MAYBE 28.918
H-Termination proof of /home/matraf/haskell/eval_FullyBlown_Fast/empty.hs
H-Termination of the given Haskell-Program with start terms could not be shown:
↳ HASKELL
↳ IFR
mainModule Main
| ((enumFrom :: Enum a => a -> [a]) :: Enum a => a -> [a]) |
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
| ((enumFrom :: Enum a => a -> [a]) :: Enum a => a -> [a]) |
module Main where
Replaced joker patterns by fresh variables and removed binding patterns.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
mainModule Main
| ((enumFrom :: Enum a => a -> [a]) :: Enum a => a -> [a]) |
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
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
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
takeWhile | p [] | = [] |
takeWhile | p (x : xs) | |
is transformed to
takeWhile | p [] | = takeWhile3 p [] |
takeWhile | p (x : xs) | = takeWhile2 p (x : xs) |
takeWhile1 | p x xs True | = x : takeWhile p xs |
takeWhile1 | p x xs False | = takeWhile0 p x xs otherwise |
takeWhile0 | p x xs True | = [] |
takeWhile2 | p (x : xs) | = takeWhile1 p x xs (p x) |
takeWhile3 | p [] | = [] |
takeWhile3 | vuz vvu | = takeWhile2 vuz vvu |
The following Function with conditions
toEnum | 0 | = False |
toEnum | 1 | = True |
is transformed to
toEnum | vvw | = toEnum3 vvw |
toEnum | vvv | = toEnum1 vvv |
toEnum1 | vvv | = toEnum0 (vvv == 1) vvv |
toEnum2 | True vvw | = False |
toEnum2 | vvx vvy | = toEnum1 vvy |
toEnum3 | vvw | = toEnum2 (vvw == 0) vvw |
toEnum3 | vvz | = toEnum1 vvz |
The following Function with conditions
toEnum | 0 | = LT |
toEnum | 1 | = EQ |
toEnum | 2 | = GT |
is transformed to
toEnum | vwz | = toEnum9 vwz |
toEnum | vwv | = toEnum7 vwv |
toEnum | vwu | = toEnum5 vwu |
toEnum5 | vwu | = toEnum4 (vwu == 2) vwu |
toEnum6 | True vwv | = EQ |
toEnum6 | vww vwx | = toEnum5 vwx |
toEnum7 | vwv | = toEnum6 (vwv == 1) vwv |
toEnum7 | vwy | = toEnum5 vwy |
toEnum8 | True vwz | = LT |
toEnum8 | vxu vxv | = toEnum7 vxv |
toEnum9 | vwz | = toEnum8 (vwz == 0) vwz |
toEnum9 | vxw | = toEnum7 vxw |
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
mainModule Main
| ((enumFrom :: Enum a => a -> [a]) :: Enum a => a -> [a]) |
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 | vxx vxy | = gcd vxx vxy |
reduce2Reduce0 | vxx vxy x y True | = x `quot` reduce2D vxx vxy :% (y `quot` reduce2D vxx vxy) |
reduce2Reduce1 | vxx vxy x y True | = error [] |
reduce2Reduce1 | vxx vxy x y False | = reduce2Reduce0 vxx vxy x y otherwise |
The bindings of the following Let/Where expression
gcd' (abs x) (abs y) |
where |
gcd' | x xz | = gcd'2 x xz |
gcd' | x y | = gcd'0 x y |
|
|
gcd'0 | x y | = gcd' y (x `rem` y) |
|
|
gcd'1 | True x xz | = x |
gcd'1 | yu yv yw | = gcd'0 yv yw |
|
|
gcd'2 | x xz | = gcd'1 (xz == 0) x xz |
gcd'2 | yx yy | = gcd'0 yx yy |
|
are unpacked to the following functions on top level
gcd0Gcd' | x xz | = gcd0Gcd'2 x xz |
gcd0Gcd' | x y | = gcd0Gcd'0 x y |
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) |
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
mainModule Main
| ((enumFrom :: Enum a => a -> [a]) :: Enum a => a -> [a]) |
module Main where
Num Reduction: All numbers are transformed to thier corresponding representation with Pos, Neg, Succ and Zero.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ Narrow
mainModule Main
| (enumFrom :: Enum a => a -> [a]) |
module Main where
Haskell To QDPs
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map0(vxz129, vxz130, :(Neg(Succ(vxz131000)), vxz1311), ba) → new_map(vxz129, vxz131000, vxz1311, Succ(vxz129), Succ(vxz131000), ba)
new_map(vxz129, vxz130, vxz131, Succ(vxz1320), Succ(vxz1330), ba) → new_map(vxz129, vxz130, vxz131, vxz1320, vxz1330, ba)
new_map(vxz129, vxz130, :(Neg(Succ(vxz131000)), vxz1311), Zero, Succ(vxz1330), ba) → new_map(vxz129, vxz131000, vxz1311, Succ(vxz129), Succ(vxz131000), ba)
new_map(vxz129, vxz130, vxz131, Zero, Zero, ba) → new_map0(vxz129, vxz130, vxz131, ba)
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_map(vxz129, vxz130, vxz131, Succ(vxz1320), Succ(vxz1330), ba) → new_map(vxz129, vxz130, vxz131, vxz1320, vxz1330, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4, 5 > 5, 6 >= 6
- new_map(vxz129, vxz130, vxz131, Zero, Zero, ba) → new_map0(vxz129, vxz130, vxz131, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 6 >= 4
- new_map(vxz129, vxz130, :(Neg(Succ(vxz131000)), vxz1311), Zero, Succ(vxz1330), ba) → new_map(vxz129, vxz131000, vxz1311, Succ(vxz129), Succ(vxz131000), ba)
The graph contains the following edges 1 >= 1, 3 > 2, 3 > 3, 3 > 5, 6 >= 6
- new_map0(vxz129, vxz130, :(Neg(Succ(vxz131000)), vxz1311), ba) → new_map(vxz129, vxz131000, vxz1311, Succ(vxz129), Succ(vxz131000), ba)
The graph contains the following edges 1 >= 1, 3 > 2, 3 > 3, 3 > 5, 4 >= 6
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map2(vxz123, :(Neg(Zero), vxz1251), ba) → new_map2(vxz123, vxz1251, ba)
new_map3(vxz123, vxz124, :(Pos(Zero), vxz1251), ba) → new_map2(vxz123, vxz1251, ba)
new_map2(vxz123, :(Pos(Zero), vxz1251), ba) → new_map2(vxz123, vxz1251, ba)
new_map1(vxz123, vxz124, vxz125, Succ(vxz1260), Succ(vxz1270), ba) → new_map1(vxz123, vxz124, vxz125, vxz1260, vxz1270, ba)
new_map1(vxz123, vxz124, vxz125, Zero, Zero, ba) → new_map3(vxz123, vxz124, vxz125, ba)
new_map1(vxz123, vxz124, :(Pos(Succ(vxz125000)), vxz1251), Zero, Succ(vxz1270), ba) → new_map1(vxz123, vxz125000, vxz1251, Succ(vxz125000), Succ(vxz123), ba)
new_map1(vxz123, vxz124, :(Neg(Succ(vxz125000)), vxz1251), Zero, Succ(vxz1270), ba) → new_map2(vxz123, vxz1251, ba)
new_map1(vxz123, vxz124, :(Pos(Zero), vxz1251), Zero, Succ(vxz1270), ba) → new_map2(vxz123, vxz1251, ba)
new_map2(vxz123, :(Pos(Succ(vxz125000)), vxz1251), ba) → new_map1(vxz123, vxz125000, vxz1251, Succ(vxz125000), Succ(vxz123), ba)
new_map3(vxz123, vxz124, :(Neg(Succ(vxz125000)), vxz1251), ba) → new_map2(vxz123, vxz1251, ba)
new_map2(vxz123, :(Neg(Succ(vxz125000)), vxz1251), ba) → new_map2(vxz123, vxz1251, ba)
new_map3(vxz123, vxz124, :(Pos(Succ(vxz125000)), vxz1251), ba) → new_map1(vxz123, vxz125000, vxz1251, Succ(vxz125000), Succ(vxz123), ba)
new_map1(vxz123, vxz124, :(Neg(Zero), vxz1251), Zero, Succ(vxz1270), ba) → new_map2(vxz123, vxz1251, ba)
new_map3(vxz123, vxz124, :(Neg(Zero), vxz1251), ba) → new_map2(vxz123, vxz1251, ba)
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_map2(vxz123, :(Pos(Succ(vxz125000)), vxz1251), ba) → new_map1(vxz123, vxz125000, vxz1251, Succ(vxz125000), Succ(vxz123), ba)
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 2 > 4, 3 >= 6
- new_map1(vxz123, vxz124, vxz125, Zero, Zero, ba) → new_map3(vxz123, vxz124, vxz125, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 6 >= 4
- new_map3(vxz123, vxz124, :(Pos(Succ(vxz125000)), vxz1251), ba) → new_map1(vxz123, vxz125000, vxz1251, Succ(vxz125000), Succ(vxz123), ba)
The graph contains the following edges 1 >= 1, 3 > 2, 3 > 3, 3 > 4, 4 >= 6
- new_map1(vxz123, vxz124, vxz125, Succ(vxz1260), Succ(vxz1270), ba) → new_map1(vxz123, vxz124, vxz125, vxz1260, vxz1270, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4, 5 > 5, 6 >= 6
- new_map1(vxz123, vxz124, :(Pos(Succ(vxz125000)), vxz1251), Zero, Succ(vxz1270), ba) → new_map1(vxz123, vxz125000, vxz1251, Succ(vxz125000), Succ(vxz123), ba)
The graph contains the following edges 1 >= 1, 3 > 2, 3 > 3, 3 > 4, 6 >= 6
- new_map2(vxz123, :(Neg(Zero), vxz1251), ba) → new_map2(vxz123, vxz1251, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
- new_map2(vxz123, :(Pos(Zero), vxz1251), ba) → new_map2(vxz123, vxz1251, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
- new_map2(vxz123, :(Neg(Succ(vxz125000)), vxz1251), ba) → new_map2(vxz123, vxz1251, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
- new_map1(vxz123, vxz124, :(Neg(Succ(vxz125000)), vxz1251), Zero, Succ(vxz1270), ba) → new_map2(vxz123, vxz1251, ba)
The graph contains the following edges 1 >= 1, 3 > 2, 6 >= 3
- new_map1(vxz123, vxz124, :(Pos(Zero), vxz1251), Zero, Succ(vxz1270), ba) → new_map2(vxz123, vxz1251, ba)
The graph contains the following edges 1 >= 1, 3 > 2, 6 >= 3
- new_map1(vxz123, vxz124, :(Neg(Zero), vxz1251), Zero, Succ(vxz1270), ba) → new_map2(vxz123, vxz1251, ba)
The graph contains the following edges 1 >= 1, 3 > 2, 6 >= 3
- new_map3(vxz123, vxz124, :(Pos(Zero), vxz1251), ba) → new_map2(vxz123, vxz1251, ba)
The graph contains the following edges 1 >= 1, 3 > 2, 4 >= 3
- new_map3(vxz123, vxz124, :(Neg(Succ(vxz125000)), vxz1251), ba) → new_map2(vxz123, vxz1251, ba)
The graph contains the following edges 1 >= 1, 3 > 2, 4 >= 3
- new_map3(vxz123, vxz124, :(Neg(Zero), vxz1251), ba) → new_map2(vxz123, vxz1251, ba)
The graph contains the following edges 1 >= 1, 3 > 2, 4 >= 3
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map5(Pos(Zero), Pos(Zero), vxz41) → new_map4(Zero, vxz41)
new_map5(Neg(Zero), Pos(Zero), :(vxz410, vxz411)) → new_map5(Neg(Zero), vxz410, vxz411)
new_map5(Pos(Succ(vxz1500)), Pos(Zero), vxz41) → new_map4(Succ(vxz1500), vxz41)
new_map5(Pos(vxz150), Neg(Succ(vxz4000)), :(vxz410, vxz411)) → new_map5(Pos(vxz150), vxz410, vxz411)
new_map5(Neg(Zero), Neg(Succ(vxz4000)), vxz41) → new_map6(vxz41)
new_map6(:(vxz410, vxz411)) → new_map5(Neg(Zero), vxz410, vxz411)
new_map5(Neg(Zero), Neg(Zero), vxz41) → new_map6(vxz41)
new_map4(vxz150, :(vxz410, vxz411)) → new_map5(Pos(vxz150), vxz410, vxz411)
new_map5(Pos(Succ(vxz1500)), Neg(Zero), vxz41) → new_map4(Succ(vxz1500), vxz41)
new_map5(Pos(Zero), Neg(Zero), vxz41) → new_map4(Zero, vxz41)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map5(Neg(Zero), Pos(Zero), :(vxz410, vxz411)) → new_map5(Neg(Zero), vxz410, vxz411)
new_map5(Neg(Zero), Neg(Succ(vxz4000)), vxz41) → new_map6(vxz41)
new_map6(:(vxz410, vxz411)) → new_map5(Neg(Zero), vxz410, vxz411)
new_map5(Neg(Zero), Neg(Zero), vxz41) → new_map6(vxz41)
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_map6(:(vxz410, vxz411)) → new_map5(Neg(Zero), vxz410, vxz411)
The graph contains the following edges 1 > 2, 1 > 3
- new_map5(Neg(Zero), Pos(Zero), :(vxz410, vxz411)) → new_map5(Neg(Zero), vxz410, vxz411)
The graph contains the following edges 1 >= 1, 3 > 2, 3 > 3
- new_map5(Neg(Zero), Neg(Succ(vxz4000)), vxz41) → new_map6(vxz41)
The graph contains the following edges 3 >= 1
- new_map5(Neg(Zero), Neg(Zero), vxz41) → new_map6(vxz41)
The graph contains the following edges 3 >= 1
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map5(Pos(Zero), Pos(Zero), vxz41) → new_map4(Zero, vxz41)
new_map5(Pos(Succ(vxz1500)), Pos(Zero), vxz41) → new_map4(Succ(vxz1500), vxz41)
new_map5(Pos(vxz150), Neg(Succ(vxz4000)), :(vxz410, vxz411)) → new_map5(Pos(vxz150), vxz410, vxz411)
new_map4(vxz150, :(vxz410, vxz411)) → new_map5(Pos(vxz150), vxz410, vxz411)
new_map5(Pos(Succ(vxz1500)), Neg(Zero), vxz41) → new_map4(Succ(vxz1500), vxz41)
new_map5(Pos(Zero), Neg(Zero), vxz41) → new_map4(Zero, vxz41)
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_map5(Pos(vxz150), Neg(Succ(vxz4000)), :(vxz410, vxz411)) → new_map5(Pos(vxz150), vxz410, vxz411)
The graph contains the following edges 1 >= 1, 3 > 2, 3 > 3
- new_map4(vxz150, :(vxz410, vxz411)) → new_map5(Pos(vxz150), vxz410, vxz411)
The graph contains the following edges 2 > 2, 2 > 3
- new_map5(Pos(Zero), Pos(Zero), vxz41) → new_map4(Zero, vxz41)
The graph contains the following edges 1 > 1, 2 > 1, 3 >= 2
- new_map5(Pos(Succ(vxz1500)), Pos(Zero), vxz41) → new_map4(Succ(vxz1500), vxz41)
The graph contains the following edges 1 > 1, 3 >= 2
- new_map5(Pos(Succ(vxz1500)), Neg(Zero), vxz41) → new_map4(Succ(vxz1500), vxz41)
The graph contains the following edges 1 > 1, 3 >= 2
- new_map5(Pos(Zero), Neg(Zero), vxz41) → new_map4(Zero, vxz41)
The graph contains the following edges 1 > 1, 2 > 1, 3 >= 2
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map7(vxz210, :(vxz910, vxz911)) → new_map8(Pos(vxz210), vxz910, vxz911)
new_map8(Neg(Zero), Neg(Zero), vxz91) → new_map9(vxz91)
new_map8(Pos(Succ(vxz2100)), Pos(Zero), vxz91) → new_map7(Succ(vxz2100), vxz91)
new_map8(Neg(Zero), Neg(Succ(vxz9000)), vxz91) → new_map9(vxz91)
new_map8(Pos(vxz210), Neg(Succ(vxz9000)), :(vxz910, vxz911)) → new_map8(Pos(vxz210), vxz910, vxz911)
new_map8(Neg(Zero), Pos(Zero), :(vxz910, vxz911)) → new_map8(Neg(Zero), vxz910, vxz911)
new_map8(Pos(Succ(vxz2100)), Neg(Zero), vxz91) → new_map7(Succ(vxz2100), vxz91)
new_map8(Pos(Zero), Neg(Zero), vxz91) → new_map7(Zero, vxz91)
new_map9(:(vxz910, vxz911)) → new_map8(Neg(Zero), vxz910, vxz911)
new_map8(Pos(Zero), Pos(Zero), vxz91) → new_map7(Zero, vxz91)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map8(Neg(Zero), Neg(Zero), vxz91) → new_map9(vxz91)
new_map8(Neg(Zero), Neg(Succ(vxz9000)), vxz91) → new_map9(vxz91)
new_map8(Neg(Zero), Pos(Zero), :(vxz910, vxz911)) → new_map8(Neg(Zero), vxz910, vxz911)
new_map9(:(vxz910, vxz911)) → new_map8(Neg(Zero), vxz910, vxz911)
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_map8(Neg(Zero), Pos(Zero), :(vxz910, vxz911)) → new_map8(Neg(Zero), vxz910, vxz911)
The graph contains the following edges 1 >= 1, 3 > 2, 3 > 3
- new_map9(:(vxz910, vxz911)) → new_map8(Neg(Zero), vxz910, vxz911)
The graph contains the following edges 1 > 2, 1 > 3
- new_map8(Neg(Zero), Neg(Zero), vxz91) → new_map9(vxz91)
The graph contains the following edges 3 >= 1
- new_map8(Neg(Zero), Neg(Succ(vxz9000)), vxz91) → new_map9(vxz91)
The graph contains the following edges 3 >= 1
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map7(vxz210, :(vxz910, vxz911)) → new_map8(Pos(vxz210), vxz910, vxz911)
new_map8(Pos(Succ(vxz2100)), Pos(Zero), vxz91) → new_map7(Succ(vxz2100), vxz91)
new_map8(Pos(vxz210), Neg(Succ(vxz9000)), :(vxz910, vxz911)) → new_map8(Pos(vxz210), vxz910, vxz911)
new_map8(Pos(Succ(vxz2100)), Neg(Zero), vxz91) → new_map7(Succ(vxz2100), vxz91)
new_map8(Pos(Zero), Neg(Zero), vxz91) → new_map7(Zero, vxz91)
new_map8(Pos(Zero), Pos(Zero), vxz91) → new_map7(Zero, vxz91)
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_map7(vxz210, :(vxz910, vxz911)) → new_map8(Pos(vxz210), vxz910, vxz911)
The graph contains the following edges 2 > 2, 2 > 3
- new_map8(Pos(vxz210), Neg(Succ(vxz9000)), :(vxz910, vxz911)) → new_map8(Pos(vxz210), vxz910, vxz911)
The graph contains the following edges 1 >= 1, 3 > 2, 3 > 3
- new_map8(Pos(Succ(vxz2100)), Pos(Zero), vxz91) → new_map7(Succ(vxz2100), vxz91)
The graph contains the following edges 1 > 1, 3 >= 2
- new_map8(Pos(Succ(vxz2100)), Neg(Zero), vxz91) → new_map7(Succ(vxz2100), vxz91)
The graph contains the following edges 1 > 1, 3 >= 2
- new_map8(Pos(Zero), Neg(Zero), vxz91) → new_map7(Zero, vxz91)
The graph contains the following edges 1 > 1, 2 > 1, 3 >= 2
- new_map8(Pos(Zero), Pos(Zero), vxz91) → new_map7(Zero, vxz91)
The graph contains the following edges 1 > 1, 2 > 1, 3 >= 2
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map11(Pos(Succ(vxz1900)), Pos(Zero), vxz81) → new_map10(Succ(vxz1900), vxz81)
new_map11(Pos(Succ(vxz1900)), Neg(Zero), vxz81) → new_map10(Succ(vxz1900), vxz81)
new_map11(Neg(Zero), Neg(Succ(vxz8000)), vxz81) → new_map12(vxz81)
new_map11(Neg(Zero), Pos(Zero), :(vxz810, vxz811)) → new_map11(Neg(Zero), vxz810, vxz811)
new_map11(Pos(Zero), Pos(Zero), vxz81) → new_map10(Zero, vxz81)
new_map12(:(vxz810, vxz811)) → new_map11(Neg(Zero), vxz810, vxz811)
new_map11(Neg(Zero), Neg(Zero), vxz81) → new_map12(vxz81)
new_map10(vxz190, :(vxz810, vxz811)) → new_map11(Pos(vxz190), vxz810, vxz811)
new_map11(Pos(Zero), Neg(Zero), vxz81) → new_map10(Zero, vxz81)
new_map11(Pos(vxz190), Neg(Succ(vxz8000)), :(vxz810, vxz811)) → new_map11(Pos(vxz190), vxz810, vxz811)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map11(Neg(Zero), Neg(Succ(vxz8000)), vxz81) → new_map12(vxz81)
new_map11(Neg(Zero), Pos(Zero), :(vxz810, vxz811)) → new_map11(Neg(Zero), vxz810, vxz811)
new_map12(:(vxz810, vxz811)) → new_map11(Neg(Zero), vxz810, vxz811)
new_map11(Neg(Zero), Neg(Zero), vxz81) → new_map12(vxz81)
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_map11(Neg(Zero), Pos(Zero), :(vxz810, vxz811)) → new_map11(Neg(Zero), vxz810, vxz811)
The graph contains the following edges 1 >= 1, 3 > 2, 3 > 3
- new_map12(:(vxz810, vxz811)) → new_map11(Neg(Zero), vxz810, vxz811)
The graph contains the following edges 1 > 2, 1 > 3
- new_map11(Neg(Zero), Neg(Succ(vxz8000)), vxz81) → new_map12(vxz81)
The graph contains the following edges 3 >= 1
- new_map11(Neg(Zero), Neg(Zero), vxz81) → new_map12(vxz81)
The graph contains the following edges 3 >= 1
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map11(Pos(Succ(vxz1900)), Pos(Zero), vxz81) → new_map10(Succ(vxz1900), vxz81)
new_map11(Pos(Succ(vxz1900)), Neg(Zero), vxz81) → new_map10(Succ(vxz1900), vxz81)
new_map11(Pos(Zero), Pos(Zero), vxz81) → new_map10(Zero, vxz81)
new_map11(Pos(Zero), Neg(Zero), vxz81) → new_map10(Zero, vxz81)
new_map10(vxz190, :(vxz810, vxz811)) → new_map11(Pos(vxz190), vxz810, vxz811)
new_map11(Pos(vxz190), Neg(Succ(vxz8000)), :(vxz810, vxz811)) → new_map11(Pos(vxz190), vxz810, vxz811)
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_map11(Pos(vxz190), Neg(Succ(vxz8000)), :(vxz810, vxz811)) → new_map11(Pos(vxz190), vxz810, vxz811)
The graph contains the following edges 1 >= 1, 3 > 2, 3 > 3
- new_map10(vxz190, :(vxz810, vxz811)) → new_map11(Pos(vxz190), vxz810, vxz811)
The graph contains the following edges 2 > 2, 2 > 3
- new_map11(Pos(Succ(vxz1900)), Pos(Zero), vxz81) → new_map10(Succ(vxz1900), vxz81)
The graph contains the following edges 1 > 1, 3 >= 2
- new_map11(Pos(Succ(vxz1900)), Neg(Zero), vxz81) → new_map10(Succ(vxz1900), vxz81)
The graph contains the following edges 1 > 1, 3 >= 2
- new_map11(Pos(Zero), Pos(Zero), vxz81) → new_map10(Zero, vxz81)
The graph contains the following edges 1 > 1, 2 > 1, 3 >= 2
- new_map11(Pos(Zero), Neg(Zero), vxz81) → new_map10(Zero, vxz81)
The graph contains the following edges 1 > 1, 2 > 1, 3 >= 2
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primMinusNat(Succ(vxz31000), Succ(vxz28000)) → new_primMinusNat(vxz31000, vxz28000)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- new_primMinusNat(Succ(vxz31000), Succ(vxz28000)) → new_primMinusNat(vxz31000, vxz28000)
The graph contains the following edges 1 > 1, 2 > 2
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_numericEnumFrom(vxz3) → new_numericEnumFrom(new_ps(vxz3))
The TRS R consists of the following rules:
new_primMinusNat0(Succ(vxz300)) → new_primMinusNat1(vxz300)
new_primPlusNat0(Succ(vxz300)) → Succ(Succ(new_primPlusNat(vxz300)))
new_primPlusNat(Succ(vxz3000)) → Succ(vxz3000)
new_primMinusNat0(Zero) → Pos(Succ(Zero))
new_primMinusNat1(Zero) → Pos(Zero)
new_primPlusNat(Zero) → Zero
new_primPlusNat0(Zero) → Succ(Zero)
new_primMinusNat1(Succ(vxz3000)) → Neg(Succ(vxz3000))
new_ps(Pos(vxz30)) → Pos(new_primPlusNat0(vxz30))
new_ps(Neg(vxz30)) → new_primMinusNat0(vxz30)
The set Q consists of the following terms:
new_primMinusNat1(Zero)
new_primPlusNat(Zero)
new_ps(Pos(x0))
new_primPlusNat0(Zero)
new_primPlusNat0(Succ(x0))
new_primMinusNat1(Succ(x0))
new_primMinusNat0(Zero)
new_primMinusNat0(Succ(x0))
new_ps(Neg(x0))
new_primPlusNat(Succ(x0))
We have to consider all minimal (P,Q,R)-chains.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented rules of the TRS R:
new_primMinusNat0(Zero) → Pos(Succ(Zero))
new_primMinusNat1(Zero) → Pos(Zero)
Used ordering: POLO with Polynomial interpretation [25]:
POL(Neg(x1)) = 2 + 2·x1
POL(Pos(x1)) = 2·x1
POL(Succ(x1)) = x1
POL(Zero) = 0
POL(new_numericEnumFrom(x1)) = x1
POL(new_primMinusNat0(x1)) = 2 + 2·x1
POL(new_primMinusNat1(x1)) = 2 + 2·x1
POL(new_primPlusNat(x1)) = x1
POL(new_primPlusNat0(x1)) = x1
POL(new_ps(x1)) = x1
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ NonTerminationProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_numericEnumFrom(vxz3) → new_numericEnumFrom(new_ps(vxz3))
The TRS R consists of the following rules:
new_primMinusNat0(Succ(vxz300)) → new_primMinusNat1(vxz300)
new_primPlusNat0(Succ(vxz300)) → Succ(Succ(new_primPlusNat(vxz300)))
new_primPlusNat(Succ(vxz3000)) → Succ(vxz3000)
new_primPlusNat(Zero) → Zero
new_primPlusNat0(Zero) → Succ(Zero)
new_primMinusNat1(Succ(vxz3000)) → Neg(Succ(vxz3000))
new_ps(Pos(vxz30)) → Pos(new_primPlusNat0(vxz30))
new_ps(Neg(vxz30)) → new_primMinusNat0(vxz30)
The set Q consists of the following terms:
new_primMinusNat1(Zero)
new_primPlusNat(Zero)
new_ps(Pos(x0))
new_primPlusNat0(Zero)
new_primPlusNat0(Succ(x0))
new_primMinusNat1(Succ(x0))
new_primMinusNat0(Zero)
new_primMinusNat0(Succ(x0))
new_ps(Neg(x0))
new_primPlusNat(Succ(x0))
We have to consider all minimal (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.
The TRS P consists of the following rules:
new_numericEnumFrom(vxz3) → new_numericEnumFrom(new_ps(vxz3))
The TRS R consists of the following rules:
new_primMinusNat0(Succ(vxz300)) → new_primMinusNat1(vxz300)
new_primPlusNat0(Succ(vxz300)) → Succ(Succ(new_primPlusNat(vxz300)))
new_primPlusNat(Succ(vxz3000)) → Succ(vxz3000)
new_primPlusNat(Zero) → Zero
new_primPlusNat0(Zero) → Succ(Zero)
new_primMinusNat1(Succ(vxz3000)) → Neg(Succ(vxz3000))
new_ps(Pos(vxz30)) → Pos(new_primPlusNat0(vxz30))
new_ps(Neg(vxz30)) → new_primMinusNat0(vxz30)
s = new_numericEnumFrom(vxz3) evaluates to t =new_numericEnumFrom(new_ps(vxz3))
Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
- Matcher: [vxz3 / new_ps(vxz3)]
- Semiunifier: [ ]
Rewriting sequence
The DP semiunifies directly so there is only one rewrite step from new_numericEnumFrom(vxz3) to new_numericEnumFrom(new_ps(vxz3)).
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_numericEnumFrom0(vxz3) → new_numericEnumFrom0(new_ps0(vxz3))
The TRS R consists of the following rules:
new_primMinusNat0(Succ(vxz300)) → new_primMinusNat1(vxz300)
new_primPlusInt(Neg(vxz300)) → new_primMinusNat0(vxz300)
new_primPlusNat0(Succ(vxz300)) → Succ(Succ(new_primPlusNat(vxz300)))
new_primPlusInt(Pos(vxz300)) → Pos(new_primPlusNat0(vxz300))
new_primPlusNat(Succ(vxz3000)) → Succ(vxz3000)
new_ps0(Integer(vxz30)) → Integer(new_primPlusInt(vxz30))
new_primMinusNat0(Zero) → Pos(Succ(Zero))
new_primMinusNat1(Zero) → Pos(Zero)
new_primPlusNat(Zero) → Zero
new_primPlusNat0(Zero) → Succ(Zero)
new_primMinusNat1(Succ(vxz3000)) → Neg(Succ(vxz3000))
The set Q consists of the following terms:
new_primMinusNat1(Zero)
new_primPlusInt(Pos(x0))
new_primPlusNat(Zero)
new_primPlusInt(Neg(x0))
new_primPlusNat0(Zero)
new_primPlusNat0(Succ(x0))
new_primMinusNat1(Succ(x0))
new_primMinusNat0(Zero)
new_ps0(Integer(x0))
new_primMinusNat0(Succ(x0))
new_primPlusNat(Succ(x0))
We have to consider all minimal (P,Q,R)-chains.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented rules of the TRS R:
new_primMinusNat0(Zero) → Pos(Succ(Zero))
new_primMinusNat1(Zero) → Pos(Zero)
Used ordering: POLO with Polynomial interpretation [25]:
POL(Integer(x1)) = x1
POL(Neg(x1)) = 2·x1
POL(Pos(x1)) = x1
POL(Succ(x1)) = x1
POL(Zero) = 1
POL(new_numericEnumFrom0(x1)) = 2·x1
POL(new_primMinusNat0(x1)) = 2·x1
POL(new_primMinusNat1(x1)) = 2·x1
POL(new_primPlusInt(x1)) = x1
POL(new_primPlusNat(x1)) = x1
POL(new_primPlusNat0(x1)) = x1
POL(new_ps0(x1)) = x1
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ NonTerminationProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_numericEnumFrom0(vxz3) → new_numericEnumFrom0(new_ps0(vxz3))
The TRS R consists of the following rules:
new_primMinusNat0(Succ(vxz300)) → new_primMinusNat1(vxz300)
new_primPlusInt(Neg(vxz300)) → new_primMinusNat0(vxz300)
new_primPlusNat0(Succ(vxz300)) → Succ(Succ(new_primPlusNat(vxz300)))
new_primPlusInt(Pos(vxz300)) → Pos(new_primPlusNat0(vxz300))
new_primPlusNat(Succ(vxz3000)) → Succ(vxz3000)
new_ps0(Integer(vxz30)) → Integer(new_primPlusInt(vxz30))
new_primPlusNat(Zero) → Zero
new_primPlusNat0(Zero) → Succ(Zero)
new_primMinusNat1(Succ(vxz3000)) → Neg(Succ(vxz3000))
The set Q consists of the following terms:
new_primMinusNat1(Zero)
new_primPlusInt(Pos(x0))
new_primPlusNat(Zero)
new_primPlusInt(Neg(x0))
new_primPlusNat0(Zero)
new_primPlusNat0(Succ(x0))
new_primMinusNat1(Succ(x0))
new_primMinusNat0(Zero)
new_ps0(Integer(x0))
new_primMinusNat0(Succ(x0))
new_primPlusNat(Succ(x0))
We have to consider all minimal (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.
The TRS P consists of the following rules:
new_numericEnumFrom0(vxz3) → new_numericEnumFrom0(new_ps0(vxz3))
The TRS R consists of the following rules:
new_primMinusNat0(Succ(vxz300)) → new_primMinusNat1(vxz300)
new_primPlusInt(Neg(vxz300)) → new_primMinusNat0(vxz300)
new_primPlusNat0(Succ(vxz300)) → Succ(Succ(new_primPlusNat(vxz300)))
new_primPlusInt(Pos(vxz300)) → Pos(new_primPlusNat0(vxz300))
new_primPlusNat(Succ(vxz3000)) → Succ(vxz3000)
new_ps0(Integer(vxz30)) → Integer(new_primPlusInt(vxz30))
new_primPlusNat(Zero) → Zero
new_primPlusNat0(Zero) → Succ(Zero)
new_primMinusNat1(Succ(vxz3000)) → Neg(Succ(vxz3000))
s = new_numericEnumFrom0(vxz3) evaluates to t =new_numericEnumFrom0(new_ps0(vxz3))
Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
- Semiunifier: [ ]
- Matcher: [vxz3 / new_ps0(vxz3)]
Rewriting sequence
The DP semiunifies directly so there is only one rewrite step from new_numericEnumFrom0(vxz3) to new_numericEnumFrom0(new_ps0(vxz3)).
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primMulNat(Succ(vxz3100)) → new_primMulNat(vxz3100)
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(vxz3100)) → new_primMulNat(vxz3100)
The graph contains the following edges 1 > 1
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_numericEnumFrom1(vxz3) → new_numericEnumFrom1(new_ps1(vxz3))
The TRS R consists of the following rules:
new_primPlusInt(Neg(vxz300)) → new_primMinusNat0(vxz300)
new_primMinusNat0(Succ(vxz300)) → new_primMinusNat1(vxz300)
new_primPlusNat0(Succ(vxz300)) → Succ(Succ(new_primPlusNat(vxz300)))
new_primPlusInt(Pos(vxz300)) → Pos(new_primPlusNat0(vxz300))
new_ps1(Float(vxz30, vxz31)) → Float(new_ps2(vxz30), new_sr(vxz31))
new_ps2(vxz30) → new_primPlusInt(vxz30)
new_primPlusNat0(Zero) → Succ(Zero)
new_primMinusNat1(Succ(vxz3000)) → Neg(Succ(vxz3000))
new_primMulNat0(Succ(vxz3100)) → new_primPlusNat0(new_primMulNat0(vxz3100))
new_primMulNat0(Zero) → Zero
new_primPlusNat(Succ(vxz3000)) → Succ(vxz3000)
new_primMinusNat0(Zero) → Pos(Succ(Zero))
new_primMulInt(Pos(vxz310)) → Pos(new_primMulNat0(vxz310))
new_primMinusNat1(Zero) → Pos(Zero)
new_primPlusNat(Zero) → Zero
new_primMulInt(Neg(vxz310)) → Neg(new_primMulNat0(vxz310))
new_sr(vxz31) → new_primMulInt(vxz31)
The set Q consists of the following terms:
new_ps1(Float(x0, x1))
new_primMulInt(Neg(x0))
new_primPlusInt(Neg(x0))
new_primPlusNat0(Succ(x0))
new_ps2(x0)
new_primMinusNat1(Succ(x0))
new_primMinusNat0(Succ(x0))
new_primPlusNat(Succ(x0))
new_primMinusNat1(Zero)
new_primPlusInt(Pos(x0))
new_sr(x0)
new_primPlusNat(Zero)
new_primMulNat0(Succ(x0))
new_primPlusNat0(Zero)
new_primMulNat0(Zero)
new_primMinusNat0(Zero)
new_primMulInt(Pos(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 rules of the TRS R:
new_primMinusNat0(Zero) → Pos(Succ(Zero))
new_primMinusNat1(Zero) → Pos(Zero)
Used ordering: POLO with Polynomial interpretation [25]:
POL(Float(x1, x2)) = 2·x1 + 2·x2
POL(Neg(x1)) = 1 + x1
POL(Pos(x1)) = x1
POL(Succ(x1)) = x1
POL(Zero) = 0
POL(new_numericEnumFrom1(x1)) = x1
POL(new_primMinusNat0(x1)) = 1 + x1
POL(new_primMinusNat1(x1)) = 1 + x1
POL(new_primMulInt(x1)) = x1
POL(new_primMulNat0(x1)) = x1
POL(new_primPlusInt(x1)) = x1
POL(new_primPlusNat(x1)) = x1
POL(new_primPlusNat0(x1)) = x1
POL(new_ps1(x1)) = x1
POL(new_ps2(x1)) = x1
POL(new_sr(x1)) = x1
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ NonTerminationProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_numericEnumFrom1(vxz3) → new_numericEnumFrom1(new_ps1(vxz3))
The TRS R consists of the following rules:
new_primPlusInt(Neg(vxz300)) → new_primMinusNat0(vxz300)
new_primMinusNat0(Succ(vxz300)) → new_primMinusNat1(vxz300)
new_primPlusNat0(Succ(vxz300)) → Succ(Succ(new_primPlusNat(vxz300)))
new_primPlusInt(Pos(vxz300)) → Pos(new_primPlusNat0(vxz300))
new_ps1(Float(vxz30, vxz31)) → Float(new_ps2(vxz30), new_sr(vxz31))
new_ps2(vxz30) → new_primPlusInt(vxz30)
new_primPlusNat0(Zero) → Succ(Zero)
new_primMinusNat1(Succ(vxz3000)) → Neg(Succ(vxz3000))
new_primMulNat0(Succ(vxz3100)) → new_primPlusNat0(new_primMulNat0(vxz3100))
new_primMulNat0(Zero) → Zero
new_primPlusNat(Succ(vxz3000)) → Succ(vxz3000)
new_primMulInt(Pos(vxz310)) → Pos(new_primMulNat0(vxz310))
new_primPlusNat(Zero) → Zero
new_primMulInt(Neg(vxz310)) → Neg(new_primMulNat0(vxz310))
new_sr(vxz31) → new_primMulInt(vxz31)
The set Q consists of the following terms:
new_ps1(Float(x0, x1))
new_primMulInt(Neg(x0))
new_primPlusInt(Neg(x0))
new_primPlusNat0(Succ(x0))
new_ps2(x0)
new_primMinusNat1(Succ(x0))
new_primMinusNat0(Succ(x0))
new_primPlusNat(Succ(x0))
new_primMinusNat1(Zero)
new_primPlusInt(Pos(x0))
new_sr(x0)
new_primPlusNat(Zero)
new_primMulNat0(Succ(x0))
new_primPlusNat0(Zero)
new_primMulNat0(Zero)
new_primMinusNat0(Zero)
new_primMulInt(Pos(x0))
We have to consider all minimal (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.
The TRS P consists of the following rules:
new_numericEnumFrom1(vxz3) → new_numericEnumFrom1(new_ps1(vxz3))
The TRS R consists of the following rules:
new_primPlusInt(Neg(vxz300)) → new_primMinusNat0(vxz300)
new_primMinusNat0(Succ(vxz300)) → new_primMinusNat1(vxz300)
new_primPlusNat0(Succ(vxz300)) → Succ(Succ(new_primPlusNat(vxz300)))
new_primPlusInt(Pos(vxz300)) → Pos(new_primPlusNat0(vxz300))
new_ps1(Float(vxz30, vxz31)) → Float(new_ps2(vxz30), new_sr(vxz31))
new_ps2(vxz30) → new_primPlusInt(vxz30)
new_primPlusNat0(Zero) → Succ(Zero)
new_primMinusNat1(Succ(vxz3000)) → Neg(Succ(vxz3000))
new_primMulNat0(Succ(vxz3100)) → new_primPlusNat0(new_primMulNat0(vxz3100))
new_primMulNat0(Zero) → Zero
new_primPlusNat(Succ(vxz3000)) → Succ(vxz3000)
new_primMulInt(Pos(vxz310)) → Pos(new_primMulNat0(vxz310))
new_primPlusNat(Zero) → Zero
new_primMulInt(Neg(vxz310)) → Neg(new_primMulNat0(vxz310))
new_sr(vxz31) → new_primMulInt(vxz31)
s = new_numericEnumFrom1(vxz3) evaluates to t =new_numericEnumFrom1(new_ps1(vxz3))
Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
- Matcher: [vxz3 / new_ps1(vxz3)]
- Semiunifier: [ ]
Rewriting sequence
The DP semiunifies directly so there is only one rewrite step from new_numericEnumFrom1(vxz3) to new_numericEnumFrom1(new_ps1(vxz3)).
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_numericEnumFrom2(vxz3) → new_numericEnumFrom2(new_ps3(vxz3))
The TRS R consists of the following rules:
new_primPlusInt(Neg(vxz300)) → new_primMinusNat0(vxz300)
new_primMinusNat0(Succ(vxz300)) → new_primMinusNat1(vxz300)
new_primPlusNat0(Succ(vxz300)) → Succ(Succ(new_primPlusNat(vxz300)))
new_primPlusInt(Pos(vxz300)) → Pos(new_primPlusNat0(vxz300))
new_ps2(vxz30) → new_primPlusInt(vxz30)
new_ps3(Double(vxz30, vxz31)) → Double(new_ps2(vxz30), new_sr(vxz31))
new_primPlusNat0(Zero) → Succ(Zero)
new_primMinusNat1(Succ(vxz3000)) → Neg(Succ(vxz3000))
new_primMulNat0(Succ(vxz3100)) → new_primPlusNat0(new_primMulNat0(vxz3100))
new_primMulNat0(Zero) → Zero
new_primPlusNat(Succ(vxz3000)) → Succ(vxz3000)
new_primMinusNat0(Zero) → Pos(Succ(Zero))
new_primMulInt(Pos(vxz310)) → Pos(new_primMulNat0(vxz310))
new_primMinusNat1(Zero) → Pos(Zero)
new_primPlusNat(Zero) → Zero
new_primMulInt(Neg(vxz310)) → Neg(new_primMulNat0(vxz310))
new_sr(vxz31) → new_primMulInt(vxz31)
The set Q consists of the following terms:
new_primMulInt(Neg(x0))
new_primPlusInt(Neg(x0))
new_primPlusNat0(Succ(x0))
new_ps2(x0)
new_primMinusNat1(Succ(x0))
new_primMinusNat0(Succ(x0))
new_primPlusNat(Succ(x0))
new_primMinusNat1(Zero)
new_primPlusInt(Pos(x0))
new_sr(x0)
new_primPlusNat(Zero)
new_primMulNat0(Succ(x0))
new_primPlusNat0(Zero)
new_primMulNat0(Zero)
new_primMinusNat0(Zero)
new_ps3(Double(x0, x1))
new_primMulInt(Pos(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 rules of the TRS R:
new_primMinusNat0(Zero) → Pos(Succ(Zero))
new_primMinusNat1(Zero) → Pos(Zero)
Used ordering: POLO with Polynomial interpretation [25]:
POL(Double(x1, x2)) = 2·x1 + 2·x2
POL(Neg(x1)) = 2 + 2·x1
POL(Pos(x1)) = 2·x1
POL(Succ(x1)) = x1
POL(Zero) = 1
POL(new_numericEnumFrom2(x1)) = x1
POL(new_primMinusNat0(x1)) = 2 + 2·x1
POL(new_primMinusNat1(x1)) = 2 + 2·x1
POL(new_primMulInt(x1)) = x1
POL(new_primMulNat0(x1)) = x1
POL(new_primPlusInt(x1)) = x1
POL(new_primPlusNat(x1)) = x1
POL(new_primPlusNat0(x1)) = x1
POL(new_ps2(x1)) = x1
POL(new_ps3(x1)) = x1
POL(new_sr(x1)) = x1
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ NonTerminationProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_numericEnumFrom2(vxz3) → new_numericEnumFrom2(new_ps3(vxz3))
The TRS R consists of the following rules:
new_primPlusInt(Neg(vxz300)) → new_primMinusNat0(vxz300)
new_primMinusNat0(Succ(vxz300)) → new_primMinusNat1(vxz300)
new_primPlusNat0(Succ(vxz300)) → Succ(Succ(new_primPlusNat(vxz300)))
new_primPlusInt(Pos(vxz300)) → Pos(new_primPlusNat0(vxz300))
new_ps2(vxz30) → new_primPlusInt(vxz30)
new_ps3(Double(vxz30, vxz31)) → Double(new_ps2(vxz30), new_sr(vxz31))
new_primPlusNat0(Zero) → Succ(Zero)
new_primMinusNat1(Succ(vxz3000)) → Neg(Succ(vxz3000))
new_primMulNat0(Succ(vxz3100)) → new_primPlusNat0(new_primMulNat0(vxz3100))
new_primMulNat0(Zero) → Zero
new_primPlusNat(Succ(vxz3000)) → Succ(vxz3000)
new_primMulInt(Pos(vxz310)) → Pos(new_primMulNat0(vxz310))
new_primPlusNat(Zero) → Zero
new_primMulInt(Neg(vxz310)) → Neg(new_primMulNat0(vxz310))
new_sr(vxz31) → new_primMulInt(vxz31)
The set Q consists of the following terms:
new_primMulInt(Neg(x0))
new_primPlusInt(Neg(x0))
new_primPlusNat0(Succ(x0))
new_ps2(x0)
new_primMinusNat1(Succ(x0))
new_primMinusNat0(Succ(x0))
new_primPlusNat(Succ(x0))
new_primMinusNat1(Zero)
new_primPlusInt(Pos(x0))
new_sr(x0)
new_primPlusNat(Zero)
new_primMulNat0(Succ(x0))
new_primPlusNat0(Zero)
new_primMulNat0(Zero)
new_primMinusNat0(Zero)
new_ps3(Double(x0, x1))
new_primMulInt(Pos(x0))
We have to consider all minimal (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.
