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

0 CpxTRS
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
↳2 CdtProblem
↳3 CdtGraphRemoveDanglingProof (BOTH BOUNDS(ID, ID))
↳4 CdtProblem
↳5 CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID))
↳6 CdtProblem
↳7 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
↳8 CdtProblem
↳9 CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication)
↳10 CdtProblem
↳11 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
↳12 CdtProblem
↳13 CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication)
↳14 CdtProblem
↳15 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
↳16 CdtProblem
↳17 CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID))
↳18 CdtProblem
↳19 CdtKnowledgeProof (⇔)
↳20 CdtProblem
↳21 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
↳22 CdtProblem
↳23 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
↳24 CdtProblem
↳25 CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication)
↳26 CdtProblem
↳27 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
↳28 CdtProblem
↳29 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
↳30 CdtProblem
↳31 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
↳32 CdtProblem
↳33 CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication)
↳34 CdtProblem
↳35 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
↳36 CdtProblem
↳37 CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication)
↳38 CdtProblem
↳39 CdtRewritingProof (BOTH BOUNDS(ID, ID))
↳40 CdtProblem
↳41 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
↳42 CdtProblem
↳43 CdtRewritingProof (BOTH BOUNDS(ID, ID))
↳44 CdtProblem
↳45 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
↳46 CdtProblem
↳47 CdtRewritingProof (BOTH BOUNDS(ID, ID))
↳48 CdtProblem
↳49 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
↳50 CdtProblem
↳51 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
↳52 CdtProblem
↳53 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
↳54 CdtProblem
↳55 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
↳56 CdtProblem
↳57 SIsEmptyProof (⇔)
↳58 BOUNDS(O(1), O(1))

(0) Obligation:

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

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

Rewrite Strategy: INNERMOST

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

Converted CpxTRS 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(s(s(z0))) → c2(HALF(z0))
LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
CONV(z0) → c8(CONVITER(z0, cons(0, nil)))
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))
CONV(z0) → c8(CONVITER(z0, cons(0, nil)))
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, CONV, CONVITER, IF

Compound Symbols:

c2, c5, c8, c9, c11

(3) CdtGraphRemoveDanglingProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 of 5 dangling nodes:

CONV(z0) → c8(CONVITER(z0, cons(0, nil)))

(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) CdtGraphRemoveTrailingProof (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) 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))

(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
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(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, conv, conviter, if

Defined Pair Symbols:

HALF, LASTBIT, IF, CONVITER

Compound Symbols:

c2, c5, c11, c9

(9) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 5 dangling nodes:

CONVITER(0, x1) → c9(IF(true, 0, 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
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(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, conv, conviter, if

Defined Pair Symbols:

HALF, LASTBIT, IF, CONVITER

Compound Symbols:

c2, c5, c11, c9

(11) 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))))

(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
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(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, zero, conv, conviter, if

Defined Pair Symbols:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11

(13) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 2 of 9 dangling nodes:

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

(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)
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(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, zero, conv, conviter, if

Defined Pair Symbols:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11

(15) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

Split RHS of tuples not part of any SCC

(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)
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(s(z0), x1) → c9(IF(false, s(z0), x1))
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))))
IF(false, 0, x1) → c(CONVITER(half(0), cons(0, x1)))
IF(false, 0, x1) → c(HALF(0))
IF(false, 0, x1) → c(LASTBIT(0))
S tuples:

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

(17) CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID) transformation)

Removed 4 trailing tuple parts

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

(19) CdtKnowledgeProof (EQUIVALENT transformation)

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

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

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

half, lastbit, zero, conv, conviter, if

Defined Pair Symbols:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11, c, c11, c

(21) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

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

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

POL(0) = [4]   
POL(CONVITER(x1, x2)) = [4]x1   
POL(HALF(x1)) = 0   
POL(IF(x1, x2, x3)) = [4]x2   
POL(LASTBIT(x1)) = 0   
POL(c) = 0   
POL(c(x1)) = 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)) = [4]   
POL(false) = 0   
POL(half(x1)) = x1   
POL(lastbit(x1)) = 0   
POL(s(x1)) = [4] + 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)
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(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) → c(CONVITER(half(0), cons(0, x1)))
IF(false, s(0), x1) → c11(CONVITER(half(s(0)), cons(s(0), x1)))
IF(false, 0, x1) → c
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) → c(CONVITER(half(0), cons(0, x1)))
IF(false, 0, x1) → c
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, zero, conv, conviter, if

