### (0) Obligation:

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

f(0) → 0
f(s(x)) → f(id(x))
id(0) → 0
id(s(x)) → s(id(x))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

### (2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(0) → 0
f(s(z0)) → f(id(z0))
id(0) → 0
id(s(z0)) → s(id(z0))
Tuples:

F(s(z0)) → c1(F(id(z0)), ID(z0))
ID(s(z0)) → c3(ID(z0))
S tuples:

F(s(z0)) → c1(F(id(z0)), ID(z0))
ID(s(z0)) → c3(ID(z0))
K tuples:none
Defined Rule Symbols:

f, id

Defined Pair Symbols:

F, ID

Compound Symbols:

c1, c3

### (3) 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(s(z0)) → c1(F(id(z0)), ID(z0))
We considered the (Usable) Rules:

id(0) → 0
id(s(z0)) → s(id(z0))
And the Tuples:

F(s(z0)) → c1(F(id(z0)), ID(z0))
ID(s(z0)) → c3(ID(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(F(x1)) = x1
POL(ID(x1)) = 0
POL(c1(x1, x2)) = x1 + x2
POL(c3(x1)) = x1
POL(id(x1)) = x1
POL(s(x1)) =  + x1

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(0) → 0
f(s(z0)) → f(id(z0))
id(0) → 0
id(s(z0)) → s(id(z0))
Tuples:

F(s(z0)) → c1(F(id(z0)), ID(z0))
ID(s(z0)) → c3(ID(z0))
S tuples:

ID(s(z0)) → c3(ID(z0))
K tuples:

F(s(z0)) → c1(F(id(z0)), ID(z0))
Defined Rule Symbols:

f, id

Defined Pair Symbols:

F, ID

Compound Symbols:

c1, c3

### (5) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

ID(s(z0)) → c3(ID(z0))
We considered the (Usable) Rules:

id(0) → 0
id(s(z0)) → s(id(z0))
And the Tuples:

F(s(z0)) → c1(F(id(z0)), ID(z0))
ID(s(z0)) → c3(ID(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(F(x1)) = x12
POL(ID(x1)) =  + x1
POL(c1(x1, x2)) = x1 + x2
POL(c3(x1)) = x1
POL(id(x1)) = x1
POL(s(x1)) =  + x1

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(0) → 0
f(s(z0)) → f(id(z0))
id(0) → 0
id(s(z0)) → s(id(z0))
Tuples:

F(s(z0)) → c1(F(id(z0)), ID(z0))
ID(s(z0)) → c3(ID(z0))
S tuples:none
K tuples:

F(s(z0)) → c1(F(id(z0)), ID(z0))
ID(s(z0)) → c3(ID(z0))
Defined Rule Symbols:

f, id

Defined Pair Symbols:

F, ID

Compound Symbols:

c1, c3

### (7) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty