Term Rewriting System R:
[N, X, Y, X1, X2, Z]
aterms(N) -> cons(recip(asqr(mark(N))), terms(s(N)))
aterms(X) -> terms(X)
asqr(0) -> 0
asqr(s(X)) -> s(add(sqr(X), dbl(X)))
asqr(X) -> sqr(X)
adbl(0) -> 0
adbl(s(X)) -> s(s(dbl(X)))
adbl(X) -> dbl(X)
aadd(0, X) -> mark(X)
aadd(s(X), Y) -> s(add(X, Y))
aadd(X1, X2) -> add(X1, X2)
afirst(0, X) -> nil
afirst(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
afirst(X1, X2) -> first(X1, X2)
mark(terms(X)) -> aterms(mark(X))
mark(sqr(X)) -> asqr(mark(X))
mark(add(X1, X2)) -> aadd(mark(X1), mark(X2))
mark(dbl(X)) -> adbl(mark(X))
mark(first(X1, X2)) -> afirst(mark(X1), mark(X2))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(recip(X)) -> recip(mark(X))
mark(s(X)) -> s(X)
mark(0) -> 0
mark(nil) -> nil

Innermost Termination of R to be shown.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

ATERMS(N) -> ASQR(mark(N))
ATERMS(N) -> MARK(N)
AADD(0, X) -> MARK(X)
AFIRST(s(X), cons(Y, Z)) -> MARK(Y)
MARK(terms(X)) -> ATERMS(mark(X))
MARK(terms(X)) -> MARK(X)
MARK(sqr(X)) -> ASQR(mark(X))
MARK(sqr(X)) -> MARK(X)
MARK(add(X1, X2)) -> AADD(mark(X1), mark(X2))
MARK(add(X1, X2)) -> MARK(X1)
MARK(add(X1, X2)) -> MARK(X2)
MARK(dbl(X)) -> ADBL(mark(X))
MARK(dbl(X)) -> MARK(X)
MARK(first(X1, X2)) -> AFIRST(mark(X1), mark(X2))
MARK(first(X1, X2)) -> MARK(X1)
MARK(first(X1, X2)) -> MARK(X2)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(recip(X)) -> MARK(X)

Furthermore, R contains one SCC.

R
DPs
→DP Problem 1
Polynomial Ordering

Dependency Pairs:

MARK(recip(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(first(X1, X2)) -> MARK(X2)
MARK(first(X1, X2)) -> MARK(X1)
AFIRST(s(X), cons(Y, Z)) -> MARK(Y)
MARK(first(X1, X2)) -> AFIRST(mark(X1), mark(X2))
MARK(dbl(X)) -> MARK(X)
MARK(add(X1, X2)) -> MARK(X2)
MARK(add(X1, X2)) -> MARK(X1)
AADD(0, X) -> MARK(X)
MARK(add(X1, X2)) -> AADD(mark(X1), mark(X2))
MARK(sqr(X)) -> MARK(X)
MARK(terms(X)) -> MARK(X)
MARK(terms(X)) -> ATERMS(mark(X))
ATERMS(N) -> MARK(N)

Rules:

aterms(N) -> cons(recip(asqr(mark(N))), terms(s(N)))
aterms(X) -> terms(X)
asqr(0) -> 0
asqr(s(X)) -> s(add(sqr(X), dbl(X)))
asqr(X) -> sqr(X)
adbl(0) -> 0
adbl(s(X)) -> s(s(dbl(X)))
adbl(X) -> dbl(X)
aadd(0, X) -> mark(X)
aadd(s(X), Y) -> s(add(X, Y))
aadd(X1, X2) -> add(X1, X2)
afirst(0, X) -> nil
afirst(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
afirst(X1, X2) -> first(X1, X2)
mark(terms(X)) -> aterms(mark(X))
mark(sqr(X)) -> asqr(mark(X))
mark(add(X1, X2)) -> aadd(mark(X1), mark(X2))
mark(dbl(X)) -> adbl(mark(X))
mark(first(X1, X2)) -> afirst(mark(X1), mark(X2))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(recip(X)) -> recip(mark(X))
mark(s(X)) -> s(X)
mark(0) -> 0
mark(nil) -> nil

Strategy:

innermost

The following dependency pair can be strictly oriented:

MARK(dbl(X)) -> MARK(X)

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

aterms(N) -> cons(recip(asqr(mark(N))), terms(s(N)))
aterms(X) -> terms(X)
mark(terms(X)) -> aterms(mark(X))
mark(sqr(X)) -> asqr(mark(X))
mark(add(X1, X2)) -> aadd(mark(X1), mark(X2))
mark(dbl(X)) -> adbl(mark(X))
mark(first(X1, X2)) -> afirst(mark(X1), mark(X2))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(recip(X)) -> recip(mark(X))
mark(s(X)) -> s(X)
mark(0) -> 0
mark(nil) -> nil
aadd(0, X) -> mark(X)
aadd(s(X), Y) -> s(add(X, Y))
aadd(X1, X2) -> add(X1, X2)
afirst(0, X) -> nil
afirst(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
afirst(X1, X2) -> first(X1, X2)
asqr(0) -> 0
asqr(s(X)) -> s(add(sqr(X), dbl(X)))
asqr(X) -> sqr(X)
adbl(0) -> 0
adbl(s(X)) -> s(s(dbl(X)))
adbl(X) -> dbl(X)

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(MARK(x1)) =  x1 POL(sqr(x1)) =  x1 POL(a__terms(x1)) =  x1 POL(a__dbl(x1)) =  1 + x1 POL(terms(x1)) =  x1 POL(mark(x1)) =  x1 POL(a__add(x1, x2)) =  x1 + x2 POL(a__first(x1, x2)) =  x1 + x2 POL(add(x1, x2)) =  x1 + x2 POL(A__ADD(x1, x2)) =  x2 POL(A__FIRST(x1, x2)) =  x2 POL(0) =  0 POL(first(x1, x2)) =  x1 + x2 POL(a__sqr(x1)) =  x1 POL(cons(x1, x2)) =  x1 POL(dbl(x1)) =  1 + x1 POL(nil) =  0 POL(s(x1)) =  0 POL(recip(x1)) =  x1 POL(A__TERMS(x1)) =  x1

resulting in one new DP problem.

R
DPs
→DP Problem 1
Polo
→DP Problem 2
Polynomial Ordering

Dependency Pairs:

MARK(recip(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(first(X1, X2)) -> MARK(X2)
MARK(first(X1, X2)) -> MARK(X1)
AFIRST(s(X), cons(Y, Z)) -> MARK(Y)
MARK(first(X1, X2)) -> AFIRST(mark(X1), mark(X2))
MARK(add(X1, X2)) -> MARK(X2)
MARK(add(X1, X2)) -> MARK(X1)
AADD(0, X) -> MARK(X)
MARK(add(X1, X2)) -> AADD(mark(X1), mark(X2))
MARK(sqr(X)) -> MARK(X)
MARK(terms(X)) -> MARK(X)
MARK(terms(X)) -> ATERMS(mark(X))
ATERMS(N) -> MARK(N)

Rules:

aterms(N) -> cons(recip(asqr(mark(N))), terms(s(N)))
aterms(X) -> terms(X)
asqr(0) -> 0
asqr(s(X)) -> s(add(sqr(X), dbl(X)))
asqr(X) -> sqr(X)
adbl(0) -> 0
adbl(s(X)) -> s(s(dbl(X)))
adbl(X) -> dbl(X)
aadd(0, X) -> mark(X)
aadd(s(X), Y) -> s(add(X, Y))
aadd(X1, X2) -> add(X1, X2)
afirst(0, X) -> nil
afirst(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
afirst(X1, X2) -> first(X1, X2)
mark(terms(X)) -> aterms(mark(X))
mark(sqr(X)) -> asqr(mark(X))
mark(add(X1, X2)) -> aadd(mark(X1), mark(X2))
mark(dbl(X)) -> adbl(mark(X))
mark(first(X1, X2)) -> afirst(mark(X1), mark(X2))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(recip(X)) -> recip(mark(X))
mark(s(X)) -> s(X)
mark(0) -> 0
mark(nil) -> nil

Strategy:

innermost

The following dependency pairs can be strictly oriented:

MARK(add(X1, X2)) -> MARK(X2)
MARK(add(X1, X2)) -> MARK(X1)
MARK(add(X1, X2)) -> AADD(mark(X1), mark(X2))

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

aterms(N) -> cons(recip(asqr(mark(N))), terms(s(N)))
aterms(X) -> terms(X)
mark(terms(X)) -> aterms(mark(X))
mark(sqr(X)) -> asqr(mark(X))
mark(add(X1, X2)) -> aadd(mark(X1), mark(X2))
mark(dbl(X)) -> adbl(mark(X))
mark(first(X1, X2)) -> afirst(mark(X1), mark(X2))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(recip(X)) -> recip(mark(X))
mark(s(X)) -> s(X)
mark(0) -> 0
mark(nil) -> nil
aadd(0, X) -> mark(X)
aadd(s(X), Y) -> s(add(X, Y))
aadd(X1, X2) -> add(X1, X2)
afirst(0, X) -> nil
afirst(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
afirst(X1, X2) -> first(X1, X2)
asqr(0) -> 0
asqr(s(X)) -> s(add(sqr(X), dbl(X)))
asqr(X) -> sqr(X)
adbl(0) -> 0
adbl(s(X)) -> s(s(dbl(X)))
adbl(X) -> dbl(X)

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(MARK(x1)) =  x1 POL(sqr(x1)) =  x1 POL(a__terms(x1)) =  x1 POL(a__dbl(x1)) =  0 POL(terms(x1)) =  x1 POL(mark(x1)) =  x1 POL(a__add(x1, x2)) =  1 + x1 + x2 POL(a__first(x1, x2)) =  x1 + x2 POL(add(x1, x2)) =  1 + x1 + x2 POL(A__ADD(x1, x2)) =  x2 POL(A__FIRST(x1, x2)) =  x2 POL(0) =  0 POL(first(x1, x2)) =  x1 + x2 POL(a__sqr(x1)) =  x1 POL(cons(x1, x2)) =  x1 POL(dbl(x1)) =  0 POL(nil) =  0 POL(s(x1)) =  0 POL(recip(x1)) =  x1 POL(A__TERMS(x1)) =  x1

resulting in one new DP problem.

R
DPs
→DP Problem 1
Polo
→DP Problem 2
Polo
...
→DP Problem 3
Dependency Graph

Dependency Pairs:

MARK(recip(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(first(X1, X2)) -> MARK(X2)
MARK(first(X1, X2)) -> MARK(X1)
AFIRST(s(X), cons(Y, Z)) -> MARK(Y)
MARK(first(X1, X2)) -> AFIRST(mark(X1), mark(X2))
AADD(0, X) -> MARK(X)
MARK(sqr(X)) -> MARK(X)
MARK(terms(X)) -> MARK(X)
MARK(terms(X)) -> ATERMS(mark(X))
ATERMS(N) -> MARK(N)

Rules:

aterms(N) -> cons(recip(asqr(mark(N))), terms(s(N)))
aterms(X) -> terms(X)
asqr(0) -> 0
asqr(s(X)) -> s(add(sqr(X), dbl(X)))
asqr(X) -> sqr(X)
adbl(0) -> 0
adbl(s(X)) -> s(s(dbl(X)))
adbl(X) -> dbl(X)
aadd(0, X) -> mark(X)
aadd(s(X), Y) -> s(add(X, Y))
aadd(X1, X2) -> add(X1, X2)
afirst(0, X) -> nil
afirst(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
afirst(X1, X2) -> first(X1, X2)
mark(terms(X)) -> aterms(mark(X))
mark(sqr(X)) -> asqr(mark(X))
mark(add(X1, X2)) -> aadd(mark(X1), mark(X2))
mark(dbl(X)) -> adbl(mark(X))
mark(first(X1, X2)) -> afirst(mark(X1), mark(X2))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(recip(X)) -> recip(mark(X))
mark(s(X)) -> s(X)
mark(0) -> 0
mark(nil) -> nil

Strategy:

innermost

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

R
DPs
→DP Problem 1
Polo
→DP Problem 2
Polo
...
→DP Problem 4
Polynomial Ordering

Dependency Pairs:

MARK(cons(X1, X2)) -> MARK(X1)
MARK(first(X1, X2)) -> MARK(X2)
MARK(first(X1, X2)) -> MARK(X1)
AFIRST(s(X), cons(Y, Z)) -> MARK(Y)
MARK(first(X1, X2)) -> AFIRST(mark(X1), mark(X2))
MARK(sqr(X)) -> MARK(X)
MARK(terms(X)) -> MARK(X)
ATERMS(N) -> MARK(N)
MARK(terms(X)) -> ATERMS(mark(X))
MARK(recip(X)) -> MARK(X)

Rules:

aterms(N) -> cons(recip(asqr(mark(N))), terms(s(N)))
aterms(X) -> terms(X)
asqr(0) -> 0
asqr(s(X)) -> s(add(sqr(X), dbl(X)))
asqr(X) -> sqr(X)
adbl(0) -> 0
adbl(s(X)) -> s(s(dbl(X)))
adbl(X) -> dbl(X)
aadd(0, X) -> mark(X)
aadd(s(X), Y) -> s(add(X, Y))
aadd(X1, X2) -> add(X1, X2)
afirst(0, X) -> nil
afirst(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
afirst(X1, X2) -> first(X1, X2)
mark(terms(X)) -> aterms(mark(X))
mark(sqr(X)) -> asqr(mark(X))
mark(add(X1, X2)) -> aadd(mark(X1), mark(X2))
mark(dbl(X)) -> adbl(mark(X))
mark(first(X1, X2)) -> afirst(mark(X1), mark(X2))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(recip(X)) -> recip(mark(X))
mark(s(X)) -> s(X)
mark(0) -> 0
mark(nil) -> nil

Strategy:

innermost

The following dependency pairs can be strictly oriented:

MARK(first(X1, X2)) -> MARK(X2)
MARK(first(X1, X2)) -> MARK(X1)
MARK(first(X1, X2)) -> AFIRST(mark(X1), mark(X2))

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

aterms(N) -> cons(recip(asqr(mark(N))), terms(s(N)))
aterms(X) -> terms(X)
mark(terms(X)) -> aterms(mark(X))
mark(sqr(X)) -> asqr(mark(X))
mark(add(X1, X2)) -> aadd(mark(X1), mark(X2))
mark(dbl(X)) -> adbl(mark(X))
mark(first(X1, X2)) -> afirst(mark(X1), mark(X2))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(recip(X)) -> recip(mark(X))
mark(s(X)) -> s(X)
mark(0) -> 0
mark(nil) -> nil
aadd(0, X) -> mark(X)
aadd(s(X), Y) -> s(add(X, Y))
aadd(X1, X2) -> add(X1, X2)
afirst(0, X) -> nil
afirst(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
afirst(X1, X2) -> first(X1, X2)
asqr(0) -> 0
asqr(s(X)) -> s(add(sqr(X), dbl(X)))
asqr(X) -> sqr(X)
adbl(0) -> 0
adbl(s(X)) -> s(s(dbl(X)))
adbl(X) -> dbl(X)

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(MARK(x1)) =  x1 POL(sqr(x1)) =  x1 POL(a__terms(x1)) =  x1 POL(a__dbl(x1)) =  0 POL(terms(x1)) =  x1 POL(mark(x1)) =  x1 POL(a__add(x1, x2)) =  x2 POL(a__first(x1, x2)) =  1 + x1 + x2 POL(add(x1, x2)) =  x2 POL(A__FIRST(x1, x2)) =  x2 POL(0) =  0 POL(first(x1, x2)) =  1 + x1 + x2 POL(a__sqr(x1)) =  x1 POL(cons(x1, x2)) =  x1 POL(dbl(x1)) =  0 POL(nil) =  0 POL(s(x1)) =  0 POL(recip(x1)) =  x1 POL(A__TERMS(x1)) =  x1

resulting in one new DP problem.

R
DPs
→DP Problem 1
Polo
→DP Problem 2
Polo
...
→DP Problem 5
Dependency Graph

Dependency Pairs:

MARK(cons(X1, X2)) -> MARK(X1)
AFIRST(s(X), cons(Y, Z)) -> MARK(Y)
MARK(sqr(X)) -> MARK(X)
MARK(terms(X)) -> MARK(X)
ATERMS(N) -> MARK(N)
MARK(terms(X)) -> ATERMS(mark(X))
MARK(recip(X)) -> MARK(X)

Rules:

aterms(N) -> cons(recip(asqr(mark(N))), terms(s(N)))
aterms(X) -> terms(X)
asqr(0) -> 0
asqr(s(X)) -> s(add(sqr(X), dbl(X)))
asqr(X) -> sqr(X)
adbl(0) -> 0
adbl(s(X)) -> s(s(dbl(X)))
adbl(X) -> dbl(X)
aadd(0, X) -> mark(X)
aadd(s(X), Y) -> s(add(X, Y))
aadd(X1, X2) -> add(X1, X2)
afirst(0, X) -> nil
afirst(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
afirst(X1, X2) -> first(X1, X2)
mark(terms(X)) -> aterms(mark(X))
mark(sqr(X)) -> asqr(mark(X))
mark(add(X1, X2)) -> aadd(mark(X1), mark(X2))
mark(dbl(X)) -> adbl(mark(X))
mark(first(X1, X2)) -> afirst(mark(X1), mark(X2))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(recip(X)) -> recip(mark(X))
mark(s(X)) -> s(X)
mark(0) -> 0
mark(nil) -> nil

Strategy:

innermost

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

R
DPs
→DP Problem 1
Polo
→DP Problem 2
Polo
...
→DP Problem 6
Polynomial Ordering

Dependency Pairs:

MARK(recip(X)) -> MARK(X)
MARK(sqr(X)) -> MARK(X)
MARK(terms(X)) -> MARK(X)
ATERMS(N) -> MARK(N)
MARK(terms(X)) -> ATERMS(mark(X))
MARK(cons(X1, X2)) -> MARK(X1)

Rules:

aterms(N) -> cons(recip(asqr(mark(N))), terms(s(N)))
aterms(X) -> terms(X)
asqr(0) -> 0
asqr(s(X)) -> s(add(sqr(X), dbl(X)))
asqr(X) -> sqr(X)
adbl(0) -> 0
adbl(s(X)) -> s(s(dbl(X)))
adbl(X) -> dbl(X)
aadd(0, X) -> mark(X)
aadd(s(X), Y) -> s(add(X, Y))
aadd(X1, X2) -> add(X1, X2)
afirst(0, X) -> nil
afirst(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
afirst(X1, X2) -> first(X1, X2)
mark(terms(X)) -> aterms(mark(X))
mark(sqr(X)) -> asqr(mark(X))
mark(add(X1, X2)) -> aadd(mark(X1), mark(X2))
mark(dbl(X)) -> adbl(mark(X))
mark(first(X1, X2)) -> afirst(mark(X1), mark(X2))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(recip(X)) -> recip(mark(X))
mark(s(X)) -> s(X)
mark(0) -> 0
mark(nil) -> nil

Strategy:

innermost

The following dependency pairs can be strictly oriented:

MARK(terms(X)) -> MARK(X)
MARK(terms(X)) -> ATERMS(mark(X))

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

aterms(N) -> cons(recip(asqr(mark(N))), terms(s(N)))
aterms(X) -> terms(X)
mark(terms(X)) -> aterms(mark(X))
mark(sqr(X)) -> asqr(mark(X))
mark(add(X1, X2)) -> aadd(mark(X1), mark(X2))
mark(dbl(X)) -> adbl(mark(X))
mark(first(X1, X2)) -> afirst(mark(X1), mark(X2))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(recip(X)) -> recip(mark(X))
mark(s(X)) -> s(X)
mark(0) -> 0
mark(nil) -> nil
aadd(0, X) -> mark(X)
aadd(s(X), Y) -> s(add(X, Y))
aadd(X1, X2) -> add(X1, X2)
afirst(0, X) -> nil
afirst(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
afirst(X1, X2) -> first(X1, X2)
asqr(0) -> 0
asqr(s(X)) -> s(add(sqr(X), dbl(X)))
asqr(X) -> sqr(X)
adbl(0) -> 0
adbl(s(X)) -> s(s(dbl(X)))
adbl(X) -> dbl(X)

