MAYBE 9.788
H-Termination proof of /home/matraf/haskell/eval_FullyBlown_Fast/empty.hs
H-Termination of the given Haskell-Program with start terms could not be shown:
↳ HASKELL
↳ IFR
mainModule Main
| ((toRational :: Float -> Ratio Integer) :: Float -> Ratio Integer) |
module Main where
If Reductions:
The following If expression
if primGEqNatS x y then Succ (primDivNatS (primMinusNatS x y) (Succ y)) else Zero
is transformed to
primDivNatS0 | x y True | = Succ (primDivNatS (primMinusNatS x y) (Succ y)) |
primDivNatS0 | x y False | = Zero |
The following If expression
if primGEqNatS x y then primModNatS (primMinusNatS x y) (Succ y) else Succ x
is transformed to
primModNatS0 | x y True | = primModNatS (primMinusNatS x y) (Succ y) |
primModNatS0 | x y False | = Succ x |
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
mainModule Main
| ((toRational :: Float -> Ratio Integer) :: Float -> Ratio Integer) |
module Main where
Replaced joker patterns by fresh variables and removed binding patterns.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
mainModule Main
| ((toRational :: Float -> Ratio Integer) :: Float -> Ratio Integer) |
module Main where
Cond Reductions:
The following Function with conditions
gcd' | x 0 | = x |
gcd' | x y | = gcd' y (x `rem` y) |
is transformed to
gcd' | x xz | = gcd'2 x xz |
gcd' | x y | = gcd'0 x y |
gcd'0 | x y | = gcd' y (x `rem` y) |
gcd'1 | True x xz | = x |
gcd'1 | yu yv yw | = gcd'0 yv yw |
gcd'2 | x xz | = gcd'1 (xz == 0) x xz |
gcd'2 | yx yy | = gcd'0 yx yy |
The following Function with conditions
gcd | 0 0 | = error [] |
gcd | x y | =
gcd' (abs x) (abs y) |
where |
gcd' | x 0 | = x |
gcd' | x y | = gcd' y (x `rem` y) |
|
|
is transformed to
gcd | yz zu | = gcd3 yz zu |
gcd | x y | = gcd0 x y |
gcd0 | x y | =
gcd' (abs x) (abs y) |
where |
gcd' | x xz | = gcd'2 x xz |
gcd' | x y | = gcd'0 x y |
|
|
gcd'0 | x y | = gcd' y (x `rem` y) |
|
|
gcd'1 | True x xz | = x |
gcd'1 | yu yv yw | = gcd'0 yv yw |
|
|
gcd'2 | x xz | = gcd'1 (xz == 0) x xz |
gcd'2 | yx yy | = gcd'0 yx yy |
|
|
gcd1 | True yz zu | = error [] |
gcd1 | zv zw zx | = gcd0 zw zx |
gcd2 | True yz zu | = gcd1 (zu == 0) yz zu |
gcd2 | zy zz vuu | = gcd0 zz vuu |
gcd3 | yz zu | = gcd2 (yz == 0) yz zu |
gcd3 | vuv vuw | = gcd0 vuv vuw |
The following Function with conditions
is transformed to
absReal1 | x True | = x |
absReal1 | x False | = absReal0 x otherwise |
absReal0 | x True | = `negate` x |
absReal2 | x | = absReal1 x (x >= 0) |
The following Function with conditions
is transformed to
undefined0 | True | = undefined |
undefined1 | | = undefined0 False |
The following Function with conditions
reduce | x y |
| | y == 0 | |
| | otherwise |
= | x `quot` d :% (y `quot` d) |
|
|
where | |
|
is transformed to
reduce2 | x y | =
reduce1 x y (y == 0) |
where | |
|
reduce0 | x y True | = x `quot` d :% (y `quot` d) |
|
|
reduce1 | x y True | = error [] |
reduce1 | x y False | = reduce0 x y otherwise |
|
|
The following Function with conditions
signumReal | x |
| | x == 0 | |
| | x > 0 | |
| | otherwise | |
|
is transformed to
signumReal | x | = signumReal3 x |
signumReal2 | x True | = 0 |
signumReal2 | x False | = signumReal1 x (x > 0) |
signumReal1 | x True | = 1 |
signumReal1 | x False | = signumReal0 x otherwise |
signumReal3 | x | = signumReal2 x (x == 0) |
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
mainModule Main
| ((toRational :: Float -> Ratio Integer) :: Float -> Ratio Integer) |
module Main where
Let/Where Reductions:
The bindings of the following Let/Where expression
reduce1 x y (y == 0) |
where | |
|
reduce0 | x y True | = x `quot` d :% (y `quot` d) |
|
|
reduce1 | x y True | = error [] |
reduce1 | x y False | = reduce0 x y otherwise |
|
are unpacked to the following functions on top level
reduce2Reduce1 | vux vuy x y True | = error [] |
reduce2Reduce1 | vux vuy x y False | = reduce2Reduce0 vux vuy x y otherwise |
reduce2Reduce0 | vux vuy x y True | = x `quot` reduce2D vux vuy :% (y `quot` reduce2D vux vuy) |
reduce2D | vux vuy | = gcd vux vuy |
The bindings of the following Let/Where expression
gcd' (abs x) (abs y) |
where |
gcd' | x xz | = gcd'2 x xz |
gcd' | x y | = gcd'0 x y |
|
|
gcd'0 | x y | = gcd' y (x `rem` y) |
|
|
gcd'1 | True x xz | = x |
gcd'1 | yu yv yw | = gcd'0 yv yw |
|
|
gcd'2 | x xz | = gcd'1 (xz == 0) x xz |
gcd'2 | yx yy | = gcd'0 yx yy |
|
are unpacked to the following functions on top level
gcd0Gcd'0 | x y | = gcd0Gcd' y (x `rem` y) |
gcd0Gcd' | x xz | = gcd0Gcd'2 x xz |
gcd0Gcd' | x y | = gcd0Gcd'0 x y |
gcd0Gcd'2 | x xz | = gcd0Gcd'1 (xz == 0) x xz |
gcd0Gcd'2 | yx yy | = gcd0Gcd'0 yx yy |
gcd0Gcd'1 | True x xz | = x |
gcd0Gcd'1 | yu yv yw | = gcd0Gcd'0 yv yw |
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
mainModule Main
| ((toRational :: Float -> Ratio Integer) :: Float -> Ratio Integer) |
module Main where
Num Reduction: All numbers are transformed to thier corresponding representation with Pos, Neg, Succ and Zero.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ Narrow
mainModule Main
| (toRational :: Float -> Ratio Integer) |
module Main where
Haskell To QDPs
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primMulNat(Succ(vuz30000)) → new_primMulNat(vuz30000)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- new_primMulNat(Succ(vuz30000)) → new_primMulNat(vuz30000)
The graph contains the following edges 1 > 1
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primMinusNatS(Succ(vuz10200), Succ(vuz1030)) → new_primMinusNatS(vuz10200, vuz1030)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- new_primMinusNatS(Succ(vuz10200), Succ(vuz1030)) → new_primMinusNatS(vuz10200, vuz1030)
The graph contains the following edges 1 > 1, 2 > 2
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS00(vuz135, vuz136, Succ(vuz1370), Zero) → new_primDivNatS(new_primMinusNatS0(Succ(vuz135), Succ(vuz136)), vuz136)
new_primDivNatS00(vuz135, vuz136, Succ(vuz1370), Succ(vuz1380)) → new_primDivNatS00(vuz135, vuz136, vuz1370, vuz1380)
new_primDivNatS0(vuz115, vuz1170) → new_primDivNatS0(vuz115, vuz1170)
new_primDivNatS01(vuz135, vuz136) → new_primDivNatS(new_primMinusNatS0(Succ(vuz135), Succ(vuz136)), vuz136)
new_primDivNatS00(vuz135, vuz136, Zero, Zero) → new_primDivNatS01(vuz135, vuz136)
new_primDivNatS(Succ(Succ(vuz13900)), vuz136) → new_primDivNatS0(vuz13900, vuz136)
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vuz1030)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz10200), Succ(vuz1030)) → new_primMinusNatS0(vuz10200, vuz1030)
new_primMinusNatS0(Succ(vuz10200), Zero) → Succ(vuz10200)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Zero, Zero)
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 4 less nodes.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vuz115, vuz1170) → new_primDivNatS0(vuz115, vuz1170)
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vuz1030)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz10200), Succ(vuz1030)) → new_primMinusNatS0(vuz10200, vuz1030)
new_primMinusNatS0(Succ(vuz10200), Zero) → Succ(vuz10200)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Zero, Zero)
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vuz115, vuz1170) → new_primDivNatS0(vuz115, vuz1170)
R is empty.
