MAYBE 123.533
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
| ((enumFromThen :: Char -> Char -> [Char]) :: Char -> Char -> [Char]) |
module Main where
If Reductions:
The following If expression
if d < c then minBound else maxBound
is transformed to
lastChar0 | True | = minBound |
lastChar0 | False | = maxBound |
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
mainModule Main
| ((enumFromThen :: Char -> Char -> [Char]) :: Char -> Char -> [Char]) |
module Main where
Replaced joker patterns by fresh variables and removed binding patterns.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
mainModule Main
| ((enumFromThen :: Char -> Char -> [Char]) :: Char -> Char -> [Char]) |
module Main where
Cond Reductions:
The following Function with conditions
is transformed to
undefined0 | True | = undefined |
undefined1 | | = undefined0 False |
The following Function with conditions
is transformed to
p1 | True | = flip (<=) m |
p1 | False | = p0 otherwise |
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) |
takeWhile0 | p x xs True | = [] |
takeWhile1 | p x xs True | = x : takeWhile p xs |
takeWhile1 | p x xs False | = takeWhile0 p x xs otherwise |
takeWhile2 | p (x : xs) | = takeWhile1 p x xs (p x) |
takeWhile3 | p [] | = [] |
takeWhile3 | vz wu | = takeWhile2 vz wu |
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
mainModule Main
| ((enumFromThen :: Char -> Char -> [Char]) :: Char -> Char -> [Char]) |
module Main where
Let/Where Reductions:
The bindings of the following Let/Where expression
takeWhile p (numericEnumFromThen n n') |
where | |
| |
|
p1 | True | = flip (<=) m |
p1 | False | = p0 otherwise |
|
| |
are unpacked to the following functions on top level
numericEnumFromThenToP1 | wv ww wx True | = flip (<=) wv |
numericEnumFromThenToP1 | wv ww wx False | = numericEnumFromThenToP0 wv ww wx otherwise |
numericEnumFromThenToP | wv ww wx | = numericEnumFromThenToP2 wv ww wx |
numericEnumFromThenToP2 | wv ww wx | = numericEnumFromThenToP1 wv ww wx (ww >= wx) |
numericEnumFromThenToP0 | wv ww wx True | = flip (>=) wv |
The bindings of the following Let/Where expression
map toEnum (enumFromThenTo (fromEnum c) (fromEnum d) (fromEnum lastChar)) |
where |
lastChar | | = lastChar0 (d < c) |
|
|
lastChar0 | True | = minBound |
lastChar0 | False | = maxBound |
|
are unpacked to the following functions on top level
enumFromThenLastChar0 | wy wz True | = minBound |
enumFromThenLastChar0 | wy wz False | = maxBound |
enumFromThenLastChar | wy wz | = enumFromThenLastChar0 wy wz (wy < wz) |
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
mainModule Main
| ((enumFromThen :: Char -> Char -> [Char]) :: Char -> Char -> [Char]) |
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
| (enumFromThen :: Char -> Char -> [Char]) |
module Main where
Haskell To QDPs
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ NonTerminationProof
↳ QDP
↳ QDP
↳ 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_map(xu230, ba) → new_map(xu230, ba)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.
The TRS P consists of the following rules:
new_map(xu230, ba) → new_map(xu230, ba)
The TRS R consists of the following rules:none
s = new_map(xu230, ba) evaluates to t =new_map(xu230, ba)
Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
- Semiunifier: [ ]
- Matcher: [ ]
Rewriting sequence
The DP semiunifies directly so there is only one rewrite step from new_map(xu230, ba) to new_map(xu230, ba).
↳ 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
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_primPlusNat(Succ(xu22800), Succ(xu2300)) → new_primPlusNat(xu22800, xu2300)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- new_primPlusNat(Succ(xu22800), Succ(xu2300)) → new_primPlusNat(xu22800, xu2300)
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
↳ Instantiation
↳ 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(xu235, xu236, ba) → new_map1(xu236, xu235, ba)
new_map1(xu2280, xu230, bb) → new_map0(Succ(new_primPlusNat0(xu2280, xu230)), xu2280, bb)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule new_map0(xu235, xu236, ba) → new_map1(xu236, xu235, ba) we obtained the following new rules:
new_map0(Succ(y_0), z0, z2) → new_map1(z0, Succ(y_0), z2)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ 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_map1(xu2280, xu230, bb) → new_map0(Succ(new_primPlusNat0(xu2280, xu230)), xu2280, bb)
new_map0(Succ(y_0), z0, z2) → new_map1(z0, Succ(y_0), z2)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule new_map1(xu2280, xu230, bb) → new_map0(Succ(new_primPlusNat0(xu2280, xu230)), xu2280, bb) we obtained the following new rules:
new_map1(z1, Succ(z0), z2) → new_map0(Succ(new_primPlusNat0(z1, Succ(z0))), z1, z2)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ MNOCProof
↳ 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(Succ(y_0), z0, z2) → new_map1(z0, Succ(y_0), z2)
new_map1(z1, Succ(z0), z2) → new_map0(Succ(new_primPlusNat0(z1, Succ(z0))), z1, z2)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
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
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ MNOCProof
↳ QDP
↳ NonTerminationProof
↳ 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(Succ(y_0), z0, z2) → new_map1(z0, Succ(y_0), z2)
new_map1(z1, Succ(z0), z2) → new_map0(Succ(new_primPlusNat0(z1, Succ(z0))), z1, z2)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
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 narrowing to the left:
The TRS P consists of the following rules:
new_map0(Succ(y_0), z0, z2) → new_map1(z0, Succ(y_0), z2)
new_map1(z1, Succ(z0), z2) → new_map0(Succ(new_primPlusNat0(z1, Succ(z0))), z1, z2)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
s = new_map1(z1, Succ(z0'), z2') evaluates to t =new_map1(z1, Succ(new_primPlusNat0(z1, Succ(z0'))), z2')
Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
- Matcher: [z0' / new_primPlusNat0(z1, Succ(z0'))]
- Semiunifier: [ ]
Rewriting sequence
new_map1(z1, Succ(z0'), z2') → new_map0(Succ(new_primPlusNat0(z1, Succ(z0'))), z1, z2')
with rule new_map1(z1', Succ(z0''), z2'') → new_map0(Succ(new_primPlusNat0(z1', Succ(z0''))), z1', z2'') at position [] and matcher [z2'' / z2', z1' / z1, z0'' / z0']
new_map0(Succ(new_primPlusNat0(z1, Succ(z0'))), z1, z2') → new_map1(z1, Succ(new_primPlusNat0(z1, Succ(z0'))), z2')
with rule new_map0(Succ(y_0), z0, z2) → new_map1(z0, Succ(y_0), z2)
Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence
All these steps are and every following step will be a correct step w.r.t to Q.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ 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(xu332, xu355, xu335, xu356, Zero, ba) → new_map3(xu332, xu355, xu335, xu356, ba)
new_map3(xu332, xu355, xu335, xu356, ba) → new_map2(xu332, new_primPlusNat0(Succ(xu335), xu356), xu335, new_primPlusNat0(Succ(xu335), xu356), new_primPlusNat0(Succ(xu335), xu356), ba)
new_map2(xu332, xu355, xu335, xu356, Succ(xu3570), ba) → new_map2(xu332, new_primPlusNat0(Succ(xu335), xu356), xu335, new_primPlusNat0(Succ(xu335), xu356), new_primPlusNat0(Succ(xu335), xu356), ba)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
new_map2(xu332, xu355, xu335, xu356, Zero, ba) → new_map3(xu332, xu355, xu335, xu356, ba)
The remaining pairs can at least be oriented weakly.
new_map3(xu332, xu355, xu335, xu356, ba) → new_map2(xu332, new_primPlusNat0(Succ(xu335), xu356), xu335, new_primPlusNat0(Succ(xu335), xu356), new_primPlusNat0(Succ(xu335), xu356), ba)
new_map2(xu332, xu355, xu335, xu356, Succ(xu3570), ba) → new_map2(xu332, new_primPlusNat0(Succ(xu335), xu356), xu335, new_primPlusNat0(Succ(xu335), xu356), new_primPlusNat0(Succ(xu335), xu356), ba)
Used ordering: Polynomial interpretation [25]:
POL(Succ(x1)) = 0
POL(Zero) = 1
POL(new_map2(x1, x2, x3, x4, x5, x6)) = x5
POL(new_map3(x1, x2, x3, x4, x5)) = 0
POL(new_primPlusNat0(x1, x2)) = 0
The following usable rules [17] were oriented:
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ 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_map3(xu332, xu355, xu335, xu356, ba) → new_map2(xu332, new_primPlusNat0(Succ(xu335), xu356), xu335, new_primPlusNat0(Succ(xu335), xu356), new_primPlusNat0(Succ(xu335), xu356), ba)
new_map2(xu332, xu355, xu335, xu356, Succ(xu3570), ba) → new_map2(xu332, new_primPlusNat0(Succ(xu335), xu356), xu335, new_primPlusNat0(Succ(xu335), xu356), new_primPlusNat0(Succ(xu335), xu356), ba)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ 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(xu332, xu355, xu335, xu356, Succ(xu3570), ba) → new_map2(xu332, new_primPlusNat0(Succ(xu335), xu356), xu335, new_primPlusNat0(Succ(xu335), xu356), new_primPlusNat0(Succ(xu335), xu356), ba)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule new_map2(xu332, xu355, xu335, xu356, Succ(xu3570), ba) → new_map2(xu332, new_primPlusNat0(Succ(xu335), xu356), xu335, new_primPlusNat0(Succ(xu335), xu356), new_primPlusNat0(Succ(xu335), xu356), ba) at position [4] we obtained the following new rules:
new_map2(y0, y1, x0, Succ(x1), Succ(y4), y5) → new_map2(y0, new_primPlusNat0(Succ(x0), Succ(x1)), x0, new_primPlusNat0(Succ(x0), Succ(x1)), Succ(Succ(new_primPlusNat0(x0, x1))), y5)
new_map2(y0, y1, x0, Zero, Succ(y4), y5) → new_map2(y0, new_primPlusNat0(Succ(x0), Zero), x0, new_primPlusNat0(Succ(x0), Zero), Succ(x0), y5)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ 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(y0, y1, x0, Zero, Succ(y4), y5) → new_map2(y0, new_primPlusNat0(Succ(x0), Zero), x0, new_primPlusNat0(Succ(x0), Zero), Succ(x0), y5)
new_map2(y0, y1, x0, Succ(x1), Succ(y4), y5) → new_map2(y0, new_primPlusNat0(Succ(x0), Succ(x1)), x0, new_primPlusNat0(Succ(x0), Succ(x1)), Succ(Succ(new_primPlusNat0(x0, x1))), y5)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_map2(y0, y1, x0, Succ(x1), Succ(y4), y5) → new_map2(y0, new_primPlusNat0(Succ(x0), Succ(x1)), x0, new_primPlusNat0(Succ(x0), Succ(x1)), Succ(Succ(new_primPlusNat0(x0, x1))), y5) at position [1] we obtained the following new rules:
new_map2(y0, y1, x0, Succ(x1), Succ(y4), y5) → new_map2(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, new_primPlusNat0(Succ(x0), Succ(x1)), Succ(Succ(new_primPlusNat0(x0, x1))), y5)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ 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(y0, y1, x0, Succ(x1), Succ(y4), y5) → new_map2(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, new_primPlusNat0(Succ(x0), Succ(x1)), Succ(Succ(new_primPlusNat0(x0, x1))), y5)
new_map2(y0, y1, x0, Zero, Succ(y4), y5) → new_map2(y0, new_primPlusNat0(Succ(x0), Zero), x0, new_primPlusNat0(Succ(x0), Zero), Succ(x0), y5)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_map2(y0, y1, x0, Zero, Succ(y4), y5) → new_map2(y0, new_primPlusNat0(Succ(x0), Zero), x0, new_primPlusNat0(Succ(x0), Zero), Succ(x0), y5) at position [1] we obtained the following new rules:
new_map2(y0, y1, x0, Zero, Succ(y4), y5) → new_map2(y0, Succ(x0), x0, new_primPlusNat0(Succ(x0), Zero), Succ(x0), y5)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ 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(y0, y1, x0, Succ(x1), Succ(y4), y5) → new_map2(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, new_primPlusNat0(Succ(x0), Succ(x1)), Succ(Succ(new_primPlusNat0(x0, x1))), y5)
new_map2(y0, y1, x0, Zero, Succ(y4), y5) → new_map2(y0, Succ(x0), x0, new_primPlusNat0(Succ(x0), Zero), Succ(x0), y5)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_map2(y0, y1, x0, Succ(x1), Succ(y4), y5) → new_map2(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, new_primPlusNat0(Succ(x0), Succ(x1)), Succ(Succ(new_primPlusNat0(x0, x1))), y5) at position [3] we obtained the following new rules:
new_map2(y0, y1, x0, Succ(x1), Succ(y4), y5) → new_map2(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y5)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ 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(y0, y1, x0, Succ(x1), Succ(y4), y5) → new_map2(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y5)
new_map2(y0, y1, x0, Zero, Succ(y4), y5) → new_map2(y0, Succ(x0), x0, new_primPlusNat0(Succ(x0), Zero), Succ(x0), y5)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ 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(y0, y1, x0, Succ(x1), Succ(y4), y5) → new_map2(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y5)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule new_map2(y0, y1, x0, Succ(x1), Succ(y4), y5) → new_map2(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y5) we obtained the following new rules:
new_map2(z0, Succ(Succ(y_0)), z2, Succ(Succ(y_1)), Succ(Succ(y_2)), z5) → new_map2(z0, Succ(Succ(new_primPlusNat0(z2, Succ(y_1)))), z2, Succ(Succ(new_primPlusNat0(z2, Succ(y_1)))), Succ(Succ(new_primPlusNat0(z2, Succ(y_1)))), z5)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ MNOCProof
↳ 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(z0, Succ(Succ(y_0)), z2, Succ(Succ(y_1)), Succ(Succ(y_2)), z5) → new_map2(z0, Succ(Succ(new_primPlusNat0(z2, Succ(y_1)))), z2, Succ(Succ(new_primPlusNat0(z2, Succ(y_1)))), Succ(Succ(new_primPlusNat0(z2, Succ(y_1)))), z5)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
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
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ MNOCProof
↳ QDP
↳ NonTerminationProof
↳ 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(z0, Succ(Succ(y_0)), z2, Succ(Succ(y_1)), Succ(Succ(y_2)), z5) → new_map2(z0, Succ(Succ(new_primPlusNat0(z2, Succ(y_1)))), z2, Succ(Succ(new_primPlusNat0(z2, Succ(y_1)))), Succ(Succ(new_primPlusNat0(z2, Succ(y_1)))), z5)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
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_map2(z0, Succ(Succ(y_0)), z2, Succ(Succ(y_1)), Succ(Succ(y_2)), z5) → new_map2(z0, Succ(Succ(new_primPlusNat0(z2, Succ(y_1)))), z2, Succ(Succ(new_primPlusNat0(z2, Succ(y_1)))), Succ(Succ(new_primPlusNat0(z2, Succ(y_1)))), z5)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
s = new_map2(z0, Succ(Succ(y_0)), z2, Succ(Succ(y_1)), Succ(Succ(y_2)), z5) evaluates to t =new_map2(z0, Succ(Succ(new_primPlusNat0(z2, Succ(y_1)))), z2, Succ(Succ(new_primPlusNat0(z2, Succ(y_1)))), Succ(Succ(new_primPlusNat0(z2, Succ(y_1)))), z5)
Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
- Matcher: [y_2 / new_primPlusNat0(z2, Succ(y_1)), y_1 / new_primPlusNat0(z2, Succ(y_1)), y_0 / new_primPlusNat0(z2, Succ(y_1))]
- Semiunifier: [ ]
Rewriting sequence
The DP semiunifies directly so there is only one rewrite step from new_map2(z0, Succ(Succ(y_0)), z2, Succ(Succ(y_1)), Succ(Succ(y_2)), z5) to new_map2(z0, Succ(Succ(new_primPlusNat0(z2, Succ(y_1)))), z2, Succ(Succ(new_primPlusNat0(z2, Succ(y_1)))), Succ(Succ(new_primPlusNat0(z2, Succ(y_1)))), z5).
↳ 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
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map4(Succ(xu2780), Zero, xu280, ba) → new_map5(xu280, xu2780, ba)
new_map4(Succ(xu2780), Succ(xu2790), xu280, ba) → new_map4(xu2780, xu2790, xu280, ba)
new_map4(Zero, Zero, xu280, ba) → new_map4(Zero, Succ(xu280), xu280, ba)
new_map5(xu2290, xu230, bb) → new_map4(Succ(xu230), Succ(xu2290), xu2290, bb)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ 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_map4(Succ(xu2780), Zero, xu280, ba) → new_map5(xu280, xu2780, ba)
new_map4(Succ(xu2780), Succ(xu2790), xu280, ba) → new_map4(xu2780, xu2790, xu280, ba)
new_map5(xu2290, xu230, bb) → new_map4(Succ(xu230), Succ(xu2290), xu2290, bb)
R is empty.
Q is empty.
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_map4(Succ(xu2780), Succ(xu2790), xu280, ba) → new_map4(xu2780, xu2790, xu280, ba)
The remaining pairs can at least be oriented weakly.
new_map4(Succ(xu2780), Zero, xu280, ba) → new_map5(xu280, xu2780, ba)
new_map5(xu2290, xu230, bb) → new_map4(Succ(xu230), Succ(xu2290), xu2290, bb)
Used ordering: Polynomial interpretation [25]:
POL(Succ(x1)) = 1 + x1
POL(Zero) = 0
POL(new_map4(x1, x2, x3, x4)) = x1
POL(new_map5(x1, x2, x3)) = 1 + x2
The following usable rules [17] were oriented:
none
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ 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_map4(Succ(xu2780), Zero, xu280, ba) → new_map5(xu280, xu2780, ba)
new_map5(xu2290, xu230, bb) → new_map4(Succ(xu230), Succ(xu2290), xu2290, bb)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 2 less nodes.
