Term Rewriting System R:
[X, Y, X1, X2]
active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(hd(cons(X, Y))) -> mark(X)
active(tl(cons(X, Y))) -> mark(Y)
active(incr(X)) -> incr(active(X))
active(hd(X)) -> hd(active(X))
active(tl(X)) -> tl(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
tl(mark(X)) -> mark(tl(X))
tl(ok(X)) -> ok(tl(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

ACTIVE(zeros) -> CONS(0, zeros)
ACTIVE(incr(cons(X, Y))) -> CONS(s(X), incr(Y))
ACTIVE(incr(cons(X, Y))) -> S(X)
ACTIVE(incr(cons(X, Y))) -> INCR(Y)
ACTIVE(incr(X)) -> INCR(active(X))
ACTIVE(incr(X)) -> ACTIVE(X)
ACTIVE(hd(X)) -> HD(active(X))
ACTIVE(hd(X)) -> ACTIVE(X)
ACTIVE(tl(X)) -> TL(active(X))
ACTIVE(tl(X)) -> ACTIVE(X)
INCR(mark(X)) -> INCR(X)
INCR(ok(X)) -> INCR(X)
HD(mark(X)) -> HD(X)
HD(ok(X)) -> HD(X)
TL(mark(X)) -> TL(X)
TL(ok(X)) -> TL(X)
PROPER(cons(X1, X2)) -> CONS(proper(X1), proper(X2))
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(incr(X)) -> INCR(proper(X))
PROPER(incr(X)) -> PROPER(X)
PROPER(s(X)) -> S(proper(X))
PROPER(s(X)) -> PROPER(X)
PROPER(hd(X)) -> HD(proper(X))
PROPER(hd(X)) -> PROPER(X)
PROPER(tl(X)) -> TL(proper(X))
PROPER(tl(X)) -> PROPER(X)
CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
S(ok(X)) -> S(X)
TOP(mark(X)) -> TOP(proper(X))
TOP(mark(X)) -> PROPER(X)
TOP(ok(X)) -> TOP(active(X))
TOP(ok(X)) -> ACTIVE(X)

Furthermore, R contains nine SCCs.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polynomial Ordering`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Remaining`

Dependency Pair:

CONS(ok(X1), ok(X2)) -> CONS(X1, X2)

Rules:

active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(hd(cons(X, Y))) -> mark(X)
active(tl(cons(X, Y))) -> mark(Y)
active(incr(X)) -> incr(active(X))
active(hd(X)) -> hd(active(X))
active(tl(X)) -> tl(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
tl(mark(X)) -> mark(tl(X))
tl(ok(X)) -> ok(tl(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

CONS(ok(X1), ok(X2)) -> CONS(X1, X2)

There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(ok(x1)) =  1 + x1 POL(CONS(x1, x2)) =  x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`           →DP Problem 10`
`             ↳Dependency Graph`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Remaining`

Dependency Pair:

Rules:

active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(hd(cons(X, Y))) -> mark(X)
active(tl(cons(X, Y))) -> mark(Y)
active(incr(X)) -> incr(active(X))
active(hd(X)) -> hd(active(X))
active(tl(X)) -> tl(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
tl(mark(X)) -> mark(tl(X))
tl(ok(X)) -> ok(tl(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polynomial Ordering`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Remaining`

Dependency Pair:

S(ok(X)) -> S(X)

Rules:

active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(hd(cons(X, Y))) -> mark(X)
active(tl(cons(X, Y))) -> mark(Y)
active(incr(X)) -> incr(active(X))
active(hd(X)) -> hd(active(X))
active(tl(X)) -> tl(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
tl(mark(X)) -> mark(tl(X))
tl(ok(X)) -> ok(tl(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

S(ok(X)) -> S(X)

There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(S(x1)) =  x1 POL(ok(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`           →DP Problem 11`
`             ↳Dependency Graph`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Remaining`

Dependency Pair:

Rules:

active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(hd(cons(X, Y))) -> mark(X)
active(tl(cons(X, Y))) -> mark(Y)
active(incr(X)) -> incr(active(X))
active(hd(X)) -> hd(active(X))
active(tl(X)) -> tl(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
tl(mark(X)) -> mark(tl(X))
tl(ok(X)) -> ok(tl(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polynomial Ordering`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Remaining`

Dependency Pairs:

INCR(ok(X)) -> INCR(X)
INCR(mark(X)) -> INCR(X)

Rules:

active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(hd(cons(X, Y))) -> mark(X)
active(tl(cons(X, Y))) -> mark(Y)
active(incr(X)) -> incr(active(X))
active(hd(X)) -> hd(active(X))
active(tl(X)) -> tl(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
tl(mark(X)) -> mark(tl(X))
tl(ok(X)) -> ok(tl(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

INCR(ok(X)) -> INCR(X)

There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(INCR(x1)) =  x1 POL(mark(x1)) =  x1 POL(ok(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`           →DP Problem 12`
`             ↳Polynomial Ordering`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Remaining`

Dependency Pair:

INCR(mark(X)) -> INCR(X)

Rules:

active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(hd(cons(X, Y))) -> mark(X)
active(tl(cons(X, Y))) -> mark(Y)
active(incr(X)) -> incr(active(X))
active(hd(X)) -> hd(active(X))
active(tl(X)) -> tl(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
tl(mark(X)) -> mark(tl(X))
tl(ok(X)) -> ok(tl(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

INCR(mark(X)) -> INCR(X)

There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(INCR(x1)) =  x1 POL(mark(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`           →DP Problem 12`
`             ↳Polo`
`             ...`
`               →DP Problem 13`
`                 ↳Dependency Graph`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Remaining`

Dependency Pair:

Rules:

active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(hd(cons(X, Y))) -> mark(X)
active(tl(cons(X, Y))) -> mark(Y)
active(incr(X)) -> incr(active(X))
active(hd(X)) -> hd(active(X))
active(tl(X)) -> tl(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
tl(mark(X)) -> mark(tl(X))
tl(ok(X)) -> ok(tl(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polynomial Ordering`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Remaining`

Dependency Pairs:

Rules:

active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(hd(cons(X, Y))) -> mark(X)
active(tl(cons(X, Y))) -> mark(Y)
active(incr(X)) -> incr(active(X))
active(hd(X)) -> hd(active(X))
active(tl(X)) -> tl(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
tl(mark(X)) -> mark(tl(X))
tl(ok(X)) -> ok(tl(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(ADX(x1)) =  x1 POL(mark(x1)) =  x1 POL(ok(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`           →DP Problem 14`
`             ↳Polynomial Ordering`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Remaining`

Dependency Pair:

Rules:

active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(hd(cons(X, Y))) -> mark(X)
active(tl(cons(X, Y))) -> mark(Y)
active(incr(X)) -> incr(active(X))
active(hd(X)) -> hd(active(X))
active(tl(X)) -> tl(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
tl(mark(X)) -> mark(tl(X))
tl(ok(X)) -> ok(tl(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(ADX(x1)) =  x1 POL(mark(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`           →DP Problem 14`
`             ↳Polo`
`             ...`
`               →DP Problem 15`
`                 ↳Dependency Graph`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Remaining`

Dependency Pair:

Rules:

active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(hd(cons(X, Y))) -> mark(X)
active(tl(cons(X, Y))) -> mark(Y)
active(incr(X)) -> incr(active(X))
active(hd(X)) -> hd(active(X))
active(tl(X)) -> tl(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
tl(mark(X)) -> mark(tl(X))
tl(ok(X)) -> ok(tl(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polynomial Ordering`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Remaining`

Dependency Pairs:

HD(ok(X)) -> HD(X)
HD(mark(X)) -> HD(X)

Rules:

active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(hd(cons(X, Y))) -> mark(X)
active(tl(cons(X, Y))) -> mark(Y)
active(incr(X)) -> incr(active(X))
active(hd(X)) -> hd(active(X))
active(tl(X)) -> tl(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
tl(mark(X)) -> mark(tl(X))
tl(ok(X)) -> ok(tl(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

HD(ok(X)) -> HD(X)

There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(HD(x1)) =  x1 POL(mark(x1)) =  x1 POL(ok(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`           →DP Problem 16`
`             ↳Polynomial Ordering`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Remaining`

Dependency Pair:

HD(mark(X)) -> HD(X)

Rules:

active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(hd(cons(X, Y))) -> mark(X)
active(tl(cons(X, Y))) -> mark(Y)
active(incr(X)) -> incr(active(X))
active(hd(X)) -> hd(active(X))
active(tl(X)) -> tl(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
tl(mark(X)) -> mark(tl(X))
tl(ok(X)) -> ok(tl(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

