from(

from(

after(0,

after(s(

activate(n

activate(

R

↳Dependency Pair Analysis

AFTER(s(N), cons(X,XS)) -> AFTER(N, activate(XS))

AFTER(s(N), cons(X,XS)) -> ACTIVATE(XS)

ACTIVATE(n_{from}(X)) -> FROM(X)

Furthermore,

R

↳DPs

→DP Problem 1

↳Narrowing Transformation

**AFTER(s( N), cons(X, XS)) -> AFTER(N, activate(XS))**

from(X) -> cons(X, n_{from}(s(X)))

from(X) -> n_{from}(X)

after(0,XS) ->XS

after(s(N), cons(X,XS)) -> after(N, activate(XS))

activate(n_{from}(X)) -> from(X)

activate(X) ->X

innermost

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

As a result of transforming the rule

two new Dependency Pairs are created:

AFTER(s(N), cons(X,XS)) -> AFTER(N, activate(XS))

AFTER(s(N), cons(X, n_{from}(X''))) -> AFTER(N, from(X''))

AFTER(s(N), cons(X,XS')) -> AFTER(N,XS')

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Nar

→DP Problem 2

↳Narrowing Transformation

**AFTER(s( N), cons(X, XS')) -> AFTER(N, XS')**

from(X) -> cons(X, n_{from}(s(X)))

from(X) -> n_{from}(X)

after(0,XS) ->XS

after(s(N), cons(X,XS)) -> after(N, activate(XS))

activate(n_{from}(X)) -> from(X)

activate(X) ->X

innermost

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

As a result of transforming the rule

two new Dependency Pairs are created:

AFTER(s(N), cons(X, n_{from}(X''))) -> AFTER(N, from(X''))

AFTER(s(N), cons(X, n_{from}(X'''))) -> AFTER(N, cons(X''', n_{from}(s(X'''))))

AFTER(s(N), cons(X, n_{from}(X'''))) -> AFTER(N, n_{from}(X'''))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Nar

→DP Problem 2

↳Nar

...

→DP Problem 3

↳Forward Instantiation Transformation

**AFTER(s( N), cons(X, n_{from}(X'''))) -> AFTER(N, cons(X''', n_{from}(s(X'''))))**

from(X) -> cons(X, n_{from}(s(X)))

from(X) -> n_{from}(X)

after(0,XS) ->XS

after(s(N), cons(X,XS)) -> after(N, activate(XS))

activate(n_{from}(X)) -> from(X)

activate(X) ->X

innermost

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

As a result of transforming the rule

two new Dependency Pairs are created:

AFTER(s(N), cons(X,XS')) -> AFTER(N,XS')

AFTER(s(s(N'')), cons(X, cons(X'',XS'''))) -> AFTER(s(N''), cons(X'',XS'''))

AFTER(s(s(N'')), cons(X, cons(X'', n_{from}(X''''')))) -> AFTER(s(N''), cons(X'', n_{from}(X''''')))

The transformation is resulting in two new DP problems:

R

↳DPs

→DP Problem 1

↳Nar

→DP Problem 2

↳Nar

...

→DP Problem 4

↳Polynomial Ordering

**AFTER(s( N), cons(X, n_{from}(X'''))) -> AFTER(N, cons(X''', n_{from}(s(X'''))))**

from(X) -> cons(X, n_{from}(s(X)))

from(X) -> n_{from}(X)

after(0,XS) ->XS

after(s(N), cons(X,XS)) -> after(N, activate(XS))

activate(n_{from}(X)) -> from(X)

activate(X) ->X

innermost

The following dependency pair can be strictly oriented:

AFTER(s(N), cons(X, n_{from}(X'''))) -> AFTER(N, cons(X''', n_{from}(s(X'''))))

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

Used ordering: Polynomial ordering with Polynomial interpretation:

_{ }^{ }POL(n__from(x)_{1})= 0 _{ }^{ }_{ }^{ }POL(AFTER(x)_{1}, x_{2})= x _{1}_{ }^{ }_{ }^{ }POL(cons(x)_{1}, x_{2})= 0 _{ }^{ }_{ }^{ }POL(s(x)_{1})= 1 + x _{1}_{ }^{ }

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳Nar

→DP Problem 2

↳Nar

...

→DP Problem 6

↳Dependency Graph

from(X) -> cons(X, n_{from}(s(X)))

from(X) -> n_{from}(X)

after(0,XS) ->XS

after(s(N), cons(X,XS)) -> after(N, activate(XS))

activate(n_{from}(X)) -> from(X)

activate(X) ->X

innermost

Using the Dependency Graph resulted in no new DP problems.

R

↳DPs

→DP Problem 1

↳Nar

→DP Problem 2

↳Nar

...

→DP Problem 5

↳Polynomial Ordering

**AFTER(s(s( N'')), cons(X, cons(X'', XS'''))) -> AFTER(s(N''), cons(X'', XS'''))**

from(X) -> cons(X, n_{from}(s(X)))

from(X) -> n_{from}(X)

after(0,XS) ->XS

after(s(N), cons(X,XS)) -> after(N, activate(XS))

activate(n_{from}(X)) -> from(X)

activate(X) ->X

innermost

The following dependency pair can be strictly oriented:

AFTER(s(s(N'')), cons(X, cons(X'',XS'''))) -> AFTER(s(N''), cons(X'',XS'''))

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

Used ordering: Polynomial ordering with Polynomial interpretation:

_{ }^{ }POL(AFTER(x)_{1}, x_{2})= x _{1}_{ }^{ }_{ }^{ }POL(cons(x)_{1}, x_{2})= 0 _{ }^{ }_{ }^{ }POL(s(x)_{1})= 1 + x _{1}_{ }^{ }

resulting in one new DP problem.

Duration:

0:00 minutes