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

Tuples:

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

S tuples:

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

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

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

A(z0) → c2(C(z0))

S tuples:

A(z0) → c2(C(z0))

K tuples:none
Defined Rule Symbols:

a, c

Defined Pair Symbols:

A

Compound Symbols:

c2

(5) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 1 dangling nodes:

A(z0) → c2(C(z0))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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