Term Rewriting System R:
[X, Y, Z, X1, X2]
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(sel(0, cons(X, Z))) -> mark(X)
active(first(0, Z)) -> 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(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) -> mark(quote(X))
active(first1(0, Z)) -> mark(nil1)
active(first1(s(X), cons(Y, Z))) -> mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) -> mark(01)
active(quote1(cons(X, Z))) -> mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) -> mark(nil1)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(sel(X, Z))) -> mark(sel1(X, Z))
active(quote1(first(X, Z))) -> mark(first1(X, Z))
active(unquote(01)) -> mark(0)
active(unquote(s1(X))) -> mark(s(unquote(X)))
active(unquote1(nil1)) -> mark(nil)
active(unquote1(cons1(X, Z))) -> mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) -> mark(cons(X, Z))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(from(X)) -> from(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(first1(X1, X2)) -> first1(active(X1), X2)
active(first1(X1, X2)) -> first1(X1, active(X2))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
active(s1(X)) -> s1(active(X))
active(unquote(X)) -> unquote(active(X))
active(unquote1(X)) -> unquote1(active(X))
active(fcons(X1, X2)) -> fcons(active(X1), X2)
active(fcons(X1, X2)) -> fcons(X1, active(X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(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))
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))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
first1(mark(X1), X2) -> mark(first1(X1, X2))
first1(X1, mark(X2)) -> mark(first1(X1, X2))
first1(ok(X1), ok(X2)) -> ok(first1(X1, X2))
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))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
unquote(mark(X)) -> mark(unquote(X))
unquote(ok(X)) -> ok(unquote(X))
unquote1(mark(X)) -> mark(unquote1(X))
unquote1(ok(X)) -> ok(unquote1(X))
fcons(mark(X1), X2) -> mark(fcons(X1, X2))
fcons(X1, mark(X2)) -> mark(fcons(X1, X2))
fcons(ok(X1), ok(X2)) -> ok(fcons(X1, X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(from(X)) -> from(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
proper(first1(X1, X2)) -> first1(proper(X1), proper(X2))
proper(nil1) -> ok(nil1)
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
proper(01) -> ok(01)
proper(quote1(X)) -> quote1(proper(X))
proper(s1(X)) -> s1(proper(X))
proper(unquote(X)) -> unquote(proper(X))
proper(unquote1(X)) -> unquote1(proper(X))
proper(fcons(X1, X2)) -> fcons(proper(X1), proper(X2))
quote(ok(X)) -> ok(quote(X))
quote1(ok(X)) -> ok(quote1(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(sel(s(X), cons(Y, Z))) -> SEL(X, Z)
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(sel1(s(X), cons(Y, Z))) -> SEL1(X, Z)
ACTIVE(sel1(0, cons(X, Z))) -> QUOTE(X)
ACTIVE(first1(s(X), cons(Y, Z))) -> CONS1(quote(Y), first1(X, Z))
ACTIVE(first1(s(X), cons(Y, Z))) -> QUOTE(Y)
ACTIVE(first1(s(X), cons(Y, Z))) -> FIRST1(X, Z)
ACTIVE(quote1(cons(X, Z))) -> CONS1(quote(X), quote1(Z))
ACTIVE(quote1(cons(X, Z))) -> QUOTE(X)
ACTIVE(quote1(cons(X, Z))) -> QUOTE1(Z)
ACTIVE(quote(s(X))) -> S1(quote(X))
ACTIVE(quote(s(X))) -> QUOTE(X)
ACTIVE(quote(sel(X, Z))) -> SEL1(X, Z)
ACTIVE(quote1(first(X, Z))) -> FIRST1(X, Z)
ACTIVE(unquote(s1(X))) -> S(unquote(X))
ACTIVE(unquote(s1(X))) -> UNQUOTE(X)
ACTIVE(unquote1(cons1(X, Z))) -> FCONS(unquote(X), unquote1(Z))
ACTIVE(unquote1(cons1(X, Z))) -> UNQUOTE(X)
ACTIVE(unquote1(cons1(X, Z))) -> UNQUOTE1(Z)
ACTIVE(fcons(X, Z)) -> CONS(X, Z)
ACTIVE(sel(X1, X2)) -> SEL(active(X1), X2)
ACTIVE(sel(X1, X2)) -> ACTIVE(X1)
ACTIVE(sel(X1, X2)) -> SEL(X1, active(X2))
ACTIVE(sel(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(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(from(X)) -> FROM(active(X))
ACTIVE(from(X)) -> ACTIVE(X)
ACTIVE(sel1(X1, X2)) -> SEL1(active(X1), X2)
ACTIVE(sel1(X1, X2)) -> ACTIVE(X1)
ACTIVE(sel1(X1, X2)) -> SEL1(X1, active(X2))
ACTIVE(sel1(X1, X2)) -> ACTIVE(X2)
ACTIVE(first1(X1, X2)) -> FIRST1(active(X1), X2)
ACTIVE(first1(X1, X2)) -> ACTIVE(X1)
ACTIVE(first1(X1, X2)) -> FIRST1(X1, active(X2))
ACTIVE(first1(X1, X2)) -> ACTIVE(X2)
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)
ACTIVE(s1(X)) -> S1(active(X))
ACTIVE(s1(X)) -> ACTIVE(X)
ACTIVE(unquote(X)) -> UNQUOTE(active(X))
ACTIVE(unquote(X)) -> ACTIVE(X)
ACTIVE(unquote1(X)) -> UNQUOTE1(active(X))
ACTIVE(unquote1(X)) -> ACTIVE(X)
ACTIVE(fcons(X1, X2)) -> FCONS(active(X1), X2)
ACTIVE(fcons(X1, X2)) -> ACTIVE(X1)
ACTIVE(fcons(X1, X2)) -> FCONS(X1, active(X2))
ACTIVE(fcons(X1, X2)) -> ACTIVE(X2)
SEL(mark(X1), X2) -> SEL(X1, X2)
SEL(X1, mark(X2)) -> SEL(X1, X2)
SEL(ok(X1), ok(X2)) -> SEL(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)
FIRST(mark(X1), X2) -> FIRST(X1, X2)
FIRST(X1, mark(X2)) -> FIRST(X1, X2)
FIRST(ok(X1), ok(X2)) -> FIRST(X1, X2)
FROM(mark(X)) -> FROM(X)
FROM(ok(X)) -> FROM(X)
SEL1(mark(X1), X2) -> SEL1(X1, X2)
SEL1(X1, mark(X2)) -> SEL1(X1, X2)
SEL1(ok(X1), ok(X2)) -> SEL1(X1, X2)
FIRST1(mark(X1), X2) -> FIRST1(X1, X2)
FIRST1(X1, mark(X2)) -> FIRST1(X1, X2)
FIRST1(ok(X1), ok(X2)) -> FIRST1(X1, X2)
CONS1(mark(X1), X2) -> CONS1(X1, X2)
CONS1(X1, mark(X2)) -> CONS1(X1, X2)
CONS1(ok(X1), ok(X2)) -> CONS1(X1, X2)
S1(mark(X)) -> S1(X)
S1(ok(X)) -> S1(X)
UNQUOTE(mark(X)) -> UNQUOTE(X)
UNQUOTE(ok(X)) -> UNQUOTE(X)
UNQUOTE1(mark(X)) -> UNQUOTE1(X)
UNQUOTE1(ok(X)) -> UNQUOTE1(X)
FCONS(mark(X1), X2) -> FCONS(X1, X2)
FCONS(X1, mark(X2)) -> FCONS(X1, X2)
FCONS(ok(X1), ok(X2)) -> FCONS(X1, X2)
PROPER(sel(X1, X2)) -> SEL(proper(X1), proper(X2))
PROPER(sel(X1, X2)) -> PROPER(X1)
PROPER(sel(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(first(X1, X2)) -> FIRST(proper(X1), proper(X2))
PROPER(first(X1, X2)) -> PROPER(X1)
PROPER(first(X1, X2)) -> PROPER(X2)
PROPER(from(X)) -> FROM(proper(X))
PROPER(from(X)) -> PROPER(X)
PROPER(sel1(X1, X2)) -> SEL1(proper(X1), proper(X2))
PROPER(sel1(X1, X2)) -> PROPER(X1)
PROPER(sel1(X1, X2)) -> PROPER(X2)
PROPER(quote(X)) -> QUOTE(proper(X))
PROPER(quote(X)) -> PROPER(X)
PROPER(first1(X1, X2)) -> FIRST1(proper(X1), proper(X2))
PROPER(first1(X1, X2)) -> PROPER(X1)
PROPER(first1(X1, X2)) -> PROPER(X2)
PROPER(cons1(X1, X2)) -> CONS1(proper(X1), proper(X2))
PROPER(cons1(X1, X2)) -> PROPER(X1)
PROPER(cons1(X1, X2)) -> PROPER(X2)
PROPER(quote1(X)) -> QUOTE1(proper(X))
PROPER(quote1(X)) -> PROPER(X)
PROPER(s1(X)) -> S1(proper(X))
PROPER(s1(X)) -> PROPER(X)
PROPER(unquote(X)) -> UNQUOTE(proper(X))
PROPER(unquote(X)) -> PROPER(X)
PROPER(unquote1(X)) -> UNQUOTE1(proper(X))
PROPER(unquote1(X)) -> PROPER(X)
PROPER(fcons(X1, X2)) -> FCONS(proper(X1), proper(X2))
PROPER(fcons(X1, X2)) -> PROPER(X1)
PROPER(fcons(X1, X2)) -> PROPER(X2)
QUOTE(ok(X)) -> QUOTE(X)
QUOTE1(ok(X)) -> QUOTE1(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 17 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
Polo
       →DP Problem 10
Polo
       →DP Problem 11
Polo
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Polo
       →DP Problem 17
Remaining


Dependency Pairs:

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


Rules:


active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(sel(0, cons(X, Z))) -> mark(X)
active(first(0, Z)) -> 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(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) -> mark(quote(X))
active(first1(0, Z)) -> mark(nil1)
active(first1(s(X), cons(Y, Z))) -> mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) -> mark(01)
active(quote1(cons(X, Z))) -> mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) -> mark(nil1)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(sel(X, Z))) -> mark(sel1(X, Z))
active(quote1(first(X, Z))) -> mark(first1(X, Z))
active(unquote(01)) -> mark(0)
active(unquote(s1(X))) -> mark(s(unquote(X)))
active(unquote1(nil1)) -> mark(nil)
active(unquote1(cons1(X, Z))) -> mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) -> mark(cons(X, Z))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(from(X)) -> from(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(first1(X1, X2)) -> first1(active(X1), X2)
active(first1(X1, X2)) -> first1(X1, active(X2))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
active(s1(X)) -> s1(active(X))
active(unquote(X)) -> unquote(active(X))
active(unquote1(X)) -> unquote1(active(X))
active(fcons(X1, X2)) -> fcons(active(X1), X2)
active(fcons(X1, X2)) -> fcons(X1, active(X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(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))
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))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
first1(mark(X1), X2) -> mark(first1(X1, X2))
first1(X1, mark(X2)) -> mark(first1(X1, X2))
first1(ok(X1), ok(X2)) -> ok(first1(X1, X2))
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))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
unquote(mark(X)) -> mark(unquote(X))
unquote(ok(X)) -> ok(unquote(X))
unquote1(mark(X)) -> mark(unquote1(X))
unquote1(ok(X)) -> ok(unquote1(X))
fcons(mark(X1), X2) -> mark(fcons(X1, X2))
fcons(X1, mark(X2)) -> mark(fcons(X1, X2))
fcons(ok(X1), ok(X2)) -> ok(fcons(X1, X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(from(X)) -> from(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
proper(first1(X1, X2)) -> first1(proper(X1), proper(X2))
proper(nil1) -> ok(nil1)
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
proper(01) -> ok(01)
proper(quote1(X)) -> quote1(proper(X))
proper(s1(X)) -> s1(proper(X))
proper(unquote(X)) -> unquote(proper(X))
proper(unquote1(X)) -> unquote1(proper(X))
proper(fcons(X1, X2)) -> fcons(proper(X1), proper(X2))
quote(ok(X)) -> ok(quote(X))
quote1(ok(X)) -> ok(quote1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


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

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(SEL(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 18
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
Polo
       →DP Problem 10
Polo
       →DP Problem 11
Polo
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Polo
       →DP Problem 17
Remaining


Dependency Pairs:

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


Rules:


active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(sel(0, cons(X, Z))) -> mark(X)
active(first(0, Z)) -> 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(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) -> mark(quote(X))
active(first1(0, Z)) -> mark(nil1)
active(first1(s(X), cons(Y, Z))) -> mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) -> mark(01)
active(quote1(cons(X, Z))) -> mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) -> mark(nil1)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(sel(X, Z))) -> mark(sel1(X, Z))
active(quote1(first(X, Z))) -> mark(first1(X, Z))
active(unquote(01)) -> mark(0)
active(unquote(s1(X))) -> mark(s(unquote(X)))
active(unquote1(nil1)) -> mark(nil)
active(unquote1(cons1(X, Z))) -> mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) -> mark(cons(X, Z))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(from(X)) -> from(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(first1(X1, X2)) -> first1(active(X1), X2)
active(first1(X1, X2)) -> first1(X1, active(X2))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
active(s1(X)) -> s1(active(X))
active(unquote(X)) -> unquote(active(X))
active(unquote1(X)) -> unquote1(active(X))
active(fcons(X1, X2)) -> fcons(active(X1), X2)
active(fcons(X1, X2)) -> fcons(X1, active(X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(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))
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))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
first1(mark(X1), X2) -> mark(first1(X1, X2))
first1(X1, mark(X2)) -> mark(first1(X1, X2))
first1(ok(X1), ok(X2)) -> ok(first1(X1, X2))
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))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
unquote(mark(X)) -> mark(unquote(X))
unquote(ok(X)) -> ok(unquote(X))
unquote1(mark(X)) -> mark(unquote1(X))
unquote1(ok(X)) -> ok(unquote1(X))
fcons(mark(X1), X2) -> mark(fcons(X1, X2))
fcons(X1, mark(X2)) -> mark(fcons(X1, X2))
fcons(ok(X1), ok(X2)) -> ok(fcons(X1, X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(from(X)) -> from(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
proper(first1(X1, X2)) -> first1(proper(X1), proper(X2))
proper(nil1) -> ok(nil1)
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
proper(01) -> ok(01)
proper(quote1(X)) -> quote1(proper(X))
proper(s1(X)) -> s1(proper(X))
proper(unquote(X)) -> unquote(proper(X))
proper(unquote1(X)) -> unquote1(proper(X))
proper(fcons(X1, X2)) -> fcons(proper(X1), proper(X2))
quote(ok(X)) -> ok(quote(X))
quote1(ok(X)) -> ok(quote1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


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

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

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
Polo
           →DP Problem 18
Polo
             ...
               →DP Problem 19
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
Polo
       →DP Problem 10
Polo
       →DP Problem 11
Polo
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Polo
       →DP Problem 17
Remaining


Dependency Pair:

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


Rules:


active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(sel(0, cons(X, Z))) -> mark(X)
active(first(0, Z)) -> 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(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) -> mark(quote(X))
active(first1(0, Z)) -> mark(nil1)
active(first1(s(X), cons(Y, Z))) -> mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) -> mark(01)
active(quote1(cons(X, Z))) -> mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) -> mark(nil1)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(sel(X, Z))) -> mark(sel1(X, Z))
active(quote1(first(X, Z))) -> mark(first1(X, Z))
active(unquote(01)) -> mark(0)
active(unquote(s1(X))) -> mark(s(unquote(X)))
active(unquote1(nil1)) -> mark(nil)
active(unquote1(cons1(X, Z))) -> mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) -> mark(cons(X, Z))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(from(X)) -> from(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(first1(X1, X2)) -> first1(active(X1), X2)
active(first1(X1, X2)) -> first1(X1, active(X2))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
active(s1(X)) -> s1(active(X))
active(unquote(X)) -> unquote(active(X))
active(unquote1(X)) -> unquote1(active(X))
active(fcons(X1, X2)) -> fcons(active(X1), X2)
active(fcons(X1, X2)) -> fcons(X1, active(X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(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))
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))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
first1(mark(X1), X2) -> mark(first1(X1, X2))
first1(X1, mark(X2)) -> mark(first1(X1, X2))
first1(ok(X1), ok(X2)) -> ok(first1(X1, X2))
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))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
unquote(mark(X)) -> mark(unquote(X))
unquote(ok(X)) -> ok(unquote(X))
unquote1(mark(X)) -> mark(unquote1(X))
unquote1(ok(X)) -> ok(unquote1(X))
fcons(mark(X1), X2) -> mark(fcons(X1, X2))
fcons(X1, mark(X2)) -> mark(fcons(X1, X2))
fcons(ok(X1), ok(X2)) -> ok(fcons(X1, X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(from(X)) -> from(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
proper(first1(X1, X2)) -> first1(proper(X1), proper(X2))
proper(nil1) -> ok(nil1)
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
proper(01) -> ok(01)
proper(quote1(X)) -> quote1(proper(X))
proper(s1(X)) -> s1(proper(X))
proper(unquote(X)) -> unquote(proper(X))
proper(unquote1(X)) -> unquote1(proper(X))
proper(fcons(X1, X2)) -> fcons(proper(X1), proper(X2))
quote(ok(X)) -> ok(quote(X))
quote1(ok(X)) -> ok(quote1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


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

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

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
Polo
           →DP Problem 18
Polo
             ...
               →DP Problem 20
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
Polo
       →DP Problem 10
Polo
       →DP Problem 11
Polo
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Polo
       →DP Problem 17
Remaining


Dependency Pair:


Rules:


active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(sel(0, cons(X, Z))) -> mark(X)
active(first(0, Z)) -> 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(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) -> mark(quote(X))
active(first1(0, Z)) -> mark(nil1)
active(first1(s(X), cons(Y, Z))) -> mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) -> mark(01)
active(quote1(cons(X, Z))) -> mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) -> mark(nil1)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(sel(X, Z))) -> mark(sel1(X, Z))
active(quote1(first(X, Z))) -> mark(first1(X, Z))
active(unquote(01)) -> mark(0)
active(unquote(s1(X))) -> mark(s(unquote(X)))
active(unquote1(nil1)) -> mark(nil)
active(unquote1(cons1(X, Z))) -> mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) -> mark(cons(X, Z))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(from(X)) -> from(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(first1(X1, X2)) -> first1(active(X1), X2)
active(first1(X1, X2)) -> first1(X1, active(X2))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
active(s1(X)) -> s1(active(X))
active(unquote(X)) -> unquote(active(X))
active(unquote1(X)) -> unquote1(active(X))
active(fcons(X1, X2)) -> fcons(active(X1), X2)
active(fcons(X1, X2)) -> fcons(X1, active(X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(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))
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))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
first1(mark(X1), X2) -> mark(first1(X1, X2))
first1(X1, mark(X2)) -> mark(first1(X1, X2))
first1(ok(X1), ok(X2)) -> ok(first1(X1, X2))
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))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
unquote(mark(X)) -> mark(unquote(X))
unquote(ok(X)) -> ok(unquote(X))
unquote1(mark(X)) -> mark(unquote1(X))
unquote1(ok(X)) -> ok(unquote1(X))
fcons(mark(X1), X2) -> mark(fcons(X1, X2))
fcons(X1, mark(X2)) -> mark(fcons(X1, X2))
fcons(ok(X1), ok(X2)) -> ok(fcons(X1, X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(from(X)) -> from(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
proper(first1(X1, X2)) -> first1(proper(X1), proper(X2))
proper(nil1) -> ok(nil1)
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
proper(01) -> ok(01)
proper(quote1(X)) -> quote1(proper(X))
proper(s1(X)) -> s1(proper(X))
proper(unquote(X)) -> unquote(proper(X))
proper(unquote1(X)) -> unquote1(proper(X))
proper(fcons(X1, X2)) -> fcons(proper(X1), proper(X2))
quote(ok(X)) -> ok(quote(X))
quote1(ok(X)) -> ok(quote1(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
Polo
       →DP Problem 10
Polo
       →DP Problem 11
Polo
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Polo
       →DP Problem 17
Remaining


Dependency Pairs:

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


Rules:


active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(sel(0, cons(X, Z))) -> mark(X)
active(first(0, Z)) -> 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(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) -> mark(quote(X))
active(first1(0, Z)) -> mark(nil1)
active(first1(s(X), cons(Y, Z))) -> mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) -> mark(01)
active(quote1(cons(X, Z))) -> mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) -> mark(nil1)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(sel(X, Z))) -> mark(sel1(X, Z))
active(quote1(first(X, Z))) -> mark(first1(X, Z))
active(unquote(01)) -> mark(0)
active(unquote(s1(X))) -> mark(s(unquote(X)))
active(unquote1(nil1)) -> mark(nil)
active(unquote1(cons1(X, Z))) -> mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) -> mark(cons(X, Z))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(from(X)) -> from(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(first1(X1, X2)) -> first1(active(X1), X2)
active(first1(X1, X2)) -> first1(X1, active(X2))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
active(s1(X)) -> s1(active(X))
active(unquote(X)) -> unquote(active(X))
active(unquote1(X)) -> unquote1(active(X))
active(fcons(X1, X2)) -> fcons(active(X1), X2)
active(fcons(X1, X2)) -> fcons(X1, active(X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(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))
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))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
first1(mark(X1), X2) -> mark(first1(X1, X2))
first1(X1, mark(X2)) -> mark(first1(X1, X2))
first1(ok(X1), ok(X2)) -> ok(first1(X1, X2))
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))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
unquote(mark(X)) -> mark(unquote(X))
unquote(ok(X)) -> ok(unquote(X))
unquote1(mark(X)) -> mark(unquote1(X))
unquote1(ok(X)) -> ok(unquote1(X))
fcons(mark(X1), X2) -> mark(fcons(X1, X2))
fcons(X1, mark(X2)) -> mark(fcons(X1, X2))
fcons(ok(X1), ok(X2)) -> ok(fcons(X1, X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(from(X)) -> from(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
proper(first1(X1, X2)) -> first1(proper(X1), proper(X2))
proper(nil1) -> ok(nil1)
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
proper(01) -> ok(01)
proper(quote1(X)) -> quote1(proper(X))
proper(s1(X)) -> s1(proper(X))
proper(unquote(X)) -> unquote(proper(X))
proper(unquote1(X)) -> unquote1(proper(X))
proper(fcons(X1, X2)) -> fcons(proper(X1), proper(X2))
quote(ok(X)) -> ok(quote(X))
quote1(ok(X)) -> ok(quote1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


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

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(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 2
Polo
           →DP Problem 21
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
Polo
       →DP Problem 10
Polo
       →DP Problem 11
Polo
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Polo
       →DP Problem 17
Remaining


Dependency Pair:

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


Rules:


active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(sel(0, cons(X, Z))) -> mark(X)
active(first(0, Z)) -> 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(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) -> mark(quote(X))
active(first1(0, Z)) -> mark(nil1)
active(first1(s(X), cons(Y, Z))) -> mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) -> mark(01)
active(quote1(cons(X, Z))) -> mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) -> mark(nil1)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(sel(X, Z))) -> mark(sel1(X, Z))
active(quote1(first(X, Z))) -> mark(first1(X, Z))
active(unquote(01)) -> mark(0)
active(unquote(s1(X))) -> mark(s(unquote(X)))
active(unquote1(nil1)) -> mark(nil)
active(unquote1(cons1(X, Z))) -> mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) -> mark(cons(X, Z))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(from(X)) -> from(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(first1(X1, X2)) -> first1(active(X1), X2)
active(first1(X1, X2)) -> first1(X1, active(X2))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
active(s1(X)) -> s1(active(X))
active(unquote(X)) -> unquote(active(X))
active(unquote1(X)) -> unquote1(active(X))
active(fcons(X1, X2)) -> fcons(active(X1), X2)
active(fcons(X1, X2)) -> fcons(X1, active(X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(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))
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))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
first1(mark(X1), X2) -> mark(first1(X1, X2))
first1(X1, mark(X2)) -> mark(first1(X1, X2))
first1(ok(X1), ok(X2)) -> ok(first1(X1, X2))
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))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
unquote(mark(X)) -> mark(unquote(X))
unquote(ok(X)) -> ok(unquote(X))
unquote1(mark(X)) -> mark(unquote1(X))
unquote1(ok(X)) -> ok(unquote1(X))
fcons(mark(X1), X2) -> mark(fcons(X1, X2))
fcons(X1, mark(X2)) -> mark(fcons(X1, X2))
fcons(ok(X1), ok(X2)) -> ok(fcons(X1, X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(from(X)) -> from(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
proper(first1(X1, X2)) -> first1(proper(X1), proper(X2))
proper(nil1) -> ok(nil1)
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
proper(01) -> ok(01)
proper(quote1(X)) -> quote1(proper(X))
proper(s1(X)) -> s1(proper(X))
proper(unquote(X)) -> unquote(proper(X))
proper(unquote1(X)) -> unquote1(proper(X))
proper(fcons(X1, X2)) -> fcons(proper(X1), proper(X2))
quote(ok(X)) -> ok(quote(X))
quote1(ok(X)) -> ok(quote1(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 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 2
Polo
           →DP Problem 21
Polo
             ...
               →DP Problem 22
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
Polo
       →DP Problem 10
Polo
       →DP Problem 11
Polo
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Polo
       →DP Problem 17
Remaining


Dependency Pair:


