terms(

sqr(0) -> 0

sqr(s(

dbl(0) -> 0

dbl(s(

add(0,

add(s(

first(0,

first(s(

half(0) -> 0

half(s(0)) -> 0

half(s(s(

half(dbl(

R

↳Dependency Pair Analysis

TERMS(N) -> SQR(N)

SQR(s(X)) -> ADD(sqr(X), dbl(X))

SQR(s(X)) -> SQR(X)

SQR(s(X)) -> DBL(X)

DBL(s(X)) -> DBL(X)

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

HALF(s(s(X))) -> HALF(X)

Furthermore,

R

↳DPs

→DP Problem 1

↳Argument Filtering and Ordering

→DP Problem 2

↳AFS

→DP Problem 3

↳AFS

→DP Problem 4

↳AFS

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

terms(N) -> cons(recip(sqr(N)))

sqr(0) -> 0

sqr(s(X)) -> s(add(sqr(X), dbl(X)))

dbl(0) -> 0

dbl(s(X)) -> s(s(dbl(X)))

add(0,X) ->X

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

first(0,X) -> nil

first(s(X), cons(Y)) -> cons(Y)

half(0) -> 0

half(s(0)) -> 0

half(s(s(X))) -> s(half(X))

half(dbl(X)) ->X

innermost

The following dependency pair can be strictly oriented:

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

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

Used ordering: Homeomorphic Embedding Order with EMB

resulting in one new DP problem.

Used Argument Filtering System:

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

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

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 5

↳Dependency Graph

→DP Problem 2

↳AFS

→DP Problem 3

↳AFS

→DP Problem 4

↳AFS

terms(N) -> cons(recip(sqr(N)))

sqr(0) -> 0

sqr(s(X)) -> s(add(sqr(X), dbl(X)))

dbl(0) -> 0

dbl(s(X)) -> s(s(dbl(X)))

add(0,X) ->X

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

first(0,X) -> nil

first(s(X), cons(Y)) -> cons(Y)

half(0) -> 0

half(s(0)) -> 0

half(s(s(X))) -> s(half(X))

half(dbl(X)) ->X

innermost

Using the Dependency Graph resulted in no new DP problems.

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳Argument Filtering and Ordering

→DP Problem 3

↳AFS

→DP Problem 4

↳AFS

**DBL(s( X)) -> DBL(X)**

terms(N) -> cons(recip(sqr(N)))

sqr(0) -> 0

sqr(s(X)) -> s(add(sqr(X), dbl(X)))

dbl(0) -> 0

dbl(s(X)) -> s(s(dbl(X)))

add(0,X) ->X

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

first(0,X) -> nil

first(s(X), cons(Y)) -> cons(Y)

half(0) -> 0

half(s(0)) -> 0

half(s(s(X))) -> s(half(X))

half(dbl(X)) ->X

innermost

The following dependency pair can be strictly oriented:

DBL(s(X)) -> DBL(X)

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

Used ordering: Homeomorphic Embedding Order with EMB

resulting in one new DP problem.

Used Argument Filtering System:

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

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

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳AFS

→DP Problem 6

↳Dependency Graph

→DP Problem 3

↳AFS

→DP Problem 4

↳AFS

terms(N) -> cons(recip(sqr(N)))

sqr(0) -> 0

sqr(s(X)) -> s(add(sqr(X), dbl(X)))

dbl(0) -> 0

dbl(s(X)) -> s(s(dbl(X)))

add(0,X) ->X

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

first(0,X) -> nil

first(s(X), cons(Y)) -> cons(Y)

half(0) -> 0

half(s(0)) -> 0

half(s(s(X))) -> s(half(X))

half(dbl(X)) ->X

innermost

Using the Dependency Graph resulted in no new DP problems.

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳AFS

→DP Problem 3

↳Argument Filtering and Ordering

→DP Problem 4

↳AFS

**HALF(s(s( X))) -> HALF(X)**

terms(N) -> cons(recip(sqr(N)))

sqr(0) -> 0

sqr(s(X)) -> s(add(sqr(X), dbl(X)))

dbl(0) -> 0

dbl(s(X)) -> s(s(dbl(X)))

add(0,X) ->X

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

first(0,X) -> nil

first(s(X), cons(Y)) -> cons(Y)

half(0) -> 0

half(s(0)) -> 0

half(s(s(X))) -> s(half(X))

half(dbl(X)) ->X

innermost

The following dependency pair can be strictly oriented:

HALF(s(s(X))) -> HALF(X)

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

Used ordering: Homeomorphic Embedding Order with EMB

resulting in one new DP problem.

Used Argument Filtering System:

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

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

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳AFS

→DP Problem 3

↳AFS

→DP Problem 7

↳Dependency Graph

→DP Problem 4

↳AFS

terms(N) -> cons(recip(sqr(N)))

sqr(0) -> 0

sqr(s(X)) -> s(add(sqr(X), dbl(X)))

dbl(0) -> 0

dbl(s(X)) -> s(s(dbl(X)))

add(0,X) ->X

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

first(0,X) -> nil

first(s(X), cons(Y)) -> cons(Y)

half(0) -> 0

half(s(0)) -> 0

half(s(s(X))) -> s(half(X))

half(dbl(X)) ->X

innermost

Using the Dependency Graph resulted in no new DP problems.

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳AFS

→DP Problem 3

↳AFS

→DP Problem 4

↳Argument Filtering and Ordering

**SQR(s( X)) -> SQR(X)**

terms(N) -> cons(recip(sqr(N)))

sqr(0) -> 0

sqr(s(X)) -> s(add(sqr(X), dbl(X)))

dbl(0) -> 0

dbl(s(X)) -> s(s(dbl(X)))

add(0,X) ->X

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

first(0,X) -> nil

first(s(X), cons(Y)) -> cons(Y)

half(0) -> 0

half(s(0)) -> 0

half(s(s(X))) -> s(half(X))

half(dbl(X)) ->X

innermost

The following dependency pair can be strictly oriented:

SQR(s(X)) -> SQR(X)

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

Used ordering: Homeomorphic Embedding Order with EMB

resulting in one new DP problem.

Used Argument Filtering System:

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

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

R

↳DPs

→DP Problem 1

↳AFS

→DP Problem 2

↳AFS

→DP Problem 3

↳AFS

→DP Problem 4

↳AFS

→DP Problem 8

↳Dependency Graph

terms(N) -> cons(recip(sqr(N)))

sqr(0) -> 0

sqr(s(X)) -> s(add(sqr(X), dbl(X)))

dbl(0) -> 0

dbl(s(X)) -> s(s(dbl(X)))

add(0,X) ->X

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

first(0,X) -> nil

first(s(X), cons(Y)) -> cons(Y)

half(0) -> 0

half(s(0)) -> 0

half(s(s(X))) -> s(half(X))

half(dbl(X)) ->X

innermost

Using the Dependency Graph resulted in no new DP problems.

Duration:

0:00 minutes