### (0) Obligation:

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

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

F(z0, z1) → c(G(z0, z1))
G(h(z0), z1) → c1(F(z0, z1))
G(h(z0), z1) → c2(G(z0, z1))
S tuples:

F(z0, z1) → c(G(z0, z1))
G(h(z0), z1) → c1(F(z0, z1))
G(h(z0), z1) → c2(G(z0, z1))
K tuples:none
Defined Rule Symbols:

f, g

Defined Pair Symbols:

F, G

Compound Symbols:

c, c1, c2

### (3) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

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

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

F(z0, z1) → c(G(z0, z1))
G(h(z0), z1) → c1(F(z0, z1))
G(h(z0), z1) → c2(G(z0, z1))
S tuples:

F(z0, z1) → c(G(z0, z1))
G(h(z0), z1) → c1(F(z0, z1))
G(h(z0), z1) → c2(G(z0, z1))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

F, G

Compound Symbols:

c, c1, c2

### (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.

F(z0, z1) → c(G(z0, z1))
G(h(z0), z1) → c1(F(z0, z1))
G(h(z0), z1) → c2(G(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

F(z0, z1) → c(G(z0, z1))
G(h(z0), z1) → c1(F(z0, z1))
G(h(z0), z1) → c2(G(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(F(x1, x2)) = [1] + [2]x1 + x2
POL(G(x1, x2)) = [2]x1 + x2
POL(c(x1)) = x1
POL(c1(x1)) = x1
POL(c2(x1)) = x1
POL(h(x1)) = [1] + x1

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

F(z0, z1) → c(G(z0, z1))
G(h(z0), z1) → c1(F(z0, z1))
G(h(z0), z1) → c2(G(z0, z1))
S tuples:none
K tuples:

F(z0, z1) → c(G(z0, z1))
G(h(z0), z1) → c1(F(z0, z1))
G(h(z0), z1) → c2(G(z0, z1))
Defined Rule Symbols:none

Defined Pair Symbols:

F, G

Compound Symbols:

c, c1, c2

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

The set S is empty