### (0) Obligation:

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

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

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

### (2) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

F(a) → c(H(a))
H(g(z0)) → c1(H(f(z0)), F(z0))
K(z0, h(z0), a) → c2(H(z0))
K(f(z0), z1, z0) → c3(F(z0))
S tuples:

F(a) → c(H(a))
H(g(z0)) → c1(H(f(z0)), F(z0))
K(z0, h(z0), a) → c2(H(z0))
K(f(z0), z1, z0) → c3(F(z0))
K tuples:none
Defined Rule Symbols:

f, h, k

Defined Pair Symbols:

F, H, K

Compound Symbols:

c, c1, c2, c3

### (3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

K(z0, h(z0), a) → c2(H(z0))
Removed 2 trailing nodes:

F(a) → c(H(a))
K(f(z0), z1, z0) → c3(F(z0))

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

H(g(z0)) → c1(H(f(z0)), F(z0))
S tuples:

H(g(z0)) → c1(H(f(z0)), F(z0))
K tuples:none
Defined Rule Symbols:

f, h, k

Defined Pair Symbols:

H

Compound Symbols:

c1

### (5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing tuple parts

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

H(g(z0)) → c1(H(f(z0)))
S tuples:

H(g(z0)) → c1(H(f(z0)))
K tuples:none
Defined Rule Symbols:

f, h, k

Defined Pair Symbols:

H

Compound Symbols:

c1

### (7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

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

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(a) → g(h(a))
Tuples:

H(g(z0)) → c1(H(f(z0)))
S tuples:

H(g(z0)) → c1(H(f(z0)))
K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

H

Compound Symbols:

c1

### (9) 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.

H(g(z0)) → c1(H(f(z0)))
We considered the (Usable) Rules:

f(a) → g(h(a))
And the Tuples:

H(g(z0)) → c1(H(f(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(H(x1)) = [4]x1
POL(a) = [4]
POL(c1(x1)) = x1
POL(f(x1)) = x1
POL(g(x1)) = [2] + x1
POL(h(x1)) = 0

### (10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(a) → g(h(a))
Tuples:

H(g(z0)) → c1(H(f(z0)))
S tuples:none
K tuples:

H(g(z0)) → c1(H(f(z0)))
Defined Rule Symbols:

f

Defined Pair Symbols:

H

Compound Symbols:

c1

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

The set S is empty