The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

0 CpxTRS
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)
↳2 CdtProblem
    ↳3 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳4 CdtProblem
    ↳5 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳6 CdtProblem
    ↳7 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳8 CdtProblem
    ↳9 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳10 CdtProblem
    ↳11 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳12 CdtProblem
    ↳13 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳14 CdtProblem
    ↳15 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳16 CdtProblem
    ↳17 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳18 CdtProblem
    ↳19 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳20 CdtProblem
    ↳21 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳22 CdtProblem
↳23 CpxTrsMatchBoundsProof (⇔, 0 ms)
↳24 BOUNDS(O(1), O(n^1))

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

a(b(x)) → b(a(a(x)))
b(c(x)) → c(b(b(x)))
c(a(x)) → a(c(c(x)))
u(a(x)) → x
v(b(x)) → x
w(c(x)) → x
a(u(x)) → x
b(v(x)) → x
c(w(x)) → x

Rewrite Strategy: INNERMOST

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(b(z0)) → b(a(a(z0)))
a(u(z0)) → z0
b(c(z0)) → c(b(b(z0)))
b(v(z0)) → z0
c(a(z0)) → a(c(c(z0)))
c(w(z0)) → z0
u(a(z0)) → z0
v(b(z0)) → z0
w(c(z0)) → z0
Tuples:

A(b(z0)) → c1(B(a(a(z0))), A(a(z0)), A(z0))
B(c(z0)) → c3(C(b(b(z0))), B(b(z0)), B(z0))
C(a(z0)) → c5(A(c(c(z0))), C(c(z0)), C(z0))
S tuples:

A(b(z0)) → c1(B(a(a(z0))), A(a(z0)), A(z0))
B(c(z0)) → c3(C(b(b(z0))), B(b(z0)), B(z0))
C(a(z0)) → c5(A(c(c(z0))), C(c(z0)), C(z0))
K tuples:none
Defined Rule Symbols:

a, b, c, u, v, w

Defined Pair Symbols:

A, B, C

Compound Symbols:

c1, c3, c5

(3) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace A(b(z0)) → c1(B(a(a(z0))), A(a(z0)), A(z0)) by

A(b(b(z0))) → c1(B(a(b(a(a(z0))))), A(a(b(z0))), A(b(z0)))
A(b(u(z0))) → c1(B(a(z0)), A(a(u(z0))), A(u(z0)))
A(b(x0)) → c1

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(b(z0)) → b(a(a(z0)))
a(u(z0)) → z0
b(c(z0)) → c(b(b(z0)))
b(v(z0)) → z0
c(a(z0)) → a(c(c(z0)))
c(w(z0)) → z0
u(a(z0)) → z0
v(b(z0)) → z0
w(c(z0)) → z0
Tuples:

B(c(z0)) → c3(C(b(b(z0))), B(b(z0)), B(z0))
C(a(z0)) → c5(A(c(c(z0))), C(c(z0)), C(z0))
A(b(b(z0))) → c1(B(a(b(a(a(z0))))), A(a(b(z0))), A(b(z0)))
A(b(u(z0))) → c1(B(a(z0)), A(a(u(z0))), A(u(z0)))
A(b(x0)) → c1
S tuples:

B(c(z0)) → c3(C(b(b(z0))), B(b(z0)), B(z0))
C(a(z0)) → c5(A(c(c(z0))), C(c(z0)), C(z0))
A(b(b(z0))) → c1(B(a(b(a(a(z0))))), A(a(b(z0))), A(b(z0)))
A(b(u(z0))) → c1(B(a(z0)), A(a(u(z0))), A(u(z0)))
A(b(x0)) → c1
K tuples:none
Defined Rule Symbols:

a, b, c, u, v, w

Defined Pair Symbols:

B, C, A

Compound Symbols:

c3, c5, c1, c1

(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing nodes:

A(b(x0)) → c1

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(b(z0)) → b(a(a(z0)))
a(u(z0)) → z0
b(c(z0)) → c(b(b(z0)))
b(v(z0)) → z0
c(a(z0)) → a(c(c(z0)))
c(w(z0)) → z0
u(a(z0)) → z0
v(b(z0)) → z0
w(c(z0)) → z0
Tuples:

B(c(z0)) → c3(C(b(b(z0))), B(b(z0)), B(z0))
C(a(z0)) → c5(A(c(c(z0))), C(c(z0)), C(z0))
A(b(b(z0))) → c1(B(a(b(a(a(z0))))), A(a(b(z0))), A(b(z0)))
A(b(u(z0))) → c1(B(a(z0)), A(a(u(z0))), A(u(z0)))
S tuples:

B(c(z0)) → c3(C(b(b(z0))), B(b(z0)), B(z0))
C(a(z0)) → c5(A(c(c(z0))), C(c(z0)), C(z0))
A(b(b(z0))) → c1(B(a(b(a(a(z0))))), A(a(b(z0))), A(b(z0)))
A(b(u(z0))) → c1(B(a(z0)), A(a(u(z0))), A(u(z0)))
K tuples:none
Defined Rule Symbols:

a, b, c, u, v, w

Defined Pair Symbols:

B, C, A

Compound Symbols:

c3, c5, c1

(7) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace B(c(z0)) → c3(C(b(b(z0))), B(b(z0)), B(z0)) by

B(c(c(z0))) → c3(C(b(c(b(b(z0))))), B(b(c(z0))), B(c(z0)))
B(c(v(z0))) → c3(C(b(z0)), B(b(v(z0))), B(v(z0)))
B(c(x0)) → c3

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(b(z0)) → b(a(a(z0)))
a(u(z0)) → z0
b(c(z0)) → c(b(b(z0)))
b(v(z0)) → z0
c(a(z0)) → a(c(c(z0)))
c(w(z0)) → z0
u(a(z0)) → z0
v(b(z0)) → z0
w(c(z0)) → z0
Tuples:

C(a(z0)) → c5(A(c(c(z0))), C(c(z0)), C(z0))
A(b(b(z0))) → c1(B(a(b(a(a(z0))))), A(a(b(z0))), A(b(z0)))
A(b(u(z0))) → c1(B(a(z0)), A(a(u(z0))), A(u(z0)))
B(c(c(z0))) → c3(C(b(c(b(b(z0))))), B(b(c(z0))), B(c(z0)))
B(c(v(z0))) → c3(C(b(z0)), B(b(v(z0))), B(v(z0)))
B(c(x0)) → c3
S tuples:

C(a(z0)) → c5(A(c(c(z0))), C(c(z0)), C(z0))
A(b(b(z0))) → c1(B(a(b(a(a(z0))))), A(a(b(z0))), A(b(z0)))
A(b(u(z0))) → c1(B(a(z0)), A(a(u(z0))), A(u(z0)))
B(c(c(z0))) → c3(C(b(c(b(b(z0))))), B(b(c(z0))), B(c(z0)))
B(c(v(z0))) → c3(C(b(z0)), B(b(v(z0))), B(v(z0)))
B(c(x0)) → c3
K tuples:none
Defined Rule Symbols:

a, b, c, u, v, w

Defined Pair Symbols:

C, A, B

Compound Symbols:

c5, c1, c3, c3

(9) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing nodes:

B(c(x0)) → c3

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(b(z0)) → b(a(a(z0)))
a(u(z0)) → z0
b(c(z0)) → c(b(b(z0)))
b(v(z0)) → z0
c(a(z0)) → a(c(c(z0)))
c(w(z0)) → z0
u(a(z0)) → z0
v(b(z0)) → z0
w(c(z0)) → z0
Tuples:

C(a(z0)) → c5(A(c(c(z0))), C(c(z0)), C(z0))
A(b(b(z0))) → c1(B(a(b(a(a(z0))))), A(a(b(z0))), A(b(z0)))
A(b(u(z0))) → c1(B(a(z0)), A(a(u(z0))), A(u(z0)))
B(c(c(z0))) → c3(C(b(c(b(b(z0))))), B(b(c(z0))), B(c(z0)))
B(c(v(z0))) → c3(C(b(z0)), B(b(v(z0))), B(v(z0)))
S tuples:

C(a(z0)) → c5(A(c(c(z0))), C(c(z0)), C(z0))
A(b(b(z0))) → c1(B(a(b(a(a(z0))))), A(a(b(z0))), A(b(z0)))
A(b(u(z0))) → c1(B(a(z0)), A(a(u(z0))), A(u(z0)))
B(c(c(z0))) → c3(C(b(c(b(b(z0))))), B(b(c(z0))), B(c(z0)))
B(c(v(z0))) → c3(C(b(z0)), B(b(v(z0))), B(v(z0)))
K tuples:none
Defined Rule Symbols:

a, b, c, u, v, w

Defined Pair Symbols:

C, A, B

Compound Symbols:

c5, c1, c3

(11) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace C(a(z0)) → c5(A(c(c(z0))), C(c(z0)), C(z0)) by

C(a(a(z0))) → c5(A(c(a(c(c(z0))))), C(c(a(z0))), C(a(z0)))
C(a(w(z0))) → c5(A(c(z0)), C(c(w(z0))), C(w(z0)))
C(a(x0)) → c5

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(b(z0)) → b(a(a(z0)))
a(u(z0)) → z0
b(c(z0)) → c(b(b(z0)))
b(v(z0)) → z0
c(a(z0)) → a(c(c(z0)))
c(w(z0)) → z0
u(a(z0)) → z0
v(b(z0)) → z0
w(c(z0)) → z0
Tuples:

A(b(b(z0))) → c1(B(a(b(a(a(z0))))), A(a(b(z0))), A(b(z0)))
A(b(u(z0))) → c1(B(a(z0)), A(a(u(z0))), A(u(z0)))
B(c(c(z0))) → c3(C(b(c(b(b(z0))))), B(b(c(z0))), B(c(z0)))
B(c(v(z0))) → c3(C(b(z0)), B(b(v(z0))), B(v(z0)))
C(a(a(z0))) → c5(A(c(a(c(c(z0))))), C(c(a(z0))), C(a(z0)))
C(a(w(z0))) → c5(A(c(z0)), C(c(w(z0))), C(w(z0)))
C(a(x0)) → c5
S tuples:

A(b(b(z0))) → c1(B(a(b(a(a(z0))))), A(a(b(z0))), A(b(z0)))
A(b(u(z0))) → c1(B(a(z0)), A(a(u(z0))), A(u(z0)))
B(c(c(z0))) → c3(C(b(c(b(b(z0))))), B(b(c(z0))), B(c(z0)))
B(c(v(z0))) → c3(C(b(z0)), B(b(v(z0))), B(v(z0)))
C(a(a(z0))) → c5(A(c(a(c(c(z0))))), C(c(a(z0))), C(a(z0)))
C(a(w(z0))) → c5(A(c(z0)), C(c(w(z0))), C(w(z0)))
C(a(x0)) → c5
K tuples:none
Defined Rule Symbols:

a, b, c, u, v, w

Defined Pair Symbols:

A, B, C

Compound Symbols:

c1, c3, c5, c5

(13) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing nodes:

C(a(x0)) → c5

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(b(z0)) → b(a(a(z0)))
a(u(z0)) → z0
b(c(z0)) → c(b(b(z0)))
b(v(z0)) → z0
c(a(z0)) → a(c(c(z0)))
c(w(z0)) → z0
u(a(z0)) → z0
v(b(z0)) → z0
w(c(z0)) → z0
Tuples:

A(b(b(z0))) → c1(B(a(b(a(a(z0))))), A(a(b(z0))), A(b(z0)))
A(b(u(z0))) → c1(B(a(z0)), A(a(u(z0))), A(u(z0)))
B(c(c(z0))) → c3(C(b(c(b(b(z0))))), B(b(c(z0))), B(c(z0)))
B(c(v(z0))) → c3(C(b(z0)), B(b(v(z0))), B(v(z0)))
C(a(a(z0))) → c5(A(c(a(c(c(z0))))), C(c(a(z0))), C(a(z0)))
C(a(w(z0))) → c5(A(c(z0)), C(c(w(z0))), C(w(z0)))
S tuples:

