f(0,

f(

f(i(

f(f(

f(g(

f(1, g(

f(2, g(

R

↳Dependency Pair Analysis

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

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

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

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

Furthermore,

R

↳DPs

→DP Problem 1

↳Argument Filtering and Ordering

**F(g( x, y), z) -> F(y, z)**

f(0,y) ->y

f(x, 0) ->x

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

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

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

f(1, g(x,y)) ->x

f(2, g(x,y)) ->y

innermost

The following dependency pairs can be strictly oriented:

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

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

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

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

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳Argument Filtering and Ordering

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

f(0,y) ->y

f(x, 0) ->x

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

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

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

f(1, g(x,y)) ->x

f(2, g(x,y)) ->y

innermost

The following dependency pair can be strictly oriented:

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

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

Used ordering: Polynomial ordering with Polynomial interpretation:

_{ }^{ }POL(F(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:

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

f(0,y) ->y

f(x, 0) ->x

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

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

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

f(1, g(x,y)) ->x

f(2, g(x,y)) ->y

innermost

Using the Dependency Graph resulted in no new DP problems.

Duration:

0:00 minutes