f(f(

f(f(

g(c(

g(d(

g(c(h(0))) -> g(d(1))

g(c(1)) -> g(d(h(0)))

g(h(

R

↳Dependency Pair Analysis

F(f(x)) -> F(c(f(x)))

F(f(x)) -> F(d(f(x)))

G(c(h(0))) -> G(d(1))

G(c(1)) -> G(d(h(0)))

G(h(x)) -> G(x)

Furthermore,

R

↳DPs

→DP Problem 1

↳Forward Instantiation Transformation

**G(h( x)) -> G(x)**

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

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

g(c(x)) ->x

g(d(x)) ->x

g(c(h(0))) -> g(d(1))

g(c(1)) -> g(d(h(0)))

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

innermost

On this DP problem, a Forward Instantiation SCC transformation can be performed.

As a result of transforming the rule

one new Dependency Pair is created:

G(h(x)) -> G(x)

G(h(h(x''))) -> G(h(x''))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 2

↳Forward Instantiation Transformation

**G(h(h( x''))) -> G(h(x''))**

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

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

g(c(x)) ->x

g(d(x)) ->x

g(c(h(0))) -> g(d(1))

g(c(1)) -> g(d(h(0)))

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

innermost

On this DP problem, a Forward Instantiation SCC transformation can be performed.

As a result of transforming the rule

one new Dependency Pair is created:

G(h(h(x''))) -> G(h(x''))

G(h(h(h(x'''')))) -> G(h(h(x'''')))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 2

↳FwdInst

...

→DP Problem 3

↳Polynomial Ordering

**G(h(h(h( x'''')))) -> G(h(h(x'''')))**

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

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

g(c(x)) ->x

g(d(x)) ->x

g(c(h(0))) -> g(d(1))

g(c(1)) -> g(d(h(0)))

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

innermost

The following dependency pair can be strictly oriented:

G(h(h(h(x'''')))) -> G(h(h(x'''')))

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(G(x)_{1})= 1 + x _{1}_{ }^{ }_{ }^{ }POL(h(x)_{1})= 1 + x _{1}_{ }^{ }

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 2

↳FwdInst

...

→DP Problem 4

↳Dependency Graph

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

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

g(c(x)) ->x

g(d(x)) ->x

g(c(h(0))) -> g(d(1))

g(c(1)) -> g(d(h(0)))

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

innermost

Using the Dependency Graph resulted in no new DP problems.

Duration:

0:00 minutes