↳ 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
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map6(Succ(xu2280), Succ(xu2290), xu230, ba) → new_map6(xu2280, xu2290, xu230, 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_map6(Succ(xu2280), Succ(xu2290), xu230, ba) → new_map6(xu2280, xu2290, xu230, ba)
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
↳ 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_map8(xu282, xu293, xu294, Zero, bb) → new_map7(xu282, xu293, xu294, Zero, Succ(xu282), bb)
new_map9(xu350, xu351, xu352, ba) → new_map8(xu350, new_primPlusNat0(Zero, xu352), new_primPlusNat0(Zero, xu352), new_primPlusNat0(Zero, xu352), ba)
new_map7(xu350, xu351, xu352, Succ(xu3530), Succ(xu3540), ba) → new_map7(xu350, xu351, xu352, xu3530, xu3540, ba)
new_map8(xu282, xu293, xu294, Succ(xu2950), bb) → new_map7(xu282, xu293, xu294, Succ(xu2950), Succ(xu282), bb)
new_map7(xu350, xu351, xu352, Succ(xu3530), Zero, ba) → new_map8(xu350, new_primPlusNat0(Zero, xu352), new_primPlusNat0(Zero, xu352), new_primPlusNat0(Zero, xu352), ba)
new_map7(xu350, xu351, xu352, Zero, Zero, ba) → new_map9(xu350, xu351, xu352, ba)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ 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_map9(xu350, xu351, xu352, ba) → new_map8(xu350, new_primPlusNat0(Zero, xu352), new_primPlusNat0(Zero, xu352), new_primPlusNat0(Zero, xu352), ba)
new_map7(xu350, xu351, xu352, Succ(xu3530), Succ(xu3540), ba) → new_map7(xu350, xu351, xu352, xu3530, xu3540, ba)
new_map8(xu282, xu293, xu294, Succ(xu2950), bb) → new_map7(xu282, xu293, xu294, Succ(xu2950), Succ(xu282), bb)
new_map7(xu350, xu351, xu352, Succ(xu3530), Zero, ba) → new_map8(xu350, new_primPlusNat0(Zero, xu352), new_primPlusNat0(Zero, xu352), new_primPlusNat0(Zero, xu352), ba)
new_map7(xu350, xu351, xu352, Zero, Zero, ba) → new_map9(xu350, xu351, xu352, ba)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ 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_map9(xu350, xu351, xu352, ba) → new_map8(xu350, new_primPlusNat0(Zero, xu352), new_primPlusNat0(Zero, xu352), new_primPlusNat0(Zero, xu352), ba)
new_map7(xu350, xu351, xu352, Succ(xu3530), Succ(xu3540), ba) → new_map7(xu350, xu351, xu352, xu3530, xu3540, ba)
new_map8(xu282, xu293, xu294, Succ(xu2950), bb) → new_map7(xu282, xu293, xu294, Succ(xu2950), Succ(xu282), bb)
new_map7(xu350, xu351, xu352, Succ(xu3530), Zero, ba) → new_map8(xu350, new_primPlusNat0(Zero, xu352), new_primPlusNat0(Zero, xu352), new_primPlusNat0(Zero, xu352), ba)
new_map7(xu350, xu351, xu352, Zero, Zero, ba) → new_map9(xu350, xu351, xu352, ba)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule new_map7(xu350, xu351, xu352, Succ(xu3530), Zero, ba) → new_map8(xu350, new_primPlusNat0(Zero, xu352), new_primPlusNat0(Zero, xu352), new_primPlusNat0(Zero, xu352), ba) at position [3] we obtained the following new rules:
new_map7(y0, y1, Succ(x0), Succ(y3), Zero, y4) → new_map8(y0, new_primPlusNat0(Zero, Succ(x0)), new_primPlusNat0(Zero, Succ(x0)), Succ(x0), y4)
new_map7(y0, y1, Zero, Succ(y3), Zero, y4) → new_map8(y0, new_primPlusNat0(Zero, Zero), new_primPlusNat0(Zero, Zero), Zero, y4)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ 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_map7(y0, y1, Zero, Succ(y3), Zero, y4) → new_map8(y0, new_primPlusNat0(Zero, Zero), new_primPlusNat0(Zero, Zero), Zero, y4)
new_map9(xu350, xu351, xu352, ba) → new_map8(xu350, new_primPlusNat0(Zero, xu352), new_primPlusNat0(Zero, xu352), new_primPlusNat0(Zero, xu352), ba)
new_map7(xu350, xu351, xu352, Succ(xu3530), Succ(xu3540), ba) → new_map7(xu350, xu351, xu352, xu3530, xu3540, ba)
new_map7(y0, y1, Succ(x0), Succ(y3), Zero, y4) → new_map8(y0, new_primPlusNat0(Zero, Succ(x0)), new_primPlusNat0(Zero, Succ(x0)), Succ(x0), y4)
new_map8(xu282, xu293, xu294, Succ(xu2950), bb) → new_map7(xu282, xu293, xu294, Succ(xu2950), Succ(xu282), bb)
new_map7(xu350, xu351, xu352, Zero, Zero, ba) → new_map9(xu350, xu351, xu352, ba)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ 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_map9(xu350, xu351, xu352, ba) → new_map8(xu350, new_primPlusNat0(Zero, xu352), new_primPlusNat0(Zero, xu352), new_primPlusNat0(Zero, xu352), ba)
new_map7(xu350, xu351, xu352, Succ(xu3530), Succ(xu3540), ba) → new_map7(xu350, xu351, xu352, xu3530, xu3540, ba)
new_map7(y0, y1, Succ(x0), Succ(y3), Zero, y4) → new_map8(y0, new_primPlusNat0(Zero, Succ(x0)), new_primPlusNat0(Zero, Succ(x0)), Succ(x0), y4)
new_map8(xu282, xu293, xu294, Succ(xu2950), bb) → new_map7(xu282, xu293, xu294, Succ(xu2950), Succ(xu282), bb)
new_map7(xu350, xu351, xu352, Zero, Zero, ba) → new_map9(xu350, xu351, xu352, ba)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_map7(y0, y1, Succ(x0), Succ(y3), Zero, y4) → new_map8(y0, new_primPlusNat0(Zero, Succ(x0)), new_primPlusNat0(Zero, Succ(x0)), Succ(x0), y4) at position [1] we obtained the following new rules:
new_map7(y0, y1, Succ(x0), Succ(y3), Zero, y4) → new_map8(y0, Succ(x0), new_primPlusNat0(Zero, Succ(x0)), Succ(x0), y4)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ 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_map9(xu350, xu351, xu352, ba) → new_map8(xu350, new_primPlusNat0(Zero, xu352), new_primPlusNat0(Zero, xu352), new_primPlusNat0(Zero, xu352), ba)
new_map7(xu350, xu351, xu352, Succ(xu3530), Succ(xu3540), ba) → new_map7(xu350, xu351, xu352, xu3530, xu3540, ba)
new_map8(xu282, xu293, xu294, Succ(xu2950), bb) → new_map7(xu282, xu293, xu294, Succ(xu2950), Succ(xu282), bb)
new_map7(y0, y1, Succ(x0), Succ(y3), Zero, y4) → new_map8(y0, Succ(x0), new_primPlusNat0(Zero, Succ(x0)), Succ(x0), y4)
new_map7(xu350, xu351, xu352, Zero, Zero, ba) → new_map9(xu350, xu351, xu352, ba)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_map7(y0, y1, Succ(x0), Succ(y3), Zero, y4) → new_map8(y0, Succ(x0), new_primPlusNat0(Zero, Succ(x0)), Succ(x0), y4) at position [2] we obtained the following new rules:
new_map7(y0, y1, Succ(x0), Succ(y3), Zero, y4) → new_map8(y0, Succ(x0), Succ(x0), Succ(x0), y4)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ 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_map9(xu350, xu351, xu352, ba) → new_map8(xu350, new_primPlusNat0(Zero, xu352), new_primPlusNat0(Zero, xu352), new_primPlusNat0(Zero, xu352), ba)
new_map7(xu350, xu351, xu352, Succ(xu3530), Succ(xu3540), ba) → new_map7(xu350, xu351, xu352, xu3530, xu3540, ba)
new_map8(xu282, xu293, xu294, Succ(xu2950), bb) → new_map7(xu282, xu293, xu294, Succ(xu2950), Succ(xu282), bb)
new_map7(y0, y1, Succ(x0), Succ(y3), Zero, y4) → new_map8(y0, Succ(x0), Succ(x0), Succ(x0), y4)
new_map7(xu350, xu351, xu352, Zero, Zero, ba) → new_map9(xu350, xu351, xu352, ba)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule new_map9(xu350, xu351, xu352, ba) → new_map8(xu350, new_primPlusNat0(Zero, xu352), new_primPlusNat0(Zero, xu352), new_primPlusNat0(Zero, xu352), ba) at position [3] we obtained the following new rules:
new_map9(y0, y1, Succ(x0), y3) → new_map8(y0, new_primPlusNat0(Zero, Succ(x0)), new_primPlusNat0(Zero, Succ(x0)), Succ(x0), y3)
new_map9(y0, y1, Zero, y3) → new_map8(y0, new_primPlusNat0(Zero, Zero), new_primPlusNat0(Zero, Zero), Zero, y3)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ 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_map9(y0, y1, Zero, y3) → new_map8(y0, new_primPlusNat0(Zero, Zero), new_primPlusNat0(Zero, Zero), Zero, y3)
new_map9(y0, y1, Succ(x0), y3) → new_map8(y0, new_primPlusNat0(Zero, Succ(x0)), new_primPlusNat0(Zero, Succ(x0)), Succ(x0), y3)
new_map7(xu350, xu351, xu352, Succ(xu3530), Succ(xu3540), ba) → new_map7(xu350, xu351, xu352, xu3530, xu3540, ba)
new_map8(xu282, xu293, xu294, Succ(xu2950), bb) → new_map7(xu282, xu293, xu294, Succ(xu2950), Succ(xu282), bb)
new_map7(xu350, xu351, xu352, Zero, Zero, ba) → new_map9(xu350, xu351, xu352, ba)
new_map7(y0, y1, Succ(x0), Succ(y3), Zero, y4) → new_map8(y0, Succ(x0), Succ(x0), Succ(x0), y4)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ 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_map9(y0, y1, Succ(x0), y3) → new_map8(y0, new_primPlusNat0(Zero, Succ(x0)), new_primPlusNat0(Zero, Succ(x0)), Succ(x0), y3)
new_map7(xu350, xu351, xu352, Succ(xu3530), Succ(xu3540), ba) → new_map7(xu350, xu351, xu352, xu3530, xu3540, ba)
new_map8(xu282, xu293, xu294, Succ(xu2950), bb) → new_map7(xu282, xu293, xu294, Succ(xu2950), Succ(xu282), bb)
new_map7(y0, y1, Succ(x0), Succ(y3), Zero, y4) → new_map8(y0, Succ(x0), Succ(x0), Succ(x0), y4)
new_map7(xu350, xu351, xu352, Zero, Zero, ba) → new_map9(xu350, xu351, xu352, ba)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ 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_map9(y0, y1, Succ(x0), y3) → new_map8(y0, new_primPlusNat0(Zero, Succ(x0)), new_primPlusNat0(Zero, Succ(x0)), Succ(x0), y3)
new_map7(xu350, xu351, xu352, Succ(xu3530), Succ(xu3540), ba) → new_map7(xu350, xu351, xu352, xu3530, xu3540, ba)
new_map8(xu282, xu293, xu294, Succ(xu2950), bb) → new_map7(xu282, xu293, xu294, Succ(xu2950), Succ(xu282), bb)
new_map7(y0, y1, Succ(x0), Succ(y3), Zero, y4) → new_map8(y0, Succ(x0), Succ(x0), Succ(x0), y4)
new_map7(xu350, xu351, xu352, Zero, Zero, ba) → new_map9(xu350, xu351, xu352, ba)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_map9(y0, y1, Succ(x0), y3) → new_map8(y0, new_primPlusNat0(Zero, Succ(x0)), new_primPlusNat0(Zero, Succ(x0)), Succ(x0), y3) at position [1] we obtained the following new rules:
new_map9(y0, y1, Succ(x0), y3) → new_map8(y0, Succ(x0), new_primPlusNat0(Zero, Succ(x0)), Succ(x0), y3)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ 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(xu350, xu351, xu352, Succ(xu3530), Succ(xu3540), ba) → new_map7(xu350, xu351, xu352, xu3530, xu3540, ba)
new_map8(xu282, xu293, xu294, Succ(xu2950), bb) → new_map7(xu282, xu293, xu294, Succ(xu2950), Succ(xu282), bb)
new_map9(y0, y1, Succ(x0), y3) → new_map8(y0, Succ(x0), new_primPlusNat0(Zero, Succ(x0)), Succ(x0), y3)
new_map7(xu350, xu351, xu352, Zero, Zero, ba) → new_map9(xu350, xu351, xu352, ba)
new_map7(y0, y1, Succ(x0), Succ(y3), Zero, y4) → new_map8(y0, Succ(x0), Succ(x0), Succ(x0), y4)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_map9(y0, y1, Succ(x0), y3) → new_map8(y0, Succ(x0), new_primPlusNat0(Zero, Succ(x0)), Succ(x0), y3) at position [2] we obtained the following new rules:
new_map9(y0, y1, Succ(x0), y3) → new_map8(y0, Succ(x0), Succ(x0), Succ(x0), y3)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ 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_map9(y0, y1, Succ(x0), y3) → new_map8(y0, Succ(x0), Succ(x0), Succ(x0), y3)
new_map7(xu350, xu351, xu352, Succ(xu3530), Succ(xu3540), ba) → new_map7(xu350, xu351, xu352, xu3530, xu3540, ba)
new_map8(xu282, xu293, xu294, Succ(xu2950), bb) → new_map7(xu282, xu293, xu294, Succ(xu2950), Succ(xu282), bb)
new_map7(y0, y1, Succ(x0), Succ(y3), Zero, y4) → new_map8(y0, Succ(x0), Succ(x0), Succ(x0), y4)
new_map7(xu350, xu351, xu352, Zero, Zero, ba) → new_map9(xu350, xu351, xu352, ba)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ 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_map9(y0, y1, Succ(x0), y3) → new_map8(y0, Succ(x0), Succ(x0), Succ(x0), y3)
new_map7(xu350, xu351, xu352, Succ(xu3530), Succ(xu3540), ba) → new_map7(xu350, xu351, xu352, xu3530, xu3540, ba)
new_map8(xu282, xu293, xu294, Succ(xu2950), bb) → new_map7(xu282, xu293, xu294, Succ(xu2950), Succ(xu282), bb)
new_map7(y0, y1, Succ(x0), Succ(y3), Zero, y4) → new_map8(y0, Succ(x0), Succ(x0), Succ(x0), y4)
new_map7(xu350, xu351, xu352, Zero, Zero, ba) → new_map9(xu350, xu351, xu352, ba)
R is empty.
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ 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_map9(y0, y1, Succ(x0), y3) → new_map8(y0, Succ(x0), Succ(x0), Succ(x0), y3)
new_map7(xu350, xu351, xu352, Succ(xu3530), Succ(xu3540), ba) → new_map7(xu350, xu351, xu352, xu3530, xu3540, ba)
new_map8(xu282, xu293, xu294, Succ(xu2950), bb) → new_map7(xu282, xu293, xu294, Succ(xu2950), Succ(xu282), bb)
new_map7(xu350, xu351, xu352, Zero, Zero, ba) → new_map9(xu350, xu351, xu352, ba)
new_map7(y0, y1, Succ(x0), Succ(y3), Zero, y4) → new_map8(y0, Succ(x0), Succ(x0), Succ(x0), y4)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule new_map8(xu282, xu293, xu294, Succ(xu2950), bb) → new_map7(xu282, xu293, xu294, Succ(xu2950), Succ(xu282), bb) we obtained the following new rules:
new_map8(z0, Succ(z2), Succ(z2), Succ(z2), z3) → new_map7(z0, Succ(z2), Succ(z2), Succ(z2), Succ(z0), z3)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ 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(z0, Succ(z2), Succ(z2), Succ(z2), z3) → new_map7(z0, Succ(z2), Succ(z2), Succ(z2), Succ(z0), z3)
new_map9(y0, y1, Succ(x0), y3) → new_map8(y0, Succ(x0), Succ(x0), Succ(x0), y3)
new_map7(xu350, xu351, xu352, Succ(xu3530), Succ(xu3540), ba) → new_map7(xu350, xu351, xu352, xu3530, xu3540, ba)
new_map7(y0, y1, Succ(x0), Succ(y3), Zero, y4) → new_map8(y0, Succ(x0), Succ(x0), Succ(x0), y4)
new_map7(xu350, xu351, xu352, Zero, Zero, ba) → new_map9(xu350, xu351, xu352, ba)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule new_map7(xu350, xu351, xu352, Succ(xu3530), Succ(xu3540), ba) → new_map7(xu350, xu351, xu352, xu3530, xu3540, ba) we obtained the following new rules:
new_map7(x0, x1, x2, Succ(Zero), Succ(Zero), x5) → new_map7(x0, x1, x2, Zero, Zero, x5)
new_map7(x0, x1, Succ(y_2), Succ(Succ(y_3)), Succ(Zero), x5) → new_map7(x0, x1, Succ(y_2), Succ(y_3), Zero, x5)
new_map7(x0, x1, x2, Succ(Succ(y_3)), Succ(Succ(y_4)), x5) → new_map7(x0, x1, x2, Succ(y_3), Succ(y_4), x5)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ 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(z0, Succ(z2), Succ(z2), Succ(z2), z3) → new_map7(z0, Succ(z2), Succ(z2), Succ(z2), Succ(z0), z3)
new_map7(x0, x1, x2, Succ(Zero), Succ(Zero), x5) → new_map7(x0, x1, x2, Zero, Zero, x5)
new_map9(y0, y1, Succ(x0), y3) → new_map8(y0, Succ(x0), Succ(x0), Succ(x0), y3)
new_map7(x0, x1, Succ(y_2), Succ(Succ(y_3)), Succ(Zero), x5) → new_map7(x0, x1, Succ(y_2), Succ(y_3), Zero, x5)
new_map7(xu350, xu351, xu352, Zero, Zero, ba) → new_map9(xu350, xu351, xu352, ba)
new_map7(y0, y1, Succ(x0), Succ(y3), Zero, y4) → new_map8(y0, Succ(x0), Succ(x0), Succ(x0), y4)
new_map7(x0, x1, x2, Succ(Succ(y_3)), Succ(Succ(y_4)), x5) → new_map7(x0, x1, x2, Succ(y_3), Succ(y_4), x5)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule new_map7(xu350, xu351, xu352, Zero, Zero, ba) → new_map9(xu350, xu351, xu352, ba) we obtained the following new rules:
new_map7(x0, x1, Succ(y_2), Zero, Zero, x3) → new_map9(x0, x1, Succ(y_2), x3)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ 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(z0, Succ(z2), Succ(z2), Succ(z2), z3) → new_map7(z0, Succ(z2), Succ(z2), Succ(z2), Succ(z0), z3)
new_map9(y0, y1, Succ(x0), y3) → new_map8(y0, Succ(x0), Succ(x0), Succ(x0), y3)
new_map7(x0, x1, x2, Succ(Zero), Succ(Zero), x5) → new_map7(x0, x1, x2, Zero, Zero, x5)
new_map7(x0, x1, Succ(y_2), Zero, Zero, x3) → new_map9(x0, x1, Succ(y_2), x3)
new_map7(x0, x1, Succ(y_2), Succ(Succ(y_3)), Succ(Zero), x5) → new_map7(x0, x1, Succ(y_2), Succ(y_3), Zero, x5)
new_map7(y0, y1, Succ(x0), Succ(y3), Zero, y4) → new_map8(y0, Succ(x0), Succ(x0), Succ(x0), y4)
new_map7(x0, x1, x2, Succ(Succ(y_3)), Succ(Succ(y_4)), x5) → new_map7(x0, x1, x2, Succ(y_3), Succ(y_4), x5)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule new_map8(z0, Succ(z2), Succ(z2), Succ(z2), z3) → new_map7(z0, Succ(z2), Succ(z2), Succ(z2), Succ(z0), z3) we obtained the following new rules:
new_map8(Zero, Succ(Succ(y_3)), Succ(Succ(y_3)), Succ(Succ(y_3)), x2) → new_map7(Zero, Succ(Succ(y_3)), Succ(Succ(y_3)), Succ(Succ(y_3)), Succ(Zero), x2)
new_map8(Succ(y_4), Succ(Succ(y_3)), Succ(Succ(y_3)), Succ(Succ(y_3)), x2) → new_map7(Succ(y_4), Succ(Succ(y_3)), Succ(Succ(y_3)), Succ(Succ(y_3)), Succ(Succ(y_4)), x2)
new_map8(Zero, Succ(Zero), Succ(Zero), Succ(Zero), x2) → new_map7(Zero, Succ(Zero), Succ(Zero), Succ(Zero), Succ(Zero), x2)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ 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(x0, x1, x2, Succ(Zero), Succ(Zero), x5) → new_map7(x0, x1, x2, Zero, Zero, x5)
new_map9(y0, y1, Succ(x0), y3) → new_map8(y0, Succ(x0), Succ(x0), Succ(x0), y3)
new_map7(x0, x1, Succ(y_2), Zero, Zero, x3) → new_map9(x0, x1, Succ(y_2), x3)
new_map8(Zero, Succ(Succ(y_3)), Succ(Succ(y_3)), Succ(Succ(y_3)), x2) → new_map7(Zero, Succ(Succ(y_3)), Succ(Succ(y_3)), Succ(Succ(y_3)), Succ(Zero), x2)
new_map7(x0, x1, Succ(y_2), Succ(Succ(y_3)), Succ(Zero), x5) → new_map7(x0, x1, Succ(y_2), Succ(y_3), Zero, x5)
new_map8(Succ(y_4), Succ(Succ(y_3)), Succ(Succ(y_3)), Succ(Succ(y_3)), x2) → new_map7(Succ(y_4), Succ(Succ(y_3)), Succ(Succ(y_3)), Succ(Succ(y_3)), Succ(Succ(y_4)), x2)
new_map7(y0, y1, Succ(x0), Succ(y3), Zero, y4) → new_map8(y0, Succ(x0), Succ(x0), Succ(x0), y4)
new_map8(Zero, Succ(Zero), Succ(Zero), Succ(Zero), x2) → new_map7(Zero, Succ(Zero), Succ(Zero), Succ(Zero), Succ(Zero), x2)
new_map7(x0, x1, x2, Succ(Succ(y_3)), Succ(Succ(y_4)), x5) → new_map7(x0, x1, x2, Succ(y_3), Succ(y_4), x5)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule new_map9(y0, y1, Succ(x0), y3) → new_map8(y0, Succ(x0), Succ(x0), Succ(x0), y3) we obtained the following new rules:
new_map9(Succ(y_0), x1, Succ(Succ(y_1)), x3) → new_map8(Succ(y_0), Succ(Succ(y_1)), Succ(Succ(y_1)), Succ(Succ(y_1)), x3)
new_map9(Zero, x1, Succ(Succ(y_0)), x3) → new_map8(Zero, Succ(Succ(y_0)), Succ(Succ(y_0)), Succ(Succ(y_0)), x3)
new_map9(Zero, x1, Succ(Zero), x3) → new_map8(Zero, Succ(Zero), Succ(Zero), Succ(Zero), x3)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ 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_map9(Succ(y_0), x1, Succ(Succ(y_1)), x3) → new_map8(Succ(y_0), Succ(Succ(y_1)), Succ(Succ(y_1)), Succ(Succ(y_1)), x3)
new_map7(x0, x1, x2, Succ(Zero), Succ(Zero), x5) → new_map7(x0, x1, x2, Zero, Zero, x5)
new_map7(x0, x1, Succ(y_2), Zero, Zero, x3) → new_map9(x0, x1, Succ(y_2), x3)
new_map9(Zero, x1, Succ(Succ(y_0)), x3) → new_map8(Zero, Succ(Succ(y_0)), Succ(Succ(y_0)), Succ(Succ(y_0)), x3)
new_map8(Zero, Succ(Succ(y_3)), Succ(Succ(y_3)), Succ(Succ(y_3)), x2) → new_map7(Zero, Succ(Succ(y_3)), Succ(Succ(y_3)), Succ(Succ(y_3)), Succ(Zero), x2)
new_map7(x0, x1, Succ(y_2), Succ(Succ(y_3)), Succ(Zero), x5) → new_map7(x0, x1, Succ(y_2), Succ(y_3), Zero, x5)
new_map8(Succ(y_4), Succ(Succ(y_3)), Succ(Succ(y_3)), Succ(Succ(y_3)), x2) → new_map7(Succ(y_4), Succ(Succ(y_3)), Succ(Succ(y_3)), Succ(Succ(y_3)), Succ(Succ(y_4)), x2)
new_map7(y0, y1, Succ(x0), Succ(y3), Zero, y4) → new_map8(y0, Succ(x0), Succ(x0), Succ(x0), y4)
new_map9(Zero, x1, Succ(Zero), x3) → new_map8(Zero, Succ(Zero), Succ(Zero), Succ(Zero), x3)
new_map7(x0, x1, x2, Succ(Succ(y_3)), Succ(Succ(y_4)), x5) → new_map7(x0, x1, x2, Succ(y_3), Succ(y_4), x5)
new_map8(Zero, Succ(Zero), Succ(Zero), Succ(Zero), x2) → new_map7(Zero, Succ(Zero), Succ(Zero), Succ(Zero), Succ(Zero), x2)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule new_map7(y0, y1, Succ(x0), Succ(y3), Zero, y4) → new_map8(y0, Succ(x0), Succ(x0), Succ(x0), y4) we obtained the following new rules:
new_map7(Zero, x1, Succ(Zero), Succ(x3), Zero, x4) → new_map8(Zero, Succ(Zero), Succ(Zero), Succ(Zero), x4)
new_map7(Zero, x1, Succ(Succ(y_0)), Succ(x3), Zero, x4) → new_map8(Zero, Succ(Succ(y_0)), Succ(Succ(y_0)), Succ(Succ(y_0)), x4)
new_map7(Succ(y_0), x1, Succ(Succ(y_1)), Succ(x3), Zero, x4) → new_map8(Succ(y_0), Succ(Succ(y_1)), Succ(Succ(y_1)), Succ(Succ(y_1)), x4)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ NonTerminationProof
↳ 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(Zero, x1, Succ(Zero), Succ(x3), Zero, x4) → new_map8(Zero, Succ(Zero), Succ(Zero), Succ(Zero), x4)
new_map7(Zero, x1, Succ(Succ(y_0)), Succ(x3), Zero, x4) → new_map8(Zero, Succ(Succ(y_0)), Succ(Succ(y_0)), Succ(Succ(y_0)), x4)
new_map8(Zero, Succ(Succ(y_3)), Succ(Succ(y_3)), Succ(Succ(y_3)), x2) → new_map7(Zero, Succ(Succ(y_3)), Succ(Succ(y_3)), Succ(Succ(y_3)), Succ(Zero), x2)
new_map8(Succ(y_4), Succ(Succ(y_3)), Succ(Succ(y_3)), Succ(Succ(y_3)), x2) → new_map7(Succ(y_4), Succ(Succ(y_3)), Succ(Succ(y_3)), Succ(Succ(y_3)), Succ(Succ(y_4)), x2)
new_map7(x0, x1, x2, Succ(Succ(y_3)), Succ(Succ(y_4)), x5) → new_map7(x0, x1, x2, Succ(y_3), Succ(y_4), x5)
new_map9(Succ(y_0), x1, Succ(Succ(y_1)), x3) → new_map8(Succ(y_0), Succ(Succ(y_1)), Succ(Succ(y_1)), Succ(Succ(y_1)), x3)
new_map7(x0, x1, x2, Succ(Zero), Succ(Zero), x5) → new_map7(x0, x1, x2, Zero, Zero, x5)
new_map7(x0, x1, Succ(y_2), Zero, Zero, x3) → new_map9(x0, x1, Succ(y_2), x3)
new_map7(Succ(y_0), x1, Succ(Succ(y_1)), Succ(x3), Zero, x4) → new_map8(Succ(y_0), Succ(Succ(y_1)), Succ(Succ(y_1)), Succ(Succ(y_1)), x4)
new_map9(Zero, x1, Succ(Succ(y_0)), x3) → new_map8(Zero, Succ(Succ(y_0)), Succ(Succ(y_0)), Succ(Succ(y_0)), x3)
new_map7(x0, x1, Succ(y_2), Succ(Succ(y_3)), Succ(Zero), x5) → new_map7(x0, x1, Succ(y_2), Succ(y_3), Zero, x5)
new_map9(Zero, x1, Succ(Zero), x3) → new_map8(Zero, Succ(Zero), Succ(Zero), Succ(Zero), x3)
new_map8(Zero, Succ(Zero), Succ(Zero), Succ(Zero), x2) → new_map7(Zero, Succ(Zero), Succ(Zero), Succ(Zero), Succ(Zero), x2)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by narrowing to the left:
The TRS P consists of the following rules:
new_map7(Zero, x1, Succ(Zero), Succ(x3), Zero, x4) → new_map8(Zero, Succ(Zero), Succ(Zero), Succ(Zero), x4)
new_map7(Zero, x1, Succ(Succ(y_0)), Succ(x3), Zero, x4) → new_map8(Zero, Succ(Succ(y_0)), Succ(Succ(y_0)), Succ(Succ(y_0)), x4)
new_map8(Zero, Succ(Succ(y_3)), Succ(Succ(y_3)), Succ(Succ(y_3)), x2) → new_map7(Zero, Succ(Succ(y_3)), Succ(Succ(y_3)), Succ(Succ(y_3)), Succ(Zero), x2)
new_map8(Succ(y_4), Succ(Succ(y_3)), Succ(Succ(y_3)), Succ(Succ(y_3)), x2) → new_map7(Succ(y_4), Succ(Succ(y_3)), Succ(Succ(y_3)), Succ(Succ(y_3)), Succ(Succ(y_4)), x2)
new_map7(x0, x1, x2, Succ(Succ(y_3)), Succ(Succ(y_4)), x5) → new_map7(x0, x1, x2, Succ(y_3), Succ(y_4), x5)
new_map9(Succ(y_0), x1, Succ(Succ(y_1)), x3) → new_map8(Succ(y_0), Succ(Succ(y_1)), Succ(Succ(y_1)), Succ(Succ(y_1)), x3)
new_map7(x0, x1, x2, Succ(Zero), Succ(Zero), x5) → new_map7(x0, x1, x2, Zero, Zero, x5)
new_map7(x0, x1, Succ(y_2), Zero, Zero, x3) → new_map9(x0, x1, Succ(y_2), x3)
new_map7(Succ(y_0), x1, Succ(Succ(y_1)), Succ(x3), Zero, x4) → new_map8(Succ(y_0), Succ(Succ(y_1)), Succ(Succ(y_1)), Succ(Succ(y_1)), x4)
new_map9(Zero, x1, Succ(Succ(y_0)), x3) → new_map8(Zero, Succ(Succ(y_0)), Succ(Succ(y_0)), Succ(Succ(y_0)), x3)
new_map7(x0, x1, Succ(y_2), Succ(Succ(y_3)), Succ(Zero), x5) → new_map7(x0, x1, Succ(y_2), Succ(y_3), Zero, x5)
new_map9(Zero, x1, Succ(Zero), x3) → new_map8(Zero, Succ(Zero), Succ(Zero), Succ(Zero), x3)
new_map8(Zero, Succ(Zero), Succ(Zero), Succ(Zero), x2) → new_map7(Zero, Succ(Zero), Succ(Zero), Succ(Zero), Succ(Zero), x2)
The TRS R consists of the following rules:none
s = new_map8(Zero, Succ(Succ(y_3')), Succ(Succ(y_3')), Succ(Succ(y_3')), x2) evaluates to t =new_map8(Zero, Succ(Succ(y_3')), Succ(Succ(y_3')), Succ(Succ(y_3')), x2)
Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
- Semiunifier: [ ]
- Matcher: [ ]
Rewriting sequence
new_map8(Zero, Succ(Succ(y_3')), Succ(Succ(y_3')), Succ(Succ(y_3')), x2) → new_map7(Zero, Succ(Succ(y_3')), Succ(Succ(y_3')), Succ(Succ(y_3')), Succ(Zero), x2)
with rule new_map8(Zero, Succ(Succ(y_3'')), Succ(Succ(y_3'')), Succ(Succ(y_3'')), x2') → new_map7(Zero, Succ(Succ(y_3'')), Succ(Succ(y_3'')), Succ(Succ(y_3'')), Succ(Zero), x2') at position [] and matcher [y_3'' / y_3', x2' / x2]
new_map7(Zero, Succ(Succ(y_3')), Succ(Succ(y_3')), Succ(Succ(y_3')), Succ(Zero), x2) → new_map7(Zero, Succ(Succ(y_3')), Succ(Succ(y_3')), Succ(y_3'), Zero, x2)
with rule new_map7(x0, x1', Succ(y_2), Succ(Succ(y_3)), Succ(Zero), x5) → new_map7(x0, x1', Succ(y_2), Succ(y_3), Zero, x5) at position [] and matcher [y_2 / Succ(y_3'), x1' / Succ(Succ(y_3')), y_3 / y_3', x5 / x2, x0 / Zero]
new_map7(Zero, Succ(Succ(y_3')), Succ(Succ(y_3')), Succ(y_3'), Zero, x2) → new_map8(Zero, Succ(Succ(y_3')), Succ(Succ(y_3')), Succ(Succ(y_3')), x2)
with rule new_map7(Zero, x1, Succ(Succ(y_0)), Succ(x3), Zero, x4) → new_map8(Zero, Succ(Succ(y_0)), Succ(Succ(y_0)), Succ(Succ(y_0)), x4)
Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence
All these steps are and every following step will be a correct step w.r.t to Q.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ 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(xu282, xu2840, xu285, bb) → new_map13(xu282, Succ(xu285), Succ(xu2840), xu2840, bb)
new_map10(xu405, xu406, xu407, Succ(xu4080), Zero, ba) → new_map11(xu405, xu407, xu406, ba)
new_map10(xu405, xu406, xu407, Zero, Zero, ba) → new_map12(xu405, xu406, xu407, ba)
new_map10(xu405, xu406, xu407, Succ(xu4080), Succ(xu4090), ba) → new_map10(xu405, xu406, xu407, xu4080, xu4090, ba)
new_map12(xu405, xu406, xu407, ba) → new_map11(xu405, xu407, xu406, ba)
new_map13(xu338, Succ(xu3390), Succ(xu3400), xu341, bc) → new_map13(xu338, xu3390, xu3400, xu341, bc)
new_map13(xu338, Succ(xu3390), Zero, xu341, bc) → new_map10(xu338, xu3390, xu341, Succ(xu3390), Succ(xu338), bc)
R is empty.
Q is empty.
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_map13(xu338, Succ(xu3390), Succ(xu3400), xu341, bc) → new_map13(xu338, xu3390, xu3400, xu341, bc)
The remaining pairs can at least be oriented weakly.
new_map11(xu282, xu2840, xu285, bb) → new_map13(xu282, Succ(xu285), Succ(xu2840), xu2840, bb)
new_map10(xu405, xu406, xu407, Succ(xu4080), Zero, ba) → new_map11(xu405, xu407, xu406, ba)
new_map10(xu405, xu406, xu407, Zero, Zero, ba) → new_map12(xu405, xu406, xu407, ba)
new_map10(xu405, xu406, xu407, Succ(xu4080), Succ(xu4090), ba) → new_map10(xu405, xu406, xu407, xu4080, xu4090, ba)
new_map12(xu405, xu406, xu407, ba) → new_map11(xu405, xu407, xu406, ba)
new_map13(xu338, Succ(xu3390), Zero, xu341, bc) → new_map10(xu338, xu3390, xu341, Succ(xu3390), Succ(xu338), bc)
Used ordering: Polynomial interpretation [25]:
POL(Succ(x1)) = 1 + x1
POL(Zero) = 0
POL(new_map10(x1, x2, x3, x4, x5, x6)) = 1 + x2
POL(new_map11(x1, x2, x3, x4)) = 1 + x3
POL(new_map12(x1, x2, x3, x4)) = 1 + x2
POL(new_map13(x1, x2, x3, x4, x5)) = x2
The following usable rules [17] were oriented:
none
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ 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(xu282, xu2840, xu285, bb) → new_map13(xu282, Succ(xu285), Succ(xu2840), xu2840, bb)
new_map10(xu405, xu406, xu407, Zero, Zero, ba) → new_map12(xu405, xu406, xu407, ba)
new_map10(xu405, xu406, xu407, Succ(xu4080), Zero, ba) → new_map11(xu405, xu407, xu406, ba)
new_map10(xu405, xu406, xu407, Succ(xu4080), Succ(xu4090), ba) → new_map10(xu405, xu406, xu407, xu4080, xu4090, ba)
new_map12(xu405, xu406, xu407, ba) → new_map11(xu405, xu407, xu406, ba)
new_map13(xu338, Succ(xu3390), Zero, xu341, bc) → new_map10(xu338, xu3390, xu341, Succ(xu3390), Succ(xu338), bc)
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 5 less nodes.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ 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_map10(xu405, xu406, xu407, Succ(xu4080), Succ(xu4090), ba) → new_map10(xu405, xu406, xu407, xu4080, xu4090, 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_map10(xu405, xu406, xu407, Succ(xu4080), Succ(xu4090), ba) → new_map10(xu405, xu406, xu407, xu4080, xu4090, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4, 5 > 5, 6 >= 6
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map14(xu343, xu344, xu345, xu346, Succ(xu3470), Succ(xu3480), ba) → new_map14(xu343, xu344, xu345, xu346, xu3470, xu3480, ba)
new_map14(xu343, xu344, xu345, xu346, Zero, Zero, ba) → new_map16(xu343, xu344, xu345, xu346, ba)
new_map15(xu282, xu290, xu2830, xu291, Zero, bb) → new_map14(xu282, xu290, xu2830, xu291, Zero, Succ(xu282), bb)
new_map14(xu343, xu344, xu345, xu346, Succ(xu3470), Zero, ba) → new_map15(xu343, new_primPlusNat0(Succ(xu345), xu346), xu345, new_primPlusNat0(Succ(xu345), xu346), new_primPlusNat0(Succ(xu345), xu346), ba)
new_map16(xu343, xu344, xu345, xu346, ba) → new_map15(xu343, new_primPlusNat0(Succ(xu345), xu346), xu345, new_primPlusNat0(Succ(xu345), xu346), new_primPlusNat0(Succ(xu345), xu346), ba)
new_map15(xu282, xu290, xu2830, xu291, Succ(xu2920), bb) → new_map14(xu282, xu290, xu2830, xu291, Succ(xu2920), Succ(xu282), bb)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map14(xu343, xu344, xu345, xu346, Succ(xu3470), Succ(xu3480), ba) → new_map14(xu343, xu344, xu345, xu346, xu3470, xu3480, ba)
new_map14(xu343, xu344, xu345, xu346, Zero, Zero, ba) → new_map16(xu343, xu344, xu345, xu346, ba)
new_map14(xu343, xu344, xu345, xu346, Succ(xu3470), Zero, ba) → new_map15(xu343, new_primPlusNat0(Succ(xu345), xu346), xu345, new_primPlusNat0(Succ(xu345), xu346), new_primPlusNat0(Succ(xu345), xu346), ba)
new_map16(xu343, xu344, xu345, xu346, ba) → new_map15(xu343, new_primPlusNat0(Succ(xu345), xu346), xu345, new_primPlusNat0(Succ(xu345), xu346), new_primPlusNat0(Succ(xu345), xu346), ba)
new_map15(xu282, xu290, xu2830, xu291, Succ(xu2920), bb) → new_map14(xu282, xu290, xu2830, xu291, Succ(xu2920), Succ(xu282), bb)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule new_map16(xu343, xu344, xu345, xu346, ba) → new_map15(xu343, new_primPlusNat0(Succ(xu345), xu346), xu345, new_primPlusNat0(Succ(xu345), xu346), new_primPlusNat0(Succ(xu345), xu346), ba) at position [4] we obtained the following new rules:
new_map16(y0, y1, x0, Succ(x1), y4) → new_map15(y0, new_primPlusNat0(Succ(x0), Succ(x1)), x0, new_primPlusNat0(Succ(x0), Succ(x1)), Succ(Succ(new_primPlusNat0(x0, x1))), y4)
new_map16(y0, y1, x0, Zero, y4) → new_map15(y0, new_primPlusNat0(Succ(x0), Zero), x0, new_primPlusNat0(Succ(x0), Zero), Succ(x0), y4)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map16(y0, y1, x0, Succ(x1), y4) → new_map15(y0, new_primPlusNat0(Succ(x0), Succ(x1)), x0, new_primPlusNat0(Succ(x0), Succ(x1)), Succ(Succ(new_primPlusNat0(x0, x1))), y4)
new_map14(xu343, xu344, xu345, xu346, Succ(xu3470), Succ(xu3480), ba) → new_map14(xu343, xu344, xu345, xu346, xu3470, xu3480, ba)
new_map14(xu343, xu344, xu345, xu346, Zero, Zero, ba) → new_map16(xu343, xu344, xu345, xu346, ba)
new_map14(xu343, xu344, xu345, xu346, Succ(xu3470), Zero, ba) → new_map15(xu343, new_primPlusNat0(Succ(xu345), xu346), xu345, new_primPlusNat0(Succ(xu345), xu346), new_primPlusNat0(Succ(xu345), xu346), ba)
new_map16(y0, y1, x0, Zero, y4) → new_map15(y0, new_primPlusNat0(Succ(x0), Zero), x0, new_primPlusNat0(Succ(x0), Zero), Succ(x0), y4)
new_map15(xu282, xu290, xu2830, xu291, Succ(xu2920), bb) → new_map14(xu282, xu290, xu2830, xu291, Succ(xu2920), Succ(xu282), bb)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_map16(y0, y1, x0, Succ(x1), y4) → new_map15(y0, new_primPlusNat0(Succ(x0), Succ(x1)), x0, new_primPlusNat0(Succ(x0), Succ(x1)), Succ(Succ(new_primPlusNat0(x0, x1))), y4) at position [1] we obtained the following new rules:
new_map16(y0, y1, x0, Succ(x1), y4) → new_map15(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, new_primPlusNat0(Succ(x0), Succ(x1)), Succ(Succ(new_primPlusNat0(x0, x1))), y4)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map14(xu343, xu344, xu345, xu346, Succ(xu3470), Succ(xu3480), ba) → new_map14(xu343, xu344, xu345, xu346, xu3470, xu3480, ba)
new_map14(xu343, xu344, xu345, xu346, Zero, Zero, ba) → new_map16(xu343, xu344, xu345, xu346, ba)
new_map14(xu343, xu344, xu345, xu346, Succ(xu3470), Zero, ba) → new_map15(xu343, new_primPlusNat0(Succ(xu345), xu346), xu345, new_primPlusNat0(Succ(xu345), xu346), new_primPlusNat0(Succ(xu345), xu346), ba)
new_map16(y0, y1, x0, Zero, y4) → new_map15(y0, new_primPlusNat0(Succ(x0), Zero), x0, new_primPlusNat0(Succ(x0), Zero), Succ(x0), y4)
new_map15(xu282, xu290, xu2830, xu291, Succ(xu2920), bb) → new_map14(xu282, xu290, xu2830, xu291, Succ(xu2920), Succ(xu282), bb)
new_map16(y0, y1, x0, Succ(x1), y4) → new_map15(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, new_primPlusNat0(Succ(x0), Succ(x1)), Succ(Succ(new_primPlusNat0(x0, x1))), y4)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_map16(y0, y1, x0, Zero, y4) → new_map15(y0, new_primPlusNat0(Succ(x0), Zero), x0, new_primPlusNat0(Succ(x0), Zero), Succ(x0), y4) at position [1] we obtained the following new rules:
new_map16(y0, y1, x0, Zero, y4) → new_map15(y0, Succ(x0), x0, new_primPlusNat0(Succ(x0), Zero), Succ(x0), y4)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map14(xu343, xu344, xu345, xu346, Succ(xu3470), Succ(xu3480), ba) → new_map14(xu343, xu344, xu345, xu346, xu3470, xu3480, ba)
new_map14(xu343, xu344, xu345, xu346, Zero, Zero, ba) → new_map16(xu343, xu344, xu345, xu346, ba)
new_map14(xu343, xu344, xu345, xu346, Succ(xu3470), Zero, ba) → new_map15(xu343, new_primPlusNat0(Succ(xu345), xu346), xu345, new_primPlusNat0(Succ(xu345), xu346), new_primPlusNat0(Succ(xu345), xu346), ba)
new_map15(xu282, xu290, xu2830, xu291, Succ(xu2920), bb) → new_map14(xu282, xu290, xu2830, xu291, Succ(xu2920), Succ(xu282), bb)
new_map16(y0, y1, x0, Zero, y4) → new_map15(y0, Succ(x0), x0, new_primPlusNat0(Succ(x0), Zero), Succ(x0), y4)
new_map16(y0, y1, x0, Succ(x1), y4) → new_map15(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, new_primPlusNat0(Succ(x0), Succ(x1)), Succ(Succ(new_primPlusNat0(x0, x1))), y4)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_map16(y0, y1, x0, Succ(x1), y4) → new_map15(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, new_primPlusNat0(Succ(x0), Succ(x1)), Succ(Succ(new_primPlusNat0(x0, x1))), y4) at position [3] we obtained the following new rules:
new_map16(y0, y1, x0, Succ(x1), y4) → new_map15(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y4)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map14(xu343, xu344, xu345, xu346, Succ(xu3470), Succ(xu3480), ba) → new_map14(xu343, xu344, xu345, xu346, xu3470, xu3480, ba)
new_map14(xu343, xu344, xu345, xu346, Zero, Zero, ba) → new_map16(xu343, xu344, xu345, xu346, ba)
new_map16(y0, y1, x0, Succ(x1), y4) → new_map15(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y4)
new_map14(xu343, xu344, xu345, xu346, Succ(xu3470), Zero, ba) → new_map15(xu343, new_primPlusNat0(Succ(xu345), xu346), xu345, new_primPlusNat0(Succ(xu345), xu346), new_primPlusNat0(Succ(xu345), xu346), ba)
new_map15(xu282, xu290, xu2830, xu291, Succ(xu2920), bb) → new_map14(xu282, xu290, xu2830, xu291, Succ(xu2920), Succ(xu282), bb)
new_map16(y0, y1, x0, Zero, y4) → new_map15(y0, Succ(x0), x0, new_primPlusNat0(Succ(x0), Zero), Succ(x0), y4)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_map16(y0, y1, x0, Zero, y4) → new_map15(y0, Succ(x0), x0, new_primPlusNat0(Succ(x0), Zero), Succ(x0), y4) at position [3] we obtained the following new rules:
new_map16(y0, y1, x0, Zero, y4) → new_map15(y0, Succ(x0), x0, Succ(x0), Succ(x0), y4)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map14(xu343, xu344, xu345, xu346, Succ(xu3470), Succ(xu3480), ba) → new_map14(xu343, xu344, xu345, xu346, xu3470, xu3480, ba)
new_map16(y0, y1, x0, Zero, y4) → new_map15(y0, Succ(x0), x0, Succ(x0), Succ(x0), y4)
new_map14(xu343, xu344, xu345, xu346, Zero, Zero, ba) → new_map16(xu343, xu344, xu345, xu346, ba)
new_map16(y0, y1, x0, Succ(x1), y4) → new_map15(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y4)
new_map14(xu343, xu344, xu345, xu346, Succ(xu3470), Zero, ba) → new_map15(xu343, new_primPlusNat0(Succ(xu345), xu346), xu345, new_primPlusNat0(Succ(xu345), xu346), new_primPlusNat0(Succ(xu345), xu346), ba)
new_map15(xu282, xu290, xu2830, xu291, Succ(xu2920), bb) → new_map14(xu282, xu290, xu2830, xu291, Succ(xu2920), Succ(xu282), bb)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule new_map14(xu343, xu344, xu345, xu346, Succ(xu3470), Zero, ba) → new_map15(xu343, new_primPlusNat0(Succ(xu345), xu346), xu345, new_primPlusNat0(Succ(xu345), xu346), new_primPlusNat0(Succ(xu345), xu346), ba) at position [4] we obtained the following new rules:
new_map14(y0, y1, x0, Zero, Succ(y4), Zero, y5) → new_map15(y0, new_primPlusNat0(Succ(x0), Zero), x0, new_primPlusNat0(Succ(x0), Zero), Succ(x0), y5)
new_map14(y0, y1, x0, Succ(x1), Succ(y4), Zero, y5) → new_map15(y0, new_primPlusNat0(Succ(x0), Succ(x1)), x0, new_primPlusNat0(Succ(x0), Succ(x1)), Succ(Succ(new_primPlusNat0(x0, x1))), y5)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map14(y0, y1, x0, Zero, Succ(y4), Zero, y5) → new_map15(y0, new_primPlusNat0(Succ(x0), Zero), x0, new_primPlusNat0(Succ(x0), Zero), Succ(x0), y5)
new_map14(xu343, xu344, xu345, xu346, Succ(xu3470), Succ(xu3480), ba) → new_map14(xu343, xu344, xu345, xu346, xu3470, xu3480, ba)
new_map14(xu343, xu344, xu345, xu346, Zero, Zero, ba) → new_map16(xu343, xu344, xu345, xu346, ba)
new_map16(y0, y1, x0, Zero, y4) → new_map15(y0, Succ(x0), x0, Succ(x0), Succ(x0), y4)
new_map16(y0, y1, x0, Succ(x1), y4) → new_map15(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y4)
new_map15(xu282, xu290, xu2830, xu291, Succ(xu2920), bb) → new_map14(xu282, xu290, xu2830, xu291, Succ(xu2920), Succ(xu282), bb)
new_map14(y0, y1, x0, Succ(x1), Succ(y4), Zero, y5) → new_map15(y0, new_primPlusNat0(Succ(x0), Succ(x1)), x0, new_primPlusNat0(Succ(x0), Succ(x1)), Succ(Succ(new_primPlusNat0(x0, x1))), y5)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_map14(y0, y1, x0, Zero, Succ(y4), Zero, y5) → new_map15(y0, new_primPlusNat0(Succ(x0), Zero), x0, new_primPlusNat0(Succ(x0), Zero), Succ(x0), y5) at position [1] we obtained the following new rules:
new_map14(y0, y1, x0, Zero, Succ(y4), Zero, y5) → new_map15(y0, Succ(x0), x0, new_primPlusNat0(Succ(x0), Zero), Succ(x0), y5)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map14(y0, y1, x0, Zero, Succ(y4), Zero, y5) → new_map15(y0, Succ(x0), x0, new_primPlusNat0(Succ(x0), Zero), Succ(x0), y5)
new_map14(xu343, xu344, xu345, xu346, Succ(xu3470), Succ(xu3480), ba) → new_map14(xu343, xu344, xu345, xu346, xu3470, xu3480, ba)
new_map16(y0, y1, x0, Zero, y4) → new_map15(y0, Succ(x0), x0, Succ(x0), Succ(x0), y4)
new_map14(xu343, xu344, xu345, xu346, Zero, Zero, ba) → new_map16(xu343, xu344, xu345, xu346, ba)
new_map16(y0, y1, x0, Succ(x1), y4) → new_map15(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y4)
new_map15(xu282, xu290, xu2830, xu291, Succ(xu2920), bb) → new_map14(xu282, xu290, xu2830, xu291, Succ(xu2920), Succ(xu282), bb)
new_map14(y0, y1, x0, Succ(x1), Succ(y4), Zero, y5) → new_map15(y0, new_primPlusNat0(Succ(x0), Succ(x1)), x0, new_primPlusNat0(Succ(x0), Succ(x1)), Succ(Succ(new_primPlusNat0(x0, x1))), y5)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_map14(y0, y1, x0, Succ(x1), Succ(y4), Zero, y5) → new_map15(y0, new_primPlusNat0(Succ(x0), Succ(x1)), x0, new_primPlusNat0(Succ(x0), Succ(x1)), Succ(Succ(new_primPlusNat0(x0, x1))), y5) at position [1] we obtained the following new rules:
new_map14(y0, y1, x0, Succ(x1), Succ(y4), Zero, y5) → new_map15(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, new_primPlusNat0(Succ(x0), Succ(x1)), Succ(Succ(new_primPlusNat0(x0, x1))), y5)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map14(xu343, xu344, xu345, xu346, Succ(xu3470), Succ(xu3480), ba) → new_map14(xu343, xu344, xu345, xu346, xu3470, xu3480, ba)
new_map14(y0, y1, x0, Zero, Succ(y4), Zero, y5) → new_map15(y0, Succ(x0), x0, new_primPlusNat0(Succ(x0), Zero), Succ(x0), y5)
new_map14(xu343, xu344, xu345, xu346, Zero, Zero, ba) → new_map16(xu343, xu344, xu345, xu346, ba)
new_map16(y0, y1, x0, Zero, y4) → new_map15(y0, Succ(x0), x0, Succ(x0), Succ(x0), y4)
new_map14(y0, y1, x0, Succ(x1), Succ(y4), Zero, y5) → new_map15(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, new_primPlusNat0(Succ(x0), Succ(x1)), Succ(Succ(new_primPlusNat0(x0, x1))), y5)
new_map16(y0, y1, x0, Succ(x1), y4) → new_map15(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y4)
new_map15(xu282, xu290, xu2830, xu291, Succ(xu2920), bb) → new_map14(xu282, xu290, xu2830, xu291, Succ(xu2920), Succ(xu282), bb)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_map14(y0, y1, x0, Zero, Succ(y4), Zero, y5) → new_map15(y0, Succ(x0), x0, new_primPlusNat0(Succ(x0), Zero), Succ(x0), y5) at position [3] we obtained the following new rules:
new_map14(y0, y1, x0, Zero, Succ(y4), Zero, y5) → new_map15(y0, Succ(x0), x0, Succ(x0), Succ(x0), y5)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map14(xu343, xu344, xu345, xu346, Succ(xu3470), Succ(xu3480), ba) → new_map14(xu343, xu344, xu345, xu346, xu3470, xu3480, ba)
new_map16(y0, y1, x0, Zero, y4) → new_map15(y0, Succ(x0), x0, Succ(x0), Succ(x0), y4)
new_map14(xu343, xu344, xu345, xu346, Zero, Zero, ba) → new_map16(xu343, xu344, xu345, xu346, ba)
new_map16(y0, y1, x0, Succ(x1), y4) → new_map15(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y4)
new_map14(y0, y1, x0, Succ(x1), Succ(y4), Zero, y5) → new_map15(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, new_primPlusNat0(Succ(x0), Succ(x1)), Succ(Succ(new_primPlusNat0(x0, x1))), y5)
new_map15(xu282, xu290, xu2830, xu291, Succ(xu2920), bb) → new_map14(xu282, xu290, xu2830, xu291, Succ(xu2920), Succ(xu282), bb)
new_map14(y0, y1, x0, Zero, Succ(y4), Zero, y5) → new_map15(y0, Succ(x0), x0, Succ(x0), Succ(x0), y5)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_map14(y0, y1, x0, Succ(x1), Succ(y4), Zero, y5) → new_map15(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, new_primPlusNat0(Succ(x0), Succ(x1)), Succ(Succ(new_primPlusNat0(x0, x1))), y5) at position [3] we obtained the following new rules:
new_map14(y0, y1, x0, Succ(x1), Succ(y4), Zero, y5) → new_map15(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y5)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map14(xu343, xu344, xu345, xu346, Succ(xu3470), Succ(xu3480), ba) → new_map14(xu343, xu344, xu345, xu346, xu3470, xu3480, ba)
new_map14(xu343, xu344, xu345, xu346, Zero, Zero, ba) → new_map16(xu343, xu344, xu345, xu346, ba)
new_map16(y0, y1, x0, Zero, y4) → new_map15(y0, Succ(x0), x0, Succ(x0), Succ(x0), y4)
new_map16(y0, y1, x0, Succ(x1), y4) → new_map15(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y4)
new_map15(xu282, xu290, xu2830, xu291, Succ(xu2920), bb) → new_map14(xu282, xu290, xu2830, xu291, Succ(xu2920), Succ(xu282), bb)
new_map14(y0, y1, x0, Succ(x1), Succ(y4), Zero, y5) → new_map15(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y5)
new_map14(y0, y1, x0, Zero, Succ(y4), Zero, y5) → new_map15(y0, Succ(x0), x0, Succ(x0), Succ(x0), y5)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule new_map15(xu282, xu290, xu2830, xu291, Succ(xu2920), bb) → new_map14(xu282, xu290, xu2830, xu291, Succ(xu2920), Succ(xu282), bb) we obtained the following new rules:
new_map15(z0, Succ(Succ(y_0)), z2, Succ(Succ(y_1)), Succ(Succ(y_2)), z4) → new_map14(z0, Succ(Succ(y_0)), z2, Succ(Succ(y_1)), Succ(Succ(y_2)), Succ(z0), z4)
new_map15(z0, Succ(z2), z2, Succ(z2), Succ(z2), z3) → new_map14(z0, Succ(z2), z2, Succ(z2), Succ(z2), Succ(z0), z3)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map15(z0, Succ(Succ(y_0)), z2, Succ(Succ(y_1)), Succ(Succ(y_2)), z4) → new_map14(z0, Succ(Succ(y_0)), z2, Succ(Succ(y_1)), Succ(Succ(y_2)), Succ(z0), z4)
new_map14(xu343, xu344, xu345, xu346, Succ(xu3470), Succ(xu3480), ba) → new_map14(xu343, xu344, xu345, xu346, xu3470, xu3480, ba)
new_map16(y0, y1, x0, Zero, y4) → new_map15(y0, Succ(x0), x0, Succ(x0), Succ(x0), y4)
new_map14(xu343, xu344, xu345, xu346, Zero, Zero, ba) → new_map16(xu343, xu344, xu345, xu346, ba)
new_map16(y0, y1, x0, Succ(x1), y4) → new_map15(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y4)
new_map15(z0, Succ(z2), z2, Succ(z2), Succ(z2), z3) → new_map14(z0, Succ(z2), z2, Succ(z2), Succ(z2), Succ(z0), z3)
new_map14(y0, y1, x0, Succ(x1), Succ(y4), Zero, y5) → new_map15(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y5)
new_map14(y0, y1, x0, Zero, Succ(y4), Zero, y5) → new_map15(y0, Succ(x0), x0, Succ(x0), Succ(x0), y5)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule new_map14(xu343, xu344, xu345, xu346, Zero, Zero, ba) → new_map16(xu343, xu344, xu345, xu346, ba) we obtained the following new rules:
new_map14(x0, x1, x2, Zero, Zero, Zero, x4) → new_map16(x0, x1, x2, Zero, x4)
new_map14(x0, x1, x2, Succ(y_3), Zero, Zero, x4) → new_map16(x0, x1, x2, Succ(y_3), x4)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map14(xu343, xu344, xu345, xu346, Succ(xu3470), Succ(xu3480), ba) → new_map14(xu343, xu344, xu345, xu346, xu3470, xu3480, ba)
new_map15(z0, Succ(Succ(y_0)), z2, Succ(Succ(y_1)), Succ(Succ(y_2)), z4) → new_map14(z0, Succ(Succ(y_0)), z2, Succ(Succ(y_1)), Succ(Succ(y_2)), Succ(z0), z4)
new_map16(y0, y1, x0, Zero, y4) → new_map15(y0, Succ(x0), x0, Succ(x0), Succ(x0), y4)
new_map16(y0, y1, x0, Succ(x1), y4) → new_map15(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y4)
new_map14(x0, x1, x2, Zero, Zero, Zero, x4) → new_map16(x0, x1, x2, Zero, x4)
new_map15(z0, Succ(z2), z2, Succ(z2), Succ(z2), z3) → new_map14(z0, Succ(z2), z2, Succ(z2), Succ(z2), Succ(z0), z3)
new_map14(y0, y1, x0, Succ(x1), Succ(y4), Zero, y5) → new_map15(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y5)
new_map14(y0, y1, x0, Zero, Succ(y4), Zero, y5) → new_map15(y0, Succ(x0), x0, Succ(x0), Succ(x0), y5)
new_map14(x0, x1, x2, Succ(y_3), Zero, Zero, x4) → new_map16(x0, x1, x2, Succ(y_3), x4)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule new_map14(xu343, xu344, xu345, xu346, Succ(xu3470), Succ(xu3480), ba) → new_map14(xu343, xu344, xu345, xu346, xu3470, xu3480, ba) we obtained the following new rules:
new_map14(x0, x1, x2, Zero, Succ(Succ(y_3)), Succ(Zero), x6) → new_map14(x0, x1, x2, Zero, Succ(y_3), Zero, x6)
new_map14(x0, x1, x2, Zero, Succ(Zero), Succ(Zero), x6) → new_map14(x0, x1, x2, Zero, Zero, Zero, x6)
new_map14(x0, x1, x2, Succ(y_3), Succ(Zero), Succ(Zero), x6) → new_map14(x0, x1, x2, Succ(y_3), Zero, Zero, x6)
new_map14(x0, x1, x2, Succ(y_3), Succ(Succ(y_4)), Succ(Zero), x6) → new_map14(x0, x1, x2, Succ(y_3), Succ(y_4), Zero, x6)
new_map14(x0, x1, x2, x3, Succ(Succ(y_4)), Succ(Succ(y_5)), x6) → new_map14(x0, x1, x2, x3, Succ(y_4), Succ(y_5), x6)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map15(z0, Succ(Succ(y_0)), z2, Succ(Succ(y_1)), Succ(Succ(y_2)), z4) → new_map14(z0, Succ(Succ(y_0)), z2, Succ(Succ(y_1)), Succ(Succ(y_2)), Succ(z0), z4)
new_map14(x0, x1, x2, Zero, Succ(Succ(y_3)), Succ(Zero), x6) → new_map14(x0, x1, x2, Zero, Succ(y_3), Zero, x6)
new_map16(y0, y1, x0, Succ(x1), y4) → new_map15(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y4)
new_map14(x0, x1, x2, Zero, Zero, Zero, x4) → new_map16(x0, x1, x2, Zero, x4)
new_map14(x0, x1, x2, Zero, Succ(Zero), Succ(Zero), x6) → new_map14(x0, x1, x2, Zero, Zero, Zero, x6)
new_map14(y0, y1, x0, Succ(x1), Succ(y4), Zero, y5) → new_map15(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y5)
new_map14(y0, y1, x0, Zero, Succ(y4), Zero, y5) → new_map15(y0, Succ(x0), x0, Succ(x0), Succ(x0), y5)
new_map14(x0, x1, x2, Succ(y_3), Succ(Succ(y_4)), Succ(Zero), x6) → new_map14(x0, x1, x2, Succ(y_3), Succ(y_4), Zero, x6)
new_map14(x0, x1, x2, x3, Succ(Succ(y_4)), Succ(Succ(y_5)), x6) → new_map14(x0, x1, x2, x3, Succ(y_4), Succ(y_5), x6)
new_map14(x0, x1, x2, Succ(y_3), Zero, Zero, x4) → new_map16(x0, x1, x2, Succ(y_3), x4)
new_map16(y0, y1, x0, Zero, y4) → new_map15(y0, Succ(x0), x0, Succ(x0), Succ(x0), y4)
new_map15(z0, Succ(z2), z2, Succ(z2), Succ(z2), z3) → new_map14(z0, Succ(z2), z2, Succ(z2), Succ(z2), Succ(z0), z3)
new_map14(x0, x1, x2, Succ(y_3), Succ(Zero), Succ(Zero), x6) → new_map14(x0, x1, x2, Succ(y_3), Zero, Zero, x6)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule new_map15(z0, Succ(Succ(y_0)), z2, Succ(Succ(y_1)), Succ(Succ(y_2)), z4) → new_map14(z0, Succ(Succ(y_0)), z2, Succ(Succ(y_1)), Succ(Succ(y_2)), Succ(z0), z4) we obtained the following new rules:
new_map15(Zero, Succ(Succ(x1)), x2, Succ(Succ(x3)), Succ(Succ(x4)), x5) → new_map14(Zero, Succ(Succ(x1)), x2, Succ(Succ(x3)), Succ(Succ(x4)), Succ(Zero), x5)
new_map15(Succ(y_5), Succ(Succ(x1)), x2, Succ(Succ(x3)), Succ(Succ(x4)), x5) → new_map14(Succ(y_5), Succ(Succ(x1)), x2, Succ(Succ(x3)), Succ(Succ(x4)), Succ(Succ(y_5)), x5)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map14(x0, x1, x2, Zero, Succ(Succ(y_3)), Succ(Zero), x6) → new_map14(x0, x1, x2, Zero, Succ(y_3), Zero, x6)
new_map16(y0, y1, x0, Succ(x1), y4) → new_map15(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y4)
new_map14(x0, x1, x2, Zero, Succ(Zero), Succ(Zero), x6) → new_map14(x0, x1, x2, Zero, Zero, Zero, x6)
new_map14(x0, x1, x2, Zero, Zero, Zero, x4) → new_map16(x0, x1, x2, Zero, x4)
new_map14(y0, y1, x0, Succ(x1), Succ(y4), Zero, y5) → new_map15(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y5)
new_map14(x0, x1, x2, Succ(y_3), Succ(Succ(y_4)), Succ(Zero), x6) → new_map14(x0, x1, x2, Succ(y_3), Succ(y_4), Zero, x6)
new_map14(y0, y1, x0, Zero, Succ(y4), Zero, y5) → new_map15(y0, Succ(x0), x0, Succ(x0), Succ(x0), y5)
new_map14(x0, x1, x2, Succ(y_3), Zero, Zero, x4) → new_map16(x0, x1, x2, Succ(y_3), x4)
new_map14(x0, x1, x2, x3, Succ(Succ(y_4)), Succ(Succ(y_5)), x6) → new_map14(x0, x1, x2, x3, Succ(y_4), Succ(y_5), x6)
new_map15(Succ(y_5), Succ(Succ(x1)), x2, Succ(Succ(x3)), Succ(Succ(x4)), x5) → new_map14(Succ(y_5), Succ(Succ(x1)), x2, Succ(Succ(x3)), Succ(Succ(x4)), Succ(Succ(y_5)), x5)
new_map16(y0, y1, x0, Zero, y4) → new_map15(y0, Succ(x0), x0, Succ(x0), Succ(x0), y4)
new_map14(x0, x1, x2, Succ(y_3), Succ(Zero), Succ(Zero), x6) → new_map14(x0, x1, x2, Succ(y_3), Zero, Zero, x6)
new_map15(z0, Succ(z2), z2, Succ(z2), Succ(z2), z3) → new_map14(z0, Succ(z2), z2, Succ(z2), Succ(z2), Succ(z0), z3)
new_map15(Zero, Succ(Succ(x1)), x2, Succ(Succ(x3)), Succ(Succ(x4)), x5) → new_map14(Zero, Succ(Succ(x1)), x2, Succ(Succ(x3)), Succ(Succ(x4)), Succ(Zero), x5)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule new_map15(z0, Succ(z2), z2, Succ(z2), Succ(z2), z3) → new_map14(z0, Succ(z2), z2, Succ(z2), Succ(z2), Succ(z0), z3) we obtained the following new rules:
new_map15(Zero, Succ(Zero), Zero, Succ(Zero), Succ(Zero), x2) → new_map14(Zero, Succ(Zero), Zero, Succ(Zero), Succ(Zero), Succ(Zero), x2)
new_map15(Zero, Succ(Succ(y_4)), Succ(y_4), Succ(Succ(y_4)), Succ(Succ(y_4)), x2) → new_map14(Zero, Succ(Succ(y_4)), Succ(y_4), Succ(Succ(y_4)), Succ(Succ(y_4)), Succ(Zero), x2)
new_map15(Succ(y_5), Succ(Succ(y_4)), Succ(y_4), Succ(Succ(y_4)), Succ(Succ(y_4)), x2) → new_map14(Succ(y_5), Succ(Succ(y_4)), Succ(y_4), Succ(Succ(y_4)), Succ(Succ(y_4)), Succ(Succ(y_5)), x2)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map15(Zero, Succ(Zero), Zero, Succ(Zero), Succ(Zero), x2) → new_map14(Zero, Succ(Zero), Zero, Succ(Zero), Succ(Zero), Succ(Zero), x2)
new_map14(x0, x1, x2, Zero, Succ(Succ(y_3)), Succ(Zero), x6) → new_map14(x0, x1, x2, Zero, Succ(y_3), Zero, x6)
new_map16(y0, y1, x0, Succ(x1), y4) → new_map15(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y4)
new_map14(x0, x1, x2, Zero, Zero, Zero, x4) → new_map16(x0, x1, x2, Zero, x4)
new_map14(x0, x1, x2, Zero, Succ(Zero), Succ(Zero), x6) → new_map14(x0, x1, x2, Zero, Zero, Zero, x6)
new_map14(y0, y1, x0, Succ(x1), Succ(y4), Zero, y5) → new_map15(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y5)
new_map15(Succ(y_5), Succ(Succ(y_4)), Succ(y_4), Succ(Succ(y_4)), Succ(Succ(y_4)), x2) → new_map14(Succ(y_5), Succ(Succ(y_4)), Succ(y_4), Succ(Succ(y_4)), Succ(Succ(y_4)), Succ(Succ(y_5)), x2)
new_map14(y0, y1, x0, Zero, Succ(y4), Zero, y5) → new_map15(y0, Succ(x0), x0, Succ(x0), Succ(x0), y5)
new_map14(x0, x1, x2, Succ(y_3), Succ(Succ(y_4)), Succ(Zero), x6) → new_map14(x0, x1, x2, Succ(y_3), Succ(y_4), Zero, x6)
new_map14(x0, x1, x2, x3, Succ(Succ(y_4)), Succ(Succ(y_5)), x6) → new_map14(x0, x1, x2, x3, Succ(y_4), Succ(y_5), x6)
new_map14(x0, x1, x2, Succ(y_3), Zero, Zero, x4) → new_map16(x0, x1, x2, Succ(y_3), x4)
new_map15(Succ(y_5), Succ(Succ(x1)), x2, Succ(Succ(x3)), Succ(Succ(x4)), x5) → new_map14(Succ(y_5), Succ(Succ(x1)), x2, Succ(Succ(x3)), Succ(Succ(x4)), Succ(Succ(y_5)), x5)
new_map16(y0, y1, x0, Zero, y4) → new_map15(y0, Succ(x0), x0, Succ(x0), Succ(x0), y4)
new_map15(Zero, Succ(Succ(y_4)), Succ(y_4), Succ(Succ(y_4)), Succ(Succ(y_4)), x2) → new_map14(Zero, Succ(Succ(y_4)), Succ(y_4), Succ(Succ(y_4)), Succ(Succ(y_4)), Succ(Zero), x2)
new_map14(x0, x1, x2, Succ(y_3), Succ(Zero), Succ(Zero), x6) → new_map14(x0, x1, x2, Succ(y_3), Zero, Zero, x6)
new_map15(Zero, Succ(Succ(x1)), x2, Succ(Succ(x3)), Succ(Succ(x4)), x5) → new_map14(Zero, Succ(Succ(x1)), x2, Succ(Succ(x3)), Succ(Succ(x4)), Succ(Zero), x5)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule new_map16(y0, y1, x0, Zero, y4) → new_map15(y0, Succ(x0), x0, Succ(x0), Succ(x0), y4) we obtained the following new rules:
new_map16(Succ(y_0), x1, Succ(y_1), Zero, x3) → new_map15(Succ(y_0), Succ(Succ(y_1)), Succ(y_1), Succ(Succ(y_1)), Succ(Succ(y_1)), x3)
new_map16(Zero, x1, Zero, Zero, x3) → new_map15(Zero, Succ(Zero), Zero, Succ(Zero), Succ(Zero), x3)
new_map16(Zero, x1, Succ(y_0), Zero, x3) → new_map15(Zero, Succ(Succ(y_0)), Succ(y_0), Succ(Succ(y_0)), Succ(Succ(y_0)), x3)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map14(x0, x1, x2, Zero, Succ(Succ(y_3)), Succ(Zero), x6) → new_map14(x0, x1, x2, Zero, Succ(y_3), Zero, x6)
new_map15(Zero, Succ(Zero), Zero, Succ(Zero), Succ(Zero), x2) → new_map14(Zero, Succ(Zero), Zero, Succ(Zero), Succ(Zero), Succ(Zero), x2)
new_map16(y0, y1, x0, Succ(x1), y4) → new_map15(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y4)
new_map14(x0, x1, x2, Zero, Succ(Zero), Succ(Zero), x6) → new_map14(x0, x1, x2, Zero, Zero, Zero, x6)
new_map14(x0, x1, x2, Zero, Zero, Zero, x4) → new_map16(x0, x1, x2, Zero, x4)
new_map16(Succ(y_0), x1, Succ(y_1), Zero, x3) → new_map15(Succ(y_0), Succ(Succ(y_1)), Succ(y_1), Succ(Succ(y_1)), Succ(Succ(y_1)), x3)
new_map14(y0, y1, x0, Succ(x1), Succ(y4), Zero, y5) → new_map15(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y5)
new_map15(Succ(y_5), Succ(Succ(y_4)), Succ(y_4), Succ(Succ(y_4)), Succ(Succ(y_4)), x2) → new_map14(Succ(y_5), Succ(Succ(y_4)), Succ(y_4), Succ(Succ(y_4)), Succ(Succ(y_4)), Succ(Succ(y_5)), x2)
new_map14(x0, x1, x2, Succ(y_3), Succ(Succ(y_4)), Succ(Zero), x6) → new_map14(x0, x1, x2, Succ(y_3), Succ(y_4), Zero, x6)
new_map14(y0, y1, x0, Zero, Succ(y4), Zero, y5) → new_map15(y0, Succ(x0), x0, Succ(x0), Succ(x0), y5)
new_map14(x0, x1, x2, Succ(y_3), Zero, Zero, x4) → new_map16(x0, x1, x2, Succ(y_3), x4)
new_map14(x0, x1, x2, x3, Succ(Succ(y_4)), Succ(Succ(y_5)), x6) → new_map14(x0, x1, x2, x3, Succ(y_4), Succ(y_5), x6)
new_map15(Succ(y_5), Succ(Succ(x1)), x2, Succ(Succ(x3)), Succ(Succ(x4)), x5) → new_map14(Succ(y_5), Succ(Succ(x1)), x2, Succ(Succ(x3)), Succ(Succ(x4)), Succ(Succ(y_5)), x5)
new_map15(Zero, Succ(Succ(y_4)), Succ(y_4), Succ(Succ(y_4)), Succ(Succ(y_4)), x2) → new_map14(Zero, Succ(Succ(y_4)), Succ(y_4), Succ(Succ(y_4)), Succ(Succ(y_4)), Succ(Zero), x2)
new_map16(Zero, x1, Zero, Zero, x3) → new_map15(Zero, Succ(Zero), Zero, Succ(Zero), Succ(Zero), x3)
new_map14(x0, x1, x2, Succ(y_3), Succ(Zero), Succ(Zero), x6) → new_map14(x0, x1, x2, Succ(y_3), Zero, Zero, x6)
new_map15(Zero, Succ(Succ(x1)), x2, Succ(Succ(x3)), Succ(Succ(x4)), x5) → new_map14(Zero, Succ(Succ(x1)), x2, Succ(Succ(x3)), Succ(Succ(x4)), Succ(Zero), x5)
new_map16(Zero, x1, Succ(y_0), Zero, x3) → new_map15(Zero, Succ(Succ(y_0)), Succ(y_0), Succ(Succ(y_0)), Succ(Succ(y_0)), x3)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule new_map14(y0, y1, x0, Zero, Succ(y4), Zero, y5) → new_map15(y0, Succ(x0), x0, Succ(x0), Succ(x0), y5) we obtained the following new rules:
new_map14(Succ(y_0), x1, Succ(y_1), Zero, Succ(x3), Zero, x4) → new_map15(Succ(y_0), Succ(Succ(y_1)), Succ(y_1), Succ(Succ(y_1)), Succ(Succ(y_1)), x4)
new_map14(Zero, x1, Zero, Zero, Succ(x3), Zero, x4) → new_map15(Zero, Succ(Zero), Zero, Succ(Zero), Succ(Zero), x4)
new_map14(Zero, x1, Succ(y_0), Zero, Succ(x3), Zero, x4) → new_map15(Zero, Succ(Succ(y_0)), Succ(y_0), Succ(Succ(y_0)), Succ(Succ(y_0)), x4)
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map15(Zero, Succ(Zero), Zero, Succ(Zero), Succ(Zero), x2) → new_map14(Zero, Succ(Zero), Zero, Succ(Zero), Succ(Zero), Succ(Zero), x2)
new_map14(x0, x1, x2, Zero, Succ(Succ(y_3)), Succ(Zero), x6) → new_map14(x0, x1, x2, Zero, Succ(y_3), Zero, x6)
new_map14(Succ(y_0), x1, Succ(y_1), Zero, Succ(x3), Zero, x4) → new_map15(Succ(y_0), Succ(Succ(y_1)), Succ(y_1), Succ(Succ(y_1)), Succ(Succ(y_1)), x4)
new_map16(y0, y1, x0, Succ(x1), y4) → new_map15(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y4)
new_map14(Zero, x1, Succ(y_0), Zero, Succ(x3), Zero, x4) → new_map15(Zero, Succ(Succ(y_0)), Succ(y_0), Succ(Succ(y_0)), Succ(Succ(y_0)), x4)
new_map14(Zero, x1, Zero, Zero, Succ(x3), Zero, x4) → new_map15(Zero, Succ(Zero), Zero, Succ(Zero), Succ(Zero), x4)
new_map14(x0, x1, x2, Zero, Zero, Zero, x4) → new_map16(x0, x1, x2, Zero, x4)
new_map14(x0, x1, x2, Zero, Succ(Zero), Succ(Zero), x6) → new_map14(x0, x1, x2, Zero, Zero, Zero, x6)
new_map16(Succ(y_0), x1, Succ(y_1), Zero, x3) → new_map15(Succ(y_0), Succ(Succ(y_1)), Succ(y_1), Succ(Succ(y_1)), Succ(Succ(y_1)), x3)
new_map14(y0, y1, x0, Succ(x1), Succ(y4), Zero, y5) → new_map15(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y5)
new_map15(Succ(y_5), Succ(Succ(y_4)), Succ(y_4), Succ(Succ(y_4)), Succ(Succ(y_4)), x2) → new_map14(Succ(y_5), Succ(Succ(y_4)), Succ(y_4), Succ(Succ(y_4)), Succ(Succ(y_4)), Succ(Succ(y_5)), x2)
new_map14(x0, x1, x2, Succ(y_3), Succ(Succ(y_4)), Succ(Zero), x6) → new_map14(x0, x1, x2, Succ(y_3), Succ(y_4), Zero, x6)
new_map14(x0, x1, x2, x3, Succ(Succ(y_4)), Succ(Succ(y_5)), x6) → new_map14(x0, x1, x2, x3, Succ(y_4), Succ(y_5), x6)
new_map14(x0, x1, x2, Succ(y_3), Zero, Zero, x4) → new_map16(x0, x1, x2, Succ(y_3), x4)
new_map15(Succ(y_5), Succ(Succ(x1)), x2, Succ(Succ(x3)), Succ(Succ(x4)), x5) → new_map14(Succ(y_5), Succ(Succ(x1)), x2, Succ(Succ(x3)), Succ(Succ(x4)), Succ(Succ(y_5)), x5)
new_map15(Zero, Succ(Succ(y_4)), Succ(y_4), Succ(Succ(y_4)), Succ(Succ(y_4)), x2) → new_map14(Zero, Succ(Succ(y_4)), Succ(y_4), Succ(Succ(y_4)), Succ(Succ(y_4)), Succ(Zero), x2)
new_map14(x0, x1, x2, Succ(y_3), Succ(Zero), Succ(Zero), x6) → new_map14(x0, x1, x2, Succ(y_3), Zero, Zero, x6)
new_map16(Zero, x1, Zero, Zero, x3) → new_map15(Zero, Succ(Zero), Zero, Succ(Zero), Succ(Zero), x3)
new_map15(Zero, Succ(Succ(x1)), x2, Succ(Succ(x3)), Succ(Succ(x4)), x5) → new_map14(Zero, Succ(Succ(x1)), x2, Succ(Succ(x3)), Succ(Succ(x4)), Succ(Zero), x5)
new_map16(Zero, x1, Succ(y_0), Zero, x3) → new_map15(Zero, Succ(Succ(y_0)), Succ(y_0), Succ(Succ(y_0)), Succ(Succ(y_0)), x3)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
new_map14(Succ(y_0), x1, Succ(y_1), Zero, Succ(x3), Zero, x4) → new_map15(Succ(y_0), Succ(Succ(y_1)), Succ(y_1), Succ(Succ(y_1)), Succ(Succ(y_1)), x4)
new_map14(Zero, x1, Succ(y_0), Zero, Succ(x3), Zero, x4) → new_map15(Zero, Succ(Succ(y_0)), Succ(y_0), Succ(Succ(y_0)), Succ(Succ(y_0)), x4)
new_map14(Zero, x1, Zero, Zero, Succ(x3), Zero, x4) → new_map15(Zero, Succ(Zero), Zero, Succ(Zero), Succ(Zero), x4)
new_map14(x0, x1, x2, Zero, Zero, Zero, x4) → new_map16(x0, x1, x2, Zero, x4)
The remaining pairs can at least be oriented weakly.
