Term Rewriting System R:
[X, Y, X1, X2]
anats -> nats
azeros -> cons(0, zeros)
azeros -> zeros
aincr(cons(X, Y)) -> cons(s(X), incr(Y))
aincr(X) -> incr(X)
ahd(cons(X, Y)) -> mark(X)
ahd(X) -> hd(X)
atl(cons(X, Y)) -> mark(Y)
atl(X) -> tl(X)
mark(nats) -> anats
mark(zeros) -> azeros
mark(incr(X)) -> aincr(mark(X))
mark(hd(X)) -> ahd(mark(X))
mark(tl(X)) -> atl(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

ANATS -> AZEROS
AHD(cons(X, Y)) -> MARK(X)
ATL(cons(X, Y)) -> MARK(Y)
MARK(nats) -> ANATS
MARK(zeros) -> AZEROS
MARK(incr(X)) -> AINCR(mark(X))
MARK(incr(X)) -> MARK(X)
MARK(hd(X)) -> AHD(mark(X))
MARK(hd(X)) -> MARK(X)
MARK(tl(X)) -> ATL(mark(X))
MARK(tl(X)) -> MARK(X)

Furthermore, R contains one SCC.

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

Dependency Pairs:

MARK(tl(X)) -> MARK(X)
ATL(cons(X, Y)) -> MARK(Y)
MARK(tl(X)) -> ATL(mark(X))
MARK(hd(X)) -> MARK(X)
MARK(hd(X)) -> AHD(mark(X))
MARK(incr(X)) -> MARK(X)
AHD(cons(X, Y)) -> MARK(X)

Rules:

anats -> nats
azeros -> cons(0, zeros)
azeros -> zeros
aincr(cons(X, Y)) -> cons(s(X), incr(Y))
aincr(X) -> incr(X)
ahd(cons(X, Y)) -> mark(X)
ahd(X) -> hd(X)
atl(cons(X, Y)) -> mark(Y)
atl(X) -> tl(X)
mark(nats) -> anats
mark(zeros) -> azeros
mark(incr(X)) -> aincr(mark(X))
mark(hd(X)) -> ahd(mark(X))
mark(tl(X)) -> atl(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)

Strategy:

innermost

The following dependency pairs can be strictly oriented:

MARK(tl(X)) -> MARK(X)
MARK(tl(X)) -> ATL(mark(X))

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

ahd(cons(X, Y)) -> mark(X)
ahd(X) -> hd(X)
atl(cons(X, Y)) -> mark(Y)
atl(X) -> tl(X)
mark(nats) -> anats
mark(zeros) -> azeros
mark(incr(X)) -> aincr(mark(X))
mark(hd(X)) -> ahd(mark(X))
mark(tl(X)) -> atl(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)
anats -> nats
azeros -> cons(0, zeros)
azeros -> zeros
aincr(cons(X, Y)) -> cons(s(X), incr(Y))
aincr(X) -> incr(X)

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(a__nats) =  0 POL(MARK(x1)) =  x1 POL(adx(x1)) =  x1 POL(a__zeros) =  0 POL(incr(x1)) =  x1 POL(A__TL(x1)) =  x1 POL(a__hd(x1)) =  x1 POL(mark(x1)) =  x1 POL(a__tl(x1)) =  1 + x1 POL(tl(x1)) =  1 + x1 POL(a__adx(x1)) =  x1 POL(A__HD(x1)) =  x1 POL(0) =  0 POL(cons(x1, x2)) =  x1 + x2 POL(hd(x1)) =  x1 POL(nats) =  0 POL(s(x1)) =  0 POL(zeros) =  0 POL(a__incr(x1)) =  x1

resulting in one new DP problem.

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

Dependency Pairs:

ATL(cons(X, Y)) -> MARK(Y)
MARK(hd(X)) -> MARK(X)
MARK(hd(X)) -> AHD(mark(X))
MARK(incr(X)) -> MARK(X)
AHD(cons(X, Y)) -> MARK(X)

Rules:

anats -> nats
azeros -> cons(0, zeros)
azeros -> zeros
aincr(cons(X, Y)) -> cons(s(X), incr(Y))
aincr(X) -> incr(X)
ahd(cons(X, Y)) -> mark(X)
ahd(X) -> hd(X)
atl(cons(X, Y)) -> mark(Y)
atl(X) -> tl(X)
mark(nats) -> anats
mark(zeros) -> azeros
mark(incr(X)) -> aincr(mark(X))
mark(hd(X)) -> ahd(mark(X))
mark(tl(X)) -> atl(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)

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(hd(X)) -> MARK(X)
AHD(cons(X, Y)) -> MARK(X)
MARK(hd(X)) -> AHD(mark(X))
MARK(incr(X)) -> MARK(X)

