Term Rewriting System R:
[X, Y, Z, X1, X2]
active(2nd(cons1(X, cons(Y, Z)))) -> mark(Y)
active(2nd(cons(X, X1))) -> mark(2nd(cons1(X, X1)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(2nd(X)) -> 2nd(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
2nd(mark(X)) -> mark(2nd(X))
2nd(ok(X)) -> ok(2nd(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))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons1(mark(X1), X2) -> mark(cons1(X1, X2))
cons1(X1, mark(X2)) -> mark(cons1(X1, X2))
cons1(ok(X1), ok(X2)) -> ok(cons1(X1, X2))
proper(2nd(X)) -> 2nd(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
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(2nd(cons(X, X1))) -> 2ND(cons1(X, X1))
ACTIVE(2nd(cons(X, X1))) -> CONS1(X, X1)
ACTIVE(from(X)) -> CONS(X, from(s(X)))
ACTIVE(from(X)) -> FROM(s(X))
ACTIVE(from(X)) -> S(X)
ACTIVE(2nd(X)) -> 2ND(active(X))
ACTIVE(2nd(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)
ACTIVE(s(X)) -> S(active(X))
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(cons1(X1, X2)) -> CONS1(active(X1), X2)
ACTIVE(cons1(X1, X2)) -> ACTIVE(X1)
ACTIVE(cons1(X1, X2)) -> CONS1(X1, active(X2))
ACTIVE(cons1(X1, X2)) -> ACTIVE(X2)
2ND(mark(X)) -> 2ND(X)
2ND(ok(X)) -> 2ND(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)
S(mark(X)) -> S(X)
S(ok(X)) -> S(X)
CONS1(mark(X1), X2) -> CONS1(X1, X2)
CONS1(X1, mark(X2)) -> CONS1(X1, X2)
CONS1(ok(X1), ok(X2)) -> CONS1(X1, X2)
PROPER(2nd(X)) -> 2ND(proper(X))
PROPER(2nd(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)
PROPER(s(X)) -> S(proper(X))
PROPER(s(X)) -> PROPER(X)
PROPER(cons1(X1, X2)) -> CONS1(proper(X1), proper(X2))
PROPER(cons1(X1, X2)) -> PROPER(X1)
PROPER(cons1(X1, X2)) -> PROPER(X2)
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 eight SCCs.


   R
DPs
       →DP Problem 1
Argument Filtering and Ordering
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
AFS
       →DP Problem 6
AFS
       →DP Problem 7
AFS
       →DP Problem 8
AFS


Dependency Pairs:

2ND(ok(X)) -> 2ND(X)
2ND(mark(X)) -> 2ND(X)


Rules:


active(2nd(cons1(X, cons(Y, Z)))) -> mark(Y)
active(2nd(cons(X, X1))) -> mark(2nd(cons1(X, X1)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(2nd(X)) -> 2nd(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
2nd(mark(X)) -> mark(2nd(X))
2nd(ok(X)) -> ok(2nd(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))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons1(mark(X1), X2) -> mark(cons1(X1, X2))
cons1(X1, mark(X2)) -> mark(cons1(X1, X2))
cons1(ok(X1), ok(X2)) -> ok(cons1(X1, X2))
proper(2nd(X)) -> 2nd(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

2ND(ok(X)) -> 2ND(X)
2ND(mark(X)) -> 2ND(X)


There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

resulting in one new DP problem.
Used Argument Filtering System:
2ND(x1) -> 2ND(x1)
ok(x1) -> ok(x1)
mark(x1) -> mark(x1)


   R
DPs
       →DP Problem 1
AFS
           →DP Problem 9
Dependency Graph
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
AFS
       →DP Problem 6
AFS
       →DP Problem 7
AFS
       →DP Problem 8
AFS


Dependency Pair:


Rules:


active(2nd(cons1(X, cons(Y, Z)))) -> mark(Y)
active(2nd(cons(X, X1))) -> mark(2nd(cons1(X, X1)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(2nd(X)) -> 2nd(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
2nd(mark(X)) -> mark(2nd(X))
2nd(ok(X)) -> ok(2nd(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))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons1(mark(X1), X2) -> mark(cons1(X1, X2))
cons1(X1, mark(X2)) -> mark(cons1(X1, X2))
cons1(ok(X1), ok(X2)) -> ok(cons1(X1, X2))
proper(2nd(X)) -> 2nd(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
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
AFS
       →DP Problem 2
Argument Filtering and Ordering
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
AFS
       →DP Problem 6
AFS
       →DP Problem 7
AFS
       →DP Problem 8
AFS


Dependency Pairs:

CONS1(ok(X1), ok(X2)) -> CONS1(X1, X2)
CONS1(mark(X1), X2) -> CONS1(X1, X2)
CONS1(X1, mark(X2)) -> CONS1(X1, X2)


Rules:


active(2nd(cons1(X, cons(Y, Z)))) -> mark(Y)
active(2nd(cons(X, X1))) -> mark(2nd(cons1(X, X1)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(2nd(X)) -> 2nd(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
2nd(mark(X)) -> mark(2nd(X))
2nd(ok(X)) -> ok(2nd(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))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons1(mark(X1), X2) -> mark(cons1(X1, X2))
cons1(X1, mark(X2)) -> mark(cons1(X1, X2))
cons1(ok(X1), ok(X2)) -> ok(cons1(X1, X2))
proper(2nd(X)) -> 2nd(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

CONS1(ok(X1), ok(X2)) -> CONS1(X1, X2)
CONS1(mark(X1), X2) -> CONS1(X1, X2)
CONS1(X1, mark(X2)) -> CONS1(X1, X2)


There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

resulting in one new DP problem.
Used Argument Filtering System:
CONS1(x1, x2) -> CONS1(x1, x2)
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
           →DP Problem 10
Dependency Graph
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
AFS
       →DP Problem 6
AFS
       →DP Problem 7
AFS
       →DP Problem 8
AFS


Dependency Pair:


Rules:


active(2nd(cons1(X, cons(Y, Z)))) -> mark(Y)
active(2nd(cons(X, X1))) -> mark(2nd(cons1(X, X1)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(2nd(X)) -> 2nd(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
2nd(mark(X)) -> mark(2nd(X))
2nd(ok(X)) -> ok(2nd(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))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons1(mark(X1), X2) -> mark(cons1(X1, X2))
cons1(X1, mark(X2)) -> mark(cons1(X1, X2))
cons1(ok(X1), ok(X2)) -> ok(cons1(X1, X2))
proper(2nd(X)) -> 2nd(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
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
AFS
       →DP Problem 2
AFS
       →DP Problem 3
Argument Filtering and Ordering
       →DP Problem 4
AFS
       →DP Problem 5
AFS
       →DP Problem 6
AFS
       →DP Problem 7
AFS
       →DP Problem 8
AFS


Dependency Pairs:

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


Rules:


active(2nd(cons1(X, cons(Y, Z)))) -> mark(Y)
active(2nd(cons(X, X1))) -> mark(2nd(cons1(X, X1)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(2nd(X)) -> 2nd(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
2nd(mark(X)) -> mark(2nd(X))
2nd(ok(X)) -> ok(2nd(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))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons1(mark(X1), X2) -> mark(cons1(X1, X2))
cons1(X1, mark(X2)) -> mark(cons1(X1, X2))
cons1(ok(X1), ok(X2)) -> ok(cons1(X1, X2))
proper(2nd(X)) -> 2nd(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

resulting in one new DP problem.
Used Argument Filtering System:
FROM(x1) -> FROM(x1)
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
           →DP Problem 11
Dependency Graph
       →DP Problem 4
AFS
       →DP Problem 5
AFS
       →DP Problem 6
AFS
       →DP Problem 7
AFS
       →DP Problem 8
AFS


Dependency Pair:


Rules:


active(2nd(cons1(X, cons(Y, Z)))) -> mark(Y)
active(2nd(cons(X, X1))) -> mark(2nd(cons1(X, X1)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(2nd(X)) -> 2nd(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
2nd(mark(X)) -> mark(2nd(X))
2nd(ok(X)) -> ok(2nd(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))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons1(mark(X1), X2) -> mark(cons1(X1, X2))
cons1(X1, mark(X2)) -> mark(cons1(X1, X2))
cons1(ok(X1), ok(X2)) -> ok(cons1(X1, X2))
proper(2nd(X)) -> 2nd(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
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
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
Argument Filtering and Ordering
       →DP Problem 5
AFS
       →DP Problem 6
AFS
       →DP Problem 7
AFS
       →DP Problem 8
AFS


Dependency Pairs:

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


Rules:


active(2nd(cons1(X, cons(Y, Z)))) -> mark(Y)
active(2nd(cons(X, X1))) -> mark(2nd(cons1(X, X1)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(2nd(X)) -> 2nd(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
2nd(mark(X)) -> mark(2nd(X))
2nd(ok(X)) -> ok(2nd(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))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons1(mark(X1), X2) -> mark(cons1(X1, X2))
cons1(X1, mark(X2)) -> mark(cons1(X1, X2))
cons1(ok(X1), ok(X2)) -> ok(cons1(X1, X2))
proper(2nd(X)) -> 2nd(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

resulting in one new DP problem.
Used Argument Filtering System:
CONS(x1, x2) -> CONS(x1, x2)
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
AFS
           →DP Problem 12
Dependency Graph
       →DP Problem 5
AFS
       →DP Problem 6
AFS
       →DP Problem 7
AFS
       →DP Problem 8
AFS


Dependency Pair:


Rules:


active(2nd(cons1(X, cons(Y, Z)))) -> mark(Y)
active(2nd(cons(X, X1))) -> mark(2nd(cons1(X, X1)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(2nd(X)) -> 2nd(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
2nd(mark(X)) -> mark(2nd(X))
2nd(ok(X)) -> ok(2nd(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))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons1(mark(X1), X2) -> mark(cons1(X1, X2))
cons1(X1, mark(X2)) -> mark(cons1(X1, X2))
cons1(ok(X1), ok(X2)) -> ok(cons1(X1, X2))
proper(2nd(X)) -> 2nd(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
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
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
Argument Filtering and Ordering
       →DP Problem 6
AFS
       →DP Problem 7
AFS
       →DP Problem 8
AFS


Dependency Pairs:

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


Rules:


active(2nd(cons1(X, cons(Y, Z)))) -> mark(Y)
active(2nd(cons(X, X1))) -> mark(2nd(cons1(X, X1)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(2nd(X)) -> 2nd(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
2nd(mark(X)) -> mark(2nd(X))
2nd(ok(X)) -> ok(2nd(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))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons1(mark(X1), X2) -> mark(cons1(X1, X2))
cons1(X1, mark(X2)) -> mark(cons1(X1, X2))
cons1(ok(X1), ok(X2)) -> ok(cons1(X1, X2))
proper(2nd(X)) -> 2nd(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

resulting in one new DP problem.
Used Argument Filtering System:
S(x1) -> S(x1)
ok(x1) -> ok(x1)
mark(x1) -> mark(x1)


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
AFS
           →DP Problem 13
Dependency Graph
       →DP Problem 6
AFS
       →DP Problem 7
AFS
       →DP Problem 8
AFS


Dependency Pair:


Rules:


active(2nd(cons1(X, cons(Y, Z)))) -> mark(Y)
active(2nd(cons(X, X1))) -> mark(2nd(cons1(X, X1)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(2nd(X)) -> 2nd(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
2nd(mark(X)) -> mark(2nd(X))
2nd(ok(X)) -> ok(2nd(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))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons1(mark(X1), X2) -> mark(cons1(X1, X2))
cons1(X1, mark(X2)) -> mark(cons1(X1, X2))
cons1(ok(X1), ok(X2)) -> ok(cons1(X1, X2))
proper(2nd(X)) -> 2nd(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
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
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
AFS
       →DP Problem 6
Argument Filtering and Ordering
       →DP Problem 7
AFS
       →DP Problem 8
AFS


Dependency Pairs:

ACTIVE(cons1(X1, X2)) -> ACTIVE(X2)
ACTIVE(cons1(X1, X2)) -> ACTIVE(X1)
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(from(X)) -> ACTIVE(X)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(2nd(X)) -> ACTIVE(X)


Rules:


active(2nd(cons1(X, cons(Y, Z)))) -> mark(Y)
active(2nd(cons(X, X1))) -> mark(2nd(cons1(X, X1)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(2nd(X)) -> 2nd(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
2nd(mark(X)) -> mark(2nd(X))
2nd(ok(X)) -> ok(2nd(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))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons1(mark(X1), X2) -> mark(cons1(X1, X2))
cons1(X1, mark(X2)) -> mark(cons1(X1, X2))
cons1(ok(X1), ok(X2)) -> ok(cons1(X1, X2))
proper(2nd(X)) -> 2nd(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

ACTIVE(cons1(X1, X2)) -> ACTIVE(X2)
ACTIVE(cons1(X1, X2)) -> ACTIVE(X1)
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(from(X)) -> ACTIVE(X)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(2nd(X)) -> ACTIVE(X)


There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

resulting in one new DP problem.
Used Argument Filtering System:
ACTIVE(x1) -> ACTIVE(x1)
s(x1) -> s(x1)
from(x1) -> from(x1)
cons(x1, x2) -> cons(x1, x2)
cons1(x1, x2) -> cons1(x1, x2)
2nd(x1) -> 2nd(x1)


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
AFS
       →DP Problem 6
AFS
           →DP Problem 14
Dependency Graph
       →DP Problem 7
AFS
       →DP Problem 8
AFS


Dependency Pair:


Rules:


active(2nd(cons1(X, cons(Y, Z)))) -> mark(Y)
active(2nd(cons(X, X1))) -> mark(2nd(cons1(X, X1)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(2nd(X)) -> 2nd(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
2nd(mark(X)) -> mark(2nd(X))
2nd(ok(X)) -> ok(2nd(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))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons1(mark(X1), X2) -> mark(cons1(X1, X2))
cons1(X1, mark(X2)) -> mark(cons1(X1, X2))
cons1(ok(X1), ok(X2)) -> ok(cons1(X1, X2))
proper(2nd(X)) -> 2nd(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
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
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
AFS
       →DP Problem 6
AFS
       →DP Problem 7
Argument Filtering and Ordering
       →DP Problem 8
AFS


Dependency Pairs:

PROPER(cons1(X1, X2)) -> PROPER(X2)
PROPER(cons1(X1, X2)) -> PROPER(X1)
PROPER(s(X)) -> PROPER(X)
PROPER(from(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(2nd(X)) -> PROPER(X)


Rules:


active(2nd(cons1(X, cons(Y, Z)))) -> mark(Y)
active(2nd(cons(X, X1))) -> mark(2nd(cons1(X, X1)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(2nd(X)) -> 2nd(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
2nd(mark(X)) -> mark(2nd(X))
2nd(ok(X)) -> ok(2nd(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))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons1(mark(X1), X2) -> mark(cons1(X1, X2))
cons1(X1, mark(X2)) -> mark(cons1(X1, X2))
cons1(ok(X1), ok(X2)) -> ok(cons1(X1, X2))
proper(2nd(X)) -> 2nd(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

PROPER(cons1(X1, X2)) -> PROPER(X2)
PROPER(cons1(X1, X2)) -> PROPER(X1)
PROPER(s(X)) -> PROPER(X)
PROPER(from(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(2nd(X)) -> PROPER(X)


There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

resulting in one new DP problem.
Used Argument Filtering System:
PROPER(x1) -> PROPER(x1)
cons(x1, x2) -> cons(x1, x2)
cons1(x1, x2) -> cons1(x1, x2)
2nd(x1) -> 2nd(x1)
from(x1) -> from(x1)
s(x1) -> s(x1)


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
AFS
       →DP Problem 6
AFS
       →DP Problem 7
AFS
           →DP Problem 15
Dependency Graph
       →DP Problem 8
AFS


Dependency Pair:


Rules:


active(2nd(cons1(X, cons(Y, Z)))) -> mark(Y)
active(2nd(cons(X, X1))) -> mark(2nd(cons1(X, X1)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(2nd(X)) -> 2nd(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
2nd(mark(X)) -> mark(2nd(X))
2nd(ok(X)) -> ok(2nd(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))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons1(mark(X1), X2) -> mark(cons1(X1, X2))
cons1(X1, mark(X2)) -> mark(cons1(X1, X2))
cons1(ok(X1), ok(X2)) -> ok(cons1(X1, X2))
proper(2nd(X)) -> 2nd(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
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
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
AFS
       →DP Problem 6
AFS
       →DP Problem 7
AFS
       →DP Problem 8
Argument Filtering and Ordering


Dependency Pairs:

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


Rules:


active(2nd(cons1(X, cons(Y, Z)))) -> mark(Y)
active(2nd(cons(X, X1))) -> mark(2nd(cons1(X, X1)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(2nd(X)) -> 2nd(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
2nd(mark(X)) -> mark(2nd(X))
2nd(ok(X)) -> ok(2nd(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))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons1(mark(X1), X2) -> mark(cons1(X1, X2))
cons1(X1, mark(X2)) -> mark(cons1(X1, X2))
cons1(ok(X1), ok(X2)) -> ok(cons1(X1, X2))
proper(2nd(X)) -> 2nd(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
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))


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

proper(2nd(X)) -> 2nd(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
active(2nd(cons1(X, cons(Y, Z)))) -> mark(Y)
active(2nd(cons(X, X1))) -> mark(2nd(cons1(X, X1)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(2nd(X)) -> 2nd(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
2nd(mark(X)) -> mark(2nd(X))
2nd(ok(X)) -> ok(2nd(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))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons1(mark(X1), X2) -> mark(cons1(X1, X2))
cons1(X1, mark(X2)) -> mark(cons1(X1, X2))
cons1(ok(X1), ok(X2)) -> ok(cons1(X1, X2))


Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
cons > {mark, proper}
ok > active > {mark, proper}
TOP > {mark, proper}
cons1 > {mark, proper}
from > {mark, proper}
2nd > {mark, proper}
s > {mark, proper}

resulting in one new DP problem.
Used Argument Filtering System:
TOP(x1) -> TOP(x1)
mark(x1) -> mark
proper(x1) -> proper
ok(x1) -> ok(x1)
active(x1) -> active(x1)
2nd(x1) -> x1
cons(x1, x2) -> x1
from(x1) -> x1
s(x1) -> x1
cons1(x1, x2) -> x1


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
AFS
       →DP Problem 6
AFS
       →DP Problem 7
AFS
       →DP Problem 8
AFS
           →DP Problem 16
Argument Filtering and Ordering


Dependency Pair:

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


Rules:


active(2nd(cons1(X, cons(Y, Z)))) -> mark(Y)
active(2nd(cons(X, X1))) -> mark(2nd(cons1(X, X1)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(2nd(X)) -> 2nd(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
2nd(mark(X)) -> mark(2nd(X))
2nd(ok(X)) -> ok(2nd(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))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons1(mark(X1), X2) -> mark(cons1(X1, X2))
cons1(X1, mark(X2)) -> mark(cons1(X1, X2))
cons1(ok(X1), ok(X2)) -> ok(cons1(X1, X2))
proper(2nd(X)) -> 2nd(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
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))


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

proper(2nd(X)) -> 2nd(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
2nd(mark(X)) -> mark(2nd(X))
2nd(ok(X)) -> ok(2nd(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))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons1(mark(X1), X2) -> mark(cons1(X1, X2))
cons1(X1, mark(X2)) -> mark(cons1(X1, X2))
cons1(ok(X1), ok(X2)) -> ok(cons1(X1, X2))


Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
cons > {ok, s} > mark
cons1 > {ok, s} > mark
from > {ok, s} > mark
2nd > {ok, s} > mark

resulting in one new DP problem.
Used Argument Filtering System:
TOP(x1) -> TOP(x1)
mark(x1) -> mark(x1)
proper(x1) -> x1
2nd(x1) -> 2nd(x1)
cons(x1, x2) -> cons(x1, x2)
from(x1) -> from(x1)
s(x1) -> s(x1)
cons1(x1, x2) -> cons1(x1, x2)
ok(x1) -> ok(x1)


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
AFS
       →DP Problem 6
AFS
       →DP Problem 7
AFS
       →DP Problem 8
AFS
           →DP Problem 16
AFS
             ...
               →DP Problem 17
Dependency Graph


Dependency Pair:


Rules:


active(2nd(cons1(X, cons(Y, Z)))) -> mark(Y)
active(2nd(cons(X, X1))) -> mark(2nd(cons1(X, X1)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(2nd(X)) -> 2nd(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(from(X)) -> from(active(X))
active(s(X)) -> s(active(X))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
2nd(mark(X)) -> mark(2nd(X))
2nd(ok(X)) -> ok(2nd(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))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons1(mark(X1), X2) -> mark(cons1(X1, X2))
cons1(X1, mark(X2)) -> mark(cons1(X1, X2))
cons1(ok(X1), ok(X2)) -> ok(cons1(X1, X2))
proper(2nd(X)) -> 2nd(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
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:
0:44 minutes