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

Furthermore,

R

↳DPs

→DP Problem 1

↳Instantiation Transformation

→DP Problem 2

↳Inst

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

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

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

innermost

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

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

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Inst

→DP Problem 3

↳Instantiation Transformation

→DP Problem 2

↳Inst

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

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

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

innermost

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

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

↳Inst

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

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

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

innermost

The following dependency pair can be strictly oriented:

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

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(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

↳Inst

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

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

innermost

Using the Dependency Graph resulted in no new DP problems.

R

↳DPs

→DP Problem 1

↳Inst

→DP Problem 2

↳Instantiation Transformation

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

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

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

innermost

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(y)) -> F(y,y)

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

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Inst

→DP Problem 2

↳Inst

→DP Problem 6

↳Forward Instantiation Transformation

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

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

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

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:

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

F(c(c(y''''')), 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

↳Inst

→DP Problem 6

↳FwdInst

...

→DP Problem 7

↳Polynomial Ordering

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

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

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

innermost

The following dependency pair can be strictly oriented:

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

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

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳Inst

→DP Problem 2

↳Inst

→DP Problem 6

↳FwdInst

...

→DP Problem 8

↳Dependency Graph

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

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

innermost

Using the Dependency Graph resulted in no new DP problems.

Duration:

0:00 minutes