f(0) -> true

f(1) -> false

f(s(

if(true,

if(false,

g(s(

g(

R

↳Dependency Pair Analysis

F(s(x)) -> F(x)

G(s(x), s(y)) -> IF(f(x), s(x), s(y))

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

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

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

Furthermore,

R

↳DPs

→DP Problem 1

↳Forward Instantiation Transformation

→DP Problem 2

↳Nar

**F(s( x)) -> F(x)**

f(0) -> true

f(1) -> false

f(s(x)) -> f(x)

if(true,x,y) ->x

if(false,x,y) ->y

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

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

F(s(s(x''))) -> F(s(x''))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 3

↳Forward Instantiation Transformation

→DP Problem 2

↳Nar

**F(s(s( x''))) -> F(s(x''))**

f(0) -> true

f(1) -> false

f(s(x)) -> f(x)

if(true,x,y) ->x

if(false,x,y) ->y

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

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

F(s(s(s(x'''')))) -> F(s(s(x'''')))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 3

↳FwdInst

...

→DP Problem 4

↳Polynomial Ordering

→DP Problem 2

↳Nar

**F(s(s(s( x'''')))) -> F(s(s(x'''')))**

f(0) -> true

f(1) -> false

f(s(x)) -> f(x)

if(true,x,y) ->x

if(false,x,y) ->y

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

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

innermost

The following dependency pair can be strictly oriented:

F(s(s(s(x'''')))) -> F(s(s(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(s(x)_{1})= 1 + x _{1}_{ }^{ }_{ }^{ }POL(F(x)_{1})= 1 + x _{1}_{ }^{ }

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 3

↳FwdInst

...

→DP Problem 5

↳Dependency Graph

→DP Problem 2

↳Nar

f(0) -> true

f(1) -> false

f(s(x)) -> f(x)

if(true,x,y) ->x

if(false,x,y) ->y

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

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

innermost

Using the Dependency Graph resulted in no new DP problems.

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 2

↳Narrowing Transformation

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

f(0) -> true

f(1) -> false

f(s(x)) -> f(x)

if(true,x,y) ->x

if(false,x,y) ->y

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

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

innermost

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

As a result of transforming the rule

two new Dependency Pairs are created:

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

G(x, c(s(y''))) -> G(x, if(f(c(s(y''))), s(c(s(y''))), s(y'')))

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

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 2

↳Nar

→DP Problem 6

↳Narrowing Transformation

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

f(0) -> true

f(1) -> false

f(s(x)) -> f(x)

if(true,x,y) ->x

if(false,x,y) ->y

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

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

innermost

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

As a result of transforming the rule

no new Dependency Pairs are created.

G(x, c(s(y''))) -> G(x, if(f(c(s(y''))), s(c(s(y''))), s(y'')))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 2

↳Nar

→DP Problem 6

↳Nar

...

→DP Problem 7

↳Narrowing Transformation

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

f(0) -> true

f(1) -> false

f(s(x)) -> f(x)

if(true,x,y) ->x

if(false,x,y) ->y

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

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

innermost

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

As a result of transforming the rule

two new Dependency Pairs are created:

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

G(x, c(c(s(y')))) -> G(x, g(s(c(c(s(y')))), if(f(c(s(y'))), s(c(s(y'))), s(y'))))

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

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 2

↳Nar

→DP Problem 6

↳Nar

...

→DP Problem 8

↳Narrowing Transformation

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

f(0) -> true

f(1) -> false

f(s(x)) -> f(x)

if(true,x,y) ->x

if(false,x,y) ->y

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

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

innermost

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

As a result of transforming the rule

no new Dependency Pairs are created.

G(x, c(c(s(y')))) -> G(x, g(s(c(c(s(y')))), if(f(c(s(y'))), s(c(s(y'))), s(y'))))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 2

↳Nar

→DP Problem 6

↳Nar

...

→DP Problem 9

↳Forward Instantiation Transformation

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

f(0) -> true

f(1) -> false

f(s(x)) -> f(x)

if(true,x,y) ->x

if(false,x,y) ->y

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

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

innermost

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

As a result of transforming the rule

two new Dependency Pairs are created:

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

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

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

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 2

↳Nar

→DP Problem 6

↳Nar

...

→DP Problem 10

↳Forward Instantiation Transformation

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

f(0) -> true

f(1) -> false

f(s(x)) -> f(x)

if(true,x,y) ->x

if(false,x,y) ->y

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

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

innermost

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

As a result of transforming the rule

three new Dependency Pairs are created:

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

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

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

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

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 2

↳Nar

→DP Problem 6

↳Nar

...

→DP Problem 11

↳Polynomial Ordering

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

f(0) -> true

f(1) -> false

f(s(x)) -> f(x)

if(true,x,y) ->x

if(false,x,y) ->y

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

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

innermost

The following dependency pair can be strictly oriented:

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

Additionally, the following usable rules for innermost w.r.t. to the implicit AFS can be oriented:

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

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

if(true,x,y) ->x

if(false,x,y) ->y

Used ordering: Polynomial ordering with Polynomial interpretation:

_{ }^{ }POL(if(x)_{1}, x_{2}, x_{3})= x _{2}+ x_{3}_{ }^{ }_{ }^{ }POL(c(x)_{1})= 1 _{ }^{ }_{ }^{ }POL(0)= 0 _{ }^{ }_{ }^{ }POL(g(x)_{1}, x_{2})= 0 _{ }^{ }_{ }^{ }POL(false)= 0 _{ }^{ }_{ }^{ }POL(G(x)_{1}, x_{2})= x _{2}_{ }^{ }_{ }^{ }POL(1)= 0 _{ }^{ }_{ }^{ }POL(true)= 0 _{ }^{ }_{ }^{ }POL(s(x)_{1})= 0 _{ }^{ }_{ }^{ }POL(f(x)_{1})= 0 _{ }^{ }

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 2

↳Nar

→DP Problem 6

↳Nar

...

→DP Problem 12

↳Instantiation Transformation

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

f(0) -> true

f(1) -> false

f(s(x)) -> f(x)

if(true,x,y) ->x

if(false,x,y) ->y

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

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

innermost

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

As a result of transforming the rule

two new Dependency Pairs are created:

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

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

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

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 2

↳Nar

→DP Problem 6

↳Nar

...

→DP Problem 13

↳Polynomial Ordering

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

f(0) -> true

f(1) -> false

f(s(x)) -> f(x)

if(true,x,y) ->x

if(false,x,y) ->y

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

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

innermost

The following dependency pairs can be strictly oriented:

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

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

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

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

G(x, c(c(c(c(c(y''''')))))) -> G(s(c(c(c(c(c(y''''')))))), c(c(c(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(G(x)_{1}, x_{2})= 1 + x _{2}_{ }^{ }_{ }^{ }POL(s(x)_{1})= 0 _{ }^{ }

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 2

↳Nar

→DP Problem 6

↳Nar

...

→DP Problem 14

↳Dependency Graph

f(0) -> true

f(1) -> false

f(s(x)) -> f(x)

if(true,x,y) ->x

if(false,x,y) ->y

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

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

innermost

Using the Dependency Graph resulted in no new DP problems.

Duration:

0:01 minutes