HD(mark(X)) -> HD(X)

There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(HD(x1)) =  x1 POL(mark(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`           →DP Problem 16`
`             ↳Polo`
`             ...`
`               →DP Problem 17`
`                 ↳Dependency Graph`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Remaining`

Dependency Pair:

Rules:

active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(hd(cons(X, Y))) -> mark(X)
active(tl(cons(X, Y))) -> mark(Y)
active(incr(X)) -> incr(active(X))
active(hd(X)) -> hd(active(X))
active(tl(X)) -> tl(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
tl(mark(X)) -> mark(tl(X))
tl(ok(X)) -> ok(tl(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polynomial Ordering`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Remaining`

Dependency Pairs:

TL(ok(X)) -> TL(X)
TL(mark(X)) -> TL(X)

Rules:

active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(hd(cons(X, Y))) -> mark(X)
active(tl(cons(X, Y))) -> mark(Y)
active(incr(X)) -> incr(active(X))
active(hd(X)) -> hd(active(X))
active(tl(X)) -> tl(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
tl(mark(X)) -> mark(tl(X))
tl(ok(X)) -> ok(tl(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

TL(ok(X)) -> TL(X)

There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(mark(x1)) =  x1 POL(TL(x1)) =  x1 POL(ok(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`           →DP Problem 18`
`             ↳Polynomial Ordering`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Remaining`

Dependency Pair:

TL(mark(X)) -> TL(X)

Rules:

active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(hd(cons(X, Y))) -> mark(X)
active(tl(cons(X, Y))) -> mark(Y)
active(incr(X)) -> incr(active(X))
active(hd(X)) -> hd(active(X))
active(tl(X)) -> tl(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
tl(mark(X)) -> mark(tl(X))
tl(ok(X)) -> ok(tl(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

TL(mark(X)) -> TL(X)

There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(mark(x1)) =  1 + x1 POL(TL(x1)) =  x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`           →DP Problem 18`
`             ↳Polo`
`             ...`
`               →DP Problem 19`
`                 ↳Dependency Graph`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Remaining`

Dependency Pair:

Rules:

active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(hd(cons(X, Y))) -> mark(X)
active(tl(cons(X, Y))) -> mark(Y)
active(incr(X)) -> incr(active(X))
active(hd(X)) -> hd(active(X))
active(tl(X)) -> tl(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
tl(mark(X)) -> mark(tl(X))
tl(ok(X)) -> ok(tl(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polynomial Ordering`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Remaining`

Dependency Pairs:

ACTIVE(tl(X)) -> ACTIVE(X)
ACTIVE(hd(X)) -> ACTIVE(X)
ACTIVE(incr(X)) -> ACTIVE(X)

Rules:

active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(hd(cons(X, Y))) -> mark(X)
active(tl(cons(X, Y))) -> mark(Y)
active(incr(X)) -> incr(active(X))
active(hd(X)) -> hd(active(X))
active(tl(X)) -> tl(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
tl(mark(X)) -> mark(tl(X))
tl(ok(X)) -> ok(tl(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

ACTIVE(tl(X)) -> ACTIVE(X)

There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(ACTIVE(x1)) =  x1 POL(adx(x1)) =  x1 POL(hd(x1)) =  x1 POL(incr(x1)) =  x1 POL(tl(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`           →DP Problem 20`
`             ↳Polynomial Ordering`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Remaining`

Dependency Pairs:

ACTIVE(hd(X)) -> ACTIVE(X)
ACTIVE(incr(X)) -> ACTIVE(X)

Rules:

active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(hd(cons(X, Y))) -> mark(X)
active(tl(cons(X, Y))) -> mark(Y)
active(incr(X)) -> incr(active(X))
active(hd(X)) -> hd(active(X))
active(tl(X)) -> tl(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
tl(mark(X)) -> mark(tl(X))
tl(ok(X)) -> ok(tl(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

ACTIVE(hd(X)) -> ACTIVE(X)

There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(ACTIVE(x1)) =  x1 POL(adx(x1)) =  x1 POL(hd(x1)) =  1 + x1 POL(incr(x1)) =  x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`           →DP Problem 20`
`             ↳Polo`
`             ...`
`               →DP Problem 21`
`                 ↳Polynomial Ordering`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Remaining`

