Termination w.r.t. Q proof of /tmp/tmpo9Fguj/3.trs

Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 QDP

↳5 MRRProof (⇔)

↳6 QDP

↳7 Narrowing (⇔)

↳8 QDP

↳9 DependencyGraphProof (⇔)

↳10 QDP

↳11 Narrowing (⇔)

↳12 QDP

↳13 Narrowing (⇔)

↳14 QDP

↳15 DependencyGraphProof (⇔)

↳16 QDP

↳17 Narrowing (⇔)

↳18 QDP

↳19 DependencyGraphProof (⇔)

↳20 QDP

↳21 QDPOrderProof (⇔)

↳22 QDP

↳23 Narrowing (⇔)

↳24 QDP

↳25 DependencyGraphProof (⇔)

↳26 QDP

↳27 RootLabelingFC2Proof (⇔)

↳28 QDP

↳29 DependencyGraphProof (⇔)

↳30 QDP

↳31 UsableRulesProof (⇔)

↳32 QDP

↳33 MRRProof (⇔)

↳34 QDP

↳35 DependencyGraphProof (⇔)

↳36 QDP

↳37 MRRProof (⇔)

↳38 QDP

↳39 DependencyGraphProof (⇔)

↳40 QDP

↳41 Narrowing (⇔)

↳42 QDP

↳43 DependencyGraphProof (⇔)

↳44 QDP

↳45 Narrowing (⇔)

↳46 QDP

↳47 DependencyGraphProof (⇔)

↳48 QDP

↳49 QDPOrderProof (⇔)

↳50 QDP

↳51 DependencyGraphProof (⇔)

↳52 TRUE

(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

a(0, b(0, x)) → b(0, a(0, x))
a(0, x) → b(0, b(0, x))
a(0, a(1, a(x, y))) → a(1, a(0, a(x, y)))
b(0, a(1, a(x, y))) → b(1, a(0, a(x, y)))
a(0, a(x, y)) → a(1, a(1, a(x, y)))

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A(0, b(0, x)) → B(0, a(0, x))
A(0, b(0, x)) → A(0, x)
A(0, x) → B(0, b(0, x))
A(0, x) → B(0, x)
A(0, a(1, a(x, y))) → A(1, a(0, a(x, y)))
A(0, a(1, a(x, y))) → A(0, a(x, y))
B(0, a(1, a(x, y))) → B(1, a(0, a(x, y)))
B(0, a(1, a(x, y))) → A(0, a(x, y))
A(0, a(x, y)) → A(1, a(1, a(x, y)))
A(0, a(x, y)) → A(1, a(x, y))

The TRS R consists of the following rules:

a(0, b(0, x)) → b(0, a(0, x))
a(0, x) → b(0, b(0, x))
a(0, a(1, a(x, y))) → a(1, a(0, a(x, y)))
b(0, a(1, a(x, y))) → b(1, a(0, a(x, y)))
a(0, a(x, y)) → a(1, a(1, a(x, y)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.

(4) Obligation:

Q DP problem:
The TRS P consists of the following rules:

B(0, a(1, a(x, y))) → A(0, a(x, y))
A(0, b(0, x)) → B(0, a(0, x))
A(0, b(0, x)) → A(0, x)
A(0, x) → B(0, b(0, x))
A(0, x) → B(0, x)
A(0, a(1, a(x, y))) → A(0, a(x, y))

The TRS R consists of the following rules:

a(0, b(0, x)) → b(0, a(0, x))
a(0, x) → b(0, b(0, x))
a(0, a(1, a(x, y))) → a(1, a(0, a(x, y)))
b(0, a(1, a(x, y))) → b(1, a(0, a(x, y)))
a(0, a(x, y)) → a(1, a(1, a(x, y)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(5) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

A(0, b(0, x)) → A(0, x)
A(0, x) → B(0, x)
A(0, a(1, a(x, y))) → A(0, a(x, y))

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 1
POL(1) = 0
POL(A(x₁, x₂)) = 1 + x₁ + x₂
POL(B(x₁, x₂)) = x₁ + x₂
POL(a(x₁, x₂)) = 1 + x₁ + x₂
POL(b(x₁, x₂)) = x₁ + x₂

(6) Obligation:

Q DP problem:
The TRS P consists of the following rules:

B(0, a(1, a(x, y))) → A(0, a(x, y))
A(0, b(0, x)) → B(0, a(0, x))
A(0, x) → B(0, b(0, x))

The TRS R consists of the following rules:

a(0, b(0, x)) → b(0, a(0, x))
a(0, x) → b(0, b(0, x))
a(0, a(1, a(x, y))) → a(1, a(0, a(x, y)))
b(0, a(1, a(x, y))) → b(1, a(0, a(x, y)))
a(0, a(x, y)) → a(1, a(1, a(x, y)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(7) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule A(0, x) → B(0, b(0, x)) at position [1] we obtained the following new rules [LPAR04]:

A(0, a(1, a(x0, x1))) → B(0, b(1, a(0, a(x0, x1))))

(8) Obligation:

Q DP problem:
The TRS P consists of the following rules:

B(0, a(1, a(x, y))) → A(0, a(x, y))
A(0, b(0, x)) → B(0, a(0, x))
A(0, a(1, a(x0, x1))) → B(0, b(1, a(0, a(x0, x1))))

The TRS R consists of the following rules:

a(0, b(0, x)) → b(0, a(0, x))
a(0, x) → b(0, b(0, x))
a(0, a(1, a(x, y))) → a(1, a(0, a(x, y)))
b(0, a(1, a(x, y))) → b(1, a(0, a(x, y)))
a(0, a(x, y)) → a(1, a(1, a(x, y)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(9) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(10) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A(0, b(0, x)) → B(0, a(0, x))
B(0, a(1, a(x, y))) → A(0, a(x, y))

The TRS R consists of the following rules:

a(0, b(0, x)) → b(0, a(0, x))
a(0, x) → b(0, b(0, x))
a(0, a(1, a(x, y))) → a(1, a(0, a(x, y)))
b(0, a(1, a(x, y))) → b(1, a(0, a(x, y)))
a(0, a(x, y)) → a(1, a(1, a(x, y)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(11) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule A(0, b(0, x)) → B(0, a(0, x)) at position [1] we obtained the following new rules [LPAR04]:

A(0, b(0, b(0, x0))) → B(0, b(0, a(0, x0)))
A(0, b(0, x0)) → B(0, b(0, b(0, x0)))
A(0, b(0, a(1, a(x0, x1)))) → B(0, a(1, a(0, a(x0, x1))))
A(0, b(0, a(x0, x1))) → B(0, a(1, a(1, a(x0, x1))))

(12) Obligation:

Q DP problem:
The TRS P consists of the following rules:

B(0, a(1, a(x, y))) → A(0, a(x, y))
A(0, b(0, b(0, x0))) → B(0, b(0, a(0, x0)))
A(0, b(0, x0)) → B(0, b(0, b(0, x0)))
A(0, b(0, a(1, a(x0, x1)))) → B(0, a(1, a(0, a(x0, x1))))
A(0, b(0, a(x0, x1))) → B(0, a(1, a(1, a(x0, x1))))

The TRS R consists of the following rules:

a(0, b(0, x)) → b(0, a(0, x))
a(0, x) → b(0, b(0, x))
a(0, a(1, a(x, y))) → a(1, a(0, a(x, y)))
b(0, a(1, a(x, y))) → b(1, a(0, a(x, y)))
a(0, a(x, y)) → a(1, a(1, a(x, y)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(13) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule B(0, a(1, a(x, y))) → A(0, a(x, y)) at position [1] we obtained the following new rules [LPAR04]:

B(0, a(1, a(0, b(0, x0)))) → A(0, b(0, a(0, x0)))
B(0, a(1, a(0, x0))) → A(0, b(0, b(0, x0)))
B(0, a(1, a(0, a(1, a(x0, x1))))) → A(0, a(1, a(0, a(x0, x1))))
B(0, a(1, a(0, a(x0, x1)))) → A(0, a(1, a(1, a(x0, x1))))

(14) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A(0, b(0, b(0, x0))) → B(0, b(0, a(0, x0)))
A(0, b(0, x0)) → B(0, b(0, b(0, x0)))
A(0, b(0, a(1, a(x0, x1)))) → B(0, a(1, a(0, a(x0, x1))))
A(0, b(0, a(x0, x1))) → B(0, a(1, a(1, a(x0, x1))))
B(0, a(1, a(0, b(0, x0)))) → A(0, b(0, a(0, x0)))
B(0, a(1, a(0, x0))) → A(0, b(0, b(0, x0)))
B(0, a(1, a(0, a(1, a(x0, x1))))) → A(0, a(1, a(0, a(x0, x1))))
B(0, a(1, a(0, a(x0, x1)))) → A(0, a(1, a(1, a(x0, x1))))

The TRS R consists of the following rules:

a(0, b(0, x)) → b(0, a(0, x))
a(0, x) → b(0, b(0, x))
a(0, a(1, a(x, y))) → a(1, a(0, a(x, y)))
b(0, a(1, a(x, y))) → b(1, a(0, a(x, y)))
a(0, a(x, y)) → a(1, a(1, a(x, y)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(15) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.

(16) Obligation:

Q DP problem:
The TRS P consists of the following rules:

B(0, a(1, a(0, b(0, x0)))) → A(0, b(0, a(0, x0)))
A(0, b(0, b(0, x0))) → B(0, b(0, a(0, x0)))
B(0, a(1, a(0, x0))) → A(0, b(0, b(0, x0)))
A(0, b(0, x0)) → B(0, b(0, b(0, x0)))
A(0, b(0, a(1, a(x0, x1)))) → B(0, a(1, a(0, a(x0, x1))))

The TRS R consists of the following rules:

a(0, b(0, x)) → b(0, a(0, x))
a(0, x) → b(0, b(0, x))
a(0, a(1, a(x, y))) → a(1, a(0, a(x, y)))
b(0, a(1, a(x, y))) → b(1, a(0, a(x, y)))
a(0, a(x, y)) → a(1, a(1, a(x, y)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(17) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule A(0, b(0, x0)) → B(0, b(0, b(0, x0))) at position [1] we obtained the following new rules [LPAR04]:

A(0, b(0, a(1, a(x0, x1)))) → B(0, b(0, b(1, a(0, a(x0, x1)))))

(18) Obligation:

Q DP problem:
The TRS P consists of the following rules:

B(0, a(1, a(0, b(0, x0)))) → A(0, b(0, a(0, x0)))
A(0, b(0, b(0, x0))) → B(0, b(0, a(0, x0)))
B(0, a(1, a(0, x0))) → A(0, b(0, b(0, x0)))
A(0, b(0, a(1, a(x0, x1)))) → B(0, a(1, a(0, a(x0, x1))))
A(0, b(0, a(1, a(x0, x1)))) → B(0, b(0, b(1, a(0, a(x0, x1)))))

The TRS R consists of the following rules:

a(0, b(0, x)) → b(0, a(0, x))
a(0, x) → b(0, b(0, x))
a(0, a(1, a(x, y))) → a(1, a(0, a(x, y)))
b(0, a(1, a(x, y))) → b(1, a(0, a(x, y)))
a(0, a(x, y)) → a(1, a(1, a(x, y)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(19) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(20) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A(0, b(0, b(0, x0))) → B(0, b(0, a(0, x0)))
B(0, a(1, a(0, b(0, x0)))) → A(0, b(0, a(0, x0)))
A(0, b(0, a(1, a(x0, x1)))) → B(0, a(1, a(0, a(x0, x1))))
B(0, a(1, a(0, x0))) → A(0, b(0, b(0, x0)))

The TRS R consists of the following rules:

a(0, b(0, x)) → b(0, a(0, x))
a(0, x) → b(0, b(0, x))
a(0, a(1, a(x, y))) → a(1, a(0, a(x, y)))
b(0, a(1, a(x, y))) → b(1, a(0, a(x, y)))
a(0, a(x, y)) → a(1, a(1, a(x, y)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(21) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A(0, b(0, b(0, x0))) → B(0, b(0, a(0, x0)))

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(A(x₁, x₂)) =
/ 1 \

\ 0 /

+
/ 1 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(0) =
/ 1 \

\ 1 /

POL(b(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(B(x₁, x₂)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 1 \

\ 0 0 /

· x₂

POL(a(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 1 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(1) =
/ 0 \

\ 1 /

The following usable rules [FROCOS05] were oriented:

a(0, a(1, a(x, y))) → a(1, a(0, a(x, y)))
b(0, a(1, a(x, y))) → b(1, a(0, a(x, y)))
a(0, a(x, y)) → a(1, a(1, a(x, y)))
a(0, b(0, x)) → b(0, a(0, x))
a(0, x) → b(0, b(0, x))

(22) Obligation:

Q DP problem:
The TRS P consists of the following rules:

B(0, a(1, a(0, b(0, x0)))) → A(0, b(0, a(0, x0)))
A(0, b(0, a(1, a(x0, x1)))) → B(0, a(1, a(0, a(x0, x1))))
B(0, a(1, a(0, x0))) → A(0, b(0, b(0, x0)))

The TRS R consists of the following rules:

a(0, b(0, x)) → b(0, a(0, x))
a(0, x) → b(0, b(0, x))
a(0, a(1, a(x, y))) → a(1, a(0, a(x, y)))
b(0, a(1, a(x, y))) → b(1, a(0, a(x, y)))
a(0, a(x, y)) → a(1, a(1, a(x, y)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(23) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule B(0, a(1, a(0, x0))) → A(0, b(0, b(0, x0))) at position [1] we obtained the following new rules [LPAR04]:

B(0, a(1, a(0, a(1, a(x0, x1))))) → A(0, b(0, b(1, a(0, a(x0, x1)))))

(24) Obligation:

Q DP problem:
The TRS P consists of the following rules:

B(0, a(1, a(0, b(0, x0)))) → A(0, b(0, a(0, x0)))
A(0, b(0, a(1, a(x0, x1)))) → B(0, a(1, a(0, a(x0, x1))))
B(0, a(1, a(0, a(1, a(x0, x1))))) → A(0, b(0, b(1, a(0, a(x0, x1)))))

The TRS R consists of the following rules:

a(0, b(0, x)) → b(0, a(0, x))
a(0, x) → b(0, b(0, x))
a(0, a(1, a(x, y))) → a(1, a(0, a(x, y)))
b(0, a(1, a(x, y))) → b(1, a(0, a(x, y)))
a(0, a(x, y)) → a(1, a(1, a(x, y)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(25) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(26) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A(0, b(0, a(1, a(x0, x1)))) → B(0, a(1, a(0, a(x0, x1))))
B(0, a(1, a(0, b(0, x0)))) → A(0, b(0, a(0, x0)))

The TRS R consists of the following rules:

a(0, b(0, x)) → b(0, a(0, x))
a(0, x) → b(0, b(0, x))
a(0, a(1, a(x, y))) → a(1, a(0, a(x, y)))
b(0, a(1, a(x, y))) → b(1, a(0, a(x, y)))
a(0, a(x, y)) → a(1, a(1, a(x, y)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(27) RootLabelingFC2Proof (EQUIVALENT transformation)

We used root labeling (second transformation) [ROOTLAB] with the following heuristic:
LabelAll: All function symbols get labeled
As Q is empty the root labeling was sound AND complete.

