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

(0) Obligation:

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

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

Tuples:

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

S tuples:

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

K tuples:none
Defined Rule Symbols:

a, c

Defined Pair Symbols:

A, C

Compound Symbols:

c2, c3

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

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

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

C(b(z0)) → c3(A(b(a(z0))), A(z0))

S tuples:

C(b(z0)) → c3(A(b(a(z0))), A(z0))

K tuples:none
Defined Rule Symbols:

a, c

Defined Pair Symbols:

C

Compound Symbols:

c3

(5) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 1 dangling nodes:

C(b(z0)) → c3(A(b(a(z0))), A(z0))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

a, c

Defined Pair Symbols:none

Compound Symbols:none

(7) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(8) BOUNDS(O(1), O(1))