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Zero, Zero)
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Zero, Zero)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ NonTerminationProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vuz115, vuz1170) → new_primDivNatS0(vuz115, vuz1170)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.
The TRS P consists of the following rules:
new_primDivNatS0(vuz115, vuz1170) → new_primDivNatS0(vuz115, vuz1170)
The TRS R consists of the following rules:none
s = new_primDivNatS0(vuz115, vuz1170) evaluates to t =new_primDivNatS0(vuz115, vuz1170)
Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
- Semiunifier: [ ]
- Matcher: [ ]
Rewriting sequence
The DP semiunifies directly so there is only one rewrite step from new_primDivNatS0(vuz115, vuz1170) to new_primDivNatS0(vuz115, vuz1170).
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS00(vuz135, vuz136, Succ(vuz1370), Succ(vuz1380)) → new_primDivNatS00(vuz135, vuz136, vuz1370, vuz1380)
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vuz1030)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz10200), Succ(vuz1030)) → new_primMinusNatS0(vuz10200, vuz1030)
new_primMinusNatS0(Succ(vuz10200), Zero) → Succ(vuz10200)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Zero, Zero)
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS00(vuz135, vuz136, Succ(vuz1370), Succ(vuz1380)) → new_primDivNatS00(vuz135, vuz136, vuz1370, vuz1380)
R is empty.
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Zero, Zero)
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Zero, Zero)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS00(vuz135, vuz136, Succ(vuz1370), Succ(vuz1380)) → new_primDivNatS00(vuz135, vuz136, vuz1370, vuz1380)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- new_primDivNatS00(vuz135, vuz136, Succ(vuz1370), Succ(vuz1380)) → new_primDivNatS00(vuz135, vuz136, vuz1370, vuz1380)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS1(Succ(vuz1070), vuz108) → new_primDivNatS1(vuz1070, vuz108)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- new_primDivNatS1(Succ(vuz1070), vuz108) → new_primDivNatS1(vuz1070, vuz108)
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
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS02(vuz520000) → new_primDivNatS2(new_primMinusNatS1(vuz520000))
new_primDivNatS2(Succ(Succ(vuz520000))) → new_primDivNatS2(new_primMinusNatS1(vuz520000))
The TRS R consists of the following rules:
new_primMinusNatS1(vuz520000) → Succ(vuz520000)
The set Q consists of the following terms:
new_primMinusNatS1(x0)
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS2(Succ(Succ(vuz520000))) → new_primDivNatS2(new_primMinusNatS1(vuz520000))
The TRS R consists of the following rules:
new_primMinusNatS1(vuz520000) → Succ(vuz520000)
The set Q consists of the following terms:
new_primMinusNatS1(x0)
We have to consider all minimal (P,Q,R)-chains.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:
new_primDivNatS2(Succ(Succ(vuz520000))) → new_primDivNatS2(new_primMinusNatS1(vuz520000))
Strictly oriented rules of the TRS R:
new_primMinusNatS1(vuz520000) → Succ(vuz520000)
Used ordering: POLO with Polynomial interpretation [25]:
POL(Succ(x1)) = 1 + 2·x1
POL(new_primDivNatS2(x1)) = x1
POL(new_primMinusNatS1(x1)) = 2 + 2·x1
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
P is empty.
R is empty.
The set Q consists of the following terms:
new_primMinusNatS1(x0)
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot(Succ(vuz5000)) → new_quot(vuz5000)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- new_quot(Succ(vuz5000)) → new_quot(vuz5000)
The graph contains the following edges 1 > 1
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Instantiation
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot1(vuz115, Succ(Succ(vuz12100)), Zero, vuz120) → new_quot2(vuz115, vuz12100, Zero)
new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Succ(vuz1190)) → new_quot0(vuz115, vuz116, vuz117, vuz1180, vuz1190)
new_quot0(vuz115, vuz116, vuz117, Zero, Succ(vuz1190)) → new_quot1(vuz115, Succ(vuz117), Succ(vuz116), Succ(vuz117))
new_quot2(vuz115, vuz116, vuz117) → new_quot1(vuz115, new_primMinusNatS0(Succ(vuz116), vuz117), vuz117, new_primMinusNatS0(Succ(vuz116), vuz117))
new_quot1(vuz115, Succ(Succ(vuz12100)), Succ(vuz1170), vuz120) → new_quot0(vuz115, vuz12100, Succ(vuz1170), vuz12100, vuz1170)
new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Zero) → new_quot1(vuz115, new_primMinusNatS0(Succ(vuz116), vuz117), vuz117, new_primMinusNatS0(Succ(vuz116), vuz117))
new_quot0(vuz115, vuz116, vuz117, Zero, Zero) → new_quot2(vuz115, vuz116, vuz117)
new_quot1(vuz115, Succ(Zero), Succ(vuz1170), vuz120) → new_quot3(vuz115, vuz1170, Zero)
new_quot3(vuz44, vuz45, vuz14) → new_quot1(vuz44, Succ(Succ(vuz45)), vuz14, Succ(Succ(vuz45)))
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vuz1030)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz10200), Succ(vuz1030)) → new_primMinusNatS0(vuz10200, vuz1030)
new_primMinusNatS0(Succ(vuz10200), Zero) → Succ(vuz10200)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Zero, Zero)
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule new_quot3(vuz44, vuz45, vuz14) → new_quot1(vuz44, Succ(Succ(vuz45)), vuz14, Succ(Succ(vuz45))) we obtained the following new rules:
