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 (BOTH BOUNDS(ID, ID), 0 ms)
↳4 CdtProblem
↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳6 CdtProblem
↳7 CdtKnowledgeProof (BOTH BOUNDS(ID, ID), 0 ms)
↳8 CdtProblem
↳9 CdtUsableRulesProof (⇔, 0 ms)
↳10 CdtProblem
↳11 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳12 CdtProblem
↳13 CdtLeafRemovalProof (ComplexityIfPolyImplication, 0 ms)
↳14 CdtProblem
↳15 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳16 CdtProblem
↳17 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳18 CdtProblem
↳19 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 46 ms)
↳20 CdtProblem
↳21 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^2))), 28 ms)
↳22 CdtProblem
↳23 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^2))), 89 ms)
↳24 CdtProblem
↳25 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)
↳26 BOUNDS(O(1), O(1))

(0) Obligation:

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

lt(0, s(x)) → true
lt(x, 0) → false
lt(s(x), s(y)) → lt(x, y)
logarithm(x) → ifa(lt(0, x), x)
ifa(true, x) → help(x, 1)
ifa(false, x) → logZeroError
help(x, y) → ifb(lt(y, x), x, y)
ifb(true, x, y) → help(half(x), s(y))
ifb(false, x, y) → y
half(0) → 0
half(s(0)) → 0
half(s(s(x))) → s(half(x))

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

lt(0, s(z0)) → true
lt(z0, 0) → false
lt(s(z0), s(z1)) → lt(z0, z1)
logarithm(z0) → ifa(lt(0, z0), z0)
ifa(true, z0) → help(z0, 1)
ifa(false, z0) → logZeroError
help(z0, z1) → ifb(lt(z1, z0), z0, z1)
ifb(true, z0, z1) → help(half(z0), s(z1))
ifb(false, z0, z1) → z1
half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
Tuples:

LT(0, s(z0)) → c
LT(z0, 0) → c1
LT(s(z0), s(z1)) → c2(LT(z0, z1))
LOGARITHM(z0) → c3(IFA(lt(0, z0), z0), LT(0, z0))
IFA(true, z0) → c4(HELP(z0, 1))
IFA(false, z0) → c5
HELP(z0, z1) → c6(IFB(lt(z1, z0), z0, z1), LT(z1, z0))
IFB(true, z0, z1) → c7(HELP(half(z0), s(z1)), HALF(z0))
IFB(false, z0, z1) → c8
HALF(0) → c9
HALF(s(0)) → c10
HALF(s(s(z0))) → c11(HALF(z0))
S tuples:

LT(0, s(z0)) → c
LT(z0, 0) → c1
LT(s(z0), s(z1)) → c2(LT(z0, z1))
LOGARITHM(z0) → c3(IFA(lt(0, z0), z0), LT(0, z0))
IFA(true, z0) → c4(HELP(z0, 1))
IFA(false, z0) → c5
HELP(z0, z1) → c6(IFB(lt(z1, z0), z0, z1), LT(z1, z0))
IFB(true, z0, z1) → c7(HELP(half(z0), s(z1)), HALF(z0))
IFB(false, z0, z1) → c8
HALF(0) → c9
HALF(s(0)) → c10
HALF(s(s(z0))) → c11(HALF(z0))
K tuples:none
Defined Rule Symbols:

lt, logarithm, ifa, help, ifb, half

Defined Pair Symbols:

LT, LOGARITHM, IFA, HELP, IFB, HALF

Compound Symbols:

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

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

Removed 6 trailing nodes:

HALF(s(0)) → c10
HALF(0) → c9
LT(0, s(z0)) → c
LT(z0, 0) → c1
IFB(false, z0, z1) → c8
IFA(false, z0) → c5

