g(f(

h(

R

↳Dependency Pair Analysis

G(f(x),y) -> H(x,y)

H(x,y) -> G(x, f(y))

Furthermore,

R

↳DPs

→DP Problem 1

↳Instantiation Transformation

**H( x, y) -> G(x, f(y))**

g(f(x),y) -> f(h(x,y))

h(x,y) -> g(x, f(y))

innermost

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

As a result of transforming the rule

one new Dependency Pair is created:

G(f(x),y) -> H(x,y)

G(f(x''), f(y'')) -> H(x'', f(y''))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Inst

→DP Problem 2

↳Instantiation Transformation

**G(f( x''), f(y'')) -> H(x'', f(y''))**

g(f(x),y) -> f(h(x,y))

h(x,y) -> g(x, f(y))

innermost

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

As a result of transforming the rule

one new Dependency Pair is created:

H(x,y) -> G(x, f(y))

H(x', f(y'''')) -> G(x', f(f(y'''')))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Inst

→DP Problem 2

↳Inst

...

→DP Problem 3

↳Instantiation Transformation

**H( x', f(y'''')) -> G(x', f(f(y'''')))**

g(f(x),y) -> f(h(x,y))

h(x,y) -> g(x, f(y))

innermost

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

As a result of transforming the rule

one new Dependency Pair is created:

G(f(x''), f(y'')) -> H(x'', f(y''))

G(f(x''''), f(f(y''''''))) -> H(x'''', f(f(y'''''')))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Inst

→DP Problem 2

↳Inst

...

→DP Problem 4

↳Instantiation Transformation

**G(f( x''''), f(f(y''''''))) -> H(x'''', f(f(y'''''')))**

g(f(x),y) -> f(h(x,y))

h(x,y) -> g(x, f(y))

innermost

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

As a result of transforming the rule

one new Dependency Pair is created:

H(x', f(y'''')) -> G(x', f(f(y'''')))

H(x'', f(f(y''''''''))) -> G(x'', f(f(f(y''''''''))))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Inst

→DP Problem 2

↳Inst

...

→DP Problem 5

↳Polynomial Ordering

**H( x'', f(f(y''''''''))) -> G(x'', f(f(f(y''''''''))))**

g(f(x),y) -> f(h(x,y))

h(x,y) -> g(x, f(y))

innermost

The following dependency pair can be strictly oriented:

G(f(x''''), f(f(y''''''))) -> H(x'''', f(f(y'''''')))

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_{2})= x _{1}_{ }^{ }_{ }^{ }POL(H(x)_{1}, x_{2})= x _{1}_{ }^{ }_{ }^{ }POL(f(x)_{1})= 1 + x _{1}_{ }^{ }

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳Inst

→DP Problem 2

↳Inst

...

→DP Problem 6

↳Dependency Graph

**H( x'', f(f(y''''''''))) -> G(x'', f(f(f(y''''''''))))**

g(f(x),y) -> f(h(x,y))

h(x,y) -> g(x, f(y))

innermost

Using the Dependency Graph resulted in no new DP problems.

Duration:

0:00 minutes