succ(x0), Zero)
new_primRemInt(Pos(x0), x1)
new_primRemInt0(Pos(x0), x1)
new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
new_primModNatS1(Succ(Zero), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
new_primRemInt0(Neg(x0), x1)
new_primRemInt0(Pos(x0), x1)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_gcd0Gcd'(vuz374, Pos(Succ(vuz37300))) → new_gcd0Gcd'(Pos(Succ(vuz37300)), new_primRemInt(vuz374, vuz37300))
new_gcd0Gcd'(Pos(x0), Neg(Succ(x1))) → new_gcd0Gcd'(Neg(Succ(x1)), Pos(new_primModNatS1(x0, x1)))
The TRS R consists of the following rules:
new_primModNatS1(Succ(Zero), Zero) → new_primModNatS1(new_primMinusNatS0(Zero, Zero), Zero)
new_primModNatS1(Succ(Succ(vuz374000)), Zero) → new_primModNatS1(new_primMinusNatS0(Succ(vuz374000), Zero), Zero)
new_primModNatS1(Succ(Zero), Succ(vuz373000)) → Succ(Zero)
new_primModNatS1(Zero, vuz37300) → Zero
new_primModNatS02(vuz391, vuz392, Succ(vuz3930), Succ(vuz3940)) → new_primModNatS02(vuz391, vuz392, vuz3930, vuz3940)
new_primModNatS1(Succ(Succ(vuz374000)), Succ(vuz373000)) → new_primModNatS02(vuz374000, vuz373000, vuz374000, vuz373000)
new_primModNatS02(vuz391, vuz392, Succ(vuz3930), Zero) → new_primModNatS01(vuz391, vuz392)
new_primModNatS01(vuz391, vuz392) → new_primModNatS1(new_primMinusNatS0(Succ(vuz391), Succ(vuz392)), Succ(vuz392))
new_primModNatS02(vuz391, vuz392, Zero, Zero) → new_primModNatS01(vuz391, vuz392)
new_primModNatS02(vuz391, vuz392, Zero, Succ(vuz3940)) → Succ(Succ(vuz391))
new_primMinusNatS0(Succ(vuz3380), Succ(vuz3390)) → new_primMinusNatS0(vuz3380, vuz3390)
new_primMinusNatS0(Zero, Succ(vuz3390)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz3380), Zero) → Succ(vuz3380)
new_primRemInt(Neg(vuz3740), vuz37300) → Neg(new_primModNatS1(vuz3740, vuz37300))
new_primRemInt(Pos(vuz3740), vuz37300) → Pos(new_primModNatS1(vuz3740, vuz37300))
The set Q consists of the following terms:
new_primModNatS02(x0, x1, Zero, Succ(x2))
new_primModNatS1(Succ(Zero), Succ(x0))
new_primModNatS1(Zero, x0)
new_primModNatS1(Succ(Succ(x0)), Zero)
new_primModNatS1(Succ(Succ(x0)), Succ(x1))
new_primModNatS01(x0, x1)
new_primModNatS02(x0, x1, Succ(x2), Zero)
new_primMinusNatS0(Zero, Succ(x0))
new_primModNatS02(x0, x1, Zero, Zero)
new_primRemInt(Neg(x0), x1)
new_primMinusNatS0(Succ(x0), Zero)
new_primRemInt(Pos(x0), x1)
new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
new_primModNatS1(Succ(Zero), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule new_gcd0Gcd'(vuz374, Pos(Succ(vuz37300))) → new_gcd0Gcd'(Pos(Succ(vuz37300)), new_primRemInt(vuz374, vuz37300)) at position [1] we obtained the following new rules:
new_gcd0Gcd'(Pos(x0), Pos(Succ(x1))) → new_gcd0Gcd'(Pos(Succ(x1)), Pos(new_primModNatS1(x0, x1)))
new_gcd0Gcd'(Neg(x0), Pos(Succ(x1))) → new_gcd0Gcd'(Pos(Succ(x1)), Neg(new_primModNatS1(x0, x1)))
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_gcd0Gcd'(Pos(x0), Pos(Succ(x1))) → new_gcd0Gcd'(Pos(Succ(x1)), Pos(new_primModNatS1(x0, x1)))
new_gcd0Gcd'(Neg(x0), Pos(Succ(x1))) → new_gcd0Gcd'(Pos(Succ(x1)), Neg(new_primModNatS1(x0, x1)))
new_gcd0Gcd'(Pos(x0), Neg(Succ(x1))) → new_gcd0Gcd'(Neg(Succ(x1)), Pos(new_primModNatS1(x0, x1)))
The TRS R consists of the following rules:
new_primModNatS1(Succ(Zero), Zero) → new_primModNatS1(new_primMinusNatS0(Zero, Zero), Zero)
new_primModNatS1(Succ(Succ(vuz374000)), Zero) → new_primModNatS1(new_primMinusNatS0(Succ(vuz374000), Zero), Zero)
new_primModNatS1(Succ(Zero), Succ(vuz373000)) → Succ(Zero)
new_primModNatS1(Zero, vuz37300) → Zero
new_primModNatS02(vuz391, vuz392, Succ(vuz3930), Succ(vuz3940)) → new_primModNatS02(vuz391, vuz392, vuz3930, vuz3940)
new_primModNatS1(Succ(Succ(vuz374000)), Succ(vuz373000)) → new_primModNatS02(vuz374000, vuz373000, vuz374000, vuz373000)
new_primModNatS02(vuz391, vuz392, Succ(vuz3930), Zero) → new_primModNatS01(vuz391, vuz392)
new_primModNatS01(vuz391, vuz392) → new_primModNatS1(new_primMinusNatS0(Succ(vuz391), Succ(vuz392)), Succ(vuz392))
new_primModNatS02(vuz391, vuz392, Zero, Zero) → new_primModNatS01(vuz391, vuz392)
new_primModNatS02(vuz391, vuz392, Zero, Succ(vuz3940)) → Succ(Succ(vuz391))
new_primMinusNatS0(Succ(vuz3380), Succ(vuz3390)) → new_primMinusNatS0(vuz3380, vuz3390)
new_primMinusNatS0(Zero, Succ(vuz3390)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz3380), Zero) → Succ(vuz3380)
new_primRemInt(Neg(vuz3740), vuz37300) → Neg(new_primModNatS1(vuz3740, vuz37300))
