Term Rewriting System R:
[X, Y, Z, X1, X2]
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(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(first(s(X), cons(Y, Z))) -> CONS(Y, first(X, Z))
ACTIVE(first(s(X), cons(Y, Z))) -> FIRST(X, Z)
ACTIVE(from(X)) -> CONS(X, from(s(X)))
ACTIVE(from(X)) -> FROM(s(X))
ACTIVE(from(X)) -> S(X)
ACTIVE(first(X1, X2)) -> FIRST(active(X1), X2)
ACTIVE(first(X1, X2)) -> ACTIVE(X1)
ACTIVE(first(X1, X2)) -> FIRST(X1, active(X2))
ACTIVE(first(X1, X2)) -> ACTIVE(X2)
ACTIVE(s(X)) -> S(active(X))
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(cons(X1, X2)) -> CONS(active(X1), X2)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(from(X)) -> FROM(active(X))
ACTIVE(from(X)) -> ACTIVE(X)
FIRST(mark(X1), X2) -> FIRST(X1, X2)
FIRST(X1, mark(X2)) -> FIRST(X1, X2)
FIRST(ok(X1), ok(X2)) -> FIRST(X1, X2)
S(mark(X)) -> S(X)
S(ok(X)) -> S(X)
CONS(mark(X1), X2) -> CONS(X1, X2)
CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
FROM(mark(X)) -> FROM(X)
FROM(ok(X)) -> FROM(X)
PROPER(first(X1, X2)) -> FIRST(proper(X1), proper(X2))
PROPER(first(X1, X2)) -> PROPER(X1)
PROPER(first(X1, X2)) -> PROPER(X2)
PROPER(s(X)) -> S(proper(X))
PROPER(s(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(from(X)) -> FROM(proper(X))
PROPER(from(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 seven 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

Dependency Pairs:

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

Rules:

active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(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 8
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

Dependency Pair:

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

Rules:

active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(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 8
Polo
...
→DP Problem 9
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

Dependency Pair:

Rules:

active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(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

Dependency Pairs:

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

Rules:

active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

FIRST(ok(X1), ok(X2)) -> FIRST(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(FIRST(x1, x2)) =  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 10
Polynomial Ordering
→DP Problem 3
Polo
→DP Problem 4
Polo
→DP Problem 5
Polo
→DP Problem 6
Polo
→DP Problem 7
Polo

Dependency Pairs:

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

Rules:

active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

FIRST(mark(X1), X2) -> FIRST(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(FIRST(x1, x2)) =  x1 POL(mark(x1)) =  1 + x1

resulting in one new DP problem.

R
DPs
→DP Problem 1
Polo
→DP Problem 2
Polo
→DP Problem 10
Polo
...
→DP Problem 11
Polynomial Ordering
→DP Problem 3
Polo
→DP Problem 4
Polo
→DP Problem 5
Polo
→DP Problem 6
Polo
→DP Problem 7
Polo

Dependency Pair:

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

Rules:

active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

FIRST(X1, mark(X2)) -> FIRST(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(FIRST(x1, x2)) =  x2 POL(mark(x1)) =  1 + x1

resulting in one new DP problem.

R
DPs
→DP Problem 1
Polo
→DP Problem 2
Polo
→DP Problem 10
Polo
...
→DP Problem 12
Dependency Graph
→DP Problem 3
Polo
→DP Problem 4
Polo
→DP Problem 5
Polo
→DP Problem 6
Polo
→DP Problem 7
Polo

Dependency Pair:

Rules:

active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(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

Dependency Pairs:

FROM(ok(X)) -> FROM(X)
FROM(mark(X)) -> FROM(X)

Rules:

active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

FROM(ok(X)) -> FROM(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(FROM(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 13
Polynomial Ordering
→DP Problem 4
Polo
→DP Problem 5
Polo
→DP Problem 6
Polo
→DP Problem 7
Polo

Dependency Pair:

FROM(mark(X)) -> FROM(X)

Rules:

active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

FROM(mark(X)) -> FROM(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(FROM(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 13
Polo
...
→DP Problem 14
Dependency Graph
→DP Problem 4
Polo
→DP Problem 5
Polo
→DP Problem 6
Polo
→DP Problem 7
Polo

Dependency Pair:

Rules:

active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(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

Dependency Pairs:

