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))

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
         ↳Size-Change Principle
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳SCP
       →DP Problem 11
         ↳SCP
       →DP Problem 12
         ↳SCP
       →DP Problem 13
         ↳SCP
       →DP Problem 14
         ↳SCP
       →DP Problem 15
         ↳SCP
       →DP Problem 16
         ↳SCP
       →DP Problem 17
         ↳Nar


Dependency Pairs:

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





We number the DPs as follows:
  1. SEL(ok(X1), ok(X2)) -> SEL(X1, X2)
  2. SEL(X1, mark(X2)) -> SEL(X1, X2)
  3. SEL(mark(X1), X2) -> SEL(X1, X2)
and get the following Size-Change Graph(s):
{3, 2, 1} , {3, 2, 1}
1>1
2>2
{3, 2, 1} , {3, 2, 1}
1=1
2>2
{3, 2, 1} , {3, 2, 1}
1>1
2=2

which lead(s) to this/these maximal multigraph(s):
{3, 2, 1} , {3, 2, 1}
1>1
2>2
{3, 2, 1} , {3, 2, 1}
1>1
2=2
{3, 2, 1} , {3, 2, 1}
1=1
2>2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳Size-Change Principle
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳SCP
       →DP Problem 11
         ↳SCP
       →DP Problem 12
         ↳SCP
       →DP Problem 13
         ↳SCP
       →DP Problem 14
         ↳SCP
       →DP Problem 15
         ↳SCP
       →DP Problem 16
         ↳SCP
       →DP Problem 17
         ↳Nar


Dependency Pairs:

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





We number the DPs as follows:
  1. CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
  2. CONS(mark(X1), X2) -> CONS(X1, X2)
and get the following Size-Change Graph(s):
{2, 1} , {2, 1}
1>1
2>2
{2, 1} , {2, 1}
1>1
2=2

which lead(s) to this/these maximal multigraph(s):
{2, 1} , {2, 1}
1>1
2>2
{2, 1} , {2, 1}
1>1
2=2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳Size-Change Principle
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳SCP
       →DP Problem 11
         ↳SCP
       →DP Problem 12
         ↳SCP
       →DP Problem 13
         ↳SCP
       →DP Problem 14
         ↳SCP
       →DP Problem 15
         ↳SCP
       →DP Problem 16
         ↳SCP
       →DP Problem 17
         ↳Nar


Dependency Pairs:

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





We number the DPs as follows:
  1. FIRST(ok(X1), ok(X2)) -> FIRST(X1, X2)
  2. FIRST(X1, mark(X2)) -> FIRST(X1, X2)
  3. FIRST(mark(X1), X2) -> FIRST(X1, X2)
and get the following Size-Change Graph(s):
{3, 2, 1} , {3, 2, 1}
1>1
2>2
{3, 2, 1} , {3, 2, 1}
1=1
2>2
{3, 2, 1} , {3, 2, 1}
1>1
2=2

which lead(s) to this/these maximal multigraph(s):
{3, 2, 1} , {3, 2, 1}
1>1
2=2
{3, 2, 1} , {3, 2, 1}
1=1
2>2
{3, 2, 1} , {3, 2, 1}
1>1
2>2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳Size-Change Principle
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳SCP
       →DP Problem 11
         ↳SCP
       →DP Problem 12
         ↳SCP
       →DP Problem 13
         ↳SCP
       →DP Problem 14
         ↳SCP
       →DP Problem 15
         ↳SCP
       →DP Problem 16
         ↳SCP
       →DP Problem 17
         ↳Nar


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))





We number the DPs as follows:
  1. FROM(ok(X)) -> FROM(X)
  2. FROM(mark(X)) -> FROM(X)
and get the following Size-Change Graph(s):
{2, 1} , {2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{2, 1} , {2, 1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳Size-Change Principle
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳SCP
       →DP Problem 11
         ↳SCP
       →DP Problem 12
         ↳SCP
       →DP Problem 13
         ↳SCP
       →DP Problem 14
         ↳SCP
       →DP Problem 15
         ↳SCP
       →DP Problem 16
         ↳SCP
       →DP Problem 17
         ↳Nar


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))





We number the DPs as follows:
  1. S(ok(X)) -> S(X)
  2. S(mark(X)) -> S(X)
and get the following Size-Change Graph(s):
{2, 1} , {2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{2, 1} , {2, 1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳Size-Change Principle
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳SCP
       →DP Problem 11
         ↳SCP
       →DP Problem 12
         ↳SCP
       →DP Problem 13
         ↳SCP
       →DP Problem 14
         ↳SCP
       →DP Problem 15
         ↳SCP
       →DP Problem 16
         ↳SCP
       →DP Problem 17
         ↳Nar


Dependency Pairs:

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





We number the DPs as follows:
  1. SEL1(ok(X1), ok(X2)) -> SEL1(X1, X2)
  2. SEL1(X1, mark(X2)) -> SEL1(X1, X2)
  3. SEL1(mark(X1), X2) -> SEL1(X1, X2)
and get the following Size-Change Graph(s):
{3, 2, 1} , {3, 2, 1}
1>1
2>2
{3, 2, 1} , {3, 2, 1}
1=1
2>2
{3, 2, 1} , {3, 2, 1}
1>1
2=2

which lead(s) to this/these maximal multigraph(s):
{3, 2, 1} , {3, 2, 1}
1>1
2=2
{3, 2, 1} , {3, 2, 1}
1=1
2>2
{3, 2, 1} , {3, 2, 1}
1>1
2>2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳Size-Change Principle
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳SCP
       →DP Problem 11
         ↳SCP
       →DP Problem 12
         ↳SCP
       →DP Problem 13
         ↳SCP
       →DP Problem 14
         ↳SCP
       →DP Problem 15
         ↳SCP
       →DP Problem 16
         ↳SCP
       →DP Problem 17
         ↳Nar


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))





We number the DPs as follows:
  1. QUOTE(ok(X)) -> QUOTE(X)
and get the following Size-Change Graph(s):
{1} , {1}
1>1

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳Size-Change Principle
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳SCP
       →DP Problem 11
         ↳SCP
       →DP Problem 12
         ↳SCP
       →DP Problem 13
         ↳SCP
       →DP Problem 14
         ↳SCP
       →DP Problem 15
         ↳SCP
       →DP Problem 16
         ↳SCP
       →DP Problem 17
         ↳Nar


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))





We number the DPs as follows:
  1. CONS1(ok(X1), ok(X2)) -> CONS1(X1, X2)
  2. CONS1(X1, mark(X2)) -> CONS1(X1, X2)
  3. CONS1(mark(X1), X2) -> CONS1(X1, X2)
and get the following Size-Change Graph(s):
{3, 2, 1} , {3, 2, 1}
1>1
2>2
{3, 2, 1} , {3, 2, 1}
1=1
2>2
{3, 2, 1} , {3, 2, 1}
1>1
2=2

which lead(s) to this/these maximal multigraph(s):
{3, 2, 1} , {3, 2, 1}
1>1
2=2
{3, 2, 1} , {3, 2, 1}
1=1
2>2
{3, 2, 1} , {3, 2, 1}
1>1
2>2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳Size-Change Principle
       →DP Problem 10
         ↳SCP
       →DP Problem 11
         ↳SCP
       →DP Problem 12
         ↳SCP
       →DP Problem 13
         ↳SCP
       →DP Problem 14
         ↳SCP
       →DP Problem 15
         ↳SCP
       →DP Problem 16
         ↳SCP
       →DP Problem 17
         ↳Nar


Dependency Pairs:

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





We number the DPs as follows:
  1. FIRST1(ok(X1), ok(X2)) -> FIRST1(X1, X2)
  2. FIRST1(X1, mark(X2)) -> FIRST1(X1, X2)
  3. FIRST1(mark(X1), X2) -> FIRST1(X1, X2)
and get the following Size-Change Graph(s):
{3, 2, 1} , {3, 2, 1}
1>1
2>2
{3, 2, 1} , {3, 2, 1}
1=1
2>2
{3, 2, 1} , {3, 2, 1}
1>1
2=2

which lead(s) to this/these maximal multigraph(s):
{3, 2, 1} , {3, 2, 1}
1>1
2>2
{3, 2, 1} , {3, 2, 1}
1>1
2=2
{3, 2, 1} , {3, 2, 1}
1=1
2>2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳Size-Change Principle
       →DP Problem 11
         ↳SCP
       →DP Problem 12
         ↳SCP
       →DP Problem 13
         ↳SCP
       →DP Problem 14
         ↳SCP
       →DP Problem 15
         ↳SCP
       →DP Problem 16
         ↳SCP
       →DP Problem 17
         ↳Nar


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))





We number the DPs as follows:
  1. QUOTE1(ok(X)) -> QUOTE1(X)
and get the following Size-Change Graph(s):
{1} , {1}
1>1

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳SCP
       →DP Problem 11
         ↳Size-Change Principle
       →DP Problem 12
         ↳SCP
       →DP Problem 13
         ↳SCP
       →DP Problem 14
         ↳SCP
       →DP Problem 15
         ↳SCP
       →DP Problem 16
         ↳SCP
       →DP Problem 17
         ↳Nar


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))





We number the DPs as follows:
  1. S1(ok(X)) -> S1(X)
  2. S1(mark(X)) -> S1(X)
and get the following Size-Change Graph(s):
{2, 1} , {2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{2, 1} , {2, 1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳SCP
       →DP Problem 11
         ↳SCP
       →DP Problem 12
         ↳Size-Change Principle
       →DP Problem 13
         ↳SCP
       →DP Problem 14
         ↳SCP
       →DP Problem 15
         ↳SCP
       →DP Problem 16
         ↳SCP
       →DP Problem 17
         ↳Nar


Dependency Pairs:

UNQUOTE(ok(X)) -> UNQUOTE(X)
UNQUOTE(mark(X)) -> UNQUOTE(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))





We number the DPs as follows:
  1. UNQUOTE(ok(X)) -> UNQUOTE(X)
  2. UNQUOTE(mark(X)) -> UNQUOTE(X)
and get the following Size-Change Graph(s):
{2, 1} , {2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{2, 1} , {2, 1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳SCP
       →DP Problem 11
         ↳SCP
       →DP Problem 12
         ↳SCP
       →DP Problem 13
         ↳Size-Change Principle
       →DP Problem 14
         ↳SCP
       →DP Problem 15
         ↳SCP
       →DP Problem 16
         ↳SCP
       →DP Problem 17
         ↳Nar


Dependency Pairs:

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





We number the DPs as follows:
  1. FCONS(ok(X1), ok(X2)) -> FCONS(X1, X2)
  2. FCONS(X1, mark(X2)) -> FCONS(X1, X2)
  3. FCONS(mark(X1), X2) -> FCONS(X1, X2)
and get the following Size-Change Graph(s):
{3, 2, 1} , {3, 2, 1}
1>1
2>2
{3, 2, 1} , {3, 2, 1}
1=1
2>2
{3, 2, 1} , {3, 2, 1}
1>1
2=2

which lead(s) to this/these maximal multigraph(s):
{3, 2, 1} , {3, 2, 1}
1>1
2=2
{3, 2, 1} , {3, 2, 1}
1=1
2>2
{3, 2, 1} , {3, 2, 1}
1>1
2>2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳SCP
       →DP Problem 11
         ↳SCP
       →DP Problem 12
         ↳SCP
       →DP Problem 13
         ↳SCP
       →DP Problem 14
         ↳Size-Change Principle
       →DP Problem 15
         ↳SCP
       →DP Problem 16
         ↳SCP
       →DP Problem 17
         ↳Nar


Dependency Pairs:

UNQUOTE1(ok(X)) -> UNQUOTE1(X)
UNQUOTE1(mark(X)) -> UNQUOTE1(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))





We number the DPs as follows:
  1. UNQUOTE1(ok(X)) -> UNQUOTE1(X)
  2. UNQUOTE1(mark(X)) -> UNQUOTE1(X)
and get the following Size-Change Graph(s):
{2, 1} , {2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{2, 1} , {2, 1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳SCP
       →DP Problem 11
         ↳SCP
       →DP Problem 12
         ↳SCP
       →DP Problem 13
         ↳SCP
       →DP Problem 14
         ↳SCP
       →DP Problem 15
         ↳Size-Change Principle
       →DP Problem 16
         ↳SCP
       →DP Problem 17
         ↳Nar


Dependency Pairs:

ACTIVE(fcons(X1, X2)) -> ACTIVE(X2)
ACTIVE(fcons(X1, X2)) -> ACTIVE(X1)
ACTIVE(unquote1(X)) -> ACTIVE(X)
ACTIVE(unquote(X)) -> ACTIVE(X)
ACTIVE(s1(X)) -> ACTIVE(X)
ACTIVE(cons1(X1, X2)) -> ACTIVE(X2)
ACTIVE(cons1(X1, X2)) -> ACTIVE(X1)
ACTIVE(first1(X1, X2)) -> ACTIVE(X2)
ACTIVE(first1(X1, X2)) -> ACTIVE(X1)
ACTIVE(sel1(X1, X2)) -> ACTIVE(X2)
ACTIVE(sel1(X1, X2)) -> ACTIVE(X1)
ACTIVE(from(X)) -> ACTIVE(X)
ACTIVE(first(X1, X2)) -> ACTIVE(X2)
ACTIVE(first(X1, X2)) -> ACTIVE(X1)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(sel(X1, X2)) -> ACTIVE(X2)
ACTIVE(sel(X1, X2)) -> ACTIVE(X1)


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))





We number the DPs as follows:
  1. ACTIVE(fcons(X1, X2)) -> ACTIVE(X2)
  2. ACTIVE(fcons(X1, X2)) -> ACTIVE(X1)
  3. ACTIVE(unquote1(X)) -> ACTIVE(X)
  4. ACTIVE(unquote(X)) -> ACTIVE(X)
  5. ACTIVE(s1(X)) -> ACTIVE(X)
  6. ACTIVE(cons1(X1, X2)) -> ACTIVE(X2)
  7. ACTIVE(cons1(X1, X2)) -> ACTIVE(X1)
  8. ACTIVE(first1(X1, X2)) -> ACTIVE(X2)
  9. ACTIVE(first1(X1, X2)) -> ACTIVE(X1)
  10. ACTIVE(sel1(X1, X2)) -> ACTIVE(X2)
  11. ACTIVE(sel1(X1, X2)) -> ACTIVE(X1)
  12. ACTIVE(from(X)) -> ACTIVE(X)
  13. ACTIVE(first(X1, X2)) -> ACTIVE(X2)
  14. ACTIVE(first(X1, X2)) -> ACTIVE(X1)
  15. ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
  16. ACTIVE(s(X)) -> ACTIVE(X)
  17. ACTIVE(sel(X1, X2)) -> ACTIVE(X2)
  18. ACTIVE(sel(X1, X2)) -> ACTIVE(X1)
and get the following Size-Change Graph(s):
{18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1} , {18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1} , {18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
first1(x1, x2) -> first1(x1, x2)
from(x1) -> from(x1)
fcons(x1, x2) -> fcons(x1, x2)
sel(x1, x2) -> sel(x1, x2)
sel1(x1, x2) -> sel1(x1, x2)
unquote1(x1) -> unquote1(x1)
s1(x1) -> s1(x1)
cons1(x1, x2) -> cons1(x1, x2)
first(x1, x2) -> first(x1, x2)
cons(x1, x2) -> cons(x1, x2)
s(x1) -> s(x1)
unquote(x1) -> unquote(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳SCP
       →DP Problem 11
         ↳SCP
       →DP Problem 12
         ↳SCP
       →DP Problem 13
         ↳SCP
       →DP Problem 14
         ↳SCP
       →DP Problem 15
         ↳SCP
       →DP Problem 16
         ↳Size-Change Principle
       →DP Problem 17
         ↳Nar


Dependency Pairs:

PROPER(fcons(X1, X2)) -> PROPER(X2)
PROPER(fcons(X1, X2)) -> PROPER(X1)
PROPER(unquote1(X)) -> PROPER(X)
PROPER(unquote(X)) -> PROPER(X)
PROPER(s1(X)) -> PROPER(X)
PROPER(quote1(X)) -> PROPER(X)
PROPER(cons1(X1, X2)) -> PROPER(X2)
PROPER(cons1(X1, X2)) -> PROPER(X1)
PROPER(first1(X1, X2)) -> PROPER(X2)
PROPER(first1(X1, X2)) -> PROPER(X1)
PROPER(quote(X)) -> PROPER(X)
PROPER(sel1(X1, X2)) -> PROPER(X2)
PROPER(sel1(X1, X2)) -> PROPER(X1)
PROPER(from(X)) -> PROPER(X)
PROPER(first(X1, X2)) -> PROPER(X2)
PROPER(first(X1, X2)) -> PROPER(X1)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(s(X)) -> PROPER(X)
PROPER(sel(X1, X2)) -> PROPER(X2)
PROPER(sel(X1, X2)) -> PROPER(X1)


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))





We number the DPs as follows:
  1. PROPER(fcons(X1, X2)) -> PROPER(X2)
  2. PROPER(fcons(X1, X2)) -> PROPER(X1)
  3. PROPER(unquote1(X)) -> PROPER(X)
  4. PROPER(unquote(X)) -> PROPER(X)
  5. PROPER(s1(X)) -> PROPER(X)
  6. PROPER(quote1(X)) -> PROPER(X)
  7. PROPER(cons1(X1, X2)) -> PROPER(X2)
  8. PROPER(cons1(X1, X2)) -> PROPER(X1)
  9. PROPER(first1(X1, X2)) -> PROPER(X2)
  10. PROPER(first1(X1, X2)) -> PROPER(X1)
  11. PROPER(quote(X)) -> PROPER(X)
  12. PROPER(sel1(X1, X2)) -> PROPER(X2)
  13. PROPER(sel1(X1, X2)) -> PROPER(X1)
  14. PROPER(from(X)) -> PROPER(X)
  15. PROPER(first(X1, X2)) -> PROPER(X2)
  16. PROPER(first(X1, X2)) -> PROPER(X1)
  17. PROPER(cons(X1, X2)) -> PROPER(X2)
  18. PROPER(cons(X1, X2)) -> PROPER(X1)
  19. PROPER(s(X)) -> PROPER(X)
  20. PROPER(sel(X1, X2)) -> PROPER(X2)
  21. PROPER(sel(X1, X2)) -> PROPER(X1)
and get the following Size-Change Graph(s):
{21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1} , {21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1} , {21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
first1(x1, x2) -> first1(x1, x2)
from(x1) -> from(x1)
fcons(x1, x2) -> fcons(x1, x2)
sel(x1, x2) -> sel(x1, x2)
sel1(x1, x2) -> sel1(x1, x2)
unquote1(x1) -> unquote1(x1)
s1(x1) -> s1(x1)
cons1(x1, x2) -> cons1(x1, x2)
first(x1, x2) -> first(x1, x2)
quote(x1) -> quote(x1)
cons(x1, x2) -> cons(x1, x2)
s(x1) -> s(x1)
quote1(x1) -> quote1(x1)
unquote(x1) -> unquote(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳SCP
       →DP Problem 11
         ↳SCP
       →DP Problem 12
         ↳SCP
       →DP Problem 13
         ↳SCP
       →DP Problem 14
         ↳SCP
       →DP Problem 15
         ↳SCP
       →DP Problem 16
         ↳SCP
       →DP Problem 17
         ↳Narrowing Transformation


Dependency Pairs:

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





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

TOP(mark(X)) -> TOP(proper(X))
18 new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳SCP
       →DP Problem 11
         ↳SCP
       →DP Problem 12
         ↳SCP
       →DP Problem 13
         ↳SCP
       →DP Problem 14
         ↳SCP
       →DP Problem 15
         ↳SCP
       →DP Problem 16
         ↳SCP
       →DP Problem 17
         ↳Nar
           →DP Problem 18
             ↳Narrowing Transformation


Dependency Pairs:

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





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

TOP(ok(X)) -> TOP(active(X))
38 new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳SCP
       →DP Problem 11
         ↳SCP
       →DP Problem 12
         ↳SCP
       →DP Problem 13
         ↳SCP
       →DP Problem 14
         ↳SCP
       →DP Problem 15
         ↳SCP
       →DP Problem 16
         ↳SCP
       →DP Problem 17
         ↳Nar
           →DP Problem 18
             ↳Nar
             ...
               →DP Problem 19
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

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




The Proof could not be continued due to a Timeout.
Termination of R could not be shown.
Duration:
1:00 minutes