new_map15(Zero, Succ(Zero), Zero, Succ(Zero), Succ(Zero), x2) → new_map14(Zero, Succ(Zero), Zero, Succ(Zero), Succ(Zero), Succ(Zero), x2)
new_map14(x0, x1, x2, Zero, Succ(Succ(y_3)), Succ(Zero), x6) → new_map14(x0, x1, x2, Zero, Succ(y_3), Zero, x6)
new_map16(y0, y1, x0, Succ(x1), y4) → new_map15(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y4)
new_map14(x0, x1, x2, Zero, Succ(Zero), Succ(Zero), x6) → new_map14(x0, x1, x2, Zero, Zero, Zero, x6)
new_map16(Succ(y_0), x1, Succ(y_1), Zero, x3) → new_map15(Succ(y_0), Succ(Succ(y_1)), Succ(y_1), Succ(Succ(y_1)), Succ(Succ(y_1)), x3)
new_map14(y0, y1, x0, Succ(x1), Succ(y4), Zero, y5) → new_map15(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y5)
new_map15(Succ(y_5), Succ(Succ(y_4)), Succ(y_4), Succ(Succ(y_4)), Succ(Succ(y_4)), x2) → new_map14(Succ(y_5), Succ(Succ(y_4)), Succ(y_4), Succ(Succ(y_4)), Succ(Succ(y_4)), Succ(Succ(y_5)), x2)
new_map14(x0, x1, x2, Succ(y_3), Succ(Succ(y_4)), Succ(Zero), x6) → new_map14(x0, x1, x2, Succ(y_3), Succ(y_4), Zero, x6)
new_map14(x0, x1, x2, x3, Succ(Succ(y_4)), Succ(Succ(y_5)), x6) → new_map14(x0, x1, x2, x3, Succ(y_4), Succ(y_5), x6)
new_map14(x0, x1, x2, Succ(y_3), Zero, Zero, x4) → new_map16(x0, x1, x2, Succ(y_3), x4)
new_map15(Succ(y_5), Succ(Succ(x1)), x2, Succ(Succ(x3)), Succ(Succ(x4)), x5) → new_map14(Succ(y_5), Succ(Succ(x1)), x2, Succ(Succ(x3)), Succ(Succ(x4)), Succ(Succ(y_5)), x5)
new_map15(Zero, Succ(Succ(y_4)), Succ(y_4), Succ(Succ(y_4)), Succ(Succ(y_4)), x2) → new_map14(Zero, Succ(Succ(y_4)), Succ(y_4), Succ(Succ(y_4)), Succ(Succ(y_4)), Succ(Zero), x2)
new_map14(x0, x1, x2, Succ(y_3), Succ(Zero), Succ(Zero), x6) → new_map14(x0, x1, x2, Succ(y_3), Zero, Zero, x6)
new_map16(Zero, x1, Zero, Zero, x3) → new_map15(Zero, Succ(Zero), Zero, Succ(Zero), Succ(Zero), x3)
new_map15(Zero, Succ(Succ(x1)), x2, Succ(Succ(x3)), Succ(Succ(x4)), x5) → new_map14(Zero, Succ(Succ(x1)), x2, Succ(Succ(x3)), Succ(Succ(x4)), Succ(Zero), x5)
new_map16(Zero, x1, Succ(y_0), Zero, x3) → new_map15(Zero, Succ(Succ(y_0)), Succ(y_0), Succ(Succ(y_0)), Succ(Succ(y_0)), x3)
Used ordering: Polynomial interpretation [25]:
POL(Succ(x1)) = 0
POL(Zero) = 1
POL(new_map14(x1, x2, x3, x4, x5, x6, x7)) = x4
POL(new_map15(x1, x2, x3, x4, x5, x6)) = 0
POL(new_map16(x1, x2, x3, x4, x5)) = 0
POL(new_primPlusNat0(x1, x2)) = 0
The following usable rules [17] were oriented:
none
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map15(Zero, Succ(Zero), Zero, Succ(Zero), Succ(Zero), x2) → new_map14(Zero, Succ(Zero), Zero, Succ(Zero), Succ(Zero), Succ(Zero), x2)
new_map14(x0, x1, x2, Zero, Succ(Succ(y_3)), Succ(Zero), x6) → new_map14(x0, x1, x2, Zero, Succ(y_3), Zero, x6)
new_map16(y0, y1, x0, Succ(x1), y4) → new_map15(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y4)
new_map14(x0, x1, x2, Zero, Succ(Zero), Succ(Zero), x6) → new_map14(x0, x1, x2, Zero, Zero, Zero, x6)
new_map16(Succ(y_0), x1, Succ(y_1), Zero, x3) → new_map15(Succ(y_0), Succ(Succ(y_1)), Succ(y_1), Succ(Succ(y_1)), Succ(Succ(y_1)), x3)
new_map14(y0, y1, x0, Succ(x1), Succ(y4), Zero, y5) → new_map15(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y5)
new_map15(Succ(y_5), Succ(Succ(y_4)), Succ(y_4), Succ(Succ(y_4)), Succ(Succ(y_4)), x2) → new_map14(Succ(y_5), Succ(Succ(y_4)), Succ(y_4), Succ(Succ(y_4)), Succ(Succ(y_4)), Succ(Succ(y_5)), x2)
new_map14(x0, x1, x2, Succ(y_3), Succ(Succ(y_4)), Succ(Zero), x6) → new_map14(x0, x1, x2, Succ(y_3), Succ(y_4), Zero, x6)
new_map14(x0, x1, x2, x3, Succ(Succ(y_4)), Succ(Succ(y_5)), x6) → new_map14(x0, x1, x2, x3, Succ(y_4), Succ(y_5), x6)
new_map14(x0, x1, x2, Succ(y_3), Zero, Zero, x4) → new_map16(x0, x1, x2, Succ(y_3), x4)
new_map15(Succ(y_5), Succ(Succ(x1)), x2, Succ(Succ(x3)), Succ(Succ(x4)), x5) → new_map14(Succ(y_5), Succ(Succ(x1)), x2, Succ(Succ(x3)), Succ(Succ(x4)), Succ(Succ(y_5)), x5)
new_map15(Zero, Succ(Succ(y_4)), Succ(y_4), Succ(Succ(y_4)), Succ(Succ(y_4)), x2) → new_map14(Zero, Succ(Succ(y_4)), Succ(y_4), Succ(Succ(y_4)), Succ(Succ(y_4)), Succ(Zero), x2)
new_map16(Zero, x1, Zero, Zero, x3) → new_map15(Zero, Succ(Zero), Zero, Succ(Zero), Succ(Zero), x3)
new_map14(x0, x1, x2, Succ(y_3), Succ(Zero), Succ(Zero), x6) → new_map14(x0, x1, x2, Succ(y_3), Zero, Zero, x6)
new_map15(Zero, Succ(Succ(x1)), x2, Succ(Succ(x3)), Succ(Succ(x4)), x5) → new_map14(Zero, Succ(Succ(x1)), x2, Succ(Succ(x3)), Succ(Succ(x4)), Succ(Zero), x5)
new_map16(Zero, x1, Succ(y_0), Zero, x3) → new_map15(Zero, Succ(Succ(y_0)), Succ(y_0), Succ(Succ(y_0)), Succ(Succ(y_0)), x3)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 6 less nodes.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ MNOCProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map16(y0, y1, x0, Succ(x1), y4) → new_map15(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y4)
new_map15(Zero, Succ(Succ(y_4)), Succ(y_4), Succ(Succ(y_4)), Succ(Succ(y_4)), x2) → new_map14(Zero, Succ(Succ(y_4)), Succ(y_4), Succ(Succ(y_4)), Succ(Succ(y_4)), Succ(Zero), x2)
new_map14(x0, x1, x2, Succ(y_3), Succ(Zero), Succ(Zero), x6) → new_map14(x0, x1, x2, Succ(y_3), Zero, Zero, x6)
new_map14(y0, y1, x0, Succ(x1), Succ(y4), Zero, y5) → new_map15(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y5)
new_map15(Succ(y_5), Succ(Succ(y_4)), Succ(y_4), Succ(Succ(y_4)), Succ(Succ(y_4)), x2) → new_map14(Succ(y_5), Succ(Succ(y_4)), Succ(y_4), Succ(Succ(y_4)), Succ(Succ(y_4)), Succ(Succ(y_5)), x2)
new_map14(x0, x1, x2, Succ(y_3), Succ(Succ(y_4)), Succ(Zero), x6) → new_map14(x0, x1, x2, Succ(y_3), Succ(y_4), Zero, x6)
new_map14(x0, x1, x2, x3, Succ(Succ(y_4)), Succ(Succ(y_5)), x6) → new_map14(x0, x1, x2, x3, Succ(y_4), Succ(y_5), x6)
new_map14(x0, x1, x2, Succ(y_3), Zero, Zero, x4) → new_map16(x0, x1, x2, Succ(y_3), x4)
new_map15(Zero, Succ(Succ(x1)), x2, Succ(Succ(x3)), Succ(Succ(x4)), x5) → new_map14(Zero, Succ(Succ(x1)), x2, Succ(Succ(x3)), Succ(Succ(x4)), Succ(Zero), x5)
new_map15(Succ(y_5), Succ(Succ(x1)), x2, Succ(Succ(x3)), Succ(Succ(x4)), x5) → new_map14(Succ(y_5), Succ(Succ(x1)), x2, Succ(Succ(x3)), Succ(Succ(x4)), Succ(Succ(y_5)), x5)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
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
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ MNOCProof
↳ 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_map16(y0, y1, x0, Succ(x1), y4) → new_map15(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y4)
new_map15(Zero, Succ(Succ(y_4)), Succ(y_4), Succ(Succ(y_4)), Succ(Succ(y_4)), x2) → new_map14(Zero, Succ(Succ(y_4)), Succ(y_4), Succ(Succ(y_4)), Succ(Succ(y_4)), Succ(Zero), x2)
new_map14(x0, x1, x2, Succ(y_3), Succ(Zero), Succ(Zero), x6) → new_map14(x0, x1, x2, Succ(y_3), Zero, Zero, x6)
new_map14(y0, y1, x0, Succ(x1), Succ(y4), Zero, y5) → new_map15(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y5)
new_map15(Succ(y_5), Succ(Succ(y_4)), Succ(y_4), Succ(Succ(y_4)), Succ(Succ(y_4)), x2) → new_map14(Succ(y_5), Succ(Succ(y_4)), Succ(y_4), Succ(Succ(y_4)), Succ(Succ(y_4)), Succ(Succ(y_5)), x2)
new_map14(x0, x1, x2, Succ(y_3), Succ(Succ(y_4)), Succ(Zero), x6) → new_map14(x0, x1, x2, Succ(y_3), Succ(y_4), Zero, x6)
new_map14(x0, x1, x2, Succ(y_3), Zero, Zero, x4) → new_map16(x0, x1, x2, Succ(y_3), x4)
new_map14(x0, x1, x2, x3, Succ(Succ(y_4)), Succ(Succ(y_5)), x6) → new_map14(x0, x1, x2, x3, Succ(y_4), Succ(y_5), x6)
new_map15(Zero, Succ(Succ(x1)), x2, Succ(Succ(x3)), Succ(Succ(x4)), x5) → new_map14(Zero, Succ(Succ(x1)), x2, Succ(Succ(x3)), Succ(Succ(x4)), Succ(Zero), x5)
new_map15(Succ(y_5), Succ(Succ(x1)), x2, Succ(Succ(x3)), Succ(Succ(x4)), x5) → new_map14(Succ(y_5), Succ(Succ(x1)), x2, Succ(Succ(x3)), Succ(Succ(x4)), Succ(Succ(y_5)), x5)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
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 narrowing to the left:
The TRS P consists of the following rules:
new_map16(y0, y1, x0, Succ(x1), y4) → new_map15(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y4)
new_map15(Zero, Succ(Succ(y_4)), Succ(y_4), Succ(Succ(y_4)), Succ(Succ(y_4)), x2) → new_map14(Zero, Succ(Succ(y_4)), Succ(y_4), Succ(Succ(y_4)), Succ(Succ(y_4)), Succ(Zero), x2)
new_map14(x0, x1, x2, Succ(y_3), Succ(Zero), Succ(Zero), x6) → new_map14(x0, x1, x2, Succ(y_3), Zero, Zero, x6)
new_map14(y0, y1, x0, Succ(x1), Succ(y4), Zero, y5) → new_map15(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y5)
new_map15(Succ(y_5), Succ(Succ(y_4)), Succ(y_4), Succ(Succ(y_4)), Succ(Succ(y_4)), x2) → new_map14(Succ(y_5), Succ(Succ(y_4)), Succ(y_4), Succ(Succ(y_4)), Succ(Succ(y_4)), Succ(Succ(y_5)), x2)
new_map14(x0, x1, x2, Succ(y_3), Succ(Succ(y_4)), Succ(Zero), x6) → new_map14(x0, x1, x2, Succ(y_3), Succ(y_4), Zero, x6)
new_map14(x0, x1, x2, Succ(y_3), Zero, Zero, x4) → new_map16(x0, x1, x2, Succ(y_3), x4)
new_map14(x0, x1, x2, x3, Succ(Succ(y_4)), Succ(Succ(y_5)), x6) → new_map14(x0, x1, x2, x3, Succ(y_4), Succ(y_5), x6)
new_map15(Zero, Succ(Succ(x1)), x2, Succ(Succ(x3)), Succ(Succ(x4)), x5) → new_map14(Zero, Succ(Succ(x1)), x2, Succ(Succ(x3)), Succ(Succ(x4)), Succ(Zero), x5)
new_map15(Succ(y_5), Succ(Succ(x1)), x2, Succ(Succ(x3)), Succ(Succ(x4)), x5) → new_map14(Succ(y_5), Succ(Succ(x1)), x2, Succ(Succ(x3)), Succ(Succ(x4)), Succ(Succ(y_5)), x5)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
s = new_map15(Zero, Succ(Succ(x1)), x2', Succ(Succ(x3)), Succ(Succ(x4)), x5) evaluates to t =new_map15(Zero, Succ(Succ(new_primPlusNat0(x2', Succ(x3)))), x2', Succ(Succ(new_primPlusNat0(x2', Succ(x3)))), Succ(Succ(new_primPlusNat0(x2', Succ(x3)))), x5)
Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
- Matcher: [x1 / new_primPlusNat0(x2', Succ(x3)), x4 / new_primPlusNat0(x2', Succ(x3)), x3 / new_primPlusNat0(x2', Succ(x3))]
- Semiunifier: [ ]
Rewriting sequence
new_map15(Zero, Succ(Succ(x1)), x2', Succ(Succ(x3)), Succ(Succ(x4)), x5) → new_map14(Zero, Succ(Succ(x1)), x2', Succ(Succ(x3)), Succ(Succ(x4)), Succ(Zero), x5)
with rule new_map15(Zero, Succ(Succ(x1')), x2'', Succ(Succ(x3')), Succ(Succ(x4')), x5') → new_map14(Zero, Succ(Succ(x1')), x2'', Succ(Succ(x3')), Succ(Succ(x4')), Succ(Zero), x5') at position [] and matcher [x2'' / x2', x1' / x1, x4' / x4, x3' / x3, x5' / x5]
new_map14(Zero, Succ(Succ(x1)), x2', Succ(Succ(x3)), Succ(Succ(x4)), Succ(Zero), x5) → new_map14(Zero, Succ(Succ(x1)), x2', Succ(Succ(x3)), Succ(x4), Zero, x5)
with rule new_map14(x0', x1', x2, Succ(y_3), Succ(Succ(y_4)), Succ(Zero), x6) → new_map14(x0', x1', x2, Succ(y_3), Succ(y_4), Zero, x6) at position [] and matcher [x6 / x5, y_4 / x4, x1' / Succ(Succ(x1)), y_3 / Succ(x3), x0' / Zero, x2 / x2']
new_map14(Zero, Succ(Succ(x1)), x2', Succ(Succ(x3)), Succ(x4), Zero, x5) → new_map15(Zero, Succ(Succ(new_primPlusNat0(x2', Succ(x3)))), x2', Succ(Succ(new_primPlusNat0(x2', Succ(x3)))), Succ(Succ(new_primPlusNat0(x2', Succ(x3)))), x5)
with rule new_map14(y0, y1, x0, Succ(x1), Succ(y4), Zero, y5) → new_map15(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y5)
Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence
All these steps are and every following step will be a correct step w.r.t to Q.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map17(xu282, Succ(xu2830), Succ(xu2840), xu285, ba) → new_map17(xu282, xu2830, xu2840, xu285, 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_map17(xu282, Succ(xu2830), Succ(xu2840), xu285, ba) → new_map17(xu282, xu2830, xu2840, xu285, ba)
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
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map18(xu204, xu205, xu206, Succ(xu2070), Succ(xu2080), ba) → new_map18(xu204, xu205, xu206, xu2070, xu2080, 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_map18(xu204, xu205, xu206, Succ(xu2070), Succ(xu2080), ba) → new_map18(xu204, xu205, xu206, xu2070, xu2080, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4, 5 > 5, 6 >= 6
↳ 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
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map19(xu139, xu140, Succ(xu1410), Succ(xu1420), ba) → new_map19(xu139, xu140, xu1410, xu1420, 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_map19(xu139, xu140, Succ(xu1410), Succ(xu1420), ba) → new_map19(xu139, xu140, xu1410, xu1420, ba)
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
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map22(xu308, xu309, xu310, ba) → new_map21(xu308, xu310, ba)
new_map20(xu308, xu309, xu310, Succ(xu3110), Succ(xu3120), ba) → new_map20(xu308, xu309, xu310, xu3110, xu3120, ba)
new_map20(xu308, xu309, xu310, Zero, Zero, ba) → new_map22(xu308, xu309, xu310, ba)
new_map23(xu196, xu249, xu250, Zero, bb) → new_map20(xu196, xu249, xu250, Zero, Succ(xu196), bb)
new_map21(xu196, xu250, bb) → new_map23(xu196, new_primPlusNat0(Zero, xu250), new_primPlusNat0(Zero, xu250), new_primPlusNat0(Zero, xu250), bb)
new_map23(xu196, xu249, xu250, Succ(xu2510), bb) → new_map20(xu196, xu249, xu250, Succ(xu2510), Succ(xu196), bb)
new_map20(xu308, xu309, xu310, Zero, Succ(xu3120), ba) → new_map21(xu308, xu310, ba)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map22(xu308, xu309, xu310, ba) → new_map21(xu308, xu310, ba)
new_map20(xu308, xu309, xu310, Succ(xu3110), Succ(xu3120), ba) → new_map20(xu308, xu309, xu310, xu3110, xu3120, ba)
new_map20(xu308, xu309, xu310, Zero, Zero, ba) → new_map22(xu308, xu309, xu310, ba)
new_map23(xu196, xu249, xu250, Zero, bb) → new_map20(xu196, xu249, xu250, Zero, Succ(xu196), bb)
new_map21(xu196, xu250, bb) → new_map23(xu196, new_primPlusNat0(Zero, xu250), new_primPlusNat0(Zero, xu250), new_primPlusNat0(Zero, xu250), bb)
new_map23(xu196, xu249, xu250, Succ(xu2510), bb) → new_map20(xu196, xu249, xu250, Succ(xu2510), Succ(xu196), bb)
new_map20(xu308, xu309, xu310, Zero, Succ(xu3120), ba) → new_map21(xu308, xu310, ba)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule new_map21(xu196, xu250, bb) → new_map23(xu196, new_primPlusNat0(Zero, xu250), new_primPlusNat0(Zero, xu250), new_primPlusNat0(Zero, xu250), bb) at position [3] we obtained the following new rules:
new_map21(y0, Zero, y2) → new_map23(y0, new_primPlusNat0(Zero, Zero), new_primPlusNat0(Zero, Zero), Zero, y2)
new_map21(y0, Succ(x0), y2) → new_map23(y0, new_primPlusNat0(Zero, Succ(x0)), new_primPlusNat0(Zero, Succ(x0)), Succ(x0), y2)
↳ 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
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map22(xu308, xu309, xu310, ba) → new_map21(xu308, xu310, ba)
new_map20(xu308, xu309, xu310, Succ(xu3110), Succ(xu3120), ba) → new_map20(xu308, xu309, xu310, xu3110, xu3120, ba)
new_map23(xu196, xu249, xu250, Zero, bb) → new_map20(xu196, xu249, xu250, Zero, Succ(xu196), bb)
new_map20(xu308, xu309, xu310, Zero, Zero, ba) → new_map22(xu308, xu309, xu310, ba)
new_map21(y0, Succ(x0), y2) → new_map23(y0, new_primPlusNat0(Zero, Succ(x0)), new_primPlusNat0(Zero, Succ(x0)), Succ(x0), y2)
new_map21(y0, Zero, y2) → new_map23(y0, new_primPlusNat0(Zero, Zero), new_primPlusNat0(Zero, Zero), Zero, y2)
new_map23(xu196, xu249, xu250, Succ(xu2510), bb) → new_map20(xu196, xu249, xu250, Succ(xu2510), Succ(xu196), bb)
new_map20(xu308, xu309, xu310, Zero, Succ(xu3120), ba) → new_map21(xu308, xu310, ba)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_map21(y0, Zero, y2) → new_map23(y0, new_primPlusNat0(Zero, Zero), new_primPlusNat0(Zero, Zero), Zero, y2) at position [1] we obtained the following new rules:
new_map21(y0, Zero, y2) → new_map23(y0, Zero, new_primPlusNat0(Zero, Zero), Zero, y2)
↳ 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
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map22(xu308, xu309, xu310, ba) → new_map21(xu308, xu310, ba)
new_map20(xu308, xu309, xu310, Succ(xu3110), Succ(xu3120), ba) → new_map20(xu308, xu309, xu310, xu3110, xu3120, ba)
new_map20(xu308, xu309, xu310, Zero, Zero, ba) → new_map22(xu308, xu309, xu310, ba)
new_map23(xu196, xu249, xu250, Zero, bb) → new_map20(xu196, xu249, xu250, Zero, Succ(xu196), bb)
new_map21(y0, Succ(x0), y2) → new_map23(y0, new_primPlusNat0(Zero, Succ(x0)), new_primPlusNat0(Zero, Succ(x0)), Succ(x0), y2)
new_map21(y0, Zero, y2) → new_map23(y0, Zero, new_primPlusNat0(Zero, Zero), Zero, y2)