The TRS P consists of the following rules:
new_numericEnumFrom2(vxz3) → new_numericEnumFrom2(new_ps3(vxz3))
The TRS R consists of the following rules:
new_primPlusInt(Neg(vxz300)) → new_primMinusNat0(vxz300)
new_primMinusNat0(Succ(vxz300)) → new_primMinusNat1(vxz300)
new_primPlusNat0(Succ(vxz300)) → Succ(Succ(new_primPlusNat(vxz300)))
new_primPlusInt(Pos(vxz300)) → Pos(new_primPlusNat0(vxz300))
new_ps2(vxz30) → new_primPlusInt(vxz30)
new_ps3(Double(vxz30, vxz31)) → Double(new_ps2(vxz30), new_sr(vxz31))
new_primPlusNat0(Zero) → Succ(Zero)
new_primMinusNat1(Succ(vxz3000)) → Neg(Succ(vxz3000))
new_primMulNat0(Succ(vxz3100)) → new_primPlusNat0(new_primMulNat0(vxz3100))
new_primMulNat0(Zero) → Zero
new_primPlusNat(Succ(vxz3000)) → Succ(vxz3000)
new_primMulInt(Pos(vxz310)) → Pos(new_primMulNat0(vxz310))
new_primPlusNat(Zero) → Zero
new_primMulInt(Neg(vxz310)) → Neg(new_primMulNat0(vxz310))
new_sr(vxz31) → new_primMulInt(vxz31)
s = new_numericEnumFrom2(vxz3) evaluates to t =new_numericEnumFrom2(new_ps3(vxz3))
Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
- Matcher: [vxz3 / new_ps3(vxz3)]
- Semiunifier: [ ]
Rewriting sequence
The DP semiunifies directly so there is only one rewrite step from new_numericEnumFrom2(vxz3) to new_numericEnumFrom2(new_ps3(vxz3)).
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primPlusNat1(Succ(vxz28000), Succ(vxz31000)) → new_primPlusNat1(vxz28000, vxz31000)
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_primPlusNat1(Succ(vxz28000), Succ(vxz31000)) → new_primPlusNat1(vxz28000, vxz31000)
The graph contains the following edges 1 > 1, 2 > 2
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primMinusNatS(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS(vxz1680, vxz1690)
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(vxz1680), Succ(vxz1690)) → new_primMinusNatS(vxz1680, vxz1690)
The graph contains the following edges 1 > 1, 2 > 2
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_gcd0Gcd'1(Succ(Zero), Succ(vxz2320), vxz236) → new_gcd0Gcd'12(Zero, Succ(vxz2320))
new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Zero) → new_gcd0Gcd'11(vxz272, vxz273)
new_gcd0Gcd'10(vxz272, vxz273, Zero, Succ(vxz2750)) → new_gcd0Gcd'12(Succ(vxz272), vxz273)
new_gcd0Gcd'10(vxz272, vxz273, Zero, Zero) → new_gcd0Gcd'11(vxz272, vxz273)
new_gcd0Gcd'1(Succ(Succ(vxz23700)), Succ(vxz2320), vxz236) → new_gcd0Gcd'10(vxz23700, Succ(vxz2320), vxz23700, vxz2320)
new_gcd0Gcd'13(vxz5200, vxz2700) → new_gcd0Gcd'1(Succ(vxz5200), vxz2700, Succ(vxz5200))
new_gcd0Gcd'1(Succ(Zero), Zero, vxz236) → new_gcd0Gcd'1(new_primMinusNatS0(Zero, Zero), Zero, new_primMinusNatS0(Zero, Zero))
new_gcd0Gcd'11(vxz263, vxz264) → new_gcd0Gcd'1(new_primMinusNatS0(Succ(vxz263), vxz264), vxz264, new_primMinusNatS0(Succ(vxz263), vxz264))
new_gcd0Gcd'12(vxz266, vxz267) → new_gcd0Gcd'13(vxz267, vxz266)
new_gcd0Gcd'1(Succ(Succ(vxz23700)), Zero, vxz236) → new_gcd0Gcd'11(vxz23700, Zero)
new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Succ(vxz2750)) → new_gcd0Gcd'10(vxz272, vxz273, vxz2740, vxz2750)
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_gcd0Gcd'1(Succ(Zero), Succ(vxz2320), vxz236) → new_gcd0Gcd'12(Zero, Succ(vxz2320))
new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Zero) → new_gcd0Gcd'11(vxz272, vxz273)
new_gcd0Gcd'10(vxz272, vxz273, Zero, Succ(vxz2750)) → new_gcd0Gcd'12(Succ(vxz272), vxz273)
new_gcd0Gcd'10(vxz272, vxz273, Zero, Zero) → new_gcd0Gcd'11(vxz272, vxz273)
new_gcd0Gcd'1(Succ(Succ(vxz23700)), Succ(vxz2320), vxz236) → new_gcd0Gcd'10(vxz23700, Succ(vxz2320), vxz23700, vxz2320)
new_gcd0Gcd'13(vxz5200, vxz2700) → new_gcd0Gcd'1(Succ(vxz5200), vxz2700, Succ(vxz5200))
new_gcd0Gcd'11(vxz263, vxz264) → new_gcd0Gcd'1(new_primMinusNatS0(Succ(vxz263), vxz264), vxz264, new_primMinusNatS0(Succ(vxz263), vxz264))
new_gcd0Gcd'12(vxz266, vxz267) → new_gcd0Gcd'13(vxz267, vxz266)
new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Succ(vxz2750)) → new_gcd0Gcd'10(vxz272, vxz273, vxz2740, vxz2750)
new_gcd0Gcd'1(Succ(Succ(vxz23700)), Zero, vxz236) → new_gcd0Gcd'11(vxz23700, Zero)
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
new_gcd0Gcd'1(Succ(Zero), Succ(vxz2320), vxz236) → new_gcd0Gcd'12(Zero, Succ(vxz2320))
The remaining pairs can at least be oriented weakly.
new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Zero) → new_gcd0Gcd'11(vxz272, vxz273)
new_gcd0Gcd'10(vxz272, vxz273, Zero, Succ(vxz2750)) → new_gcd0Gcd'12(Succ(vxz272), vxz273)
new_gcd0Gcd'10(vxz272, vxz273, Zero, Zero) → new_gcd0Gcd'11(vxz272, vxz273)
new_gcd0Gcd'1(Succ(Succ(vxz23700)), Succ(vxz2320), vxz236) → new_gcd0Gcd'10(vxz23700, Succ(vxz2320), vxz23700, vxz2320)
new_gcd0Gcd'13(vxz5200, vxz2700) → new_gcd0Gcd'1(Succ(vxz5200), vxz2700, Succ(vxz5200))
new_gcd0Gcd'11(vxz263, vxz264) → new_gcd0Gcd'1(new_primMinusNatS0(Succ(vxz263), vxz264), vxz264, new_primMinusNatS0(Succ(vxz263), vxz264))
new_gcd0Gcd'12(vxz266, vxz267) → new_gcd0Gcd'13(vxz267, vxz266)
new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Succ(vxz2750)) → new_gcd0Gcd'10(vxz272, vxz273, vxz2740, vxz2750)
new_gcd0Gcd'1(Succ(Succ(vxz23700)), Zero, vxz236) → new_gcd0Gcd'11(vxz23700, Zero)
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( new_primMinusNatS0(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
Tuple symbols:
M( new_gcd0Gcd'10(x1, ..., x4) ) = | 0 | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
M( new_gcd0Gcd'11(x1, x2) ) = | 0 | + | | · | x1 | + | | · | x2 |
M( new_gcd0Gcd'1(x1, ..., x3) ) = | 0 | + | | · | x1 | + | | · | x2 | + | | · | x3 |
M( new_gcd0Gcd'12(x1, x2) ) = | 0 | + | | · | x1 | + | | · | x2 |
M( new_gcd0Gcd'13(x1, x2) ) = | 0 | + | | · | x1 | + | | · | x2 |
Matrix type:
We used a basic matrix type which is not further parametrizeable.
As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
The following usable rules [17] were oriented:
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Zero) → new_gcd0Gcd'11(vxz272, vxz273)
new_gcd0Gcd'10(vxz272, vxz273, Zero, Succ(vxz2750)) → new_gcd0Gcd'12(Succ(vxz272), vxz273)
new_gcd0Gcd'10(vxz272, vxz273, Zero, Zero) → new_gcd0Gcd'11(vxz272, vxz273)
new_gcd0Gcd'13(vxz5200, vxz2700) → new_gcd0Gcd'1(Succ(vxz5200), vxz2700, Succ(vxz5200))
new_gcd0Gcd'1(Succ(Succ(vxz23700)), Succ(vxz2320), vxz236) → new_gcd0Gcd'10(vxz23700, Succ(vxz2320), vxz23700, vxz2320)
new_gcd0Gcd'12(vxz266, vxz267) → new_gcd0Gcd'13(vxz267, vxz266)
new_gcd0Gcd'11(vxz263, vxz264) → new_gcd0Gcd'1(new_primMinusNatS0(Succ(vxz263), vxz264), vxz264, new_primMinusNatS0(Succ(vxz263), vxz264))
new_gcd0Gcd'1(Succ(Succ(vxz23700)), Zero, vxz236) → new_gcd0Gcd'11(vxz23700, Zero)
new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Succ(vxz2750)) → new_gcd0Gcd'10(vxz272, vxz273, vxz2740, vxz2750)
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
new_gcd0Gcd'1(Succ(Succ(vxz23700)), Zero, vxz236) → new_gcd0Gcd'11(vxz23700, Zero)
The remaining pairs can at least be oriented weakly.
new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Zero) → new_gcd0Gcd'11(vxz272, vxz273)
new_gcd0Gcd'10(vxz272, vxz273, Zero, Succ(vxz2750)) → new_gcd0Gcd'12(Succ(vxz272), vxz273)
new_gcd0Gcd'10(vxz272, vxz273, Zero, Zero) → new_gcd0Gcd'11(vxz272, vxz273)
new_gcd0Gcd'13(vxz5200, vxz2700) → new_gcd0Gcd'1(Succ(vxz5200), vxz2700, Succ(vxz5200))
new_gcd0Gcd'1(Succ(Succ(vxz23700)), Succ(vxz2320), vxz236) → new_gcd0Gcd'10(vxz23700, Succ(vxz2320), vxz23700, vxz2320)
new_gcd0Gcd'12(vxz266, vxz267) → new_gcd0Gcd'13(vxz267, vxz266)
new_gcd0Gcd'11(vxz263, vxz264) → new_gcd0Gcd'1(new_primMinusNatS0(Succ(vxz263), vxz264), vxz264, new_primMinusNatS0(Succ(vxz263), vxz264))
new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Succ(vxz2750)) → new_gcd0Gcd'10(vxz272, vxz273, vxz2740, vxz2750)
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( new_primMinusNatS0(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
Tuple symbols:
M( new_gcd0Gcd'10(x1, ..., x4) ) = | 0 | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
M( new_gcd0Gcd'11(x1, x2) ) = | 0 | + | | · | x1 | + | | · | x2 |
M( new_gcd0Gcd'1(x1, ..., x3) ) = | 0 | + | | · | x1 | + | | · | x2 | + | | · | x3 |
M( new_gcd0Gcd'12(x1, x2) ) = | 0 | + | | · | x1 | + | | · | x2 |
M( new_gcd0Gcd'13(x1, x2) ) = | 0 | + | | · | x1 | + | | · | x2 |
Matrix type:
We used a basic matrix type which is not further parametrizeable.
As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
The following usable rules [17] were oriented:
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Zero) → new_gcd0Gcd'11(vxz272, vxz273)
new_gcd0Gcd'10(vxz272, vxz273, Zero, Succ(vxz2750)) → new_gcd0Gcd'12(Succ(vxz272), vxz273)
new_gcd0Gcd'10(vxz272, vxz273, Zero, Zero) → new_gcd0Gcd'11(vxz272, vxz273)
new_gcd0Gcd'1(Succ(Succ(vxz23700)), Succ(vxz2320), vxz236) → new_gcd0Gcd'10(vxz23700, Succ(vxz2320), vxz23700, vxz2320)
new_gcd0Gcd'13(vxz5200, vxz2700) → new_gcd0Gcd'1(Succ(vxz5200), vxz2700, Succ(vxz5200))
new_gcd0Gcd'11(vxz263, vxz264) → new_gcd0Gcd'1(new_primMinusNatS0(Succ(vxz263), vxz264), vxz264, new_primMinusNatS0(Succ(vxz263), vxz264))
new_gcd0Gcd'12(vxz266, vxz267) → new_gcd0Gcd'13(vxz267, vxz266)
new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Succ(vxz2750)) → new_gcd0Gcd'10(vxz272, vxz273, vxz2740, vxz2750)
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Zero) → new_gcd0Gcd'11(vxz272, vxz273)
new_gcd0Gcd'10(vxz272, vxz273, Zero, Zero) → new_gcd0Gcd'11(vxz272, vxz273)
The remaining pairs can at least be oriented weakly.
new_gcd0Gcd'10(vxz272, vxz273, Zero, Succ(vxz2750)) → new_gcd0Gcd'12(Succ(vxz272), vxz273)
new_gcd0Gcd'1(Succ(Succ(vxz23700)), Succ(vxz2320), vxz236) → new_gcd0Gcd'10(vxz23700, Succ(vxz2320), vxz23700, vxz2320)
new_gcd0Gcd'13(vxz5200, vxz2700) → new_gcd0Gcd'1(Succ(vxz5200), vxz2700, Succ(vxz5200))
new_gcd0Gcd'11(vxz263, vxz264) → new_gcd0Gcd'1(new_primMinusNatS0(Succ(vxz263), vxz264), vxz264, new_primMinusNatS0(Succ(vxz263), vxz264))
new_gcd0Gcd'12(vxz266, vxz267) → new_gcd0Gcd'13(vxz267, vxz266)
new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Succ(vxz2750)) → new_gcd0Gcd'10(vxz272, vxz273, vxz2740, vxz2750)
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( new_primMinusNatS0(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
Tuple symbols:
M( new_gcd0Gcd'10(x1, ..., x4) ) = | 1 | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
M( new_gcd0Gcd'11(x1, x2) ) = | 0 | + | | · | x1 | + | | · | x2 |
M( new_gcd0Gcd'1(x1, ..., x3) ) = | 0 | + | | · | x1 | + | | · | x2 | + | | · | x3 |
M( new_gcd0Gcd'12(x1, x2) ) = | 0 | + | | · | x1 | + | | · | x2 |
M( new_gcd0Gcd'13(x1, x2) ) = | 0 | + | | · | x1 | + | | · | x2 |
Matrix type:
We used a basic matrix type which is not further parametrizeable.
As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
The following usable rules [17] were oriented:
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_gcd0Gcd'10(vxz272, vxz273, Zero, Succ(vxz2750)) → new_gcd0Gcd'12(Succ(vxz272), vxz273)
new_gcd0Gcd'13(vxz5200, vxz2700) → new_gcd0Gcd'1(Succ(vxz5200), vxz2700, Succ(vxz5200))
new_gcd0Gcd'1(Succ(Succ(vxz23700)), Succ(vxz2320), vxz236) → new_gcd0Gcd'10(vxz23700, Succ(vxz2320), vxz23700, vxz2320)
new_gcd0Gcd'12(vxz266, vxz267) → new_gcd0Gcd'13(vxz267, vxz266)
new_gcd0Gcd'11(vxz263, vxz264) → new_gcd0Gcd'1(new_primMinusNatS0(Succ(vxz263), vxz264), vxz264, new_primMinusNatS0(Succ(vxz263), vxz264))
new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Succ(vxz2750)) → new_gcd0Gcd'10(vxz272, vxz273, vxz2740, vxz2750)
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_gcd0Gcd'10(vxz272, vxz273, Zero, Succ(vxz2750)) → new_gcd0Gcd'12(Succ(vxz272), vxz273)
new_gcd0Gcd'1(Succ(Succ(vxz23700)), Succ(vxz2320), vxz236) → new_gcd0Gcd'10(vxz23700, Succ(vxz2320), vxz23700, vxz2320)
new_gcd0Gcd'13(vxz5200, vxz2700) → new_gcd0Gcd'1(Succ(vxz5200), vxz2700, Succ(vxz5200))
new_gcd0Gcd'12(vxz266, vxz267) → new_gcd0Gcd'13(vxz267, vxz266)
new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Succ(vxz2750)) → new_gcd0Gcd'10(vxz272, vxz273, vxz2740, vxz2750)
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_gcd0Gcd'10(vxz272, vxz273, Zero, Succ(vxz2750)) → new_gcd0Gcd'12(Succ(vxz272), vxz273)
new_gcd0Gcd'1(Succ(Succ(vxz23700)), Succ(vxz2320), vxz236) → new_gcd0Gcd'10(vxz23700, Succ(vxz2320), vxz23700, vxz2320)
new_gcd0Gcd'13(vxz5200, vxz2700) → new_gcd0Gcd'1(Succ(vxz5200), vxz2700, Succ(vxz5200))
new_gcd0Gcd'12(vxz266, vxz267) → new_gcd0Gcd'13(vxz267, vxz266)
new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Succ(vxz2750)) → new_gcd0Gcd'10(vxz272, vxz273, vxz2740, vxz2750)
R is empty.
