### (0) Obligation:

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

f(g(x)) → g(g(f(x)))
f(g(x)) → g(g(g(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(g(z0)) → g(g(f(z0)))
f(g(z0)) → g(g(g(z0)))
Tuples:

F(g(z0)) → c(F(z0))
F(g(z0)) → c1
S tuples:

F(g(z0)) → c(F(z0))
F(g(z0)) → c1
K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c, c1

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

Removed 1 trailing nodes:

F(g(z0)) → c1

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(g(z0)) → g(g(f(z0)))
f(g(z0)) → g(g(g(z0)))
Tuples:

F(g(z0)) → c(F(z0))
S tuples:

F(g(z0)) → c(F(z0))
K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c

### (5) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

f(g(z0)) → g(g(f(z0)))
f(g(z0)) → g(g(g(z0)))

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

F(g(z0)) → c(F(z0))
S tuples:

F(g(z0)) → c(F(z0))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

F

Compound Symbols:

c

### (7) 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(g(z0)) → c(F(z0))
We considered the (Usable) Rules:none
And the Tuples:

F(g(z0)) → c(F(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(F(x1)) = [5]x1
POL(c(x1)) = x1
POL(g(x1)) = [1] + x1

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

F(g(z0)) → c(F(z0))
S tuples:none
K tuples:

F(g(z0)) → c(F(z0))
Defined Rule Symbols:none

Defined Pair Symbols:

F

Compound Symbols:

c

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

The set S is empty