new_quot3(z0, z1, Zero) → new_quot1(z0, Succ(Succ(z1)), Zero, Succ(Succ(z1)))
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot1(vuz115, Succ(Succ(vuz12100)), Zero, vuz120) → new_quot2(vuz115, vuz12100, Zero)
new_quot3(z0, z1, Zero) → new_quot1(z0, Succ(Succ(z1)), Zero, Succ(Succ(z1)))
new_quot0(vuz115, vuz116, vuz117, Zero, Succ(vuz1190)) → new_quot1(vuz115, Succ(vuz117), Succ(vuz116), Succ(vuz117))
new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Succ(vuz1190)) → new_quot0(vuz115, vuz116, vuz117, vuz1180, vuz1190)
new_quot2(vuz115, vuz116, vuz117) → new_quot1(vuz115, new_primMinusNatS0(Succ(vuz116), vuz117), vuz117, new_primMinusNatS0(Succ(vuz116), vuz117))
new_quot1(vuz115, Succ(Succ(vuz12100)), Succ(vuz1170), vuz120) → new_quot0(vuz115, vuz12100, Succ(vuz1170), vuz12100, vuz1170)
new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Zero) → new_quot1(vuz115, new_primMinusNatS0(Succ(vuz116), vuz117), vuz117, new_primMinusNatS0(Succ(vuz116), vuz117))
new_quot1(vuz115, Succ(Zero), Succ(vuz1170), vuz120) → new_quot3(vuz115, vuz1170, Zero)
new_quot0(vuz115, vuz116, vuz117, Zero, Zero) → new_quot2(vuz115, vuz116, vuz117)
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vuz1030)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz10200), Succ(vuz1030)) → new_primMinusNatS0(vuz10200, vuz1030)
new_primMinusNatS0(Succ(vuz10200), Zero) → Succ(vuz10200)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Zero, Zero)
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule new_quot2(vuz115, vuz116, vuz117) → new_quot1(vuz115, new_primMinusNatS0(Succ(vuz116), vuz117), vuz117, new_primMinusNatS0(Succ(vuz116), vuz117)) at position [1] we obtained the following new rules:
new_quot2(y0, x0, Zero) → new_quot1(y0, Succ(x0), Zero, new_primMinusNatS0(Succ(x0), Zero))
new_quot2(y0, x0, Succ(x1)) → new_quot1(y0, new_primMinusNatS0(x0, x1), Succ(x1), new_primMinusNatS0(Succ(x0), Succ(x1)))
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot2(y0, x0, Zero) → new_quot1(y0, Succ(x0), Zero, new_primMinusNatS0(Succ(x0), Zero))
new_quot1(vuz115, Succ(Succ(vuz12100)), Zero, vuz120) → new_quot2(vuz115, vuz12100, Zero)
new_quot3(z0, z1, Zero) → new_quot1(z0, Succ(Succ(z1)), Zero, Succ(Succ(z1)))
new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Succ(vuz1190)) → new_quot0(vuz115, vuz116, vuz117, vuz1180, vuz1190)
new_quot0(vuz115, vuz116, vuz117, Zero, Succ(vuz1190)) → new_quot1(vuz115, Succ(vuz117), Succ(vuz116), Succ(vuz117))
new_quot1(vuz115, Succ(Succ(vuz12100)), Succ(vuz1170), vuz120) → new_quot0(vuz115, vuz12100, Succ(vuz1170), vuz12100, vuz1170)
new_quot2(y0, x0, Succ(x1)) → new_quot1(y0, new_primMinusNatS0(x0, x1), Succ(x1), new_primMinusNatS0(Succ(x0), Succ(x1)))
new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Zero) → new_quot1(vuz115, new_primMinusNatS0(Succ(vuz116), vuz117), vuz117, new_primMinusNatS0(Succ(vuz116), vuz117))
new_quot0(vuz115, vuz116, vuz117, Zero, Zero) → new_quot2(vuz115, vuz116, vuz117)
new_quot1(vuz115, Succ(Zero), Succ(vuz1170), vuz120) → new_quot3(vuz115, vuz1170, Zero)
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vuz1030)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz10200), Succ(vuz1030)) → new_primMinusNatS0(vuz10200, vuz1030)
new_primMinusNatS0(Succ(vuz10200), Zero) → Succ(vuz10200)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Zero, Zero)
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 2 less nodes.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot1(vuz115, Succ(Succ(vuz12100)), Zero, vuz120) → new_quot2(vuz115, vuz12100, Zero)
new_quot2(y0, x0, Zero) → new_quot1(y0, Succ(x0), Zero, new_primMinusNatS0(Succ(x0), Zero))
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vuz1030)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz10200), Succ(vuz1030)) → new_primMinusNatS0(vuz10200, vuz1030)
new_primMinusNatS0(Succ(vuz10200), Zero) → Succ(vuz10200)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Zero, Zero)
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot1(vuz115, Succ(Succ(vuz12100)), Zero, vuz120) → new_quot2(vuz115, vuz12100, Zero)
new_quot2(y0, x0, Zero) → new_quot1(y0, Succ(x0), Zero, new_primMinusNatS0(Succ(x0), Zero))
The TRS R consists of the following rules:
new_primMinusNatS0(Succ(vuz10200), Zero) → Succ(vuz10200)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Zero, Zero)
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_quot2(y0, x0, Zero) → new_quot1(y0, Succ(x0), Zero, new_primMinusNatS0(Succ(x0), Zero)) at position [3] we obtained the following new rules:
new_quot2(y0, x0, Zero) → new_quot1(y0, Succ(x0), Zero, Succ(x0))
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot1(vuz115, Succ(Succ(vuz12100)), Zero, vuz120) → new_quot2(vuz115, vuz12100, Zero)
new_quot2(y0, x0, Zero) → new_quot1(y0, Succ(x0), Zero, Succ(x0))
The TRS R consists of the following rules:
new_primMinusNatS0(Succ(vuz10200), Zero) → Succ(vuz10200)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Zero, Zero)
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot1(vuz115, Succ(Succ(vuz12100)), Zero, vuz120) → new_quot2(vuz115, vuz12100, Zero)
new_quot2(y0, x0, Zero) → new_quot1(y0, Succ(x0), Zero, Succ(x0))
R is empty.
