f(

g(0) -> s(0)

g(s(

sel(0, cons(

sel(s(

R

↳Dependency Pair Analysis

F(X) -> F(g(X))

F(X) -> G(X)

G(s(X)) -> G(X)

SEL(s(X), cons(Y,Z)) -> SEL(X,Z)

Furthermore,

R

↳DPs

→DP Problem 1

↳Argument Filtering and Ordering

→DP Problem 2

↳AFS

→DP Problem 3

↳Remaining

**G(s( X)) -> G(X)**

f(X) -> cons(X, f(g(X)))

g(0) -> s(0)

g(s(X)) -> s(s(g(X)))

sel(0, cons(X,Y)) ->X

sel(s(X), cons(Y,Z)) -> sel(X,Z)

innermost

The following dependency pair can be strictly oriented:

G(s(X)) -> G(X)

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}

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

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 4

↳Dependency Graph

→DP Problem 2

↳AFS

→DP Problem 3

↳Remaining

f(X) -> cons(X, f(g(X)))

g(0) -> s(0)

g(s(X)) -> s(s(g(X)))

sel(0, cons(X,Y)) ->X

sel(s(X), cons(Y,Z)) -> sel(X,Z)

innermost

Using the Dependency Graph resulted in no new DP problems.

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳Argument Filtering and Ordering

→DP Problem 3

↳Remaining

**SEL(s( X), cons(Y, Z)) -> SEL(X, Z)**

f(X) -> cons(X, f(g(X)))

g(0) -> s(0)

g(s(X)) -> s(s(g(X)))

sel(0, cons(X,Y)) ->X

sel(s(X), cons(Y,Z)) -> sel(X,Z)

innermost

The following dependency pair can be strictly oriented:

SEL(s(X), cons(Y,Z)) -> SEL(X,Z)

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

Used ordering: Polynomial ordering with Polynomial interpretation:

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

resulting in one new DP problem.

Used Argument Filtering System:

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

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

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

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳AFS

→DP Problem 5

↳Dependency Graph

→DP Problem 3

↳Remaining

f(X) -> cons(X, f(g(X)))

g(0) -> s(0)

g(s(X)) -> s(s(g(X)))

sel(0, cons(X,Y)) ->X

sel(s(X), cons(Y,Z)) -> sel(X,Z)

innermost

Using the Dependency Graph resulted in no new DP problems.

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳AFS

→DP Problem 3

↳Remaining Obligation(s)

The following remains to be proven:

**F( X) -> F(g(X))**

f(X) -> cons(X, f(g(X)))

g(0) -> s(0)

g(s(X)) -> s(s(g(X)))

sel(0, cons(X,Y)) ->X

sel(s(X), cons(Y,Z)) -> sel(X,Z)

innermost

Duration:

0:00 minutes