Termination w.r.t. Q proof of /tmp/tmpUW3DoY/Ex5_7_Luc97

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 PisEmptyProof (⇔)

    ↳19 TRUE

  ↳20 QDP

    ↳21 QDPOrderProof (⇔)

    ↳22 QDP

    ↳23 QDPOrderProof (⇔)

    ↳24 QDP

    ↳25 PisEmptyProof (⇔)

    ↳26 TRUE

  ↳27 QDP

    ↳28 QDPOrderProof (⇔)

    ↳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 QDPOrderProof (⇔)

    ↳84 QDP

    ↳85 QDPOrderProof (⇔)

    ↳86 QDP

    ↳87 QDPOrderProof (⇔)

    ↳88 QDP

    ↳89 PisEmptyProof (⇔)

    ↳90 TRUE

  ↳91 QDP

    ↳92 QDPOrderProof (⇔)

    ↳93 QDP

    ↳94 QDPOrderProof (⇔)

    ↳95 QDP

    ↳96 QDPOrderProof (⇔)

    ↳97 QDP

    ↳98 QDPOrderProof (⇔)

    ↳99 QDP

    ↳100 PisEmptyProof (⇔)

    ↳101 TRUE

  ↳102 QDP

(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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(dbl(s(X))) → S(s(dbl(X)))
ACTIVE(dbl(s(X))) → S(dbl(X))
ACTIVE(dbl(s(X))) → DBL(X)
ACTIVE(dbls(cons(X, Y))) → CONS(dbl(X), dbls(Y))
ACTIVE(dbls(cons(X, Y))) → DBL(X)
ACTIVE(dbls(cons(X, Y))) → DBLS(Y)
ACTIVE(sel(s(X), cons(Y, Z))) → SEL(X, Z)
ACTIVE(indx(cons(X, Y), Z)) → CONS(sel(X, Z), indx(Y, Z))
ACTIVE(indx(cons(X, Y), Z)) → SEL(X, Z)
ACTIVE(indx(cons(X, Y), Z)) → INDX(Y, Z)
ACTIVE(from(X)) → CONS(X, from(s(X)))
ACTIVE(from(X)) → FROM(s(X))
ACTIVE(from(X)) → S(X)
ACTIVE(dbl1(s(X))) → S1(s1(dbl1(X)))
ACTIVE(dbl1(s(X))) → S1(dbl1(X))
ACTIVE(dbl1(s(X))) → DBL1(X)
ACTIVE(sel1(s(X), cons(Y, Z))) → SEL1(X, Z)
ACTIVE(quote(s(X))) → S1(quote(X))
ACTIVE(quote(s(X))) → QUOTE(X)
ACTIVE(quote(dbl(X))) → DBL1(X)
ACTIVE(quote(sel(X, Y))) → SEL1(X, Y)
ACTIVE(dbl(X)) → DBL(active(X))
ACTIVE(dbl(X)) → ACTIVE(X)
ACTIVE(dbls(X)) → DBLS(active(X))
ACTIVE(dbls(X)) → ACTIVE(X)
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(indx(X1, X2)) → INDX(active(X1), X2)
ACTIVE(indx(X1, X2)) → ACTIVE(X1)
ACTIVE(dbl1(X)) → DBL1(active(X))
ACTIVE(dbl1(X)) → ACTIVE(X)
ACTIVE(s1(X)) → S1(active(X))
ACTIVE(s1(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(quote(X)) → QUOTE(active(X))
ACTIVE(quote(X)) → ACTIVE(X)
DBL(mark(X)) → DBL(X)
DBLS(mark(X)) → DBLS(X)
SEL(mark(X1), X2) → SEL(X1, X2)
SEL(X1, mark(X2)) → SEL(X1, X2)
INDX(mark(X1), X2) → INDX(X1, X2)
DBL1(mark(X)) → DBL1(X)
S1(mark(X)) → S1(X)
SEL1(mark(X1), X2) → SEL1(X1, X2)
SEL1(X1, mark(X2)) → SEL1(X1, X2)
QUOTE(mark(X)) → QUOTE(X)
PROPER(dbl(X)) → DBL(proper(X))
PROPER(dbl(X)) → PROPER(X)
PROPER(s(X)) → S(proper(X))
PROPER(s(X)) → PROPER(X)
PROPER(dbls(X)) → DBLS(proper(X))
PROPER(dbls(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(sel(X1, X2)) → SEL(proper(X1), proper(X2))
PROPER(sel(X1, X2)) → PROPER(X1)
PROPER(sel(X1, X2)) → PROPER(X2)
PROPER(indx(X1, X2)) → INDX(proper(X1), proper(X2))
PROPER(indx(X1, X2)) → PROPER(X1)
PROPER(indx(X1, X2)) → PROPER(X2)
PROPER(from(X)) → FROM(proper(X))
PROPER(from(X)) → PROPER(X)
PROPER(dbl1(X)) → DBL1(proper(X))
PROPER(dbl1(X)) → PROPER(X)
PROPER(s1(X)) → S1(proper(X))
PROPER(s1(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)
DBL(ok(X)) → DBL(X)
S(ok(X)) → S(X)
DBLS(ok(X)) → DBLS(X)
CONS(ok(X1), ok(X2)) → CONS(X1, X2)
SEL(ok(X1), ok(X2)) → SEL(X1, X2)
INDX(ok(X1), ok(X2)) → INDX(X1, X2)
FROM(ok(X)) → FROM(X)
DBL1(ok(X)) → DBL1(X)
S1(ok(X)) → S1(X)
SEL1(ok(X1), ok(X2)) → SEL1(X1, X2)
QUOTE(ok(X)) → QUOTE(X)
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(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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 14 SCCs with 44 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

FROM(ok(X)) → FROM(X)

The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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.

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) = x1
ok(x1) = ok(x1)
active(x1) = active(x1)
dbl(x1) = x1
0 = 0
mark(x1) = mark
s(x1) = s(x1)
dbls(x1) = dbls(x1)
nil = nil
cons(x1, x2) = cons(x1, x2)
sel(x1, x2) = sel(x1, x2)
indx(x1, x2) = x1
from(x1) = x1
dbl1(x1) = dbl1(x1)
01 = 01
s1(x1) = s1(x1)
sel1(x1, x2) = sel1(x1, x2)
quote(x1) = quote(x1)
proper(x1) = proper(x1)
top(x1) = top

Lexicographic path order with status [LPO].
Precedence:

01 > ok₁
top > active₁ > mark > proper₁ > 0 > ok₁
top > active₁ > mark > proper₁ > s₁ > s1₁ > ok₁
top > active₁ > mark > proper₁ > s₁ > sel1₂ > ok₁
top > active₁ > mark > proper₁ > nil > ok₁
top > active₁ > mark > proper₁ > cons₂ > dbls₁ > ok₁
top > active₁ > mark > proper₁ > cons₂ > sel1₂ > ok₁
top > active₁ > mark > proper₁ > sel₂ > sel1₂ > ok₁
top > active₁ > mark > proper₁ > dbl1₁ > s1₁ > ok₁
top > active₁ > mark > proper₁ > quote₁ > s1₁ > ok₁
top > active₁ > mark > proper₁ > quote₁ > sel1₂ > ok₁

Status:

sel1₂: [2,1]
s1₁: [1]
01: []
sel₂: [1,2]
quote₁: [1]
ok₁: [1]
dbls₁: [1]
0: []
cons₂: [2,1]
active₁: [1]
mark: []
s₁: [1]
proper₁: [1]
top: []
dbl1₁: [1]
nil: []

The following usable rules [FROCOS05] were oriented:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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:

CONS(ok(X1), ok(X2)) → CONS(X1, X2)

The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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.

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)
active(x1) = x1
dbl(x1) = x1
0 = 0
mark(x1) = mark
s(x1) = s(x1)
dbls(x1) = x1
nil = nil
cons(x1, x2) = cons(x1, x2)
sel(x1, x2) = sel(x1, x2)
indx(x1, x2) = x2
from(x1) = x1
dbl1(x1) = x1
01 = 01
s1(x1) = x1
sel1(x1, x2) = x1
quote(x1) = x1
proper(x1) = proper(x1)
top(x1) = top

Lexicographic path order with status [LPO].
Precedence:

CONS₁ > mark
proper₁ > 0 > 01 > mark
proper₁ > s₁ > sel₂ > ok₁ > mark
proper₁ > nil > mark
proper₁ > cons₂ > ok₁ > mark
top > mark

Status:

sel₂: [1,2]
01: []
cons₂: [1,2]
CONS₁: [1]
mark: []
ok₁: [1]
s₁: [1]
proper₁: [1]
top: []
0: []
nil: []

The following usable rules [FROCOS05] were oriented:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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:

S(ok(X)) → S(X)

The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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.

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) = x1
ok(x1) = ok(x1)
active(x1) = active(x1)
dbl(x1) = x1
0 = 0
mark(x1) = mark
s(x1) = s(x1)
dbls(x1) = dbls(x1)
nil = nil
cons(x1, x2) = cons(x1, x2)
sel(x1, x2) = sel(x1, x2)
indx(x1, x2) = x1
from(x1) = x1
dbl1(x1) = dbl1(x1)
01 = 01
s1(x1) = s1(x1)
sel1(x1, x2) = sel1(x1, x2)
quote(x1) = quote(x1)
proper(x1) = proper(x1)
top(x1) = top

Lexicographic path order with status [LPO].
Precedence:

01 > ok₁
top > active₁ > mark > proper₁ > 0 > ok₁
top > active₁ > mark > proper₁ > s₁ > s1₁ > ok₁
top > active₁ > mark > proper₁ > s₁ > sel1₂ > ok₁
top > active₁ > mark > proper₁ > nil > ok₁
top > active₁ > mark > proper₁ > cons₂ > dbls₁ > ok₁
top > active₁ > mark > proper₁ > cons₂ > sel1₂ > ok₁
top > active₁ > mark > proper₁ > sel₂ > sel1₂ > ok₁
top > active₁ > mark > proper₁ > dbl1₁ > s1₁ > ok₁
top > active₁ > mark > proper₁ > quote₁ > s1₁ > ok₁
top > active₁ > mark > proper₁ > quote₁ > sel1₂ > ok₁

Status:

sel1₂: [2,1]
s1₁: [1]
01: []
sel₂: [1,2]
quote₁: [1]
ok₁: [1]
dbls₁: [1]
0: []
cons₂: [2,1]
active₁: [1]
mark: []
s₁: [1]
proper₁: [1]
top: []
dbl1₁: [1]
nil: []

The following usable rules [FROCOS05] were oriented:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(17) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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) PisEmptyProof (EQUIVALENT transformation)

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

(19) TRUE

(20) Obligation:

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

QUOTE(ok(X)) → QUOTE(X)
QUOTE(mark(X)) → QUOTE(X)

The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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.

(21) 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) = x1
ok(x1) = ok(x1)
mark(x1) = x1
active(x1) = active(x1)
dbl(x1) = dbl(x1)
0 = 0
s(x1) = x1
dbls(x1) = dbls(x1)
nil = nil
cons(x1, x2) = cons(x1, x2)
sel(x1, x2) = sel(x1, x2)
indx(x1, x2) = indx(x1, x2)
from(x1) = x1
dbl1(x1) = dbl1(x1)
01 = 01
s1(x1) = x1
sel1(x1, x2) = x2
quote(x1) = x1
proper(x1) = proper(x1)
top(x1) = top

Lexicographic path order with status [LPO].
Precedence:

active₁ > dbl₁ > 0 > ok₁ > top
active₁ > dbl₁ > 0 > 01 > top
active₁ > nil > ok₁ > top
active₁ > cons₂ > dbls₁ > ok₁ > top
active₁ > cons₂ > indx₂ > ok₁ > top
active₁ > sel₂ > ok₁ > top
active₁ > dbl1₁ > ok₁ > top
active₁ > dbl1₁ > 01 > top
proper₁ > dbl₁ > 0 > ok₁ > top
proper₁ > dbl₁ > 0 > 01 > top
proper₁ > nil > ok₁ > top
proper₁ > cons₂ > dbls₁ > ok₁ > top
proper₁ > cons₂ > indx₂ > ok₁ > top
proper₁ > sel₂ > ok₁ > top
proper₁ > dbl1₁ > ok₁ > top
proper₁ > dbl1₁ > 01 > top

Status:

01: []
sel₂: [1,2]
ok₁: [1]
dbls₁: [1]
0: []
indx₂: [2,1]
cons₂: [1,2]
active₁: [1]
dbl₁: [1]
proper₁: [1]
top: []
dbl1₁: [1]
nil: []

The following usable rules [FROCOS05] were oriented:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(22) Obligation:

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

QUOTE(mark(X)) → QUOTE(X)

The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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.

QUOTE(mark(X)) → QUOTE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
QUOTE(x1) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
dbl(x1) = x1
0 = 0
s(x1) = s(x1)
dbls(x1) = x1
nil = nil
cons(x1, x2) = cons(x1, x2)
sel(x1, x2) = sel(x1, x2)
indx(x1, x2) = indx(x1, x2)
from(x1) = x1
dbl1(x1) = dbl1(x1)
01 = 01
s1(x1) = s1(x1)
sel1(x1, x2) = sel1(x1, x2)
quote(x1) = x1
proper(x1) = x1
ok(x1) = ok
top(x1) = top

Lexicographic path order with status [LPO].
Precedence:

nil > mark₁
nil > ok
top > active₁ > 0 > mark₁
top > active₁ > 0 > 01 > ok
top > active₁ > s₁ > sel₂ > mark₁
top > active₁ > s₁ > dbl1₁ > mark₁
top > active₁ > s₁ > sel1₂ > mark₁
top > active₁ > s₁ > ok
top > active₁ > cons₂ > sel₂ > mark₁
top > active₁ > cons₂ > indx₂ > mark₁
top > active₁ > cons₂ > sel1₂ > mark₁
top > active₁ > cons₂ > ok
top > active₁ > s1₁ > mark₁
top > active₁ > s1₁ > ok

Status:

sel1₂: [2,1]
s1₁: [1]
01: []
sel₂: [2,1]
mark₁: [1]
0: []
indx₂: [2,1]
cons₂: [1,2]
active₁: [1]
ok: []
s₁: [1]
top: []
dbl1₁: [1]
nil: []

The following usable rules [FROCOS05] were oriented:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(24) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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) PisEmptyProof (EQUIVALENT transformation)

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

(26) TRUE

(27) 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(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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.

(28) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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(x1)
mark(x1) = x1
ok(x1) = ok(x1)
active(x1) = active(x1)
dbl(x1) = dbl(x1)
0 = 0
s(x1) = x1
dbls(x1) = x1
nil = nil
cons(x1, x2) = cons(x1, x2)
sel(x1, x2) = x2
indx(x1, x2) = x2
from(x1) = from(x1)
dbl1(x1) = x1
01 = 01
s1(x1) = x1
sel1(x1, x2) = x2
quote(x1) = x1
proper(x1) = proper(x1)
top(x1) = top

Lexicographic path order with status [LPO].
Precedence:

active₁ > dbl₁ > ok₁ > top
active₁ > dbl₁ > 0
active₁ > nil > ok₁ > top
active₁ > cons₂ > ok₁ > top
active₁ > from₁ > ok₁ > top
active₁ > 01
proper₁ > dbl₁ > ok₁ > top
proper₁ > dbl₁ > 0
proper₁ > nil > ok₁ > top
proper₁ > cons₂ > ok₁ > top
proper₁ > from₁ > ok₁ > top
proper₁ > 01

Status:

01: []
active₁: [1]
cons₂: [2,1]
from₁: [1]
SEL1₁: [1]
dbl₁: [1]
ok₁: [1]
proper₁: [1]
top: []
0: []
nil: []

The following usable rules [FROCOS05] were oriented:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(29) 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)

The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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.

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) = SEL1(x1)
mark(x1) = mark(x1)
active(x1) = active(x1)
dbl(x1) = dbl(x1)
0 = 0
s(x1) = s(x1)
dbls(x1) = dbls(x1)
nil = nil
cons(x1, x2) = cons(x1, x2)
sel(x1, x2) = sel(x1, x2)
indx(x1, x2) = indx(x1, x2)
from(x1) = from(x1)
dbl1(x1) = dbl1(x1)
01 = 01
s1(x1) = s1(x1)
sel1(x1, x2) = sel1(x1, x2)
quote(x1) = x1
proper(x1) = x1
ok(x1) = x1
top(x1) = top

Lexicographic path order with status [LPO].
Precedence:

SEL1₁ > top
active₁ > dbl₁ > mark₁ > top
active₁ > dbl₁ > 0 > top
active₁ > dbl₁ > s₁ > top
active₁ > dbls₁ > mark₁ > top
active₁ > dbls₁ > nil > top
active₁ > dbls₁ > cons₂ > top
active₁ > sel₂ > mark₁ > top
active₁ > indx₂ > mark₁ > top
active₁ > indx₂ > nil > top
active₁ > indx₂ > cons₂ > top
active₁ > from₁ > top
active₁ > dbl1₁ > s1₁ > mark₁ > top
active₁ > 01 > top
active₁ > sel1₂ > mark₁ > top

Status:

sel₂: [1,2]
01: []
s1₁: [1]
sel1₂: [2,1]
from₁: [1]
mark₁: [1]
dbls₁: [1]
0: []
indx₂: [2,1]
active₁: [1]
cons₂: [1,2]
SEL1₁: [1]
dbl₁: [1]
s₁: [1]
top: []
nil: []
dbl1₁: [1]

The following usable rules [FROCOS05] were oriented:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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:

SEL1(X1, mark(X2)) → SEL1(X1, X2)

The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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.

SEL1(X1, mark(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(x1, x2)
mark(x1) = mark(x1)
active(x1) = active(x1)
dbl(x1) = x1
0 = 0
s(x1) = x1
dbls(x1) = dbls(x1)
nil = nil
cons(x1, x2) = cons(x1, x2)
sel(x1, x2) = sel(x1, x2)
indx(x1, x2) = indx(x1, x2)
from(x1) = from(x1)
dbl1(x1) = x1
01 = 01
s1(x1) = x1
sel1(x1, x2) = sel1(x1, x2)
quote(x1) = x1
proper(x1) = x1
ok(x1) = ok
top(x1) = top

Lexicographic path order with status [LPO].
Precedence:

active₁ > 0 > ok > cons₂
active₁ > 0 > ok > from₁
active₁ > dbls₁ > mark₁ > SEL1₂
active₁ > dbls₁ > mark₁ > top
active₁ > dbls₁ > nil > ok > cons₂
active₁ > dbls₁ > nil > ok > from₁
active₁ > sel₂ > sel1₂ > mark₁ > SEL1₂
active₁ > sel₂ > sel1₂ > mark₁ > top
active₁ > sel₂ > sel1₂ > ok > cons₂
active₁ > sel₂ > sel1₂ > ok > from₁
active₁ > indx₂ > mark₁ > SEL1₂
active₁ > indx₂ > mark₁ > top
active₁ > indx₂ > nil > ok > cons₂
active₁ > indx₂ > nil > ok > from₁
active₁ > 01 > ok > cons₂
active₁ > 01 > ok > from₁

Status:

sel1₂: [1,2]
01: []
sel₂: [2,1]
from₁: [1]
SEL1₂: [1,2]
mark₁: [1]
dbls₁: [1]
0: []
indx₂: [2,1]
cons₂: [1,2]
active₁: [1]
ok: []
top: []
nil: []

The following usable rules [FROCOS05] were oriented:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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) = x1
ok(x1) = ok(x1)
mark(x1) = x1
active(x1) = active(x1)
dbl(x1) = dbl(x1)
0 = 0
s(x1) = x1
dbls(x1) = dbls(x1)
nil = nil
cons(x1, x2) = cons(x1, x2)
sel(x1, x2) = sel(x1, x2)
indx(x1, x2) = indx(x1, x2)
from(x1) = x1
dbl1(x1) = dbl1(x1)
01 = 01
s1(x1) = x1
sel1(x1, x2) = x2
quote(x1) = x1
proper(x1) = proper(x1)
top(x1) = top

Lexicographic path order with status [LPO].
Precedence:

active₁ > dbl₁ > 0 > ok₁ > top
active₁ > dbl₁ > 0 > 01 > top
active₁ > nil > ok₁ > top
active₁ > cons₂ > dbls₁ > ok₁ > top
active₁ > cons₂ > indx₂ > ok₁ > top
active₁ > sel₂ > ok₁ > top
active₁ > dbl1₁ > ok₁ > top
active₁ > dbl1₁ > 01 > top
proper₁ > dbl₁ > 0 > ok₁ > top
proper₁ > dbl₁ > 0 > 01 > top
proper₁ > nil > ok₁ > top
proper₁ > cons₂ > dbls₁ > ok₁ > top
proper₁ > cons₂ > indx₂ > ok₁ > top
proper₁ > sel₂ > ok₁ > top
proper₁ > dbl1₁ > ok₁ > top
proper₁ > dbl1₁ > 01 > top

Status:

01: []
sel₂: [1,2]
ok₁: [1]
dbls₁: [1]
0: []
indx₂: [2,1]
cons₂: [1,2]
active₁: [1]
dbl₁: [1]
proper₁: [1]
top: []
dbl1₁: [1]
nil: []

The following usable rules [FROCOS05] were oriented:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
dbl(x1) = x1
0 = 0
s(x1) = s(x1)
dbls(x1) = x1
nil = nil
cons(x1, x2) = cons(x1, x2)
sel(x1, x2) = sel(x1, x2)
indx(x1, x2) = indx(x1, x2)
from(x1) = x1
dbl1(x1) = dbl1(x1)
01 = 01
s1(x1) = s1(x1)
sel1(x1, x2) = sel1(x1, x2)
quote(x1) = x1
proper(x1) = x1
ok(x1) = ok
top(x1) = top

Lexicographic path order with status [LPO].
Precedence:

nil > mark₁
nil > ok
top > active₁ > 0 > mark₁
top > active₁ > 0 > 01 > ok
top > active₁ > s₁ > sel₂ > mark₁
top > active₁ > s₁ > dbl1₁ > mark₁
top > active₁ > s₁ > sel1₂ > mark₁
top > active₁ > s₁ > ok
top > active₁ > cons₂ > sel₂ > mark₁
top > active₁ > cons₂ > indx₂ > mark₁
top > active₁ > cons₂ > sel1₂ > mark₁
top > active₁ > cons₂ > ok
top > active₁ > s1₁ > mark₁
top > active₁ > s1₁ > ok

Status:

sel1₂: [2,1]
s1₁: [1]
01: []
sel₂: [2,1]
mark₁: [1]
0: []
indx₂: [2,1]
cons₂: [1,2]
active₁: [1]
ok: []
s₁: [1]
top: []
dbl1₁: [1]
nil: []

The following usable rules [FROCOS05] were oriented:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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:

DBL1(ok(X)) → DBL1(X)
DBL1(mark(X)) → DBL1(X)

The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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.