Rules:

anats -> nats
azeros -> cons(0, zeros)
azeros -> zeros
aincr(cons(X, Y)) -> cons(s(X), incr(Y))
aincr(X) -> incr(X)
ahd(cons(X, Y)) -> mark(X)
ahd(X) -> hd(X)
atl(cons(X, Y)) -> mark(Y)
atl(X) -> tl(X)
mark(nats) -> anats
mark(zeros) -> azeros
mark(incr(X)) -> aincr(mark(X))
mark(hd(X)) -> ahd(mark(X))
mark(tl(X)) -> atl(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)

Strategy:

innermost

The following dependency pairs can be strictly oriented:

MARK(hd(X)) -> MARK(X)
MARK(hd(X)) -> AHD(mark(X))

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

ahd(cons(X, Y)) -> mark(X)
ahd(X) -> hd(X)
atl(cons(X, Y)) -> mark(Y)
atl(X) -> tl(X)
mark(nats) -> anats
mark(zeros) -> azeros
mark(incr(X)) -> aincr(mark(X))
mark(hd(X)) -> ahd(mark(X))
mark(tl(X)) -> atl(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)
anats -> nats
azeros -> cons(0, zeros)
azeros -> zeros
aincr(cons(X, Y)) -> cons(s(X), incr(Y))
aincr(X) -> incr(X)

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(a__nats) =  0 POL(MARK(x1)) =  x1 POL(adx(x1)) =  x1 POL(a__zeros) =  0 POL(incr(x1)) =  x1 POL(a__hd(x1)) =  1 + x1 POL(mark(x1)) =  x1 POL(a__tl(x1)) =  x1 POL(tl(x1)) =  x1 POL(a__adx(x1)) =  x1 POL(A__HD(x1)) =  x1 POL(0) =  0 POL(cons(x1, x2)) =  x1 + x2 POL(hd(x1)) =  1 + x1 POL(nats) =  0 POL(s(x1)) =  0 POL(zeros) =  0 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:

AHD(cons(X, Y)) -> MARK(X)
MARK(incr(X)) -> MARK(X)

Rules:

anats -> nats
azeros -> cons(0, zeros)
azeros -> zeros
aincr(cons(X, Y)) -> cons(s(X), incr(Y))
aincr(X) -> incr(X)
ahd(cons(X, Y)) -> mark(X)
ahd(X) -> hd(X)
atl(cons(X, Y)) -> mark(Y)
atl(X) -> tl(X)
mark(nats) -> anats
mark(zeros) -> azeros
mark(incr(X)) -> aincr(mark(X))
mark(hd(X)) -> ahd(mark(X))
mark(tl(X)) -> atl(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)

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(incr(X)) -> MARK(X)

Rules:

anats -> nats
azeros -> cons(0, zeros)
azeros -> zeros
aincr(cons(X, Y)) -> cons(s(X), incr(Y))
aincr(X) -> incr(X)
ahd(cons(X, Y)) -> mark(X)
ahd(X) -> hd(X)
atl(cons(X, Y)) -> mark(Y)
atl(X) -> tl(X)
mark(nats) -> anats
mark(zeros) -> azeros
mark(incr(X)) -> aincr(mark(X))
mark(hd(X)) -> ahd(mark(X))
mark(tl(X)) -> atl(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)

Strategy:

innermost

The following dependency pair can be strictly oriented:

MARK(incr(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(adx(x1)) =  x1 POL(incr(x1)) =  1 + x1

resulting in one new DP problem.

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

Dependency Pair:

Rules:

anats -> nats
azeros -> cons(0, zeros)
azeros -> zeros
aincr(cons(X, Y)) -> cons(s(X), incr(Y))
aincr(X) -> incr(X)
ahd(cons(X, Y)) -> mark(X)
ahd(X) -> hd(X)
atl(cons(X, Y)) -> mark(Y)
atl(X) -> tl(X)
mark(nats) -> anats
mark(zeros) -> azeros
mark(incr(X)) -> aincr(mark(X))
mark(hd(X)) -> ahd(mark(X))
mark(tl(X)) -> atl(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)

Strategy:

innermost

The following dependency pair can be strictly oriented:

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(adx(x1)) =  1 + x1

resulting in one new DP problem.

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

Dependency Pair:

Rules:

anats -> nats
azeros -> cons(0, zeros)
azeros -> zeros
aincr(cons(X, Y)) -> cons(s(X), incr(Y))
aincr(X) -> incr(X)
ahd(cons(X, Y)) -> mark(X)
ahd(X) -> hd(X)
atl(cons(X, Y)) -> mark(Y)
atl(X) -> tl(X)
mark(nats) -> anats
mark(zeros) -> azeros
mark(incr(X)) -> aincr(mark(X))
mark(hd(X)) -> ahd(mark(X))
mark(tl(X)) -> atl(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:00 minutes