YES 14.914
H-Termination proof of /home/matraf/haskell/eval_FullyBlown_Fast/empty.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:
↳ HASKELL
↳ IFR
mainModule Main
| ((realToFrac :: Float -> Float) :: Float -> Float) |
module Main where
If Reductions:
The following If expression
if primGEqNatS x y then Succ (primDivNatS (primMinusNatS x y) (Succ y)) else Zero
is transformed to
primDivNatS0 | x y True | = Succ (primDivNatS (primMinusNatS x y) (Succ y)) |
primDivNatS0 | x y False | = Zero |
The following If expression
if primGEqNatS x y then primModNatS (primMinusNatS x y) (Succ y) else Succ x
is transformed to
primModNatS0 | x y True | = primModNatS (primMinusNatS x y) (Succ y) |
primModNatS0 | x y False | = Succ x |
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
mainModule Main
| ((realToFrac :: Float -> Float) :: Float -> Float) |
module Main where
Replaced joker patterns by fresh variables and removed binding patterns.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
mainModule Main
| ((realToFrac :: Float -> Float) :: Float -> Float) |
module Main where
Cond Reductions:
The following Function with conditions
gcd' | x 0 | = x |
gcd' | x y | = gcd' y (x `rem` y) |
is transformed to
gcd' | x xz | = gcd'2 x xz |
gcd' | x y | = gcd'0 x y |
gcd'0 | x y | = gcd' y (x `rem` y) |
gcd'1 | True x xz | = x |
gcd'1 | yu yv yw | = gcd'0 yv yw |
gcd'2 | x xz | = gcd'1 (xz == 0) x xz |
gcd'2 | yx yy | = gcd'0 yx yy |
The following Function with conditions
gcd | 0 0 | = error [] |
gcd | x y | =
gcd' (abs x) (abs y) |
where |
gcd' | x 0 | = x |
gcd' | x y | = gcd' y (x `rem` y) |
|
|
is transformed to
gcd | yz zu | = gcd3 yz zu |
gcd | x y | = gcd0 x y |
gcd0 | x y | =
gcd' (abs x) (abs y) |
where |
gcd' | x xz | = gcd'2 x xz |
gcd' | x y | = gcd'0 x y |
|
|
gcd'0 | x y | = gcd' y (x `rem` y) |
|
|
gcd'1 | True x xz | = x |
gcd'1 | yu yv yw | = gcd'0 yv yw |
|
|
gcd'2 | x xz | = gcd'1 (xz == 0) x xz |
gcd'2 | yx yy | = gcd'0 yx yy |
|
|
gcd1 | True yz zu | = error [] |
gcd1 | zv zw zx | = gcd0 zw zx |
gcd2 | True yz zu | = gcd1 (zu == 0) yz zu |
gcd2 | zy zz vuu | = gcd0 zz vuu |
gcd3 | yz zu | = gcd2 (yz == 0) yz zu |
gcd3 | vuv vuw | = gcd0 vuv vuw |
The following Function with conditions
is transformed to
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
| ((realToFrac :: Float -> Float) :: Float -> Float) |
module Main where
Let/Where Reductions:
The bindings of the following Let/Where expression
reduce1 x y (y == 0) |
where | |
|
reduce0 | x y True | = x `quot` d :% (y `quot` d) |
|
|
reduce1 | x y True | = error [] |
reduce1 | x y False | = reduce0 x y otherwise |
|
are unpacked to the following functions on top level
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 |
reduce2Reduce0 | vux vuy x y True | = x `quot` reduce2D vux vuy :% (y `quot` reduce2D vux vuy) |
The bindings of the following Let/Where expression
gcd' (abs x) (abs y) |
where |
gcd' | x xz | = gcd'2 x xz |
gcd' | x y | = gcd'0 x y |
|
|
gcd'0 | x y | = gcd' y (x `rem` y) |
|
|
gcd'1 | True x xz | = x |
gcd'1 | yu yv yw | = gcd'0 yv yw |
|
|
gcd'2 | x xz | = gcd'1 (xz == 0) x xz |
gcd'2 | yx yy | = gcd'0 yx yy |
|
are unpacked to the following functions on top level
gcd0Gcd'1 | True x xz | = x |
gcd0Gcd'1 | yu yv yw | = gcd0Gcd'0 yv yw |
gcd0Gcd'2 | x xz | = gcd0Gcd'1 (xz == 0) x xz |
gcd0Gcd'2 | yx yy | = gcd0Gcd'0 yx yy |
gcd0Gcd'0 | x y | = gcd0Gcd' y (x `rem` y) |
gcd0Gcd' | x xz | = gcd0Gcd'2 x xz |
gcd0Gcd' | x y | = gcd0Gcd'0 x y |
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
mainModule Main
| ((realToFrac :: Float -> Float) :: Float -> Float) |
module Main where
Num Reduction: All numbers are transformed to thier corresponding representation with Pos, Neg, Succ and Zero.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
mainModule Main
| (realToFrac :: Float -> Float) |
module Main where
Haskell To QDPs
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primMulNat(Succ(vuz30000)) → new_primMulNat(vuz30000)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- new_primMulNat(Succ(vuz30000)) → new_primMulNat(vuz30000)
The graph contains the following edges 1 > 1
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primMinusNatS(Succ(vuz12200), Succ(vuz1230)) → new_primMinusNatS(vuz12200, vuz1230)
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(vuz12200), Succ(vuz1230)) → new_primMinusNatS(vuz12200, vuz1230)
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
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vuz175, vuz176, Zero, Zero) → new_primDivNatS00(vuz175, vuz176)
new_primDivNatS(Succ(Succ(vuz830000)), Succ(vuz840)) → new_primDivNatS0(vuz830000, vuz840, vuz830000, vuz840)
new_primDivNatS0(vuz175, vuz176, Succ(vuz1770), Succ(vuz1780)) → new_primDivNatS0(vuz175, vuz176, vuz1770, vuz1780)
new_primDivNatS00(vuz175, vuz176) → new_primDivNatS(new_primMinusNatS2(Succ(vuz175), Succ(vuz176)), Succ(vuz176))
new_primDivNatS(Succ(Succ(vuz830000)), Zero) → new_primDivNatS(new_primMinusNatS0(vuz830000), Zero)
new_primDivNatS0(vuz175, vuz176, Succ(vuz1770), Zero) → new_primDivNatS(new_primMinusNatS2(Succ(vuz175), Succ(vuz176)), Succ(vuz176))
new_primDivNatS(Succ(Zero), Zero) → new_primDivNatS(new_primMinusNatS1, Zero)
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12200), Succ(vuz1230)) → new_primMinusNatS2(vuz12200, vuz1230)
new_primMinusNatS2(Zero, Succ(vuz1230)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12200), Zero) → Succ(vuz12200)
new_primMinusNatS1 → Zero
new_primMinusNatS0(vuz830000) → Succ(vuz830000)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS0(x0)
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS1
new_primMinusNatS2(Succ(x0), Zero)
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 1 less node.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS(Succ(Succ(vuz830000)), Zero) → new_primDivNatS(new_primMinusNatS0(vuz830000), Zero)
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12200), Succ(vuz1230)) → new_primMinusNatS2(vuz12200, vuz1230)
new_primMinusNatS2(Zero, Succ(vuz1230)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12200), Zero) → Succ(vuz12200)
new_primMinusNatS1 → Zero
new_primMinusNatS0(vuz830000) → Succ(vuz830000)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS0(x0)
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS1
new_primMinusNatS2(Succ(x0), 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
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS(Succ(Succ(vuz830000)), Zero) → new_primDivNatS(new_primMinusNatS0(vuz830000), Zero)
The TRS R consists of the following rules:
new_primMinusNatS0(vuz830000) → Succ(vuz830000)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS0(x0)
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS1
new_primMinusNatS2(Succ(x0), 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_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS1
new_primMinusNatS2(Succ(x0), Zero)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS(Succ(Succ(vuz830000)), Zero) → new_primDivNatS(new_primMinusNatS0(vuz830000), Zero)
The TRS R consists of the following rules:
new_primMinusNatS0(vuz830000) → Succ(vuz830000)
The set Q consists of the following terms:
new_primMinusNatS0(x0)
We have to consider all minimal (P,Q,R)-chains.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:
new_primDivNatS(Succ(Succ(vuz830000)), Zero) → new_primDivNatS(new_primMinusNatS0(vuz830000), Zero)
Strictly oriented rules of the TRS R:
new_primMinusNatS0(vuz830000) → Succ(vuz830000)
Used ordering: POLO with Polynomial interpretation [25]:
POL(Succ(x1)) = 1 + 2·x1
POL(Zero) = 0
POL(new_primDivNatS(x1, x2)) = x1 + x2
POL(new_primMinusNatS0(x1)) = 2 + 2·x1
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
P is empty.
R is empty.
The set Q consists of the following terms:
new_primMinusNatS0(x0)
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vuz175, vuz176, Zero, Zero) → new_primDivNatS00(vuz175, vuz176)
new_primDivNatS0(vuz175, vuz176, Succ(vuz1770), Succ(vuz1780)) → new_primDivNatS0(vuz175, vuz176, vuz1770, vuz1780)
new_primDivNatS(Succ(Succ(vuz830000)), Succ(vuz840)) → new_primDivNatS0(vuz830000, vuz840, vuz830000, vuz840)
new_primDivNatS00(vuz175, vuz176) → new_primDivNatS(new_primMinusNatS2(Succ(vuz175), Succ(vuz176)), Succ(vuz176))
new_primDivNatS0(vuz175, vuz176, Succ(vuz1770), Zero) → new_primDivNatS(new_primMinusNatS2(Succ(vuz175), Succ(vuz176)), Succ(vuz176))
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12200), Succ(vuz1230)) → new_primMinusNatS2(vuz12200, vuz1230)
new_primMinusNatS2(Zero, Succ(vuz1230)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12200), Zero) → Succ(vuz12200)
new_primMinusNatS1 → Zero
new_primMinusNatS0(vuz830000) → Succ(vuz830000)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS0(x0)
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS1
new_primMinusNatS2(Succ(x0), 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
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vuz175, vuz176, Zero, Zero) → new_primDivNatS00(vuz175, vuz176)
new_primDivNatS0(vuz175, vuz176, Succ(vuz1770), Succ(vuz1780)) → new_primDivNatS0(vuz175, vuz176, vuz1770, vuz1780)
new_primDivNatS(Succ(Succ(vuz830000)), Succ(vuz840)) → new_primDivNatS0(vuz830000, vuz840, vuz830000, vuz840)
new_primDivNatS00(vuz175, vuz176) → new_primDivNatS(new_primMinusNatS2(Succ(vuz175), Succ(vuz176)), Succ(vuz176))
new_primDivNatS0(vuz175, vuz176, Succ(vuz1770), Zero) → new_primDivNatS(new_primMinusNatS2(Succ(vuz175), Succ(vuz176)), Succ(vuz176))
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12200), Succ(vuz1230)) → new_primMinusNatS2(vuz12200, vuz1230)
new_primMinusNatS2(Zero, Succ(vuz1230)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12200), Zero) → Succ(vuz12200)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS0(x0)
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS1
new_primMinusNatS2(Succ(x0), 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(x0)
new_primMinusNatS1
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vuz175, vuz176, Zero, Zero) → new_primDivNatS00(vuz175, vuz176)
new_primDivNatS(Succ(Succ(vuz830000)), Succ(vuz840)) → new_primDivNatS0(vuz830000, vuz840, vuz830000, vuz840)
new_primDivNatS0(vuz175, vuz176, Succ(vuz1770), Succ(vuz1780)) → new_primDivNatS0(vuz175, vuz176, vuz1770, vuz1780)
new_primDivNatS00(vuz175, vuz176) → new_primDivNatS(new_primMinusNatS2(Succ(vuz175), Succ(vuz176)), Succ(vuz176))
new_primDivNatS0(vuz175, vuz176, Succ(vuz1770), Zero) → new_primDivNatS(new_primMinusNatS2(Succ(vuz175), Succ(vuz176)), Succ(vuz176))
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12200), Succ(vuz1230)) → new_primMinusNatS2(vuz12200, vuz1230)
new_primMinusNatS2(Zero, Succ(vuz1230)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12200), Zero) → Succ(vuz12200)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Zero)
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_primDivNatS00(vuz175, vuz176) → new_primDivNatS(new_primMinusNatS2(Succ(vuz175), Succ(vuz176)), Succ(vuz176)) at position [0] we obtained the following new rules:
new_primDivNatS00(vuz175, vuz176) → new_primDivNatS(new_primMinusNatS2(vuz175, vuz176), Succ(vuz176))
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vuz175, vuz176, Zero, Zero) → new_primDivNatS00(vuz175, vuz176)
new_primDivNatS0(vuz175, vuz176, Succ(vuz1770), Succ(vuz1780)) → new_primDivNatS0(vuz175, vuz176, vuz1770, vuz1780)
new_primDivNatS(Succ(Succ(vuz830000)), Succ(vuz840)) → new_primDivNatS0(vuz830000, vuz840, vuz830000, vuz840)
new_primDivNatS00(vuz175, vuz176) → new_primDivNatS(new_primMinusNatS2(vuz175, vuz176), Succ(vuz176))
new_primDivNatS0(vuz175, vuz176, Succ(vuz1770), Zero) → new_primDivNatS(new_primMinusNatS2(Succ(vuz175), Succ(vuz176)), Succ(vuz176))
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12200), Succ(vuz1230)) → new_primMinusNatS2(vuz12200, vuz1230)
new_primMinusNatS2(Zero, Succ(vuz1230)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12200), Zero) → Succ(vuz12200)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Zero)
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_primDivNatS0(vuz175, vuz176, Succ(vuz1770), Zero) → new_primDivNatS(new_primMinusNatS2(Succ(vuz175), Succ(vuz176)), Succ(vuz176)) at position [0] we obtained the following new rules:
new_primDivNatS0(vuz175, vuz176, Succ(vuz1770), Zero) → new_primDivNatS(new_primMinusNatS2(vuz175, vuz176), Succ(vuz176))
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vuz175, vuz176, Zero, Zero) → new_primDivNatS00(vuz175, vuz176)
new_primDivNatS0(vuz175, vuz176, Succ(vuz1770), Zero) → new_primDivNatS(new_primMinusNatS2(vuz175, vuz176), Succ(vuz176))
new_primDivNatS(Succ(Succ(vuz830000)), Succ(vuz840)) → new_primDivNatS0(vuz830000, vuz840, vuz830000, vuz840)
new_primDivNatS0(vuz175, vuz176, Succ(vuz1770), Succ(vuz1780)) → new_primDivNatS0(vuz175, vuz176, vuz1770, vuz1780)
new_primDivNatS00(vuz175, vuz176) → new_primDivNatS(new_primMinusNatS2(vuz175, vuz176), Succ(vuz176))
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12200), Succ(vuz1230)) → new_primMinusNatS2(vuz12200, vuz1230)
new_primMinusNatS2(Zero, Succ(vuz1230)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12200), Zero) → Succ(vuz12200)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), 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_primDivNatS0(vuz175, vuz176, Zero, Zero) → new_primDivNatS00(vuz175, vuz176)
new_primDivNatS0(vuz175, vuz176, Succ(vuz1770), Zero) → new_primDivNatS(new_primMinusNatS2(vuz175, vuz176), Succ(vuz176))
new_primDivNatS(Succ(Succ(vuz830000)), Succ(vuz840)) → new_primDivNatS0(vuz830000, vuz840, vuz830000, vuz840)
The remaining pairs can at least be oriented weakly.
new_primDivNatS0(vuz175, vuz176, Succ(vuz1770), Succ(vuz1780)) → new_primDivNatS0(vuz175, vuz176, vuz1770, vuz1780)
new_primDivNatS00(vuz175, vuz176) → new_primDivNatS(new_primMinusNatS2(vuz175, vuz176), Succ(vuz176))
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_primMinusNatS2(x1, x2)) = x1
The following usable rules [17] were oriented:
new_primMinusNatS2(Succ(vuz12200), Zero) → Succ(vuz12200)
new_primMinusNatS2(Zero, Succ(vuz1230)) → Zero
new_primMinusNatS2(Succ(vuz12200), Succ(vuz1230)) → new_primMinusNatS2(vuz12200, vuz1230)
new_primMinusNatS2(Zero, 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
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vuz175, vuz176, Succ(vuz1770), Succ(vuz1780)) → new_primDivNatS0(vuz175, vuz176, vuz1770, vuz1780)
new_primDivNatS00(vuz175, vuz176) → new_primDivNatS(new_primMinusNatS2(vuz175, vuz176), Succ(vuz176))
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12200), Succ(vuz1230)) → new_primMinusNatS2(vuz12200, vuz1230)
new_primMinusNatS2(Zero, Succ(vuz1230)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12200), Zero) → Succ(vuz12200)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), 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
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vuz175, vuz176, Succ(vuz1770), Succ(vuz1780)) → new_primDivNatS0(vuz175, vuz176, vuz1770, vuz1780)
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12200), Succ(vuz1230)) → new_primMinusNatS2(vuz12200, vuz1230)
new_primMinusNatS2(Zero, Succ(vuz1230)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12200), Zero) → Succ(vuz12200)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), 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
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vuz175, vuz176, Succ(vuz1770), Succ(vuz1780)) → new_primDivNatS0(vuz175, vuz176, vuz1770, vuz1780)
R is empty.
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), 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_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), 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
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vuz175, vuz176, Succ(vuz1770), Succ(vuz1780)) → new_primDivNatS0(vuz175, vuz176, vuz1770, vuz1780)
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(vuz175, vuz176, Succ(vuz1770), Succ(vuz1780)) → new_primDivNatS0(vuz175, vuz176, vuz1770, vuz1780)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_quot0 → new_quot(new_primMinusNatS1, new_primMinusNatS1)
new_quot(Succ(Succ(vuz11200)), vuz111) → new_quot(new_primMinusNatS2(Succ(vuz11200), Zero), new_primMinusNatS2(Succ(vuz11200), Zero))
new_quot(Succ(Zero), vuz111) → new_quot0
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12200), Succ(vuz1230)) → new_primMinusNatS2(vuz12200, vuz1230)
new_primMinusNatS2(Zero, Succ(vuz1230)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12200), Zero) → Succ(vuz12200)
new_primMinusNatS1 → Zero
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS1
new_primMinusNatS2(Succ(x0), 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
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_quot(Succ(Succ(vuz11200)), vuz111) → new_quot(new_primMinusNatS2(Succ(vuz11200), Zero), new_primMinusNatS2(Succ(vuz11200), Zero))
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12200), Succ(vuz1230)) → new_primMinusNatS2(vuz12200, vuz1230)
new_primMinusNatS2(Zero, Succ(vuz1230)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12200), Zero) → Succ(vuz12200)
new_primMinusNatS1 → Zero
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS1
new_primMinusNatS2(Succ(x0), 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
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_quot(Succ(Succ(vuz11200)), vuz111) → new_quot(new_primMinusNatS2(Succ(vuz11200), Zero), new_primMinusNatS2(Succ(vuz11200), Zero))
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12200), Zero) → Succ(vuz12200)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS1
new_primMinusNatS2(Succ(x0), 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_primMinusNatS1
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_quot(Succ(Succ(vuz11200)), vuz111) → new_quot(new_primMinusNatS2(Succ(vuz11200), Zero), new_primMinusNatS2(Succ(vuz11200), Zero))
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12200), Zero) → Succ(vuz12200)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Zero)
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_quot(Succ(Succ(vuz11200)), vuz111) → new_quot(new_primMinusNatS2(Succ(vuz11200), Zero), new_primMinusNatS2(Succ(vuz11200), Zero))
Used ordering: POLO with Polynomial interpretation [25]:
POL(Succ(x1)) = 1 + 2·x1
POL(Zero) = 0
POL(new_primMinusNatS2(x1, x2)) = x1 + 2·x2
POL(new_quot(x1, x2)) = x1 + x2
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
P is empty.
