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 CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID))
↳8 CdtProblem
↳9 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
↳10 CdtProblem
↳11 CdtRewritingProof (BOTH BOUNDS(ID, ID))
↳12 CdtProblem
↳13 CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
↳14 CdtProblem
↳15 CdtUnreachableProof (⇔)
↳16 CdtProblem
↳17 SIsEmptyProof (⇔)
↳18 BOUNDS(O(1), O(1))

(0) Obligation:

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

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

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

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

a, c

Defined Pair Symbols:

A, C

Compound Symbols:

c2, c3

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

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

A(c(z0)) → c2(C(c(b(z0))), C(a(z0)), A(z0))
A(c(c(z0))) → c2(C(c(b(c(c(a(z0)))))), C(a(c(z0))), A(c(z0)))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

a, c

Defined Pair Symbols:

C, A

Compound Symbols:

c3, c2

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

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

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

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

a, c

Defined Pair Symbols:

C, A

Compound Symbols:

c3, c2, c2

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

Removed 1 trailing tuple parts

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

a, c

Defined Pair Symbols:

C, A

Compound Symbols:

c3, c2, c2

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

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

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

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

a, c

Defined Pair Symbols:

C, A

Compound Symbols:

c3, c2, c2, c2

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

Used rewriting to replace A(c(b(z0))) → c2(C(a(z0)), C(a(b(z0)))) by A(c(b(z0))) → c2(C(a(z0)), C(b(b(z0))))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

a, c

Defined Pair Symbols:

C, A

Compound Symbols:

c3, c2, c2, c2

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

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

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

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

a, c

Defined Pair Symbols:

A, C

Compound Symbols:

c2, c2, c2, c3

(15) CdtUnreachableProof (EQUIVALENT transformation)

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

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

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

a, c

Defined Pair Symbols:none

Compound Symbols:none

(17) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(18) BOUNDS(O(1), O(1))