half(0) -> 0

half(s(0)) -> 0

half(s(s(

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

log(s(

R

↳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

↳Argument Filtering and 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))))

innermost

The following dependency pair can be strictly oriented:

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

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

Used ordering: Homeomorphic Embedding Order with EMB

resulting in one new DP problem.

Used Argument Filtering System:

HALF(x) -> HALF(_{1}x)_{1}

s(x) -> s(_{1}x)_{1}

R

↳DPs

→DP Problem 1

↳AFS

→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))))

innermost

Using the Dependency Graph resulted in no new DP problems.

R

↳DPs

→DP Problem 1

↳AFS

→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))))

innermost

Duration:

0:00 minutes