ackin(s(

u21(ackout(

R

↳Dependency Pair Analysis

ACKIN(s(X), s(Y)) -> U21(ackin(s(X),Y),X)

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

U21(ackout(X),Y) -> ACKIN(Y,X)

Furthermore,

R

↳DPs

→DP Problem 1

↳Argument Filtering and Ordering

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

ackin(s(X), s(Y)) -> u21(ackin(s(X),Y),X)

u21(ackout(X),Y) -> u22(ackin(Y,X))

innermost

The following dependency pair can be strictly oriented:

ACKIN(s(X), s(Y)) -> U21(ackin(s(X),Y),X)

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

ackin(s(X), s(Y)) -> u21(ackin(s(X),Y),X)

u21(ackout(X),Y) -> u22(ackin(Y,X))

Used ordering: Polynomial ordering with Polynomial interpretation:

_{ }^{ }POL(u22(x)_{1})= x _{1}_{ }^{ }_{ }^{ }POL(U21(x)_{1}, x_{2})= x _{1}+ x_{2}_{ }^{ }_{ }^{ }POL(s(x)_{1})= x _{1}_{ }^{ }_{ }^{ }POL(ACKIN(x)_{1}, x_{2})= 1 + x _{1}+ x_{2}_{ }^{ }_{ }^{ }POL(ackout(x)_{1})= 1 + x _{1}_{ }^{ }

resulting in one new DP problem.

Used Argument Filtering System:

ACKIN(x,_{1}x) -> ACKIN(_{2}x,_{1}x)_{2}

U21(x,_{1}x) -> U21(_{2}x,_{1}x)_{2}

s(x) -> s(_{1}x)_{1}

ackin(x,_{1}x) ->_{2}x_{2}

ackout(x) -> ackout(_{1}x)_{1}

u21(x,_{1}x) ->_{2}x_{1}

u22(x) -> u22(_{1}x)_{1}

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳Dependency Graph

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

ackin(s(X), s(Y)) -> u21(ackin(s(X),Y),X)

u21(ackout(X),Y) -> u22(ackin(Y,X))

innermost

Using the Dependency Graph the DP problem was split into 1 DP problems.

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳DGraph

...

→DP Problem 3

↳Argument Filtering and Ordering

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

ackin(s(X), s(Y)) -> u21(ackin(s(X),Y),X)

u21(ackout(X),Y) -> u22(ackin(Y,X))

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

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

resulting in one new DP problem.

Used Argument Filtering System:

ACKIN(x,_{1}x) -> ACKIN(_{2}x,_{1}x)_{2}

s(x) -> s(_{1}x)_{1}

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳DGraph

...

→DP Problem 4

↳Dependency Graph

ackin(s(X), s(Y)) -> u21(ackin(s(X),Y),X)

u21(ackout(X),Y) -> u22(ackin(Y,X))

innermost

Using the Dependency Graph resulted in no new DP problems.

Duration:

0:00 minutes