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

Rules:

active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(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(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 15
Polynomial Ordering
→DP Problem 5
Polo
→DP Problem 6
Polo
→DP Problem 7
Polo

Dependency Pair:

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

Rules:

active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(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

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 15
Polo
...
→DP Problem 16
Dependency Graph
→DP Problem 5
Polo
→DP Problem 6
Polo
→DP Problem 7
Polo

Dependency Pair:

Rules:

active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(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

Dependency Pairs:

ACTIVE(from(X)) -> ACTIVE(X)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(first(X1, X2)) -> ACTIVE(X2)
ACTIVE(first(X1, X2)) -> ACTIVE(X1)

Rules:

active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

ACTIVE(first(X1, X2)) -> ACTIVE(X2)
ACTIVE(first(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(from(x1)) =  x1 POL(ACTIVE(x1)) =  x1 POL(first(x1, x2)) =  1 + x1 + x2 POL(cons(x1, x2)) =  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 17
Polynomial Ordering
→DP Problem 6
Polo
→DP Problem 7
Polo

Dependency Pairs:

ACTIVE(from(X)) -> ACTIVE(X)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(s(X)) -> ACTIVE(X)

Rules:

active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(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(from(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(from(x1)) =  1 + x1 POL(ACTIVE(x1)) =  x1 POL(cons(x1, x2)) =  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 17
Polo
...
→DP Problem 18
Polynomial Ordering
→DP Problem 6
Polo
→DP Problem 7
Polo

Dependency Pairs:

ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(s(X)) -> ACTIVE(X)

Rules:

active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(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(cons(x1, x2)) =  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 17
Polo
...
→DP Problem 19
Polynomial Ordering
→DP Problem 6
Polo
→DP Problem 7
Polo

Dependency Pair:

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

Rules:

active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(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 17
Polo
...
→DP Problem 20
Dependency Graph
→DP Problem 6
Polo
→DP Problem 7
Polo

Dependency Pair:

Rules:

active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(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

Dependency Pairs:

PROPER(from(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(s(X)) -> PROPER(X)
PROPER(first(X1, X2)) -> PROPER(X2)
PROPER(first(X1, X2)) -> PROPER(X1)

Rules:

active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(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(first(X1, X2)) -> PROPER(X2)
PROPER(first(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(from(x1)) =  x1 POL(first(x1, x2)) =  1 + x1 + x2 POL(PROPER(x1)) =  x1 POL(cons(x1, x2)) =  x1 + x2 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 21
Polynomial Ordering
→DP Problem 7
Polo

Dependency Pairs:

PROPER(from(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(s(X)) -> PROPER(X)

Rules:

active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(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(from(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(from(x1)) =  1 + x1 POL(PROPER(x1)) =  x1 POL(cons(x1, x2)) =  x1 + x2 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 21
Polo
...
→DP Problem 22
Polynomial Ordering
→DP Problem 7
Polo

Dependency Pairs:

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

Rules:

active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(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(PROPER(x1)) =  x1 POL(cons(x1, x2)) =  1 + x1 + x2 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 21
Polo
...
→DP Problem 23
Polynomial Ordering
→DP Problem 7
Polo

Dependency Pair:

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

Rules:

active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(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 21
Polo
...
→DP Problem 24
Dependency Graph
→DP Problem 7
Polo

Dependency Pair:

Rules:

active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(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

Dependency Pairs:

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

Rules:

active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

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

active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(from(x1)) =  1 + x1 POL(active(x1)) =  x1 POL(proper(x1)) =  x1 POL(first(x1, x2)) =  x1 + x2 POL(0) =  1 POL(cons(x1, x2)) =  x1 POL(nil) =  0 POL(s(x1)) =  1 + x1 POL(mark(x1)) =  1 + x1 POL(ok(x1)) =  x1 POL(TOP(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 25
Polynomial Ordering

Dependency Pair:

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

Rules:

active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

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

active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(from(x1)) =  x1 POL(active(x1)) =  x1 POL(first(x1, x2)) =  x1 POL(0) =  0 POL(cons(x1, x2)) =  x2 POL(nil) =  0 POL(s(x1)) =  x1 POL(mark(x1)) =  0 POL(ok(x1)) =  1 + x1 POL(TOP(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 25
Polo
...
→DP Problem 26
Dependency Graph

Dependency Pair:

Rules:

active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(nil) -> ok(nil)
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(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.

Innermost Termination of R successfully shown.
Duration:
0:01 minutes