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(x, y, z), u, f(x, y, v)) → f(x, y, f(z, u, v))
f(x, y, y) → y
f(x, y, g(y)) → x
f(x, x, y) → x
f(g(x), x, y) → y

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(z0, z1, z2), z3, f(z0, z1, z4)) → f(z0, z1, f(z2, z3, z4))
f(z0, z1, z1) → z1
f(z0, z1, g(z1)) → z0
f(z0, z0, z1) → z0
f(g(z0), z0, z1) → z1

Tuples:

F(f(z0, z1, z2), z3, f(z0, z1, z4)) → c(F(z0, z1, f(z2, z3, z4)), F(z2, z3, z4))

S tuples:

F(f(z0, z1, z2), z3, f(z0, z1, z4)) → c(F(z0, z1, f(z2, z3, z4)), F(z2, z3, z4))

K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

F(f(z0, z1, z2), z3, f(z0, z1, z4)) → c(F(z0, z1, f(z2, z3, z4)), F(z2, z3, z4))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(z0, z1, z2), z3, f(z0, z1, z4)) → f(z0, z1, f(z2, z3, z4))
f(z0, z1, z1) → z1
f(z0, z1, g(z1)) → z0
f(z0, z0, z1) → z0
f(g(z0), z0, z1) → z1

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))