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_gcd0Gcd'10(vxz272, vxz273, Zero, Succ(vxz2750)) → new_gcd0Gcd'12(Succ(vxz272), vxz273)
new_gcd0Gcd'13(vxz5200, vxz2700) → new_gcd0Gcd'1(Succ(vxz5200), vxz2700, Succ(vxz5200))
new_gcd0Gcd'1(Succ(Succ(vxz23700)), Succ(vxz2320), vxz236) → new_gcd0Gcd'10(vxz23700, Succ(vxz2320), vxz23700, vxz2320)
new_gcd0Gcd'12(vxz266, vxz267) → new_gcd0Gcd'13(vxz267, vxz266)
new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Succ(vxz2750)) → new_gcd0Gcd'10(vxz272, vxz273, vxz2740, vxz2750)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule new_gcd0Gcd'1(Succ(Succ(vxz23700)), Succ(vxz2320), vxz236) → new_gcd0Gcd'10(vxz23700, Succ(vxz2320), vxz23700, vxz2320) we obtained the following new rules:
new_gcd0Gcd'1(Succ(Succ(x0)), Succ(x1), Succ(Succ(x0))) → new_gcd0Gcd'10(x0, Succ(x1), x0, x1)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ NonInfProof
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_gcd0Gcd'1(Succ(Succ(x0)), Succ(x1), Succ(Succ(x0))) → new_gcd0Gcd'10(x0, Succ(x1), x0, x1)
new_gcd0Gcd'10(vxz272, vxz273, Zero, Succ(vxz2750)) → new_gcd0Gcd'12(Succ(vxz272), vxz273)
new_gcd0Gcd'13(vxz5200, vxz2700) → new_gcd0Gcd'1(Succ(vxz5200), vxz2700, Succ(vxz5200))
new_gcd0Gcd'12(vxz266, vxz267) → new_gcd0Gcd'13(vxz267, vxz266)
new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Succ(vxz2750)) → new_gcd0Gcd'10(vxz272, vxz273, vxz2740, vxz2750)
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_gcd0Gcd'10(vxz272, vxz273, Zero, Succ(vxz2750)) → new_gcd0Gcd'12(Succ(vxz272), vxz273) the following chains were created:
- We consider the chain new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Succ(vxz2750)) → new_gcd0Gcd'10(vxz272, vxz273, vxz2740, vxz2750), new_gcd0Gcd'10(vxz272, vxz273, Zero, Succ(vxz2750)) → new_gcd0Gcd'12(Succ(vxz272), vxz273) which results in the following constraint:
(1) (new_gcd0Gcd'10(x7, x8, x9, x10)=new_gcd0Gcd'10(x11, x12, Zero, Succ(x13)) ⇒ new_gcd0Gcd'10(x11, x12, Zero, Succ(x13))≥new_gcd0Gcd'12(Succ(x11), x12))
We simplified constraint (1) using rules (I), (II), (III) which results in the following new constraint:
(2) (new_gcd0Gcd'10(x7, x8, Zero, Succ(x13))≥new_gcd0Gcd'12(Succ(x7), x8))
- We consider the chain new_gcd0Gcd'1(Succ(Succ(x0)), Succ(x1), Succ(Succ(x0))) → new_gcd0Gcd'10(x0, Succ(x1), x0, x1), new_gcd0Gcd'10(vxz272, vxz273, Zero, Succ(vxz2750)) → new_gcd0Gcd'12(Succ(vxz272), vxz273) which results in the following constraint:
(3) (new_gcd0Gcd'10(x14, Succ(x15), x14, x15)=new_gcd0Gcd'10(x16, x17, Zero, Succ(x18)) ⇒ new_gcd0Gcd'10(x16, x17, Zero, Succ(x18))≥new_gcd0Gcd'12(Succ(x16), x17))
We simplified constraint (3) using rules (I), (II), (III) which results in the following new constraint:
(4) (new_gcd0Gcd'10(Zero, Succ(Succ(x18)), Zero, Succ(x18))≥new_gcd0Gcd'12(Succ(Zero), Succ(Succ(x18))))
For Pair new_gcd0Gcd'13(vxz5200, vxz2700) → new_gcd0Gcd'1(Succ(vxz5200), vxz2700, Succ(vxz5200)) the following chains were created:
- We consider the chain new_gcd0Gcd'12(vxz266, vxz267) → new_gcd0Gcd'13(vxz267, vxz266), new_gcd0Gcd'13(vxz5200, vxz2700) → new_gcd0Gcd'1(Succ(vxz5200), vxz2700, Succ(vxz5200)) which results in the following constraint:
(5) (new_gcd0Gcd'13(x25, x24)=new_gcd0Gcd'13(x26, x27) ⇒ new_gcd0Gcd'13(x26, x27)≥new_gcd0Gcd'1(Succ(x26), x27, Succ(x26)))
We simplified constraint (5) using rules (I), (II), (III) which results in the following new constraint:
(6) (new_gcd0Gcd'13(x25, x24)≥new_gcd0Gcd'1(Succ(x25), x24, Succ(x25)))
For Pair new_gcd0Gcd'12(vxz266, vxz267) → new_gcd0Gcd'13(vxz267, vxz266) the following chains were created:
- We consider the chain new_gcd0Gcd'10(vxz272, vxz273, Zero, Succ(vxz2750)) → new_gcd0Gcd'12(Succ(vxz272), vxz273), new_gcd0Gcd'12(vxz266, vxz267) → new_gcd0Gcd'13(vxz267, vxz266) which results in the following constraint:
(7) (new_gcd0Gcd'12(Succ(x34), x35)=new_gcd0Gcd'12(x37, x38) ⇒ new_gcd0Gcd'12(x37, x38)≥new_gcd0Gcd'13(x38, x37))
We simplified constraint (7) using rules (I), (II), (III) which results in the following new constraint:
(8) (new_gcd0Gcd'12(Succ(x34), x35)≥new_gcd0Gcd'13(x35, Succ(x34)))
For Pair new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Succ(vxz2750)) → new_gcd0Gcd'10(vxz272, vxz273, vxz2740, vxz2750) the following chains were created:
- We consider the chain new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Succ(vxz2750)) → new_gcd0Gcd'10(vxz272, vxz273, vxz2740, vxz2750), new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Succ(vxz2750)) → new_gcd0Gcd'10(vxz272, vxz273, vxz2740, vxz2750) which results in the following constraint:
(9) (new_gcd0Gcd'10(x56, x57, x58, x59)=new_gcd0Gcd'10(x60, x61, Succ(x62), Succ(x63)) ⇒ new_gcd0Gcd'10(x60, x61, Succ(x62), Succ(x63))≥new_gcd0Gcd'10(x60, x61, x62, x63))
We simplified constraint (9) using rules (I), (II), (III) which results in the following new constraint:
(10) (new_gcd0Gcd'10(x56, x57, Succ(x62), Succ(x63))≥new_gcd0Gcd'10(x56, x57, x62, x63))
- We consider the chain new_gcd0Gcd'1(Succ(Succ(x0)), Succ(x1), Succ(Succ(x0))) → new_gcd0Gcd'10(x0, Succ(x1), x0, x1), new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Succ(vxz2750)) → new_gcd0Gcd'10(vxz272, vxz273, vxz2740, vxz2750) which results in the following constraint:
(11) (new_gcd0Gcd'10(x64, Succ(x65), x64, x65)=new_gcd0Gcd'10(x66, x67, Succ(x68), Succ(x69)) ⇒ new_gcd0Gcd'10(x66, x67, Succ(x68), Succ(x69))≥new_gcd0Gcd'10(x66, x67, x68, x69))
We simplified constraint (11) using rules (I), (II), (III) which results in the following new constraint:
(12) (new_gcd0Gcd'10(Succ(x68), Succ(Succ(x69)), Succ(x68), Succ(x69))≥new_gcd0Gcd'10(Succ(x68), Succ(Succ(x69)), x68, x69))
For Pair new_gcd0Gcd'1(Succ(Succ(x0)), Succ(x1), Succ(Succ(x0))) → new_gcd0Gcd'10(x0, Succ(x1), x0, x1) the following chains were created:
- We consider the chain new_gcd0Gcd'13(vxz5200, vxz2700) → new_gcd0Gcd'1(Succ(vxz5200), vxz2700, Succ(vxz5200)), new_gcd0Gcd'1(Succ(Succ(x0)), Succ(x1), Succ(Succ(x0))) → new_gcd0Gcd'10(x0, Succ(x1), x0, x1) which results in the following constraint:
(13) (new_gcd0Gcd'1(Succ(x73), x74, Succ(x73))=new_gcd0Gcd'1(Succ(Succ(x75)), Succ(x76), Succ(Succ(x75))) ⇒ new_gcd0Gcd'1(Succ(Succ(x75)), Succ(x76), Succ(Succ(x75)))≥new_gcd0Gcd'10(x75, Succ(x76), x75, x76))
We simplified constraint (13) using rules (I), (II), (III) which results in the following new constraint:
(14) (new_gcd0Gcd'1(Succ(Succ(x75)), Succ(x76), Succ(Succ(x75)))≥new_gcd0Gcd'10(x75, Succ(x76), x75, x76))
To summarize, we get the following constraints P≥ for the following pairs.
- new_gcd0Gcd'10(vxz272, vxz273, Zero, Succ(vxz2750)) → new_gcd0Gcd'12(Succ(vxz272), vxz273)
- (new_gcd0Gcd'10(x7, x8, Zero, Succ(x13))≥new_gcd0Gcd'12(Succ(x7), x8))
- (new_gcd0Gcd'10(Zero, Succ(Succ(x18)), Zero, Succ(x18))≥new_gcd0Gcd'12(Succ(Zero), Succ(Succ(x18))))
- new_gcd0Gcd'13(vxz5200, vxz2700) → new_gcd0Gcd'1(Succ(vxz5200), vxz2700, Succ(vxz5200))
- (new_gcd0Gcd'13(x25, x24)≥new_gcd0Gcd'1(Succ(x25), x24, Succ(x25)))
- new_gcd0Gcd'12(vxz266, vxz267) → new_gcd0Gcd'13(vxz267, vxz266)
- (new_gcd0Gcd'12(Succ(x34), x35)≥new_gcd0Gcd'13(x35, Succ(x34)))
- new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Succ(vxz2750)) → new_gcd0Gcd'10(vxz272, vxz273, vxz2740, vxz2750)
- (new_gcd0Gcd'10(x56, x57, Succ(x62), Succ(x63))≥new_gcd0Gcd'10(x56, x57, x62, x63))
- (new_gcd0Gcd'10(Succ(x68), Succ(Succ(x69)), Succ(x68), Succ(x69))≥new_gcd0Gcd'10(Succ(x68), Succ(Succ(x69)), x68, x69))
- new_gcd0Gcd'1(Succ(Succ(x0)), Succ(x1), Succ(Succ(x0))) → new_gcd0Gcd'10(x0, Succ(x1), x0, x1)
- (new_gcd0Gcd'1(Succ(Succ(x75)), Succ(x76), Succ(Succ(x75)))≥new_gcd0Gcd'10(x75, Succ(x76), x75, x76))
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_gcd0Gcd'1(x1, x2, x3)) = -1 - x1 + x2 + x3
POL(new_gcd0Gcd'10(x1, x2, x3, x4)) = -1 + x1 - x3 + x4
POL(new_gcd0Gcd'12(x1, x2)) = -1 + x1
POL(new_gcd0Gcd'13(x1, x2)) = -1 + x2
The following pairs are in P>:
new_gcd0Gcd'1(Succ(Succ(x0)), Succ(x1), Succ(Succ(x0))) → new_gcd0Gcd'10(x0, Succ(x1), x0, x1)
The following pairs are in Pbound:
new_gcd0Gcd'10(vxz272, vxz273, Zero, Succ(vxz2750)) → new_gcd0Gcd'12(Succ(vxz272), vxz273)
new_gcd0Gcd'13(vxz5200, vxz2700) → new_gcd0Gcd'1(Succ(vxz5200), vxz2700, Succ(vxz5200))
new_gcd0Gcd'12(vxz266, vxz267) → new_gcd0Gcd'13(vxz267, vxz266)
new_gcd0Gcd'1(Succ(Succ(x0)), Succ(x1), Succ(Succ(x0))) → new_gcd0Gcd'10(x0, Succ(x1), x0, x1)
There are no usable rules
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ NonInfProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_gcd0Gcd'10(vxz272, vxz273, Zero, Succ(vxz2750)) → new_gcd0Gcd'12(Succ(vxz272), vxz273)
new_gcd0Gcd'13(vxz5200, vxz2700) → new_gcd0Gcd'1(Succ(vxz5200), vxz2700, Succ(vxz5200))
new_gcd0Gcd'12(vxz266, vxz267) → new_gcd0Gcd'13(vxz267, vxz266)
new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Succ(vxz2750)) → new_gcd0Gcd'10(vxz272, vxz273, vxz2740, vxz2750)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 3 less nodes.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ NonInfProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Succ(vxz2750)) → new_gcd0Gcd'10(vxz272, vxz273, vxz2740, vxz2750)
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_gcd0Gcd'10(vxz272, vxz273, Succ(vxz2740), Succ(vxz2750)) → new_gcd0Gcd'10(vxz272, vxz273, vxz2740, vxz2750)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vxz168, vxz169, Zero, Zero) → new_primDivNatS00(vxz168, vxz169)
new_primDivNatS(Succ(Succ(vxz13500)), Succ(vxz136000)) → new_primDivNatS0(vxz13500, vxz136000, vxz13500, vxz136000)
new_primDivNatS0(vxz168, vxz169, Succ(vxz1700), Succ(vxz1710)) → new_primDivNatS0(vxz168, vxz169, vxz1700, vxz1710)
new_primDivNatS00(vxz168, vxz169) → new_primDivNatS(new_primMinusNatS0(vxz168, vxz169), Succ(vxz169))
new_primDivNatS0(vxz168, vxz169, Succ(vxz1700), Zero) → new_primDivNatS(new_primMinusNatS0(vxz168, vxz169), Succ(vxz169))
new_primDivNatS(Succ(Zero), Zero) → new_primDivNatS(new_primMinusNatS2, Zero)
new_primDivNatS(Succ(Succ(vxz13500)), Zero) → new_primDivNatS(new_primMinusNatS1(vxz13500), Zero)
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS1(vxz13500) → Succ(vxz13500)
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)
new_primMinusNatS2 → Zero
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS2
new_primMinusNatS1(x0)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 1 less node.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS(Succ(Succ(vxz13500)), Zero) → new_primDivNatS(new_primMinusNatS1(vxz13500), Zero)
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS1(vxz13500) → Succ(vxz13500)
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)
new_primMinusNatS2 → Zero
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS2
new_primMinusNatS1(x0)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS(Succ(Succ(vxz13500)), Zero) → new_primDivNatS(new_primMinusNatS1(vxz13500), Zero)
The TRS R consists of the following rules:
new_primMinusNatS1(vxz13500) → Succ(vxz13500)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS2
new_primMinusNatS1(x0)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS2
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS(Succ(Succ(vxz13500)), Zero) → new_primDivNatS(new_primMinusNatS1(vxz13500), Zero)
The TRS R consists of the following rules:
new_primMinusNatS1(vxz13500) → Succ(vxz13500)
The set Q consists of the following terms:
new_primMinusNatS1(x0)
We have to consider all minimal (P,Q,R)-chains.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:
new_primDivNatS(Succ(Succ(vxz13500)), Zero) → new_primDivNatS(new_primMinusNatS1(vxz13500), Zero)
Strictly oriented rules of the TRS R:
new_primMinusNatS1(vxz13500) → Succ(vxz13500)
Used ordering: POLO with Polynomial interpretation [25]:
POL(Succ(x1)) = 1 + 2·x1
POL(Zero) = 0
POL(new_primDivNatS(x1, x2)) = x1 + x2
POL(new_primMinusNatS1(x1)) = 2 + 2·x1
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
P is empty.
R is empty.
The set Q consists of the following terms:
new_primMinusNatS1(x0)
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vxz168, vxz169, Zero, Zero) → new_primDivNatS00(vxz168, vxz169)
new_primDivNatS0(vxz168, vxz169, Succ(vxz1700), Succ(vxz1710)) → new_primDivNatS0(vxz168, vxz169, vxz1700, vxz1710)
new_primDivNatS(Succ(Succ(vxz13500)), Succ(vxz136000)) → new_primDivNatS0(vxz13500, vxz136000, vxz13500, vxz136000)
new_primDivNatS00(vxz168, vxz169) → new_primDivNatS(new_primMinusNatS0(vxz168, vxz169), Succ(vxz169))
new_primDivNatS0(vxz168, vxz169, Succ(vxz1700), Zero) → new_primDivNatS(new_primMinusNatS0(vxz168, vxz169), Succ(vxz169))
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS1(vxz13500) → Succ(vxz13500)
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)
new_primMinusNatS2 → Zero
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS2
new_primMinusNatS1(x0)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vxz168, vxz169, Zero, Zero) → new_primDivNatS00(vxz168, vxz169)
new_primDivNatS0(vxz168, vxz169, Succ(vxz1700), Succ(vxz1710)) → new_primDivNatS0(vxz168, vxz169, vxz1700, vxz1710)
new_primDivNatS(Succ(Succ(vxz13500)), Succ(vxz136000)) → new_primDivNatS0(vxz13500, vxz136000, vxz13500, vxz136000)
new_primDivNatS00(vxz168, vxz169) → new_primDivNatS(new_primMinusNatS0(vxz168, vxz169), Succ(vxz169))
new_primDivNatS0(vxz168, vxz169, Succ(vxz1700), Zero) → new_primDivNatS(new_primMinusNatS0(vxz168, vxz169), Succ(vxz169))
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS2
new_primMinusNatS1(x0)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
new_primMinusNatS2
new_primMinusNatS1(x0)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vxz168, vxz169, Zero, Zero) → new_primDivNatS00(vxz168, vxz169)
new_primDivNatS00(vxz168, vxz169) → new_primDivNatS(new_primMinusNatS0(vxz168, vxz169), Succ(vxz169))
new_primDivNatS(Succ(Succ(vxz13500)), Succ(vxz136000)) → new_primDivNatS0(vxz13500, vxz136000, vxz13500, vxz136000)
new_primDivNatS0(vxz168, vxz169, Succ(vxz1700), Succ(vxz1710)) → new_primDivNatS0(vxz168, vxz169, vxz1700, vxz1710)
new_primDivNatS0(vxz168, vxz169, Succ(vxz1700), Zero) → new_primDivNatS(new_primMinusNatS0(vxz168, vxz169), Succ(vxz169))
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
new_primDivNatS00(vxz168, vxz169) → new_primDivNatS(new_primMinusNatS0(vxz168, vxz169), Succ(vxz169))
new_primDivNatS(Succ(Succ(vxz13500)), Succ(vxz136000)) → new_primDivNatS0(vxz13500, vxz136000, vxz13500, vxz136000)
new_primDivNatS0(vxz168, vxz169, Succ(vxz1700), Zero) → new_primDivNatS(new_primMinusNatS0(vxz168, vxz169), Succ(vxz169))
The remaining pairs can at least be oriented weakly.
new_primDivNatS0(vxz168, vxz169, Zero, Zero) → new_primDivNatS00(vxz168, vxz169)
new_primDivNatS0(vxz168, vxz169, Succ(vxz1700), Succ(vxz1710)) → new_primDivNatS0(vxz168, vxz169, vxz1700, vxz1710)
Used ordering: Polynomial interpretation [25]:
POL(Succ(x1)) = 1 + x1
POL(Zero) = 0
POL(new_primDivNatS(x1, x2)) = x1
POL(new_primDivNatS0(x1, x2, x3, x4)) = 1 + x1
POL(new_primDivNatS00(x1, x2)) = 1 + x1
POL(new_primMinusNatS0(x1, x2)) = x1
The following usable rules [17] were oriented:
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vxz168, vxz169, Zero, Zero) → new_primDivNatS00(vxz168, vxz169)
new_primDivNatS0(vxz168, vxz169, Succ(vxz1700), Succ(vxz1710)) → new_primDivNatS0(vxz168, vxz169, vxz1700, vxz1710)
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vxz168, vxz169, Succ(vxz1700), Succ(vxz1710)) → new_primDivNatS0(vxz168, vxz169, vxz1700, vxz1710)
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vxz168, vxz169, Succ(vxz1700), Succ(vxz1710)) → new_primDivNatS0(vxz168, vxz169, vxz1700, vxz1710)
R is empty.
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primDivNatS0(vxz168, vxz169, Succ(vxz1700), Succ(vxz1710)) → new_primDivNatS0(vxz168, vxz169, vxz1700, vxz1710)
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(vxz168, vxz169, Succ(vxz1700), Succ(vxz1710)) → new_primDivNatS0(vxz168, vxz169, vxz1700, vxz1710)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot3(vxz285, vxz286, vxz287) → new_quot0(vxz285, new_primMinusNatS0(Succ(vxz286), vxz287), vxz287, new_primMinusNatS0(Succ(vxz286), vxz287))
new_quot1(vxz285, vxz286, vxz287, Succ(vxz2880), Zero) → new_quot0(vxz285, new_primMinusNatS0(Succ(vxz286), vxz287), vxz287, new_primMinusNatS0(Succ(vxz286), vxz287))
new_quot0(vxz249, Succ(Succ(vxz25500)), Succ(vxz2510), vxz254) → new_quot1(vxz249, vxz25500, Succ(vxz2510), vxz25500, vxz2510)
new_quot0(vxz249, Succ(Zero), Succ(vxz2510), vxz254) → new_quot2(vxz249, Succ(vxz2510), Zero)
new_quot1(vxz285, vxz286, vxz287, Zero, Succ(vxz2890)) → new_quot2(vxz285, vxz287, Succ(vxz286))
new_quot0(vxz249, Succ(Zero), Zero, vxz254) → new_quot0(vxz249, new_primMinusNatS0(Zero, Zero), Zero, new_primMinusNatS0(Zero, Zero))
new_quot1(vxz285, vxz286, vxz287, Succ(vxz2880), Succ(vxz2890)) → new_quot1(vxz285, vxz286, vxz287, vxz2880, vxz2890)
new_quot(vxz79, vxz8000, vxz3000) → new_quot0(vxz79, Succ(vxz8000), vxz3000, Succ(vxz8000))
new_quot2(vxz79, vxz8000, vxz3000) → new_quot0(vxz79, Succ(vxz8000), vxz3000, Succ(vxz8000))
new_quot0(vxz249, Succ(Succ(vxz25500)), Zero, vxz254) → new_quot0(vxz249, new_primMinusNatS0(Succ(vxz25500), Zero), Zero, new_primMinusNatS0(Succ(vxz25500), Zero))
new_quot1(vxz285, vxz286, vxz287, Zero, Zero) → new_quot3(vxz285, vxz286, vxz287)
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 2 less nodes.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot0(vxz249, Succ(Succ(vxz25500)), Zero, vxz254) → new_quot0(vxz249, new_primMinusNatS0(Succ(vxz25500), Zero), Zero, new_primMinusNatS0(Succ(vxz25500), Zero))
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot0(vxz249, Succ(Succ(vxz25500)), Zero, vxz254) → new_quot0(vxz249, new_primMinusNatS0(Succ(vxz25500), Zero), Zero, new_primMinusNatS0(Succ(vxz25500), Zero))
The TRS R consists of the following rules:
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
We have to consider all minimal (P,Q,R)-chains.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:
new_quot0(vxz249, Succ(Succ(vxz25500)), Zero, vxz254) → new_quot0(vxz249, new_primMinusNatS0(Succ(vxz25500), Zero), Zero, new_primMinusNatS0(Succ(vxz25500), Zero))
Used ordering: POLO with Polynomial interpretation [25]:
POL(Succ(x1)) = 1 + 2·x1
POL(Zero) = 0
POL(new_primMinusNatS0(x1, x2)) = x1 + 2·x2
POL(new_quot0(x1, x2, x3, x4)) = x1 + x2 + x3 + x4
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
P is empty.
The TRS R consists of the following rules:
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot3(vxz285, vxz286, vxz287) → new_quot0(vxz285, new_primMinusNatS0(Succ(vxz286), vxz287), vxz287, new_primMinusNatS0(Succ(vxz286), vxz287))
new_quot1(vxz285, vxz286, vxz287, Succ(vxz2880), Zero) → new_quot0(vxz285, new_primMinusNatS0(Succ(vxz286), vxz287), vxz287, new_primMinusNatS0(Succ(vxz286), vxz287))
new_quot1(vxz285, vxz286, vxz287, Zero, Succ(vxz2890)) → new_quot2(vxz285, vxz287, Succ(vxz286))
new_quot0(vxz249, Succ(Succ(vxz25500)), Succ(vxz2510), vxz254) → new_quot1(vxz249, vxz25500, Succ(vxz2510), vxz25500, vxz2510)
new_quot0(vxz249, Succ(Zero), Succ(vxz2510), vxz254) → new_quot2(vxz249, Succ(vxz2510), Zero)
new_quot1(vxz285, vxz286, vxz287, Succ(vxz2880), Succ(vxz2890)) → new_quot1(vxz285, vxz286, vxz287, vxz2880, vxz2890)
new_quot2(vxz79, vxz8000, vxz3000) → new_quot0(vxz79, Succ(vxz8000), vxz3000, Succ(vxz8000))
new_quot1(vxz285, vxz286, vxz287, Zero, Zero) → new_quot3(vxz285, vxz286, vxz287)
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule new_quot2(vxz79, vxz8000, vxz3000) → new_quot0(vxz79, Succ(vxz8000), vxz3000, Succ(vxz8000)) we obtained the following new rules:
new_quot2(z0, z2, Succ(z1)) → new_quot0(z0, Succ(z2), Succ(z1), Succ(z2))
new_quot2(z0, Succ(z1), Zero) → new_quot0(z0, Succ(Succ(z1)), Zero, Succ(Succ(z1)))
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot2(z0, z2, Succ(z1)) → new_quot0(z0, Succ(z2), Succ(z1), Succ(z2))
new_quot3(vxz285, vxz286, vxz287) → new_quot0(vxz285, new_primMinusNatS0(Succ(vxz286), vxz287), vxz287, new_primMinusNatS0(Succ(vxz286), vxz287))
new_quot2(z0, Succ(z1), Zero) → new_quot0(z0, Succ(Succ(z1)), Zero, Succ(Succ(z1)))
new_quot1(vxz285, vxz286, vxz287, Succ(vxz2880), Zero) → new_quot0(vxz285, new_primMinusNatS0(Succ(vxz286), vxz287), vxz287, new_primMinusNatS0(Succ(vxz286), vxz287))
new_quot0(vxz249, Succ(Zero), Succ(vxz2510), vxz254) → new_quot2(vxz249, Succ(vxz2510), Zero)
new_quot0(vxz249, Succ(Succ(vxz25500)), Succ(vxz2510), vxz254) → new_quot1(vxz249, vxz25500, Succ(vxz2510), vxz25500, vxz2510)
new_quot1(vxz285, vxz286, vxz287, Zero, Succ(vxz2890)) → new_quot2(vxz285, vxz287, Succ(vxz286))
new_quot1(vxz285, vxz286, vxz287, Succ(vxz2880), Succ(vxz2890)) → new_quot1(vxz285, vxz286, vxz287, vxz2880, vxz2890)
new_quot1(vxz285, vxz286, vxz287, Zero, Zero) → new_quot3(vxz285, vxz286, vxz287)
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot3(vxz285, vxz286, vxz287) → new_quot0(vxz285, new_primMinusNatS0(Succ(vxz286), vxz287), vxz287, new_primMinusNatS0(Succ(vxz286), vxz287))
new_quot2(z0, z2, Succ(z1)) → new_quot0(z0, Succ(z2), Succ(z1), Succ(z2))
new_quot1(vxz285, vxz286, vxz287, Succ(vxz2880), Zero) → new_quot0(vxz285, new_primMinusNatS0(Succ(vxz286), vxz287), vxz287, new_primMinusNatS0(Succ(vxz286), vxz287))
new_quot1(vxz285, vxz286, vxz287, Zero, Succ(vxz2890)) → new_quot2(vxz285, vxz287, Succ(vxz286))
new_quot0(vxz249, Succ(Succ(vxz25500)), Succ(vxz2510), vxz254) → new_quot1(vxz249, vxz25500, Succ(vxz2510), vxz25500, vxz2510)
new_quot1(vxz285, vxz286, vxz287, Succ(vxz2880), Succ(vxz2890)) → new_quot1(vxz285, vxz286, vxz287, vxz2880, vxz2890)
new_quot1(vxz285, vxz286, vxz287, Zero, Zero) → new_quot3(vxz285, vxz286, vxz287)
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
new_quot3(vxz285, vxz286, vxz287) → new_quot0(vxz285, new_primMinusNatS0(Succ(vxz286), vxz287), vxz287, new_primMinusNatS0(Succ(vxz286), vxz287))
new_quot1(vxz285, vxz286, vxz287, Succ(vxz2880), Zero) → new_quot0(vxz285, new_primMinusNatS0(Succ(vxz286), vxz287), vxz287, new_primMinusNatS0(Succ(vxz286), vxz287))
The remaining pairs can at least be oriented weakly.
new_quot2(z0, z2, Succ(z1)) → new_quot0(z0, Succ(z2), Succ(z1), Succ(z2))
new_quot1(vxz285, vxz286, vxz287, Zero, Succ(vxz2890)) → new_quot2(vxz285, vxz287, Succ(vxz286))
new_quot0(vxz249, Succ(Succ(vxz25500)), Succ(vxz2510), vxz254) → new_quot1(vxz249, vxz25500, Succ(vxz2510), vxz25500, vxz2510)
new_quot1(vxz285, vxz286, vxz287, Succ(vxz2880), Succ(vxz2890)) → new_quot1(vxz285, vxz286, vxz287, vxz2880, vxz2890)
new_quot1(vxz285, vxz286, vxz287, Zero, Zero) → new_quot3(vxz285, vxz286, vxz287)
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( new_primMinusNatS0(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
Tuple symbols:
M( new_quot2(x1, ..., x3) ) = | 1 | + | | · | x1 | + | | · | x2 | + | | · | x3 |
M( new_quot1(x1, ..., x5) ) = | 1 | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 | + | | · | x5 |
M( new_quot0(x1, ..., x4) ) = | 0 | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
M( new_quot3(x1, ..., x3) ) = | 1 | + | | · | 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_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot2(z0, z2, Succ(z1)) → new_quot0(z0, Succ(z2), Succ(z1), Succ(z2))
new_quot0(vxz249, Succ(Succ(vxz25500)), Succ(vxz2510), vxz254) → new_quot1(vxz249, vxz25500, Succ(vxz2510), vxz25500, vxz2510)
new_quot1(vxz285, vxz286, vxz287, Zero, Succ(vxz2890)) → new_quot2(vxz285, vxz287, Succ(vxz286))
new_quot1(vxz285, vxz286, vxz287, Succ(vxz2880), Succ(vxz2890)) → new_quot1(vxz285, vxz286, vxz287, vxz2880, vxz2890)
new_quot1(vxz285, vxz286, vxz287, Zero, Zero) → new_quot3(vxz285, vxz286, vxz287)
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot2(z0, z2, Succ(z1)) → new_quot0(z0, Succ(z2), Succ(z1), Succ(z2))
new_quot1(vxz285, vxz286, vxz287, Zero, Succ(vxz2890)) → new_quot2(vxz285, vxz287, Succ(vxz286))
new_quot0(vxz249, Succ(Succ(vxz25500)), Succ(vxz2510), vxz254) → new_quot1(vxz249, vxz25500, Succ(vxz2510), vxz25500, vxz2510)
new_quot1(vxz285, vxz286, vxz287, Succ(vxz2880), Succ(vxz2890)) → new_quot1(vxz285, vxz286, vxz287, vxz2880, vxz2890)
The TRS R consists of the following rules:
new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_primMinusNatS0(Zero, Zero) → Zero
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot2(z0, z2, Succ(z1)) → new_quot0(z0, Succ(z2), Succ(z1), Succ(z2))
new_quot1(vxz285, vxz286, vxz287, Zero, Succ(vxz2890)) → new_quot2(vxz285, vxz287, Succ(vxz286))
new_quot0(vxz249, Succ(Succ(vxz25500)), Succ(vxz2510), vxz254) → new_quot1(vxz249, vxz25500, Succ(vxz2510), vxz25500, vxz2510)
new_quot1(vxz285, vxz286, vxz287, Succ(vxz2880), Succ(vxz2890)) → new_quot1(vxz285, vxz286, vxz287, vxz2880, vxz2890)
R is empty.
The set Q consists of the following terms:
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_primMinusNatS0(Zero, Zero)
new_primMinusNatS0(Zero, Succ(x0))
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot2(z0, z2, Succ(z1)) → new_quot0(z0, Succ(z2), Succ(z1), Succ(z2))
new_quot0(vxz249, Succ(Succ(vxz25500)), Succ(vxz2510), vxz254) → new_quot1(vxz249, vxz25500, Succ(vxz2510), vxz25500, vxz2510)
new_quot1(vxz285, vxz286, vxz287, Zero, Succ(vxz2890)) → new_quot2(vxz285, vxz287, Succ(vxz286))
new_quot1(vxz285, vxz286, vxz287, Succ(vxz2880), Succ(vxz2890)) → new_quot1(vxz285, vxz286, vxz287, vxz2880, vxz2890)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule new_quot0(vxz249, Succ(Succ(vxz25500)), Succ(vxz2510), vxz254) → new_quot1(vxz249, vxz25500, Succ(vxz2510), vxz25500, vxz2510) we obtained the following new rules:
new_quot0(z0, Succ(Succ(x1)), Succ(z2), Succ(Succ(x1))) → new_quot1(z0, x1, Succ(z2), x1, z2)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ NonInfProof
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot2(z0, z2, Succ(z1)) → new_quot0(z0, Succ(z2), Succ(z1), Succ(z2))
new_quot1(vxz285, vxz286, vxz287, Zero, Succ(vxz2890)) → new_quot2(vxz285, vxz287, Succ(vxz286))
new_quot1(vxz285, vxz286, vxz287, Succ(vxz2880), Succ(vxz2890)) → new_quot1(vxz285, vxz286, vxz287, vxz2880, vxz2890)
new_quot0(z0, Succ(Succ(x1)), Succ(z2), Succ(Succ(x1))) → new_quot1(z0, x1, Succ(z2), x1, z2)
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_quot2(z0, z2, Succ(z1)) → new_quot0(z0, Succ(z2), Succ(z1), Succ(z2)) the following chains were created:
- We consider the chain new_quot1(vxz285, vxz286, vxz287, Zero, Succ(vxz2890)) → new_quot2(vxz285, vxz287, Succ(vxz286)), new_quot2(z0, z2, Succ(z1)) → new_quot0(z0, Succ(z2), Succ(z1), Succ(z2)) which results in the following constraint:
(1) (new_quot2(x3, x5, Succ(x4))=new_quot2(x7, x8, Succ(x9)) ⇒ new_quot2(x7, x8, Succ(x9))≥new_quot0(x7, Succ(x8), Succ(x9), Succ(x8)))
We simplified constraint (1) using rules (I), (II), (III) which results in the following new constraint:
(2) (new_quot2(x3, x5, Succ(x4))≥new_quot0(x3, Succ(x5), Succ(x4), Succ(x5)))
For Pair new_quot1(vxz285, vxz286, vxz287, Zero, Succ(vxz2890)) → new_quot2(vxz285, vxz287, Succ(vxz286)) the following chains were created:
- We consider the chain new_quot1(vxz285, vxz286, vxz287, Succ(vxz2880), Succ(vxz2890)) → new_quot1(vxz285, vxz286, vxz287, vxz2880, vxz2890), new_quot1(vxz285, vxz286, vxz287, Zero, Succ(vxz2890)) → new_quot2(vxz285, vxz287, Succ(vxz286)) which results in the following constraint:
(3) (new_quot1(x25, x26, x27, x28, x29)=new_quot1(x30, x31, x32, Zero, Succ(x33)) ⇒ new_quot1(x30, x31, x32, Zero, Succ(x33))≥new_quot2(x30, x32, Succ(x31)))
We simplified constraint (3) using rules (I), (II), (III) which results in the following new constraint:
(4) (new_quot1(x25, x26, x27, Zero, Succ(x33))≥new_quot2(x25, x27, Succ(x26)))
- We consider the chain new_quot0(z0, Succ(Succ(x1)), Succ(z2), Succ(Succ(x1))) → new_quot1(z0, x1, Succ(z2), x1, z2), new_quot1(vxz285, vxz286, vxz287, Zero, Succ(vxz2890)) → new_quot2(vxz285, vxz287, Succ(vxz286)) which results in the following constraint:
(5) (new_quot1(x34, x35, Succ(x36), x35, x36)=new_quot1(x37, x38, x39, Zero, Succ(x40)) ⇒ new_quot1(x37, x38, x39, Zero, Succ(x40))≥new_quot2(x37, x39, Succ(x38)))
We simplified constraint (5) using rules (I), (II), (III) which results in the following new constraint:
(6) (new_quot1(x34, Zero, Succ(Succ(x40)), Zero, Succ(x40))≥new_quot2(x34, Succ(Succ(x40)), Succ(Zero)))
For Pair new_quot1(vxz285, vxz286, vxz287, Succ(vxz2880), Succ(vxz2890)) → new_quot1(vxz285, vxz286, vxz287, vxz2880, vxz2890) the following chains were created:
- We consider the chain new_quot1(vxz285, vxz286, vxz287, Succ(vxz2880), Succ(vxz2890)) → new_quot1(vxz285, vxz286, vxz287, vxz2880, vxz2890), new_quot1(vxz285, vxz286, vxz287, Succ(vxz2880), Succ(vxz2890)) → new_quot1(vxz285, vxz286, vxz287, vxz2880, vxz2890) which results in the following constraint:
(7) (new_quot1(x48, x49, x50, x51, x52)=new_quot1(x53, x54, x55, Succ(x56), Succ(x57)) ⇒ new_quot1(x53, x54, x55, Succ(x56), Succ(x57))≥new_quot1(x53, x54, x55, x56, x57))
We simplified constraint (7) using rules (I), (II), (III) which results in the following new constraint:
(8) (new_quot1(x48, x49, x50, Succ(x56), Succ(x57))≥new_quot1(x48, x49, x50, x56, x57))
- We consider the chain new_quot0(z0, Succ(Succ(x1)), Succ(z2), Succ(Succ(x1))) → new_quot1(z0, x1, Succ(z2), x1, z2), new_quot1(vxz285, vxz286, vxz287, Succ(vxz2880), Succ(vxz2890)) → new_quot1(vxz285, vxz286, vxz287, vxz2880, vxz2890) which results in the following constraint:
(9) (new_quot1(x58, x59, Succ(x60), x59, x60)=new_quot1(x61, x62, x63, Succ(x64), Succ(x65)) ⇒ new_quot1(x61, x62, x63, Succ(x64), Succ(x65))≥new_quot1(x61, x62, x63, x64, x65))
We simplified constraint (9) using rules (I), (II), (III) which results in the following new constraint:
(10) (new_quot1(x58, Succ(x64), Succ(Succ(x65)), Succ(x64), Succ(x65))≥new_quot1(x58, Succ(x64), Succ(Succ(x65)), x64, x65))
For Pair new_quot0(z0, Succ(Succ(x1)), Succ(z2), Succ(Succ(x1))) → new_quot1(z0, x1, Succ(z2), x1, z2) the following chains were created:
- We consider the chain new_quot2(z0, z2, Succ(z1)) → new_quot0(z0, Succ(z2), Succ(z1), Succ(z2)), new_quot0(z0, Succ(Succ(x1)), Succ(z2), Succ(Succ(x1))) → new_quot1(z0, x1, Succ(z2), x1, z2) which results in the following constraint:
(11) (new_quot0(x66, Succ(x67), Succ(x68), Succ(x67))=new_quot0(x69, Succ(Succ(x70)), Succ(x71), Succ(Succ(x70))) ⇒ new_quot0(x69, Succ(Succ(x70)), Succ(x71), Succ(Succ(x70)))≥new_quot1(x69, x70, Succ(x71), x70, x71))
We simplified constraint (11) using rules (I), (II), (III) which results in the following new constraint:
(12) (new_quot0(x66, Succ(Succ(x70)), Succ(x68), Succ(Succ(x70)))≥new_quot1(x66, x70, Succ(x68), x70, x68))
To summarize, we get the following constraints P≥ for the following pairs.
- new_quot2(z0, z2, Succ(z1)) → new_quot0(z0, Succ(z2), Succ(z1), Succ(z2))
- (new_quot2(x3, x5, Succ(x4))≥new_quot0(x3, Succ(x5), Succ(x4), Succ(x5)))
- new_quot1(vxz285, vxz286, vxz287, Zero, Succ(vxz2890)) → new_quot2(vxz285, vxz287, Succ(vxz286))
- (new_quot1(x25, x26, x27, Zero, Succ(x33))≥new_quot2(x25, x27, Succ(x26)))
- (new_quot1(x34, Zero, Succ(Succ(x40)), Zero, Succ(x40))≥new_quot2(x34, Succ(Succ(x40)), Succ(Zero)))
- new_quot1(vxz285, vxz286, vxz287, Succ(vxz2880), Succ(vxz2890)) → new_quot1(vxz285, vxz286, vxz287, vxz2880, vxz2890)
- (new_quot1(x48, x49, x50, Succ(x56), Succ(x57))≥new_quot1(x48, x49, x50, x56, x57))
- (new_quot1(x58, Succ(x64), Succ(Succ(x65)), Succ(x64), Succ(x65))≥new_quot1(x58, Succ(x64), Succ(Succ(x65)), x64, x65))
- new_quot0(z0, Succ(Succ(x1)), Succ(z2), Succ(Succ(x1))) → new_quot1(z0, x1, Succ(z2), x1, z2)
- (new_quot0(x66, Succ(Succ(x70)), Succ(x68), Succ(Succ(x70)))≥new_quot1(x66, x70, Succ(x68), x70, x68))
The constraints for P> respective Pbound are constructed from P≥ where we just replace every occurence of "t ≥ s" in P≥ by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation [18]:
POL(Succ(x1)) = 1 + x1
POL(Zero) = 0
POL(c) = -1
POL(new_quot0(x1, x2, x3, x4)) = -1 + x2 + x3 - x4
POL(new_quot1(x1, x2, x3, x4, x5)) = -1 + x2 - x4 + x5
POL(new_quot2(x1, x2, x3)) = -1 + x3
The following pairs are in P>:
new_quot0(z0, Succ(Succ(x1)), Succ(z2), Succ(Succ(x1))) → new_quot1(z0, x1, Succ(z2), x1, z2)
The following pairs are in Pbound:
new_quot2(z0, z2, Succ(z1)) → new_quot0(z0, Succ(z2), Succ(z1), Succ(z2))
new_quot1(vxz285, vxz286, vxz287, Zero, Succ(vxz2890)) → new_quot2(vxz285, vxz287, Succ(vxz286))
new_quot0(z0, Succ(Succ(x1)), Succ(z2), Succ(Succ(x1))) → new_quot1(z0, x1, Succ(z2), x1, z2)
There are no usable rules
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ NonInfProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot2(z0, z2, Succ(z1)) → new_quot0(z0, Succ(z2), Succ(z1), Succ(z2))
new_quot1(vxz285, vxz286, vxz287, Zero, Succ(vxz2890)) → new_quot2(vxz285, vxz287, Succ(vxz286))
new_quot1(vxz285, vxz286, vxz287, Succ(vxz2880), Succ(vxz2890)) → new_quot1(vxz285, vxz286, vxz287, vxz2880, vxz2890)
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 2 less nodes.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ NonInfProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_quot1(vxz285, vxz286, vxz287, Succ(vxz2880), Succ(vxz2890)) → new_quot1(vxz285, vxz286, vxz287, vxz2880, vxz2890)
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(vxz285, vxz286, vxz287, Succ(vxz2880), Succ(vxz2890)) → new_quot1(vxz285, vxz286, vxz287, vxz2880, vxz2890)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4, 5 > 5
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ MNOCProof
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_numericEnumFrom3(vxz3, ba) → new_numericEnumFrom3(new_ps4(vxz3, ba), ba)
The TRS R consists of the following rules:
new_primDivNatS1(Succ(Zero), Zero) → Succ(new_primDivNatS1(new_primMinusNatS2, Zero))
new_quot14(vxz79, Pos(Zero), vxz80, vxz30) → new_quot19(vxz79, vxz80, vxz30)
new_quot19(vxz79, vxz80, Pos(Zero)) → new_quot15(vxz79)
new_gcd0Gcd'19(vxz263, vxz264) → new_gcd0Gcd'17(new_primMinusNatS0(Succ(vxz263), vxz264), vxz264, new_primMinusNatS0(Succ(vxz263), vxz264))
new_quot18(vxz79, Pos(Succ(vxz8000))) → new_quot8(vxz79, vxz8000)
new_primQuotInt0(Pos(vxz510), vxz52, vxz2700) → new_primQuotInt1(vxz510, new_gcd0(vxz52, vxz2700))
new_quot18(vxz79, Pos(Zero)) → Integer(new_primQuotInt4(vxz79))
new_reduce2Reduce10(vxz30, vxz310, vxz30, vxz29, Neg(Succ(vxz3100))) → new_reduce2Reduce1(vxz30, vxz310, vxz30, vxz29)
new_quot13(vxz249, Succ(Succ(vxz25500)), Succ(vxz2510), vxz254) → new_quot9(vxz249, vxz25500, Succ(vxz2510), vxz25500, vxz2510)
new_quot15(vxz79) → error([])
new_primQuotInt5(Neg(vxz790), vxz8000) → new_primQuotInt3(vxz790, Pos(Succ(vxz8000)))
new_ps5(Neg(vxz280), Pos(vxz310)) → new_primPlusInt2(vxz280, vxz310)
new_primQuotInt2(Pos(vxz530), vxz54, vxz2700) → new_primQuotInt1(vxz530, new_gcd(vxz54, vxz2700))
new_gcd(Neg(Succ(vxz5400)), vxz2700) → new_gcd0Gcd'14(vxz2700, vxz5400)
new_primMinusNatS0(Zero, Zero) → Zero
new_gcd0Gcd'18(vxz272, vxz273, Zero, Zero) → new_gcd0Gcd'19(vxz272, vxz273)
new_quot17(vxz79, vxz80, Neg(Succ(vxz3000))) → new_quot16(vxz79, vxz3000, vxz80)
new_reduce2D(vxz137, vxz2700) → new_gcd0(vxz137, vxz2700)
new_primMulInt(Neg(vxz310)) → Neg(new_primMulNat0(vxz310))
new_primPlusInt4(Neg(vxz390), Neg(vxz3100)) → new_primPlusInt3(vxz390, vxz3100)
new_primQuotInt1(vxz135, Pos(Succ(vxz13600))) → Pos(new_primDivNatS1(vxz135, vxz13600))
new_primPlusInt1(Succ(vxz2800), Succ(vxz3100)) → new_primMinusNat2(vxz2800, Succ(vxz3100))
new_reduce(vxz28, vxz31, Pos(Succ(vxz2700))) → :%(new_primQuotInt0(new_ps5(vxz28, vxz31), new_ps5(vxz28, vxz31), vxz2700), new_primQuotInt1(Succ(vxz2700), new_reduce2D(new_ps5(vxz28, vxz31), vxz2700)))
new_quot11(vxz79, vxz8000, vxz3000) → new_quot12(vxz79, vxz8000, vxz3000)
new_primQuotInt5(Pos(vxz790), vxz8000) → new_primQuotInt(Pos(vxz790), vxz8000)
new_primPlusNat4(Zero, Zero) → Zero
new_primDivNatS1(Succ(Succ(vxz13500)), Zero) → Succ(new_primDivNatS1(new_primMinusNatS1(vxz13500), Zero))
new_primQuotInt3(vxz207, Pos(Zero)) → new_error0
new_ps4(:%(vxz30, vxz31), ty_Int) → new_reduce(new_sr(vxz30), vxz31, new_sr(vxz31))
new_primMinusNat3(Zero, Succ(vxz28000)) → Neg(Succ(vxz28000))
new_primPlusInt4(Pos(vxz390), Pos(vxz3100)) → new_primPlusInt0(vxz390, vxz3100)
new_gcd0Gcd'17(Succ(Zero), Zero, vxz236) → new_gcd0Gcd'17(new_primMinusNatS0(Zero, Zero), Zero, new_primMinusNatS0(Zero, Zero))
new_ps4(:%(vxz30, Integer(vxz310)), ty_Integer) → new_reduce2Reduce10(vxz30, vxz310, new_primMulInt(vxz310), new_primMulInt(vxz310), new_primMulInt(vxz310))
new_primPlusInt1(Succ(vxz2800), Zero) → Pos(Succ(vxz2800))
new_gcd0Gcd'112(vxz2700) → new_gcd0Gcd'17(Zero, vxz2700, Zero)
new_primMinusNat2(vxz3100, Zero) → Pos(Succ(vxz3100))
new_quot18(vxz79, Neg(Succ(vxz8000))) → Integer(new_primQuotInt5(vxz79, vxz8000))
new_primDivNatS01(vxz168, vxz169, Zero, Succ(vxz1710)) → Zero
new_primPlusNat(Zero) → Zero
new_primMinusNatS1(vxz13500) → Succ(vxz13500)
new_reduce2Reduce11(vxz30, vxz310, vxz30, vxz29) → error([])
new_sr(vxz31) → new_primMulInt(vxz31)
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)
new_primQuotInt6(Neg(vxz790)) → new_error0
new_ps5(Neg(vxz280), Neg(vxz310)) → new_primPlusInt3(vxz280, vxz310)
new_primQuotInt0(Neg(vxz510), vxz52, vxz2700) → new_primQuotInt3(vxz510, new_gcd0(vxz52, vxz2700))
new_gcd0Gcd'111(vxz2700, vxz5200) → new_gcd0Gcd'15(vxz5200, vxz2700)
new_quot9(vxz285, vxz286, vxz287, Zero, Zero) → new_quot10(vxz285, vxz286, vxz287)
new_primPlusNat2(vxz3100) → Succ(vxz3100)
new_quot16(vxz79, vxz3000, Pos(Zero)) → new_quot13(vxz79, Zero, vxz3000, Zero)
new_quot6(vxz39, vxz310, vxz40, vxz30) → new_quot14(new_primPlusInt4(vxz39, vxz310), new_primPlusInt4(vxz39, vxz310), new_primPlusInt4(vxz39, vxz310), vxz30)
new_primPlusNat4(Succ(vxz28000), Succ(vxz31000)) → Succ(Succ(new_primPlusNat4(vxz28000, vxz31000)))
new_error0 → error([])
new_primPlusInt4(Pos(vxz390), Neg(vxz3100)) → new_primPlusInt1(vxz390, vxz3100)
new_quot14(vxz79, Neg(Succ(vxz8100)), vxz80, vxz30) → new_quot20(vxz79, vxz80, vxz30)
new_quot14(vxz79, Neg(Zero), vxz80, vxz30) → new_quot19(vxz79, vxz80, vxz30)
new_quot9(vxz285, vxz286, vxz287, Succ(vxz2880), Zero) → new_quot10(vxz285, vxz286, vxz287)
new_gcd0(Neg(Succ(vxz5200)), vxz2700) → new_gcd0Gcd'14(vxz2700, vxz5200)
new_primPlusInt3(vxz280, vxz310) → Neg(new_primPlusNat3(vxz280, vxz310))
new_primQuotInt1(vxz135, Neg(Zero)) → new_error0
new_primPlusNat3(Zero, Succ(vxz3100)) → new_primPlusNat2(vxz3100)
new_quot19(vxz79, vxz80, Neg(Zero)) → new_quot15(vxz79)
new_primPlusInt2(vxz280, Succ(vxz3100)) → new_primMinusNat2(vxz3100, vxz280)
new_primPlusInt4(Neg(vxz390), Pos(vxz3100)) → new_primPlusInt2(vxz390, vxz3100)
new_quot19(vxz79, vxz80, Neg(Succ(vxz3000))) → new_quot17(vxz79, vxz80, Neg(Succ(vxz3000)))
new_primPlusNat3(Succ(vxz2800), Zero) → new_primPlusNat(Succ(vxz2800))
new_reduce(vxz28, vxz31, Pos(Zero)) → new_error
new_reduce2Reduce1(vxz30, vxz310, vxz30, vxz29) → :%(new_quot4(vxz30, vxz310, vxz30), new_quot5(vxz29, vxz30, vxz310, vxz30))
new_quot16(vxz79, vxz3000, Neg(Succ(vxz8000))) → new_quot12(vxz79, vxz8000, vxz3000)
new_quot4(Integer(vxz300), vxz310, vxz30) → new_quot6(new_primMulInt(vxz300), vxz310, new_primMulInt(vxz300), vxz30)
new_quot17(vxz79, vxz80, Pos(Succ(vxz3000))) → new_quot16(vxz79, vxz3000, vxz80)
new_gcd0Gcd'16(vxz2700) → Pos(Succ(vxz2700))
new_primQuotInt1(vxz135, Pos(Zero)) → new_error0
new_gcd0Gcd'15(vxz5200, vxz2700) → new_gcd0Gcd'17(Succ(vxz5200), vxz2700, Succ(vxz5200))
new_primPlusInt1(Zero, Succ(vxz3100)) → new_primMinusNat1(new_primPlusNat2(vxz3100))
new_gcd0Gcd'14(vxz2700, vxz5200) → new_gcd0Gcd'15(vxz5200, vxz2700)
new_primMulNat0(Succ(vxz3100)) → new_primPlusNat0(new_primMulNat0(vxz3100))
new_quot13(vxz249, Succ(Zero), Zero, vxz254) → new_quot13(vxz249, new_primMinusNatS0(Zero, Zero), Zero, new_primMinusNatS0(Zero, Zero))
new_primMinusNat3(Succ(vxz31000), Succ(vxz28000)) → new_primMinusNat3(vxz31000, vxz28000)
new_primQuotInt1(vxz135, Neg(Succ(vxz13600))) → Neg(new_primDivNatS1(vxz135, vxz13600))
new_reduce2Reduce10(vxz30, vxz310, vxz30, vxz29, Neg(Zero)) → new_reduce2Reduce11(vxz30, vxz310, vxz30, vxz29)
new_quot18(vxz79, Neg(Zero)) → Integer(new_primQuotInt6(vxz79))
new_gcd0(Neg(Zero), vxz2700) → new_gcd0Gcd'16(vxz2700)
new_primMulInt(Pos(vxz310)) → Pos(new_primMulNat0(vxz310))
new_reduce(vxz28, vxz31, Neg(Succ(vxz2700))) → :%(new_primQuotInt2(new_ps5(vxz28, vxz31), new_ps5(vxz28, vxz31), vxz2700), new_primQuotInt3(Succ(vxz2700), new_reduce2D0(new_ps5(vxz28, vxz31), vxz2700)))
new_ps5(Pos(vxz280), Pos(vxz310)) → new_primPlusInt0(vxz280, vxz310)
new_primPlusNat4(Zero, Succ(vxz31000)) → Succ(vxz31000)
new_primPlusNat4(Succ(vxz28000), Zero) → Succ(vxz28000)
new_primQuotInt2(Neg(vxz530), vxz54, vxz2700) → new_primQuotInt3(vxz530, new_gcd(vxz54, vxz2700))
new_quot14(vxz79, Pos(Succ(vxz8100)), vxz80, vxz30) → new_quot20(vxz79, vxz80, vxz30)
new_quot13(vxz249, Succ(Zero), Succ(vxz2510), vxz254) → new_quot11(vxz249, Succ(vxz2510), Zero)
new_reduce2Reduce10(vxz30, vxz310, vxz30, vxz29, Pos(Zero)) → new_reduce2Reduce11(vxz30, vxz310, vxz30, vxz29)
new_quot17(vxz79, vxz80, Pos(Zero)) → new_quot18(vxz79, vxz80)
new_primQuotInt3(vxz207, Neg(Succ(vxz22600))) → Pos(new_primDivNatS1(vxz207, vxz22600))
new_gcd0Gcd'18(vxz272, vxz273, Zero, Succ(vxz2750)) → new_gcd0Gcd'110(Succ(vxz272), vxz273)
new_primQuotInt3(vxz207, Neg(Zero)) → new_error0
new_gcd0Gcd'18(vxz272, vxz273, Succ(vxz2740), Succ(vxz2750)) → new_gcd0Gcd'18(vxz272, vxz273, vxz2740, vxz2750)
new_primDivNatS01(vxz168, vxz169, Succ(vxz1700), Succ(vxz1710)) → new_primDivNatS01(vxz168, vxz169, vxz1700, vxz1710)
new_primPlusNat(Succ(vxz3000)) → Succ(vxz3000)
new_primPlusNat3(Succ(vxz2800), Succ(vxz3100)) → Succ(Succ(new_primPlusNat4(vxz2800, vxz3100)))
new_primDivNatS01(vxz168, vxz169, Zero, Zero) → new_primDivNatS02(vxz168, vxz169)
new_quot17(vxz79, vxz80, Neg(Zero)) → new_quot18(vxz79, vxz80)
new_reduce2D0(vxz227, vxz2700) → new_gcd(vxz227, vxz2700)
new_primPlusInt2(vxz280, Zero) → new_primMinusNat1(vxz280)
new_primQuotInt4(Neg(vxz790)) → new_error0
new_quot7(vxz29, vxz48, vxz310, vxz47, vxz30) → new_quot14(vxz29, new_primPlusInt4(vxz48, vxz310), new_primPlusInt4(vxz48, vxz310), vxz30)
new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_reduce(vxz28, vxz31, Neg(Zero)) → new_error
new_primQuotInt4(Pos(vxz790)) → new_error0
new_gcd0Gcd'17(Succ(Zero), Succ(vxz2320), vxz236) → new_gcd0Gcd'110(Zero, Succ(vxz2320))
new_quot9(vxz285, vxz286, vxz287, Succ(vxz2880), Succ(vxz2890)) → new_quot9(vxz285, vxz286, vxz287, vxz2880, vxz2890)
new_quot13(vxz249, Zero, vxz251, vxz254) → new_quot8(vxz249, vxz251)
new_quot16(vxz79, vxz3000, Pos(Succ(vxz8000))) → new_quot11(vxz79, vxz8000, vxz3000)
new_quot13(vxz249, Succ(Succ(vxz25500)), Zero, vxz254) → new_quot13(vxz249, new_primMinusNatS0(Succ(vxz25500), Zero), Zero, new_primMinusNatS0(Succ(vxz25500), Zero))
new_primQuotInt(Pos(vxz790), vxz8000) → Pos(new_primDivNatS1(vxz790, vxz8000))
new_primQuotInt6(Pos(vxz790)) → new_error0
new_primPlusInt1(Zero, Zero) → new_primMinusNat1(Zero)
new_gcd(Pos(Succ(vxz5400)), vxz2700) → new_gcd0Gcd'111(vxz2700, vxz5400)
new_primDivNatS01(vxz168, vxz169, Succ(vxz1700), Zero) → new_primDivNatS02(vxz168, vxz169)
new_quot12(vxz79, vxz8000, vxz3000) → new_quot13(vxz79, Succ(vxz8000), vxz3000, Succ(vxz8000))
new_primMulNat0(Zero) → Zero
new_primDivNatS02(vxz168, vxz169) → Succ(new_primDivNatS1(new_primMinusNatS0(vxz168, vxz169), Succ(vxz169)))
new_quot19(vxz79, vxz80, Pos(Succ(vxz3000))) → new_quot17(vxz79, vxz80, Pos(Succ(vxz3000)))
new_gcd0(Pos(Zero), vxz2700) → new_gcd0Gcd'112(vxz2700)
new_gcd(Neg(Zero), vxz2700) → new_gcd0Gcd'16(vxz2700)
new_primDivNatS1(Succ(Succ(vxz13500)), Succ(vxz136000)) → new_primDivNatS01(vxz13500, vxz136000, vxz13500, vxz136000)
new_quot8(vxz79, vxz8000) → Integer(new_primQuotInt(vxz79, vxz8000))
new_primDivNatS1(Zero, vxz13600) → Zero
new_gcd0Gcd'17(Succ(Succ(vxz23700)), Zero, vxz236) → new_gcd0Gcd'19(vxz23700, Zero)
new_primPlusNat0(Succ(vxz300)) → Succ(Succ(new_primPlusNat(vxz300)))
new_primPlusInt0(vxz280, vxz310) → Pos(new_primPlusNat3(vxz280, vxz310))
new_primPlusNat0(Zero) → Succ(Zero)
new_gcd0Gcd'110(vxz266, vxz267) → new_gcd0Gcd'15(vxz267, vxz266)
new_ps5(Pos(vxz280), Neg(vxz310)) → new_primPlusInt1(vxz280, vxz310)
new_error → error([])
new_reduce2Reduce10(vxz30, vxz310, vxz30, vxz29, Pos(Succ(vxz3100))) → new_reduce2Reduce1(vxz30, vxz310, vxz30, vxz29)
new_gcd(Pos(Zero), vxz2700) → new_gcd0Gcd'112(vxz2700)
new_gcd0Gcd'17(Succ(Succ(vxz23700)), Succ(vxz2320), vxz236) → new_gcd0Gcd'18(vxz23700, Succ(vxz2320), vxz23700, vxz2320)
new_quot5(vxz29, Integer(vxz300), vxz310, vxz30) → new_quot7(vxz29, new_primMulInt(vxz300), vxz310, new_primMulInt(vxz300), vxz30)
new_primMinusNat3(Succ(vxz31000), Zero) → Pos(Succ(vxz31000))
new_primMinusNat1(Zero) → Pos(Zero)
new_quot16(vxz79, vxz3000, Neg(Zero)) → new_quot8(vxz79, vxz3000)
new_primQuotInt(Neg(vxz790), vxz8000) → Neg(new_primDivNatS1(vxz790, vxz8000))
new_primMinusNat3(Zero, Zero) → Pos(Zero)
new_gcd0Gcd'18(vxz272, vxz273, Succ(vxz2740), Zero) → new_gcd0Gcd'19(vxz272, vxz273)
new_primMinusNat1(Succ(vxz3000)) → Neg(Succ(vxz3000))
new_primMinusNatS2 → Zero
new_gcd0(Pos(Succ(vxz5200)), vxz2700) → new_gcd0Gcd'111(vxz2700, vxz5200)
new_primPlusNat3(Zero, Zero) → new_primPlusNat(Zero)
new_quot9(vxz285, vxz286, vxz287, Zero, Succ(vxz2890)) → new_quot11(vxz285, vxz287, Succ(vxz286))
new_quot10(vxz285, vxz286, vxz287) → new_quot13(vxz285, new_primMinusNatS0(Succ(vxz286), vxz287), vxz287, new_primMinusNatS0(Succ(vxz286), vxz287))
new_gcd0Gcd'17(Zero, vxz232, vxz236) → Pos(Succ(vxz232))
new_quot20(vxz79, vxz80, vxz30) → new_quot17(vxz79, vxz80, vxz30)
new_primDivNatS1(Succ(Zero), Succ(vxz136000)) → Zero
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primQuotInt3(vxz207, Pos(Succ(vxz22600))) → Neg(new_primDivNatS1(vxz207, vxz22600))
new_primMinusNat2(vxz3100, Succ(vxz2800)) → new_primMinusNat3(vxz3100, vxz2800)
The set Q consists of the following terms:
new_gcd0Gcd'16(x0)
new_gcd0Gcd'17(Zero, x0, x1)
new_ps5(Neg(x0), Neg(x1))
new_primMinusNatS2
new_reduce2D0(x0, x1)
new_gcd0Gcd'111(x0, x1)
new_gcd0Gcd'14(x0, x1)
new_primQuotInt2(Pos(x0), x1, x2)
new_quot19(x0, x1, Neg(Succ(x2)))
new_primPlusNat3(Succ(x0), Succ(x1))
new_reduce(x0, x1, Pos(Zero))
new_primPlusInt4(Neg(x0), Neg(x1))
new_primPlusNat3(Zero, Zero)
new_quot16(x0, x1, Pos(Succ(x2)))
new_primQuotInt4(Pos(x0))
new_gcd(Pos(Succ(x0)), x1)
new_quot4(Integer(x0), x1, x2)
new_quot10(x0, x1, x2)
new_primPlusInt2(x0, Zero)
new_primMinusNatS0(Zero, Succ(x0))
new_primPlusNat4(Succ(x0), Succ(x1))
new_reduce(x0, x1, Neg(Succ(x2)))
new_primDivNatS02(x0, x1)
new_reduce(x0, x1, Pos(Succ(x2)))
new_primPlusNat2(x0)
new_primMulInt(Neg(x0))
new_primQuotInt5(Neg(x0), x1)
new_gcd0(Pos(Succ(x0)), x1)
new_error
new_quot13(x0, Zero, x1, x2)
new_primPlusNat(Succ(x0))
new_primQuotInt3(x0, Pos(Succ(x1)))
new_quot16(x0, x1, Neg(Zero))
new_primDivNatS01(x0, x1, Succ(x2), Zero)
new_quot17(x0, x1, Neg(Zero))
new_ps4(:%(x0, x1), ty_Int)
new_quot9(x0, x1, x2, Succ(x3), Succ(x4))
new_quot11(x0, x1, x2)
new_quot16(x0, x1, Pos(Zero))
new_quot14(x0, Neg(Succ(x1)), x2, x3)
new_quot17(x0, x1, Pos(Succ(x2)))
new_primDivNatS01(x0, x1, Zero, Zero)
new_quot13(x0, Succ(Succ(x1)), Succ(x2), x3)
new_primPlusNat4(Zero, Zero)
new_primQuotInt6(Neg(x0))
new_primQuotInt(Neg(x0), x1)
new_primQuotInt4(Neg(x0))
new_primQuotInt1(x0, Pos(Succ(x1)))
new_primMinusNat1(Zero)
new_reduce2Reduce10(x0, x1, x0, x2, Neg(Succ(x3)))
new_reduce(x0, x1, Neg(Zero))
new_quot13(x0, Succ(Zero), Succ(x1), x2)
new_primQuotInt6(Pos(x0))
new_primPlusInt2(x0, Succ(x1))
new_quot15(x0)
new_gcd0Gcd'17(Succ(Succ(x0)), Zero, x1)
new_quot17(x0, x1, Pos(Zero))
new_primMinusNatS1(x0)
new_gcd0(Neg(Succ(x0)), x1)
new_quot18(x0, Neg(Zero))
new_ps5(Pos(x0), Pos(x1))
new_primQuotInt3(x0, Neg(Zero))
new_gcd0Gcd'18(x0, x1, Succ(x2), Succ(x3))
new_primQuotInt3(x0, Pos(Zero))
new_gcd0Gcd'18(x0, x1, Zero, Zero)
new_primPlusNat3(Zero, Succ(x0))
new_primMinusNat2(x0, Succ(x1))
new_sr(x0)
new_gcd0Gcd'15(x0, x1)
new_reduce2Reduce10(x0, x1, x0, x2, Pos(Zero))
new_gcd0Gcd'17(Succ(Zero), Succ(x0), x1)
new_primQuotInt1(x0, Neg(Zero))
new_primDivNatS1(Zero, x0)
new_quot9(x0, x1, x2, Zero, Succ(x3))
new_quot12(x0, x1, x2)
new_primPlusInt0(x0, x1)
new_primPlusInt1(Succ(x0), Zero)
new_primDivNatS1(Succ(Succ(x0)), Zero)
new_gcd0(Pos(Zero), x0)
new_primQuotInt2(Neg(x0), x1, x2)
new_primPlusNat0(Succ(x0))
new_gcd0Gcd'18(x0, x1, Succ(x2), Zero)
new_primMinusNat3(Zero, Succ(x0))
new_primPlusInt3(x0, x1)
new_primPlusNat4(Succ(x0), Zero)
new_primPlusInt1(Succ(x0), Succ(x1))
new_reduce2Reduce10(x0, x1, x0, x2, Pos(Succ(x3)))
new_quot19(x0, x1, Neg(Zero))
new_gcd0Gcd'112(x0)
new_quot18(x0, Pos(Zero))
new_primMinusNat2(x0, Zero)
new_gcd0Gcd'18(x0, x1, Zero, Succ(x2))
new_reduce2Reduce1(x0, x1, x0, x2)
new_primPlusInt4(Pos(x0), Neg(x1))
new_primPlusInt4(Neg(x0), Pos(x1))
new_primMulNat0(Zero)
new_primMinusNatS0(Zero, Zero)
new_primQuotInt0(Neg(x0), x1, x2)
new_quot19(x0, x1, Pos(Succ(x2)))
new_quot18(x0, Neg(Succ(x1)))
new_quot16(x0, x1, Neg(Succ(x2)))
new_quot17(x0, x1, Neg(Succ(x2)))
new_primDivNatS1(Succ(Zero), Zero)
new_gcd(Neg(Zero), x0)
new_gcd0Gcd'19(x0, x1)
new_primDivNatS01(x0, x1, Succ(x2), Succ(x3))
new_primMinusNat3(Succ(x0), Succ(x1))
new_error0
new_quot9(x0, x1, x2, Zero, Zero)
new_primQuotInt5(Pos(x0), x1)
new_quot7(x0, x1, x2, x3, x4)
new_reduce2Reduce11(x0, x1, x0, x2)
new_primMulNat0(Succ(x0))
new_primPlusNat0(Zero)
new_gcd0Gcd'17(Succ(Zero), Zero, x0)
new_primMinusNat3(Zero, Zero)
new_gcd0Gcd'17(Succ(Succ(x0)), Succ(x1), x2)
new_reduce2Reduce10(x0, x1, x0, x2, Neg(Zero))
new_primPlusNat4(Zero, Succ(x0))
new_primQuotInt(Pos(x0), x1)
new_ps4(:%(x0, Integer(x1)), ty_Integer)
new_primQuotInt3(x0, Neg(Succ(x1)))
new_quot8(x0, x1)
new_primMinusNatS0(Succ(x0), Zero)
new_primMinusNat3(Succ(x0), Zero)
new_primPlusNat3(Succ(x0), Zero)
new_primPlusInt4(Pos(x0), Pos(x1))
new_quot5(x0, Integer(x1), x2, x3)
new_primPlusInt1(Zero, Zero)
new_primPlusNat(Zero)
new_quot6(x0, x1, x2, x3)
new_quot14(x0, Neg(Zero), x1, x2)
new_primQuotInt1(x0, Pos(Zero))
new_gcd0(Neg(Zero), x0)
new_reduce2D(x0, x1)
new_primMulInt(Pos(x0))
new_primDivNatS1(Succ(Zero), Succ(x0))
new_gcd0Gcd'110(x0, x1)
new_quot18(x0, Pos(Succ(x1)))
new_primMinusNat1(Succ(x0))
new_primDivNatS1(Succ(Succ(x0)), Succ(x1))
new_quot9(x0, x1, x2, Succ(x3), Zero)
new_quot14(x0, Pos(Succ(x1)), x2, x3)
new_gcd(Neg(Succ(x0)), x1)
new_primDivNatS01(x0, x1, Zero, Succ(x2))
new_primQuotInt0(Pos(x0), x1, x2)
new_ps5(Pos(x0), Neg(x1))
new_ps5(Neg(x0), Pos(x1))
new_primPlusInt1(Zero, Succ(x0))
new_gcd(Pos(Zero), x0)
new_primMinusNatS0(Succ(x0), Succ(x1))
new_quot14(x0, Pos(Zero), x1, x2)
new_quot13(x0, Succ(Zero), Zero, x1)
new_primQuotInt1(x0, Neg(Succ(x1)))
new_quot19(x0, x1, Pos(Zero))
new_quot13(x0, Succ(Succ(x1)), Zero, x2)
new_quot20(x0, x1, x2)
We have to consider all minimal (P,Q,R)-chains.