Rules:


active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(sel(0, cons(X, Z))) -> mark(X)
active(first(0, Z)) -> 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(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) -> mark(quote(X))
active(first1(0, Z)) -> mark(nil1)
active(first1(s(X), cons(Y, Z))) -> mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) -> mark(01)
active(quote1(cons(X, Z))) -> mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) -> mark(nil1)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(sel(X, Z))) -> mark(sel1(X, Z))
active(quote1(first(X, Z))) -> mark(first1(X, Z))
active(unquote(01)) -> mark(0)
active(unquote(s1(X))) -> mark(s(unquote(X)))
active(unquote1(nil1)) -> mark(nil)
active(unquote1(cons1(X, Z))) -> mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) -> mark(cons(X, Z))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(from(X)) -> from(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(first1(X1, X2)) -> first1(active(X1), X2)
active(first1(X1, X2)) -> first1(X1, active(X2))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
active(s1(X)) -> s1(active(X))
active(unquote(X)) -> unquote(active(X))
active(unquote1(X)) -> unquote1(active(X))
active(fcons(X1, X2)) -> fcons(active(X1), X2)
active(fcons(X1, X2)) -> fcons(X1, active(X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(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))
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))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
first1(mark(X1), X2) -> mark(first1(X1, X2))
first1(X1, mark(X2)) -> mark(first1(X1, X2))
first1(ok(X1), ok(X2)) -> ok(first1(X1, X2))
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))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
unquote(mark(X)) -> mark(unquote(X))
unquote(ok(X)) -> ok(unquote(X))
unquote1(mark(X)) -> mark(unquote1(X))
unquote1(ok(X)) -> ok(unquote1(X))
fcons(mark(X1), X2) -> mark(fcons(X1, X2))
fcons(X1, mark(X2)) -> mark(fcons(X1, X2))
fcons(ok(X1), ok(X2)) -> ok(fcons(X1, X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(from(X)) -> from(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
proper(first1(X1, X2)) -> first1(proper(X1), proper(X2))
proper(nil1) -> ok(nil1)
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
proper(01) -> ok(01)
proper(quote1(X)) -> quote1(proper(X))
proper(s1(X)) -> s1(proper(X))
proper(unquote(X)) -> unquote(proper(X))
proper(unquote1(X)) -> unquote1(proper(X))
proper(fcons(X1, X2)) -> fcons(proper(X1), proper(X2))
quote(ok(X)) -> ok(quote(X))
quote1(ok(X)) -> ok(quote1(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
Polo
       →DP Problem 10
Polo
       →DP Problem 11
Polo
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Polo
       →DP Problem 17
Remaining


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(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(sel(0, cons(X, Z))) -> mark(X)
active(first(0, Z)) -> 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(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) -> mark(quote(X))
active(first1(0, Z)) -> mark(nil1)
active(first1(s(X), cons(Y, Z))) -> mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) -> mark(01)
active(quote1(cons(X, Z))) -> mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) -> mark(nil1)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(sel(X, Z))) -> mark(sel1(X, Z))
active(quote1(first(X, Z))) -> mark(first1(X, Z))
active(unquote(01)) -> mark(0)
active(unquote(s1(X))) -> mark(s(unquote(X)))
active(unquote1(nil1)) -> mark(nil)
active(unquote1(cons1(X, Z))) -> mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) -> mark(cons(X, Z))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(from(X)) -> from(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(first1(X1, X2)) -> first1(active(X1), X2)
active(first1(X1, X2)) -> first1(X1, active(X2))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
active(s1(X)) -> s1(active(X))
active(unquote(X)) -> unquote(active(X))
active(unquote1(X)) -> unquote1(active(X))
active(fcons(X1, X2)) -> fcons(active(X1), X2)
active(fcons(X1, X2)) -> fcons(X1, active(X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(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))
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))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
first1(mark(X1), X2) -> mark(first1(X1, X2))
first1(X1, mark(X2)) -> mark(first1(X1, X2))
first1(ok(X1), ok(X2)) -> ok(first1(X1, X2))
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))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
unquote(mark(X)) -> mark(unquote(X))
unquote(ok(X)) -> ok(unquote(X))
unquote1(mark(X)) -> mark(unquote1(X))
unquote1(ok(X)) -> ok(unquote1(X))
fcons(mark(X1), X2) -> mark(fcons(X1, X2))
fcons(X1, mark(X2)) -> mark(fcons(X1, X2))
fcons(ok(X1), ok(X2)) -> ok(fcons(X1, X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(from(X)) -> from(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
proper(first1(X1, X2)) -> first1(proper(X1), proper(X2))
proper(nil1) -> ok(nil1)
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
proper(01) -> ok(01)
proper(quote1(X)) -> quote1(proper(X))
proper(s1(X)) -> s1(proper(X))
proper(unquote(X)) -> unquote(proper(X))
proper(unquote1(X)) -> unquote1(proper(X))
proper(fcons(X1, X2)) -> fcons(proper(X1), proper(X2))
quote(ok(X)) -> ok(quote(X))
quote1(ok(X)) -> ok(quote1(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 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 3
Polo
           →DP Problem 23
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
Polo
       →DP Problem 10
Polo
       →DP Problem 11
Polo
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Polo
       →DP Problem 17
Remaining


Dependency Pairs:

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


Rules:


active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(sel(0, cons(X, Z))) -> mark(X)
active(first(0, Z)) -> 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(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) -> mark(quote(X))
active(first1(0, Z)) -> mark(nil1)
active(first1(s(X), cons(Y, Z))) -> mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) -> mark(01)
active(quote1(cons(X, Z))) -> mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) -> mark(nil1)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(sel(X, Z))) -> mark(sel1(X, Z))
active(quote1(first(X, Z))) -> mark(first1(X, Z))
active(unquote(01)) -> mark(0)
active(unquote(s1(X))) -> mark(s(unquote(X)))
active(unquote1(nil1)) -> mark(nil)
active(unquote1(cons1(X, Z))) -> mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) -> mark(cons(X, Z))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(from(X)) -> from(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(first1(X1, X2)) -> first1(active(X1), X2)
active(first1(X1, X2)) -> first1(X1, active(X2))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
active(s1(X)) -> s1(active(X))
active(unquote(X)) -> unquote(active(X))
active(unquote1(X)) -> unquote1(active(X))
active(fcons(X1, X2)) -> fcons(active(X1), X2)
active(fcons(X1, X2)) -> fcons(X1, active(X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(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))
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))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
first1(mark(X1), X2) -> mark(first1(X1, X2))
first1(X1, mark(X2)) -> mark(first1(X1, X2))
first1(ok(X1), ok(X2)) -> ok(first1(X1, X2))
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))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
unquote(mark(X)) -> mark(unquote(X))
unquote(ok(X)) -> ok(unquote(X))
unquote1(mark(X)) -> mark(unquote1(X))
unquote1(ok(X)) -> ok(unquote1(X))
fcons(mark(X1), X2) -> mark(fcons(X1, X2))
fcons(X1, mark(X2)) -> mark(fcons(X1, X2))
fcons(ok(X1), ok(X2)) -> ok(fcons(X1, X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(from(X)) -> from(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
proper(first1(X1, X2)) -> first1(proper(X1), proper(X2))
proper(nil1) -> ok(nil1)
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
proper(01) -> ok(01)
proper(quote1(X)) -> quote1(proper(X))
proper(s1(X)) -> s1(proper(X))
proper(unquote(X)) -> unquote(proper(X))
proper(unquote1(X)) -> unquote1(proper(X))
proper(fcons(X1, X2)) -> fcons(proper(X1), proper(X2))
quote(ok(X)) -> ok(quote(X))
quote1(ok(X)) -> ok(quote1(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 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 3
Polo
           →DP Problem 23
Polo
             ...
               →DP Problem 24
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
Polo
       →DP Problem 10
Polo
       →DP Problem 11
Polo
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Polo
       →DP Problem 17
Remaining


Dependency Pair:

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


Rules:


active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(sel(0, cons(X, Z))) -> mark(X)
active(first(0, Z)) -> 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(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) -> mark(quote(X))
active(first1(0, Z)) -> mark(nil1)
active(first1(s(X), cons(Y, Z))) -> mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) -> mark(01)
active(quote1(cons(X, Z))) -> mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) -> mark(nil1)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(sel(X, Z))) -> mark(sel1(X, Z))
active(quote1(first(X, Z))) -> mark(first1(X, Z))
active(unquote(01)) -> mark(0)
active(unquote(s1(X))) -> mark(s(unquote(X)))
active(unquote1(nil1)) -> mark(nil)
active(unquote1(cons1(X, Z))) -> mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) -> mark(cons(X, Z))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(from(X)) -> from(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(first1(X1, X2)) -> first1(active(X1), X2)
active(first1(X1, X2)) -> first1(X1, active(X2))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
active(s1(X)) -> s1(active(X))
active(unquote(X)) -> unquote(active(X))
active(unquote1(X)) -> unquote1(active(X))
active(fcons(X1, X2)) -> fcons(active(X1), X2)
active(fcons(X1, X2)) -> fcons(X1, active(X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(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))
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))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
first1(mark(X1), X2) -> mark(first1(X1, X2))
first1(X1, mark(X2)) -> mark(first1(X1, X2))
first1(ok(X1), ok(X2)) -> ok(first1(X1, X2))
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))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
unquote(mark(X)) -> mark(unquote(X))
unquote(ok(X)) -> ok(unquote(X))
unquote1(mark(X)) -> mark(unquote1(X))
unquote1(ok(X)) -> ok(unquote1(X))
fcons(mark(X1), X2) -> mark(fcons(X1, X2))
fcons(X1, mark(X2)) -> mark(fcons(X1, X2))
fcons(ok(X1), ok(X2)) -> ok(fcons(X1, X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(from(X)) -> from(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
proper(first1(X1, X2)) -> first1(proper(X1), proper(X2))
proper(nil1) -> ok(nil1)
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
proper(01) -> ok(01)
proper(quote1(X)) -> quote1(proper(X))
proper(s1(X)) -> s1(proper(X))
proper(unquote(X)) -> unquote(proper(X))
proper(unquote1(X)) -> unquote1(proper(X))
proper(fcons(X1, X2)) -> fcons(proper(X1), proper(X2))
quote(ok(X)) -> ok(quote(X))
quote1(ok(X)) -> ok(quote1(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 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 3
Polo
           →DP Problem 23
Polo
             ...
               →DP Problem 25
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
Polo
       →DP Problem 10
Polo
       →DP Problem 11
Polo
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Polo
       →DP Problem 17
Remaining


Dependency Pair:


Rules:


active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(sel(0, cons(X, Z))) -> mark(X)
active(first(0, Z)) -> 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(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) -> mark(quote(X))
active(first1(0, Z)) -> mark(nil1)
active(first1(s(X), cons(Y, Z))) -> mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) -> mark(01)
active(quote1(cons(X, Z))) -> mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) -> mark(nil1)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(sel(X, Z))) -> mark(sel1(X, Z))
active(quote1(first(X, Z))) -> mark(first1(X, Z))
active(unquote(01)) -> mark(0)
active(unquote(s1(X))) -> mark(s(unquote(X)))
active(unquote1(nil1)) -> mark(nil)
active(unquote1(cons1(X, Z))) -> mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) -> mark(cons(X, Z))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(from(X)) -> from(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(first1(X1, X2)) -> first1(active(X1), X2)
active(first1(X1, X2)) -> first1(X1, active(X2))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
active(s1(X)) -> s1(active(X))
active(unquote(X)) -> unquote(active(X))
active(unquote1(X)) -> unquote1(active(X))
active(fcons(X1, X2)) -> fcons(active(X1), X2)
active(fcons(X1, X2)) -> fcons(X1, active(X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(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))
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))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
first1(mark(X1), X2) -> mark(first1(X1, X2))
first1(X1, mark(X2)) -> mark(first1(X1, X2))
first1(ok(X1), ok(X2)) -> ok(first1(X1, X2))
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))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
unquote(mark(X)) -> mark(unquote(X))
unquote(ok(X)) -> ok(unquote(X))
unquote1(mark(X)) -> mark(unquote1(X))
unquote1(ok(X)) -> ok(unquote1(X))
fcons(mark(X1), X2) -> mark(fcons(X1, X2))
fcons(X1, mark(X2)) -> mark(fcons(X1, X2))
fcons(ok(X1), ok(X2)) -> ok(fcons(X1, X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(from(X)) -> from(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
proper(first1(X1, X2)) -> first1(proper(X1), proper(X2))
proper(nil1) -> ok(nil1)
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
proper(01) -> ok(01)
proper(quote1(X)) -> quote1(proper(X))
proper(s1(X)) -> s1(proper(X))
proper(unquote(X)) -> unquote(proper(X))
proper(unquote1(X)) -> unquote1(proper(X))
proper(fcons(X1, X2)) -> fcons(proper(X1), proper(X2))
quote(ok(X)) -> ok(quote(X))
quote1(ok(X)) -> ok(quote1(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
Polo
       →DP Problem 10
Polo
       →DP Problem 11
Polo
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Polo
       →DP Problem 17
Remaining


Dependency Pairs:

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


Rules:


active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(sel(0, cons(X, Z))) -> mark(X)
active(first(0, Z)) -> 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(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) -> mark(quote(X))
active(first1(0, Z)) -> mark(nil1)
active(first1(s(X), cons(Y, Z))) -> mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) -> mark(01)
active(quote1(cons(X, Z))) -> mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) -> mark(nil1)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(sel(X, Z))) -> mark(sel1(X, Z))
active(quote1(first(X, Z))) -> mark(first1(X, Z))
active(unquote(01)) -> mark(0)
active(unquote(s1(X))) -> mark(s(unquote(X)))
active(unquote1(nil1)) -> mark(nil)
active(unquote1(cons1(X, Z))) -> mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) -> mark(cons(X, Z))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(from(X)) -> from(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(first1(X1, X2)) -> first1(active(X1), X2)
active(first1(X1, X2)) -> first1(X1, active(X2))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
active(s1(X)) -> s1(active(X))
active(unquote(X)) -> unquote(active(X))
active(unquote1(X)) -> unquote1(active(X))
active(fcons(X1, X2)) -> fcons(active(X1), X2)
active(fcons(X1, X2)) -> fcons(X1, active(X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(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))
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))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
first1(mark(X1), X2) -> mark(first1(X1, X2))
first1(X1, mark(X2)) -> mark(first1(X1, X2))
first1(ok(X1), ok(X2)) -> ok(first1(X1, X2))
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))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
unquote(mark(X)) -> mark(unquote(X))
unquote(ok(X)) -> ok(unquote(X))
unquote1(mark(X)) -> mark(unquote1(X))
unquote1(ok(X)) -> ok(unquote1(X))
fcons(mark(X1), X2) -> mark(fcons(X1, X2))
fcons(X1, mark(X2)) -> mark(fcons(X1, X2))
fcons(ok(X1), ok(X2)) -> ok(fcons(X1, X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(from(X)) -> from(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
proper(first1(X1, X2)) -> first1(proper(X1), proper(X2))
proper(nil1) -> ok(nil1)
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
proper(01) -> ok(01)
proper(quote1(X)) -> quote1(proper(X))
proper(s1(X)) -> s1(proper(X))
proper(unquote(X)) -> unquote(proper(X))
proper(unquote1(X)) -> unquote1(proper(X))
proper(fcons(X1, X2)) -> fcons(proper(X1), proper(X2))
quote(ok(X)) -> ok(quote(X))
quote1(ok(X)) -> ok(quote1(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 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 4
Polo
           →DP Problem 26
Polynomial Ordering
       →DP Problem 5
Polo
       →DP Problem 6
Polo
       →DP Problem 7
Polo
       →DP Problem 8
Polo
       →DP Problem 9
Polo
       →DP Problem 10
Polo
       →DP Problem 11
Polo
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Polo
       →DP Problem 17
Remaining


Dependency Pair:

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


Rules:


active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(sel(0, cons(X, Z))) -> mark(X)
active(first(0, Z)) -> 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(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) -> mark(quote(X))
active(first1(0, Z)) -> mark(nil1)
active(first1(s(X), cons(Y, Z))) -> mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) -> mark(01)
active(quote1(cons(X, Z))) -> mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) -> mark(nil1)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(sel(X, Z))) -> mark(sel1(X, Z))
active(quote1(first(X, Z))) -> mark(first1(X, Z))
active(unquote(01)) -> mark(0)
active(unquote(s1(X))) -> mark(s(unquote(X)))
active(unquote1(nil1)) -> mark(nil)
active(unquote1(cons1(X, Z))) -> mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) -> mark(cons(X, Z))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(from(X)) -> from(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(first1(X1, X2)) -> first1(active(X1), X2)
active(first1(X1, X2)) -> first1(X1, active(X2))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
active(s1(X)) -> s1(active(X))
active(unquote(X)) -> unquote(active(X))
active(unquote1(X)) -> unquote1(active(X))
active(fcons(X1, X2)) -> fcons(active(X1), X2)
active(fcons(X1, X2)) -> fcons(X1, active(X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(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))
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))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
first1(mark(X1), X2) -> mark(first1(X1, X2))
first1(X1, mark(X2)) -> mark(first1(X1, X2))
first1(ok(X1), ok(X2)) -> ok(first1(X1, X2))
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))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
unquote(mark(X)) -> mark(unquote(X))
unquote(ok(X)) -> ok(unquote(X))
unquote1(mark(X)) -> mark(unquote1(X))
unquote1(ok(X)) -> ok(unquote1(X))
fcons(mark(X1), X2) -> mark(fcons(X1, X2))
fcons(X1, mark(X2)) -> mark(fcons(X1, X2))
fcons(ok(X1), ok(X2)) -> ok(fcons(X1, X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(from(X)) -> from(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
proper(first1(X1, X2)) -> first1(proper(X1), proper(X2))
proper(nil1) -> ok(nil1)
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
proper(01) -> ok(01)
proper(quote1(X)) -> quote1(proper(X))
proper(s1(X)) -> s1(proper(X))
proper(unquote(X)) -> unquote(proper(X))
proper(unquote1(X)) -> unquote1(proper(X))
proper(fcons(X1, X2)) -> fcons(proper(X1), proper(X2))
quote(ok(X)) -> ok(quote(X))
quote1(ok(X)) -> ok(quote1(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 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 4
Polo
           →DP Problem 26
Polo
             ...
               →DP Problem 27
Dependency Graph
       →DP Problem 5
Polo
       →DP Problem 6
Polo
       →DP Problem 7
Polo
       →DP Problem 8
Polo
       →DP Problem 9
Polo
       →DP Problem 10
Polo
       →DP Problem 11
Polo
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Polo
       →DP Problem 17
Remaining


Dependency Pair:


Rules:


active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(sel(0, cons(X, Z))) -> mark(X)
active(first(0, Z)) -> 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(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) -> mark(quote(X))
active(first1(0, Z)) -> mark(nil1)
active(first1(s(X), cons(Y, Z))) -> mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) -> mark(01)
active(quote1(cons(X, Z))) -> mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) -> mark(nil1)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(sel(X, Z))) -> mark(sel1(X, Z))
active(quote1(first(X, Z))) -> mark(first1(X, Z))
active(unquote(01)) -> mark(0)
active(unquote(s1(X))) -> mark(s(unquote(X)))
active(unquote1(nil1)) -> mark(nil)
active(unquote1(cons1(X, Z))) -> mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) -> mark(cons(X, Z))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(from(X)) -> from(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(first1(X1, X2)) -> first1(active(X1), X2)
active(first1(X1, X2)) -> first1(X1, active(X2))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
active(s1(X)) -> s1(active(X))
active(unquote(X)) -> unquote(active(X))
active(unquote1(X)) -> unquote1(active(X))
active(fcons(X1, X2)) -> fcons(active(X1), X2)
active(fcons(X1, X2)) -> fcons(X1, active(X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(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))
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))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
first1(mark(X1), X2) -> mark(first1(X1, X2))
first1(X1, mark(X2)) -> mark(first1(X1, X2))
first1(ok(X1), ok(X2)) -> ok(first1(X1, X2))
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))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
unquote(mark(X)) -> mark(unquote(X))
unquote(ok(X)) -> ok(unquote(X))
unquote1(mark(X)) -> mark(unquote1(X))
unquote1(ok(X)) -> ok(unquote1(X))
fcons(mark(X1), X2) -> mark(fcons(X1, X2))
fcons(X1, mark(X2)) -> mark(fcons(X1, X2))
fcons(ok(X1), ok(X2)) -> ok(fcons(X1, X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(from(X)) -> from(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
proper(first1(X1, X2)) -> first1(proper(X1), proper(X2))
proper(nil1) -> ok(nil1)
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
proper(01) -> ok(01)
proper(quote1(X)) -> quote1(proper(X))
proper(s1(X)) -> s1(proper(X))
proper(unquote(X)) -> unquote(proper(X))
proper(unquote1(X)) -> unquote1(proper(X))
proper(fcons(X1, X2)) -> fcons(proper(X1), proper(X2))
quote(ok(X)) -> ok(quote(X))
quote1(ok(X)) -> ok(quote1(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
Polo
       →DP Problem 10
Polo
       →DP Problem 11
Polo
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Polo
       →DP Problem 17
Remaining


Dependency Pairs:

SEL1(ok(X1), ok(X2)) -> SEL1(X1, X2)
SEL1(mark(X1), X2) -> SEL1(X1, X2)
SEL1(X1, mark(X2)) -> SEL1(X1, X2)


Rules:


active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(sel(0, cons(X, Z))) -> mark(X)
active(first(0, Z)) -> 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(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) -> mark(quote(X))
active(first1(0, Z)) -> mark(nil1)
active(first1(s(X), cons(Y, Z))) -> mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) -> mark(01)
active(quote1(cons(X, Z))) -> mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) -> mark(nil1)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(sel(X, Z))) -> mark(sel1(X, Z))
active(quote1(first(X, Z))) -> mark(first1(X, Z))
active(unquote(01)) -> mark(0)
active(unquote(s1(X))) -> mark(s(unquote(X)))
active(unquote1(nil1)) -> mark(nil)
active(unquote1(cons1(X, Z))) -> mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) -> mark(cons(X, Z))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(from(X)) -> from(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(first1(X1, X2)) -> first1(active(X1), X2)
active(first1(X1, X2)) -> first1(X1, active(X2))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
active(s1(X)) -> s1(active(X))
active(unquote(X)) -> unquote(active(X))
active(unquote1(X)) -> unquote1(active(X))
active(fcons(X1, X2)) -> fcons(active(X1), X2)
active(fcons(X1, X2)) -> fcons(X1, active(X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(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))
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))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
first1(mark(X1), X2) -> mark(first1(X1, X2))
first1(X1, mark(X2)) -> mark(first1(X1, X2))
first1(ok(X1), ok(X2)) -> ok(first1(X1, X2))
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))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
unquote(mark(X)) -> mark(unquote(X))
unquote(ok(X)) -> ok(unquote(X))
unquote1(mark(X)) -> mark(unquote1(X))
unquote1(ok(X)) -> ok(unquote1(X))
fcons(mark(X1), X2) -> mark(fcons(X1, X2))
fcons(X1, mark(X2)) -> mark(fcons(X1, X2))
fcons(ok(X1), ok(X2)) -> ok(fcons(X1, X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(from(X)) -> from(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
proper(first1(X1, X2)) -> first1(proper(X1), proper(X2))
proper(nil1) -> ok(nil1)
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
proper(01) -> ok(01)
proper(quote1(X)) -> quote1(proper(X))
proper(s1(X)) -> s1(proper(X))
proper(unquote(X)) -> unquote(proper(X))
proper(unquote1(X)) -> unquote1(proper(X))
proper(fcons(X1, X2)) -> fcons(proper(X1), proper(X2))
quote(ok(X)) -> ok(quote(X))
quote1(ok(X)) -> ok(quote1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

SEL1(ok(X1), ok(X2)) -> SEL1(X1, X2)


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

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(mark(x1))=  x1  
  POL(ok(x1))=  1 + x1  
  POL(SEL1(x1, x2))=  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 28
Polynomial Ordering
       →DP Problem 6
Polo
       →DP Problem 7
Polo
       →DP Problem 8
Polo
       →DP Problem 9
Polo
       →DP Problem 10
Polo
       →DP Problem 11
Polo
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Polo
       →DP Problem 17
Remaining


Dependency Pairs:

SEL1(mark(X1), X2) -> SEL1(X1, X2)
SEL1(X1, mark(X2)) -> SEL1(X1, X2)


Rules:


active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(sel(0, cons(X, Z))) -> mark(X)
active(first(0, Z)) -> 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(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) -> mark(quote(X))
active(first1(0, Z)) -> mark(nil1)
active(first1(s(X), cons(Y, Z))) -> mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) -> mark(01)
active(quote1(cons(X, Z))) -> mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) -> mark(nil1)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(sel(X, Z))) -> mark(sel1(X, Z))
active(quote1(first(X, Z))) -> mark(first1(X, Z))
active(unquote(01)) -> mark(0)
active(unquote(s1(X))) -> mark(s(unquote(X)))
active(unquote1(nil1)) -> mark(nil)
active(unquote1(cons1(X, Z))) -> mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) -> mark(cons(X, Z))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(from(X)) -> from(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(first1(X1, X2)) -> first1(active(X1), X2)
active(first1(X1, X2)) -> first1(X1, active(X2))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
active(s1(X)) -> s1(active(X))
active(unquote(X)) -> unquote(active(X))
active(unquote1(X)) -> unquote1(active(X))
active(fcons(X1, X2)) -> fcons(active(X1), X2)
active(fcons(X1, X2)) -> fcons(X1, active(X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(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))
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))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
first1(mark(X1), X2) -> mark(first1(X1, X2))
first1(X1, mark(X2)) -> mark(first1(X1, X2))
first1(ok(X1), ok(X2)) -> ok(first1(X1, X2))
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))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
unquote(mark(X)) -> mark(unquote(X))
unquote(ok(X)) -> ok(unquote(X))
unquote1(mark(X)) -> mark(unquote1(X))
unquote1(ok(X)) -> ok(unquote1(X))
fcons(mark(X1), X2) -> mark(fcons(X1, X2))
fcons(X1, mark(X2)) -> mark(fcons(X1, X2))
fcons(ok(X1), ok(X2)) -> ok(fcons(X1, X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(from(X)) -> from(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
proper(first1(X1, X2)) -> first1(proper(X1), proper(X2))
proper(nil1) -> ok(nil1)
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
proper(01) -> ok(01)
proper(quote1(X)) -> quote1(proper(X))
proper(s1(X)) -> s1(proper(X))
proper(unquote(X)) -> unquote(proper(X))
proper(unquote1(X)) -> unquote1(proper(X))
proper(fcons(X1, X2)) -> fcons(proper(X1), proper(X2))
quote(ok(X)) -> ok(quote(X))
quote1(ok(X)) -> ok(quote1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

SEL1(mark(X1), X2) -> SEL1(X1, X2)


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

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(mark(x1))=  1 + x1  
  POL(SEL1(x1, x2))=  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 28
Polo
             ...
               →DP Problem 29
Polynomial Ordering
       →DP Problem 6
Polo
       →DP Problem 7
Polo
       →DP Problem 8
Polo
       →DP Problem 9
Polo
       →DP Problem 10
Polo
       →DP Problem 11
Polo
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Polo
       →DP Problem 17
Remaining


Dependency Pair:

SEL1(X1, mark(X2)) -> SEL1(X1, X2)


Rules:


active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(sel(0, cons(X, Z))) -> mark(X)
active(first(0, Z)) -> 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(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) -> mark(quote(X))
active(first1(0, Z)) -> mark(nil1)
active(first1(s(X), cons(Y, Z))) -> mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) -> mark(01)
active(quote1(cons(X, Z))) -> mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) -> mark(nil1)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(sel(X, Z))) -> mark(sel1(X, Z))
active(quote1(first(X, Z))) -> mark(first1(X, Z))
active(unquote(01)) -> mark(0)
active(unquote(s1(X))) -> mark(s(unquote(X)))
active(unquote1(nil1)) -> mark(nil)
active(unquote1(cons1(X, Z))) -> mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) -> mark(cons(X, Z))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(from(X)) -> from(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(first1(X1, X2)) -> first1(active(X1), X2)
active(first1(X1, X2)) -> first1(X1, active(X2))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
active(s1(X)) -> s1(active(X))
active(unquote(X)) -> unquote(active(X))
active(unquote1(X)) -> unquote1(active(X))
active(fcons(X1, X2)) -> fcons(active(X1), X2)
active(fcons(X1, X2)) -> fcons(X1, active(X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(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))
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))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
first1(mark(X1), X2) -> mark(first1(X1, X2))
first1(X1, mark(X2)) -> mark(first1(X1, X2))
first1(ok(X1), ok(X2)) -> ok(first1(X1, X2))
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))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
unquote(mark(X)) -> mark(unquote(X))
unquote(ok(X)) -> ok(unquote(X))
unquote1(mark(X)) -> mark(unquote1(X))
unquote1(ok(X)) -> ok(unquote1(X))
fcons(mark(X1), X2) -> mark(fcons(X1, X2))
fcons(X1, mark(X2)) -> mark(fcons(X1, X2))
fcons(ok(X1), ok(X2)) -> ok(fcons(X1, X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(from(X)) -> from(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
proper(first1(X1, X2)) -> first1(proper(X1), proper(X2))
proper(nil1) -> ok(nil1)
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
proper(01) -> ok(01)
proper(quote1(X)) -> quote1(proper(X))
proper(s1(X)) -> s1(proper(X))
proper(unquote(X)) -> unquote(proper(X))
proper(unquote1(X)) -> unquote1(proper(X))
proper(fcons(X1, X2)) -> fcons(proper(X1), proper(X2))
quote(ok(X)) -> ok(quote(X))
quote1(ok(X)) -> ok(quote1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

SEL1(X1, mark(X2)) -> SEL1(X1, X2)


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

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(mark(x1))=  1 + x1  
  POL(SEL1(x1, x2))=  x2  

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 28
Polo
             ...
               →DP Problem 30
Dependency Graph
       →DP Problem 6
Polo
       →DP Problem 7
Polo
       →DP Problem 8
Polo
       →DP Problem 9
Polo
       →DP Problem 10
Polo
       →DP Problem 11
Polo
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Polo
       →DP Problem 17
Remaining


Dependency Pair:


Rules:


active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(sel(0, cons(X, Z))) -> mark(X)
active(first(0, Z)) -> 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(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) -> mark(quote(X))
active(first1(0, Z)) -> mark(nil1)
active(first1(s(X), cons(Y, Z))) -> mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) -> mark(01)
active(quote1(cons(X, Z))) -> mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) -> mark(nil1)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(sel(X, Z))) -> mark(sel1(X, Z))
active(quote1(first(X, Z))) -> mark(first1(X, Z))
active(unquote(01)) -> mark(0)
active(unquote(s1(X))) -> mark(s(unquote(X)))
active(unquote1(nil1)) -> mark(nil)
active(unquote1(cons1(X, Z))) -> mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) -> mark(cons(X, Z))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(from(X)) -> from(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(first1(X1, X2)) -> first1(active(X1), X2)
active(first1(X1, X2)) -> first1(X1, active(X2))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
active(s1(X)) -> s1(active(X))
active(unquote(X)) -> unquote(active(X))
active(unquote1(X)) -> unquote1(active(X))
active(fcons(X1, X2)) -> fcons(active(X1), X2)
active(fcons(X1, X2)) -> fcons(X1, active(X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(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))
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))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
first1(mark(X1), X2) -> mark(first1(X1, X2))
first1(X1, mark(X2)) -> mark(first1(X1, X2))
first1(ok(X1), ok(X2)) -> ok(first1(X1, X2))
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))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
unquote(mark(X)) -> mark(unquote(X))
unquote(ok(X)) -> ok(unquote(X))
unquote1(mark(X)) -> mark(unquote1(X))
unquote1(ok(X)) -> ok(unquote1(X))
fcons(mark(X1), X2) -> mark(fcons(X1, X2))
fcons(X1, mark(X2)) -> mark(fcons(X1, X2))
fcons(ok(X1), ok(X2)) -> ok(fcons(X1, X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(from(X)) -> from(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
proper(first1(X1, X2)) -> first1(proper(X1), proper(X2))
proper(nil1) -> ok(nil1)
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
proper(01) -> ok(01)
proper(quote1(X)) -> quote1(proper(X))
proper(s1(X)) -> s1(proper(X))
proper(unquote(X)) -> unquote(proper(X))
proper(unquote1(X)) -> unquote1(proper(X))
proper(fcons(X1, X2)) -> fcons(proper(X1), proper(X2))
quote(ok(X)) -> ok(quote(X))
quote1(ok(X)) -> ok(quote1(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
Polo
       →DP Problem 10
Polo
       →DP Problem 11
Polo
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Polo
       →DP Problem 17
Remaining


Dependency Pair:

QUOTE(ok(X)) -> QUOTE(X)


Rules:


active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(sel(0, cons(X, Z))) -> mark(X)
active(first(0, Z)) -> 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(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) -> mark(quote(X))
active(first1(0, Z)) -> mark(nil1)
active(first1(s(X), cons(Y, Z))) -> mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) -> mark(01)
active(quote1(cons(X, Z))) -> mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) -> mark(nil1)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(sel(X, Z))) -> mark(sel1(X, Z))
active(quote1(first(X, Z))) -> mark(first1(X, Z))
active(unquote(01)) -> mark(0)
active(unquote(s1(X))) -> mark(s(unquote(X)))
active(unquote1(nil1)) -> mark(nil)
active(unquote1(cons1(X, Z))) -> mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) -> mark(cons(X, Z))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(from(X)) -> from(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(first1(X1, X2)) -> first1(active(X1), X2)
active(first1(X1, X2)) -> first1(X1, active(X2))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
active(s1(X)) -> s1(active(X))
active(unquote(X)) -> unquote(active(X))
active(unquote1(X)) -> unquote1(active(X))
active(fcons(X1, X2)) -> fcons(active(X1), X2)
active(fcons(X1, X2)) -> fcons(X1, active(X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(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))
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))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
first1(mark(X1), X2) -> mark(first1(X1, X2))
first1(X1, mark(X2)) -> mark(first1(X1, X2))
first1(ok(X1), ok(X2)) -> ok(first1(X1, X2))
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))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
unquote(mark(X)) -> mark(unquote(X))
unquote(ok(X)) -> ok(unquote(X))
unquote1(mark(X)) -> mark(unquote1(X))
unquote1(ok(X)) -> ok(unquote1(X))
fcons(mark(X1), X2) -> mark(fcons(X1, X2))
fcons(X1, mark(X2)) -> mark(fcons(X1, X2))
fcons(ok(X1), ok(X2)) -> ok(fcons(X1, X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(from(X)) -> from(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
proper(first1(X1, X2)) -> first1(proper(X1), proper(X2))
proper(nil1) -> ok(nil1)
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
proper(01) -> ok(01)
proper(quote1(X)) -> quote1(proper(X))
proper(s1(X)) -> s1(proper(X))
proper(unquote(X)) -> unquote(proper(X))
proper(unquote1(X)) -> unquote1(proper(X))
proper(fcons(X1, X2)) -> fcons(proper(X1), proper(X2))
quote(ok(X)) -> ok(quote(X))
quote1(ok(X)) -> ok(quote1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

QUOTE(ok(X)) -> QUOTE(X)


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

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(QUOTE(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 31
Dependency Graph
       →DP Problem 7
Polo
       →DP Problem 8
Polo
       →DP Problem 9
Polo
       →DP Problem 10
Polo
       →DP Problem 11
Polo
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Polo
       →DP Problem 17
Remaining


Dependency Pair:


Rules:


active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(sel(0, cons(X, Z))) -> mark(X)
active(first(0, Z)) -> 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(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) -> mark(quote(X))
active(first1(0, Z)) -> mark(nil1)
active(first1(s(X), cons(Y, Z))) -> mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) -> mark(01)
active(quote1(cons(X, Z))) -> mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) -> mark(nil1)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(sel(X, Z))) -> mark(sel1(X, Z))
active(quote1(first(X, Z))) -> mark(first1(X, Z))
active(unquote(01)) -> mark(0)
active(unquote(s1(X))) -> mark(s(unquote(X)))
active(unquote1(nil1)) -> mark(nil)
active(unquote1(cons1(X, Z))) -> mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) -> mark(cons(X, Z))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(from(X)) -> from(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(first1(X1, X2)) -> first1(active(X1), X2)
active(first1(X1, X2)) -> first1(X1, active(X2))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
active(s1(X)) -> s1(active(X))
active(unquote(X)) -> unquote(active(X))
active(unquote1(X)) -> unquote1(active(X))
active(fcons(X1, X2)) -> fcons(active(X1), X2)
active(fcons(X1, X2)) -> fcons(X1, active(X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(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))
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))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
first1(mark(X1), X2) -> mark(first1(X1, X2))
first1(X1, mark(X2)) -> mark(first1(X1, X2))
first1(ok(X1), ok(X2)) -> ok(first1(X1, X2))
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))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
unquote(mark(X)) -> mark(unquote(X))
unquote(ok(X)) -> ok(unquote(X))
unquote1(mark(X)) -> mark(unquote1(X))
unquote1(ok(X)) -> ok(unquote1(X))
fcons(mark(X1), X2) -> mark(fcons(X1, X2))
fcons(X1, mark(X2)) -> mark(fcons(X1, X2))
fcons(ok(X1), ok(X2)) -> ok(fcons(X1, X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(from(X)) -> from(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
proper(first1(X1, X2)) -> first1(proper(X1), proper(X2))
proper(nil1) -> ok(nil1)
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
proper(01) -> ok(01)
proper(quote1(X)) -> quote1(proper(X))
proper(s1(X)) -> s1(proper(X))
proper(unquote(X)) -> unquote(proper(X))
proper(unquote1(X)) -> unquote1(proper(X))
proper(fcons(X1, X2)) -> fcons(proper(X1), proper(X2))
quote(ok(X)) -> ok(quote(X))
quote1(ok(X)) -> ok(quote1(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
Polo
       →DP Problem 10
Polo
       →DP Problem 11
Polo
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Polo
       →DP Problem 17
Remaining


Dependency Pairs:

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


Rules:


active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(sel(0, cons(X, Z))) -> mark(X)
active(first(0, Z)) -> 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(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) -> mark(quote(X))
active(first1(0, Z)) -> mark(nil1)
active(first1(s(X), cons(Y, Z))) -> mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) -> mark(01)
active(quote1(cons(X, Z))) -> mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) -> mark(nil1)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(sel(X, Z))) -> mark(sel1(X, Z))
active(quote1(first(X, Z))) -> mark(first1(X, Z))
active(unquote(01)) -> mark(0)
active(unquote(s1(X))) -> mark(s(unquote(X)))
active(unquote1(nil1)) -> mark(nil)
active(unquote1(cons1(X, Z))) -> mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) -> mark(cons(X, Z))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(from(X)) -> from(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(first1(X1, X2)) -> first1(active(X1), X2)
active(first1(X1, X2)) -> first1(X1, active(X2))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
active(s1(X)) -> s1(active(X))
active(unquote(X)) -> unquote(active(X))
active(unquote1(X)) -> unquote1(active(X))
active(fcons(X1, X2)) -> fcons(active(X1), X2)
active(fcons(X1, X2)) -> fcons(X1, active(X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(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))
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))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
first1(mark(X1), X2) -> mark(first1(X1, X2))
first1(X1, mark(X2)) -> mark(first1(X1, X2))
first1(ok(X1), ok(X2)) -> ok(first1(X1, X2))
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))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
unquote(mark(X)) -> mark(unquote(X))
unquote(ok(X)) -> ok(unquote(X))
unquote1(mark(X)) -> mark(unquote1(X))
unquote1(ok(X)) -> ok(unquote1(X))
fcons(mark(X1), X2) -> mark(fcons(X1, X2))
fcons(X1, mark(X2)) -> mark(fcons(X1, X2))
fcons(ok(X1), ok(X2)) -> ok(fcons(X1, X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(from(X)) -> from(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
proper(first1(X1, X2)) -> first1(proper(X1), proper(X2))
proper(nil1) -> ok(nil1)
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
proper(01) -> ok(01)
proper(quote1(X)) -> quote1(proper(X))
proper(s1(X)) -> s1(proper(X))
proper(unquote(X)) -> unquote(proper(X))
proper(unquote1(X)) -> unquote1(proper(X))
proper(fcons(X1, X2)) -> fcons(proper(X1), proper(X2))
quote(ok(X)) -> ok(quote(X))
quote1(ok(X)) -> ok(quote1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


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

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(CONS1(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 3
Polo
       →DP Problem 4
Polo
       →DP Problem 5
Polo
       →DP Problem 6
Polo
       →DP Problem 7
Polo
           →DP Problem 32
Polynomial Ordering
       →DP Problem 8
Polo
       →DP Problem 9
Polo
       →DP Problem 10
Polo
       →DP Problem 11
Polo
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Polo
       →DP Problem 17
Remaining


Dependency Pairs:

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


Rules:


active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(sel(0, cons(X, Z))) -> mark(X)
active(first(0, Z)) -> 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(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) -> mark(quote(X))
active(first1(0, Z)) -> mark(nil1)
active(first1(s(X), cons(Y, Z))) -> mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) -> mark(01)
active(quote1(cons(X, Z))) -> mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) -> mark(nil1)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(sel(X, Z))) -> mark(sel1(X, Z))
active(quote1(first(X, Z))) -> mark(first1(X, Z))
active(unquote(01)) -> mark(0)
active(unquote(s1(X))) -> mark(s(unquote(X)))
active(unquote1(nil1)) -> mark(nil)
active(unquote1(cons1(X, Z))) -> mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) -> mark(cons(X, Z))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(from(X)) -> from(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(first1(X1, X2)) -> first1(active(X1), X2)
active(first1(X1, X2)) -> first1(X1, active(X2))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
active(s1(X)) -> s1(active(X))
active(unquote(X)) -> unquote(active(X))
active(unquote1(X)) -> unquote1(active(X))
active(fcons(X1, X2)) -> fcons(active(X1), X2)
active(fcons(X1, X2)) -> fcons(X1, active(X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(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))
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))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
first1(mark(X1), X2) -> mark(first1(X1, X2))
first1(X1, mark(X2)) -> mark(first1(X1, X2))
first1(ok(X1), ok(X2)) -> ok(first1(X1, X2))
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))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
unquote(mark(X)) -> mark(unquote(X))
unquote(ok(X)) -> ok(unquote(X))
unquote1(mark(X)) -> mark(unquote1(X))
unquote1(ok(X)) -> ok(unquote1(X))
fcons(mark(X1), X2) -> mark(fcons(X1, X2))
fcons(X1, mark(X2)) -> mark(fcons(X1, X2))
fcons(ok(X1), ok(X2)) -> ok(fcons(X1, X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(from(X)) -> from(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
proper(first1(X1, X2)) -> first1(proper(X1), proper(X2))
proper(nil1) -> ok(nil1)
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
proper(01) -> ok(01)
proper(quote1(X)) -> quote1(proper(X))
proper(s1(X)) -> s1(proper(X))
proper(unquote(X)) -> unquote(proper(X))
proper(unquote1(X)) -> unquote1(proper(X))
proper(fcons(X1, X2)) -> fcons(proper(X1), proper(X2))
quote(ok(X)) -> ok(quote(X))
quote1(ok(X)) -> ok(quote1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


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

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(CONS1(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 3
Polo
       →DP Problem 4
Polo
       →DP Problem 5
Polo
       →DP Problem 6
Polo
       →DP Problem 7
Polo
           →DP Problem 32
Polo
             ...
               →DP Problem 33
Polynomial Ordering
       →DP Problem 8
Polo
       →DP Problem 9
Polo
       →DP Problem 10
Polo
       →DP Problem 11
Polo
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Polo
       →DP Problem 17
Remaining


Dependency Pair:

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


Rules:


active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(sel(0, cons(X, Z))) -> mark(X)
active(first(0, Z)) -> 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(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) -> mark(quote(X))
active(first1(0, Z)) -> mark(nil1)
active(first1(s(X), cons(Y, Z))) -> mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) -> mark(01)
active(quote1(cons(X, Z))) -> mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) -> mark(nil1)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(sel(X, Z))) -> mark(sel1(X, Z))
active(quote1(first(X, Z))) -> mark(first1(X, Z))
active(unquote(01)) -> mark(0)
active(unquote(s1(X))) -> mark(s(unquote(X)))
active(unquote1(nil1)) -> mark(nil)
active(unquote1(cons1(X, Z))) -> mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) -> mark(cons(X, Z))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(from(X)) -> from(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(first1(X1, X2)) -> first1(active(X1), X2)
active(first1(X1, X2)) -> first1(X1, active(X2))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
active(s1(X)) -> s1(active(X))
active(unquote(X)) -> unquote(active(X))
active(unquote1(X)) -> unquote1(active(X))
active(fcons(X1, X2)) -> fcons(active(X1), X2)
active(fcons(X1, X2)) -> fcons(X1, active(X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(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))
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))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
first1(mark(X1), X2) -> mark(first1(X1, X2))
first1(X1, mark(X2)) -> mark(first1(X1, X2))
first1(ok(X1), ok(X2)) -> ok(first1(X1, X2))
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))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
unquote(mark(X)) -> mark(unquote(X))
unquote(ok(X)) -> ok(unquote(X))
unquote1(mark(X)) -> mark(unquote1(X))
unquote1(ok(X)) -> ok(unquote1(X))
fcons(mark(X1), X2) -> mark(fcons(X1, X2))
fcons(X1, mark(X2)) -> mark(fcons(X1, X2))
fcons(ok(X1), ok(X2)) -> ok(fcons(X1, X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(from(X)) -> from(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
proper(first1(X1, X2)) -> first1(proper(X1), proper(X2))
proper(nil1) -> ok(nil1)
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
proper(01) -> ok(01)
proper(quote1(X)) -> quote1(proper(X))
proper(s1(X)) -> s1(proper(X))
proper(unquote(X)) -> unquote(proper(X))
proper(unquote1(X)) -> unquote1(proper(X))
proper(fcons(X1, X2)) -> fcons(proper(X1), proper(X2))
quote(ok(X)) -> ok(quote(X))
quote1(ok(X)) -> ok(quote1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


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

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(CONS1(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 3
Polo
       →DP Problem 4
Polo
       →DP Problem 5
Polo
       →DP Problem 6
Polo
       →DP Problem 7
Polo
           →DP Problem 32
Polo
             ...
               →DP Problem 34
Dependency Graph
       →DP Problem 8
Polo
       →DP Problem 9
Polo
       →DP Problem 10
Polo
       →DP Problem 11
Polo
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Polo
       →DP Problem 17
Remaining


Dependency Pair:


Rules:


active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(sel(0, cons(X, Z))) -> mark(X)
active(first(0, Z)) -> 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(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) -> mark(quote(X))
active(first1(0, Z)) -> mark(nil1)
active(first1(s(X), cons(Y, Z))) -> mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) -> mark(01)
active(quote1(cons(X, Z))) -> mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) -> mark(nil1)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(sel(X, Z))) -> mark(sel1(X, Z))
active(quote1(first(X, Z))) -> mark(first1(X, Z))
active(unquote(01)) -> mark(0)
active(unquote(s1(X))) -> mark(s(unquote(X)))
active(unquote1(nil1)) -> mark(nil)
active(unquote1(cons1(X, Z))) -> mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) -> mark(cons(X, Z))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(from(X)) -> from(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(first1(X1, X2)) -> first1(active(X1), X2)
active(first1(X1, X2)) -> first1(X1, active(X2))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
active(s1(X)) -> s1(active(X))
active(unquote(X)) -> unquote(active(X))
active(unquote1(X)) -> unquote1(active(X))
active(fcons(X1, X2)) -> fcons(active(X1), X2)
active(fcons(X1, X2)) -> fcons(X1, active(X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(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))
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))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
first1(mark(X1), X2) -> mark(first1(X1, X2))
first1(X1, mark(X2)) -> mark(first1(X1, X2))
first1(ok(X1), ok(X2)) -> ok(first1(X1, X2))
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))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
unquote(mark(X)) -> mark(unquote(X))
unquote(ok(X)) -> ok(unquote(X))
unquote1(mark(X)) -> mark(unquote1(X))
unquote1(ok(X)) -> ok(unquote1(X))
fcons(mark(X1), X2) -> mark(fcons(X1, X2))
fcons(X1, mark(X2)) -> mark(fcons(X1, X2))
fcons(ok(X1), ok(X2)) -> ok(fcons(X1, X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(from(X)) -> from(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
proper(first1(X1, X2)) -> first1(proper(X1), proper(X2))
proper(nil1) -> ok(nil1)
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
proper(01) -> ok(01)
proper(quote1(X)) -> quote1(proper(X))
proper(s1(X)) -> s1(proper(X))
proper(unquote(X)) -> unquote(proper(X))
proper(unquote1(X)) -> unquote1(proper(X))
proper(fcons(X1, X2)) -> fcons(proper(X1), proper(X2))
quote(ok(X)) -> ok(quote(X))
quote1(ok(X)) -> ok(quote1(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
Polo
       →DP Problem 10
Polo
       →DP Problem 11
Polo
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Polo
       →DP Problem 17
Remaining


Dependency Pairs:

FIRST1(ok(X1), ok(X2)) -> FIRST1(X1, X2)
FIRST1(mark(X1), X2) -> FIRST1(X1, X2)
FIRST1(X1, mark(X2)) -> FIRST1(X1, X2)


Rules:


active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(sel(0, cons(X, Z))) -> mark(X)
active(first(0, Z)) -> 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(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) -> mark(quote(X))
active(first1(0, Z)) -> mark(nil1)
active(first1(s(X), cons(Y, Z))) -> mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) -> mark(01)
active(quote1(cons(X, Z))) -> mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) -> mark(nil1)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(sel(X, Z))) -> mark(sel1(X, Z))
active(quote1(first(X, Z))) -> mark(first1(X, Z))
active(unquote(01)) -> mark(0)
active(unquote(s1(X))) -> mark(s(unquote(X)))
active(unquote1(nil1)) -> mark(nil)
active(unquote1(cons1(X, Z))) -> mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) -> mark(cons(X, Z))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(from(X)) -> from(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(first1(X1, X2)) -> first1(active(X1), X2)
active(first1(X1, X2)) -> first1(X1, active(X2))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
active(s1(X)) -> s1(active(X))
active(unquote(X)) -> unquote(active(X))
active(unquote1(X)) -> unquote1(active(X))
active(fcons(X1, X2)) -> fcons(active(X1), X2)
active(fcons(X1, X2)) -> fcons(X1, active(X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(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))
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))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
first1(mark(X1), X2) -> mark(first1(X1, X2))
first1(X1, mark(X2)) -> mark(first1(X1, X2))
first1(ok(X1), ok(X2)) -> ok(first1(X1, X2))
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))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
unquote(mark(X)) -> mark(unquote(X))
unquote(ok(X)) -> ok(unquote(X))
unquote1(mark(X)) -> mark(unquote1(X))
unquote1(ok(X)) -> ok(unquote1(X))
fcons(mark(X1), X2) -> mark(fcons(X1, X2))
fcons(X1, mark(X2)) -> mark(fcons(X1, X2))
fcons(ok(X1), ok(X2)) -> ok(fcons(X1, X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(from(X)) -> from(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
proper(first1(X1, X2)) -> first1(proper(X1), proper(X2))
proper(nil1) -> ok(nil1)
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
proper(01) -> ok(01)
proper(quote1(X)) -> quote1(proper(X))
proper(s1(X)) -> s1(proper(X))
proper(unquote(X)) -> unquote(proper(X))
proper(unquote1(X)) -> unquote1(proper(X))
proper(fcons(X1, X2)) -> fcons(proper(X1), proper(X2))
quote(ok(X)) -> ok(quote(X))
quote1(ok(X)) -> ok(quote1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

FIRST1(ok(X1), ok(X2)) -> FIRST1(X1, X2)


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

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(FIRST1(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 3
Polo
       →DP Problem 4
Polo
       →DP Problem 5
Polo
       →DP Problem 6
Polo
       →DP Problem 7
Polo
       →DP Problem 8
Polo
           →DP Problem 35
Polynomial Ordering
       →DP Problem 9
Polo
       →DP Problem 10
Polo
       →DP Problem 11
Polo
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Polo
       →DP Problem 17
Remaining


Dependency Pairs:

FIRST1(mark(X1), X2) -> FIRST1(X1, X2)
FIRST1(X1, mark(X2)) -> FIRST1(X1, X2)


Rules:


active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(sel(0, cons(X, Z))) -> mark(X)
active(first(0, Z)) -> 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(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) -> mark(quote(X))
active(first1(0, Z)) -> mark(nil1)
active(first1(s(X), cons(Y, Z))) -> mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) -> mark(01)
active(quote1(cons(X, Z))) -> mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) -> mark(nil1)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(sel(X, Z))) -> mark(sel1(X, Z))
active(quote1(first(X, Z))) -> mark(first1(X, Z))
active(unquote(01)) -> mark(0)
active(unquote(s1(X))) -> mark(s(unquote(X)))
active(unquote1(nil1)) -> mark(nil)
active(unquote1(cons1(X, Z))) -> mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) -> mark(cons(X, Z))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(from(X)) -> from(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(first1(X1, X2)) -> first1(active(X1), X2)
active(first1(X1, X2)) -> first1(X1, active(X2))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
active(s1(X)) -> s1(active(X))
active(unquote(X)) -> unquote(active(X))
active(unquote1(X)) -> unquote1(active(X))
active(fcons(X1, X2)) -> fcons(active(X1), X2)
active(fcons(X1, X2)) -> fcons(X1, active(X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(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))
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))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
first1(mark(X1), X2) -> mark(first1(X1, X2))
first1(X1, mark(X2)) -> mark(first1(X1, X2))
first1(ok(X1), ok(X2)) -> ok(first1(X1, X2))
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))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
unquote(mark(X)) -> mark(unquote(X))
unquote(ok(X)) -> ok(unquote(X))
unquote1(mark(X)) -> mark(unquote1(X))
unquote1(ok(X)) -> ok(unquote1(X))
fcons(mark(X1), X2) -> mark(fcons(X1, X2))
fcons(X1, mark(X2)) -> mark(fcons(X1, X2))
fcons(ok(X1), ok(X2)) -> ok(fcons(X1, X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(from(X)) -> from(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
proper(first1(X1, X2)) -> first1(proper(X1), proper(X2))
proper(nil1) -> ok(nil1)
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
proper(01) -> ok(01)
proper(quote1(X)) -> quote1(proper(X))
proper(s1(X)) -> s1(proper(X))
proper(unquote(X)) -> unquote(proper(X))
proper(unquote1(X)) -> unquote1(proper(X))
proper(fcons(X1, X2)) -> fcons(proper(X1), proper(X2))
quote(ok(X)) -> ok(quote(X))
quote1(ok(X)) -> ok(quote1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

FIRST1(mark(X1), X2) -> FIRST1(X1, X2)


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

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(FIRST1(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 3
Polo
       →DP Problem 4
Polo
       →DP Problem 5
Polo
       →DP Problem 6
Polo
       →DP Problem 7
Polo
       →DP Problem 8
Polo
           →DP Problem 35
Polo
             ...
               →DP Problem 36
Polynomial Ordering
       →DP Problem 9
Polo
       →DP Problem 10
Polo
       →DP Problem 11
Polo
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Polo
       →DP Problem 17
Remaining


Dependency Pair:

FIRST1(X1, mark(X2)) -> FIRST1(X1, X2)


Rules:


active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(sel(0, cons(X, Z))) -> mark(X)
active(first(0, Z)) -> 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(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) -> mark(quote(X))
active(first1(0, Z)) -> mark(nil1)
active(first1(s(X), cons(Y, Z))) -> mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) -> mark(01)
active(quote1(cons(X, Z))) -> mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) -> mark(nil1)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(sel(X, Z))) -> mark(sel1(X, Z))
active(quote1(first(X, Z))) -> mark(first1(X, Z))
active(unquote(01)) -> mark(0)
active(unquote(s1(X))) -> mark(s(unquote(X)))
active(unquote1(nil1)) -> mark(nil)
active(unquote1(cons1(X, Z))) -> mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) -> mark(cons(X, Z))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(from(X)) -> from(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(first1(X1, X2)) -> first1(active(X1), X2)
active(first1(X1, X2)) -> first1(X1, active(X2))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
active(s1(X)) -> s1(active(X))
active(unquote(X)) -> unquote(active(X))
active(unquote1(X)) -> unquote1(active(X))
active(fcons(X1, X2)) -> fcons(active(X1), X2)
active(fcons(X1, X2)) -> fcons(X1, active(X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(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))
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))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
first1(mark(X1), X2) -> mark(first1(X1, X2))
first1(X1, mark(X2)) -> mark(first1(X1, X2))
first1(ok(X1), ok(X2)) -> ok(first1(X1, X2))
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))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
unquote(mark(X)) -> mark(unquote(X))
unquote(ok(X)) -> ok(unquote(X))
unquote1(mark(X)) -> mark(unquote1(X))
unquote1(ok(X)) -> ok(unquote1(X))
fcons(mark(X1), X2) -> mark(fcons(X1, X2))
fcons(X1, mark(X2)) -> mark(fcons(X1, X2))
fcons(ok(X1), ok(X2)) -> ok(fcons(X1, X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(from(X)) -> from(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
proper(first1(X1, X2)) -> first1(proper(X1), proper(X2))
proper(nil1) -> ok(nil1)
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
proper(01) -> ok(01)
proper(quote1(X)) -> quote1(proper(X))
proper(s1(X)) -> s1(proper(X))
proper(unquote(X)) -> unquote(proper(X))
proper(unquote1(X)) -> unquote1(proper(X))
proper(fcons(X1, X2)) -> fcons(proper(X1), proper(X2))
quote(ok(X)) -> ok(quote(X))
quote1(ok(X)) -> ok(quote1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

FIRST1(X1, mark(X2)) -> FIRST1(X1, X2)


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

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(FIRST1(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 3
Polo
       →DP Problem 4
Polo
       →DP Problem 5
Polo
       →DP Problem 6
Polo
       →DP Problem 7
Polo
       →DP Problem 8
Polo
           →DP Problem 35
Polo
             ...
               →DP Problem 37
Dependency Graph
       →DP Problem 9
Polo
       →DP Problem 10
Polo
       →DP Problem 11
Polo
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Polo
       →DP Problem 17
Remaining


Dependency Pair:


Rules:


active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(sel(0, cons(X, Z))) -> mark(X)
active(first(0, Z)) -> 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(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) -> mark(quote(X))
active(first1(0, Z)) -> mark(nil1)
active(first1(s(X), cons(Y, Z))) -> mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) -> mark(01)
active(quote1(cons(X, Z))) -> mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) -> mark(nil1)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(sel(X, Z))) -> mark(sel1(X, Z))
active(quote1(first(X, Z))) -> mark(first1(X, Z))
active(unquote(01)) -> mark(0)
active(unquote(s1(X))) -> mark(s(unquote(X)))
active(unquote1(nil1)) -> mark(nil)
active(unquote1(cons1(X, Z))) -> mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) -> mark(cons(X, Z))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(from(X)) -> from(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(first1(X1, X2)) -> first1(active(X1), X2)
active(first1(X1, X2)) -> first1(X1, active(X2))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
active(s1(X)) -> s1(active(X))
active(unquote(X)) -> unquote(active(X))
active(unquote1(X)) -> unquote1(active(X))
active(fcons(X1, X2)) -> fcons(active(X1), X2)
active(fcons(X1, X2)) -> fcons(X1, active(X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(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))
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))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
first1(mark(X1), X2) -> mark(first1(X1, X2))
first1(X1, mark(X2)) -> mark(first1(X1, X2))
first1(ok(X1), ok(X2)) -> ok(first1(X1, X2))
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))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
unquote(mark(X)) -> mark(unquote(X))
unquote(ok(X)) -> ok(unquote(X))
unquote1(mark(X)) -> mark(unquote1(X))
unquote1(ok(X)) -> ok(unquote1(X))
fcons(mark(X1), X2) -> mark(fcons(X1, X2))
fcons(X1, mark(X2)) -> mark(fcons(X1, X2))
fcons(ok(X1), ok(X2)) -> ok(fcons(X1, X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(from(X)) -> from(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
proper(first1(X1, X2)) -> first1(proper(X1), proper(X2))
proper(nil1) -> ok(nil1)
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
proper(01) -> ok(01)
proper(quote1(X)) -> quote1(proper(X))
proper(s1(X)) -> s1(proper(X))
proper(unquote(X)) -> unquote(proper(X))
proper(unquote1(X)) -> unquote1(proper(X))
proper(fcons(X1, X2)) -> fcons(proper(X1), proper(X2))
quote(ok(X)) -> ok(quote(X))
quote1(ok(X)) -> ok(quote1(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
Polynomial Ordering
       →DP Problem 10
Polo
       →DP Problem 11
Polo
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Polo
       →DP Problem 17
Remaining


Dependency Pair:

QUOTE1(ok(X)) -> QUOTE1(X)


Rules:


active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(sel(0, cons(X, Z))) -> mark(X)
active(first(0, Z)) -> 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(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) -> mark(quote(X))
active(first1(0, Z)) -> mark(nil1)
active(first1(s(X), cons(Y, Z))) -> mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) -> mark(01)
active(quote1(cons(X, Z))) -> mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) -> mark(nil1)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(sel(X, Z))) -> mark(sel1(X, Z))
active(quote1(first(X, Z))) -> mark(first1(X, Z))
active(unquote(01)) -> mark(0)
active(unquote(s1(X))) -> mark(s(unquote(X)))
active(unquote1(nil1)) -> mark(nil)
active(unquote1(cons1(X, Z))) -> mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) -> mark(cons(X, Z))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(from(X)) -> from(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(first1(X1, X2)) -> first1(active(X1), X2)
active(first1(X1, X2)) -> first1(X1, active(X2))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
active(s1(X)) -> s1(active(X))
active(unquote(X)) -> unquote(active(X))
active(unquote1(X)) -> unquote1(active(X))
active(fcons(X1, X2)) -> fcons(active(X1), X2)
active(fcons(X1, X2)) -> fcons(X1, active(X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(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))
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))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
first1(mark(X1), X2) -> mark(first1(X1, X2))
first1(X1, mark(X2)) -> mark(first1(X1, X2))
first1(ok(X1), ok(X2)) -> ok(first1(X1, X2))
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))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
unquote(mark(X)) -> mark(unquote(X))
unquote(ok(X)) -> ok(unquote(X))
unquote1(mark(X)) -> mark(unquote1(X))
unquote1(ok(X)) -> ok(unquote1(X))
fcons(mark(X1), X2) -> mark(fcons(X1, X2))
fcons(X1, mark(X2)) -> mark(fcons(X1, X2))
fcons(ok(X1), ok(X2)) -> ok(fcons(X1, X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(from(X)) -> from(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
proper(first1(X1, X2)) -> first1(proper(X1), proper(X2))
proper(nil1) -> ok(nil1)
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
proper(01) -> ok(01)
proper(quote1(X)) -> quote1(proper(X))
proper(s1(X)) -> s1(proper(X))
proper(unquote(X)) -> unquote(proper(X))
proper(unquote1(X)) -> unquote1(proper(X))
proper(fcons(X1, X2)) -> fcons(proper(X1), proper(X2))
quote(ok(X)) -> ok(quote(X))
quote1(ok(X)) -> ok(quote1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

QUOTE1(ok(X)) -> QUOTE1(X)


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

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(QUOTE1(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 7
Polo
       →DP Problem 8
Polo
       →DP Problem 9
Polo
           →DP Problem 38
Dependency Graph
       →DP Problem 10
Polo
       →DP Problem 11
Polo
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Polo
       →DP Problem 17
Remaining


Dependency Pair:


Rules:


active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(sel(0, cons(X, Z))) -> mark(X)
active(first(0, Z)) -> 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(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) -> mark(quote(X))
active(first1(0, Z)) -> mark(nil1)
active(first1(s(X), cons(Y, Z))) -> mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) -> mark(01)
active(quote1(cons(X, Z))) -> mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) -> mark(nil1)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(sel(X, Z))) -> mark(sel1(X, Z))
active(quote1(first(X, Z))) -> mark(first1(X, Z))
active(unquote(01)) -> mark(0)
active(unquote(s1(X))) -> mark(s(unquote(X)))
active(unquote1(nil1)) -> mark(nil)
active(unquote1(cons1(X, Z))) -> mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) -> mark(cons(X, Z))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(from(X)) -> from(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(first1(X1, X2)) -> first1(active(X1), X2)
active(first1(X1, X2)) -> first1(X1, active(X2))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
active(s1(X)) -> s1(active(X))
active(unquote(X)) -> unquote(active(X))
active(unquote1(X)) -> unquote1(active(X))
active(fcons(X1, X2)) -> fcons(active(X1), X2)
active(fcons(X1, X2)) -> fcons(X1, active(X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(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))
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))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
first1(mark(X1), X2) -> mark(first1(X1, X2))
first1(X1, mark(X2)) -> mark(first1(X1, X2))
first1(ok(X1), ok(X2)) -> ok(first1(X1, X2))
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))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
unquote(mark(X)) -> mark(unquote(X))
unquote(ok(X)) -> ok(unquote(X))
unquote1(mark(X)) -> mark(unquote1(X))
unquote1(ok(X)) -> ok(unquote1(X))
fcons(mark(X1), X2) -> mark(fcons(X1, X2))
fcons(X1, mark(X2)) -> mark(fcons(X1, X2))
fcons(ok(X1), ok(X2)) -> ok(fcons(X1, X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(from(X)) -> from(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
proper(first1(X1, X2)) -> first1(proper(X1), proper(X2))
proper(nil1) -> ok(nil1)
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
proper(01) -> ok(01)
proper(quote1(X)) -> quote1(proper(X))
proper(s1(X)) -> s1(proper(X))
proper(unquote(X)) -> unquote(proper(X))
proper(unquote1(X)) -> unquote1(proper(X))
proper(fcons(X1, X2)) -> fcons(proper(X1), proper(X2))
quote(ok(X)) -> ok(quote(X))
quote1(ok(X)) -> ok(quote1(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
Polo
       →DP Problem 10
Polynomial Ordering
       →DP Problem 11
Polo
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Polo
       →DP Problem 17
Remaining


Dependency Pairs:

S1(ok(X)) -> S1(X)
S1(mark(X)) -> S1(X)


Rules:


active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(sel(0, cons(X, Z))) -> mark(X)
active(first(0, Z)) -> 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(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) -> mark(quote(X))
active(first1(0, Z)) -> mark(nil1)
active(first1(s(X), cons(Y, Z))) -> mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) -> mark(01)
active(quote1(cons(X, Z))) -> mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) -> mark(nil1)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(sel(X, Z))) -> mark(sel1(X, Z))
active(quote1(first(X, Z))) -> mark(first1(X, Z))
active(unquote(01)) -> mark(0)
active(unquote(s1(X))) -> mark(s(unquote(X)))
active(unquote1(nil1)) -> mark(nil)
active(unquote1(cons1(X, Z))) -> mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) -> mark(cons(X, Z))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(from(X)) -> from(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(first1(X1, X2)) -> first1(active(X1), X2)
active(first1(X1, X2)) -> first1(X1, active(X2))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
active(s1(X)) -> s1(active(X))
active(unquote(X)) -> unquote(active(X))
active(unquote1(X)) -> unquote1(active(X))
active(fcons(X1, X2)) -> fcons(active(X1), X2)
active(fcons(X1, X2)) -> fcons(X1, active(X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(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))
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))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
first1(mark(X1), X2) -> mark(first1(X1, X2))
first1(X1, mark(X2)) -> mark(first1(X1, X2))
first1(ok(X1), ok(X2)) -> ok(first1(X1, X2))
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))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
unquote(mark(X)) -> mark(unquote(X))
unquote(ok(X)) -> ok(unquote(X))
unquote1(mark(X)) -> mark(unquote1(X))
unquote1(ok(X)) -> ok(unquote1(X))
fcons(mark(X1), X2) -> mark(fcons(X1, X2))
fcons(X1, mark(X2)) -> mark(fcons(X1, X2))
fcons(ok(X1), ok(X2)) -> ok(fcons(X1, X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(from(X)) -> from(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
proper(first1(X1, X2)) -> first1(proper(X1), proper(X2))
proper(nil1) -> ok(nil1)
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
proper(01) -> ok(01)
proper(quote1(X)) -> quote1(proper(X))
proper(s1(X)) -> s1(proper(X))
proper(unquote(X)) -> unquote(proper(X))
proper(unquote1(X)) -> unquote1(proper(X))
proper(fcons(X1, X2)) -> fcons(proper(X1), proper(X2))
quote(ok(X)) -> ok(quote(X))
quote1(ok(X)) -> ok(quote1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

S1(ok(X)) -> S1(X)


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

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(S1(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 7
Polo
       →DP Problem 8
Polo
       →DP Problem 9
Polo
       →DP Problem 10
Polo
           →DP Problem 39
Polynomial Ordering
       →DP Problem 11
Polo
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Polo
       →DP Problem 17
Remaining


Dependency Pair:

S1(mark(X)) -> S1(X)


Rules:


active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(sel(0, cons(X, Z))) -> mark(X)
active(first(0, Z)) -> 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(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) -> mark(quote(X))
active(first1(0, Z)) -> mark(nil1)
active(first1(s(X), cons(Y, Z))) -> mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) -> mark(01)
active(quote1(cons(X, Z))) -> mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) -> mark(nil1)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(sel(X, Z))) -> mark(sel1(X, Z))
active(quote1(first(X, Z))) -> mark(first1(X, Z))
active(unquote(01)) -> mark(0)
active(unquote(s1(X))) -> mark(s(unquote(X)))
active(unquote1(nil1)) -> mark(nil)
active(unquote1(cons1(X, Z))) -> mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) -> mark(cons(X, Z))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(from(X)) -> from(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(first1(X1, X2)) -> first1(active(X1), X2)
active(first1(X1, X2)) -> first1(X1, active(X2))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
active(s1(X)) -> s1(active(X))
active(unquote(X)) -> unquote(active(X))
active(unquote1(X)) -> unquote1(active(X))
active(fcons(X1, X2)) -> fcons(active(X1), X2)
active(fcons(X1, X2)) -> fcons(X1, active(X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(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))
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))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
first1(mark(X1), X2) -> mark(first1(X1, X2))
first1(X1, mark(X2)) -> mark(first1(X1, X2))
first1(ok(X1), ok(X2)) -> ok(first1(X1, X2))
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))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
unquote(mark(X)) -> mark(unquote(X))
unquote(ok(X)) -> ok(unquote(X))
unquote1(mark(X)) -> mark(unquote1(X))
unquote1(ok(X)) -> ok(unquote1(X))
fcons(mark(X1), X2) -> mark(fcons(X1, X2))
fcons(X1, mark(X2)) -> mark(fcons(X1, X2))
fcons(ok(X1), ok(X2)) -> ok(fcons(X1, X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(from(X)) -> from(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
proper(first1(X1, X2)) -> first1(proper(X1), proper(X2))
proper(nil1) -> ok(nil1)
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
proper(01) -> ok(01)
proper(quote1(X)) -> quote1(proper(X))
proper(s1(X)) -> s1(proper(X))
proper(unquote(X)) -> unquote(proper(X))
proper(unquote1(X)) -> unquote1(proper(X))
proper(fcons(X1, X2)) -> fcons(proper(X1), proper(X2))
quote(ok(X)) -> ok(quote(X))
quote1(ok(X)) -> ok(quote1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

S1(mark(X)) -> S1(X)


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

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(S1(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 7
Polo
       →DP Problem 8
Polo
       →DP Problem 9
Polo
       →DP Problem 10
Polo
           →DP Problem 39
Polo
             ...
               →DP Problem 40
Dependency Graph
       →DP Problem 11
Polo
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Polo
       →DP Problem 17
Remaining


Dependency Pair:


Rules:


active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(sel(0, cons(X, Z))) -> mark(X)
active(first(0, Z)) -> 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(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) -> mark(quote(X))
active(first1(0, Z)) -> mark(nil1)
active(first1(s(X), cons(Y, Z))) -> mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) -> mark(01)
active(quote1(cons(X, Z))) -> mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) -> mark(nil1)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(sel(X, Z))) -> mark(sel1(X, Z))
active(quote1(first(X, Z))) -> mark(first1(X, Z))
active(unquote(01)) -> mark(0)
active(unquote(s1(X))) -> mark(s(unquote(X)))
active(unquote1(nil1)) -> mark(nil)
active(unquote1(cons1(X, Z))) -> mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) -> mark(cons(X, Z))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(from(X)) -> from(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(first1(X1, X2)) -> first1(active(X1), X2)
active(first1(X1, X2)) -> first1(X1, active(X2))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
active(s1(X)) -> s1(active(X))
active(unquote(X)) -> unquote(active(X))
active(unquote1(X)) -> unquote1(active(X))
active(fcons(X1, X2)) -> fcons(active(X1), X2)
active(fcons(X1, X2)) -> fcons(X1, active(X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(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))
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))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
first1(mark(X1), X2) -> mark(first1(X1, X2))
first1(X1, mark(X2)) -> mark(first1(X1, X2))
first1(ok(X1), ok(X2)) -> ok(first1(X1, X2))
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))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
unquote(mark(X)) -> mark(unquote(X))
unquote(ok(X)) -> ok(unquote(X))
unquote1(mark(X)) -> mark(unquote1(X))
unquote1(ok(X)) -> ok(unquote1(X))
fcons(mark(X1), X2) -> mark(fcons(X1, X2))
fcons(X1, mark(X2)) -> mark(fcons(X1, X2))
fcons(ok(X1), ok(X2)) -> ok(fcons(X1, X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(from(X)) -> from(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
proper(first1(X1, X2)) -> first1(proper(X1), proper(X2))
proper(nil1) -> ok(nil1)
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
proper(01) -> ok(01)
proper(quote1(X)) -> quote1(proper(X))
proper(s1(X)) -> s1(proper(X))
proper(unquote(X)) -> unquote(proper(X))
proper(unquote1(X)) -> unquote1(proper(X))
proper(fcons(X1, X2)) -> fcons(proper(X1), proper(X2))
quote(ok(X)) -> ok(quote(X))
quote1(ok(X)) -> ok(quote1(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
Polo
       →DP Problem 10
Polo
       →DP Problem 11
Polynomial Ordering
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Polo
       →DP Problem 17
Remaining


Dependency Pairs:

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


Rules:


active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(sel(0, cons(X, Z))) -> mark(X)
active(first(0, Z)) -> 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(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) -> mark(quote(X))
active(first1(0, Z)) -> mark(nil1)
active(first1(s(X), cons(Y, Z))) -> mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) -> mark(01)
active(quote1(cons(X, Z))) -> mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) -> mark(nil1)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(sel(X, Z))) -> mark(sel1(X, Z))
active(quote1(first(X, Z))) -> mark(first1(X, Z))
active(unquote(01)) -> mark(0)
active(unquote(s1(X))) -> mark(s(unquote(X)))
active(unquote1(nil1)) -> mark(nil)
active(unquote1(cons1(X, Z))) -> mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) -> mark(cons(X, Z))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(from(X)) -> from(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(first1(X1, X2)) -> first1(active(X1), X2)
active(first1(X1, X2)) -> first1(X1, active(X2))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
active(s1(X)) -> s1(active(X))
active(unquote(X)) -> unquote(active(X))
active(unquote1(X)) -> unquote1(active(X))
active(fcons(X1, X2)) -> fcons(active(X1), X2)
active(fcons(X1, X2)) -> fcons(X1, active(X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(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))
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))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
first1(mark(X1), X2) -> mark(first1(X1, X2))
first1(X1, mark(X2)) -> mark(first1(X1, X2))
first1(ok(X1), ok(X2)) -> ok(first1(X1, X2))
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))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
unquote(mark(X)) -> mark(unquote(X))
unquote(ok(X)) -> ok(unquote(X))
unquote1(mark(X)) -> mark(unquote1(X))
unquote1(ok(X)) -> ok(unquote1(X))
fcons(mark(X1), X2) -> mark(fcons(X1, X2))
fcons(X1, mark(X2)) -> mark(fcons(X1, X2))
fcons(ok(X1), ok(X2)) -> ok(fcons(X1, X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(from(X)) -> from(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
proper(first1(X1, X2)) -> first1(proper(X1), proper(X2))
proper(nil1) -> ok(nil1)
proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2))
proper(01) -> ok(01)
proper(quote1(X)) -> quote1(proper(X))
proper(s1(X)) -> s1(proper(X))
proper(unquote(X)) -> unquote(proper(X))
proper(unquote1(X)) -> unquote1(proper(X))
proper(fcons(X1, X2)) -> fcons(proper(X1), proper(X2))
quote(ok(X)) -> ok(quote(X))
quote1(ok(X)) -> ok(quote1(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


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

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(S(x1))=  x1  
  POL(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 7
Polo
       →DP Problem 8
Polo
       →DP Problem 9
Polo
       →DP Problem 10
Polo
       →DP Problem 11
Polo
           →DP Problem 41
Polynomial Ordering
       →DP Problem 12
Polo
       →DP Problem 13
Polo
       →DP Problem 14
Polo
       →DP Problem 15
Polo
       →DP Problem 16
Polo
       →DP Problem 17
Remaining


Dependency Pair:

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


Rules:


active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(sel(0, cons(X, Z))) -> mark(X)
active(first(0, Z)) -> 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(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) -> mark(quote(X))
active(first1(0, Z)) -> mark(nil1)
active(first1(s(X), cons(Y, Z))) -> mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) -> mark(01)
active(quote1(cons(X, Z))) -> mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) -> mark(nil1)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(sel(X, Z))) -> mark(sel1(X, Z))
active(quote1(first(X, Z))) -> mark(first1(X, Z))
active(unquote(01)) -> mark(0)
active(unquote(s1(X))) -> mark(s(unquote(X)))
active(unquote1(nil1)) -> mark(nil)
active(unquote1(cons1(X, Z))) -> mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) -> mark(cons(X, Z))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(from(X)) -> from(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(first1(X1, X2)) -> first1(active(X1), X2)
active(first1(X1, X2)) -> first1(X1, active(X2))
active(cons1(X1, X2)) -> cons1(active(X1), X2)
active(cons1(X1, X2)) -> cons1(X1, active(X2))
active(s1(X)) -> s1(active(X))
active(unquote(X)) -> unquote(active(X))
active(unquote1(X)) -> unquote1(active(X))
active(fcons(X1, X2)) -> fcons(active(X1), X2)
active(fcons(X1, X2)) -> fcons(X1, active(X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(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))
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))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
first1(mark(X1), X2) -> mark(first1(X1, X2))
first1(X1, mark(X2)) -> mark(first1(X1, X2))
first1(ok(X1), ok(X2)) -> ok(first1(X1, X2))
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))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
unquote(mark(X)) -> mark(unquote(X))
unquote(ok(X)) -> ok(unquote(X))
unquote1(mark(X)) -> mark(unquote1(X))
unquote1(ok(X)) -> ok(unquote1(X))
fcons(mark(X1), X2) -> mark(fcons(X1, X2))
fcons(X1, mark(X2)) -> mark(fcons(X1, X2))
fcons(ok(X1), ok(X2)) -> ok(fcons(X1, X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(from(X)) -> from(proper(X))
proper(sel1(X1, X2)) ->