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(a(x0)), p(x1, p(a(x2), x3))) → p(x2, p(a(a(b(x1))), p(a(a(x0)), x3)))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(a(a(z0)), p(z1, p(a(z2), z3))) → p(z2, p(a(a(b(z1))), p(a(a(z0)), z3)))

Tuples:

P(a(a(z0)), p(z1, p(a(z2), z3))) → c(P(z2, p(a(a(b(z1))), p(a(a(z0)), z3))), P(a(a(b(z1))), p(a(a(z0)), z3)), P(a(a(z0)), z3))

S tuples:

P(a(a(z0)), p(z1, p(a(z2), z3))) → c(P(z2, p(a(a(b(z1))), p(a(a(z0)), z3))), P(a(a(b(z1))), p(a(a(z0)), z3)), P(a(a(z0)), z3))

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(a(z0)), p(z1, p(a(z2), z3))) → c(P(z2, p(a(a(b(z1))), p(a(a(z0)), z3))), P(a(a(b(z1))), p(a(a(z0)), z3)), P(a(a(z0)), z3))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(a(a(z0)), p(z1, p(a(z2), z3))) → p(z2, p(a(a(b(z1))), p(a(a(z0)), z3)))

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))