g(f(

g(h(

g(

R

↳Dependency Pair Analysis

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

G(h(x,y),z) -> G(x, f(y,z))

G(x, h(y,z)) -> G(x,y)

Furthermore,

R

↳DPs

→DP Problem 1

↳Argument Filtering and Ordering

**G( x, h(y, z)) -> G(x, y)**

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

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

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

The following dependency pairs can be strictly oriented:

G(x, h(y,z)) -> G(x,y)

G(h(x,y),z) -> G(x, f(y,z))

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

Used ordering: Polynomial ordering with Polynomial interpretation:

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

resulting in one new DP problem.

Used Argument Filtering System:

G(x,_{1}x) -> G(_{2}x,_{1}x)_{2}

h(x,_{1}x) -> h(_{2}x,_{1}x)_{2}

f(x,_{1}x) -> f(_{2}x,_{1}x)_{2}

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳Argument Filtering and Ordering

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

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

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

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

The following dependency pair can be strictly oriented:

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

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

Used ordering: Polynomial ordering with Polynomial interpretation:

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

resulting in one new DP problem.

Used Argument Filtering System:

G(x,_{1}x) -> G(_{2}x,_{1}x)_{2}

f(x,_{1}x) -> f(_{2}x,_{1}x)_{2}

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳AFS

...

→DP Problem 3

↳Dependency Graph

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

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

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

Using the Dependency Graph resulted in no new DP problems.

Duration:

0:00 minutes