DBL1(ok(X)) → DBL1(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
DBL1(x1) = x1
ok(x1) = ok(x1)
mark(x1) = x1
active(x1) = active(x1)
dbl(x1) = dbl(x1)
0 = 0
s(x1) = x1
dbls(x1) = dbls(x1)
nil = nil
cons(x1, x2) = cons(x1, x2)
sel(x1, x2) = sel(x1, x2)
indx(x1, x2) = indx(x1, x2)
from(x1) = x1
dbl1(x1) = dbl1(x1)
01 = 01
s1(x1) = x1
sel1(x1, x2) = x2
quote(x1) = x1
proper(x1) = proper(x1)
top(x1) = top

Lexicographic path order with status [LPO].
Precedence:

active₁ > dbl₁ > 0 > ok₁ > top
active₁ > dbl₁ > 0 > 01 > top
active₁ > nil > ok₁ > top
active₁ > cons₂ > dbls₁ > ok₁ > top
active₁ > cons₂ > indx₂ > ok₁ > top
active₁ > sel₂ > ok₁ > top
active₁ > dbl1₁ > ok₁ > top
active₁ > dbl1₁ > 01 > top
proper₁ > dbl₁ > 0 > ok₁ > top
proper₁ > dbl₁ > 0 > 01 > top
proper₁ > nil > ok₁ > top
proper₁ > cons₂ > dbls₁ > ok₁ > top
proper₁ > cons₂ > indx₂ > ok₁ > top
proper₁ > sel₂ > ok₁ > top
proper₁ > dbl1₁ > ok₁ > top
proper₁ > dbl1₁ > 01 > top

Status:

01: []
sel₂: [1,2]
ok₁: [1]
dbls₁: [1]
0: []
indx₂: [2,1]
cons₂: [1,2]
active₁: [1]
dbl₁: [1]
proper₁: [1]
top: []
dbl1₁: [1]
nil: []

The following usable rules [FROCOS05] were oriented:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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:

DBL1(mark(X)) → DBL1(X)

The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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.

DBL1(mark(X)) → DBL1(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
DBL1(x1) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
dbl(x1) = x1
0 = 0
s(x1) = s(x1)
dbls(x1) = x1
nil = nil
cons(x1, x2) = cons(x1, x2)
sel(x1, x2) = sel(x1, x2)
indx(x1, x2) = indx(x1, x2)
from(x1) = x1
dbl1(x1) = dbl1(x1)
01 = 01
s1(x1) = s1(x1)
sel1(x1, x2) = sel1(x1, x2)
quote(x1) = x1
proper(x1) = x1
ok(x1) = ok
top(x1) = top

Lexicographic path order with status [LPO].
Precedence:

nil > mark₁
nil > ok
top > active₁ > 0 > mark₁
top > active₁ > 0 > 01 > ok
top > active₁ > s₁ > sel₂ > mark₁
top > active₁ > s₁ > dbl1₁ > mark₁
top > active₁ > s₁ > sel1₂ > mark₁
top > active₁ > s₁ > ok
top > active₁ > cons₂ > sel₂ > mark₁
top > active₁ > cons₂ > indx₂ > mark₁
top > active₁ > cons₂ > sel1₂ > mark₁
top > active₁ > cons₂ > ok
top > active₁ > s1₁ > mark₁
top > active₁ > s1₁ > ok

Status:

sel1₂: [2,1]
s1₁: [1]
01: []
sel₂: [2,1]
mark₁: [1]
0: []
indx₂: [2,1]
cons₂: [1,2]
active₁: [1]
ok: []
s₁: [1]
top: []
dbl1₁: [1]
nil: []

The following usable rules [FROCOS05] were oriented:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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:

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

The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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.

INDX(ok(X1), ok(X2)) → INDX(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
INDX(x1, x2) = INDX(x1)
ok(x1) = ok(x1)
mark(x1) = x1
active(x1) = active(x1)
dbl(x1) = x1
0 = 0
s(x1) = x1
dbls(x1) = x1
nil = nil
cons(x1, x2) = cons(x1, x2)
sel(x1, x2) = sel(x2)
indx(x1, x2) = x2
from(x1) = x1
dbl1(x1) = x1
01 = 01
s1(x1) = x1
sel1(x1, x2) = sel1(x2)
quote(x1) = x1
proper(x1) = proper(x1)
top(x1) = top

Lexicographic path order with status [LPO].
Precedence:

top > active₁ > 0
top > active₁ > nil > ok₁
top > active₁ > cons₂ > ok₁
top > active₁ > sel₁ > ok₁
top > active₁ > 01
top > active₁ > sel1₁ > ok₁
top > proper₁ > 0
top > proper₁ > nil > ok₁
top > proper₁ > cons₂ > ok₁
top > proper₁ > sel₁ > ok₁
top > proper₁ > 01
top > proper₁ > sel1₁ > ok₁

Status:

01: []
active₁: [1]
cons₂: [1,2]
sel₁: [1]
sel1₁: [1]
INDX₁: [1]
ok₁: [1]
proper₁: [1]
top: []
0: []
nil: []

The following usable rules [FROCOS05] were oriented:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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:

INDX(mark(X1), X2) → INDX(X1, X2)

The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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.

INDX(mark(X1), X2) → INDX(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
INDX(x1, x2) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
dbl(x1) = x1
0 = 0
s(x1) = x1
dbls(x1) = x1
nil = nil
cons(x1, x2) = cons(x1, x2)
sel(x1, x2) = sel(x1, x2)
indx(x1, x2) = indx(x1, x2)
from(x1) = from(x1)
dbl1(x1) = x1
01 = 01
s1(x1) = s1(x1)
sel1(x1, x2) = sel1(x1, x2)
quote(x1) = quote(x1)
proper(x1) = proper(x1)
ok(x1) = ok
top(x1) = top

Lexicographic path order with status [LPO].
Precedence:

top > active₁ > 0 > 01 > ok > indx₂ > mark₁
top > active₁ > 0 > 01 > ok > indx₂ > nil
top > active₁ > 0 > 01 > ok > indx₂ > cons₂
top > active₁ > 0 > 01 > ok > from₁
top > active₁ > 0 > 01 > ok > quote₁ > mark₁
top > active₁ > sel₂ > ok > indx₂ > mark₁
top > active₁ > sel₂ > ok > indx₂ > nil
top > active₁ > sel₂ > ok > indx₂ > cons₂
top > active₁ > sel₂ > ok > from₁
top > active₁ > sel₂ > ok > quote₁ > mark₁
top > active₁ > s1₁ > mark₁
top > active₁ > sel1₂ > ok > indx₂ > mark₁
top > active₁ > sel1₂ > ok > indx₂ > nil
top > active₁ > sel1₂ > ok > indx₂ > cons₂
top > active₁ > sel1₂ > ok > from₁
top > active₁ > sel1₂ > ok > quote₁ > mark₁
top > proper₁ > sel₂ > ok > indx₂ > mark₁
top > proper₁ > sel₂ > ok > indx₂ > nil
top > proper₁ > sel₂ > ok > indx₂ > cons₂
top > proper₁ > sel₂ > ok > from₁
top > proper₁ > sel₂ > ok > quote₁ > mark₁
top > proper₁ > 01 > ok > indx₂ > mark₁
top > proper₁ > 01 > ok > indx₂ > nil
top > proper₁ > 01 > ok > indx₂ > cons₂
top > proper₁ > 01 > ok > from₁
top > proper₁ > 01 > ok > quote₁ > mark₁
top > proper₁ > s1₁ > mark₁
top > proper₁ > sel1₂ > ok > indx₂ > mark₁
top > proper₁ > sel1₂ > ok > indx₂ > nil
top > proper₁ > sel1₂ > ok > indx₂ > cons₂
top > proper₁ > sel1₂ > ok > from₁
top > proper₁ > sel1₂ > ok > quote₁ > mark₁

Status:

sel1₂: [2,1]
s1₁: [1]
01: []
sel₂: [1,2]
from₁: [1]
quote₁: [1]
mark₁: [1]
0: []
indx₂: [2,1]
cons₂: [1,2]
active₁: [1]
ok: []
proper₁: [1]
top: []
nil: []

The following usable rules [FROCOS05] were oriented:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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:

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(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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.

