Termination Proof using AProVE

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

Innermost Termination of R to be shown.

     ↳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.

     ↳DPs

       →DP Problem 1

         ↳Polynomial Ordering

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(SEL(x₁, x₂)) = x₁

POL(mark(x₁)) = x₁

POL(ok(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

           →DP Problem 18

             ↳Polynomial Ordering

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(SEL(x₁, x₂)) = x₁

POL(mark(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

           →DP Problem 18

             ↳Polo

...

               →DP Problem 19

                 ↳Polynomial Ordering

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pair:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(SEL(x₁, x₂)) = x₂

POL(mark(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

           →DP Problem 18

             ↳Polo

...

               →DP Problem 20

                 ↳Dependency Graph

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polynomial Ordering

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(mark(x₁)) = 0

POL(ok(x₁)) = 1 + x₁

POL(CONS(x₁, x₂)) = x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

           →DP Problem 21

             ↳Polynomial Ordering

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pair:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(mark(x₁)) = 1 + x₁

POL(CONS(x₁, x₂)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

           →DP Problem 21

             ↳Polo

...

               →DP Problem 22

                 ↳Dependency Graph

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polynomial Ordering

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(FIRST(x₁, x₂)) = x₁

POL(mark(x₁)) = x₁

POL(ok(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

           →DP Problem 23

             ↳Polynomial Ordering

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(FIRST(x₁, x₂)) = x₁

POL(mark(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

           →DP Problem 23

             ↳Polo

...

               →DP Problem 24

                 ↳Polynomial Ordering

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pair:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(FIRST(x₁, x₂)) = x₂

POL(mark(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

           →DP Problem 23

             ↳Polo

...

               →DP Problem 25

                 ↳Dependency Graph

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polynomial Ordering

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(FROM(x₁)) = x₁

POL(mark(x₁)) = x₁

POL(ok(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

           →DP Problem 26

             ↳Polynomial Ordering

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pair:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(FROM(x₁)) = x₁

POL(mark(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

           →DP Problem 26

             ↳Polo

...

               →DP Problem 27

                 ↳Dependency Graph

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polynomial Ordering

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(mark(x₁)) = x₁

POL(ok(x₁)) = 1 + x₁

POL(SEL1(x₁, x₂)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

           →DP Problem 28

             ↳Polynomial Ordering

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(mark(x₁)) = 1 + x₁

POL(SEL1(x₁, x₂)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

           →DP Problem 28

             ↳Polo

...

               →DP Problem 29

                 ↳Polynomial Ordering

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pair:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(mark(x₁)) = 1 + x₁

POL(SEL1(x₁, x₂)) = x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

           →DP Problem 28

             ↳Polo

...

               →DP Problem 30

                 ↳Dependency Graph

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polynomial Ordering

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pair:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(QUOTE(x₁)) = x₁

POL(ok(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

           →DP Problem 31

             ↳Dependency Graph

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polynomial Ordering

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(CONS1(x₁, x₂)) = x₁

POL(mark(x₁)) = x₁

POL(ok(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

           →DP Problem 32

             ↳Polynomial Ordering

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(CONS1(x₁, x₂)) = x₁

POL(mark(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

           →DP Problem 32

             ↳Polo

...

               →DP Problem 33

                 ↳Polynomial Ordering

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pair:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(CONS1(x₁, x₂)) = x₂

POL(mark(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

           →DP Problem 32

             ↳Polo

...

               →DP Problem 34

                 ↳Dependency Graph

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polynomial Ordering

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(FIRST1(x₁, x₂)) = x₁

POL(mark(x₁)) = x₁

POL(ok(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

           →DP Problem 35

             ↳Polynomial Ordering

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(FIRST1(x₁, x₂)) = x₁

POL(mark(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

           →DP Problem 35

             ↳Polo

...

               →DP Problem 36

                 ↳Polynomial Ordering

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pair:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(FIRST1(x₁, x₂)) = x₂

POL(mark(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

           →DP Problem 35

             ↳Polo

...

               →DP Problem 37

                 ↳Dependency Graph

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polynomial Ordering

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pair:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(QUOTE1(x₁)) = x₁

POL(ok(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

           →DP Problem 38

             ↳Dependency Graph

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polynomial Ordering

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(S1(x₁)) = x₁

POL(mark(x₁)) = x₁

POL(ok(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

           →DP Problem 39

             ↳Polynomial Ordering

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pair:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(S1(x₁)) = x₁

POL(mark(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

           →DP Problem 39

             ↳Polo

...

               →DP Problem 40

                 ↳Dependency Graph

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polynomial Ordering

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(S(x₁)) = x₁

POL(mark(x₁)) = x₁

POL(ok(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

           →DP Problem 41

             ↳Polynomial Ordering

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pair:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(S(x₁)) = x₁

POL(mark(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

           →DP Problem 41

             ↳Polo

...

               →DP Problem 42

                 ↳Dependency Graph

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polynomial Ordering

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

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))

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(FCONS(x₁, x₂)) = x₁

POL(mark(x₁)) = x₁

POL(ok(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

           →DP Problem 43

             ↳Polynomial Ordering

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pairs:

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))

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(FCONS(x₁, x₂)) = x₁

POL(mark(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

           →DP Problem 43

             ↳Polo

...

               →DP Problem 44

                 ↳Polynomial Ordering

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pair:

FCONS(X1, mark(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))

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(FCONS(x₁, x₂)) = x₂

POL(mark(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

           →DP Problem 43

             ↳Polo

...

               →DP Problem 45

                 ↳Dependency Graph

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polynomial Ordering

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pairs:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(UNQUOTE(x₁)) = x₁

POL(mark(x₁)) = 1 + x₁

POL(ok(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

           →DP Problem 46

             ↳Polynomial Ordering

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pair:

UNQUOTE(ok(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))

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(UNQUOTE(x₁)) = x₁

POL(ok(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

           →DP Problem 46

             ↳Polo

...

               →DP Problem 47

                 ↳Dependency Graph

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polynomial Ordering

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pairs:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(mark(x₁)) = 1 + x₁

POL(ok(x₁)) = x₁

POL(UNQUOTE1(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

           →DP Problem 48

             ↳Polynomial Ordering

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pair:

UNQUOTE1(ok(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))

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(ok(x₁)) = 1 + x₁

POL(UNQUOTE1(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

           →DP Problem 48

             ↳Polo

...

               →DP Problem 49

                 ↳Dependency Graph

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polynomial Ordering

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

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))

Strategy:

innermost

The following dependency pair can be strictly oriented:

ACTIVE(s1(X)) -> ACTIVE(X)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(first1(x₁, x₂)) = x₁ + x₂

POL(from(x₁)) = x₁

POL(fcons(x₁, x₂)) = x₁ + x₂

POL(sel(x₁, x₂)) = x₁ + x₂

POL(sel1(x₁, x₂)) = x₁ + x₂

POL(unquote1(x₁)) = x₁

POL(s1(x₁)) = 1 + x₁

POL(ACTIVE(x₁)) = x₁

POL(cons1(x₁, x₂)) = x₁ + x₂

POL(first(x₁, x₂)) = x₁ + x₂

POL(cons(x₁, x₂)) = x₁

POL(s(x₁)) = x₁

POL(unquote(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

           →DP Problem 50

             ↳Polynomial Ordering

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

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(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))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

ACTIVE(cons1(X1, X2)) -> ACTIVE(X2)
ACTIVE(cons1(X1, X2)) -> ACTIVE(X1)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(first1(x₁, x₂)) = x₁ + x₂

POL(from(x₁)) = x₁

POL(ACTIVE(x₁)) = x₁

POL(fcons(x₁, x₂)) = x₁ + x₂

POL(cons1(x₁, x₂)) = 1 + x₁ + x₂

POL(first(x₁, x₂)) = x₁ + x₂

POL(cons(x₁, x₂)) = x₁

POL(s(x₁)) = x₁

POL(sel(x₁, x₂)) = x₁ + x₂

POL(sel1(x₁, x₂)) = x₁ + x₂

POL(unquote1(x₁)) = x₁

POL(unquote(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

           →DP Problem 50

             ↳Polo

...

               →DP Problem 51

                 ↳Polynomial Ordering

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

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(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))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

ACTIVE(first1(X1, X2)) -> ACTIVE(X2)
ACTIVE(first1(X1, X2)) -> ACTIVE(X1)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(first1(x₁, x₂)) = 1 + x₁ + x₂

POL(from(x₁)) = x₁

POL(ACTIVE(x₁)) = x₁

POL(fcons(x₁, x₂)) = x₁ + x₂

POL(first(x₁, x₂)) = x₁ + x₂

POL(cons(x₁, x₂)) = x₁

POL(s(x₁)) = x₁

POL(sel(x₁, x₂)) = x₁ + x₂

POL(sel1(x₁, x₂)) = x₁ + x₂

POL(unquote1(x₁)) = x₁

POL(unquote(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

           →DP Problem 50

             ↳Polo

...

               →DP Problem 52

                 ↳Polynomial Ordering

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

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(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))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

ACTIVE(sel1(X1, X2)) -> ACTIVE(X2)
ACTIVE(sel1(X1, X2)) -> ACTIVE(X1)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(from(x₁)) = x₁

POL(ACTIVE(x₁)) = x₁

POL(fcons(x₁, x₂)) = x₁ + x₂

POL(first(x₁, x₂)) = x₁ + x₂

POL(cons(x₁, x₂)) = x₁

POL(s(x₁)) = x₁

POL(sel(x₁, x₂)) = x₁ + x₂

POL(sel1(x₁, x₂)) = 1 + x₁ + x₂

POL(unquote1(x₁)) = x₁

POL(unquote(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

           →DP Problem 50

             ↳Polo

...

               →DP Problem 53

                 ↳Polynomial Ordering

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

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(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))

Strategy:

innermost

The following dependency pair can be strictly oriented:

ACTIVE(from(X)) -> ACTIVE(X)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(from(x₁)) = 1 + x₁

POL(ACTIVE(x₁)) = x₁

POL(fcons(x₁, x₂)) = x₁ + x₂

POL(first(x₁, x₂)) = x₁ + x₂

POL(cons(x₁, x₂)) = x₁

POL(s(x₁)) = x₁

POL(sel(x₁, x₂)) = x₁ + x₂

POL(unquote1(x₁)) = x₁

POL(unquote(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

           →DP Problem 50

             ↳Polo

...

               →DP Problem 54

                 ↳Polynomial Ordering

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pairs:

Rules:

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

Strategy:

innermost

The following dependency pairs can be strictly oriented:

ACTIVE(first(X1, X2)) -> ACTIVE(X2)
ACTIVE(first(X1, X2)) -> ACTIVE(X1)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(ACTIVE(x₁)) = x₁

POL(fcons(x₁, x₂)) = x₁ + x₂

POL(first(x₁, x₂)) = 1 + x₁ + x₂

POL(cons(x₁, x₂)) = x₁

POL(s(x₁)) = x₁

POL(sel(x₁, x₂)) = x₁ + x₂

POL(unquote1(x₁)) = x₁

POL(unquote(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

           →DP Problem 50

             ↳Polo

...

               →DP Problem 55

                 ↳Polynomial Ordering

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pairs:

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(ACTIVE(x₁)) = x₁

POL(fcons(x₁, x₂)) = x₁ + x₂

POL(cons(x₁, x₂)) = 1 + x₁

POL(s(x₁)) = x₁

POL(sel(x₁, x₂)) = x₁ + x₂

POL(unquote1(x₁)) = x₁

POL(unquote(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

           →DP Problem 50

             ↳Polo

...

               →DP Problem 56

                 ↳Polynomial Ordering

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

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(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))

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(ACTIVE(x₁)) = x₁

POL(fcons(x₁, x₂)) = x₁ + x₂

POL(s(x₁)) = 1 + x₁

POL(sel(x₁, x₂)) = x₁ + x₂

POL(unquote1(x₁)) = x₁

POL(unquote(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

           →DP Problem 50

             ↳Polo

...

               →DP Problem 57

                 ↳Polynomial Ordering

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pairs:

Rules:

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

Strategy:

innermost

The following dependency pairs can be strictly oriented:

ACTIVE(sel(X1, X2)) -> ACTIVE(X2)
ACTIVE(sel(X1, X2)) -> ACTIVE(X1)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(ACTIVE(x₁)) = x₁

POL(fcons(x₁, x₂)) = x₁ + x₂

POL(sel(x₁, x₂)) = 1 + x₁ + x₂

POL(unquote1(x₁)) = x₁

POL(unquote(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

           →DP Problem 50

             ↳Polo

...

               →DP Problem 58

                 ↳Polynomial Ordering

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pairs:

ACTIVE(fcons(X1, X2)) -> ACTIVE(X2)
ACTIVE(fcons(X1, X2)) -> ACTIVE(X1)
ACTIVE(unquote1(X)) -> ACTIVE(X)
ACTIVE(unquote(X)) -> 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))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

ACTIVE(fcons(X1, X2)) -> ACTIVE(X2)
ACTIVE(fcons(X1, X2)) -> ACTIVE(X1)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(ACTIVE(x₁)) = x₁

POL(fcons(x₁, x₂)) = 1 + x₁ + x₂

POL(unquote1(x₁)) = x₁

POL(unquote(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

           →DP Problem 50

             ↳Polo

...

               →DP Problem 59

                 ↳Polynomial Ordering

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pairs:

ACTIVE(unquote1(X)) -> ACTIVE(X)
ACTIVE(unquote(X)) -> 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))

Strategy:

innermost

The following dependency pair can be strictly oriented:

ACTIVE(unquote1(X)) -> ACTIVE(X)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(ACTIVE(x₁)) = x₁

POL(unquote1(x₁)) = 1 + x₁

POL(unquote(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

           →DP Problem 50

             ↳Polo

...

               →DP Problem 60

                 ↳Polynomial Ordering

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pair:

ACTIVE(unquote(X)) -> 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))

Strategy:

innermost

The following dependency pair can be strictly oriented:

ACTIVE(unquote(X)) -> ACTIVE(X)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(ACTIVE(x₁)) = x₁

POL(unquote(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

           →DP Problem 50

             ↳Polo

...

               →DP Problem 61

                 ↳Dependency Graph

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polynomial Ordering

       →DP Problem 17

         ↳Remaining

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))

Strategy:

innermost

The following dependency pair can be strictly oriented:

PROPER(s1(X)) -> PROPER(X)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(first1(x₁, x₂)) = x₁ + x₂

POL(from(x₁)) = x₁

POL(fcons(x₁, x₂)) = x₁ + x₂

POL(PROPER(x₁)) = x₁

POL(sel(x₁, x₂)) = x₁ + x₂

POL(sel1(x₁, x₂)) = x₁ + x₂

POL(unquote1(x₁)) = x₁

POL(s1(x₁)) = 1 + x₁

POL(cons1(x₁, x₂)) = x₁ + x₂

POL(first(x₁, x₂)) = x₁ + x₂

POL(quote(x₁)) = x₁

POL(cons(x₁, x₂)) = x₁ + x₂

POL(s(x₁)) = x₁

POL(quote1(x₁)) = x₁

POL(unquote(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

           →DP Problem 62

             ↳Polynomial Ordering

       →DP Problem 17

         ↳Remaining

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(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))

Strategy:

innermost

The following dependency pair can be strictly oriented:

PROPER(quote1(X)) -> PROPER(X)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(first1(x₁, x₂)) = x₁ + x₂

POL(from(x₁)) = x₁

POL(fcons(x₁, x₂)) = x₁ + x₂

POL(PROPER(x₁)) = x₁

POL(sel(x₁, x₂)) = x₁ + x₂

POL(sel1(x₁, x₂)) = x₁ + x₂

POL(unquote1(x₁)) = x₁

POL(cons1(x₁, x₂)) = x₁ + x₂

POL(first(x₁, x₂)) = x₁ + x₂

POL(quote(x₁)) = x₁

POL(cons(x₁, x₂)) = x₁ + x₂

POL(s(x₁)) = x₁

POL(quote1(x₁)) = 1 + x₁

POL(unquote(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

           →DP Problem 62

             ↳Polo

...

               →DP Problem 63

                 ↳Polynomial Ordering

       →DP Problem 17

         ↳Remaining

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(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))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

PROPER(cons1(X1, X2)) -> PROPER(X2)
PROPER(cons1(X1, X2)) -> PROPER(X1)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(first1(x₁, x₂)) = x₁ + x₂

POL(from(x₁)) = x₁

POL(fcons(x₁, x₂)) = x₁ + x₂

POL(PROPER(x₁)) = x₁

POL(sel(x₁, x₂)) = x₁ + x₂

POL(sel1(x₁, x₂)) = x₁ + x₂

POL(unquote1(x₁)) = x₁

POL(cons1(x₁, x₂)) = 1 + x₁ + x₂

POL(first(x₁, x₂)) = x₁ + x₂

POL(quote(x₁)) = x₁

POL(cons(x₁, x₂)) = x₁ + x₂

POL(s(x₁)) = x₁

POL(unquote(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

           →DP Problem 62

             ↳Polo

...

               →DP Problem 64

                 ↳Polynomial Ordering

       →DP Problem 17

         ↳Remaining

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(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))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

PROPER(first1(X1, X2)) -> PROPER(X2)
PROPER(first1(X1, X2)) -> PROPER(X1)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(first1(x₁, x₂)) = 1 + x₁ + x₂

POL(from(x₁)) = x₁

POL(fcons(x₁, x₂)) = x₁ + x₂

POL(first(x₁, x₂)) = x₁ + x₂

POL(quote(x₁)) = x₁

POL(PROPER(x₁)) = x₁

POL(cons(x₁, x₂)) = x₁ + x₂

POL(s(x₁)) = x₁

POL(sel(x₁, x₂)) = x₁ + x₂

POL(sel1(x₁, x₂)) = x₁ + x₂

POL(unquote1(x₁)) = x₁

POL(unquote(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

           →DP Problem 62

             ↳Polo

...

               →DP Problem 65

                 ↳Polynomial Ordering

       →DP Problem 17

         ↳Remaining

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(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))

Strategy:

innermost

The following dependency pair can be strictly oriented:

PROPER(quote(X)) -> PROPER(X)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(from(x₁)) = x₁

POL(fcons(x₁, x₂)) = x₁ + x₂

POL(first(x₁, x₂)) = x₁ + x₂

POL(quote(x₁)) = 1 + x₁

POL(PROPER(x₁)) = x₁

POL(cons(x₁, x₂)) = x₁ + x₂

POL(s(x₁)) = x₁

POL(sel(x₁, x₂)) = x₁ + x₂

POL(sel1(x₁, x₂)) = x₁ + x₂

POL(unquote1(x₁)) = x₁

POL(unquote(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

           →DP Problem 62

             ↳Polo

...

               →DP Problem 66

                 ↳Polynomial Ordering

       →DP Problem 17

         ↳Remaining

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(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))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

PROPER(sel1(X1, X2)) -> PROPER(X2)
PROPER(sel1(X1, X2)) -> PROPER(X1)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(from(x₁)) = x₁

POL(fcons(x₁, x₂)) = x₁ + x₂

POL(first(x₁, x₂)) = x₁ + x₂

POL(PROPER(x₁)) = x₁

POL(cons(x₁, x₂)) = x₁ + x₂

POL(s(x₁)) = x₁

POL(sel(x₁, x₂)) = x₁ + x₂

POL(sel1(x₁, x₂)) = 1 + x₁ + x₂

POL(unquote1(x₁)) = x₁

POL(unquote(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

           →DP Problem 62

             ↳Polo

...

               →DP Problem 67

                 ↳Polynomial Ordering

       →DP Problem 17

         ↳Remaining

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(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))

Strategy:

innermost

The following dependency pair can be strictly oriented:

PROPER(from(X)) -> PROPER(X)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(from(x₁)) = 1 + x₁

POL(fcons(x₁, x₂)) = x₁ + x₂

POL(first(x₁, x₂)) = x₁ + x₂

POL(PROPER(x₁)) = x₁

POL(cons(x₁, x₂)) = x₁ + x₂

POL(s(x₁)) = x₁

POL(sel(x₁, x₂)) = x₁ + x₂

POL(unquote1(x₁)) = x₁

POL(unquote(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

           →DP Problem 62

             ↳Polo

...

               →DP Problem 68

                 ↳Polynomial Ordering

       →DP Problem 17

         ↳Remaining

Dependency Pairs:

Rules:

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

Strategy:

innermost

The following dependency pairs can be strictly oriented:

PROPER(first(X1, X2)) -> PROPER(X2)
PROPER(first(X1, X2)) -> PROPER(X1)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(fcons(x₁, x₂)) = x₁ + x₂

POL(first(x₁, x₂)) = 1 + x₁ + x₂

POL(PROPER(x₁)) = x₁

POL(cons(x₁, x₂)) = x₁ + x₂

POL(s(x₁)) = x₁

POL(sel(x₁, x₂)) = x₁ + x₂

POL(unquote1(x₁)) = x₁

POL(unquote(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

           →DP Problem 62

             ↳Polo

...

               →DP Problem 69

                 ↳Polynomial Ordering

       →DP Problem 17

         ↳Remaining

Dependency Pairs:

Rules:

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

Strategy:

innermost

The following dependency pairs can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(fcons(x₁, x₂)) = x₁ + x₂

POL(PROPER(x₁)) = x₁

POL(cons(x₁, x₂)) = 1 + x₁ + x₂

POL(s(x₁)) = x₁

POL(sel(x₁, x₂)) = x₁ + x₂

POL(unquote1(x₁)) = x₁

POL(unquote(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

           →DP Problem 62

             ↳Polo

...

               →DP Problem 70

                 ↳Polynomial Ordering

       →DP Problem 17

         ↳Remaining

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(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))

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(fcons(x₁, x₂)) = x₁ + x₂

POL(PROPER(x₁)) = x₁

POL(s(x₁)) = 1 + x₁

POL(sel(x₁, x₂)) = x₁ + x₂

POL(unquote1(x₁)) = x₁

POL(unquote(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

           →DP Problem 62

             ↳Polo

...

               →DP Problem 71

                 ↳Polynomial Ordering

       →DP Problem 17

         ↳Remaining

Dependency Pairs:

Rules:

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

Strategy:

innermost

The following dependency pairs can be strictly oriented:

PROPER(sel(X1, X2)) -> PROPER(X2)
PROPER(sel(X1, X2)) -> PROPER(X1)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(fcons(x₁, x₂)) = x₁ + x₂

POL(PROPER(x₁)) = x₁

POL(sel(x₁, x₂)) = 1 + x₁ + x₂

POL(unquote1(x₁)) = x₁

POL(unquote(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

           →DP Problem 62

             ↳Polo

...

               →DP Problem 72

                 ↳Polynomial Ordering

       →DP Problem 17

         ↳Remaining

Dependency Pairs:

PROPER(fcons(X1, X2)) -> PROPER(X2)
PROPER(fcons(X1, X2)) -> PROPER(X1)
PROPER(unquote1(X)) -> PROPER(X)
PROPER(unquote(X)) -> 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))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

PROPER(fcons(X1, X2)) -> PROPER(X2)
PROPER(fcons(X1, X2)) -> PROPER(X1)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(fcons(x₁, x₂)) = 1 + x₁ + x₂

POL(PROPER(x₁)) = x₁

POL(unquote1(x₁)) = x₁

POL(unquote(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

           →DP Problem 62

             ↳Polo

...

               →DP Problem 73

                 ↳Polynomial Ordering

       →DP Problem 17

         ↳Remaining

Dependency Pairs:

PROPER(unquote1(X)) -> PROPER(X)
PROPER(unquote(X)) -> 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))

Strategy:

innermost

The following dependency pair can be strictly oriented:

PROPER(unquote1(X)) -> PROPER(X)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(PROPER(x₁)) = x₁

POL(unquote1(x₁)) = 1 + x₁

POL(unquote(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

           →DP Problem 62

             ↳Polo

...

               →DP Problem 74

                 ↳Polynomial Ordering

       →DP Problem 17

         ↳Remaining

Dependency Pair:

PROPER(unquote(X)) -> 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))

Strategy:

innermost

The following dependency pair can be strictly oriented:

PROPER(unquote(X)) -> PROPER(X)

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(PROPER(x₁)) = x₁

POL(unquote(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

           →DP Problem 62

             ↳Polo

...

               →DP Problem 75

                 ↳Dependency Graph

       →DP Problem 17

         ↳Remaining

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Polo

       →DP Problem 3

         ↳Polo

       →DP Problem 4

         ↳Polo

       →DP Problem 5

         ↳Polo

       →DP Problem 6

         ↳Polo

       →DP Problem 7

         ↳Polo

       →DP Problem 8

         ↳Polo

       →DP Problem 9

         ↳Polo

       →DP Problem 10

         ↳Polo

       →DP Problem 11

         ↳Polo

       →DP Problem 12

         ↳Polo

       →DP Problem 13

         ↳Polo

       →DP Problem 14

         ↳Polo

       →DP Problem 15

         ↳Polo

       →DP Problem 16

         ↳Polo

       →DP Problem 17

         ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

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

Rules:

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

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:17 minutes

POL(SEL(x₁, x₂))	= x₁
POL(mark(x₁))	= x₁
POL(ok(x₁))	= 1 + x₁

POL(mark(x₁))	= 1 + x₁
POL(CONS(x₁, x₂))	= x₁

POL(FIRST(x₁, x₂))	= x₁
POL(mark(x₁))	= x₁
POL(ok(x₁))	= 1 + x₁

POL(FIRST(x₁, x₂))	= x₂
POL(mark(x₁))	= 1 + x₁

POL(FROM(x₁))	= x₁
POL(mark(x₁))	= x₁
POL(ok(x₁))	= 1 + x₁

POL(mark(x₁))	= 1 + x₁
POL(SEL1(x₁, x₂))	= x₁

POL(CONS1(x₁, x₂))	= x₁
POL(mark(x₁))	= x₁
POL(ok(x₁))	= 1 + x₁

POL(FIRST1(x₁, x₂))	= x₁
POL(mark(x₁))	= x₁
POL(ok(x₁))	= 1 + x₁

POL(S1(x₁))	= x₁
POL(mark(x₁))	= x₁
POL(ok(x₁))	= 1 + x₁

POL(S(x₁))	= x₁
POL(mark(x₁))	= x₁
POL(ok(x₁))	= 1 + x₁

POL(FCONS(x₁, x₂))	= x₁
POL(mark(x₁))	= x₁
POL(ok(x₁))	= 1 + x₁

POL(UNQUOTE(x₁))	= x₁
POL(mark(x₁))	= 1 + x₁
POL(ok(x₁))	= x₁

POL(first1(x₁, x₂))	= x₁ + x₂
POL(from(x₁))	= x₁
POL(fcons(x₁, x₂))	= x₁ + x₂
POL(sel(x₁, x₂))	= x₁ + x₂
POL(sel1(x₁, x₂))	= x₁ + x₂
POL(unquote1(x₁))	= x₁
POL(s1(x₁))	= 1 + x₁
POL(ACTIVE(x₁))	= x₁
POL(cons1(x₁, x₂))	= x₁ + x₂
POL(first(x₁, x₂))	= x₁ + x₂
POL(cons(x₁, x₂))	= x₁
POL(s(x₁))	= x₁
POL(unquote(x₁))	= x₁

POL(first1(x₁, x₂))	= 1 + x₁ + x₂
POL(from(x₁))	= x₁
POL(ACTIVE(x₁))	= x₁
POL(fcons(x₁, x₂))	= x₁ + x₂
POL(first(x₁, x₂))	= x₁ + x₂
POL(cons(x₁, x₂))	= x₁
POL(s(x₁))	= x₁
POL(sel(x₁, x₂))	= x₁ + x₂
POL(sel1(x₁, x₂))	= x₁ + x₂
POL(unquote1(x₁))	= x₁
POL(unquote(x₁))	= x₁

POL(fcons(x₁, x₂))	= x₁ + x₂
POL(PROPER(x₁))	= x₁
POL(cons(x₁, x₂))	= 1 + x₁ + x₂
POL(s(x₁))	= x₁
POL(sel(x₁, x₂))	= x₁ + x₂
POL(unquote1(x₁))	= x₁
POL(unquote(x₁))	= x₁