Termination w.r.t. Q proof of /home/cern_httpd/provide/research/cycsrs/tpdb/TPDB-d9b80194f163/SRS_Standard/Secret_07

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

0 QTRS

↳1 FlatCCProof (⇔, 0 ms)

↳2 QTRS

↳3 RootLabelingProof (⇔, 0 ms)

↳4 QTRS

↳5 QTRSRRRProof (⇔, 138 ms)

↳6 QTRS

↳7 DependencyPairsProof (⇔, 51 ms)

↳8 QDP

↳9 DependencyGraphProof (⇔, 0 ms)

↳10 QDP

↳11 QDPOrderProof (⇔, 5479 ms)

↳12 QDP

↳13 DependencyGraphProof (⇔, 0 ms)

↳14 TRUE

(0) Obligation:

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

a(b(x)) → c(d(x))
d(d(x)) → b(e(x))
b(x) → d(c(x))
d(x) → x
e(c(x)) → d(a(x))
a(x) → e(d(x))

Q is empty.

(1) FlatCCProof (EQUIVALENT transformation)

We used flat context closure [ROOTLAB] As Q is empty the flat context closure was sound AND complete.

(2) Obligation:

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

a(a(b(x))) → a(c(d(x)))
b(a(b(x))) → b(c(d(x)))
c(a(b(x))) → c(c(d(x)))
d(a(b(x))) → d(c(d(x)))
e(a(b(x))) → e(c(d(x)))
a(d(d(x))) → a(b(e(x)))
b(d(d(x))) → b(b(e(x)))
c(d(d(x))) → c(b(e(x)))
d(d(d(x))) → d(b(e(x)))
e(d(d(x))) → e(b(e(x)))
a(b(x)) → a(d(c(x)))
b(b(x)) → b(d(c(x)))
c(b(x)) → c(d(c(x)))
d(b(x)) → d(d(c(x)))
e(b(x)) → e(d(c(x)))
a(d(x)) → a(x)
b(d(x)) → b(x)
c(d(x)) → c(x)
d(d(x)) → d(x)
e(d(x)) → e(x)
a(e(c(x))) → a(d(a(x)))
b(e(c(x))) → b(d(a(x)))
c(e(c(x))) → c(d(a(x)))
d(e(c(x))) → d(d(a(x)))
e(e(c(x))) → e(d(a(x)))
a(a(x)) → a(e(d(x)))
b(a(x)) → b(e(d(x)))
c(a(x)) → c(e(d(x)))
d(a(x)) → d(e(d(x)))
e(a(x)) → e(e(d(x)))

Q is empty.

(3) RootLabelingProof (EQUIVALENT transformation)

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

(4) Obligation:

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

a_{a_1}(a_{b_1}(b_{a_1}(x))) → a_{c_1}(c_{d_1}(d_{a_1}(x)))
a_{a_1}(a_{b_1}(b_{b_1}(x))) → a_{c_1}(c_{d_1}(d_{b_1}(x)))
a_{a_1}(a_{b_1}(b_{c_1}(x))) → a_{c_1}(c_{d_1}(d_{c_1}(x)))
a_{a_1}(a_{b_1}(b_{d_1}(x))) → a_{c_1}(c_{d_1}(d_{d_1}(x)))
a_{a_1}(a_{b_1}(b_{e_1}(x))) → a_{c_1}(c_{d_1}(d_{e_1}(x)))
b_{a_1}(a_{b_1}(b_{a_1}(x))) → b_{c_1}(c_{d_1}(d_{a_1}(x)))
b_{a_1}(a_{b_1}(b_{b_1}(x))) → b_{c_1}(c_{d_1}(d_{b_1}(x)))
b_{a_1}(a_{b_1}(b_{c_1}(x))) → b_{c_1}(c_{d_1}(d_{c_1}(x)))
b_{a_1}(a_{b_1}(b_{d_1}(x))) → b_{c_1}(c_{d_1}(d_{d_1}(x)))
b_{a_1}(a_{b_1}(b_{e_1}(x))) → b_{c_1}(c_{d_1}(d_{e_1}(x)))
c_{a_1}(a_{b_1}(b_{a_1}(x))) → c_{c_1}(c_{d_1}(d_{a_1}(x)))
c_{a_1}(a_{b_1}(b_{b_1}(x))) → c_{c_1}(c_{d_1}(d_{b_1}(x)))
c_{a_1}(a_{b_1}(b_{c_1}(x))) → c_{c_1}(c_{d_1}(d_{c_1}(x)))
c_{a_1}(a_{b_1}(b_{d_1}(x))) → c_{c_1}(c_{d_1}(d_{d_1}(x)))
c_{a_1}(a_{b_1}(b_{e_1}(x))) → c_{c_1}(c_{d_1}(d_{e_1}(x)))
d_{a_1}(a_{b_1}(b_{a_1}(x))) → d_{c_1}(c_{d_1}(d_{a_1}(x)))
d_{a_1}(a_{b_1}(b_{b_1}(x))) → d_{c_1}(c_{d_1}(d_{b_1}(x)))
d_{a_1}(a_{b_1}(b_{c_1}(x))) → d_{c_1}(c_{d_1}(d_{c_1}(x)))
d_{a_1}(a_{b_1}(b_{d_1}(x))) → d_{c_1}(c_{d_1}(d_{d_1}(x)))
d_{a_1}(a_{b_1}(b_{e_1}(x))) → d_{c_1}(c_{d_1}(d_{e_1}(x)))
e_{a_1}(a_{b_1}(b_{a_1}(x))) → e_{c_1}(c_{d_1}(d_{a_1}(x)))
e_{a_1}(a_{b_1}(b_{b_1}(x))) → e_{c_1}(c_{d_1}(d_{b_1}(x)))
e_{a_1}(a_{b_1}(b_{c_1}(x))) → e_{c_1}(c_{d_1}(d_{c_1}(x)))
e_{a_1}(a_{b_1}(b_{d_1}(x))) → e_{c_1}(c_{d_1}(d_{d_1}(x)))
e_{a_1}(a_{b_1}(b_{e_1}(x))) → e_{c_1}(c_{d_1}(d_{e_1}(x)))
a_{d_1}(d_{d_1}(d_{a_1}(x))) → a_{b_1}(b_{e_1}(e_{a_1}(x)))
a_{d_1}(d_{d_1}(d_{b_1}(x))) → a_{b_1}(b_{e_1}(e_{b_1}(x)))
a_{d_1}(d_{d_1}(d_{c_1}(x))) → a_{b_1}(b_{e_1}(e_{c_1}(x)))
a_{d_1}(d_{d_1}(d_{d_1}(x))) → a_{b_1}(b_{e_1}(e_{d_1}(x)))
a_{d_1}(d_{d_1}(d_{e_1}(x))) → a_{b_1}(b_{e_1}(e_{e_1}(x)))
b_{d_1}(d_{d_1}(d_{a_1}(x))) → b_{b_1}(b_{e_1}(e_{a_1}(x)))
b_{d_1}(d_{d_1}(d_{b_1}(x))) → b_{b_1}(b_{e_1}(e_{b_1}(x)))
b_{d_1}(d_{d_1}(d_{c_1}(x))) → b_{b_1}(b_{e_1}(e_{c_1}(x)))
b_{d_1}(d_{d_1}(d_{d_1}(x))) → b_{b_1}(b_{e_1}(e_{d_1}(x)))
b_{d_1}(d_{d_1}(d_{e_1}(x))) → b_{b_1}(b_{e_1}(e_{e_1}(x)))
c_{d_1}(d_{d_1}(d_{a_1}(x))) → c_{b_1}(b_{e_1}(e_{a_1}(x)))
c_{d_1}(d_{d_1}(d_{b_1}(x))) → c_{b_1}(b_{e_1}(e_{b_1}(x)))
c_{d_1}(d_{d_1}(d_{c_1}(x))) → c_{b_1}(b_{e_1}(e_{c_1}(x)))
c_{d_1}(d_{d_1}(d_{d_1}(x))) → c_{b_1}(b_{e_1}(e_{d_1}(x)))
c_{d_1}(d_{d_1}(d_{e_1}(x))) → c_{b_1}(b_{e_1}(e_{e_1}(x)))
d_{d_1}(d_{d_1}(d_{a_1}(x))) → d_{b_1}(b_{e_1}(e_{a_1}(x)))
d_{d_1}(d_{d_1}(d_{b_1}(x))) → d_{b_1}(b_{e_1}(e_{b_1}(x)))
d_{d_1}(d_{d_1}(d_{c_1}(x))) → d_{b_1}(b_{e_1}(e_{c_1}(x)))
d_{d_1}(d_{d_1}(d_{d_1}(x))) → d_{b_1}(b_{e_1}(e_{d_1}(x)))
d_{d_1}(d_{d_1}(d_{e_1}(x))) → d_{b_1}(b_{e_1}(e_{e_1}(x)))
e_{d_1}(d_{d_1}(d_{a_1}(x))) → e_{b_1}(b_{e_1}(e_{a_1}(x)))
e_{d_1}(d_{d_1}(d_{b_1}(x))) → e_{b_1}(b_{e_1}(e_{b_1}(x)))
e_{d_1}(d_{d_1}(d_{c_1}(x))) → e_{b_1}(b_{e_1}(e_{c_1}(x)))
e_{d_1}(d_{d_1}(d_{d_1}(x))) → e_{b_1}(b_{e_1}(e_{d_1}(x)))
e_{d_1}(d_{d_1}(d_{e_1}(x))) → e_{b_1}(b_{e_1}(e_{e_1}(x)))
a_{b_1}(b_{a_1}(x)) → a_{d_1}(d_{c_1}(c_{a_1}(x)))
a_{b_1}(b_{b_1}(x)) → a_{d_1}(d_{c_1}(c_{b_1}(x)))
a_{b_1}(b_{c_1}(x)) → a_{d_1}(d_{c_1}(c_{c_1}(x)))
a_{b_1}(b_{d_1}(x)) → a_{d_1}(d_{c_1}(c_{d_1}(x)))
a_{b_1}(b_{e_1}(x)) → a_{d_1}(d_{c_1}(c_{e_1}(x)))
b_{b_1}(b_{a_1}(x)) → b_{d_1}(d_{c_1}(c_{a_1}(x)))
b_{b_1}(b_{b_1}(x)) → b_{d_1}(d_{c_1}(c_{b_1}(x)))
b_{b_1}(b_{c_1}(x)) → b_{d_1}(d_{c_1}(c_{c_1}(x)))
b_{b_1}(b_{d_1}(x)) → b_{d_1}(d_{c_1}(c_{d_1}(x)))
b_{b_1}(b_{e_1}(x)) → b_{d_1}(d_{c_1}(c_{e_1}(x)))
c_{b_1}(b_{a_1}(x)) → c_{d_1}(d_{c_1}(c_{a_1}(x)))
c_{b_1}(b_{b_1}(x)) → c_{d_1}(d_{c_1}(c_{b_1}(x)))
c_{b_1}(b_{c_1}(x)) → c_{d_1}(d_{c_1}(c_{c_1}(x)))
c_{b_1}(b_{d_1}(x)) → c_{d_1}(d_{c_1}(c_{d_1}(x)))
c_{b_1}(b_{e_1}(x)) → c_{d_1}(d_{c_1}(c_{e_1}(x)))
d_{b_1}(b_{a_1}(x)) → d_{d_1}(d_{c_1}(c_{a_1}(x)))
d_{b_1}(b_{b_1}(x)) → d_{d_1}(d_{c_1}(c_{b_1}(x)))
d_{b_1}(b_{c_1}(x)) → d_{d_1}(d_{c_1}(c_{c_1}(x)))
d_{b_1}(b_{d_1}(x)) → d_{d_1}(d_{c_1}(c_{d_1}(x)))
d_{b_1}(b_{e_1}(x)) → d_{d_1}(d_{c_1}(c_{e_1}(x)))
e_{b_1}(b_{a_1}(x)) → e_{d_1}(d_{c_1}(c_{a_1}(x)))
e_{b_1}(b_{b_1}(x)) → e_{d_1}(d_{c_1}(c_{b_1}(x)))
e_{b_1}(b_{c_1}(x)) → e_{d_1}(d_{c_1}(c_{c_1}(x)))
e_{b_1}(b_{d_1}(x)) → e_{d_1}(d_{c_1}(c_{d_1}(x)))
e_{b_1}(b_{e_1}(x)) → e_{d_1}(d_{c_1}(c_{e_1}(x)))
a_{d_1}(d_{a_1}(x)) → a_{a_1}(x)
a_{d_1}(d_{b_1}(x)) → a_{b_1}(x)
a_{d_1}(d_{c_1}(x)) → a_{c_1}(x)
a_{d_1}(d_{d_1}(x)) → a_{d_1}(x)
a_{d_1}(d_{e_1}(x)) → a_{e_1}(x)
b_{d_1}(d_{a_1}(x)) → b_{a_1}(x)
b_{d_1}(d_{b_1}(x)) → b_{b_1}(x)
b_{d_1}(d_{c_1}(x)) → b_{c_1}(x)
b_{d_1}(d_{d_1}(x)) → b_{d_1}(x)
b_{d_1}(d_{e_1}(x)) → b_{e_1}(x)
c_{d_1}(d_{a_1}(x)) → c_{a_1}(x)
c_{d_1}(d_{b_1}(x)) → c_{b_1}(x)
c_{d_1}(d_{c_1}(x)) → c_{c_1}(x)
c_{d_1}(d_{d_1}(x)) → c_{d_1}(x)
c_{d_1}(d_{e_1}(x)) → c_{e_1}(x)
d_{d_1}(d_{a_1}(x)) → d_{a_1}(x)
d_{d_1}(d_{b_1}(x)) → d_{b_1}(x)
d_{d_1}(d_{c_1}(x)) → d_{c_1}(x)
d_{d_1}(d_{d_1}(x)) → d_{d_1}(x)
d_{d_1}(d_{e_1}(x)) → d_{e_1}(x)
e_{d_1}(d_{a_1}(x)) → e_{a_1}(x)
e_{d_1}(d_{b_1}(x)) → e_{b_1}(x)
e_{d_1}(d_{c_1}(x)) → e_{c_1}(x)
e_{d_1}(d_{d_1}(x)) → e_{d_1}(x)
e_{d_1}(d_{e_1}(x)) → e_{e_1}(x)
a_{e_1}(e_{c_1}(c_{a_1}(x))) → a_{d_1}(d_{a_1}(a_{a_1}(x)))
a_{e_1}(e_{c_1}(c_{b_1}(x))) → a_{d_1}(d_{a_1}(a_{b_1}(x)))
a_{e_1}(e_{c_1}(c_{c_1}(x))) → a_{d_1}(d_{a_1}(a_{c_1}(x)))
a_{e_1}(e_{c_1}(c_{d_1}(x))) → a_{d_1}(d_{a_1}(a_{d_1}(x)))
a_{e_1}(e_{c_1}(c_{e_1}(x))) → a_{d_1}(d_{a_1}(a_{e_1}(x)))
b_{e_1}(e_{c_1}(c_{a_1}(x))) → b_{d_1}(d_{a_1}(a_{a_1}(x)))
b_{e_1}(e_{c_1}(c_{b_1}(x))) → b_{d_1}(d_{a_1}(a_{b_1}(x)))
b_{e_1}(e_{c_1}(c_{c_1}(x))) → b_{d_1}(d_{a_1}(a_{c_1}(x)))
b_{e_1}(e_{c_1}(c_{d_1}(x))) → b_{d_1}(d_{a_1}(a_{d_1}(x)))
b_{e_1}(e_{c_1}(c_{e_1}(x))) → b_{d_1}(d_{a_1}(a_{e_1}(x)))
c_{e_1}(e_{c_1}(c_{a_1}(x))) → c_{d_1}(d_{a_1}(a_{a_1}(x)))
c_{e_1}(e_{c_1}(c_{b_1}(x))) → c_{d_1}(d_{a_1}(a_{b_1}(x)))
c_{e_1}(e_{c_1}(c_{c_1}(x))) → c_{d_1}(d_{a_1}(a_{c_1}(x)))
c_{e_1}(e_{c_1}(c_{d_1}(x))) → c_{d_1}(d_{a_1}(a_{d_1}(x)))
c_{e_1}(e_{c_1}(c_{e_1}(x))) → c_{d_1}(d_{a_1}(a_{e_1}(x)))
d_{e_1}(e_{c_1}(c_{a_1}(x))) → d_{d_1}(d_{a_1}(a_{a_1}(x)))
d_{e_1}(e_{c_1}(c_{b_1}(x))) → d_{d_1}(d_{a_1}(a_{b_1}(x)))
d_{e_1}(e_{c_1}(c_{c_1}(x))) → d_{d_1}(d_{a_1}(a_{c_1}(x)))
d_{e_1}(e_{c_1}(c_{d_1}(x))) → d_{d_1}(d_{a_1}(a_{d_1}(x)))
d_{e_1}(e_{c_1}(c_{e_1}(x))) → d_{d_1}(d_{a_1}(a_{e_1}(x)))
e_{e_1}(e_{c_1}(c_{a_1}(x))) → e_{d_1}(d_{a_1}(a_{a_1}(x)))
e_{e_1}(e_{c_1}(c_{b_1}(x))) → e_{d_1}(d_{a_1}(a_{b_1}(x)))
e_{e_1}(e_{c_1}(c_{c_1}(x))) → e_{d_1}(d_{a_1}(a_{c_1}(x)))
e_{e_1}(e_{c_1}(c_{d_1}(x))) → e_{d_1}(d_{a_1}(a_{d_1}(x)))
e_{e_1}(e_{c_1}(c_{e_1}(x))) → e_{d_1}(d_{a_1}(a_{e_1}(x)))
a_{a_1}(a_{a_1}(x)) → a_{e_1}(e_{d_1}(d_{a_1}(x)))
a_{a_1}(a_{b_1}(x)) → a_{e_1}(e_{d_1}(d_{b_1}(x)))
a_{a_1}(a_{c_1}(x)) → a_{e_1}(e_{d_1}(d_{c_1}(x)))
a_{a_1}(a_{d_1}(x)) → a_{e_1}(e_{d_1}(d_{d_1}(x)))
a_{a_1}(a_{e_1}(x)) → a_{e_1}(e_{d_1}(d_{e_1}(x)))
b_{a_1}(a_{a_1}(x)) → b_{e_1}(e_{d_1}(d_{a_1}(x)))
b_{a_1}(a_{b_1}(x)) → b_{e_1}(e_{d_1}(d_{b_1}(x)))
b_{a_1}(a_{c_1}(x)) → b_{e_1}(e_{d_1}(d_{c_1}(x)))
b_{a_1}(a_{d_1}(x)) → b_{e_1}(e_{d_1}(d_{d_1}(x)))
b_{a_1}(a_{e_1}(x)) → b_{e_1}(e_{d_1}(d_{e_1}(x)))
c_{a_1}(a_{a_1}(x)) → c_{e_1}(e_{d_1}(d_{a_1}(x)))
c_{a_1}(a_{b_1}(x)) → c_{e_1}(e_{d_1}(d_{b_1}(x)))
c_{a_1}(a_{c_1}(x)) → c_{e_1}(e_{d_1}(d_{c_1}(x)))
c_{a_1}(a_{d_1}(x)) → c_{e_1}(e_{d_1}(d_{d_1}(x)))
c_{a_1}(a_{e_1}(x)) → c_{e_1}(e_{d_1}(d_{e_1}(x)))
d_{a_1}(a_{a_1}(x)) → d_{e_1}(e_{d_1}(d_{a_1}(x)))
d_{a_1}(a_{b_1}(x)) → d_{e_1}(e_{d_1}(d_{b_1}(x)))
d_{a_1}(a_{c_1}(x)) → d_{e_1}(e_{d_1}(d_{c_1}(x)))
d_{a_1}(a_{d_1}(x)) → d_{e_1}(e_{d_1}(d_{d_1}(x)))
d_{a_1}(a_{e_1}(x)) → d_{e_1}(e_{d_1}(d_{e_1}(x)))
e_{a_1}(a_{a_1}(x)) → e_{e_1}(e_{d_1}(d_{a_1}(x)))
e_{a_1}(a_{b_1}(x)) → e_{e_1}(e_{d_1}(d_{b_1}(x)))
e_{a_1}(a_{c_1}(x)) → e_{e_1}(e_{d_1}(d_{c_1}(x)))
e_{a_1}(a_{d_1}(x)) → e_{e_1}(e_{d_1}(d_{d_1}(x)))
e_{a_1}(a_{e_1}(x)) → e_{e_1}(e_{d_1}(d_{e_1}(x)))