SEL(mark(X1), 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(x1)
mark(x1) = mark(x1)
ok(x1) = ok(x1)
active(x1) = active(x1)
dbl(x1) = dbl(x1)
0 = 0
s(x1) = s(x1)
dbls(x1) = x1
nil = nil
cons(x1, x2) = cons(x1, x2)
sel(x1, x2) = sel(x1, x2)
indx(x1, x2) = indx(x1, x2)
from(x1) = from(x1)
dbl1(x1) = x1
01 = 01
s1(x1) = s1(x1)
sel1(x1, x2) = sel1(x1, x2)
quote(x1) = x1
proper(x1) = proper(x1)
top(x1) = top

Lexicographic path order with status [LPO].
Precedence:

SEL₁ > mark₁
active₁ > dbl₁ > 0 > ok₁ > mark₁
active₁ > dbl₁ > s₁ > sel1₂ > ok₁ > mark₁
active₁ > nil > ok₁ > mark₁
active₁ > cons₂ > indx₂ > ok₁ > mark₁
active₁ > cons₂ > sel1₂ > ok₁ > mark₁
active₁ > sel₂ > sel1₂ > ok₁ > mark₁
active₁ > from₁ > ok₁ > mark₁
active₁ > 01 > ok₁ > mark₁
active₁ > s1₁ > ok₁ > mark₁
proper₁ > dbl₁ > 0 > ok₁ > mark₁
proper₁ > dbl₁ > s₁ > sel1₂ > ok₁ > mark₁
proper₁ > nil > ok₁ > mark₁
proper₁ > cons₂ > indx₂ > ok₁ > mark₁
proper₁ > cons₂ > sel1₂ > ok₁ > mark₁
proper₁ > sel₂ > sel1₂ > ok₁ > mark₁
proper₁ > from₁ > ok₁ > mark₁
proper₁ > 01 > ok₁ > mark₁
proper₁ > s1₁ > ok₁ > mark₁
top > mark₁

Status:

sel₂: [1,2]
01: []
s1₁: [1]
sel1₂: [2,1]
from₁: [1]
mark₁: [1]
ok₁: [1]
SEL₁: [1]
0: []
indx₂: [2,1]
active₁: [1]
cons₂: [2,1]
dbl₁: [1]
s₁: [1]
proper₁: [1]
top: []
nil: []

The following usable rules [FROCOS05] were oriented:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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:

SEL(X1, mark(X2)) → SEL(X1, X2)

The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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.

SEL(X1, mark(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) = x2
mark(x1) = mark(x1)
active(x1) = active(x1)
dbl(x1) = dbl(x1)
0 = 0
s(x1) = s(x1)
dbls(x1) = x1
nil = nil
cons(x1, x2) = cons(x1, x2)
sel(x1, x2) = sel(x1, x2)
indx(x1, x2) = indx(x1, x2)
from(x1) = from(x1)
dbl1(x1) = dbl1(x1)
01 = 01
s1(x1) = s1(x1)
sel1(x1, x2) = sel1(x1, x2)
quote(x1) = x1
proper(x1) = proper(x1)
ok(x1) = ok
top(x1) = top

Lexicographic path order with status [LPO].
Precedence:

top > active₁ > dbl₁ > mark₁ > nil
top > active₁ > dbl₁ > 0 > 01 > nil
top > active₁ > dbl₁ > ok > nil
top > active₁ > s₁ > mark₁ > nil
top > active₁ > s₁ > ok > nil
top > active₁ > sel₂ > mark₁ > nil
top > active₁ > sel₂ > ok > nil
top > active₁ > indx₂ > mark₁ > nil
top > active₁ > indx₂ > cons₂ > nil
top > active₁ > indx₂ > ok > nil
top > active₁ > from₁ > mark₁ > nil
top > active₁ > from₁ > cons₂ > nil
top > active₁ > from₁ > ok > nil
top > active₁ > dbl1₁ > mark₁ > nil
top > active₁ > dbl1₁ > 01 > nil
top > active₁ > dbl1₁ > ok > nil
top > active₁ > s1₁ > mark₁ > nil
top > active₁ > s1₁ > ok > nil
top > active₁ > sel1₂ > mark₁ > nil
top > active₁ > sel1₂ > ok > nil
top > proper₁ > dbl₁ > mark₁ > nil
top > proper₁ > dbl₁ > 0 > 01 > nil
top > proper₁ > dbl₁ > ok > nil
top > proper₁ > s₁ > mark₁ > nil
top > proper₁ > s₁ > ok > nil
top > proper₁ > sel₂ > mark₁ > nil
top > proper₁ > sel₂ > ok > nil
top > proper₁ > indx₂ > mark₁ > nil
top > proper₁ > indx₂ > cons₂ > nil
top > proper₁ > indx₂ > ok > nil
top > proper₁ > from₁ > mark₁ > nil
top > proper₁ > from₁ > cons₂ > nil
top > proper₁ > from₁ > ok > nil
top > proper₁ > dbl1₁ > mark₁ > nil
top > proper₁ > dbl1₁ > 01 > nil
top > proper₁ > dbl1₁ > ok > nil
top > proper₁ > s1₁ > mark₁ > nil
top > proper₁ > s1₁ > ok > nil
top > proper₁ > sel1₂ > mark₁ > nil
top > proper₁ > sel1₂ > ok > nil

Status:

sel₂: [1,2]
01: []
s1₁: [1]
sel1₂: [1,2]
from₁: [1]
mark₁: [1]
0: []
indx₂: [2,1]
active₁: [1]
cons₂: [2,1]
dbl₁: [1]
s₁: [1]
ok: []
proper₁: [1]
top: []
nil: []
dbl1₁: [1]

The following usable rules [FROCOS05] were oriented:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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:

DBLS(ok(X)) → DBLS(X)
DBLS(mark(X)) → DBLS(X)

The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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.

DBLS(ok(X)) → DBLS(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
DBLS(x1) = x1
ok(x1) = ok(x1)
mark(x1) = x1
active(x1) = active(x1)
dbl(x1) = dbl(x1)
0 = 0
s(x1) = x1
dbls(x1) = dbls(x1)
nil = nil
cons(x1, x2) = cons(x1, x2)
sel(x1, x2) = sel(x1, x2)
indx(x1, x2) = indx(x1, x2)
from(x1) = x1
dbl1(x1) = dbl1(x1)
01 = 01
s1(x1) = x1
sel1(x1, x2) = x2
quote(x1) = x1
proper(x1) = proper(x1)
top(x1) = top

Lexicographic path order with status [LPO].
Precedence:

active₁ > dbl₁ > 0 > ok₁ > top
active₁ > dbl₁ > 0 > 01 > top
active₁ > nil > ok₁ > top
active₁ > cons₂ > dbls₁ > ok₁ > top
active₁ > cons₂ > indx₂ > ok₁ > top
active₁ > sel₂ > ok₁ > top
active₁ > dbl1₁ > ok₁ > top
active₁ > dbl1₁ > 01 > top
proper₁ > dbl₁ > 0 > ok₁ > top
proper₁ > dbl₁ > 0 > 01 > top
proper₁ > nil > ok₁ > top
proper₁ > cons₂ > dbls₁ > ok₁ > top
proper₁ > cons₂ > indx₂ > ok₁ > top
proper₁ > sel₂ > ok₁ > top
proper₁ > dbl1₁ > ok₁ > top
proper₁ > dbl1₁ > 01 > top

Status:

01: []
sel₂: [1,2]
ok₁: [1]
dbls₁: [1]
0: []
indx₂: [2,1]
cons₂: [1,2]
active₁: [1]
dbl₁: [1]
proper₁: [1]
top: []
dbl1₁: [1]
nil: []

The following usable rules [FROCOS05] were oriented:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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:

DBLS(mark(X)) → DBLS(X)

The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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.