new_primRemInt(Pos(vuz3740), vuz37300) → Pos(new_primModNatS1(vuz3740, vuz37300))
The set Q consists of the following terms:
new_primModNatS02(x0, x1, Zero, Succ(x2))
new_primModNatS1(Succ(Zero), Succ(x0))
new_primModNatS1(Zero, x0)
new_primModNatS1(Succ(Succ(x0)), Zero)
new_primModNatS1(Succ(Succ(x0)), Succ(x1))
new_primModNatS01(x0, x1)
new_primModNatS02(x0, x1, Succ(x2), Zero)
new_primMinusNatS0(Zero, Succ(x0))
new_primModNatS02(x0, x1, Zero, Zero)
new_primRemInt(Neg(x0), x1)
new_primMinusNatS0(Succ(x0), Zero)
new_primRemInt(Pos(x0), x1)
new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
new_primModNatS1(Succ(Zero), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_gcd0Gcd'(Pos(x0), Pos(Succ(x1))) → new_gcd0Gcd'(Pos(Succ(x1)), Pos(new_primModNatS1(x0, x1)))
The TRS R consists of the following rules:
new_primModNatS1(Succ(Zero), Zero) → new_primModNatS1(new_primMinusNatS0(Zero, Zero), Zero)
new_primModNatS1(Succ(Succ(vuz374000)), Zero) → new_primModNatS1(new_primMinusNatS0(Succ(vuz374000), Zero), Zero)
new_primModNatS1(Succ(Zero), Succ(vuz373000)) → Succ(Zero)
new_primModNatS1(Zero, vuz37300) → Zero
new_primModNatS02(vuz391, vuz392, Succ(vuz3930), Succ(vuz3940)) → new_primModNatS02(vuz391, vuz392, vuz3930, vuz3940)
new_primModNatS1(Succ(Succ(vuz374000)), Succ(vuz373000)) → new_primModNatS02(vuz374000, vuz373000, vuz374000, vuz373000)
new_primModNatS02(vuz391, vuz392, Succ(vuz3930), Zero) → new_primModNatS01(vuz391, vuz392)
new_primModNatS01(vuz391, vuz392) → new_primModNatS1(new_primMinusNatS0(Succ(vuz391), Succ(vuz392)), Succ(vuz392))
new_primModNatS02(vuz391, vuz392, Zero, Zero) → new_primModNatS01(vuz391, vuz392)
new_primModNatS02(vuz391, vuz392, Zero, Succ(vuz3940)) → Succ(Succ(vuz391))
new_primMinusNatS0(Succ(vuz3380), Succ(vuz3390)) → new_primMinusNatS0(vuz3380, vuz3390)
new_primMinusNatS0(Zero, Succ(vuz3390)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz3380), Zero) → Succ(vuz3380)
new_primRemInt(Neg(vuz3740), vuz37300) → Neg(new_primModNatS1(vuz3740, vuz37300))
new_primRemInt(Pos(vuz3740), vuz37300) → Pos(new_primModNatS1(vuz3740, vuz37300))
The set Q consists of the following terms:
new_primModNatS02(x0, x1, Zero, Succ(x2))
new_primModNatS1(Succ(Zero), Succ(x0))
new_primModNatS1(Zero, x0)
new_primModNatS1(Succ(Succ(x0)), Zero)
new_primModNatS1(Succ(Succ(x0)), Succ(x1))
new_primModNatS01(x0, x1)
new_primModNatS02(x0, x1, Succ(x2), Zero)
new_primMinusNatS0(Zero, Succ(x0))
new_primModNatS02(x0, x1, Zero, Zero)
new_primRemInt(Neg(x0), x1)
new_primMinusNatS0(Succ(x0), Zero)
new_primRemInt(Pos(x0), x1)
new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
new_primModNatS1(Succ(Zero), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_gcd0Gcd'(Pos(x0), Pos(Succ(x1))) → new_gcd0Gcd'(Pos(Succ(x1)), Pos(new_primModNatS1(x0, x1)))
The TRS R consists of the following rules:
new_primModNatS1(Succ(Zero), Zero) → new_primModNatS1(new_primMinusNatS0(Zero, Zero), Zero)
new_primModNatS1(Succ(Succ(vuz374000)), Zero) → new_primModNatS1(new_primMinusNatS0(Succ(vuz374000), Zero), Zero)
new_primModNatS1(Succ(Zero), Succ(vuz373000)) → Succ(Zero)
new_primModNatS1(Zero, vuz37300) → Zero
new_primModNatS02(vuz391, vuz392, Succ(vuz3930), Succ(vuz3940)) → new_primModNatS02(vuz391, vuz392, vuz3930, vuz3940)
new_primModNatS1(Succ(Succ(vuz374000)), Succ(vuz373000)) → new_primModNatS02(vuz374000, vuz373000, vuz374000, vuz373000)
new_primModNatS02(vuz391, vuz392, Succ(vuz3930), Zero) → new_primModNatS01(vuz391, vuz392)
new_primModNatS01(vuz391, vuz392) → new_primModNatS1(new_primMinusNatS0(Succ(vuz391), Succ(vuz392)), Succ(vuz392))
new_primModNatS02(vuz391, vuz392, Zero, Zero) → new_primModNatS01(vuz391, vuz392)
new_primModNatS02(vuz391, vuz392, Zero, Succ(vuz3940)) → Succ(Succ(vuz391))
new_primMinusNatS0(Succ(vuz3380), Succ(vuz3390)) → new_primMinusNatS0(vuz3380, vuz3390)
new_primMinusNatS0(Zero, Succ(vuz3390)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz3380), Zero) → Succ(vuz3380)
The set Q consists of the following terms:
new_primModNatS02(x0, x1, Zero, Succ(x2))
new_primModNatS1(Succ(Zero), Succ(x0))
new_primModNatS1(Zero, x0)
new_primModNatS1(Succ(Succ(x0)), Zero)
new_primModNatS1(Succ(Succ(x0)), Succ(x1))
new_primModNatS01(x0, x1)
new_primModNatS02(x0, x1, Succ(x2), Zero)
new_primMinusNatS0(Zero, Succ(x0))
new_primModNatS02(x0, x1, Zero, Zero)
