Term Rewriting System R:
[X, L, X1, X2]
aincr(nil) -> nil
aincr(cons(X, L)) -> cons(s(mark(X)), incr(L))
aincr(X) -> incr(X)
anats -> nats
azeros -> cons(0, zeros)
azeros -> zeros
atail(cons(X, L)) -> mark(L)
atail(X) -> tail(X)
mark(incr(X)) -> aincr(mark(X))
mark(nats) -> anats
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
mark(0) -> 0

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

AINCR(cons(X, L)) -> MARK(X)
ANATS -> AZEROS
ATAIL(cons(X, L)) -> MARK(L)
MARK(incr(X)) -> AINCR(mark(X))
MARK(incr(X)) -> MARK(X)
MARK(nats) -> ANATS
MARK(zeros) -> AZEROS
MARK(tail(X)) -> ATAIL(mark(X))
MARK(tail(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(s(X)) -> MARK(X)

Furthermore, R contains one SCC.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polynomial Ordering`

Dependency Pairs:

MARK(s(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(tail(X)) -> MARK(X)
ATAIL(cons(X, L)) -> MARK(L)
MARK(tail(X)) -> ATAIL(mark(X))
MARK(nats) -> ANATS
MARK(incr(X)) -> MARK(X)
MARK(incr(X)) -> AINCR(mark(X))
AINCR(cons(X, L)) -> MARK(X)

Rules:

aincr(nil) -> nil
aincr(cons(X, L)) -> cons(s(mark(X)), incr(L))
aincr(X) -> incr(X)
anats -> nats
azeros -> cons(0, zeros)
azeros -> zeros
atail(cons(X, L)) -> mark(L)
atail(X) -> tail(X)
mark(incr(X)) -> aincr(mark(X))
mark(nats) -> anats
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
mark(0) -> 0

Strategy:

innermost

The following dependency pair can be strictly oriented:

MARK(nats) -> ANATS

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

anats -> nats
atail(cons(X, L)) -> mark(L)
atail(X) -> tail(X)
mark(incr(X)) -> aincr(mark(X))
mark(nats) -> anats
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
mark(0) -> 0
aincr(nil) -> nil
aincr(cons(X, L)) -> cons(s(mark(X)), incr(L))
aincr(X) -> incr(X)
azeros -> cons(0, zeros)
azeros -> zeros

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(a__nats) =  1 POL(MARK(x1)) =  x1 POL(adx(x1)) =  x1 POL(A__NATS) =  0 POL(a__zeros) =  0 POL(A__ADX(x1)) =  x1 POL(tail(x1)) =  x1 POL(incr(x1)) =  x1 POL(mark(x1)) =  x1 POL(A__TAIL(x1)) =  x1 POL(a__adx(x1)) =  x1 POL(A__INCR(x1)) =  x1 POL(0) =  0 POL(cons(x1, x2)) =  x1 + x2 POL(nats) =  1 POL(nil) =  0 POL(a__tail(x1)) =  x1 POL(s(x1)) =  x1 POL(head(x1)) =  x1 POL(zeros) =  0 POL(a__head(x1)) =  x1 POL(A__HEAD(x1)) =  x1 POL(a__incr(x1)) =  x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`           →DP Problem 2`
`             ↳Dependency Graph`

Dependency Pairs:

MARK(s(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(tail(X)) -> MARK(X)
ATAIL(cons(X, L)) -> MARK(L)
MARK(tail(X)) -> ATAIL(mark(X))
MARK(incr(X)) -> MARK(X)
MARK(incr(X)) -> AINCR(mark(X))
AINCR(cons(X, L)) -> MARK(X)

Rules:

aincr(nil) -> nil
aincr(cons(X, L)) -> cons(s(mark(X)), incr(L))
aincr(X) -> incr(X)
anats -> nats
azeros -> cons(0, zeros)
azeros -> zeros
atail(cons(X, L)) -> mark(L)
atail(X) -> tail(X)
mark(incr(X)) -> aincr(mark(X))
mark(nats) -> anats
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
mark(0) -> 0

