half(0) -> 0

half(s(0)) -> 0

half(s(s(

s(log(0)) -> s(0)

log(s(

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↳Dependency Pair Analysis

HALF(s(s(x))) -> S(half(x))

HALF(s(s(x))) -> HALF(x)

S(log(0)) -> S(0)

LOG(s(x)) -> S(log(half(s(x))))

LOG(s(x)) -> LOG(half(s(x)))

LOG(s(x)) -> HALF(s(x))

Furthermore,

R

↳DPs

→DP Problem 1

↳Polynomial Ordering

→DP Problem 2

↳Remaining

**HALF(s(s( x))) -> HALF(x)**

half(0) -> 0

half(s(0)) -> 0

half(s(s(x))) -> s(half(x))

s(log(0)) -> s(0)

log(s(x)) -> s(log(half(s(x))))

The following dependency pair can be strictly oriented:

HALF(s(s(x))) -> HALF(x)

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

_{ }^{ }POL(HALF(x)_{1})= 1 + x _{1}_{ }^{ }_{ }^{ }POL(s(x)_{1})= 1 + x _{1}_{ }^{ }

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 3

↳Dependency Graph

→DP Problem 2

↳Remaining

half(0) -> 0

half(s(0)) -> 0

half(s(s(x))) -> s(half(x))

s(log(0)) -> s(0)

log(s(x)) -> s(log(half(s(x))))

Using the Dependency Graph resulted in no new DP problems.

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳Remaining Obligation(s)

The following remains to be proven:

**LOG(s( x)) -> LOG(half(s(x)))**

half(0) -> 0

half(s(0)) -> 0

half(s(s(x))) -> s(half(x))

s(log(0)) -> s(0)

log(s(x)) -> s(log(half(s(x))))

Duration:

0:00 minutes