new_primRemInt(Neg(x0), x1)
new_primMinusNatS0(Succ(x0), Zero)
new_primRemInt(Pos(x0), x1)
new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
new_primModNatS1(Succ(Zero), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
new_primRemInt(Neg(x0), x1)
new_primRemInt(Pos(x0), x1)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_gcd0Gcd'(Pos(x0), Pos(Succ(x1))) → new_gcd0Gcd'(Pos(Succ(x1)), Pos(new_primModNatS1(x0, x1)))
The TRS R consists of the following rules:
new_primModNatS1(Succ(Zero), Zero) → new_primModNatS1(new_primMinusNatS0(Zero, Zero), Zero)
new_primModNatS1(Succ(Succ(vuz374000)), Zero) → new_primModNatS1(new_primMinusNatS0(Succ(vuz374000), Zero), Zero)
new_primModNatS1(Succ(Zero), Succ(vuz373000)) → Succ(Zero)
new_primModNatS1(Zero, vuz37300) → Zero
new_primModNatS02(vuz391, vuz392, Succ(vuz3930), Succ(vuz3940)) → new_primModNatS02(vuz391, vuz392, vuz3930, vuz3940)
new_primModNatS1(Succ(Succ(vuz374000)), Succ(vuz373000)) → new_primModNatS02(vuz374000, vuz373000, vuz374000, vuz373000)
new_primModNatS02(vuz391, vuz392, Succ(vuz3930), Zero) → new_primModNatS01(vuz391, vuz392)
new_primModNatS01(vuz391, vuz392) → new_primModNatS1(new_primMinusNatS0(Succ(vuz391), Succ(vuz392)), Succ(vuz392))
new_primModNatS02(vuz391, vuz392, Zero, Zero) → new_primModNatS01(vuz391, vuz392)
new_primModNatS02(vuz391, vuz392, Zero, Succ(vuz3940)) → Succ(Succ(vuz391))
new_primMinusNatS0(Succ(vuz3380), Succ(vuz3390)) → new_primMinusNatS0(vuz3380, vuz3390)
new_primMinusNatS0(Zero, Succ(vuz3390)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz3380), Zero) → Succ(vuz3380)
The set Q consists of the following terms:
new_primModNatS02(x0, x1, Zero, Succ(x2))
new_primModNatS1(Succ(Zero), Succ(x0))
new_primModNatS1(Zero, x0)
new_primModNatS1(Succ(Succ(x0)), Zero)
new_primModNatS1(Succ(Succ(x0)), Succ(x1))
new_primModNatS01(x0, x1)
new_primModNatS02(x0, x1, Succ(x2), Zero)
new_primMinusNatS0(Zero, Succ(x0))
new_primModNatS02(x0, x1, Zero, Zero)
new_primMinusNatS0(Succ(x0), Zero)
new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
new_primModNatS1(Succ(Zero), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule new_gcd0Gcd'(Pos(x0), Pos(Succ(x1))) → new_gcd0Gcd'(Pos(Succ(x1)), Pos(new_primModNatS1(x0, x1))) we obtained the following new rules:
new_gcd0Gcd'(Pos(Succ(z1)), Pos(Succ(x1))) → new_gcd0Gcd'(Pos(Succ(x1)), Pos(new_primModNatS1(Succ(z1), x1)))
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ MNOCProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_gcd0Gcd'(Pos(Succ(z1)), Pos(Succ(x1))) → new_gcd0Gcd'(Pos(Succ(x1)), Pos(new_primModNatS1(Succ(z1), x1)))
The TRS R consists of the following rules:
new_primModNatS1(Succ(Zero), Zero) → new_primModNatS1(new_primMinusNatS0(Zero, Zero), Zero)
new_primModNatS1(Succ(Succ(vuz374000)), Zero) → new_primModNatS1(new_primMinusNatS0(Succ(vuz374000), Zero), Zero)
new_primModNatS1(Succ(Zero), Succ(vuz373000)) → Succ(Zero)
new_primModNatS1(Zero, vuz37300) → Zero
new_primModNatS02(vuz391, vuz392, Succ(vuz3930), Succ(vuz3940)) → new_primModNatS02(vuz391, vuz392, vuz3930, vuz3940)
new_primModNatS1(Succ(Succ(vuz374000)), Succ(vuz373000)) → new_primModNatS02(vuz374000, vuz373000, vuz374000, vuz373000)
new_primModNatS02(vuz391, vuz392, Succ(vuz3930), Zero) → new_primModNatS01(vuz391, vuz392)
new_primModNatS01(vuz391, vuz392) → new_primModNatS1(new_primMinusNatS0(Succ(vuz391), Succ(vuz392)), Succ(vuz392))
new_primModNatS02(vuz391, vuz392, Zero, Zero) → new_primModNatS01(vuz391, vuz392)
new_primModNatS02(vuz391, vuz392, Zero, Succ(vuz3940)) → Succ(Succ(vuz391))
new_primMinusNatS0(Succ(vuz3380), Succ(vuz3390)) → new_primMinusNatS0(vuz3380, vuz3390)
new_primMinusNatS0(Zero, Succ(vuz3390)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz3380), Zero) → Succ(vuz3380)
The set Q consists of the following terms:
new_primModNatS02(x0, x1, Zero, Succ(x2))
new_primModNatS1(Succ(Zero), Succ(x0))
new_primModNatS1(Zero, x0)
new_primModNatS1(Succ(Succ(x0)), Zero)
new_primModNatS1(Succ(Succ(x0)), Succ(x1))
new_primModNatS01(x0, x1)
new_primModNatS02(x0, x1, Succ(x2), Zero)
new_primMinusNatS0(Zero, Succ(x0))
new_primModNatS02(x0, x1, Zero, Zero)
new_primMinusNatS0(Succ(x0), Zero)
new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
new_primModNatS1(Succ(Zero), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
We have to consider all minimal (P,Q,R)-chains.
We use the modular non-overlap check [17] to decrease Q to the empty set.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ MNOCProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_gcd0Gcd'(Pos(Succ(z1)), Pos(Succ(x1))) → new_gcd0Gcd'(Pos(Succ(x1)), Pos(new_primModNatS1(Succ(z1), x1)))
The TRS R consists of the following rules:
new_primModNatS1(Succ(Zero), Zero) → new_primModNatS1(new_primMinusNatS0(Zero, Zero), Zero)
new_primModNatS1(Succ(Succ(vuz374000)), Zero) → new_primModNatS1(new_primMinusNatS0(Succ(vuz374000), Zero), Zero)
new_primModNatS1(Succ(Zero), Succ(vuz373000)) → Succ(Zero)
new_primModNatS1(Zero, vuz37300) → Zero
new_primModNatS02(vuz391, vuz392, Succ(vuz3930), Succ(vuz3940)) → new_primModNatS02(vuz391, vuz392, vuz3930, vuz3940)
new_primModNatS1(Succ(Succ(vuz374000)), Succ(vuz373000)) → new_primModNatS02(vuz374000, vuz373000, vuz374000, vuz373000)
new_primModNatS02(vuz391, vuz392, Succ(vuz3930), Zero) → new_primModNatS01(vuz391, vuz392)
new_primModNatS01(vuz391, vuz392) → new_primModNatS1(new_primMinusNatS0(Succ(vuz391), Succ(vuz392)), Succ(vuz392))
new_primModNatS02(vuz391, vuz392, Zero, Zero) → new_primModNatS01(vuz391, vuz392)
new_primModNatS02(vuz391, vuz392, Zero, Succ(vuz3940)) → Succ(Succ(vuz391))
new_primMinusNatS0(Succ(vuz3380), Succ(vuz3390)) → new_primMinusNatS0(vuz3380, vuz3390)
new_primMinusNatS0(Zero, Succ(vuz3390)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz3380), Zero) → Succ(vuz3380)
Q is empty.
We have to consider all (P,Q,R)-chains.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_gcd0Gcd'(Pos(x0), Neg(Succ(x1))) → new_gcd0Gcd'(Neg(Succ(x1)), Pos(new_primModNatS1(x0, x1)))
new_gcd0Gcd'(Neg(x0), Pos(Succ(x1))) → new_gcd0Gcd'(Pos(Succ(x1)), Neg(new_primModNatS1(x0, x1)))
The TRS R consists of the following rules:
new_primModNatS1(Succ(Zero), Zero) → new_primModNatS1(new_primMinusNatS0(Zero, Zero), Zero)
new_primModNatS1(Succ(Succ(vuz374000)), Zero) → new_primModNatS1(new_primMinusNatS0(Succ(vuz374000), Zero), Zero)
new_primModNatS1(Succ(Zero), Succ(vuz373000)) → Succ(Zero)
new_primModNatS1(Zero, vuz37300) → Zero
new_primModNatS02(vuz391, vuz392, Succ(vuz3930), Succ(vuz3940)) → new_primModNatS02(vuz391, vuz392, vuz3930, vuz3940)
new_primModNatS1(Succ(Succ(vuz374000)), Succ(vuz373000)) → new_primModNatS02(vuz374000, vuz373000, vuz374000, vuz373000)
new_primModNatS02(vuz391, vuz392, Succ(vuz3930), Zero) → new_primModNatS01(vuz391, vuz392)
new_primModNatS01(vuz391, vuz392) → new_primModNatS1(new_primMinusNatS0(Succ(vuz391), Succ(vuz392)), Succ(vuz392))
new_primModNatS02(vuz391, vuz392, Zero, Zero) → new_primModNatS01(vuz391, vuz392)
new_primModNatS02(vuz391, vuz392, Zero, Succ(vuz3940)) → Succ(Succ(vuz391))
new_primMinusNatS0(Succ(vuz3380), Succ(vuz3390)) → new_primMinusNatS0(vuz3380, vuz3390)
new_primMinusNatS0(Zero, Succ(vuz3390)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz3380), Zero) → Succ(vuz3380)
new_primRemInt(Neg(vuz3740), vuz37300) → Neg(new_primModNatS1(vuz3740, vuz37300))
new_primRemInt(Pos(vuz3740), vuz37300) → Pos(new_primModNatS1(vuz3740, vuz37300))
The set Q consists of the following terms:
new_primModNatS02(x0, x1, Zero, Succ(x2))
new_primModNatS1(Succ(Zero), Succ(x0))
new_primModNatS1(Zero, x0)
new_primModNatS1(Succ(Succ(x0)), Zero)
new_primModNatS1(Succ(Succ(x0)), Succ(x1))
new_primModNatS01(x0, x1)
new_primModNatS02(x0, x1, Succ(x2), Zero)
new_primMinusNatS0(Zero, Succ(x0))
new_primModNatS02(x0, x1, Zero, Zero)
new_primRemInt(Neg(x0), x1)
new_primMinusNatS0(Succ(x0), Zero)
new_primRemInt(Pos(x0), x1)
new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
new_primModNatS1(Succ(Zero), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_gcd0Gcd'(Pos(x0), Neg(Succ(x1))) → new_gcd0Gcd'(Neg(Succ(x1)), Pos(new_primModNatS1(x0, x1)))
new_gcd0Gcd'(Neg(x0), Pos(Succ(x1))) → new_gcd0Gcd'(Pos(Succ(x1)), Neg(new_primModNatS1(x0, x1)))
The TRS R consists of the following rules:
new_primModNatS1(Succ(Zero), Zero) → new_primModNatS1(new_primMinusNatS0(Zero, Zero), Zero)
new_primModNatS1(Succ(Succ(vuz374000)), Zero) → new_primModNatS1(new_primMinusNatS0(Succ(vuz374000), Zero), Zero)
new_primModNatS1(Succ(Zero), Succ(vuz373000)) → Succ(Zero)
new_primModNatS1(Zero, vuz37300) → Zero
new_primModNatS02(vuz391, vuz392, Succ(vuz3930), Succ(vuz3940)) → new_primModNatS02(vuz391, vuz392, vuz3930, vuz3940)
new_primModNatS1(Succ(Succ(vuz374000)), Succ(vuz373000)) → new_primModNatS02(vuz374000, vuz373000, vuz374000, vuz373000)
new_primModNatS02(vuz391, vuz392, Succ(vuz3930), Zero) → new_primModNatS01(vuz391, vuz392)
new_primModNatS01(vuz391, vuz392) → new_primModNatS1(new_primMinusNatS0(Succ(vuz391), Succ(vuz392)), Succ(vuz392))
new_primModNatS02(vuz391, vuz392, Zero, Zero) → new_primModNatS01(vuz391, vuz392)
new_primModNatS02(vuz391, vuz392, Zero, Succ(vuz3940)) → Succ(Succ(vuz391))
new_primMinusNatS0(Succ(vuz3380), Succ(vuz3390)) → new_primMinusNatS0(vuz3380, vuz3390)
new_primMinusNatS0(Zero, Succ(vuz3390)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz3380), Zero) → Succ(vuz3380)
The set Q consists of the following terms:
new_primModNatS02(x0, x1, Zero, Succ(x2))
new_primModNatS1(Succ(Zero), Succ(x0))
new_primModNatS1(Zero, x0)
new_primModNatS1(Succ(Succ(x0)), Zero)
new_primModNatS1(Succ(Succ(x0)), Succ(x1))
new_primModNatS01(x0, x1)
new_primModNatS02(x0, x1, Succ(x2), Zero)
new_primMinusNatS0(Zero, Succ(x0))
new_primModNatS02(x0, x1, Zero, Zero)
new_primRemInt(Neg(x0), x1)
new_primMinusNatS0(Succ(x0), Zero)
new_primRemInt(Pos(x0), x1)
new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
new_primModNatS1(Succ(Zero), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
new_primRemInt(Neg(x0), x1)
new_primRemInt(Pos(x0), x1)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_gcd0Gcd'(Neg(x0), Pos(Succ(x1))) → new_gcd0Gcd'(Pos(Succ(x1)), Neg(new_primModNatS1(x0, x1)))
new_gcd0Gcd'(Pos(x0), Neg(Succ(x1))) → new_gcd0Gcd'(Neg(Succ(x1)), Pos(new_primModNatS1(x0, x1)))
The TRS R consists of the following rules:
new_primModNatS1(Succ(Zero), Zero) → new_primModNatS1(new_primMinusNatS0(Zero, Zero), Zero)
new_primModNatS1(Succ(Succ(vuz374000)), Zero) → new_primModNatS1(new_primMinusNatS0(Succ(vuz374000), Zero), Zero)
new_primModNatS1(Succ(Zero), Succ(vuz373000)) → Succ(Zero)
new_primModNatS1(Zero, vuz37300) → Zero
new_primModNatS02(vuz391, vuz392, Succ(vuz3930), Succ(vuz3940)) → new_primModNatS02(vuz391, vuz392, vuz3930, vuz3940)
new_primModNatS1(Succ(Succ(vuz374000)), Succ(vuz373000)) → new_primModNatS02(vuz374000, vuz373000, vuz374000, vuz373000)
new_primModNatS02(vuz391, vuz392, Succ(vuz3930), Zero) → new_primModNatS01(vuz391, vuz392)
new_primModNatS01(vuz391, vuz392) → new_primModNatS1(new_primMinusNatS0(Succ(vuz391), Succ(vuz392)), Succ(vuz392))
new_primModNatS02(vuz391, vuz392, Zero, Zero) → new_primModNatS01(vuz391, vuz392)
new_primModNatS02(vuz391, vuz392, Zero, Succ(vuz3940)) → Succ(Succ(vuz391))
new_primMinusNatS0(Succ(vuz3380), Succ(vuz3390)) → new_primMinusNatS0(vuz3380, vuz3390)
new_primMinusNatS0(Zero, Succ(vuz3390)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz3380), Zero) → Succ(vuz3380)
The set Q consists of the following terms:
new_primModNatS02(x0, x1, Zero, Succ(x2))
new_primModNatS1(Succ(Zero), Succ(x0))
new_primModNatS1(Zero, x0)
new_primModNatS1(Succ(Succ(x0)), Zero)
new_primModNatS1(Succ(Succ(x0)), Succ(x1))
new_primModNatS01(x0, x1)
new_primModNatS02(x0, x1, Succ(x2), Zero)
new_primMinusNatS0(Zero, Succ(x0))
new_primModNatS02(x0, x1, Zero, Zero)
new_primMinusNatS0(Succ(x0), Zero)
new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
new_primModNatS1(Succ(Zero), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule new_gcd0Gcd'(Pos(x0), Neg(Succ(x1))) → new_gcd0Gcd'(Neg(Succ(x1)), Pos(new_primModNatS1(x0, x1))) we obtained the following new rules:
new_gcd0Gcd'(Pos(Succ(z1)), Neg(Succ(x1))) → new_gcd0Gcd'(Neg(Succ(x1)), Pos(new_primModNatS1(Succ(z1), x1)))
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_gcd0Gcd'(Neg(x0), Pos(Succ(x1))) → new_gcd0Gcd'(Pos(Succ(x1)), Neg(new_primModNatS1(x0, x1)))
new_gcd0Gcd'(Pos(Succ(z1)), Neg(Succ(x1))) → new_gcd0Gcd'(Neg(Succ(x1)), Pos(new_primModNatS1(Succ(z1), x1)))