Q is empty.

(5) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(a_{a_1}(x₁)) = 2 + x₁
POL(a_{b_1}(x₁)) = 5 + x₁
POL(a_{c_1}(x₁)) = 6 + x₁
POL(a_{d_1}(x₁)) = 5 + x₁
POL(a_{e_1}(x₁)) = x₁
POL(b_{a_1}(x₁)) = 9 + x₁
POL(b_{b_1}(x₁)) = 10 + x₁
POL(b_{c_1}(x₁)) = 13 + x₁
POL(b_{d_1}(x₁)) = 10 + x₁
POL(b_{e_1}(x₁)) = 5 + x₁
POL(c_{a_1}(x₁)) = 3 + x₁
POL(c_{b_1}(x₁)) = 5 + x₁
POL(c_{c_1}(x₁)) = 7 + x₁
POL(c_{d_1}(x₁)) = 5 + x₁
POL(c_{e_1}(x₁)) = x₁
POL(d_{a_1}(x₁)) = x₁
POL(d_{b_1}(x₁)) = 5 + x₁
POL(d_{c_1}(x₁)) = 5 + x₁
POL(d_{d_1}(x₁)) = 5 + x₁
POL(d_{e_1}(x₁)) = x₁
POL(e_{a_1}(x₁)) = x₁
POL(e_{b_1}(x₁)) = x₁
POL(e_{c_1}(x₁)) = 5 + x₁
POL(e_{d_1}(x₁)) = x₁
POL(e_{e_1}(x₁)) = x₁