Strategy:

innermost

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`

Dependency Pairs:

MARK(cons(X1, X2)) -> MARK(X1)
MARK(tail(X)) -> MARK(X)
ATAIL(cons(X, L)) -> MARK(L)
MARK(tail(X)) -> ATAIL(mark(X))
MARK(incr(X)) -> MARK(X)
AINCR(cons(X, L)) -> MARK(X)
MARK(incr(X)) -> AINCR(mark(X))
MARK(s(X)) -> MARK(X)

Rules:

aincr(nil) -> nil
aincr(cons(X, L)) -> cons(s(mark(X)), incr(L))
aincr(X) -> incr(X)
anats -> nats
azeros -> cons(0, zeros)
azeros -> zeros
atail(cons(X, L)) -> mark(L)
atail(X) -> tail(X)
mark(incr(X)) -> aincr(mark(X))
mark(nats) -> anats
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
mark(0) -> 0

Strategy:

innermost

The following dependency pairs can be strictly oriented:

MARK(tail(X)) -> MARK(X)
MARK(tail(X)) -> ATAIL(mark(X))

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

anats -> nats
atail(cons(X, L)) -> mark(L)
atail(X) -> tail(X)
mark(incr(X)) -> aincr(mark(X))
mark(nats) -> anats
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
mark(0) -> 0
aincr(nil) -> nil
aincr(cons(X, L)) -> cons(s(mark(X)), incr(L))
aincr(X) -> incr(X)
azeros -> cons(0, zeros)
azeros -> zeros

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(a__nats) =  0 POL(MARK(x1)) =  x1 POL(adx(x1)) =  x1 POL(a__zeros) =  0 POL(A__ADX(x1)) =  x1 POL(tail(x1)) =  1 + x1 POL(incr(x1)) =  x1 POL(mark(x1)) =  x1 POL(A__TAIL(x1)) =  x1 POL(a__adx(x1)) =  x1 POL(A__INCR(x1)) =  x1 POL(0) =  0 POL(cons(x1, x2)) =  x1 + x2 POL(nats) =  0 POL(nil) =  0 POL(a__tail(x1)) =  1 + x1 POL(s(x1)) =  x1 POL(head(x1)) =  x1 POL(zeros) =  0 POL(a__head(x1)) =  x1 POL(A__HEAD(x1)) =  x1 POL(a__incr(x1)) =  x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`           →DP Problem 2`
`             ↳DGraph`
`             ...`
`               →DP Problem 4`
`                 ↳Dependency Graph`

Dependency Pairs:

MARK(cons(X1, X2)) -> MARK(X1)
ATAIL(cons(X, L)) -> MARK(L)
MARK(incr(X)) -> MARK(X)
AINCR(cons(X, L)) -> MARK(X)
MARK(incr(X)) -> AINCR(mark(X))
MARK(s(X)) -> MARK(X)

Rules:

aincr(nil) -> nil
aincr(cons(X, L)) -> cons(s(mark(X)), incr(L))
aincr(X) -> incr(X)
anats -> nats
azeros -> cons(0, zeros)
azeros -> zeros
atail(cons(X, L)) -> mark(L)
atail(X) -> tail(X)
mark(incr(X)) -> aincr(mark(X))
mark(nats) -> anats
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
mark(0) -> 0

Strategy:

innermost

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 5`
`                 ↳Polynomial Ordering`

Dependency Pairs:

MARK(s(X)) -> MARK(X)
MARK(incr(X)) -> MARK(X)
AINCR(cons(X, L)) -> MARK(X)
MARK(incr(X)) -> AINCR(mark(X))
MARK(cons(X1, X2)) -> MARK(X1)

Rules:

aincr(nil) -> nil
aincr(cons(X, L)) -> cons(s(mark(X)), incr(L))
aincr(X) -> incr(X)
anats -> nats
azeros -> cons(0, zeros)
azeros -> zeros
atail(cons(X, L)) -> mark(L)
atail(X) -> tail(X)
mark(incr(X)) -> aincr(mark(X))
mark(nats) -> anats
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
mark(0) -> 0

Strategy:

innermost

The following dependency pairs can be strictly oriented:

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

anats -> nats
atail(cons(X, L)) -> mark(L)
atail(X) -> tail(X)
mark(incr(X)) -> aincr(mark(X))
mark(nats) -> anats
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
mark(0) -> 0
aincr(nil) -> nil
aincr(cons(X, L)) -> cons(s(mark(X)), incr(L))
aincr(X) -> incr(X)
azeros -> cons(0, zeros)
azeros -> zeros

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(a__nats) =  0 POL(MARK(x1)) =  x1 POL(adx(x1)) =  x1 POL(a__zeros) =  0 POL(A__ADX(x1)) =  x1 POL(tail(x1)) =  x1 POL(incr(x1)) =  x1 POL(mark(x1)) =  x1 POL(a__adx(x1)) =  x1 POL(A__INCR(x1)) =  x1 POL(0) =  0 POL(cons(x1, x2)) =  x1 + x2 POL(nats) =  0 POL(nil) =  0 POL(a__tail(x1)) =  x1 POL(s(x1)) =  x1 POL(head(x1)) =  1 + x1 POL(zeros) =  0 POL(a__head(x1)) =  1 + x1 POL(A__HEAD(x1)) =  x1 POL(a__incr(x1)) =  x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`           →DP Problem 2`
`             ↳DGraph`
`             ...`
`               →DP Problem 6`
`                 ↳Dependency Graph`

Dependency Pairs:

MARK(s(X)) -> MARK(X)
MARK(incr(X)) -> MARK(X)
AINCR(cons(X, L)) -> MARK(X)
MARK(incr(X)) -> AINCR(mark(X))
MARK(cons(X1, X2)) -> MARK(X1)

Rules:

aincr(nil) -> nil
aincr(cons(X, L)) -> cons(s(mark(X)), incr(L))
aincr(X) -> incr(X)
anats -> nats
azeros -> cons(0, zeros)
azeros -> zeros
atail(cons(X, L)) -> mark(L)
atail(X) -> tail(X)
mark(incr(X)) -> aincr(mark(X))
mark(nats) -> anats
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
mark(0) -> 0

Strategy:

innermost

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 7`
`                 ↳Polynomial Ordering`

Dependency Pairs:

MARK(cons(X1, X2)) -> MARK(X1)
MARK(incr(X)) -> MARK(X)
AINCR(cons(X, L)) -> MARK(X)
MARK(incr(X)) -> AINCR(mark(X))
MARK(s(X)) -> MARK(X)

Rules:

aincr(nil) -> nil
aincr(cons(X, L)) -> cons(s(mark(X)), incr(L))
aincr(X) -> incr(X)
anats -> nats
azeros -> cons(0, zeros)
azeros -> zeros
atail(cons(X, L)) -> mark(L)
atail(X) -> tail(X)
mark(incr(X)) -> aincr(mark(X))
mark(nats) -> anats
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
mark(0) -> 0

Strategy:

innermost

The following dependency pairs can be strictly oriented:

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

anats -> nats
atail(cons(X, L)) -> mark(L)
atail(X) -> tail(X)
mark(incr(X)) -> aincr(mark(X))
mark(nats) -> anats
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
mark(0) -> 0
aincr(nil) -> nil
aincr(cons(X, L)) -> cons(s(mark(X)), incr(L))
aincr(X) -> incr(X)
azeros -> cons(0, zeros)
azeros -> zeros

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(a__nats) =  1 POL(MARK(x1)) =  x1 POL(adx(x1)) =  1 + x1 POL(a__zeros) =  0 POL(A__ADX(x1)) =  1 + x1 POL(tail(x1)) =  x1 POL(incr(x1)) =  x1 POL(mark(x1)) =  x1 POL(a__adx(x1)) =  1 + x1 POL(A__INCR(x1)) =  x1 POL(0) =  0 POL(cons(x1, x2)) =  x1 + x2 POL(nats) =  1 POL(nil) =  0 POL(a__tail(x1)) =  x1 POL(s(x1)) =  x1 POL(head(x1)) =  x1 POL(zeros) =  0 POL(a__head(x1)) =  x1 POL(a__incr(x1)) =  x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`           →DP Problem 2`
`             ↳DGraph`
`             ...`
`               →DP Problem 8`
`                 ↳Narrowing Transformation`

