f(c(s(

f(c(s(

g(c(

g(c(s(

R

↳Dependency Pair Analysis

F(c(s(x),y)) -> F(c(x, s(y)))

F(c(s(x), s(y))) -> G(c(x,y))

G(c(x, s(y))) -> G(c(s(x),y))

G(c(s(x), s(y))) -> F(c(x,y))

Furthermore,

R

↳DPs

→DP Problem 1

↳Polynomial Ordering

**G(c(s( x), s(y))) -> F(c(x, y))**

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

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

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

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

The following dependency pairs can be strictly oriented:

G(c(s(x), s(y))) -> F(c(x,y))

F(c(s(x), s(y))) -> G(c(x,y))

Additionally, the following rules can be oriented:

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

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

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

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

Used ordering: Polynomial ordering with Polynomial interpretation:

_{ }^{ }POL(c(x)_{1}, x_{2})= x _{1}+ x_{2}_{ }^{ }_{ }^{ }POL(g(x)_{1})= 0 _{ }^{ }_{ }^{ }POL(G(x)_{1})= x _{1}_{ }^{ }_{ }^{ }POL(s(x)_{1})= 1 + x _{1}_{ }^{ }_{ }^{ }POL(f(x)_{1})= 0 _{ }^{ }_{ }^{ }POL(F(x)_{1})= x _{1}_{ }^{ }

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳Dependency Graph

**G(c( x, s(y))) -> G(c(s(x), y))**

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

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

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

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

Using the Dependency Graph the DP problem was split into 2 DP problems.

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳DGraph

...

→DP Problem 3

↳Polynomial Ordering

**G(c( x, s(y))) -> G(c(s(x), y))**

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

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

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

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

The following dependency pair can be strictly oriented:

G(c(x, s(y))) -> G(c(s(x),y))

Additionally, the following rules can be oriented:

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

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

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

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

Used ordering: Polynomial ordering with Polynomial interpretation:

_{ }^{ }POL(c(x)_{1}, x_{2})= x _{2}_{ }^{ }_{ }^{ }POL(g(x)_{1})= 0 _{ }^{ }_{ }^{ }POL(G(x)_{1})= x _{1}_{ }^{ }_{ }^{ }POL(s(x)_{1})= 1 + x _{1}_{ }^{ }_{ }^{ }POL(f(x)_{1})= 0 _{ }^{ }

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳DGraph

...

→DP Problem 5

↳Dependency Graph

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

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

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

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

Using the Dependency Graph resulted in no new DP problems.

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳DGraph

...

→DP Problem 4

↳Polynomial Ordering

**F(c(s( x), y)) -> F(c(x, s(y)))**

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

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

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

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

The following dependency pair can be strictly oriented:

F(c(s(x),y)) -> F(c(x, s(y)))

Additionally, the following rules can be oriented:

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

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

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

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

Used ordering: Polynomial ordering with Polynomial interpretation:

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

resulting in one new DP problem.

Duration:

0:00 minutes