Term Rewriting System R:
[x, y]
natsactive -> nats
hdactive(x) -> hd(x)
hdactive(cons(x, y)) -> mark(x)
zerosactive -> cons(0, zeros)
zerosactive -> zeros
tlactive(x) -> tl(x)
tlactive(cons(x, y)) -> mark(y)
incractive(cons(x, y)) -> cons(s(x), incr(y))
incractive(x) -> incr(x)
mark(nats) -> natsactive
mark(zeros) -> zerosactive
mark(incr(x)) -> incractive(mark(x))
mark(hd(x)) -> hdactive(mark(x))
mark(tl(x)) -> tlactive(mark(x))
mark(0) -> 0
mark(s(x)) -> s(x)
mark(cons(x, y)) -> cons(x, y)

Termination of R to be shown.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

NATSACTIVE -> ZEROSACTIVE
HDACTIVE(cons(x, y)) -> MARK(x)
TLACTIVE(cons(x, y)) -> MARK(y)
MARK(nats) -> NATSACTIVE
MARK(zeros) -> ZEROSACTIVE
MARK(incr(x)) -> INCRACTIVE(mark(x))
MARK(incr(x)) -> MARK(x)
MARK(hd(x)) -> HDACTIVE(mark(x))
MARK(hd(x)) -> MARK(x)
MARK(tl(x)) -> TLACTIVE(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)
TLACTIVE(cons(x, y)) -> MARK(y)
MARK(tl(x)) -> TLACTIVE(mark(x))
MARK(hd(x)) -> MARK(x)
MARK(hd(x)) -> HDACTIVE(mark(x))
MARK(incr(x)) -> MARK(x)
HDACTIVE(cons(x, y)) -> MARK(x)

Rules:

natsactive -> nats
hdactive(x) -> hd(x)
hdactive(cons(x, y)) -> mark(x)
zerosactive -> cons(0, zeros)
zerosactive -> zeros
tlactive(x) -> tl(x)
tlactive(cons(x, y)) -> mark(y)
incractive(cons(x, y)) -> cons(s(x), incr(y))
incractive(x) -> incr(x)
mark(nats) -> natsactive
mark(zeros) -> zerosactive
mark(incr(x)) -> incractive(mark(x))
mark(hd(x)) -> hdactive(mark(x))
mark(tl(x)) -> tlactive(mark(x))
mark(0) -> 0
mark(s(x)) -> s(x)
mark(cons(x, y)) -> cons(x, y)

The following dependency pairs can be strictly oriented:

MARK(hd(x)) -> MARK(x)
MARK(hd(x)) -> HDACTIVE(mark(x))

Additionally, the following rules can be oriented:

natsactive -> nats
zerosactive -> cons(0, zeros)
zerosactive -> zeros
hdactive(x) -> hd(x)
hdactive(cons(x, y)) -> mark(x)
mark(nats) -> natsactive
mark(zeros) -> zerosactive
mark(incr(x)) -> incractive(mark(x))
mark(hd(x)) -> hdactive(mark(x))
mark(tl(x)) -> tlactive(mark(x))
mark(0) -> 0
mark(s(x)) -> s(x)
mark(cons(x, y)) -> cons(x, y)
tlactive(x) -> tl(x)
tlactive(cons(x, y)) -> mark(y)
incractive(cons(x, y)) -> cons(s(x), incr(y))
incractive(x) -> incr(x)

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(zeros_active) =  0 POL(incr_active(x1)) =  x1 POL(hd_active(x1)) =  1 + x1 POL(MARK(x1)) =  x1 POL(TL_ACTIVE(x1)) =  x1 POL(incr(x1)) =  x1 POL(tl(x1)) =  x1 POL(mark(x1)) =  x1 POL(HD_ACTIVE(x1)) =  x1 POL(add(x1)) =  x1 POL(add_active(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(nats_active) =  0 POL(tl_active(x1)) =  x1

resulting in one new DP problem.

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

Dependency Pairs:

MARK(tl(x)) -> MARK(x)
TLACTIVE(cons(x, y)) -> MARK(y)
MARK(tl(x)) -> TLACTIVE(mark(x))
MARK(incr(x)) -> MARK(x)
HDACTIVE(cons(x, y)) -> MARK(x)

Rules:

natsactive -> nats
hdactive(x) -> hd(x)
hdactive(cons(x, y)) -> mark(x)
zerosactive -> cons(0, zeros)
zerosactive -> zeros
tlactive(x) -> tl(x)
tlactive(cons(x, y)) -> mark(y)
incractive(cons(x, y)) -> cons(s(x), incr(y))
incractive(x) -> incr(x)
mark(nats) -> natsactive
mark(zeros) -> zerosactive
mark(incr(x)) -> incractive(mark(x))
mark(hd(x)) -> hdactive(mark(x))
mark(tl(x)) -> tlactive(mark(x))
mark(0) -> 0
mark(s(x)) -> s(x)
mark(cons(x, y)) -> cons(x, y)

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:

TLACTIVE(cons(x, y)) -> MARK(y)
MARK(tl(x)) -> TLACTIVE(mark(x))
MARK(incr(x)) -> MARK(x)
MARK(tl(x)) -> MARK(x)

Rules:

natsactive -> nats
hdactive(x) -> hd(x)
hdactive(cons(x, y)) -> mark(x)
zerosactive -> cons(0, zeros)
zerosactive -> zeros
tlactive(x) -> tl(x)
tlactive(cons(x, y)) -> mark(y)
incractive(cons(x, y)) -> cons(s(x), incr(y))
incractive(x) -> incr(x)
mark(nats) -> natsactive
mark(zeros) -> zerosactive
mark(incr(x)) -> incractive(mark(x))
mark(hd(x)) -> hdactive(mark(x))
mark(tl(x)) -> tlactive(mark(x))
mark(0) -> 0
mark(s(x)) -> s(x)
mark(cons(x, y)) -> cons(x, y)

The following dependency pairs can be strictly oriented:

MARK(tl(x)) -> TLACTIVE(mark(x))
MARK(tl(x)) -> MARK(x)

Additionally, the following rules can be oriented:

natsactive -> nats
zerosactive -> cons(0, zeros)
zerosactive -> zeros
hdactive(x) -> hd(x)
hdactive(cons(x, y)) -> mark(x)
mark(nats) -> natsactive
mark(zeros) -> zerosactive
mark(incr(x)) -> incractive(mark(x))
mark(hd(x)) -> hdactive(mark(x))
mark(tl(x)) -> tlactive(mark(x))
mark(0) -> 0
mark(s(x)) -> s(x)
mark(cons(x, y)) -> cons(x, y)
tlactive(x) -> tl(x)
tlactive(cons(x, y)) -> mark(y)
incractive(cons(x, y)) -> cons(s(x), incr(y))
incractive(x) -> incr(x)

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(zeros_active) =  0 POL(incr_active(x1)) =  x1 POL(hd_active(x1)) =  x1 POL(MARK(x1)) =  x1 POL(TL_ACTIVE(x1)) =  x1 POL(incr(x1)) =  x1 POL(tl(x1)) =  1 + x1 POL(mark(x1)) =  x1 POL(add(x1)) =  x1 POL(add_active(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(nats_active) =  0 POL(tl_active(x1)) =  1 + x1

resulting in one new DP problem.

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

Dependency Pairs:

TLACTIVE(cons(x, y)) -> MARK(y)
MARK(incr(x)) -> MARK(x)

Rules:

natsactive -> nats
hdactive(x) -> hd(x)
hdactive(cons(x, y)) -> mark(x)
zerosactive -> cons(0, zeros)
zerosactive -> zeros
tlactive(x) -> tl(x)
tlactive(cons(x, y)) -> mark(y)
incractive(cons(x, y)) -> cons(s(x), incr(y))
incractive(x) -> incr(x)
mark(nats) -> natsactive
mark(zeros) -> zerosactive
mark(incr(x)) -> incractive(mark(x))
mark(hd(x)) -> hdactive(mark(x))
mark(tl(x)) -> tlactive(mark(x))
mark(0) -> 0
mark(s(x)) -> s(x)
mark(cons(x, y)) -> cons(x, y)

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:

natsactive -> nats
hdactive(x) -> hd(x)
hdactive(cons(x, y)) -> mark(x)
zerosactive -> cons(0, zeros)
zerosactive -> zeros
tlactive(x) -> tl(x)
tlactive(cons(x, y)) -> mark(y)
incractive(cons(x, y)) -> cons(s(x), incr(y))
incractive(x) -> incr(x)
mark(nats) -> natsactive
mark(zeros) -> zerosactive
mark(incr(x)) -> incractive(mark(x))
mark(hd(x)) -> hdactive(mark(x))
mark(tl(x)) -> tlactive(mark(x))
mark(0) -> 0
mark(s(x)) -> s(x)
mark(cons(x, y)) -> cons(x, y)

The following dependency pair can be strictly oriented:

Additionally, the following rules can be oriented:

natsactive -> nats
zerosactive -> cons(0, zeros)
zerosactive -> zeros
hdactive(x) -> hd(x)
hdactive(cons(x, y)) -> mark(x)
mark(nats) -> natsactive
mark(zeros) -> zerosactive
mark(incr(x)) -> incractive(mark(x))
mark(hd(x)) -> hdactive(mark(x))
mark(tl(x)) -> tlactive(mark(x))
mark(0) -> 0
mark(s(x)) -> s(x)
mark(cons(x, y)) -> cons(x, y)
tlactive(x) -> tl(x)
tlactive(cons(x, y)) -> mark(y)
incractive(cons(x, y)) -> cons(s(x), incr(y))
incractive(x) -> incr(x)

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(zeros_active) =  0 POL(incr_active(x1)) =  x1 POL(hd_active(x1)) =  x1 POL(MARK(x1)) =  1 + x1 POL(incr(x1)) =  x1 POL(tl(x1)) =  x1 POL(mark(x1)) =  x1 POL(add(x1)) =  1 + x1 POL(add_active(x1)) =  1 + x1 POL(0) =  0 POL(cons(x1, x2)) =  x1 + x2 POL(hd(x1)) =  x1 POL(nats) =  1 POL(s(x1)) =  0 POL(zeros) =  0 POL(nats_active) =  1 POL(tl_active(x1)) =  x1

resulting in one new DP problem.

R
DPs
→DP Problem 1
Polo
→DP Problem 2
DGraph
...
→DP Problem 6
Remaining Obligation(s)

The following remains to be proven:
Dependency Pair:

MARK(incr(x)) -> MARK(x)

Rules:

natsactive -> nats
hdactive(x) -> hd(x)
hdactive(cons(x, y)) -> mark(x)
zerosactive -> cons(0, zeros)
zerosactive -> zeros
tlactive(x) -> tl(x)
tlactive(cons(x, y)) -> mark(y)
incractive(cons(x, y)) -> cons(s(x), incr(y))
incractive(x) -> incr(x)
mark(nats) -> natsactive
mark(zeros) -> zerosactive
mark(incr(x)) -> incractive(mark(x))
mark(hd(x)) -> hdactive(mark(x))
mark(tl(x)) -> tlactive(mark(x))
mark(0) -> 0
mark(s(x)) -> s(x)
mark(cons(x, y)) -> cons(x, y)