Dependency Pairs:

MARK(cons(X1, X2)) -> MARK(X1)
MARK(incr(X)) -> MARK(X)
AINCR(cons(X, L)) -> MARK(X)
MARK(incr(X)) -> AINCR(mark(X))
MARK(s(X)) -> MARK(X)

Rules:

aincr(nil) -> nil
aincr(cons(X, L)) -> cons(s(mark(X)), incr(L))
aincr(X) -> incr(X)
anats -> nats
azeros -> cons(0, zeros)
azeros -> zeros
atail(cons(X, L)) -> mark(L)
atail(X) -> tail(X)
mark(incr(X)) -> aincr(mark(X))
mark(nats) -> anats
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
mark(0) -> 0

Strategy:

innermost

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MARK(incr(X)) -> AINCR(mark(X))
10 new Dependency Pairs are created:

MARK(incr(incr(X''))) -> AINCR(aincr(mark(X'')))
MARK(incr(nats)) -> AINCR(anats)
MARK(incr(zeros)) -> AINCR(azeros)
MARK(incr(tail(X''))) -> AINCR(atail(mark(X'')))
MARK(incr(nil)) -> AINCR(nil)
MARK(incr(cons(X1', X2'))) -> AINCR(cons(mark(X1'), X2'))
MARK(incr(s(X''))) -> AINCR(s(mark(X'')))
MARK(incr(0)) -> AINCR(0)

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`           →DP Problem 2`
`             ↳DGraph`
`             ...`
`               →DP Problem 9`
`                 ↳Narrowing Transformation`

Dependency Pairs:

MARK(incr(cons(X1', X2'))) -> AINCR(cons(mark(X1'), X2'))
MARK(incr(tail(X''))) -> AINCR(atail(mark(X'')))
MARK(incr(zeros)) -> AINCR(azeros)
MARK(incr(nats)) -> AINCR(anats)
MARK(incr(incr(X''))) -> AINCR(aincr(mark(X'')))
MARK(s(X)) -> MARK(X)
AINCR(cons(X, L)) -> MARK(X)
MARK(incr(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)

Rules:

aincr(nil) -> nil
aincr(cons(X, L)) -> cons(s(mark(X)), incr(L))
aincr(X) -> incr(X)
anats -> nats
azeros -> cons(0, zeros)
azeros -> zeros
atail(cons(X, L)) -> mark(L)
atail(X) -> tail(X)
mark(incr(X)) -> aincr(mark(X))
mark(nats) -> anats
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
mark(0) -> 0

Strategy:

innermost

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

10 new Dependency Pairs are created:

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`           →DP Problem 2`
`             ↳DGraph`
`             ...`
`               →DP Problem 10`
`                 ↳Remaining Obligation(s)`

The following remains to be proven:
Dependency Pairs:

MARK(incr(tail(X''))) -> AINCR(atail(mark(X'')))
MARK(incr(zeros)) -> AINCR(azeros)
MARK(incr(nats)) -> AINCR(anats)
MARK(incr(incr(X''))) -> AINCR(aincr(mark(X'')))
MARK(s(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(incr(X)) -> MARK(X)
AINCR(cons(X, L)) -> MARK(X)
MARK(incr(cons(X1', X2'))) -> AINCR(cons(mark(X1'), X2'))

Rules:

aincr(nil) -> nil
aincr(cons(X, L)) -> cons(s(mark(X)), incr(L))
aincr(X) -> incr(X)
anats -> nats
azeros -> cons(0, zeros)
azeros -> zeros
atail(cons(X, L)) -> mark(L)
atail(X) -> tail(X)
mark(incr(X)) -> aincr(mark(X))
mark(nats) -> anats
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
mark(0) -> 0

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:12 minutes