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

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

Tuples:

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

S tuples:

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

K tuples:none
Defined Rule Symbols:

p

Defined Pair Symbols:

P

Compound Symbols:

c

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

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

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

p

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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