The TRS R consists of the following rules:
new_primModNatS1(Succ(Zero), Zero) → new_primModNatS1(new_primMinusNatS0(Zero, Zero), Zero)
new_primModNatS1(Succ(Succ(vuz374000)), Zero) → new_primModNatS1(new_primMinusNatS0(Succ(vuz374000), Zero), Zero)
new_primModNatS1(Succ(Zero), Succ(vuz373000)) → Succ(Zero)
new_primModNatS1(Zero, vuz37300) → Zero
new_primModNatS02(vuz391, vuz392, Succ(vuz3930), Succ(vuz3940)) → new_primModNatS02(vuz391, vuz392, vuz3930, vuz3940)
new_primModNatS1(Succ(Succ(vuz374000)), Succ(vuz373000)) → new_primModNatS02(vuz374000, vuz373000, vuz374000, vuz373000)
new_primModNatS02(vuz391, vuz392, Succ(vuz3930), Zero) → new_primModNatS01(vuz391, vuz392)
new_primModNatS01(vuz391, vuz392) → new_primModNatS1(new_primMinusNatS0(Succ(vuz391), Succ(vuz392)), Succ(vuz392))
new_primModNatS02(vuz391, vuz392, Zero, Zero) → new_primModNatS01(vuz391, vuz392)
new_primModNatS02(vuz391, vuz392, Zero, Succ(vuz3940)) → Succ(Succ(vuz391))
new_primMinusNatS0(Succ(vuz3380), Succ(vuz3390)) → new_primMinusNatS0(vuz3380, vuz3390)
new_primMinusNatS0(Zero, Succ(vuz3390)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz3380), Zero) → Succ(vuz3380)
The set Q consists of the following terms:
new_primModNatS02(x0, x1, Zero, Succ(x2))
new_primModNatS1(Succ(Zero), Succ(x0))
new_primModNatS1(Zero, x0)
new_primModNatS1(Succ(Succ(x0)), Zero)
new_primModNatS1(Succ(Succ(x0)), Succ(x1))
new_primModNatS01(x0, x1)
new_primModNatS02(x0, x1, Succ(x2), Zero)
new_primMinusNatS0(Zero, Succ(x0))
new_primModNatS02(x0, x1, Zero, Zero)
new_primMinusNatS0(Succ(x0), Zero)
new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
new_primModNatS1(Succ(Zero), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule new_gcd0Gcd'(Neg(x0), Pos(Succ(x1))) → new_gcd0Gcd'(Pos(Succ(x1)), Neg(new_primModNatS1(x0, x1))) we obtained the following new rules:
new_gcd0Gcd'(Neg(Succ(z1)), Pos(Succ(x1))) → new_gcd0Gcd'(Pos(Succ(x1)), Neg(new_primModNatS1(Succ(z1), x1)))
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ MNOCProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_gcd0Gcd'(Neg(Succ(z1)), Pos(Succ(x1))) → new_gcd0Gcd'(Pos(Succ(x1)), Neg(new_primModNatS1(Succ(z1), x1)))
new_gcd0Gcd'(Pos(Succ(z1)), Neg(Succ(x1))) → new_gcd0Gcd'(Neg(Succ(x1)), Pos(new_primModNatS1(Succ(z1), x1)))
The TRS R consists of the following rules:
new_primModNatS1(Succ(Zero), Zero) → new_primModNatS1(new_primMinusNatS0(Zero, Zero), Zero)
new_primModNatS1(Succ(Succ(vuz374000)), Zero) → new_primModNatS1(new_primMinusNatS0(Succ(vuz374000), Zero), Zero)
new_primModNatS1(Succ(Zero), Succ(vuz373000)) → Succ(Zero)
new_primModNatS1(Zero, vuz37300) → Zero
new_primModNatS02(vuz391, vuz392, Succ(vuz3930), Succ(vuz3940)) → new_primModNatS02(vuz391, vuz392, vuz3930, vuz3940)
new_primModNatS1(Succ(Succ(vuz374000)), Succ(vuz373000)) → new_primModNatS02(vuz374000, vuz373000, vuz374000, vuz373000)
new_primModNatS02(vuz391, vuz392, Succ(vuz3930), Zero) → new_primModNatS01(vuz391, vuz392)
new_primModNatS01(vuz391, vuz392) → new_primModNatS1(new_primMinusNatS0(Succ(vuz391), Succ(vuz392)), Succ(vuz392))
new_primModNatS02(vuz391, vuz392, Zero, Zero) → new_primModNatS01(vuz391, vuz392)
new_primModNatS02(vuz391, vuz392, Zero, Succ(vuz3940)) → Succ(Succ(vuz391))
new_primMinusNatS0(Succ(vuz3380), Succ(vuz3390)) → new_primMinusNatS0(vuz3380, vuz3390)
new_primMinusNatS0(Zero, Succ(vuz3390)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz3380), Zero) → Succ(vuz3380)
The set Q consists of the following terms:
new_primModNatS02(x0, x1, Zero, Succ(x2))
new_primModNatS1(Succ(Zero), Succ(x0))
new_primModNatS1(Zero, x0)
new_primModNatS1(Succ(Succ(x0)), Zero)
new_primModNatS1(Succ(Succ(x0)), Succ(x1))
new_primModNatS01(x0, x1)
new_primModNatS02(x0, x1, Succ(x2), Zero)
new_primMinusNatS0(Zero, Succ(x0))
new_primModNatS02(x0, x1, Zero, Zero)
new_primMinusNatS0(Succ(x0), Zero)
new_primModNatS02(x0, x1, Succ(x2), Succ(x3))
new_primModNatS1(Succ(Zero), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
We have to consider all minimal (P,Q,R)-chains.
