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

(0) Obligation:

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

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

Tuples:

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

S tuples:

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

Tuples:

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

S tuples:

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

C(b(b(z0))) → c3(A(a(c(z0))), C(b(z0)))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(a(z0)) → a(b(b(c(z0))))
a(b(z0)) → z0
c(b(z0)) → a(c(z0))

Tuples:

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

S tuples:

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

K tuples:none
Defined Rule Symbols:

a, c

Defined Pair Symbols:

A, C

Compound Symbols:

c1, c3

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

Use forward instantiation to replace A(a(z0)) → c1(C(z0)) by

A(a(b(b(y0)))) → c1(C(b(b(y0))))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(a(z0)) → a(b(b(c(z0))))
a(b(z0)) → z0
c(b(z0)) → a(c(z0))

Tuples:

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

S tuples:

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

K tuples:none
Defined Rule Symbols:

a, c

Defined Pair Symbols:

C, A

Compound Symbols:

c3, c1

(9) CdtUnreachableProof (EQUIVALENT transformation)

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

A(a(b(b(y0)))) → c1(C(b(b(y0))))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(a(z0)) → a(b(b(c(z0))))
a(b(z0)) → z0
c(b(z0)) → a(c(z0))

Tuples:

C(b(b(z0))) → c3(A(a(c(z0))), C(b(z0)))

S tuples:

C(b(b(z0))) → c3(A(a(c(z0))), C(b(z0)))

K tuples:none
Defined Rule Symbols:

a, c

Defined Pair Symbols:

C

Compound Symbols:

c3

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

Removed 1 trailing tuple parts

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(a(z0)) → a(b(b(c(z0))))
a(b(z0)) → z0
c(b(z0)) → a(c(z0))

Tuples:

C(b(b(z0))) → c3(C(b(z0)))

S tuples:

C(b(b(z0))) → c3(C(b(z0)))

K tuples:none
Defined Rule Symbols:

a, c

Defined Pair Symbols:

C

Compound Symbols:

c3

(13) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

C(b(b(z0))) → c3(C(b(z0)))

We considered the (Usable) Rules:none
And the Tuples:

C(b(b(z0))) → c3(C(b(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(C(x₁)) = x₁   
POL(b(x₁)) = [1] + x₁   
POL(c3(x₁)) = x₁   

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(a(z0)) → a(b(b(c(z0))))
a(b(z0)) → z0
c(b(z0)) → a(c(z0))

Tuples:

C(b(b(z0))) → c3(C(b(z0)))

S tuples:none
K tuples:

C(b(b(z0))) → c3(C(b(z0)))

Defined Rule Symbols:

a, c

Defined Pair Symbols:

C

Compound Symbols:

c3

(15) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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