times(

times(

times(

plus(

plus(

R

↳Dependency Pair Analysis

TIMES(x, plus(y, s(z))) -> PLUS(times(x, plus(y, times(s(z), 0))), times(x, s(z)))

TIMES(x, plus(y, s(z))) -> TIMES(x, plus(y, times(s(z), 0)))

TIMES(x, plus(y, s(z))) -> PLUS(y, times(s(z), 0))

TIMES(x, plus(y, s(z))) -> TIMES(s(z), 0)

TIMES(x, plus(y, s(z))) -> TIMES(x, s(z))

TIMES(x, s(y)) -> PLUS(times(x,y),x)

TIMES(x, s(y)) -> TIMES(x,y)

PLUS(x, s(y)) -> PLUS(x,y)

Furthermore,

R

↳DPs

→DP Problem 1

↳Forward Instantiation Transformation

→DP Problem 2

↳FwdInst

**PLUS( x, s(y)) -> PLUS(x, y)**

times(x, plus(y, s(z))) -> plus(times(x, plus(y, times(s(z), 0))), times(x, s(z)))

times(x, 0) -> 0

times(x, s(y)) -> plus(times(x,y),x)

plus(x, 0) ->x

plus(x, s(y)) -> s(plus(x,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:

PLUS(x, s(y)) -> PLUS(x,y)

PLUS(x'', s(s(y''))) -> PLUS(x'', s(y''))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 3

↳Forward Instantiation Transformation

→DP Problem 2

↳FwdInst

**PLUS( x'', s(s(y''))) -> PLUS(x'', s(y''))**

times(x, plus(y, s(z))) -> plus(times(x, plus(y, times(s(z), 0))), times(x, s(z)))

times(x, 0) -> 0

times(x, s(y)) -> plus(times(x,y),x)

plus(x, 0) ->x

plus(x, s(y)) -> s(plus(x,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:

PLUS(x'', s(s(y''))) -> PLUS(x'', s(y''))

PLUS(x'''', s(s(s(y'''')))) -> PLUS(x'''', s(s(y'''')))

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

↳FwdInst

**PLUS( x'''', s(s(s(y'''')))) -> PLUS(x'''', s(s(y'''')))**

times(x, plus(y, s(z))) -> plus(times(x, plus(y, times(s(z), 0))), times(x, s(z)))

times(x, 0) -> 0

times(x, s(y)) -> plus(times(x,y),x)

plus(x, 0) ->x

plus(x, s(y)) -> s(plus(x,y))

innermost

The following dependency pair can be strictly oriented:

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

↳FwdInst

times(x, plus(y, s(z))) -> plus(times(x, plus(y, times(s(z), 0))), times(x, s(z)))

times(x, 0) -> 0

times(x, s(y)) -> plus(times(x,y),x)

plus(x, 0) ->x

plus(x, s(y)) -> s(plus(x,y))

innermost

Using the Dependency Graph resulted in no new DP problems.

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 2

↳Forward Instantiation Transformation

**TIMES( x, s(y)) -> TIMES(x, y)**

times(x, plus(y, s(z))) -> plus(times(x, plus(y, times(s(z), 0))), times(x, s(z)))

times(x, 0) -> 0

times(x, s(y)) -> plus(times(x,y),x)

plus(x, 0) ->x

plus(x, s(y)) -> s(plus(x,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:

TIMES(x, s(y)) -> TIMES(x,y)

TIMES(x'', s(s(y''))) -> TIMES(x'', s(y''))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 2

↳FwdInst

→DP Problem 6

↳Forward Instantiation Transformation

**TIMES( x'', s(s(y''))) -> TIMES(x'', s(y''))**

times(x, plus(y, s(z))) -> plus(times(x, plus(y, times(s(z), 0))), times(x, s(z)))

times(x, 0) -> 0

times(x, s(y)) -> plus(times(x,y),x)

plus(x, 0) ->x

plus(x, s(y)) -> s(plus(x,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:

TIMES(x'', s(s(y''))) -> TIMES(x'', s(y''))

TIMES(x'''', s(s(s(y'''')))) -> TIMES(x'''', s(s(y'''')))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 2

↳FwdInst

→DP Problem 6

↳FwdInst

...

→DP Problem 7

↳Polynomial Ordering

**TIMES( x'''', s(s(s(y'''')))) -> TIMES(x'''', s(s(y'''')))**

times(x, plus(y, s(z))) -> plus(times(x, plus(y, times(s(z), 0))), times(x, s(z)))

times(x, 0) -> 0

times(x, s(y)) -> plus(times(x,y),x)

plus(x, 0) ->x

plus(x, s(y)) -> s(plus(x,y))

innermost

The following dependency pair can be strictly oriented:

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

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳FwdInst

→DP Problem 2

↳FwdInst

→DP Problem 6

↳FwdInst

...

→DP Problem 8

↳Dependency Graph

times(x, plus(y, s(z))) -> plus(times(x, plus(y, times(s(z), 0))), times(x, s(z)))

times(x, 0) -> 0

times(x, s(y)) -> plus(times(x,y),x)

plus(x, 0) ->x

plus(x, s(y)) -> s(plus(x,y))

innermost

Using the Dependency Graph resulted in no new DP problems.

Duration:

0:00 minutes