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

a_{a_1}(a_{b_1}(b_{a_1}(x))) → a_{c_1}(c_{d_1}(d_{a_1}(x)))
a_{a_1}(a_{b_1}(b_{b_1}(x))) → a_{c_1}(c_{d_1}(d_{b_1}(x)))
a_{a_1}(a_{b_1}(b_{c_1}(x))) → a_{c_1}(c_{d_1}(d_{c_1}(x)))
a_{a_1}(a_{b_1}(b_{d_1}(x))) → a_{c_1}(c_{d_1}(d_{d_1}(x)))
a_{a_1}(a_{b_1}(b_{e_1}(x))) → a_{c_1}(c_{d_1}(d_{e_1}(x)))
b_{a_1}(a_{b_1}(b_{a_1}(x))) → b_{c_1}(c_{d_1}(d_{a_1}(x)))
b_{a_1}(a_{b_1}(b_{b_1}(x))) → b_{c_1}(c_{d_1}(d_{b_1}(x)))
b_{a_1}(a_{b_1}(b_{c_1}(x))) → b_{c_1}(c_{d_1}(d_{c_1}(x)))
b_{a_1}(a_{b_1}(b_{d_1}(x))) → b_{c_1}(c_{d_1}(d_{d_1}(x)))
b_{a_1}(a_{b_1}(b_{e_1}(x))) → b_{c_1}(c_{d_1}(d_{e_1}(x)))
c_{a_1}(a_{b_1}(b_{a_1}(x))) → c_{c_1}(c_{d_1}(d_{a_1}(x)))
c_{a_1}(a_{b_1}(b_{b_1}(x))) → c_{c_1}(c_{d_1}(d_{b_1}(x)))
c_{a_1}(a_{b_1}(b_{c_1}(x))) → c_{c_1}(c_{d_1}(d_{c_1}(x)))
c_{a_1}(a_{b_1}(b_{d_1}(x))) → c_{c_1}(c_{d_1}(d_{d_1}(x)))
c_{a_1}(a_{b_1}(b_{e_1}(x))) → c_{c_1}(c_{d_1}(d_{e_1}(x)))
d_{a_1}(a_{b_1}(b_{a_1}(x))) → d_{c_1}(c_{d_1}(d_{a_1}(x)))
d_{a_1}(a_{b_1}(b_{c_1}(x))) → d_{c_1}(c_{d_1}(d_{c_1}(x)))
e_{a_1}(a_{b_1}(b_{a_1}(x))) → e_{c_1}(c_{d_1}(d_{a_1}(x)))
e_{a_1}(a_{b_1}(b_{c_1}(x))) → e_{c_1}(c_{d_1}(d_{c_1}(x)))
a_{d_1}(d_{d_1}(d_{b_1}(x))) → a_{b_1}(b_{e_1}(e_{b_1}(x)))
a_{d_1}(d_{d_1}(d_{d_1}(x))) → a_{b_1}(b_{e_1}(e_{d_1}(x)))
b_{d_1}(d_{d_1}(d_{b_1}(x))) → b_{b_1}(b_{e_1}(e_{b_1}(x)))
b_{d_1}(d_{d_1}(d_{d_1}(x))) → b_{b_1}(b_{e_1}(e_{d_1}(x)))
c_{d_1}(d_{d_1}(d_{b_1}(x))) → c_{b_1}(b_{e_1}(e_{b_1}(x)))
c_{d_1}(d_{d_1}(d_{d_1}(x))) → c_{b_1}(b_{e_1}(e_{d_1}(x)))
d_{d_1}(d_{d_1}(d_{b_1}(x))) → d_{b_1}(b_{e_1}(e_{b_1}(x)))
d_{d_1}(d_{d_1}(d_{d_1}(x))) → d_{b_1}(b_{e_1}(e_{d_1}(x)))
e_{d_1}(d_{d_1}(d_{b_1}(x))) → e_{b_1}(b_{e_1}(e_{b_1}(x)))
e_{d_1}(d_{d_1}(d_{d_1}(x))) → e_{b_1}(b_{e_1}(e_{d_1}(x)))
a_{b_1}(b_{a_1}(x)) → a_{d_1}(d_{c_1}(c_{a_1}(x)))
a_{b_1}(b_{c_1}(x)) → a_{d_1}(d_{c_1}(c_{c_1}(x)))
b_{b_1}(b_{a_1}(x)) → b_{d_1}(d_{c_1}(c_{a_1}(x)))
b_{b_1}(b_{c_1}(x)) → b_{d_1}(d_{c_1}(c_{c_1}(x)))
c_{b_1}(b_{a_1}(x)) → c_{d_1}(d_{c_1}(c_{a_1}(x)))
c_{b_1}(b_{c_1}(x)) → c_{d_1}(d_{c_1}(c_{c_1}(x)))
d_{b_1}(b_{a_1}(x)) → d_{d_1}(d_{c_1}(c_{a_1}(x)))
d_{b_1}(b_{c_1}(x)) → d_{d_1}(d_{c_1}(c_{c_1}(x)))
e_{b_1}(b_{a_1}(x)) → e_{d_1}(d_{c_1}(c_{a_1}(x)))
e_{b_1}(b_{c_1}(x)) → e_{d_1}(d_{c_1}(c_{c_1}(x)))
a_{d_1}(d_{a_1}(x)) → a_{a_1}(x)
a_{d_1}(d_{b_1}(x)) → a_{b_1}(x)
a_{d_1}(d_{c_1}(x)) → a_{c_1}(x)
a_{d_1}(d_{d_1}(x)) → a_{d_1}(x)
a_{d_1}(d_{e_1}(x)) → a_{e_1}(x)
b_{d_1}(d_{a_1}(x)) → b_{a_1}(x)
b_{d_1}(d_{b_1}(x)) → b_{b_1}(x)
b_{d_1}(d_{c_1}(x)) → b_{c_1}(x)
b_{d_1}(d_{d_1}(x)) → b_{d_1}(x)
b_{d_1}(d_{e_1}(x)) → b_{e_1}(x)
c_{d_1}(d_{a_1}(x)) → c_{a_1}(x)
c_{d_1}(d_{b_1}(x)) → c_{b_1}(x)
c_{d_1}(d_{c_1}(x)) → c_{c_1}(x)
c_{d_1}(d_{d_1}(x)) → c_{d_1}(x)
c_{d_1}(d_{e_1}(x)) → c_{e_1}(x)
d_{d_1}(d_{a_1}(x)) → d_{a_1}(x)
d_{d_1}(d_{b_1}(x)) → d_{b_1}(x)
d_{d_1}(d_{c_1}(x)) → d_{c_1}(x)
d_{d_1}(d_{d_1}(x)) → d_{d_1}(x)
d_{d_1}(d_{e_1}(x)) → d_{e_1}(x)
e_{d_1}(d_{b_1}(x)) → e_{b_1}(x)
e_{d_1}(d_{d_1}(x)) → e_{d_1}(x)
a_{e_1}(e_{c_1}(c_{a_1}(x))) → a_{d_1}(d_{a_1}(a_{a_1}(x)))
a_{e_1}(e_{c_1}(c_{c_1}(x))) → a_{d_1}(d_{a_1}(a_{c_1}(x)))
b_{e_1}(e_{c_1}(c_{a_1}(x))) → b_{d_1}(d_{a_1}(a_{a_1}(x)))
b_{e_1}(e_{c_1}(c_{c_1}(x))) → b_{d_1}(d_{a_1}(a_{c_1}(x)))
c_{e_1}(e_{c_1}(c_{a_1}(x))) → c_{d_1}(d_{a_1}(a_{a_1}(x)))
c_{e_1}(e_{c_1}(c_{c_1}(x))) → c_{d_1}(d_{a_1}(a_{c_1}(x)))
d_{e_1}(e_{c_1}(c_{a_1}(x))) → d_{d_1}(d_{a_1}(a_{a_1}(x)))
d_{e_1}(e_{c_1}(c_{c_1}(x))) → d_{d_1}(d_{a_1}(a_{c_1}(x)))
e_{e_1}(e_{c_1}(c_{a_1}(x))) → e_{d_1}(d_{a_1}(a_{a_1}(x)))
e_{e_1}(e_{c_1}(c_{b_1}(x))) → e_{d_1}(d_{a_1}(a_{b_1}(x)))
e_{e_1}(e_{c_1}(c_{c_1}(x))) → e_{d_1}(d_{a_1}(a_{c_1}(x)))
e_{e_1}(e_{c_1}(c_{d_1}(x))) → e_{d_1}(d_{a_1}(a_{d_1}(x)))
e_{e_1}(e_{c_1}(c_{e_1}(x))) → e_{d_1}(d_{a_1}(a_{e_1}(x)))
a_{a_1}(a_{a_1}(x)) → a_{e_1}(e_{d_1}(d_{a_1}(x)))
a_{a_1}(a_{b_1}(x)) → a_{e_1}(e_{d_1}(d_{b_1}(x)))
a_{a_1}(a_{c_1}(x)) → a_{e_1}(e_{d_1}(d_{c_1}(x)))
a_{a_1}(a_{d_1}(x)) → a_{e_1}(e_{d_1}(d_{d_1}(x)))
a_{a_1}(a_{e_1}(x)) → a_{e_1}(e_{d_1}(d_{e_1}(x)))
b_{a_1}(a_{a_1}(x)) → b_{e_1}(e_{d_1}(d_{a_1}(x)))
b_{a_1}(a_{b_1}(x)) → b_{e_1}(e_{d_1}(d_{b_1}(x)))
b_{a_1}(a_{c_1}(x)) → b_{e_1}(e_{d_1}(d_{c_1}(x)))
b_{a_1}(a_{d_1}(x)) → b_{e_1}(e_{d_1}(d_{d_1}(x)))
b_{a_1}(a_{e_1}(x)) → b_{e_1}(e_{d_1}(d_{e_1}(x)))
c_{a_1}(a_{a_1}(x)) → c_{e_1}(e_{d_1}(d_{a_1}(x)))
c_{a_1}(a_{b_1}(x)) → c_{e_1}(e_{d_1}(d_{b_1}(x)))
c_{a_1}(a_{c_1}(x)) → c_{e_1}(e_{d_1}(d_{c_1}(x)))
c_{a_1}(a_{d_1}(x)) → c_{e_1}(e_{d_1}(d_{d_1}(x)))
c_{a_1}(a_{e_1}(x)) → c_{e_1}(e_{d_1}(d_{e_1}(x)))
d_{a_1}(a_{a_1}(x)) → d_{e_1}(e_{d_1}(d_{a_1}(x)))
d_{a_1}(a_{c_1}(x)) → d_{e_1}(e_{d_1}(d_{c_1}(x)))
e_{a_1}(a_{a_1}(x)) → e_{e_1}(e_{d_1}(d_{a_1}(x)))
e_{a_1}(a_{c_1}(x)) → e_{e_1}(e_{d_1}(d_{c_1}(x)))

(6) Obligation:

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

d_{a_1}(a_{b_1}(b_{b_1}(x))) → d_{c_1}(c_{d_1}(d_{b_1}(x)))
d_{a_1}(a_{b_1}(b_{d_1}(x))) → d_{c_1}(c_{d_1}(d_{d_1}(x)))
d_{a_1}(a_{b_1}(b_{e_1}(x))) → d_{c_1}(c_{d_1}(d_{e_1}(x)))
e_{a_1}(a_{b_1}(b_{b_1}(x))) → e_{c_1}(c_{d_1}(d_{b_1}(x)))
e_{a_1}(a_{b_1}(b_{d_1}(x))) → e_{c_1}(c_{d_1}(d_{d_1}(x)))
e_{a_1}(a_{b_1}(b_{e_1}(x))) → e_{c_1}(c_{d_1}(d_{e_1}(x)))
a_{d_1}(d_{d_1}(d_{a_1}(x))) → a_{b_1}(b_{e_1}(e_{a_1}(x)))
a_{d_1}(d_{d_1}(d_{c_1}(x))) → a_{b_1}(b_{e_1}(e_{c_1}(x)))
a_{d_1}(d_{d_1}(d_{e_1}(x))) → a_{b_1}(b_{e_1}(e_{e_1}(x)))
b_{d_1}(d_{d_1}(d_{a_1}(x))) → b_{b_1}(b_{e_1}(e_{a_1}(x)))
b_{d_1}(d_{d_1}(d_{c_1}(x))) → b_{b_1}(b_{e_1}(e_{c_1}(x)))
b_{d_1}(d_{d_1}(d_{e_1}(x))) → b_{b_1}(b_{e_1}(e_{e_1}(x)))
c_{d_1}(d_{d_1}(d_{a_1}(x))) → c_{b_1}(b_{e_1}(e_{a_1}(x)))
c_{d_1}(d_{d_1}(d_{c_1}(x))) → c_{b_1}(b_{e_1}(e_{c_1}(x)))
c_{d_1}(d_{d_1}(d_{e_1}(x))) → c_{b_1}(b_{e_1}(e_{e_1}(x)))
d_{d_1}(d_{d_1}(d_{a_1}(x))) → d_{b_1}(b_{e_1}(e_{a_1}(x)))
d_{d_1}(d_{d_1}(d_{c_1}(x))) → d_{b_1}(b_{e_1}(e_{c_1}(x)))
d_{d_1}(d_{d_1}(d_{e_1}(x))) → d_{b_1}(b_{e_1}(e_{e_1}(x)))
e_{d_1}(d_{d_1}(d_{a_1}(x))) → e_{b_1}(b_{e_1}(e_{a_1}(x)))
e_{d_1}(d_{d_1}(d_{c_1}(x))) → e_{b_1}(b_{e_1}(e_{c_1}(x)))
e_{d_1}(d_{d_1}(d_{e_1}(x))) → e_{b_1}(b_{e_1}(e_{e_1}(x)))
a_{b_1}(b_{b_1}(x)) → a_{d_1}(d_{c_1}(c_{b_1}(x)))
a_{b_1}(b_{d_1}(x)) → a_{d_1}(d_{c_1}(c_{d_1}(x)))
a_{b_1}(b_{e_1}(x)) → a_{d_1}(d_{c_1}(c_{e_1}(x)))
b_{b_1}(b_{b_1}(x)) → b_{d_1}(d_{c_1}(c_{b_1}(x)))
b_{b_1}(b_{d_1}(x)) → b_{d_1}(d_{c_1}(c_{d_1}(x)))
b_{b_1}(b_{e_1}(x)) → b_{d_1}(d_{c_1}(c_{e_1}(x)))
c_{b_1}(b_{b_1}(x)) → c_{d_1}(d_{c_1}(c_{b_1}(x)))
c_{b_1}(b_{d_1}(x)) → c_{d_1}(d_{c_1}(c_{d_1}(x)))
c_{b_1}(b_{e_1}(x)) → c_{d_1}(d_{c_1}(c_{e_1}(x)))
d_{b_1}(b_{b_1}(x)) → d_{d_1}(d_{c_1}(c_{b_1}(x)))
d_{b_1}(b_{d_1}(x)) → d_{d_1}(d_{c_1}(c_{d_1}(x)))
d_{b_1}(b_{e_1}(x)) → d_{d_1}(d_{c_1}(c_{e_1}(x)))
e_{b_1}(b_{b_1}(x)) → e_{d_1}(d_{c_1}(c_{b_1}(x)))
e_{b_1}(b_{d_1}(x)) → e_{d_1}(d_{c_1}(c_{d_1}(x)))
e_{b_1}(b_{e_1}(x)) → e_{d_1}(d_{c_1}(c_{e_1}(x)))
e_{d_1}(d_{a_1}(x)) → e_{a_1}(x)
e_{d_1}(d_{c_1}(x)) → e_{c_1}(x)
e_{d_1}(d_{e_1}(x)) → e_{e_1}(x)
a_{e_1}(e_{c_1}(c_{b_1}(x))) → a_{d_1}(d_{a_1}(a_{b_1}(x)))
a_{e_1}(e_{c_1}(c_{d_1}(x))) → a_{d_1}(d_{a_1}(a_{d_1}(x)))
a_{e_1}(e_{c_1}(c_{e_1}(x))) → a_{d_1}(d_{a_1}(a_{e_1}(x)))
b_{e_1}(e_{c_1}(c_{b_1}(x))) → b_{d_1}(d_{a_1}(a_{b_1}(x)))
b_{e_1}(e_{c_1}(c_{d_1}(x))) → b_{d_1}(d_{a_1}(a_{d_1}(x)))
b_{e_1}(e_{c_1}(c_{e_1}(x))) → b_{d_1}(d_{a_1}(a_{e_1}(x)))
c_{e_1}(e_{c_1}(c_{b_1}(x))) → c_{d_1}(d_{a_1}(a_{b_1}(x)))
c_{e_1}(e_{c_1}(c_{d_1}(x))) → c_{d_1}(d_{a_1}(a_{d_1}(x)))
c_{e_1}(e_{c_1}(c_{e_1}(x))) → c_{d_1}(d_{a_1}(a_{e_1}(x)))
d_{e_1}(e_{c_1}(c_{b_1}(x))) → d_{d_1}(d_{a_1}(a_{b_1}(x)))
d_{e_1}(e_{c_1}(c_{d_1}(x))) → d_{d_1}(d_{a_1}(a_{d_1}(x)))
d_{e_1}(e_{c_1}(c_{e_1}(x))) → d_{d_1}(d_{a_1}(a_{e_1}(x)))
d_{a_1}(a_{b_1}(x)) → d_{e_1}(e_{d_1}(d_{b_1}(x)))
d_{a_1}(a_{d_1}(x)) → d_{e_1}(e_{d_1}(d_{d_1}(x)))
d_{a_1}(a_{e_1}(x)) → d_{e_1}(e_{d_1}(d_{e_1}(x)))
e_{a_1}(a_{b_1}(x)) → e_{e_1}(e_{d_1}(d_{b_1}(x)))
e_{a_1}(a_{d_1}(x)) → e_{e_1}(e_{d_1}(d_{d_1}(x)))
e_{a_1}(a_{e_1}(x)) → e_{e_1}(e_{d_1}(d_{e_1}(x)))

Q is empty.