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12200), Zero) → Succ(vuz12200)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Zero)
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_quot1(vuz155, vuz156, vuz157, Succ(vuz1580), Zero) → new_quot2(vuz155, new_primMinusNatS2(Succ(vuz156), vuz157), vuz157, new_primMinusNatS2(Succ(vuz156), vuz157))
new_quot2(vuz155, Succ(Succ(vuz16500)), Zero, vuz164) → new_quot3(vuz155, vuz16500, Zero)
new_quot3(vuz155, vuz156, vuz157) → new_quot2(vuz155, new_primMinusNatS2(Succ(vuz156), vuz157), vuz157, new_primMinusNatS2(Succ(vuz156), vuz157))
new_quot1(vuz155, vuz156, vuz157, Succ(vuz1580), Succ(vuz1590)) → new_quot1(vuz155, vuz156, vuz157, vuz1580, vuz1590)
new_quot2(vuz155, Succ(Zero), Succ(vuz1570), vuz164) → new_quot4(vuz155, vuz1570, Zero)
new_quot2(vuz155, Succ(Zero), Zero, vuz164) → new_quot2(vuz155, new_primMinusNatS2(Zero, Zero), Zero, new_primMinusNatS2(Zero, Zero))
new_quot4(vuz78, vuz79, vuz48) → new_quot2(vuz78, Succ(Succ(vuz79)), vuz48, Succ(Succ(vuz79)))
new_quot2(vuz155, Succ(Succ(vuz16500)), Succ(vuz1570), vuz164) → new_quot1(vuz155, vuz16500, Succ(vuz1570), vuz16500, vuz1570)
new_quot1(vuz155, vuz156, vuz157, Zero, Succ(vuz1590)) → new_quot2(vuz155, Succ(vuz157), Succ(vuz156), Succ(vuz157))
new_quot1(vuz155, vuz156, vuz157, Zero, Zero) → new_quot3(vuz155, vuz156, vuz157)
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12200), Succ(vuz1230)) → new_primMinusNatS2(vuz12200, vuz1230)
new_primMinusNatS2(Zero, Succ(vuz1230)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12200), Zero) → Succ(vuz12200)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), 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
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_quot1(vuz155, vuz156, vuz157, Succ(vuz1580), Zero) → new_quot2(vuz155, new_primMinusNatS2(Succ(vuz156), vuz157), vuz157, new_primMinusNatS2(Succ(vuz156), vuz157))
new_quot2(vuz155, Succ(Succ(vuz16500)), Zero, vuz164) → new_quot3(vuz155, vuz16500, Zero)
new_quot3(vuz155, vuz156, vuz157) → new_quot2(vuz155, new_primMinusNatS2(Succ(vuz156), vuz157), vuz157, new_primMinusNatS2(Succ(vuz156), vuz157))
new_quot1(vuz155, vuz156, vuz157, Succ(vuz1580), Succ(vuz1590)) → new_quot1(vuz155, vuz156, vuz157, vuz1580, vuz1590)
new_quot2(vuz155, Succ(Zero), Succ(vuz1570), vuz164) → new_quot4(vuz155, vuz1570, Zero)
new_quot4(vuz78, vuz79, vuz48) → new_quot2(vuz78, Succ(Succ(vuz79)), vuz48, Succ(Succ(vuz79)))
new_quot2(vuz155, Succ(Succ(vuz16500)), Succ(vuz1570), vuz164) → new_quot1(vuz155, vuz16500, Succ(vuz1570), vuz16500, vuz1570)
new_quot1(vuz155, vuz156, vuz157, Zero, Succ(vuz1590)) → new_quot2(vuz155, Succ(vuz157), Succ(vuz156), Succ(vuz157))
new_quot1(vuz155, vuz156, vuz157, Zero, Zero) → new_quot3(vuz155, vuz156, vuz157)
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12200), Succ(vuz1230)) → new_primMinusNatS2(vuz12200, vuz1230)
new_primMinusNatS2(Zero, Succ(vuz1230)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12200), Zero) → Succ(vuz12200)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Zero)
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule new_quot4(vuz78, vuz79, vuz48) → new_quot2(vuz78, Succ(Succ(vuz79)), vuz48, Succ(Succ(vuz79))) we obtained the following new rules:
new_quot4(z0, z1, Zero) → new_quot2(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
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_quot2(vuz155, Succ(Succ(vuz16500)), Zero, vuz164) → new_quot3(vuz155, vuz16500, Zero)
new_quot1(vuz155, vuz156, vuz157, Succ(vuz1580), Zero) → new_quot2(vuz155, new_primMinusNatS2(Succ(vuz156), vuz157), vuz157, new_primMinusNatS2(Succ(vuz156), vuz157))
new_quot3(vuz155, vuz156, vuz157) → new_quot2(vuz155, new_primMinusNatS2(Succ(vuz156), vuz157), vuz157, new_primMinusNatS2(Succ(vuz156), vuz157))
new_quot2(vuz155, Succ(Zero), Succ(vuz1570), vuz164) → new_quot4(vuz155, vuz1570, Zero)
new_quot1(vuz155, vuz156, vuz157, Succ(vuz1580), Succ(vuz1590)) → new_quot1(vuz155, vuz156, vuz157, vuz1580, vuz1590)
new_quot4(z0, z1, Zero) → new_quot2(z0, Succ(Succ(z1)), Zero, Succ(Succ(z1)))
new_quot2(vuz155, Succ(Succ(vuz16500)), Succ(vuz1570), vuz164) → new_quot1(vuz155, vuz16500, Succ(vuz1570), vuz16500, vuz1570)
new_quot1(vuz155, vuz156, vuz157, Zero, Succ(vuz1590)) → new_quot2(vuz155, Succ(vuz157), Succ(vuz156), Succ(vuz157))
new_quot1(vuz155, vuz156, vuz157, Zero, Zero) → new_quot3(vuz155, vuz156, vuz157)
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12200), Succ(vuz1230)) → new_primMinusNatS2(vuz12200, vuz1230)
new_primMinusNatS2(Zero, Succ(vuz1230)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12200), Zero) → Succ(vuz12200)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Zero)
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule new_quot3(vuz155, vuz156, vuz157) → new_quot2(vuz155, new_primMinusNatS2(Succ(vuz156), vuz157), vuz157, new_primMinusNatS2(Succ(vuz156), vuz157)) at position [1] we obtained the following new rules:
new_quot3(y0, x0, Zero) → new_quot2(y0, Succ(x0), Zero, new_primMinusNatS2(Succ(x0), Zero))
new_quot3(y0, x0, Succ(x1)) → new_quot2(y0, new_primMinusNatS2(x0, x1), Succ(x1), new_primMinusNatS2(Succ(x0), Succ(x1)))
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_quot1(vuz155, vuz156, vuz157, Succ(vuz1580), Zero) → new_quot2(vuz155, new_primMinusNatS2(Succ(vuz156), vuz157), vuz157, new_primMinusNatS2(Succ(vuz156), vuz157))
new_quot2(vuz155, Succ(Succ(vuz16500)), Zero, vuz164) → new_quot3(vuz155, vuz16500, Zero)
new_quot3(y0, x0, Succ(x1)) → new_quot2(y0, new_primMinusNatS2(x0, x1), Succ(x1), new_primMinusNatS2(Succ(x0), Succ(x1)))
new_quot1(vuz155, vuz156, vuz157, Succ(vuz1580), Succ(vuz1590)) → new_quot1(vuz155, vuz156, vuz157, vuz1580, vuz1590)
new_quot2(vuz155, Succ(Zero), Succ(vuz1570), vuz164) → new_quot4(vuz155, vuz1570, Zero)
new_quot3(y0, x0, Zero) → new_quot2(y0, Succ(x0), Zero, new_primMinusNatS2(Succ(x0), Zero))
new_quot4(z0, z1, Zero) → new_quot2(z0, Succ(Succ(z1)), Zero, Succ(Succ(z1)))
new_quot2(vuz155, Succ(Succ(vuz16500)), Succ(vuz1570), vuz164) → new_quot1(vuz155, vuz16500, Succ(vuz1570), vuz16500, vuz1570)
new_quot1(vuz155, vuz156, vuz157, Zero, Succ(vuz1590)) → new_quot2(vuz155, Succ(vuz157), Succ(vuz156), Succ(vuz157))
new_quot1(vuz155, vuz156, vuz157, Zero, Zero) → new_quot3(vuz155, vuz156, vuz157)
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12200), Succ(vuz1230)) → new_primMinusNatS2(vuz12200, vuz1230)
new_primMinusNatS2(Zero, Succ(vuz1230)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12200), Zero) → Succ(vuz12200)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), 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
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_quot2(vuz155, Succ(Succ(vuz16500)), Zero, vuz164) → new_quot3(vuz155, vuz16500, Zero)
new_quot3(y0, x0, Zero) → new_quot2(y0, Succ(x0), Zero, new_primMinusNatS2(Succ(x0), Zero))
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12200), Succ(vuz1230)) → new_primMinusNatS2(vuz12200, vuz1230)
new_primMinusNatS2(Zero, Succ(vuz1230)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12200), Zero) → Succ(vuz12200)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), 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
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_quot2(vuz155, Succ(Succ(vuz16500)), Zero, vuz164) → new_quot3(vuz155, vuz16500, Zero)
new_quot3(y0, x0, Zero) → new_quot2(y0, Succ(x0), Zero, new_primMinusNatS2(Succ(x0), Zero))
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12200), Zero) → Succ(vuz12200)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Zero)
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_quot3(y0, x0, Zero) → new_quot2(y0, Succ(x0), Zero, new_primMinusNatS2(Succ(x0), Zero)) at position [3] we obtained the following new rules:
new_quot3(y0, x0, Zero) → new_quot2(y0, Succ(x0), Zero, Succ(x0))
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_quot2(vuz155, Succ(Succ(vuz16500)), Zero, vuz164) → new_quot3(vuz155, vuz16500, Zero)
new_quot3(y0, x0, Zero) → new_quot2(y0, Succ(x0), Zero, Succ(x0))
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12200), Zero) → Succ(vuz12200)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), 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
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_quot2(vuz155, Succ(Succ(vuz16500)), Zero, vuz164) → new_quot3(vuz155, vuz16500, Zero)
new_quot3(y0, x0, Zero) → new_quot2(y0, Succ(x0), Zero, Succ(x0))
R is empty.
