f(a) -> f(c(a))

f(c(

f(c(a)) -> f(d(b))

f(a) -> f(d(a))

f(d(

f(c(b)) -> f(d(a))

e(g(

R

↳Dependency Pair Analysis

F(a) -> F(c(a))

F(c(a)) -> F(d(b))

F(a) -> F(d(a))

F(c(b)) -> F(d(a))

E(g(X)) -> E(X)

Furthermore,

R

↳DPs

→DP Problem 1

↳Argument Filtering and Ordering

**E(g( X)) -> E(X)**

f(a) -> f(c(a))

f(c(X)) ->X

f(c(a)) -> f(d(b))

f(a) -> f(d(a))

f(d(X)) ->X

f(c(b)) -> f(d(a))

e(g(X)) -> e(X)

The following dependency pair can be strictly oriented:

E(g(X)) -> E(X)

The following rules can be oriented:

f(a) -> f(c(a))

f(c(X)) ->X

f(c(a)) -> f(d(b))

f(a) -> f(d(a))

f(d(X)) ->X

f(c(b)) -> f(d(a))

e(g(X)) -> e(X)

Used ordering: Polynomial ordering with Polynomial interpretation:

_{ }^{ }POL(c(x)_{1})= x _{1}_{ }^{ }_{ }^{ }POL(E(x)_{1})= x _{1}_{ }^{ }_{ }^{ }POL(g(x)_{1})= 1 + x _{1}_{ }^{ }_{ }^{ }POL(e(x)_{1})= x _{1}_{ }^{ }_{ }^{ }POL(b)= 0 _{ }^{ }_{ }^{ }POL(d(x)_{1})= x _{1}_{ }^{ }_{ }^{ }POL(a)= 0 _{ }^{ }_{ }^{ }POL(f(x)_{1})= x _{1}_{ }^{ }

resulting in one new DP problem.

Used Argument Filtering System:

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

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

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

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

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

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

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳Dependency Graph

f(a) -> f(c(a))

f(c(X)) ->X

f(c(a)) -> f(d(b))

f(a) -> f(d(a))

f(d(X)) ->X

f(c(b)) -> f(d(a))

e(g(X)) -> e(X)

Using the Dependency Graph resulted in no new DP problems.

Duration:

0:00 minutes