### (0) Obligation:

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

f(X) → g(n__h(n__f(X)))
h(X) → n__h(X)
f(X) → n__f(X)
activate(n__h(X)) → h(activate(X))
activate(n__f(X)) → f(activate(X))
activate(X) → 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(z0) → g(n__h(n__f(z0)))
f(z0) → n__f(z0)
h(z0) → n__h(z0)
activate(n__h(z0)) → h(activate(z0))
activate(n__f(z0)) → f(activate(z0))
activate(z0) → z0
Tuples:

F(z0) → c
F(z0) → c1
H(z0) → c2
ACTIVATE(n__h(z0)) → c3(H(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__f(z0)) → c4(F(activate(z0)), ACTIVATE(z0))
ACTIVATE(z0) → c5
S tuples:

F(z0) → c
F(z0) → c1
H(z0) → c2
ACTIVATE(n__h(z0)) → c3(H(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__f(z0)) → c4(F(activate(z0)), ACTIVATE(z0))
ACTIVATE(z0) → c5
K tuples:none
Defined Rule Symbols:

f, h, activate

Defined Pair Symbols:

F, H, ACTIVATE

Compound Symbols:

c, c1, c2, c3, c4, c5

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

Removed 4 trailing nodes:

F(z0) → c
F(z0) → c1
ACTIVATE(z0) → c5
H(z0) → c2

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(z0) → g(n__h(n__f(z0)))
f(z0) → n__f(z0)
h(z0) → n__h(z0)
activate(n__h(z0)) → h(activate(z0))
activate(n__f(z0)) → f(activate(z0))
activate(z0) → z0
Tuples:

ACTIVATE(n__h(z0)) → c3(H(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__f(z0)) → c4(F(activate(z0)), ACTIVATE(z0))
S tuples:

ACTIVATE(n__h(z0)) → c3(H(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__f(z0)) → c4(F(activate(z0)), ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

f, h, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c3, c4

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

Removed 2 trailing tuple parts

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(z0) → g(n__h(n__f(z0)))
f(z0) → n__f(z0)
h(z0) → n__h(z0)
activate(n__h(z0)) → h(activate(z0))
activate(n__f(z0)) → f(activate(z0))
activate(z0) → z0
Tuples:

ACTIVATE(n__h(z0)) → c3(ACTIVATE(z0))
ACTIVATE(n__f(z0)) → c4(ACTIVATE(z0))
S tuples:

ACTIVATE(n__h(z0)) → c3(ACTIVATE(z0))
ACTIVATE(n__f(z0)) → c4(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

f, h, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c3, c4

### (7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

f(z0) → g(n__h(n__f(z0)))
f(z0) → n__f(z0)
h(z0) → n__h(z0)
activate(n__h(z0)) → h(activate(z0))
activate(n__f(z0)) → f(activate(z0))
activate(z0) → z0

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

ACTIVATE(n__h(z0)) → c3(ACTIVATE(z0))
ACTIVATE(n__f(z0)) → c4(ACTIVATE(z0))
S tuples:

ACTIVATE(n__h(z0)) → c3(ACTIVATE(z0))
ACTIVATE(n__f(z0)) → c4(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c3, c4

### (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.

ACTIVATE(n__h(z0)) → c3(ACTIVATE(z0))
ACTIVATE(n__f(z0)) → c4(ACTIVATE(z0))
We considered the (Usable) Rules:none
And the Tuples:

ACTIVATE(n__h(z0)) → c3(ACTIVATE(z0))
ACTIVATE(n__f(z0)) → c4(ACTIVATE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVATE(x1)) = [4]x1
POL(c3(x1)) = x1
POL(c4(x1)) = x1
POL(n__f(x1)) = [4] + x1
POL(n__h(x1)) = [1] + x1

### (10) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

ACTIVATE(n__h(z0)) → c3(ACTIVATE(z0))
ACTIVATE(n__f(z0)) → c4(ACTIVATE(z0))
S tuples:none
K tuples:

ACTIVATE(n__h(z0)) → c3(ACTIVATE(z0))
ACTIVATE(n__f(z0)) → c4(ACTIVATE(z0))
Defined Rule Symbols:none

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c3, c4

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

The set S is empty