min(

min(s(

quot(0, s(

quot(s(

log(s(0)) -> 0

log(s(s(

R

↳Dependency Pair Analysis

MIN(s(X), s(Y)) -> MIN(X,Y)

QUOT(s(X), s(Y)) -> QUOT(min(X,Y), s(Y))

QUOT(s(X), s(Y)) -> MIN(X,Y)

LOG(s(s(X))) -> LOG(s(quot(X, s(s(0)))))

LOG(s(s(X))) -> QUOT(X, s(s(0)))

Furthermore,

R

↳DPs

→DP Problem 1

↳Forward Instantiation Transformation

→DP Problem 2

↳Nar

→DP Problem 3

↳Nar

**MIN(s( X), s(Y)) -> MIN(X, Y)**

min(X, 0) ->X

min(s(X), s(Y)) -> min(X,Y)

quot(0, s(Y)) -> 0

quot(s(X), s(Y)) -> s(quot(min(X,Y), s(Y)))

log(s(0)) -> 0

log(s(s(X))) -> s(log(s(quot(X, s(s(0))))))

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:

MIN(s(X), s(Y)) -> MIN(X,Y)

MIN(s(s(X'')), s(s(Y''))) -> MIN(s(X''), s(Y''))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 4

↳Forward Instantiation Transformation

→DP Problem 2

↳Nar

→DP Problem 3

↳Nar

**MIN(s(s( X'')), s(s(Y''))) -> MIN(s(X''), s(Y''))**

min(X, 0) ->X

min(s(X), s(Y)) -> min(X,Y)

quot(0, s(Y)) -> 0

quot(s(X), s(Y)) -> s(quot(min(X,Y), s(Y)))

log(s(0)) -> 0

log(s(s(X))) -> s(log(s(quot(X, s(s(0))))))

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:

MIN(s(s(X'')), s(s(Y''))) -> MIN(s(X''), s(Y''))

MIN(s(s(s(X''''))), s(s(s(Y'''')))) -> MIN(s(s(X'''')), s(s(Y'''')))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 4

↳FwdInst

...

→DP Problem 5

↳Polynomial Ordering

→DP Problem 2

↳Nar

→DP Problem 3

↳Nar

**MIN(s(s(s( X''''))), s(s(s(Y'''')))) -> MIN(s(s(X'''')), s(s(Y'''')))**

min(X, 0) ->X

min(s(X), s(Y)) -> min(X,Y)

quot(0, s(Y)) -> 0

quot(s(X), s(Y)) -> s(quot(min(X,Y), s(Y)))

log(s(0)) -> 0

log(s(s(X))) -> s(log(s(quot(X, s(s(0))))))

innermost

The following dependency pair can be strictly oriented:

MIN(s(s(s(X''''))), s(s(s(Y'''')))) -> MIN(s(s(X'''')), s(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(MIN(x)_{1}, x_{2})= 1 + x _{1}_{ }^{ }_{ }^{ }POL(s(x)_{1})= 1 + x _{1}_{ }^{ }

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 4

↳FwdInst

...

→DP Problem 6

↳Dependency Graph

→DP Problem 2

↳Nar

→DP Problem 3

↳Nar

min(X, 0) ->X

min(s(X), s(Y)) -> min(X,Y)

quot(0, s(Y)) -> 0

quot(s(X), s(Y)) -> s(quot(min(X,Y), s(Y)))

log(s(0)) -> 0

log(s(s(X))) -> s(log(s(quot(X, s(s(0))))))

innermost

Using the Dependency Graph resulted in no new DP problems.

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 2

↳Narrowing Transformation

→DP Problem 3

↳Nar

**QUOT(s( X), s(Y)) -> QUOT(min(X, Y), s(Y))**

min(X, 0) ->X

min(s(X), s(Y)) -> min(X,Y)

quot(0, s(Y)) -> 0

quot(s(X), s(Y)) -> s(quot(min(X,Y), s(Y)))

log(s(0)) -> 0

log(s(s(X))) -> s(log(s(quot(X, s(s(0))))))

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:

QUOT(s(X), s(Y)) -> QUOT(min(X,Y), s(Y))

QUOT(s(X''), s(0)) -> QUOT(X'', s(0))

QUOT(s(s(X'')), s(s(Y''))) -> QUOT(min(X'',Y''), s(s(Y'')))

The transformation is resulting in two new DP problems:

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 2

↳Nar

→DP Problem 7

↳Forward Instantiation Transformation

→DP Problem 8

↳Nar

→DP Problem 3

↳Nar

**QUOT(s( X''), s(0)) -> QUOT(X'', s(0))**

min(X, 0) ->X

min(s(X), s(Y)) -> min(X,Y)

quot(0, s(Y)) -> 0

quot(s(X), s(Y)) -> s(quot(min(X,Y), s(Y)))

log(s(0)) -> 0

log(s(s(X))) -> s(log(s(quot(X, s(s(0))))))

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:

QUOT(s(X''), s(0)) -> QUOT(X'', s(0))

QUOT(s(s(X'''')), s(0)) -> QUOT(s(X''''), s(0))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 2

↳Nar

→DP Problem 7

↳FwdInst

...