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(MARK(x1)) =  x1 POL(sqr(x1)) =  x1 POL(a__terms(x1)) =  1 + x1 POL(a__dbl(x1)) =  0 POL(terms(x1)) =  1 + x1 POL(mark(x1)) =  x1 POL(a__add(x1, x2)) =  x2 POL(a__first(x1, x2)) =  x2 POL(add(x1, x2)) =  x2 POL(0) =  0 POL(first(x1, x2)) =  x2 POL(cons(x1, x2)) =  x1 POL(a__sqr(x1)) =  x1 POL(dbl(x1)) =  0 POL(nil) =  0 POL(s(x1)) =  0 POL(recip(x1)) =  x1 POL(A__TERMS(x1)) =  x1

resulting in one new DP problem.

R
DPs
→DP Problem 1
Polo
→DP Problem 2
Polo
...
→DP Problem 7
Dependency Graph

Dependency Pairs:

MARK(recip(X)) -> MARK(X)
MARK(sqr(X)) -> MARK(X)
ATERMS(N) -> MARK(N)
MARK(cons(X1, X2)) -> MARK(X1)

Rules:

aterms(N) -> cons(recip(asqr(mark(N))), terms(s(N)))
aterms(X) -> terms(X)
asqr(0) -> 0
asqr(s(X)) -> s(add(sqr(X), dbl(X)))
asqr(X) -> sqr(X)
adbl(0) -> 0
adbl(s(X)) -> s(s(dbl(X)))
adbl(X) -> dbl(X)
aadd(0, X) -> mark(X)
aadd(s(X), Y) -> s(add(X, Y))
aadd(X1, X2) -> add(X1, X2)
afirst(0, X) -> nil
afirst(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
afirst(X1, X2) -> first(X1, X2)
mark(terms(X)) -> aterms(mark(X))
mark(sqr(X)) -> asqr(mark(X))
mark(add(X1, X2)) -> aadd(mark(X1), mark(X2))
mark(dbl(X)) -> adbl(mark(X))
mark(first(X1, X2)) -> afirst(mark(X1), mark(X2))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(recip(X)) -> recip(mark(X))
mark(s(X)) -> s(X)
mark(0) -> 0
mark(nil) -> nil

Strategy:

innermost

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

R
DPs
→DP Problem 1
Polo
→DP Problem 2
Polo
...
→DP Problem 8
Polynomial Ordering

Dependency Pairs:

MARK(cons(X1, X2)) -> MARK(X1)
MARK(sqr(X)) -> MARK(X)
MARK(recip(X)) -> MARK(X)

Rules:

aterms(N) -> cons(recip(asqr(mark(N))), terms(s(N)))
aterms(X) -> terms(X)
asqr(0) -> 0
asqr(s(X)) -> s(add(sqr(X), dbl(X)))
asqr(X) -> sqr(X)
adbl(0) -> 0
adbl(s(X)) -> s(s(dbl(X)))
adbl(X) -> dbl(X)
aadd(0, X) -> mark(X)
aadd(s(X), Y) -> s(add(X, Y))
aadd(X1, X2) -> add(X1, X2)
afirst(0, X) -> nil
afirst(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
afirst(X1, X2) -> first(X1, X2)
mark(terms(X)) -> aterms(mark(X))
mark(sqr(X)) -> asqr(mark(X))
mark(add(X1, X2)) -> aadd(mark(X1), mark(X2))
mark(dbl(X)) -> adbl(mark(X))
mark(first(X1, X2)) -> afirst(mark(X1), mark(X2))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(recip(X)) -> recip(mark(X))
mark(s(X)) -> s(X)
mark(0) -> 0
mark(nil) -> nil

Strategy:

innermost

The following dependency pair can be strictly oriented:

MARK(cons(X1, X2)) -> MARK(X1)

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(MARK(x1)) =  x1 POL(cons(x1, x2)) =  1 + x1 POL(sqr(x1)) =  x1 POL(recip(x1)) =  x1

resulting in one new DP problem.

R
DPs
→DP Problem 1
Polo
→DP Problem 2
Polo
...
→DP Problem 9
Polynomial Ordering

Dependency Pairs:

MARK(sqr(X)) -> MARK(X)
MARK(recip(X)) -> MARK(X)

Rules:

aterms(N) -> cons(recip(asqr(mark(N))), terms(s(N)))
aterms(X) -> terms(X)
asqr(0) -> 0
asqr(s(X)) -> s(add(sqr(X), dbl(X)))
asqr(X) -> sqr(X)
adbl(0) -> 0
adbl(s(X)) -> s(s(dbl(X)))
adbl(X) -> dbl(X)
aadd(0, X) -> mark(X)
aadd(s(X), Y) -> s(add(X, Y))
aadd(X1, X2) -> add(X1, X2)
afirst(0, X) -> nil
afirst(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
afirst(X1, X2) -> first(X1, X2)
mark(terms(X)) -> aterms(mark(X))
mark(sqr(X)) -> asqr(mark(X))
mark(add(X1, X2)) -> aadd(mark(X1), mark(X2))
mark(dbl(X)) -> adbl(mark(X))
mark(first(X1, X2)) -> afirst(mark(X1), mark(X2))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(recip(X)) -> recip(mark(X))
mark(s(X)) -> s(X)
mark(0) -> 0
mark(nil) -> nil

Strategy:

innermost

The following dependency pair can be strictly oriented:

MARK(sqr(X)) -> MARK(X)

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(MARK(x1)) =  x1 POL(sqr(x1)) =  1 + x1 POL(recip(x1)) =  x1

resulting in one new DP problem.

R
DPs
→DP Problem 1
Polo
→DP Problem 2
Polo
...
→DP Problem 10
Polynomial Ordering

Dependency Pair:

MARK(recip(X)) -> MARK(X)

Rules:

aterms(N) -> cons(recip(asqr(mark(N))), terms(s(N)))
aterms(X) -> terms(X)
asqr(0) -> 0
asqr(s(X)) -> s(add(sqr(X), dbl(X)))
asqr(X) -> sqr(X)
adbl(0) -> 0
adbl(s(X)) -> s(s(dbl(X)))
adbl(X) -> dbl(X)
aadd(0, X) -> mark(X)
aadd(s(X), Y) -> s(add(X, Y))
aadd(X1, X2) -> add(X1, X2)
afirst(0, X) -> nil
afirst(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
afirst(X1, X2) -> first(X1, X2)
mark(terms(X)) -> aterms(mark(X))
mark(sqr(X)) -> asqr(mark(X))
mark(add(X1, X2)) -> aadd(mark(X1), mark(X2))
mark(dbl(X)) -> adbl(mark(X))
mark(first(X1, X2)) -> afirst(mark(X1), mark(X2))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(recip(X)) -> recip(mark(X))
mark(s(X)) -> s(X)
mark(0) -> 0
mark(nil) -> nil

Strategy:

innermost

The following dependency pair can be strictly oriented:

MARK(recip(X)) -> MARK(X)

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(MARK(x1)) =  x1 POL(recip(x1)) =  1 + x1

resulting in one new DP problem.

R
DPs
→DP Problem 1
Polo
→DP Problem 2
Polo
...
→DP Problem 11
Dependency Graph

Dependency Pair:

Rules:

aterms(N) -> cons(recip(asqr(mark(N))), terms(s(N)))
aterms(X) -> terms(X)
asqr(0) -> 0
asqr(s(X)) -> s(add(sqr(X), dbl(X)))
asqr(X) -> sqr(X)
adbl(0) -> 0
adbl(s(X)) -> s(s(dbl(X)))
adbl(X) -> dbl(X)
aadd(0, X) -> mark(X)
aadd(s(X), Y) -> s(add(X, Y))
aadd(X1, X2) -> add(X1, X2)
afirst(0, X) -> nil
afirst(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
afirst(X1, X2) -> first(X1, X2)
mark(terms(X)) -> aterms(mark(X))
mark(sqr(X)) -> asqr(mark(X))
mark(add(X1, X2)) -> aadd(mark(X1), mark(X2))
mark(dbl(X)) -> adbl(mark(X))
mark(first(X1, X2)) -> afirst(mark(X1), mark(X2))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(recip(X)) -> recip(mark(X))
mark(s(X)) -> s(X)
mark(0) -> 0
mark(nil) -> nil

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:16 minutes