(7) DependencyPairsProof (EQUIVALENT transformation)

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

(8) Obligation:

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

D_{A_1}(a_{b_1}(b_{b_1}(x))) → C_{D_1}(d_{b_1}(x))
D_{A_1}(a_{b_1}(b_{b_1}(x))) → D_{B_1}(x)
D_{A_1}(a_{b_1}(b_{d_1}(x))) → C_{D_1}(d_{d_1}(x))
D_{A_1}(a_{b_1}(b_{d_1}(x))) → D_{D_1}(x)
D_{A_1}(a_{b_1}(b_{e_1}(x))) → C_{D_1}(d_{e_1}(x))
D_{A_1}(a_{b_1}(b_{e_1}(x))) → D_{E_1}(x)
E_{A_1}(a_{b_1}(b_{b_1}(x))) → C_{D_1}(d_{b_1}(x))
E_{A_1}(a_{b_1}(b_{b_1}(x))) → D_{B_1}(x)
E_{A_1}(a_{b_1}(b_{d_1}(x))) → C_{D_1}(d_{d_1}(x))
E_{A_1}(a_{b_1}(b_{d_1}(x))) → D_{D_1}(x)
E_{A_1}(a_{b_1}(b_{e_1}(x))) → C_{D_1}(d_{e_1}(x))
E_{A_1}(a_{b_1}(b_{e_1}(x))) → D_{E_1}(x)
A_{D_1}(d_{d_1}(d_{a_1}(x))) → A_{B_1}(b_{e_1}(e_{a_1}(x)))
A_{D_1}(d_{d_1}(d_{a_1}(x))) → B_{E_1}(e_{a_1}(x))
A_{D_1}(d_{d_1}(d_{a_1}(x))) → E_{A_1}(x)
A_{D_1}(d_{d_1}(d_{c_1}(x))) → A_{B_1}(b_{e_1}(e_{c_1}(x)))
A_{D_1}(d_{d_1}(d_{c_1}(x))) → B_{E_1}(e_{c_1}(x))
A_{D_1}(d_{d_1}(d_{e_1}(x))) → A_{B_1}(b_{e_1}(e_{e_1}(x)))
A_{D_1}(d_{d_1}(d_{e_1}(x))) → B_{E_1}(e_{e_1}(x))
B_{D_1}(d_{d_1}(d_{a_1}(x))) → B_{B_1}(b_{e_1}(e_{a_1}(x)))
B_{D_1}(d_{d_1}(d_{a_1}(x))) → B_{E_1}(e_{a_1}(x))
B_{D_1}(d_{d_1}(d_{a_1}(x))) → E_{A_1}(x)
B_{D_1}(d_{d_1}(d_{c_1}(x))) → B_{B_1}(b_{e_1}(e_{c_1}(x)))
B_{D_1}(d_{d_1}(d_{c_1}(x))) → B_{E_1}(e_{c_1}(x))
B_{D_1}(d_{d_1}(d_{e_1}(x))) → B_{B_1}(b_{e_1}(e_{e_1}(x)))
B_{D_1}(d_{d_1}(d_{e_1}(x))) → B_{E_1}(e_{e_1}(x))
C_{D_1}(d_{d_1}(d_{a_1}(x))) → C_{B_1}(b_{e_1}(e_{a_1}(x)))
C_{D_1}(d_{d_1}(d_{a_1}(x))) → B_{E_1}(e_{a_1}(x))
C_{D_1}(d_{d_1}(d_{a_1}(x))) → E_{A_1}(x)
C_{D_1}(d_{d_1}(d_{c_1}(x))) → C_{B_1}(b_{e_1}(e_{c_1}(x)))
C_{D_1}(d_{d_1}(d_{c_1}(x))) → B_{E_1}(e_{c_1}(x))
C_{D_1}(d_{d_1}(d_{e_1}(x))) → C_{B_1}(b_{e_1}(e_{e_1}(x)))
C_{D_1}(d_{d_1}(d_{e_1}(x))) → B_{E_1}(e_{e_1}(x))
D_{D_1}(d_{d_1}(d_{a_1}(x))) → D_{B_1}(b_{e_1}(e_{a_1}(x)))
D_{D_1}(d_{d_1}(d_{a_1}(x))) → B_{E_1}(e_{a_1}(x))
D_{D_1}(d_{d_1}(d_{a_1}(x))) → E_{A_1}(x)
D_{D_1}(d_{d_1}(d_{c_1}(x))) → D_{B_1}(b_{e_1}(e_{c_1}(x)))
D_{D_1}(d_{d_1}(d_{c_1}(x))) → B_{E_1}(e_{c_1}(x))
D_{D_1}(d_{d_1}(d_{e_1}(x))) → D_{B_1}(b_{e_1}(e_{e_1}(x)))
D_{D_1}(d_{d_1}(d_{e_1}(x))) → B_{E_1}(e_{e_1}(x))
E_{D_1}(d_{d_1}(d_{a_1}(x))) → E_{B_1}(b_{e_1}(e_{a_1}(x)))
E_{D_1}(d_{d_1}(d_{a_1}(x))) → B_{E_1}(e_{a_1}(x))
E_{D_1}(d_{d_1}(d_{a_1}(x))) → E_{A_1}(x)
E_{D_1}(d_{d_1}(d_{c_1}(x))) → E_{B_1}(b_{e_1}(e_{c_1}(x)))
E_{D_1}(d_{d_1}(d_{c_1}(x))) → B_{E_1}(e_{c_1}(x))
E_{D_1}(d_{d_1}(d_{e_1}(x))) → E_{B_1}(b_{e_1}(e_{e_1}(x)))
E_{D_1}(d_{d_1}(d_{e_1}(x))) → B_{E_1}(e_{e_1}(x))
A_{B_1}(b_{b_1}(x)) → A_{D_1}(d_{c_1}(c_{b_1}(x)))
A_{B_1}(b_{b_1}(x)) → C_{B_1}(x)
A_{B_1}(b_{d_1}(x)) → A_{D_1}(d_{c_1}(c_{d_1}(x)))
A_{B_1}(b_{d_1}(x)) → C_{D_1}(x)
A_{B_1}(b_{e_1}(x)) → A_{D_1}(d_{c_1}(c_{e_1}(x)))
A_{B_1}(b_{e_1}(x)) → C_{E_1}(x)
B_{B_1}(b_{b_1}(x)) → B_{D_1}(d_{c_1}(c_{b_1}(x)))
B_{B_1}(b_{b_1}(x)) → C_{B_1}(x)
B_{B_1}(b_{d_1}(x)) → B_{D_1}(d_{c_1}(c_{d_1}(x)))
B_{B_1}(b_{d_1}(x)) → C_{D_1}(x)
B_{B_1}(b_{e_1}(x)) → B_{D_1}(d_{c_1}(c_{e_1}(x)))
B_{B_1}(b_{e_1}(x)) → C_{E_1}(x)
C_{B_1}(b_{b_1}(x)) → C_{D_1}(d_{c_1}(c_{b_1}(x)))
C_{B_1}(b_{b_1}(x)) → C_{B_1}(x)
C_{B_1}(b_{d_1}(x)) → C_{D_1}(d_{c_1}(c_{d_1}(x)))
C_{B_1}(b_{d_1}(x)) → C_{D_1}(x)
C_{B_1}(b_{e_1}(x)) → C_{D_1}(d_{c_1}(c_{e_1}(x)))
C_{B_1}(b_{e_1}(x)) → C_{E_1}(x)
D_{B_1}(b_{b_1}(x)) → D_{D_1}(d_{c_1}(c_{b_1}(x)))
D_{B_1}(b_{b_1}(x)) → C_{B_1}(x)
D_{B_1}(b_{d_1}(x)) → D_{D_1}(d_{c_1}(c_{d_1}(x)))
D_{B_1}(b_{d_1}(x)) → C_{D_1}(x)
D_{B_1}(b_{e_1}(x)) → D_{D_1}(d_{c_1}(c_{e_1}(x)))
D_{B_1}(b_{e_1}(x)) → C_{E_1}(x)
E_{B_1}(b_{b_1}(x)) → E_{D_1}(d_{c_1}(c_{b_1}(x)))
E_{B_1}(b_{b_1}(x)) → C_{B_1}(x)
E_{B_1}(b_{d_1}(x)) → E_{D_1}(d_{c_1}(c_{d_1}(x)))
E_{B_1}(b_{d_1}(x)) → C_{D_1}(x)
E_{B_1}(b_{e_1}(x)) → E_{D_1}(d_{c_1}(c_{e_1}(x)))
E_{B_1}(b_{e_1}(x)) → C_{E_1}(x)
E_{D_1}(d_{a_1}(x)) → E_{A_1}(x)
A_{E_1}(e_{c_1}(c_{b_1}(x))) → A_{D_1}(d_{a_1}(a_{b_1}(x)))
A_{E_1}(e_{c_1}(c_{b_1}(x))) → D_{A_1}(a_{b_1}(x))
A_{E_1}(e_{c_1}(c_{b_1}(x))) → A_{B_1}(x)
A_{E_1}(e_{c_1}(c_{d_1}(x))) → A_{D_1}(d_{a_1}(a_{d_1}(x)))
A_{E_1}(e_{c_1}(c_{d_1}(x))) → D_{A_1}(a_{d_1}(x))
A_{E_1}(e_{c_1}(c_{d_1}(x))) → A_{D_1}(x)
A_{E_1}(e_{c_1}(c_{e_1}(x))) → A_{D_1}(d_{a_1}(a_{e_1}(x)))
A_{E_1}(e_{c_1}(c_{e_1}(x))) → D_{A_1}(a_{e_1}(x))
A_{E_1}(e_{c_1}(c_{e_1}(x))) → A_{E_1}(x)
B_{E_1}(e_{c_1}(c_{b_1}(x))) → B_{D_1}(d_{a_1}(a_{b_1}(x)))
B_{E_1}(e_{c_1}(c_{b_1}(x))) → D_{A_1}(a_{b_1}(x))
B_{E_1}(e_{c_1}(c_{b_1}(x))) → A_{B_1}(x)
B_{E_1}(e_{c_1}(c_{d_1}(x))) → B_{D_1}(d_{a_1}(a_{d_1}(x)))
B_{E_1}(e_{c_1}(c_{d_1}(x))) → D_{A_1}(a_{d_1}(x))
B_{E_1}(e_{c_1}(c_{d_1}(x))) → A_{D_1}(x)
B_{E_1}(e_{c_1}(c_{e_1}(x))) → B_{D_1}(d_{a_1}(a_{e_1}(x)))
B_{E_1}(e_{c_1}(c_{e_1}(x))) → D_{A_1}(a_{e_1}(x))
B_{E_1}(e_{c_1}(c_{e_1}(x))) → A_{E_1}(x)
C_{E_1}(e_{c_1}(c_{b_1}(x))) → C_{D_1}(d_{a_1}(a_{b_1}(x)))
C_{E_1}(e_{c_1}(c_{b_1}(x))) → D_{A_1}(a_{b_1}(x))
C_{E_1}(e_{c_1}(c_{b_1}(x))) → A_{B_1}(x)
C_{E_1}(e_{c_1}(c_{d_1}(x))) → C_{D_1}(d_{a_1}(a_{d_1}(x)))
C_{E_1}(e_{c_1}(c_{d_1}(x))) → D_{A_1}(a_{d_1}(x))
C_{E_1}(e_{c_1}(c_{d_1}(x))) → A_{D_1}(x)
C_{E_1}(e_{c_1}(c_{e_1}(x))) → C_{D_1}(d_{a_1}(a_{e_1}(x)))
C_{E_1}(e_{c_1}(c_{e_1}(x))) → D_{A_1}(a_{e_1}(x))
C_{E_1}(e_{c_1}(c_{e_1}(x))) → A_{E_1}(x)
D_{E_1}(e_{c_1}(c_{b_1}(x))) → D_{D_1}(d_{a_1}(a_{b_1}(x)))
D_{E_1}(e_{c_1}(c_{b_1}(x))) → D_{A_1}(a_{b_1}(x))
D_{E_1}(e_{c_1}(c_{b_1}(x))) → A_{B_1}(x)
D_{E_1}(e_{c_1}(c_{d_1}(x))) → D_{D_1}(d_{a_1}(a_{d_1}(x)))
D_{E_1}(e_{c_1}(c_{d_1}(x))) → D_{A_1}(a_{d_1}(x))
D_{E_1}(e_{c_1}(c_{d_1}(x))) → A_{D_1}(x)
D_{E_1}(e_{c_1}(c_{e_1}(x))) → D_{D_1}(d_{a_1}(a_{e_1}(x)))
D_{E_1}(e_{c_1}(c_{e_1}(x))) → D_{A_1}(a_{e_1}(x))
D_{E_1}(e_{c_1}(c_{e_1}(x))) → A_{E_1}(x)
D_{A_1}(a_{b_1}(x)) → D_{E_1}(e_{d_1}(d_{b_1}(x)))
D_{A_1}(a_{b_1}(x)) → E_{D_1}(d_{b_1}(x))
D_{A_1}(a_{b_1}(x)) → D_{B_1}(x)
D_{A_1}(a_{d_1}(x)) → D_{E_1}(e_{d_1}(d_{d_1}(x)))
D_{A_1}(a_{d_1}(x)) → E_{D_1}(d_{d_1}(x))
D_{A_1}(a_{d_1}(x)) → D_{D_1}(x)
D_{A_1}(a_{e_1}(x)) → D_{E_1}(e_{d_1}(d_{e_1}(x)))
D_{A_1}(a_{e_1}(x)) → E_{D_1}(d_{e_1}(x))
D_{A_1}(a_{e_1}(x)) → D_{E_1}(x)
E_{A_1}(a_{b_1}(x)) → E_{D_1}(d_{b_1}(x))
E_{A_1}(a_{b_1}(x)) → D_{B_1}(x)
E_{A_1}(a_{d_1}(x)) → E_{D_1}(d_{d_1}(x))
E_{A_1}(a_{d_1}(x)) → D_{D_1}(x)
E_{A_1}(a_{e_1}(x)) → E_{D_1}(d_{e_1}(x))
E_{A_1}(a_{e_1}(x)) → D_{E_1}(x)

