Term Rewriting System R:
[X, L, X1, X2]
active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(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(incr(cons(X, L))) -> CONS(s(X), incr(L))
ACTIVE(incr(cons(X, L))) -> S(X)
ACTIVE(incr(cons(X, L))) -> INCR(L)
ACTIVE(zeros) -> CONS(0, zeros)
ACTIVE(incr(X)) -> INCR(active(X))
ACTIVE(incr(X)) -> ACTIVE(X)
ACTIVE(cons(X1, X2)) -> CONS(active(X1), X2)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(s(X)) -> S(active(X))
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(tail(X)) -> TAIL(active(X))
ACTIVE(tail(X)) -> ACTIVE(X)
INCR(mark(X)) -> INCR(X)
INCR(ok(X)) -> INCR(X)
CONS(mark(X1), X2) -> CONS(X1, X2)
CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
S(mark(X)) -> S(X)
S(ok(X)) -> S(X)
TAIL(mark(X)) -> TAIL(X)
TAIL(ok(X)) -> TAIL(X)
PROPER(incr(X)) -> INCR(proper(X))
PROPER(incr(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> CONS(proper(X1), proper(X2))
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(s(X)) -> S(proper(X))
PROPER(s(X)) -> PROPER(X)
PROPER(tail(X)) -> TAIL(proper(X))
PROPER(tail(X)) -> PROPER(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 Pairs:

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

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(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 w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(mark(x1)) =  0 POL(ok(x1)) =  1 + x1 POL(CONS(x1, x2)) =  x2

resulting in one new DP problem.

R
DPs
→DP Problem 1
Polo
→DP Problem 10
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(mark(X1), X2) -> CONS(X1, X2)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

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)) =  1 + x1 POL(CONS(x1, x2)) =  x1

resulting in one new DP problem.

R
DPs
→DP Problem 1
Polo
→DP Problem 10
Polo
...
→DP Problem 11
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(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(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 Pairs:

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

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

S(mark(X)) -> S(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(S(x1)) =  x1 POL(mark(x1)) =  1 + x1 POL(ok(x1)) =  x1

resulting in one new DP problem.

R
DPs
→DP Problem 1
Polo
→DP Problem 2
Polo
→DP Problem 12
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(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(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 w.r.t. to the implicit AFS 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 12
Polo
...
→DP Problem 13
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(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(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(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(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 w.r.t. to the implicit AFS 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 14
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(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(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 w.r.t. to the implicit AFS 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 14
Polo
...
→DP Problem 15
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(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(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(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(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 w.r.t. to the implicit AFS 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 16
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(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(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 w.r.t. to the implicit AFS 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 16
Polo
...
→DP Problem 17
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(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(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:

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(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 w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(HEAD(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 18
Polynomial Ordering
→DP Problem 6
Polo
→DP Problem 7
Polo
→DP Problem 8
Polo
→DP Problem 9
Remaining

Dependency Pair:

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(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 w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(HEAD(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 18
Polo
...
→DP Problem 19
Dependency Graph
→DP Problem 6
Polo
→DP Problem 7
Polo
→DP Problem 8
Polo
→DP Problem 9
Remaining

Dependency Pair:

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(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:

TAIL(ok(X)) -> TAIL(X)
TAIL(mark(X)) -> TAIL(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

TAIL(ok(X)) -> TAIL(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(TAIL(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 6
Polo
→DP Problem 20
Polynomial Ordering
→DP Problem 7
Polo
→DP Problem 8
Polo
→DP Problem 9
Remaining

Dependency Pair:

TAIL(mark(X)) -> TAIL(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

TAIL(mark(X)) -> TAIL(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(TAIL(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 6
Polo
→DP Problem 20
Polo
...
→DP Problem 21
Dependency Graph
→DP Problem 7
Polo
→DP Problem 8
Polo
→DP Problem 9
Remaining

Dependency Pair:

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(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(tail(X)) -> ACTIVE(X)
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(incr(X)) -> ACTIVE(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