→DP Problem 9

↳Polynomial Ordering

→DP Problem 8

↳Nar

→DP Problem 3

↳Nar

**QUOT(s(s( X'''')), s(0)) -> QUOT(s(X''''), s(0))**

min(X, 0) ->X

min(s(X), s(Y)) -> min(X,Y)

quot(0, s(Y)) -> 0

quot(s(X), s(Y)) -> s(quot(min(X,Y), s(Y)))

log(s(0)) -> 0

log(s(s(X))) -> s(log(s(quot(X, s(s(0))))))

innermost

The following dependency pair can be strictly oriented:

QUOT(s(s(X'''')), s(0)) -> QUOT(s(X''''), s(0))

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

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 2

↳Nar

→DP Problem 7

↳FwdInst

...

→DP Problem 12

↳Dependency Graph

→DP Problem 8

↳Nar

→DP Problem 3

↳Nar

min(X, 0) ->X

min(s(X), s(Y)) -> min(X,Y)

quot(0, s(Y)) -> 0

quot(s(X), s(Y)) -> s(quot(min(X,Y), s(Y)))

log(s(0)) -> 0

log(s(s(X))) -> s(log(s(quot(X, s(s(0))))))

innermost

Using the Dependency Graph resulted in no new DP problems.

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 2

↳Nar

→DP Problem 7

↳FwdInst

→DP Problem 8

↳Narrowing Transformation

→DP Problem 3

↳Nar

**QUOT(s(s( X'')), s(s(Y''))) -> QUOT(min(X'', Y''), s(s(Y'')))**

min(X, 0) ->X

min(s(X), s(Y)) -> min(X,Y)

quot(0, s(Y)) -> 0

quot(s(X), s(Y)) -> s(quot(min(X,Y), s(Y)))

log(s(0)) -> 0

log(s(s(X))) -> s(log(s(quot(X, s(s(0))))))

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:

QUOT(s(s(X'')), s(s(Y''))) -> QUOT(min(X'',Y''), s(s(Y'')))

QUOT(s(s(X''')), s(s(0))) -> QUOT(X''', s(s(0)))

QUOT(s(s(s(X'))), s(s(s(Y')))) -> QUOT(min(X',Y'), s(s(s(Y'))))

The transformation is resulting in two new DP problems:

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 2

↳Nar

→DP Problem 7

↳FwdInst

→DP Problem 8

↳Nar

...

→DP Problem 10

↳Polynomial Ordering

→DP Problem 3

↳Nar

**QUOT(s(s( X''')), s(s(0))) -> QUOT(X''', s(s(0)))**

min(X, 0) ->X

min(s(X), s(Y)) -> min(X,Y)

quot(0, s(Y)) -> 0

quot(s(X), s(Y)) -> s(quot(min(X,Y), s(Y)))

log(s(0)) -> 0

log(s(s(X))) -> s(log(s(quot(X, s(s(0))))))

innermost

The following dependency pair can be strictly oriented:

QUOT(s(s(X''')), s(s(0))) -> QUOT(X''', s(s(0)))

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

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 2

↳Nar

→DP Problem 7

↳FwdInst

→DP Problem 8

↳Nar

...

→DP Problem 11

↳Polynomial Ordering

→DP Problem 3

↳Nar

**QUOT(s(s(s( X'))), s(s(s(Y')))) -> QUOT(min(X', Y'), s(s(s(Y'))))**

min(X, 0) ->X

min(s(X), s(Y)) -> min(X,Y)

quot(0, s(Y)) -> 0

quot(s(X), s(Y)) -> s(quot(min(X,Y), s(Y)))

log(s(0)) -> 0

log(s(s(X))) -> s(log(s(quot(X, s(s(0))))))

innermost

The following dependency pair can be strictly oriented:

QUOT(s(s(s(X'))), s(s(s(Y')))) -> QUOT(min(X',Y'), s(s(s(Y'))))

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

min(X, 0) ->X

min(s(X), s(Y)) -> min(X,Y)

