ackin(0,

ackin(s(

ackin(s(

u11(ackout(

u21(ackout(

u22(ackout(

R

↳Dependency Pair Analysis

ACKIN(s(X), 0) -> U11(ackin(X, s(0)))

ACKIN(s(X), 0) -> ACKIN(X, s(0))

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

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

U21(ackout(X),Y) -> U22(ackin(Y,X))

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

Furthermore,

R

↳DPs

→DP Problem 1

↳Polynomial Ordering

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

ackin(0,X) -> ackout(s(X))

ackin(s(X), 0) -> u11(ackin(X, s(0)))

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

u11(ackout(X)) -> ackout(X)

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

u22(ackout(X)) -> ackout(X)

The following dependency pairs can be strictly oriented:

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

ACKIN(s(X), 0) -> ACKIN(X, s(0))

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(0)= 0 _{ }^{ }_{ }^{ }POL(u11(x)_{1})= 0 _{ }^{ }_{ }^{ }POL(u22(x)_{1})= 0 _{ }^{ }_{ }^{ }POL(U21(x)_{1}, x_{2})= x _{2}_{ }^{ }_{ }^{ }POL(ackin(x)_{1}, x_{2})= 0 _{ }^{ }_{ }^{ }POL(u21(x)_{1}, x_{2})= 0 _{ }^{ }_{ }^{ }POL(s(x)_{1})= 1 + x _{1}_{ }^{ }_{ }^{ }POL(ACKIN(x)_{1}, x_{2})= x _{1}_{ }^{ }_{ }^{ }POL(ackout(x)_{1})= 0 _{ }^{ }

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳Dependency Graph

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

ackin(0,X) -> ackout(s(X))

ackin(s(X), 0) -> u11(ackin(X, s(0)))

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

u11(ackout(X)) -> ackout(X)

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

u22(ackout(X)) -> ackout(X)

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

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳DGraph

...

→DP Problem 3

↳Polynomial Ordering

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

ackin(0,X) -> ackout(s(X))

ackin(s(X), 0) -> u11(ackin(X, s(0)))

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

u11(ackout(X)) -> ackout(X)

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

u22(ackout(X)) -> ackout(X)

The following dependency pair can be strictly oriented:

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

resulting in one new DP problem.

R

↳DPs

→DP Problem 1

↳Polo

→DP Problem 2

↳DGraph

...

→DP Problem 4

↳Dependency Graph

ackin(0,X) -> ackout(s(X))

ackin(s(X), 0) -> u11(ackin(X, s(0)))

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

u11(ackout(X)) -> ackout(X)

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

u22(ackout(X)) -> ackout(X)

Using the Dependency Graph resulted in no new DP problems.

Duration:

0:00 minutes