### (0) Obligation:

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

odd(S(x)) → even(x)
even(S(x)) → odd(x)
odd(0) → 0
even(0) → S(0)

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

### (2) Obligation:

Complexity Dependency Tuples Problem
Rules:

odd(S(z0)) → even(z0)
odd(0) → 0
even(S(z0)) → odd(z0)
even(0) → S(0)
Tuples:

ODD(S(z0)) → c(EVEN(z0))
ODD(0) → c1
EVEN(S(z0)) → c2(ODD(z0))
EVEN(0) → c3
S tuples:

ODD(S(z0)) → c(EVEN(z0))
ODD(0) → c1
EVEN(S(z0)) → c2(ODD(z0))
EVEN(0) → c3
K tuples:none
Defined Rule Symbols:

odd, even

Defined Pair Symbols:

ODD, EVEN

Compound Symbols:

c, c1, c2, c3

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

Removed 2 trailing nodes:

EVEN(0) → c3
ODD(0) → c1

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

odd(S(z0)) → even(z0)
odd(0) → 0
even(S(z0)) → odd(z0)
even(0) → S(0)
Tuples:

ODD(S(z0)) → c(EVEN(z0))
EVEN(S(z0)) → c2(ODD(z0))
S tuples:

ODD(S(z0)) → c(EVEN(z0))
EVEN(S(z0)) → c2(ODD(z0))
K tuples:none
Defined Rule Symbols:

odd, even

Defined Pair Symbols:

ODD, EVEN

Compound Symbols:

c, c2

### (5) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

odd(S(z0)) → even(z0)
odd(0) → 0
even(S(z0)) → odd(z0)
even(0) → S(0)

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

ODD(S(z0)) → c(EVEN(z0))
EVEN(S(z0)) → c2(ODD(z0))
S tuples:

ODD(S(z0)) → c(EVEN(z0))
EVEN(S(z0)) → c2(ODD(z0))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

ODD, EVEN

Compound Symbols:

c, c2

### (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.

ODD(S(z0)) → c(EVEN(z0))
EVEN(S(z0)) → c2(ODD(z0))
We considered the (Usable) Rules:none
And the Tuples:

ODD(S(z0)) → c(EVEN(z0))
EVEN(S(z0)) → c2(ODD(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(EVEN(x1)) = [4] + [4]x1
POL(ODD(x1)) = [2] + [4]x1
POL(S(x1)) = [5] + x1
POL(c(x1)) = x1
POL(c2(x1)) = x1

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

ODD(S(z0)) → c(EVEN(z0))
EVEN(S(z0)) → c2(ODD(z0))
S tuples:none
K tuples:

ODD(S(z0)) → c(EVEN(z0))
EVEN(S(z0)) → c2(ODD(z0))
Defined Rule Symbols:none

Defined Pair Symbols:

ODD, EVEN

Compound Symbols:

c, c2

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

The set S is empty