We use the modular non-overlap check [17] to decrease Q to the empty set.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ MNOCProof
↳ QDP
↳ NonTerminationProof
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_numericEnumFrom3(vxz3, ba) → new_numericEnumFrom3(new_ps4(vxz3, ba), ba)
The TRS R consists of the following rules:
new_primDivNatS1(Succ(Zero), Zero) → Succ(new_primDivNatS1(new_primMinusNatS2, Zero))
new_quot14(vxz79, Pos(Zero), vxz80, vxz30) → new_quot19(vxz79, vxz80, vxz30)
new_quot19(vxz79, vxz80, Pos(Zero)) → new_quot15(vxz79)
new_gcd0Gcd'19(vxz263, vxz264) → new_gcd0Gcd'17(new_primMinusNatS0(Succ(vxz263), vxz264), vxz264, new_primMinusNatS0(Succ(vxz263), vxz264))
new_quot18(vxz79, Pos(Succ(vxz8000))) → new_quot8(vxz79, vxz8000)
new_primQuotInt0(Pos(vxz510), vxz52, vxz2700) → new_primQuotInt1(vxz510, new_gcd0(vxz52, vxz2700))
new_quot18(vxz79, Pos(Zero)) → Integer(new_primQuotInt4(vxz79))
new_reduce2Reduce10(vxz30, vxz310, vxz30, vxz29, Neg(Succ(vxz3100))) → new_reduce2Reduce1(vxz30, vxz310, vxz30, vxz29)
new_quot13(vxz249, Succ(Succ(vxz25500)), Succ(vxz2510), vxz254) → new_quot9(vxz249, vxz25500, Succ(vxz2510), vxz25500, vxz2510)
new_quot15(vxz79) → error([])
new_primQuotInt5(Neg(vxz790), vxz8000) → new_primQuotInt3(vxz790, Pos(Succ(vxz8000)))
new_ps5(Neg(vxz280), Pos(vxz310)) → new_primPlusInt2(vxz280, vxz310)
new_primQuotInt2(Pos(vxz530), vxz54, vxz2700) → new_primQuotInt1(vxz530, new_gcd(vxz54, vxz2700))
new_gcd(Neg(Succ(vxz5400)), vxz2700) → new_gcd0Gcd'14(vxz2700, vxz5400)
new_primMinusNatS0(Zero, Zero) → Zero
new_gcd0Gcd'18(vxz272, vxz273, Zero, Zero) → new_gcd0Gcd'19(vxz272, vxz273)
new_quot17(vxz79, vxz80, Neg(Succ(vxz3000))) → new_quot16(vxz79, vxz3000, vxz80)
new_reduce2D(vxz137, vxz2700) → new_gcd0(vxz137, vxz2700)
new_primMulInt(Neg(vxz310)) → Neg(new_primMulNat0(vxz310))
new_primPlusInt4(Neg(vxz390), Neg(vxz3100)) → new_primPlusInt3(vxz390, vxz3100)
new_primQuotInt1(vxz135, Pos(Succ(vxz13600))) → Pos(new_primDivNatS1(vxz135, vxz13600))
new_primPlusInt1(Succ(vxz2800), Succ(vxz3100)) → new_primMinusNat2(vxz2800, Succ(vxz3100))
new_reduce(vxz28, vxz31, Pos(Succ(vxz2700))) → :%(new_primQuotInt0(new_ps5(vxz28, vxz31), new_ps5(vxz28, vxz31), vxz2700), new_primQuotInt1(Succ(vxz2700), new_reduce2D(new_ps5(vxz28, vxz31), vxz2700)))
new_quot11(vxz79, vxz8000, vxz3000) → new_quot12(vxz79, vxz8000, vxz3000)
new_primQuotInt5(Pos(vxz790), vxz8000) → new_primQuotInt(Pos(vxz790), vxz8000)
new_primPlusNat4(Zero, Zero) → Zero
new_primDivNatS1(Succ(Succ(vxz13500)), Zero) → Succ(new_primDivNatS1(new_primMinusNatS1(vxz13500), Zero))
new_primQuotInt3(vxz207, Pos(Zero)) → new_error0
new_ps4(:%(vxz30, vxz31), ty_Int) → new_reduce(new_sr(vxz30), vxz31, new_sr(vxz31))
new_primMinusNat3(Zero, Succ(vxz28000)) → Neg(Succ(vxz28000))
new_primPlusInt4(Pos(vxz390), Pos(vxz3100)) → new_primPlusInt0(vxz390, vxz3100)
new_gcd0Gcd'17(Succ(Zero), Zero, vxz236) → new_gcd0Gcd'17(new_primMinusNatS0(Zero, Zero), Zero, new_primMinusNatS0(Zero, Zero))
new_ps4(:%(vxz30, Integer(vxz310)), ty_Integer) → new_reduce2Reduce10(vxz30, vxz310, new_primMulInt(vxz310), new_primMulInt(vxz310), new_primMulInt(vxz310))
new_primPlusInt1(Succ(vxz2800), Zero) → Pos(Succ(vxz2800))
new_gcd0Gcd'112(vxz2700) → new_gcd0Gcd'17(Zero, vxz2700, Zero)
new_primMinusNat2(vxz3100, Zero) → Pos(Succ(vxz3100))
new_quot18(vxz79, Neg(Succ(vxz8000))) → Integer(new_primQuotInt5(vxz79, vxz8000))
new_primDivNatS01(vxz168, vxz169, Zero, Succ(vxz1710)) → Zero
new_primPlusNat(Zero) → Zero
new_primMinusNatS1(vxz13500) → Succ(vxz13500)
new_reduce2Reduce11(vxz30, vxz310, vxz30, vxz29) → error([])
new_sr(vxz31) → new_primMulInt(vxz31)
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)
new_primQuotInt6(Neg(vxz790)) → new_error0
new_ps5(Neg(vxz280), Neg(vxz310)) → new_primPlusInt3(vxz280, vxz310)
new_primQuotInt0(Neg(vxz510), vxz52, vxz2700) → new_primQuotInt3(vxz510, new_gcd0(vxz52, vxz2700))
new_gcd0Gcd'111(vxz2700, vxz5200) → new_gcd0Gcd'15(vxz5200, vxz2700)
new_quot9(vxz285, vxz286, vxz287, Zero, Zero) → new_quot10(vxz285, vxz286, vxz287)
new_primPlusNat2(vxz3100) → Succ(vxz3100)
new_quot16(vxz79, vxz3000, Pos(Zero)) → new_quot13(vxz79, Zero, vxz3000, Zero)
new_quot6(vxz39, vxz310, vxz40, vxz30) → new_quot14(new_primPlusInt4(vxz39, vxz310), new_primPlusInt4(vxz39, vxz310), new_primPlusInt4(vxz39, vxz310), vxz30)
new_primPlusNat4(Succ(vxz28000), Succ(vxz31000)) → Succ(Succ(new_primPlusNat4(vxz28000, vxz31000)))
new_error0 → error([])
new_primPlusInt4(Pos(vxz390), Neg(vxz3100)) → new_primPlusInt1(vxz390, vxz3100)
new_quot14(vxz79, Neg(Succ(vxz8100)), vxz80, vxz30) → new_quot20(vxz79, vxz80, vxz30)
new_quot14(vxz79, Neg(Zero), vxz80, vxz30) → new_quot19(vxz79, vxz80, vxz30)
new_quot9(vxz285, vxz286, vxz287, Succ(vxz2880), Zero) → new_quot10(vxz285, vxz286, vxz287)
new_gcd0(Neg(Succ(vxz5200)), vxz2700) → new_gcd0Gcd'14(vxz2700, vxz5200)
new_primPlusInt3(vxz280, vxz310) → Neg(new_primPlusNat3(vxz280, vxz310))
new_primQuotInt1(vxz135, Neg(Zero)) → new_error0
new_primPlusNat3(Zero, Succ(vxz3100)) → new_primPlusNat2(vxz3100)
new_quot19(vxz79, vxz80, Neg(Zero)) → new_quot15(vxz79)
new_primPlusInt2(vxz280, Succ(vxz3100)) → new_primMinusNat2(vxz3100, vxz280)
new_primPlusInt4(Neg(vxz390), Pos(vxz3100)) → new_primPlusInt2(vxz390, vxz3100)
new_quot19(vxz79, vxz80, Neg(Succ(vxz3000))) → new_quot17(vxz79, vxz80, Neg(Succ(vxz3000)))
new_primPlusNat3(Succ(vxz2800), Zero) → new_primPlusNat(Succ(vxz2800))
new_reduce(vxz28, vxz31, Pos(Zero)) → new_error
new_reduce2Reduce1(vxz30, vxz310, vxz30, vxz29) → :%(new_quot4(vxz30, vxz310, vxz30), new_quot5(vxz29, vxz30, vxz310, vxz30))
new_quot16(vxz79, vxz3000, Neg(Succ(vxz8000))) → new_quot12(vxz79, vxz8000, vxz3000)
new_quot4(Integer(vxz300), vxz310, vxz30) → new_quot6(new_primMulInt(vxz300), vxz310, new_primMulInt(vxz300), vxz30)
new_quot17(vxz79, vxz80, Pos(Succ(vxz3000))) → new_quot16(vxz79, vxz3000, vxz80)
new_gcd0Gcd'16(vxz2700) → Pos(Succ(vxz2700))
new_primQuotInt1(vxz135, Pos(Zero)) → new_error0
new_gcd0Gcd'15(vxz5200, vxz2700) → new_gcd0Gcd'17(Succ(vxz5200), vxz2700, Succ(vxz5200))
new_primPlusInt1(Zero, Succ(vxz3100)) → new_primMinusNat1(new_primPlusNat2(vxz3100))
new_gcd0Gcd'14(vxz2700, vxz5200) → new_gcd0Gcd'15(vxz5200, vxz2700)
new_primMulNat0(Succ(vxz3100)) → new_primPlusNat0(new_primMulNat0(vxz3100))
new_quot13(vxz249, Succ(Zero), Zero, vxz254) → new_quot13(vxz249, new_primMinusNatS0(Zero, Zero), Zero, new_primMinusNatS0(Zero, Zero))
new_primMinusNat3(Succ(vxz31000), Succ(vxz28000)) → new_primMinusNat3(vxz31000, vxz28000)
new_primQuotInt1(vxz135, Neg(Succ(vxz13600))) → Neg(new_primDivNatS1(vxz135, vxz13600))
new_reduce2Reduce10(vxz30, vxz310, vxz30, vxz29, Neg(Zero)) → new_reduce2Reduce11(vxz30, vxz310, vxz30, vxz29)
new_quot18(vxz79, Neg(Zero)) → Integer(new_primQuotInt6(vxz79))
new_gcd0(Neg(Zero), vxz2700) → new_gcd0Gcd'16(vxz2700)
new_primMulInt(Pos(vxz310)) → Pos(new_primMulNat0(vxz310))
new_reduce(vxz28, vxz31, Neg(Succ(vxz2700))) → :%(new_primQuotInt2(new_ps5(vxz28, vxz31), new_ps5(vxz28, vxz31), vxz2700), new_primQuotInt3(Succ(vxz2700), new_reduce2D0(new_ps5(vxz28, vxz31), vxz2700)))
new_ps5(Pos(vxz280), Pos(vxz310)) → new_primPlusInt0(vxz280, vxz310)
new_primPlusNat4(Zero, Succ(vxz31000)) → Succ(vxz31000)
new_primPlusNat4(Succ(vxz28000), Zero) → Succ(vxz28000)
new_primQuotInt2(Neg(vxz530), vxz54, vxz2700) → new_primQuotInt3(vxz530, new_gcd(vxz54, vxz2700))
new_quot14(vxz79, Pos(Succ(vxz8100)), vxz80, vxz30) → new_quot20(vxz79, vxz80, vxz30)
new_quot13(vxz249, Succ(Zero), Succ(vxz2510), vxz254) → new_quot11(vxz249, Succ(vxz2510), Zero)
new_reduce2Reduce10(vxz30, vxz310, vxz30, vxz29, Pos(Zero)) → new_reduce2Reduce11(vxz30, vxz310, vxz30, vxz29)
new_quot17(vxz79, vxz80, Pos(Zero)) → new_quot18(vxz79, vxz80)
new_primQuotInt3(vxz207, Neg(Succ(vxz22600))) → Pos(new_primDivNatS1(vxz207, vxz22600))
new_gcd0Gcd'18(vxz272, vxz273, Zero, Succ(vxz2750)) → new_gcd0Gcd'110(Succ(vxz272), vxz273)
new_primQuotInt3(vxz207, Neg(Zero)) → new_error0
new_gcd0Gcd'18(vxz272, vxz273, Succ(vxz2740), Succ(vxz2750)) → new_gcd0Gcd'18(vxz272, vxz273, vxz2740, vxz2750)
new_primDivNatS01(vxz168, vxz169, Succ(vxz1700), Succ(vxz1710)) → new_primDivNatS01(vxz168, vxz169, vxz1700, vxz1710)
new_primPlusNat(Succ(vxz3000)) → Succ(vxz3000)
new_primPlusNat3(Succ(vxz2800), Succ(vxz3100)) → Succ(Succ(new_primPlusNat4(vxz2800, vxz3100)))
new_primDivNatS01(vxz168, vxz169, Zero, Zero) → new_primDivNatS02(vxz168, vxz169)
new_quot17(vxz79, vxz80, Neg(Zero)) → new_quot18(vxz79, vxz80)
new_reduce2D0(vxz227, vxz2700) → new_gcd(vxz227, vxz2700)
new_primPlusInt2(vxz280, Zero) → new_primMinusNat1(vxz280)
new_primQuotInt4(Neg(vxz790)) → new_error0
new_quot7(vxz29, vxz48, vxz310, vxz47, vxz30) → new_quot14(vxz29, new_primPlusInt4(vxz48, vxz310), new_primPlusInt4(vxz48, vxz310), vxz30)
new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_reduce(vxz28, vxz31, Neg(Zero)) → new_error
new_primQuotInt4(Pos(vxz790)) → new_error0
new_gcd0Gcd'17(Succ(Zero), Succ(vxz2320), vxz236) → new_gcd0Gcd'110(Zero, Succ(vxz2320))
new_quot9(vxz285, vxz286, vxz287, Succ(vxz2880), Succ(vxz2890)) → new_quot9(vxz285, vxz286, vxz287, vxz2880, vxz2890)
new_quot13(vxz249, Zero, vxz251, vxz254) → new_quot8(vxz249, vxz251)
new_quot16(vxz79, vxz3000, Pos(Succ(vxz8000))) → new_quot11(vxz79, vxz8000, vxz3000)
new_quot13(vxz249, Succ(Succ(vxz25500)), Zero, vxz254) → new_quot13(vxz249, new_primMinusNatS0(Succ(vxz25500), Zero), Zero, new_primMinusNatS0(Succ(vxz25500), Zero))
new_primQuotInt(Pos(vxz790), vxz8000) → Pos(new_primDivNatS1(vxz790, vxz8000))
new_primQuotInt6(Pos(vxz790)) → new_error0
new_primPlusInt1(Zero, Zero) → new_primMinusNat1(Zero)
new_gcd(Pos(Succ(vxz5400)), vxz2700) → new_gcd0Gcd'111(vxz2700, vxz5400)
new_primDivNatS01(vxz168, vxz169, Succ(vxz1700), Zero) → new_primDivNatS02(vxz168, vxz169)
new_quot12(vxz79, vxz8000, vxz3000) → new_quot13(vxz79, Succ(vxz8000), vxz3000, Succ(vxz8000))
new_primMulNat0(Zero) → Zero
new_primDivNatS02(vxz168, vxz169) → Succ(new_primDivNatS1(new_primMinusNatS0(vxz168, vxz169), Succ(vxz169)))
new_quot19(vxz79, vxz80, Pos(Succ(vxz3000))) → new_quot17(vxz79, vxz80, Pos(Succ(vxz3000)))
new_gcd0(Pos(Zero), vxz2700) → new_gcd0Gcd'112(vxz2700)
new_gcd(Neg(Zero), vxz2700) → new_gcd0Gcd'16(vxz2700)
new_primDivNatS1(Succ(Succ(vxz13500)), Succ(vxz136000)) → new_primDivNatS01(vxz13500, vxz136000, vxz13500, vxz136000)
new_quot8(vxz79, vxz8000) → Integer(new_primQuotInt(vxz79, vxz8000))
new_primDivNatS1(Zero, vxz13600) → Zero
new_gcd0Gcd'17(Succ(Succ(vxz23700)), Zero, vxz236) → new_gcd0Gcd'19(vxz23700, Zero)
new_primPlusNat0(Succ(vxz300)) → Succ(Succ(new_primPlusNat(vxz300)))
new_primPlusInt0(vxz280, vxz310) → Pos(new_primPlusNat3(vxz280, vxz310))
new_primPlusNat0(Zero) → Succ(Zero)
new_gcd0Gcd'110(vxz266, vxz267) → new_gcd0Gcd'15(vxz267, vxz266)
new_ps5(Pos(vxz280), Neg(vxz310)) → new_primPlusInt1(vxz280, vxz310)
new_error → error([])
new_reduce2Reduce10(vxz30, vxz310, vxz30, vxz29, Pos(Succ(vxz3100))) → new_reduce2Reduce1(vxz30, vxz310, vxz30, vxz29)
new_gcd(Pos(Zero), vxz2700) → new_gcd0Gcd'112(vxz2700)
new_gcd0Gcd'17(Succ(Succ(vxz23700)), Succ(vxz2320), vxz236) → new_gcd0Gcd'18(vxz23700, Succ(vxz2320), vxz23700, vxz2320)
new_quot5(vxz29, Integer(vxz300), vxz310, vxz30) → new_quot7(vxz29, new_primMulInt(vxz300), vxz310, new_primMulInt(vxz300), vxz30)
new_primMinusNat3(Succ(vxz31000), Zero) → Pos(Succ(vxz31000))
new_primMinusNat1(Zero) → Pos(Zero)
new_quot16(vxz79, vxz3000, Neg(Zero)) → new_quot8(vxz79, vxz3000)
new_primQuotInt(Neg(vxz790), vxz8000) → Neg(new_primDivNatS1(vxz790, vxz8000))
new_primMinusNat3(Zero, Zero) → Pos(Zero)
new_gcd0Gcd'18(vxz272, vxz273, Succ(vxz2740), Zero) → new_gcd0Gcd'19(vxz272, vxz273)
new_primMinusNat1(Succ(vxz3000)) → Neg(Succ(vxz3000))
new_primMinusNatS2 → Zero
new_gcd0(Pos(Succ(vxz5200)), vxz2700) → new_gcd0Gcd'111(vxz2700, vxz5200)
new_primPlusNat3(Zero, Zero) → new_primPlusNat(Zero)
new_quot9(vxz285, vxz286, vxz287, Zero, Succ(vxz2890)) → new_quot11(vxz285, vxz287, Succ(vxz286))
new_quot10(vxz285, vxz286, vxz287) → new_quot13(vxz285, new_primMinusNatS0(Succ(vxz286), vxz287), vxz287, new_primMinusNatS0(Succ(vxz286), vxz287))
new_gcd0Gcd'17(Zero, vxz232, vxz236) → Pos(Succ(vxz232))
new_quot20(vxz79, vxz80, vxz30) → new_quot17(vxz79, vxz80, vxz30)
new_primDivNatS1(Succ(Zero), Succ(vxz136000)) → Zero
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primQuotInt3(vxz207, Pos(Succ(vxz22600))) → Neg(new_primDivNatS1(vxz207, vxz22600))
new_primMinusNat2(vxz3100, Succ(vxz2800)) → new_primMinusNat3(vxz3100, vxz2800)
Q is empty.
We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.
The TRS P consists of the following rules:
new_numericEnumFrom3(vxz3, ba) → new_numericEnumFrom3(new_ps4(vxz3, ba), ba)
The TRS R consists of the following rules:
new_primDivNatS1(Succ(Zero), Zero) → Succ(new_primDivNatS1(new_primMinusNatS2, Zero))
new_quot14(vxz79, Pos(Zero), vxz80, vxz30) → new_quot19(vxz79, vxz80, vxz30)
new_quot19(vxz79, vxz80, Pos(Zero)) → new_quot15(vxz79)
new_gcd0Gcd'19(vxz263, vxz264) → new_gcd0Gcd'17(new_primMinusNatS0(Succ(vxz263), vxz264), vxz264, new_primMinusNatS0(Succ(vxz263), vxz264))
new_quot18(vxz79, Pos(Succ(vxz8000))) → new_quot8(vxz79, vxz8000)
new_primQuotInt0(Pos(vxz510), vxz52, vxz2700) → new_primQuotInt1(vxz510, new_gcd0(vxz52, vxz2700))
new_quot18(vxz79, Pos(Zero)) → Integer(new_primQuotInt4(vxz79))
new_reduce2Reduce10(vxz30, vxz310, vxz30, vxz29, Neg(Succ(vxz3100))) → new_reduce2Reduce1(vxz30, vxz310, vxz30, vxz29)
new_quot13(vxz249, Succ(Succ(vxz25500)), Succ(vxz2510), vxz254) → new_quot9(vxz249, vxz25500, Succ(vxz2510), vxz25500, vxz2510)
new_quot15(vxz79) → error([])
new_primQuotInt5(Neg(vxz790), vxz8000) → new_primQuotInt3(vxz790, Pos(Succ(vxz8000)))
new_ps5(Neg(vxz280), Pos(vxz310)) → new_primPlusInt2(vxz280, vxz310)
new_primQuotInt2(Pos(vxz530), vxz54, vxz2700) → new_primQuotInt1(vxz530, new_gcd(vxz54, vxz2700))
new_gcd(Neg(Succ(vxz5400)), vxz2700) → new_gcd0Gcd'14(vxz2700, vxz5400)
new_primMinusNatS0(Zero, Zero) → Zero
new_gcd0Gcd'18(vxz272, vxz273, Zero, Zero) → new_gcd0Gcd'19(vxz272, vxz273)
new_quot17(vxz79, vxz80, Neg(Succ(vxz3000))) → new_quot16(vxz79, vxz3000, vxz80)
new_reduce2D(vxz137, vxz2700) → new_gcd0(vxz137, vxz2700)
new_primMulInt(Neg(vxz310)) → Neg(new_primMulNat0(vxz310))
new_primPlusInt4(Neg(vxz390), Neg(vxz3100)) → new_primPlusInt3(vxz390, vxz3100)
new_primQuotInt1(vxz135, Pos(Succ(vxz13600))) → Pos(new_primDivNatS1(vxz135, vxz13600))
new_primPlusInt1(Succ(vxz2800), Succ(vxz3100)) → new_primMinusNat2(vxz2800, Succ(vxz3100))
new_reduce(vxz28, vxz31, Pos(Succ(vxz2700))) → :%(new_primQuotInt0(new_ps5(vxz28, vxz31), new_ps5(vxz28, vxz31), vxz2700), new_primQuotInt1(Succ(vxz2700), new_reduce2D(new_ps5(vxz28, vxz31), vxz2700)))
new_quot11(vxz79, vxz8000, vxz3000) → new_quot12(vxz79, vxz8000, vxz3000)
new_primQuotInt5(Pos(vxz790), vxz8000) → new_primQuotInt(Pos(vxz790), vxz8000)
new_primPlusNat4(Zero, Zero) → Zero
new_primDivNatS1(Succ(Succ(vxz13500)), Zero) → Succ(new_primDivNatS1(new_primMinusNatS1(vxz13500), Zero))
new_primQuotInt3(vxz207, Pos(Zero)) → new_error0
new_ps4(:%(vxz30, vxz31), ty_Int) → new_reduce(new_sr(vxz30), vxz31, new_sr(vxz31))
new_primMinusNat3(Zero, Succ(vxz28000)) → Neg(Succ(vxz28000))
new_primPlusInt4(Pos(vxz390), Pos(vxz3100)) → new_primPlusInt0(vxz390, vxz3100)
new_gcd0Gcd'17(Succ(Zero), Zero, vxz236) → new_gcd0Gcd'17(new_primMinusNatS0(Zero, Zero), Zero, new_primMinusNatS0(Zero, Zero))
new_ps4(:%(vxz30, Integer(vxz310)), ty_Integer) → new_reduce2Reduce10(vxz30, vxz310, new_primMulInt(vxz310), new_primMulInt(vxz310), new_primMulInt(vxz310))
new_primPlusInt1(Succ(vxz2800), Zero) → Pos(Succ(vxz2800))
new_gcd0Gcd'112(vxz2700) → new_gcd0Gcd'17(Zero, vxz2700, Zero)
new_primMinusNat2(vxz3100, Zero) → Pos(Succ(vxz3100))
new_quot18(vxz79, Neg(Succ(vxz8000))) → Integer(new_primQuotInt5(vxz79, vxz8000))
new_primDivNatS01(vxz168, vxz169, Zero, Succ(vxz1710)) → Zero
new_primPlusNat(Zero) → Zero
new_primMinusNatS1(vxz13500) → Succ(vxz13500)
new_reduce2Reduce11(vxz30, vxz310, vxz30, vxz29) → error([])
new_sr(vxz31) → new_primMulInt(vxz31)
new_primMinusNatS0(Succ(vxz1680), Zero) → Succ(vxz1680)
new_primQuotInt6(Neg(vxz790)) → new_error0
new_ps5(Neg(vxz280), Neg(vxz310)) → new_primPlusInt3(vxz280, vxz310)
new_primQuotInt0(Neg(vxz510), vxz52, vxz2700) → new_primQuotInt3(vxz510, new_gcd0(vxz52, vxz2700))
new_gcd0Gcd'111(vxz2700, vxz5200) → new_gcd0Gcd'15(vxz5200, vxz2700)
new_quot9(vxz285, vxz286, vxz287, Zero, Zero) → new_quot10(vxz285, vxz286, vxz287)
new_primPlusNat2(vxz3100) → Succ(vxz3100)
new_quot16(vxz79, vxz3000, Pos(Zero)) → new_quot13(vxz79, Zero, vxz3000, Zero)
new_quot6(vxz39, vxz310, vxz40, vxz30) → new_quot14(new_primPlusInt4(vxz39, vxz310), new_primPlusInt4(vxz39, vxz310), new_primPlusInt4(vxz39, vxz310), vxz30)
new_primPlusNat4(Succ(vxz28000), Succ(vxz31000)) → Succ(Succ(new_primPlusNat4(vxz28000, vxz31000)))
new_error0 → error([])
new_primPlusInt4(Pos(vxz390), Neg(vxz3100)) → new_primPlusInt1(vxz390, vxz3100)
new_quot14(vxz79, Neg(Succ(vxz8100)), vxz80, vxz30) → new_quot20(vxz79, vxz80, vxz30)
new_quot14(vxz79, Neg(Zero), vxz80, vxz30) → new_quot19(vxz79, vxz80, vxz30)
new_quot9(vxz285, vxz286, vxz287, Succ(vxz2880), Zero) → new_quot10(vxz285, vxz286, vxz287)
new_gcd0(Neg(Succ(vxz5200)), vxz2700) → new_gcd0Gcd'14(vxz2700, vxz5200)
new_primPlusInt3(vxz280, vxz310) → Neg(new_primPlusNat3(vxz280, vxz310))
new_primQuotInt1(vxz135, Neg(Zero)) → new_error0
new_primPlusNat3(Zero, Succ(vxz3100)) → new_primPlusNat2(vxz3100)
new_quot19(vxz79, vxz80, Neg(Zero)) → new_quot15(vxz79)
new_primPlusInt2(vxz280, Succ(vxz3100)) → new_primMinusNat2(vxz3100, vxz280)
new_primPlusInt4(Neg(vxz390), Pos(vxz3100)) → new_primPlusInt2(vxz390, vxz3100)
new_quot19(vxz79, vxz80, Neg(Succ(vxz3000))) → new_quot17(vxz79, vxz80, Neg(Succ(vxz3000)))
new_primPlusNat3(Succ(vxz2800), Zero) → new_primPlusNat(Succ(vxz2800))
new_reduce(vxz28, vxz31, Pos(Zero)) → new_error
new_reduce2Reduce1(vxz30, vxz310, vxz30, vxz29) → :%(new_quot4(vxz30, vxz310, vxz30), new_quot5(vxz29, vxz30, vxz310, vxz30))
new_quot16(vxz79, vxz3000, Neg(Succ(vxz8000))) → new_quot12(vxz79, vxz8000, vxz3000)
new_quot4(Integer(vxz300), vxz310, vxz30) → new_quot6(new_primMulInt(vxz300), vxz310, new_primMulInt(vxz300), vxz30)
new_quot17(vxz79, vxz80, Pos(Succ(vxz3000))) → new_quot16(vxz79, vxz3000, vxz80)
new_gcd0Gcd'16(vxz2700) → Pos(Succ(vxz2700))
new_primQuotInt1(vxz135, Pos(Zero)) → new_error0
new_gcd0Gcd'15(vxz5200, vxz2700) → new_gcd0Gcd'17(Succ(vxz5200), vxz2700, Succ(vxz5200))
new_primPlusInt1(Zero, Succ(vxz3100)) → new_primMinusNat1(new_primPlusNat2(vxz3100))
new_gcd0Gcd'14(vxz2700, vxz5200) → new_gcd0Gcd'15(vxz5200, vxz2700)
new_primMulNat0(Succ(vxz3100)) → new_primPlusNat0(new_primMulNat0(vxz3100))
new_quot13(vxz249, Succ(Zero), Zero, vxz254) → new_quot13(vxz249, new_primMinusNatS0(Zero, Zero), Zero, new_primMinusNatS0(Zero, Zero))
new_primMinusNat3(Succ(vxz31000), Succ(vxz28000)) → new_primMinusNat3(vxz31000, vxz28000)
new_primQuotInt1(vxz135, Neg(Succ(vxz13600))) → Neg(new_primDivNatS1(vxz135, vxz13600))
new_reduce2Reduce10(vxz30, vxz310, vxz30, vxz29, Neg(Zero)) → new_reduce2Reduce11(vxz30, vxz310, vxz30, vxz29)
new_quot18(vxz79, Neg(Zero)) → Integer(new_primQuotInt6(vxz79))
new_gcd0(Neg(Zero), vxz2700) → new_gcd0Gcd'16(vxz2700)
new_primMulInt(Pos(vxz310)) → Pos(new_primMulNat0(vxz310))
new_reduce(vxz28, vxz31, Neg(Succ(vxz2700))) → :%(new_primQuotInt2(new_ps5(vxz28, vxz31), new_ps5(vxz28, vxz31), vxz2700), new_primQuotInt3(Succ(vxz2700), new_reduce2D0(new_ps5(vxz28, vxz31), vxz2700)))
new_ps5(Pos(vxz280), Pos(vxz310)) → new_primPlusInt0(vxz280, vxz310)
new_primPlusNat4(Zero, Succ(vxz31000)) → Succ(vxz31000)
new_primPlusNat4(Succ(vxz28000), Zero) → Succ(vxz28000)
new_primQuotInt2(Neg(vxz530), vxz54, vxz2700) → new_primQuotInt3(vxz530, new_gcd(vxz54, vxz2700))
new_quot14(vxz79, Pos(Succ(vxz8100)), vxz80, vxz30) → new_quot20(vxz79, vxz80, vxz30)
new_quot13(vxz249, Succ(Zero), Succ(vxz2510), vxz254) → new_quot11(vxz249, Succ(vxz2510), Zero)
new_reduce2Reduce10(vxz30, vxz310, vxz30, vxz29, Pos(Zero)) → new_reduce2Reduce11(vxz30, vxz310, vxz30, vxz29)
new_quot17(vxz79, vxz80, Pos(Zero)) → new_quot18(vxz79, vxz80)
new_primQuotInt3(vxz207, Neg(Succ(vxz22600))) → Pos(new_primDivNatS1(vxz207, vxz22600))
new_gcd0Gcd'18(vxz272, vxz273, Zero, Succ(vxz2750)) → new_gcd0Gcd'110(Succ(vxz272), vxz273)
new_primQuotInt3(vxz207, Neg(Zero)) → new_error0
new_gcd0Gcd'18(vxz272, vxz273, Succ(vxz2740), Succ(vxz2750)) → new_gcd0Gcd'18(vxz272, vxz273, vxz2740, vxz2750)
new_primDivNatS01(vxz168, vxz169, Succ(vxz1700), Succ(vxz1710)) → new_primDivNatS01(vxz168, vxz169, vxz1700, vxz1710)
new_primPlusNat(Succ(vxz3000)) → Succ(vxz3000)
new_primPlusNat3(Succ(vxz2800), Succ(vxz3100)) → Succ(Succ(new_primPlusNat4(vxz2800, vxz3100)))
new_primDivNatS01(vxz168, vxz169, Zero, Zero) → new_primDivNatS02(vxz168, vxz169)
new_quot17(vxz79, vxz80, Neg(Zero)) → new_quot18(vxz79, vxz80)
new_reduce2D0(vxz227, vxz2700) → new_gcd(vxz227, vxz2700)
new_primPlusInt2(vxz280, Zero) → new_primMinusNat1(vxz280)
new_primQuotInt4(Neg(vxz790)) → new_error0
new_quot7(vxz29, vxz48, vxz310, vxz47, vxz30) → new_quot14(vxz29, new_primPlusInt4(vxz48, vxz310), new_primPlusInt4(vxz48, vxz310), vxz30)
new_primMinusNatS0(Zero, Succ(vxz1690)) → Zero
new_reduce(vxz28, vxz31, Neg(Zero)) → new_error
new_primQuotInt4(Pos(vxz790)) → new_error0
new_gcd0Gcd'17(Succ(Zero), Succ(vxz2320), vxz236) → new_gcd0Gcd'110(Zero, Succ(vxz2320))
new_quot9(vxz285, vxz286, vxz287, Succ(vxz2880), Succ(vxz2890)) → new_quot9(vxz285, vxz286, vxz287, vxz2880, vxz2890)
new_quot13(vxz249, Zero, vxz251, vxz254) → new_quot8(vxz249, vxz251)
new_quot16(vxz79, vxz3000, Pos(Succ(vxz8000))) → new_quot11(vxz79, vxz8000, vxz3000)
new_quot13(vxz249, Succ(Succ(vxz25500)), Zero, vxz254) → new_quot13(vxz249, new_primMinusNatS0(Succ(vxz25500), Zero), Zero, new_primMinusNatS0(Succ(vxz25500), Zero))
new_primQuotInt(Pos(vxz790), vxz8000) → Pos(new_primDivNatS1(vxz790, vxz8000))
new_primQuotInt6(Pos(vxz790)) → new_error0
new_primPlusInt1(Zero, Zero) → new_primMinusNat1(Zero)
new_gcd(Pos(Succ(vxz5400)), vxz2700) → new_gcd0Gcd'111(vxz2700, vxz5400)
new_primDivNatS01(vxz168, vxz169, Succ(vxz1700), Zero) → new_primDivNatS02(vxz168, vxz169)
new_quot12(vxz79, vxz8000, vxz3000) → new_quot13(vxz79, Succ(vxz8000), vxz3000, Succ(vxz8000))
new_primMulNat0(Zero) → Zero
new_primDivNatS02(vxz168, vxz169) → Succ(new_primDivNatS1(new_primMinusNatS0(vxz168, vxz169), Succ(vxz169)))
new_quot19(vxz79, vxz80, Pos(Succ(vxz3000))) → new_quot17(vxz79, vxz80, Pos(Succ(vxz3000)))
new_gcd0(Pos(Zero), vxz2700) → new_gcd0Gcd'112(vxz2700)
new_gcd(Neg(Zero), vxz2700) → new_gcd0Gcd'16(vxz2700)
new_primDivNatS1(Succ(Succ(vxz13500)), Succ(vxz136000)) → new_primDivNatS01(vxz13500, vxz136000, vxz13500, vxz136000)
new_quot8(vxz79, vxz8000) → Integer(new_primQuotInt(vxz79, vxz8000))
new_primDivNatS1(Zero, vxz13600) → Zero
new_gcd0Gcd'17(Succ(Succ(vxz23700)), Zero, vxz236) → new_gcd0Gcd'19(vxz23700, Zero)
new_primPlusNat0(Succ(vxz300)) → Succ(Succ(new_primPlusNat(vxz300)))
new_primPlusInt0(vxz280, vxz310) → Pos(new_primPlusNat3(vxz280, vxz310))
new_primPlusNat0(Zero) → Succ(Zero)
new_gcd0Gcd'110(vxz266, vxz267) → new_gcd0Gcd'15(vxz267, vxz266)
new_ps5(Pos(vxz280), Neg(vxz310)) → new_primPlusInt1(vxz280, vxz310)
new_error → error([])
new_reduce2Reduce10(vxz30, vxz310, vxz30, vxz29, Pos(Succ(vxz3100))) → new_reduce2Reduce1(vxz30, vxz310, vxz30, vxz29)
new_gcd(Pos(Zero), vxz2700) → new_gcd0Gcd'112(vxz2700)
new_gcd0Gcd'17(Succ(Succ(vxz23700)), Succ(vxz2320), vxz236) → new_gcd0Gcd'18(vxz23700, Succ(vxz2320), vxz23700, vxz2320)
new_quot5(vxz29, Integer(vxz300), vxz310, vxz30) → new_quot7(vxz29, new_primMulInt(vxz300), vxz310, new_primMulInt(vxz300), vxz30)
new_primMinusNat3(Succ(vxz31000), Zero) → Pos(Succ(vxz31000))
new_primMinusNat1(Zero) → Pos(Zero)
new_quot16(vxz79, vxz3000, Neg(Zero)) → new_quot8(vxz79, vxz3000)
new_primQuotInt(Neg(vxz790), vxz8000) → Neg(new_primDivNatS1(vxz790, vxz8000))
new_primMinusNat3(Zero, Zero) → Pos(Zero)
new_gcd0Gcd'18(vxz272, vxz273, Succ(vxz2740), Zero) → new_gcd0Gcd'19(vxz272, vxz273)
new_primMinusNat1(Succ(vxz3000)) → Neg(Succ(vxz3000))
new_primMinusNatS2 → Zero
new_gcd0(Pos(Succ(vxz5200)), vxz2700) → new_gcd0Gcd'111(vxz2700, vxz5200)
new_primPlusNat3(Zero, Zero) → new_primPlusNat(Zero)
new_quot9(vxz285, vxz286, vxz287, Zero, Succ(vxz2890)) → new_quot11(vxz285, vxz287, Succ(vxz286))
new_quot10(vxz285, vxz286, vxz287) → new_quot13(vxz285, new_primMinusNatS0(Succ(vxz286), vxz287), vxz287, new_primMinusNatS0(Succ(vxz286), vxz287))
new_gcd0Gcd'17(Zero, vxz232, vxz236) → Pos(Succ(vxz232))
new_quot20(vxz79, vxz80, vxz30) → new_quot17(vxz79, vxz80, vxz30)
new_primDivNatS1(Succ(Zero), Succ(vxz136000)) → Zero
new_primMinusNatS0(Succ(vxz1680), Succ(vxz1690)) → new_primMinusNatS0(vxz1680, vxz1690)
new_primQuotInt3(vxz207, Pos(Succ(vxz22600))) → Neg(new_primDivNatS1(vxz207, vxz22600))
new_primMinusNat2(vxz3100, Succ(vxz2800)) → new_primMinusNat3(vxz3100, vxz2800)
s = new_numericEnumFrom3(vxz3, ba) evaluates to t =new_numericEnumFrom3(new_ps4(vxz3, ba), ba)
Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
- Matcher: [vxz3 / new_ps4(vxz3, ba)]
- Semiunifier: [ ]
Rewriting sequence
The DP semiunifies directly so there is only one rewrite step from new_numericEnumFrom3(vxz3, ba) to new_numericEnumFrom3(new_ps4(vxz3, ba), ba).
Haskell To QDPs