Termination w.r.t. Q proof of /tmp/tmpXkNy3a/Ex6_15_AEL02

Termination w.r.t. Q of the given QTRS could not be shown:

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 AND

  ↳5 QDP

    ↳6 QDPOrderProof (⇔)

    ↳7 QDP

    ↳8 PisEmptyProof (⇔)

    ↳9 TRUE

  ↳10 QDP

    ↳11 QDPOrderProof (⇔)

    ↳12 QDP

    ↳13 PisEmptyProof (⇔)

    ↳14 TRUE

  ↳15 QDP

    ↳16 QDPOrderProof (⇔)

    ↳17 QDP

    ↳18 QDPOrderProof (⇔)

    ↳19 QDP

    ↳20 PisEmptyProof (⇔)

    ↳21 TRUE

  ↳22 QDP

    ↳23 QDPOrderProof (⇔)

    ↳24 QDP

    ↳25 QDPOrderProof (⇔)

    ↳26 QDP

    ↳27 PisEmptyProof (⇔)

    ↳28 TRUE

  ↳29 QDP

    ↳30 QDPOrderProof (⇔)

    ↳31 QDP

    ↳32 QDPOrderProof (⇔)

    ↳33 QDP

    ↳34 PisEmptyProof (⇔)

    ↳35 TRUE

  ↳36 QDP

    ↳37 QDPOrderProof (⇔)

    ↳38 QDP

    ↳39 QDPOrderProof (⇔)

    ↳40 QDP

    ↳41 PisEmptyProof (⇔)

    ↳42 TRUE

  ↳43 QDP

    ↳44 QDPOrderProof (⇔)

    ↳45 QDP

    ↳46 QDPOrderProof (⇔)

    ↳47 QDP

    ↳48 PisEmptyProof (⇔)

    ↳49 TRUE

  ↳50 QDP

    ↳51 QDPOrderProof (⇔)

    ↳52 QDP

    ↳53 QDPOrderProof (⇔)

    ↳54 QDP

    ↳55 PisEmptyProof (⇔)

    ↳56 TRUE

  ↳57 QDP

    ↳58 QDPOrderProof (⇔)

    ↳59 QDP

    ↳60 QDPOrderProof (⇔)

    ↳61 QDP

    ↳62 PisEmptyProof (⇔)

    ↳63 TRUE

  ↳64 QDP

    ↳65 QDPOrderProof (⇔)

    ↳66 QDP

    ↳67 QDPOrderProof (⇔)

    ↳68 QDP

    ↳69 PisEmptyProof (⇔)

    ↳70 TRUE

  ↳71 QDP

    ↳72 QDPOrderProof (⇔)

    ↳73 QDP

    ↳74 QDPOrderProof (⇔)

    ↳75 QDP

    ↳76 PisEmptyProof (⇔)

    ↳77 TRUE

  ↳78 QDP

    ↳79 QDPOrderProof (⇔)

    ↳80 QDP

    ↳81 QDPOrderProof (⇔)

    ↳82 QDP

    ↳83 PisEmptyProof (⇔)

    ↳84 TRUE

  ↳85 QDP

    ↳86 QDPOrderProof (⇔)

    ↳87 QDP

    ↳88 QDPOrderProof (⇔)

    ↳89 QDP

    ↳90 PisEmptyProof (⇔)

    ↳91 TRUE

  ↳92 QDP

    ↳93 QDPOrderProof (⇔)

    ↳94 QDP

    ↳95 QDPOrderProof (⇔)

    ↳96 QDP

    ↳97 PisEmptyProof (⇔)

    ↳98 TRUE

  ↳99 QDP

    ↳100 QDPOrderProof (⇔)

    ↳101 QDP

    ↳102 QDPOrderProof (⇔)

    ↳103 QDP

    ↳104 QDPOrderProof (⇔)

    ↳105 QDP

    ↳106 QDPOrderProof (⇔)

    ↳107 QDP

    ↳108 QDPOrderProof (⇔)

    ↳109 QDP

    ↳110 PisEmptyProof (⇔)

    ↳111 TRUE

  ↳112 QDP

    ↳113 QDPOrderProof (⇔)

    ↳114 QDP

    ↳115 QDPOrderProof (⇔)

    ↳116 QDP

    ↳117 QDPOrderProof (⇔)

    ↳118 QDP

    ↳119 QDPOrderProof (⇔)

    ↳120 QDP

    ↳121 PisEmptyProof (⇔)

    ↳122 TRUE

  ↳123 QDP

(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

Q DP problem:
The TRS P consists of the following rules:

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)
S(mark(X)) → S(X)
CONS(mark(X1), X2) → CONS(X1, X2)
FIRST(mark(X1), X2) → FIRST(X1, X2)
FIRST(X1, mark(X2)) → FIRST(X1, X2)
FROM(mark(X)) → FROM(X)
SEL1(mark(X1), X2) → SEL1(X1, X2)
SEL1(X1, mark(X2)) → SEL1(X1, X2)
FIRST1(mark(X1), X2) → FIRST1(X1, X2)
FIRST1(X1, mark(X2)) → FIRST1(X1, X2)
CONS1(mark(X1), X2) → CONS1(X1, X2)
CONS1(X1, mark(X2)) → CONS1(X1, X2)
S1(mark(X)) → S1(X)
UNQUOTE(mark(X)) → UNQUOTE(X)
UNQUOTE1(mark(X)) → UNQUOTE1(X)
FCONS(mark(X1), X2) → FCONS(X1, X2)
FCONS(X1, mark(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)
SEL(ok(X1), ok(X2)) → SEL(X1, X2)
S(ok(X)) → S(X)
CONS(ok(X1), ok(X2)) → CONS(X1, X2)
FIRST(ok(X1), ok(X2)) → FIRST(X1, X2)
FROM(ok(X)) → FROM(X)
SEL1(ok(X1), ok(X2)) → SEL1(X1, X2)
QUOTE(ok(X)) → QUOTE(X)
FIRST1(ok(X1), ok(X2)) → FIRST1(X1, X2)
CONS1(ok(X1), ok(X2)) → CONS1(X1, X2)
QUOTE1(ok(X)) → QUOTE1(X)
S1(ok(X)) → S1(X)
UNQUOTE(ok(X)) → UNQUOTE(X)
UNQUOTE1(ok(X)) → UNQUOTE1(X)
FCONS(ok(X1), ok(X2)) → FCONS(X1, X2)
TOP(mark(X)) → TOP(proper(X))
TOP(mark(X)) → PROPER(X)
TOP(ok(X)) → TOP(active(X))
TOP(ok(X)) → ACTIVE(X)

The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 17 SCCs with 58 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

Q DP problem:
The TRS P consists of the following rules:

QUOTE1(ok(X)) → QUOTE1(X)

The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

QUOTE1(ok(X)) → QUOTE1(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
QUOTE1(x1) = QUOTE1(x1)
ok(x1) = ok(x1)
active(x1) = active(x1)
sel(x1, x2) = sel(x1)
s(x1) = s(x1)
cons(x1, x2) = cons(x1, x2)
mark(x1) = mark
0 = 0
first(x1, x2) = first(x1, x2)
nil = nil
from(x1) = from(x1)
sel1(x1, x2) = sel1(x2)
quote(x1) = quote(x1)
first1(x1, x2) = first1(x1)
nil1 = nil1
cons1(x1, x2) = cons1(x1)
01 = 01
quote1(x1) = quote1(x1)
s1(x1) = s1(x1)
unquote(x1) = x1
unquote1(x1) = x1
fcons(x1, x2) = fcons(x1)
proper(x1) = proper(x1)
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

QUOTE1₁ > [ok₁, sel₁, mark, 0, nil, sel1₁, quote₁, first1₁, nil1, cons1₁, 01, s1₁, fcons₁, top]
proper₁ > [active₁, s₁, cons₂, first₂, from₁, quote1₁] > [ok₁, sel₁, mark, 0, nil, sel1₁, quote₁, first1₁, nil1, cons1₁, 01, s1₁, fcons₁, top]

Status:

QUOTE1₁: [1]
ok₁: [1]
active₁: [1]
sel₁: [1]
s₁: [1]
cons₂: [2,1]
mark: []
0: []
first₂: [2,1]
nil: []
from₁: [1]
sel1₁: [1]
quote₁: [1]
first1₁: [1]
nil1: []
cons1₁: [1]
01: []
quote1₁: [1]
s1₁: [1]
fcons₁: [1]
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(7) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(8) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(9) TRUE

(10) Obligation:

Q DP problem:
The TRS P consists of the following rules:

QUOTE(ok(X)) → QUOTE(X)

The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

QUOTE(ok(X)) → QUOTE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
QUOTE(x1) = QUOTE(x1)
ok(x1) = ok(x1)
active(x1) = active(x1)
sel(x1, x2) = sel(x1)
s(x1) = s(x1)
cons(x1, x2) = cons(x1, x2)
mark(x1) = mark
0 = 0
first(x1, x2) = first(x1, x2)
nil = nil
from(x1) = from(x1)
sel1(x1, x2) = sel1(x2)
quote(x1) = quote(x1)
first1(x1, x2) = first1(x1)
nil1 = nil1
cons1(x1, x2) = cons1(x1)
01 = 01
quote1(x1) = quote1(x1)
s1(x1) = s1(x1)
unquote(x1) = x1
unquote1(x1) = x1
fcons(x1, x2) = fcons(x1)
proper(x1) = proper(x1)
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

QUOTE₁ > [ok₁, sel₁, mark, 0, nil, sel1₁, quote₁, first1₁, nil1, cons1₁, 01, s1₁, fcons₁, top]
proper₁ > [active₁, s₁, cons₂, first₂, from₁, quote1₁] > [ok₁, sel₁, mark, 0, nil, sel1₁, quote₁, first1₁, nil1, cons1₁, 01, s1₁, fcons₁, top]

Status:

QUOTE₁: [1]
ok₁: [1]
active₁: [1]
sel₁: [1]
s₁: [1]
cons₂: [2,1]
mark: []
0: []
first₂: [2,1]
nil: []
from₁: [1]
sel1₁: [1]
quote₁: [1]
first1₁: [1]
nil1: []
cons1₁: [1]
01: []
quote1₁: [1]
s1₁: [1]
fcons₁: [1]
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(12) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(13) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(14) TRUE

(15) Obligation:

Q DP problem:
The TRS P consists of the following rules:

FCONS(X1, mark(X2)) → FCONS(X1, X2)
FCONS(mark(X1), X2) → FCONS(X1, X2)
FCONS(ok(X1), ok(X2)) → FCONS(X1, X2)

The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(16) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FCONS(X1, mark(X2)) → FCONS(X1, X2)
FCONS(ok(X1), ok(X2)) → FCONS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
FCONS(x1, x2) = FCONS(x2)
mark(x1) = mark(x1)
ok(x1) = ok(x1)
active(x1) = active(x1)
sel(x1, x2) = sel(x1, x2)
s(x1) = x1
cons(x1, x2) = cons(x1, x2)
0 = 0
first(x1, x2) = first(x1, x2)
nil = nil
from(x1) = from(x1)
sel1(x1, x2) = sel1(x1, x2)
quote(x1) = x1
first1(x1, x2) = first1(x1, x2)
nil1 = nil1
cons1(x1, x2) = cons1(x1, x2)
01 = 01
quote1(x1) = quote1(x1)
s1(x1) = s1(x1)
unquote(x1) = unquote(x1)
unquote1(x1) = unquote1(x1)
fcons(x1, x2) = fcons(x1, x2)
proper(x1) = proper(x1)
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

FCONS₁ > [mark₁, ok₁, nil, nil1, s1₁, top]
proper₁ > [active₁, cons₂, from₁, 01, quote1₁, unquote1₁] > [sel₂, sel1₂] > [mark₁, ok₁, nil, nil1, s1₁, top]
proper₁ > [active₁, cons₂, from₁, 01, quote1₁, unquote1₁] > [first₂, first1₂, cons1₂, unquote₁, fcons₂] > 0 > [mark₁, ok₁, nil, nil1, s1₁, top]

Status:

FCONS₁: [1]
mark₁: [1]
ok₁: [1]
active₁: [1]
sel₂: [1,2]
cons₂: [1,2]
0: []
first₂: [1,2]
nil: []
from₁: [1]
sel1₂: [1,2]
first1₂: [2,1]
nil1: []
cons1₂: [2,1]
01: []
quote1₁: [1]
s1₁: [1]
unquote₁: [1]
unquote1₁: [1]
fcons₂: [2,1]
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(17) Obligation:

Q DP problem:
The TRS P consists of the following rules:

FCONS(mark(X1), X2) → FCONS(X1, X2)

The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(18) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FCONS(mark(X1), X2) → FCONS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
FCONS(x1, x2) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
sel(x1, x2) = sel(x1, x2)
s(x1) = s(x1)
cons(x1, x2) = cons(x1, x2)
0 = 0
first(x1, x2) = first(x1, x2)
nil = nil
from(x1) = from(x1)
sel1(x1, x2) = sel1(x1, x2)
quote(x1) = x1
first1(x1, x2) = first1(x1, x2)
nil1 = nil1
cons1(x1, x2) = cons1(x1, x2)
01 = 01
quote1(x1) = x1
s1(x1) = s1(x1)
unquote(x1) = x1
unquote1(x1) = x1
fcons(x1, x2) = fcons(x1, x2)
proper(x1) = x1
ok(x1) = x1
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

[active₁, sel₂, cons₂, first₂, sel1₂, first1₂, cons1₂] > 0 > [mark₁, s₁, nil, from₁, nil1, 01, s1₁, top]
[active₁, sel₂, cons₂, first₂, sel1₂, first1₂, cons1₂] > fcons₂ > [mark₁, s₁, nil, from₁, nil1, 01, s1₁, top]

Status:

mark₁: [1]
active₁: [1]
sel₂: [1,2]
s₁: [1]
cons₂: [1,2]
0: []
first₂: [1,2]
nil: []
from₁: [1]
sel1₂: [2,1]
first1₂: [1,2]
nil1: []
cons1₂: [2,1]
01: []
s1₁: [1]
fcons₂: [1,2]
top: []

The following usable rules [FROCOS05] were oriented:

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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(19) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(20) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(21) TRUE

(22) Obligation:

Q DP problem:
The TRS P consists of the following rules:

UNQUOTE1(ok(X)) → UNQUOTE1(X)
UNQUOTE1(mark(X)) → UNQUOTE1(X)

The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(23) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

UNQUOTE1(ok(X)) → UNQUOTE1(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
UNQUOTE1(x1) = UNQUOTE1(x1)
ok(x1) = ok(x1)
mark(x1) = x1
active(x1) = active(x1)
sel(x1, x2) = sel(x2)
s(x1) = s(x1)
cons(x1, x2) = cons(x1, x2)
0 = 0
first(x1, x2) = first(x2)
nil = nil
from(x1) = from(x1)
sel1(x1, x2) = sel1(x2)
quote(x1) = x1
first1(x1, x2) = first1(x2)
nil1 = nil1
cons1(x1, x2) = cons1(x1, x2)
01 = 01
quote1(x1) = quote1(x1)
s1(x1) = s1(x1)
unquote(x1) = unquote(x1)
unquote1(x1) = unquote1(x1)
fcons(x1, x2) = fcons(x1, x2)
proper(x1) = proper(x1)
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

UNQUOTE1₁ > [ok₁, sel₁, nil, sel1₁, first1₁, nil1, s1₁]
[0, 01, proper₁] > [active₁, unquote₁, fcons₂] > [s₁, cons₂, first₁, from₁, quote1₁] > [ok₁, sel₁, nil, sel1₁, first1₁, nil1, s1₁]
[0, 01, proper₁] > [active₁, unquote₁, fcons₂] > cons1₂ > unquote1₁ > [ok₁, sel₁, nil, sel1₁, first1₁, nil1, s1₁]
top > [active₁, unquote₁, fcons₂] > [s₁, cons₂, first₁, from₁, quote1₁] > [ok₁, sel₁, nil, sel1₁, first1₁, nil1, s1₁]
top > [active₁, unquote₁, fcons₂] > cons1₂ > unquote1₁ > [ok₁, sel₁, nil, sel1₁, first1₁, nil1, s1₁]

Status:

UNQUOTE1₁: [1]
ok₁: [1]
active₁: [1]
sel₁: [1]
s₁: [1]
cons₂: [1,2]
0: []
first₁: [1]
nil: []
from₁: [1]
sel1₁: [1]
first1₁: [1]
nil1: []
cons1₂: [1,2]
01: []
quote1₁: [1]
s1₁: [1]
unquote₁: [1]
unquote1₁: [1]
fcons₂: [2,1]
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(24) Obligation:

Q DP problem:
The TRS P consists of the following rules:

UNQUOTE1(mark(X)) → UNQUOTE1(X)

The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(25) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

UNQUOTE1(mark(X)) → UNQUOTE1(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
UNQUOTE1(x1) = UNQUOTE1(x1)
mark(x1) = mark(x1)
active(x1) = active(x1)
sel(x1, x2) = sel(x1, x2)
s(x1) = s(x1)
cons(x1, x2) = cons(x1, x2)
0 = 0
first(x1, x2) = first(x1, x2)
nil = nil
from(x1) = from(x1)
sel1(x1, x2) = sel1(x1, x2)
quote(x1) = x1
first1(x1, x2) = first1(x1, x2)
nil1 = nil1
cons1(x1, x2) = cons1(x1, x2)
01 = 01
quote1(x1) = quote1(x1)
s1(x1) = s1(x1)
unquote(x1) = x1
unquote1(x1) = x1
fcons(x1, x2) = fcons(x1, x2)
proper(x1) = x1
ok(x1) = x1
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

UNQUOTE1₁ > [mark₁, s₁, nil, nil1, 01, s1₁, top]
[active₁, cons₂, first₂] > [sel₂, sel1₂] > [mark₁, s₁, nil, nil1, 01, s1₁, top]
[active₁, cons₂, first₂] > 0 > [mark₁, s₁, nil, nil1, 01, s1₁, top]
[active₁, cons₂, first₂] > from₁ > [mark₁, s₁, nil, nil1, 01, s1₁, top]
[active₁, cons₂, first₂] > first1₂ > [mark₁, s₁, nil, nil1, 01, s1₁, top]
[active₁, cons₂, first₂] > [cons1₂, fcons₂] > [mark₁, s₁, nil, nil1, 01, s1₁, top]
[active₁, cons₂, first₂] > quote1₁ > [mark₁, s₁, nil, nil1, 01, s1₁, top]

Status:

UNQUOTE1₁: [1]
mark₁: [1]
active₁: [1]
sel₂: [1,2]
s₁: [1]
cons₂: [1,2]
0: []
first₂: [2,1]
nil: []
from₁: [1]
sel1₂: [2,1]
first1₂: [1,2]
nil1: []
cons1₂: [2,1]
01: []
quote1₁: [1]
s1₁: [1]
fcons₂: [2,1]
top: []

The following usable rules [FROCOS05] were oriented:

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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(26) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(27) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(28) TRUE

(29) Obligation:

Q DP problem:
The TRS P consists of the following rules:

UNQUOTE(ok(X)) → UNQUOTE(X)
UNQUOTE(mark(X)) → UNQUOTE(X)

The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(30) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

UNQUOTE(ok(X)) → UNQUOTE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
UNQUOTE(x1) = UNQUOTE(x1)
ok(x1) = ok(x1)
mark(x1) = x1
active(x1) = active(x1)
sel(x1, x2) = sel(x2)
s(x1) = s(x1)
cons(x1, x2) = cons(x1, x2)
0 = 0
first(x1, x2) = first(x2)
nil = nil
from(x1) = from(x1)
sel1(x1, x2) = sel1(x2)
quote(x1) = x1
first1(x1, x2) = first1(x2)
nil1 = nil1
cons1(x1, x2) = cons1(x1, x2)
01 = 01
quote1(x1) = quote1(x1)
s1(x1) = s1(x1)
unquote(x1) = unquote(x1)
unquote1(x1) = unquote1(x1)
fcons(x1, x2) = fcons(x1, x2)
proper(x1) = proper(x1)
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

UNQUOTE₁ > [ok₁, sel₁, nil, sel1₁, first1₁, nil1, s1₁]
[0, 01, proper₁] > [active₁, unquote₁, fcons₂] > [s₁, cons₂, first₁, from₁, quote1₁] > [ok₁, sel₁, nil, sel1₁, first1₁, nil1, s1₁]
[0, 01, proper₁] > [active₁, unquote₁, fcons₂] > cons1₂ > unquote1₁ > [ok₁, sel₁, nil, sel1₁, first1₁, nil1, s1₁]
top > [active₁, unquote₁, fcons₂] > [s₁, cons₂, first₁, from₁, quote1₁] > [ok₁, sel₁, nil, sel1₁, first1₁, nil1, s1₁]
top > [active₁, unquote₁, fcons₂] > cons1₂ > unquote1₁ > [ok₁, sel₁, nil, sel1₁, first1₁, nil1, s1₁]

Status:

UNQUOTE₁: [1]
ok₁: [1]
active₁: [1]
sel₁: [1]
s₁: [1]
cons₂: [1,2]
0: []
first₁: [1]
nil: []
from₁: [1]
sel1₁: [1]
first1₁: [1]
nil1: []
cons1₂: [1,2]
01: []
quote1₁: [1]
s1₁: [1]
unquote₁: [1]
unquote1₁: [1]
fcons₂: [2,1]
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(31) Obligation:

Q DP problem:
The TRS P consists of the following rules:

UNQUOTE(mark(X)) → UNQUOTE(X)

The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(32) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

UNQUOTE(mark(X)) → UNQUOTE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
UNQUOTE(x1) = UNQUOTE(x1)
mark(x1) = mark(x1)
active(x1) = active(x1)
sel(x1, x2) = sel(x1, x2)
s(x1) = s(x1)
cons(x1, x2) = cons(x1, x2)
0 = 0
first(x1, x2) = first(x1, x2)
nil = nil
from(x1) = from(x1)
sel1(x1, x2) = sel1(x1, x2)
quote(x1) = x1
first1(x1, x2) = first1(x1, x2)
nil1 = nil1
cons1(x1, x2) = cons1(x1, x2)
01 = 01
quote1(x1) = quote1(x1)
s1(x1) = s1(x1)
unquote(x1) = x1
unquote1(x1) = x1
fcons(x1, x2) = fcons(x1, x2)
proper(x1) = x1
ok(x1) = x1
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

UNQUOTE₁ > [mark₁, s₁, nil, nil1, 01, s1₁, top]
[active₁, cons₂, first₂] > [sel₂, sel1₂] > [mark₁, s₁, nil, nil1, 01, s1₁, top]
[active₁, cons₂, first₂] > 0 > [mark₁, s₁, nil, nil1, 01, s1₁, top]
[active₁, cons₂, first₂] > from₁ > [mark₁, s₁, nil, nil1, 01, s1₁, top]
[active₁, cons₂, first₂] > first1₂ > [mark₁, s₁, nil, nil1, 01, s1₁, top]
[active₁, cons₂, first₂] > [cons1₂, fcons₂] > [mark₁, s₁, nil, nil1, 01, s1₁, top]
[active₁, cons₂, first₂] > quote1₁ > [mark₁, s₁, nil, nil1, 01, s1₁, top]

Status:

UNQUOTE₁: [1]
mark₁: [1]
active₁: [1]
sel₂: [1,2]
s₁: [1]
cons₂: [1,2]
0: []
first₂: [2,1]
nil: []
from₁: [1]
sel1₂: [2,1]
first1₂: [1,2]
nil1: []
cons1₂: [2,1]
01: []
quote1₁: [1]
s1₁: [1]
fcons₂: [2,1]
top: []

The following usable rules [FROCOS05] were oriented:

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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(33) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(34) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(35) TRUE

(36) Obligation:

Q DP problem:
The TRS P consists of the following rules:

S1(ok(X)) → S1(X)
S1(mark(X)) → S1(X)

The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(37) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

S1(ok(X)) → S1(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
S1(x1) = S1(x1)
ok(x1) = ok(x1)
mark(x1) = x1
active(x1) = active(x1)
sel(x1, x2) = sel(x2)
s(x1) = s(x1)
cons(x1, x2) = cons(x1, x2)
0 = 0
first(x1, x2) = first(x2)
nil = nil
from(x1) = from(x1)
sel1(x1, x2) = sel1(x2)
quote(x1) = x1
first1(x1, x2) = first1(x2)
nil1 = nil1
cons1(x1, x2) = cons1(x1, x2)
01 = 01
quote1(x1) = quote1(x1)
s1(x1) = s1(x1)
unquote(x1) = unquote(x1)
unquote1(x1) = unquote1(x1)
fcons(x1, x2) = fcons(x1, x2)
proper(x1) = proper(x1)
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

S1₁ > [ok₁, sel₁, nil, sel1₁, first1₁, nil1, s1₁]
[0, 01, proper₁] > [active₁, unquote₁, fcons₂] > [s₁, cons₂, first₁, from₁, quote1₁] > [ok₁, sel₁, nil, sel1₁, first1₁, nil1, s1₁]
[0, 01, proper₁] > [active₁, unquote₁, fcons₂] > cons1₂ > unquote1₁ > [ok₁, sel₁, nil, sel1₁, first1₁, nil1, s1₁]
top > [active₁, unquote₁, fcons₂] > [s₁, cons₂, first₁, from₁, quote1₁] > [ok₁, sel₁, nil, sel1₁, first1₁, nil1, s1₁]
top > [active₁, unquote₁, fcons₂] > cons1₂ > unquote1₁ > [ok₁, sel₁, nil, sel1₁, first1₁, nil1, s1₁]

Status:

S1₁: [1]
ok₁: [1]
active₁: [1]
sel₁: [1]
s₁: [1]
cons₂: [1,2]
0: []
first₁: [1]
nil: []
from₁: [1]
sel1₁: [1]
first1₁: [1]
nil1: []
cons1₂: [1,2]
01: []
quote1₁: [1]
s1₁: [1]
unquote₁: [1]
unquote1₁: [1]
fcons₂: [2,1]
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(38) Obligation:

Q DP problem:
The TRS P consists of the following rules:

S1(mark(X)) → S1(X)

The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(39) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

S1(mark(X)) → S1(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
S1(x1) = S1(x1)
mark(x1) = mark(x1)
active(x1) = active(x1)
sel(x1, x2) = sel(x1, x2)
s(x1) = s(x1)
cons(x1, x2) = cons(x1, x2)
0 = 0
first(x1, x2) = first(x1, x2)
nil = nil
from(x1) = from(x1)
sel1(x1, x2) = sel1(x1, x2)
quote(x1) = x1
first1(x1, x2) = first1(x1, x2)
nil1 = nil1
cons1(x1, x2) = cons1(x1, x2)
01 = 01
quote1(x1) = quote1(x1)
s1(x1) = s1(x1)
unquote(x1) = x1
unquote1(x1) = x1
fcons(x1, x2) = fcons(x1, x2)
proper(x1) = x1
ok(x1) = x1
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

S1₁ > [mark₁, s₁, nil, nil1, 01, s1₁, top]
[active₁, cons₂, first₂] > [sel₂, sel1₂] > [mark₁, s₁, nil, nil1, 01, s1₁, top]
[active₁, cons₂, first₂] > 0 > [mark₁, s₁, nil, nil1, 01, s1₁, top]
[active₁, cons₂, first₂] > from₁ > [mark₁, s₁, nil, nil1, 01, s1₁, top]
[active₁, cons₂, first₂] > first1₂ > [mark₁, s₁, nil, nil1, 01, s1₁, top]
[active₁, cons₂, first₂] > [cons1₂, fcons₂] > [mark₁, s₁, nil, nil1, 01, s1₁, top]
[active₁, cons₂, first₂] > quote1₁ > [mark₁, s₁, nil, nil1, 01, s1₁, top]

Status:

S1₁: [1]
mark₁: [1]
active₁: [1]
sel₂: [1,2]
s₁: [1]
cons₂: [1,2]
0: []
first₂: [2,1]
nil: []
from₁: [1]
sel1₂: [2,1]
first1₂: [1,2]
nil1: []
cons1₂: [2,1]
01: []
quote1₁: [1]
s1₁: [1]
fcons₂: [2,1]
top: []

The following usable rules [FROCOS05] were oriented:

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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(40) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(41) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(42) TRUE

(43) Obligation:

Q DP problem:
The TRS P consists of the following rules:

CONS1(X1, mark(X2)) → CONS1(X1, X2)
CONS1(mark(X1), X2) → CONS1(X1, X2)
CONS1(ok(X1), ok(X2)) → CONS1(X1, X2)

The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(44) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS1(X1, mark(X2)) → CONS1(X1, X2)
CONS1(ok(X1), ok(X2)) → CONS1(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS1(x1, x2) = CONS1(x2)
mark(x1) = mark(x1)
ok(x1) = ok(x1)
active(x1) = active(x1)
sel(x1, x2) = sel(x1, x2)
s(x1) = x1
cons(x1, x2) = cons(x1, x2)
0 = 0
first(x1, x2) = first(x1, x2)
nil = nil
from(x1) = from(x1)
sel1(x1, x2) = sel1(x1, x2)
quote(x1) = x1
first1(x1, x2) = first1(x1, x2)
nil1 = nil1
cons1(x1, x2) = cons1(x1, x2)
01 = 01
quote1(x1) = quote1(x1)
s1(x1) = s1(x1)
unquote(x1) = unquote(x1)
unquote1(x1) = unquote1(x1)
fcons(x1, x2) = fcons(x1, x2)
proper(x1) = proper(x1)
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

CONS1₁ > [mark₁, ok₁, nil, nil1, s1₁, top]
proper₁ > [active₁, cons₂, from₁, 01, quote1₁, unquote1₁] > [sel₂, sel1₂] > [mark₁, ok₁, nil, nil1, s1₁, top]
proper₁ > [active₁, cons₂, from₁, 01, quote1₁, unquote1₁] > [first₂, first1₂, cons1₂, unquote₁, fcons₂] > 0 > [mark₁, ok₁, nil, nil1, s1₁, top]

Status:

CONS1₁: [1]
mark₁: [1]
ok₁: [1]
active₁: [1]
sel₂: [1,2]
cons₂: [1,2]
0: []
first₂: [1,2]
nil: []
from₁: [1]
sel1₂: [1,2]
first1₂: [2,1]
nil1: []
cons1₂: [2,1]
01: []
quote1₁: [1]
s1₁: [1]
unquote₁: [1]
unquote1₁: [1]
fcons₂: [2,1]
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(45) Obligation:

Q DP problem:
The TRS P consists of the following rules:

CONS1(mark(X1), X2) → CONS1(X1, X2)

The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(46) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS1(mark(X1), X2) → CONS1(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS1(x1, x2) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
sel(x1, x2) = sel(x1, x2)
s(x1) = s(x1)
cons(x1, x2) = cons(x1, x2)
0 = 0
first(x1, x2) = first(x1, x2)
nil = nil
from(x1) = from(x1)
sel1(x1, x2) = sel1(x1, x2)
quote(x1) = x1
first1(x1, x2) = first1(x1, x2)
nil1 = nil1
cons1(x1, x2) = cons1(x1, x2)
01 = 01
quote1(x1) = x1
s1(x1) = s1(x1)
unquote(x1) = x1
unquote1(x1) = x1
fcons(x1, x2) = fcons(x1, x2)
proper(x1) = x1
ok(x1) = x1
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

[active₁, sel₂, cons₂, first₂, sel1₂, first1₂, cons1₂] > 0 > [mark₁, s₁, nil, from₁, nil1, 01, s1₁, top]
[active₁, sel₂, cons₂, first₂, sel1₂, first1₂, cons1₂] > fcons₂ > [mark₁, s₁, nil, from₁, nil1, 01, s1₁, top]

Status:

mark₁: [1]
active₁: [1]
sel₂: [1,2]
s₁: [1]
cons₂: [1,2]
0: []
first₂: [1,2]
nil: []
from₁: [1]
sel1₂: [2,1]
first1₂: [1,2]
nil1: []
cons1₂: [2,1]
01: []
s1₁: [1]
fcons₂: [1,2]
top: []

The following usable rules [FROCOS05] were oriented:

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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(47) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(48) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(49) TRUE

(50) Obligation:

Q DP problem:
The TRS P consists of the following rules:

FIRST1(X1, mark(X2)) → FIRST1(X1, X2)
FIRST1(mark(X1), X2) → FIRST1(X1, X2)
FIRST1(ok(X1), ok(X2)) → FIRST1(X1, X2)

The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(51) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FIRST1(X1, mark(X2)) → FIRST1(X1, X2)
FIRST1(ok(X1), ok(X2)) → FIRST1(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
FIRST1(x1, x2) = FIRST1(x2)
mark(x1) = mark(x1)
ok(x1) = ok(x1)
active(x1) = active(x1)
sel(x1, x2) = sel(x1, x2)
s(x1) = x1
cons(x1, x2) = cons(x1, x2)
0 = 0
first(x1, x2) = first(x1, x2)
nil = nil
from(x1) = from(x1)
sel1(x1, x2) = sel1(x1, x2)
quote(x1) = x1
first1(x1, x2) = first1(x1, x2)
nil1 = nil1
cons1(x1, x2) = cons1(x1, x2)
01 = 01
quote1(x1) = quote1(x1)
s1(x1) = s1(x1)
unquote(x1) = unquote(x1)
unquote1(x1) = unquote1(x1)
fcons(x1, x2) = fcons(x1, x2)
proper(x1) = proper(x1)
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

FIRST1₁ > [mark₁, ok₁, nil, nil1, s1₁, top]
proper₁ > [active₁, cons₂, from₁, 01, quote1₁, unquote1₁] > [sel₂, sel1₂] > [mark₁, ok₁, nil, nil1, s1₁, top]
proper₁ > [active₁, cons₂, from₁, 01, quote1₁, unquote1₁] > [first₂, first1₂, cons1₂, unquote₁, fcons₂] > 0 > [mark₁, ok₁, nil, nil1, s1₁, top]

Status:

FIRST1₁: [1]
mark₁: [1]
ok₁: [1]
active₁: [1]
sel₂: [1,2]
cons₂: [1,2]
0: []
first₂: [1,2]
nil: []
from₁: [1]
sel1₂: [1,2]
first1₂: [2,1]
nil1: []
cons1₂: [2,1]
01: []
quote1₁: [1]
s1₁: [1]
unquote₁: [1]
unquote1₁: [1]
fcons₂: [2,1]
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(52) Obligation:

Q DP problem:
The TRS P consists of the following rules:

FIRST1(mark(X1), X2) → FIRST1(X1, X2)

The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(53) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FIRST1(mark(X1), X2) → FIRST1(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
FIRST1(x1, x2) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
sel(x1, x2) = sel(x1, x2)
s(x1) = s(x1)
cons(x1, x2) = cons(x1, x2)
0 = 0
first(x1, x2) = first(x1, x2)
nil = nil
from(x1) = from(x1)
sel1(x1, x2) = sel1(x1, x2)
quote(x1) = x1
first1(x1, x2) = first1(x1, x2)
nil1 = nil1
cons1(x1, x2) = cons1(x1, x2)
01 = 01
quote1(x1) = x1
s1(x1) = s1(x1)
unquote(x1) = x1
unquote1(x1) = x1
fcons(x1, x2) = fcons(x1, x2)
proper(x1) = x1
ok(x1) = x1
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

[active₁, sel₂, cons₂, first₂, sel1₂, first1₂, cons1₂] > 0 > [mark₁, s₁, nil, from₁, nil1, 01, s1₁, top]
[active₁, sel₂, cons₂, first₂, sel1₂, first1₂, cons1₂] > fcons₂ > [mark₁, s₁, nil, from₁, nil1, 01, s1₁, top]

Status:

mark₁: [1]
active₁: [1]
sel₂: [1,2]
s₁: [1]
cons₂: [1,2]
0: []
first₂: [1,2]
nil: []
from₁: [1]
sel1₂: [2,1]
first1₂: [1,2]
nil1: []
cons1₂: [2,1]
01: []
s1₁: [1]
fcons₂: [1,2]
top: []

The following usable rules [FROCOS05] were oriented:

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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(54) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(55) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(56) TRUE

(57) Obligation:

Q DP problem:
The TRS P consists of the following rules:

SEL1(X1, mark(X2)) → SEL1(X1, X2)
SEL1(mark(X1), X2) → SEL1(X1, X2)
SEL1(ok(X1), ok(X2)) → SEL1(X1, X2)

The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(58) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SEL1(X1, mark(X2)) → SEL1(X1, X2)
SEL1(ok(X1), ok(X2)) → SEL1(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
SEL1(x1, x2) = SEL1(x2)
mark(x1) = mark(x1)
ok(x1) = ok(x1)
active(x1) = active(x1)
sel(x1, x2) = sel(x1, x2)
s(x1) = x1
cons(x1, x2) = cons(x1, x2)
0 = 0
first(x1, x2) = first(x1, x2)
nil = nil
from(x1) = from(x1)
sel1(x1, x2) = sel1(x1, x2)
quote(x1) = x1
first1(x1, x2) = first1(x1, x2)
nil1 = nil1
cons1(x1, x2) = cons1(x1, x2)
01 = 01
quote1(x1) = quote1(x1)
s1(x1) = s1(x1)
unquote(x1) = unquote(x1)
unquote1(x1) = unquote1(x1)
fcons(x1, x2) = fcons(x1, x2)
proper(x1) = proper(x1)
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

SEL1₁ > [mark₁, ok₁, nil, nil1, s1₁, top]
proper₁ > [active₁, cons₂, from₁, 01, quote1₁, unquote1₁] > [sel₂, sel1₂] > [mark₁, ok₁, nil, nil1, s1₁, top]
proper₁ > [active₁, cons₂, from₁, 01, quote1₁, unquote1₁] > [first₂, first1₂, cons1₂, unquote₁, fcons₂] > 0 > [mark₁, ok₁, nil, nil1, s1₁, top]

Status:

SEL1₁: [1]
mark₁: [1]
ok₁: [1]
active₁: [1]
sel₂: [1,2]
cons₂: [1,2]
0: []
first₂: [1,2]
nil: []
from₁: [1]
sel1₂: [1,2]
first1₂: [2,1]
nil1: []
cons1₂: [2,1]
01: []
quote1₁: [1]
s1₁: [1]
unquote₁: [1]
unquote1₁: [1]
fcons₂: [2,1]
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(59) Obligation:

Q DP problem:
The TRS P consists of the following rules:

SEL1(mark(X1), X2) → SEL1(X1, X2)

The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(60) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SEL1(mark(X1), X2) → SEL1(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
SEL1(x1, x2) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
sel(x1, x2) = sel(x1, x2)
s(x1) = s(x1)
cons(x1, x2) = cons(x1, x2)
0 = 0
first(x1, x2) = first(x1, x2)
nil = nil
from(x1) = from(x1)
sel1(x1, x2) = sel1(x1, x2)
quote(x1) = x1
first1(x1, x2) = first1(x1, x2)
nil1 = nil1
cons1(x1, x2) = cons1(x1, x2)
01 = 01
quote1(x1) = x1
s1(x1) = s1(x1)
unquote(x1) = x1
unquote1(x1) = x1
fcons(x1, x2) = fcons(x1, x2)
proper(x1) = x1
ok(x1) = x1
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

[active₁, sel₂, cons₂, first₂, sel1₂, first1₂, cons1₂] > 0 > [mark₁, s₁, nil, from₁, nil1, 01, s1₁, top]
[active₁, sel₂, cons₂, first₂, sel1₂, first1₂, cons1₂] > fcons₂ > [mark₁, s₁, nil, from₁, nil1, 01, s1₁, top]

Status:

mark₁: [1]
active₁: [1]
sel₂: [1,2]
s₁: [1]
cons₂: [1,2]
0: []
first₂: [1,2]
nil: []
from₁: [1]
sel1₂: [2,1]
first1₂: [1,2]
nil1: []
cons1₂: [2,1]
01: []
s1₁: [1]
fcons₂: [1,2]
top: []

The following usable rules [FROCOS05] were oriented:

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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(61) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(62) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(63) TRUE

(64) Obligation:

Q DP problem:
The TRS P consists of the following rules:

FROM(ok(X)) → FROM(X)
FROM(mark(X)) → FROM(X)

The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(65) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FROM(ok(X)) → FROM(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
FROM(x1) = FROM(x1)
ok(x1) = ok(x1)
mark(x1) = x1
active(x1) = active(x1)
sel(x1, x2) = sel(x2)
s(x1) = s(x1)
cons(x1, x2) = cons(x1, x2)
0 = 0
first(x1, x2) = first(x2)
nil = nil
from(x1) = from(x1)
sel1(x1, x2) = sel1(x2)
quote(x1) = x1
first1(x1, x2) = first1(x2)
nil1 = nil1
cons1(x1, x2) = cons1(x1, x2)
01 = 01
quote1(x1) = quote1(x1)
s1(x1) = s1(x1)
unquote(x1) = unquote(x1)
unquote1(x1) = unquote1(x1)
fcons(x1, x2) = fcons(x1, x2)
proper(x1) = proper(x1)
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

FROM₁ > [ok₁, sel₁, nil, sel1₁, first1₁, nil1, s1₁]
[0, 01, proper₁] > [active₁, unquote₁, fcons₂] > [s₁, cons₂, first₁, from₁, quote1₁] > [ok₁, sel₁, nil, sel1₁, first1₁, nil1, s1₁]
[0, 01, proper₁] > [active₁, unquote₁, fcons₂] > cons1₂ > unquote1₁ > [ok₁, sel₁, nil, sel1₁, first1₁, nil1, s1₁]
top > [active₁, unquote₁, fcons₂] > [s₁, cons₂, first₁, from₁, quote1₁] > [ok₁, sel₁, nil, sel1₁, first1₁, nil1, s1₁]
top > [active₁, unquote₁, fcons₂] > cons1₂ > unquote1₁ > [ok₁, sel₁, nil, sel1₁, first1₁, nil1, s1₁]

Status:

FROM₁: [1]
ok₁: [1]
active₁: [1]
sel₁: [1]
s₁: [1]
cons₂: [1,2]
0: []
first₁: [1]
nil: []
from₁: [1]
sel1₁: [1]
first1₁: [1]
nil1: []
cons1₂: [1,2]
01: []
quote1₁: [1]
s1₁: [1]
unquote₁: [1]
unquote1₁: [1]
fcons₂: [2,1]
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(66) Obligation:

Q DP problem:
The TRS P consists of the following rules:

FROM(mark(X)) → FROM(X)

The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(67) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FROM(mark(X)) → FROM(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
FROM(x1) = FROM(x1)
mark(x1) = mark(x1)
active(x1) = active(x1)
sel(x1, x2) = sel(x1, x2)
s(x1) = s(x1)
cons(x1, x2) = cons(x1, x2)
0 = 0
first(x1, x2) = first(x1, x2)
nil = nil
from(x1) = from(x1)
sel1(x1, x2) = sel1(x1, x2)
quote(x1) = x1
first1(x1, x2) = first1(x1, x2)
nil1 = nil1
cons1(x1, x2) = cons1(x1, x2)
01 = 01
quote1(x1) = quote1(x1)
s1(x1) = s1(x1)
unquote(x1) = x1
unquote1(x1) = x1
fcons(x1, x2) = fcons(x1, x2)
proper(x1) = x1
ok(x1) = x1
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

FROM₁ > [mark₁, s₁, nil, nil1, 01, s1₁, top]
[active₁, cons₂, first₂] > [sel₂, sel1₂] > [mark₁, s₁, nil, nil1, 01, s1₁, top]
[active₁, cons₂, first₂] > 0 > [mark₁, s₁, nil, nil1, 01, s1₁, top]
[active₁, cons₂, first₂] > from₁ > [mark₁, s₁, nil, nil1, 01, s1₁, top]
[active₁, cons₂, first₂] > first1₂ > [mark₁, s₁, nil, nil1, 01, s1₁, top]
[active₁, cons₂, first₂] > [cons1₂, fcons₂] > [mark₁, s₁, nil, nil1, 01, s1₁, top]
[active₁, cons₂, first₂] > quote1₁ > [mark₁, s₁, nil, nil1, 01, s1₁, top]

Status:

FROM₁: [1]
mark₁: [1]
active₁: [1]
sel₂: [1,2]
s₁: [1]
cons₂: [1,2]
0: []
first₂: [2,1]
nil: []
from₁: [1]
sel1₂: [2,1]
first1₂: [1,2]
nil1: []
cons1₂: [2,1]
01: []
quote1₁: [1]
s1₁: [1]
fcons₂: [2,1]
top: []

The following usable rules [FROCOS05] were oriented:

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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(68) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(69) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(70) TRUE

(71) Obligation:

Q DP problem:
The TRS P consists of the following rules:

FIRST(X1, mark(X2)) → FIRST(X1, X2)
FIRST(mark(X1), X2) → FIRST(X1, X2)
FIRST(ok(X1), ok(X2)) → FIRST(X1, X2)

The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(72) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FIRST(X1, mark(X2)) → FIRST(X1, X2)
FIRST(ok(X1), ok(X2)) → FIRST(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
FIRST(x1, x2) = FIRST(x2)
mark(x1) = mark(x1)
ok(x1) = ok(x1)
active(x1) = active(x1)
sel(x1, x2) = sel(x1, x2)
s(x1) = x1
cons(x1, x2) = cons(x1, x2)
0 = 0
first(x1, x2) = first(x1, x2)
nil = nil
from(x1) = from(x1)
sel1(x1, x2) = sel1(x1, x2)
quote(x1) = x1
first1(x1, x2) = first1(x1, x2)
nil1 = nil1
cons1(x1, x2) = cons1(x1, x2)
01 = 01
quote1(x1) = quote1(x1)
s1(x1) = s1(x1)
unquote(x1) = unquote(x1)
unquote1(x1) = unquote1(x1)
fcons(x1, x2) = fcons(x1, x2)
proper(x1) = proper(x1)
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

FIRST₁ > [mark₁, ok₁, nil, nil1, s1₁, top]
proper₁ > [active₁, cons₂, from₁, 01, quote1₁, unquote1₁] > [sel₂, sel1₂] > [mark₁, ok₁, nil, nil1, s1₁, top]
proper₁ > [active₁, cons₂, from₁, 01, quote1₁, unquote1₁] > [first₂, first1₂, cons1₂, unquote₁, fcons₂] > 0 > [mark₁, ok₁, nil, nil1, s1₁, top]

Status:

FIRST₁: [1]
mark₁: [1]
ok₁: [1]
active₁: [1]
sel₂: [1,2]
cons₂: [1,2]
0: []
first₂: [1,2]
nil: []
from₁: [1]
sel1₂: [1,2]
first1₂: [2,1]
nil1: []
cons1₂: [2,1]
01: []
quote1₁: [1]
s1₁: [1]
unquote₁: [1]
unquote1₁: [1]
fcons₂: [2,1]
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(73) Obligation:

Q DP problem:
The TRS P consists of the following rules:

FIRST(mark(X1), X2) → FIRST(X1, X2)

The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(74) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FIRST(mark(X1), X2) → FIRST(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
FIRST(x1, x2) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
sel(x1, x2) = sel(x1, x2)
s(x1) = s(x1)
cons(x1, x2) = cons(x1, x2)
0 = 0
first(x1, x2) = first(x1, x2)
nil = nil
from(x1) = from(x1)
sel1(x1, x2) = sel1(x1, x2)
quote(x1) = x1
first1(x1, x2) = first1(x1, x2)
nil1 = nil1
cons1(x1, x2) = cons1(x1, x2)
01 = 01
quote1(x1) = x1
s1(x1) = s1(x1)
unquote(x1) = x1
unquote1(x1) = x1
fcons(x1, x2) = fcons(x1, x2)
proper(x1) = x1
ok(x1) = x1
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

[active₁, sel₂, cons₂, first₂, sel1₂, first1₂, cons1₂] > 0 > [mark₁, s₁, nil, from₁, nil1, 01, s1₁, top]
[active₁, sel₂, cons₂, first₂, sel1₂, first1₂, cons1₂] > fcons₂ > [mark₁, s₁, nil, from₁, nil1, 01, s1₁, top]

Status:

mark₁: [1]
active₁: [1]
sel₂: [1,2]
s₁: [1]
cons₂: [1,2]
0: []
first₂: [1,2]
nil: []
from₁: [1]
sel1₂: [2,1]
first1₂: [1,2]
nil1: []
cons1₂: [2,1]
01: []
s1₁: [1]
fcons₂: [1,2]
top: []

The following usable rules [FROCOS05] were oriented:

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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(75) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(76) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(77) TRUE

(78) Obligation:

Q DP problem:
The TRS P consists of the following rules:

CONS(ok(X1), ok(X2)) → CONS(X1, X2)
CONS(mark(X1), X2) → CONS(X1, X2)

The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(79) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(ok(X1), ok(X2)) → CONS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS(x1, x2) = CONS(x2)
ok(x1) = ok(x1)
mark(x1) = mark
active(x1) = active(x1)
sel(x1, x2) = sel(x1)
s(x1) = s(x1)
cons(x1, x2) = cons(x1, x2)
0 = 0
first(x1, x2) = first(x2)
nil = nil
from(x1) = from(x1)
sel1(x1, x2) = sel1(x2)
quote(x1) = quote(x1)
first1(x1, x2) = first1(x2)
nil1 = nil1
cons1(x1, x2) = cons1(x2)
01 = 01
quote1(x1) = quote1(x1)
s1(x1) = s1(x1)
unquote(x1) = unquote(x1)
unquote1(x1) = unquote1(x1)
fcons(x1, x2) = fcons(x1, x2)
proper(x1) = proper(x1)
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

active₁ > [mark, sel₁, first₁, sel1₁, first1₁, cons1₁, 01] > CONS₁ > [ok₁, nil, quote₁, nil1, s1₁, unquote₁, top]
active₁ > [mark, sel₁, first₁, sel1₁, first1₁, cons1₁, 01] > 0 > [ok₁, nil, quote₁, nil1, s1₁, unquote₁, top]
active₁ > [mark, sel₁, first₁, sel1₁, first1₁, cons1₁, 01] > unquote1₁ > [ok₁, nil, quote₁, nil1, s1₁, unquote₁, top]
active₁ > [mark, sel₁, first₁, sel1₁, first1₁, cons1₁, 01] > fcons₂ > [s₁, cons₂, from₁, quote1₁] > [ok₁, nil, quote₁, nil1, s1₁, unquote₁, top]
proper₁ > [mark, sel₁, first₁, sel1₁, first1₁, cons1₁, 01] > CONS₁ > [ok₁, nil, quote₁, nil1, s1₁, unquote₁, top]
proper₁ > [mark, sel₁, first₁, sel1₁, first1₁, cons1₁, 01] > 0 > [ok₁, nil, quote₁, nil1, s1₁, unquote₁, top]
proper₁ > [mark, sel₁, first₁, sel1₁, first1₁, cons1₁, 01] > unquote1₁ > [ok₁, nil, quote₁, nil1, s1₁, unquote₁, top]
proper₁ > [mark, sel₁, first₁, sel1₁, first1₁, cons1₁, 01] > fcons₂ > [s₁, cons₂, from₁, quote1₁] > [ok₁, nil, quote₁, nil1, s1₁, unquote₁, top]

Status:

CONS₁: [1]
ok₁: [1]
mark: []
active₁: [1]
sel₁: [1]
s₁: [1]
cons₂: [2,1]
0: []
first₁: [1]
nil: []
from₁: [1]
sel1₁: [1]
quote₁: [1]
first1₁: [1]
nil1: []
cons1₁: [1]
01: []
quote1₁: [1]
s1₁: [1]
unquote₁: [1]
unquote1₁: [1]
fcons₂: [2,1]
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(80) Obligation:

Q DP problem:
The TRS P consists of the following rules:

CONS(mark(X1), X2) → CONS(X1, X2)

The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(81) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(mark(X1), X2) → CONS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS(x1, x2) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
sel(x1, x2) = sel(x1, x2)
s(x1) = s(x1)
cons(x1, x2) = cons(x1, x2)
0 = 0
first(x1, x2) = first(x1, x2)
nil = nil
from(x1) = from(x1)
sel1(x1, x2) = sel1(x1, x2)
quote(x1) = x1
first1(x1, x2) = first1(x1, x2)
nil1 = nil1
cons1(x1, x2) = cons1(x1, x2)
01 = 01
quote1(x1) = x1
s1(x1) = s1(x1)
unquote(x1) = x1
unquote1(x1) = x1
fcons(x1, x2) = fcons(x1, x2)
proper(x1) = x1
ok(x1) = x1
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

[active₁, sel₂, cons₂, first₂, sel1₂, first1₂, cons1₂] > 0 > [mark₁, s₁, nil, from₁, nil1, 01, s1₁, top]
[active₁, sel₂, cons₂, first₂, sel1₂, first1₂, cons1₂] > fcons₂ > [mark₁, s₁, nil, from₁, nil1, 01, s1₁, top]

Status:

mark₁: [1]
active₁: [1]
sel₂: [1,2]
s₁: [1]
cons₂: [1,2]
0: []
first₂: [1,2]
nil: []
from₁: [1]
sel1₂: [2,1]
first1₂: [1,2]
nil1: []
cons1₂: [2,1]
01: []
s1₁: [1]
fcons₂: [1,2]
top: []

The following usable rules [FROCOS05] were oriented:

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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(82) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(83) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(84) TRUE

(85) Obligation:

Q DP problem:
The TRS P consists of the following rules:

S(ok(X)) → S(X)
S(mark(X)) → S(X)

The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(86) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

S(ok(X)) → S(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
S(x1) = S(x1)
ok(x1) = ok(x1)
mark(x1) = x1
active(x1) = active(x1)
sel(x1, x2) = sel(x2)
s(x1) = s(x1)
cons(x1, x2) = cons(x1, x2)
0 = 0
first(x1, x2) = first(x2)
nil = nil
from(x1) = from(x1)
sel1(x1, x2) = sel1(x2)
quote(x1) = x1
first1(x1, x2) = first1(x2)
nil1 = nil1
cons1(x1, x2) = cons1(x1, x2)
01 = 01
quote1(x1) = quote1(x1)
s1(x1) = s1(x1)
unquote(x1) = unquote(x1)
unquote1(x1) = unquote1(x1)
fcons(x1, x2) = fcons(x1, x2)
proper(x1) = proper(x1)
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

S₁ > [ok₁, sel₁, nil, sel1₁, first1₁, nil1, s1₁]
[0, 01, proper₁] > [active₁, unquote₁, fcons₂] > [s₁, cons₂, first₁, from₁, quote1₁] > [ok₁, sel₁, nil, sel1₁, first1₁, nil1, s1₁]
[0, 01, proper₁] > [active₁, unquote₁, fcons₂] > cons1₂ > unquote1₁ > [ok₁, sel₁, nil, sel1₁, first1₁, nil1, s1₁]
top > [active₁, unquote₁, fcons₂] > [s₁, cons₂, first₁, from₁, quote1₁] > [ok₁, sel₁, nil, sel1₁, first1₁, nil1, s1₁]
top > [active₁, unquote₁, fcons₂] > cons1₂ > unquote1₁ > [ok₁, sel₁, nil, sel1₁, first1₁, nil1, s1₁]

Status:

S₁: [1]
ok₁: [1]
active₁: [1]
sel₁: [1]
s₁: [1]
cons₂: [1,2]
0: []
first₁: [1]
nil: []
from₁: [1]
sel1₁: [1]
first1₁: [1]
nil1: []
cons1₂: [1,2]
01: []
quote1₁: [1]
s1₁: [1]
unquote₁: [1]
unquote1₁: [1]
fcons₂: [2,1]
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(87) Obligation:

Q DP problem:
The TRS P consists of the following rules:

S(mark(X)) → S(X)

The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(88) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

S(mark(X)) → S(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
S(x1) = S(x1)
mark(x1) = mark(x1)
active(x1) = active(x1)
sel(x1, x2) = sel(x1, x2)
s(x1) = s(x1)
cons(x1, x2) = cons(x1, x2)
0 = 0
first(x1, x2) = first(x1, x2)
nil = nil
from(x1) = from(x1)
sel1(x1, x2) = sel1(x1, x2)
quote(x1) = x1
first1(x1, x2) = first1(x1, x2)
nil1 = nil1
cons1(x1, x2) = cons1(x1, x2)
01 = 01
quote1(x1) = quote1(x1)
s1(x1) = s1(x1)
unquote(x1) = x1
unquote1(x1) = x1
fcons(x1, x2) = fcons(x1, x2)
proper(x1) = x1
ok(x1) = x1
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

S₁ > [mark₁, s₁, nil, nil1, 01, s1₁, top]
[active₁, cons₂, first₂] > [sel₂, sel1₂] > [mark₁, s₁, nil, nil1, 01, s1₁, top]
[active₁, cons₂, first₂] > 0 > [mark₁, s₁, nil, nil1, 01, s1₁, top]
[active₁, cons₂, first₂] > from₁ > [mark₁, s₁, nil, nil1, 01, s1₁, top]
[active₁, cons₂, first₂] > first1₂ > [mark₁, s₁, nil, nil1, 01, s1₁, top]
[active₁, cons₂, first₂] > [cons1₂, fcons₂] > [mark₁, s₁, nil, nil1, 01, s1₁, top]
[active₁, cons₂, first₂] > quote1₁ > [mark₁, s₁, nil, nil1, 01, s1₁, top]

Status:

S₁: [1]
mark₁: [1]
active₁: [1]
sel₂: [1,2]
s₁: [1]
cons₂: [1,2]
0: []
first₂: [2,1]
nil: []
from₁: [1]
sel1₂: [2,1]
first1₂: [1,2]
nil1: []
cons1₂: [2,1]
01: []
quote1₁: [1]
s1₁: [1]
fcons₂: [2,1]
top: []

The following usable rules [FROCOS05] were oriented:

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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(89) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(90) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(91) TRUE

(92) Obligation:

Q DP problem:
The TRS P consists of the following rules:

SEL(X1, mark(X2)) → SEL(X1, X2)
SEL(mark(X1), X2) → SEL(X1, X2)
SEL(ok(X1), ok(X2)) → SEL(X1, X2)

The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(93) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SEL(X1, mark(X2)) → SEL(X1, X2)
SEL(ok(X1), ok(X2)) → SEL(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
SEL(x1, x2) = SEL(x2)
mark(x1) = mark(x1)
ok(x1) = ok(x1)
active(x1) = active(x1)
sel(x1, x2) = sel(x1, x2)
s(x1) = x1
cons(x1, x2) = cons(x1, x2)
0 = 0
first(x1, x2) = first(x1, x2)
nil = nil
from(x1) = from(x1)
sel1(x1, x2) = sel1(x1, x2)
quote(x1) = x1
first1(x1, x2) = first1(x1, x2)
nil1 = nil1
cons1(x1, x2) = cons1(x1, x2)
01 = 01
quote1(x1) = quote1(x1)
s1(x1) = s1(x1)
unquote(x1) = unquote(x1)
unquote1(x1) = unquote1(x1)
fcons(x1, x2) = fcons(x1, x2)
proper(x1) = proper(x1)
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

SEL₁ > [mark₁, ok₁, nil, nil1, s1₁, top]
proper₁ > [active₁, cons₂, from₁, 01, quote1₁, unquote1₁] > [sel₂, sel1₂] > [mark₁, ok₁, nil, nil1, s1₁, top]
proper₁ > [active₁, cons₂, from₁, 01, quote1₁, unquote1₁] > [first₂, first1₂, cons1₂, unquote₁, fcons₂] > 0 > [mark₁, ok₁, nil, nil1, s1₁, top]

Status:

SEL₁: [1]
mark₁: [1]
ok₁: [1]
active₁: [1]
sel₂: [1,2]
cons₂: [1,2]
0: []
first₂: [1,2]
nil: []
from₁: [1]
sel1₂: [1,2]
first1₂: [2,1]
nil1: []
cons1₂: [2,1]
01: []
quote1₁: [1]
s1₁: [1]
unquote₁: [1]
unquote1₁: [1]
fcons₂: [2,1]
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(94) Obligation:

Q DP problem:
The TRS P consists of the following rules:

SEL(mark(X1), X2) → SEL(X1, X2)

The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(95) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SEL(mark(X1), X2) → SEL(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
SEL(x1, x2) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
sel(x1, x2) = sel(x1, x2)
s(x1) = s(x1)
cons(x1, x2) = cons(x1, x2)
0 = 0
first(x1, x2) = first(x1, x2)
nil = nil
from(x1) = from(x1)
sel1(x1, x2) = sel1(x1, x2)
quote(x1) = x1
first1(x1, x2) = first1(x1, x2)
nil1 = nil1
cons1(x1, x2) = cons1(x1, x2)
01 = 01
quote1(x1) = x1
s1(x1) = s1(x1)
unquote(x1) = x1
unquote1(x1) = x1
fcons(x1, x2) = fcons(x1, x2)
proper(x1) = x1
ok(x1) = x1
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

[active₁, sel₂, cons₂, first₂, sel1₂, first1₂, cons1₂] > 0 > [mark₁, s₁, nil, from₁, nil1, 01, s1₁, top]
[active₁, sel₂, cons₂, first₂, sel1₂, first1₂, cons1₂] > fcons₂ > [mark₁, s₁, nil, from₁, nil1, 01, s1₁, top]

Status:

mark₁: [1]
active₁: [1]
sel₂: [1,2]
s₁: [1]
cons₂: [1,2]
0: []
first₂: [1,2]
nil: []
from₁: [1]
sel1₂: [2,1]
first1₂: [1,2]
nil1: []
cons1₂: [2,1]
01: []
s1₁: [1]
fcons₂: [1,2]
top: []

The following usable rules [FROCOS05] were oriented:

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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(96) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(97) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(98) TRUE

(99) Obligation:

Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(100) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(sel(X1, X2)) → PROPER(X2)
PROPER(sel(X1, X2)) → PROPER(X1)
PROPER(s(X)) → PROPER(X)
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(first(X1, X2)) → PROPER(X1)
PROPER(first(X1, X2)) → PROPER(X2)
PROPER(from(X)) → PROPER(X)
PROPER(sel1(X1, X2)) → PROPER(X1)
PROPER(sel1(X1, X2)) → PROPER(X2)
PROPER(first1(X1, X2)) → PROPER(X1)
PROPER(first1(X1, X2)) → PROPER(X2)
PROPER(cons1(X1, X2)) → PROPER(X1)
PROPER(cons1(X1, X2)) → PROPER(X2)
PROPER(s1(X)) → PROPER(X)
PROPER(fcons(X1, X2)) → PROPER(X1)
PROPER(fcons(X1, X2)) → PROPER(X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1) = x1
sel(x1, x2) = sel(x1, x2)
s(x1) = s(x1)
cons(x1, x2) = cons(x1, x2)
first(x1, x2) = first(x1, x2)
from(x1) = from(x1)
sel1(x1, x2) = sel1(x1, x2)
quote(x1) = x1
first1(x1, x2) = first1(x1, x2)
cons1(x1, x2) = cons1(x1, x2)
quote1(x1) = x1
s1(x1) = s1(x1)
unquote(x1) = x1
unquote1(x1) = x1
fcons(x1, x2) = fcons(x1, x2)
active(x1) = active(x1)
mark(x1) = mark
0 = 0
nil = nil
nil1 = nil1
01 = 01
proper(x1) = x1
ok(x1) = ok
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

active₁ > mark > [sel₂, cons₂, first₂, from₁, first1₂, cons1₂] > [s₁, sel1₂, s1₁, fcons₂, nil, nil1, 01, ok, top]
active₁ > 0 > [s₁, sel1₂, s1₁, fcons₂, nil, nil1, 01, ok, top]

Status:

sel₂: [2,1]
s₁: [1]
cons₂: [1,2]
first₂: [2,1]
from₁: [1]
sel1₂: [1,2]
first1₂: [2,1]
cons1₂: [2,1]
s1₁: [1]
fcons₂: [2,1]
active₁: [1]
mark: []
0: []
nil: []
nil1: []
01: []
ok: []
top: []

The following usable rules [FROCOS05] were oriented:

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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(101) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PROPER(quote(X)) → PROPER(X)
PROPER(quote1(X)) → PROPER(X)
PROPER(unquote(X)) → PROPER(X)
PROPER(unquote1(X)) → PROPER(X)

The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(102) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(unquote1(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1) = PROPER(x1)
quote(x1) = x1
quote1(x1) = x1
unquote(x1) = x1
unquote1(x1) = unquote1(x1)
active(x1) = active(x1)
sel(x1, x2) = sel(x1, x2)
s(x1) = s(x1)
cons(x1, x2) = cons(x1, x2)
mark(x1) = mark
0 = 0
first(x1, x2) = first(x1, x2)
nil = nil
from(x1) = from(x1)
sel1(x1, x2) = sel1(x1, x2)
first1(x1, x2) = first1
nil1 = nil1
cons1(x1, x2) = cons1(x1, x2)
01 = 01
s1(x1) = s1(x1)
fcons(x1, x2) = fcons(x1, x2)
proper(x1) = proper(x1)
ok(x1) = ok
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

first1 > [unquote1₁, active₁, s₁, cons₂, 0, from₁, proper₁] > [PROPER₁, sel₂, mark, first₂, nil, sel1₂, nil1, cons1₂, s1₁, fcons₂, ok, top]
01 > [unquote1₁, active₁, s₁, cons₂, 0, from₁, proper₁] > [PROPER₁, sel₂, mark, first₂, nil, sel1₂, nil1, cons1₂, s1₁, fcons₂, ok, top]

Status:

PROPER₁: [1]
unquote1₁: [1]
active₁: [1]
sel₂: [2,1]
s₁: [1]
cons₂: [2,1]
mark: []
0: []
first₂: [2,1]
nil: []
from₁: [1]
sel1₂: [2,1]
first1: []
nil1: []
cons1₂: [2,1]
01: []
s1₁: [1]
fcons₂: [2,1]
proper₁: [1]
ok: []
top: []

The following usable rules [FROCOS05] were oriented:

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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(103) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PROPER(quote(X)) → PROPER(X)
PROPER(quote1(X)) → PROPER(X)
PROPER(unquote(X)) → PROPER(X)

The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(104) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(unquote(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1) = PROPER(x1)
quote(x1) = x1
quote1(x1) = x1
unquote(x1) = unquote(x1)
active(x1) = active(x1)
sel(x1, x2) = sel(x1, x2)
s(x1) = s(x1)
cons(x1, x2) = cons(x1, x2)
mark(x1) = mark
0 = 0
first(x1, x2) = first(x1, x2)
nil = nil
from(x1) = from(x1)
sel1(x1, x2) = sel1(x1, x2)
first1(x1, x2) = first1(x1, x2)
nil1 = nil1
cons1(x1, x2) = cons1(x1, x2)
01 = 01
s1(x1) = s1(x1)
unquote1(x1) = unquote1(x1)
fcons(x1, x2) = fcons(x1, x2)
proper(x1) = proper(x1)
ok(x1) = ok
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

01 > [active₁, s₁, cons₂, 0, from₁, unquote1₁, proper₁] > [PROPER₁, unquote₁, sel₂, mark, first₂, nil, sel1₂, first1₂, nil1, cons1₂, s1₁, fcons₂, ok, top]

Status:

PROPER₁: [1]
unquote₁: [1]
active₁: [1]
sel₂: [2,1]
s₁: [1]
cons₂: [2,1]
mark: []
0: []
first₂: [2,1]
nil: []
from₁: [1]
sel1₂: [2,1]
first1₂: [2,1]
nil1: []
cons1₂: [2,1]
01: []
s1₁: [1]
unquote1₁: [1]
fcons₂: [2,1]
proper₁: [1]
ok: []
top: []

The following usable rules [FROCOS05] were oriented:

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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(105) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PROPER(quote(X)) → PROPER(X)
PROPER(quote1(X)) → PROPER(X)

The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(106) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(quote(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1) = PROPER(x1)
quote(x1) = quote(x1)
quote1(x1) = x1
active(x1) = active(x1)
sel(x1, x2) = sel(x1, x2)
s(x1) = s(x1)
cons(x1, x2) = cons(x1, x2)
mark(x1) = mark
0 = 0
first(x1, x2) = first(x1, x2)
nil = nil
from(x1) = from(x1)
sel1(x1, x2) = sel1(x1, x2)
first1(x1, x2) = first1(x1, x2)
nil1 = nil1
cons1(x1, x2) = cons1(x1, x2)
01 = 01
s1(x1) = s1(x1)
unquote(x1) = unquote(x1)
unquote1(x1) = unquote1(x1)
fcons(x1, x2) = fcons(x1, x2)
proper(x1) = x1
ok(x1) = ok
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

quote₁ > PROPER₁ > [sel₂, mark, first₂, nil, sel1₂, first1₂, nil1, cons1₂, s1₁, unquote₁, fcons₂, ok, top]
01 > [active₁, s₁, cons₂, 0, from₁, unquote1₁] > [sel₂, mark, first₂, nil, sel1₂, first1₂, nil1, cons1₂, s1₁, unquote₁, fcons₂, ok, top]

Status:

PROPER₁: [1]
quote₁: [1]
active₁: [1]
sel₂: [2,1]
s₁: [1]
cons₂: [2,1]
mark: []
0: []
first₂: [2,1]
nil: []
from₁: [1]
sel1₂: [2,1]
first1₂: [2,1]
nil1: []
cons1₂: [2,1]
01: []
s1₁: [1]
unquote₁: [1]
unquote1₁: [1]
fcons₂: [2,1]
ok: []
top: []

The following usable rules [FROCOS05] were oriented:

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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(107) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PROPER(quote1(X)) → PROPER(X)

The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(108) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(quote1(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1) = PROPER(x1)
quote1(x1) = quote1(x1)
active(x1) = active(x1)
sel(x1, x2) = sel(x1, x2)
s(x1) = s(x1)
cons(x1, x2) = cons(x1, x2)
mark(x1) = mark
0 = 0
first(x1, x2) = first
nil = nil
from(x1) = from(x1)
sel1(x1, x2) = sel1(x1, x2)
quote(x1) = quote(x1)
first1(x1, x2) = first1(x1, x2)
nil1 = nil1
cons1(x1, x2) = cons1(x1, x2)
01 = 01
s1(x1) = s1(x1)
unquote(x1) = unquote(x1)
unquote1(x1) = unquote1(x1)
fcons(x1, x2) = fcons(x1, x2)
proper(x1) = x1
ok(x1) = ok
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

PROPER₁ > [sel₂, mark, nil, sel1₂, quote₁, first1₂, nil1, cons1₂, 01, s1₁, unquote₁, fcons₂, ok, top]
0 > [sel₂, mark, nil, sel1₂, quote₁, first1₂, nil1, cons1₂, 01, s1₁, unquote₁, fcons₂, ok, top]
first > [quote1₁, active₁, s₁, cons₂, from₁, unquote1₁] > [sel₂, mark, nil, sel1₂, quote₁, first1₂, nil1, cons1₂, 01, s1₁, unquote₁, fcons₂, ok, top]

Status:

PROPER₁: [1]
quote1₁: [1]
active₁: [1]
sel₂: [2,1]
s₁: [1]
cons₂: [2,1]
mark: []
0: []
first: []
nil: []
from₁: [1]
sel1₂: [2,1]
quote₁: [1]
first1₂: [2,1]
nil1: []
cons1₂: [2,1]
01: []
s1₁: [1]
unquote₁: [1]
unquote1₁: [1]
fcons₂: [2,1]
ok: []
top: []

The following usable rules [FROCOS05] were oriented:

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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(109) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(110) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(111) TRUE

(112) Obligation:

Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(113) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(sel(X1, X2)) → ACTIVE(X2)
ACTIVE(sel(X1, X2)) → ACTIVE(X1)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(first(X1, X2)) → ACTIVE(X1)
ACTIVE(first(X1, X2)) → ACTIVE(X2)
ACTIVE(from(X)) → ACTIVE(X)
ACTIVE(sel1(X1, X2)) → ACTIVE(X1)
ACTIVE(sel1(X1, X2)) → ACTIVE(X2)
ACTIVE(first1(X1, X2)) → ACTIVE(X1)
ACTIVE(first1(X1, X2)) → ACTIVE(X2)
ACTIVE(cons1(X1, X2)) → ACTIVE(X1)
ACTIVE(cons1(X1, X2)) → ACTIVE(X2)
ACTIVE(s1(X)) → ACTIVE(X)
ACTIVE(fcons(X1, X2)) → ACTIVE(X1)
ACTIVE(fcons(X1, X2)) → ACTIVE(X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1) = ACTIVE(x1)
sel(x1, x2) = sel(x1, x2)
s(x1) = x1
cons(x1, x2) = cons(x1, x2)
first(x1, x2) = first(x1, x2)
from(x1) = from(x1)
sel1(x1, x2) = sel1(x1, x2)
first1(x1, x2) = first1(x1, x2)
cons1(x1, x2) = cons1(x1, x2)
s1(x1) = s1(x1)
unquote(x1) = x1
unquote1(x1) = x1
fcons(x1, x2) = fcons(x1, x2)
active(x1) = x1
mark(x1) = mark
0 = 0
nil = nil
quote(x1) = quote(x1)
nil1 = nil1
01 = 01
quote1(x1) = quote1(x1)
proper(x1) = x1
ok(x1) = ok
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

sel₂ > [cons₂, sel1₂, s1₁, mark, nil, quote₁, nil1, 01, ok, top]
[first₂, first1₂, cons1₂, quote1₁] > [cons₂, sel1₂, s1₁, mark, nil, quote₁, nil1, 01, ok, top]
from₁ > [cons₂, sel1₂, s1₁, mark, nil, quote₁, nil1, 01, ok, top]
fcons₂ > ACTIVE₁ > [cons₂, sel1₂, s1₁, mark, nil, quote₁, nil1, 01, ok, top]
0 > [cons₂, sel1₂, s1₁, mark, nil, quote₁, nil1, 01, ok, top]

Status:

ACTIVE₁: [1]
sel₂: [2,1]
cons₂: [1,2]
first₂: [1,2]
from₁: [1]
sel1₂: [1,2]
first1₂: [1,2]
cons1₂: [1,2]
s1₁: [1]
fcons₂: [2,1]
mark: []
0: []
nil: []
quote₁: [1]
nil1: []
01: []
quote1₁: [1]
ok: []
top: []

The following usable rules [FROCOS05] were oriented:

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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(114) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(unquote(X)) → ACTIVE(X)
ACTIVE(unquote1(X)) → ACTIVE(X)

The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(115) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(s(X)) → ACTIVE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1) = ACTIVE(x1)
s(x1) = s(x1)
unquote(x1) = x1
unquote1(x1) = x1
active(x1) = x1
sel(x1, x2) = sel(x1, x2)
cons(x1, x2) = cons(x1, x2)
mark(x1) = mark
0 = 0
first(x1, x2) = first(x1, x2)
nil = nil
from(x1) = from(x1)
sel1(x1, x2) = sel1(x1, x2)
quote(x1) = quote
first1(x1, x2) = first1(x1, x2)
nil1 = nil1
cons1(x1, x2) = cons1(x1, x2)
01 = 01
quote1(x1) = quote1(x1)
s1(x1) = s1(x1)
fcons(x1, x2) = fcons(x1, x2)
proper(x1) = x1
ok(x1) = ok
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

s₁ > ACTIVE₁ > [sel₂, mark, first₂, nil, sel1₂, quote, first1₂, nil1, cons1₂, s1₁, fcons₂, ok, top]
s₁ > [cons₂, from₁, quote1₁] > [sel₂, mark, first₂, nil, sel1₂, quote, first1₂, nil1, cons1₂, s1₁, fcons₂, ok, top]
0 > 01 > [sel₂, mark, first₂, nil, sel1₂, quote, first1₂, nil1, cons1₂, s1₁, fcons₂, ok, top]

Status:

ACTIVE₁: [1]
s₁: [1]
sel₂: [2,1]
cons₂: [2,1]
mark: []
0: []
first₂: [2,1]
nil: []
from₁: [1]
sel1₂: [2,1]
quote: []
first1₂: [2,1]
nil1: []
cons1₂: [2,1]
01: []
quote1₁: [1]
s1₁: [1]
fcons₂: [2,1]
ok: []
top: []

The following usable rules [FROCOS05] were oriented:

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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(116) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(unquote(X)) → ACTIVE(X)
ACTIVE(unquote1(X)) → ACTIVE(X)

The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(117) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(unquote(X)) → ACTIVE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1) = x1
unquote(x1) = unquote(x1)
unquote1(x1) = x1
active(x1) = active(x1)
sel(x1, x2) = sel(x1, x2)
s(x1) = s(x1)
cons(x1, x2) = cons(x1, x2)
mark(x1) = mark
0 = 0
first(x1, x2) = first(x1, x2)
nil = nil
from(x1) = from(x1)
sel1(x1, x2) = sel1(x1, x2)
quote(x1) = quote(x1)
first1(x1, x2) = first1(x1, x2)
nil1 = nil1
cons1(x1, x2) = cons1(x1, x2)
01 = 01
quote1(x1) = quote1(x1)
s1(x1) = s1(x1)
fcons(x1, x2) = fcons(x1, x2)
proper(x1) = proper(x1)
ok(x1) = ok
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

01 > [active₁, s₁, cons₂, 0, from₁, quote1₁, proper₁] > [unquote₁, sel₂, mark, first₂, nil, sel1₂, quote₁, first1₂, nil1, cons1₂, s1₁, fcons₂, ok, top]

Status:

unquote₁: [1]
active₁: [1]
sel₂: [2,1]
s₁: [1]
cons₂: [2,1]
mark: []
0: []
first₂: [2,1]
nil: []
from₁: [1]
sel1₂: [2,1]
quote₁: [1]
first1₂: [2,1]
nil1: []
cons1₂: [2,1]
01: []
quote1₁: [1]
s1₁: [1]
fcons₂: [2,1]
proper₁: [1]
ok: []
top: []

The following usable rules [FROCOS05] were oriented:

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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(118) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(unquote1(X)) → ACTIVE(X)

The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(119) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(unquote1(X)) → ACTIVE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1) = ACTIVE(x1)
unquote1(x1) = unquote1(x1)
active(x1) = x1
sel(x1, x2) = sel(x1, x2)
s(x1) = s(x1)
cons(x1, x2) = cons(x1, x2)
mark(x1) = mark
0 = 0
first(x1, x2) = first(x1, x2)
nil = nil
from(x1) = from(x1)
sel1(x1, x2) = sel1(x1, x2)
quote(x1) = quote(x1)
first1(x1, x2) = first1(x1, x2)
nil1 = nil1
cons1(x1, x2) = cons1(x1, x2)
01 = 01
quote1(x1) = quote1(x1)
s1(x1) = s1(x1)
unquote(x1) = unquote(x1)
fcons(x1, x2) = fcons(x1, x2)
proper(x1) = x1
ok(x1) = ok
top(x1) = top

Lexicographic path order with status [LPO].
Quasi-Precedence:

ACTIVE₁ > [unquote1₁, sel₂, mark, nil, sel1₂, quote₁, first1₂, nil1, cons1₂, 01, s1₁, unquote₁, fcons₂, ok, top]
0 > [unquote1₁, sel₂, mark, nil, sel1₂, quote₁, first1₂, nil1, cons1₂, 01, s1₁, unquote₁, fcons₂, ok, top]
first₂ > [s₁, cons₂, from₁, quote1₁] > [unquote1₁, sel₂, mark, nil, sel1₂, quote₁, first1₂, nil1, cons1₂, 01, s1₁, unquote₁, fcons₂, ok, top]

Status:

ACTIVE₁: [1]
unquote1₁: [1]
sel₂: [2,1]
s₁: [1]
cons₂: [2,1]
mark: []
0: []
first₂: [2,1]
nil: []
from₁: [1]
sel1₂: [2,1]
quote₁: [1]
first1₂: [2,1]
nil1: []
cons1₂: [2,1]
01: []
quote1₁: [1]
s1₁: [1]
unquote₁: [1]
fcons₂: [2,1]
ok: []
top: []

The following usable rules [FROCOS05] were oriented:

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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(120) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(121) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(122) TRUE

(123) Obligation:

Q DP problem:
The TRS P consists of the following rules:

TOP(ok(X)) → TOP(active(X))
TOP(mark(X)) → TOP(proper(X))

The TRS R consists of the following 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))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(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))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.