(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 Cpx (relative) TRS 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))
F(z0, z1, z1) → c1
F(z0, z1, g(z1)) → c2
F(z0, z0, z1) → c3
F(g(z0), z0, z1) → c4
S tuples:

F(f(z0, z1, z2), z3, f(z0, z1, z4)) → c(F(z0, z1, f(z2, z3, z4)), F(z2, z3, z4))
F(z0, z1, z1) → c1
F(z0, z1, g(z1)) → c2
F(z0, z0, z1) → c3
F(g(z0), z0, z1) → c4
K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c, c1, c2, c3, c4

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 4 trailing nodes:

F(z0, z1, z1) → c1
F(z0, z0, z1) → c3
F(z0, z1, g(z1)) → c2
F(g(z0), z0, z1) → c4

(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:

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

(5) 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))

(6) 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

(7) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty

(8) BOUNDS(1, 1)