g(0, f(

g(

g(s(

g(f(

R

↳Dependency Pair Analysis

G(x, s(y)) -> G(f(x,y), 0)

G(s(x),y) -> G(f(x,y), 0)

G(f(x,y), 0) -> G(x, 0)

G(f(x,y), 0) -> G(y, 0)

Furthermore,

R

↳DPs

→DP Problem 1

↳Polynomial Ordering

**G(f( x, y), 0) -> G(y, 0)**

g(0, f(x,x)) ->x

g(x, s(y)) -> g(f(x,y), 0)

g(s(x),y) -> g(f(x,y), 0)

g(f(x,y), 0) -> f(g(x, 0), g(y, 0))

innermost

The following dependency pair can be strictly oriented:

G(s(x),y) -> G(f(x,y), 0)

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

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳Polynomial Ordering

**G(f( x, y), 0) -> G(y, 0)**

g(0, f(x,x)) ->x

g(x, s(y)) -> g(f(x,y), 0)

g(s(x),y) -> g(f(x,y), 0)

g(f(x,y), 0) -> f(g(x, 0), g(y, 0))

innermost

The following dependency pairs can be strictly oriented:

G(f(x,y), 0) -> G(y, 0)

G(f(x,y), 0) -> G(x, 0)

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(0)= 0 _{ }^{ }_{ }^{ }POL(G(x)_{1}, x_{2})= x _{1}_{ }^{ }_{ }^{ }POL(f(x)_{1}, x_{2})= 1 + x _{1}+ x_{2}_{ }^{ }

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳Polo

...

→DP Problem 3

↳Dependency Graph

g(0, f(x,x)) ->x

g(x, s(y)) -> g(f(x,y), 0)

g(s(x),y) -> g(f(x,y), 0)

g(f(x,y), 0) -> f(g(x, 0), g(y, 0))

innermost

Using the Dependency Graph resulted in no new DP problems.

Duration:

0:00 minutes