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 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
↳10 CdtProblem
↳11 CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
↳12 CdtProblem
↳13 CdtUnreachableProof (⇔)
↳14 CdtProblem
↳15 SIsEmptyProof (⇔)
↳16 BOUNDS(O(1), O(1))

(0) Obligation:

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

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

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

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

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

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

a, b

Defined Pair Symbols:

A, B

Compound Symbols:

c2, c3

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

Use narrowing to replace B(b(c(x0))) → c3(A(c(c(c(a(x0))))), A(a(x0)), A(x0)) by

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

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

a, b

Defined Pair Symbols:

A, B

Compound Symbols:

c2, c3, c3

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

Use narrowing to replace B(b(c(x0))) → c3(A(c(c(c(b(a(x0)))))), A(a(x0)), A(x0)) by

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

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

a, b

Defined Pair Symbols:

A, B

Compound Symbols:

c2, c3, c3

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

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

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

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

a, b

Defined Pair Symbols:

A, B

Compound Symbols:

c2, c3, c3, c3

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

Use forward instantiation to replace A(z0) → c2(B(z0)) by

A(b(c(y0))) → c2(B(b(c(y0))))
A(b(c(b(c(y0))))) → c2(B(b(c(b(c(y0))))))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

a, b

Defined Pair Symbols:

B, A

Compound Symbols:

c3, c3, c3, c2

(13) CdtUnreachableProof (EQUIVALENT transformation)

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

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

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

a, b

Defined Pair Symbols:none

Compound Symbols:none

(15) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(16) BOUNDS(O(1), O(1))