The TRS R consists of the following rules:

d_{a_1}(a_{b_1}(b_{b_1}(x))) → d_{c_1}(c_{d_1}(d_{b_1}(x)))
d_{a_1}(a_{b_1}(b_{d_1}(x))) → d_{c_1}(c_{d_1}(d_{d_1}(x)))
d_{a_1}(a_{b_1}(b_{e_1}(x))) → d_{c_1}(c_{d_1}(d_{e_1}(x)))
e_{a_1}(a_{b_1}(b_{b_1}(x))) → e_{c_1}(c_{d_1}(d_{b_1}(x)))
e_{a_1}(a_{b_1}(b_{d_1}(x))) → e_{c_1}(c_{d_1}(d_{d_1}(x)))
e_{a_1}(a_{b_1}(b_{e_1}(x))) → e_{c_1}(c_{d_1}(d_{e_1}(x)))
a_{d_1}(d_{d_1}(d_{a_1}(x))) → a_{b_1}(b_{e_1}(e_{a_1}(x)))
a_{d_1}(d_{d_1}(d_{c_1}(x))) → a_{b_1}(b_{e_1}(e_{c_1}(x)))
a_{d_1}(d_{d_1}(d_{e_1}(x))) → a_{b_1}(b_{e_1}(e_{e_1}(x)))
b_{d_1}(d_{d_1}(d_{a_1}(x))) → b_{b_1}(b_{e_1}(e_{a_1}(x)))
b_{d_1}(d_{d_1}(d_{c_1}(x))) → b_{b_1}(b_{e_1}(e_{c_1}(x)))
b_{d_1}(d_{d_1}(d_{e_1}(x))) → b_{b_1}(b_{e_1}(e_{e_1}(x)))
c_{d_1}(d_{d_1}(d_{a_1}(x))) → c_{b_1}(b_{e_1}(e_{a_1}(x)))
c_{d_1}(d_{d_1}(d_{c_1}(x))) → c_{b_1}(b_{e_1}(e_{c_1}(x)))
c_{d_1}(d_{d_1}(d_{e_1}(x))) → c_{b_1}(b_{e_1}(e_{e_1}(x)))
d_{d_1}(d_{d_1}(d_{a_1}(x))) → d_{b_1}(b_{e_1}(e_{a_1}(x)))
d_{d_1}(d_{d_1}(d_{c_1}(x))) → d_{b_1}(b_{e_1}(e_{c_1}(x)))
d_{d_1}(d_{d_1}(d_{e_1}(x))) → d_{b_1}(b_{e_1}(e_{e_1}(x)))
e_{d_1}(d_{d_1}(d_{a_1}(x))) → e_{b_1}(b_{e_1}(e_{a_1}(x)))
e_{d_1}(d_{d_1}(d_{c_1}(x))) → e_{b_1}(b_{e_1}(e_{c_1}(x)))
e_{d_1}(d_{d_1}(d_{e_1}(x))) → e_{b_1}(b_{e_1}(e_{e_1}(x)))
a_{b_1}(b_{b_1}(x)) → a_{d_1}(d_{c_1}(c_{b_1}(x)))
a_{b_1}(b_{d_1}(x)) → a_{d_1}(d_{c_1}(c_{d_1}(x)))
a_{b_1}(b_{e_1}(x)) → a_{d_1}(d_{c_1}(c_{e_1}(x)))
b_{b_1}(b_{b_1}(x)) → b_{d_1}(d_{c_1}(c_{b_1}(x)))
b_{b_1}(b_{d_1}(x)) → b_{d_1}(d_{c_1}(c_{d_1}(x)))
b_{b_1}(b_{e_1}(x)) → b_{d_1}(d_{c_1}(c_{e_1}(x)))
c_{b_1}(b_{b_1}(x)) → c_{d_1}(d_{c_1}(c_{b_1}(x)))
c_{b_1}(b_{d_1}(x)) → c_{d_1}(d_{c_1}(c_{d_1}(x)))
c_{b_1}(b_{e_1}(x)) → c_{d_1}(d_{c_1}(c_{e_1}(x)))
d_{b_1}(b_{b_1}(x)) → d_{d_1}(d_{c_1}(c_{b_1}(x)))
d_{b_1}(b_{d_1}(x)) → d_{d_1}(d_{c_1}(c_{d_1}(x)))
d_{b_1}(b_{e_1}(x)) → d_{d_1}(d_{c_1}(c_{e_1}(x)))
e_{b_1}(b_{b_1}(x)) → e_{d_1}(d_{c_1}(c_{b_1}(x)))
e_{b_1}(b_{d_1}(x)) → e_{d_1}(d_{c_1}(c_{d_1}(x)))
e_{b_1}(b_{e_1}(x)) → e_{d_1}(d_{c_1}(c_{e_1}(x)))
e_{d_1}(d_{a_1}(x)) → e_{a_1}(x)
e_{d_1}(d_{c_1}(x)) → e_{c_1}(x)
e_{d_1}(d_{e_1}(x)) → e_{e_1}(x)
a_{e_1}(e_{c_1}(c_{b_1}(x))) → a_{d_1}(d_{a_1}(a_{b_1}(x)))
a_{e_1}(e_{c_1}(c_{d_1}(x))) → a_{d_1}(d_{a_1}(a_{d_1}(x)))
a_{e_1}(e_{c_1}(c_{e_1}(x))) → a_{d_1}(d_{a_1}(a_{e_1}(x)))
b_{e_1}(e_{c_1}(c_{b_1}(x))) → b_{d_1}(d_{a_1}(a_{b_1}(x)))
b_{e_1}(e_{c_1}(c_{d_1}(x))) → b_{d_1}(d_{a_1}(a_{d_1}(x)))
b_{e_1}(e_{c_1}(c_{e_1}(x))) → b_{d_1}(d_{a_1}(a_{e_1}(x)))
c_{e_1}(e_{c_1}(c_{b_1}(x))) → c_{d_1}(d_{a_1}(a_{b_1}(x)))
c_{e_1}(e_{c_1}(c_{d_1}(x))) → c_{d_1}(d_{a_1}(a_{d_1}(x)))
c_{e_1}(e_{c_1}(c_{e_1}(x))) → c_{d_1}(d_{a_1}(a_{e_1}(x)))
d_{e_1}(e_{c_1}(c_{b_1}(x))) → d_{d_1}(d_{a_1}(a_{b_1}(x)))
d_{e_1}(e_{c_1}(c_{d_1}(x))) → d_{d_1}(d_{a_1}(a_{d_1}(x)))
d_{e_1}(e_{c_1}(c_{e_1}(x))) → d_{d_1}(d_{a_1}(a_{e_1}(x)))
d_{a_1}(a_{b_1}(x)) → d_{e_1}(e_{d_1}(d_{b_1}(x)))
d_{a_1}(a_{d_1}(x)) → d_{e_1}(e_{d_1}(d_{d_1}(x)))
d_{a_1}(a_{e_1}(x)) → d_{e_1}(e_{d_1}(d_{e_1}(x)))
e_{a_1}(a_{b_1}(x)) → e_{e_1}(e_{d_1}(d_{b_1}(x)))
e_{a_1}(a_{d_1}(x)) → e_{e_1}(e_{d_1}(d_{d_1}(x)))
e_{a_1}(a_{e_1}(x)) → e_{e_1}(e_{d_1}(d_{e_1}(x)))

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 21 less nodes.

(10) Obligation:

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

C_{D_1}(d_{d_1}(d_{a_1}(x))) → C_{B_1}(b_{e_1}(e_{a_1}(x)))
C_{B_1}(b_{b_1}(x)) → C_{B_1}(x)
C_{B_1}(b_{d_1}(x)) → C_{D_1}(x)
C_{D_1}(d_{d_1}(d_{a_1}(x))) → B_{E_1}(e_{a_1}(x))
B_{E_1}(e_{c_1}(c_{b_1}(x))) → B_{D_1}(d_{a_1}(a_{b_1}(x)))
B_{D_1}(d_{d_1}(d_{a_1}(x))) → B_{B_1}(b_{e_1}(e_{a_1}(x)))
B_{B_1}(b_{b_1}(x)) → C_{B_1}(x)
C_{B_1}(b_{e_1}(x)) → C_{E_1}(x)
C_{E_1}(e_{c_1}(c_{b_1}(x))) → C_{D_1}(d_{a_1}(a_{b_1}(x)))
C_{D_1}(d_{d_1}(d_{a_1}(x))) → E_{A_1}(x)
E_{A_1}(a_{b_1}(b_{b_1}(x))) → C_{D_1}(d_{b_1}(x))
C_{D_1}(d_{d_1}(d_{c_1}(x))) → C_{B_1}(b_{e_1}(e_{c_1}(x)))
C_{D_1}(d_{d_1}(d_{c_1}(x))) → B_{E_1}(e_{c_1}(x))
B_{E_1}(e_{c_1}(c_{b_1}(x))) → D_{A_1}(a_{b_1}(x))
D_{A_1}(a_{b_1}(b_{b_1}(x))) → C_{D_1}(d_{b_1}(x))
C_{D_1}(d_{d_1}(d_{e_1}(x))) → C_{B_1}(b_{e_1}(e_{e_1}(x)))
D_{A_1}(a_{b_1}(b_{b_1}(x))) → D_{B_1}(x)
D_{B_1}(b_{b_1}(x)) → C_{B_1}(x)
D_{B_1}(b_{d_1}(x)) → C_{D_1}(x)
D_{B_1}(b_{e_1}(x)) → C_{E_1}(x)
C_{E_1}(e_{c_1}(c_{b_1}(x))) → D_{A_1}(a_{b_1}(x))
D_{A_1}(a_{b_1}(b_{d_1}(x))) → C_{D_1}(d_{d_1}(x))
D_{A_1}(a_{b_1}(b_{d_1}(x))) → D_{D_1}(x)
D_{D_1}(d_{d_1}(d_{a_1}(x))) → D_{B_1}(b_{e_1}(e_{a_1}(x)))
D_{D_1}(d_{d_1}(d_{a_1}(x))) → B_{E_1}(e_{a_1}(x))
B_{E_1}(e_{c_1}(c_{b_1}(x))) → A_{B_1}(x)
A_{B_1}(b_{b_1}(x)) → C_{B_1}(x)
A_{B_1}(b_{d_1}(x)) → C_{D_1}(x)
A_{B_1}(b_{e_1}(x)) → C_{E_1}(x)
C_{E_1}(e_{c_1}(c_{b_1}(x))) → A_{B_1}(x)
C_{E_1}(e_{c_1}(c_{d_1}(x))) → C_{D_1}(d_{a_1}(a_{d_1}(x)))
C_{E_1}(e_{c_1}(c_{d_1}(x))) → D_{A_1}(a_{d_1}(x))
D_{A_1}(a_{b_1}(b_{e_1}(x))) → C_{D_1}(d_{e_1}(x))
D_{A_1}(a_{b_1}(b_{e_1}(x))) → D_{E_1}(x)
D_{E_1}(e_{c_1}(c_{b_1}(x))) → D_{D_1}(d_{a_1}(a_{b_1}(x)))
D_{D_1}(d_{d_1}(d_{a_1}(x))) → E_{A_1}(x)
E_{A_1}(a_{b_1}(b_{b_1}(x))) → D_{B_1}(x)
E_{A_1}(a_{b_1}(b_{d_1}(x))) → C_{D_1}(d_{d_1}(x))
E_{A_1}(a_{b_1}(b_{d_1}(x))) → D_{D_1}(x)
D_{D_1}(d_{d_1}(d_{c_1}(x))) → D_{B_1}(b_{e_1}(e_{c_1}(x)))
D_{D_1}(d_{d_1}(d_{c_1}(x))) → B_{E_1}(e_{c_1}(x))
B_{E_1}(e_{c_1}(c_{d_1}(x))) → B_{D_1}(d_{a_1}(a_{d_1}(x)))
B_{D_1}(d_{d_1}(d_{a_1}(x))) → B_{E_1}(e_{a_1}(x))
B_{E_1}(e_{c_1}(c_{d_1}(x))) → D_{A_1}(a_{d_1}(x))
D_{A_1}(a_{b_1}(x)) → D_{E_1}(e_{d_1}(d_{b_1}(x)))
D_{E_1}(e_{c_1}(c_{b_1}(x))) → D_{A_1}(a_{b_1}(x))
D_{A_1}(a_{b_1}(x)) → E_{D_1}(d_{b_1}(x))
E_{D_1}(d_{d_1}(d_{a_1}(x))) → E_{B_1}(b_{e_1}(e_{a_1}(x)))
E_{B_1}(b_{b_1}(x)) → C_{B_1}(x)
E_{B_1}(b_{d_1}(x)) → C_{D_1}(x)
E_{B_1}(b_{e_1}(x)) → C_{E_1}(x)
C_{E_1}(e_{c_1}(c_{d_1}(x))) → A_{D_1}(x)
A_{D_1}(d_{d_1}(d_{a_1}(x))) → A_{B_1}(b_{e_1}(e_{a_1}(x)))
A_{D_1}(d_{d_1}(d_{a_1}(x))) → B_{E_1}(e_{a_1}(x))
B_{E_1}(e_{c_1}(c_{d_1}(x))) → A_{D_1}(x)
A_{D_1}(d_{d_1}(d_{a_1}(x))) → E_{A_1}(x)
E_{A_1}(a_{b_1}(b_{e_1}(x))) → C_{D_1}(d_{e_1}(x))
E_{A_1}(a_{b_1}(b_{e_1}(x))) → D_{E_1}(x)
D_{E_1}(e_{c_1}(c_{b_1}(x))) → A_{B_1}(x)
D_{E_1}(e_{c_1}(c_{d_1}(x))) → D_{D_1}(d_{a_1}(a_{d_1}(x)))
D_{D_1}(d_{d_1}(d_{e_1}(x))) → D_{B_1}(b_{e_1}(e_{e_1}(x)))
D_{E_1}(e_{c_1}(c_{d_1}(x))) → D_{A_1}(a_{d_1}(x))
D_{A_1}(a_{b_1}(x)) → D_{B_1}(x)
D_{A_1}(a_{d_1}(x)) → D_{E_1}(e_{d_1}(d_{d_1}(x)))
D_{E_1}(e_{c_1}(c_{d_1}(x))) → A_{D_1}(x)
A_{D_1}(d_{d_1}(d_{c_1}(x))) → A_{B_1}(b_{e_1}(e_{c_1}(x)))
A_{D_1}(d_{d_1}(d_{c_1}(x))) → B_{E_1}(e_{c_1}(x))
B_{E_1}(e_{c_1}(c_{e_1}(x))) → B_{D_1}(d_{a_1}(a_{e_1}(x)))
B_{D_1}(d_{d_1}(d_{a_1}(x))) → E_{A_1}(x)
E_{A_1}(a_{b_1}(x)) → E_{D_1}(d_{b_1}(x))
E_{D_1}(d_{d_1}(d_{a_1}(x))) → B_{E_1}(e_{a_1}(x))
B_{E_1}(e_{c_1}(c_{e_1}(x))) → D_{A_1}(a_{e_1}(x))
D_{A_1}(a_{d_1}(x)) → E_{D_1}(d_{d_1}(x))
E_{D_1}(d_{d_1}(d_{a_1}(x))) → E_{A_1}(x)
E_{A_1}(a_{b_1}(x)) → D_{B_1}(x)
E_{A_1}(a_{d_1}(x)) → E_{D_1}(d_{d_1}(x))
E_{D_1}(d_{d_1}(d_{c_1}(x))) → E_{B_1}(b_{e_1}(e_{c_1}(x)))
E_{D_1}(d_{d_1}(d_{c_1}(x))) → B_{E_1}(e_{c_1}(x))
B_{E_1}(e_{c_1}(c_{e_1}(x))) → A_{E_1}(x)
A_{E_1}(e_{c_1}(c_{b_1}(x))) → A_{D_1}(d_{a_1}(a_{b_1}(x)))
A_{D_1}(d_{d_1}(d_{e_1}(x))) → A_{B_1}(b_{e_1}(e_{e_1}(x)))
A_{E_1}(e_{c_1}(c_{b_1}(x))) → D_{A_1}(a_{b_1}(x))
D_{A_1}(a_{d_1}(x)) → D_{D_1}(x)
A_{E_1}(e_{c_1}(c_{b_1}(x))) → A_{B_1}(x)
A_{E_1}(e_{c_1}(c_{d_1}(x))) → A_{D_1}(d_{a_1}(a_{d_1}(x)))
A_{E_1}(e_{c_1}(c_{d_1}(x))) → D_{A_1}(a_{d_1}(x))
A_{E_1}(e_{c_1}(c_{d_1}(x))) → A_{D_1}(x)
A_{E_1}(e_{c_1}(c_{e_1}(x))) → A_{D_1}(d_{a_1}(a_{e_1}(x)))
A_{E_1}(e_{c_1}(c_{e_1}(x))) → D_{A_1}(a_{e_1}(x))
D_{A_1}(a_{e_1}(x)) → D_{E_1}(e_{d_1}(d_{e_1}(x)))
D_{E_1}(e_{c_1}(c_{e_1}(x))) → D_{D_1}(d_{a_1}(a_{e_1}(x)))
D_{E_1}(e_{c_1}(c_{e_1}(x))) → D_{A_1}(a_{e_1}(x))
D_{A_1}(a_{e_1}(x)) → E_{D_1}(d_{e_1}(x))
E_{D_1}(d_{d_1}(d_{e_1}(x))) → E_{B_1}(b_{e_1}(e_{e_1}(x)))
D_{A_1}(a_{e_1}(x)) → D_{E_1}(x)
D_{E_1}(e_{c_1}(c_{e_1}(x))) → A_{E_1}(x)
A_{E_1}(e_{c_1}(c_{e_1}(x))) → A_{E_1}(x)
E_{A_1}(a_{d_1}(x)) → D_{D_1}(x)
E_{A_1}(a_{e_1}(x)) → E_{D_1}(d_{e_1}(x))
E_{A_1}(a_{e_1}(x)) → D_{E_1}(x)
B_{D_1}(d_{d_1}(d_{c_1}(x))) → B_{B_1}(b_{e_1}(e_{c_1}(x)))
B_{B_1}(b_{d_1}(x)) → C_{D_1}(x)
B_{B_1}(b_{e_1}(x)) → C_{E_1}(x)
C_{E_1}(e_{c_1}(c_{e_1}(x))) → C_{D_1}(d_{a_1}(a_{e_1}(x)))
C_{E_1}(e_{c_1}(c_{e_1}(x))) → D_{A_1}(a_{e_1}(x))
C_{E_1}(e_{c_1}(c_{e_1}(x))) → A_{E_1}(x)
B_{D_1}(d_{d_1}(d_{c_1}(x))) → B_{E_1}(e_{c_1}(x))
B_{D_1}(d_{d_1}(d_{e_1}(x))) → B_{B_1}(b_{e_1}(e_{e_1}(x)))