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

lt(0, s(z0)) → true
lt(z0, 0) → false
lt(s(z0), s(z1)) → lt(z0, z1)
logarithm(z0) → ifa(lt(0, z0), z0)
ifa(true, z0) → help(z0, 1)
ifa(false, z0) → logZeroError
help(z0, z1) → ifb(lt(z1, z0), z0, z1)
ifb(true, z0, z1) → help(half(z0), s(z1))
ifb(false, z0, z1) → z1
half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
Tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1))
LOGARITHM(z0) → c3(IFA(lt(0, z0), z0), LT(0, z0))
IFA(true, z0) → c4(HELP(z0, 1))
HELP(z0, z1) → c6(IFB(lt(z1, z0), z0, z1), LT(z1, z0))
IFB(true, z0, z1) → c7(HELP(half(z0), s(z1)), HALF(z0))
HALF(s(s(z0))) → c11(HALF(z0))
S tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1))
LOGARITHM(z0) → c3(IFA(lt(0, z0), z0), LT(0, z0))
IFA(true, z0) → c4(HELP(z0, 1))
HELP(z0, z1) → c6(IFB(lt(z1, z0), z0, z1), LT(z1, z0))
IFB(true, z0, z1) → c7(HELP(half(z0), s(z1)), HALF(z0))
HALF(s(s(z0))) → c11(HALF(z0))
K tuples:none
Defined Rule Symbols:

lt, logarithm, ifa, help, ifb, half

Defined Pair Symbols:

LT, LOGARITHM, IFA, HELP, IFB, HALF

Compound Symbols:

c2, c3, c4, c6, c7, c11

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

Removed 1 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

lt(0, s(z0)) → true
lt(z0, 0) → false
lt(s(z0), s(z1)) → lt(z0, z1)
logarithm(z0) → ifa(lt(0, z0), z0)
ifa(true, z0) → help(z0, 1)
ifa(false, z0) → logZeroError
help(z0, z1) → ifb(lt(z1, z0), z0, z1)
ifb(true, z0, z1) → help(half(z0), s(z1))
ifb(false, z0, z1) → z1
half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
Tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1))
IFA(true, z0) → c4(HELP(z0, 1))
HELP(z0, z1) → c6(IFB(lt(z1, z0), z0, z1), LT(z1, z0))
IFB(true, z0, z1) → c7(HELP(half(z0), s(z1)), HALF(z0))
HALF(s(s(z0))) → c11(HALF(z0))
LOGARITHM(z0) → c3(IFA(lt(0, z0), z0))
S tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1))
IFA(true, z0) → c4(HELP(z0, 1))
HELP(z0, z1) → c6(IFB(lt(z1, z0), z0, z1), LT(z1, z0))
IFB(true, z0, z1) → c7(HELP(half(z0), s(z1)), HALF(z0))
HALF(s(s(z0))) → c11(HALF(z0))
LOGARITHM(z0) → c3(IFA(lt(0, z0), z0))
K tuples:none
Defined Rule Symbols:

lt, logarithm, ifa, help, ifb, half

Defined Pair Symbols:

LT, IFA, HELP, IFB, HALF, LOGARITHM

Compound Symbols:

c2, c4, c6, c7, c11, c3

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

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

LOGARITHM(z0) → c3(IFA(lt(0, z0), z0))
IFA(true, z0) → c4(HELP(z0, 1))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

lt(0, s(z0)) → true
lt(z0, 0) → false
lt(s(z0), s(z1)) → lt(z0, z1)
logarithm(z0) → ifa(lt(0, z0), z0)
ifa(true, z0) → help(z0, 1)
ifa(false, z0) → logZeroError
help(z0, z1) → ifb(lt(z1, z0), z0, z1)
ifb(true, z0, z1) → help(half(z0), s(z1))
ifb(false, z0, z1) → z1
half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
Tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1))
IFA(true, z0) → c4(HELP(z0, 1))
HELP(z0, z1) → c6(IFB(lt(z1, z0), z0, z1), LT(z1, z0))
IFB(true, z0, z1) → c7(HELP(half(z0), s(z1)), HALF(z0))
HALF(s(s(z0))) → c11(HALF(z0))
LOGARITHM(z0) → c3(IFA(lt(0, z0), z0))
S tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1))
HELP(z0, z1) → c6(IFB(lt(z1, z0), z0, z1), LT(z1, z0))
IFB(true, z0, z1) → c7(HELP(half(z0), s(z1)), HALF(z0))
HALF(s(s(z0))) → c11(HALF(z0))
K tuples:

LOGARITHM(z0) → c3(IFA(lt(0, z0), z0))
IFA(true, z0) → c4(HELP(z0, 1))
Defined Rule Symbols:

lt, logarithm, ifa, help, ifb, half

Defined Pair Symbols:

LT, IFA, HELP, IFB, HALF, LOGARITHM

Compound Symbols:

c2, c4, c6, c7, c11, c3

(9) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

logarithm(z0) → ifa(lt(0, z0), z0)
ifa(true, z0) → help(z0, 1)
ifa(false, z0) → logZeroError
help(z0, z1) → ifb(lt(z1, z0), z0, z1)
ifb(true, z0, z1) → help(half(z0), s(z1))
ifb(false, z0, z1) → z1

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

lt(0, s(z0)) → true
lt(z0, 0) → false
lt(s(z0), s(z1)) → lt(z0, z1)
half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
Tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1))
IFA(true, z0) → c4(HELP(z0, 1))
HELP(z0, z1) → c6(IFB(lt(z1, z0), z0, z1), LT(z1, z0))
IFB(true, z0, z1) → c7(HELP(half(z0), s(z1)), HALF(z0))
HALF(s(s(z0))) → c11(HALF(z0))
LOGARITHM(z0) → c3(IFA(lt(0, z0), z0))
S tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1))
HELP(z0, z1) → c6(IFB(lt(z1, z0), z0, z1), LT(z1, z0))
IFB(true, z0, z1) → c7(HELP(half(z0), s(z1)), HALF(z0))
HALF(s(s(z0))) → c11(HALF(z0))
K tuples:

LOGARITHM(z0) → c3(IFA(lt(0, z0), z0))
IFA(true, z0) → c4(HELP(z0, 1))
Defined Rule Symbols:

lt, half

Defined Pair Symbols:

LT, IFA, HELP, IFB, HALF, LOGARITHM

Compound Symbols:

c2, c4, c6, c7, c11, c3

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

Use narrowing to replace HELP(z0, z1) → c6(IFB(lt(z1, z0), z0, z1), LT(z1, z0)) by

HELP(s(z0), 0) → c6(IFB(true, s(z0), 0), LT(0, s(z0)))
HELP(0, z0) → c6(IFB(false, 0, z0), LT(z0, 0))
HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1)))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

lt(0, s(z0)) → true
lt(z0, 0) → false
lt(s(z0), s(z1)) → lt(z0, z1)
half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
Tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1))
IFA(true, z0) → c4(HELP(z0, 1))
IFB(true, z0, z1) → c7(HELP(half(z0), s(z1)), HALF(z0))
HALF(s(s(z0))) → c11(HALF(z0))
LOGARITHM(z0) → c3(IFA(lt(0, z0), z0))
HELP(s(z0), 0) → c6(IFB(true, s(z0), 0), LT(0, s(z0)))
HELP(0, z0) → c6(IFB(false, 0, z0), LT(z0, 0))
HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1)))
S tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1))
IFB(true, z0, z1) → c7(HELP(half(z0), s(z1)), HALF(z0))
HALF(s(s(z0))) → c11(HALF(z0))
HELP(s(z0), 0) → c6(IFB(true, s(z0), 0), LT(0, s(z0)))
HELP(0, z0) → c6(IFB(false, 0, z0), LT(z0, 0))
HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1)))
K tuples:

LOGARITHM(z0) → c3(IFA(lt(0, z0), z0))
IFA(true, z0) → c4(HELP(z0, 1))
Defined Rule Symbols:

lt, half

Defined Pair Symbols:

LT, IFA, IFB, HALF, LOGARITHM, HELP

Compound Symbols:

c2, c4, c7, c11, c3, c6

(13) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

HELP(s(z0), 0) → c6(IFB(true, s(z0), 0), LT(0, s(z0)))
Removed 3 trailing nodes:

HELP(0, z0) → c6(IFB(false, 0, z0), LT(z0, 0))
IFA(true, z0) → c4(HELP(z0, 1))
LOGARITHM(z0) → c3(IFA(lt(0, z0), z0))

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

lt(0, s(z0)) → true
lt(z0, 0) → false
lt(s(z0), s(z1)) → lt(z0, z1)
half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
Tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1))
IFB(true, z0, z1) → c7(HELP(half(z0), s(z1)), HALF(z0))
HALF(s(s(z0))) → c11(HALF(z0))
HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1)))
S tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1))
IFB(true, z0, z1) → c7(HELP(half(z0), s(z1)), HALF(z0))
HALF(s(s(z0))) → c11(HALF(z0))
HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:

lt, half

Defined Pair Symbols:

LT, IFB, HALF, HELP

Compound Symbols:

c2, c7, c11, c6

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

Use narrowing to replace IFB(true, z0, z1) → c7(HELP(half(z0), s(z1)), HALF(z0)) by

IFB(true, 0, x1) → c7(HELP(0, s(x1)), HALF(0))
IFB(true, s(0), x1) → c7(HELP(0, s(x1)), HALF(s(0)))
IFB(true, s(s(z0)), x1) → c7(HELP(s(half(z0)), s(x1)), HALF(s(s(z0))))

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

lt(0, s(z0)) → true
lt(z0, 0) → false
lt(s(z0), s(z1)) → lt(z0, z1)
half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
Tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1))
HALF(s(s(z0))) → c11(HALF(z0))
HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1)))
IFB(true, 0, x1) → c7(HELP(0, s(x1)), HALF(0))
IFB(true, s(0), x1) → c7(HELP(0, s(x1)), HALF(s(0)))
IFB(true, s(s(z0)), x1) → c7(HELP(s(half(z0)), s(x1)), HALF(s(s(z0))))
S tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1))
HALF(s(s(z0))) → c11(HALF(z0))
HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1)))
IFB(true, 0, x1) → c7(HELP(0, s(x1)), HALF(0))
IFB(true, s(0), x1) → c7(HELP(0, s(x1)), HALF(s(0)))
IFB(true, s(s(z0)), x1) → c7(HELP(s(half(z0)), s(x1)), HALF(s(s(z0))))
K tuples:none
Defined Rule Symbols:

lt, half

Defined Pair Symbols:

LT, HALF, HELP, IFB

Compound Symbols:

c2, c11, c6, c7

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

Removed 2 trailing nodes:

IFB(true, 0, x1) → c7(HELP(0, s(x1)), HALF(0))
IFB(true, s(0), x1) → c7(HELP(0, s(x1)), HALF(s(0)))

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

lt(0, s(z0)) → true
lt(z0, 0) → false
lt(s(z0), s(z1)) → lt(z0, z1)
half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
Tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1))
HALF(s(s(z0))) → c11(HALF(z0))
HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1)))
IFB(true, s(s(z0)), x1) → c7(HELP(s(half(z0)), s(x1)), HALF(s(s(z0))))
S tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1))
HALF(s(s(z0))) → c11(HALF(z0))
HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1)))
IFB(true, s(s(z0)), x1) → c7(HELP(s(half(z0)), s(x1)), HALF(s(s(z0))))
K tuples:none
Defined Rule Symbols:

lt, half

Defined Pair Symbols:

LT, HALF, HELP, IFB

Compound Symbols:

c2, c11, c6, c7

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

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

HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1)))
IFB(true, s(s(z0)), x1) → c7(HELP(s(half(z0)), s(x1)), HALF(s(s(z0))))
We considered the (Usable) Rules:

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

LT(s(z0), s(z1)) → c2(LT(z0, z1))
HALF(s(s(z0))) → c11(HALF(z0))
HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1)))
IFB(true, s(s(z0)), x1) → c7(HELP(s(half(z0)), s(x1)), HALF(s(s(z0))))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(HALF(x1)) = 0   
POL(HELP(x1, x2)) = [1] + [2]x1   
POL(IFB(x1, x2, x3)) = [2]x2   
POL(LT(x1, x2)) = 0   
POL(c11(x1)) = x1   
POL(c2(x1)) = x1   
POL(c6(x1, x2)) = x1 + x2   
POL(c7(x1, x2)) = x1 + x2   
POL(false) = 0   
POL(half(x1)) = x1   
POL(lt(x1, x2)) = 0   
POL(s(x1)) = [1] + x1   
POL(true) = 0   

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

lt(0, s(z0)) → true
lt(z0, 0) → false
lt(s(z0), s(z1)) → lt(z0, z1)
half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
Tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1))
HALF(s(s(z0))) → c11(HALF(z0))
HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1)))
IFB(true, s(s(z0)), x1) → c7(HELP(s(half(z0)), s(x1)), HALF(s(s(z0))))
S tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1))
HALF(s(s(z0))) → c11(HALF(z0))
K tuples:

HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1)))
IFB(true, s(s(z0)), x1) → c7(HELP(s(half(z0)), s(x1)), HALF(s(s(z0))))
Defined Rule Symbols:

lt, half

Defined Pair Symbols:

LT, HALF, HELP, IFB

Compound Symbols:

c2, c11, c6, c7

(21) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

HALF(s(s(z0))) → c11(HALF(z0))
We considered the (Usable) Rules:

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

LT(s(z0), s(z1)) → c2(LT(z0, z1))
HALF(s(s(z0))) → c11(HALF(z0))
HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1)))
IFB(true, s(s(z0)), x1) → c7(HELP(s(half(z0)), s(x1)), HALF(s(s(z0))))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(HALF(x1)) = [3]x1   
POL(HELP(x1, x2)) = [2] + x1 + x12   
POL(IFB(x1, x2, x3)) = x2 + x22   
POL(LT(x1, x2)) = 0   
POL(c11(x1)) = x1   
POL(c2(x1)) = x1   
POL(c6(x1, x2)) = x1 + x2   
POL(c7(x1, x2)) = x1 + x2   
POL(false) = 0   
POL(half(x1)) = x1   
POL(lt(x1, x2)) = 0   
POL(s(x1)) = [2] + x1   
POL(true) = 0   

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

lt(0, s(z0)) → true
lt(z0, 0) → false
lt(s(z0), s(z1)) → lt(z0, z1)
half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
Tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1))
HALF(s(s(z0))) → c11(HALF(z0))
HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1)))
IFB(true, s(s(z0)), x1) → c7(HELP(s(half(z0)), s(x1)), HALF(s(s(z0))))
S tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1))
K tuples:

HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1)))
IFB(true, s(s(z0)), x1) → c7(HELP(s(half(z0)), s(x1)), HALF(s(s(z0))))
HALF(s(s(z0))) → c11(HALF(z0))
Defined Rule Symbols:

lt, half

Defined Pair Symbols:

LT, HALF, HELP, IFB

Compound Symbols:

c2, c11, c6, c7

(23) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

LT(s(z0), s(z1)) → c2(LT(z0, z1))
We considered the (Usable) Rules:

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

LT(s(z0), s(z1)) → c2(LT(z0, z1))
HALF(s(s(z0))) → c11(HALF(z0))
HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1)))
IFB(true, s(s(z0)), x1) → c7(HELP(s(half(z0)), s(x1)), HALF(s(s(z0))))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(HALF(x1)) = [2]x1   
POL(HELP(x1, x2)) = [2]x1 + [2]x12   
POL(IFB(x1, x2, x3)) = [2]x22   
POL(LT(x1, x2)) = [2]x2   
POL(c11(x1)) = x1   
POL(c2(x1)) = x1   
POL(c6(x1, x2)) = x1 + x2   
POL(c7(x1, x2)) = x1 + x2   
POL(false) = 0   
POL(half(x1)) = [1] + x1   
POL(lt(x1, x2)) = 0   
POL(s(x1)) = [2] + x1   
POL(true) = 0   

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

lt(0, s(z0)) → true
lt(z0, 0) → false
lt(s(z0), s(z1)) → lt(z0, z1)
half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
Tuples:

LT(s(z0), s(z1)) → c2(LT(z0, z1))
HALF(s(s(z0))) → c11(HALF(z0))
HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1)))
IFB(true, s(s(z0)), x1) → c7(HELP(s(half(z0)), s(x1)), HALF(s(s(z0))))
S tuples:none
K tuples:

HELP(s(z1), s(z0)) → c6(IFB(lt(z0, z1), s(z1), s(z0)), LT(s(z0), s(z1)))
IFB(true, s(s(z0)), x1) → c7(HELP(s(half(z0)), s(x1)), HALF(s(s(z0))))
HALF(s(s(z0))) → c11(HALF(z0))
LT(s(z0), s(z1)) → c2(LT(z0, z1))
Defined Rule Symbols:

lt, half

Defined Pair Symbols:

LT, HALF, HELP, IFB

Compound Symbols:

c2, c11, c6, c7

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

The set S is empty

(26) BOUNDS(O(1), O(1))