(28) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{a,a}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{a,a}(x0, x1))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{a,0}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{a,0}(x0, x1))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{a,b}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{a,b}(x0, x1))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{a,1}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{a,1}(x0, x1))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{0,a}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{0,a}(x0, x1))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{0,0}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{0,0}(x0, x1))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{0,b}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{0,b}(x0, x1))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{0,1}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{0,1}(x0, x1))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{b,a}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{b,a}(x0, x1))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{b,0}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{b,0}(x0, x1))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{b,b}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{b,b}(x0, x1))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{b,1}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{b,1}(x0, x1))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,a}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{1,a}(x0, x1))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,0}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{1,0}(x0, x1))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,b}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{1,b}(x0, x1))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,1}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{1,1}(x0, x1))))
B_{0,a}(0, a_{1,a}(1, a_{0,b}(0, b_{0,a}(0, x0)))) → A_{0,b}(0, b_{0,a}(0, a_{0,a}(0, x0)))
B_{0,a}(0, a_{1,a}(1, a_{0,b}(0, b_{0,0}(0, x0)))) → A_{0,b}(0, b_{0,a}(0, a_{0,0}(0, x0)))
B_{0,a}(0, a_{1,a}(1, a_{0,b}(0, b_{0,b}(0, x0)))) → A_{0,b}(0, b_{0,a}(0, a_{0,b}(0, x0)))
B_{0,a}(0, a_{1,a}(1, a_{0,b}(0, b_{0,1}(0, x0)))) → A_{0,b}(0, b_{0,a}(0, a_{0,1}(0, x0)))
A_{a,a}(a_{0,b}(0, b_{0,a}(0, _x0)), flat1) → A_{b,a}(b_{0,a}(0, a_{0,a}(0, _x0)), flat1)
A_{a,0}(a_{0,b}(0, b_{0,a}(0, _x0)), flat1) → A_{b,0}(b_{0,a}(0, a_{0,a}(0, _x0)), flat1)
A_{a,b}(a_{0,b}(0, b_{0,a}(0, _x0)), flat1) → A_{b,b}(b_{0,a}(0, a_{0,a}(0, _x0)), flat1)
A_{a,1}(a_{0,b}(0, b_{0,a}(0, _x0)), flat1) → A_{b,1}(b_{0,a}(0, a_{0,a}(0, _x0)), flat1)
A_{a,a}(a_{0,b}(0, b_{0,0}(0, _x0)), flat1) → A_{b,a}(b_{0,a}(0, a_{0,0}(0, _x0)), flat1)
A_{a,0}(a_{0,b}(0, b_{0,0}(0, _x0)), flat1) → A_{b,0}(b_{0,a}(0, a_{0,0}(0, _x0)), flat1)
A_{a,b}(a_{0,b}(0, b_{0,0}(0, _x0)), flat1) → A_{b,b}(b_{0,a}(0, a_{0,0}(0, _x0)), flat1)
A_{a,1}(a_{0,b}(0, b_{0,0}(0, _x0)), flat1) → A_{b,1}(b_{0,a}(0, a_{0,0}(0, _x0)), flat1)
A_{a,a}(a_{0,b}(0, b_{0,b}(0, _x0)), flat1) → A_{b,a}(b_{0,a}(0, a_{0,b}(0, _x0)), flat1)
A_{a,0}(a_{0,b}(0, b_{0,b}(0, _x0)), flat1) → A_{b,0}(b_{0,a}(0, a_{0,b}(0, _x0)), flat1)
A_{a,b}(a_{0,b}(0, b_{0,b}(0, _x0)), flat1) → A_{b,b}(b_{0,a}(0, a_{0,b}(0, _x0)), flat1)
A_{a,1}(a_{0,b}(0, b_{0,b}(0, _x0)), flat1) → A_{b,1}(b_{0,a}(0, a_{0,b}(0, _x0)), flat1)
A_{a,a}(a_{0,b}(0, b_{0,1}(0, _x0)), flat1) → A_{b,a}(b_{0,a}(0, a_{0,1}(0, _x0)), flat1)
A_{a,0}(a_{0,b}(0, b_{0,1}(0, _x0)), flat1) → A_{b,0}(b_{0,a}(0, a_{0,1}(0, _x0)), flat1)
A_{a,b}(a_{0,b}(0, b_{0,1}(0, _x0)), flat1) → A_{b,b}(b_{0,a}(0, a_{0,1}(0, _x0)), flat1)
A_{a,1}(a_{0,b}(0, b_{0,1}(0, _x0)), flat1) → A_{b,1}(b_{0,a}(0, a_{0,1}(0, _x0)), flat1)
A_{a,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → A_{a,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
A_{a,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → A_{a,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
A_{a,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → A_{a,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
A_{a,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → A_{a,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
A_{0,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → A_{0,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
A_{0,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → A_{0,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
A_{0,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → A_{0,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
A_{0,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → A_{0,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
A_{b,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → A_{b,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
A_{b,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → A_{b,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
A_{b,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → A_{b,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
A_{b,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → A_{b,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
A_{1,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → A_{1,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
A_{1,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → A_{1,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
A_{1,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → A_{1,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
A_{1,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → A_{1,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
B_{a,a}(a_{0,b}(0, b_{0,a}(0, _x0)), flat1) → B_{b,a}(b_{0,a}(0, a_{0,a}(0, _x0)), flat1)
B_{a,0}(a_{0,b}(0, b_{0,a}(0, _x0)), flat1) → B_{b,0}(b_{0,a}(0, a_{0,a}(0, _x0)), flat1)
B_{a,b}(a_{0,b}(0, b_{0,a}(0, _x0)), flat1) → B_{b,b}(b_{0,a}(0, a_{0,a}(0, _x0)), flat1)
B_{a,1}(a_{0,b}(0, b_{0,a}(0, _x0)), flat1) → B_{b,1}(b_{0,a}(0, a_{0,a}(0, _x0)), flat1)
B_{a,a}(a_{0,b}(0, b_{0,0}(0, _x0)), flat1) → B_{b,a}(b_{0,a}(0, a_{0,0}(0, _x0)), flat1)
B_{a,0}(a_{0,b}(0, b_{0,0}(0, _x0)), flat1) → B_{b,0}(b_{0,a}(0, a_{0,0}(0, _x0)), flat1)
B_{a,b}(a_{0,b}(0, b_{0,0}(0, _x0)), flat1) → B_{b,b}(b_{0,a}(0, a_{0,0}(0, _x0)), flat1)
B_{a,1}(a_{0,b}(0, b_{0,0}(0, _x0)), flat1) → B_{b,1}(b_{0,a}(0, a_{0,0}(0, _x0)), flat1)
B_{a,a}(a_{0,b}(0, b_{0,b}(0, _x0)), flat1) → B_{b,a}(b_{0,a}(0, a_{0,b}(0, _x0)), flat1)
B_{a,0}(a_{0,b}(0, b_{0,b}(0, _x0)), flat1) → B_{b,0}(b_{0,a}(0, a_{0,b}(0, _x0)), flat1)
B_{a,b}(a_{0,b}(0, b_{0,b}(0, _x0)), flat1) → B_{b,b}(b_{0,a}(0, a_{0,b}(0, _x0)), flat1)
B_{a,1}(a_{0,b}(0, b_{0,b}(0, _x0)), flat1) → B_{b,1}(b_{0,a}(0, a_{0,b}(0, _x0)), flat1)
B_{a,a}(a_{0,b}(0, b_{0,1}(0, _x0)), flat1) → B_{b,a}(b_{0,a}(0, a_{0,1}(0, _x0)), flat1)
B_{a,0}(a_{0,b}(0, b_{0,1}(0, _x0)), flat1) → B_{b,0}(b_{0,a}(0, a_{0,1}(0, _x0)), flat1)
B_{a,b}(a_{0,b}(0, b_{0,1}(0, _x0)), flat1) → B_{b,b}(b_{0,a}(0, a_{0,1}(0, _x0)), flat1)
B_{a,1}(a_{0,b}(0, b_{0,1}(0, _x0)), flat1) → B_{b,1}(b_{0,a}(0, a_{0,1}(0, _x0)), flat1)
B_{a,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → B_{a,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
B_{a,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → B_{a,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
B_{a,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → B_{a,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
B_{a,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → B_{a,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
B_{0,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → B_{0,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
B_{0,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → B_{0,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
B_{0,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → B_{0,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
B_{0,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → B_{0,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
B_{b,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → B_{b,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
B_{b,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → B_{b,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
B_{b,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → B_{b,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
B_{b,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → B_{b,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
B_{1,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → B_{1,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
B_{1,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → B_{1,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
B_{1,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → B_{1,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
B_{1,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → B_{1,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
A_{a,a}(a_{0,a}(0, _x0), flat1) → A_{b,a}(b_{0,b}(0, b_{0,a}(0, _x0)), flat1)
A_{a,0}(a_{0,a}(0, _x0), flat1) → A_{b,0}(b_{0,b}(0, b_{0,a}(0, _x0)), flat1)
A_{a,b}(a_{0,a}(0, _x0), flat1) → A_{b,b}(b_{0,b}(0, b_{0,a}(0, _x0)), flat1)
A_{a,1}(a_{0,a}(0, _x0), flat1) → A_{b,1}(b_{0,b}(0, b_{0,a}(0, _x0)), flat1)
A_{a,a}(a_{0,0}(0, _x0), flat1) → A_{b,a}(b_{0,b}(0, b_{0,0}(0, _x0)), flat1)
A_{a,0}(a_{0,0}(0, _x0), flat1) → A_{b,0}(b_{0,b}(0, b_{0,0}(0, _x0)), flat1)
A_{a,b}(a_{0,0}(0, _x0), flat1) → A_{b,b}(b_{0,b}(0, b_{0,0}(0, _x0)), flat1)
A_{a,1}(a_{0,0}(0, _x0), flat1) → A_{b,1}(b_{0,b}(0, b_{0,0}(0, _x0)), flat1)
A_{a,a}(a_{0,b}(0, _x0), flat1) → A_{b,a}(b_{0,b}(0, b_{0,b}(0, _x0)), flat1)
A_{a,0}(a_{0,b}(0, _x0), flat1) → A_{b,0}(b_{0,b}(0, b_{0,b}(0, _x0)), flat1)
A_{a,b}(a_{0,b}(0, _x0), flat1) → A_{b,b}(b_{0,b}(0, b_{0,b}(0, _x0)), flat1)
A_{a,1}(a_{0,b}(0, _x0), flat1) → A_{b,1}(b_{0,b}(0, b_{0,b}(0, _x0)), flat1)
A_{a,a}(a_{0,1}(0, _x0), flat1) → A_{b,a}(b_{0,b}(0, b_{0,1}(0, _x0)), flat1)
A_{a,0}(a_{0,1}(0, _x0), flat1) → A_{b,0}(b_{0,b}(0, b_{0,1}(0, _x0)), flat1)
A_{a,b}(a_{0,1}(0, _x0), flat1) → A_{b,b}(b_{0,b}(0, b_{0,1}(0, _x0)), flat1)
A_{a,1}(a_{0,1}(0, _x0), flat1) → A_{b,1}(b_{0,b}(0, b_{0,1}(0, _x0)), flat1)
A_{a,a}(flat0, a_{0,a}(0, _x0)) → A_{a,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
A_{a,a}(flat0, a_{0,0}(0, _x0)) → A_{a,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
A_{a,a}(flat0, a_{0,b}(0, _x0)) → A_{a,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
A_{a,a}(flat0, a_{0,1}(0, _x0)) → A_{a,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
A_{0,a}(flat0, a_{0,a}(0, _x0)) → A_{0,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
A_{0,a}(flat0, a_{0,0}(0, _x0)) → A_{0,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
A_{0,a}(flat0, a_{0,b}(0, _x0)) → A_{0,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
A_{0,a}(flat0, a_{0,1}(0, _x0)) → A_{0,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
A_{b,a}(flat0, a_{0,a}(0, _x0)) → A_{b,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
A_{b,a}(flat0, a_{0,0}(0, _x0)) → A_{b,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
A_{b,a}(flat0, a_{0,b}(0, _x0)) → A_{b,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
A_{b,a}(flat0, a_{0,1}(0, _x0)) → A_{b,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
A_{1,a}(flat0, a_{0,a}(0, _x0)) → A_{1,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
A_{1,a}(flat0, a_{0,0}(0, _x0)) → A_{1,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
A_{1,a}(flat0, a_{0,b}(0, _x0)) → A_{1,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
A_{1,a}(flat0, a_{0,1}(0, _x0)) → A_{1,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
B_{a,a}(a_{0,a}(0, _x0), flat1) → B_{b,a}(b_{0,b}(0, b_{0,a}(0, _x0)), flat1)
B_{a,0}(a_{0,a}(0, _x0), flat1) → B_{b,0}(b_{0,b}(0, b_{0,a}(0, _x0)), flat1)
B_{a,b}(a_{0,a}(0, _x0), flat1) → B_{b,b}(b_{0,b}(0, b_{0,a}(0, _x0)), flat1)
B_{a,1}(a_{0,a}(0, _x0), flat1) → B_{b,1}(b_{0,b}(0, b_{0,a}(0, _x0)), flat1)
B_{a,a}(a_{0,0}(0, _x0), flat1) → B_{b,a}(b_{0,b}(0, b_{0,0}(0, _x0)), flat1)
B_{a,0}(a_{0,0}(0, _x0), flat1) → B_{b,0}(b_{0,b}(0, b_{0,0}(0, _x0)), flat1)
B_{a,b}(a_{0,0}(0, _x0), flat1) → B_{b,b}(b_{0,b}(0, b_{0,0}(0, _x0)), flat1)
B_{a,1}(a_{0,0}(0, _x0), flat1) → B_{b,1}(b_{0,b}(0, b_{0,0}(0, _x0)), flat1)
B_{a,a}(a_{0,b}(0, _x0), flat1) → B_{b,a}(b_{0,b}(0, b_{0,b}(0, _x0)), flat1)
B_{a,0}(a_{0,b}(0, _x0), flat1) → B_{b,0}(b_{0,b}(0, b_{0,b}(0, _x0)), flat1)
B_{a,b}(a_{0,b}(0, _x0), flat1) → B_{b,b}(b_{0,b}(0, b_{0,b}(0, _x0)), flat1)
B_{a,1}(a_{0,b}(0, _x0), flat1) → B_{b,1}(b_{0,b}(0, b_{0,b}(0, _x0)), flat1)
B_{a,a}(a_{0,1}(0, _x0), flat1) → B_{b,a}(b_{0,b}(0, b_{0,1}(0, _x0)), flat1)
B_{a,0}(a_{0,1}(0, _x0), flat1) → B_{b,0}(b_{0,b}(0, b_{0,1}(0, _x0)), flat1)
B_{a,b}(a_{0,1}(0, _x0), flat1) → B_{b,b}(b_{0,b}(0, b_{0,1}(0, _x0)), flat1)
B_{a,1}(a_{0,1}(0, _x0), flat1) → B_{b,1}(b_{0,b}(0, b_{0,1}(0, _x0)), flat1)
B_{a,a}(flat0, a_{0,a}(0, _x0)) → B_{a,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
B_{a,a}(flat0, a_{0,0}(0, _x0)) → B_{a,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
B_{a,a}(flat0, a_{0,b}(0, _x0)) → B_{a,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
B_{a,a}(flat0, a_{0,1}(0, _x0)) → B_{a,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
B_{0,a}(flat0, a_{0,a}(0, _x0)) → B_{0,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
B_{0,a}(flat0, a_{0,0}(0, _x0)) → B_{0,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
B_{0,a}(flat0, a_{0,b}(0, _x0)) → B_{0,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
B_{0,a}(flat0, a_{0,1}(0, _x0)) → B_{0,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
B_{b,a}(flat0, a_{0,a}(0, _x0)) → B_{b,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
B_{b,a}(flat0, a_{0,0}(0, _x0)) → B_{b,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
B_{b,a}(flat0, a_{0,b}(0, _x0)) → B_{b,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
B_{b,a}(flat0, a_{0,1}(0, _x0)) → B_{b,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
B_{1,a}(flat0, a_{0,a}(0, _x0)) → B_{1,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
B_{1,a}(flat0, a_{0,0}(0, _x0)) → B_{1,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
B_{1,a}(flat0, a_{0,b}(0, _x0)) → B_{1,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
B_{1,a}(flat0, a_{0,1}(0, _x0)) → B_{1,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))

