f(

f(s(

R

↳Dependency Pair Analysis

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

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

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

Furthermore,

R

↳DPs

→DP Problem 1

↳Instantiation Transformation

→DP Problem 2

↳FwdInst

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

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

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

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

As a result of transforming the rule

one new Dependency Pair is created:

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

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

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Inst

→DP Problem 3

↳Instantiation Transformation

→DP Problem 2

↳FwdInst

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

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

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

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

As a result of transforming the rule

one new Dependency Pair is created:

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

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

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Inst

→DP Problem 3

↳Inst

...

→DP Problem 4

↳Polynomial Ordering

→DP Problem 2

↳FwdInst

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

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

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

The following dependency pair can be strictly oriented:

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

There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

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

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳Inst

→DP Problem 3

↳Inst

...

→DP Problem 5

↳Dependency Graph

→DP Problem 2

↳FwdInst

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

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

Using the Dependency Graph resulted in no new DP problems.

R

↳DPs

→DP Problem 1

↳Inst

→DP Problem 2

↳Forward Instantiation Transformation

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

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

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

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:

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

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

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Inst

→DP Problem 2

↳FwdInst

→DP Problem 6

↳Instantiation Transformation

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

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

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

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

As a result of transforming the rule

one new Dependency Pair is created:

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

F(c(y'''''), c(c(y''0))) -> F(c(y''0), c(y''0))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Inst

→DP Problem 2

↳FwdInst

→DP Problem 6

↳Inst

...

→DP Problem 7

↳Polynomial Ordering

**F(c( y'''''), c(c(y''0))) -> F(c(y''0), c(y''0))**

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

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

The following dependency pair can be strictly oriented:

F(c(y'''''), c(c(y''0))) -> F(c(y''0), c(y''0))

There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

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

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳Inst

→DP Problem 2

↳FwdInst

→DP Problem 6

↳Inst

...

→DP Problem 8

↳Dependency Graph

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

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

Using the Dependency Graph resulted in no new DP problems.

Duration:

0:00 minutes