The TRS R consists of the following rules:

d_{a_1}(a_{b_1}(b_{b_1}(x))) → d_{c_1}(c_{d_1}(d_{b_1}(x)))
d_{a_1}(a_{b_1}(b_{d_1}(x))) → d_{c_1}(c_{d_1}(d_{d_1}(x)))
d_{a_1}(a_{b_1}(b_{e_1}(x))) → d_{c_1}(c_{d_1}(d_{e_1}(x)))
e_{a_1}(a_{b_1}(b_{b_1}(x))) → e_{c_1}(c_{d_1}(d_{b_1}(x)))
e_{a_1}(a_{b_1}(b_{d_1}(x))) → e_{c_1}(c_{d_1}(d_{d_1}(x)))
e_{a_1}(a_{b_1}(b_{e_1}(x))) → e_{c_1}(c_{d_1}(d_{e_1}(x)))
a_{d_1}(d_{d_1}(d_{a_1}(x))) → a_{b_1}(b_{e_1}(e_{a_1}(x)))
a_{d_1}(d_{d_1}(d_{c_1}(x))) → a_{b_1}(b_{e_1}(e_{c_1}(x)))
a_{d_1}(d_{d_1}(d_{e_1}(x))) → a_{b_1}(b_{e_1}(e_{e_1}(x)))
b_{d_1}(d_{d_1}(d_{a_1}(x))) → b_{b_1}(b_{e_1}(e_{a_1}(x)))
b_{d_1}(d_{d_1}(d_{c_1}(x))) → b_{b_1}(b_{e_1}(e_{c_1}(x)))
b_{d_1}(d_{d_1}(d_{e_1}(x))) → b_{b_1}(b_{e_1}(e_{e_1}(x)))
c_{d_1}(d_{d_1}(d_{a_1}(x))) → c_{b_1}(b_{e_1}(e_{a_1}(x)))
c_{d_1}(d_{d_1}(d_{c_1}(x))) → c_{b_1}(b_{e_1}(e_{c_1}(x)))
c_{d_1}(d_{d_1}(d_{e_1}(x))) → c_{b_1}(b_{e_1}(e_{e_1}(x)))
d_{d_1}(d_{d_1}(d_{a_1}(x))) → d_{b_1}(b_{e_1}(e_{a_1}(x)))
d_{d_1}(d_{d_1}(d_{c_1}(x))) → d_{b_1}(b_{e_1}(e_{c_1}(x)))
d_{d_1}(d_{d_1}(d_{e_1}(x))) → d_{b_1}(b_{e_1}(e_{e_1}(x)))
e_{d_1}(d_{d_1}(d_{a_1}(x))) → e_{b_1}(b_{e_1}(e_{a_1}(x)))
e_{d_1}(d_{d_1}(d_{c_1}(x))) → e_{b_1}(b_{e_1}(e_{c_1}(x)))
e_{d_1}(d_{d_1}(d_{e_1}(x))) → e_{b_1}(b_{e_1}(e_{e_1}(x)))
a_{b_1}(b_{b_1}(x)) → a_{d_1}(d_{c_1}(c_{b_1}(x)))
a_{b_1}(b_{d_1}(x)) → a_{d_1}(d_{c_1}(c_{d_1}(x)))
a_{b_1}(b_{e_1}(x)) → a_{d_1}(d_{c_1}(c_{e_1}(x)))
b_{b_1}(b_{b_1}(x)) → b_{d_1}(d_{c_1}(c_{b_1}(x)))
b_{b_1}(b_{d_1}(x)) → b_{d_1}(d_{c_1}(c_{d_1}(x)))
b_{b_1}(b_{e_1}(x)) → b_{d_1}(d_{c_1}(c_{e_1}(x)))
c_{b_1}(b_{b_1}(x)) → c_{d_1}(d_{c_1}(c_{b_1}(x)))
c_{b_1}(b_{d_1}(x)) → c_{d_1}(d_{c_1}(c_{d_1}(x)))
c_{b_1}(b_{e_1}(x)) → c_{d_1}(d_{c_1}(c_{e_1}(x)))
d_{b_1}(b_{b_1}(x)) → d_{d_1}(d_{c_1}(c_{b_1}(x)))
d_{b_1}(b_{d_1}(x)) → d_{d_1}(d_{c_1}(c_{d_1}(x)))
d_{b_1}(b_{e_1}(x)) → d_{d_1}(d_{c_1}(c_{e_1}(x)))
e_{b_1}(b_{b_1}(x)) → e_{d_1}(d_{c_1}(c_{b_1}(x)))
e_{b_1}(b_{d_1}(x)) → e_{d_1}(d_{c_1}(c_{d_1}(x)))
e_{b_1}(b_{e_1}(x)) → e_{d_1}(d_{c_1}(c_{e_1}(x)))
e_{d_1}(d_{a_1}(x)) → e_{a_1}(x)
e_{d_1}(d_{c_1}(x)) → e_{c_1}(x)
e_{d_1}(d_{e_1}(x)) → e_{e_1}(x)
a_{e_1}(e_{c_1}(c_{b_1}(x))) → a_{d_1}(d_{a_1}(a_{b_1}(x)))
a_{e_1}(e_{c_1}(c_{d_1}(x))) → a_{d_1}(d_{a_1}(a_{d_1}(x)))
a_{e_1}(e_{c_1}(c_{e_1}(x))) → a_{d_1}(d_{a_1}(a_{e_1}(x)))
b_{e_1}(e_{c_1}(c_{b_1}(x))) → b_{d_1}(d_{a_1}(a_{b_1}(x)))
b_{e_1}(e_{c_1}(c_{d_1}(x))) → b_{d_1}(d_{a_1}(a_{d_1}(x)))
b_{e_1}(e_{c_1}(c_{e_1}(x))) → b_{d_1}(d_{a_1}(a_{e_1}(x)))
c_{e_1}(e_{c_1}(c_{b_1}(x))) → c_{d_1}(d_{a_1}(a_{b_1}(x)))
c_{e_1}(e_{c_1}(c_{d_1}(x))) → c_{d_1}(d_{a_1}(a_{d_1}(x)))
c_{e_1}(e_{c_1}(c_{e_1}(x))) → c_{d_1}(d_{a_1}(a_{e_1}(x)))
d_{e_1}(e_{c_1}(c_{b_1}(x))) → d_{d_1}(d_{a_1}(a_{b_1}(x)))
d_{e_1}(e_{c_1}(c_{d_1}(x))) → d_{d_1}(d_{a_1}(a_{d_1}(x)))
d_{e_1}(e_{c_1}(c_{e_1}(x))) → d_{d_1}(d_{a_1}(a_{e_1}(x)))
d_{a_1}(a_{b_1}(x)) → d_{e_1}(e_{d_1}(d_{b_1}(x)))
d_{a_1}(a_{d_1}(x)) → d_{e_1}(e_{d_1}(d_{d_1}(x)))
d_{a_1}(a_{e_1}(x)) → d_{e_1}(e_{d_1}(d_{e_1}(x)))
e_{a_1}(a_{b_1}(x)) → e_{e_1}(e_{d_1}(d_{b_1}(x)))
e_{a_1}(a_{d_1}(x)) → e_{e_1}(e_{d_1}(d_{d_1}(x)))
e_{a_1}(a_{e_1}(x)) → e_{e_1}(e_{d_1}(d_{e_1}(x)))

