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

0 CpxTRS
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
↳2 CdtProblem
↳3 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
↳4 CdtProblem
↳5 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
↳6 CdtProblem
↳7 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
↳8 CdtProblem
↳9 CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
↳10 CdtProblem
↳11 CdtUnreachableProof (⇔)
↳12 CdtProblem
↳13 SIsEmptyProof (⇔)
↳14 BOUNDS(O(1), O(1))

(0) Obligation:

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

a(x1) → x1
a(b(b(x1))) → b(b(a(b(c(x1)))))
b(c(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(z0) → z0
a(b(b(z0))) → b(b(a(b(c(z0)))))
b(c(z0)) → a(z0)
Tuples:

A(b(b(z0))) → c2(B(b(a(b(c(z0))))), B(a(b(c(z0)))), A(b(c(z0))), B(c(z0)))
B(c(z0)) → c3(A(z0))
S tuples:

A(b(b(z0))) → c2(B(b(a(b(c(z0))))), B(a(b(c(z0)))), A(b(c(z0))), B(c(z0)))
B(c(z0)) → c3(A(z0))
K tuples:none
Defined Rule Symbols:

a, b

Defined Pair Symbols:

A, B

Compound Symbols:

c2, c3

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

Use narrowing to replace A(b(b(z0))) → c2(B(b(a(b(c(z0))))), B(a(b(c(z0)))), A(b(c(z0))), B(c(z0))) by

A(b(b(x0))) → c2(B(b(b(c(x0)))), B(a(b(c(x0)))), A(b(c(x0))), B(c(x0)))
A(b(b(z0))) → c2(B(b(a(a(z0)))), B(a(b(c(z0)))), A(b(c(z0))), B(c(z0)))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

B(c(z0)) → c3(A(z0))
A(b(b(x0))) → c2(B(b(b(c(x0)))), B(a(b(c(x0)))), A(b(c(x0))), B(c(x0)))
A(b(b(z0))) → c2(B(b(a(a(z0)))), B(a(b(c(z0)))), A(b(c(z0))), B(c(z0)))
S tuples:

B(c(z0)) → c3(A(z0))
A(b(b(x0))) → c2(B(b(b(c(x0)))), B(a(b(c(x0)))), A(b(c(x0))), B(c(x0)))
A(b(b(z0))) → c2(B(b(a(a(z0)))), B(a(b(c(z0)))), A(b(c(z0))), B(c(z0)))
K tuples:none
Defined Rule Symbols:

a, b

Defined Pair Symbols:

B, A

Compound Symbols:

c3, c2

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

Use narrowing to replace A(b(b(x0))) → c2(B(b(b(c(x0)))), B(a(b(c(x0)))), A(b(c(x0))), B(c(x0))) by

A(b(b(z0))) → c2(B(b(a(z0))), B(a(b(c(z0)))), A(b(c(z0))), B(c(z0)))
A(b(b(x0))) → c2(B(a(b(c(x0)))))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

B(c(z0)) → c3(A(z0))
A(b(b(z0))) → c2(B(b(a(a(z0)))), B(a(b(c(z0)))), A(b(c(z0))), B(c(z0)))
A(b(b(z0))) → c2(B(b(a(z0))), B(a(b(c(z0)))), A(b(c(z0))), B(c(z0)))
A(b(b(x0))) → c2(B(a(b(c(x0)))))
S tuples:

B(c(z0)) → c3(A(z0))
A(b(b(z0))) → c2(B(b(a(a(z0)))), B(a(b(c(z0)))), A(b(c(z0))), B(c(z0)))
A(b(b(z0))) → c2(B(b(a(z0))), B(a(b(c(z0)))), A(b(c(z0))), B(c(z0)))
A(b(b(x0))) → c2(B(a(b(c(x0)))))
K tuples:none
Defined Rule Symbols:

a, b

Defined Pair Symbols:

B, A

Compound Symbols:

c3, c2, c2

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

Use narrowing to replace A(b(b(z0))) → c2(B(b(a(a(z0)))), B(a(b(c(z0)))), A(b(c(z0))), B(c(z0))) by