The TRS R consists of the following rules:

a_{0,a}(0, a_{1,a}(1, a_{a,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{a,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{a,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{a,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,1}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,1}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,1}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,1}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,1}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,1}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,1}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,1}(x, y)))
a_{0,a}(0, a_{a,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,a}(x, y)))
a_{0,a}(0, a_{a,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,0}(x, y)))
a_{0,a}(0, a_{a,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,b}(x, y)))
a_{0,a}(0, a_{a,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,1}(x, y)))
a_{0,a}(0, a_{0,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,a}(x, y)))
a_{0,a}(0, a_{0,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,0}(x, y)))
a_{0,a}(0, a_{0,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,b}(x, y)))
a_{0,a}(0, a_{0,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,1}(x, y)))
a_{0,a}(0, a_{b,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,a}(x, y)))
a_{0,a}(0, a_{b,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,0}(x, y)))
a_{0,a}(0, a_{b,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,b}(x, y)))
a_{0,a}(0, a_{b,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,1}(x, y)))
a_{0,a}(0, a_{1,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,a}(x, y)))
a_{0,a}(0, a_{1,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,0}(x, y)))
a_{0,a}(0, a_{1,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,b}(x, y)))
a_{0,a}(0, a_{1,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,1}(x, y)))
a_{a,a}(a_{0,b}(0, b_{0,a}(0, _x0)), flat1) → a_{b,a}(b_{0,a}(0, a_{0,a}(0, _x0)), flat1)
a_{a,0}(a_{0,b}(0, b_{0,a}(0, _x0)), flat1) → a_{b,0}(b_{0,a}(0, a_{0,a}(0, _x0)), flat1)
a_{a,b}(a_{0,b}(0, b_{0,a}(0, _x0)), flat1) → a_{b,b}(b_{0,a}(0, a_{0,a}(0, _x0)), flat1)
a_{a,1}(a_{0,b}(0, b_{0,a}(0, _x0)), flat1) → a_{b,1}(b_{0,a}(0, a_{0,a}(0, _x0)), flat1)
a_{a,a}(a_{0,b}(0, b_{0,0}(0, _x0)), flat1) → a_{b,a}(b_{0,a}(0, a_{0,0}(0, _x0)), flat1)
a_{a,0}(a_{0,b}(0, b_{0,0}(0, _x0)), flat1) → a_{b,0}(b_{0,a}(0, a_{0,0}(0, _x0)), flat1)
a_{a,b}(a_{0,b}(0, b_{0,0}(0, _x0)), flat1) → a_{b,b}(b_{0,a}(0, a_{0,0}(0, _x0)), flat1)
a_{a,1}(a_{0,b}(0, b_{0,0}(0, _x0)), flat1) → a_{b,1}(b_{0,a}(0, a_{0,0}(0, _x0)), flat1)
a_{a,a}(a_{0,b}(0, b_{0,b}(0, _x0)), flat1) → a_{b,a}(b_{0,a}(0, a_{0,b}(0, _x0)), flat1)
a_{a,0}(a_{0,b}(0, b_{0,b}(0, _x0)), flat1) → a_{b,0}(b_{0,a}(0, a_{0,b}(0, _x0)), flat1)
a_{a,b}(a_{0,b}(0, b_{0,b}(0, _x0)), flat1) → a_{b,b}(b_{0,a}(0, a_{0,b}(0, _x0)), flat1)
a_{a,1}(a_{0,b}(0, b_{0,b}(0, _x0)), flat1) → a_{b,1}(b_{0,a}(0, a_{0,b}(0, _x0)), flat1)
a_{a,a}(a_{0,b}(0, b_{0,1}(0, _x0)), flat1) → a_{b,a}(b_{0,a}(0, a_{0,1}(0, _x0)), flat1)
a_{a,0}(a_{0,b}(0, b_{0,1}(0, _x0)), flat1) → a_{b,0}(b_{0,a}(0, a_{0,1}(0, _x0)), flat1)
a_{a,b}(a_{0,b}(0, b_{0,1}(0, _x0)), flat1) → a_{b,b}(b_{0,a}(0, a_{0,1}(0, _x0)), flat1)
a_{a,1}(a_{0,b}(0, b_{0,1}(0, _x0)), flat1) → a_{b,1}(b_{0,a}(0, a_{0,1}(0, _x0)), flat1)
a_{a,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → a_{a,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
a_{a,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → a_{a,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
a_{a,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → a_{a,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
a_{a,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → a_{a,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
a_{b,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → a_{b,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
a_{b,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → a_{b,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
a_{b,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → a_{b,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
a_{b,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → a_{b,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
a_{1,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → a_{1,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
a_{1,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → a_{1,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
a_{1,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → a_{1,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
a_{1,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → a_{1,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
b_{a,a}(a_{0,b}(0, b_{0,a}(0, _x0)), flat1) → b_{b,a}(b_{0,a}(0, a_{0,a}(0, _x0)), flat1)
b_{a,0}(a_{0,b}(0, b_{0,a}(0, _x0)), flat1) → b_{b,0}(b_{0,a}(0, a_{0,a}(0, _x0)), flat1)
b_{a,b}(a_{0,b}(0, b_{0,a}(0, _x0)), flat1) → b_{b,b}(b_{0,a}(0, a_{0,a}(0, _x0)), flat1)
b_{a,1}(a_{0,b}(0, b_{0,a}(0, _x0)), flat1) → b_{b,1}(b_{0,a}(0, a_{0,a}(0, _x0)), flat1)
b_{a,a}(a_{0,b}(0, b_{0,0}(0, _x0)), flat1) → b_{b,a}(b_{0,a}(0, a_{0,0}(0, _x0)), flat1)
b_{a,0}(a_{0,b}(0, b_{0,0}(0, _x0)), flat1) → b_{b,0}(b_{0,a}(0, a_{0,0}(0, _x0)), flat1)
b_{a,b}(a_{0,b}(0, b_{0,0}(0, _x0)), flat1) → b_{b,b}(b_{0,a}(0, a_{0,0}(0, _x0)), flat1)
b_{a,1}(a_{0,b}(0, b_{0,0}(0, _x0)), flat1) → b_{b,1}(b_{0,a}(0, a_{0,0}(0, _x0)), flat1)
b_{a,a}(a_{0,b}(0, b_{0,b}(0, _x0)), flat1) → b_{b,a}(b_{0,a}(0, a_{0,b}(0, _x0)), flat1)
b_{a,0}(a_{0,b}(0, b_{0,b}(0, _x0)), flat1) → b_{b,0}(b_{0,a}(0, a_{0,b}(0, _x0)), flat1)
b_{a,b}(a_{0,b}(0, b_{0,b}(0, _x0)), flat1) → b_{b,b}(b_{0,a}(0, a_{0,b}(0, _x0)), flat1)
b_{a,1}(a_{0,b}(0, b_{0,b}(0, _x0)), flat1) → b_{b,1}(b_{0,a}(0, a_{0,b}(0, _x0)), flat1)
b_{a,a}(a_{0,b}(0, b_{0,1}(0, _x0)), flat1) → b_{b,a}(b_{0,a}(0, a_{0,1}(0, _x0)), flat1)
b_{a,0}(a_{0,b}(0, b_{0,1}(0, _x0)), flat1) → b_{b,0}(b_{0,a}(0, a_{0,1}(0, _x0)), flat1)
b_{a,b}(a_{0,b}(0, b_{0,1}(0, _x0)), flat1) → b_{b,b}(b_{0,a}(0, a_{0,1}(0, _x0)), flat1)
b_{a,1}(a_{0,b}(0, b_{0,1}(0, _x0)), flat1) → b_{b,1}(b_{0,a}(0, a_{0,1}(0, _x0)), flat1)
b_{a,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → b_{a,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
b_{a,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → b_{a,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
b_{a,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → b_{a,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
b_{a,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → b_{a,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
b_{b,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → b_{b,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
b_{b,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → b_{b,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
b_{b,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → b_{b,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
b_{b,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → b_{b,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
a_{a,a}(a_{0,a}(0, _x0), flat1) → a_{b,a}(b_{0,b}(0, b_{0,a}(0, _x0)), flat1)
a_{a,0}(a_{0,a}(0, _x0), flat1) → a_{b,0}(b_{0,b}(0, b_{0,a}(0, _x0)), flat1)
a_{a,b}(a_{0,a}(0, _x0), flat1) → a_{b,b}(b_{0,b}(0, b_{0,a}(0, _x0)), flat1)
a_{a,1}(a_{0,a}(0, _x0), flat1) → a_{b,1}(b_{0,b}(0, b_{0,a}(0, _x0)), flat1)
a_{a,a}(a_{0,0}(0, _x0), flat1) → a_{b,a}(b_{0,b}(0, b_{0,0}(0, _x0)), flat1)
a_{a,0}(a_{0,0}(0, _x0), flat1) → a_{b,0}(b_{0,b}(0, b_{0,0}(0, _x0)), flat1)
a_{a,b}(a_{0,0}(0, _x0), flat1) → a_{b,b}(b_{0,b}(0, b_{0,0}(0, _x0)), flat1)
a_{a,1}(a_{0,0}(0, _x0), flat1) → a_{b,1}(b_{0,b}(0, b_{0,0}(0, _x0)), flat1)
a_{a,a}(a_{0,b}(0, _x0), flat1) → a_{b,a}(b_{0,b}(0, b_{0,b}(0, _x0)), flat1)
a_{a,0}(a_{0,b}(0, _x0), flat1) → a_{b,0}(b_{0,b}(0, b_{0,b}(0, _x0)), flat1)
a_{a,b}(a_{0,b}(0, _x0), flat1) → a_{b,b}(b_{0,b}(0, b_{0,b}(0, _x0)), flat1)
a_{a,1}(a_{0,b}(0, _x0), flat1) → a_{b,1}(b_{0,b}(0, b_{0,b}(0, _x0)), flat1)
a_{a,a}(a_{0,1}(0, _x0), flat1) → a_{b,a}(b_{0,b}(0, b_{0,1}(0, _x0)), flat1)
a_{a,0}(a_{0,1}(0, _x0), flat1) → a_{b,0}(b_{0,b}(0, b_{0,1}(0, _x0)), flat1)
a_{a,b}(a_{0,1}(0, _x0), flat1) → a_{b,b}(b_{0,b}(0, b_{0,1}(0, _x0)), flat1)
a_{a,1}(a_{0,1}(0, _x0), flat1) → a_{b,1}(b_{0,b}(0, b_{0,1}(0, _x0)), flat1)
a_{a,a}(flat0, a_{0,a}(0, _x0)) → a_{a,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
a_{a,a}(flat0, a_{0,0}(0, _x0)) → a_{a,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
a_{a,a}(flat0, a_{0,b}(0, _x0)) → a_{a,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
a_{a,a}(flat0, a_{0,1}(0, _x0)) → a_{a,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
a_{0,a}(flat0, a_{0,a}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
a_{0,a}(flat0, a_{0,0}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
a_{0,a}(flat0, a_{0,1}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
a_{b,a}(flat0, a_{0,a}(0, _x0)) → a_{b,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
a_{b,a}(flat0, a_{0,0}(0, _x0)) → a_{b,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
a_{b,a}(flat0, a_{0,b}(0, _x0)) → a_{b,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
a_{b,a}(flat0, a_{0,1}(0, _x0)) → a_{b,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
a_{1,a}(flat0, a_{0,a}(0, _x0)) → a_{1,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
a_{1,a}(flat0, a_{0,0}(0, _x0)) → a_{1,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
a_{1,a}(flat0, a_{0,b}(0, _x0)) → a_{1,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
a_{1,a}(flat0, a_{0,1}(0, _x0)) → a_{1,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
b_{a,a}(a_{0,a}(0, _x0), flat1) → b_{b,a}(b_{0,b}(0, b_{0,a}(0, _x0)), flat1)
b_{a,0}(a_{0,a}(0, _x0), flat1) → b_{b,0}(b_{0,b}(0, b_{0,a}(0, _x0)), flat1)
b_{a,b}(a_{0,a}(0, _x0), flat1) → b_{b,b}(b_{0,b}(0, b_{0,a}(0, _x0)), flat1)
b_{a,1}(a_{0,a}(0, _x0), flat1) → b_{b,1}(b_{0,b}(0, b_{0,a}(0, _x0)), flat1)
b_{a,a}(a_{0,0}(0, _x0), flat1) → b_{b,a}(b_{0,b}(0, b_{0,0}(0, _x0)), flat1)
b_{a,0}(a_{0,0}(0, _x0), flat1) → b_{b,0}(b_{0,b}(0, b_{0,0}(0, _x0)), flat1)
b_{a,b}(a_{0,0}(0, _x0), flat1) → b_{b,b}(b_{0,b}(0, b_{0,0}(0, _x0)), flat1)
b_{a,1}(a_{0,0}(0, _x0), flat1) → b_{b,1}(b_{0,b}(0, b_{0,0}(0, _x0)), flat1)
b_{a,a}(a_{0,b}(0, _x0), flat1) → b_{b,a}(b_{0,b}(0, b_{0,b}(0, _x0)), flat1)
b_{a,0}(a_{0,b}(0, _x0), flat1) → b_{b,0}(b_{0,b}(0, b_{0,b}(0, _x0)), flat1)
b_{a,b}(a_{0,b}(0, _x0), flat1) → b_{b,b}(b_{0,b}(0, b_{0,b}(0, _x0)), flat1)
b_{a,1}(a_{0,b}(0, _x0), flat1) → b_{b,1}(b_{0,b}(0, b_{0,b}(0, _x0)), flat1)
b_{a,a}(a_{0,1}(0, _x0), flat1) → b_{b,a}(b_{0,b}(0, b_{0,1}(0, _x0)), flat1)
b_{a,0}(a_{0,1}(0, _x0), flat1) → b_{b,0}(b_{0,b}(0, b_{0,1}(0, _x0)), flat1)
b_{a,b}(a_{0,1}(0, _x0), flat1) → b_{b,b}(b_{0,b}(0, b_{0,1}(0, _x0)), flat1)
b_{a,1}(a_{0,1}(0, _x0), flat1) → b_{b,1}(b_{0,b}(0, b_{0,1}(0, _x0)), flat1)
b_{a,a}(flat0, a_{0,a}(0, _x0)) → b_{a,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
b_{a,a}(flat0, a_{0,0}(0, _x0)) → b_{a,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
b_{a,a}(flat0, a_{0,b}(0, _x0)) → b_{a,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
b_{a,a}(flat0, a_{0,1}(0, _x0)) → b_{a,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
b_{0,a}(flat0, a_{0,a}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
b_{0,a}(flat0, a_{0,0}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
b_{0,a}(flat0, a_{0,1}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
b_{b,a}(flat0, a_{0,a}(0, _x0)) → b_{b,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
b_{b,a}(flat0, a_{0,0}(0, _x0)) → b_{b,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
b_{b,a}(flat0, a_{0,b}(0, _x0)) → b_{b,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
b_{b,a}(flat0, a_{0,1}(0, _x0)) → b_{b,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
b_{1,a}(flat0, a_{0,a}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
b_{1,a}(flat0, a_{0,0}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
b_{1,a}(flat0, a_{0,1}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(29) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 137 less nodes.

