Runtime Complexity (innermost) proof of /tmp/tmpfUFApF/thiemann10.xml

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), 9 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)), 54 ms)

↳24 CdtProblem

↳25 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳26 CdtProblem

↳27 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 22 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 CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID), 0 ms)

↳40 CdtProblem

↳41 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳42 CdtProblem

↳43 CdtRewritingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳44 CdtProblem

↳45 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 19 ms)

↳46 CdtProblem

↳47 CdtInstantiationProof (BOTH BOUNDS(ID, ID), 0 ms)

↳48 CdtProblem

↳49 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳50 CdtProblem

↳51 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 0 ms)

↳52 CdtProblem

↳53 CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID), 0 ms)

↳54 CdtProblem

↳55 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

↳56 CdtProblem

↳57 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳58 CdtProblem

↳59 CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)), 3337 ms)

↳60 CdtProblem

↳61 CdtInstantiationProof (BOTH BOUNDS(ID, ID), 0 ms)

↳62 CdtProblem

↳63 CdtRewritingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳64 CdtProblem

↳65 CdtInstantiationProof (BOTH BOUNDS(ID, ID), 0 ms)

↳66 CdtProblem

↳67 CdtRewritingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳68 CdtProblem

↳69 CdtRewritingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳70 CdtProblem

↳71 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 19 ms)

↳72 CdtProblem

↳73 CdtRewritingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳74 CdtProblem

↳75 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 0 ms)

↳76 CdtProblem

↳77 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 15 ms)

↳78 CdtProblem

↳79 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 8 ms)

↳80 CdtProblem

↳81 CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)), 371 ms)

↳82 CdtProblem

↳83 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)

↳84 BOUNDS(1, 1)

(0) Obligation:

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

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

(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)) = 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) = 0
POL(half(x₁)) = x₁
POL(lastbit(x₁)) = [2]
POL(s(x₁)) = [1] + 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) = 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) = 0
POL(half(x₁)) = x₁
POL(lastbit(x₁)) = 0
POL(s(x₁)) = [1] + 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(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(s(0)) → 0
half(s(s(z0))) → s(half(z0))
half(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) = [1]
POL(CONVITER(x₁, x₂)) = x₁
POL(HALF(x₁)) = 0
POL(IF(x₁, x₂, x₃)) = 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)) = [1] + 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) CdtInstantiationProof (BOTH BOUNDS(ID, ID) transformation)

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

CONVITER(s(y0), cons(y1, x1)) → c9(IF(false, s(y0), cons(y1, x1)))
CONVITER(s(s(y0)), cons(y1, x1)) → c9(IF(false, s(s(y0)), cons(y1, x1)))
CONVITER(s(z0), cons(0, x0)) → c9(IF(false, s(z0), cons(0, x0)))
CONVITER(s(z0), cons(s(0), x0)) → c9(IF(false, s(z0), cons(s(0), x0)))
CONVITER(s(0), cons(y0, x0)) → c9(IF(false, s(0), cons(y0, x0)))

(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))
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))))))
CONVITER(s(y0), cons(y1, x1)) → c9(IF(false, s(y0), cons(y1, x1)))
CONVITER(s(s(y0)), cons(y1, x1)) → c9(IF(false, s(s(y0)), cons(y1, x1)))
CONVITER(s(z0), cons(0, x0)) → c9(IF(false, s(z0), cons(0, x0)))
CONVITER(s(z0), cons(s(0), x0)) → c9(IF(false, s(z0), cons(s(0), x0)))
CONVITER(s(0), cons(y0, x0)) → c9(IF(false, s(0), cons(y0, x0)))

S tuples:

LASTBIT(s(s(z0))) → c5(LASTBIT(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)))))
CONVITER(s(y0), cons(y1, x1)) → c9(IF(false, s(y0), cons(y1, x1)))
CONVITER(s(s(y0)), cons(y1, x1)) → c9(IF(false, s(s(y0)), cons(y1, x1)))
CONVITER(s(z0), cons(0, x0)) → c9(IF(false, s(z0), cons(0, x0)))
CONVITER(s(z0), cons(s(0), x0)) → c9(IF(false, s(z0), cons(s(0), x0)))
CONVITER(s(0), cons(y0, x0)) → c9(IF(false, s(0), cons(y0, 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))))))

Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

LASTBIT, IF, HALF, CONVITER

Compound Symbols:

c5, c11, c2, c11, c9

(49) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing tuple parts

(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:

LASTBIT(s(s(z0))) → c5(LASTBIT(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)))))
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))))))
CONVITER(s(y0), cons(y1, x1)) → c9(IF(false, s(y0), cons(y1, x1)))
CONVITER(s(s(y0)), cons(y1, x1)) → c9(IF(false, s(s(y0)), cons(y1, x1)))
CONVITER(s(z0), cons(0, x0)) → c9(IF(false, s(z0), cons(0, x0)))
CONVITER(s(z0), cons(s(0), x0)) → c9(IF(false, s(z0), cons(s(0), x0)))
CONVITER(s(0), cons(y0, x0)) → c9

