### (0) Obligation:

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

f(c(n, x)) → c(f(g(c(n, x))), f(h(c(n, x))))
g(c(0, x)) → x
h(c(1, x)) → x

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

### (2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(c(z0, z1)) → c(f(g(c(z0, z1))), f(h(c(z0, z1))))
g(c(0, z0)) → z0
h(c(1, z0)) → z0
Tuples:

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

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

f, g, h

Defined Pair Symbols:

F

Compound Symbols:

c1

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

Removed 2 trailing tuple parts

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(c(z0, z1)) → c(f(g(c(z0, z1))), f(h(c(z0, z1))))
g(c(0, z0)) → z0
h(c(1, z0)) → z0
Tuples:

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

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

f, g, h

Defined Pair Symbols:

F

Compound Symbols:

c1

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

Use narrowing to replace F(c(z0, z1)) → c1(F(g(c(z0, z1))), F(h(c(z0, z1)))) by

F(c(0, z0)) → c1(F(z0), F(h(c(0, z0))))
F(c(x0, x1)) → c1(F(h(c(x0, x1))))

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(c(z0, z1)) → c(f(g(c(z0, z1))), f(h(c(z0, z1))))
g(c(0, z0)) → z0
h(c(1, z0)) → z0
Tuples:

F(c(0, z0)) → c1(F(z0), F(h(c(0, z0))))
F(c(x0, x1)) → c1(F(h(c(x0, x1))))
S tuples:

F(c(0, z0)) → c1(F(z0), F(h(c(0, z0))))
F(c(x0, x1)) → c1(F(h(c(x0, x1))))
K tuples:none
Defined Rule Symbols:

f, g, h

Defined Pair Symbols:

F

Compound Symbols:

c1, c1

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

Removed 1 trailing tuple parts

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(c(z0, z1)) → c(f(g(c(z0, z1))), f(h(c(z0, z1))))
g(c(0, z0)) → z0
h(c(1, z0)) → z0
Tuples:

F(c(x0, x1)) → c1(F(h(c(x0, x1))))
F(c(0, z0)) → c1(F(z0))
S tuples:

F(c(x0, x1)) → c1(F(h(c(x0, x1))))
F(c(0, z0)) → c1(F(z0))
K tuples:none
Defined Rule Symbols:

f, g, h

Defined Pair Symbols:

F

Compound Symbols:

c1

### (9) CdtPolyRedPairProof (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(c(0, z0)) → c1(F(z0))
We considered the (Usable) Rules:

h(c(1, z0)) → z0
And the Tuples:

F(c(x0, x1)) → c1(F(h(c(x0, x1))))
F(c(0, z0)) → c1(F(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 
POL(1) = 0
POL(F(x1)) = x1
POL(c(x1, x2)) =  + x2
POL(c1(x1)) = x1
POL(h(x1)) = x1

### (10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(c(z0, z1)) → c(f(g(c(z0, z1))), f(h(c(z0, z1))))
g(c(0, z0)) → z0
h(c(1, z0)) → z0
Tuples:

F(c(x0, x1)) → c1(F(h(c(x0, x1))))
F(c(0, z0)) → c1(F(z0))
S tuples:

F(c(x0, x1)) → c1(F(h(c(x0, x1))))
K tuples:

F(c(0, z0)) → c1(F(z0))
Defined Rule Symbols:

f, g, h

Defined Pair Symbols:

F

Compound Symbols:

c1

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

Use narrowing to replace F(c(x0, x1)) → c1(F(h(c(x0, x1)))) by

F(c(1, z0)) → c1(F(z0))
F(c(x0, x1)) → c1

### (12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(c(z0, z1)) → c(f(g(c(z0, z1))), f(h(c(z0, z1))))
g(c(0, z0)) → z0
h(c(1, z0)) → z0
Tuples:

F(c(0, z0)) → c1(F(z0))
F(c(1, z0)) → c1(F(z0))
F(c(x0, x1)) → c1
S tuples:

F(c(1, z0)) → c1(F(z0))
F(c(x0, x1)) → c1
K tuples:

F(c(0, z0)) → c1(F(z0))
Defined Rule Symbols:

f, g, h

Defined Pair Symbols:

F

Compound Symbols:

c1, c1

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

Removed 1 trailing nodes:

F(c(x0, x1)) → c1

### (14) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(c(z0, z1)) → c(f(g(c(z0, z1))), f(h(c(z0, z1))))
g(c(0, z0)) → z0
h(c(1, z0)) → z0
Tuples:

F(c(0, z0)) → c1(F(z0))
F(c(1, z0)) → c1(F(z0))
S tuples:

F(c(1, z0)) → c1(F(z0))
K tuples:

F(c(0, z0)) → c1(F(z0))
Defined Rule Symbols:

f, g, h

Defined Pair Symbols:

F

Compound Symbols:

c1

### (15) CdtPolyRedPairProof (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(c(1, z0)) → c1(F(z0))
We considered the (Usable) Rules:none
And the Tuples:

F(c(0, z0)) → c1(F(z0))
F(c(1, z0)) → c1(F(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(1) = 
POL(F(x1)) = x1
POL(c(x1, x2)) =  + x2
POL(c1(x1)) = x1

### (16) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(c(z0, z1)) → c(f(g(c(z0, z1))), f(h(c(z0, z1))))
g(c(0, z0)) → z0
h(c(1, z0)) → z0
Tuples:

F(c(0, z0)) → c1(F(z0))
F(c(1, z0)) → c1(F(z0))
S tuples:none
K tuples:

F(c(0, z0)) → c1(F(z0))
F(c(1, z0)) → c1(F(z0))
Defined Rule Symbols:

f, g, h

Defined Pair Symbols:

F

Compound Symbols:

c1

### (17) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty