f(c(s(

f(c(s(

g(c(

g(c(s(

R

↳Dependency Pair Analysis

F(c(s(x),y)) -> F(c(x, s(y)))

F(c(s(x), s(y))) -> G(c(x,y))

G(c(x, s(y))) -> G(c(s(x),y))

G(c(s(x), s(y))) -> F(c(x,y))

Furthermore,

R

↳DPs

→DP Problem 1

↳Argument Filtering and Ordering

**G(c(s( x), s(y))) -> F(c(x, y))**

f(c(s(x),y)) -> f(c(x, s(y)))

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

g(c(x, s(y))) -> g(c(s(x),y))

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

innermost

The following dependency pairs can be strictly oriented:

G(c(s(x), s(y))) -> F(c(x,y))

F(c(s(x), s(y))) -> G(c(x,y))

There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

_{ }^{ }POL(c(x)_{1}, x_{2})= x _{1}+ x_{2}_{ }^{ }_{ }^{ }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.

Used Argument Filtering System:

F(x) -> F(_{1}x)_{1}

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

s(x) -> s(_{1}x)_{1}

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

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳Dependency Graph

**G(c( x, s(y))) -> G(c(s(x), y))**

f(c(s(x),y)) -> f(c(x, s(y)))

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

g(c(x, s(y))) -> g(c(s(x),y))

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

innermost

Using the Dependency Graph the DP problem was split into 2 DP problems.

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳DGraph

...

→DP Problem 3

↳Argument Filtering and Ordering

**G(c( x, s(y))) -> G(c(s(x), y))**

f(c(s(x),y)) -> f(c(x, s(y)))

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

g(c(x, s(y))) -> g(c(s(x),y))

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

innermost

The following dependency pair can be strictly oriented:

G(c(x, s(y))) -> G(c(s(x),y))

There are no usable rules for innermost 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}_{ }^{ }

resulting in one new DP problem.

Used Argument Filtering System:

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

c(x,_{1}x) ->_{2}x_{2}

s(x) -> s(_{1}x)_{1}

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳DGraph

...

→DP Problem 5

↳Dependency Graph

f(c(s(x),y)) -> f(c(x, s(y)))

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

g(c(x, s(y))) -> g(c(s(x),y))

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

innermost

Using the Dependency Graph resulted in no new DP problems.

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳DGraph

...

→DP Problem 4

↳Argument Filtering and Ordering

**F(c(s( x), y)) -> F(c(x, s(y)))**

f(c(s(x),y)) -> f(c(x, s(y)))

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

g(c(x, s(y))) -> g(c(s(x),y))

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

innermost

The following dependency pair can be strictly oriented:

F(c(s(x),y)) -> F(c(x, s(y)))

There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

_{ }^{ }POL(s(x)_{1})= 1 + x _{1}_{ }^{ }_{ }^{ }POL(F(x)_{1})= x _{1}_{ }^{ }

resulting in one new DP problem.

Used Argument Filtering System:

F(x) -> F(_{1}x)_{1}

c(x,_{1}x) ->_{2}x_{1}

s(x) -> s(_{1}x)_{1}

Duration:

0:00 minutes