### (0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

g(f(x), y) → f(h(x, y))
h(x, y) → g(x, f(y))

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

### (2) Obligation:

Complexity Dependency Tuples Problem
Rules:

g(f(z0), z1) → f(h(z0, z1))
h(z0, z1) → g(z0, f(z1))
Tuples:

G(f(z0), z1) → c(H(z0, z1))
H(z0, z1) → c1(G(z0, f(z1)))
S tuples:

G(f(z0), z1) → c(H(z0, z1))
H(z0, z1) → c1(G(z0, f(z1)))
K tuples:none
Defined Rule Symbols:

g, h

Defined Pair Symbols:

G, H

Compound Symbols:

c, c1

### (3) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

g(f(z0), z1) → f(h(z0, z1))
h(z0, z1) → g(z0, f(z1))

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

G(f(z0), z1) → c(H(z0, z1))
H(z0, z1) → c1(G(z0, f(z1)))
S tuples:

G(f(z0), z1) → c(H(z0, z1))
H(z0, z1) → c1(G(z0, f(z1)))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

G, H

Compound Symbols:

c, c1

### (5) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

G(f(z0), z1) → c(H(z0, z1))
H(z0, z1) → c1(G(z0, f(z1)))
We considered the (Usable) Rules:none
And the Tuples:

G(f(z0), z1) → c(H(z0, z1))
H(z0, z1) → c1(G(z0, f(z1)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(G(x1, x2)) = [2] + [4]x1
POL(H(x1, x2)) = [4] + [4]x1
POL(c(x1)) = x1
POL(c1(x1)) = x1
POL(f(x1)) = [2] + x1

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

G(f(z0), z1) → c(H(z0, z1))
H(z0, z1) → c1(G(z0, f(z1)))
S tuples:none
K tuples:

G(f(z0), z1) → c(H(z0, z1))
H(z0, z1) → c1(G(z0, f(z1)))
Defined Rule Symbols:none

Defined Pair Symbols:

G, H

Compound Symbols:

c, c1

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

The set S is empty