new_map23(xu196, xu249, xu250, Succ(xu2510), bb) → new_map20(xu196, xu249, xu250, Succ(xu2510), Succ(xu196), bb)
new_map20(xu308, xu309, xu310, Zero, Succ(xu3120), ba) → new_map21(xu308, xu310, ba)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_map21(y0, Succ(x0), y2) → new_map23(y0, new_primPlusNat0(Zero, Succ(x0)), new_primPlusNat0(Zero, Succ(x0)), Succ(x0), y2) at position [1] we obtained the following new rules:
new_map21(y0, Succ(x0), y2) → new_map23(y0, Succ(x0), new_primPlusNat0(Zero, Succ(x0)), Succ(x0), y2)
↳ 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
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map22(xu308, xu309, xu310, ba) → new_map21(xu308, xu310, ba)
new_map20(xu308, xu309, xu310, Succ(xu3110), Succ(xu3120), ba) → new_map20(xu308, xu309, xu310, xu3110, xu3120, ba)
new_map23(xu196, xu249, xu250, Zero, bb) → new_map20(xu196, xu249, xu250, Zero, Succ(xu196), bb)
new_map20(xu308, xu309, xu310, Zero, Zero, ba) → new_map22(xu308, xu309, xu310, ba)
new_map21(y0, Succ(x0), y2) → new_map23(y0, Succ(x0), new_primPlusNat0(Zero, Succ(x0)), Succ(x0), y2)
new_map21(y0, Zero, y2) → new_map23(y0, Zero, new_primPlusNat0(Zero, Zero), Zero, y2)
new_map23(xu196, xu249, xu250, Succ(xu2510), bb) → new_map20(xu196, xu249, xu250, Succ(xu2510), Succ(xu196), bb)
new_map20(xu308, xu309, xu310, Zero, Succ(xu3120), ba) → new_map21(xu308, xu310, ba)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_map21(y0, Zero, y2) → new_map23(y0, Zero, new_primPlusNat0(Zero, Zero), Zero, y2) at position [2] we obtained the following new rules:
new_map21(y0, Zero, y2) → new_map23(y0, Zero, Zero, Zero, y2)
↳ 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
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map22(xu308, xu309, xu310, ba) → new_map21(xu308, xu310, ba)
new_map20(xu308, xu309, xu310, Succ(xu3110), Succ(xu3120), ba) → new_map20(xu308, xu309, xu310, xu3110, xu3120, ba)
new_map20(xu308, xu309, xu310, Zero, Zero, ba) → new_map22(xu308, xu309, xu310, ba)
new_map23(xu196, xu249, xu250, Zero, bb) → new_map20(xu196, xu249, xu250, Zero, Succ(xu196), bb)
new_map21(y0, Succ(x0), y2) → new_map23(y0, Succ(x0), new_primPlusNat0(Zero, Succ(x0)), Succ(x0), y2)
new_map23(xu196, xu249, xu250, Succ(xu2510), bb) → new_map20(xu196, xu249, xu250, Succ(xu2510), Succ(xu196), bb)
new_map21(y0, Zero, y2) → new_map23(y0, Zero, Zero, Zero, y2)
new_map20(xu308, xu309, xu310, Zero, Succ(xu3120), ba) → new_map21(xu308, xu310, ba)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map22(xu308, xu309, xu310, ba) → new_map21(xu308, xu310, ba)
new_map20(xu308, xu309, xu310, Succ(xu3110), Succ(xu3120), ba) → new_map20(xu308, xu309, xu310, xu3110, xu3120, ba)
new_map20(xu308, xu309, xu310, Zero, Zero, ba) → new_map22(xu308, xu309, xu310, ba)
new_map23(xu196, xu249, xu250, Zero, bb) → new_map20(xu196, xu249, xu250, Zero, Succ(xu196), bb)
new_map21(y0, Succ(x0), y2) → new_map23(y0, Succ(x0), new_primPlusNat0(Zero, Succ(x0)), Succ(x0), y2)
new_map23(xu196, xu249, xu250, Succ(xu2510), bb) → new_map20(xu196, xu249, xu250, Succ(xu2510), Succ(xu196), bb)
new_map21(y0, Zero, y2) → new_map23(y0, Zero, Zero, Zero, y2)
new_map20(xu308, xu309, xu310, Zero, Succ(xu3120), ba) → new_map21(xu308, xu310, ba)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_map21(y0, Succ(x0), y2) → new_map23(y0, Succ(x0), new_primPlusNat0(Zero, Succ(x0)), Succ(x0), y2) at position [2] we obtained the following new rules:
new_map21(y0, Succ(x0), y2) → new_map23(y0, Succ(x0), Succ(x0), Succ(x0), y2)
↳ 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
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map22(xu308, xu309, xu310, ba) → new_map21(xu308, xu310, ba)
new_map20(xu308, xu309, xu310, Succ(xu3110), Succ(xu3120), ba) → new_map20(xu308, xu309, xu310, xu3110, xu3120, ba)
new_map23(xu196, xu249, xu250, Zero, bb) → new_map20(xu196, xu249, xu250, Zero, Succ(xu196), bb)
new_map20(xu308, xu309, xu310, Zero, Zero, ba) → new_map22(xu308, xu309, xu310, ba)
new_map21(y0, Succ(x0), y2) → new_map23(y0, Succ(x0), Succ(x0), Succ(x0), y2)
new_map23(xu196, xu249, xu250, Succ(xu2510), bb) → new_map20(xu196, xu249, xu250, Succ(xu2510), Succ(xu196), bb)
new_map20(xu308, xu309, xu310, Zero, Succ(xu3120), ba) → new_map21(xu308, xu310, ba)
new_map21(y0, Zero, y2) → new_map23(y0, Zero, Zero, Zero, y2)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map22(xu308, xu309, xu310, ba) → new_map21(xu308, xu310, ba)
new_map20(xu308, xu309, xu310, Succ(xu3110), Succ(xu3120), ba) → new_map20(xu308, xu309, xu310, xu3110, xu3120, ba)
new_map20(xu308, xu309, xu310, Zero, Zero, ba) → new_map22(xu308, xu309, xu310, ba)
new_map23(xu196, xu249, xu250, Zero, bb) → new_map20(xu196, xu249, xu250, Zero, Succ(xu196), bb)
new_map21(y0, Succ(x0), y2) → new_map23(y0, Succ(x0), Succ(x0), Succ(x0), y2)
new_map23(xu196, xu249, xu250, Succ(xu2510), bb) → new_map20(xu196, xu249, xu250, Succ(xu2510), Succ(xu196), bb)
new_map21(y0, Zero, y2) → new_map23(y0, Zero, Zero, Zero, y2)
new_map20(xu308, xu309, xu310, Zero, Succ(xu3120), ba) → new_map21(xu308, xu310, ba)
R is empty.
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(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
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map22(xu308, xu309, xu310, ba) → new_map21(xu308, xu310, ba)
new_map20(xu308, xu309, xu310, Succ(xu3110), Succ(xu3120), ba) → new_map20(xu308, xu309, xu310, xu3110, xu3120, ba)
new_map23(xu196, xu249, xu250, Zero, bb) → new_map20(xu196, xu249, xu250, Zero, Succ(xu196), bb)
new_map20(xu308, xu309, xu310, Zero, Zero, ba) → new_map22(xu308, xu309, xu310, ba)
new_map21(y0, Succ(x0), y2) → new_map23(y0, Succ(x0), Succ(x0), Succ(x0), y2)
new_map23(xu196, xu249, xu250, Succ(xu2510), bb) → new_map20(xu196, xu249, xu250, Succ(xu2510), Succ(xu196), bb)
new_map20(xu308, xu309, xu310, Zero, Succ(xu3120), ba) → new_map21(xu308, xu310, ba)
new_map21(y0, Zero, y2) → new_map23(y0, Zero, Zero, Zero, y2)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule new_map23(xu196, xu249, xu250, Zero, bb) → new_map20(xu196, xu249, xu250, Zero, Succ(xu196), bb) we obtained the following new rules:
new_map23(z0, Zero, Zero, Zero, z1) → new_map20(z0, Zero, Zero, Zero, Succ(z0), 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
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map22(xu308, xu309, xu310, ba) → new_map21(xu308, xu310, ba)
new_map20(xu308, xu309, xu310, Succ(xu3110), Succ(xu3120), ba) → new_map20(xu308, xu309, xu310, xu3110, xu3120, ba)
new_map20(xu308, xu309, xu310, Zero, Zero, ba) → new_map22(xu308, xu309, xu310, ba)
new_map23(z0, Zero, Zero, Zero, z1) → new_map20(z0, Zero, Zero, Zero, Succ(z0), z1)
new_map21(y0, Succ(x0), y2) → new_map23(y0, Succ(x0), Succ(x0), Succ(x0), y2)
new_map23(xu196, xu249, xu250, Succ(xu2510), bb) → new_map20(xu196, xu249, xu250, Succ(xu2510), Succ(xu196), bb)
new_map21(y0, Zero, y2) → new_map23(y0, Zero, Zero, Zero, y2)
new_map20(xu308, xu309, xu310, Zero, Succ(xu3120), ba) → new_map21(xu308, xu310, ba)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule new_map23(xu196, xu249, xu250, Succ(xu2510), bb) → new_map20(xu196, xu249, xu250, Succ(xu2510), Succ(xu196), bb) we obtained the following new rules:
new_map23(z0, Succ(z1), Succ(z1), Succ(z1), z2) → new_map20(z0, Succ(z1), Succ(z1), Succ(z1), Succ(z0), 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
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map23(z0, Succ(z1), Succ(z1), Succ(z1), z2) → new_map20(z0, Succ(z1), Succ(z1), Succ(z1), Succ(z0), z2)
new_map22(xu308, xu309, xu310, ba) → new_map21(xu308, xu310, ba)
new_map20(xu308, xu309, xu310, Succ(xu3110), Succ(xu3120), ba) → new_map20(xu308, xu309, xu310, xu3110, xu3120, ba)
new_map20(xu308, xu309, xu310, Zero, Zero, ba) → new_map22(xu308, xu309, xu310, ba)
new_map23(z0, Zero, Zero, Zero, z1) → new_map20(z0, Zero, Zero, Zero, Succ(z0), z1)
new_map21(y0, Succ(x0), y2) → new_map23(y0, Succ(x0), Succ(x0), Succ(x0), y2)
new_map20(xu308, xu309, xu310, Zero, Succ(xu3120), ba) → new_map21(xu308, xu310, ba)
new_map21(y0, Zero, y2) → new_map23(y0, Zero, Zero, Zero, y2)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule new_map22(xu308, xu309, xu310, ba) → new_map21(xu308, xu310, ba) we obtained the following new rules:
new_map22(x0, x1, Succ(y_1), x3) → new_map21(x0, Succ(y_1), x3)
new_map22(x0, x1, Zero, x3) → new_map21(x0, Zero, x3)
↳ 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
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map23(z0, Succ(z1), Succ(z1), Succ(z1), z2) → new_map20(z0, Succ(z1), Succ(z1), Succ(z1), Succ(z0), z2)
new_map22(x0, x1, Succ(y_1), x3) → new_map21(x0, Succ(y_1), x3)
new_map20(xu308, xu309, xu310, Succ(xu3110), Succ(xu3120), ba) → new_map20(xu308, xu309, xu310, xu3110, xu3120, ba)
new_map20(xu308, xu309, xu310, Zero, Zero, ba) → new_map22(xu308, xu309, xu310, ba)
new_map23(z0, Zero, Zero, Zero, z1) → new_map20(z0, Zero, Zero, Zero, Succ(z0), z1)
new_map22(x0, x1, Zero, x3) → new_map21(x0, Zero, x3)
new_map21(y0, Succ(x0), y2) → new_map23(y0, Succ(x0), Succ(x0), Succ(x0), y2)
new_map21(y0, Zero, y2) → new_map23(y0, Zero, Zero, Zero, y2)
new_map20(xu308, xu309, xu310, Zero, Succ(xu3120), ba) → new_map21(xu308, xu310, ba)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule new_map20(xu308, xu309, xu310, Succ(xu3110), Succ(xu3120), ba) → new_map20(xu308, xu309, xu310, xu3110, xu3120, ba) we obtained the following new rules:
new_map20(x0, x1, x2, Succ(Succ(y_3)), Succ(Succ(y_4)), x5) → new_map20(x0, x1, x2, Succ(y_3), Succ(y_4), x5)
new_map20(x0, x1, x2, Succ(Zero), Succ(Succ(y_3)), x5) → new_map20(x0, x1, x2, Zero, Succ(y_3), x5)
new_map20(x0, x1, x2, Succ(Zero), Succ(Zero), x5) → new_map20(x0, x1, x2, Zero, Zero, x5)
↳ 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
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map23(z0, Succ(z1), Succ(z1), Succ(z1), z2) → new_map20(z0, Succ(z1), Succ(z1), Succ(z1), Succ(z0), z2)
new_map20(x0, x1, x2, Succ(Succ(y_3)), Succ(Succ(y_4)), x5) → new_map20(x0, x1, x2, Succ(y_3), Succ(y_4), x5)
new_map22(x0, x1, Succ(y_1), x3) → new_map21(x0, Succ(y_1), x3)
new_map20(xu308, xu309, xu310, Zero, Zero, ba) → new_map22(xu308, xu309, xu310, ba)
new_map23(z0, Zero, Zero, Zero, z1) → new_map20(z0, Zero, Zero, Zero, Succ(z0), z1)
new_map20(x0, x1, x2, Succ(Zero), Succ(Zero), x5) → new_map20(x0, x1, x2, Zero, Zero, x5)
new_map20(x0, x1, x2, Succ(Zero), Succ(Succ(y_3)), x5) → new_map20(x0, x1, x2, Zero, Succ(y_3), x5)
new_map22(x0, x1, Zero, x3) → new_map21(x0, Zero, x3)
new_map21(y0, Succ(x0), y2) → new_map23(y0, Succ(x0), Succ(x0), Succ(x0), y2)
new_map20(xu308, xu309, xu310, Zero, Succ(xu3120), ba) → new_map21(xu308, xu310, ba)
new_map21(y0, Zero, y2) → new_map23(y0, Zero, Zero, Zero, y2)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule new_map20(xu308, xu309, xu310, Zero, Zero, ba) → new_map22(xu308, xu309, xu310, ba) we obtained the following new rules:
new_map20(x0, x1, Zero, Zero, Zero, x3) → new_map22(x0, x1, Zero, x3)
new_map20(x0, x1, Succ(y_2), Zero, Zero, x3) → new_map22(x0, x1, Succ(y_2), x3)
↳ 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
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map23(z0, Succ(z1), Succ(z1), Succ(z1), z2) → new_map20(z0, Succ(z1), Succ(z1), Succ(z1), Succ(z0), z2)
new_map20(x0, x1, Zero, Zero, Zero, x3) → new_map22(x0, x1, Zero, x3)
new_map20(x0, x1, Succ(y_2), Zero, Zero, x3) → new_map22(x0, x1, Succ(y_2), x3)
new_map22(x0, x1, Zero, x3) → new_map21(x0, Zero, x3)
new_map21(y0, Succ(x0), y2) → new_map23(y0, Succ(x0), Succ(x0), Succ(x0), y2)
new_map21(y0, Zero, y2) → new_map23(y0, Zero, Zero, Zero, y2)
new_map20(x0, x1, x2, Succ(Succ(y_3)), Succ(Succ(y_4)), x5) → new_map20(x0, x1, x2, Succ(y_3), Succ(y_4), x5)
new_map22(x0, x1, Succ(y_1), x3) → new_map21(x0, Succ(y_1), x3)
new_map23(z0, Zero, Zero, Zero, z1) → new_map20(z0, Zero, Zero, Zero, Succ(z0), z1)
new_map20(x0, x1, x2, Succ(Zero), Succ(Zero), x5) → new_map20(x0, x1, x2, Zero, Zero, x5)
new_map20(x0, x1, x2, Succ(Zero), Succ(Succ(y_3)), x5) → new_map20(x0, x1, x2, Zero, Succ(y_3), x5)
new_map20(xu308, xu309, xu310, Zero, Succ(xu3120), ba) → new_map21(xu308, xu310, ba)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule new_map20(xu308, xu309, xu310, Zero, Succ(xu3120), ba) → new_map21(xu308, xu310, ba) we obtained the following new rules:
new_map20(x0, x1, Zero, Zero, Succ(x3), x4) → new_map21(x0, Zero, x4)
new_map20(x0, x1, Succ(y_1), Zero, Succ(x3), x4) → new_map21(x0, Succ(y_1), 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
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map20(x0, x1, Zero, Zero, Zero, x3) → new_map22(x0, x1, Zero, x3)
new_map23(z0, Succ(z1), Succ(z1), Succ(z1), z2) → new_map20(z0, Succ(z1), Succ(z1), Succ(z1), Succ(z0), z2)
new_map20(x0, x1, Zero, Zero, Succ(x3), x4) → new_map21(x0, Zero, x4)
new_map20(x0, x1, Succ(y_2), Zero, Zero, x3) → new_map22(x0, x1, Succ(y_2), x3)
new_map22(x0, x1, Zero, x3) → new_map21(x0, Zero, x3)
new_map20(x0, x1, Succ(y_1), Zero, Succ(x3), x4) → new_map21(x0, Succ(y_1), x4)
new_map21(y0, Succ(x0), y2) → new_map23(y0, Succ(x0), Succ(x0), Succ(x0), y2)
new_map21(y0, Zero, y2) → new_map23(y0, Zero, Zero, Zero, y2)
new_map20(x0, x1, x2, Succ(Succ(y_3)), Succ(Succ(y_4)), x5) → new_map20(x0, x1, x2, Succ(y_3), Succ(y_4), x5)
new_map22(x0, x1, Succ(y_1), x3) → new_map21(x0, Succ(y_1), x3)
new_map23(z0, Zero, Zero, Zero, z1) → new_map20(z0, Zero, Zero, Zero, Succ(z0), z1)
new_map20(x0, x1, x2, Succ(Zero), Succ(Succ(y_3)), x5) → new_map20(x0, x1, x2, Zero, Succ(y_3), x5)
new_map20(x0, x1, x2, Succ(Zero), Succ(Zero), x5) → new_map20(x0, x1, x2, Zero, Zero, x5)
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 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
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Instantiation
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map20(x0, x1, Zero, Zero, Succ(x3), x4) → new_map21(x0, Zero, x4)
new_map23(z0, Zero, Zero, Zero, z1) → new_map20(z0, Zero, Zero, Zero, Succ(z0), z1)
new_map21(y0, Zero, y2) → new_map23(y0, Zero, Zero, Zero, y2)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule new_map20(x0, x1, Zero, Zero, Succ(x3), x4) → new_map21(x0, Zero, x4) we obtained the following new rules:
new_map20(z0, Zero, Zero, Zero, Succ(z0), z1) → new_map21(z0, Zero, 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
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Instantiation
↳ QDP
↳ NonTerminationProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map20(z0, Zero, Zero, Zero, Succ(z0), z1) → new_map21(z0, Zero, z1)
new_map23(z0, Zero, Zero, Zero, z1) → new_map20(z0, Zero, Zero, Zero, Succ(z0), z1)
new_map21(y0, Zero, y2) → new_map23(y0, Zero, Zero, Zero, y2)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by narrowing to the left:
The TRS P consists of the following rules:
new_map20(z0, Zero, Zero, Zero, Succ(z0), z1) → new_map21(z0, Zero, z1)
new_map23(z0, Zero, Zero, Zero, z1) → new_map20(z0, Zero, Zero, Zero, Succ(z0), z1)
new_map21(y0, Zero, y2) → new_map23(y0, Zero, Zero, Zero, y2)
The TRS R consists of the following rules:none
s = new_map21(y0, Zero, y2) evaluates to t =new_map21(y0, Zero, y2)
Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
- Semiunifier: [ ]
- Matcher: [ ]
Rewriting sequence
new_map21(y0, Zero, y2) → new_map23(y0, Zero, Zero, Zero, y2)
with rule new_map21(y0', Zero, y2') → new_map23(y0', Zero, Zero, Zero, y2') at position [] and matcher [y2' / y2, y0' / y0]
new_map23(y0, Zero, Zero, Zero, y2) → new_map20(y0, Zero, Zero, Zero, Succ(y0), y2)
with rule new_map23(z0', Zero, Zero, Zero, z1') → new_map20(z0', Zero, Zero, Zero, Succ(z0'), z1') at position [] and matcher [z1' / y2, z0' / y0]
new_map20(y0, Zero, Zero, Zero, Succ(y0), y2) → new_map21(y0, Zero, y2)
with rule new_map20(z0, Zero, Zero, Zero, Succ(z0), z1) → new_map21(z0, Zero, z1)
Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence
All these steps are and every following step will be a correct step w.r.t to Q.