(30) Obligation:

Q DP problem:
The TRS P consists of the following rules:

B_{0,a}(0, a_{1,a}(1, a_{0,b}(0, b_{0,a}(0, x0)))) → A_{0,b}(0, b_{0,a}(0, a_{0,a}(0, x0)))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{a,b}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{a,b}(x0, x1))))
B_{0,a}(0, a_{1,a}(1, a_{0,b}(0, b_{0,b}(0, x0)))) → A_{0,b}(0, b_{0,a}(0, a_{0,b}(0, x0)))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{0,a}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{0,a}(x0, x1))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{0,b}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{0,b}(x0, x1))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{b,a}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{b,a}(x0, x1))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{b,0}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{b,0}(x0, x1))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{b,b}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{b,b}(x0, x1))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{b,1}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{b,1}(x0, x1))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,a}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{1,a}(x0, x1))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,b}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{1,b}(x0, x1))))

The TRS R consists of the following rules:

a_{0,a}(0, a_{1,a}(1, a_{a,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{a,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{a,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{a,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,1}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,1}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,1}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,1}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,1}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,1}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,1}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,1}(x, y)))
a_{0,a}(0, a_{a,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,a}(x, y)))
a_{0,a}(0, a_{a,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,0}(x, y)))
a_{0,a}(0, a_{a,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,b}(x, y)))
a_{0,a}(0, a_{a,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,1}(x, y)))
a_{0,a}(0, a_{0,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,a}(x, y)))
a_{0,a}(0, a_{0,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,0}(x, y)))
a_{0,a}(0, a_{0,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,b}(x, y)))
a_{0,a}(0, a_{0,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,1}(x, y)))
a_{0,a}(0, a_{b,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,a}(x, y)))
a_{0,a}(0, a_{b,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,0}(x, y)))
a_{0,a}(0, a_{b,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,b}(x, y)))
a_{0,a}(0, a_{b,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,1}(x, y)))
a_{0,a}(0, a_{1,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,a}(x, y)))
a_{0,a}(0, a_{1,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,0}(x, y)))
a_{0,a}(0, a_{1,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,b}(x, y)))
a_{0,a}(0, a_{1,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,1}(x, y)))
a_{a,a}(a_{0,b}(0, b_{0,a}(0, _x0)), flat1) → a_{b,a}(b_{0,a}(0, a_{0,a}(0, _x0)), flat1)
a_{a,0}(a_{0,b}(0, b_{0,a}(0, _x0)), flat1) → a_{b,0}(b_{0,a}(0, a_{0,a}(0, _x0)), flat1)
a_{a,b}(a_{0,b}(0, b_{0,a}(0, _x0)), flat1) → a_{b,b}(b_{0,a}(0, a_{0,a}(0, _x0)), flat1)
a_{a,1}(a_{0,b}(0, b_{0,a}(0, _x0)), flat1) → a_{b,1}(b_{0,a}(0, a_{0,a}(0, _x0)), flat1)
a_{a,a}(a_{0,b}(0, b_{0,0}(0, _x0)), flat1) → a_{b,a}(b_{0,a}(0, a_{0,0}(0, _x0)), flat1)
a_{a,0}(a_{0,b}(0, b_{0,0}(0, _x0)), flat1) → a_{b,0}(b_{0,a}(0, a_{0,0}(0, _x0)), flat1)
a_{a,b}(a_{0,b}(0, b_{0,0}(0, _x0)), flat1) → a_{b,b}(b_{0,a}(0, a_{0,0}(0, _x0)), flat1)
a_{a,1}(a_{0,b}(0, b_{0,0}(0, _x0)), flat1) → a_{b,1}(b_{0,a}(0, a_{0,0}(0, _x0)), flat1)
a_{a,a}(a_{0,b}(0, b_{0,b}(0, _x0)), flat1) → a_{b,a}(b_{0,a}(0, a_{0,b}(0, _x0)), flat1)
a_{a,0}(a_{0,b}(0, b_{0,b}(0, _x0)), flat1) → a_{b,0}(b_{0,a}(0, a_{0,b}(0, _x0)), flat1)
a_{a,b}(a_{0,b}(0, b_{0,b}(0, _x0)), flat1) → a_{b,b}(b_{0,a}(0, a_{0,b}(0, _x0)), flat1)
a_{a,1}(a_{0,b}(0, b_{0,b}(0, _x0)), flat1) → a_{b,1}(b_{0,a}(0, a_{0,b}(0, _x0)), flat1)
a_{a,a}(a_{0,b}(0, b_{0,1}(0, _x0)), flat1) → a_{b,a}(b_{0,a}(0, a_{0,1}(0, _x0)), flat1)
a_{a,0}(a_{0,b}(0, b_{0,1}(0, _x0)), flat1) → a_{b,0}(b_{0,a}(0, a_{0,1}(0, _x0)), flat1)
a_{a,b}(a_{0,b}(0, b_{0,1}(0, _x0)), flat1) → a_{b,b}(b_{0,a}(0, a_{0,1}(0, _x0)), flat1)
a_{a,1}(a_{0,b}(0, b_{0,1}(0, _x0)), flat1) → a_{b,1}(b_{0,a}(0, a_{0,1}(0, _x0)), flat1)
a_{a,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → a_{a,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
a_{a,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → a_{a,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
a_{a,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → a_{a,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
a_{a,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → a_{a,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
a_{b,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → a_{b,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
a_{b,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → a_{b,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
a_{b,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → a_{b,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
a_{b,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → a_{b,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
a_{1,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → a_{1,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
a_{1,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → a_{1,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
a_{1,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → a_{1,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
a_{1,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → a_{1,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
b_{a,a}(a_{0,b}(0, b_{0,a}(0, _x0)), flat1) → b_{b,a}(b_{0,a}(0, a_{0,a}(0, _x0)), flat1)
b_{a,0}(a_{0,b}(0, b_{0,a}(0, _x0)), flat1) → b_{b,0}(b_{0,a}(0, a_{0,a}(0, _x0)), flat1)
b_{a,b}(a_{0,b}(0, b_{0,a}(0, _x0)), flat1) → b_{b,b}(b_{0,a}(0, a_{0,a}(0, _x0)), flat1)
b_{a,1}(a_{0,b}(0, b_{0,a}(0, _x0)), flat1) → b_{b,1}(b_{0,a}(0, a_{0,a}(0, _x0)), flat1)
b_{a,a}(a_{0,b}(0, b_{0,0}(0, _x0)), flat1) → b_{b,a}(b_{0,a}(0, a_{0,0}(0, _x0)), flat1)
b_{a,0}(a_{0,b}(0, b_{0,0}(0, _x0)), flat1) → b_{b,0}(b_{0,a}(0, a_{0,0}(0, _x0)), flat1)
b_{a,b}(a_{0,b}(0, b_{0,0}(0, _x0)), flat1) → b_{b,b}(b_{0,a}(0, a_{0,0}(0, _x0)), flat1)
b_{a,1}(a_{0,b}(0, b_{0,0}(0, _x0)), flat1) → b_{b,1}(b_{0,a}(0, a_{0,0}(0, _x0)), flat1)
b_{a,a}(a_{0,b}(0, b_{0,b}(0, _x0)), flat1) → b_{b,a}(b_{0,a}(0, a_{0,b}(0, _x0)), flat1)
b_{a,0}(a_{0,b}(0, b_{0,b}(0, _x0)), flat1) → b_{b,0}(b_{0,a}(0, a_{0,b}(0, _x0)), flat1)
b_{a,b}(a_{0,b}(0, b_{0,b}(0, _x0)), flat1) → b_{b,b}(b_{0,a}(0, a_{0,b}(0, _x0)), flat1)
b_{a,1}(a_{0,b}(0, b_{0,b}(0, _x0)), flat1) → b_{b,1}(b_{0,a}(0, a_{0,b}(0, _x0)), flat1)
b_{a,a}(a_{0,b}(0, b_{0,1}(0, _x0)), flat1) → b_{b,a}(b_{0,a}(0, a_{0,1}(0, _x0)), flat1)
b_{a,0}(a_{0,b}(0, b_{0,1}(0, _x0)), flat1) → b_{b,0}(b_{0,a}(0, a_{0,1}(0, _x0)), flat1)
b_{a,b}(a_{0,b}(0, b_{0,1}(0, _x0)), flat1) → b_{b,b}(b_{0,a}(0, a_{0,1}(0, _x0)), flat1)
b_{a,1}(a_{0,b}(0, b_{0,1}(0, _x0)), flat1) → b_{b,1}(b_{0,a}(0, a_{0,1}(0, _x0)), flat1)
b_{a,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → b_{a,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
b_{a,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → b_{a,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
b_{a,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → b_{a,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
b_{a,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → b_{a,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
b_{b,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → b_{b,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
b_{b,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → b_{b,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
b_{b,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → b_{b,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
b_{b,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → b_{b,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
a_{a,a}(a_{0,a}(0, _x0), flat1) → a_{b,a}(b_{0,b}(0, b_{0,a}(0, _x0)), flat1)
a_{a,0}(a_{0,a}(0, _x0), flat1) → a_{b,0}(b_{0,b}(0, b_{0,a}(0, _x0)), flat1)
a_{a,b}(a_{0,a}(0, _x0), flat1) → a_{b,b}(b_{0,b}(0, b_{0,a}(0, _x0)), flat1)
a_{a,1}(a_{0,a}(0, _x0), flat1) → a_{b,1}(b_{0,b}(0, b_{0,a}(0, _x0)), flat1)
a_{a,a}(a_{0,0}(0, _x0), flat1) → a_{b,a}(b_{0,b}(0, b_{0,0}(0, _x0)), flat1)
a_{a,0}(a_{0,0}(0, _x0), flat1) → a_{b,0}(b_{0,b}(0, b_{0,0}(0, _x0)), flat1)
a_{a,b}(a_{0,0}(0, _x0), flat1) → a_{b,b}(b_{0,b}(0, b_{0,0}(0, _x0)), flat1)
a_{a,1}(a_{0,0}(0, _x0), flat1) → a_{b,1}(b_{0,b}(0, b_{0,0}(0, _x0)), flat1)
a_{a,a}(a_{0,b}(0, _x0), flat1) → a_{b,a}(b_{0,b}(0, b_{0,b}(0, _x0)), flat1)
a_{a,0}(a_{0,b}(0, _x0), flat1) → a_{b,0}(b_{0,b}(0, b_{0,b}(0, _x0)), flat1)
a_{a,b}(a_{0,b}(0, _x0), flat1) → a_{b,b}(b_{0,b}(0, b_{0,b}(0, _x0)), flat1)
a_{a,1}(a_{0,b}(0, _x0), flat1) → a_{b,1}(b_{0,b}(0, b_{0,b}(0, _x0)), flat1)
a_{a,a}(a_{0,1}(0, _x0), flat1) → a_{b,a}(b_{0,b}(0, b_{0,1}(0, _x0)), flat1)
a_{a,0}(a_{0,1}(0, _x0), flat1) → a_{b,0}(b_{0,b}(0, b_{0,1}(0, _x0)), flat1)
a_{a,b}(a_{0,1}(0, _x0), flat1) → a_{b,b}(b_{0,b}(0, b_{0,1}(0, _x0)), flat1)
a_{a,1}(a_{0,1}(0, _x0), flat1) → a_{b,1}(b_{0,b}(0, b_{0,1}(0, _x0)), flat1)
a_{a,a}(flat0, a_{0,a}(0, _x0)) → a_{a,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
a_{a,a}(flat0, a_{0,0}(0, _x0)) → a_{a,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
a_{a,a}(flat0, a_{0,b}(0, _x0)) → a_{a,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
a_{a,a}(flat0, a_{0,1}(0, _x0)) → a_{a,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
a_{0,a}(flat0, a_{0,a}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
a_{0,a}(flat0, a_{0,0}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
a_{0,a}(flat0, a_{0,1}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
a_{b,a}(flat0, a_{0,a}(0, _x0)) → a_{b,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
a_{b,a}(flat0, a_{0,0}(0, _x0)) → a_{b,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
a_{b,a}(flat0, a_{0,b}(0, _x0)) → a_{b,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
a_{b,a}(flat0, a_{0,1}(0, _x0)) → a_{b,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
a_{1,a}(flat0, a_{0,a}(0, _x0)) → a_{1,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
a_{1,a}(flat0, a_{0,0}(0, _x0)) → a_{1,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
a_{1,a}(flat0, a_{0,b}(0, _x0)) → a_{1,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
a_{1,a}(flat0, a_{0,1}(0, _x0)) → a_{1,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
b_{a,a}(a_{0,a}(0, _x0), flat1) → b_{b,a}(b_{0,b}(0, b_{0,a}(0, _x0)), flat1)
b_{a,0}(a_{0,a}(0, _x0), flat1) → b_{b,0}(b_{0,b}(0, b_{0,a}(0, _x0)), flat1)
b_{a,b}(a_{0,a}(0, _x0), flat1) → b_{b,b}(b_{0,b}(0, b_{0,a}(0, _x0)), flat1)
b_{a,1}(a_{0,a}(0, _x0), flat1) → b_{b,1}(b_{0,b}(0, b_{0,a}(0, _x0)), flat1)
b_{a,a}(a_{0,0}(0, _x0), flat1) → b_{b,a}(b_{0,b}(0, b_{0,0}(0, _x0)), flat1)
b_{a,0}(a_{0,0}(0, _x0), flat1) → b_{b,0}(b_{0,b}(0, b_{0,0}(0, _x0)), flat1)
b_{a,b}(a_{0,0}(0, _x0), flat1) → b_{b,b}(b_{0,b}(0, b_{0,0}(0, _x0)), flat1)
b_{a,1}(a_{0,0}(0, _x0), flat1) → b_{b,1}(b_{0,b}(0, b_{0,0}(0, _x0)), flat1)
b_{a,a}(a_{0,b}(0, _x0), flat1) → b_{b,a}(b_{0,b}(0, b_{0,b}(0, _x0)), flat1)
b_{a,0}(a_{0,b}(0, _x0), flat1) → b_{b,0}(b_{0,b}(0, b_{0,b}(0, _x0)), flat1)
b_{a,b}(a_{0,b}(0, _x0), flat1) → b_{b,b}(b_{0,b}(0, b_{0,b}(0, _x0)), flat1)
b_{a,1}(a_{0,b}(0, _x0), flat1) → b_{b,1}(b_{0,b}(0, b_{0,b}(0, _x0)), flat1)
b_{a,a}(a_{0,1}(0, _x0), flat1) → b_{b,a}(b_{0,b}(0, b_{0,1}(0, _x0)), flat1)
b_{a,0}(a_{0,1}(0, _x0), flat1) → b_{b,0}(b_{0,b}(0, b_{0,1}(0, _x0)), flat1)
b_{a,b}(a_{0,1}(0, _x0), flat1) → b_{b,b}(b_{0,b}(0, b_{0,1}(0, _x0)), flat1)
b_{a,1}(a_{0,1}(0, _x0), flat1) → b_{b,1}(b_{0,b}(0, b_{0,1}(0, _x0)), flat1)
b_{a,a}(flat0, a_{0,a}(0, _x0)) → b_{a,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
b_{a,a}(flat0, a_{0,0}(0, _x0)) → b_{a,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
b_{a,a}(flat0, a_{0,b}(0, _x0)) → b_{a,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
b_{a,a}(flat0, a_{0,1}(0, _x0)) → b_{a,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
b_{0,a}(flat0, a_{0,a}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
b_{0,a}(flat0, a_{0,0}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
b_{0,a}(flat0, a_{0,1}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
b_{b,a}(flat0, a_{0,a}(0, _x0)) → b_{b,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
b_{b,a}(flat0, a_{0,0}(0, _x0)) → b_{b,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
b_{b,a}(flat0, a_{0,b}(0, _x0)) → b_{b,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
b_{b,a}(flat0, a_{0,1}(0, _x0)) → b_{b,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
b_{1,a}(flat0, a_{0,a}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
b_{1,a}(flat0, a_{0,0}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
b_{1,a}(flat0, a_{0,1}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(31) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(32) Obligation:

Q DP problem:
The TRS P consists of the following rules:

B_{0,a}(0, a_{1,a}(1, a_{0,b}(0, b_{0,a}(0, x0)))) → A_{0,b}(0, b_{0,a}(0, a_{0,a}(0, x0)))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{a,b}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{a,b}(x0, x1))))
B_{0,a}(0, a_{1,a}(1, a_{0,b}(0, b_{0,b}(0, x0)))) → A_{0,b}(0, b_{0,a}(0, a_{0,b}(0, x0)))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{0,a}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{0,a}(x0, x1))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{0,b}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{0,b}(x0, x1))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{b,a}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{b,a}(x0, x1))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{b,0}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{b,0}(x0, x1))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{b,b}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{b,b}(x0, x1))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{b,1}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{b,1}(x0, x1))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,a}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{1,a}(x0, x1))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,b}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{1,b}(x0, x1))))

The TRS R consists of the following rules:

a_{0,a}(0, a_{1,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,b}(x, y)))
a_{1,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → a_{1,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
a_{1,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → a_{1,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
a_{1,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → a_{1,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
a_{1,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → a_{1,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
a_{1,a}(flat0, a_{0,a}(0, _x0)) → a_{1,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
a_{1,a}(flat0, a_{0,0}(0, _x0)) → a_{1,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
a_{1,a}(flat0, a_{0,b}(0, _x0)) → a_{1,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
a_{1,a}(flat0, a_{0,1}(0, _x0)) → a_{1,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
b_{0,a}(0, a_{1,a}(1, a_{a,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,1}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,1}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,1}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,1}(x, y)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
b_{0,a}(flat0, a_{0,a}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
b_{0,a}(flat0, a_{0,0}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
b_{0,a}(flat0, a_{0,1}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
a_{0,a}(0, a_{1,a}(1, a_{a,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{a,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{a,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{a,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,1}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,1}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,1}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,1}(x, y)))
a_{0,a}(0, a_{a,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,a}(x, y)))
a_{0,a}(0, a_{a,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,0}(x, y)))
a_{0,a}(0, a_{a,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,b}(x, y)))
a_{0,a}(0, a_{a,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,1}(x, y)))
a_{0,a}(0, a_{0,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,a}(x, y)))
a_{0,a}(0, a_{0,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,0}(x, y)))
a_{0,a}(0, a_{0,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,b}(x, y)))
a_{0,a}(0, a_{0,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,1}(x, y)))
a_{0,a}(0, a_{b,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,a}(x, y)))
a_{0,a}(0, a_{b,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,0}(x, y)))
a_{0,a}(0, a_{b,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,b}(x, y)))
a_{0,a}(0, a_{b,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,1}(x, y)))
a_{0,a}(0, a_{1,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,a}(x, y)))
a_{0,a}(0, a_{1,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,0}(x, y)))
a_{0,a}(0, a_{1,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,1}(x, y)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
a_{0,a}(flat0, a_{0,a}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
a_{0,a}(flat0, a_{0,0}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
a_{0,a}(flat0, a_{0,1}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
a_{b,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → a_{b,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
a_{b,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → a_{b,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
a_{b,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → a_{b,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
a_{b,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → a_{b,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
a_{b,a}(flat0, a_{0,a}(0, _x0)) → a_{b,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
a_{b,a}(flat0, a_{0,0}(0, _x0)) → a_{b,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
a_{b,a}(flat0, a_{0,b}(0, _x0)) → a_{b,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
a_{b,a}(flat0, a_{0,1}(0, _x0)) → a_{b,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
a_{a,1}(a_{0,b}(0, b_{0,a}(0, _x0)), flat1) → a_{b,1}(b_{0,a}(0, a_{0,a}(0, _x0)), flat1)
a_{a,1}(a_{0,b}(0, b_{0,0}(0, _x0)), flat1) → a_{b,1}(b_{0,a}(0, a_{0,0}(0, _x0)), flat1)
a_{a,1}(a_{0,b}(0, b_{0,b}(0, _x0)), flat1) → a_{b,1}(b_{0,a}(0, a_{0,b}(0, _x0)), flat1)
a_{a,1}(a_{0,b}(0, b_{0,1}(0, _x0)), flat1) → a_{b,1}(b_{0,a}(0, a_{0,1}(0, _x0)), flat1)
a_{a,1}(a_{0,a}(0, _x0), flat1) → a_{b,1}(b_{0,b}(0, b_{0,a}(0, _x0)), flat1)
a_{a,1}(a_{0,0}(0, _x0), flat1) → a_{b,1}(b_{0,b}(0, b_{0,0}(0, _x0)), flat1)
a_{a,1}(a_{0,b}(0, _x0), flat1) → a_{b,1}(b_{0,b}(0, b_{0,b}(0, _x0)), flat1)
a_{a,1}(a_{0,1}(0, _x0), flat1) → a_{b,1}(b_{0,b}(0, b_{0,1}(0, _x0)), flat1)
a_{a,b}(a_{0,b}(0, b_{0,a}(0, _x0)), flat1) → a_{b,b}(b_{0,a}(0, a_{0,a}(0, _x0)), flat1)
a_{a,b}(a_{0,b}(0, b_{0,0}(0, _x0)), flat1) → a_{b,b}(b_{0,a}(0, a_{0,0}(0, _x0)), flat1)
a_{a,b}(a_{0,b}(0, b_{0,b}(0, _x0)), flat1) → a_{b,b}(b_{0,a}(0, a_{0,b}(0, _x0)), flat1)
a_{a,b}(a_{0,b}(0, b_{0,1}(0, _x0)), flat1) → a_{b,b}(b_{0,a}(0, a_{0,1}(0, _x0)), flat1)
a_{a,b}(a_{0,a}(0, _x0), flat1) → a_{b,b}(b_{0,b}(0, b_{0,a}(0, _x0)), flat1)
a_{a,b}(a_{0,0}(0, _x0), flat1) → a_{b,b}(b_{0,b}(0, b_{0,0}(0, _x0)), flat1)
a_{a,b}(a_{0,b}(0, _x0), flat1) → a_{b,b}(b_{0,b}(0, b_{0,b}(0, _x0)), flat1)
a_{a,b}(a_{0,1}(0, _x0), flat1) → a_{b,b}(b_{0,b}(0, b_{0,1}(0, _x0)), flat1)
a_{a,0}(a_{0,b}(0, b_{0,a}(0, _x0)), flat1) → a_{b,0}(b_{0,a}(0, a_{0,a}(0, _x0)), flat1)
a_{a,0}(a_{0,b}(0, b_{0,0}(0, _x0)), flat1) → a_{b,0}(b_{0,a}(0, a_{0,0}(0, _x0)), flat1)
a_{a,0}(a_{0,b}(0, b_{0,b}(0, _x0)), flat1) → a_{b,0}(b_{0,a}(0, a_{0,b}(0, _x0)), flat1)
a_{a,0}(a_{0,b}(0, b_{0,1}(0, _x0)), flat1) → a_{b,0}(b_{0,a}(0, a_{0,1}(0, _x0)), flat1)
a_{a,0}(a_{0,a}(0, _x0), flat1) → a_{b,0}(b_{0,b}(0, b_{0,a}(0, _x0)), flat1)
a_{a,0}(a_{0,0}(0, _x0), flat1) → a_{b,0}(b_{0,b}(0, b_{0,0}(0, _x0)), flat1)
a_{a,0}(a_{0,b}(0, _x0), flat1) → a_{b,0}(b_{0,b}(0, b_{0,b}(0, _x0)), flat1)
a_{a,0}(a_{0,1}(0, _x0), flat1) → a_{b,0}(b_{0,b}(0, b_{0,1}(0, _x0)), flat1)
a_{a,a}(a_{0,b}(0, b_{0,a}(0, _x0)), flat1) → a_{b,a}(b_{0,a}(0, a_{0,a}(0, _x0)), flat1)
a_{a,a}(a_{0,b}(0, b_{0,0}(0, _x0)), flat1) → a_{b,a}(b_{0,a}(0, a_{0,0}(0, _x0)), flat1)
a_{a,a}(a_{0,b}(0, b_{0,b}(0, _x0)), flat1) → a_{b,a}(b_{0,a}(0, a_{0,b}(0, _x0)), flat1)
a_{a,a}(a_{0,b}(0, b_{0,1}(0, _x0)), flat1) → a_{b,a}(b_{0,a}(0, a_{0,1}(0, _x0)), flat1)
a_{a,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → a_{a,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
a_{a,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → a_{a,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
a_{a,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → a_{a,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
a_{a,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → a_{a,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
a_{a,a}(a_{0,a}(0, _x0), flat1) → a_{b,a}(b_{0,b}(0, b_{0,a}(0, _x0)), flat1)
a_{a,a}(a_{0,0}(0, _x0), flat1) → a_{b,a}(b_{0,b}(0, b_{0,0}(0, _x0)), flat1)
a_{a,a}(a_{0,b}(0, _x0), flat1) → a_{b,a}(b_{0,b}(0, b_{0,b}(0, _x0)), flat1)
a_{a,a}(a_{0,1}(0, _x0), flat1) → a_{b,a}(b_{0,b}(0, b_{0,1}(0, _x0)), flat1)
a_{a,a}(flat0, a_{0,a}(0, _x0)) → a_{a,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
a_{a,a}(flat0, a_{0,0}(0, _x0)) → a_{a,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
a_{a,a}(flat0, a_{0,b}(0, _x0)) → a_{a,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
a_{a,a}(flat0, a_{0,1}(0, _x0)) → a_{a,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
b_{1,a}(flat0, a_{0,a}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
b_{1,a}(flat0, a_{0,0}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
b_{1,a}(flat0, a_{0,1}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(33) MRRProof (EQUIVALENT transformation)

a_{1,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → a_{1,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
a_{1,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → a_{1,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
a_{1,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → a_{1,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
a_{1,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → a_{1,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
a_{1,a}(flat0, a_{0,a}(0, _x0)) → a_{1,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
a_{1,a}(flat0, a_{0,0}(0, _x0)) → a_{1,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
a_{1,a}(flat0, a_{0,b}(0, _x0)) → a_{1,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
a_{1,a}(flat0, a_{0,1}(0, _x0)) → a_{1,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
a_{a,1}(a_{0,b}(0, b_{0,a}(0, _x0)), flat1) → a_{b,1}(b_{0,a}(0, a_{0,a}(0, _x0)), flat1)
a_{a,1}(a_{0,b}(0, b_{0,0}(0, _x0)), flat1) → a_{b,1}(b_{0,a}(0, a_{0,0}(0, _x0)), flat1)
a_{a,1}(a_{0,b}(0, b_{0,b}(0, _x0)), flat1) → a_{b,1}(b_{0,a}(0, a_{0,b}(0, _x0)), flat1)
a_{a,1}(a_{0,b}(0, b_{0,1}(0, _x0)), flat1) → a_{b,1}(b_{0,a}(0, a_{0,1}(0, _x0)), flat1)
a_{a,1}(a_{0,a}(0, _x0), flat1) → a_{b,1}(b_{0,b}(0, b_{0,a}(0, _x0)), flat1)
a_{a,1}(a_{0,0}(0, _x0), flat1) → a_{b,1}(b_{0,b}(0, b_{0,0}(0, _x0)), flat1)
a_{a,1}(a_{0,b}(0, _x0), flat1) → a_{b,1}(b_{0,b}(0, b_{0,b}(0, _x0)), flat1)
a_{a,1}(a_{0,1}(0, _x0), flat1) → a_{b,1}(b_{0,b}(0, b_{0,1}(0, _x0)), flat1)
a_{a,b}(a_{0,b}(0, b_{0,a}(0, _x0)), flat1) → a_{b,b}(b_{0,a}(0, a_{0,a}(0, _x0)), flat1)
a_{a,b}(a_{0,b}(0, b_{0,0}(0, _x0)), flat1) → a_{b,b}(b_{0,a}(0, a_{0,0}(0, _x0)), flat1)
a_{a,b}(a_{0,b}(0, b_{0,b}(0, _x0)), flat1) → a_{b,b}(b_{0,a}(0, a_{0,b}(0, _x0)), flat1)
a_{a,b}(a_{0,b}(0, b_{0,1}(0, _x0)), flat1) → a_{b,b}(b_{0,a}(0, a_{0,1}(0, _x0)), flat1)
a_{a,b}(a_{0,a}(0, _x0), flat1) → a_{b,b}(b_{0,b}(0, b_{0,a}(0, _x0)), flat1)
a_{a,b}(a_{0,0}(0, _x0), flat1) → a_{b,b}(b_{0,b}(0, b_{0,0}(0, _x0)), flat1)
a_{a,b}(a_{0,b}(0, _x0), flat1) → a_{b,b}(b_{0,b}(0, b_{0,b}(0, _x0)), flat1)
a_{a,b}(a_{0,1}(0, _x0), flat1) → a_{b,b}(b_{0,b}(0, b_{0,1}(0, _x0)), flat1)
a_{a,0}(a_{0,b}(0, b_{0,a}(0, _x0)), flat1) → a_{b,0}(b_{0,a}(0, a_{0,a}(0, _x0)), flat1)
a_{a,0}(a_{0,b}(0, b_{0,0}(0, _x0)), flat1) → a_{b,0}(b_{0,a}(0, a_{0,0}(0, _x0)), flat1)
a_{a,0}(a_{0,b}(0, b_{0,b}(0, _x0)), flat1) → a_{b,0}(b_{0,a}(0, a_{0,b}(0, _x0)), flat1)
a_{a,0}(a_{0,b}(0, b_{0,1}(0, _x0)), flat1) → a_{b,0}(b_{0,a}(0, a_{0,1}(0, _x0)), flat1)
a_{a,0}(a_{0,a}(0, _x0), flat1) → a_{b,0}(b_{0,b}(0, b_{0,a}(0, _x0)), flat1)
a_{a,0}(a_{0,0}(0, _x0), flat1) → a_{b,0}(b_{0,b}(0, b_{0,0}(0, _x0)), flat1)
a_{a,0}(a_{0,b}(0, _x0), flat1) → a_{b,0}(b_{0,b}(0, b_{0,b}(0, _x0)), flat1)
a_{a,0}(a_{0,1}(0, _x0), flat1) → a_{b,0}(b_{0,b}(0, b_{0,1}(0, _x0)), flat1)
a_{a,a}(a_{0,b}(0, b_{0,a}(0, _x0)), flat1) → a_{b,a}(b_{0,a}(0, a_{0,a}(0, _x0)), flat1)
a_{a,a}(a_{0,b}(0, b_{0,0}(0, _x0)), flat1) → a_{b,a}(b_{0,a}(0, a_{0,0}(0, _x0)), flat1)
a_{a,a}(a_{0,b}(0, b_{0,b}(0, _x0)), flat1) → a_{b,a}(b_{0,a}(0, a_{0,b}(0, _x0)), flat1)
a_{a,a}(a_{0,b}(0, b_{0,1}(0, _x0)), flat1) → a_{b,a}(b_{0,a}(0, a_{0,1}(0, _x0)), flat1)
a_{a,a}(a_{0,a}(0, _x0), flat1) → a_{b,a}(b_{0,b}(0, b_{0,a}(0, _x0)), flat1)
a_{a,a}(a_{0,0}(0, _x0), flat1) → a_{b,a}(b_{0,b}(0, b_{0,0}(0, _x0)), flat1)
a_{a,a}(a_{0,b}(0, _x0), flat1) → a_{b,a}(b_{0,b}(0, b_{0,b}(0, _x0)), flat1)
a_{a,a}(a_{0,1}(0, _x0), flat1) → a_{b,a}(b_{0,b}(0, b_{0,1}(0, _x0)), flat1)

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 1
POL(1) = 0
POL(A_{0,b}(x₁, x₂)) = 1 + x₁ + x₂
POL(B_{0,a}(x₁, x₂)) = x₁ + x₂
POL(a_{0,0}(x₁, x₂)) = 1 + x₁ + x₂
POL(a_{0,1}(x₁, x₂)) = 1 + x₁ + x₂
POL(a_{0,a}(x₁, x₂)) = 1 + x₁ + x₂
POL(a_{0,b}(x₁, x₂)) = 1 + x₁ + x₂
POL(a_{1,0}(x₁, x₂)) = x₁ + x₂
POL(a_{1,1}(x₁, x₂)) = x₁ + x₂
POL(a_{1,a}(x₁, x₂)) = 1 + x₁ + x₂
POL(a_{1,b}(x₁, x₂)) = x₁ + x₂
POL(a_{a,0}(x₁, x₂)) = 1 + x₁ + x₂
POL(a_{a,1}(x₁, x₂)) = 1 + x₁ + x₂
POL(a_{a,a}(x₁, x₂)) = 1 + x₁ + x₂
POL(a_{a,b}(x₁, x₂)) = 1 + x₁ + x₂
POL(a_{b,0}(x₁, x₂)) = x₁ + x₂
POL(a_{b,1}(x₁, x₂)) = x₁ + x₂
POL(a_{b,a}(x₁, x₂)) = x₁ + x₂
POL(a_{b,b}(x₁, x₂)) = x₁ + x₂
POL(b_{0,0}(x₁, x₂)) = x₁ + x₂
POL(b_{0,1}(x₁, x₂)) = x₁ + x₂
POL(b_{0,a}(x₁, x₂)) = x₁ + x₂
POL(b_{0,b}(x₁, x₂)) = x₁ + x₂
POL(b_{1,a}(x₁, x₂)) = x₁ + x₂
POL(b_{1,b}(x₁, x₂)) = x₁ + x₂

(34) Obligation:

Q DP problem:
The TRS P consists of the following rules:

B_{0,a}(0, a_{1,a}(1, a_{0,b}(0, b_{0,a}(0, x0)))) → A_{0,b}(0, b_{0,a}(0, a_{0,a}(0, x0)))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{a,b}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{a,b}(x0, x1))))
B_{0,a}(0, a_{1,a}(1, a_{0,b}(0, b_{0,b}(0, x0)))) → A_{0,b}(0, b_{0,a}(0, a_{0,b}(0, x0)))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{0,a}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{0,a}(x0, x1))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{0,b}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{0,b}(x0, x1))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{b,a}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{b,a}(x0, x1))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{b,0}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{b,0}(x0, x1))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{b,b}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{b,b}(x0, x1))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{b,1}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{b,1}(x0, x1))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,a}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{1,a}(x0, x1))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,b}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{1,b}(x0, x1))))

The TRS R consists of the following rules:

a_{0,a}(0, a_{1,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,1}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,1}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,1}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,1}(x, y)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
b_{0,a}(flat0, a_{0,a}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
b_{0,a}(flat0, a_{0,0}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
b_{0,a}(flat0, a_{0,1}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
a_{0,a}(0, a_{1,a}(1, a_{a,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{a,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{a,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{a,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,1}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,1}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,1}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,1}(x, y)))
a_{0,a}(0, a_{a,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,a}(x, y)))
a_{0,a}(0, a_{a,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,0}(x, y)))
a_{0,a}(0, a_{a,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,b}(x, y)))
a_{0,a}(0, a_{a,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,1}(x, y)))
a_{0,a}(0, a_{0,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,a}(x, y)))
a_{0,a}(0, a_{0,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,0}(x, y)))
a_{0,a}(0, a_{0,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,b}(x, y)))
a_{0,a}(0, a_{0,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,1}(x, y)))
a_{0,a}(0, a_{b,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,a}(x, y)))
a_{0,a}(0, a_{b,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,0}(x, y)))
a_{0,a}(0, a_{b,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,b}(x, y)))
a_{0,a}(0, a_{b,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,1}(x, y)))
a_{0,a}(0, a_{1,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,a}(x, y)))
a_{0,a}(0, a_{1,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,0}(x, y)))
a_{0,a}(0, a_{1,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,1}(x, y)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
a_{0,a}(flat0, a_{0,a}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
a_{0,a}(flat0, a_{0,0}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
a_{0,a}(flat0, a_{0,1}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
a_{b,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → a_{b,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
a_{b,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → a_{b,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
a_{b,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → a_{b,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
a_{b,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → a_{b,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
a_{b,a}(flat0, a_{0,a}(0, _x0)) → a_{b,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
a_{b,a}(flat0, a_{0,0}(0, _x0)) → a_{b,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
a_{b,a}(flat0, a_{0,b}(0, _x0)) → a_{b,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
a_{b,a}(flat0, a_{0,1}(0, _x0)) → a_{b,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
a_{a,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → a_{a,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
a_{a,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → a_{a,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
a_{a,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → a_{a,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
a_{a,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → a_{a,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
a_{a,a}(flat0, a_{0,a}(0, _x0)) → a_{a,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
a_{a,a}(flat0, a_{0,0}(0, _x0)) → a_{a,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
a_{a,a}(flat0, a_{0,b}(0, _x0)) → a_{a,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
a_{a,a}(flat0, a_{0,1}(0, _x0)) → a_{a,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
b_{1,a}(flat0, a_{0,a}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
b_{1,a}(flat0, a_{0,0}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
b_{1,a}(flat0, a_{0,1}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(35) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.

(36) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{a,b}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{a,b}(x0, x1))))
B_{0,a}(0, a_{1,a}(1, a_{0,b}(0, b_{0,a}(0, x0)))) → A_{0,b}(0, b_{0,a}(0, a_{0,a}(0, x0)))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{0,a}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{0,a}(x0, x1))))
B_{0,a}(0, a_{1,a}(1, a_{0,b}(0, b_{0,b}(0, x0)))) → A_{0,b}(0, b_{0,a}(0, a_{0,b}(0, x0)))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{0,b}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{0,b}(x0, x1))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{b,b}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{b,b}(x0, x1))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,a}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{1,a}(x0, x1))))

The TRS R consists of the following rules:

a_{0,a}(0, a_{1,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,1}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,1}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,1}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,1}(x, y)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
b_{0,a}(flat0, a_{0,a}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
b_{0,a}(flat0, a_{0,0}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
b_{0,a}(flat0, a_{0,1}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
a_{0,a}(0, a_{1,a}(1, a_{a,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{a,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{a,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{a,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,1}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,1}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,1}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,1}(x, y)))
a_{0,a}(0, a_{a,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,a}(x, y)))
a_{0,a}(0, a_{a,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,0}(x, y)))
a_{0,a}(0, a_{a,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,b}(x, y)))
a_{0,a}(0, a_{a,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,1}(x, y)))
a_{0,a}(0, a_{0,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,a}(x, y)))
a_{0,a}(0, a_{0,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,0}(x, y)))
a_{0,a}(0, a_{0,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,b}(x, y)))
a_{0,a}(0, a_{0,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,1}(x, y)))
a_{0,a}(0, a_{b,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,a}(x, y)))
a_{0,a}(0, a_{b,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,0}(x, y)))
a_{0,a}(0, a_{b,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,b}(x, y)))
a_{0,a}(0, a_{b,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,1}(x, y)))
a_{0,a}(0, a_{1,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,a}(x, y)))
a_{0,a}(0, a_{1,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,0}(x, y)))
a_{0,a}(0, a_{1,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,1}(x, y)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
a_{0,a}(flat0, a_{0,a}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
a_{0,a}(flat0, a_{0,0}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
a_{0,a}(flat0, a_{0,1}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
a_{b,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → a_{b,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
a_{b,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → a_{b,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
a_{b,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → a_{b,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
a_{b,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → a_{b,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
a_{b,a}(flat0, a_{0,a}(0, _x0)) → a_{b,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
a_{b,a}(flat0, a_{0,0}(0, _x0)) → a_{b,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
a_{b,a}(flat0, a_{0,b}(0, _x0)) → a_{b,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
a_{b,a}(flat0, a_{0,1}(0, _x0)) → a_{b,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
a_{a,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → a_{a,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
a_{a,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → a_{a,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
a_{a,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → a_{a,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
a_{a,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → a_{a,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
a_{a,a}(flat0, a_{0,a}(0, _x0)) → a_{a,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
a_{a,a}(flat0, a_{0,0}(0, _x0)) → a_{a,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
a_{a,a}(flat0, a_{0,b}(0, _x0)) → a_{a,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
a_{a,a}(flat0, a_{0,1}(0, _x0)) → a_{a,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
b_{1,a}(flat0, a_{0,a}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
b_{1,a}(flat0, a_{0,0}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
b_{1,a}(flat0, a_{0,1}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(37) MRRProof (EQUIVALENT transformation)

a_{b,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → a_{b,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
a_{b,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → a_{b,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
a_{b,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → a_{b,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
a_{b,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → a_{b,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
a_{b,a}(flat0, a_{0,a}(0, _x0)) → a_{b,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
a_{b,a}(flat0, a_{0,0}(0, _x0)) → a_{b,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
a_{b,a}(flat0, a_{0,b}(0, _x0)) → a_{b,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
a_{b,a}(flat0, a_{0,1}(0, _x0)) → a_{b,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
a_{a,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → a_{a,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
a_{a,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → a_{a,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
a_{a,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → a_{a,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
a_{a,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → a_{a,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
a_{a,a}(flat0, a_{0,a}(0, _x0)) → a_{a,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
a_{a,a}(flat0, a_{0,0}(0, _x0)) → a_{a,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
a_{a,a}(flat0, a_{0,b}(0, _x0)) → a_{a,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
a_{a,a}(flat0, a_{0,1}(0, _x0)) → a_{a,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(1) = 0
POL(A_{0,b}(x₁, x₂)) = x₁ + x₂
POL(B_{0,a}(x₁, x₂)) = x₁ + x₂
POL(a_{0,0}(x₁, x₂)) = x₁ + x₂
POL(a_{0,1}(x₁, x₂)) = x₁ + x₂
POL(a_{0,a}(x₁, x₂)) = x₁ + x₂
POL(a_{0,b}(x₁, x₂)) = x₁ + x₂
POL(a_{1,0}(x₁, x₂)) = x₁ + x₂
POL(a_{1,1}(x₁, x₂)) = x₁ + x₂
POL(a_{1,a}(x₁, x₂)) = x₁ + x₂
POL(a_{1,b}(x₁, x₂)) = x₁ + x₂
POL(a_{a,0}(x₁, x₂)) = x₁ + x₂
POL(a_{a,1}(x₁, x₂)) = x₁ + x₂
POL(a_{a,a}(x₁, x₂)) = 1 + x₁ + x₂
POL(a_{a,b}(x₁, x₂)) = x₁ + x₂
POL(a_{b,0}(x₁, x₂)) = x₁ + x₂
POL(a_{b,1}(x₁, x₂)) = x₁ + x₂
POL(a_{b,a}(x₁, x₂)) = 1 + x₁ + x₂
POL(a_{b,b}(x₁, x₂)) = x₁ + x₂
POL(b_{0,0}(x₁, x₂)) = x₁ + x₂
POL(b_{0,1}(x₁, x₂)) = x₁ + x₂
POL(b_{0,a}(x₁, x₂)) = x₁ + x₂
POL(b_{0,b}(x₁, x₂)) = x₁ + x₂
POL(b_{1,a}(x₁, x₂)) = x₁ + x₂
POL(b_{1,b}(x₁, x₂)) = x₁ + x₂