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Zero, Zero)
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Zero, Zero)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot1(vuz115, Succ(Succ(vuz12100)), Zero, vuz120) → new_quot2(vuz115, vuz12100, Zero)
new_quot2(y0, x0, Zero) → new_quot1(y0, Succ(x0), Zero, Succ(x0))
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule new_quot1(vuz115, Succ(Succ(vuz12100)), Zero, vuz120) → new_quot2(vuz115, vuz12100, Zero) we obtained the following new rules:
new_quot1(z0, Succ(Succ(x1)), Zero, Succ(Succ(x1))) → new_quot2(z0, x1, Zero)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot1(z0, Succ(Succ(x1)), Zero, Succ(Succ(x1))) → new_quot2(z0, x1, Zero)
new_quot2(y0, x0, Zero) → new_quot1(y0, Succ(x0), Zero, Succ(x0))
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:
new_quot1(z0, Succ(Succ(x1)), Zero, Succ(Succ(x1))) → new_quot2(z0, x1, Zero)
Used ordering: POLO with Polynomial interpretation [25]:
POL(Succ(x1)) = 1 + x1
POL(Zero) = 0
POL(new_quot1(x1, x2, x3, x4)) = x1 + x2 + 2·x3 + x4
POL(new_quot2(x1, x2, x3)) = 2 + x1 + 2·x2 + 2·x3
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot2(y0, x0, Zero) → new_quot1(y0, Succ(x0), Zero, Succ(x0))
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 1 less node.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Succ(vuz1190)) → new_quot0(vuz115, vuz116, vuz117, vuz1180, vuz1190)
new_quot0(vuz115, vuz116, vuz117, Zero, Succ(vuz1190)) → new_quot1(vuz115, Succ(vuz117), Succ(vuz116), Succ(vuz117))
new_quot1(vuz115, Succ(Succ(vuz12100)), Succ(vuz1170), vuz120) → new_quot0(vuz115, vuz12100, Succ(vuz1170), vuz12100, vuz1170)
new_quot2(y0, x0, Succ(x1)) → new_quot1(y0, new_primMinusNatS0(x0, x1), Succ(x1), new_primMinusNatS0(Succ(x0), Succ(x1)))
new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Zero) → new_quot1(vuz115, new_primMinusNatS0(Succ(vuz116), vuz117), vuz117, new_primMinusNatS0(Succ(vuz116), vuz117))
new_quot0(vuz115, vuz116, vuz117, Zero, Zero) → new_quot2(vuz115, vuz116, vuz117)
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vuz1030)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz10200), Succ(vuz1030)) → new_primMinusNatS0(vuz10200, vuz1030)
new_primMinusNatS0(Succ(vuz10200), Zero) → Succ(vuz10200)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Zero, Zero)
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_quot2(y0, x0, Succ(x1)) → new_quot1(y0, new_primMinusNatS0(x0, x1), Succ(x1), new_primMinusNatS0(Succ(x0), Succ(x1))) at position [3] we obtained the following new rules:
new_quot2(y0, x0, Succ(x1)) → new_quot1(y0, new_primMinusNatS0(x0, x1), Succ(x1), new_primMinusNatS0(x0, x1))
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot0(vuz115, vuz116, vuz117, Zero, Succ(vuz1190)) → new_quot1(vuz115, Succ(vuz117), Succ(vuz116), Succ(vuz117))
new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Succ(vuz1190)) → new_quot0(vuz115, vuz116, vuz117, vuz1180, vuz1190)
new_quot2(y0, x0, Succ(x1)) → new_quot1(y0, new_primMinusNatS0(x0, x1), Succ(x1), new_primMinusNatS0(x0, x1))
new_quot1(vuz115, Succ(Succ(vuz12100)), Succ(vuz1170), vuz120) → new_quot0(vuz115, vuz12100, Succ(vuz1170), vuz12100, vuz1170)
new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Zero) → new_quot1(vuz115, new_primMinusNatS0(Succ(vuz116), vuz117), vuz117, new_primMinusNatS0(Succ(vuz116), vuz117))
new_quot0(vuz115, vuz116, vuz117, Zero, Zero) → new_quot2(vuz115, vuz116, vuz117)
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vuz1030)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz10200), Succ(vuz1030)) → new_primMinusNatS0(vuz10200, vuz1030)
new_primMinusNatS0(Succ(vuz10200), Zero) → Succ(vuz10200)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Zero, Zero)
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
new_quot2(y0, x0, Succ(x1)) → new_quot1(y0, new_primMinusNatS0(x0, x1), Succ(x1), new_primMinusNatS0(x0, x1))
The remaining pairs can at least be oriented weakly.
new_quot0(vuz115, vuz116, vuz117, Zero, Succ(vuz1190)) → new_quot1(vuz115, Succ(vuz117), Succ(vuz116), Succ(vuz117))
new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Succ(vuz1190)) → new_quot0(vuz115, vuz116, vuz117, vuz1180, vuz1190)
new_quot1(vuz115, Succ(Succ(vuz12100)), Succ(vuz1170), vuz120) → new_quot0(vuz115, vuz12100, Succ(vuz1170), vuz12100, vuz1170)
new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Zero) → new_quot1(vuz115, new_primMinusNatS0(Succ(vuz116), vuz117), vuz117, new_primMinusNatS0(Succ(vuz116), vuz117))
new_quot0(vuz115, vuz116, vuz117, Zero, Zero) → new_quot2(vuz115, vuz116, vuz117)
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( new_primMinusNatS0(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
Tuple symbols:
M( new_quot1(x1, ..., x4) ) = | 1 | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
M( new_quot2(x1, ..., x3) ) = | 1 | + | | · | x1 | + | | · | x2 | + | | · | x3 |
M( new_quot0(x1, ..., x5) ) = | 1 | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 | + | | · | x5 |
Matrix type:
We used a basic matrix type which is not further parametrizeable.
As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
The following usable rules [17] were oriented:
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz10200), Succ(vuz1030)) → new_primMinusNatS0(vuz10200, vuz1030)
new_primMinusNatS0(Zero, Succ(vuz1030)) → Zero
new_primMinusNatS0(Succ(vuz10200), Zero) → Succ(vuz10200)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Succ(vuz1190)) → new_quot0(vuz115, vuz116, vuz117, vuz1180, vuz1190)
new_quot0(vuz115, vuz116, vuz117, Zero, Succ(vuz1190)) → new_quot1(vuz115, Succ(vuz117), Succ(vuz116), Succ(vuz117))
new_quot1(vuz115, Succ(Succ(vuz12100)), Succ(vuz1170), vuz120) → new_quot0(vuz115, vuz12100, Succ(vuz1170), vuz12100, vuz1170)
new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Zero) → new_quot1(vuz115, new_primMinusNatS0(Succ(vuz116), vuz117), vuz117, new_primMinusNatS0(Succ(vuz116), vuz117))
new_quot0(vuz115, vuz116, vuz117, Zero, Zero) → new_quot2(vuz115, vuz116, vuz117)
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vuz1030)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz10200), Succ(vuz1030)) → new_primMinusNatS0(vuz10200, vuz1030)
new_primMinusNatS0(Succ(vuz10200), Zero) → Succ(vuz10200)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Zero, Zero)
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Succ(vuz1190)) → new_quot0(vuz115, vuz116, vuz117, vuz1180, vuz1190)
new_quot0(vuz115, vuz116, vuz117, Zero, Succ(vuz1190)) → new_quot1(vuz115, Succ(vuz117), Succ(vuz116), Succ(vuz117))
new_quot1(vuz115, Succ(Succ(vuz12100)), Succ(vuz1170), vuz120) → new_quot0(vuz115, vuz12100, Succ(vuz1170), vuz12100, vuz1170)
new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Zero) → new_quot1(vuz115, new_primMinusNatS0(Succ(vuz116), vuz117), vuz117, new_primMinusNatS0(Succ(vuz116), vuz117))
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vuz1030)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz10200), Succ(vuz1030)) → new_primMinusNatS0(vuz10200, vuz1030)
new_primMinusNatS0(Succ(vuz10200), Zero) → Succ(vuz10200)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Zero, Zero)
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Zero) → new_quot1(vuz115, new_primMinusNatS0(Succ(vuz116), vuz117), vuz117, new_primMinusNatS0(Succ(vuz116), vuz117))
The remaining pairs can at least be oriented weakly.
new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Succ(vuz1190)) → new_quot0(vuz115, vuz116, vuz117, vuz1180, vuz1190)
new_quot0(vuz115, vuz116, vuz117, Zero, Succ(vuz1190)) → new_quot1(vuz115, Succ(vuz117), Succ(vuz116), Succ(vuz117))
new_quot1(vuz115, Succ(Succ(vuz12100)), Succ(vuz1170), vuz120) → new_quot0(vuz115, vuz12100, Succ(vuz1170), vuz12100, vuz1170)
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( new_primMinusNatS0(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
Tuple symbols:
M( new_quot1(x1, ..., x4) ) = | 0 | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
M( new_quot0(x1, ..., x5) ) = | 1 | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 | + | | · | x5 |
Matrix type:
We used a basic matrix type which is not further parametrizeable.
