half(0) -> 0

half(s(s(

log(s(0)) -> 0

log(s(s(

R

↳Dependency Pair Analysis

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

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

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

Furthermore,

R

↳DPs

→DP Problem 1

↳Argument Filtering and Ordering

→DP Problem 2

↳Nar

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

half(0) -> 0

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

log(s(0)) -> 0

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

The following dependency pair can be strictly oriented:

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

There are no usable rules using the Ce-refinement 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

↳Nar

half(0) -> 0

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

log(s(0)) -> 0

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

Using the Dependency Graph resulted in no new DP problems.

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳Narrowing Transformation

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

half(0) -> 0

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

log(s(0)) -> 0

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

On this DP problem, a Narrowing SCC transformation can be performed.

As a result of transforming the rule

two new Dependency Pairs are created:

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

LOG(s(s(0))) -> LOG(s(0))

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

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳Nar

→DP Problem 4

↳Narrowing Transformation

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

half(0) -> 0

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

log(s(0)) -> 0

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

On this DP problem, a Narrowing SCC transformation can be performed.

As a result of transforming the rule

two new Dependency Pairs are created:

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

LOG(s(s(s(s(0))))) -> LOG(s(s(0)))

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

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳Nar

→DP Problem 4

↳Nar

...

→DP Problem 5

↳Argument Filtering and Ordering

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

half(0) -> 0

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

log(s(0)) -> 0

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

The following dependency pair can be strictly oriented:

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

The following usable rules using the Ce-refinement can be oriented:

half(0) -> 0

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

Used ordering: Homeomorphic Embedding Order with EMB

resulting in one new DP problem.

Used Argument Filtering System:

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

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

half(x) ->_{1}x_{1}

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳Nar

→DP Problem 4

↳Nar

...

→DP Problem 6

↳Dependency Graph

half(0) -> 0

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

log(s(0)) -> 0

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

Using the Dependency Graph resulted in no new DP problems.

Duration:

0:00 minutes