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), 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_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Zero)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_quot2(vuz155, Succ(Succ(vuz16500)), Zero, vuz164) → new_quot3(vuz155, vuz16500, Zero)
new_quot3(y0, x0, Zero) → new_quot2(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_quot2(vuz155, Succ(Succ(vuz16500)), Zero, vuz164) → new_quot3(vuz155, vuz16500, Zero) we obtained the following new rules:
new_quot2(z0, Succ(Succ(x1)), Zero, Succ(Succ(x1))) → new_quot3(z0, x1, Zero)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_quot3(y0, x0, Zero) → new_quot2(y0, Succ(x0), Zero, Succ(x0))
new_quot2(z0, Succ(Succ(x1)), Zero, Succ(Succ(x1))) → new_quot3(z0, x1, Zero)
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_quot2(z0, Succ(Succ(x1)), Zero, Succ(Succ(x1))) → new_quot3(z0, x1, Zero)
Used ordering: POLO with Polynomial interpretation [25]:
POL(Succ(x1)) = 1 + x1
POL(Zero) = 0
POL(new_quot2(x1, x2, x3, x4)) = x1 + x2 + 2·x3 + x4
POL(new_quot3(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
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_quot3(y0, x0, Zero) → new_quot2(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
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_quot1(vuz155, vuz156, vuz157, Succ(vuz1580), Zero) → new_quot2(vuz155, new_primMinusNatS2(Succ(vuz156), vuz157), vuz157, new_primMinusNatS2(Succ(vuz156), vuz157))
new_quot3(y0, x0, Succ(x1)) → new_quot2(y0, new_primMinusNatS2(x0, x1), Succ(x1), new_primMinusNatS2(Succ(x0), Succ(x1)))
new_quot1(vuz155, vuz156, vuz157, Succ(vuz1580), Succ(vuz1590)) → new_quot1(vuz155, vuz156, vuz157, vuz1580, vuz1590)
new_quot2(vuz155, Succ(Succ(vuz16500)), Succ(vuz1570), vuz164) → new_quot1(vuz155, vuz16500, Succ(vuz1570), vuz16500, vuz1570)
new_quot1(vuz155, vuz156, vuz157, Zero, Succ(vuz1590)) → new_quot2(vuz155, Succ(vuz157), Succ(vuz156), Succ(vuz157))
new_quot1(vuz155, vuz156, vuz157, Zero, Zero) → new_quot3(vuz155, vuz156, vuz157)
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12200), Succ(vuz1230)) → new_primMinusNatS2(vuz12200, vuz1230)
new_primMinusNatS2(Zero, Succ(vuz1230)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12200), Zero) → Succ(vuz12200)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Zero)
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_quot3(y0, x0, Succ(x1)) → new_quot2(y0, new_primMinusNatS2(x0, x1), Succ(x1), new_primMinusNatS2(Succ(x0), Succ(x1))) at position [3] we obtained the following new rules:
new_quot3(y0, x0, Succ(x1)) → new_quot2(y0, new_primMinusNatS2(x0, x1), Succ(x1), new_primMinusNatS2(x0, x1))
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_quot1(vuz155, vuz156, vuz157, Succ(vuz1580), Zero) → new_quot2(vuz155, new_primMinusNatS2(Succ(vuz156), vuz157), vuz157, new_primMinusNatS2(Succ(vuz156), vuz157))
new_quot1(vuz155, vuz156, vuz157, Succ(vuz1580), Succ(vuz1590)) → new_quot1(vuz155, vuz156, vuz157, vuz1580, vuz1590)
new_quot3(y0, x0, Succ(x1)) → new_quot2(y0, new_primMinusNatS2(x0, x1), Succ(x1), new_primMinusNatS2(x0, x1))
new_quot2(vuz155, Succ(Succ(vuz16500)), Succ(vuz1570), vuz164) → new_quot1(vuz155, vuz16500, Succ(vuz1570), vuz16500, vuz1570)
new_quot1(vuz155, vuz156, vuz157, Zero, Succ(vuz1590)) → new_quot2(vuz155, Succ(vuz157), Succ(vuz156), Succ(vuz157))
new_quot1(vuz155, vuz156, vuz157, Zero, Zero) → new_quot3(vuz155, vuz156, vuz157)
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12200), Succ(vuz1230)) → new_primMinusNatS2(vuz12200, vuz1230)
new_primMinusNatS2(Zero, Succ(vuz1230)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12200), Zero) → Succ(vuz12200)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), 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_quot3(y0, x0, Succ(x1)) → new_quot2(y0, new_primMinusNatS2(x0, x1), Succ(x1), new_primMinusNatS2(x0, x1))
The remaining pairs can at least be oriented weakly.
new_quot1(vuz155, vuz156, vuz157, Succ(vuz1580), Zero) → new_quot2(vuz155, new_primMinusNatS2(Succ(vuz156), vuz157), vuz157, new_primMinusNatS2(Succ(vuz156), vuz157))
new_quot1(vuz155, vuz156, vuz157, Succ(vuz1580), Succ(vuz1590)) → new_quot1(vuz155, vuz156, vuz157, vuz1580, vuz1590)
new_quot2(vuz155, Succ(Succ(vuz16500)), Succ(vuz1570), vuz164) → new_quot1(vuz155, vuz16500, Succ(vuz1570), vuz16500, vuz1570)
new_quot1(vuz155, vuz156, vuz157, Zero, Succ(vuz1590)) → new_quot2(vuz155, Succ(vuz157), Succ(vuz156), Succ(vuz157))
new_quot1(vuz155, vuz156, vuz157, Zero, Zero) → new_quot3(vuz155, vuz156, vuz157)
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( new_primMinusNatS2(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
Tuple symbols:
M( new_quot1(x1, ..., x5) ) = | 0 | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 | + | | · | x5 |
M( new_quot2(x1, ..., x4) ) = | 0 | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
M( new_quot3(x1, ..., x3) ) = | 0 | + | | · | x1 | + | | · | x2 | + | | · | x3 |
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_primMinusNatS2(Succ(vuz12200), Zero) → Succ(vuz12200)
new_primMinusNatS2(Succ(vuz12200), Succ(vuz1230)) → new_primMinusNatS2(vuz12200, vuz1230)
new_primMinusNatS2(Zero, Succ(vuz1230)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_quot1(vuz155, vuz156, vuz157, Succ(vuz1580), Zero) → new_quot2(vuz155, new_primMinusNatS2(Succ(vuz156), vuz157), vuz157, new_primMinusNatS2(Succ(vuz156), vuz157))
new_quot1(vuz155, vuz156, vuz157, Succ(vuz1580), Succ(vuz1590)) → new_quot1(vuz155, vuz156, vuz157, vuz1580, vuz1590)
new_quot2(vuz155, Succ(Succ(vuz16500)), Succ(vuz1570), vuz164) → new_quot1(vuz155, vuz16500, Succ(vuz1570), vuz16500, vuz1570)
new_quot1(vuz155, vuz156, vuz157, Zero, Succ(vuz1590)) → new_quot2(vuz155, Succ(vuz157), Succ(vuz156), Succ(vuz157))
new_quot1(vuz155, vuz156, vuz157, Zero, Zero) → new_quot3(vuz155, vuz156, vuz157)
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12200), Succ(vuz1230)) → new_primMinusNatS2(vuz12200, vuz1230)
new_primMinusNatS2(Zero, Succ(vuz1230)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12200), Zero) → Succ(vuz12200)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), 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
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_quot1(vuz155, vuz156, vuz157, Succ(vuz1580), Zero) → new_quot2(vuz155, new_primMinusNatS2(Succ(vuz156), vuz157), vuz157, new_primMinusNatS2(Succ(vuz156), vuz157))
new_quot1(vuz155, vuz156, vuz157, Succ(vuz1580), Succ(vuz1590)) → new_quot1(vuz155, vuz156, vuz157, vuz1580, vuz1590)
new_quot2(vuz155, Succ(Succ(vuz16500)), Succ(vuz1570), vuz164) → new_quot1(vuz155, vuz16500, Succ(vuz1570), vuz16500, vuz1570)
new_quot1(vuz155, vuz156, vuz157, Zero, Succ(vuz1590)) → new_quot2(vuz155, Succ(vuz157), Succ(vuz156), Succ(vuz157))
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12200), Succ(vuz1230)) → new_primMinusNatS2(vuz12200, vuz1230)
new_primMinusNatS2(Zero, Succ(vuz1230)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12200), Zero) → Succ(vuz12200)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), 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_quot1(vuz155, vuz156, vuz157, Succ(vuz1580), Zero) → new_quot2(vuz155, new_primMinusNatS2(Succ(vuz156), vuz157), vuz157, new_primMinusNatS2(Succ(vuz156), vuz157))
The remaining pairs can at least be oriented weakly.
new_quot1(vuz155, vuz156, vuz157, Succ(vuz1580), Succ(vuz1590)) → new_quot1(vuz155, vuz156, vuz157, vuz1580, vuz1590)
new_quot2(vuz155, Succ(Succ(vuz16500)), Succ(vuz1570), vuz164) → new_quot1(vuz155, vuz16500, Succ(vuz1570), vuz16500, vuz1570)
new_quot1(vuz155, vuz156, vuz157, Zero, Succ(vuz1590)) → new_quot2(vuz155, Succ(vuz157), Succ(vuz156), Succ(vuz157))
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( new_primMinusNatS2(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
Tuple symbols:
M( new_quot1(x1, ..., x5) ) = | 1 | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 | + | | · | x5 |
M( new_quot2(x1, ..., x4) ) = | 0 | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
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_primMinusNatS2(Succ(vuz12200), Zero) → Succ(vuz12200)
new_primMinusNatS2(Succ(vuz12200), Succ(vuz1230)) → new_primMinusNatS2(vuz12200, vuz1230)
new_primMinusNatS2(Zero, Succ(vuz1230)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_quot1(vuz155, vuz156, vuz157, Succ(vuz1580), Succ(vuz1590)) → new_quot1(vuz155, vuz156, vuz157, vuz1580, vuz1590)
new_quot2(vuz155, Succ(Succ(vuz16500)), Succ(vuz1570), vuz164) → new_quot1(vuz155, vuz16500, Succ(vuz1570), vuz16500, vuz1570)
new_quot1(vuz155, vuz156, vuz157, Zero, Succ(vuz1590)) → new_quot2(vuz155, Succ(vuz157), Succ(vuz156), Succ(vuz157))
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12200), Succ(vuz1230)) → new_primMinusNatS2(vuz12200, vuz1230)
new_primMinusNatS2(Zero, Succ(vuz1230)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12200), Zero) → Succ(vuz12200)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), 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
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_quot1(vuz155, vuz156, vuz157, Succ(vuz1580), Succ(vuz1590)) → new_quot1(vuz155, vuz156, vuz157, vuz1580, vuz1590)
new_quot2(vuz155, Succ(Succ(vuz16500)), Succ(vuz1570), vuz164) → new_quot1(vuz155, vuz16500, Succ(vuz1570), vuz16500, vuz1570)
new_quot1(vuz155, vuz156, vuz157, Zero, Succ(vuz1590)) → new_quot2(vuz155, Succ(vuz157), Succ(vuz156), Succ(vuz157))
R is empty.