As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
The following usable rules [17] were oriented:
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz10200), Succ(vuz1030)) → new_primMinusNatS0(vuz10200, vuz1030)
new_primMinusNatS0(Zero, Succ(vuz1030)) → Zero
new_primMinusNatS0(Succ(vuz10200), Zero) → Succ(vuz10200)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot0(vuz115, vuz116, vuz117, Zero, Succ(vuz1190)) → new_quot1(vuz115, Succ(vuz117), Succ(vuz116), Succ(vuz117))
new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Succ(vuz1190)) → new_quot0(vuz115, vuz116, vuz117, vuz1180, vuz1190)
new_quot1(vuz115, Succ(Succ(vuz12100)), Succ(vuz1170), vuz120) → new_quot0(vuz115, vuz12100, Succ(vuz1170), vuz12100, vuz1170)
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vuz1030)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz10200), Succ(vuz1030)) → new_primMinusNatS0(vuz10200, vuz1030)
new_primMinusNatS0(Succ(vuz10200), Zero) → Succ(vuz10200)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Zero, Zero)
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot0(vuz115, vuz116, vuz117, Zero, Succ(vuz1190)) → new_quot1(vuz115, Succ(vuz117), Succ(vuz116), Succ(vuz117))
new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Succ(vuz1190)) → new_quot0(vuz115, vuz116, vuz117, vuz1180, vuz1190)
new_quot1(vuz115, Succ(Succ(vuz12100)), Succ(vuz1170), vuz120) → new_quot0(vuz115, vuz12100, Succ(vuz1170), vuz12100, vuz1170)
R is empty.
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Zero, Zero)
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Zero, Zero)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Succ(vuz1190)) → new_quot0(vuz115, vuz116, vuz117, vuz1180, vuz1190)
new_quot0(vuz115, vuz116, vuz117, Zero, Succ(vuz1190)) → new_quot1(vuz115, Succ(vuz117), Succ(vuz116), Succ(vuz117))
new_quot1(vuz115, Succ(Succ(vuz12100)), Succ(vuz1170), vuz120) → new_quot0(vuz115, vuz12100, Succ(vuz1170), vuz12100, vuz1170)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule new_quot1(vuz115, Succ(Succ(vuz12100)), Succ(vuz1170), vuz120) → new_quot0(vuz115, vuz12100, Succ(vuz1170), vuz12100, vuz1170) we obtained the following new rules:
new_quot1(z0, Succ(Succ(x1)), Succ(z1), Succ(Succ(x1))) → new_quot0(z0, x1, Succ(z1), x1, z1)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ NonInfProof
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot1(z0, Succ(Succ(x1)), Succ(z1), Succ(Succ(x1))) → new_quot0(z0, x1, Succ(z1), x1, z1)
new_quot0(vuz115, vuz116, vuz117, Zero, Succ(vuz1190)) → new_quot1(vuz115, Succ(vuz117), Succ(vuz116), Succ(vuz117))
new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Succ(vuz1190)) → new_quot0(vuz115, vuz116, vuz117, vuz1180, vuz1190)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The DP Problem is simplified using the Induction Calculus [18] with the following steps:
Note that final constraints are written in bold face.
For Pair new_quot0(vuz115, vuz116, vuz117, Zero, Succ(vuz1190)) → new_quot1(vuz115, Succ(vuz117), Succ(vuz116), Succ(vuz117)) the following chains were created:
- We consider the chain new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Succ(vuz1190)) → new_quot0(vuz115, vuz116, vuz117, vuz1180, vuz1190), new_quot0(vuz115, vuz116, vuz117, Zero, Succ(vuz1190)) → new_quot1(vuz115, Succ(vuz117), Succ(vuz116), Succ(vuz117)) which results in the following constraint:
(1) (new_quot0(x4, x5, x6, x7, x8)=new_quot0(x9, x10, x11, Zero, Succ(x12)) ⇒ new_quot0(x9, x10, x11, Zero, Succ(x12))≥new_quot1(x9, Succ(x11), Succ(x10), Succ(x11)))
We simplified constraint (1) using rules (I), (II), (III) which results in the following new constraint:
(2) (new_quot0(x4, x5, x6, Zero, Succ(x12))≥new_quot1(x4, Succ(x6), Succ(x5), Succ(x6)))
- We consider the chain new_quot1(z0, Succ(Succ(x1)), Succ(z1), Succ(Succ(x1))) → new_quot0(z0, x1, Succ(z1), x1, z1), new_quot0(vuz115, vuz116, vuz117, Zero, Succ(vuz1190)) → new_quot1(vuz115, Succ(vuz117), Succ(vuz116), Succ(vuz117)) which results in the following constraint:
(3) (new_quot0(x13, x14, Succ(x15), x14, x15)=new_quot0(x16, x17, x18, Zero, Succ(x19)) ⇒ new_quot0(x16, x17, x18, Zero, Succ(x19))≥new_quot1(x16, Succ(x18), Succ(x17), Succ(x18)))
We simplified constraint (3) using rules (I), (II), (III) which results in the following new constraint:
(4) (new_quot0(x13, Zero, Succ(Succ(x19)), Zero, Succ(x19))≥new_quot1(x13, Succ(Succ(Succ(x19))), Succ(Zero), Succ(Succ(Succ(x19)))))
For Pair new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Succ(vuz1190)) → new_quot0(vuz115, vuz116, vuz117, vuz1180, vuz1190) the following chains were created:
- We consider the chain new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Succ(vuz1190)) → new_quot0(vuz115, vuz116, vuz117, vuz1180, vuz1190), new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Succ(vuz1190)) → new_quot0(vuz115, vuz116, vuz117, vuz1180, vuz1190) which results in the following constraint:
(5) (new_quot0(x24, x25, x26, x27, x28)=new_quot0(x29, x30, x31, Succ(x32), Succ(x33)) ⇒ new_quot0(x29, x30, x31, Succ(x32), Succ(x33))≥new_quot0(x29, x30, x31, x32, x33))
We simplified constraint (5) using rules (I), (II), (III) which results in the following new constraint:
(6) (new_quot0(x24, x25, x26, Succ(x32), Succ(x33))≥new_quot0(x24, x25, x26, x32, x33))
- We consider the chain new_quot1(z0, Succ(Succ(x1)), Succ(z1), Succ(Succ(x1))) → new_quot0(z0, x1, Succ(z1), x1, z1), new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Succ(vuz1190)) → new_quot0(vuz115, vuz116, vuz117, vuz1180, vuz1190) which results in the following constraint:
(7) (new_quot0(x34, x35, Succ(x36), x35, x36)=new_quot0(x37, x38, x39, Succ(x40), Succ(x41)) ⇒ new_quot0(x37, x38, x39, Succ(x40), Succ(x41))≥new_quot0(x37, x38, x39, x40, x41))
We simplified constraint (7) using rules (I), (II), (III) which results in the following new constraint:
(8) (new_quot0(x34, Succ(x40), Succ(Succ(x41)), Succ(x40), Succ(x41))≥new_quot0(x34, Succ(x40), Succ(Succ(x41)), x40, x41))
For Pair new_quot1(z0, Succ(Succ(x1)), Succ(z1), Succ(Succ(x1))) → new_quot0(z0, x1, Succ(z1), x1, z1) the following chains were created:
- We consider the chain new_quot0(vuz115, vuz116, vuz117, Zero, Succ(vuz1190)) → new_quot1(vuz115, Succ(vuz117), Succ(vuz116), Succ(vuz117)), new_quot1(z0, Succ(Succ(x1)), Succ(z1), Succ(Succ(x1))) → new_quot0(z0, x1, Succ(z1), x1, z1) which results in the following constraint:
(9) (new_quot1(x42, Succ(x44), Succ(x43), Succ(x44))=new_quot1(x46, Succ(Succ(x47)), Succ(x48), Succ(Succ(x47))) ⇒ new_quot1(x46, Succ(Succ(x47)), Succ(x48), Succ(Succ(x47)))≥new_quot0(x46, x47, Succ(x48), x47, x48))
We simplified constraint (9) using rules (I), (II), (III) which results in the following new constraint:
(10) (new_quot1(x42, Succ(Succ(x47)), Succ(x43), Succ(Succ(x47)))≥new_quot0(x42, x47, Succ(x43), x47, x43))
To summarize, we get the following constraints P≥ for the following pairs.
- new_quot0(vuz115, vuz116, vuz117, Zero, Succ(vuz1190)) → new_quot1(vuz115, Succ(vuz117), Succ(vuz116), Succ(vuz117))
- (new_quot0(x4, x5, x6, Zero, Succ(x12))≥new_quot1(x4, Succ(x6), Succ(x5), Succ(x6)))
- (new_quot0(x13, Zero, Succ(Succ(x19)), Zero, Succ(x19))≥new_quot1(x13, Succ(Succ(Succ(x19))), Succ(Zero), Succ(Succ(Succ(x19)))))
- new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Succ(vuz1190)) → new_quot0(vuz115, vuz116, vuz117, vuz1180, vuz1190)
- (new_quot0(x24, x25, x26, Succ(x32), Succ(x33))≥new_quot0(x24, x25, x26, x32, x33))
- (new_quot0(x34, Succ(x40), Succ(Succ(x41)), Succ(x40), Succ(x41))≥new_quot0(x34, Succ(x40), Succ(Succ(x41)), x40, x41))
- new_quot1(z0, Succ(Succ(x1)), Succ(z1), Succ(Succ(x1))) → new_quot0(z0, x1, Succ(z1), x1, z1)
- (new_quot1(x42, Succ(Succ(x47)), Succ(x43), Succ(Succ(x47)))≥new_quot0(x42, x47, Succ(x43), x47, x43))
The constraints for P> respective Pbound are constructed from P≥ where we just replace every occurence of "t ≥ s" in P≥ by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation [18]:
POL(Succ(x1)) = 1 + x1
POL(Zero) = 0
POL(c) = -1
POL(new_quot0(x1, x2, x3, x4, x5)) = -1 + x1 + x2 - x4 + x5
POL(new_quot1(x1, x2, x3, x4)) = -1 + x1 + x2 + x3 - x4
The following pairs are in P>:
new_quot1(z0, Succ(Succ(x1)), Succ(z1), Succ(Succ(x1))) → new_quot0(z0, x1, Succ(z1), x1, z1)
The following pairs are in Pbound:
new_quot0(vuz115, vuz116, vuz117, Zero, Succ(vuz1190)) → new_quot1(vuz115, Succ(vuz117), Succ(vuz116), Succ(vuz117))
new_quot1(z0, Succ(Succ(x1)), Succ(z1), Succ(Succ(x1))) → new_quot0(z0, x1, Succ(z1), x1, z1)
There are no usable rules
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ NonInfProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Succ(vuz1190)) → new_quot0(vuz115, vuz116, vuz117, vuz1180, vuz1190)
new_quot0(vuz115, vuz116, vuz117, Zero, Succ(vuz1190)) → new_quot1(vuz115, Succ(vuz117), Succ(vuz116), Succ(vuz117))
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ NonInfProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Succ(vuz1190)) → new_quot0(vuz115, vuz116, vuz117, vuz1180, vuz1190)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- new_quot0(vuz115, vuz116, vuz117, Succ(vuz1180), Succ(vuz1190)) → new_quot0(vuz115, vuz116, vuz117, vuz1180, vuz1190)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4, 5 > 5
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot4(vuz46, Succ(vuz470)) → new_quot4(vuz46, vuz470)
new_quot5(vuz46, Succ(vuz470)) → new_quot4(vuz46, vuz470)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot4(vuz46, Succ(vuz470)) → new_quot4(vuz46, vuz470)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- new_quot4(vuz46, Succ(vuz470)) → new_quot4(vuz46, vuz470)
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
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot6(vuz89, Succ(Succ(vuz9000)), Succ(Zero), Succ(vuz920), Zero) → new_quot6(vuz89, vuz9000, Succ(Zero), vuz9000, Zero)
new_quot8(vuz89, Zero, Succ(Succ(vuz9100))) → new_quot7(vuz89, Zero, vuz9100, Zero)
new_quot6(vuz89, vuz90, Succ(vuz910), Zero, Succ(vuz930)) → new_quot6(vuz89, vuz910, Succ(vuz90), vuz910, vuz90)
new_quot6(vuz89, Succ(vuz900), Succ(Succ(vuz9100)), Succ(vuz920), Zero) → new_quot7(vuz89, new_primMinusNatS0(vuz900, vuz9100), vuz9100, new_primMinusNatS0(vuz900, vuz9100))
new_quot8(vuz89, Succ(vuz900), Succ(Succ(vuz9100))) → new_quot7(vuz89, new_primMinusNatS0(vuz900, vuz9100), vuz9100, new_primMinusNatS0(vuz900, vuz9100))
new_quot8(vuz89, Succ(Succ(vuz9000)), Succ(Zero)) → new_quot6(vuz89, vuz9000, Succ(Zero), vuz9000, Zero)