Used ordering: Polynomial ordering with Polynomial interpretation:

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

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 2

↳Nar

→DP Problem 3

↳Narrowing Transformation

**LOG(s(s( X))) -> LOG(s(quot(X, s(s(0)))))**

min(X, 0) ->X

min(s(X), s(Y)) -> min(X,Y)

quot(0, s(Y)) -> 0

quot(s(X), s(Y)) -> s(quot(min(X,Y), s(Y)))

log(s(0)) -> 0

log(s(s(X))) -> s(log(s(quot(X, s(s(0))))))

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:

LOG(s(s(X))) -> LOG(s(quot(X, s(s(0)))))

LOG(s(s(0))) -> LOG(s(0))

LOG(s(s(s(X'')))) -> LOG(s(s(quot(min(X'', s(0)), s(s(0))))))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 2

↳Nar

→DP Problem 3

↳Nar

→DP Problem 15

↳Narrowing Transformation

**LOG(s(s(s( X'')))) -> LOG(s(s(quot(min(X'', s(0)), s(s(0))))))**

min(X, 0) ->X

min(s(X), s(Y)) -> min(X,Y)

quot(0, s(Y)) -> 0

quot(s(X), s(Y)) -> s(quot(min(X,Y), s(Y)))

log(s(0)) -> 0

log(s(s(X))) -> s(log(s(quot(X, s(s(0))))))

innermost

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

As a result of transforming the rule

one new Dependency Pair is created:

LOG(s(s(s(X'')))) -> LOG(s(s(quot(min(X'', s(0)), s(s(0))))))

LOG(s(s(s(s(X'))))) -> LOG(s(s(quot(min(X', 0), s(s(0))))))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 2

↳Nar

→DP Problem 3

↳Nar

→DP Problem 15

↳Nar

...

→DP Problem 16

↳Rewriting Transformation

**LOG(s(s(s(s( X'))))) -> LOG(s(s(quot(min(X', 0), s(s(0))))))**

min(X, 0) ->X

min(s(X), s(Y)) -> min(X,Y)

quot(0, s(Y)) -> 0

quot(s(X), s(Y)) -> s(quot(min(X,Y), s(Y)))

log(s(0)) -> 0

log(s(s(X))) -> s(log(s(quot(X, s(s(0))))))

innermost

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

As a result of transforming the rule

one new Dependency Pair is created:

LOG(s(s(s(s(X'))))) -> LOG(s(s(quot(min(X', 0), s(s(0))))))

LOG(s(s(s(s(X'))))) -> LOG(s(s(quot(X', s(s(0))))))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 2

↳Nar

→DP Problem 3

↳Nar

→DP Problem 15

↳Nar

...

→DP Problem 17

↳Polynomial Ordering

**LOG(s(s(s(s( X'))))) -> LOG(s(s(quot(X', s(s(0))))))**

min(X, 0) ->X

min(s(X), s(Y)) -> min(X,Y)

quot(0, s(Y)) -> 0

quot(s(X), s(Y)) -> s(quot(min(X,Y), s(Y)))

log(s(0)) -> 0

log(s(s(X))) -> s(log(s(quot(X, s(s(0))))))

innermost

The following dependency pair can be strictly oriented:

LOG(s(s(s(s(X'))))) -> LOG(s(s(quot(X', s(s(0))))))

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

quot(0, s(Y)) -> 0

quot(s(X), s(Y)) -> s(quot(min(X,Y), s(Y)))

min(X, 0) ->X

min(s(X), s(Y)) -> min(X,Y)

Used ordering: Polynomial ordering with Polynomial interpretation:

_{ }^{ }POL(0)= 0 _{ }^{ }_{ }^{ }POL(quot(x)_{1}, x_{2})= x _{1}_{ }^{ }_{ }^{ }POL(min(x)_{1}, x_{2})= x _{1}_{ }^{ }_{ }^{ }POL(s(x)_{1})= 1 + x _{1}_{ }^{ }_{ }^{ }POL(LOG(x)_{1})= 1 + x _{1}_{ }^{ }

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 2

↳Nar

→DP Problem 3

↳Nar

→DP Problem 15

↳Nar

...

→DP Problem 18

↳Dependency Graph

min(X, 0) ->X

min(s(X), s(Y)) -> min(X,Y)

quot(0, s(Y)) -> 0

quot(s(X), s(Y)) -> s(quot(min(X,Y), s(Y)))

log(s(0)) -> 0

log(s(s(X))) -> s(log(s(quot(X, s(s(0))))))

innermost

Using the Dependency Graph resulted in no new DP problems.

Duration:

0:00 minutes