A(b(b(z0))) → c1(B(a(b(a(a(z0))))), A(a(b(z0))), A(b(z0)))
A(b(u(z0))) → c1(B(a(z0)), A(a(u(z0))), A(u(z0)))
B(c(c(z0))) → c3(C(b(c(b(b(z0))))), B(b(c(z0))), B(c(z0)))
B(c(v(z0))) → c3(C(b(z0)), B(b(v(z0))), B(v(z0)))
C(a(a(z0))) → c5(A(c(a(c(c(z0))))), C(c(a(z0))), C(a(z0)))
C(a(w(z0))) → c5(A(c(z0)), C(c(w(z0))), C(w(z0)))
K tuples:none
Defined Rule Symbols:

a, b, c, u, v, w

Defined Pair Symbols:

A, B, C

Compound Symbols:

c1, c3, c5

(15) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace A(b(b(z0))) → c1(B(a(b(a(a(z0))))), A(a(b(z0))), A(b(z0))) by

A(b(b(x0))) → c1(B(b(a(a(a(a(x0)))))), A(a(b(x0))), A(b(x0)))
A(b(b(b(z0)))) → c1(B(a(b(a(b(a(a(z0))))))), A(a(b(b(z0)))), A(b(b(z0))))
A(b(b(u(z0)))) → c1(B(a(b(a(z0)))), A(a(b(u(z0)))), A(b(u(z0))))
A(b(b(x0))) → c1(A(a(b(x0))))

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(b(z0)) → b(a(a(z0)))
a(u(z0)) → z0
b(c(z0)) → c(b(b(z0)))
b(v(z0)) → z0
c(a(z0)) → a(c(c(z0)))
c(w(z0)) → z0
u(a(z0)) → z0
v(b(z0)) → z0
w(c(z0)) → z0
Tuples:

A(b(u(z0))) → c1(B(a(z0)), A(a(u(z0))), A(u(z0)))
B(c(c(z0))) → c3(C(b(c(b(b(z0))))), B(b(c(z0))), B(c(z0)))
B(c(v(z0))) → c3(C(b(z0)), B(b(v(z0))), B(v(z0)))
C(a(a(z0))) → c5(A(c(a(c(c(z0))))), C(c(a(z0))), C(a(z0)))
C(a(w(z0))) → c5(A(c(z0)), C(c(w(z0))), C(w(z0)))
A(b(b(x0))) → c1(B(b(a(a(a(a(x0)))))), A(a(b(x0))), A(b(x0)))
A(b(b(b(z0)))) → c1(B(a(b(a(b(a(a(z0))))))), A(a(b(b(z0)))), A(b(b(z0))))
A(b(b(u(z0)))) → c1(B(a(b(a(z0)))), A(a(b(u(z0)))), A(b(u(z0))))
A(b(b(x0))) → c1(A(a(b(x0))))
S tuples:

A(b(u(z0))) → c1(B(a(z0)), A(a(u(z0))), A(u(z0)))
B(c(c(z0))) → c3(C(b(c(b(b(z0))))), B(b(c(z0))), B(c(z0)))
B(c(v(z0))) → c3(C(b(z0)), B(b(v(z0))), B(v(z0)))
C(a(a(z0))) → c5(A(c(a(c(c(z0))))), C(c(a(z0))), C(a(z0)))
C(a(w(z0))) → c5(A(c(z0)), C(c(w(z0))), C(w(z0)))
A(b(b(x0))) → c1(B(b(a(a(a(a(x0)))))), A(a(b(x0))), A(b(x0)))
A(b(b(b(z0)))) → c1(B(a(b(a(b(a(a(z0))))))), A(a(b(b(z0)))), A(b(b(z0))))
A(b(b(u(z0)))) → c1(B(a(b(a(z0)))), A(a(b(u(z0)))), A(b(u(z0))))
A(b(b(x0))) → c1(A(a(b(x0))))
K tuples:none
Defined Rule Symbols:

a, b, c, u, v, w

Defined Pair Symbols:

A, B, C

Compound Symbols:

c1, c3, c5, c1

(17) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace A(b(u(z0))) → c1(B(a(z0)), A(a(u(z0))), A(u(z0))) by

A(b(u(b(z0)))) → c1(B(b(a(a(z0)))), A(a(u(b(z0)))), A(u(b(z0))))
A(b(u(u(z0)))) → c1(B(z0), A(a(u(u(z0)))), A(u(u(z0))))
A(b(u(x0))) → c1(A(a(u(x0))))