↳ 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
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map23(z0, Succ(z1), Succ(z1), Succ(z1), z2) → new_map20(z0, Succ(z1), Succ(z1), Succ(z1), Succ(z0), z2)
new_map20(x0, x1, x2, Succ(Succ(y_3)), Succ(Succ(y_4)), x5) → new_map20(x0, x1, x2, Succ(y_3), Succ(y_4), x5)
new_map22(x0, x1, Succ(y_1), x3) → new_map21(x0, Succ(y_1), x3)
new_map20(x0, x1, x2, Succ(Zero), Succ(Zero), x5) → new_map20(x0, x1, x2, Zero, Zero, x5)
new_map20(x0, x1, x2, Succ(Zero), Succ(Succ(y_3)), x5) → new_map20(x0, x1, x2, Zero, Succ(y_3), x5)
new_map20(x0, x1, Succ(y_2), Zero, Zero, x3) → new_map22(x0, x1, Succ(y_2), x3)
new_map20(x0, x1, Succ(y_1), Zero, Succ(x3), x4) → new_map21(x0, Succ(y_1), x4)
new_map21(y0, Succ(x0), y2) → new_map23(y0, Succ(x0), Succ(x0), Succ(x0), y2)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule new_map23(z0, Succ(z1), Succ(z1), Succ(z1), z2) → new_map20(z0, Succ(z1), Succ(z1), Succ(z1), Succ(z0), z2) we obtained the following new rules:
new_map23(Succ(y_3), Succ(Zero), Succ(Zero), Succ(Zero), x2) → new_map20(Succ(y_3), Succ(Zero), Succ(Zero), Succ(Zero), Succ(Succ(y_3)), x2)
new_map23(Succ(y_4), Succ(Succ(y_3)), Succ(Succ(y_3)), Succ(Succ(y_3)), x2) → new_map20(Succ(y_4), Succ(Succ(y_3)), Succ(Succ(y_3)), Succ(Succ(y_3)), Succ(Succ(y_4)), x2)
new_map23(Zero, Succ(Zero), Succ(Zero), Succ(Zero), x2) → new_map20(Zero, Succ(Zero), Succ(Zero), Succ(Zero), Succ(Zero), x2)
↳ 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
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map23(Succ(y_3), Succ(Zero), Succ(Zero), Succ(Zero), x2) → new_map20(Succ(y_3), Succ(Zero), Succ(Zero), Succ(Zero), Succ(Succ(y_3)), x2)
new_map20(x0, x1, x2, Succ(Succ(y_3)), Succ(Succ(y_4)), x5) → new_map20(x0, x1, x2, Succ(y_3), Succ(y_4), x5)
new_map22(x0, x1, Succ(y_1), x3) → new_map21(x0, Succ(y_1), x3)
new_map23(Succ(y_4), Succ(Succ(y_3)), Succ(Succ(y_3)), Succ(Succ(y_3)), x2) → new_map20(Succ(y_4), Succ(Succ(y_3)), Succ(Succ(y_3)), Succ(Succ(y_3)), Succ(Succ(y_4)), x2)
new_map20(x0, x1, x2, Succ(Zero), Succ(Succ(y_3)), x5) → new_map20(x0, x1, x2, Zero, Succ(y_3), x5)
new_map20(x0, x1, x2, Succ(Zero), Succ(Zero), x5) → new_map20(x0, x1, x2, Zero, Zero, x5)
new_map20(x0, x1, Succ(y_2), Zero, Zero, x3) → new_map22(x0, x1, Succ(y_2), x3)
new_map20(x0, x1, Succ(y_1), Zero, Succ(x3), x4) → new_map21(x0, Succ(y_1), x4)
new_map21(y0, Succ(x0), y2) → new_map23(y0, Succ(x0), Succ(x0), Succ(x0), y2)
new_map23(Zero, Succ(Zero), Succ(Zero), Succ(Zero), x2) → new_map20(Zero, Succ(Zero), Succ(Zero), Succ(Zero), Succ(Zero), x2)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule new_map21(y0, Succ(x0), y2) → new_map23(y0, Succ(x0), Succ(x0), Succ(x0), y2) we obtained the following new rules:
new_map21(Succ(y_0), Succ(Succ(y_1)), x2) → new_map23(Succ(y_0), Succ(Succ(y_1)), Succ(Succ(y_1)), Succ(Succ(y_1)), x2)
new_map21(Succ(y_0), Succ(Zero), x2) → new_map23(Succ(y_0), Succ(Zero), Succ(Zero), Succ(Zero), x2)
new_map21(Zero, Succ(Zero), x2) → new_map23(Zero, Succ(Zero), Succ(Zero), Succ(Zero), x2)
↳ 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
↳ UsableRulesProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ NonTerminationProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map21(Succ(y_0), Succ(Zero), x2) → new_map23(Succ(y_0), Succ(Zero), Succ(Zero), Succ(Zero), x2)
new_map20(x0, x1, Succ(y_2), Zero, Zero, x3) → new_map22(x0, x1, Succ(y_2), x3)
new_map20(x0, x1, Succ(y_1), Zero, Succ(x3), x4) → new_map21(x0, Succ(y_1), x4)
new_map23(Zero, Succ(Zero), Succ(Zero), Succ(Zero), x2) → new_map20(Zero, Succ(Zero), Succ(Zero), Succ(Zero), Succ(Zero), x2)
new_map23(Succ(y_3), Succ(Zero), Succ(Zero), Succ(Zero), x2) → new_map20(Succ(y_3), Succ(Zero), Succ(Zero), Succ(Zero), Succ(Succ(y_3)), x2)
new_map20(x0, x1, x2, Succ(Succ(y_3)), Succ(Succ(y_4)), x5) → new_map20(x0, x1, x2, Succ(y_3), Succ(y_4), x5)
new_map22(x0, x1, Succ(y_1), x3) → new_map21(x0, Succ(y_1), x3)
new_map21(Succ(y_0), Succ(Succ(y_1)), x2) → new_map23(Succ(y_0), Succ(Succ(y_1)), Succ(Succ(y_1)), Succ(Succ(y_1)), x2)
new_map23(Succ(y_4), Succ(Succ(y_3)), Succ(Succ(y_3)), Succ(Succ(y_3)), x2) → new_map20(Succ(y_4), Succ(Succ(y_3)), Succ(Succ(y_3)), Succ(Succ(y_3)), Succ(Succ(y_4)), x2)
new_map20(x0, x1, x2, Succ(Zero), Succ(Succ(y_3)), x5) → new_map20(x0, x1, x2, Zero, Succ(y_3), x5)
new_map20(x0, x1, x2, Succ(Zero), Succ(Zero), x5) → new_map20(x0, x1, x2, Zero, Zero, x5)
new_map21(Zero, Succ(Zero), x2) → new_map23(Zero, Succ(Zero), Succ(Zero), Succ(Zero), x2)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by narrowing to the left:
The TRS P consists of the following rules:
new_map21(Succ(y_0), Succ(Zero), x2) → new_map23(Succ(y_0), Succ(Zero), Succ(Zero), Succ(Zero), x2)
new_map20(x0, x1, Succ(y_2), Zero, Zero, x3) → new_map22(x0, x1, Succ(y_2), x3)
new_map20(x0, x1, Succ(y_1), Zero, Succ(x3), x4) → new_map21(x0, Succ(y_1), x4)
new_map23(Zero, Succ(Zero), Succ(Zero), Succ(Zero), x2) → new_map20(Zero, Succ(Zero), Succ(Zero), Succ(Zero), Succ(Zero), x2)
new_map23(Succ(y_3), Succ(Zero), Succ(Zero), Succ(Zero), x2) → new_map20(Succ(y_3), Succ(Zero), Succ(Zero), Succ(Zero), Succ(Succ(y_3)), x2)
new_map20(x0, x1, x2, Succ(Succ(y_3)), Succ(Succ(y_4)), x5) → new_map20(x0, x1, x2, Succ(y_3), Succ(y_4), x5)
new_map22(x0, x1, Succ(y_1), x3) → new_map21(x0, Succ(y_1), x3)
new_map21(Succ(y_0), Succ(Succ(y_1)), x2) → new_map23(Succ(y_0), Succ(Succ(y_1)), Succ(Succ(y_1)), Succ(Succ(y_1)), x2)
new_map23(Succ(y_4), Succ(Succ(y_3)), Succ(Succ(y_3)), Succ(Succ(y_3)), x2) → new_map20(Succ(y_4), Succ(Succ(y_3)), Succ(Succ(y_3)), Succ(Succ(y_3)), Succ(Succ(y_4)), x2)
new_map20(x0, x1, x2, Succ(Zero), Succ(Succ(y_3)), x5) → new_map20(x0, x1, x2, Zero, Succ(y_3), x5)
new_map20(x0, x1, x2, Succ(Zero), Succ(Zero), x5) → new_map20(x0, x1, x2, Zero, Zero, x5)
new_map21(Zero, Succ(Zero), x2) → new_map23(Zero, Succ(Zero), Succ(Zero), Succ(Zero), x2)
The TRS R consists of the following rules:none
s = new_map23(Succ(y_3'), Succ(Zero), Succ(Zero), Succ(Zero), x2) evaluates to t =new_map23(Succ(y_3'), Succ(Zero), Succ(Zero), Succ(Zero), x2)
Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
- Semiunifier: [ ]
- Matcher: [ ]
Rewriting sequence
new_map23(Succ(y_3'), Succ(Zero), Succ(Zero), Succ(Zero), x2) → new_map20(Succ(y_3'), Succ(Zero), Succ(Zero), Succ(Zero), Succ(Succ(y_3')), x2)
with rule new_map23(Succ(y_3''), Succ(Zero), Succ(Zero), Succ(Zero), x2') → new_map20(Succ(y_3''), Succ(Zero), Succ(Zero), Succ(Zero), Succ(Succ(y_3'')), x2') at position [] and matcher [y_3'' / y_3', x2' / x2]
new_map20(Succ(y_3'), Succ(Zero), Succ(Zero), Succ(Zero), Succ(Succ(y_3')), x2) → new_map20(Succ(y_3'), Succ(Zero), Succ(Zero), Zero, Succ(y_3'), x2)
with rule new_map20(x0, x1', x2', Succ(Zero), Succ(Succ(y_3)), x5) → new_map20(x0, x1', x2', Zero, Succ(y_3), x5) at position [] and matcher [x1' / Succ(Zero), y_3 / y_3', x5 / x2, x0 / Succ(y_3'), x2' / Succ(Zero)]
new_map20(Succ(y_3'), Succ(Zero), Succ(Zero), Zero, Succ(y_3'), x2) → new_map21(Succ(y_3'), Succ(Zero), x2)
with rule new_map20(x0, x1, Succ(y_1), Zero, Succ(x3), x4) → new_map21(x0, Succ(y_1), x4) at position [] and matcher [x1 / Succ(Zero), x4 / x2, x3 / y_3', y_1 / Zero, x0 / Succ(y_3')]
new_map21(Succ(y_3'), Succ(Zero), x2) → new_map23(Succ(y_3'), Succ(Zero), Succ(Zero), Succ(Zero), x2)
with rule new_map21(Succ(y_0), Succ(Zero), x2) → new_map23(Succ(y_0), Succ(Zero), Succ(Zero), Succ(Zero), x2)
Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence
All these steps are and every following step will be a correct step w.r.t to Q.
↳ 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
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map24(xu386, xu387, xu388, Zero, Zero, ba) → new_map26(xu386, xu387, xu388, ba)
new_map24(xu386, xu387, xu388, Succ(xu3890), Succ(xu3900), ba) → new_map24(xu386, xu387, xu388, xu3890, xu3900, ba)
new_map25(xu273, xu2750, xu276, bb) → new_map27(xu273, Succ(xu276), Succ(xu2750), xu2750, bb)
new_map27(xu332, Zero, Zero, xu335, bc) → new_map27(xu332, Zero, Succ(xu335), xu335, bc)
new_map24(xu386, xu387, xu388, Zero, Succ(xu3900), ba) → new_map25(xu386, xu388, xu387, ba)
new_map27(xu332, Succ(xu3330), Succ(xu3340), xu335, bc) → new_map27(xu332, xu3330, xu3340, xu335, bc)
new_map26(xu386, xu387, xu388, ba) → new_map25(xu386, xu388, xu387, ba)
new_map27(xu332, Succ(xu3330), Zero, xu335, bc) → new_map24(xu332, xu3330, xu335, Succ(xu3330), Succ(xu332), bc)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map24(xu386, xu387, xu388, Zero, Zero, ba) → new_map26(xu386, xu387, xu388, ba)
new_map24(xu386, xu387, xu388, Succ(xu3890), Succ(xu3900), ba) → new_map24(xu386, xu387, xu388, xu3890, xu3900, ba)
new_map25(xu273, xu2750, xu276, bb) → new_map27(xu273, Succ(xu276), Succ(xu2750), xu2750, bb)
new_map24(xu386, xu387, xu388, Zero, Succ(xu3900), ba) → new_map25(xu386, xu388, xu387, ba)
new_map27(xu332, Succ(xu3330), Succ(xu3340), xu335, bc) → new_map27(xu332, xu3330, xu3340, xu335, bc)
new_map26(xu386, xu387, xu388, ba) → new_map25(xu386, xu388, xu387, ba)
new_map27(xu332, Succ(xu3330), Zero, xu335, bc) → new_map24(xu332, xu3330, xu335, Succ(xu3330), Succ(xu332), bc)
R is empty.
Q is empty.
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_map27(xu332, Succ(xu3330), Succ(xu3340), xu335, bc) → new_map27(xu332, xu3330, xu3340, xu335, bc)
The remaining pairs can at least be oriented weakly.