S tuples:

LASTBIT(s(s(z0))) → c5(LASTBIT(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)))))
CONVITER(s(y0), cons(y1, x1)) → c9(IF(false, s(y0), cons(y1, x1)))
CONVITER(s(s(y0)), cons(y1, x1)) → c9(IF(false, s(s(y0)), cons(y1, x1)))
CONVITER(s(z0), cons(0, x0)) → c9(IF(false, s(z0), cons(0, x0)))
CONVITER(s(z0), cons(s(0), x0)) → c9(IF(false, s(z0), cons(s(0), x0)))
CONVITER(s(0), cons(y0, x0)) → c9

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, IF, HALF, CONVITER

Compound Symbols:

c5, c11, c2, c11, c9, c9

(51) 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(0), cons(y0, x0)) → c9

We considered the (Usable) Rules:

half(s(s(z0))) → s(half(z0))

And the Tuples:

LASTBIT(s(s(z0))) → c5(LASTBIT(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)))))
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))))))
CONVITER(s(y0), cons(y1, x1)) → c9(IF(false, s(y0), cons(y1, x1)))
CONVITER(s(s(y0)), cons(y1, x1)) → c9(IF(false, s(s(y0)), cons(y1, x1)))
CONVITER(s(z0), cons(0, x0)) → c9(IF(false, s(z0), cons(0, x0)))
CONVITER(s(z0), cons(s(0), x0)) → c9(IF(false, s(z0), cons(s(0), x0)))
CONVITER(s(0), cons(y0, x0)) → c9

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(CONVITER(x₁, x₂)) = [1] + x₁
POL(HALF(x₁)) = 0
POL(IF(x₁, x₂, x₃)) = [1] + x₂ + x₃
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) = 0
POL(c9(x₁)) = x₁
POL(cons(x₁, x₂)) = 0
POL(false) = 0
POL(half(x₁)) = [1]
POL(lastbit(x₁)) = 0
POL(s(x₁)) = [1]

(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:

LASTBIT(s(s(z0))) → c5(LASTBIT(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)))))
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))))))
CONVITER(s(y0), cons(y1, x1)) → c9(IF(false, s(y0), cons(y1, x1)))
CONVITER(s(s(y0)), cons(y1, x1)) → c9(IF(false, s(s(y0)), cons(y1, x1)))
CONVITER(s(z0), cons(0, x0)) → c9(IF(false, s(z0), cons(0, x0)))
CONVITER(s(z0), cons(s(0), x0)) → c9(IF(false, s(z0), cons(s(0), x0)))
CONVITER(s(0), cons(y0, x0)) → c9

S tuples:

LASTBIT(s(s(z0))) → c5(LASTBIT(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)))))
CONVITER(s(y0), cons(y1, x1)) → c9(IF(false, s(y0), cons(y1, x1)))
CONVITER(s(s(y0)), cons(y1, x1)) → c9(IF(false, s(s(y0)), cons(y1, x1)))
CONVITER(s(z0), cons(0, x0)) → c9(IF(false, s(z0), cons(0, x0)))
CONVITER(s(z0), cons(s(0), x0)) → c9(IF(false, 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))))))
CONVITER(s(0), cons(y0, x0)) → c9

Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

LASTBIT, IF, HALF, CONVITER

Compound Symbols:

c5, c11, c2, c11, c9, c9

(53) 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))))

(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:

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))))))
CONVITER(s(y0), cons(y1, x1)) → c9(IF(false, s(y0), cons(y1, x1)))
CONVITER(s(s(y0)), cons(y1, x1)) → c9(IF(false, s(s(y0)), cons(y1, x1)))
CONVITER(s(z0), cons(0, x0)) → c9(IF(false, s(z0), cons(0, x0)))
CONVITER(s(z0), cons(s(0), x0)) → c9(IF(false, s(z0), cons(s(0), x0)))
CONVITER(s(0), cons(y0, x0)) → c9
LASTBIT(s(s(s(s(y0))))) → c5(LASTBIT(s(s(y0))))

S tuples:

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)))))
CONVITER(s(y0), cons(y1, x1)) → c9(IF(false, s(y0), cons(y1, x1)))
CONVITER(s(s(y0)), cons(y1, x1)) → c9(IF(false, s(s(y0)), cons(y1, x1)))
CONVITER(s(z0), cons(0, x0)) → c9(IF(false, s(z0), cons(0, x0)))
CONVITER(s(z0), cons(s(0), x0)) → c9(IF(false, s(z0), cons(s(0), x0)))
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))))))
CONVITER(s(0), cons(y0, x0)) → c9

Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

IF, HALF, CONVITER, LASTBIT

Compound Symbols:

c11, c2, c11, c9, c9, c5

(55) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing nodes:

CONVITER(s(0), cons(y0, x0)) → c9

(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(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))))))
CONVITER(s(y0), cons(y1, x1)) → c9(IF(false, s(y0), cons(y1, x1)))
CONVITER(s(s(y0)), cons(y1, x1)) → c9(IF(false, s(s(y0)), cons(y1, x1)))
CONVITER(s(z0), cons(0, x0)) → c9(IF(false, s(z0), cons(0, x0)))
CONVITER(s(z0), cons(s(0), x0)) → c9(IF(false, s(z0), cons(s(0), x0)))
LASTBIT(s(s(s(s(y0))))) → c5(LASTBIT(s(s(y0))))

S tuples:

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)))))
CONVITER(s(y0), cons(y1, x1)) → c9(IF(false, s(y0), cons(y1, x1)))
CONVITER(s(s(y0)), cons(y1, x1)) → c9(IF(false, s(s(y0)), cons(y1, x1)))
CONVITER(s(z0), cons(0, x0)) → c9(IF(false, s(z0), cons(0, x0)))
CONVITER(s(z0), cons(s(0), x0)) → c9(IF(false, s(z0), cons(s(0), x0)))
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))))))

Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

IF, HALF, CONVITER, LASTBIT

Compound Symbols:

c11, c2, c11, c9, c5

(57) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 4 trailing tuple parts

(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))))))
CONVITER(s(y0), cons(y1, x1)) → c9(IF(false, s(y0), cons(y1, x1)))
CONVITER(s(s(y0)), cons(y1, x1)) → c9(IF(false, s(s(y0)), cons(y1, x1)))
CONVITER(s(z0), cons(0, x0)) → c9(IF(false, s(z0), cons(0, x0)))
CONVITER(s(z0), cons(s(0), x0)) → c9(IF(false, s(z0), cons(s(0), x0)))
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:

HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
CONVITER(s(y0), cons(y1, x1)) → c9(IF(false, s(y0), cons(y1, x1)))
CONVITER(s(s(y0)), cons(y1, x1)) → c9(IF(false, s(s(y0)), cons(y1, x1)))
CONVITER(s(z0), cons(0, x0)) → c9(IF(false, s(z0), cons(0, x0)))
CONVITER(s(z0), cons(s(0), x0)) → c9(IF(false, s(z0), cons(s(0), x0)))
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))))))

Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

IF, HALF, CONVITER, LASTBIT

Compound Symbols:

c11, c2, c9, c5, c11

(59) CdtRuleRemovalProof (UPPER BOUND(ADD(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(s(0)) → 0
half(s(s(z0))) → s(half(z0))
half(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))))
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))))))
CONVITER(s(y0), cons(y1, x1)) → c9(IF(false, s(y0), cons(y1, x1)))
CONVITER(s(s(y0)), cons(y1, x1)) → c9(IF(false, s(s(y0)), cons(y1, x1)))
CONVITER(s(z0), cons(0, x0)) → c9(IF(false, s(z0), cons(0, x0)))
CONVITER(s(z0), cons(s(0), x0)) → c9(IF(false, s(z0), cons(s(0), x0)))
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)) = x₂ + x₂² + [2]x₁²
POL(HALF(x₁)) = [1]
POL(IF(x₁, x₂, x₃)) = x₁ + [2]x₃² + [2]x₁·x₃ + [2]x₂²
POL(LASTBIT(x₁)) = [1] + 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)) = [1] + x₁

(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))))))
CONVITER(s(y0), cons(y1, x1)) → c9(IF(false, s(y0), cons(y1, x1)))
CONVITER(s(s(y0)), cons(y1, x1)) → c9(IF(false, s(s(y0)), cons(y1, x1)))
CONVITER(s(z0), cons(0, x0)) → c9(IF(false, s(z0), cons(0, x0)))
CONVITER(s(z0), cons(s(0), x0)) → c9(IF(false, s(z0), cons(s(0), x0)))
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:

HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
CONVITER(s(y0), cons(y1, x1)) → c9(IF(false, s(y0), cons(y1, x1)))
CONVITER(s(s(y0)), cons(y1, x1)) → c9(IF(false, s(s(y0)), cons(y1, x1)))
CONVITER(s(z0), cons(0, x0)) → c9(IF(false, s(z0), cons(0, x0)))
CONVITER(s(z0), cons(s(0), x0)) → c9(IF(false, s(z0), cons(s(0), x0)))
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))))))
LASTBIT(s(s(s(s(y0))))) → c5(LASTBIT(s(s(y0))))

Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

IF, HALF, CONVITER, LASTBIT

Compound Symbols:

c11, c2, c9, c5, c11

(61) CdtInstantiationProof (BOTH BOUNDS(ID, ID) transformation)

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

IF(false, s(s(z0)), cons(x1, x2)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(x1, x2))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(0, x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(0, x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(s(0), x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(s(0), x1))), HALF(s(s(z0))), LASTBIT(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))))
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))))))
CONVITER(s(y0), cons(y1, x1)) → c9(IF(false, s(y0), cons(y1, x1)))
CONVITER(s(s(y0)), cons(y1, x1)) → c9(IF(false, s(s(y0)), cons(y1, x1)))
CONVITER(s(z0), cons(0, x0)) → c9(IF(false, s(z0), cons(0, x0)))
CONVITER(s(z0), cons(s(0), x0)) → c9(IF(false, s(z0), cons(s(0), x0)))
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)))
IF(false, s(s(z0)), cons(x1, x2)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(x1, x2))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(0, x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(0, x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(s(0), x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(s(0), x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))

S tuples:

HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
CONVITER(s(y0), cons(y1, x1)) → c9(IF(false, s(y0), cons(y1, x1)))
CONVITER(s(s(y0)), cons(y1, x1)) → c9(IF(false, s(s(y0)), cons(y1, x1)))
CONVITER(s(z0), cons(0, x0)) → c9(IF(false, s(z0), cons(0, x0)))
CONVITER(s(z0), cons(s(0), x0)) → c9(IF(false, s(z0), cons(s(0), x0)))
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(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(z0)), cons(x1, x2)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(x1, x2))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(0, x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(0, x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(s(0), x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(s(0), x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))

Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

IF, HALF, CONVITER, LASTBIT

Compound Symbols:

c11, c2, c9, c5, c11

(63) 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))))))

(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))))
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))))))
CONVITER(s(y0), cons(y1, x1)) → c9(IF(false, s(y0), cons(y1, x1)))
CONVITER(s(s(y0)), cons(y1, x1)) → c9(IF(false, s(s(y0)), cons(y1, x1)))
CONVITER(s(z0), cons(0, x0)) → c9(IF(false, s(z0), cons(0, x0)))
CONVITER(s(z0), cons(s(0), x0)) → c9(IF(false, s(z0), cons(s(0), x0)))
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)))
IF(false, s(s(z0)), cons(x1, x2)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(x1, x2))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(0, x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(0, x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(s(0), x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(s(0), x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
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:

HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
CONVITER(s(y0), cons(y1, x1)) → c9(IF(false, s(y0), cons(y1, x1)))
CONVITER(s(s(y0)), cons(y1, x1)) → c9(IF(false, s(s(y0)), cons(y1, x1)))
CONVITER(s(z0), cons(0, x0)) → c9(IF(false, s(z0), cons(0, x0)))
CONVITER(s(z0), cons(s(0), x0)) → c9(IF(false, s(z0), cons(s(0), x0)))
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(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(z0)), cons(x1, x2)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(x1, x2))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(0, x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(0, x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(s(0), x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(s(0), x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))

Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

IF, HALF, CONVITER, LASTBIT

Compound Symbols:

c11, c2, c9, c5, c11

(65) CdtInstantiationProof (BOTH BOUNDS(ID, ID) transformation)

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

IF(false, s(s(z0)), cons(x1, x2)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(x1, x2))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(0, x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(0, x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(s(0), x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(s(0), x1))), HALF(s(s(z0))), LASTBIT(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:

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))))))
CONVITER(s(y0), cons(y1, x1)) → c9(IF(false, s(y0), cons(y1, x1)))
CONVITER(s(s(y0)), cons(y1, x1)) → c9(IF(false, s(s(y0)), cons(y1, x1)))
CONVITER(s(z0), cons(0, x0)) → c9(IF(false, s(z0), cons(0, x0)))
CONVITER(s(z0), cons(s(0), x0)) → c9(IF(false, s(z0), cons(s(0), x0)))
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)))
IF(false, s(s(z0)), cons(x1, x2)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(x1, x2))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(0, x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(0, x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(s(0), x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(s(0), x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
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:

HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
CONVITER(s(y0), cons(y1, x1)) → c9(IF(false, s(y0), cons(y1, x1)))
CONVITER(s(s(y0)), cons(y1, x1)) → c9(IF(false, s(s(y0)), cons(y1, x1)))
CONVITER(s(z0), cons(0, x0)) → c9(IF(false, s(z0), cons(0, x0)))
CONVITER(s(z0), cons(s(0), x0)) → c9(IF(false, s(z0), cons(s(0), x0)))
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(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(z0)), cons(x1, x2)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(x1, x2))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(0, x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(0, x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(s(0), x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(s(0), x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))

Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

HALF, IF, CONVITER, LASTBIT

Compound Symbols:

c2, c11, c9, c5, c11

(67) 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))))))

(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:

HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
CONVITER(s(y0), cons(y1, x1)) → c9(IF(false, s(y0), cons(y1, x1)))
CONVITER(s(s(y0)), cons(y1, x1)) → c9(IF(false, s(s(y0)), cons(y1, x1)))
CONVITER(s(z0), cons(0, x0)) → c9(IF(false, s(z0), cons(0, x0)))
CONVITER(s(z0), cons(s(0), x0)) → c9(IF(false, s(z0), cons(s(0), x0)))
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)))
IF(false, s(s(z0)), cons(x1, x2)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(x1, x2))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(0, x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(0, x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(s(0), x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(s(0), x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
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))))))
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:

HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
CONVITER(s(y0), cons(y1, x1)) → c9(IF(false, s(y0), cons(y1, x1)))
CONVITER(s(s(y0)), cons(y1, x1)) → c9(IF(false, s(s(y0)), cons(y1, x1)))
CONVITER(s(z0), cons(0, x0)) → c9(IF(false, s(z0), cons(0, x0)))
CONVITER(s(z0), cons(s(0), x0)) → c9(IF(false, s(z0), cons(s(0), x0)))
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(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(z0)), cons(x1, x2)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(x1, x2))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(0, x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(0, x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(s(0), x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(s(0), x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))

Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

HALF, CONVITER, LASTBIT, IF

Compound Symbols:

c2, c9, c5, c11, c11

(69) 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)))

(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:

HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
CONVITER(s(y0), cons(y1, x1)) → c9(IF(false, s(y0), cons(y1, x1)))
CONVITER(s(s(y0)), cons(y1, x1)) → c9(IF(false, s(s(y0)), cons(y1, x1)))
CONVITER(s(z0), cons(0, x0)) → c9(IF(false, s(z0), cons(0, x0)))
CONVITER(s(z0), cons(s(0), x0)) → c9(IF(false, s(z0), cons(s(0), x0)))
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)))
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)))
IF(false, s(s(z0)), cons(x1, x2)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(x1, x2))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(0, x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(0, x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(s(0), x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(s(0), x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
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))))))
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:

HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
CONVITER(s(y0), cons(y1, x1)) → c9(IF(false, s(y0), cons(y1, x1)))
CONVITER(s(s(y0)), cons(y1, x1)) → c9(IF(false, s(s(y0)), cons(y1, x1)))
CONVITER(s(z0), cons(0, x0)) → c9(IF(false, s(z0), cons(0, x0)))
CONVITER(s(z0), cons(s(0), x0)) → c9(IF(false, s(z0), cons(s(0), x0)))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)))
IF(false, s(s(0)), z0) → c11(CONVITER(s(half(0)), cons(0, z0)))

K tuples:

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(z0)), cons(x1, x2)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(x1, x2))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(0, x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(0, x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(s(0), x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(s(0), x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))

Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

HALF, CONVITER, LASTBIT, IF

Compound Symbols:

c2, c9, c5, c11, c11

(71) 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)))

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(s(s(y0))))) → c2(HALF(s(s(y0))))
CONVITER(s(y0), cons(y1, x1)) → c9(IF(false, s(y0), cons(y1, x1)))
CONVITER(s(s(y0)), cons(y1, x1)) → c9(IF(false, s(s(y0)), cons(y1, x1)))
CONVITER(s(z0), cons(0, x0)) → c9(IF(false, s(z0), cons(0, x0)))
CONVITER(s(z0), cons(s(0), x0)) → c9(IF(false, s(z0), cons(s(0), x0)))
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)))
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)))
IF(false, s(s(z0)), cons(x1, x2)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(x1, x2))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(0, x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(0, x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(s(0), x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(s(0), x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
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))))))
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) = [1]
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) = 0
POL(half(x₁)) = x₁
POL(lastbit(x₁)) = 0
POL(s(x₁)) = [1] + x₁

(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:

HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
CONVITER(s(y0), cons(y1, x1)) → c9(IF(false, s(y0), cons(y1, x1)))
CONVITER(s(s(y0)), cons(y1, x1)) → c9(IF(false, s(s(y0)), cons(y1, x1)))
CONVITER(s(z0), cons(0, x0)) → c9(IF(false, s(z0), cons(0, x0)))
CONVITER(s(z0), cons(s(0), x0)) → c9(IF(false, s(z0), cons(s(0), x0)))
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)))
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)))
IF(false, s(s(z0)), cons(x1, x2)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(x1, x2))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(0, x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(0, x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(s(0), x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(s(0), x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
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))))))
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:

HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
CONVITER(s(y0), cons(y1, x1)) → c9(IF(false, s(y0), cons(y1, x1)))
CONVITER(s(s(y0)), cons(y1, x1)) → c9(IF(false, s(s(y0)), cons(y1, x1)))
CONVITER(s(z0), cons(0, x0)) → c9(IF(false, s(z0), cons(0, x0)))
CONVITER(s(z0), cons(s(0), x0)) → c9(IF(false, s(z0), cons(s(0), x0)))
IF(false, s(s(s(0))), x1) → c11(CONVITER(half(s(s(s(0)))), cons(s(0), x1)))

