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:

f(f(s(x), 0), f(y, z)) → f(f(y, z), f(y, s(z)))
f(f(s(x), s(y)), f(z, w)) → f(f(x, y), f(z, w))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(s(z0), 0), f(z1, z2)) → f(f(z1, z2), f(z1, s(z2)))
f(f(s(z0), s(z1)), f(z2, z3)) → f(f(z0, z1), f(z2, z3))

Tuples:

F(f(s(z0), 0), f(z1, z2)) → c(F(f(z1, z2), f(z1, s(z2))), F(z1, z2), F(z1, s(z2)))
F(f(s(z0), s(z1)), f(z2, z3)) → c1(F(f(z0, z1), f(z2, z3)), F(z0, z1), F(z2, z3))

S tuples:

F(f(s(z0), 0), f(z1, z2)) → c(F(f(z1, z2), f(z1, s(z2))), F(z1, z2), F(z1, s(z2)))
F(f(s(z0), s(z1)), f(z2, z3)) → c1(F(f(z0, z1), f(z2, z3)), F(z0, z1), F(z2, z3))

K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c, c1

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

F(f(s(z0), 0), f(z1, z2)) → c(F(f(z1, z2), f(z1, s(z2))), F(z1, z2), F(z1, s(z2)))
F(f(s(z0), s(z1)), f(z2, z3)) → c1(F(f(z0, z1), f(z2, z3)), F(z0, z1), F(z2, z3))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(s(z0), 0), f(z1, z2)) → f(f(z1, z2), f(z1, s(z2)))
f(f(s(z0), s(z1)), f(z2, z3)) → f(f(z0, z1), f(z2, z3))

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

f

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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