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

Tuples:

A(a(z0)) → c2(C(z0))
C(b(z0)) → c4(C(a(z0)), A(z0))

S tuples:

A(a(z0)) → c2(C(z0))
C(b(z0)) → c4(C(a(z0)), A(z0))

K tuples:none
Defined Rule Symbols:

a, c

Defined Pair Symbols:

A, C

Compound Symbols:

c2, c4

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

A(a(z0)) → c2(C(z0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

C(b(z0)) → c4(C(a(z0)), A(z0))

S tuples:

C(b(z0)) → c4(C(a(z0)), A(z0))

K tuples:none
Defined Rule Symbols:

a, c

Defined Pair Symbols:

C

Compound Symbols:

c4

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

Removed 1 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

C(b(z0)) → c4(C(a(z0)))

S tuples:

C(b(z0)) → c4(C(a(z0)))

K tuples:none
Defined Rule Symbols:

a, c

Defined Pair Symbols:

C

Compound Symbols:

c4

(7) 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(z0)) → c4(C(a(z0)))

We considered the (Usable) Rules:

a(z0) → z0

And the Tuples:

C(b(z0)) → c4(C(a(z0)))

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

POL(C(x₁)) = [4]x₁   
POL(a(x₁)) = x₁   
POL(b(x₁)) = [4] + x₁   
POL(c4(x₁)) = x₁   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

C(b(z0)) → c4(C(a(z0)))

S tuples:none
K tuples:

C(b(z0)) → c4(C(a(z0)))

Defined Rule Symbols:

a, c

Defined Pair Symbols:

C

Compound Symbols:

c4

(9) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(10) BOUNDS(O(1), O(1))