Runtime Complexity (innermost) proof of /tmp/tmpmcG9b2/thiemann10.xml

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), 0 ms)

↳2 CdtProblem

↳3 CdtLeafRemovalProof (ComplexityIfPolyImplication, 0 ms)

↳4 CdtProblem

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

↳6 CdtProblem

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

↳8 CdtProblem

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

↳10 CdtProblem

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

↳12 CdtProblem

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

↳14 CdtProblem

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

↳16 CdtProblem

↳17 CdtKnowledgeProof (⇔, 0 ms)

↳18 CdtProblem

↳19 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 203 ms)

↳20 CdtProblem

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

↳22 CdtProblem

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

↳24 CdtProblem

↳25 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 237 ms)

↳26 CdtProblem

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

↳28 CdtProblem

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

↳30 CdtProblem

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

↳32 CdtProblem

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

↳34 CdtProblem

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

↳36 CdtProblem

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

↳38 CdtProblem

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

↳40 CdtProblem

↳41 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 1144 ms)

↳42 CdtProblem

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

↳44 CdtProblem

↳45 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 99 ms)

↳46 CdtProblem

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

↳48 CdtProblem

↳49 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 62 ms)

↳50 CdtProblem

↳51 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))), 937 ms)

↳52 CdtProblem

↳53 SIsEmptyProof (⇔, 0 ms)

↳54 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) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading 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) 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) 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))
CONVITER(x0, x1) → c9

(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))
CONVITER(x0, x1) → c9

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))
CONVITER(x0, x1) → c9

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, c9

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

Removed 2 trailing nodes:

CONVITER(0, x1) → c9(IF(true, 0, x1))
CONVITER(x0, x1) → c9

(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))))
IF(false, x0, x1) → c11

(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))))
IF(false, x0, x1) → c11

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))))
IF(false, x0, x1) → c11

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, c11

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

Removed 3 trailing nodes:

IF(false, s(0), x1) → c11(CONVITER(0, cons(lastbit(s(0)), x1)), HALF(s(0)), LASTBIT(s(0)))
IF(false, x0, x1) → c11
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) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 4 trailing tuple parts

(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(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, zero, conv, conviter, if

Defined Pair Symbols:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11, c11

(17) CdtKnowledgeProof (EQUIVALENT 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)))

(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) → 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, zero, conv, conviter, if

Defined Pair Symbols:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11, c11

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

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

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

(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) → 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, zero, conv, conviter, if

Defined Pair Symbols:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11, c11

(21) 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))))
IF(false, s(s(x0)), x1) → c11

(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(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))))
IF(false, s(s(x0)), x1) → c11

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))))
IF(false, s(s(x0)), x1) → c11

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, zero, conv, conviter, if

Defined Pair Symbols:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11, c11, c11

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

Removed 1 trailing nodes:

IF(false, s(s(x0)), x1) → c11

(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) → 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, zero, conv, conviter, if

Defined Pair Symbols:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11, c11

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

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

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

(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) → 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, zero, conv, conviter, if

Defined Pair Symbols:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11, c11

(27) 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))))))
IF(false, s(s(x0)), x1) → c11

(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, 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))))))
IF(false, s(s(x0)), x1) → c11

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, zero, conv, conviter, if

Defined Pair Symbols:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11, c11, c11

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

Removed 1 trailing nodes:

IF(false, s(s(x0)), x1) → c11

(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) → 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(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

(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)))
IF(false, 0, x0) → c11

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

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(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, c11

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

Removed 2 trailing nodes:

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

(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)))
IF(false, s(0), x0) → c11

(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)))
IF(false, s(0), x0) → c11

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)))
IF(false, s(0), x0) → c11

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, c11

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

Removed 2 trailing nodes:

IF(false, s(0), x0) → c11
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.

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

We considered the (Usable) Rules:

half(s(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))

K tuples:

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

Defined Rule Symbols:

half, lastbit, zero, conv, conviter, if

Defined Pair Symbols:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11

(51) 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))
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) = 0
POL(CONVITER(x₁, x₂)) = [2] + x₁ + x₁²
POL(HALF(x₁)) = x₁
POL(IF(x₁, x₂, x₃)) = [2] + x₂ + x₂²
POL(LASTBIT(x₁)) = [2] + [2]x₁
POL(c11(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c2(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(cons(x₁, x₂)) = 0
POL(false) = 0
POL(half(x₁)) = x₁
POL(lastbit(x₁)) = 0
POL(s(x₁)) = [2] + x₁

(52) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Defined Rule Symbols:

half, lastbit, zero, conv, conviter, if

Defined Pair Symbols:

HALF, LASTBIT, CONVITER, IF

Compound Symbols:

c2, c5, c9, c11

(53) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(0) Obligation:

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

(2) Obligation:

(3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

(4) Obligation:

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

(6) Obligation:

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

(8) Obligation:

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

(10) Obligation:

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

(12) Obligation:

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

(14) Obligation:

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

(16) Obligation:

(17) CdtKnowledgeProof (EQUIVALENT transformation)

(18) Obligation:

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

(20) Obligation:

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

(22) Obligation:

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

(24) Obligation:

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

(26) Obligation:

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

(28) Obligation:

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

(30) Obligation:

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

(32) Obligation:

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

(34) Obligation:

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

(36) Obligation:

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

(38) Obligation:

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

(40) Obligation:

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

(42) Obligation:

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

(44) Obligation:

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

(46) Obligation:

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

(48) Obligation:

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

(50) Obligation:

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

(52) Obligation:

(53) SIsEmptyProof (EQUIVALENT transformation)

(54) BOUNDS(O(1), O(1))