s(a) -> a

s(s(

s(f(

s(g(

f(

f(a,

f(g(

g(a, a) -> a

R

↳Dependency Pair Analysis

S(f(x,y)) -> F(s(y), s(x))

S(f(x,y)) -> S(y)

S(f(x,y)) -> S(x)

S(g(x,y)) -> G(s(x), s(y))

S(g(x,y)) -> S(x)

S(g(x,y)) -> S(y)

F(g(x,y), g(u,v)) -> G(f(x,u), f(y,v))

F(g(x,y), g(u,v)) -> F(x,u)

F(g(x,y), g(u,v)) -> F(y,v)

Furthermore,

R

↳DPs

→DP Problem 1

↳Argument Filtering and Ordering

→DP Problem 2

↳AFS

**F(g( x, y), g(u, v)) -> F(y, v)**

s(a) -> a

s(s(x)) ->x

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

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

f(x, a) ->x

f(a,y) ->y

f(g(x,y), g(u,v)) -> g(f(x,u), f(y,v))

g(a, a) -> a

innermost

The following dependency pairs can be strictly oriented:

F(g(x,y), g(u,v)) -> F(y,v)

F(g(x,y), g(u,v)) -> F(x,u)

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

Used ordering: Polynomial ordering with Polynomial interpretation:

_{ }^{ }POL(g(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:

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

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

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 3

↳Dependency Graph

→DP Problem 2

↳AFS

s(a) -> a

s(s(x)) ->x

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

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

f(x, a) ->x

f(a,y) ->y

f(g(x,y), g(u,v)) -> g(f(x,u), f(y,v))

g(a, a) -> a

innermost

Using the Dependency Graph resulted in no new DP problems.

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳Argument Filtering and Ordering

**S(g( x, y)) -> S(y)**

s(a) -> a

s(s(x)) ->x

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

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

f(x, a) ->x

f(a,y) ->y

f(g(x,y), g(u,v)) -> g(f(x,u), f(y,v))

g(a, a) -> a

innermost

The following dependency pairs can be strictly oriented:

S(f(x,y)) -> S(x)

S(f(x,y)) -> S(y)

There are no usable rules for innermost 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(S(x)_{1})= x _{1}_{ }^{ }_{ }^{ }POL(f(x)_{1}, x_{2})= 1 + x _{1}+ x_{2}_{ }^{ }

resulting in one new DP problem.

Used Argument Filtering System:

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

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

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

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳AFS

→DP Problem 4

↳Argument Filtering and Ordering

**S(g( x, y)) -> S(y)**

s(a) -> a

s(s(x)) ->x

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

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

f(x, a) ->x

f(a,y) ->y

f(g(x,y), g(u,v)) -> g(f(x,u), f(y,v))

g(a, a) -> a

innermost

The following dependency pairs can be strictly oriented:

S(g(x,y)) -> S(y)

S(g(x,y)) -> S(x)

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

Used ordering: Polynomial ordering with Polynomial interpretation:

_{ }^{ }POL(g(x)_{1}, x_{2})= 1 + x _{1}+ x_{2}_{ }^{ }_{ }^{ }POL(S(x)_{1})= x _{1}_{ }^{ }

resulting in one new DP problem.

Used Argument Filtering System:

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

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

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳AFS

→DP Problem 4

↳AFS

...

→DP Problem 5

↳Dependency Graph

s(a) -> a

s(s(x)) ->x

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

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

f(x, a) ->x

f(a,y) ->y

f(g(x,y), g(u,v)) -> g(f(x,u), f(y,v))

g(a, a) -> a

innermost

Using the Dependency Graph resulted in no new DP problems.

Duration:

0:00 minutes