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), 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_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Zero)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_quot1(vuz155, vuz156, vuz157, Succ(vuz1580), Succ(vuz1590)) → new_quot1(vuz155, vuz156, vuz157, vuz1580, vuz1590)
new_quot2(vuz155, Succ(Succ(vuz16500)), Succ(vuz1570), vuz164) → new_quot1(vuz155, vuz16500, Succ(vuz1570), vuz16500, vuz1570)
new_quot1(vuz155, vuz156, vuz157, Zero, Succ(vuz1590)) → new_quot2(vuz155, Succ(vuz157), Succ(vuz156), Succ(vuz157))
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule new_quot2(vuz155, Succ(Succ(vuz16500)), Succ(vuz1570), vuz164) → new_quot1(vuz155, vuz16500, Succ(vuz1570), vuz16500, vuz1570) we obtained the following new rules:
new_quot2(z0, Succ(Succ(x1)), Succ(z1), Succ(Succ(x1))) → new_quot1(z0, x1, Succ(z1), x1, z1)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ 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
Q DP problem:
The TRS P consists of the following rules:
new_quot1(vuz155, vuz156, vuz157, Succ(vuz1580), Succ(vuz1590)) → new_quot1(vuz155, vuz156, vuz157, vuz1580, vuz1590)
new_quot2(z0, Succ(Succ(x1)), Succ(z1), Succ(Succ(x1))) → new_quot1(z0, x1, Succ(z1), x1, z1)
new_quot1(vuz155, vuz156, vuz157, Zero, Succ(vuz1590)) → new_quot2(vuz155, Succ(vuz157), Succ(vuz156), Succ(vuz157))
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_quot1(vuz155, vuz156, vuz157, Succ(vuz1580), Succ(vuz1590)) → new_quot1(vuz155, vuz156, vuz157, vuz1580, vuz1590) the following chains were created:
- We consider the chain new_quot1(vuz155, vuz156, vuz157, Succ(vuz1580), Succ(vuz1590)) → new_quot1(vuz155, vuz156, vuz157, vuz1580, vuz1590), new_quot1(vuz155, vuz156, vuz157, Succ(vuz1580), Succ(vuz1590)) → new_quot1(vuz155, vuz156, vuz157, vuz1580, vuz1590) which results in the following constraint:
(1) (new_quot1(x0, x1, x2, x3, x4)=new_quot1(x5, x6, x7, Succ(x8), Succ(x9)) ⇒ new_quot1(x5, x6, x7, Succ(x8), Succ(x9))≥new_quot1(x5, x6, x7, x8, x9))
We simplified constraint (1) using rules (I), (II), (III) which results in the following new constraint:
(2) (new_quot1(x0, x1, x2, Succ(x8), Succ(x9))≥new_quot1(x0, x1, x2, x8, x9))
- We consider the chain new_quot2(z0, Succ(Succ(x1)), Succ(z1), Succ(Succ(x1))) → new_quot1(z0, x1, Succ(z1), x1, z1), new_quot1(vuz155, vuz156, vuz157, Succ(vuz1580), Succ(vuz1590)) → new_quot1(vuz155, vuz156, vuz157, vuz1580, vuz1590) which results in the following constraint:
(3) (new_quot1(x14, x15, Succ(x16), x15, x16)=new_quot1(x17, x18, x19, Succ(x20), Succ(x21)) ⇒ new_quot1(x17, x18, x19, Succ(x20), Succ(x21))≥new_quot1(x17, x18, x19, x20, x21))
We simplified constraint (3) using rules (I), (II), (III) which results in the following new constraint:
(4) (new_quot1(x14, Succ(x20), Succ(Succ(x21)), Succ(x20), Succ(x21))≥new_quot1(x14, Succ(x20), Succ(Succ(x21)), x20, x21))
For Pair new_quot1(vuz155, vuz156, vuz157, Zero, Succ(vuz1590)) → new_quot2(vuz155, Succ(vuz157), Succ(vuz156), Succ(vuz157)) the following chains were created:
- We consider the chain new_quot1(vuz155, vuz156, vuz157, Succ(vuz1580), Succ(vuz1590)) → new_quot1(vuz155, vuz156, vuz157, vuz1580, vuz1590), new_quot1(vuz155, vuz156, vuz157, Zero, Succ(vuz1590)) → new_quot2(vuz155, Succ(vuz157), Succ(vuz156), Succ(vuz157)) which results in the following constraint:
(5) (new_quot1(x22, x23, x24, x25, x26)=new_quot1(x27, x28, x29, Zero, Succ(x30)) ⇒ new_quot1(x27, x28, x29, Zero, Succ(x30))≥new_quot2(x27, Succ(x29), Succ(x28), Succ(x29)))
We simplified constraint (5) using rules (I), (II), (III) which results in the following new constraint:
(6) (new_quot1(x22, x23, x24, Zero, Succ(x30))≥new_quot2(x22, Succ(x24), Succ(x23), Succ(x24)))
- We consider the chain new_quot2(z0, Succ(Succ(x1)), Succ(z1), Succ(Succ(x1))) → new_quot1(z0, x1, Succ(z1), x1, z1), new_quot1(vuz155, vuz156, vuz157, Zero, Succ(vuz1590)) → new_quot2(vuz155, Succ(vuz157), Succ(vuz156), Succ(vuz157)) which results in the following constraint:
(7) (new_quot1(x35, x36, Succ(x37), x36, x37)=new_quot1(x38, x39, x40, Zero, Succ(x41)) ⇒ new_quot1(x38, x39, x40, Zero, Succ(x41))≥new_quot2(x38, Succ(x40), Succ(x39), Succ(x40)))
We simplified constraint (7) using rules (I), (II), (III) which results in the following new constraint:
(8) (new_quot1(x35, Zero, Succ(Succ(x41)), Zero, Succ(x41))≥new_quot2(x35, Succ(Succ(Succ(x41))), Succ(Zero), Succ(Succ(Succ(x41)))))
For Pair new_quot2(z0, Succ(Succ(x1)), Succ(z1), Succ(Succ(x1))) → new_quot1(z0, x1, Succ(z1), x1, z1) the following chains were created:
- We consider the chain new_quot1(vuz155, vuz156, vuz157, Zero, Succ(vuz1590)) → new_quot2(vuz155, Succ(vuz157), Succ(vuz156), Succ(vuz157)), new_quot2(z0, Succ(Succ(x1)), Succ(z1), Succ(Succ(x1))) → new_quot1(z0, x1, Succ(z1), x1, z1) which results in the following constraint:
(9) (new_quot2(x47, Succ(x49), Succ(x48), Succ(x49))=new_quot2(x51, Succ(Succ(x52)), Succ(x53), Succ(Succ(x52))) ⇒ new_quot2(x51, Succ(Succ(x52)), Succ(x53), Succ(Succ(x52)))≥new_quot1(x51, x52, Succ(x53), x52, x53))
We simplified constraint (9) using rules (I), (II), (III) which results in the following new constraint:
(10) (new_quot2(x47, Succ(Succ(x52)), Succ(x48), Succ(Succ(x52)))≥new_quot1(x47, x52, Succ(x48), x52, x48))
To summarize, we get the following constraints P≥ for the following pairs.