new_map24(xu386, xu387, xu388, Zero, Zero, ba) → new_map26(xu386, xu387, xu388, ba)
new_map24(xu386, xu387, xu388, Succ(xu3890), Succ(xu3900), ba) → new_map24(xu386, xu387, xu388, xu3890, xu3900, ba)
new_map25(xu273, xu2750, xu276, bb) → new_map27(xu273, Succ(xu276), Succ(xu2750), xu2750, bb)
new_map24(xu386, xu387, xu388, Zero, Succ(xu3900), ba) → new_map25(xu386, xu388, xu387, ba)
new_map26(xu386, xu387, xu388, ba) → new_map25(xu386, xu388, xu387, ba)
new_map27(xu332, Succ(xu3330), Zero, xu335, bc) → new_map24(xu332, xu3330, xu335, Succ(xu3330), Succ(xu332), bc)
Used ordering: Polynomial interpretation [25]:
POL(Succ(x1)) = 1 + x1
POL(Zero) = 0
POL(new_map24(x1, x2, x3, x4, x5, x6)) = 1 + x2
POL(new_map25(x1, x2, x3, x4)) = 1 + x3
POL(new_map26(x1, x2, x3, x4)) = 1 + x2
POL(new_map27(x1, x2, x3, x4, x5)) = x2
The following usable rules [17] were oriented:
none
↳ 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
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map24(xu386, xu387, xu388, Zero, Zero, ba) → new_map26(xu386, xu387, xu388, ba)
new_map24(xu386, xu387, xu388, Succ(xu3890), Succ(xu3900), ba) → new_map24(xu386, xu387, xu388, xu3890, xu3900, ba)
new_map25(xu273, xu2750, xu276, bb) → new_map27(xu273, Succ(xu276), Succ(xu2750), xu2750, bb)
new_map24(xu386, xu387, xu388, Zero, Succ(xu3900), ba) → new_map25(xu386, xu388, xu387, ba)
new_map26(xu386, xu387, xu388, ba) → new_map25(xu386, xu388, xu387, ba)
new_map27(xu332, Succ(xu3330), Zero, xu335, bc) → new_map24(xu332, xu3330, xu335, Succ(xu3330), Succ(xu332), bc)
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 5 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
↳ DependencyGraphProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map24(xu386, xu387, xu388, Succ(xu3890), Succ(xu3900), ba) → new_map24(xu386, xu387, xu388, xu3890, xu3900, 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_map24(xu386, xu387, xu388, Succ(xu3890), Succ(xu3900), ba) → new_map24(xu386, xu387, xu388, xu3890, xu3900, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4, 5 > 5, 6 >= 6
↳ 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
↳ Narrowing
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map31(xu196, xu240, xu1980000, xu241, Zero, bb) → new_map28(xu196, xu240, xu1980000, xu241, Zero, Succ(xu196), bb)
new_map28(xu301, xu302, xu303, xu304, Succ(xu3050), Succ(xu3060), ba) → new_map28(xu301, xu302, xu303, xu304, xu3050, xu3060, ba)
new_map31(xu196, xu240, xu1980000, xu241, Succ(xu2420), bb) → new_map28(xu196, xu240, xu1980000, xu241, Succ(xu2420), Succ(xu196), bb)
new_map28(xu301, xu302, xu303, xu304, Zero, Succ(xu3060), ba) → new_map29(xu301, xu303, xu304, ba)
new_map28(xu301, xu302, xu303, xu304, Zero, Zero, ba) → new_map30(xu301, xu302, xu303, xu304, ba)
new_map30(xu301, xu302, xu303, xu304, ba) → new_map29(xu301, xu303, xu304, ba)
new_map29(xu196, xu1980000, xu241, bb) → new_map31(xu196, new_primPlusNat0(Succ(xu1980000), xu241), xu1980000, new_primPlusNat0(Succ(xu1980000), xu241), new_primPlusNat0(Succ(xu1980000), xu241), bb)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule new_map29(xu196, xu1980000, xu241, bb) → new_map31(xu196, new_primPlusNat0(Succ(xu1980000), xu241), xu1980000, new_primPlusNat0(Succ(xu1980000), xu241), new_primPlusNat0(Succ(xu1980000), xu241), bb) at position [4] we obtained the following new rules:
new_map29(y0, x0, Zero, y3) → new_map31(y0, new_primPlusNat0(Succ(x0), Zero), x0, new_primPlusNat0(Succ(x0), Zero), Succ(x0), y3)
new_map29(y0, x0, Succ(x1), y3) → new_map31(y0, new_primPlusNat0(Succ(x0), Succ(x1)), x0, new_primPlusNat0(Succ(x0), Succ(x1)), Succ(Succ(new_primPlusNat0(x0, x1))), y3)
↳ 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
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map31(xu196, xu240, xu1980000, xu241, Zero, bb) → new_map28(xu196, xu240, xu1980000, xu241, Zero, Succ(xu196), bb)
new_map29(y0, x0, Zero, y3) → new_map31(y0, new_primPlusNat0(Succ(x0), Zero), x0, new_primPlusNat0(Succ(x0), Zero), Succ(x0), y3)
new_map31(xu196, xu240, xu1980000, xu241, Succ(xu2420), bb) → new_map28(xu196, xu240, xu1980000, xu241, Succ(xu2420), Succ(xu196), bb)
new_map28(xu301, xu302, xu303, xu304, Succ(xu3050), Succ(xu3060), ba) → new_map28(xu301, xu302, xu303, xu304, xu3050, xu3060, ba)
new_map28(xu301, xu302, xu303, xu304, Zero, Succ(xu3060), ba) → new_map29(xu301, xu303, xu304, ba)
new_map28(xu301, xu302, xu303, xu304, Zero, Zero, ba) → new_map30(xu301, xu302, xu303, xu304, ba)
new_map29(y0, x0, Succ(x1), y3) → new_map31(y0, new_primPlusNat0(Succ(x0), Succ(x1)), x0, new_primPlusNat0(Succ(x0), Succ(x1)), Succ(Succ(new_primPlusNat0(x0, x1))), y3)
new_map30(xu301, xu302, xu303, xu304, ba) → new_map29(xu301, xu303, xu304, ba)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map29(y0, x0, Zero, y3) → new_map31(y0, new_primPlusNat0(Succ(x0), Zero), x0, new_primPlusNat0(Succ(x0), Zero), Succ(x0), y3)
new_map31(xu196, xu240, xu1980000, xu241, Succ(xu2420), bb) → new_map28(xu196, xu240, xu1980000, xu241, Succ(xu2420), Succ(xu196), bb)
new_map28(xu301, xu302, xu303, xu304, Succ(xu3050), Succ(xu3060), ba) → new_map28(xu301, xu302, xu303, xu304, xu3050, xu3060, ba)
new_map28(xu301, xu302, xu303, xu304, Zero, Succ(xu3060), ba) → new_map29(xu301, xu303, xu304, ba)
new_map28(xu301, xu302, xu303, xu304, Zero, Zero, ba) → new_map30(xu301, xu302, xu303, xu304, ba)
new_map30(xu301, xu302, xu303, xu304, ba) → new_map29(xu301, xu303, xu304, ba)
new_map29(y0, x0, Succ(x1), y3) → new_map31(y0, new_primPlusNat0(Succ(x0), Succ(x1)), x0, new_primPlusNat0(Succ(x0), Succ(x1)), Succ(Succ(new_primPlusNat0(x0, x1))), y3)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_map29(y0, x0, Zero, y3) → new_map31(y0, new_primPlusNat0(Succ(x0), Zero), x0, new_primPlusNat0(Succ(x0), Zero), Succ(x0), y3) at position [1] we obtained the following new rules:
new_map29(y0, x0, Zero, y3) → new_map31(y0, Succ(x0), x0, new_primPlusNat0(Succ(x0), Zero), Succ(x0), y3)
↳ 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
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map28(xu301, xu302, xu303, xu304, Succ(xu3050), Succ(xu3060), ba) → new_map28(xu301, xu302, xu303, xu304, xu3050, xu3060, ba)
new_map31(xu196, xu240, xu1980000, xu241, Succ(xu2420), bb) → new_map28(xu196, xu240, xu1980000, xu241, Succ(xu2420), Succ(xu196), bb)
new_map28(xu301, xu302, xu303, xu304, Zero, Succ(xu3060), ba) → new_map29(xu301, xu303, xu304, ba)
new_map28(xu301, xu302, xu303, xu304, Zero, Zero, ba) → new_map30(xu301, xu302, xu303, xu304, ba)
new_map29(y0, x0, Succ(x1), y3) → new_map31(y0, new_primPlusNat0(Succ(x0), Succ(x1)), x0, new_primPlusNat0(Succ(x0), Succ(x1)), Succ(Succ(new_primPlusNat0(x0, x1))), y3)
new_map30(xu301, xu302, xu303, xu304, ba) → new_map29(xu301, xu303, xu304, ba)
new_map29(y0, x0, Zero, y3) → new_map31(y0, Succ(x0), x0, new_primPlusNat0(Succ(x0), Zero), Succ(x0), y3)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_map29(y0, x0, Succ(x1), y3) → new_map31(y0, new_primPlusNat0(Succ(x0), Succ(x1)), x0, new_primPlusNat0(Succ(x0), Succ(x1)), Succ(Succ(new_primPlusNat0(x0, x1))), y3) at position [1] we obtained the following new rules:
new_map29(y0, x0, Succ(x1), y3) → new_map31(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, new_primPlusNat0(Succ(x0), Succ(x1)), Succ(Succ(new_primPlusNat0(x0, x1))), y3)
↳ 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
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map29(y0, x0, Succ(x1), y3) → new_map31(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, new_primPlusNat0(Succ(x0), Succ(x1)), Succ(Succ(new_primPlusNat0(x0, x1))), y3)
new_map31(xu196, xu240, xu1980000, xu241, Succ(xu2420), bb) → new_map28(xu196, xu240, xu1980000, xu241, Succ(xu2420), Succ(xu196), bb)
new_map28(xu301, xu302, xu303, xu304, Succ(xu3050), Succ(xu3060), ba) → new_map28(xu301, xu302, xu303, xu304, xu3050, xu3060, ba)
new_map28(xu301, xu302, xu303, xu304, Zero, Succ(xu3060), ba) → new_map29(xu301, xu303, xu304, ba)
new_map28(xu301, xu302, xu303, xu304, Zero, Zero, ba) → new_map30(xu301, xu302, xu303, xu304, ba)
new_map30(xu301, xu302, xu303, xu304, ba) → new_map29(xu301, xu303, xu304, ba)
new_map29(y0, x0, Zero, y3) → new_map31(y0, Succ(x0), x0, new_primPlusNat0(Succ(x0), Zero), Succ(x0), y3)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_map29(y0, x0, Zero, y3) → new_map31(y0, Succ(x0), x0, new_primPlusNat0(Succ(x0), Zero), Succ(x0), y3) at position [3] we obtained the following new rules:
new_map29(y0, x0, Zero, y3) → new_map31(y0, Succ(x0), x0, Succ(x0), Succ(x0), y3)
↳ 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
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map29(y0, x0, Zero, y3) → new_map31(y0, Succ(x0), x0, Succ(x0), Succ(x0), y3)
new_map29(y0, x0, Succ(x1), y3) → new_map31(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, new_primPlusNat0(Succ(x0), Succ(x1)), Succ(Succ(new_primPlusNat0(x0, x1))), y3)
new_map28(xu301, xu302, xu303, xu304, Succ(xu3050), Succ(xu3060), ba) → new_map28(xu301, xu302, xu303, xu304, xu3050, xu3060, ba)
new_map31(xu196, xu240, xu1980000, xu241, Succ(xu2420), bb) → new_map28(xu196, xu240, xu1980000, xu241, Succ(xu2420), Succ(xu196), bb)
new_map28(xu301, xu302, xu303, xu304, Zero, Succ(xu3060), ba) → new_map29(xu301, xu303, xu304, ba)
new_map28(xu301, xu302, xu303, xu304, Zero, Zero, ba) → new_map30(xu301, xu302, xu303, xu304, ba)
new_map30(xu301, xu302, xu303, xu304, ba) → new_map29(xu301, xu303, xu304, ba)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule new_map29(y0, x0, Succ(x1), y3) → new_map31(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, new_primPlusNat0(Succ(x0), Succ(x1)), Succ(Succ(new_primPlusNat0(x0, x1))), y3) at position [3] we obtained the following new rules:
new_map29(y0, x0, Succ(x1), y3) → new_map31(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y3)
↳ 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
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map29(y0, x0, Zero, y3) → new_map31(y0, Succ(x0), x0, Succ(x0), Succ(x0), y3)
new_map31(xu196, xu240, xu1980000, xu241, Succ(xu2420), bb) → new_map28(xu196, xu240, xu1980000, xu241, Succ(xu2420), Succ(xu196), bb)
new_map28(xu301, xu302, xu303, xu304, Succ(xu3050), Succ(xu3060), ba) → new_map28(xu301, xu302, xu303, xu304, xu3050, xu3060, ba)
new_map28(xu301, xu302, xu303, xu304, Zero, Succ(xu3060), ba) → new_map29(xu301, xu303, xu304, ba)
new_map28(xu301, xu302, xu303, xu304, Zero, Zero, ba) → new_map30(xu301, xu302, xu303, xu304, ba)
new_map30(xu301, xu302, xu303, xu304, ba) → new_map29(xu301, xu303, xu304, ba)
new_map29(y0, x0, Succ(x1), y3) → new_map31(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y3)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
new_map29(y0, x0, Zero, y3) → new_map31(y0, Succ(x0), x0, Succ(x0), Succ(x0), y3)
The remaining pairs can at least be oriented weakly.
new_map31(xu196, xu240, xu1980000, xu241, Succ(xu2420), bb) → new_map28(xu196, xu240, xu1980000, xu241, Succ(xu2420), Succ(xu196), bb)
new_map28(xu301, xu302, xu303, xu304, Succ(xu3050), Succ(xu3060), ba) → new_map28(xu301, xu302, xu303, xu304, xu3050, xu3060, ba)
new_map28(xu301, xu302, xu303, xu304, Zero, Succ(xu3060), ba) → new_map29(xu301, xu303, xu304, ba)
new_map28(xu301, xu302, xu303, xu304, Zero, Zero, ba) → new_map30(xu301, xu302, xu303, xu304, ba)
new_map30(xu301, xu302, xu303, xu304, ba) → new_map29(xu301, xu303, xu304, ba)
new_map29(y0, x0, Succ(x1), y3) → new_map31(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y3)
Used ordering: Polynomial interpretation [25]:
POL(Succ(x1)) = 0
POL(Zero) = 1
POL(new_map28(x1, x2, x3, x4, x5, x6, x7)) = x4
POL(new_map29(x1, x2, x3, x4)) = x3
POL(new_map30(x1, x2, x3, x4, x5)) = x4
POL(new_map31(x1, x2, x3, x4, x5, x6)) = x4
POL(new_primPlusNat0(x1, x2)) = 0
The following usable rules [17] were oriented:
none
↳ 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
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Instantiation
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map28(xu301, xu302, xu303, xu304, Succ(xu3050), Succ(xu3060), ba) → new_map28(xu301, xu302, xu303, xu304, xu3050, xu3060, ba)
new_map31(xu196, xu240, xu1980000, xu241, Succ(xu2420), bb) → new_map28(xu196, xu240, xu1980000, xu241, Succ(xu2420), Succ(xu196), bb)
new_map28(xu301, xu302, xu303, xu304, Zero, Succ(xu3060), ba) → new_map29(xu301, xu303, xu304, ba)
new_map28(xu301, xu302, xu303, xu304, Zero, Zero, ba) → new_map30(xu301, xu302, xu303, xu304, ba)
new_map30(xu301, xu302, xu303, xu304, ba) → new_map29(xu301, xu303, xu304, ba)
new_map29(y0, x0, Succ(x1), y3) → new_map31(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y3)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule new_map31(xu196, xu240, xu1980000, xu241, Succ(xu2420), bb) → new_map28(xu196, xu240, xu1980000, xu241, Succ(xu2420), Succ(xu196), bb) we obtained the following new rules:
new_map31(z0, Succ(Succ(y_0)), z1, Succ(Succ(y_1)), Succ(Succ(y_2)), z3) → new_map28(z0, Succ(Succ(y_0)), z1, Succ(Succ(y_1)), Succ(Succ(y_2)), Succ(z0), z3)
↳ 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
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map28(xu301, xu302, xu303, xu304, Succ(xu3050), Succ(xu3060), ba) → new_map28(xu301, xu302, xu303, xu304, xu3050, xu3060, ba)
new_map28(xu301, xu302, xu303, xu304, Zero, Succ(xu3060), ba) → new_map29(xu301, xu303, xu304, ba)
new_map28(xu301, xu302, xu303, xu304, Zero, Zero, ba) → new_map30(xu301, xu302, xu303, xu304, ba)
new_map30(xu301, xu302, xu303, xu304, ba) → new_map29(xu301, xu303, xu304, ba)
new_map29(y0, x0, Succ(x1), y3) → new_map31(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y3)
new_map31(z0, Succ(Succ(y_0)), z1, Succ(Succ(y_1)), Succ(Succ(y_2)), z3) → new_map28(z0, Succ(Succ(y_0)), z1, Succ(Succ(y_1)), Succ(Succ(y_2)), Succ(z0), z3)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule new_map28(xu301, xu302, xu303, xu304, Succ(xu3050), Succ(xu3060), ba) → new_map28(xu301, xu302, xu303, xu304, xu3050, xu3060, ba) we obtained the following new rules:
new_map28(x0, x1, x2, x3, Succ(Zero), Succ(Zero), x6) → new_map28(x0, x1, x2, x3, Zero, Zero, x6)
new_map28(x0, x1, x2, x3, Succ(Zero), Succ(Succ(y_4)), x6) → new_map28(x0, x1, x2, x3, Zero, Succ(y_4), x6)
new_map28(x0, x1, x2, x3, Succ(Succ(y_4)), Succ(Succ(y_5)), x6) → new_map28(x0, x1, x2, x3, Succ(y_4), Succ(y_5), x6)
↳ 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
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map28(x0, x1, x2, x3, Succ(Zero), Succ(Zero), x6) → new_map28(x0, x1, x2, x3, Zero, Zero, x6)
new_map28(xu301, xu302, xu303, xu304, Zero, Succ(xu3060), ba) → new_map29(xu301, xu303, xu304, ba)
new_map28(x0, x1, x2, x3, Succ(Zero), Succ(Succ(y_4)), x6) → new_map28(x0, x1, x2, x3, Zero, Succ(y_4), x6)
new_map28(xu301, xu302, xu303, xu304, Zero, Zero, ba) → new_map30(xu301, xu302, xu303, xu304, ba)
new_map28(x0, x1, x2, x3, Succ(Succ(y_4)), Succ(Succ(y_5)), x6) → new_map28(x0, x1, x2, x3, Succ(y_4), Succ(y_5), x6)
new_map30(xu301, xu302, xu303, xu304, ba) → new_map29(xu301, xu303, xu304, ba)
new_map29(y0, x0, Succ(x1), y3) → new_map31(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y3)
new_map31(z0, Succ(Succ(y_0)), z1, Succ(Succ(y_1)), Succ(Succ(y_2)), z3) → new_map28(z0, Succ(Succ(y_0)), z1, Succ(Succ(y_1)), Succ(Succ(y_2)), Succ(z0), z3)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule new_map28(xu301, xu302, xu303, xu304, Zero, Succ(xu3060), ba) → new_map29(xu301, xu303, xu304, ba) we obtained the following new rules:
new_map28(x0, x1, x2, Succ(y_2), Zero, Succ(x4), x5) → new_map29(x0, x2, Succ(y_2), x5)
↳ 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
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map28(x0, x1, x2, Succ(y_2), Zero, Succ(x4), x5) → new_map29(x0, x2, Succ(y_2), x5)
new_map28(x0, x1, x2, x3, Succ(Zero), Succ(Zero), x6) → new_map28(x0, x1, x2, x3, Zero, Zero, x6)
new_map28(xu301, xu302, xu303, xu304, Zero, Zero, ba) → new_map30(xu301, xu302, xu303, xu304, ba)
new_map28(x0, x1, x2, x3, Succ(Zero), Succ(Succ(y_4)), x6) → new_map28(x0, x1, x2, x3, Zero, Succ(y_4), x6)
new_map30(xu301, xu302, xu303, xu304, ba) → new_map29(xu301, xu303, xu304, ba)
new_map28(x0, x1, x2, x3, Succ(Succ(y_4)), Succ(Succ(y_5)), x6) → new_map28(x0, x1, x2, x3, Succ(y_4), Succ(y_5), x6)
new_map29(y0, x0, Succ(x1), y3) → new_map31(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y3)
new_map31(z0, Succ(Succ(y_0)), z1, Succ(Succ(y_1)), Succ(Succ(y_2)), z3) → new_map28(z0, Succ(Succ(y_0)), z1, Succ(Succ(y_1)), Succ(Succ(y_2)), Succ(z0), z3)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule new_map30(xu301, xu302, xu303, xu304, ba) → new_map29(xu301, xu303, xu304, ba) we obtained the following new rules:
new_map30(x0, x1, x2, Succ(y_2), x4) → new_map29(x0, x2, Succ(y_2), 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
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map30(x0, x1, x2, Succ(y_2), x4) → new_map29(x0, x2, Succ(y_2), x4)
new_map28(x0, x1, x2, x3, Succ(Zero), Succ(Zero), x6) → new_map28(x0, x1, x2, x3, Zero, Zero, x6)
new_map28(x0, x1, x2, Succ(y_2), Zero, Succ(x4), x5) → new_map29(x0, x2, Succ(y_2), x5)
new_map28(x0, x1, x2, x3, Succ(Zero), Succ(Succ(y_4)), x6) → new_map28(x0, x1, x2, x3, Zero, Succ(y_4), x6)
new_map28(xu301, xu302, xu303, xu304, Zero, Zero, ba) → new_map30(xu301, xu302, xu303, xu304, ba)
new_map28(x0, x1, x2, x3, Succ(Succ(y_4)), Succ(Succ(y_5)), x6) → new_map28(x0, x1, x2, x3, Succ(y_4), Succ(y_5), x6)
new_map29(y0, x0, Succ(x1), y3) → new_map31(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y3)
new_map31(z0, Succ(Succ(y_0)), z1, Succ(Succ(y_1)), Succ(Succ(y_2)), z3) → new_map28(z0, Succ(Succ(y_0)), z1, Succ(Succ(y_1)), Succ(Succ(y_2)), Succ(z0), z3)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule new_map28(xu301, xu302, xu303, xu304, Zero, Zero, ba) → new_map30(xu301, xu302, xu303, xu304, ba) we obtained the following new rules:
new_map28(x0, x1, x2, Succ(y_3), Zero, Zero, x4) → new_map30(x0, x1, x2, Succ(y_3), 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
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map30(x0, x1, x2, Succ(y_2), x4) → new_map29(x0, x2, Succ(y_2), x4)
new_map28(x0, x1, x2, Succ(y_2), Zero, Succ(x4), x5) → new_map29(x0, x2, Succ(y_2), x5)
new_map28(x0, x1, x2, x3, Succ(Zero), Succ(Zero), x6) → new_map28(x0, x1, x2, x3, Zero, Zero, x6)
new_map28(x0, x1, x2, Succ(y_3), Zero, Zero, x4) → new_map30(x0, x1, x2, Succ(y_3), x4)
new_map28(x0, x1, x2, x3, Succ(Zero), Succ(Succ(y_4)), x6) → new_map28(x0, x1, x2, x3, Zero, Succ(y_4), x6)
new_map28(x0, x1, x2, x3, Succ(Succ(y_4)), Succ(Succ(y_5)), x6) → new_map28(x0, x1, x2, x3, Succ(y_4), Succ(y_5), x6)
new_map29(y0, x0, Succ(x1), y3) → new_map31(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y3)
new_map31(z0, Succ(Succ(y_0)), z1, Succ(Succ(y_1)), Succ(Succ(y_2)), z3) → new_map28(z0, Succ(Succ(y_0)), z1, Succ(Succ(y_1)), Succ(Succ(y_2)), Succ(z0), z3)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule new_map31(z0, Succ(Succ(y_0)), z1, Succ(Succ(y_1)), Succ(Succ(y_2)), z3) → new_map28(z0, Succ(Succ(y_0)), z1, Succ(Succ(y_1)), Succ(Succ(y_2)), Succ(z0), z3) we obtained the following new rules:
new_map31(Succ(y_5), Succ(Succ(x1)), x2, Succ(Succ(x3)), Succ(Succ(x4)), x5) → new_map28(Succ(y_5), Succ(Succ(x1)), x2, Succ(Succ(x3)), Succ(Succ(x4)), Succ(Succ(y_5)), x5)
↳ 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
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ MNOCProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map30(x0, x1, x2, Succ(y_2), x4) → new_map29(x0, x2, Succ(y_2), x4)
new_map28(x0, x1, x2, x3, Succ(Zero), Succ(Zero), x6) → new_map28(x0, x1, x2, x3, Zero, Zero, x6)
new_map28(x0, x1, x2, Succ(y_2), Zero, Succ(x4), x5) → new_map29(x0, x2, Succ(y_2), x5)
new_map28(x0, x1, x2, x3, Succ(Zero), Succ(Succ(y_4)), x6) → new_map28(x0, x1, x2, x3, Zero, Succ(y_4), x6)
new_map28(x0, x1, x2, Succ(y_3), Zero, Zero, x4) → new_map30(x0, x1, x2, Succ(y_3), x4)
new_map28(x0, x1, x2, x3, Succ(Succ(y_4)), Succ(Succ(y_5)), x6) → new_map28(x0, x1, x2, x3, Succ(y_4), Succ(y_5), x6)
new_map31(Succ(y_5), Succ(Succ(x1)), x2, Succ(Succ(x3)), Succ(Succ(x4)), x5) → new_map28(Succ(y_5), Succ(Succ(x1)), x2, Succ(Succ(x3)), Succ(Succ(x4)), Succ(Succ(y_5)), x5)
new_map29(y0, x0, Succ(x1), y3) → new_map31(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y3)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
The set Q consists of the following terms:
new_primPlusNat0(Zero, Zero)
new_primPlusNat0(Succ(x0), Zero)
new_primPlusNat0(Zero, Succ(x0))
new_primPlusNat0(Succ(x0), Succ(x1))
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
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ MNOCProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map30(x0, x1, x2, Succ(y_2), x4) → new_map29(x0, x2, Succ(y_2), x4)
new_map28(x0, x1, x2, Succ(y_2), Zero, Succ(x4), x5) → new_map29(x0, x2, Succ(y_2), x5)
new_map28(x0, x1, x2, x3, Succ(Zero), Succ(Zero), x6) → new_map28(x0, x1, x2, x3, Zero, Zero, x6)
new_map28(x0, x1, x2, Succ(y_3), Zero, Zero, x4) → new_map30(x0, x1, x2, Succ(y_3), x4)
new_map28(x0, x1, x2, x3, Succ(Zero), Succ(Succ(y_4)), x6) → new_map28(x0, x1, x2, x3, Zero, Succ(y_4), x6)
new_map28(x0, x1, x2, x3, Succ(Succ(y_4)), Succ(Succ(y_5)), x6) → new_map28(x0, x1, x2, x3, Succ(y_4), Succ(y_5), x6)
new_map29(y0, x0, Succ(x1), y3) → new_map31(y0, Succ(Succ(new_primPlusNat0(x0, x1))), x0, Succ(Succ(new_primPlusNat0(x0, x1))), Succ(Succ(new_primPlusNat0(x0, x1))), y3)
new_map31(Succ(y_5), Succ(Succ(x1)), x2, Succ(Succ(x3)), Succ(Succ(x4)), x5) → new_map28(Succ(y_5), Succ(Succ(x1)), x2, Succ(Succ(x3)), Succ(Succ(x4)), Succ(Succ(y_5)), x5)
The TRS R consists of the following rules:
new_primPlusNat0(Zero, Zero) → Zero
new_primPlusNat0(Succ(xu22800), Zero) → Succ(xu22800)
new_primPlusNat0(Zero, Succ(xu2300)) → Succ(xu2300)
new_primPlusNat0(Succ(xu22800), Succ(xu2300)) → Succ(Succ(new_primPlusNat0(xu22800, xu2300)))
Q is empty.
We have to consider all (P,Q,R)-chains.
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map32(xu273, Succ(xu2740), Succ(xu2750), xu276, ba) → new_map32(xu273, xu2740, xu2750, xu276, 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_map32(xu273, Succ(xu2740), Succ(xu2750), xu276, ba) → new_map32(xu273, xu2740, xu2750, xu276, ba)
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
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map33(xu196, xu197, xu198, Succ(xu1990), Succ(xu2000), ba) → new_map33(xu196, xu197, xu198, xu1990, xu2000, 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_map33(xu196, xu197, xu198, Succ(xu1990), Succ(xu2000), ba) → new_map33(xu196, xu197, xu198, xu1990, xu2000, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4, 5 > 5, 6 >= 6
↳ 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
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map34(xu115, xu116, Succ(xu1170), Succ(xu1180), ba) → new_map34(xu115, xu116, xu1170, xu1180, 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_map34(xu115, xu116, Succ(xu1170), Succ(xu1180), ba) → new_map34(xu115, xu116, xu1170, xu1180, ba)
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
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ Narrow
Q DP problem:
The TRS P consists of the following rules:
new_map35(xu29, xu30, Succ(xu310), Succ(xu320), ba) → new_map35(xu29, xu30, xu310, xu320, 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_map35(xu29, xu30, Succ(xu310), Succ(xu320), ba) → new_map35(xu29, xu30, xu310, xu320, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4, 5 >= 5
Haskell To QDPs
↳ HASKELL
↳ IFR
↳ HASKELL
↳ BR
↳ HASKELL
↳ COR
↳ HASKELL
↳ LetRed
↳ HASKELL
↳ NumRed
↳ HASKELL
↳ Narrow
↳ Narrow
↳ QDP
↳ PisEmptyProof
Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.