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:

p(a(x0), p(b(a(x1)), x2)) → p(x1, p(a(b(a(x1))), x2))
a(b(a(x0))) → b(a(b(x0)))

Rewrite Strategy: INNERMOST

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(a(z0), p(b(a(z1)), z2)) → p(z1, p(a(b(a(z1))), z2))
a(b(a(z0))) → b(a(b(z0)))

Tuples:

P(a(z0), p(b(a(z1)), z2)) → c(P(z1, p(a(b(a(z1))), z2)), P(a(b(a(z1))), z2), A(b(a(z1))), A(z1))
A(b(a(z0))) → c1(A(b(z0)))

S tuples:

P(a(z0), p(b(a(z1)), z2)) → c(P(z1, p(a(b(a(z1))), z2)), P(a(b(a(z1))), z2), A(b(a(z1))), A(z1))
A(b(a(z0))) → c1(A(b(z0)))

K tuples:none
Defined Rule Symbols:

p, a

Defined Pair Symbols:

P, A

Compound Symbols:

c, c1

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

P(a(z0), p(b(a(z1)), z2)) → c(P(z1, p(a(b(a(z1))), z2)), P(a(b(a(z1))), z2), A(b(a(z1))), A(z1))
A(b(a(z0))) → c1(A(b(z0)))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(a(z0), p(b(a(z1)), z2)) → p(z1, p(a(b(a(z1))), z2))
a(b(a(z0))) → b(a(b(z0)))

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

p, a

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(6) BOUNDS(O(1), O(1))