K tuples:

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(z0)), cons(x1, x2)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(x1, x2))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(0, x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(0, x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(s(0), x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(s(0), x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(0)), z0) → c11(CONVITER(s(half(0)), cons(0, z0)))

Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

HALF, CONVITER, LASTBIT, IF

Compound Symbols:

c2, c9, c5, c11, c11

(73) 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)))

(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:

HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
CONVITER(s(y0), cons(y1, x1)) → c9(IF(false, s(y0), cons(y1, x1)))
CONVITER(s(s(y0)), cons(y1, x1)) → c9(IF(false, s(s(y0)), cons(y1, x1)))
CONVITER(s(z0), cons(0, x0)) → c9(IF(false, s(z0), cons(0, x0)))
CONVITER(s(z0), cons(s(0), x0)) → c9(IF(false, s(z0), cons(s(0), x0)))
LASTBIT(s(s(s(s(y0))))) → c5(LASTBIT(s(s(y0))))
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)))
IF(false, s(s(z0)), cons(x1, x2)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(x1, x2))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(0, x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(0, x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(s(0), x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(s(0), x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
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))))))
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)))
IF(false, s(s(s(0))), z0) → c11(CONVITER(s(half(s(0))), cons(s(0), z0)))

S tuples:

HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
CONVITER(s(y0), cons(y1, x1)) → c9(IF(false, s(y0), cons(y1, x1)))
CONVITER(s(s(y0)), cons(y1, x1)) → c9(IF(false, s(s(y0)), cons(y1, x1)))
CONVITER(s(z0), cons(0, x0)) → c9(IF(false, s(z0), cons(0, x0)))
CONVITER(s(z0), cons(s(0), x0)) → c9(IF(false, 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(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(z0)), cons(x1, x2)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(x1, x2))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(0, x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(0, x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(s(0), x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(s(0), x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(0)), z0) → c11(CONVITER(s(half(0)), cons(0, z0)))

Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

HALF, CONVITER, LASTBIT, IF

Compound Symbols:

c2, c9, c5, c11, c11

(75) 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), cons(s(0), x0)) → c9(IF(false, s(z0), cons(s(0), x0)))

We considered the (Usable) Rules:

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

And the Tuples:

HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
CONVITER(s(y0), cons(y1, x1)) → c9(IF(false, s(y0), cons(y1, x1)))
CONVITER(s(s(y0)), cons(y1, x1)) → c9(IF(false, s(s(y0)), cons(y1, x1)))
CONVITER(s(z0), cons(0, x0)) → c9(IF(false, s(z0), cons(0, x0)))
CONVITER(s(z0), cons(s(0), x0)) → c9(IF(false, s(z0), cons(s(0), x0)))
LASTBIT(s(s(s(s(y0))))) → c5(LASTBIT(s(s(y0))))
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)))
IF(false, s(s(z0)), cons(x1, x2)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(x1, x2))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(0, x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(0, x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(s(0), x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(s(0), x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
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))))))
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)))
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)) = [1] + x₁ + x₂
POL(HALF(x₁)) = 0
POL(IF(x₁, x₂, x₃)) = [1] + x₁ + 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)) = [1] + x₁
POL(false) = [1]
POL(half(x₁)) = x₁
POL(lastbit(x₁)) = [1]
POL(s(x₁)) = [1] + x₁

(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:

HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
CONVITER(s(y0), cons(y1, x1)) → c9(IF(false, s(y0), cons(y1, x1)))
CONVITER(s(s(y0)), cons(y1, x1)) → c9(IF(false, s(s(y0)), cons(y1, x1)))
CONVITER(s(z0), cons(0, x0)) → c9(IF(false, s(z0), cons(0, x0)))
CONVITER(s(z0), cons(s(0), x0)) → c9(IF(false, s(z0), cons(s(0), x0)))
LASTBIT(s(s(s(s(y0))))) → c5(LASTBIT(s(s(y0))))
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)))
IF(false, s(s(z0)), cons(x1, x2)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(x1, x2))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(0, x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(0, x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(s(0), x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(s(0), x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
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))))))
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)))
IF(false, s(s(s(0))), z0) → c11(CONVITER(s(half(s(0))), cons(s(0), z0)))

S tuples:

HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
CONVITER(s(y0), cons(y1, x1)) → c9(IF(false, s(y0), cons(y1, x1)))
CONVITER(s(s(y0)), cons(y1, x1)) → c9(IF(false, s(s(y0)), cons(y1, x1)))
CONVITER(s(z0), cons(0, x0)) → c9(IF(false, s(z0), cons(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(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(z0)), cons(x1, x2)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(x1, x2))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(0, x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(0, x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(s(0), x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(s(0), x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(0)), z0) → c11(CONVITER(s(half(0)), cons(0, z0)))
CONVITER(s(z0), cons(s(0), x0)) → c9(IF(false, s(z0), cons(s(0), x0)))

Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

HALF, CONVITER, LASTBIT, IF

Compound Symbols:

c2, c9, c5, c11, c11

(77) 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)))

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(s(s(y0))))) → c2(HALF(s(s(y0))))
CONVITER(s(y0), cons(y1, x1)) → c9(IF(false, s(y0), cons(y1, x1)))
CONVITER(s(s(y0)), cons(y1, x1)) → c9(IF(false, s(s(y0)), cons(y1, x1)))
CONVITER(s(z0), cons(0, x0)) → c9(IF(false, s(z0), cons(0, x0)))
CONVITER(s(z0), cons(s(0), x0)) → c9(IF(false, s(z0), cons(s(0), x0)))
LASTBIT(s(s(s(s(y0))))) → c5(LASTBIT(s(s(y0))))
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)))
IF(false, s(s(z0)), cons(x1, x2)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(x1, x2))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(0, x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(0, x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(s(0), x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(s(0), x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
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))))))
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)))
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)) = x₁
POL(HALF(x₁)) = 0
POL(IF(x₁, x₂, x₃)) = x₂
POL(LASTBIT(x₁)) = [1]
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)) = [1] + x₁

(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:

HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
CONVITER(s(y0), cons(y1, x1)) → c9(IF(false, s(y0), cons(y1, x1)))
CONVITER(s(s(y0)), cons(y1, x1)) → c9(IF(false, s(s(y0)), cons(y1, x1)))
CONVITER(s(z0), cons(0, x0)) → c9(IF(false, s(z0), cons(0, x0)))
CONVITER(s(z0), cons(s(0), x0)) → c9(IF(false, s(z0), cons(s(0), x0)))
LASTBIT(s(s(s(s(y0))))) → c5(LASTBIT(s(s(y0))))
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)))
IF(false, s(s(z0)), cons(x1, x2)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(x1, x2))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(0, x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(0, x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(s(0), x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(s(0), x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
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))))))
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)))
IF(false, s(s(s(0))), z0) → c11(CONVITER(s(half(s(0))), cons(s(0), z0)))

S tuples:

HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
CONVITER(s(y0), cons(y1, x1)) → c9(IF(false, s(y0), cons(y1, x1)))
CONVITER(s(s(y0)), cons(y1, x1)) → c9(IF(false, s(s(y0)), cons(y1, x1)))
CONVITER(s(z0), cons(0, x0)) → c9(IF(false, s(z0), cons(0, x0)))

K tuples:

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(z0)), cons(x1, x2)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(x1, x2))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(0, x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(0, x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(s(0), x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(s(0), x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(0)), z0) → c11(CONVITER(s(half(0)), cons(0, z0)))
CONVITER(s(z0), cons(s(0), x0)) → c9(IF(false, s(z0), cons(s(0), x0)))
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:

HALF, CONVITER, LASTBIT, IF

Compound Symbols:

c2, c9, c5, c11, c11

(79) 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(y0), cons(y1, x1)) → c9(IF(false, s(y0), cons(y1, x1)))
CONVITER(s(s(y0)), cons(y1, x1)) → c9(IF(false, s(s(y0)), cons(y1, x1)))
CONVITER(s(z0), cons(0, x0)) → c9(IF(false, s(z0), cons(0, x0)))

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(s(s(y0))))) → c2(HALF(s(s(y0))))
CONVITER(s(y0), cons(y1, x1)) → c9(IF(false, s(y0), cons(y1, x1)))
CONVITER(s(s(y0)), cons(y1, x1)) → c9(IF(false, s(s(y0)), cons(y1, x1)))
CONVITER(s(z0), cons(0, x0)) → c9(IF(false, s(z0), cons(0, x0)))
CONVITER(s(z0), cons(s(0), x0)) → c9(IF(false, s(z0), cons(s(0), x0)))
LASTBIT(s(s(s(s(y0))))) → c5(LASTBIT(s(s(y0))))
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)))
IF(false, s(s(z0)), cons(x1, x2)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(x1, x2))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(0, x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(0, x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(s(0), x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(s(0), x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
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))))))
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)))
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)) = [1] + x₁ + x₂
POL(HALF(x₁)) = 0
POL(IF(x₁, x₂, x₃)) = [1] + 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)) = [1]
POL(false) = 0
POL(half(x₁)) = x₁
POL(lastbit(x₁)) = 0
POL(s(x₁)) = [1] + 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:

HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
CONVITER(s(y0), cons(y1, x1)) → c9(IF(false, s(y0), cons(y1, x1)))
CONVITER(s(s(y0)), cons(y1, x1)) → c9(IF(false, s(s(y0)), cons(y1, x1)))
CONVITER(s(z0), cons(0, x0)) → c9(IF(false, s(z0), cons(0, x0)))
CONVITER(s(z0), cons(s(0), x0)) → c9(IF(false, s(z0), cons(s(0), x0)))
LASTBIT(s(s(s(s(y0))))) → c5(LASTBIT(s(s(y0))))
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)))
IF(false, s(s(z0)), cons(x1, x2)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(x1, x2))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(0, x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(0, x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(s(0), x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(s(0), x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
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))))))
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)))
IF(false, s(s(s(0))), z0) → c11(CONVITER(s(half(s(0))), cons(s(0), z0)))

S tuples:

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

K tuples:

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(z0)), cons(x1, x2)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(x1, x2))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(0, x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(0, x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(s(0), x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(s(0), x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(0)), z0) → c11(CONVITER(s(half(0)), cons(0, z0)))
CONVITER(s(z0), cons(s(0), x0)) → c9(IF(false, s(z0), cons(s(0), x0)))
IF(false, s(s(s(0))), z0) → c11(CONVITER(s(half(s(0))), cons(s(0), z0)))
CONVITER(s(y0), cons(y1, x1)) → c9(IF(false, s(y0), cons(y1, x1)))
CONVITER(s(s(y0)), cons(y1, x1)) → c9(IF(false, s(s(y0)), cons(y1, x1)))
CONVITER(s(z0), cons(0, x0)) → c9(IF(false, s(z0), cons(0, x0)))

Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

HALF, CONVITER, LASTBIT, IF

Compound Symbols:

c2, c9, c5, c11, c11

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

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(s(s(y0))))) → c2(HALF(s(s(y0))))
CONVITER(s(y0), cons(y1, x1)) → c9(IF(false, s(y0), cons(y1, x1)))
CONVITER(s(s(y0)), cons(y1, x1)) → c9(IF(false, s(s(y0)), cons(y1, x1)))
CONVITER(s(z0), cons(0, x0)) → c9(IF(false, s(z0), cons(0, x0)))
CONVITER(s(z0), cons(s(0), x0)) → c9(IF(false, s(z0), cons(s(0), x0)))
LASTBIT(s(s(s(s(y0))))) → c5(LASTBIT(s(s(y0))))
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)))
IF(false, s(s(z0)), cons(x1, x2)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(x1, x2))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(0, x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(0, x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(s(0), x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(s(0), x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
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))))))
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)))
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)) = [2]x₁·x₂ + [2]x₁²
POL(HALF(x₁)) = [1] + x₁
POL(IF(x₁, x₂, x₃)) = [2]x₁ + [2]x₂·x₃ + x₁·x₂ + [2]x₂²
POL(LASTBIT(x₁)) = [1]
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)) = [2] + x₂
POL(false) = 0
POL(half(x₁)) = x₁
POL(lastbit(x₁)) = [2]
POL(s(x₁)) = [2] + 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:

HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))
CONVITER(s(y0), cons(y1, x1)) → c9(IF(false, s(y0), cons(y1, x1)))
CONVITER(s(s(y0)), cons(y1, x1)) → c9(IF(false, s(s(y0)), cons(y1, x1)))
CONVITER(s(z0), cons(0, x0)) → c9(IF(false, s(z0), cons(0, x0)))
CONVITER(s(z0), cons(s(0), x0)) → c9(IF(false, s(z0), cons(s(0), x0)))
LASTBIT(s(s(s(s(y0))))) → c5(LASTBIT(s(s(y0))))
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)))
IF(false, s(s(z0)), cons(x1, x2)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(x1, x2))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(0, x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(0, x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(s(0), x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(s(0), x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
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))))))
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)))
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(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(z0)), cons(x1, x2)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(x1, x2))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(0, x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(0, x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(z0)), cons(s(0), x1)) → c11(CONVITER(s(half(z0)), cons(lastbit(z0), cons(s(0), x1))), HALF(s(s(z0))), LASTBIT(s(s(z0))))
IF(false, s(s(0)), z0) → c11(CONVITER(s(half(0)), cons(0, z0)))
CONVITER(s(z0), cons(s(0), x0)) → c9(IF(false, s(z0), cons(s(0), x0)))
IF(false, s(s(s(0))), z0) → c11(CONVITER(s(half(s(0))), cons(s(0), z0)))
CONVITER(s(y0), cons(y1, x1)) → c9(IF(false, s(y0), cons(y1, x1)))
CONVITER(s(s(y0)), cons(y1, x1)) → c9(IF(false, s(s(y0)), cons(y1, x1)))
CONVITER(s(z0), cons(0, x0)) → c9(IF(false, s(z0), cons(0, x0)))
HALF(s(s(s(s(y0))))) → c2(HALF(s(s(y0))))

Defined Rule Symbols:

half, lastbit

Defined Pair Symbols:

HALF, CONVITER, LASTBIT, IF

Compound Symbols:

c2, c9, c5, c11, c11

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

The set S is empty