We use the modular non-overlap check [17] to decrease Q to the empty set.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ MNOCProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_gcd0Gcd'(Pos(Succ(z1)), Neg(Succ(x1))) → new_gcd0Gcd'(Neg(Succ(x1)), Pos(new_primModNatS1(Succ(z1), x1)))
new_gcd0Gcd'(Neg(Succ(z1)), Pos(Succ(x1))) → new_gcd0Gcd'(Pos(Succ(x1)), Neg(new_primModNatS1(Succ(z1), x1)))
The TRS R consists of the following rules:
new_primModNatS1(Succ(Zero), Zero) → new_primModNatS1(new_primMinusNatS0(Zero, Zero), Zero)
new_primModNatS1(Succ(Succ(vuz374000)), Zero) → new_primModNatS1(new_primMinusNatS0(Succ(vuz374000), Zero), Zero)
new_primModNatS1(Succ(Zero), Succ(vuz373000)) → Succ(Zero)
new_primModNatS1(Zero, vuz37300) → Zero
new_primModNatS02(vuz391, vuz392, Succ(vuz3930), Succ(vuz3940)) → new_primModNatS02(vuz391, vuz392, vuz3930, vuz3940)
new_primModNatS1(Succ(Succ(vuz374000)), Succ(vuz373000)) → new_primModNatS02(vuz374000, vuz373000, vuz374000, vuz373000)
new_primModNatS02(vuz391, vuz392, Succ(vuz3930), Zero) → new_primModNatS01(vuz391, vuz392)
new_primModNatS01(vuz391, vuz392) → new_primModNatS1(new_primMinusNatS0(Succ(vuz391), Succ(vuz392)), Succ(vuz392))
new_primModNatS02(vuz391, vuz392, Zero, Zero) → new_primModNatS01(vuz391, vuz392)
new_primModNatS02(vuz391, vuz392, Zero, Succ(vuz3940)) → Succ(Succ(vuz391))
new_primMinusNatS0(Succ(vuz3380), Succ(vuz3390)) → new_primMinusNatS0(vuz3380, vuz3390)
new_primMinusNatS0(Zero, Succ(vuz3390)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz3380), Zero) → Succ(vuz3380)
Q is empty.
We have to consider all (P,Q,R)-chains.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_gcd2(Succ(vuz2870), Succ(vuz3030), vuz68) → new_gcd2(vuz2870, vuz3030, vuz68)
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_gcd2(Succ(vuz2870), Succ(vuz3030), vuz68) → new_gcd2(vuz2870, vuz3030, vuz68)
The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_gcd20(Succ(vuz3500), Succ(vuz3660), vuz144) → new_gcd20(vuz3500, vuz3660, vuz144)
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_gcd20(Succ(vuz3500), Succ(vuz3660), vuz144) → new_gcd20(vuz3500, vuz3660, vuz144)
The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vuz338, vuz339, Zero, Zero) → new_primDivNatS00(vuz338, vuz339)
new_primDivNatS0(vuz338, vuz339, Succ(vuz3400), Succ(vuz3410)) → new_primDivNatS0(vuz338, vuz339, vuz3400, vuz3410)
new_primDivNatS00(vuz338, vuz339) → new_primDivNatS(new_primMinusNatS0(vuz338, vuz339), Succ(vuz339))
new_primDivNatS(Succ(Succ(vuz28000)), Succ(vuz281000)) → new_primDivNatS0(vuz28000, vuz281000, vuz28000, vuz281000)
new_primDivNatS0(vuz338, vuz339, Succ(vuz3400), Zero) → new_primDivNatS(new_primMinusNatS0(vuz338, vuz339), Succ(vuz339))
new_primDivNatS(Succ(Zero), Zero) → new_primDivNatS(new_primMinusNatS2, Zero)
new_primDivNatS(Succ(Succ(vuz28000)), Zero) → new_primDivNatS(new_primMinusNatS1(vuz28000), Zero)
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vuz3390)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS1(vuz28000) → Succ(vuz28000)
new_primMinusNatS0(Succ(vuz3380), Succ(vuz3390)) → new_primMinusNatS0(vuz3380, vuz3390)
new_primMinusNatS0(Succ(vuz3380), Zero) → Succ(vuz3380)
new_primMinusNatS2 → Zero
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS2
new_primMinusNatS1(x0)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
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
↳ QDP
↳ QDP
↳ 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(vuz28000)), Zero) → new_primDivNatS(new_primMinusNatS1(vuz28000), Zero)
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vuz3390)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS1(vuz28000) → Succ(vuz28000)
new_primMinusNatS0(Succ(vuz3380), Succ(vuz3390)) → new_primMinusNatS0(vuz3380, vuz3390)
new_primMinusNatS0(Succ(vuz3380), Zero) → Succ(vuz3380)
new_primMinusNatS2 → Zero
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS2
new_primMinusNatS1(x0)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ 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(vuz28000)), Zero) → new_primDivNatS(new_primMinusNatS1(vuz28000), Zero)
The TRS R consists of the following rules:
new_primMinusNatS1(vuz28000) → Succ(vuz28000)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS2
new_primMinusNatS1(x0)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS2
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ 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(vuz28000)), Zero) → new_primDivNatS(new_primMinusNatS1(vuz28000), Zero)
The TRS R consists of the following rules:
new_primMinusNatS1(vuz28000) → Succ(vuz28000)
The set Q consists of the following terms:
new_primMinusNatS1(x0)
We have to consider all minimal (P,Q,R)-chains.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:
new_primDivNatS(Succ(Succ(vuz28000)), Zero) → new_primDivNatS(new_primMinusNatS1(vuz28000), Zero)
Strictly oriented rules of the TRS R:
new_primMinusNatS1(vuz28000) → Succ(vuz28000)
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_primMinusNatS1(x1)) = 2 + 2·x1
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ 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_primMinusNatS1(x0)
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vuz338, vuz339, Zero, Zero) → new_primDivNatS00(vuz338, vuz339)
new_primDivNatS0(vuz338, vuz339, Succ(vuz3400), Succ(vuz3410)) → new_primDivNatS0(vuz338, vuz339, vuz3400, vuz3410)
new_primDivNatS(Succ(Succ(vuz28000)), Succ(vuz281000)) → new_primDivNatS0(vuz28000, vuz281000, vuz28000, vuz281000)
