plus(plus(

times(

R

↳Dependency Pair Analysis

PLUS(plus(X,Y),Z) -> PLUS(X, plus(Y,Z))

PLUS(plus(X,Y),Z) -> PLUS(Y,Z)

TIMES(X, s(Y)) -> PLUS(X, times(Y,X))

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

Furthermore,

R

↳DPs

→DP Problem 1

↳Polynomial Ordering

→DP Problem 2

↳FwdInst

**PLUS(plus( X, Y), Z) -> PLUS(Y, Z)**

plus(plus(X,Y),Z) -> plus(X, plus(Y,Z))

times(X, s(Y)) -> plus(X, times(Y,X))

The following dependency pairs can be strictly oriented:

PLUS(plus(X,Y),Z) -> PLUS(Y,Z)

PLUS(plus(X,Y),Z) -> PLUS(X, plus(Y,Z))

There are no usable rules 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 _{1}+ x_{2}_{ }^{ }_{ }^{ }POL(PLUS(x)_{1}, x_{2})= x _{1}_{ }^{ }

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 3

↳Dependency Graph

→DP Problem 2

↳FwdInst

plus(plus(X,Y),Z) -> plus(X, plus(Y,Z))

times(X, s(Y)) -> plus(X, times(Y,X))

Using the Dependency Graph resulted in no new DP problems.

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳Forward Instantiation Transformation

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

plus(plus(X,Y),Z) -> plus(X, plus(Y,Z))

times(X, s(Y)) -> plus(X, times(Y,X))

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(Y,X)

TIMES(s(Y''), s(Y0)) -> TIMES(Y0, s(Y''))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳FwdInst

→DP Problem 4

↳Forward Instantiation Transformation

**TIMES(s( Y''), s(Y0)) -> TIMES(Y0, s(Y''))**

plus(plus(X,Y),Z) -> plus(X, plus(Y,Z))

times(X, s(Y)) -> plus(X, times(Y,X))

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(s(Y''), s(Y0)) -> TIMES(Y0, s(Y''))

TIMES(s(Y''''), s(s(Y'''''))) -> TIMES(s(Y'''''), s(Y''''))

The transformation is resulting in one new DP problem:

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳FwdInst

→DP Problem 4

↳FwdInst

...

→DP Problem 5

↳Polynomial Ordering

**TIMES(s( Y''''), s(s(Y'''''))) -> TIMES(s(Y'''''), s(Y''''))**

plus(plus(X,Y),Z) -> plus(X, plus(Y,Z))

times(X, s(Y)) -> plus(X, times(Y,X))

The following dependency pair can be strictly oriented:

TIMES(s(Y''''), s(s(Y'''''))) -> TIMES(s(Y'''''), s(Y''''))

There are no usable rules 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 _{1}+ x_{2}_{ }^{ }_{ }^{ }POL(s(x)_{1})= 1 + x _{1}_{ }^{ }

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳FwdInst

→DP Problem 4

↳FwdInst

...

→DP Problem 6

↳Dependency Graph

plus(plus(X,Y),Z) -> plus(X, plus(Y,Z))

times(X, s(Y)) -> plus(X, times(Y,X))

Using the Dependency Graph resulted in no new DP problems.

Duration:

0:00 minutes