ACTIVE(cons(X1, X2)) -> ACTIVE(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(ACTIVE(x1)) =  x1 POL(adx(x1)) =  x1 POL(cons(x1, x2)) =  1 + x1 POL(tail(x1)) =  x1 POL(incr(x1)) =  x1 POL(s(x1)) =  x1 POL(head(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 22
Polynomial Ordering
→DP Problem 8
Polo
→DP Problem 9
Remaining

Dependency Pairs:

ACTIVE(tail(X)) -> ACTIVE(X)
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(incr(X)) -> ACTIVE(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(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 w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(ACTIVE(x1)) =  x1 POL(adx(x1)) =  x1 POL(tail(x1)) =  x1 POL(incr(x1)) =  1 + x1 POL(s(x1)) =  x1 POL(head(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 22
Polo
...
→DP Problem 23
Polynomial Ordering
→DP Problem 8
Polo
→DP Problem 9
Remaining

Dependency Pairs:

ACTIVE(tail(X)) -> ACTIVE(X)
ACTIVE(s(X)) -> ACTIVE(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

ACTIVE(tail(X)) -> ACTIVE(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(ACTIVE(x1)) =  x1 POL(adx(x1)) =  x1 POL(tail(x1)) =  1 + x1 POL(s(x1)) =  x1 POL(head(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 22
Polo
...
→DP Problem 24
Polynomial Ordering
→DP Problem 8
Polo
→DP Problem 9
Remaining

Dependency Pairs:

ACTIVE(s(X)) -> ACTIVE(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(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 w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(ACTIVE(x1)) =  x1 POL(adx(x1)) =  x1 POL(s(x1)) =  x1 POL(head(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 22
Polo
...
→DP Problem 25
Polynomial Ordering
→DP Problem 8
Polo
→DP Problem 9
Remaining

Dependency Pairs:

ACTIVE(s(X)) -> ACTIVE(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(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 w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(ACTIVE(x1)) =  x1 POL(adx(x1)) =  1 + 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 22
Polo
...
→DP Problem 26
Polynomial Ordering
→DP Problem 8
Polo
→DP Problem 9
Remaining

Dependency Pair:

ACTIVE(s(X)) -> ACTIVE(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

ACTIVE(s(X)) -> ACTIVE(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(ACTIVE(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 22
Polo
...
→DP Problem 27
Dependency Graph
→DP Problem 8
Polo
→DP Problem 9
Remaining

Dependency Pair:

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(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(tail(X)) -> PROPER(X)
PROPER(s(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(incr(X)) -> PROPER(X)

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(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 w.r.t. to the implicit AFS 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(tail(x1)) =  x1 POL(incr(x1)) =  x1 POL(s(x1)) =  x1 POL(head(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 28
Polynomial Ordering
→DP Problem 9
Remaining

Dependency Pairs:

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

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(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 w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(adx(x1)) =  x1 POL(PROPER(x1)) =  x1 POL(tail(x1)) =  x1 POL(incr(x1)) =  1 + x1 POL(s(x1)) =  x1 POL(head(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 28
Polo
...
→DP Problem 29
Polynomial Ordering
→DP Problem 9
Remaining

Dependency Pairs:

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

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

PROPER(tail(X)) -> PROPER(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(adx(x1)) =  x1 POL(PROPER(x1)) =  x1 POL(tail(x1)) =  1 + x1 POL(s(x1)) =  x1 POL(head(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 28
Polo
...
→DP Problem 30
Polynomial Ordering
→DP Problem 9
Remaining

Dependency Pairs:

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

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(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 w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(adx(x1)) =  x1 POL(PROPER(x1)) =  x1 POL(s(x1)) =  x1 POL(head(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 28
Polo
...
→DP Problem 31
Polynomial Ordering
→DP Problem 9
Remaining

Dependency Pairs:

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

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(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 w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(adx(x1)) =  1 + x1 POL(PROPER(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 28
Polo
...
→DP Problem 32
Polynomial Ordering
→DP Problem 9
Remaining

Dependency Pair:

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

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(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 w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(PROPER(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 28
Polo
...
→DP Problem 33
Dependency Graph
→DP Problem 9
Remaining

Dependency Pair:

Rules:

active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(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(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(zeros) -> mark(cons(0, zeros))
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(tail(X)) -> tail(proper(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