new_primDivNatS00(vuz338, vuz339) → new_primDivNatS(new_primMinusNatS0(vuz338, vuz339), Succ(vuz339))
new_primDivNatS0(vuz338, vuz339, Succ(vuz3400), Zero) → new_primDivNatS(new_primMinusNatS0(vuz338, vuz339), Succ(vuz339))
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vuz3390)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS1(vuz28000) → Succ(vuz28000)
new_primMinusNatS0(Succ(vuz3380), Succ(vuz3390)) → new_primMinusNatS0(vuz3380, vuz3390)
new_primMinusNatS0(Succ(vuz3380), Zero) → Succ(vuz3380)
new_primMinusNatS2 → Zero
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS2
new_primMinusNatS1(x0)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vuz338, vuz339, Zero, Zero) → new_primDivNatS00(vuz338, vuz339)
new_primDivNatS0(vuz338, vuz339, Succ(vuz3400), Succ(vuz3410)) → new_primDivNatS0(vuz338, vuz339, vuz3400, vuz3410)
new_primDivNatS(Succ(Succ(vuz28000)), Succ(vuz281000)) → new_primDivNatS0(vuz28000, vuz281000, vuz28000, vuz281000)
new_primDivNatS00(vuz338, vuz339) → new_primDivNatS(new_primMinusNatS0(vuz338, vuz339), Succ(vuz339))
new_primDivNatS0(vuz338, vuz339, Succ(vuz3400), Zero) → new_primDivNatS(new_primMinusNatS0(vuz338, vuz339), Succ(vuz339))
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vuz3390)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz3380), Succ(vuz3390)) → new_primMinusNatS0(vuz3380, vuz3390)
new_primMinusNatS0(Succ(vuz3380), Zero) → Succ(vuz3380)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS2
new_primMinusNatS1(x0)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
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
new_primMinusNatS1(x0)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vuz338, vuz339, Zero, Zero) → new_primDivNatS00(vuz338, vuz339)
new_primDivNatS00(vuz338, vuz339) → new_primDivNatS(new_primMinusNatS0(vuz338, vuz339), Succ(vuz339))
new_primDivNatS(Succ(Succ(vuz28000)), Succ(vuz281000)) → new_primDivNatS0(vuz28000, vuz281000, vuz28000, vuz281000)
new_primDivNatS0(vuz338, vuz339, Succ(vuz3400), Succ(vuz3410)) → new_primDivNatS0(vuz338, vuz339, vuz3400, vuz3410)
new_primDivNatS0(vuz338, vuz339, Succ(vuz3400), Zero) → new_primDivNatS(new_primMinusNatS0(vuz338, vuz339), Succ(vuz339))
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vuz3390)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz3380), Succ(vuz3390)) → new_primMinusNatS0(vuz3380, vuz3390)
new_primMinusNatS0(Succ(vuz3380), Zero) → Succ(vuz3380)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
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(vuz338, vuz339, Zero, Zero) → new_primDivNatS00(vuz338, vuz339)
new_primDivNatS(Succ(Succ(vuz28000)), Succ(vuz281000)) → new_primDivNatS0(vuz28000, vuz281000, vuz28000, vuz281000)
new_primDivNatS0(vuz338, vuz339, Succ(vuz3400), Zero) → new_primDivNatS(new_primMinusNatS0(vuz338, vuz339), Succ(vuz339))
The remaining pairs can at least be oriented weakly.
new_primDivNatS00(vuz338, vuz339) → new_primDivNatS(new_primMinusNatS0(vuz338, vuz339), Succ(vuz339))
new_primDivNatS0(vuz338, vuz339, Succ(vuz3400), Succ(vuz3410)) → new_primDivNatS0(vuz338, vuz339, vuz3400, vuz3410)
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)) = x1
POL(new_primMinusNatS0(x1, x2)) = x1
The following usable rules [17] were oriented:
new_primMinusNatS0(Succ(vuz3380), Succ(vuz3390)) → new_primMinusNatS0(vuz3380, vuz3390)
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz3380), Zero) → Succ(vuz3380)
new_primMinusNatS0(Zero, Succ(vuz3390)) → Zero
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vuz338, vuz339, Succ(vuz3400), Succ(vuz3410)) → new_primDivNatS0(vuz338, vuz339, vuz3400, vuz3410)
new_primDivNatS00(vuz338, vuz339) → new_primDivNatS(new_primMinusNatS0(vuz338, vuz339), Succ(vuz339))
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vuz3390)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz3380), Succ(vuz3390)) → new_primMinusNatS0(vuz3380, vuz3390)
new_primMinusNatS0(Succ(vuz3380), Zero) → Succ(vuz3380)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vuz338, vuz339, Succ(vuz3400), Succ(vuz3410)) → new_primDivNatS0(vuz338, vuz339, vuz3400, vuz3410)
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vuz3390)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz3380), Succ(vuz3390)) → new_primMinusNatS0(vuz3380, vuz3390)
new_primMinusNatS0(Succ(vuz3380), Zero) → Succ(vuz3380)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vuz338, vuz339, Succ(vuz3400), Succ(vuz3410)) → new_primDivNatS0(vuz338, vuz339, vuz3400, vuz3410)
R is empty.
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ 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(vuz338, vuz339, Succ(vuz3400), Succ(vuz3410)) → new_primDivNatS0(vuz338, vuz339, vuz3400, vuz3410)
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(vuz338, vuz339, Succ(vuz3400), Succ(vuz3410)) → new_primDivNatS0(vuz338, vuz339, vuz3400, vuz3410)
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
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primQuotInt(Succ(vuz1690), Succ(vuz2050), vuz68) → new_primQuotInt(vuz1690, vuz2050, vuz68)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- new_primQuotInt(Succ(vuz1690), Succ(vuz2050), vuz68) → new_primQuotInt(vuz1690, vuz2050, vuz68)
The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primQuotInt0(Succ(vuz1850), Succ(vuz1990), vuz144) → new_primQuotInt0(vuz1850, vuz1990, vuz144)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- new_primQuotInt0(Succ(vuz1850), Succ(vuz1990), vuz144) → new_primQuotInt0(vuz1850, vuz1990, vuz144)
The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3
Haskell To QDPs