- new_quot1(vuz155, vuz156, vuz157, Succ(vuz1580), Succ(vuz1590)) → new_quot1(vuz155, vuz156, vuz157, vuz1580, vuz1590)
- (new_quot1(x0, x1, x2, Succ(x8), Succ(x9))≥new_quot1(x0, x1, x2, x8, x9))
- (new_quot1(x14, Succ(x20), Succ(Succ(x21)), Succ(x20), Succ(x21))≥new_quot1(x14, Succ(x20), Succ(Succ(x21)), x20, x21))
- new_quot1(vuz155, vuz156, vuz157, Zero, Succ(vuz1590)) → new_quot2(vuz155, Succ(vuz157), Succ(vuz156), Succ(vuz157))
- (new_quot1(x22, x23, x24, Zero, Succ(x30))≥new_quot2(x22, Succ(x24), Succ(x23), Succ(x24)))
- (new_quot1(x35, Zero, Succ(Succ(x41)), Zero, Succ(x41))≥new_quot2(x35, Succ(Succ(Succ(x41))), Succ(Zero), Succ(Succ(Succ(x41)))))
- new_quot2(z0, Succ(Succ(x1)), Succ(z1), Succ(Succ(x1))) → new_quot1(z0, x1, Succ(z1), x1, z1)
- (new_quot2(x47, Succ(Succ(x52)), Succ(x48), Succ(Succ(x52)))≥new_quot1(x47, x52, Succ(x48), x52, x48))
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_quot1(x1, x2, x3, x4, x5)) = x1 + x2 - x4 + x5
POL(new_quot2(x1, x2, x3, x4)) = x1 - x2 + x3 + x4
The following pairs are in P>:
new_quot2(z0, Succ(Succ(x1)), Succ(z1), Succ(Succ(x1))) → new_quot1(z0, x1, Succ(z1), x1, z1)
The following pairs are in Pbound:
new_quot1(vuz155, vuz156, vuz157, Zero, Succ(vuz1590)) → new_quot2(vuz155, Succ(vuz157), Succ(vuz156), Succ(vuz157))
new_quot2(z0, Succ(Succ(x1)), Succ(z1), Succ(Succ(x1))) → new_quot1(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
↳ DependencyGraphProof
↳ 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
Q DP problem:
The TRS P consists of the following rules:
new_quot1(vuz155, vuz156, vuz157, Succ(vuz1580), Succ(vuz1590)) → new_quot1(vuz155, vuz156, vuz157, vuz1580, vuz1590)
new_quot1(vuz155, vuz156, vuz157, Zero, Succ(vuz1590)) → new_quot2(vuz155, Succ(vuz157), Succ(vuz156), Succ(vuz157))
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
↳ DependencyGraphProof
↳ 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
Q DP problem:
The TRS P consists of the following rules:
new_quot1(vuz155, vuz156, vuz157, Succ(vuz1580), Succ(vuz1590)) → new_quot1(vuz155, vuz156, vuz157, vuz1580, vuz1590)
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_quot1(vuz155, vuz156, vuz157, Succ(vuz1580), Succ(vuz1590)) → new_quot1(vuz155, vuz156, vuz157, vuz1580, vuz1590)
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
↳ DependencyGraphProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_quot6(vuz141, Succ(Zero), Zero, vuz152) → new_quot6(vuz141, new_primMinusNatS2(Zero, Zero), Zero, new_primMinusNatS2(Zero, Zero))
new_quot6(vuz141, Succ(Succ(vuz15300)), Zero, vuz152) → new_quot7(vuz141, vuz15300, Zero)
new_quot7(vuz141, vuz142, vuz143) → new_quot6(vuz141, new_primMinusNatS2(Succ(vuz142), vuz143), vuz143, new_primMinusNatS2(Succ(vuz142), vuz143))
new_quot5(vuz141, vuz142, vuz143, Zero, Zero) → new_quot7(vuz141, vuz142, vuz143)
new_quot5(vuz141, vuz142, vuz143, Succ(vuz1440), Succ(vuz1450)) → new_quot5(vuz141, vuz142, vuz143, vuz1440, vuz1450)
new_quot5(vuz141, vuz142, vuz143, Zero, Succ(vuz1450)) → new_quot6(vuz141, Succ(vuz143), Succ(vuz142), Succ(vuz143))
new_quot6(vuz141, Succ(Succ(vuz15300)), Succ(vuz1430), vuz152) → new_quot5(vuz141, vuz15300, Succ(vuz1430), vuz15300, vuz1430)
new_quot5(vuz141, vuz142, vuz143, Succ(vuz1440), Zero) → new_quot6(vuz141, new_primMinusNatS2(Succ(vuz142), vuz143), vuz143, new_primMinusNatS2(Succ(vuz142), vuz143))
new_quot8(vuz97, vuz98) → new_quot6(vuz97, Succ(vuz98), Zero, Succ(vuz98))
new_quot6(vuz141, Succ(Zero), Succ(vuz1430), vuz152) → new_quot8(vuz141, Succ(vuz1430))
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12200), Succ(vuz1230)) → new_primMinusNatS2(vuz12200, vuz1230)
new_primMinusNatS2(Zero, Succ(vuz1230)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12200), Zero) → Succ(vuz12200)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), 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
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_quot6(vuz141, Succ(Succ(vuz15300)), Zero, vuz152) → new_quot7(vuz141, vuz15300, Zero)
new_quot7(vuz141, vuz142, vuz143) → new_quot6(vuz141, new_primMinusNatS2(Succ(vuz142), vuz143), vuz143, new_primMinusNatS2(Succ(vuz142), vuz143))
new_quot5(vuz141, vuz142, vuz143, Zero, Zero) → new_quot7(vuz141, vuz142, vuz143)
new_quot5(vuz141, vuz142, vuz143, Succ(vuz1440), Succ(vuz1450)) → new_quot5(vuz141, vuz142, vuz143, vuz1440, vuz1450)
new_quot5(vuz141, vuz142, vuz143, Zero, Succ(vuz1450)) → new_quot6(vuz141, Succ(vuz143), Succ(vuz142), Succ(vuz143))
new_quot6(vuz141, Succ(Succ(vuz15300)), Succ(vuz1430), vuz152) → new_quot5(vuz141, vuz15300, Succ(vuz1430), vuz15300, vuz1430)
new_quot5(vuz141, vuz142, vuz143, Succ(vuz1440), Zero) → new_quot6(vuz141, new_primMinusNatS2(Succ(vuz142), vuz143), vuz143, new_primMinusNatS2(Succ(vuz142), vuz143))
new_quot8(vuz97, vuz98) → new_quot6(vuz97, Succ(vuz98), Zero, Succ(vuz98))
new_quot6(vuz141, Succ(Zero), Succ(vuz1430), vuz152) → new_quot8(vuz141, Succ(vuz1430))
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12200), Succ(vuz1230)) → new_primMinusNatS2(vuz12200, vuz1230)
new_primMinusNatS2(Zero, Succ(vuz1230)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12200), Zero) → Succ(vuz12200)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Zero)
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule new_quot7(vuz141, vuz142, vuz143) → new_quot6(vuz141, new_primMinusNatS2(Succ(vuz142), vuz143), vuz143, new_primMinusNatS2(Succ(vuz142), vuz143)) at position [1] we obtained the following new rules:
new_quot7(y0, x0, Zero) → new_quot6(y0, Succ(x0), Zero, new_primMinusNatS2(Succ(x0), Zero))
new_quot7(y0, x0, Succ(x1)) → new_quot6(y0, new_primMinusNatS2(x0, x1), Succ(x1), new_primMinusNatS2(Succ(x0), Succ(x1)))
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_quot6(vuz141, Succ(Succ(vuz15300)), Zero, vuz152) → new_quot7(vuz141, vuz15300, Zero)
new_quot5(vuz141, vuz142, vuz143, Zero, Zero) → new_quot7(vuz141, vuz142, vuz143)
new_quot7(y0, x0, Succ(x1)) → new_quot6(y0, new_primMinusNatS2(x0, x1), Succ(x1), new_primMinusNatS2(Succ(x0), Succ(x1)))
new_quot5(vuz141, vuz142, vuz143, Succ(vuz1440), Succ(vuz1450)) → new_quot5(vuz141, vuz142, vuz143, vuz1440, vuz1450)
new_quot6(vuz141, Succ(Succ(vuz15300)), Succ(vuz1430), vuz152) → new_quot5(vuz141, vuz15300, Succ(vuz1430), vuz15300, vuz1430)
new_quot5(vuz141, vuz142, vuz143, Zero, Succ(vuz1450)) → new_quot6(vuz141, Succ(vuz143), Succ(vuz142), Succ(vuz143))
new_quot8(vuz97, vuz98) → new_quot6(vuz97, Succ(vuz98), Zero, Succ(vuz98))
new_quot5(vuz141, vuz142, vuz143, Succ(vuz1440), Zero) → new_quot6(vuz141, new_primMinusNatS2(Succ(vuz142), vuz143), vuz143, new_primMinusNatS2(Succ(vuz142), vuz143))
new_quot7(y0, x0, Zero) → new_quot6(y0, Succ(x0), Zero, new_primMinusNatS2(Succ(x0), Zero))
new_quot6(vuz141, Succ(Zero), Succ(vuz1430), vuz152) → new_quot8(vuz141, Succ(vuz1430))
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12200), Succ(vuz1230)) → new_primMinusNatS2(vuz12200, vuz1230)
new_primMinusNatS2(Zero, Succ(vuz1230)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12200), Zero) → Succ(vuz12200)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), 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
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_quot6(vuz141, Succ(Succ(vuz15300)), Zero, vuz152) → new_quot7(vuz141, vuz15300, Zero)
new_quot7(y0, x0, Zero) → new_quot6(y0, Succ(x0), Zero, new_primMinusNatS2(Succ(x0), Zero))
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12200), Succ(vuz1230)) → new_primMinusNatS2(vuz12200, vuz1230)
new_primMinusNatS2(Zero, Succ(vuz1230)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12200), Zero) → Succ(vuz12200)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), 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
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_quot6(vuz141, Succ(Succ(vuz15300)), Zero, vuz152) → new_quot7(vuz141, vuz15300, Zero)
new_quot7(y0, x0, Zero) → new_quot6(y0, Succ(x0), Zero, new_primMinusNatS2(Succ(x0), Zero))
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12200), Zero) → Succ(vuz12200)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Zero)
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_quot7(y0, x0, Zero) → new_quot6(y0, Succ(x0), Zero, new_primMinusNatS2(Succ(x0), Zero)) at position [3] we obtained the following new rules:
new_quot7(y0, x0, Zero) → new_quot6(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
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_quot6(vuz141, Succ(Succ(vuz15300)), Zero, vuz152) → new_quot7(vuz141, vuz15300, Zero)
new_quot7(y0, x0, Zero) → new_quot6(y0, Succ(x0), Zero, Succ(x0))
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12200), Zero) → Succ(vuz12200)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), 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
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_quot6(vuz141, Succ(Succ(vuz15300)), Zero, vuz152) → new_quot7(vuz141, vuz15300, Zero)
new_quot7(y0, x0, Zero) → new_quot6(y0, Succ(x0), Zero, Succ(x0))
R is empty.
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), 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_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Zero)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_quot6(vuz141, Succ(Succ(vuz15300)), Zero, vuz152) → new_quot7(vuz141, vuz15300, Zero)
new_quot7(y0, x0, Zero) → new_quot6(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_quot6(vuz141, Succ(Succ(vuz15300)), Zero, vuz152) → new_quot7(vuz141, vuz15300, Zero) we obtained the following new rules:
new_quot6(z0, Succ(Succ(x1)), Zero, Succ(Succ(x1))) → new_quot7(z0, x1, Zero)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_quot6(z0, Succ(Succ(x1)), Zero, Succ(Succ(x1))) → new_quot7(z0, x1, Zero)
new_quot7(y0, x0, Zero) → new_quot6(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_quot6(z0, Succ(Succ(x1)), Zero, Succ(Succ(x1))) → new_quot7(z0, x1, Zero)
Used ordering: POLO with Polynomial interpretation [25]:
POL(Succ(x1)) = 1 + x1
POL(Zero) = 0
POL(new_quot6(x1, x2, x3, x4)) = x1 + x2 + 2·x3 + x4
POL(new_quot7(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
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_quot7(y0, x0, Zero) → new_quot6(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
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Rewriting
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_quot5(vuz141, vuz142, vuz143, Zero, Zero) → new_quot7(vuz141, vuz142, vuz143)
new_quot5(vuz141, vuz142, vuz143, Succ(vuz1440), Succ(vuz1450)) → new_quot5(vuz141, vuz142, vuz143, vuz1440, vuz1450)
new_quot7(y0, x0, Succ(x1)) → new_quot6(y0, new_primMinusNatS2(x0, x1), Succ(x1), new_primMinusNatS2(Succ(x0), Succ(x1)))
new_quot5(vuz141, vuz142, vuz143, Zero, Succ(vuz1450)) → new_quot6(vuz141, Succ(vuz143), Succ(vuz142), Succ(vuz143))
new_quot6(vuz141, Succ(Succ(vuz15300)), Succ(vuz1430), vuz152) → new_quot5(vuz141, vuz15300, Succ(vuz1430), vuz15300, vuz1430)
new_quot5(vuz141, vuz142, vuz143, Succ(vuz1440), Zero) → new_quot6(vuz141, new_primMinusNatS2(Succ(vuz142), vuz143), vuz143, new_primMinusNatS2(Succ(vuz142), vuz143))
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12200), Succ(vuz1230)) → new_primMinusNatS2(vuz12200, vuz1230)
new_primMinusNatS2(Zero, Succ(vuz1230)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12200), Zero) → Succ(vuz12200)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Zero)
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_quot7(y0, x0, Succ(x1)) → new_quot6(y0, new_primMinusNatS2(x0, x1), Succ(x1), new_primMinusNatS2(Succ(x0), Succ(x1))) at position [3] we obtained the following new rules:
new_quot7(y0, x0, Succ(x1)) → new_quot6(y0, new_primMinusNatS2(x0, x1), Succ(x1), new_primMinusNatS2(x0, x1))
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_quot5(vuz141, vuz142, vuz143, Zero, Zero) → new_quot7(vuz141, vuz142, vuz143)
new_quot5(vuz141, vuz142, vuz143, Succ(vuz1440), Succ(vuz1450)) → new_quot5(vuz141, vuz142, vuz143, vuz1440, vuz1450)
new_quot6(vuz141, Succ(Succ(vuz15300)), Succ(vuz1430), vuz152) → new_quot5(vuz141, vuz15300, Succ(vuz1430), vuz15300, vuz1430)
new_quot5(vuz141, vuz142, vuz143, Zero, Succ(vuz1450)) → new_quot6(vuz141, Succ(vuz143), Succ(vuz142), Succ(vuz143))
new_quot5(vuz141, vuz142, vuz143, Succ(vuz1440), Zero) → new_quot6(vuz141, new_primMinusNatS2(Succ(vuz142), vuz143), vuz143, new_primMinusNatS2(Succ(vuz142), vuz143))
new_quot7(y0, x0, Succ(x1)) → new_quot6(y0, new_primMinusNatS2(x0, x1), Succ(x1), new_primMinusNatS2(x0, x1))
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12200), Succ(vuz1230)) → new_primMinusNatS2(vuz12200, vuz1230)
new_primMinusNatS2(Zero, Succ(vuz1230)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12200), Zero) → Succ(vuz12200)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), 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_quot7(y0, x0, Succ(x1)) → new_quot6(y0, new_primMinusNatS2(x0, x1), Succ(x1), new_primMinusNatS2(x0, x1))
The remaining pairs can at least be oriented weakly.
new_quot5(vuz141, vuz142, vuz143, Zero, Zero) → new_quot7(vuz141, vuz142, vuz143)
new_quot5(vuz141, vuz142, vuz143, Succ(vuz1440), Succ(vuz1450)) → new_quot5(vuz141, vuz142, vuz143, vuz1440, vuz1450)
new_quot6(vuz141, Succ(Succ(vuz15300)), Succ(vuz1430), vuz152) → new_quot5(vuz141, vuz15300, Succ(vuz1430), vuz15300, vuz1430)
new_quot5(vuz141, vuz142, vuz143, Zero, Succ(vuz1450)) → new_quot6(vuz141, Succ(vuz143), Succ(vuz142), Succ(vuz143))
new_quot5(vuz141, vuz142, vuz143, Succ(vuz1440), Zero) → new_quot6(vuz141, new_primMinusNatS2(Succ(vuz142), vuz143), vuz143, new_primMinusNatS2(Succ(vuz142), vuz143))
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( new_primMinusNatS2(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
Tuple symbols:
M( new_quot6(x1, ..., x4) ) = | 0 | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
M( new_quot5(x1, ..., x5) ) = | 0 | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 | + | | · | x5 |
M( new_quot7(x1, ..., x3) ) = | 0 | + | | · | x1 | + | | · | x2 | + | | · | x3 |
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_primMinusNatS2(Succ(vuz12200), Succ(vuz1230)) → new_primMinusNatS2(vuz12200, vuz1230)
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12200), Zero) → Succ(vuz12200)
new_primMinusNatS2(Zero, Succ(vuz1230)) → Zero
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_quot5(vuz141, vuz142, vuz143, Zero, Zero) → new_quot7(vuz141, vuz142, vuz143)
new_quot5(vuz141, vuz142, vuz143, Succ(vuz1440), Succ(vuz1450)) → new_quot5(vuz141, vuz142, vuz143, vuz1440, vuz1450)
new_quot5(vuz141, vuz142, vuz143, Zero, Succ(vuz1450)) → new_quot6(vuz141, Succ(vuz143), Succ(vuz142), Succ(vuz143))
new_quot6(vuz141, Succ(Succ(vuz15300)), Succ(vuz1430), vuz152) → new_quot5(vuz141, vuz15300, Succ(vuz1430), vuz15300, vuz1430)
new_quot5(vuz141, vuz142, vuz143, Succ(vuz1440), Zero) → new_quot6(vuz141, new_primMinusNatS2(Succ(vuz142), vuz143), vuz143, new_primMinusNatS2(Succ(vuz142), vuz143))
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12200), Succ(vuz1230)) → new_primMinusNatS2(vuz12200, vuz1230)
new_primMinusNatS2(Zero, Succ(vuz1230)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12200), Zero) → Succ(vuz12200)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), 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
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_quot5(vuz141, vuz142, vuz143, Succ(vuz1440), Succ(vuz1450)) → new_quot5(vuz141, vuz142, vuz143, vuz1440, vuz1450)
new_quot6(vuz141, Succ(Succ(vuz15300)), Succ(vuz1430), vuz152) → new_quot5(vuz141, vuz15300, Succ(vuz1430), vuz15300, vuz1430)
new_quot5(vuz141, vuz142, vuz143, Zero, Succ(vuz1450)) → new_quot6(vuz141, Succ(vuz143), Succ(vuz142), Succ(vuz143))
new_quot5(vuz141, vuz142, vuz143, Succ(vuz1440), Zero) → new_quot6(vuz141, new_primMinusNatS2(Succ(vuz142), vuz143), vuz143, new_primMinusNatS2(Succ(vuz142), vuz143))
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12200), Succ(vuz1230)) → new_primMinusNatS2(vuz12200, vuz1230)
new_primMinusNatS2(Zero, Succ(vuz1230)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12200), Zero) → Succ(vuz12200)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), 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_quot5(vuz141, vuz142, vuz143, Succ(vuz1440), Zero) → new_quot6(vuz141, new_primMinusNatS2(Succ(vuz142), vuz143), vuz143, new_primMinusNatS2(Succ(vuz142), vuz143))
The remaining pairs can at least be oriented weakly.
new_quot5(vuz141, vuz142, vuz143, Succ(vuz1440), Succ(vuz1450)) → new_quot5(vuz141, vuz142, vuz143, vuz1440, vuz1450)
new_quot6(vuz141, Succ(Succ(vuz15300)), Succ(vuz1430), vuz152) → new_quot5(vuz141, vuz15300, Succ(vuz1430), vuz15300, vuz1430)
new_quot5(vuz141, vuz142, vuz143, Zero, Succ(vuz1450)) → new_quot6(vuz141, Succ(vuz143), Succ(vuz142), Succ(vuz143))
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( new_primMinusNatS2(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
Tuple symbols:
M( new_quot6(x1, ..., x4) ) = | 0 | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
M( new_quot5(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_primMinusNatS2(Succ(vuz12200), Succ(vuz1230)) → new_primMinusNatS2(vuz12200, vuz1230)
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12200), Zero) → Succ(vuz12200)
new_primMinusNatS2(Zero, Succ(vuz1230)) → Zero
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_quot5(vuz141, vuz142, vuz143, Succ(vuz1440), Succ(vuz1450)) → new_quot5(vuz141, vuz142, vuz143, vuz1440, vuz1450)
new_quot5(vuz141, vuz142, vuz143, Zero, Succ(vuz1450)) → new_quot6(vuz141, Succ(vuz143), Succ(vuz142), Succ(vuz143))
new_quot6(vuz141, Succ(Succ(vuz15300)), Succ(vuz1430), vuz152) → new_quot5(vuz141, vuz15300, Succ(vuz1430), vuz15300, vuz1430)
The TRS R consists of the following rules:
new_primMinusNatS2(Succ(vuz12200), Succ(vuz1230)) → new_primMinusNatS2(vuz12200, vuz1230)
new_primMinusNatS2(Zero, Succ(vuz1230)) → Zero
new_primMinusNatS2(Zero, Zero) → Zero
new_primMinusNatS2(Succ(vuz12200), Zero) → Succ(vuz12200)
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), 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
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_quot5(vuz141, vuz142, vuz143, Succ(vuz1440), Succ(vuz1450)) → new_quot5(vuz141, vuz142, vuz143, vuz1440, vuz1450)
new_quot5(vuz141, vuz142, vuz143, Zero, Succ(vuz1450)) → new_quot6(vuz141, Succ(vuz143), Succ(vuz142), Succ(vuz143))
new_quot6(vuz141, Succ(Succ(vuz15300)), Succ(vuz1430), vuz152) → new_quot5(vuz141, vuz15300, Succ(vuz1430), vuz15300, vuz1430)
R is empty.
The set Q consists of the following terms:
new_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), 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_primMinusNatS2(Zero, Succ(x0))
new_primMinusNatS2(Succ(x0), Succ(x1))
new_primMinusNatS2(Zero, Zero)
new_primMinusNatS2(Succ(x0), Zero)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_quot5(vuz141, vuz142, vuz143, Succ(vuz1440), Succ(vuz1450)) → new_quot5(vuz141, vuz142, vuz143, vuz1440, vuz1450)
new_quot6(vuz141, Succ(Succ(vuz15300)), Succ(vuz1430), vuz152) → new_quot5(vuz141, vuz15300, Succ(vuz1430), vuz15300, vuz1430)
new_quot5(vuz141, vuz142, vuz143, Zero, Succ(vuz1450)) → new_quot6(vuz141, Succ(vuz143), Succ(vuz142), Succ(vuz143))
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule new_quot6(vuz141, Succ(Succ(vuz15300)), Succ(vuz1430), vuz152) → new_quot5(vuz141, vuz15300, Succ(vuz1430), vuz15300, vuz1430) we obtained the following new rules:
new_quot6(z0, Succ(Succ(x1)), Succ(z1), Succ(Succ(x1))) → new_quot5(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
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ NonInfProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
new_quot6(z0, Succ(Succ(x1)), Succ(z1), Succ(Succ(x1))) → new_quot5(z0, x1, Succ(z1), x1, z1)
new_quot5(vuz141, vuz142, vuz143, Succ(vuz1440), Succ(vuz1450)) → new_quot5(vuz141, vuz142, vuz143, vuz1440, vuz1450)
new_quot5(vuz141, vuz142, vuz143, Zero, Succ(vuz1450)) → new_quot6(vuz141, Succ(vuz143), Succ(vuz142), Succ(vuz143))
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_quot5(vuz141, vuz142, vuz143, Succ(vuz1440), Succ(vuz1450)) → new_quot5(vuz141, vuz142, vuz143, vuz1440, vuz1450) the following chains were created:
- We consider the chain new_quot5(vuz141, vuz142, vuz143, Succ(vuz1440), Succ(vuz1450)) → new_quot5(vuz141, vuz142, vuz143, vuz1440, vuz1450), new_quot5(vuz141, vuz142, vuz143, Succ(vuz1440), Succ(vuz1450)) → new_quot5(vuz141, vuz142, vuz143, vuz1440, vuz1450) which results in the following constraint:
(1) (new_quot5(x0, x1, x2, x3, x4)=new_quot5(x5, x6, x7, Succ(x8), Succ(x9)) ⇒ new_quot5(x5, x6, x7, Succ(x8), Succ(x9))≥new_quot5(x5, x6, x7, x8, x9))
We simplified constraint (1) using rules (I), (II), (III) which results in the following new constraint:
(2) (new_quot5(x0, x1, x2, Succ(x8), Succ(x9))≥new_quot5(x0, x1, x2, x8, x9))
- We consider the chain new_quot6(z0, Succ(Succ(x1)), Succ(z1), Succ(Succ(x1))) → new_quot5(z0, x1, Succ(z1), x1, z1), new_quot5(vuz141, vuz142, vuz143, Succ(vuz1440), Succ(vuz1450)) → new_quot5(vuz141, vuz142, vuz143, vuz1440, vuz1450) which results in the following constraint:
(3) (new_quot5(x14, x15, Succ(x16), x15, x16)=new_quot5(x17, x18, x19, Succ(x20), Succ(x21)) ⇒ new_quot5(x17, x18, x19, Succ(x20), Succ(x21))≥new_quot5(x17, x18, x19, x20, x21))
We simplified constraint (3) using rules (I), (II), (III) which results in the following new constraint:
(4) (new_quot5(x14, Succ(x20), Succ(Succ(x21)), Succ(x20), Succ(x21))≥new_quot5(x14, Succ(x20), Succ(Succ(x21)), x20, x21))
For Pair new_quot5(vuz141, vuz142, vuz143, Zero, Succ(vuz1450)) → new_quot6(vuz141, Succ(vuz143), Succ(vuz142), Succ(vuz143)) the following chains were created:
- We consider the chain new_quot5(vuz141, vuz142, vuz143, Succ(vuz1440), Succ(vuz1450)) → new_quot5(vuz141, vuz142, vuz143, vuz1440, vuz1450), new_quot5(vuz141, vuz142, vuz143, Zero, Succ(vuz1450)) → new_quot6(vuz141, Succ(vuz143), Succ(vuz142), Succ(vuz143)) which results in the following constraint:
(5) (new_quot5(x22, x23, x24, x25, x26)=new_quot5(x27, x28, x29, Zero, Succ(x30)) ⇒ new_quot5(x27, x28, x29, Zero, Succ(x30))≥new_quot6(x27, Succ(x29), Succ(x28), Succ(x29)))
We simplified constraint (5) using rules (I), (II), (III) which results in the following new constraint:
(6) (new_quot5(x22, x23, x24, Zero, Succ(x30))≥new_quot6(x22, Succ(x24), Succ(x23), Succ(x24)))
- We consider the chain new_quot6(z0, Succ(Succ(x1)), Succ(z1), Succ(Succ(x1))) → new_quot5(z0, x1, Succ(z1), x1, z1), new_quot5(vuz141, vuz142, vuz143, Zero, Succ(vuz1450)) → new_quot6(vuz141, Succ(vuz143), Succ(vuz142), Succ(vuz143)) which results in the following constraint:
(7) (new_quot5(x35, x36, Succ(x37), x36, x37)=new_quot5(x38, x39, x40, Zero, Succ(x41)) ⇒ new_quot5(x38, x39, x40, Zero, Succ(x41))≥new_quot6(x38, Succ(x40), Succ(x39), Succ(x40)))
We simplified constraint (7) using rules (I), (II), (III) which results in the following new constraint:
(8) (new_quot5(x35, Zero, Succ(Succ(x41)), Zero, Succ(x41))≥new_quot6(x35, Succ(Succ(Succ(x41))), Succ(Zero), Succ(Succ(Succ(x41)))))
For Pair new_quot6(z0, Succ(Succ(x1)), Succ(z1), Succ(Succ(x1))) → new_quot5(z0, x1, Succ(z1), x1, z1) the following chains were created:
- We consider the chain new_quot5(vuz141, vuz142, vuz143, Zero, Succ(vuz1450)) → new_quot6(vuz141, Succ(vuz143), Succ(vuz142), Succ(vuz143)), new_quot6(z0, Succ(Succ(x1)), Succ(z1), Succ(Succ(x1))) → new_quot5(z0, x1, Succ(z1), x1, z1) which results in the following constraint:
(9) (new_quot6(x47, Succ(x49), Succ(x48), Succ(x49))=new_quot6(x51, Succ(Succ(x52)), Succ(x53), Succ(Succ(x52))) ⇒ new_quot6(x51, Succ(Succ(x52)), Succ(x53), Succ(Succ(x52)))≥new_quot5(x51, x52, Succ(x53), x52, x53))
We simplified constraint (9) using rules (I), (II), (III) which results in the following new constraint:
(10) (new_quot6(x47, Succ(Succ(x52)), Succ(x48), Succ(Succ(x52)))≥new_quot5(x47, x52, Succ(x48), x52, x48))
To summarize, we get the following constraints P≥ for the following pairs.
- new_quot5(vuz141, vuz142, vuz143, Succ(vuz1440), Succ(vuz1450)) → new_quot5(vuz141, vuz142, vuz143, vuz1440, vuz1450)
- (new_quot5(x0, x1, x2, Succ(x8), Succ(x9))≥new_quot5(x0, x1, x2, x8, x9))
- (new_quot5(x14, Succ(x20), Succ(Succ(x21)), Succ(x20), Succ(x21))≥new_quot5(x14, Succ(x20), Succ(Succ(x21)), x20, x21))
- new_quot5(vuz141, vuz142, vuz143, Zero, Succ(vuz1450)) → new_quot6(vuz141, Succ(vuz143), Succ(vuz142), Succ(vuz143))
- (new_quot5(x22, x23, x24, Zero, Succ(x30))≥new_quot6(x22, Succ(x24), Succ(x23), Succ(x24)))
- (new_quot5(x35, Zero, Succ(Succ(x41)), Zero, Succ(x41))≥new_quot6(x35, Succ(Succ(Succ(x41))), Succ(Zero), Succ(Succ(Succ(x41)))))
- new_quot6(z0, Succ(Succ(x1)), Succ(z1), Succ(Succ(x1))) → new_quot5(z0, x1, Succ(z1), x1, z1)
- (new_quot6(x47, Succ(Succ(x52)), Succ(x48), Succ(Succ(x52)))≥new_quot5(x47, x52, Succ(x48), x52, x48))
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_quot5(x1, x2, x3, x4, x5)) = 1 + x1 + x2 - x4 + x5
POL(new_quot6(x1, x2, x3, x4)) = 1 + x1 + x2 + x3 - x4
The following pairs are in P>:
new_quot6(z0, Succ(Succ(x1)), Succ(z1), Succ(Succ(x1))) → new_quot5(z0, x1, Succ(z1), x1, z1)
The following pairs are in Pbound:
new_quot5(vuz141, vuz142, vuz143, Zero, Succ(vuz1450)) → new_quot6(vuz141, Succ(vuz143), Succ(vuz142), Succ(vuz143))
new_quot6(z0, Succ(Succ(x1)), Succ(z1), Succ(Succ(x1))) → new_quot5(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
↳ DependencyGraphProof
↳ 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
Q DP problem:
The TRS P consists of the following rules:
new_quot5(vuz141, vuz142, vuz143, Succ(vuz1440), Succ(vuz1450)) → new_quot5(vuz141, vuz142, vuz143, vuz1440, vuz1450)
new_quot5(vuz141, vuz142, vuz143, Zero, Succ(vuz1450)) → new_quot6(vuz141, Succ(vuz143), Succ(vuz142), Succ(vuz143))
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
↳ DependencyGraphProof
↳ 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
Q DP problem:
The TRS P consists of the following rules:
new_quot5(vuz141, vuz142, vuz143, Succ(vuz1440), Succ(vuz1450)) → new_quot5(vuz141, vuz142, vuz143, vuz1440, vuz1450)
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_quot5(vuz141, vuz142, vuz143, Succ(vuz1440), Succ(vuz1450)) → new_quot5(vuz141, vuz142, vuz143, vuz1440, vuz1450)
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
↳ QDPSizeChangeProof
Q DP problem:
The TRS P consists of the following rules:
new_quot9(Zero, Succ(vuz5000)) → new_quot9(Zero, 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_quot9(Zero, Succ(vuz5000)) → new_quot9(Zero, vuz5000)
The graph contains the following edges 1 >= 1, 2 > 2