Defined Pair Symbols:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11, c, c11, c

(23) 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))))

(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)
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(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) → c(CONVITER(half(0), cons(0, x1)))
IF(false, s(0), x1) → c11(CONVITER(half(s(0)), cons(s(0), x1)))
IF(false, 0, x1) → c
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) → c(CONVITER(half(0), cons(0, x1)))
IF(false, 0, x1) → c
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, zero, conv, conviter, if

Defined Pair Symbols:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11, c, c11, c

(25) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 11 dangling nodes:

IF(false, 0, x1) → c

(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)
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(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) → c(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) → c(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, zero, conv, conviter, if

Defined Pair Symbols:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11, c, c11

(27) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

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

half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
lastbit(0) → 0
lastbit(s(0)) → s(0)
lastbit(s(s(z0))) → lastbit(z0)
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) → c(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)) = [4]x1   
POL(HALF(x1)) = 0   
POL(IF(x1, x2, x3)) = [4]x2   
POL(LASTBIT(x1)) = 0   
POL(c(x1)) = 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) = [2]   
POL(half(x1)) = x1   
POL(lastbit(x1)) = 0   
POL(s(x1)) = [4] + x1   

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
lastbit(0) → 0
lastbit(s(0)) → s(0)
lastbit(s(s(z0))) → lastbit(z0)
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(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) → c(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) → c(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, zero, conv, conviter, if

Defined Pair Symbols:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11, c, 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)
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(s(z0), x1) → c9(IF(false, s(z0), x1))
IF(false, 0, x1) → c(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) → c(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, zero, conv, conviter, if

Defined Pair Symbols:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c, c11, c11

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

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

IF(false, 0, x0) → c(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)
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(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) → c(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) → c(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, zero, conv, conviter, if

Defined Pair Symbols:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11, c11, c

(33) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 13 dangling nodes:

IF(false, 0, x0) → c(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)
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(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, zero, conv, conviter, if

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)
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(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, zero, conv, conviter, if

Defined Pair Symbols:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11, c11

(37) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 12 dangling 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)
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(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, zero, conv, conviter, if

Defined Pair Symbols:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11

(39) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)

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

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

half, lastbit, zero, conv, conviter, if

Defined Pair Symbols:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11

(41) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

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

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

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

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

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

half, lastbit, zero, conv, conviter, if

Defined Pair Symbols:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11

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

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

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

half, lastbit, zero, conv, conviter, if

Defined Pair Symbols:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11

(45) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

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

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

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

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

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

half, lastbit, zero, conv, conviter, if

Defined Pair Symbols:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11

(47) CdtRewritingProof (BOTH BOUNDS(ID, ID) transformation)

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

(48) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

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

half, lastbit, zero, conv, conviter, if

Defined Pair Symbols:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11

(49) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

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

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

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

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

(50) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

HALF(s(s(z0))) → c2(HALF(z0))
LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
CONVITER(s(z0), x1) → c9(IF(false, s(z0), x1))
K tuples:

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

Defined Pair Symbols:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11

(51) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

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

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

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

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

(52) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

HALF(s(s(z0))) → c2(HALF(z0))
LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
K tuples:

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

half, lastbit, zero, conv, conviter, if

Defined Pair Symbols:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11

(53) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

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

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

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

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

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

LASTBIT(s(s(z0))) → c5(LASTBIT(z0))
K tuples:

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

half, lastbit, zero, conv, conviter, if

Defined Pair Symbols:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11

(55) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

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

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

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

POL(0) = [1]   
POL(CONVITER(x1, x2)) = [2]x2 + x12   
POL(HALF(x1)) = 0   
POL(IF(x1, x2, x3)) = [2]x3 + x22   
POL(LASTBIT(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   

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

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

half, lastbit, zero, conv, conviter, if

Defined Pair Symbols:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11

(57) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(58) BOUNDS(O(1), O(1))