+(0,

+(s(

+(s(

R

↳Dependency Pair Analysis

+'(s(x), s(y)) -> +'(s(x), +(y, 0))

+'(s(x), s(y)) -> +'(y, 0)

Furthermore,

R

↳DPs

→DP Problem 1

↳Narrowing Transformation

**+'(s( x), s(y)) -> +'(s(x), +(y, 0))**

+(0,y) ->y

+(s(x), 0) -> s(x)

+(s(x), s(y)) -> s(+(s(x), +(y, 0)))

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:

+'(s(x), s(y)) -> +'(s(x), +(y, 0))

+'(s(x), s(0)) -> +'(s(x), 0)

+'(s(x), s(s(x''))) -> +'(s(x), s(x''))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Nar

→DP Problem 2

↳Forward Instantiation Transformation

**+'(s( x), s(s(x''))) -> +'(s(x), s(x''))**

+(0,y) ->y

+(s(x), 0) -> s(x)

+(s(x), s(y)) -> s(+(s(x), +(y, 0)))

innermost

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

As a result of transforming the rule

one new Dependency Pair is created:

+'(s(x), s(s(x''))) -> +'(s(x), s(x''))

+'(s(x0), s(s(s(x'''')))) -> +'(s(x0), s(s(x'''')))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Nar

→DP Problem 2

↳FwdInst

...

→DP Problem 3

↳Polynomial Ordering

**+'(s( x0), s(s(s(x'''')))) -> +'(s(x0), s(s(x'''')))**

+(0,y) ->y

+(s(x), 0) -> s(x)

+(s(x), s(y)) -> s(+(s(x), +(y, 0)))

innermost

The following dependency pair can be strictly oriented:

+'(s(x0), s(s(s(x'''')))) -> +'(s(x0), s(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(s(x)_{1})= 1 + x _{1}_{ }^{ }_{ }^{ }POL(+'(x)_{1}, x_{2})= 1 + x _{2}_{ }^{ }

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳Nar

→DP Problem 2

↳FwdInst

...

→DP Problem 4

↳Dependency Graph

+(0,y) ->y

+(s(x), 0) -> s(x)

+(s(x), s(y)) -> s(+(s(x), +(y, 0)))

innermost

Using the Dependency Graph resulted in no new DP problems.

Duration:

0:00 minutes