g(s(

g(0) -> 0

f(0) -> s(0)

f(s(

R

↳Dependency Pair Analysis

G(s(x)) -> F(x)

F(s(x)) -> G(x)

Furthermore,

R

↳DPs

→DP Problem 1

↳Forward Instantiation Transformation

**F(s( x)) -> G(x)**

g(s(x)) -> f(x)

g(0) -> 0

f(0) -> s(0)

f(s(x)) -> s(s(g(x)))

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:

G(s(x)) -> F(x)

G(s(s(x''))) -> F(s(x''))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 2

↳Forward Instantiation Transformation

**G(s(s( x''))) -> F(s(x''))**

g(s(x)) -> f(x)

g(0) -> 0

f(0) -> s(0)

f(s(x)) -> s(s(g(x)))

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:

F(s(x)) -> G(x)

F(s(s(s(x'''')))) -> G(s(s(x'''')))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 2

↳FwdInst

...

→DP Problem 3

↳Forward Instantiation Transformation

**F(s(s(s( x'''')))) -> G(s(s(x'''')))**

g(s(x)) -> f(x)

g(0) -> 0

f(0) -> s(0)

f(s(x)) -> s(s(g(x)))

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:

G(s(s(x''))) -> F(s(x''))

G(s(s(s(s(x''''''))))) -> F(s(s(s(x''''''))))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 2

↳FwdInst

...

→DP Problem 4

↳Forward Instantiation Transformation

**G(s(s(s(s( x''''''))))) -> F(s(s(s(x''''''))))**

g(s(x)) -> f(x)

g(0) -> 0

f(0) -> s(0)

f(s(x)) -> s(s(g(x)))

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:

F(s(s(s(x'''')))) -> G(s(s(x'''')))

F(s(s(s(s(s(x'''''''')))))) -> G(s(s(s(s(x'''''''')))))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 2

↳FwdInst

...

→DP Problem 5

↳Polynomial Ordering

**F(s(s(s(s(s( x'''''''')))))) -> G(s(s(s(s(x'''''''')))))**

g(s(x)) -> f(x)

g(0) -> 0

f(0) -> s(0)

f(s(x)) -> s(s(g(x)))

innermost

The following dependency pairs can be strictly oriented:

F(s(s(s(s(s(x'''''''')))))) -> G(s(s(s(s(x'''''''')))))

G(s(s(s(s(x''''''))))) -> F(s(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(G(x)_{1})= x _{1}_{ }^{ }_{ }^{ }POL(s(x)_{1})= 1 + x _{1}_{ }^{ }_{ }^{ }POL(F(x)_{1})= x _{1}_{ }^{ }

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 2

↳FwdInst

...

→DP Problem 6

↳Dependency Graph

g(s(x)) -> f(x)

g(0) -> 0

f(0) -> s(0)

f(s(x)) -> s(s(g(x)))

innermost

Using the Dependency Graph resulted in no new DP problems.

Duration:

0:00 minutes