Dependency Pairs:

ACTIVE(incr(X)) -> ACTIVE(X)

Rules:

active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(hd(cons(X, Y))) -> mark(X)
active(tl(cons(X, Y))) -> mark(Y)
active(incr(X)) -> incr(active(X))
active(hd(X)) -> hd(active(X))
active(tl(X)) -> tl(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
tl(mark(X)) -> mark(tl(X))
tl(ok(X)) -> ok(tl(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

ACTIVE(incr(X)) -> ACTIVE(X)

There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(ACTIVE(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`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`           →DP Problem 20`
`             ↳Polo`
`             ...`
`               →DP Problem 22`
`                 ↳Polynomial Ordering`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Remaining`

Dependency Pair:

Rules:

active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(hd(cons(X, Y))) -> mark(X)
active(tl(cons(X, Y))) -> mark(Y)
active(incr(X)) -> incr(active(X))
active(hd(X)) -> hd(active(X))
active(tl(X)) -> tl(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
tl(mark(X)) -> mark(tl(X))
tl(ok(X)) -> ok(tl(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(ACTIVE(x1)) =  x1 POL(adx(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`           →DP Problem 20`
`             ↳Polo`
`             ...`
`               →DP Problem 23`
`                 ↳Dependency Graph`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Remaining`

Dependency Pair:

Rules:

active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(hd(cons(X, Y))) -> mark(X)
active(tl(cons(X, Y))) -> mark(Y)
active(incr(X)) -> incr(active(X))
active(hd(X)) -> hd(active(X))
active(tl(X)) -> tl(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
tl(mark(X)) -> mark(tl(X))
tl(ok(X)) -> ok(tl(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polynomial Ordering`
`       →DP Problem 9`
`         ↳Remaining`

Dependency Pairs:

PROPER(tl(X)) -> PROPER(X)
PROPER(hd(X)) -> PROPER(X)
PROPER(s(X)) -> PROPER(X)
PROPER(incr(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)

Rules:

active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(hd(cons(X, Y))) -> mark(X)
active(tl(cons(X, Y))) -> mark(Y)
active(incr(X)) -> incr(active(X))
active(hd(X)) -> hd(active(X))
active(tl(X)) -> tl(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
tl(mark(X)) -> mark(tl(X))
tl(ok(X)) -> ok(tl(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

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

There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(adx(x1)) =  x1 POL(PROPER(x1)) =  x1 POL(cons(x1, x2)) =  1 + x1 + x2 POL(hd(x1)) =  x1 POL(incr(x1)) =  x1 POL(s(x1)) =  x1 POL(tl(x1)) =  x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`           →DP Problem 24`
`             ↳Polynomial Ordering`
`       →DP Problem 9`
`         ↳Remaining`

Dependency Pairs:

PROPER(tl(X)) -> PROPER(X)
PROPER(hd(X)) -> PROPER(X)
PROPER(s(X)) -> PROPER(X)
PROPER(incr(X)) -> PROPER(X)

Rules:

active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(hd(cons(X, Y))) -> mark(X)
active(tl(cons(X, Y))) -> mark(Y)
active(incr(X)) -> incr(active(X))
active(hd(X)) -> hd(active(X))
active(tl(X)) -> tl(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
tl(mark(X)) -> mark(tl(X))
tl(ok(X)) -> ok(tl(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(adx(x1)) =  1 + x1 POL(PROPER(x1)) =  x1 POL(hd(x1)) =  x1 POL(incr(x1)) =  x1 POL(s(x1)) =  x1 POL(tl(x1)) =  x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`           →DP Problem 24`
`             ↳Polo`
`             ...`
`               →DP Problem 25`
`                 ↳Polynomial Ordering`
`       →DP Problem 9`
`         ↳Remaining`

Dependency Pairs:

PROPER(tl(X)) -> PROPER(X)
PROPER(hd(X)) -> PROPER(X)
PROPER(s(X)) -> PROPER(X)
PROPER(incr(X)) -> PROPER(X)

Rules:

active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(hd(cons(X, Y))) -> mark(X)
active(tl(cons(X, Y))) -> mark(Y)
active(incr(X)) -> incr(active(X))
active(hd(X)) -> hd(active(X))
active(tl(X)) -> tl(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
tl(mark(X)) -> mark(tl(X))
tl(ok(X)) -> ok(tl(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

PROPER(tl(X)) -> PROPER(X)

There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(PROPER(x1)) =  x1 POL(hd(x1)) =  x1 POL(incr(x1)) =  x1 POL(s(x1)) =  x1 POL(tl(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`           →DP Problem 24`
`             ↳Polo`
`             ...`
`               →DP Problem 26`
`                 ↳Polynomial Ordering`
`       →DP Problem 9`
`         ↳Remaining`

Dependency Pairs:

PROPER(hd(X)) -> PROPER(X)
PROPER(s(X)) -> PROPER(X)
PROPER(incr(X)) -> PROPER(X)

Rules:

active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(hd(cons(X, Y))) -> mark(X)
active(tl(cons(X, Y))) -> mark(Y)
active(incr(X)) -> incr(active(X))
active(hd(X)) -> hd(active(X))
active(tl(X)) -> tl(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
tl(mark(X)) -> mark(tl(X))
tl(ok(X)) -> ok(tl(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

PROPER(hd(X)) -> PROPER(X)

There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(PROPER(x1)) =  x1 POL(hd(x1)) =  1 + x1 POL(incr(x1)) =  x1 POL(s(x1)) =  x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`           →DP Problem 24`
`             ↳Polo`
`             ...`
`               →DP Problem 27`
`                 ↳Polynomial Ordering`
`       →DP Problem 9`
`         ↳Remaining`

Dependency Pairs:

PROPER(s(X)) -> PROPER(X)
PROPER(incr(X)) -> PROPER(X)

Rules:

active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(hd(cons(X, Y))) -> mark(X)
active(tl(cons(X, Y))) -> mark(Y)
active(incr(X)) -> incr(active(X))
active(hd(X)) -> hd(active(X))
active(tl(X)) -> tl(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
tl(mark(X)) -> mark(tl(X))
tl(ok(X)) -> ok(tl(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

PROPER(s(X)) -> PROPER(X)

There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(PROPER(x1)) =  x1 POL(incr(x1)) =  x1 POL(s(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`           →DP Problem 24`
`             ↳Polo`
`             ...`
`               →DP Problem 28`
`                 ↳Polynomial Ordering`
`       →DP Problem 9`
`         ↳Remaining`

Dependency Pair:

PROPER(incr(X)) -> PROPER(X)

Rules:

active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(hd(cons(X, Y))) -> mark(X)
active(tl(cons(X, Y))) -> mark(Y)
active(incr(X)) -> incr(active(X))
active(hd(X)) -> hd(active(X))
active(tl(X)) -> tl(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
tl(mark(X)) -> mark(tl(X))
tl(ok(X)) -> ok(tl(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

PROPER(incr(X)) -> PROPER(X)

There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(PROPER(x1)) =  x1 POL(incr(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`           →DP Problem 24`
`             ↳Polo`
`             ...`
`               →DP Problem 29`
`                 ↳Dependency Graph`
`       →DP Problem 9`
`         ↳Remaining`

Dependency Pair:

Rules:

active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(hd(cons(X, Y))) -> mark(X)
active(tl(cons(X, Y))) -> mark(Y)
active(incr(X)) -> incr(active(X))
active(hd(X)) -> hd(active(X))
active(tl(X)) -> tl(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
tl(mark(X)) -> mark(tl(X))
tl(ok(X)) -> ok(tl(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`       →DP Problem 2`
`         ↳Polo`
`       →DP Problem 3`
`         ↳Polo`
`       →DP Problem 4`
`         ↳Polo`
`       →DP Problem 5`
`         ↳Polo`
`       →DP Problem 6`
`         ↳Polo`
`       →DP Problem 7`
`         ↳Polo`
`       →DP Problem 8`
`         ↳Polo`
`       →DP Problem 9`
`         ↳Remaining Obligation(s)`

The following remains to be proven:
Dependency Pairs:

TOP(ok(X)) -> TOP(active(X))
TOP(mark(X)) -> TOP(proper(X))

Rules:

active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(hd(cons(X, Y))) -> mark(X)
active(tl(cons(X, Y))) -> mark(Y)
active(incr(X)) -> incr(active(X))
active(hd(X)) -> hd(active(X))
active(tl(X)) -> tl(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
tl(mark(X)) -> mark(tl(X))
tl(ok(X)) -> ok(tl(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:01 minutes