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

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

C_{B_1}(b_{b_1}(x)) → C_{B_1}(x)
C_{B_1}(b_{d_1}(x)) → C_{D_1}(x)
C_{D_1}(d_{d_1}(d_{a_1}(x))) → B_{E_1}(e_{a_1}(x))
B_{D_1}(d_{d_1}(d_{a_1}(x))) → B_{B_1}(b_{e_1}(e_{a_1}(x)))
B_{B_1}(b_{b_1}(x)) → C_{B_1}(x)
C_{B_1}(b_{e_1}(x)) → C_{E_1}(x)
C_{E_1}(e_{c_1}(c_{b_1}(x))) → C_{D_1}(d_{a_1}(a_{b_1}(x)))
C_{D_1}(d_{d_1}(d_{a_1}(x))) → E_{A_1}(x)
E_{A_1}(a_{b_1}(b_{b_1}(x))) → C_{D_1}(d_{b_1}(x))
C_{D_1}(d_{d_1}(d_{c_1}(x))) → B_{E_1}(e_{c_1}(x))
D_{A_1}(a_{b_1}(b_{b_1}(x))) → C_{D_1}(d_{b_1}(x))
D_{A_1}(a_{b_1}(b_{b_1}(x))) → D_{B_1}(x)
D_{B_1}(b_{b_1}(x)) → C_{B_1}(x)
D_{B_1}(b_{d_1}(x)) → C_{D_1}(x)
D_{B_1}(b_{e_1}(x)) → C_{E_1}(x)
C_{E_1}(e_{c_1}(c_{b_1}(x))) → D_{A_1}(a_{b_1}(x))
D_{A_1}(a_{b_1}(b_{d_1}(x))) → C_{D_1}(d_{d_1}(x))
D_{A_1}(a_{b_1}(b_{d_1}(x))) → D_{D_1}(x)
D_{D_1}(d_{d_1}(d_{a_1}(x))) → D_{B_1}(b_{e_1}(e_{a_1}(x)))
D_{D_1}(d_{d_1}(d_{a_1}(x))) → B_{E_1}(e_{a_1}(x))
B_{E_1}(e_{c_1}(c_{b_1}(x))) → A_{B_1}(x)
A_{B_1}(b_{b_1}(x)) → C_{B_1}(x)
A_{B_1}(b_{d_1}(x)) → C_{D_1}(x)
C_{E_1}(e_{c_1}(c_{b_1}(x))) → A_{B_1}(x)
C_{E_1}(e_{c_1}(c_{d_1}(x))) → C_{D_1}(d_{a_1}(a_{d_1}(x)))
C_{E_1}(e_{c_1}(c_{d_1}(x))) → D_{A_1}(a_{d_1}(x))
D_{A_1}(a_{b_1}(b_{e_1}(x))) → C_{D_1}(d_{e_1}(x))
D_{A_1}(a_{b_1}(b_{e_1}(x))) → D_{E_1}(x)
D_{D_1}(d_{d_1}(d_{a_1}(x))) → E_{A_1}(x)
E_{A_1}(a_{b_1}(b_{b_1}(x))) → D_{B_1}(x)
E_{A_1}(a_{b_1}(b_{d_1}(x))) → C_{D_1}(d_{d_1}(x))
E_{A_1}(a_{b_1}(b_{d_1}(x))) → D_{D_1}(x)
D_{D_1}(d_{d_1}(d_{c_1}(x))) → D_{B_1}(b_{e_1}(e_{c_1}(x)))
D_{D_1}(d_{d_1}(d_{c_1}(x))) → B_{E_1}(e_{c_1}(x))
B_{D_1}(d_{d_1}(d_{a_1}(x))) → B_{E_1}(e_{a_1}(x))
D_{E_1}(e_{c_1}(c_{b_1}(x))) → D_{A_1}(a_{b_1}(x))
E_{B_1}(b_{b_1}(x)) → C_{B_1}(x)
E_{B_1}(b_{d_1}(x)) → C_{D_1}(x)
E_{B_1}(b_{e_1}(x)) → C_{E_1}(x)
C_{E_1}(e_{c_1}(c_{d_1}(x))) → A_{D_1}(x)
A_{D_1}(d_{d_1}(d_{a_1}(x))) → A_{B_1}(b_{e_1}(e_{a_1}(x)))
A_{D_1}(d_{d_1}(d_{a_1}(x))) → B_{E_1}(e_{a_1}(x))
B_{E_1}(e_{c_1}(c_{d_1}(x))) → A_{D_1}(x)
E_{A_1}(a_{b_1}(b_{e_1}(x))) → C_{D_1}(d_{e_1}(x))
E_{A_1}(a_{b_1}(b_{e_1}(x))) → D_{E_1}(x)
D_{E_1}(e_{c_1}(c_{b_1}(x))) → A_{B_1}(x)
D_{D_1}(d_{d_1}(d_{e_1}(x))) → D_{B_1}(b_{e_1}(e_{e_1}(x)))
D_{E_1}(e_{c_1}(c_{d_1}(x))) → D_{A_1}(a_{d_1}(x))
D_{A_1}(a_{b_1}(x)) → D_{B_1}(x)
D_{E_1}(e_{c_1}(c_{d_1}(x))) → A_{D_1}(x)
A_{D_1}(d_{d_1}(d_{c_1}(x))) → A_{B_1}(b_{e_1}(e_{c_1}(x)))
A_{D_1}(d_{d_1}(d_{c_1}(x))) → B_{E_1}(e_{c_1}(x))
B_{D_1}(d_{d_1}(d_{a_1}(x))) → E_{A_1}(x)
E_{A_1}(a_{b_1}(x)) → E_{D_1}(d_{b_1}(x))
E_{D_1}(d_{d_1}(d_{a_1}(x))) → B_{E_1}(e_{a_1}(x))
E_{D_1}(d_{d_1}(d_{a_1}(x))) → E_{A_1}(x)
E_{A_1}(a_{b_1}(x)) → D_{B_1}(x)
E_{A_1}(a_{d_1}(x)) → E_{D_1}(d_{d_1}(x))
E_{D_1}(d_{d_1}(d_{c_1}(x))) → B_{E_1}(e_{c_1}(x))
A_{E_1}(e_{c_1}(c_{b_1}(x))) → A_{D_1}(d_{a_1}(a_{b_1}(x)))
A_{D_1}(d_{d_1}(d_{e_1}(x))) → A_{B_1}(b_{e_1}(e_{e_1}(x)))
A_{E_1}(e_{c_1}(c_{b_1}(x))) → D_{A_1}(a_{b_1}(x))
A_{E_1}(e_{c_1}(c_{b_1}(x))) → A_{B_1}(x)
A_{E_1}(e_{c_1}(c_{d_1}(x))) → A_{D_1}(d_{a_1}(a_{d_1}(x)))
A_{E_1}(e_{c_1}(c_{d_1}(x))) → D_{A_1}(a_{d_1}(x))
A_{E_1}(e_{c_1}(c_{d_1}(x))) → A_{D_1}(x)
A_{E_1}(e_{c_1}(c_{e_1}(x))) → A_{D_1}(d_{a_1}(a_{e_1}(x)))
A_{E_1}(e_{c_1}(c_{e_1}(x))) → D_{A_1}(a_{e_1}(x))
D_{E_1}(e_{c_1}(c_{e_1}(x))) → D_{A_1}(a_{e_1}(x))
D_{E_1}(e_{c_1}(c_{e_1}(x))) → A_{E_1}(x)
A_{E_1}(e_{c_1}(c_{e_1}(x))) → A_{E_1}(x)
E_{A_1}(a_{d_1}(x)) → D_{D_1}(x)
E_{A_1}(a_{e_1}(x)) → E_{D_1}(d_{e_1}(x))
E_{A_1}(a_{e_1}(x)) → D_{E_1}(x)
B_{D_1}(d_{d_1}(d_{c_1}(x))) → B_{B_1}(b_{e_1}(e_{c_1}(x)))
B_{B_1}(b_{d_1}(x)) → C_{D_1}(x)
C_{E_1}(e_{c_1}(c_{e_1}(x))) → C_{D_1}(d_{a_1}(a_{e_1}(x)))
C_{E_1}(e_{c_1}(c_{e_1}(x))) → D_{A_1}(a_{e_1}(x))
C_{E_1}(e_{c_1}(c_{e_1}(x))) → A_{E_1}(x)
B_{D_1}(d_{d_1}(d_{c_1}(x))) → B_{E_1}(e_{c_1}(x))
B_{D_1}(d_{d_1}(d_{e_1}(x))) → B_{B_1}(b_{e_1}(e_{e_1}(x)))

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( B_{E_1}(x₁) ) = max{0, 2x₁ - 2}

POL( D_{E_1}(x₁) ) = 2x₁

POL( A_{B_1}(x₁) ) = 2x₁

POL( b_{e_1}(x₁) ) = x₁

POL( A_{D_1}(x₁) ) = max{0, 2x₁ - 1}

POL( d_{e_1}(x₁) ) = x₁

POL( B_{B_1}(x₁) ) = 2x₁

POL( B_{D_1}(x₁) ) = 2x₁

POL( C_{B_1}(x₁) ) = 2x₁ + 2

POL( C_{D_1}(x₁) ) = 2x₁

POL( D_{B_1}(x₁) ) = 2x₁ + 2

POL( D_{D_1}(x₁) ) = 2x₁ + 2

POL( E_{B_1}(x₁) ) = 2x₁ + 2

POL( b_{d_1}(x₁) ) = x₁ + 1

POL( d_{b_1}(x₁) ) = x₁ + 2

POL( D_{A_1}(x₁) ) = 2x₁

POL( a_{b_1}(x₁) ) = x₁ + 2

POL( a_{d_1}(x₁) ) = x₁ + 1

POL( E_{D_1}(x₁) ) = 2x₁

POL( d_{d_1}(x₁) ) = x₁ + 1

POL( d_{a_1}(x₁) ) = x₁

POL( c_{b_1}(x₁) ) = x₁ + 2

POL( d_{c_1}(x₁) ) = x₁ + 1

POL( e_{a_1}(x₁) ) = x₁

POL( b_{b_1}(x₁) ) = x₁ + 2

POL( e_{c_1}(x₁) ) = x₁ + 1

POL( c_{d_1}(x₁) ) = x₁ + 1

POL( e_{e_1}(x₁) ) = max{0, -2}

POL( e_{d_1}(x₁) ) = x₁

POL( a_{e_1}(x₁) ) = x₁

POL( e_{b_1}(x₁) ) = x₁ + 1

POL( c_{e_1}(x₁) ) = x₁

POL( C_{E_1}(x₁) ) = 2x₁

POL( E_{A_1}(x₁) ) = 2x₁ + 1

POL( A_{E_1}(x₁) ) = 2x₁

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

e_{a_1}(a_{b_1}(b_{b_1}(x))) → e_{c_1}(c_{d_1}(d_{b_1}(x)))
e_{a_1}(a_{b_1}(b_{d_1}(x))) → e_{c_1}(c_{d_1}(d_{d_1}(x)))
e_{a_1}(a_{b_1}(b_{e_1}(x))) → e_{c_1}(c_{d_1}(d_{e_1}(x)))
e_{a_1}(a_{b_1}(x)) → e_{e_1}(e_{d_1}(d_{b_1}(x)))
e_{a_1}(a_{d_1}(x)) → e_{e_1}(e_{d_1}(d_{d_1}(x)))
e_{a_1}(a_{e_1}(x)) → e_{e_1}(e_{d_1}(d_{e_1}(x)))
b_{e_1}(e_{c_1}(c_{b_1}(x))) → b_{d_1}(d_{a_1}(a_{b_1}(x)))
b_{e_1}(e_{c_1}(c_{d_1}(x))) → b_{d_1}(d_{a_1}(a_{d_1}(x)))
b_{e_1}(e_{c_1}(c_{e_1}(x))) → b_{d_1}(d_{a_1}(a_{e_1}(x)))
a_{b_1}(b_{b_1}(x)) → a_{d_1}(d_{c_1}(c_{b_1}(x)))
a_{b_1}(b_{d_1}(x)) → a_{d_1}(d_{c_1}(c_{d_1}(x)))
a_{b_1}(b_{e_1}(x)) → a_{d_1}(d_{c_1}(c_{e_1}(x)))
d_{a_1}(a_{b_1}(b_{b_1}(x))) → d_{c_1}(c_{d_1}(d_{b_1}(x)))
d_{a_1}(a_{b_1}(b_{d_1}(x))) → d_{c_1}(c_{d_1}(d_{d_1}(x)))
d_{a_1}(a_{b_1}(b_{e_1}(x))) → d_{c_1}(c_{d_1}(d_{e_1}(x)))
d_{a_1}(a_{b_1}(x)) → d_{e_1}(e_{d_1}(d_{b_1}(x)))
d_{a_1}(a_{d_1}(x)) → d_{e_1}(e_{d_1}(d_{d_1}(x)))
d_{a_1}(a_{e_1}(x)) → d_{e_1}(e_{d_1}(d_{e_1}(x)))
d_{b_1}(b_{b_1}(x)) → d_{d_1}(d_{c_1}(c_{b_1}(x)))
d_{b_1}(b_{d_1}(x)) → d_{d_1}(d_{c_1}(c_{d_1}(x)))
d_{b_1}(b_{e_1}(x)) → d_{d_1}(d_{c_1}(c_{e_1}(x)))
d_{d_1}(d_{d_1}(d_{a_1}(x))) → d_{b_1}(b_{e_1}(e_{a_1}(x)))
d_{d_1}(d_{d_1}(d_{c_1}(x))) → d_{b_1}(b_{e_1}(e_{c_1}(x)))
d_{d_1}(d_{d_1}(d_{e_1}(x))) → d_{b_1}(b_{e_1}(e_{e_1}(x)))
a_{d_1}(d_{d_1}(d_{a_1}(x))) → a_{b_1}(b_{e_1}(e_{a_1}(x)))
a_{d_1}(d_{d_1}(d_{c_1}(x))) → a_{b_1}(b_{e_1}(e_{c_1}(x)))
a_{d_1}(d_{d_1}(d_{e_1}(x))) → a_{b_1}(b_{e_1}(e_{e_1}(x)))
d_{e_1}(e_{c_1}(c_{b_1}(x))) → d_{d_1}(d_{a_1}(a_{b_1}(x)))
d_{e_1}(e_{c_1}(c_{d_1}(x))) → d_{d_1}(d_{a_1}(a_{d_1}(x)))
d_{e_1}(e_{c_1}(c_{e_1}(x))) → d_{d_1}(d_{a_1}(a_{e_1}(x)))
e_{d_1}(d_{d_1}(d_{a_1}(x))) → e_{b_1}(b_{e_1}(e_{a_1}(x)))
e_{d_1}(d_{d_1}(d_{c_1}(x))) → e_{b_1}(b_{e_1}(e_{c_1}(x)))
e_{d_1}(d_{d_1}(d_{e_1}(x))) → e_{b_1}(b_{e_1}(e_{e_1}(x)))
e_{d_1}(d_{a_1}(x)) → e_{a_1}(x)
e_{d_1}(d_{c_1}(x)) → e_{c_1}(x)
e_{d_1}(d_{e_1}(x)) → e_{e_1}(x)
a_{e_1}(e_{c_1}(c_{b_1}(x))) → a_{d_1}(d_{a_1}(a_{b_1}(x)))
a_{e_1}(e_{c_1}(c_{d_1}(x))) → a_{d_1}(d_{a_1}(a_{d_1}(x)))
a_{e_1}(e_{c_1}(c_{e_1}(x))) → a_{d_1}(d_{a_1}(a_{e_1}(x)))
c_{b_1}(b_{b_1}(x)) → c_{d_1}(d_{c_1}(c_{b_1}(x)))
c_{b_1}(b_{d_1}(x)) → c_{d_1}(d_{c_1}(c_{d_1}(x)))
c_{b_1}(b_{e_1}(x)) → c_{d_1}(d_{c_1}(c_{e_1}(x)))
c_{d_1}(d_{d_1}(d_{a_1}(x))) → c_{b_1}(b_{e_1}(e_{a_1}(x)))
c_{d_1}(d_{d_1}(d_{c_1}(x))) → c_{b_1}(b_{e_1}(e_{c_1}(x)))
c_{d_1}(d_{d_1}(d_{e_1}(x))) → c_{b_1}(b_{e_1}(e_{e_1}(x)))
b_{d_1}(d_{d_1}(d_{a_1}(x))) → b_{b_1}(b_{e_1}(e_{a_1}(x)))
b_{d_1}(d_{d_1}(d_{c_1}(x))) → b_{b_1}(b_{e_1}(e_{c_1}(x)))
b_{d_1}(d_{d_1}(d_{e_1}(x))) → b_{b_1}(b_{e_1}(e_{e_1}(x)))
c_{e_1}(e_{c_1}(c_{b_1}(x))) → c_{d_1}(d_{a_1}(a_{b_1}(x)))
c_{e_1}(e_{c_1}(c_{d_1}(x))) → c_{d_1}(d_{a_1}(a_{d_1}(x)))
c_{e_1}(e_{c_1}(c_{e_1}(x))) → c_{d_1}(d_{a_1}(a_{e_1}(x)))
e_{b_1}(b_{b_1}(x)) → e_{d_1}(d_{c_1}(c_{b_1}(x)))
e_{b_1}(b_{d_1}(x)) → e_{d_1}(d_{c_1}(c_{d_1}(x)))
e_{b_1}(b_{e_1}(x)) → e_{d_1}(d_{c_1}(c_{e_1}(x)))
b_{b_1}(b_{b_1}(x)) → b_{d_1}(d_{c_1}(c_{b_1}(x)))
b_{b_1}(b_{d_1}(x)) → b_{d_1}(d_{c_1}(c_{d_1}(x)))
b_{b_1}(b_{e_1}(x)) → b_{d_1}(d_{c_1}(c_{e_1}(x)))

(12) Obligation:

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

C_{D_1}(d_{d_1}(d_{a_1}(x))) → C_{B_1}(b_{e_1}(e_{a_1}(x)))
B_{E_1}(e_{c_1}(c_{b_1}(x))) → B_{D_1}(d_{a_1}(a_{b_1}(x)))
C_{D_1}(d_{d_1}(d_{c_1}(x))) → C_{B_1}(b_{e_1}(e_{c_1}(x)))
B_{E_1}(e_{c_1}(c_{b_1}(x))) → D_{A_1}(a_{b_1}(x))
C_{D_1}(d_{d_1}(d_{e_1}(x))) → C_{B_1}(b_{e_1}(e_{e_1}(x)))
A_{B_1}(b_{e_1}(x)) → C_{E_1}(x)
D_{E_1}(e_{c_1}(c_{b_1}(x))) → D_{D_1}(d_{a_1}(a_{b_1}(x)))
B_{E_1}(e_{c_1}(c_{d_1}(x))) → B_{D_1}(d_{a_1}(a_{d_1}(x)))
B_{E_1}(e_{c_1}(c_{d_1}(x))) → D_{A_1}(a_{d_1}(x))
D_{A_1}(a_{b_1}(x)) → D_{E_1}(e_{d_1}(d_{b_1}(x)))
D_{A_1}(a_{b_1}(x)) → E_{D_1}(d_{b_1}(x))
E_{D_1}(d_{d_1}(d_{a_1}(x))) → E_{B_1}(b_{e_1}(e_{a_1}(x)))
A_{D_1}(d_{d_1}(d_{a_1}(x))) → E_{A_1}(x)
D_{E_1}(e_{c_1}(c_{d_1}(x))) → D_{D_1}(d_{a_1}(a_{d_1}(x)))
D_{A_1}(a_{d_1}(x)) → D_{E_1}(e_{d_1}(d_{d_1}(x)))
B_{E_1}(e_{c_1}(c_{e_1}(x))) → B_{D_1}(d_{a_1}(a_{e_1}(x)))
B_{E_1}(e_{c_1}(c_{e_1}(x))) → D_{A_1}(a_{e_1}(x))
D_{A_1}(a_{d_1}(x)) → E_{D_1}(d_{d_1}(x))
E_{D_1}(d_{d_1}(d_{c_1}(x))) → E_{B_1}(b_{e_1}(e_{c_1}(x)))
B_{E_1}(e_{c_1}(c_{e_1}(x))) → A_{E_1}(x)
D_{A_1}(a_{d_1}(x)) → D_{D_1}(x)
D_{A_1}(a_{e_1}(x)) → D_{E_1}(e_{d_1}(d_{e_1}(x)))
D_{E_1}(e_{c_1}(c_{e_1}(x))) → D_{D_1}(d_{a_1}(a_{e_1}(x)))
D_{A_1}(a_{e_1}(x)) → E_{D_1}(d_{e_1}(x))
E_{D_1}(d_{d_1}(d_{e_1}(x))) → E_{B_1}(b_{e_1}(e_{e_1}(x)))
D_{A_1}(a_{e_1}(x)) → D_{E_1}(x)
B_{B_1}(b_{e_1}(x)) → C_{E_1}(x)

The TRS R consists of the following rules:

d_{a_1}(a_{b_1}(b_{b_1}(x))) → d_{c_1}(c_{d_1}(d_{b_1}(x)))
d_{a_1}(a_{b_1}(b_{d_1}(x))) → d_{c_1}(c_{d_1}(d_{d_1}(x)))
d_{a_1}(a_{b_1}(b_{e_1}(x))) → d_{c_1}(c_{d_1}(d_{e_1}(x)))
e_{a_1}(a_{b_1}(b_{b_1}(x))) → e_{c_1}(c_{d_1}(d_{b_1}(x)))
e_{a_1}(a_{b_1}(b_{d_1}(x))) → e_{c_1}(c_{d_1}(d_{d_1}(x)))
e_{a_1}(a_{b_1}(b_{e_1}(x))) → e_{c_1}(c_{d_1}(d_{e_1}(x)))
a_{d_1}(d_{d_1}(d_{a_1}(x))) → a_{b_1}(b_{e_1}(e_{a_1}(x)))
a_{d_1}(d_{d_1}(d_{c_1}(x))) → a_{b_1}(b_{e_1}(e_{c_1}(x)))
a_{d_1}(d_{d_1}(d_{e_1}(x))) → a_{b_1}(b_{e_1}(e_{e_1}(x)))
b_{d_1}(d_{d_1}(d_{a_1}(x))) → b_{b_1}(b_{e_1}(e_{a_1}(x)))
b_{d_1}(d_{d_1}(d_{c_1}(x))) → b_{b_1}(b_{e_1}(e_{c_1}(x)))
b_{d_1}(d_{d_1}(d_{e_1}(x))) → b_{b_1}(b_{e_1}(e_{e_1}(x)))
c_{d_1}(d_{d_1}(d_{a_1}(x))) → c_{b_1}(b_{e_1}(e_{a_1}(x)))
c_{d_1}(d_{d_1}(d_{c_1}(x))) → c_{b_1}(b_{e_1}(e_{c_1}(x)))
c_{d_1}(d_{d_1}(d_{e_1}(x))) → c_{b_1}(b_{e_1}(e_{e_1}(x)))
d_{d_1}(d_{d_1}(d_{a_1}(x))) → d_{b_1}(b_{e_1}(e_{a_1}(x)))
d_{d_1}(d_{d_1}(d_{c_1}(x))) → d_{b_1}(b_{e_1}(e_{c_1}(x)))
d_{d_1}(d_{d_1}(d_{e_1}(x))) → d_{b_1}(b_{e_1}(e_{e_1}(x)))
e_{d_1}(d_{d_1}(d_{a_1}(x))) → e_{b_1}(b_{e_1}(e_{a_1}(x)))
e_{d_1}(d_{d_1}(d_{c_1}(x))) → e_{b_1}(b_{e_1}(e_{c_1}(x)))
e_{d_1}(d_{d_1}(d_{e_1}(x))) → e_{b_1}(b_{e_1}(e_{e_1}(x)))
a_{b_1}(b_{b_1}(x)) → a_{d_1}(d_{c_1}(c_{b_1}(x)))
a_{b_1}(b_{d_1}(x)) → a_{d_1}(d_{c_1}(c_{d_1}(x)))
a_{b_1}(b_{e_1}(x)) → a_{d_1}(d_{c_1}(c_{e_1}(x)))
b_{b_1}(b_{b_1}(x)) → b_{d_1}(d_{c_1}(c_{b_1}(x)))
b_{b_1}(b_{d_1}(x)) → b_{d_1}(d_{c_1}(c_{d_1}(x)))
b_{b_1}(b_{e_1}(x)) → b_{d_1}(d_{c_1}(c_{e_1}(x)))
c_{b_1}(b_{b_1}(x)) → c_{d_1}(d_{c_1}(c_{b_1}(x)))
c_{b_1}(b_{d_1}(x)) → c_{d_1}(d_{c_1}(c_{d_1}(x)))
c_{b_1}(b_{e_1}(x)) → c_{d_1}(d_{c_1}(c_{e_1}(x)))
d_{b_1}(b_{b_1}(x)) → d_{d_1}(d_{c_1}(c_{b_1}(x)))
d_{b_1}(b_{d_1}(x)) → d_{d_1}(d_{c_1}(c_{d_1}(x)))
d_{b_1}(b_{e_1}(x)) → d_{d_1}(d_{c_1}(c_{e_1}(x)))
e_{b_1}(b_{b_1}(x)) → e_{d_1}(d_{c_1}(c_{b_1}(x)))
e_{b_1}(b_{d_1}(x)) → e_{d_1}(d_{c_1}(c_{d_1}(x)))
e_{b_1}(b_{e_1}(x)) → e_{d_1}(d_{c_1}(c_{e_1}(x)))
e_{d_1}(d_{a_1}(x)) → e_{a_1}(x)
e_{d_1}(d_{c_1}(x)) → e_{c_1}(x)
e_{d_1}(d_{e_1}(x)) → e_{e_1}(x)
a_{e_1}(e_{c_1}(c_{b_1}(x))) → a_{d_1}(d_{a_1}(a_{b_1}(x)))
a_{e_1}(e_{c_1}(c_{d_1}(x))) → a_{d_1}(d_{a_1}(a_{d_1}(x)))
a_{e_1}(e_{c_1}(c_{e_1}(x))) → a_{d_1}(d_{a_1}(a_{e_1}(x)))
b_{e_1}(e_{c_1}(c_{b_1}(x))) → b_{d_1}(d_{a_1}(a_{b_1}(x)))
b_{e_1}(e_{c_1}(c_{d_1}(x))) → b_{d_1}(d_{a_1}(a_{d_1}(x)))
b_{e_1}(e_{c_1}(c_{e_1}(x))) → b_{d_1}(d_{a_1}(a_{e_1}(x)))
c_{e_1}(e_{c_1}(c_{b_1}(x))) → c_{d_1}(d_{a_1}(a_{b_1}(x)))
c_{e_1}(e_{c_1}(c_{d_1}(x))) → c_{d_1}(d_{a_1}(a_{d_1}(x)))
c_{e_1}(e_{c_1}(c_{e_1}(x))) → c_{d_1}(d_{a_1}(a_{e_1}(x)))
d_{e_1}(e_{c_1}(c_{b_1}(x))) → d_{d_1}(d_{a_1}(a_{b_1}(x)))
d_{e_1}(e_{c_1}(c_{d_1}(x))) → d_{d_1}(d_{a_1}(a_{d_1}(x)))
d_{e_1}(e_{c_1}(c_{e_1}(x))) → d_{d_1}(d_{a_1}(a_{e_1}(x)))
d_{a_1}(a_{b_1}(x)) → d_{e_1}(e_{d_1}(d_{b_1}(x)))
d_{a_1}(a_{d_1}(x)) → d_{e_1}(e_{d_1}(d_{d_1}(x)))
d_{a_1}(a_{e_1}(x)) → d_{e_1}(e_{d_1}(d_{e_1}(x)))
e_{a_1}(a_{b_1}(x)) → e_{e_1}(e_{d_1}(d_{b_1}(x)))
e_{a_1}(a_{d_1}(x)) → e_{e_1}(e_{d_1}(d_{d_1}(x)))
e_{a_1}(a_{e_1}(x)) → e_{e_1}(e_{d_1}(d_{e_1}(x)))

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

(13) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 27 less nodes.

(0) Obligation:

(1) FlatCCProof (EQUIVALENT transformation)

(2) Obligation:

(3) RootLabelingProof (EQUIVALENT transformation)

(4) Obligation:

(5) QTRSRRRProof (EQUIVALENT transformation)

(6) Obligation:

(7) DependencyPairsProof (EQUIVALENT transformation)

(8) Obligation:

(9) DependencyGraphProof (EQUIVALENT transformation)

(10) Obligation:

(11) QDPOrderProof (EQUIVALENT transformation)

(12) Obligation:

(13) DependencyGraphProof (EQUIVALENT transformation)

(14) TRUE