The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).

0 CpxTRS
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)
↳2 CdtProblem
↳3 CdtLeafRemovalProof (ComplexityIfPolyImplication, 0 ms)
↳4 CdtProblem
↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳6 CdtProblem
↳7 CdtUsableRulesProof (⇔, 0 ms)
↳8 CdtProblem
↳9 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳10 CdtProblem
↳11 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳12 CdtProblem
↳13 CdtUsableRulesProof (⇔, 0 ms)
↳14 CdtProblem
↳15 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳16 CdtProblem
↳17 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳18 CdtProblem
↳19 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳20 CdtProblem
↳21 CdtKnowledgeProof (BOTH BOUNDS(ID, ID), 0 ms)
↳22 CdtProblem
↳23 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 128 ms)
↳24 CdtProblem
↳25 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳26 CdtProblem
↳27 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 296 ms)
↳28 CdtProblem
↳29 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳30 CdtProblem
↳31 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳32 CdtProblem
↳33 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳34 CdtProblem
↳35 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳36 CdtProblem
↳37 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳38 CdtProblem
↳39 CdtRewritingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳40 CdtProblem
↳41 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 1083 ms)
↳42 CdtProblem
↳43 CdtRewritingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳44 CdtProblem
↳45 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 97 ms)
↳46 CdtProblem
↳47 CdtRewritingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳48 CdtProblem
↳49 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 76 ms)
↳50 CdtProblem
↳51 CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)), 835 ms)
↳52 CdtProblem
↳53 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)
↳54 BOUNDS(1, 1)

(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)

Removed 1 leading nodes:

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

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

(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, 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)))

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

(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:

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(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(s(0)), z0) → c11(CONVITER(s(half(0)), cons(0, z0)), HALF(s(s(0))), LASTBIT(s(s(0))))
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(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(0)), z0) → c11(CONVITER(s(half(0)), cons(0, z0)), HALF(s(s(0))), LASTBIT(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:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11

(41) CdtRuleRemovalProof (UPPER BOUND(ADD(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)), HALF(s(s(0))), LASTBIT(s(s(0))))
We considered the (Usable) Rules:

half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
half(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(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(s(0)), z0) → c11(CONVITER(s(half(0)), cons(0, z0)), HALF(s(s(0))), LASTBIT(s(s(0))))
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)) = [4]x2   
POL(LASTBIT(x1)) = 0   
POL(c11(x1, x2, x3)) = x1 + x2 + x3   
POL(c2(x1)) = x1   
POL(c5(x1)) = x1   
POL(c9(x1)) = x1   
POL(cons(x1, x2)) = [5]   
POL(false) = 0   
POL(half(x1)) = x1   
POL(lastbit(x1)) = 0   
POL(s(x1)) = [4] + x1   

(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:

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

half, lastbit

Defined Pair Symbols:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11

(43) 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)), HALF(s(s(s(0)))), LASTBIT(s(s(s(0))))) by IF(false, s(s(s(0))), z0) → c11(CONVITER(s(half(s(0))), cons(s(0), z0)), HALF(s(s(s(0)))), LASTBIT(s(s(s(0)))))

(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:

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

half, lastbit

Defined Pair Symbols:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11

(45) CdtRuleRemovalProof (UPPER BOUND(ADD(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(0))), z0) → c11(CONVITER(s(half(s(0))), cons(s(0), z0)), HALF(s(s(s(0)))), LASTBIT(s(s(s(0)))))
We considered the (Usable) Rules:

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

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

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

half, lastbit

Defined Pair Symbols:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11

(47) 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))))))

(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:

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

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

half, lastbit

Defined Pair Symbols:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11

(49) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
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(s(0)) → 0
half(s(s(z0))) → s(half(z0))
half(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(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(s(0)), z0) → c11(CONVITER(s(half(0)), cons(0, z0)), HALF(s(s(0))), LASTBIT(s(s(0))))
IF(false, s(s(s(0))), z0) → c11(CONVITER(s(half(s(0))), cons(s(0), z0)), HALF(s(s(s(0)))), 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] + [4]x1 + [4]x2   
POL(HALF(x1)) = 0   
POL(IF(x1, x2, x3)) = x1 + [4]x2 + [2]x3   
POL(LASTBIT(x1)) = 0   
POL(c11(x1, x2, x3)) = x1 + x2 + x3   
POL(c2(x1)) = x1   
POL(c5(x1)) = x1   
POL(c9(x1)) = x1   
POL(cons(x1, x2)) = [3]   
POL(false) = [1]   
POL(half(x1)) = x1   
POL(lastbit(x1)) = [4]   
POL(s(x1)) = [4] + x1   

(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:

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

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

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11

(51) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)) transformation)

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

HALF(s(s(z0))) → c2(HALF(z0))
LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
We considered the (Usable) Rules:

half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
half(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(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(s(0)), z0) → c11(CONVITER(s(half(0)), cons(0, z0)), HALF(s(s(0))), LASTBIT(s(s(0))))
IF(false, s(s(s(0))), z0) → c11(CONVITER(s(half(s(0))), cons(s(0), z0)), HALF(s(s(s(0)))), 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)) = [2] + x1 + x12   
POL(HALF(x1)) = x1   
POL(IF(x1, x2, x3)) = [2] + x2 + x22   
POL(LASTBIT(x1)) = [2] + [2]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   

(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:

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

half, lastbit

Defined Pair Symbols:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11

(53) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty

(54) BOUNDS(1, 1)