+(

+(

+(0,

+(s(

+(

f(g(f(

f(g(h(

f(h(

R

↳Dependency Pair Analysis

+'(x, s(y)) -> +'(x,y)

+'(s(x),y) -> +'(x,y)

+'(x, +(y,z)) -> +'(+(x,y),z)

+'(x, +(y,z)) -> +'(x,y)

F(g(f(x))) -> F(h(s(0),x))

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

F(h(x, h(y,z))) -> F(h(+(x,y),z))

F(h(x, h(y,z))) -> +'(x,y)

Furthermore,

R

↳DPs

→DP Problem 1

↳Polynomial Ordering

→DP Problem 2

↳Polo

**+'( x, +(y, z)) -> +'(x, y)**

+(x, 0) ->x

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

+(0,y) ->y

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

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

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

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

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

innermost

The following dependency pairs can be strictly oriented:

+'(x, +(y,z)) -> +'(x,y)

+'(x, +(y,z)) -> +'(+(x,y),z)

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(s(x)_{1})= x _{1}_{ }^{ }_{ }^{ }POL(+(x)_{1}, x_{2})= 1 + x _{1}+ x_{2}_{ }^{ }_{ }^{ }POL(+'(x)_{1}, x_{2})= x _{2}_{ }^{ }

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 3

↳Polynomial Ordering

→DP Problem 2

↳Polo

**+'(s( x), y) -> +'(x, y)**

+(x, 0) ->x

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

+(0,y) ->y

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

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

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

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

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

innermost

The following dependency pair can be strictly oriented:

+'(s(x),y) -> +'(x,y)

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(s(x)_{1})= 1 + x _{1}_{ }^{ }_{ }^{ }POL(+'(x)_{1}, x_{2})= x _{1}_{ }^{ }

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 3

↳Polo

...

→DP Problem 4

↳Polynomial Ordering

→DP Problem 2

↳Polo

**+'( x, s(y)) -> +'(x, y)**

+(x, 0) ->x

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

+(0,y) ->y

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

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

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

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

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

innermost

The following dependency pair can be strictly oriented:

+'(x, s(y)) -> +'(x,y)

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(s(x)_{1})= 1 + x _{1}_{ }^{ }_{ }^{ }POL(+'(x)_{1}, x_{2})= x _{2}_{ }^{ }

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 3

↳Polo

...

→DP Problem 5

↳Dependency Graph

→DP Problem 2

↳Polo

+(x, 0) ->x

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

+(0,y) ->y

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

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

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

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

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

innermost

Using the Dependency Graph resulted in no new DP problems.

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳Polynomial Ordering

**F(h( x, h(y, z))) -> F(h(+(x, y), z))**

+(x, 0) ->x

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

+(0,y) ->y

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

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

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

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

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

innermost

The following dependency pair can be strictly oriented:

F(h(x, h(y,z))) -> F(h(+(x,y),z))

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

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳Polo

→DP Problem 6

↳Dependency Graph

+(x, 0) ->x

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

+(0,y) ->y

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

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

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

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

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

innermost

Using the Dependency Graph resulted in no new DP problems.

Duration:

0:00 minutes