new_quot6(vuz89, Zero, Succ(Succ(vuz9100)), Succ(vuz920), Zero) → new_quot7(vuz89, Zero, vuz9100, Zero)
new_quot6(vuz89, vuz90, vuz91, Zero, Zero) → new_quot8(vuz89, vuz90, vuz91)
new_quot7(vuz89, Succ(Succ(vuz11300)), vuz9100, vuz112) → new_quot6(vuz89, vuz11300, Succ(Succ(vuz9100)), vuz11300, Succ(vuz9100))
new_quot6(vuz89, vuz90, vuz91, Succ(vuz920), Succ(vuz930)) → new_quot6(vuz89, vuz90, vuz91, vuz920, vuz930)
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vuz1030)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz10200), Succ(vuz1030)) → new_primMinusNatS0(vuz10200, vuz1030)
new_primMinusNatS0(Succ(vuz10200), Zero) → Succ(vuz10200)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Zero, Zero)
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot6(vuz89, Succ(Succ(vuz9000)), Succ(Zero), Succ(vuz920), Zero) → new_quot6(vuz89, vuz9000, Succ(Zero), vuz9000, Zero)
new_quot6(vuz89, vuz90, Succ(vuz910), Zero, Succ(vuz930)) → new_quot6(vuz89, vuz910, Succ(vuz90), vuz910, vuz90)
new_quot6(vuz89, Succ(vuz900), Succ(Succ(vuz9100)), Succ(vuz920), Zero) → new_quot7(vuz89, new_primMinusNatS0(vuz900, vuz9100), vuz9100, new_primMinusNatS0(vuz900, vuz9100))
new_quot8(vuz89, Succ(vuz900), Succ(Succ(vuz9100))) → new_quot7(vuz89, new_primMinusNatS0(vuz900, vuz9100), vuz9100, new_primMinusNatS0(vuz900, vuz9100))
new_quot8(vuz89, Succ(Succ(vuz9000)), Succ(Zero)) → new_quot6(vuz89, vuz9000, Succ(Zero), vuz9000, Zero)
new_quot6(vuz89, vuz90, vuz91, Zero, Zero) → new_quot8(vuz89, vuz90, vuz91)
new_quot7(vuz89, Succ(Succ(vuz11300)), vuz9100, vuz112) → new_quot6(vuz89, vuz11300, Succ(Succ(vuz9100)), vuz11300, Succ(vuz9100))
new_quot6(vuz89, vuz90, vuz91, Succ(vuz920), Succ(vuz930)) → new_quot6(vuz89, vuz90, vuz91, vuz920, vuz930)
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vuz1030)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz10200), Succ(vuz1030)) → new_primMinusNatS0(vuz10200, vuz1030)
new_primMinusNatS0(Succ(vuz10200), Zero) → Succ(vuz10200)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Zero, Zero)
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
new_quot6(vuz89, Succ(Succ(vuz9000)), Succ(Zero), Succ(vuz920), Zero) → new_quot6(vuz89, vuz9000, Succ(Zero), vuz9000, Zero)
new_quot6(vuz89, Succ(vuz900), Succ(Succ(vuz9100)), Succ(vuz920), Zero) → new_quot7(vuz89, new_primMinusNatS0(vuz900, vuz9100), vuz9100, new_primMinusNatS0(vuz900, vuz9100))
new_quot8(vuz89, Succ(vuz900), Succ(Succ(vuz9100))) → new_quot7(vuz89, new_primMinusNatS0(vuz900, vuz9100), vuz9100, new_primMinusNatS0(vuz900, vuz9100))
new_quot8(vuz89, Succ(Succ(vuz9000)), Succ(Zero)) → new_quot6(vuz89, vuz9000, Succ(Zero), vuz9000, Zero)
The remaining pairs can at least be oriented weakly.
new_quot6(vuz89, vuz90, Succ(vuz910), Zero, Succ(vuz930)) → new_quot6(vuz89, vuz910, Succ(vuz90), vuz910, vuz90)
new_quot6(vuz89, vuz90, vuz91, Zero, Zero) → new_quot8(vuz89, vuz90, vuz91)
new_quot7(vuz89, Succ(Succ(vuz11300)), vuz9100, vuz112) → new_quot6(vuz89, vuz11300, Succ(Succ(vuz9100)), vuz11300, Succ(vuz9100))
new_quot6(vuz89, vuz90, vuz91, Succ(vuz920), Succ(vuz930)) → new_quot6(vuz89, vuz90, vuz91, vuz920, vuz930)
Used ordering: Polynomial interpretation [25]:
POL(Succ(x1)) = 1 + x1
POL(Zero) = 1
POL(new_primMinusNatS0(x1, x2)) = x1
POL(new_quot6(x1, x2, x3, x4, x5)) = x2 + x3
POL(new_quot7(x1, x2, x3, x4)) = x2 + x3
POL(new_quot8(x1, x2, x3)) = x2 + x3
The following usable rules [17] were oriented:
new_primMinusNatS0(Succ(vuz10200), Zero) → Succ(vuz10200)
new_primMinusNatS0(Zero, Succ(vuz1030)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz10200), Succ(vuz1030)) → new_primMinusNatS0(vuz10200, vuz1030)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot6(vuz89, vuz90, Succ(vuz910), Zero, Succ(vuz930)) → new_quot6(vuz89, vuz910, Succ(vuz90), vuz910, vuz90)
new_quot6(vuz89, vuz90, vuz91, Zero, Zero) → new_quot8(vuz89, vuz90, vuz91)
new_quot7(vuz89, Succ(Succ(vuz11300)), vuz9100, vuz112) → new_quot6(vuz89, vuz11300, Succ(Succ(vuz9100)), vuz11300, Succ(vuz9100))
new_quot6(vuz89, vuz90, vuz91, Succ(vuz920), Succ(vuz930)) → new_quot6(vuz89, vuz90, vuz91, vuz920, vuz930)
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vuz1030)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz10200), Succ(vuz1030)) → new_primMinusNatS0(vuz10200, vuz1030)
new_primMinusNatS0(Succ(vuz10200), Zero) → Succ(vuz10200)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Zero, Zero)
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot6(vuz89, vuz90, Succ(vuz910), Zero, Succ(vuz930)) → new_quot6(vuz89, vuz910, Succ(vuz90), vuz910, vuz90)
new_quot6(vuz89, vuz90, vuz91, Succ(vuz920), Succ(vuz930)) → new_quot6(vuz89, vuz90, vuz91, vuz920, vuz930)
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vuz1030)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vuz10200), Succ(vuz1030)) → new_primMinusNatS0(vuz10200, vuz1030)
new_primMinusNatS0(Succ(vuz10200), Zero) → Succ(vuz10200)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Zero, Zero)
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot6(vuz89, vuz90, Succ(vuz910), Zero, Succ(vuz930)) → new_quot6(vuz89, vuz910, Succ(vuz90), vuz910, vuz90)
new_quot6(vuz89, vuz90, vuz91, Succ(vuz920), Succ(vuz930)) → new_quot6(vuz89, vuz90, vuz91, vuz920, vuz930)
R is empty.
