### (0) Obligation:

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

not(true) → false
not(false) → true
odd(0) → false
odd(s(x)) → not(odd(x))
+(x, 0) → x
+(x, s(y)) → s(+(x, y))
+(s(x), y) → s(+(x, y))

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

### (2) Obligation:

Complexity Dependency Tuples Problem
Rules:

not(true) → false
not(false) → true
odd(0) → false
odd(s(z0)) → not(odd(z0))
+(z0, 0) → z0
+(z0, s(z1)) → s(+(z0, z1))
+(s(z0), z1) → s(+(z0, z1))
Tuples:

NOT(true) → c
NOT(false) → c1
ODD(0) → c2
ODD(s(z0)) → c3(NOT(odd(z0)), ODD(z0))
+'(z0, 0) → c4
+'(z0, s(z1)) → c5(+'(z0, z1))
+'(s(z0), z1) → c6(+'(z0, z1))
S tuples:

NOT(true) → c
NOT(false) → c1
ODD(0) → c2
ODD(s(z0)) → c3(NOT(odd(z0)), ODD(z0))
+'(z0, 0) → c4
+'(z0, s(z1)) → c5(+'(z0, z1))
+'(s(z0), z1) → c6(+'(z0, z1))
K tuples:none
Defined Rule Symbols:

not, odd, +

Defined Pair Symbols:

NOT, ODD, +'

Compound Symbols:

c, c1, c2, c3, c4, c5, c6

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

Removed 4 trailing nodes:

NOT(true) → c
NOT(false) → c1
ODD(0) → c2
+'(z0, 0) → c4

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

not(true) → false
not(false) → true
odd(0) → false
odd(s(z0)) → not(odd(z0))
+(z0, 0) → z0
+(z0, s(z1)) → s(+(z0, z1))
+(s(z0), z1) → s(+(z0, z1))
Tuples:

ODD(s(z0)) → c3(NOT(odd(z0)), ODD(z0))
+'(z0, s(z1)) → c5(+'(z0, z1))
+'(s(z0), z1) → c6(+'(z0, z1))
S tuples:

ODD(s(z0)) → c3(NOT(odd(z0)), ODD(z0))
+'(z0, s(z1)) → c5(+'(z0, z1))
+'(s(z0), z1) → c6(+'(z0, z1))
K tuples:none
Defined Rule Symbols:

not, odd, +

Defined Pair Symbols:

ODD, +'

Compound Symbols:

c3, c5, c6

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

Removed 1 trailing tuple parts

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:

not(true) → false
not(false) → true
odd(0) → false
odd(s(z0)) → not(odd(z0))
+(z0, 0) → z0
+(z0, s(z1)) → s(+(z0, z1))
+(s(z0), z1) → s(+(z0, z1))
Tuples:

+'(z0, s(z1)) → c5(+'(z0, z1))
+'(s(z0), z1) → c6(+'(z0, z1))
ODD(s(z0)) → c3(ODD(z0))
S tuples:

+'(z0, s(z1)) → c5(+'(z0, z1))
+'(s(z0), z1) → c6(+'(z0, z1))
ODD(s(z0)) → c3(ODD(z0))
K tuples:none
Defined Rule Symbols:

not, odd, +

Defined Pair Symbols:

+', ODD

Compound Symbols:

c5, c6, c3

### (7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

not(true) → false
not(false) → true
odd(0) → false
odd(s(z0)) → not(odd(z0))
+(z0, 0) → z0
+(z0, s(z1)) → s(+(z0, z1))
+(s(z0), z1) → s(+(z0, z1))

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

+'(z0, s(z1)) → c5(+'(z0, z1))
+'(s(z0), z1) → c6(+'(z0, z1))
ODD(s(z0)) → c3(ODD(z0))
S tuples:

+'(z0, s(z1)) → c5(+'(z0, z1))
+'(s(z0), z1) → c6(+'(z0, z1))
ODD(s(z0)) → c3(ODD(z0))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

+', ODD

Compound Symbols:

c5, c6, c3

### (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.

+'(z0, s(z1)) → c5(+'(z0, z1))
+'(s(z0), z1) → c6(+'(z0, z1))
ODD(s(z0)) → c3(ODD(z0))
We considered the (Usable) Rules:none
And the Tuples:

+'(z0, s(z1)) → c5(+'(z0, z1))
+'(s(z0), z1) → c6(+'(z0, z1))
ODD(s(z0)) → c3(ODD(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(+'(x1, x2)) = [5]x1 + [4]x2
POL(ODD(x1)) = [4]x1
POL(c3(x1)) = x1
POL(c5(x1)) = x1
POL(c6(x1)) = x1
POL(s(x1)) = [4] + x1

### (10) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

+'(z0, s(z1)) → c5(+'(z0, z1))
+'(s(z0), z1) → c6(+'(z0, z1))
ODD(s(z0)) → c3(ODD(z0))
S tuples:none
K tuples:

+'(z0, s(z1)) → c5(+'(z0, z1))
+'(s(z0), z1) → c6(+'(z0, z1))
ODD(s(z0)) → c3(ODD(z0))
Defined Rule Symbols:none

Defined Pair Symbols:

+', ODD

Compound Symbols:

c5, c6, c3

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

The set S is empty