DBLS(mark(X)) → DBLS(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
DBLS(x1) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
dbl(x1) = x1
0 = 0
s(x1) = s(x1)
dbls(x1) = x1
nil = nil
cons(x1, x2) = cons(x1, x2)
sel(x1, x2) = sel(x1, x2)
indx(x1, x2) = indx(x1, x2)
from(x1) = x1
dbl1(x1) = dbl1(x1)
01 = 01
s1(x1) = s1(x1)
sel1(x1, x2) = sel1(x1, x2)
quote(x1) = x1
proper(x1) = x1
ok(x1) = ok
top(x1) = top

Lexicographic path order with status [LPO].
Precedence:

nil > mark₁
nil > ok
top > active₁ > 0 > mark₁
top > active₁ > 0 > 01 > ok
top > active₁ > s₁ > sel₂ > mark₁
top > active₁ > s₁ > dbl1₁ > mark₁
top > active₁ > s₁ > sel1₂ > mark₁
top > active₁ > s₁ > ok
top > active₁ > cons₂ > sel₂ > mark₁
top > active₁ > cons₂ > indx₂ > mark₁
top > active₁ > cons₂ > sel1₂ > mark₁
top > active₁ > cons₂ > ok
top > active₁ > s1₁ > mark₁
top > active₁ > s1₁ > ok

Status:

sel1₂: [2,1]
s1₁: [1]
01: []
sel₂: [2,1]
mark₁: [1]
0: []
indx₂: [2,1]
cons₂: [1,2]
active₁: [1]
ok: []
s₁: [1]
top: []
dbl1₁: [1]
nil: []

The following usable rules [FROCOS05] were oriented:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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:

DBL(ok(X)) → DBL(X)
DBL(mark(X)) → DBL(X)

The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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.

DBL(ok(X)) → DBL(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
DBL(x1) = x1
ok(x1) = ok(x1)
mark(x1) = x1
active(x1) = active(x1)
dbl(x1) = dbl(x1)
0 = 0
s(x1) = x1
dbls(x1) = dbls(x1)
nil = nil
cons(x1, x2) = cons(x1, x2)
sel(x1, x2) = sel(x1, x2)
indx(x1, x2) = indx(x1, x2)
from(x1) = x1
dbl1(x1) = dbl1(x1)
01 = 01
s1(x1) = x1
sel1(x1, x2) = x2
quote(x1) = x1
proper(x1) = proper(x1)
top(x1) = top

Lexicographic path order with status [LPO].
Precedence:

active₁ > dbl₁ > 0 > ok₁ > top
active₁ > dbl₁ > 0 > 01 > top
active₁ > nil > ok₁ > top
active₁ > cons₂ > dbls₁ > ok₁ > top
active₁ > cons₂ > indx₂ > ok₁ > top
active₁ > sel₂ > ok₁ > top
active₁ > dbl1₁ > ok₁ > top
active₁ > dbl1₁ > 01 > top
proper₁ > dbl₁ > 0 > ok₁ > top
proper₁ > dbl₁ > 0 > 01 > top
proper₁ > nil > ok₁ > top
proper₁ > cons₂ > dbls₁ > ok₁ > top
proper₁ > cons₂ > indx₂ > ok₁ > top
proper₁ > sel₂ > ok₁ > top
proper₁ > dbl1₁ > ok₁ > top
proper₁ > dbl1₁ > 01 > top

Status:

01: []
sel₂: [1,2]
ok₁: [1]
dbls₁: [1]
0: []
indx₂: [2,1]
cons₂: [1,2]
active₁: [1]
dbl₁: [1]
proper₁: [1]
top: []
dbl1₁: [1]
nil: []

The following usable rules [FROCOS05] were oriented:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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:

DBL(mark(X)) → DBL(X)

The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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.

DBL(mark(X)) → DBL(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
DBL(x1) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
dbl(x1) = x1
0 = 0
s(x1) = s(x1)
dbls(x1) = x1
nil = nil
cons(x1, x2) = cons(x1, x2)
sel(x1, x2) = sel(x1, x2)
indx(x1, x2) = indx(x1, x2)
from(x1) = x1
dbl1(x1) = dbl1(x1)
01 = 01
s1(x1) = s1(x1)
sel1(x1, x2) = sel1(x1, x2)
quote(x1) = x1
proper(x1) = x1
ok(x1) = ok
top(x1) = top

Lexicographic path order with status [LPO].
Precedence:

nil > mark₁
nil > ok
top > active₁ > 0 > mark₁
top > active₁ > 0 > 01 > ok
top > active₁ > s₁ > sel₂ > mark₁
top > active₁ > s₁ > dbl1₁ > mark₁
top > active₁ > s₁ > sel1₂ > mark₁
top > active₁ > s₁ > ok
top > active₁ > cons₂ > sel₂ > mark₁
top > active₁ > cons₂ > indx₂ > mark₁
top > active₁ > cons₂ > sel1₂ > mark₁
top > active₁ > cons₂ > ok
top > active₁ > s1₁ > mark₁
top > active₁ > s1₁ > ok

Status:

sel1₂: [2,1]
s1₁: [1]
01: []
sel₂: [2,1]
mark₁: [1]
0: []
indx₂: [2,1]
cons₂: [1,2]
active₁: [1]
ok: []
s₁: [1]
top: []
dbl1₁: [1]
nil: []

The following usable rules [FROCOS05] were oriented:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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:

PROPER(s(X)) → PROPER(X)
PROPER(dbl(X)) → PROPER(X)
PROPER(dbls(X)) → PROPER(X)
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(sel(X1, X2)) → PROPER(X1)
PROPER(sel(X1, X2)) → PROPER(X2)
PROPER(indx(X1, X2)) → PROPER(X1)
PROPER(indx(X1, X2)) → PROPER(X2)
PROPER(from(X)) → PROPER(X)
PROPER(dbl1(X)) → PROPER(X)
PROPER(s1(X)) → PROPER(X)
PROPER(sel1(X1, X2)) → PROPER(X1)
PROPER(sel1(X1, X2)) → PROPER(X2)
PROPER(quote(X)) → PROPER(X)

The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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.

PROPER(dbl(X)) → PROPER(X)
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(sel(X1, X2)) → PROPER(X1)
PROPER(sel(X1, X2)) → PROPER(X2)
PROPER(indx(X1, X2)) → PROPER(X1)
PROPER(indx(X1, X2)) → PROPER(X2)
PROPER(sel1(X1, X2)) → PROPER(X1)
PROPER(sel1(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) = PROPER(x1)
s(x1) = x1
dbl(x1) = dbl(x1)
dbls(x1) = x1
cons(x1, x2) = cons(x1, x2)
sel(x1, x2) = sel(x1, x2)
indx(x1, x2) = indx(x1, x2)
from(x1) = x1
dbl1(x1) = x1
s1(x1) = x1
sel1(x1, x2) = sel1(x1, x2)
quote(x1) = x1
active(x1) = active(x1)
0 = 0
mark(x1) = mark(x1)
nil = nil
01 = 01
proper(x1) = x1
ok(x1) = ok
top(x1) = top

Lexicographic path order with status [LPO].
Precedence:

active₁ > dbl₁ > ok > mark₁
active₁ > sel₂ > PROPER₁ > mark₁
active₁ > sel₂ > sel1₂ > ok > mark₁
active₁ > indx₂ > cons₂ > ok > mark₁
active₁ > indx₂ > nil > ok > mark₁
0 > 01 > ok > mark₁
top > mark₁

Status:

01: []
sel1₂: [1,2]
sel₂: [1,2]
PROPER₁: [1]
mark₁: [1]
0: []
indx₂: [1,2]
active₁: [1]
cons₂: [1,2]
dbl₁: [1]
ok: []
top: []
nil: []

The following usable rules [FROCOS05] were oriented:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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:

PROPER(s(X)) → PROPER(X)
PROPER(dbls(X)) → PROPER(X)
PROPER(from(X)) → PROPER(X)
PROPER(dbl1(X)) → PROPER(X)
PROPER(s1(X)) → PROPER(X)
PROPER(quote(X)) → PROPER(X)

The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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.

PROPER(dbl1(X)) → PROPER(X)
PROPER(s1(X)) → PROPER(X)
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) = x1
s(x1) = x1
dbls(x1) = x1
from(x1) = x1
dbl1(x1) = dbl1(x1)
s1(x1) = s1(x1)
quote(x1) = quote(x1)
active(x1) = active(x1)
dbl(x1) = x1
0 = 0
mark(x1) = x1
nil = nil
cons(x1, x2) = cons(x1, x2)
sel(x1, x2) = x2
indx(x1, x2) = indx(x1, x2)
01 = 01
sel1(x1, x2) = x2
proper(x1) = proper(x1)
ok(x1) = ok
top(x1) = top

Lexicographic path order with status [LPO].
Precedence:

0 > 01 > ok > active₁ > quote₁ > dbl1₁ > top
0 > 01 > ok > active₁ > quote₁ > s1₁ > top
0 > 01 > ok > active₁ > indx₂ > cons₂ > top
nil > ok > active₁ > quote₁ > dbl1₁ > top
nil > ok > active₁ > quote₁ > s1₁ > top
nil > ok > active₁ > indx₂ > cons₂ > top
proper₁ > quote₁ > dbl1₁ > top
proper₁ > quote₁ > s1₁ > top
proper₁ > indx₂ > cons₂ > top

Status:

s1₁: [1]
01: []
quote₁: [1]
0: []
indx₂: [2,1]
active₁: [1]
cons₂: [1,2]
ok: []
proper₁: [1]
top: []
dbl1₁: [1]
nil: []

The following usable rules [FROCOS05] were oriented:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(82) Obligation:

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

PROPER(s(X)) → PROPER(X)
PROPER(dbls(X)) → PROPER(X)
PROPER(from(X)) → PROPER(X)

The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(from(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1) = x1
s(x1) = x1
dbls(x1) = x1
from(x1) = from(x1)
active(x1) = active(x1)
dbl(x1) = x1
0 = 0
mark(x1) = x1
nil = nil
cons(x1, x2) = cons(x1, x2)
sel(x1, x2) = x2
indx(x1, x2) = indx(x1, x2)
dbl1(x1) = x1
01 = 01
s1(x1) = s1(x1)
sel1(x1, x2) = x2
quote(x1) = x1
proper(x1) = proper(x1)
ok(x1) = ok
top(x1) = top

Lexicographic path order with status [LPO].
Precedence:

top > active₁ > from₁ > ok > s1₁
top > active₁ > 0 > s1₁
top > active₁ > cons₂ > ok > s1₁
top > active₁ > indx₂ > ok > s1₁
top > active₁ > 01 > ok > s1₁
top > proper₁ > from₁ > ok > s1₁
top > proper₁ > 0 > s1₁
top > proper₁ > nil > ok > s1₁
top > proper₁ > cons₂ > ok > s1₁
top > proper₁ > indx₂ > ok > s1₁
top > proper₁ > 01 > ok > s1₁

Status:

indx₂: [1,2]
01: []
s1₁: [1]
from₁: [1]
active₁: [1]
cons₂: [2,1]
ok: []
proper₁: [1]
top: []
0: []
nil: []

The following usable rules [FROCOS05] were oriented:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(84) Obligation:

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

PROPER(s(X)) → PROPER(X)
PROPER(dbls(X)) → PROPER(X)

The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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.

(85) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(s(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1) = x1
s(x1) = s(x1)
dbls(x1) = x1
active(x1) = active(x1)
dbl(x1) = x1
0 = 0
mark(x1) = mark
nil = nil
cons(x1, x2) = cons(x1, x2)
sel(x1, x2) = sel
indx(x1, x2) = indx
from(x1) = from(x1)
dbl1(x1) = x1
01 = 01
s1(x1) = x1
sel1(x1, x2) = sel1
quote(x1) = quote
proper(x1) = proper(x1)
ok(x1) = ok
top(x1) = top

Lexicographic path order with status [LPO].
Precedence:

nil > mark > top > active₁
nil > ok > top > active₁
sel > sel1 > proper₁ > s₁ > mark > top > active₁
sel > sel1 > proper₁ > s₁ > ok > top > active₁
sel > sel1 > proper₁ > 0 > mark > top > active₁
sel > sel1 > proper₁ > 0 > ok > top > active₁
sel > sel1 > proper₁ > cons₂ > mark > top > active₁
sel > sel1 > proper₁ > cons₂ > ok > top > active₁
sel > sel1 > proper₁ > indx > mark > top > active₁
sel > sel1 > proper₁ > indx > ok > top > active₁
sel > sel1 > proper₁ > from₁ > mark > top > active₁
sel > sel1 > proper₁ > from₁ > ok > top > active₁
sel > sel1 > proper₁ > quote > mark > top > active₁
sel > sel1 > proper₁ > quote > ok > top > active₁
01 > ok > top > active₁

Status:

01: []
from₁: [1]
sel1: []
sel: []
0: []
cons₂: [1,2]
active₁: [1]
quote: []
mark: []
s₁: [1]
ok: []
proper₁: [1]
top: []
nil: []
indx: []

The following usable rules [FROCOS05] were oriented:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(86) Obligation:

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

PROPER(dbls(X)) → PROPER(X)

The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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.

(87) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(dbls(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1) = x1
dbls(x1) = dbls(x1)
active(x1) = active(x1)
dbl(x1) = x1
0 = 0
mark(x1) = mark
s(x1) = s(x1)
nil = nil
cons(x1, x2) = x1
sel(x1, x2) = sel(x1)
indx(x1, x2) = indx(x1, x2)
from(x1) = x1
dbl1(x1) = dbl1
01 = 01
s1(x1) = s1
sel1(x1, x2) = sel1(x2)
quote(x1) = quote(x1)
proper(x1) = x1
ok(x1) = x1
top(x1) = top

Lexicographic path order with status [LPO].
Precedence:

dbl1 > active₁ > dbls₁ > mark > top
dbl1 > active₁ > dbls₁ > nil
dbl1 > active₁ > 0
dbl1 > active₁ > s₁
dbl1 > active₁ > sel₁ > mark > top
dbl1 > active₁ > indx₂ > mark > top
dbl1 > active₁ > indx₂ > nil
dbl1 > active₁ > quote₁ > 01
dbl1 > active₁ > quote₁ > sel1₁ > mark > top
s1 > active₁ > dbls₁ > mark > top
s1 > active₁ > dbls₁ > nil
s1 > active₁ > 0
s1 > active₁ > s₁
s1 > active₁ > sel₁ > mark > top
s1 > active₁ > indx₂ > mark > top
s1 > active₁ > indx₂ > nil
s1 > active₁ > quote₁ > 01
s1 > active₁ > quote₁ > sel1₁ > mark > top

Status:

01: []
quote₁: [1]
dbls₁: [1]
0: []
dbl1: []
indx₂: [2,1]
sel1₁: [1]
s1: []
sel₁: [1]
active₁: [1]
mark: []
s₁: [1]
top: []
nil: []

The following usable rules [FROCOS05] were oriented:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(88) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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.

(89) PisEmptyProof (EQUIVALENT transformation)

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

(90) TRUE

(91) Obligation:

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

ACTIVE(dbls(X)) → ACTIVE(X)
ACTIVE(dbl(X)) → ACTIVE(X)
ACTIVE(sel(X1, X2)) → ACTIVE(X1)
ACTIVE(sel(X1, X2)) → ACTIVE(X2)
ACTIVE(indx(X1, X2)) → ACTIVE(X1)
ACTIVE(dbl1(X)) → ACTIVE(X)
ACTIVE(s1(X)) → ACTIVE(X)
ACTIVE(sel1(X1, X2)) → ACTIVE(X1)
ACTIVE(sel1(X1, X2)) → ACTIVE(X2)
ACTIVE(quote(X)) → ACTIVE(X)

The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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.

(92) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(dbls(X)) → ACTIVE(X)
ACTIVE(sel(X1, X2)) → ACTIVE(X1)
ACTIVE(sel(X1, X2)) → ACTIVE(X2)
ACTIVE(dbl1(X)) → ACTIVE(X)
ACTIVE(sel1(X1, X2)) → ACTIVE(X1)
ACTIVE(sel1(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)
dbls(x1) = dbls(x1)
dbl(x1) = x1
sel(x1, x2) = sel(x1, x2)
indx(x1, x2) = x1
dbl1(x1) = dbl1(x1)
s1(x1) = x1
sel1(x1, x2) = sel1(x1, x2)
quote(x1) = x1
active(x1) = active(x1)
0 = 0
mark(x1) = mark
s(x1) = s(x1)
nil = nil
cons(x1, x2) = cons
from(x1) = from
01 = 01
proper(x1) = proper(x1)
ok(x1) = x1
top(x1) = top

Lexicographic path order with status [LPO].
Precedence:

active₁ > 0 > mark > proper₁ > dbls₁ > ACTIVE₁ > cons
active₁ > 0 > mark > proper₁ > dbls₁ > nil > cons
active₁ > 0 > mark > proper₁ > s₁ > sel₂ > cons
active₁ > 0 > mark > proper₁ > s₁ > dbl1₁ > cons
active₁ > 0 > mark > proper₁ > s₁ > sel1₂ > cons
active₁ > 0 > mark > proper₁ > from > cons
active₁ > 0 > mark > proper₁ > 01 > cons
top > proper₁ > dbls₁ > ACTIVE₁ > cons
top > proper₁ > dbls₁ > nil > cons
top > proper₁ > s₁ > sel₂ > cons
top > proper₁ > s₁ > dbl1₁ > cons
top > proper₁ > s₁ > sel1₂ > cons
top > proper₁ > from > cons
top > proper₁ > 01 > cons

Status:

01: []
sel1₂: [2,1]
sel₂: [1,2]
dbls₁: [1]
0: []
ACTIVE₁: [1]
from: []
active₁: [1]
cons: []
mark: []
s₁: [1]
proper₁: [1]
top: []
nil: []
dbl1₁: [1]

The following usable rules [FROCOS05] were oriented:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(93) Obligation:

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

ACTIVE(dbl(X)) → ACTIVE(X)
ACTIVE(indx(X1, X2)) → ACTIVE(X1)
ACTIVE(s1(X)) → ACTIVE(X)
ACTIVE(quote(X)) → ACTIVE(X)

The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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.

(94) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(indx(X1, X2)) → ACTIVE(X1)
ACTIVE(quote(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)
dbl(x1) = x1
indx(x1, x2) = indx(x1, x2)
s1(x1) = x1
quote(x1) = quote(x1)
active(x1) = active(x1)
0 = 0
mark(x1) = x1
s(x1) = x1
dbls(x1) = x1
nil = nil
cons(x1, x2) = cons(x1, x2)
sel(x1, x2) = sel(x2)
from(x1) = from(x1)
dbl1(x1) = x1
01 = 01
sel1(x1, x2) = sel1(x2)
proper(x1) = x1
ok(x1) = ok
top(x1) = top

Lexicographic path order with status [LPO].
Precedence:

top > active₁ > indx₂ > ACTIVE₁
top > active₁ > indx₂ > nil > ok
top > active₁ > indx₂ > cons₂ > sel₁ > ok
top > active₁ > quote₁ > ACTIVE₁
top > active₁ > quote₁ > 01 > ok
top > active₁ > quote₁ > sel1₁ > ok
top > active₁ > 0 > ok
top > active₁ > from₁ > ok

Status:

01: []
from₁: [1]
quote₁: [1]
0: []
ACTIVE₁: [1]
indx₂: [1,2]
sel1₁: [1]
sel₁: [1]
cons₂: [1,2]
active₁: [1]
ok: []
top: []
nil: []

The following usable rules [FROCOS05] were oriented:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(95) Obligation:

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

ACTIVE(dbl(X)) → ACTIVE(X)
ACTIVE(s1(X)) → ACTIVE(X)

The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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.

(96) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(dbl(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
dbl(x1) = dbl(x1)
s1(x1) = x1
active(x1) = active(x1)
0 = 0
mark(x1) = x1
s(x1) = x1
dbls(x1) = dbls(x1)
nil = nil
cons(x1, x2) = cons(x1, x2)
sel(x1, x2) = x2
indx(x1, x2) = indx(x2)
from(x1) = from(x1)
dbl1(x1) = dbl1(x1)
01 = 01
sel1(x1, x2) = sel1(x2)
quote(x1) = x1
proper(x1) = proper(x1)
ok(x1) = ok
top(x1) = top

Lexicographic path order with status [LPO].
Precedence:

proper₁ > ok > active₁ > 0 > top
proper₁ > ok > active₁ > nil > top
proper₁ > ok > active₁ > indx₁ > cons₂ > dbls₁ > dbl₁ > top
proper₁ > ok > active₁ > from₁ > top
proper₁ > ok > active₁ > dbl1₁ > top
proper₁ > ok > active₁ > 01 > top
proper₁ > ok > active₁ > sel1₁ > top

Status:

01: []
from₁: [1]
dbls₁: [1]
0: []
indx₁: [1]
sel1₁: [1]
cons₂: [1,2]
active₁: [1]
dbl₁: [1]
ok: []
proper₁: [1]
top: []
dbl1₁: [1]
nil: []

The following usable rules [FROCOS05] were oriented:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(97) Obligation:

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

ACTIVE(s1(X)) → ACTIVE(X)

The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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.

(98) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(s1(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
s1(x1) = s1(x1)
active(x1) = active(x1)
dbl(x1) = dbl
0 = 0
mark(x1) = mark
s(x1) = s
dbls(x1) = dbls
nil = nil
cons(x1, x2) = cons(x1)
sel(x1, x2) = x2
indx(x1, x2) = indx
from(x1) = x1
dbl1(x1) = dbl1(x1)
01 = 01
sel1(x1, x2) = sel1(x1)
quote(x1) = quote(x1)
proper(x1) = proper(x1)
ok(x1) = ok
top(x1) = top

Lexicographic path order with status [LPO].
Precedence:

active₁ > dbls > nil > ok > quote₁ > mark
active₁ > dbls > proper₁ > s > s1₁ > ok > quote₁ > mark
active₁ > dbls > proper₁ > s > dbl > 0 > ok > quote₁ > mark
active₁ > dbls > proper₁ > s > dbl > dbl1₁ > ok > quote₁ > mark
active₁ > dbls > proper₁ > s > sel1₁ > ok > quote₁ > mark
active₁ > dbls > proper₁ > indx > cons₁ > ok > quote₁ > mark
01 > ok > quote₁ > mark
top > proper₁ > s > s1₁ > ok > quote₁ > mark
top > proper₁ > s > dbl > 0 > ok > quote₁ > mark
top > proper₁ > s > dbl > dbl1₁ > ok > quote₁ > mark
top > proper₁ > s > sel1₁ > ok > quote₁ > mark
top > proper₁ > indx > cons₁ > ok > quote₁ > mark

Status:

s1₁: [1]
01: []
cons₁: [1]
quote₁: [1]
dbl: []
s: []
0: []
active₁: [1]
sel1₁: [1]
mark: []
dbls: []
ok: []
proper₁: [1]
top: []
nil: []
dbl1₁: [1]
indx: []

The following usable rules [FROCOS05] were oriented:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(99) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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) PisEmptyProof (EQUIVALENT transformation)

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

(101) TRUE

(102) 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(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
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.