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Zero, Zero)
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Succ(x0))
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Zero, Zero)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ NonInfProof
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot6(vuz89, vuz90, Succ(vuz910), Zero, Succ(vuz930)) → new_quot6(vuz89, vuz910, Succ(vuz90), vuz910, vuz90)
new_quot6(vuz89, vuz90, vuz91, Succ(vuz920), Succ(vuz930)) → new_quot6(vuz89, vuz90, vuz91, vuz920, vuz930)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The DP Problem is simplified using the Induction Calculus [18] with the following steps:
Note that final constraints are written in bold face.
For Pair new_quot6(vuz89, vuz90, Succ(vuz910), Zero, Succ(vuz930)) → new_quot6(vuz89, vuz910, Succ(vuz90), vuz910, vuz90) the following chains were created:
- We consider the chain new_quot6(vuz89, vuz90, Succ(vuz910), Zero, Succ(vuz930)) → new_quot6(vuz89, vuz910, Succ(vuz90), vuz910, vuz90), new_quot6(vuz89, vuz90, Succ(vuz910), Zero, Succ(vuz930)) → new_quot6(vuz89, vuz910, Succ(vuz90), vuz910, vuz90) which results in the following constraint:
(1) (new_quot6(x0, x2, Succ(x1), x2, x1)=new_quot6(x4, x5, Succ(x6), Zero, Succ(x7)) ⇒ new_quot6(x4, x5, Succ(x6), Zero, Succ(x7))≥new_quot6(x4, x6, Succ(x5), x6, x5))
We simplified constraint (1) using rules (I), (II), (III) which results in the following new constraint:
(2) (new_quot6(x0, Zero, Succ(Succ(x7)), Zero, Succ(x7))≥new_quot6(x0, Succ(x7), Succ(Zero), Succ(x7), Zero))
- We consider the chain new_quot6(vuz89, vuz90, vuz91, Succ(vuz920), Succ(vuz930)) → new_quot6(vuz89, vuz90, vuz91, vuz920, vuz930), new_quot6(vuz89, vuz90, Succ(vuz910), Zero, Succ(vuz930)) → new_quot6(vuz89, vuz910, Succ(vuz90), vuz910, vuz90) which results in the following constraint:
(3) (new_quot6(x8, x9, x10, x11, x12)=new_quot6(x13, x14, Succ(x15), Zero, Succ(x16)) ⇒ new_quot6(x13, x14, Succ(x15), Zero, Succ(x16))≥new_quot6(x13, x15, Succ(x14), x15, x14))
We simplified constraint (3) using rules (I), (II), (III) which results in the following new constraint:
(4) (new_quot6(x8, x9, Succ(x15), Zero, Succ(x16))≥new_quot6(x8, x15, Succ(x9), x15, x9))
For Pair new_quot6(vuz89, vuz90, vuz91, Succ(vuz920), Succ(vuz930)) → new_quot6(vuz89, vuz90, vuz91, vuz920, vuz930) the following chains were created:
- We consider the chain new_quot6(vuz89, vuz90, Succ(vuz910), Zero, Succ(vuz930)) → new_quot6(vuz89, vuz910, Succ(vuz90), vuz910, vuz90), new_quot6(vuz89, vuz90, vuz91, Succ(vuz920), Succ(vuz930)) → new_quot6(vuz89, vuz90, vuz91, vuz920, vuz930) which results in the following constraint:
(5) (new_quot6(x17, x19, Succ(x18), x19, x18)=new_quot6(x21, x22, x23, Succ(x24), Succ(x25)) ⇒ new_quot6(x21, x22, x23, Succ(x24), Succ(x25))≥new_quot6(x21, x22, x23, x24, x25))
We simplified constraint (5) using rules (I), (II), (III) which results in the following new constraint:
(6) (new_quot6(x17, Succ(x24), Succ(Succ(x25)), Succ(x24), Succ(x25))≥new_quot6(x17, Succ(x24), Succ(Succ(x25)), x24, x25))
- We consider the chain new_quot6(vuz89, vuz90, vuz91, Succ(vuz920), Succ(vuz930)) → new_quot6(vuz89, vuz90, vuz91, vuz920, vuz930), new_quot6(vuz89, vuz90, vuz91, Succ(vuz920), Succ(vuz930)) → new_quot6(vuz89, vuz90, vuz91, vuz920, vuz930) which results in the following constraint:
(7) (new_quot6(x26, x27, x28, x29, x30)=new_quot6(x31, x32, x33, Succ(x34), Succ(x35)) ⇒ new_quot6(x31, x32, x33, Succ(x34), Succ(x35))≥new_quot6(x31, x32, x33, x34, x35))
We simplified constraint (7) using rules (I), (II), (III) which results in the following new constraint:
(8) (new_quot6(x26, x27, x28, Succ(x34), Succ(x35))≥new_quot6(x26, x27, x28, x34, x35))
To summarize, we get the following constraints P≥ for the following pairs.
- new_quot6(vuz89, vuz90, Succ(vuz910), Zero, Succ(vuz930)) → new_quot6(vuz89, vuz910, Succ(vuz90), vuz910, vuz90)
- (new_quot6(x0, Zero, Succ(Succ(x7)), Zero, Succ(x7))≥new_quot6(x0, Succ(x7), Succ(Zero), Succ(x7), Zero))
- (new_quot6(x8, x9, Succ(x15), Zero, Succ(x16))≥new_quot6(x8, x15, Succ(x9), x15, x9))
- new_quot6(vuz89, vuz90, vuz91, Succ(vuz920), Succ(vuz930)) → new_quot6(vuz89, vuz90, vuz91, vuz920, vuz930)
- (new_quot6(x17, Succ(x24), Succ(Succ(x25)), Succ(x24), Succ(x25))≥new_quot6(x17, Succ(x24), Succ(Succ(x25)), x24, x25))
- (new_quot6(x26, x27, x28, Succ(x34), Succ(x35))≥new_quot6(x26, x27, x28, x34, x35))
The constraints for P> respective Pbound are constructed from P≥ where we just replace every occurence of "t ≥ s" in P≥ by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation [18]:
POL(Succ(x1)) = 1 + x1
POL(Zero) = 0
POL(c) = -1
POL(new_quot6(x1, x2, x3, x4, x5)) = -1 + x1 + x2 - x4 + x5
The following pairs are in P>:
new_quot6(vuz89, vuz90, Succ(vuz910), Zero, Succ(vuz930)) → new_quot6(vuz89, vuz910, Succ(vuz90), vuz910, vuz90)
The following pairs are in Pbound:
new_quot6(vuz89, vuz90, Succ(vuz910), Zero, Succ(vuz930)) → new_quot6(vuz89, vuz910, Succ(vuz90), vuz910, vuz90)
There are no usable rules
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ NonInfProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot6(vuz89, vuz90, vuz91, Succ(vuz920), Succ(vuz930)) → new_quot6(vuz89, vuz90, vuz91, vuz920, vuz930)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- new_quot6(vuz89, vuz90, vuz91, Succ(vuz920), Succ(vuz930)) → new_quot6(vuz89, vuz90, vuz91, vuz920, vuz930)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4, 5 > 5
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot9(Zero, Succ(vuz1600)) → new_quot9(Zero, vuz1600)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- new_quot9(Zero, Succ(vuz1600)) → new_quot9(Zero, vuz1600)
The graph contains the following edges 1 >= 1, 2 > 2
Haskell To QDPs