A(b(b(x0))) → c2(B(b(a(x0))), B(a(b(c(x0)))), A(b(c(x0))), B(c(x0)))
A(b(b(b(b(z0))))) → c2(B(b(a(b(b(a(b(c(z0)))))))), B(a(b(c(b(b(z0)))))), A(b(c(b(b(z0))))), B(c(b(b(z0)))))
A(b(b(x0))) → c2(B(a(b(c(x0)))))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

B(c(z0)) → c3(A(z0))
A(b(b(z0))) → c2(B(b(a(z0))), B(a(b(c(z0)))), A(b(c(z0))), B(c(z0)))
A(b(b(x0))) → c2(B(a(b(c(x0)))))
A(b(b(b(b(z0))))) → c2(B(b(a(b(b(a(b(c(z0)))))))), B(a(b(c(b(b(z0)))))), A(b(c(b(b(z0))))), B(c(b(b(z0)))))
S tuples:

B(c(z0)) → c3(A(z0))
A(b(b(z0))) → c2(B(b(a(z0))), B(a(b(c(z0)))), A(b(c(z0))), B(c(z0)))
A(b(b(x0))) → c2(B(a(b(c(x0)))))
A(b(b(b(b(z0))))) → c2(B(b(a(b(b(a(b(c(z0)))))))), B(a(b(c(b(b(z0)))))), A(b(c(b(b(z0))))), B(c(b(b(z0)))))
K tuples:none
Defined Rule Symbols:

a, b

Defined Pair Symbols:

B, A

Compound Symbols:

c3, c2, c2

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

Use forward instantiation to replace B(c(z0)) → c3(A(z0)) by

B(c(b(b(y0)))) → c3(A(b(b(y0))))
B(c(b(b(b(b(y0)))))) → c3(A(b(b(b(b(y0))))))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

A(b(b(z0))) → c2(B(b(a(z0))), B(a(b(c(z0)))), A(b(c(z0))), B(c(z0)))
A(b(b(x0))) → c2(B(a(b(c(x0)))))
A(b(b(b(b(z0))))) → c2(B(b(a(b(b(a(b(c(z0)))))))), B(a(b(c(b(b(z0)))))), A(b(c(b(b(z0))))), B(c(b(b(z0)))))
B(c(b(b(y0)))) → c3(A(b(b(y0))))
B(c(b(b(b(b(y0)))))) → c3(A(b(b(b(b(y0))))))
S tuples:

A(b(b(z0))) → c2(B(b(a(z0))), B(a(b(c(z0)))), A(b(c(z0))), B(c(z0)))
A(b(b(x0))) → c2(B(a(b(c(x0)))))
A(b(b(b(b(z0))))) → c2(B(b(a(b(b(a(b(c(z0)))))))), B(a(b(c(b(b(z0)))))), A(b(c(b(b(z0))))), B(c(b(b(z0)))))
B(c(b(b(y0)))) → c3(A(b(b(y0))))
B(c(b(b(b(b(y0)))))) → c3(A(b(b(b(b(y0))))))
K tuples:none
Defined Rule Symbols:

a, b

Defined Pair Symbols:

A, B

Compound Symbols:

c2, c2, c3

(11) CdtUnreachableProof (EQUIVALENT transformation)

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

A(b(b(z0))) → c2(B(b(a(z0))), B(a(b(c(z0)))), A(b(c(z0))), B(c(z0)))
A(b(b(x0))) → c2(B(a(b(c(x0)))))
A(b(b(b(b(z0))))) → c2(B(b(a(b(b(a(b(c(z0)))))))), B(a(b(c(b(b(z0)))))), A(b(c(b(b(z0))))), B(c(b(b(z0)))))
B(c(b(b(y0)))) → c3(A(b(b(y0))))
B(c(b(b(b(b(y0)))))) → c3(A(b(b(b(b(y0))))))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

a, b

Defined Pair Symbols:none

Compound Symbols:none

(13) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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