f(1) -> f(g(1))

f(f(

g(0) -> g(f(0))

g(g(

R

↳Dependency Pair Analysis

F(1) -> F(g(1))

F(1) -> G(1)

G(0) -> G(f(0))

G(0) -> F(0)

Furthermore,

R

↳DPs

→DP Problem 1

↳Argument Filtering and Ordering

→DP Problem 2

↳AFS

**F(1) -> F(g(1))**

f(1) -> f(g(1))

f(f(x)) -> f(x)

g(0) -> g(f(0))

g(g(x)) -> g(x)

innermost

The following dependency pair can be strictly oriented:

F(1) -> F(g(1))

The following usable rules for innermost can be oriented:

g(0) -> g(f(0))

g(g(x)) -> g(x)

f(1) -> f(g(1))

f(f(x)) -> f(x)

Used ordering: Polynomial ordering with Polynomial interpretation:

_{ }^{ }POL(g)= 0 _{ }^{ }_{ }^{ }POL(1)= 1 _{ }^{ }_{ }^{ }POL(F(x)_{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}

g(x) -> g_{1}

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

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 3

↳Dependency Graph

→DP Problem 2

↳AFS

f(1) -> f(g(1))

f(f(x)) -> f(x)

g(0) -> g(f(0))

g(g(x)) -> g(x)

innermost

Using the Dependency Graph resulted in no new DP problems.

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳Argument Filtering and Ordering

**G(0) -> G(f(0))**

f(1) -> f(g(1))

f(f(x)) -> f(x)

g(0) -> g(f(0))

g(g(x)) -> g(x)

innermost

The following dependency pair can be strictly oriented:

G(0) -> G(f(0))

The following usable rules for innermost can be oriented:

f(1) -> f(g(1))

f(f(x)) -> f(x)

g(0) -> g(f(0))

g(g(x)) -> g(x)

Used ordering: Polynomial ordering with Polynomial interpretation:

_{ }^{ }POL(0)= 1 _{ }^{ }_{ }^{ }POL(g(x)_{1})= x _{1}_{ }^{ }_{ }^{ }POL(G(x)_{1})= x _{1}_{ }^{ }_{ }^{ }POL(f)= 0 _{ }^{ }

resulting in one new DP problem.

Used Argument Filtering System:

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

f(x) -> f_{1}

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

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳AFS

→DP Problem 4

↳Dependency Graph

f(1) -> f(g(1))

f(f(x)) -> f(x)

g(0) -> g(f(0))

g(g(x)) -> g(x)

innermost

Using the Dependency Graph resulted in no new DP problems.

Duration:

0:00 minutes