(0) Obligation:

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

a(a(x1)) → b(a(b(c(c(x1)))))
c(a(x1)) → x1
c(b(x1)) → a(x1)

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(a(z0)) → b(a(b(c(c(z0)))))
c(a(z0)) → z0
c(b(z0)) → a(z0)
Tuples:

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

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

a, c

Defined Pair Symbols:

A, C

Compound Symbols:

c1, c3

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

Removed 1 trailing tuple parts

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(a(z0)) → b(a(b(c(c(z0)))))
c(a(z0)) → z0
c(b(z0)) → a(z0)
Tuples:

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

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

a, c

Defined Pair Symbols:

C, A

Compound Symbols:

c3, c1

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

Use narrowing to replace A(a(z0)) → c1(C(c(z0)), C(z0)) by

A(a(b(z0))) → c1(C(a(z0)), C(b(z0)))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(a(z0)) → b(a(b(c(c(z0)))))
c(a(z0)) → z0
c(b(z0)) → a(z0)
Tuples:

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

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

a, c

Defined Pair Symbols:

C, A

Compound Symbols:

c3, c1

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

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

A(a(b(a(z0)))) → c1(C(b(a(b(c(c(z0)))))), C(b(a(z0))))
A(a(b(x0))) → c1(C(b(x0)))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(a(z0)) → b(a(b(c(c(z0)))))
c(a(z0)) → z0
c(b(z0)) → a(z0)
Tuples:

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

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

a, c

Defined Pair Symbols:

C, A

Compound Symbols:

c3, c1, c1

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

Use forward instantiation to replace C(b(z0)) → c3(A(z0)) by

C(b(a(b(a(y0))))) → c3(A(a(b(a(y0)))))
C(b(a(b(y0)))) → c3(A(a(b(y0))))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(a(z0)) → b(a(b(c(c(z0)))))
c(a(z0)) → z0
c(b(z0)) → a(z0)
Tuples:

A(a(b(a(z0)))) → c1(C(b(a(b(c(c(z0)))))), C(b(a(z0))))
A(a(b(x0))) → c1(C(b(x0)))
C(b(a(b(a(y0))))) → c3(A(a(b(a(y0)))))
C(b(a(b(y0)))) → c3(A(a(b(y0))))
S tuples:

A(a(b(a(z0)))) → c1(C(b(a(b(c(c(z0)))))), C(b(a(z0))))
A(a(b(x0))) → c1(C(b(x0)))
C(b(a(b(a(y0))))) → c3(A(a(b(a(y0)))))
C(b(a(b(y0)))) → c3(A(a(b(y0))))
K tuples:none
Defined Rule Symbols:

a, c

Defined Pair Symbols:

A, C

Compound Symbols:

c1, c1, c3

(11) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

A(a(b(a(z0)))) → c1(C(b(a(b(c(c(z0)))))), C(b(a(z0))))
A(a(b(x0))) → c1(C(b(x0)))
C(b(a(b(a(y0))))) → c3(A(a(b(a(y0)))))
C(b(a(b(y0)))) → c3(A(a(b(y0))))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(a(z0)) → b(a(b(c(c(z0)))))
c(a(z0)) → z0
c(b(z0)) → a(z0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

a, c

Defined Pair Symbols:none

Compound Symbols:none

(13) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(14) BOUNDS(O(1), O(1))