(38) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{a,b}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{a,b}(x0, x1))))
B_{0,a}(0, a_{1,a}(1, a_{0,b}(0, b_{0,a}(0, x0)))) → A_{0,b}(0, b_{0,a}(0, a_{0,a}(0, x0)))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{0,a}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{0,a}(x0, x1))))
B_{0,a}(0, a_{1,a}(1, a_{0,b}(0, b_{0,b}(0, x0)))) → A_{0,b}(0, b_{0,a}(0, a_{0,b}(0, x0)))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{0,b}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{0,b}(x0, x1))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{b,b}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{b,b}(x0, x1))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,a}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{1,a}(x0, x1))))

The TRS R consists of the following rules:

a_{0,a}(0, a_{1,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,1}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,1}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,1}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,1}(x, y)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
b_{0,a}(flat0, a_{0,a}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
b_{0,a}(flat0, a_{0,0}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
b_{0,a}(flat0, a_{0,1}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
a_{0,a}(0, a_{1,a}(1, a_{a,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{a,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{a,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{a,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,1}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,1}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,1}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,1}(x, y)))
a_{0,a}(0, a_{a,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,a}(x, y)))
a_{0,a}(0, a_{a,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,0}(x, y)))
a_{0,a}(0, a_{a,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,b}(x, y)))
a_{0,a}(0, a_{a,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,1}(x, y)))
a_{0,a}(0, a_{0,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,a}(x, y)))
a_{0,a}(0, a_{0,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,0}(x, y)))
a_{0,a}(0, a_{0,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,b}(x, y)))
a_{0,a}(0, a_{0,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,1}(x, y)))
a_{0,a}(0, a_{b,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,a}(x, y)))
a_{0,a}(0, a_{b,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,0}(x, y)))
a_{0,a}(0, a_{b,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,b}(x, y)))
a_{0,a}(0, a_{b,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,1}(x, y)))
a_{0,a}(0, a_{1,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,a}(x, y)))
a_{0,a}(0, a_{1,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,0}(x, y)))
a_{0,a}(0, a_{1,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,1}(x, y)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
a_{0,a}(flat0, a_{0,a}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
a_{0,a}(flat0, a_{0,0}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
a_{0,a}(flat0, a_{0,1}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
b_{1,a}(flat0, a_{0,a}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
b_{1,a}(flat0, a_{0,0}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
b_{1,a}(flat0, a_{0,1}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(39) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

(40) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{0,a}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{0,a}(x0, x1))))
B_{0,a}(0, a_{1,a}(1, a_{0,b}(0, b_{0,a}(0, x0)))) → A_{0,b}(0, b_{0,a}(0, a_{0,a}(0, x0)))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{0,b}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{0,b}(x0, x1))))
B_{0,a}(0, a_{1,a}(1, a_{0,b}(0, b_{0,b}(0, x0)))) → A_{0,b}(0, b_{0,a}(0, a_{0,b}(0, x0)))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,a}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{1,a}(x0, x1))))

The TRS R consists of the following rules:

a_{0,a}(0, a_{1,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,1}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,1}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,1}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,1}(x, y)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
b_{0,a}(flat0, a_{0,a}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
b_{0,a}(flat0, a_{0,0}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
b_{0,a}(flat0, a_{0,1}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
a_{0,a}(0, a_{1,a}(1, a_{a,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{a,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{a,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{a,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,1}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,1}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,1}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,1}(x, y)))
a_{0,a}(0, a_{a,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,a}(x, y)))
a_{0,a}(0, a_{a,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,0}(x, y)))
a_{0,a}(0, a_{a,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,b}(x, y)))
a_{0,a}(0, a_{a,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,1}(x, y)))
a_{0,a}(0, a_{0,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,a}(x, y)))
a_{0,a}(0, a_{0,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,0}(x, y)))
a_{0,a}(0, a_{0,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,b}(x, y)))
a_{0,a}(0, a_{0,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,1}(x, y)))
a_{0,a}(0, a_{b,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,a}(x, y)))
a_{0,a}(0, a_{b,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,0}(x, y)))
a_{0,a}(0, a_{b,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,b}(x, y)))
a_{0,a}(0, a_{b,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,1}(x, y)))
a_{0,a}(0, a_{1,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,a}(x, y)))
a_{0,a}(0, a_{1,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,0}(x, y)))
a_{0,a}(0, a_{1,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,1}(x, y)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
a_{0,a}(flat0, a_{0,a}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
a_{0,a}(flat0, a_{0,0}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
a_{0,a}(flat0, a_{0,1}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
b_{1,a}(flat0, a_{0,a}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
b_{1,a}(flat0, a_{0,0}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
b_{1,a}(flat0, a_{0,1}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(41) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule B_{0,a}(0, a_{1,a}(1, a_{0,b}(0, b_{0,b}(0, x0)))) → A_{0,b}(0, b_{0,a}(0, a_{0,b}(0, x0))) at position [1] we obtained the following new rules [LPAR04]:

B_{0,a}(0, a_{1,a}(1, a_{0,b}(0, b_{0,b}(0, b_{0,a}(0, x1))))) → A_{0,b}(0, b_{0,b}(0, b_{0,a}(0, a_{0,a}(0, x1))))
B_{0,a}(0, a_{1,a}(1, a_{0,b}(0, b_{0,b}(0, b_{0,0}(0, x1))))) → A_{0,b}(0, b_{0,b}(0, b_{0,a}(0, a_{0,0}(0, x1))))
B_{0,a}(0, a_{1,a}(1, a_{0,b}(0, b_{0,b}(0, b_{0,b}(0, x1))))) → A_{0,b}(0, b_{0,b}(0, b_{0,a}(0, a_{0,b}(0, x1))))
B_{0,a}(0, a_{1,a}(1, a_{0,b}(0, b_{0,b}(0, b_{0,1}(0, x1))))) → A_{0,b}(0, b_{0,b}(0, b_{0,a}(0, a_{0,1}(0, x1))))
B_{0,a}(0, a_{1,a}(1, a_{0,b}(0, b_{0,b}(0, x1)))) → A_{0,b}(0, b_{0,b}(0, b_{0,b}(0, b_{0,b}(0, x1))))

(42) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{0,a}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{0,a}(x0, x1))))
B_{0,a}(0, a_{1,a}(1, a_{0,b}(0, b_{0,a}(0, x0)))) → A_{0,b}(0, b_{0,a}(0, a_{0,a}(0, x0)))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{0,b}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{0,b}(x0, x1))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,a}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{1,a}(x0, x1))))
B_{0,a}(0, a_{1,a}(1, a_{0,b}(0, b_{0,b}(0, b_{0,a}(0, x1))))) → A_{0,b}(0, b_{0,b}(0, b_{0,a}(0, a_{0,a}(0, x1))))
B_{0,a}(0, a_{1,a}(1, a_{0,b}(0, b_{0,b}(0, b_{0,0}(0, x1))))) → A_{0,b}(0, b_{0,b}(0, b_{0,a}(0, a_{0,0}(0, x1))))
B_{0,a}(0, a_{1,a}(1, a_{0,b}(0, b_{0,b}(0, b_{0,b}(0, x1))))) → A_{0,b}(0, b_{0,b}(0, b_{0,a}(0, a_{0,b}(0, x1))))
B_{0,a}(0, a_{1,a}(1, a_{0,b}(0, b_{0,b}(0, b_{0,1}(0, x1))))) → A_{0,b}(0, b_{0,b}(0, b_{0,a}(0, a_{0,1}(0, x1))))
B_{0,a}(0, a_{1,a}(1, a_{0,b}(0, b_{0,b}(0, x1)))) → A_{0,b}(0, b_{0,b}(0, b_{0,b}(0, b_{0,b}(0, x1))))

The TRS R consists of the following rules:

a_{0,a}(0, a_{1,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,1}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,1}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,1}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,1}(x, y)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
b_{0,a}(flat0, a_{0,a}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
b_{0,a}(flat0, a_{0,0}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
b_{0,a}(flat0, a_{0,1}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
a_{0,a}(0, a_{1,a}(1, a_{a,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{a,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{a,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{a,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,1}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,1}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,1}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,1}(x, y)))
a_{0,a}(0, a_{a,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,a}(x, y)))
a_{0,a}(0, a_{a,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,0}(x, y)))
a_{0,a}(0, a_{a,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,b}(x, y)))
a_{0,a}(0, a_{a,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,1}(x, y)))
a_{0,a}(0, a_{0,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,a}(x, y)))
a_{0,a}(0, a_{0,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,0}(x, y)))
a_{0,a}(0, a_{0,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,b}(x, y)))
a_{0,a}(0, a_{0,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,1}(x, y)))
a_{0,a}(0, a_{b,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,a}(x, y)))
a_{0,a}(0, a_{b,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,0}(x, y)))
a_{0,a}(0, a_{b,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,b}(x, y)))
a_{0,a}(0, a_{b,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,1}(x, y)))
a_{0,a}(0, a_{1,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,a}(x, y)))
a_{0,a}(0, a_{1,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,0}(x, y)))
a_{0,a}(0, a_{1,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,1}(x, y)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
a_{0,a}(flat0, a_{0,a}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
a_{0,a}(flat0, a_{0,0}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
a_{0,a}(flat0, a_{0,1}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
b_{1,a}(flat0, a_{0,a}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
b_{1,a}(flat0, a_{0,0}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
b_{1,a}(flat0, a_{0,1}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(43) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes.

(44) Obligation:

Q DP problem:
The TRS P consists of the following rules:

B_{0,a}(0, a_{1,a}(1, a_{0,b}(0, b_{0,a}(0, x0)))) → A_{0,b}(0, b_{0,a}(0, a_{0,a}(0, x0)))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{0,a}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{0,a}(x0, x1))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{0,b}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{0,b}(x0, x1))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,a}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{1,a}(x0, x1))))

The TRS R consists of the following rules:

a_{0,a}(0, a_{1,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,1}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,1}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,1}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,1}(x, y)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
b_{0,a}(flat0, a_{0,a}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
b_{0,a}(flat0, a_{0,0}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
b_{0,a}(flat0, a_{0,1}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
a_{0,a}(0, a_{1,a}(1, a_{a,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{a,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{a,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{a,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,1}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,1}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,1}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,1}(x, y)))
a_{0,a}(0, a_{a,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,a}(x, y)))
a_{0,a}(0, a_{a,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,0}(x, y)))
a_{0,a}(0, a_{a,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,b}(x, y)))
a_{0,a}(0, a_{a,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,1}(x, y)))
a_{0,a}(0, a_{0,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,a}(x, y)))
a_{0,a}(0, a_{0,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,0}(x, y)))
a_{0,a}(0, a_{0,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,b}(x, y)))
a_{0,a}(0, a_{0,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,1}(x, y)))
a_{0,a}(0, a_{b,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,a}(x, y)))
a_{0,a}(0, a_{b,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,0}(x, y)))
a_{0,a}(0, a_{b,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,b}(x, y)))
a_{0,a}(0, a_{b,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,1}(x, y)))
a_{0,a}(0, a_{1,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,a}(x, y)))
a_{0,a}(0, a_{1,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,0}(x, y)))
a_{0,a}(0, a_{1,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,1}(x, y)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
a_{0,a}(flat0, a_{0,a}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
a_{0,a}(flat0, a_{0,0}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
a_{0,a}(flat0, a_{0,1}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
b_{1,a}(flat0, a_{0,a}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
b_{1,a}(flat0, a_{0,0}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
b_{1,a}(flat0, a_{0,1}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(45) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,a}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{1,a}(x0, x1)))) at position [1,1] we obtained the following new rules [LPAR04]:

A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{a,a}(x0, x1))))) → B_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{0,a}(0, a_{a,a}(x0, x1)))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{a,0}(x0, x1))))) → B_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{0,a}(0, a_{a,0}(x0, x1)))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{a,b}(x0, x1))))) → B_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{0,a}(0, a_{a,b}(x0, x1)))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{a,1}(x0, x1))))) → B_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{0,a}(0, a_{a,1}(x0, x1)))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{0,a}(x0, x1))))) → B_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{0,a}(0, a_{0,a}(x0, x1)))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{0,0}(x0, x1))))) → B_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{0,a}(0, a_{0,0}(x0, x1)))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{0,b}(x0, x1))))) → B_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{0,a}(0, a_{0,b}(x0, x1)))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{0,1}(x0, x1))))) → B_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{0,a}(0, a_{0,1}(x0, x1)))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{b,a}(x0, x1))))) → B_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{0,a}(0, a_{b,a}(x0, x1)))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{b,0}(x0, x1))))) → B_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{0,a}(0, a_{b,0}(x0, x1)))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{b,b}(x0, x1))))) → B_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{0,a}(0, a_{b,b}(x0, x1)))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{b,1}(x0, x1))))) → B_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{0,a}(0, a_{b,1}(x0, x1)))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{1,a}(x0, x1))))) → B_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{0,a}(0, a_{1,a}(x0, x1)))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{1,0}(x0, x1))))) → B_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{0,a}(0, a_{1,0}(x0, x1)))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{1,b}(x0, x1))))) → B_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{0,a}(0, a_{1,b}(x0, x1)))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{1,1}(x0, x1))))) → B_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{0,a}(0, a_{1,1}(x0, x1)))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,a}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{1,a}(1, a_{1,a}(x0, x1)))))

(46) Obligation:

Q DP problem:
The TRS P consists of the following rules:

B_{0,a}(0, a_{1,a}(1, a_{0,b}(0, b_{0,a}(0, x0)))) → A_{0,b}(0, b_{0,a}(0, a_{0,a}(0, x0)))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{0,a}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{0,a}(x0, x1))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{0,b}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{0,b}(x0, x1))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{a,a}(x0, x1))))) → B_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{0,a}(0, a_{a,a}(x0, x1)))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{a,0}(x0, x1))))) → B_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{0,a}(0, a_{a,0}(x0, x1)))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{a,b}(x0, x1))))) → B_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{0,a}(0, a_{a,b}(x0, x1)))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{a,1}(x0, x1))))) → B_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{0,a}(0, a_{a,1}(x0, x1)))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{0,a}(x0, x1))))) → B_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{0,a}(0, a_{0,a}(x0, x1)))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{0,0}(x0, x1))))) → B_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{0,a}(0, a_{0,0}(x0, x1)))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{0,b}(x0, x1))))) → B_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{0,a}(0, a_{0,b}(x0, x1)))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{0,1}(x0, x1))))) → B_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{0,a}(0, a_{0,1}(x0, x1)))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{b,a}(x0, x1))))) → B_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{0,a}(0, a_{b,a}(x0, x1)))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{b,0}(x0, x1))))) → B_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{0,a}(0, a_{b,0}(x0, x1)))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{b,b}(x0, x1))))) → B_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{0,a}(0, a_{b,b}(x0, x1)))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{b,1}(x0, x1))))) → B_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{0,a}(0, a_{b,1}(x0, x1)))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{1,a}(x0, x1))))) → B_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{0,a}(0, a_{1,a}(x0, x1)))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{1,0}(x0, x1))))) → B_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{0,a}(0, a_{1,0}(x0, x1)))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{1,b}(x0, x1))))) → B_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{0,a}(0, a_{1,b}(x0, x1)))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{1,1}(x0, x1))))) → B_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{0,a}(0, a_{1,1}(x0, x1)))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{1,a}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{1,a}(1, a_{1,a}(1, a_{1,a}(x0, x1)))))

The TRS R consists of the following rules:

a_{0,a}(0, a_{1,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,1}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,1}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,1}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,1}(x, y)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
b_{0,a}(flat0, a_{0,a}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
b_{0,a}(flat0, a_{0,0}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
b_{0,a}(flat0, a_{0,1}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
a_{0,a}(0, a_{1,a}(1, a_{a,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{a,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{a,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{a,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,1}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,1}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,1}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,1}(x, y)))
a_{0,a}(0, a_{a,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,a}(x, y)))
a_{0,a}(0, a_{a,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,0}(x, y)))
a_{0,a}(0, a_{a,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,b}(x, y)))
a_{0,a}(0, a_{a,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,1}(x, y)))
a_{0,a}(0, a_{0,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,a}(x, y)))
a_{0,a}(0, a_{0,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,0}(x, y)))
a_{0,a}(0, a_{0,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,b}(x, y)))
a_{0,a}(0, a_{0,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,1}(x, y)))
a_{0,a}(0, a_{b,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,a}(x, y)))
a_{0,a}(0, a_{b,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,0}(x, y)))
a_{0,a}(0, a_{b,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,b}(x, y)))
a_{0,a}(0, a_{b,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,1}(x, y)))
a_{0,a}(0, a_{1,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,a}(x, y)))
a_{0,a}(0, a_{1,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,0}(x, y)))
a_{0,a}(0, a_{1,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,1}(x, y)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
a_{0,a}(flat0, a_{0,a}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
a_{0,a}(flat0, a_{0,0}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
a_{0,a}(flat0, a_{0,1}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
b_{1,a}(flat0, a_{0,a}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
b_{1,a}(flat0, a_{0,0}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
b_{1,a}(flat0, a_{0,1}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(47) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 17 less nodes.

(48) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{0,a}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{0,a}(x0, x1))))
B_{0,a}(0, a_{1,a}(1, a_{0,b}(0, b_{0,a}(0, x0)))) → A_{0,b}(0, b_{0,a}(0, a_{0,a}(0, x0)))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{0,b}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{0,b}(x0, x1))))

The TRS R consists of the following rules:

a_{0,a}(0, a_{1,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,1}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,1}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,1}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,1}(x, y)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
b_{0,a}(flat0, a_{0,a}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
b_{0,a}(flat0, a_{0,0}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
b_{0,a}(flat0, a_{0,1}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
a_{0,a}(0, a_{1,a}(1, a_{a,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{a,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{a,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{a,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,1}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,1}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,1}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,1}(x, y)))
a_{0,a}(0, a_{a,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,a}(x, y)))
a_{0,a}(0, a_{a,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,0}(x, y)))
a_{0,a}(0, a_{a,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,b}(x, y)))
a_{0,a}(0, a_{a,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,1}(x, y)))
a_{0,a}(0, a_{0,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,a}(x, y)))
a_{0,a}(0, a_{0,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,0}(x, y)))
a_{0,a}(0, a_{0,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,b}(x, y)))
a_{0,a}(0, a_{0,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,1}(x, y)))
a_{0,a}(0, a_{b,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,a}(x, y)))
a_{0,a}(0, a_{b,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,0}(x, y)))
a_{0,a}(0, a_{b,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,b}(x, y)))
a_{0,a}(0, a_{b,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,1}(x, y)))
a_{0,a}(0, a_{1,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,a}(x, y)))
a_{0,a}(0, a_{1,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,0}(x, y)))
a_{0,a}(0, a_{1,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,1}(x, y)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
a_{0,a}(flat0, a_{0,a}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
a_{0,a}(flat0, a_{0,0}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
a_{0,a}(flat0, a_{0,1}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
b_{1,a}(flat0, a_{0,a}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
b_{1,a}(flat0, a_{0,0}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
b_{1,a}(flat0, a_{0,1}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(49) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

B_{0,a}(0, a_{1,a}(1, a_{0,b}(0, b_{0,a}(0, x0)))) → A_{0,b}(0, b_{0,a}(0, a_{0,a}(0, x0)))

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(A_{0,b}(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 1 \

\ 0 0 /

· x₂

POL(0) =
/ 1 \

\ 1 /

POL(b_{0,a}(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 1 0 \

\ 1 1 /

· x₁ +
/ 1 0 \

\ 1 0 /

· x₂

POL(a_{1,a}(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 1 1 \

\ 0 0 /

· x₁ +
/ 0 1 \

\ 0 0 /

· x₂

POL(1) =
/ 1 \

\ 1 /

POL(a_{0,a}(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 1 1 \

\ 1 1 /

· x₁ +
/ 1 0 \

\ 0 1 /

· x₂

POL(B_{0,a}(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 1 0 \

\ 0 0 /

· x₂

POL(a_{0,b}(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 1 1 /

· x₁ +
/ 0 0 \

\ 1 0 /

· x₂

POL(a_{a,0}(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(a_{a,b}(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(a_{0,1}(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(b_{0,b}(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(b_{0,1}(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(a_{a,a}(x₁, x₂)) =
/ 1 \

\ 1 /

+
/ 1 1 \

\ 1 1 /

· x₁ +
/ 1 0 \

\ 1 0 /

· x₂

POL(a_{0,0}(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(b_{0,0}(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(a_{1,1}(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 1 0 \

\ 1 0 /

· x₂

POL(b_{1,a}(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 1 /

· x₂

POL(a_{1,0}(x₁, x₂)) =
/ 1 \

\ 0 /

+
/ 1 1 \

\ 1 1 /

· x₁ +
/ 1 1 \

\ 1 1 /

· x₂

POL(a_{1,b}(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 1 0 \

\ 1 1 /

· x₁ +
/ 1 0 \

\ 1 1 /

· x₂

POL(a_{b,1}(x₁, x₂)) =
/ 1 \

\ 0 /

+
/ 1 1 \

\ 1 0 /

· x₁ +
/ 1 1 \

\ 1 0 /

· x₂

POL(a_{b,b}(x₁, x₂)) =
/ 1 \

\ 0 /

+
/ 1 0 \

\ 1 0 /

· x₁ +
/ 0 1 \

\ 0 1 /

· x₂

POL(a_{b,0}(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(a_{b,a}(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(a_{a,1}(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 1 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(b_{1,b}(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

The following usable rules [FROCOS05] were oriented:

a_{0,a}(0, a_{1,a}(1, a_{a,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{a,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,b}(x, y)))
b_{0,a}(flat0, a_{0,1}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
a_{0,a}(0, a_{1,a}(1, a_{a,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,a}(x, y)))
b_{0,a}(flat0, a_{0,0}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
b_{0,a}(flat0, a_{0,a}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
b_{0,a}(0, a_{1,a}(1, a_{1,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,1}(x, y)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
b_{0,a}(0, a_{1,a}(1, a_{1,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,1}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,1}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,1}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,a}(x, y)))
a_{0,a}(0, a_{1,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,b}(x, y)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
a_{0,a}(0, a_{b,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,1}(x, y)))
a_{0,a}(0, a_{1,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,a}(x, y)))
a_{0,a}(0, a_{1,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,0}(x, y)))
a_{0,a}(0, a_{1,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,1}(x, y)))
a_{0,a}(0, a_{0,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,1}(x, y)))
a_{0,a}(0, a_{b,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,a}(x, y)))
a_{0,a}(0, a_{b,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,0}(x, y)))
a_{0,a}(0, a_{b,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,b}(x, y)))
a_{0,a}(0, a_{a,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,1}(x, y)))
a_{0,a}(0, a_{0,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,a}(x, y)))
a_{0,a}(0, a_{0,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,0}(x, y)))
a_{0,a}(0, a_{0,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,b}(x, y)))
a_{0,a}(0, a_{a,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,1}(x, y)))
a_{0,a}(0, a_{a,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,b}(x, y)))
a_{0,a}(0, a_{a,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,1}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,1}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{a,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,1}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,0}(x, y)))
a_{0,a}(flat0, a_{0,b}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
a_{0,a}(flat0, a_{0,1}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
a_{0,a}(flat0, a_{0,a}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
a_{0,a}(flat0, a_{0,0}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
b_{1,a}(flat0, a_{0,1}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
b_{1,a}(flat0, a_{0,a}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
b_{1,a}(flat0, a_{0,0}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))

(50) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{0,a}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{0,a}(x0, x1))))
A_{0,b}(0, b_{0,a}(0, a_{1,a}(1, a_{0,b}(x0, x1)))) → B_{0,a}(0, a_{1,a}(1, a_{0,a}(0, a_{0,b}(x0, x1))))

The TRS R consists of the following rules:

a_{0,a}(0, a_{1,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{a,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{a,1}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{0,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{0,1}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{b,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{b,1}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,a}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,a}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,0}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,0}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,b}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,b}(x, y)))
b_{0,a}(0, a_{1,a}(1, a_{1,1}(x, y))) → b_{1,a}(1, a_{0,a}(0, a_{1,1}(x, y)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → b_{0,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
b_{0,a}(flat0, a_{0,a}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
b_{0,a}(flat0, a_{0,0}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
b_{0,a}(flat0, a_{0,b}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
b_{0,a}(flat0, a_{0,1}(0, _x0)) → b_{0,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
a_{0,a}(0, a_{1,a}(1, a_{a,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{a,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{a,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{a,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{a,1}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{0,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{0,1}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{b,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{b,1}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,a}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,a}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,0}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,0}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,b}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,b}(x, y)))
a_{0,a}(0, a_{1,a}(1, a_{1,1}(x, y))) → a_{1,a}(1, a_{0,a}(0, a_{1,1}(x, y)))
a_{0,a}(0, a_{a,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,a}(x, y)))
a_{0,a}(0, a_{a,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,0}(x, y)))
a_{0,a}(0, a_{a,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,b}(x, y)))
a_{0,a}(0, a_{a,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{a,1}(x, y)))
a_{0,a}(0, a_{0,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,a}(x, y)))
a_{0,a}(0, a_{0,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,0}(x, y)))
a_{0,a}(0, a_{0,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,b}(x, y)))
a_{0,a}(0, a_{0,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{0,1}(x, y)))
a_{0,a}(0, a_{b,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,a}(x, y)))
a_{0,a}(0, a_{b,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,0}(x, y)))
a_{0,a}(0, a_{b,b}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,b}(x, y)))
a_{0,a}(0, a_{b,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{b,1}(x, y)))
a_{0,a}(0, a_{1,a}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,a}(x, y)))
a_{0,a}(0, a_{1,0}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,0}(x, y)))
a_{0,a}(0, a_{1,1}(x, y)) → a_{1,a}(1, a_{1,a}(1, a_{1,1}(x, y)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → a_{0,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
a_{0,a}(flat0, a_{0,a}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
a_{0,a}(flat0, a_{0,0}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
a_{0,a}(flat0, a_{0,b}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
a_{0,a}(flat0, a_{0,1}(0, _x0)) → a_{0,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,a}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,a}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,0}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,0}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,b}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,b}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, b_{0,1}(0, _x0))) → b_{1,b}(flat0, b_{0,a}(0, a_{0,1}(0, _x0)))
b_{1,a}(flat0, a_{0,a}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,a}(0, _x0)))
b_{1,a}(flat0, a_{0,0}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,0}(0, _x0)))
b_{1,a}(flat0, a_{0,b}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,b}(0, _x0)))
b_{1,a}(flat0, a_{0,1}(0, _x0)) → b_{1,b}(flat0, b_{0,b}(0, b_{0,1}(0, _x0)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(51) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(0) Obligation:

(1) DependencyPairsProof (EQUIVALENT transformation)

(2) Obligation:

(3) DependencyGraphProof (EQUIVALENT transformation)

(4) Obligation:

(5) MRRProof (EQUIVALENT transformation)

(6) Obligation:

(7) Narrowing (EQUIVALENT transformation)

(8) Obligation:

(9) DependencyGraphProof (EQUIVALENT transformation)

(10) Obligation:

(11) Narrowing (EQUIVALENT transformation)

(12) Obligation:

(13) Narrowing (EQUIVALENT transformation)

(14) Obligation:

(15) DependencyGraphProof (EQUIVALENT transformation)

(16) Obligation:

(17) Narrowing (EQUIVALENT transformation)

(18) Obligation:

(19) DependencyGraphProof (EQUIVALENT transformation)

(20) Obligation:

(21) QDPOrderProof (EQUIVALENT transformation)

(22) Obligation:

(23) Narrowing (EQUIVALENT transformation)

(24) Obligation:

(25) DependencyGraphProof (EQUIVALENT transformation)

(26) Obligation:

(27) RootLabelingFC2Proof (EQUIVALENT transformation)

(28) Obligation:

(29) DependencyGraphProof (EQUIVALENT transformation)

(30) Obligation:

(31) UsableRulesProof (EQUIVALENT transformation)

(32) Obligation:

(33) MRRProof (EQUIVALENT transformation)

(34) Obligation:

(35) DependencyGraphProof (EQUIVALENT transformation)

(36) Obligation:

(37) MRRProof (EQUIVALENT transformation)

(38) Obligation:

(39) DependencyGraphProof (EQUIVALENT transformation)

(40) Obligation:

(41) Narrowing (EQUIVALENT transformation)

(42) Obligation:

(43) DependencyGraphProof (EQUIVALENT transformation)

(44) Obligation:

(45) Narrowing (EQUIVALENT transformation)

(46) Obligation:

(47) DependencyGraphProof (EQUIVALENT transformation)

(48) Obligation:

(49) QDPOrderProof (EQUIVALENT transformation)

(50) Obligation:

(51) DependencyGraphProof (EQUIVALENT transformation)

(52) TRUE