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(b(z0)) → b(a(a(z0)))
a(u(z0)) → z0
b(c(z0)) → c(b(b(z0)))
b(v(z0)) → z0
c(a(z0)) → a(c(c(z0)))
c(w(z0)) → z0
u(a(z0)) → z0
v(b(z0)) → z0
w(c(z0)) → z0
Tuples:

B(c(c(z0))) → c3(C(b(c(b(b(z0))))), B(b(c(z0))), B(c(z0)))
B(c(v(z0))) → c3(C(b(z0)), B(b(v(z0))), B(v(z0)))
C(a(a(z0))) → c5(A(c(a(c(c(z0))))), C(c(a(z0))), C(a(z0)))
C(a(w(z0))) → c5(A(c(z0)), C(c(w(z0))), C(w(z0)))
A(b(b(x0))) → c1(B(b(a(a(a(a(x0)))))), A(a(b(x0))), A(b(x0)))
A(b(b(b(z0)))) → c1(B(a(b(a(b(a(a(z0))))))), A(a(b(b(z0)))), A(b(b(z0))))
A(b(b(u(z0)))) → c1(B(a(b(a(z0)))), A(a(b(u(z0)))), A(b(u(z0))))
A(b(b(x0))) → c1(A(a(b(x0))))
A(b(u(b(z0)))) → c1(B(b(a(a(z0)))), A(a(u(b(z0)))), A(u(b(z0))))
A(b(u(u(z0)))) → c1(B(z0), A(a(u(u(z0)))), A(u(u(z0))))
A(b(u(x0))) → c1(A(a(u(x0))))
S tuples:

B(c(c(z0))) → c3(C(b(c(b(b(z0))))), B(b(c(z0))), B(c(z0)))
B(c(v(z0))) → c3(C(b(z0)), B(b(v(z0))), B(v(z0)))
C(a(a(z0))) → c5(A(c(a(c(c(z0))))), C(c(a(z0))), C(a(z0)))
C(a(w(z0))) → c5(A(c(z0)), C(c(w(z0))), C(w(z0)))
A(b(b(x0))) → c1(B(b(a(a(a(a(x0)))))), A(a(b(x0))), A(b(x0)))
A(b(b(b(z0)))) → c1(B(a(b(a(b(a(a(z0))))))), A(a(b(b(z0)))), A(b(b(z0))))
A(b(b(u(z0)))) → c1(B(a(b(a(z0)))), A(a(b(u(z0)))), A(b(u(z0))))
A(b(b(x0))) → c1(A(a(b(x0))))
A(b(u(b(z0)))) → c1(B(b(a(a(z0)))), A(a(u(b(z0)))), A(u(b(z0))))
A(b(u(u(z0)))) → c1(B(z0), A(a(u(u(z0)))), A(u(u(z0))))
A(b(u(x0))) → c1(A(a(u(x0))))
K tuples:none
Defined Rule Symbols:

a, b, c, u, v, w

Defined Pair Symbols:

B, C, A

Compound Symbols:

c3, c5, c1, c1

(19) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace B(c(c(z0))) → c3(C(b(c(b(b(z0))))), B(b(c(z0))), B(c(z0))) by

B(c(c(x0))) → c3(C(c(b(b(b(b(x0)))))), B(b(c(x0))), B(c(x0)))
B(c(c(c(z0)))) → c3(C(b(c(b(c(b(b(z0))))))), B(b(c(c(z0)))), B(c(c(z0))))
B(c(c(v(z0)))) → c3(C(b(c(b(z0)))), B(b(c(v(z0)))), B(c(v(z0))))
B(c(c(x0))) → c3(B(b(c(x0))))

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(b(z0)) → b(a(a(z0)))
a(u(z0)) → z0
b(c(z0)) → c(b(b(z0)))
b(v(z0)) → z0
c(a(z0)) → a(c(c(z0)))
c(w(z0)) → z0
u(a(z0)) → z0
v(b(z0)) → z0
w(c(z0)) → z0
Tuples:

B(c(v(z0))) → c3(C(b(z0)), B(b(v(z0))), B(v(z0)))
C(a(a(z0))) → c5(A(c(a(c(c(z0))))), C(c(a(z0))), C(a(z0)))
C(a(w(z0))) → c5(A(c(z0)), C(c(w(z0))), C(w(z0)))
A(b(b(x0))) → c1(B(b(a(a(a(a(x0)))))), A(a(b(x0))), A(b(x0)))
A(b(b(b(z0)))) → c1(B(a(b(a(b(a(a(z0))))))), A(a(b(b(z0)))), A(b(b(z0))))
A(b(b(u(z0)))) → c1(B(a(b(a(z0)))), A(a(b(u(z0)))), A(b(u(z0))))
A(b(b(x0))) → c1(A(a(b(x0))))
A(b(u(b(z0)))) → c1(B(b(a(a(z0)))), A(a(u(b(z0)))), A(u(b(z0))))
A(b(u(u(z0)))) → c1(B(z0), A(a(u(u(z0)))), A(u(u(z0))))
A(b(u(x0))) → c1(A(a(u(x0))))
B(c(c(x0))) → c3(C(c(b(b(b(b(x0)))))), B(b(c(x0))), B(c(x0)))
B(c(c(c(z0)))) → c3(C(b(c(b(c(b(b(z0))))))), B(b(c(c(z0)))), B(c(c(z0))))
B(c(c(v(z0)))) → c3(C(b(c(b(z0)))), B(b(c(v(z0)))), B(c(v(z0))))
B(c(c(x0))) → c3(B(b(c(x0))))
S tuples:

B(c(v(z0))) → c3(C(b(z0)), B(b(v(z0))), B(v(z0)))
C(a(a(z0))) → c5(A(c(a(c(c(z0))))), C(c(a(z0))), C(a(z0)))
C(a(w(z0))) → c5(A(c(z0)), C(c(w(z0))), C(w(z0)))
A(b(b(x0))) → c1(B(b(a(a(a(a(x0)))))), A(a(b(x0))), A(b(x0)))
A(b(b(b(z0)))) → c1(B(a(b(a(b(a(a(z0))))))), A(a(b(b(z0)))), A(b(b(z0))))
A(b(b(u(z0)))) → c1(B(a(b(a(z0)))), A(a(b(u(z0)))), A(b(u(z0))))
A(b(b(x0))) → c1(A(a(b(x0))))
A(b(u(b(z0)))) → c1(B(b(a(a(z0)))), A(a(u(b(z0)))), A(u(b(z0))))
A(b(u(u(z0)))) → c1(B(z0), A(a(u(u(z0)))), A(u(u(z0))))
A(b(u(x0))) → c1(A(a(u(x0))))
B(c(c(x0))) → c3(C(c(b(b(b(b(x0)))))), B(b(c(x0))), B(c(x0)))
B(c(c(c(z0)))) → c3(C(b(c(b(c(b(b(z0))))))), B(b(c(c(z0)))), B(c(c(z0))))
B(c(c(v(z0)))) → c3(C(b(c(b(z0)))), B(b(c(v(z0)))), B(c(v(z0))))
B(c(c(x0))) → c3(B(b(c(x0))))
K tuples:none
Defined Rule Symbols:

a, b, c, u, v, w

Defined Pair Symbols:

B, C, A

Compound Symbols:

c3, c5, c1, c1, c3

(21) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace B(c(v(z0))) → c3(C(b(z0)), B(b(v(z0))), B(v(z0))) by

B(c(v(c(z0)))) → c3(C(c(b(b(z0)))), B(b(v(c(z0)))), B(v(c(z0))))
B(c(v(v(z0)))) → c3(C(z0), B(b(v(v(z0)))), B(v(v(z0))))
B(c(v(x0))) → c3(B(b(v(x0))))

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(b(z0)) → b(a(a(z0)))
a(u(z0)) → z0
b(c(z0)) → c(b(b(z0)))
b(v(z0)) → z0
c(a(z0)) → a(c(c(z0)))
c(w(z0)) → z0
u(a(z0)) → z0
v(b(z0)) → z0
w(c(z0)) → z0
Tuples:

C(a(a(z0))) → c5(A(c(a(c(c(z0))))), C(c(a(z0))), C(a(z0)))
C(a(w(z0))) → c5(A(c(z0)), C(c(w(z0))), C(w(z0)))
A(b(b(x0))) → c1(B(b(a(a(a(a(x0)))))), A(a(b(x0))), A(b(x0)))
A(b(b(b(z0)))) → c1(B(a(b(a(b(a(a(z0))))))), A(a(b(b(z0)))), A(b(b(z0))))
A(b(b(u(z0)))) → c1(B(a(b(a(z0)))), A(a(b(u(z0)))), A(b(u(z0))))
A(b(b(x0))) → c1(A(a(b(x0))))
A(b(u(b(z0)))) → c1(B(b(a(a(z0)))), A(a(u(b(z0)))), A(u(b(z0))))
A(b(u(u(z0)))) → c1(B(z0), A(a(u(u(z0)))), A(u(u(z0))))
A(b(u(x0))) → c1(A(a(u(x0))))
B(c(c(x0))) → c3(C(c(b(b(b(b(x0)))))), B(b(c(x0))), B(c(x0)))
B(c(c(c(z0)))) → c3(C(b(c(b(c(b(b(z0))))))), B(b(c(c(z0)))), B(c(c(z0))))
B(c(c(v(z0)))) → c3(C(b(c(b(z0)))), B(b(c(v(z0)))), B(c(v(z0))))
B(c(c(x0))) → c3(B(b(c(x0))))
B(c(v(c(z0)))) → c3(C(c(b(b(z0)))), B(b(v(c(z0)))), B(v(c(z0))))
B(c(v(v(z0)))) → c3(C(z0), B(b(v(v(z0)))), B(v(v(z0))))
B(c(v(x0))) → c3(B(b(v(x0))))
S tuples:

C(a(a(z0))) → c5(A(c(a(c(c(z0))))), C(c(a(z0))), C(a(z0)))
C(a(w(z0))) → c5(A(c(z0)), C(c(w(z0))), C(w(z0)))
A(b(b(x0))) → c1(B(b(a(a(a(a(x0)))))), A(a(b(x0))), A(b(x0)))
A(b(b(b(z0)))) → c1(B(a(b(a(b(a(a(z0))))))), A(a(b(b(z0)))), A(b(b(z0))))
A(b(b(u(z0)))) → c1(B(a(b(a(z0)))), A(a(b(u(z0)))), A(b(u(z0))))
A(b(b(x0))) → c1(A(a(b(x0))))
A(b(u(b(z0)))) → c1(B(b(a(a(z0)))), A(a(u(b(z0)))), A(u(b(z0))))
A(b(u(u(z0)))) → c1(B(z0), A(a(u(u(z0)))), A(u(u(z0))))
A(b(u(x0))) → c1(A(a(u(x0))))
B(c(c(x0))) → c3(C(c(b(b(b(b(x0)))))), B(b(c(x0))), B(c(x0)))
B(c(c(c(z0)))) → c3(C(b(c(b(c(b(b(z0))))))), B(b(c(c(z0)))), B(c(c(z0))))
B(c(c(v(z0)))) → c3(C(b(c(b(z0)))), B(b(c(v(z0)))), B(c(v(z0))))
B(c(c(x0))) → c3(B(b(c(x0))))
B(c(v(c(z0)))) → c3(C(c(b(b(z0)))), B(b(v(c(z0)))), B(v(c(z0))))
B(c(v(v(z0)))) → c3(C(z0), B(b(v(v(z0)))), B(v(v(z0))))
B(c(v(x0))) → c3(B(b(v(x0))))
K tuples:none
Defined Rule Symbols:

a, b, c, u, v, w

Defined Pair Symbols:

C, A, B

Compound Symbols:

c5, c1, c1, c3, c3

(23) CpxTrsMatchBoundsProof (EQUIVALENT transformation)

A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 0.
The certificate found is represented by the following graph.
Start state: 1179
Accept states: [1180, 1181, 1182, 1183, 1184, 1185]
Transitions:
1179→1180[a_1|0]
1179→1181[b_1|0]
1179→1182[c_1|0]
1179→1183[u_1|0]
1179→1184[v_1|0]
1179→1185[w_1|0]

(24) BOUNDS(O(1), O(n^1))