### (0) Obligation:

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

half(0) → 0
half(s(0)) → 0
half(s(s(x))) → s(half(x))
lastbit(0) → 0
lastbit(s(0)) → s(0)
lastbit(s(s(x))) → lastbit(x)
zero(0) → true
zero(s(x)) → false
conv(x) → conviter(x, cons(0, nil))
conviter(x, l) → if(zero(x), x, l)
if(true, x, l) → l
if(false, x, l) → conviter(half(x), cons(lastbit(x), l))

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

### (2) Obligation:

Complexity Dependency Tuples Problem
Rules:

half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
lastbit(0) → 0
lastbit(s(0)) → s(0)
lastbit(s(s(z0))) → lastbit(z0)
zero(0) → true
zero(s(z0)) → false
conv(z0) → conviter(z0, cons(0, nil))
conviter(z0, z1) → if(zero(z0), z0, z1)
if(true, z0, z1) → z1
if(false, z0, z1) → conviter(half(z0), cons(lastbit(z0), z1))
Tuples:

HALF(0) → c
HALF(s(0)) → c1
HALF(s(s(z0))) → c2(HALF(z0))
LASTBIT(0) → c3
LASTBIT(s(0)) → c4
LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
ZERO(0) → c6
ZERO(s(z0)) → c7
CONV(z0) → c8(CONVITER(z0, cons(0, nil)))
CONVITER(z0, z1) → c9(IF(zero(z0), z0, z1), ZERO(z0))
IF(true, z0, z1) → c10
IF(false, z0, z1) → c11(CONVITER(half(z0), cons(lastbit(z0), z1)), HALF(z0), LASTBIT(z0))
S tuples:

HALF(0) → c
HALF(s(0)) → c1
HALF(s(s(z0))) → c2(HALF(z0))
LASTBIT(0) → c3
LASTBIT(s(0)) → c4
LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
ZERO(0) → c6
ZERO(s(z0)) → c7
CONV(z0) → c8(CONVITER(z0, cons(0, nil)))
CONVITER(z0, z1) → c9(IF(zero(z0), z0, z1), ZERO(z0))
IF(true, z0, z1) → c10
IF(false, z0, z1) → c11(CONVITER(half(z0), cons(lastbit(z0), z1)), HALF(z0), LASTBIT(z0))
K tuples:none
Defined Rule Symbols:

half, lastbit, zero, conv, conviter, if

Defined Pair Symbols:

HALF, LASTBIT, ZERO, CONV, CONVITER, IF

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11

### (3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

CONV(z0) → c8(CONVITER(z0, cons(0, nil)))
Removed 7 trailing nodes:

IF(true, z0, z1) → c10
HALF(0) → c
HALF(s(0)) → c1
LASTBIT(0) → c3
LASTBIT(s(0)) → c4
ZERO(0) → c6
ZERO(s(z0)) → c7

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
lastbit(0) → 0
lastbit(s(0)) → s(0)
lastbit(s(s(z0))) → lastbit(z0)
zero(0) → true
zero(s(z0)) → false
conv(z0) → conviter(z0, cons(0, nil))
conviter(z0, z1) → if(zero(z0), z0, z1)
if(true, z0, z1) → z1
if(false, z0, z1) → conviter(half(z0), cons(lastbit(z0), z1))
Tuples:

HALF(s(s(z0))) → c2(HALF(z0))
LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
CONVITER(z0, z1) → c9(IF(zero(z0), z0, z1), ZERO(z0))
IF(false, z0, z1) → c11(CONVITER(half(z0), cons(lastbit(z0), z1)), HALF(z0), LASTBIT(z0))
S tuples:

HALF(s(s(z0))) → c2(HALF(z0))
LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
CONVITER(z0, z1) → c9(IF(zero(z0), z0, z1), ZERO(z0))
IF(false, z0, z1) → c11(CONVITER(half(z0), cons(lastbit(z0), z1)), HALF(z0), LASTBIT(z0))
K tuples:none
Defined Rule Symbols:

half, lastbit, zero, conv, conviter, if

Defined Pair Symbols:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11

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

Removed 1 trailing tuple parts

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:

half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
lastbit(0) → 0
lastbit(s(0)) → s(0)
lastbit(s(s(z0))) → lastbit(z0)
zero(0) → true
zero(s(z0)) → false
conv(z0) → conviter(z0, cons(0, nil))
conviter(z0, z1) → if(zero(z0), z0, z1)
if(true, z0, z1) → z1
if(false, z0, z1) → conviter(half(z0), cons(lastbit(z0), z1))
Tuples:

HALF(s(s(z0))) → c2(HALF(z0))
LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
IF(false, z0, z1) → c11(CONVITER(half(z0), cons(lastbit(z0), z1)), HALF(z0), LASTBIT(z0))
CONVITER(z0, z1) → c9(IF(zero(z0), z0, z1))
S tuples:

HALF(s(s(z0))) → c2(HALF(z0))
LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
IF(false, z0, z1) → c11(CONVITER(half(z0), cons(lastbit(z0), z1)), HALF(z0), LASTBIT(z0))
CONVITER(z0, z1) → c9(IF(zero(z0), z0, z1))
K tuples:none
Defined Rule Symbols:

half, lastbit, zero, conv, conviter, if

Defined Pair Symbols:

HALF, LASTBIT, IF, CONVITER

Compound Symbols:

c2, c5, c11, c9

### (7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

conv(z0) → conviter(z0, cons(0, nil))
conviter(z0, z1) → if(zero(z0), z0, z1)
if(true, z0, z1) → z1
if(false, z0, z1) → conviter(half(z0), cons(lastbit(z0), z1))

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:

half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
lastbit(0) → 0
lastbit(s(0)) → s(0)
lastbit(s(s(z0))) → lastbit(z0)
zero(0) → true
zero(s(z0)) → false
Tuples:

HALF(s(s(z0))) → c2(HALF(z0))
LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
IF(false, z0, z1) → c11(CONVITER(half(z0), cons(lastbit(z0), z1)), HALF(z0), LASTBIT(z0))
CONVITER(z0, z1) → c9(IF(zero(z0), z0, z1))
S tuples:

HALF(s(s(z0))) → c2(HALF(z0))
LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
IF(false, z0, z1) → c11(CONVITER(half(z0), cons(lastbit(z0), z1)), HALF(z0), LASTBIT(z0))
CONVITER(z0, z1) → c9(IF(zero(z0), z0, z1))
K tuples:none
Defined Rule Symbols:

half, lastbit, zero

Defined Pair Symbols:

HALF, LASTBIT, IF, CONVITER

Compound Symbols:

c2, c5, c11, c9

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

Use narrowing to replace CONVITER(z0, z1) → c9(IF(zero(z0), z0, z1)) by

CONVITER(0, x1) → c9(IF(true, 0, x1))
CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))

### (10) Obligation:

Complexity Dependency Tuples Problem
Rules:

half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
lastbit(0) → 0
lastbit(s(0)) → s(0)
lastbit(s(s(z0))) → lastbit(z0)
zero(0) → true
zero(s(z0)) → false
Tuples:

HALF(s(s(z0))) → c2(HALF(z0))
LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
IF(false, z0, z1) → c11(CONVITER(half(z0), cons(lastbit(z0), z1)), HALF(z0), LASTBIT(z0))
CONVITER(0, x1) → c9(IF(true, 0, x1))
CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
S tuples:

HALF(s(s(z0))) → c2(HALF(z0))
LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
IF(false, z0, z1) → c11(CONVITER(half(z0), cons(lastbit(z0), z1)), HALF(z0), LASTBIT(z0))
CONVITER(0, x1) → c9(IF(true, 0, x1))
CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
K tuples:none
Defined Rule Symbols:

half, lastbit, zero

Defined Pair Symbols:

HALF, LASTBIT, IF, CONVITER

Compound Symbols:

c2, c5, c11, c9

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

Removed 1 trailing nodes:

CONVITER(0, x1) → c9(IF(true, 0, x1))

### (12) Obligation:

Complexity Dependency Tuples Problem
Rules:

half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
lastbit(0) → 0
lastbit(s(0)) → s(0)
lastbit(s(s(z0))) → lastbit(z0)
zero(0) → true
zero(s(z0)) → false
Tuples:

HALF(s(s(z0))) → c2(HALF(z0))
LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
IF(false, z0, z1) → c11(CONVITER(half(z0), cons(lastbit(z0), z1)), HALF(z0), LASTBIT(z0))
CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
S tuples:

HALF(s(s(z0))) → c2(HALF(z0))
LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
IF(false, z0, z1) → c11(CONVITER(half(z0), cons(lastbit(z0), z1)), HALF(z0), LASTBIT(z0))
CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
K tuples:none
Defined Rule Symbols:

half, lastbit, zero

Defined Pair Symbols:

HALF, LASTBIT, IF, CONVITER

Compound Symbols:

c2, c5, c11, c9

### (13) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

zero(0) → true
zero(s(z0)) → false

### (14) Obligation:

Complexity Dependency Tuples Problem
Rules:

half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
lastbit(0) → 0
lastbit(s(0)) → s(0)
lastbit(s(s(z0))) → lastbit(z0)
Tuples:

HALF(s(s(z0))) → c2(HALF(z0))
LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
IF(false, z0, z1) → c11(CONVITER(half(z0), cons(lastbit(z0), z1)), HALF(z0), LASTBIT(z0))
CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
S tuples:

HALF(s(s(z0))) → c2(HALF(z0))
LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
IF(false, z0, z1) → c11(CONVITER(half(z0), cons(lastbit(z0), z1)), HALF(z0), LASTBIT(z0))
CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
K tuples:none
Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

HALF, LASTBIT, IF, CONVITER

Compound Symbols:

c2, c5, c11, c9

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

Use narrowing to replace IF(false, z0, z1) → c11(CONVITER(half(z0), cons(lastbit(z0), z1)), HALF(z0), LASTBIT(z0)) by

IF(false, 0, x1) → c11(CONVITER(half(0), cons(0, x1)), HALF(0), LASTBIT(0))
IF(false, s(0), x1) → c11(CONVITER(half(s(0)), cons(s(0), x1)), HALF(s(0)), LASTBIT(s(0)))
IF(false, s(s(z0)), x1) → c11(CONVITER(half(s(s(z0))), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, 0, x1) → c11(CONVITER(0, cons(lastbit(0), x1)), HALF(0), LASTBIT(0))
IF(false, s(0), x1) → c11(CONVITER(0, cons(lastbit(s(0)), x1)), HALF(s(0)), LASTBIT(s(0)))
IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(s(s(z0))), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))

### (16) Obligation:

Complexity Dependency Tuples Problem
Rules:

half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
lastbit(0) → 0
lastbit(s(0)) → s(0)
lastbit(s(s(z0))) → lastbit(z0)
Tuples:

HALF(s(s(z0))) → c2(HALF(z0))
LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
IF(false, 0, x1) → c11(CONVITER(half(0), cons(0, x1)), HALF(0), LASTBIT(0))
IF(false, s(0), x1) → c11(CONVITER(half(s(0)), cons(s(0), x1)), HALF(s(0)), LASTBIT(s(0)))
IF(false, s(s(z0)), x1) → c11(CONVITER(half(s(s(z0))), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, 0, x1) → c11(CONVITER(0, cons(lastbit(0), x1)), HALF(0), LASTBIT(0))
IF(false, s(0), x1) → c11(CONVITER(0, cons(lastbit(s(0)), x1)), HALF(s(0)), LASTBIT(s(0)))
IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(s(s(z0))), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
S tuples:

HALF(s(s(z0))) → c2(HALF(z0))
LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
IF(false, 0, x1) → c11(CONVITER(half(0), cons(0, x1)), HALF(0), LASTBIT(0))
IF(false, s(0), x1) → c11(CONVITER(half(s(0)), cons(s(0), x1)), HALF(s(0)), LASTBIT(s(0)))
IF(false, s(s(z0)), x1) → c11(CONVITER(half(s(s(z0))), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, 0, x1) → c11(CONVITER(0, cons(lastbit(0), x1)), HALF(0), LASTBIT(0))
IF(false, s(0), x1) → c11(CONVITER(0, cons(lastbit(s(0)), x1)), HALF(s(0)), LASTBIT(s(0)))
IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(s(s(z0))), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
K tuples:none
Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11

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

Removed 2 trailing nodes:

IF(false, s(0), x1) → c11(CONVITER(0, cons(lastbit(s(0)), x1)), HALF(s(0)), LASTBIT(s(0)))
IF(false, 0, x1) → c11(CONVITER(0, cons(lastbit(0), x1)), HALF(0), LASTBIT(0))

### (18) Obligation:

Complexity Dependency Tuples Problem
Rules:

half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
lastbit(0) → 0
lastbit(s(0)) → s(0)
lastbit(s(s(z0))) → lastbit(z0)
Tuples:

HALF(s(s(z0))) → c2(HALF(z0))
LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
IF(false, 0, x1) → c11(CONVITER(half(0), cons(0, x1)), HALF(0), LASTBIT(0))
IF(false, s(0), x1) → c11(CONVITER(half(s(0)), cons(s(0), x1)), HALF(s(0)), LASTBIT(s(0)))
IF(false, s(s(z0)), x1) → c11(CONVITER(half(s(s(z0))), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(s(s(z0))), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
S tuples:

HALF(s(s(z0))) → c2(HALF(z0))
LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
IF(false, 0, x1) → c11(CONVITER(half(0), cons(0, x1)), HALF(0), LASTBIT(0))
IF(false, s(0), x1) → c11(CONVITER(half(s(0)), cons(s(0), x1)), HALF(s(0)), LASTBIT(s(0)))
IF(false, s(s(z0)), x1) → c11(CONVITER(half(s(s(z0))), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(s(s(z0))), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
K tuples:none
Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11

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

Removed 4 trailing tuple parts

### (20) Obligation:

Complexity Dependency Tuples Problem
Rules:

half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
lastbit(0) → 0
lastbit(s(0)) → s(0)
lastbit(s(s(z0))) → lastbit(z0)
Tuples:

HALF(s(s(z0))) → c2(HALF(z0))
LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
IF(false, s(s(z0)), x1) → c11(CONVITER(half(s(s(z0))), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(s(s(z0))), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, 0, x1) → c11(CONVITER(half(0), cons(0, x1)))
IF(false, s(0), x1) → c11(CONVITER(half(s(0)), cons(s(0), x1)))
S tuples:

HALF(s(s(z0))) → c2(HALF(z0))
LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
IF(false, s(s(z0)), x1) → c11(CONVITER(half(s(s(z0))), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(s(s(z0))), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, 0, x1) → c11(CONVITER(half(0), cons(0, x1)))
IF(false, s(0), x1) → c11(CONVITER(half(s(0)), cons(s(0), x1)))
K tuples:none
Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11, c11

### (21) CdtKnowledgeProof (BOTH BOUNDS(ID, ID) transformation)

The following tuples could be moved from S to K by knowledge propagation:

IF(false, 0, x1) → c11(CONVITER(half(0), cons(0, x1)))

### (22) Obligation:

Complexity Dependency Tuples Problem
Rules:

half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
lastbit(0) → 0
lastbit(s(0)) → s(0)
lastbit(s(s(z0))) → lastbit(z0)
Tuples:

HALF(s(s(z0))) → c2(HALF(z0))
LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
IF(false, s(s(z0)), x1) → c11(CONVITER(half(s(s(z0))), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(s(s(z0))), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, 0, x1) → c11(CONVITER(half(0), cons(0, x1)))
IF(false, s(0), x1) → c11(CONVITER(half(s(0)), cons(s(0), x1)))
S tuples:

HALF(s(s(z0))) → c2(HALF(z0))
LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
IF(false, s(s(z0)), x1) → c11(CONVITER(half(s(s(z0))), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(s(s(z0))), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(0), x1) → c11(CONVITER(half(s(0)), cons(s(0), x1)))
K tuples:

IF(false, 0, x1) → c11(CONVITER(half(0), cons(0, x1)))
Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11, c11

### (23) 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.

IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(s(s(z0))), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
We considered the (Usable) Rules:

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

HALF(s(s(z0))) → c2(HALF(z0))
LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
IF(false, s(s(z0)), x1) → c11(CONVITER(half(s(s(z0))), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(s(s(z0))), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, 0, x1) → c11(CONVITER(half(0), cons(0, x1)))
IF(false, s(0), x1) → c11(CONVITER(half(s(0)), cons(s(0), x1)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(CONVITER(x1, x2)) = [4]x1
POL(HALF(x1)) = 0
POL(IF(x1, x2, x3)) = [2]x1 + [4]x2
POL(LASTBIT(x1)) = 0
POL(c11(x1)) = x1
POL(c11(x1, x2, x3)) = x1 + x2 + x3
POL(c2(x1)) = x1
POL(c5(x1)) = x1
POL(c9(x1)) = x1
POL(cons(x1, x2)) = 0
POL(false) = 0
POL(half(x1)) = x1
POL(lastbit(x1)) = 0
POL(s(x1)) = [2] + x1

### (24) Obligation:

Complexity Dependency Tuples Problem
Rules:

half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
lastbit(0) → 0
lastbit(s(0)) → s(0)
lastbit(s(s(z0))) → lastbit(z0)
Tuples:

HALF(s(s(z0))) → c2(HALF(z0))
LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
IF(false, s(s(z0)), x1) → c11(CONVITER(half(s(s(z0))), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(s(s(z0))), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, 0, x1) → c11(CONVITER(half(0), cons(0, x1)))
IF(false, s(0), x1) → c11(CONVITER(half(s(0)), cons(s(0), x1)))
S tuples:

HALF(s(s(z0))) → c2(HALF(z0))
LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
IF(false, s(s(z0)), x1) → c11(CONVITER(half(s(s(z0))), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(0), x1) → c11(CONVITER(half(s(0)), cons(s(0), x1)))
K tuples:

IF(false, 0, x1) → c11(CONVITER(half(0), cons(0, x1)))
IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(s(s(z0))), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11, c11

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

Use narrowing to replace IF(false, s(s(z0)), x1) → c11(CONVITER(half(s(s(z0))), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0)))) by

IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1)), HALF(s(s(0))), LASTBIT(s(s(0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)), HALF(s(s(s(0)))), LASTBIT(s(s(s(0)))))
IF(false, s(s(s(s(z0)))), x1) → c11(CONVITER(half(s(s(s(s(z0))))), cons(lastbit(z0), x1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))

### (26) Obligation:

Complexity Dependency Tuples Problem
Rules:

half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
lastbit(0) → 0
lastbit(s(0)) → s(0)
lastbit(s(s(z0))) → lastbit(z0)
Tuples:

HALF(s(s(z0))) → c2(HALF(z0))
LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(s(s(z0))), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, 0, x1) → c11(CONVITER(half(0), cons(0, x1)))
IF(false, s(0), x1) → c11(CONVITER(half(s(0)), cons(s(0), x1)))
IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1)), HALF(s(s(0))), LASTBIT(s(s(0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)), HALF(s(s(s(0)))), LASTBIT(s(s(s(0)))))
IF(false, s(s(s(s(z0)))), x1) → c11(CONVITER(half(s(s(s(s(z0))))), cons(lastbit(z0), x1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
S tuples:

HALF(s(s(z0))) → c2(HALF(z0))
LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
IF(false, s(0), x1) → c11(CONVITER(half(s(0)), cons(s(0), x1)))
IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1)), HALF(s(s(0))), LASTBIT(s(s(0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)), HALF(s(s(s(0)))), LASTBIT(s(s(s(0)))))
IF(false, s(s(s(s(z0)))), x1) → c11(CONVITER(half(s(s(s(s(z0))))), cons(lastbit(z0), x1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
K tuples:

IF(false, 0, x1) → c11(CONVITER(half(0), cons(0, x1)))
IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(s(s(z0))), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11, c11

### (27) 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.

IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
We considered the (Usable) Rules:

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

HALF(s(s(z0))) → c2(HALF(z0))
LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(s(s(z0))), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, 0, x1) → c11(CONVITER(half(0), cons(0, x1)))
IF(false, s(0), x1) → c11(CONVITER(half(s(0)), cons(s(0), x1)))
IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1)), HALF(s(s(0))), LASTBIT(s(s(0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)), HALF(s(s(s(0)))), LASTBIT(s(s(s(0)))))
IF(false, s(s(s(s(z0)))), x1) → c11(CONVITER(half(s(s(s(s(z0))))), cons(lastbit(z0), x1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [4]
POL(CONVITER(x1, x2)) = [4]x1
POL(HALF(x1)) = 0
POL(IF(x1, x2, x3)) = [3]x1 + [4]x2
POL(LASTBIT(x1)) = 0
POL(c11(x1)) = x1
POL(c11(x1, x2, x3)) = x1 + x2 + x3
POL(c2(x1)) = x1
POL(c5(x1)) = x1
POL(c9(x1)) = x1
POL(cons(x1, x2)) = 0
POL(false) = 0
POL(half(x1)) = x1
POL(lastbit(x1)) = 0
POL(s(x1)) = [4] + x1

### (28) Obligation:

Complexity Dependency Tuples Problem
Rules:

half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
lastbit(0) → 0
lastbit(s(0)) → s(0)
lastbit(s(s(z0))) → lastbit(z0)
Tuples:

HALF(s(s(z0))) → c2(HALF(z0))
LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(s(s(z0))), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, 0, x1) → c11(CONVITER(half(0), cons(0, x1)))
IF(false, s(0), x1) → c11(CONVITER(half(s(0)), cons(s(0), x1)))
IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1)), HALF(s(s(0))), LASTBIT(s(s(0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)), HALF(s(s(s(0)))), LASTBIT(s(s(s(0)))))
IF(false, s(s(s(s(z0)))), x1) → c11(CONVITER(half(s(s(s(s(z0))))), cons(lastbit(z0), x1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
S tuples:

HALF(s(s(z0))) → c2(HALF(z0))
LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
IF(false, s(0), x1) → c11(CONVITER(half(s(0)), cons(s(0), x1)))
IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1)), HALF(s(s(0))), LASTBIT(s(s(0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)), HALF(s(s(s(0)))), LASTBIT(s(s(s(0)))))
IF(false, s(s(s(s(z0)))), x1) → c11(CONVITER(half(s(s(s(s(z0))))), cons(lastbit(z0), x1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
K tuples:

IF(false, 0, x1) → c11(CONVITER(half(0), cons(0, x1)))
IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(s(s(z0))), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11, c11

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

Use narrowing to replace IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(s(s(z0))), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0)))) by

IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(0)), x1) → c11(CONVITER(s(0), cons(lastbit(s(s(0))), x1)), HALF(s(s(0))), LASTBIT(s(s(0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(s(0), cons(lastbit(s(s(s(0)))), x1)), HALF(s(s(s(0)))), LASTBIT(s(s(s(0)))))
IF(false, s(s(s(s(z0)))), x1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(s(s(s(s(z0))))), x1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))

### (30) Obligation:

Complexity Dependency Tuples Problem
Rules:

half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
lastbit(0) → 0
lastbit(s(0)) → s(0)
lastbit(s(s(z0))) → lastbit(z0)
Tuples:

HALF(s(s(z0))) → c2(HALF(z0))
LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
IF(false, 0, x1) → c11(CONVITER(half(0), cons(0, x1)))
IF(false, s(0), x1) → c11(CONVITER(half(s(0)), cons(s(0), x1)))
IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1)), HALF(s(s(0))), LASTBIT(s(s(0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)), HALF(s(s(s(0)))), LASTBIT(s(s(s(0)))))
IF(false, s(s(s(s(z0)))), x1) → c11(CONVITER(half(s(s(s(s(z0))))), cons(lastbit(z0), x1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(0)), x1) → c11(CONVITER(s(0), cons(lastbit(s(s(0))), x1)), HALF(s(s(0))), LASTBIT(s(s(0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(s(0), cons(lastbit(s(s(s(0)))), x1)), HALF(s(s(s(0)))), LASTBIT(s(s(s(0)))))
IF(false, s(s(s(s(z0)))), x1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(s(s(s(s(z0))))), x1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
S tuples:

HALF(s(s(z0))) → c2(HALF(z0))
LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
IF(false, s(0), x1) → c11(CONVITER(half(s(0)), cons(s(0), x1)))
IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1)), HALF(s(s(0))), LASTBIT(s(s(0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)), HALF(s(s(s(0)))), LASTBIT(s(s(s(0)))))
IF(false, s(s(s(s(z0)))), x1) → c11(CONVITER(half(s(s(s(s(z0))))), cons(lastbit(z0), x1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
K tuples:

IF(false, 0, x1) → c11(CONVITER(half(0), cons(0, x1)))
IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(s(s(z0))), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11, c11

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

Use narrowing to replace IF(false, 0, x1) → c11(CONVITER(half(0), cons(0, x1))) by

IF(false, 0, x0) → c11(CONVITER(0, cons(0, x0)))

### (32) Obligation:

Complexity Dependency Tuples Problem
Rules:

half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
lastbit(0) → 0
lastbit(s(0)) → s(0)
lastbit(s(s(z0))) → lastbit(z0)
Tuples:

HALF(s(s(z0))) → c2(HALF(z0))
LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
IF(false, s(0), x1) → c11(CONVITER(half(s(0)), cons(s(0), x1)))
IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1)), HALF(s(s(0))), LASTBIT(s(s(0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)), HALF(s(s(s(0)))), LASTBIT(s(s(s(0)))))
IF(false, s(s(s(s(z0)))), x1) → c11(CONVITER(half(s(s(s(s(z0))))), cons(lastbit(z0), x1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(0)), x1) → c11(CONVITER(s(0), cons(lastbit(s(s(0))), x1)), HALF(s(s(0))), LASTBIT(s(s(0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(s(0), cons(lastbit(s(s(s(0)))), x1)), HALF(s(s(s(0)))), LASTBIT(s(s(s(0)))))
IF(false, s(s(s(s(z0)))), x1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(s(s(s(s(z0))))), x1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
IF(false, 0, x0) → c11(CONVITER(0, cons(0, x0)))
S tuples:

HALF(s(s(z0))) → c2(HALF(z0))
LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
IF(false, s(0), x1) → c11(CONVITER(half(s(0)), cons(s(0), x1)))
IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1)), HALF(s(s(0))), LASTBIT(s(s(0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)), HALF(s(s(s(0)))), LASTBIT(s(s(s(0)))))
IF(false, s(s(s(s(z0)))), x1) → c11(CONVITER(half(s(s(s(s(z0))))), cons(lastbit(z0), x1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
K tuples:

IF(false, 0, x1) → c11(CONVITER(half(0), cons(0, x1)))
IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(s(s(z0))), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11, c11

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

Removed 1 trailing nodes:

IF(false, 0, x0) → c11(CONVITER(0, cons(0, x0)))

### (34) Obligation:

Complexity Dependency Tuples Problem
Rules:

half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
lastbit(0) → 0
lastbit(s(0)) → s(0)
lastbit(s(s(z0))) → lastbit(z0)
Tuples:

HALF(s(s(z0))) → c2(HALF(z0))
LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
IF(false, s(0), x1) → c11(CONVITER(half(s(0)), cons(s(0), x1)))
IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1)), HALF(s(s(0))), LASTBIT(s(s(0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)), HALF(s(s(s(0)))), LASTBIT(s(s(s(0)))))
IF(false, s(s(s(s(z0)))), x1) → c11(CONVITER(half(s(s(s(s(z0))))), cons(lastbit(z0), x1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(0)), x1) → c11(CONVITER(s(0), cons(lastbit(s(s(0))), x1)), HALF(s(s(0))), LASTBIT(s(s(0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(s(0), cons(lastbit(s(s(s(0)))), x1)), HALF(s(s(s(0)))), LASTBIT(s(s(s(0)))))
IF(false, s(s(s(s(z0)))), x1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(s(s(s(s(z0))))), x1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
S tuples:

HALF(s(s(z0))) → c2(HALF(z0))
LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
IF(false, s(0), x1) → c11(CONVITER(half(s(0)), cons(s(0), x1)))
IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1)), HALF(s(s(0))), LASTBIT(s(s(0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)), HALF(s(s(s(0)))), LASTBIT(s(s(s(0)))))
IF(false, s(s(s(s(z0)))), x1) → c11(CONVITER(half(s(s(s(s(z0))))), cons(lastbit(z0), x1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
K tuples:

IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11, c11

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

Use narrowing to replace IF(false, s(0), x1) → c11(CONVITER(half(s(0)), cons(s(0), x1))) by

IF(false, s(0), x0) → c11(CONVITER(0, cons(s(0), x0)))

### (36) Obligation:

Complexity Dependency Tuples Problem
Rules:

half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
lastbit(0) → 0
lastbit(s(0)) → s(0)
lastbit(s(s(z0))) → lastbit(z0)
Tuples:

HALF(s(s(z0))) → c2(HALF(z0))
LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1)), HALF(s(s(0))), LASTBIT(s(s(0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)), HALF(s(s(s(0)))), LASTBIT(s(s(s(0)))))
IF(false, s(s(s(s(z0)))), x1) → c11(CONVITER(half(s(s(s(s(z0))))), cons(lastbit(z0), x1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(0)), x1) → c11(CONVITER(s(0), cons(lastbit(s(s(0))), x1)), HALF(s(s(0))), LASTBIT(s(s(0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(s(0), cons(lastbit(s(s(s(0)))), x1)), HALF(s(s(s(0)))), LASTBIT(s(s(s(0)))))
IF(false, s(s(s(s(z0)))), x1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(s(s(s(s(z0))))), x1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
IF(false, s(0), x0) → c11(CONVITER(0, cons(s(0), x0)))
S tuples:

HALF(s(s(z0))) → c2(HALF(z0))
LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1)), HALF(s(s(0))), LASTBIT(s(s(0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)), HALF(s(s(s(0)))), LASTBIT(s(s(s(0)))))
IF(false, s(s(s(s(z0)))), x1) → c11(CONVITER(half(s(s(s(s(z0))))), cons(lastbit(z0), x1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
IF(false, s(0), x0) → c11(CONVITER(0, cons(s(0), x0)))
K tuples:

IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11, c11

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

Removed 1 trailing nodes:

IF(false, s(0), x0) → c11(CONVITER(0, cons(s(0), x0)))

### (38) Obligation:

Complexity Dependency Tuples Problem
Rules:

half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
lastbit(0) → 0
lastbit(s(0)) → s(0)
lastbit(s(s(z0))) → lastbit(z0)
Tuples:

HALF(s(s(z0))) → c2(HALF(z0))
LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1)), HALF(s(s(0))), LASTBIT(s(s(0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)), HALF(s(s(s(0)))), LASTBIT(s(s(s(0)))))
IF(false, s(s(s(s(z0)))), x1) → c11(CONVITER(half(s(s(s(s(z0))))), cons(lastbit(z0), x1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(0)), x1) → c11(CONVITER(s(0), cons(lastbit(s(s(0))), x1)), HALF(s(s(0))), LASTBIT(s(s(0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(s(0), cons(lastbit(s(s(s(0)))), x1)), HALF(s(s(s(0)))), LASTBIT(s(s(s(0)))))
IF(false, s(s(s(s(z0)))), x1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(s(s(s(s(z0))))), x1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
S tuples:

HALF(s(s(z0))) → c2(HALF(z0))
LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1)), HALF(s(s(0))), LASTBIT(s(s(0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)), HALF(s(s(s(0)))), LASTBIT(s(s(s(0)))))
IF(false, s(s(s(s(z0)))), x1) → c11(CONVITER(half(s(s(s(s(z0))))), cons(lastbit(z0), x1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
K tuples:

IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11

### (39) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)

Use forward instantiation to replace HALF(s(s(z0))) → c2(HALF(z0)) by

HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))

### (40) Obligation:

Complexity Dependency Tuples Problem
Rules:

half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
lastbit(0) → 0
lastbit(s(0)) → s(0)
lastbit(s(s(z0))) → lastbit(z0)
Tuples:

LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1)), HALF(s(s(0))), LASTBIT(s(s(0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)), HALF(s(s(s(0)))), LASTBIT(s(s(s(0)))))
IF(false, s(s(s(s(z0)))), x1) → c11(CONVITER(half(s(s(s(s(z0))))), cons(lastbit(z0), x1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(0)), x1) → c11(CONVITER(s(0), cons(lastbit(s(s(0))), x1)), HALF(s(s(0))), LASTBIT(s(s(0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(s(0), cons(lastbit(s(s(s(0)))), x1)), HALF(s(s(s(0)))), LASTBIT(s(s(s(0)))))
IF(false, s(s(s(s(z0)))), x1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(s(s(s(s(z0))))), x1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
S tuples:

LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1)), HALF(s(s(0))), LASTBIT(s(s(0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)), HALF(s(s(s(0)))), LASTBIT(s(s(s(0)))))
IF(false, s(s(s(s(z0)))), x1) → c11(CONVITER(half(s(s(s(s(z0))))), cons(lastbit(z0), x1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
K tuples:

IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

LASTBIT, CONVITER, IF, HALF

Compound Symbols:

c5, c9, c11, c2

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

Removed 4 trailing tuple parts

### (42) Obligation:

Complexity Dependency Tuples Problem
Rules:

half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
lastbit(0) → 0
lastbit(s(0)) → s(0)
lastbit(s(s(z0))) → lastbit(z0)
Tuples:

LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
IF(false, s(s(s(s(z0)))), x1) → c11(CONVITER(half(s(s(s(s(z0))))), cons(lastbit(z0), x1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(s(s(z0)))), x1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(s(s(s(s(z0))))), x1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1)), LASTBIT(s(s(0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)), LASTBIT(s(s(s(0)))))
IF(false, s(s(0)), x1) → c11(CONVITER(s(0), cons(lastbit(s(s(0))), x1)), LASTBIT(s(s(0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(s(0), cons(lastbit(s(s(s(0)))), x1)), LASTBIT(s(s(s(0)))))
S tuples:

LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
IF(false, s(s(s(s(z0)))), x1) → c11(CONVITER(half(s(s(s(s(z0))))), cons(lastbit(z0), x1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1)), LASTBIT(s(s(0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)), LASTBIT(s(s(s(0)))))
K tuples:

IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

LASTBIT, CONVITER, IF, HALF

Compound Symbols:

c5, c9, c11, c2, c11

### (43) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)

Used rewriting to replace IF(false, s(s(s(s(z0)))), x1) → c11(CONVITER(half(s(s(s(s(z0))))), cons(lastbit(z0), x1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0)))))) by IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(half(s(s(z0)))), cons(lastbit(z0), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))

### (44) Obligation:

Complexity Dependency Tuples Problem
Rules:

half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
lastbit(0) → 0
lastbit(s(0)) → s(0)
lastbit(s(s(z0))) → lastbit(z0)
Tuples:

LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(s(s(z0)))), x1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(s(s(s(s(z0))))), x1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1)), LASTBIT(s(s(0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)), LASTBIT(s(s(s(0)))))
IF(false, s(s(0)), x1) → c11(CONVITER(s(0), cons(lastbit(s(s(0))), x1)), LASTBIT(s(s(0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(s(0), cons(lastbit(s(s(s(0)))), x1)), LASTBIT(s(s(s(0)))))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(half(s(s(z0)))), cons(lastbit(z0), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
S tuples:

LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1)), LASTBIT(s(s(0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)), LASTBIT(s(s(s(0)))))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(half(s(s(z0)))), cons(lastbit(z0), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
K tuples:

IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

LASTBIT, CONVITER, IF, HALF

Compound Symbols:

c5, c9, c11, c2, c11

### (45) 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.

IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(half(s(s(z0)))), cons(lastbit(z0), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
We considered the (Usable) Rules:

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

LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(s(s(z0)))), x1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(s(s(s(s(z0))))), x1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1)), LASTBIT(s(s(0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)), LASTBIT(s(s(s(0)))))
IF(false, s(s(0)), x1) → c11(CONVITER(s(0), cons(lastbit(s(s(0))), x1)), LASTBIT(s(s(0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(s(0), cons(lastbit(s(s(s(0)))), x1)), LASTBIT(s(s(s(0)))))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(half(s(s(z0)))), cons(lastbit(z0), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(CONVITER(x1, x2)) = [4]x1
POL(HALF(x1)) = 0
POL(IF(x1, x2, x3)) = x1 + [4]x2
POL(LASTBIT(x1)) = 0
POL(c11(x1, x2)) = x1 + x2
POL(c11(x1, x2, x3)) = x1 + x2 + x3
POL(c2(x1)) = x1
POL(c5(x1)) = x1
POL(c9(x1)) = x1
POL(cons(x1, x2)) = 0
POL(false) = 0
POL(half(x1)) = x1
POL(lastbit(x1)) = 0
POL(s(x1)) = [2] + x1

### (46) Obligation:

Complexity Dependency Tuples Problem
Rules:

half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
lastbit(0) → 0
lastbit(s(0)) → s(0)
lastbit(s(s(z0))) → lastbit(z0)
Tuples:

LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(s(s(z0)))), x1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(s(s(s(s(z0))))), x1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1)), LASTBIT(s(s(0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)), LASTBIT(s(s(s(0)))))
IF(false, s(s(0)), x1) → c11(CONVITER(s(0), cons(lastbit(s(s(0))), x1)), LASTBIT(s(s(0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(s(0), cons(lastbit(s(s(s(0)))), x1)), LASTBIT(s(s(s(0)))))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(half(s(s(z0)))), cons(lastbit(z0), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
S tuples:

LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1)), LASTBIT(s(s(0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)), LASTBIT(s(s(s(0)))))
K tuples:

IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(half(s(s(z0)))), cons(lastbit(z0), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

LASTBIT, CONVITER, IF, HALF

Compound Symbols:

c5, c9, c11, c2, c11

### (47) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^3))) transformation)

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

HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
We considered the (Usable) Rules:

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

LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(s(s(z0)))), x1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(s(s(s(s(z0))))), x1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1)), LASTBIT(s(s(0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)), LASTBIT(s(s(s(0)))))
IF(false, s(s(0)), x1) → c11(CONVITER(s(0), cons(lastbit(s(s(0))), x1)), LASTBIT(s(s(0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(s(0), cons(lastbit(s(s(s(0)))), x1)), LASTBIT(s(s(s(0)))))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(half(s(s(z0)))), cons(lastbit(z0), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(CONVITER(x1, x2)) = [1] + x12
POL(HALF(x1)) = [1] + x1
POL(IF(x1, x2, x3)) = [1] + x1·x22
POL(LASTBIT(x1)) = 0
POL(c11(x1, x2)) = x1 + x2
POL(c11(x1, x2, x3)) = x1 + x2 + x3
POL(c2(x1)) = x1
POL(c5(x1)) = x1
POL(c9(x1)) = x1
POL(cons(x1, x2)) = 0
POL(false) = [1]
POL(half(x1)) = x1
POL(lastbit(x1)) = 0
POL(s(x1)) = [1] + x1

### (48) Obligation:

Complexity Dependency Tuples Problem
Rules:

half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
lastbit(0) → 0
lastbit(s(0)) → s(0)
lastbit(s(s(z0))) → lastbit(z0)
Tuples:

LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(s(s(z0)))), x1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(s(s(s(s(z0))))), x1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1)), LASTBIT(s(s(0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)), LASTBIT(s(s(s(0)))))
IF(false, s(s(0)), x1) → c11(CONVITER(s(0), cons(lastbit(s(s(0))), x1)), LASTBIT(s(s(0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(s(0), cons(lastbit(s(s(s(0)))), x1)), LASTBIT(s(s(s(0)))))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(half(s(s(z0)))), cons(lastbit(z0), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
S tuples:

LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1)), LASTBIT(s(s(0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)), LASTBIT(s(s(s(0)))))
K tuples:

IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(half(s(s(z0)))), cons(lastbit(z0), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

LASTBIT, CONVITER, IF, HALF

Compound Symbols:

c5, c9, c11, c2, c11

### (49) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)

Use forward instantiation to replace LASTBIT(s(s(z0))) → c5(LASTBIT(z0)) by

LASTBIT(s(s(s(s(y0))))) → c5(LASTBIT(s(s(y0))))

### (50) Obligation:

Complexity Dependency Tuples Problem
Rules:

half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
lastbit(0) → 0
lastbit(s(0)) → s(0)
lastbit(s(s(z0))) → lastbit(z0)
Tuples:

CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(s(s(z0)))), x1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(s(s(s(s(z0))))), x1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1)), LASTBIT(s(s(0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)), LASTBIT(s(s(s(0)))))
IF(false, s(s(0)), x1) → c11(CONVITER(s(0), cons(lastbit(s(s(0))), x1)), LASTBIT(s(s(0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(s(0), cons(lastbit(s(s(s(0)))), x1)), LASTBIT(s(s(s(0)))))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(half(s(s(z0)))), cons(lastbit(z0), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
LASTBIT(s(s(s(s(y0))))) → c5(LASTBIT(s(s(y0))))
S tuples:

CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1)), LASTBIT(s(s(0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)), LASTBIT(s(s(s(0)))))
LASTBIT(s(s(s(s(y0))))) → c5(LASTBIT(s(s(y0))))
K tuples:

IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(half(s(s(z0)))), cons(lastbit(z0), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

CONVITER, IF, HALF, LASTBIT

Compound Symbols:

c9, c11, c2, c11, c5

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

Removed 4 trailing tuple parts

### (52) Obligation:

Complexity Dependency Tuples Problem
Rules:

half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
lastbit(0) → 0
lastbit(s(0)) → s(0)
lastbit(s(s(z0))) → lastbit(z0)
Tuples:

CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(s(s(z0)))), x1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(s(s(s(s(z0))))), x1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(half(s(s(z0)))), cons(lastbit(z0), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
LASTBIT(s(s(s(s(y0))))) → c5(LASTBIT(s(s(y0))))
IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1)))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)))
IF(false, s(s(0)), x1) → c11(CONVITER(s(0), cons(lastbit(s(s(0))), x1)))
IF(false, s(s(s(0))), x1) → c11(CONVITER(s(0), cons(lastbit(s(s(s(0)))), x1)))
S tuples:

CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
LASTBIT(s(s(s(s(y0))))) → c5(LASTBIT(s(s(y0))))
IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1)))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)))
K tuples:

IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(half(s(s(z0)))), cons(lastbit(z0), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

CONVITER, IF, HALF, LASTBIT

Compound Symbols:

c9, c11, c2, c5, c11

### (53) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

LASTBIT(s(s(s(s(y0))))) → c5(LASTBIT(s(s(y0))))
We considered the (Usable) Rules:

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

CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(s(s(z0)))), x1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(s(s(s(s(z0))))), x1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(half(s(s(z0)))), cons(lastbit(z0), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
LASTBIT(s(s(s(s(y0))))) → c5(LASTBIT(s(s(y0))))
IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1)))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)))
IF(false, s(s(0)), x1) → c11(CONVITER(s(0), cons(lastbit(s(s(0))), x1)))
IF(false, s(s(s(0))), x1) → c11(CONVITER(s(0), cons(lastbit(s(s(s(0)))), x1)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(CONVITER(x1, x2)) = x12
POL(HALF(x1)) = 0
POL(IF(x1, x2, x3)) = x22
POL(LASTBIT(x1)) = [2]x1
POL(c11(x1)) = x1
POL(c11(x1, x2, x3)) = x1 + x2 + x3
POL(c2(x1)) = x1
POL(c5(x1)) = x1
POL(c9(x1)) = x1
POL(cons(x1, x2)) = 0
POL(false) = 0
POL(half(x1)) = x1
POL(lastbit(x1)) = 0
POL(s(x1)) = [2] + x1

### (54) Obligation:

Complexity Dependency Tuples Problem
Rules:

half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
lastbit(0) → 0
lastbit(s(0)) → s(0)
lastbit(s(s(z0))) → lastbit(z0)
Tuples:

CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(s(s(z0)))), x1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(s(s(s(s(z0))))), x1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(half(s(s(z0)))), cons(lastbit(z0), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
LASTBIT(s(s(s(s(y0))))) → c5(LASTBIT(s(s(y0))))
IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1)))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)))
IF(false, s(s(0)), x1) → c11(CONVITER(s(0), cons(lastbit(s(s(0))), x1)))
IF(false, s(s(s(0))), x1) → c11(CONVITER(s(0), cons(lastbit(s(s(s(0)))), x1)))
S tuples:

CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1)))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)))
K tuples:

IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(half(s(s(z0)))), cons(lastbit(z0), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
LASTBIT(s(s(s(s(y0))))) → c5(LASTBIT(s(s(y0))))
Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

CONVITER, IF, HALF, LASTBIT

Compound Symbols:

c9, c11, c2, c5, c11

### (55) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)

Use forward instantiation to replace CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1)) by

CONVITER(s(s(y0)), z1) → c9(IF(false, s(s(y0)), z1))
CONVITER(s(s(s(s(y0)))), z1) → c9(IF(false, s(s(s(s(y0)))), z1))
CONVITER(s(s(0)), z1) → c9(IF(false, s(s(0)), z1))
CONVITER(s(s(s(0))), z1) → c9(IF(false, s(s(s(0))), z1))

### (56) Obligation:

Complexity Dependency Tuples Problem
Rules:

half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
lastbit(0) → 0
lastbit(s(0)) → s(0)
lastbit(s(s(z0))) → lastbit(z0)
Tuples:

IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(s(s(z0)))), x1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(s(s(s(s(z0))))), x1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(half(s(s(z0)))), cons(lastbit(z0), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
LASTBIT(s(s(s(s(y0))))) → c5(LASTBIT(s(s(y0))))
IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1)))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)))
IF(false, s(s(0)), x1) → c11(CONVITER(s(0), cons(lastbit(s(s(0))), x1)))
IF(false, s(s(s(0))), x1) → c11(CONVITER(s(0), cons(lastbit(s(s(s(0)))), x1)))
CONVITER(s(s(y0)), z1) → c9(IF(false, s(s(y0)), z1))
CONVITER(s(s(s(s(y0)))), z1) → c9(IF(false, s(s(s(s(y0)))), z1))
CONVITER(s(s(0)), z1) → c9(IF(false, s(s(0)), z1))
CONVITER(s(s(s(0))), z1) → c9(IF(false, s(s(s(0))), z1))
S tuples:

IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1)))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)))
CONVITER(s(s(y0)), z1) → c9(IF(false, s(s(y0)), z1))
CONVITER(s(s(s(s(y0)))), z1) → c9(IF(false, s(s(s(s(y0)))), z1))
CONVITER(s(s(0)), z1) → c9(IF(false, s(s(0)), z1))
CONVITER(s(s(s(0))), z1) → c9(IF(false, s(s(s(0))), z1))
K tuples:

IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(half(s(s(z0)))), cons(lastbit(z0), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
LASTBIT(s(s(s(s(y0))))) → c5(LASTBIT(s(s(y0))))
Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

IF, HALF, LASTBIT, CONVITER

Compound Symbols:

c11, c2, c5, c11, c9

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

Removed 2 trailing nodes:

IF(false, s(s(0)), x1) → c11(CONVITER(s(0), cons(lastbit(s(s(0))), x1)))
IF(false, s(s(s(0))), x1) → c11(CONVITER(s(0), cons(lastbit(s(s(s(0)))), x1)))

### (58) Obligation:

Complexity Dependency Tuples Problem
Rules:

half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
lastbit(0) → 0
lastbit(s(0)) → s(0)
lastbit(s(s(z0))) → lastbit(z0)
Tuples:

IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(s(s(z0)))), x1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(s(s(s(s(z0))))), x1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(half(s(s(z0)))), cons(lastbit(z0), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
LASTBIT(s(s(s(s(y0))))) → c5(LASTBIT(s(s(y0))))
IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1)))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)))
CONVITER(s(s(y0)), z1) → c9(IF(false, s(s(y0)), z1))
CONVITER(s(s(s(s(y0)))), z1) → c9(IF(false, s(s(s(s(y0)))), z1))
CONVITER(s(s(0)), z1) → c9(IF(false, s(s(0)), z1))
CONVITER(s(s(s(0))), z1) → c9(IF(false, s(s(s(0))), z1))
S tuples:

IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1)))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)))
CONVITER(s(s(y0)), z1) → c9(IF(false, s(s(y0)), z1))
CONVITER(s(s(s(s(y0)))), z1) → c9(IF(false, s(s(s(s(y0)))), z1))
CONVITER(s(s(0)), z1) → c9(IF(false, s(s(0)), z1))
CONVITER(s(s(s(0))), z1) → c9(IF(false, s(s(s(0))), z1))
K tuples:

IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(half(s(s(z0)))), cons(lastbit(z0), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
LASTBIT(s(s(s(s(y0))))) → c5(LASTBIT(s(s(y0))))
Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

IF, HALF, LASTBIT, CONVITER

Compound Symbols:

c11, c2, c5, c11, c9

### (59) CdtInstantiationProof (BOTH BOUNDS(ID, ID) transformation)

Use instantiation to replace CONVITER(s(s(y0)), z1) → c9(IF(false, s(s(y0)), z1)) by

CONVITER(s(s(z0)), cons(y1, x1)) → c9(IF(false, s(s(z0)), cons(y1, x1)))
CONVITER(s(s(z0)), cons(0, x0)) → c9(IF(false, s(s(z0)), cons(0, x0)))
CONVITER(s(s(z0)), cons(s(0), x0)) → c9(IF(false, s(s(z0)), cons(s(0), x0)))

### (60) Obligation:

Complexity Dependency Tuples Problem
Rules:

half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
lastbit(0) → 0
lastbit(s(0)) → s(0)
lastbit(s(s(z0))) → lastbit(z0)
Tuples:

IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(s(s(z0)))), x1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(s(s(s(s(z0))))), x1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(half(s(s(z0)))), cons(lastbit(z0), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
LASTBIT(s(s(s(s(y0))))) → c5(LASTBIT(s(s(y0))))
IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1)))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)))
CONVITER(s(s(s(s(y0)))), z1) → c9(IF(false, s(s(s(s(y0)))), z1))
CONVITER(s(s(0)), z1) → c9(IF(false, s(s(0)), z1))
CONVITER(s(s(s(0))), z1) → c9(IF(false, s(s(s(0))), z1))
CONVITER(s(s(z0)), cons(y1, x1)) → c9(IF(false, s(s(z0)), cons(y1, x1)))
CONVITER(s(s(z0)), cons(0, x0)) → c9(IF(false, s(s(z0)), cons(0, x0)))
CONVITER(s(s(z0)), cons(s(0), x0)) → c9(IF(false, s(s(z0)), cons(s(0), x0)))
S tuples:

IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1)))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)))
CONVITER(s(s(s(s(y0)))), z1) → c9(IF(false, s(s(s(s(y0)))), z1))
CONVITER(s(s(0)), z1) → c9(IF(false, s(s(0)), z1))
CONVITER(s(s(s(0))), z1) → c9(IF(false, s(s(s(0))), z1))
CONVITER(s(s(z0)), cons(y1, x1)) → c9(IF(false, s(s(z0)), cons(y1, x1)))
CONVITER(s(s(z0)), cons(0, x0)) → c9(IF(false, s(s(z0)), cons(0, x0)))
CONVITER(s(s(z0)), cons(s(0), x0)) → c9(IF(false, s(s(z0)), cons(s(0), x0)))
K tuples:

IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(half(s(s(z0)))), cons(lastbit(z0), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
LASTBIT(s(s(s(s(y0))))) → c5(LASTBIT(s(s(y0))))
Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

IF, HALF, LASTBIT, CONVITER

Compound Symbols:

c11, c2, c5, c11, c9

### (61) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)

Used rewriting to replace IF(false, s(s(s(s(z0)))), x1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(s(s(s(s(z0))))), x1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0)))))) by IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(s(s(z0))), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))

### (62) Obligation:

Complexity Dependency Tuples Problem
Rules:

half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
lastbit(0) → 0
lastbit(s(0)) → s(0)
lastbit(s(s(z0))) → lastbit(z0)
Tuples:

IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(half(s(s(z0)))), cons(lastbit(z0), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
LASTBIT(s(s(s(s(y0))))) → c5(LASTBIT(s(s(y0))))
IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1)))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)))
CONVITER(s(s(s(s(y0)))), z1) → c9(IF(false, s(s(s(s(y0)))), z1))
CONVITER(s(s(0)), z1) → c9(IF(false, s(s(0)), z1))
CONVITER(s(s(s(0))), z1) → c9(IF(false, s(s(s(0))), z1))
CONVITER(s(s(z0)), cons(y1, x1)) → c9(IF(false, s(s(z0)), cons(y1, x1)))
CONVITER(s(s(z0)), cons(0, x0)) → c9(IF(false, s(s(z0)), cons(0, x0)))
CONVITER(s(s(z0)), cons(s(0), x0)) → c9(IF(false, s(s(z0)), cons(s(0), x0)))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(s(s(z0))), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
S tuples:

IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1)))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)))
CONVITER(s(s(s(s(y0)))), z1) → c9(IF(false, s(s(s(s(y0)))), z1))
CONVITER(s(s(0)), z1) → c9(IF(false, s(s(0)), z1))
CONVITER(s(s(s(0))), z1) → c9(IF(false, s(s(s(0))), z1))
CONVITER(s(s(z0)), cons(y1, x1)) → c9(IF(false, s(s(z0)), cons(y1, x1)))
CONVITER(s(s(z0)), cons(0, x0)) → c9(IF(false, s(s(z0)), cons(0, x0)))
CONVITER(s(s(z0)), cons(s(0), x0)) → c9(IF(false, s(s(z0)), cons(s(0), x0)))
K tuples:

IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(half(s(s(z0)))), cons(lastbit(z0), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
LASTBIT(s(s(s(s(y0))))) → c5(LASTBIT(s(s(y0))))
Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

IF, HALF, LASTBIT, CONVITER

Compound Symbols:

c11, c2, c5, c11, c9

### (63) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)

Use forward instantiation to replace HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0)))) by

HALF(s(s(s(s(s(s(y0))))))) → c2(HALF(s(s(s(s(y0))))))

### (64) Obligation:

Complexity Dependency Tuples Problem
Rules:

half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
lastbit(0) → 0
lastbit(s(0)) → s(0)
lastbit(s(s(z0))) → lastbit(z0)
Tuples:

IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(half(s(s(z0)))), cons(lastbit(z0), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
LASTBIT(s(s(s(s(y0))))) → c5(LASTBIT(s(s(y0))))
IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1)))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)))
CONVITER(s(s(s(s(y0)))), z1) → c9(IF(false, s(s(s(s(y0)))), z1))
CONVITER(s(s(0)), z1) → c9(IF(false, s(s(0)), z1))
CONVITER(s(s(s(0))), z1) → c9(IF(false, s(s(s(0))), z1))
CONVITER(s(s(z0)), cons(y1, x1)) → c9(IF(false, s(s(z0)), cons(y1, x1)))
CONVITER(s(s(z0)), cons(0, x0)) → c9(IF(false, s(s(z0)), cons(0, x0)))
CONVITER(s(s(z0)), cons(s(0), x0)) → c9(IF(false, s(s(z0)), cons(s(0), x0)))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(s(s(z0))), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
HALF(s(s(s(s(s(s(y0))))))) → c2(HALF(s(s(s(s(y0))))))
S tuples:

IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1)))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)))
CONVITER(s(s(s(s(y0)))), z1) → c9(IF(false, s(s(s(s(y0)))), z1))
CONVITER(s(s(0)), z1) → c9(IF(false, s(s(0)), z1))
CONVITER(s(s(s(0))), z1) → c9(IF(false, s(s(s(0))), z1))
CONVITER(s(s(z0)), cons(y1, x1)) → c9(IF(false, s(s(z0)), cons(y1, x1)))
CONVITER(s(s(z0)), cons(0, x0)) → c9(IF(false, s(s(z0)), cons(0, x0)))
CONVITER(s(s(z0)), cons(s(0), x0)) → c9(IF(false, s(s(z0)), cons(s(0), x0)))
K tuples:

IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(half(s(s(z0)))), cons(lastbit(z0), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
LASTBIT(s(s(s(s(y0))))) → c5(LASTBIT(s(s(y0))))
HALF(s(s(s(s(s(s(y0))))))) → c2(HALF(s(s(s(s(y0))))))
Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

IF, LASTBIT, CONVITER, HALF

Compound Symbols:

c11, c5, c11, c9, c2

### (65) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)

Used rewriting to replace IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(half(s(s(z0)))), cons(lastbit(z0), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0)))))) by IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(z0), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))

### (66) Obligation:

Complexity Dependency Tuples Problem
Rules:

half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
lastbit(0) → 0
lastbit(s(0)) → s(0)
lastbit(s(s(z0))) → lastbit(z0)
Tuples:

IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
LASTBIT(s(s(s(s(y0))))) → c5(LASTBIT(s(s(y0))))
IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1)))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)))
CONVITER(s(s(s(s(y0)))), z1) → c9(IF(false, s(s(s(s(y0)))), z1))
CONVITER(s(s(0)), z1) → c9(IF(false, s(s(0)), z1))
CONVITER(s(s(s(0))), z1) → c9(IF(false, s(s(s(0))), z1))
CONVITER(s(s(z0)), cons(y1, x1)) → c9(IF(false, s(s(z0)), cons(y1, x1)))
CONVITER(s(s(z0)), cons(0, x0)) → c9(IF(false, s(s(z0)), cons(0, x0)))
CONVITER(s(s(z0)), cons(s(0), x0)) → c9(IF(false, s(s(z0)), cons(s(0), x0)))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(s(s(z0))), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
HALF(s(s(s(s(s(s(y0))))))) → c2(HALF(s(s(s(s(y0))))))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(z0), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
S tuples:

IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1)))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)))
CONVITER(s(s(s(s(y0)))), z1) → c9(IF(false, s(s(s(s(y0)))), z1))
CONVITER(s(s(0)), z1) → c9(IF(false, s(s(0)), z1))
CONVITER(s(s(s(0))), z1) → c9(IF(false, s(s(s(0))), z1))
CONVITER(s(s(z0)), cons(y1, x1)) → c9(IF(false, s(s(z0)), cons(y1, x1)))
CONVITER(s(s(z0)), cons(0, x0)) → c9(IF(false, s(s(z0)), cons(0, x0)))
CONVITER(s(s(z0)), cons(s(0), x0)) → c9(IF(false, s(s(z0)), cons(s(0), x0)))
K tuples:

IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(half(s(s(z0)))), cons(lastbit(z0), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
LASTBIT(s(s(s(s(y0))))) → c5(LASTBIT(s(s(y0))))
HALF(s(s(s(s(s(s(y0))))))) → c2(HALF(s(s(s(s(y0))))))
Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

IF, LASTBIT, CONVITER, HALF

Compound Symbols:

c11, c5, c11, c9, c2

### (67) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)

Used rewriting to replace IF(false, s(s(0)), x1) → c11(CONVITER(half(s(s(0))), cons(0, x1))) by IF(false, s(s(0)), z0) → c11(CONVITER(s(half(0)), cons(0, z0)))

### (68) Obligation:

Complexity Dependency Tuples Problem
Rules:

half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
lastbit(0) → 0
lastbit(s(0)) → s(0)
lastbit(s(s(z0))) → lastbit(z0)
Tuples:

IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
LASTBIT(s(s(s(s(y0))))) → c5(LASTBIT(s(s(y0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)))
CONVITER(s(s(s(s(y0)))), z1) → c9(IF(false, s(s(s(s(y0)))), z1))
CONVITER(s(s(0)), z1) → c9(IF(false, s(s(0)), z1))
CONVITER(s(s(s(0))), z1) → c9(IF(false, s(s(s(0))), z1))
CONVITER(s(s(z0)), cons(y1, x1)) → c9(IF(false, s(s(z0)), cons(y1, x1)))
CONVITER(s(s(z0)), cons(0, x0)) → c9(IF(false, s(s(z0)), cons(0, x0)))
CONVITER(s(s(z0)), cons(s(0), x0)) → c9(IF(false, s(s(z0)), cons(s(0), x0)))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(s(s(z0))), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
HALF(s(s(s(s(s(s(y0))))))) → c2(HALF(s(s(s(s(y0))))))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(z0), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
IF(false, s(s(0)), z0) → c11(CONVITER(s(half(0)), cons(0, z0)))
S tuples:

IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)))
CONVITER(s(s(s(s(y0)))), z1) → c9(IF(false, s(s(s(s(y0)))), z1))
CONVITER(s(s(0)), z1) → c9(IF(false, s(s(0)), z1))
CONVITER(s(s(s(0))), z1) → c9(IF(false, s(s(s(0))), z1))
CONVITER(s(s(z0)), cons(y1, x1)) → c9(IF(false, s(s(z0)), cons(y1, x1)))
CONVITER(s(s(z0)), cons(0, x0)) → c9(IF(false, s(s(z0)), cons(0, x0)))
CONVITER(s(s(z0)), cons(s(0), x0)) → c9(IF(false, s(s(z0)), cons(s(0), x0)))
IF(false, s(s(0)), z0) → c11(CONVITER(s(half(0)), cons(0, z0)))
K tuples:

IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(half(s(s(z0)))), cons(lastbit(z0), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
LASTBIT(s(s(s(s(y0))))) → c5(LASTBIT(s(s(y0))))
HALF(s(s(s(s(s(s(y0))))))) → c2(HALF(s(s(s(s(y0))))))
Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

IF, LASTBIT, CONVITER, HALF

Compound Symbols:

c11, c5, c11, c9, c2

### (69) 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.

IF(false, s(s(0)), z0) → c11(CONVITER(s(half(0)), cons(0, z0)))
We considered the (Usable) Rules:

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

IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
LASTBIT(s(s(s(s(y0))))) → c5(LASTBIT(s(s(y0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)))
CONVITER(s(s(s(s(y0)))), z1) → c9(IF(false, s(s(s(s(y0)))), z1))
CONVITER(s(s(0)), z1) → c9(IF(false, s(s(0)), z1))
CONVITER(s(s(s(0))), z1) → c9(IF(false, s(s(s(0))), z1))
CONVITER(s(s(z0)), cons(y1, x1)) → c9(IF(false, s(s(z0)), cons(y1, x1)))
CONVITER(s(s(z0)), cons(0, x0)) → c9(IF(false, s(s(z0)), cons(0, x0)))
CONVITER(s(s(z0)), cons(s(0), x0)) → c9(IF(false, s(s(z0)), cons(s(0), x0)))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(s(s(z0))), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
HALF(s(s(s(s(s(s(y0))))))) → c2(HALF(s(s(s(s(y0))))))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(z0), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
IF(false, s(s(0)), z0) → c11(CONVITER(s(half(0)), cons(0, z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(CONVITER(x1, x2)) = x1
POL(HALF(x1)) = 0
POL(IF(x1, x2, x3)) = x2
POL(LASTBIT(x1)) = 0
POL(c11(x1)) = x1
POL(c11(x1, x2, x3)) = x1 + x2 + x3
POL(c2(x1)) = x1
POL(c5(x1)) = x1
POL(c9(x1)) = x1
POL(cons(x1, x2)) = 0
POL(false) = [2]
POL(half(x1)) = x1
POL(lastbit(x1)) = [4] + [4]x1
POL(s(x1)) = [1] + x1

### (70) Obligation:

Complexity Dependency Tuples Problem
Rules:

half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
lastbit(0) → 0
lastbit(s(0)) → s(0)
lastbit(s(s(z0))) → lastbit(z0)
Tuples:

IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
LASTBIT(s(s(s(s(y0))))) → c5(LASTBIT(s(s(y0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)))
CONVITER(s(s(s(s(y0)))), z1) → c9(IF(false, s(s(s(s(y0)))), z1))
CONVITER(s(s(0)), z1) → c9(IF(false, s(s(0)), z1))
CONVITER(s(s(s(0))), z1) → c9(IF(false, s(s(s(0))), z1))
CONVITER(s(s(z0)), cons(y1, x1)) → c9(IF(false, s(s(z0)), cons(y1, x1)))
CONVITER(s(s(z0)), cons(0, x0)) → c9(IF(false, s(s(z0)), cons(0, x0)))
CONVITER(s(s(z0)), cons(s(0), x0)) → c9(IF(false, s(s(z0)), cons(s(0), x0)))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(s(s(z0))), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
HALF(s(s(s(s(s(s(y0))))))) → c2(HALF(s(s(s(s(y0))))))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(z0), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
IF(false, s(s(0)), z0) → c11(CONVITER(s(half(0)), cons(0, z0)))
S tuples:

IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)))
CONVITER(s(s(s(s(y0)))), z1) → c9(IF(false, s(s(s(s(y0)))), z1))
CONVITER(s(s(0)), z1) → c9(IF(false, s(s(0)), z1))
CONVITER(s(s(s(0))), z1) → c9(IF(false, s(s(s(0))), z1))
CONVITER(s(s(z0)), cons(y1, x1)) → c9(IF(false, s(s(z0)), cons(y1, x1)))
CONVITER(s(s(z0)), cons(0, x0)) → c9(IF(false, s(s(z0)), cons(0, x0)))
CONVITER(s(s(z0)), cons(s(0), x0)) → c9(IF(false, s(s(z0)), cons(s(0), x0)))
K tuples:

IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(half(s(s(z0)))), cons(lastbit(z0), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
LASTBIT(s(s(s(s(y0))))) → c5(LASTBIT(s(s(y0))))
HALF(s(s(s(s(s(s(y0))))))) → c2(HALF(s(s(s(s(y0))))))
IF(false, s(s(0)), z0) → c11(CONVITER(s(half(0)), cons(0, z0)))
Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

IF, LASTBIT, CONVITER, HALF

Compound Symbols:

c11, c5, c11, c9, c2

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

Use narrowing to replace IF(false, s(s(0)), z0) → c11(CONVITER(s(half(0)), cons(0, z0))) by

IF(false, s(s(0)), x0) → c11(CONVITER(s(0), cons(0, x0)))

### (72) Obligation:

Complexity Dependency Tuples Problem
Rules:

half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
lastbit(0) → 0
lastbit(s(0)) → s(0)
lastbit(s(s(z0))) → lastbit(z0)
Tuples:

IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
LASTBIT(s(s(s(s(y0))))) → c5(LASTBIT(s(s(y0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)))
CONVITER(s(s(s(s(y0)))), z1) → c9(IF(false, s(s(s(s(y0)))), z1))
CONVITER(s(s(0)), z1) → c9(IF(false, s(s(0)), z1))
CONVITER(s(s(s(0))), z1) → c9(IF(false, s(s(s(0))), z1))
CONVITER(s(s(z0)), cons(y1, x1)) → c9(IF(false, s(s(z0)), cons(y1, x1)))
CONVITER(s(s(z0)), cons(0, x0)) → c9(IF(false, s(s(z0)), cons(0, x0)))
CONVITER(s(s(z0)), cons(s(0), x0)) → c9(IF(false, s(s(z0)), cons(s(0), x0)))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(s(s(z0))), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
HALF(s(s(s(s(s(s(y0))))))) → c2(HALF(s(s(s(s(y0))))))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(z0), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
IF(false, s(s(0)), x0) → c11(CONVITER(s(0), cons(0, x0)))
S tuples:

IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)))
CONVITER(s(s(s(s(y0)))), z1) → c9(IF(false, s(s(s(s(y0)))), z1))
CONVITER(s(s(0)), z1) → c9(IF(false, s(s(0)), z1))
CONVITER(s(s(s(0))), z1) → c9(IF(false, s(s(s(0))), z1))
CONVITER(s(s(z0)), cons(y1, x1)) → c9(IF(false, s(s(z0)), cons(y1, x1)))
CONVITER(s(s(z0)), cons(0, x0)) → c9(IF(false, s(s(z0)), cons(0, x0)))
CONVITER(s(s(z0)), cons(s(0), x0)) → c9(IF(false, s(s(z0)), cons(s(0), x0)))
K tuples:

IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(half(s(s(z0)))), cons(lastbit(z0), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
LASTBIT(s(s(s(s(y0))))) → c5(LASTBIT(s(s(y0))))
HALF(s(s(s(s(s(s(y0))))))) → c2(HALF(s(s(s(s(y0))))))
IF(false, s(s(0)), z0) → c11(CONVITER(s(half(0)), cons(0, z0)))
Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

IF, LASTBIT, CONVITER, HALF

Compound Symbols:

c11, c5, c11, c9, c2

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

Removed 1 trailing nodes:

IF(false, s(s(0)), x0) → c11(CONVITER(s(0), cons(0, x0)))

### (74) Obligation:

Complexity Dependency Tuples Problem
Rules:

half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
lastbit(0) → 0
lastbit(s(0)) → s(0)
lastbit(s(s(z0))) → lastbit(z0)
Tuples:

IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
LASTBIT(s(s(s(s(y0))))) → c5(LASTBIT(s(s(y0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)))
CONVITER(s(s(s(s(y0)))), z1) → c9(IF(false, s(s(s(s(y0)))), z1))
CONVITER(s(s(0)), z1) → c9(IF(false, s(s(0)), z1))
CONVITER(s(s(s(0))), z1) → c9(IF(false, s(s(s(0))), z1))
CONVITER(s(s(z0)), cons(y1, x1)) → c9(IF(false, s(s(z0)), cons(y1, x1)))
CONVITER(s(s(z0)), cons(0, x0)) → c9(IF(false, s(s(z0)), cons(0, x0)))
CONVITER(s(s(z0)), cons(s(0), x0)) → c9(IF(false, s(s(z0)), cons(s(0), x0)))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(s(s(z0))), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
HALF(s(s(s(s(s(s(y0))))))) → c2(HALF(s(s(s(s(y0))))))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(z0), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
S tuples:

IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)))
CONVITER(s(s(s(s(y0)))), z1) → c9(IF(false, s(s(s(s(y0)))), z1))
CONVITER(s(s(0)), z1) → c9(IF(false, s(s(0)), z1))
CONVITER(s(s(s(0))), z1) → c9(IF(false, s(s(s(0))), z1))
CONVITER(s(s(z0)), cons(y1, x1)) → c9(IF(false, s(s(z0)), cons(y1, x1)))
CONVITER(s(s(z0)), cons(0, x0)) → c9(IF(false, s(s(z0)), cons(0, x0)))
CONVITER(s(s(z0)), cons(s(0), x0)) → c9(IF(false, s(s(z0)), cons(s(0), x0)))
K tuples:

IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
LASTBIT(s(s(s(s(y0))))) → c5(LASTBIT(s(s(y0))))
HALF(s(s(s(s(s(s(y0))))))) → c2(HALF(s(s(s(s(y0))))))
Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

IF, LASTBIT, CONVITER, HALF

Compound Symbols:

c11, c5, c11, c9, c2

### (75) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID) transformation)

Use forward instantiation to replace LASTBIT(s(s(s(s(y0))))) → c5(LASTBIT(s(s(y0)))) by

LASTBIT(s(s(s(s(s(s(y0))))))) → c5(LASTBIT(s(s(s(s(y0))))))

### (76) Obligation:

Complexity Dependency Tuples Problem
Rules:

half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
lastbit(0) → 0
lastbit(s(0)) → s(0)
lastbit(s(s(z0))) → lastbit(z0)
Tuples:

IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)))
CONVITER(s(s(s(s(y0)))), z1) → c9(IF(false, s(s(s(s(y0)))), z1))
CONVITER(s(s(0)), z1) → c9(IF(false, s(s(0)), z1))
CONVITER(s(s(s(0))), z1) → c9(IF(false, s(s(s(0))), z1))
CONVITER(s(s(z0)), cons(y1, x1)) → c9(IF(false, s(s(z0)), cons(y1, x1)))
CONVITER(s(s(z0)), cons(0, x0)) → c9(IF(false, s(s(z0)), cons(0, x0)))
CONVITER(s(s(z0)), cons(s(0), x0)) → c9(IF(false, s(s(z0)), cons(s(0), x0)))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(s(s(z0))), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
HALF(s(s(s(s(s(s(y0))))))) → c2(HALF(s(s(s(s(y0))))))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(z0), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
LASTBIT(s(s(s(s(s(s(y0))))))) → c5(LASTBIT(s(s(s(s(y0))))))
S tuples:

IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)))
CONVITER(s(s(s(s(y0)))), z1) → c9(IF(false, s(s(s(s(y0)))), z1))
CONVITER(s(s(0)), z1) → c9(IF(false, s(s(0)), z1))
CONVITER(s(s(s(0))), z1) → c9(IF(false, s(s(s(0))), z1))
CONVITER(s(s(z0)), cons(y1, x1)) → c9(IF(false, s(s(z0)), cons(y1, x1)))
CONVITER(s(s(z0)), cons(0, x0)) → c9(IF(false, s(s(z0)), cons(0, x0)))
CONVITER(s(s(z0)), cons(s(0), x0)) → c9(IF(false, s(s(z0)), cons(s(0), x0)))
K tuples:

IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
HALF(s(s(s(s(s(s(y0))))))) → c2(HALF(s(s(s(s(y0))))))
LASTBIT(s(s(s(s(s(s(y0))))))) → c5(LASTBIT(s(s(s(s(y0))))))
Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

IF, CONVITER, HALF, LASTBIT

Compound Symbols:

c11, c11, c9, c2, c5

### (77) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)

Used rewriting to replace IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1))) by IF(false, s(s(s(0))), z0) → c11(CONVITER(s(half(s(0))), cons(s(0), z0)))

### (78) Obligation:

Complexity Dependency Tuples Problem
Rules:

half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
lastbit(0) → 0
lastbit(s(0)) → s(0)
lastbit(s(s(z0))) → lastbit(z0)
Tuples:

IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
CONVITER(s(s(s(s(y0)))), z1) → c9(IF(false, s(s(s(s(y0)))), z1))
CONVITER(s(s(0)), z1) → c9(IF(false, s(s(0)), z1))
CONVITER(s(s(s(0))), z1) → c9(IF(false, s(s(s(0))), z1))
CONVITER(s(s(z0)), cons(y1, x1)) → c9(IF(false, s(s(z0)), cons(y1, x1)))
CONVITER(s(s(z0)), cons(0, x0)) → c9(IF(false, s(s(z0)), cons(0, x0)))
CONVITER(s(s(z0)), cons(s(0), x0)) → c9(IF(false, s(s(z0)), cons(s(0), x0)))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(s(s(z0))), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
HALF(s(s(s(s(s(s(y0))))))) → c2(HALF(s(s(s(s(y0))))))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(z0), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
LASTBIT(s(s(s(s(s(s(y0))))))) → c5(LASTBIT(s(s(s(s(y0))))))
IF(false, s(s(s(0))), z0) → c11(CONVITER(s(half(s(0))), cons(s(0), z0)))
S tuples:

CONVITER(s(s(s(s(y0)))), z1) → c9(IF(false, s(s(s(s(y0)))), z1))
CONVITER(s(s(0)), z1) → c9(IF(false, s(s(0)), z1))
CONVITER(s(s(s(0))), z1) → c9(IF(false, s(s(s(0))), z1))
CONVITER(s(s(z0)), cons(y1, x1)) → c9(IF(false, s(s(z0)), cons(y1, x1)))
CONVITER(s(s(z0)), cons(0, x0)) → c9(IF(false, s(s(z0)), cons(0, x0)))
CONVITER(s(s(z0)), cons(s(0), x0)) → c9(IF(false, s(s(z0)), cons(s(0), x0)))
IF(false, s(s(s(0))), z0) → c11(CONVITER(s(half(s(0))), cons(s(0), z0)))
K tuples:

IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
HALF(s(s(s(s(s(s(y0))))))) → c2(HALF(s(s(s(s(y0))))))
LASTBIT(s(s(s(s(s(s(y0))))))) → c5(LASTBIT(s(s(s(s(y0))))))
Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

IF, CONVITER, HALF, LASTBIT

Compound Symbols:

c11, c9, c2, c5, c11

### (79) 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.

CONVITER(s(s(z0)), cons(y1, x1)) → c9(IF(false, s(s(z0)), cons(y1, x1)))
CONVITER(s(s(z0)), cons(0, x0)) → c9(IF(false, s(s(z0)), cons(0, x0)))
CONVITER(s(s(z0)), cons(s(0), x0)) → c9(IF(false, s(s(z0)), cons(s(0), x0)))
We considered the (Usable) Rules:

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

IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
CONVITER(s(s(s(s(y0)))), z1) → c9(IF(false, s(s(s(s(y0)))), z1))
CONVITER(s(s(0)), z1) → c9(IF(false, s(s(0)), z1))
CONVITER(s(s(s(0))), z1) → c9(IF(false, s(s(s(0))), z1))
CONVITER(s(s(z0)), cons(y1, x1)) → c9(IF(false, s(s(z0)), cons(y1, x1)))
CONVITER(s(s(z0)), cons(0, x0)) → c9(IF(false, s(s(z0)), cons(0, x0)))
CONVITER(s(s(z0)), cons(s(0), x0)) → c9(IF(false, s(s(z0)), cons(s(0), x0)))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(s(s(z0))), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
HALF(s(s(s(s(s(s(y0))))))) → c2(HALF(s(s(s(s(y0))))))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(z0), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
LASTBIT(s(s(s(s(s(s(y0))))))) → c5(LASTBIT(s(s(s(s(y0))))))
IF(false, s(s(s(0))), z0) → c11(CONVITER(s(half(s(0))), cons(s(0), z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(CONVITER(x1, x2)) = [4]x1 + [4]x2
POL(HALF(x1)) = 0
POL(IF(x1, x2, x3)) = [4]x1 + [4]x2
POL(LASTBIT(x1)) = 0
POL(c11(x1)) = x1
POL(c11(x1, x2, x3)) = x1 + x2 + x3
POL(c2(x1)) = x1
POL(c5(x1)) = x1
POL(c9(x1)) = x1
POL(cons(x1, x2)) = [4]
POL(false) = 0
POL(half(x1)) = x1
POL(lastbit(x1)) = 0
POL(s(x1)) = [4] + x1

### (80) Obligation:

Complexity Dependency Tuples Problem
Rules:

half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
lastbit(0) → 0
lastbit(s(0)) → s(0)
lastbit(s(s(z0))) → lastbit(z0)
Tuples:

IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
CONVITER(s(s(s(s(y0)))), z1) → c9(IF(false, s(s(s(s(y0)))), z1))
CONVITER(s(s(0)), z1) → c9(IF(false, s(s(0)), z1))
CONVITER(s(s(s(0))), z1) → c9(IF(false, s(s(s(0))), z1))
CONVITER(s(s(z0)), cons(y1, x1)) → c9(IF(false, s(s(z0)), cons(y1, x1)))
CONVITER(s(s(z0)), cons(0, x0)) → c9(IF(false, s(s(z0)), cons(0, x0)))
CONVITER(s(s(z0)), cons(s(0), x0)) → c9(IF(false, s(s(z0)), cons(s(0), x0)))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(s(s(z0))), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
HALF(s(s(s(s(s(s(y0))))))) → c2(HALF(s(s(s(s(y0))))))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(z0), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
LASTBIT(s(s(s(s(s(s(y0))))))) → c5(LASTBIT(s(s(s(s(y0))))))
IF(false, s(s(s(0))), z0) → c11(CONVITER(s(half(s(0))), cons(s(0), z0)))
S tuples:

CONVITER(s(s(s(s(y0)))), z1) → c9(IF(false, s(s(s(s(y0)))), z1))
CONVITER(s(s(0)), z1) → c9(IF(false, s(s(0)), z1))
CONVITER(s(s(s(0))), z1) → c9(IF(false, s(s(s(0))), z1))
IF(false, s(s(s(0))), z0) → c11(CONVITER(s(half(s(0))), cons(s(0), z0)))
K tuples:

IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
HALF(s(s(s(s(s(s(y0))))))) → c2(HALF(s(s(s(s(y0))))))
LASTBIT(s(s(s(s(s(s(y0))))))) → c5(LASTBIT(s(s(s(s(y0))))))
CONVITER(s(s(z0)), cons(y1, x1)) → c9(IF(false, s(s(z0)), cons(y1, x1)))
CONVITER(s(s(z0)), cons(0, x0)) → c9(IF(false, s(s(z0)), cons(0, x0)))
CONVITER(s(s(z0)), cons(s(0), x0)) → c9(IF(false, s(s(z0)), cons(s(0), x0)))
Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

IF, CONVITER, HALF, LASTBIT

Compound Symbols:

c11, c9, c2, c5, c11

### (81) 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.

CONVITER(s(s(s(s(y0)))), z1) → c9(IF(false, s(s(s(s(y0)))), z1))
CONVITER(s(s(0)), z1) → c9(IF(false, s(s(0)), z1))
CONVITER(s(s(s(0))), z1) → c9(IF(false, s(s(s(0))), z1))
IF(false, s(s(s(0))), z0) → c11(CONVITER(s(half(s(0))), cons(s(0), z0)))
We considered the (Usable) Rules:

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

IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
CONVITER(s(s(s(s(y0)))), z1) → c9(IF(false, s(s(s(s(y0)))), z1))
CONVITER(s(s(0)), z1) → c9(IF(false, s(s(0)), z1))
CONVITER(s(s(s(0))), z1) → c9(IF(false, s(s(s(0))), z1))
CONVITER(s(s(z0)), cons(y1, x1)) → c9(IF(false, s(s(z0)), cons(y1, x1)))
CONVITER(s(s(z0)), cons(0, x0)) → c9(IF(false, s(s(z0)), cons(0, x0)))
CONVITER(s(s(z0)), cons(s(0), x0)) → c9(IF(false, s(s(z0)), cons(s(0), x0)))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(s(s(z0))), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
HALF(s(s(s(s(s(s(y0))))))) → c2(HALF(s(s(s(s(y0))))))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(z0), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
LASTBIT(s(s(s(s(s(s(y0))))))) → c5(LASTBIT(s(s(s(s(y0))))))
IF(false, s(s(s(0))), z0) → c11(CONVITER(s(half(s(0))), cons(s(0), z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(CONVITER(x1, x2)) = [3] + [4]x1
POL(HALF(x1)) = 0
POL(IF(x1, x2, x3)) = [5]x1 + [4]x2
POL(LASTBIT(x1)) = 0
POL(c11(x1)) = x1
POL(c11(x1, x2, x3)) = x1 + x2 + x3
POL(c2(x1)) = x1
POL(c5(x1)) = x1
POL(c9(x1)) = x1
POL(cons(x1, x2)) = 0
POL(false) = 0
POL(half(x1)) = [3] + x1
POL(lastbit(x1)) = [3] + [3]x1
POL(s(x1)) = [4] + x1

### (82) Obligation:

Complexity Dependency Tuples Problem
Rules:

half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
lastbit(0) → 0
lastbit(s(0)) → s(0)
lastbit(s(s(z0))) → lastbit(z0)
Tuples:

IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
CONVITER(s(s(s(s(y0)))), z1) → c9(IF(false, s(s(s(s(y0)))), z1))
CONVITER(s(s(0)), z1) → c9(IF(false, s(s(0)), z1))
CONVITER(s(s(s(0))), z1) → c9(IF(false, s(s(s(0))), z1))
CONVITER(s(s(z0)), cons(y1, x1)) → c9(IF(false, s(s(z0)), cons(y1, x1)))
CONVITER(s(s(z0)), cons(0, x0)) → c9(IF(false, s(s(z0)), cons(0, x0)))
CONVITER(s(s(z0)), cons(s(0), x0)) → c9(IF(false, s(s(z0)), cons(s(0), x0)))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(s(s(z0))), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
HALF(s(s(s(s(s(s(y0))))))) → c2(HALF(s(s(s(s(y0))))))
IF(false, s(s(s(s(z0)))), z1) → c11(CONVITER(s(s(half(z0))), cons(lastbit(z0), z1)), HALF(s(s(s(s(z0))))), LASTBIT(s(s(s(s(z0))))))
LASTBIT(s(s(s(s(s(s(y0))))))) → c5(LASTBIT(s(s(s(s(y0))))))
IF(false, s(s(s(0))), z0) → c11(CONVITER(s(half(s(0))), cons(s(0), z0)))
S tuples:none
K tuples:

IF(false, s(s(z0)), x1) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), x1)), HALF(s(s(z0))), LASTBIT(s(s(z0))))
HALF(s(s(s(s(s(s(y0))))))) → c2(HALF(s(s(s(s(y0))))))
LASTBIT(s(s(s(s(s(s(y0))))))) → c5(LASTBIT(s(s(s(s(y0))))))
CONVITER(s(s(z0)), cons(y1, x1)) → c9(IF(false, s(s(z0)), cons(y1, x1)))
CONVITER(s(s(z0)), cons(0, x0)) → c9(IF(false, s(s(z0)), cons(0, x0)))
CONVITER(s(s(z0)), cons(s(0), x0)) → c9(IF(false, s(s(z0)), cons(s(0), x0)))
CONVITER(s(s(s(s(y0)))), z1) → c9(IF(false, s(s(s(s(y0)))), z1))
CONVITER(s(s(0)), z1) → c9(IF(false, s(s(0)), z1))
CONVITER(s(s(s(0))), z1) → c9(IF(false, s(s(s(0))), z1))
IF(false, s(s(s(0))), z0) → c11(CONVITER(s(half(s(0))), cons(s(0), z0)))
Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

IF, CONVITER, HALF, LASTBIT

Compound Symbols:

c11, c9, c2, c5, c11

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

The set S is empty