Termination w.r.t. Q proof of /tmp/tmpFAnHQR/PALINDROME_nokinds_noand

Termination w.r.t. Q of the given QTRS could be proven:

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 PisEmptyProof (⇔)

    ↳24 TRUE

  ↳25 QDP

    ↳26 QDPOrderProof (⇔)

    ↳27 QDP

    ↳28 PisEmptyProof (⇔)

    ↳29 TRUE

  ↳30 QDP

    ↳31 QDPOrderProof (⇔)

    ↳32 QDP

    ↳33 QDPOrderProof (⇔)

    ↳34 QDP

    ↳35 PisEmptyProof (⇔)

    ↳36 TRUE

  ↳37 QDP

    ↳38 QDPOrderProof (⇔)

    ↳39 QDP

    ↳40 QDPOrderProof (⇔)

    ↳41 QDP

    ↳42 PisEmptyProof (⇔)

    ↳43 TRUE

  ↳44 QDP

    ↳45 QDPOrderProof (⇔)

    ↳46 QDP

    ↳47 QDPOrderProof (⇔)

    ↳48 QDP

    ↳49 PisEmptyProof (⇔)

    ↳50 TRUE

  ↳51 QDP

    ↳52 QDPOrderProof (⇔)

    ↳53 QDP

    ↳54 QDPOrderProof (⇔)

    ↳55 QDP

    ↳56 PisEmptyProof (⇔)

    ↳57 TRUE

  ↳58 QDP

    ↳59 QDPOrderProof (⇔)

    ↳60 QDP

    ↳61 QDPOrderProof (⇔)

    ↳62 QDP

    ↳63 PisEmptyProof (⇔)

    ↳64 TRUE

  ↳65 QDP

    ↳66 QDPOrderProof (⇔)

    ↳67 QDP

    ↳68 QDPOrderProof (⇔)

    ↳69 QDP

    ↳70 PisEmptyProof (⇔)

    ↳71 TRUE

  ↳72 QDP

    ↳73 QDPOrderProof (⇔)

    ↳74 QDP

    ↳75 QDPOrderProof (⇔)

    ↳76 QDP

    ↳77 PisEmptyProof (⇔)

    ↳78 TRUE

  ↳79 QDP

    ↳80 QDPOrderProof (⇔)

    ↳81 QDP

    ↳82 QDPOrderProof (⇔)

    ↳83 QDP

    ↳84 PisEmptyProof (⇔)

    ↳85 TRUE

  ↳86 QDP

    ↳87 QDPOrderProof (⇔)

    ↳88 QDP

    ↳89 QDPOrderProof (⇔)

    ↳90 QDP

    ↳91 PisEmptyProof (⇔)

    ↳92 TRUE

  ↳93 QDP

    ↳94 QDPOrderProof (⇔)

    ↳95 QDP

    ↳96 QDPOrderProof (⇔)

    ↳97 QDP

    ↳98 PisEmptyProof (⇔)

    ↳99 TRUE

  ↳100 QDP

    ↳101 QDPOrderProof (⇔)

    ↳102 QDP

    ↳103 QDPOrderProof (⇔)

    ↳104 QDP

    ↳105 PisEmptyProof (⇔)

    ↳106 TRUE

  ↳107 QDP

    ↳108 QDPOrderProof (⇔)

    ↳109 QDP

    ↳110 QDPOrderProof (⇔)

    ↳111 QDP

    ↳112 PisEmptyProof (⇔)

    ↳113 TRUE

  ↳114 QDP

    ↳115 QDPOrderProof (⇔)

    ↳116 QDP

    ↳117 QDPOrderProof (⇔)

    ↳118 QDP

    ↳119 PisEmptyProof (⇔)

    ↳120 TRUE

  ↳121 QDP

    ↳122 QDPOrderProof (⇔)

    ↳123 QDP

    ↳124 QDPOrderProof (⇔)

    ↳125 QDP

    ↳126 QDPOrderProof (⇔)

    ↳127 QDP

    ↳128 QDPOrderProof (⇔)

    ↳129 QDP

    ↳130 QDPOrderProof (⇔)

    ↳131 QDP

    ↳132 QDPOrderProof (⇔)

    ↳133 QDP

    ↳134 QDPOrderProof (⇔)

    ↳135 QDP

    ↳136 QDPOrderProof (⇔)

    ↳137 QDP

    ↳138 QDPOrderProof (⇔)

    ↳139 QDP

    ↳140 PisEmptyProof (⇔)

    ↳141 TRUE

  ↳142 QDP

    ↳143 QDPOrderProof (⇔)

    ↳144 QDP

    ↳145 PisEmptyProof (⇔)

    ↳146 TRUE

  ↳147 QDP

    ↳148 QDPOrderProof (⇔)

    ↳149 QDP

    ↳150 QDPOrderProof (⇔)

    ↳151 QDP

    ↳152 PisEmptyProof (⇔)

    ↳153 TRUE

(0) Obligation:

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

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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(__(__(X, Y), Z)) → __¹(X, __(Y, Z))
ACTIVE(__(__(X, Y), Z)) → __¹(Y, Z)
ACTIVE(U21(tt, V2)) → U22¹(isList(V2))
ACTIVE(U21(tt, V2)) → ISLIST(V2)
ACTIVE(U41(tt, V2)) → U42¹(isNeList(V2))
ACTIVE(U41(tt, V2)) → ISNELIST(V2)
ACTIVE(U51(tt, V2)) → U52¹(isList(V2))
ACTIVE(U51(tt, V2)) → ISLIST(V2)
ACTIVE(U71(tt, P)) → U72¹(isPal(P))
ACTIVE(U71(tt, P)) → ISPAL(P)
ACTIVE(isList(V)) → U11¹(isNeList(V))
ACTIVE(isList(V)) → ISNELIST(V)
ACTIVE(isList(__(V1, V2))) → U21¹(isList(V1), V2)
ACTIVE(isList(__(V1, V2))) → ISLIST(V1)
ACTIVE(isNeList(V)) → U31¹(isQid(V))
ACTIVE(isNeList(V)) → ISQID(V)
ACTIVE(isNeList(__(V1, V2))) → U41¹(isList(V1), V2)
ACTIVE(isNeList(__(V1, V2))) → ISLIST(V1)
ACTIVE(isNeList(__(V1, V2))) → U51¹(isNeList(V1), V2)
ACTIVE(isNeList(__(V1, V2))) → ISNELIST(V1)
ACTIVE(isNePal(V)) → U61¹(isQid(V))
ACTIVE(isNePal(V)) → ISQID(V)
ACTIVE(isNePal(__(I, __(P, I)))) → U71¹(isQid(I), P)
ACTIVE(isNePal(__(I, __(P, I)))) → ISQID(I)
ACTIVE(isPal(V)) → U81¹(isNePal(V))
ACTIVE(isPal(V)) → ISNEPAL(V)
ACTIVE(__(X1, X2)) → __¹(active(X1), X2)
ACTIVE(__(X1, X2)) → ACTIVE(X1)
ACTIVE(__(X1, X2)) → __¹(X1, active(X2))
ACTIVE(__(X1, X2)) → ACTIVE(X2)
ACTIVE(U11(X)) → U11¹(active(X))
ACTIVE(U11(X)) → ACTIVE(X)
ACTIVE(U21(X1, X2)) → U21¹(active(X1), X2)
ACTIVE(U21(X1, X2)) → ACTIVE(X1)
ACTIVE(U22(X)) → U22¹(active(X))
ACTIVE(U22(X)) → ACTIVE(X)
ACTIVE(U31(X)) → U31¹(active(X))
ACTIVE(U31(X)) → ACTIVE(X)
ACTIVE(U41(X1, X2)) → U41¹(active(X1), X2)
ACTIVE(U41(X1, X2)) → ACTIVE(X1)
ACTIVE(U42(X)) → U42¹(active(X))
ACTIVE(U42(X)) → ACTIVE(X)
ACTIVE(U51(X1, X2)) → U51¹(active(X1), X2)
ACTIVE(U51(X1, X2)) → ACTIVE(X1)
ACTIVE(U52(X)) → U52¹(active(X))
ACTIVE(U52(X)) → ACTIVE(X)
ACTIVE(U61(X)) → U61¹(active(X))
ACTIVE(U61(X)) → ACTIVE(X)
ACTIVE(U71(X1, X2)) → U71¹(active(X1), X2)
ACTIVE(U71(X1, X2)) → ACTIVE(X1)
ACTIVE(U72(X)) → U72¹(active(X))
ACTIVE(U72(X)) → ACTIVE(X)
ACTIVE(U81(X)) → U81¹(active(X))
ACTIVE(U81(X)) → ACTIVE(X)
__¹(mark(X1), X2) → __¹(X1, X2)
__¹(X1, mark(X2)) → __¹(X1, X2)
U11¹(mark(X)) → U11¹(X)
U21¹(mark(X1), X2) → U21¹(X1, X2)
U22¹(mark(X)) → U22¹(X)
U31¹(mark(X)) → U31¹(X)
U41¹(mark(X1), X2) → U41¹(X1, X2)
U42¹(mark(X)) → U42¹(X)
U51¹(mark(X1), X2) → U51¹(X1, X2)
U52¹(mark(X)) → U52¹(X)
U61¹(mark(X)) → U61¹(X)
U71¹(mark(X1), X2) → U71¹(X1, X2)
U72¹(mark(X)) → U72¹(X)
U81¹(mark(X)) → U81¹(X)
PROPER(__(X1, X2)) → __¹(proper(X1), proper(X2))
PROPER(__(X1, X2)) → PROPER(X1)
PROPER(__(X1, X2)) → PROPER(X2)
PROPER(U11(X)) → U11¹(proper(X))
PROPER(U11(X)) → PROPER(X)
PROPER(U21(X1, X2)) → U21¹(proper(X1), proper(X2))
PROPER(U21(X1, X2)) → PROPER(X1)
PROPER(U21(X1, X2)) → PROPER(X2)
PROPER(U22(X)) → U22¹(proper(X))
PROPER(U22(X)) → PROPER(X)
PROPER(isList(X)) → ISLIST(proper(X))
PROPER(isList(X)) → PROPER(X)
PROPER(U31(X)) → U31¹(proper(X))
PROPER(U31(X)) → PROPER(X)
PROPER(U41(X1, X2)) → U41¹(proper(X1), proper(X2))
PROPER(U41(X1, X2)) → PROPER(X1)
PROPER(U41(X1, X2)) → PROPER(X2)
PROPER(U42(X)) → U42¹(proper(X))
PROPER(U42(X)) → PROPER(X)
PROPER(isNeList(X)) → ISNELIST(proper(X))
PROPER(isNeList(X)) → PROPER(X)
PROPER(U51(X1, X2)) → U51¹(proper(X1), proper(X2))
PROPER(U51(X1, X2)) → PROPER(X1)
PROPER(U51(X1, X2)) → PROPER(X2)
PROPER(U52(X)) → U52¹(proper(X))
PROPER(U52(X)) → PROPER(X)
PROPER(U61(X)) → U61¹(proper(X))
PROPER(U61(X)) → PROPER(X)
PROPER(U71(X1, X2)) → U71¹(proper(X1), proper(X2))
PROPER(U71(X1, X2)) → PROPER(X1)
PROPER(U71(X1, X2)) → PROPER(X2)
PROPER(U72(X)) → U72¹(proper(X))
PROPER(U72(X)) → PROPER(X)
PROPER(isPal(X)) → ISPAL(proper(X))
PROPER(isPal(X)) → PROPER(X)
PROPER(U81(X)) → U81¹(proper(X))
PROPER(U81(X)) → PROPER(X)
PROPER(isQid(X)) → ISQID(proper(X))
PROPER(isQid(X)) → PROPER(X)
PROPER(isNePal(X)) → ISNEPAL(proper(X))
PROPER(isNePal(X)) → PROPER(X)
__¹(ok(X1), ok(X2)) → __¹(X1, X2)
U11¹(ok(X)) → U11¹(X)
U21¹(ok(X1), ok(X2)) → U21¹(X1, X2)
U22¹(ok(X)) → U22¹(X)
ISLIST(ok(X)) → ISLIST(X)
U31¹(ok(X)) → U31¹(X)
U41¹(ok(X1), ok(X2)) → U41¹(X1, X2)
U42¹(ok(X)) → U42¹(X)
ISNELIST(ok(X)) → ISNELIST(X)
U51¹(ok(X1), ok(X2)) → U51¹(X1, X2)
U52¹(ok(X)) → U52¹(X)
U61¹(ok(X)) → U61¹(X)
U71¹(ok(X1), ok(X2)) → U71¹(X1, X2)
U72¹(ok(X)) → U72¹(X)
ISPAL(ok(X)) → ISPAL(X)
U81¹(ok(X)) → U81¹(X)
ISQID(ok(X)) → ISQID(X)
ISNEPAL(ok(X)) → ISNEPAL(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(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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 21 SCCs with 60 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

ISNEPAL(ok(X)) → ISNEPAL(X)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

ISNEPAL(ok(X)) → ISNEPAL(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNEPAL(x1) = ISNEPAL(x1)
ok(x1) = ok(x1)
active(x1) = x1
__(x1, x2) = __(x1, x2)
mark(x1) = mark
nil = nil
U11(x1) = U11(x1)
tt = tt
U21(x1, x2) = U21(x1)
U22(x1) = U22(x1)
isList(x1) = isList(x1)
U31(x1) = x1
U41(x1, x2) = x1
U42(x1) = U42(x1)
isNeList(x1) = isNeList(x1)
U51(x1, x2) = x2
U52(x1) = U52(x1)
U61(x1) = x1
U71(x1, x2) = U71(x1)
U72(x1) = U72(x1)
isPal(x1) = isPal(x1)
U81(x1) = U81(x1)
isQid(x1) = isQid(x1)
isNePal(x1) = x1
a = a
e = e
i = i
o = o
u = u
proper(x1) = proper(x1)
top(x1) = top

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

[_{_₂}, isNeList₁, proper₁] > [ISNEPAL₁, ok₁, U11₁, U21₁, U22₁, isList₁, U42₁, U52₁, U71₁, U72₁, isPal₁, U81₁, isQid₁] > tt > mark
[_{_₂}, isNeList₁, proper₁] > [ISNEPAL₁, ok₁, U11₁, U21₁, U22₁, isList₁, U42₁, U52₁, U71₁, U72₁, isPal₁, U81₁, isQid₁] > top > mark
[_{_₂}, isNeList₁, proper₁] > nil > mark
[_{_₂}, isNeList₁, proper₁] > a > mark
[_{_₂}, isNeList₁, proper₁] > e > tt > mark
[_{_₂}, isNeList₁, proper₁] > i > mark
[_{_₂}, isNeList₁, proper₁] > u > mark
o > [ISNEPAL₁, ok₁, U11₁, U21₁, U22₁, isList₁, U42₁, U52₁, U71₁, U72₁, isPal₁, U81₁, isQid₁] > tt > mark
o > [ISNEPAL₁, ok₁, U11₁, U21₁, U22₁, isList₁, U42₁, U52₁, U71₁, U72₁, isPal₁, U81₁, isQid₁] > top > mark

Status:

ISNEPAL₁: [1]
ok₁: [1]
__₂: [2,1]
mark: []
nil: []
U11₁: [1]
tt: []
U21₁: [1]
U22₁: [1]
isList₁: [1]
U42₁: [1]
isNeList₁: [1]
U52₁: [1]
U71₁: [1]
U72₁: [1]
isPal₁: [1]
U81₁: [1]
isQid₁: [1]
a: []
e: []
i: []
o: []
u: []
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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:

ISQID(ok(X)) → ISQID(X)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

ISQID(ok(X)) → ISQID(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISQID(x1) = ISQID(x1)
ok(x1) = ok(x1)
active(x1) = x1
__(x1, x2) = __(x1, x2)
mark(x1) = mark
nil = nil
U11(x1) = U11(x1)
tt = tt
U21(x1, x2) = U21(x1)
U22(x1) = U22(x1)
isList(x1) = isList(x1)
U31(x1) = x1
U41(x1, x2) = x1
U42(x1) = U42(x1)
isNeList(x1) = isNeList(x1)
U51(x1, x2) = x2
U52(x1) = U52(x1)
U61(x1) = x1
U71(x1, x2) = U71(x1)
U72(x1) = U72(x1)
isPal(x1) = isPal(x1)
U81(x1) = U81(x1)
isQid(x1) = isQid(x1)
isNePal(x1) = x1
a = a
e = e
i = i
o = o
u = u
proper(x1) = proper(x1)
top(x1) = top

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

[_{_₂}, isNeList₁, proper₁] > [ISQID₁, ok₁, U11₁, U21₁, U22₁, isList₁, U42₁, U52₁, U71₁, U72₁, isPal₁, U81₁, isQid₁] > tt > mark
[_{_₂}, isNeList₁, proper₁] > [ISQID₁, ok₁, U11₁, U21₁, U22₁, isList₁, U42₁, U52₁, U71₁, U72₁, isPal₁, U81₁, isQid₁] > top > mark
[_{_₂}, isNeList₁, proper₁] > nil > mark
[_{_₂}, isNeList₁, proper₁] > a > mark
[_{_₂}, isNeList₁, proper₁] > e > tt > mark
[_{_₂}, isNeList₁, proper₁] > i > mark
[_{_₂}, isNeList₁, proper₁] > u > mark
o > [ISQID₁, ok₁, U11₁, U21₁, U22₁, isList₁, U42₁, U52₁, U71₁, U72₁, isPal₁, U81₁, isQid₁] > tt > mark
o > [ISQID₁, ok₁, U11₁, U21₁, U22₁, isList₁, U42₁, U52₁, U71₁, U72₁, isPal₁, U81₁, isQid₁] > top > mark

Status:

ISQID₁: [1]
ok₁: [1]
__₂: [2,1]
mark: []
nil: []
U11₁: [1]
tt: []
U21₁: [1]
U22₁: [1]
isList₁: [1]
U42₁: [1]
isNeList₁: [1]
U52₁: [1]
U71₁: [1]
U72₁: [1]
isPal₁: [1]
U81₁: [1]
isQid₁: [1]
a: []
e: []
i: []
o: []
u: []
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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:

ISPAL(ok(X)) → ISPAL(X)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

ISPAL(ok(X)) → ISPAL(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISPAL(x1) = ISPAL(x1)
ok(x1) = ok(x1)
active(x1) = x1
__(x1, x2) = __(x1, x2)
mark(x1) = mark
nil = nil
U11(x1) = U11(x1)
tt = tt
U21(x1, x2) = U21(x1)
U22(x1) = U22(x1)
isList(x1) = isList(x1)
U31(x1) = x1
U41(x1, x2) = x1
U42(x1) = U42(x1)
isNeList(x1) = isNeList(x1)
U51(x1, x2) = x2
U52(x1) = U52(x1)
U61(x1) = x1
U71(x1, x2) = U71(x1)
U72(x1) = U72(x1)
isPal(x1) = isPal(x1)
U81(x1) = U81(x1)
isQid(x1) = isQid(x1)
isNePal(x1) = x1
a = a
e = e
i = i
o = o
u = u
proper(x1) = proper(x1)
top(x1) = top

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

[_{_₂}, isNeList₁, proper₁] > [ISPAL₁, ok₁, U11₁, U21₁, U22₁, isList₁, U42₁, U52₁, U71₁, U72₁, isPal₁, U81₁, isQid₁] > tt > mark
[_{_₂}, isNeList₁, proper₁] > [ISPAL₁, ok₁, U11₁, U21₁, U22₁, isList₁, U42₁, U52₁, U71₁, U72₁, isPal₁, U81₁, isQid₁] > top > mark
[_{_₂}, isNeList₁, proper₁] > nil > mark
[_{_₂}, isNeList₁, proper₁] > a > mark
[_{_₂}, isNeList₁, proper₁] > e > tt > mark
[_{_₂}, isNeList₁, proper₁] > i > mark
[_{_₂}, isNeList₁, proper₁] > u > mark
o > [ISPAL₁, ok₁, U11₁, U21₁, U22₁, isList₁, U42₁, U52₁, U71₁, U72₁, isPal₁, U81₁, isQid₁] > tt > mark
o > [ISPAL₁, ok₁, U11₁, U21₁, U22₁, isList₁, U42₁, U52₁, U71₁, U72₁, isPal₁, U81₁, isQid₁] > top > mark

Status:

ISPAL₁: [1]
ok₁: [1]
__₂: [2,1]
mark: []
nil: []
U11₁: [1]
tt: []
U21₁: [1]
U22₁: [1]
isList₁: [1]
U42₁: [1]
isNeList₁: [1]
U52₁: [1]
U71₁: [1]
U72₁: [1]
isPal₁: [1]
U81₁: [1]
isQid₁: [1]
a: []
e: []
i: []
o: []
u: []
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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:

ISNELIST(ok(X)) → ISNELIST(X)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

ISNELIST(ok(X)) → ISNELIST(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNELIST(x1) = ISNELIST(x1)
ok(x1) = ok(x1)
active(x1) = x1
__(x1, x2) = __(x1, x2)
mark(x1) = mark
nil = nil
U11(x1) = U11(x1)
tt = tt
U21(x1, x2) = U21(x1)
U22(x1) = U22(x1)
isList(x1) = isList(x1)
U31(x1) = x1
U41(x1, x2) = x1
U42(x1) = U42(x1)
isNeList(x1) = isNeList(x1)
U51(x1, x2) = x2
U52(x1) = U52(x1)
U61(x1) = x1
U71(x1, x2) = U71(x1)
U72(x1) = U72(x1)
isPal(x1) = isPal(x1)
U81(x1) = U81(x1)
isQid(x1) = isQid(x1)
isNePal(x1) = x1
a = a
e = e
i = i
o = o
u = u
proper(x1) = proper(x1)
top(x1) = top

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

[_{_₂}, isNeList₁, proper₁] > [ISNELIST₁, ok₁, U11₁, U21₁, U22₁, isList₁, U42₁, U52₁, U71₁, U72₁, isPal₁, U81₁, isQid₁] > tt > mark
[_{_₂}, isNeList₁, proper₁] > [ISNELIST₁, ok₁, U11₁, U21₁, U22₁, isList₁, U42₁, U52₁, U71₁, U72₁, isPal₁, U81₁, isQid₁] > top > mark
[_{_₂}, isNeList₁, proper₁] > nil > mark
[_{_₂}, isNeList₁, proper₁] > a > mark
[_{_₂}, isNeList₁, proper₁] > e > tt > mark
[_{_₂}, isNeList₁, proper₁] > i > mark
[_{_₂}, isNeList₁, proper₁] > u > mark
o > [ISNELIST₁, ok₁, U11₁, U21₁, U22₁, isList₁, U42₁, U52₁, U71₁, U72₁, isPal₁, U81₁, isQid₁] > tt > mark
o > [ISNELIST₁, ok₁, U11₁, U21₁, U22₁, isList₁, U42₁, U52₁, U71₁, U72₁, isPal₁, U81₁, isQid₁] > top > mark

Status:

ISNELIST₁: [1]
ok₁: [1]
__₂: [2,1]
mark: []
nil: []
U11₁: [1]
tt: []
U21₁: [1]
U22₁: [1]
isList₁: [1]
U42₁: [1]
isNeList₁: [1]
U52₁: [1]
U71₁: [1]
U72₁: [1]
isPal₁: [1]
U81₁: [1]
isQid₁: [1]
a: []
e: []
i: []
o: []
u: []
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(22) Obligation:

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

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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) PisEmptyProof (EQUIVALENT transformation)

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

(24) TRUE

(25) Obligation:

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

ISLIST(ok(X)) → ISLIST(X)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(26) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISLIST(ok(X)) → ISLIST(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISLIST(x1) = ISLIST(x1)
ok(x1) = ok(x1)
active(x1) = x1
__(x1, x2) = __(x1, x2)
mark(x1) = mark
nil = nil
U11(x1) = U11(x1)
tt = tt
U21(x1, x2) = U21(x1)
U22(x1) = U22(x1)
isList(x1) = isList(x1)
U31(x1) = x1
U41(x1, x2) = x1
U42(x1) = U42(x1)
isNeList(x1) = isNeList(x1)
U51(x1, x2) = x2
U52(x1) = U52(x1)
U61(x1) = x1
U71(x1, x2) = U71(x1)
U72(x1) = U72(x1)
isPal(x1) = isPal(x1)
U81(x1) = U81(x1)
isQid(x1) = isQid(x1)
isNePal(x1) = x1
a = a
e = e
i = i
o = o
u = u
proper(x1) = proper(x1)
top(x1) = top

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

[_{_₂}, isNeList₁, proper₁] > [ISLIST₁, ok₁, U11₁, U21₁, U22₁, isList₁, U42₁, U52₁, U71₁, U72₁, isPal₁, U81₁, isQid₁] > tt > mark
[_{_₂}, isNeList₁, proper₁] > [ISLIST₁, ok₁, U11₁, U21₁, U22₁, isList₁, U42₁, U52₁, U71₁, U72₁, isPal₁, U81₁, isQid₁] > top > mark
[_{_₂}, isNeList₁, proper₁] > nil > mark
[_{_₂}, isNeList₁, proper₁] > a > mark
[_{_₂}, isNeList₁, proper₁] > e > tt > mark
[_{_₂}, isNeList₁, proper₁] > i > mark
[_{_₂}, isNeList₁, proper₁] > u > mark
o > [ISLIST₁, ok₁, U11₁, U21₁, U22₁, isList₁, U42₁, U52₁, U71₁, U72₁, isPal₁, U81₁, isQid₁] > tt > mark
o > [ISLIST₁, ok₁, U11₁, U21₁, U22₁, isList₁, U42₁, U52₁, U71₁, U72₁, isPal₁, U81₁, isQid₁] > top > mark

Status:

ISLIST₁: [1]
ok₁: [1]
__₂: [2,1]
mark: []
nil: []
U11₁: [1]
tt: []
U21₁: [1]
U22₁: [1]
isList₁: [1]
U42₁: [1]
isNeList₁: [1]
U52₁: [1]
U71₁: [1]
U72₁: [1]
isPal₁: [1]
U81₁: [1]
isQid₁: [1]
a: []
e: []
i: []
o: []
u: []
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(27) Obligation:

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

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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) PisEmptyProof (EQUIVALENT transformation)

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

(29) TRUE

(30) Obligation:

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

U81¹(ok(X)) → U81¹(X)
U81¹(mark(X)) → U81¹(X)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(31) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U81¹(ok(X)) → U81¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U81¹(x1) = U81¹(x1)
ok(x1) = ok(x1)
mark(x1) = x1
active(x1) = active(x1)
__(x1, x2) = __(x1, x2)
nil = nil
U11(x1) = U11(x1)
tt = tt
U21(x1, x2) = U21(x1, x2)
U22(x1) = U22(x1)
isList(x1) = x1
U31(x1) = x1
U41(x1, x2) = x2
U42(x1) = U42(x1)
isNeList(x1) = x1
U51(x1, x2) = U51(x1, x2)
U52(x1) = U52(x1)
U61(x1) = U61(x1)
U71(x1, x2) = U71(x1, x2)
U72(x1) = U72(x1)
isPal(x1) = isPal(x1)
U81(x1) = U81(x1)
isQid(x1) = x1
isNePal(x1) = isNePal(x1)
a = a
e = e
i = i
o = o
u = u
proper(x1) = proper(x1)
top(x1) = top

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

i > [nil, tt, U21₂, U22₁, U42₁, U52₁, U81₁] > [U81^1₁, ok₁, U61₁, U72₁, isPal₁, a, e]
top > active₁ > [_{_₂}, U51₂, isNePal₁, u, proper₁] > U11₁ > [nil, tt, U21₂, U22₁, U42₁, U52₁, U81₁] > [U81^1₁, ok₁, U61₁, U72₁, isPal₁, a, e]
top > active₁ > [_{_₂}, U51₂, isNePal₁, u, proper₁] > U71₂ > [U81^1₁, ok₁, U61₁, U72₁, isPal₁, a, e]
top > active₁ > [_{_₂}, U51₂, isNePal₁, u, proper₁] > o > [nil, tt, U21₂, U22₁, U42₁, U52₁, U81₁] > [U81^1₁, ok₁, U61₁, U72₁, isPal₁, a, e]

Status:

U81^1₁: [1]
ok₁: [1]
active₁: [1]
__₂: [2,1]
nil: []
U11₁: [1]
tt: []
U21₂: [1,2]
U22₁: [1]
U42₁: [1]
U51₂: [1,2]
U52₁: [1]
U61₁: [1]
U71₂: [1,2]
U72₁: [1]
isPal₁: [1]
U81₁: [1]
isNePal₁: [1]
a: []
e: []
i: []
o: []
u: []
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(32) Obligation:

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

U81¹(mark(X)) → U81¹(X)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(33) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U81¹(mark(X)) → U81¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U81¹(x1) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
__(x1, x2) = __(x1, x2)
nil = nil
U11(x1) = U11(x1)
tt = tt
U21(x1, x2) = U21(x1, x2)
U22(x1) = U22(x1)
isList(x1) = x1
U31(x1) = x1
U41(x1, x2) = U41(x1, x2)
U42(x1) = x1
isNeList(x1) = isNeList(x1)
U51(x1, x2) = U51(x1, x2)
U52(x1) = U52(x1)
U61(x1) = x1
U71(x1, x2) = U71(x1, x2)
U72(x1) = x1
isPal(x1) = isPal(x1)
U81(x1) = U81(x1)
isQid(x1) = isQid(x1)
isNePal(x1) = isNePal(x1)
a = a
e = e
i = i
o = o
u = u
proper(x1) = x1
ok(x1) = x1
top(x1) = top

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

a > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > [U21₂, U22₁] > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
a > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > U41₂ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
a > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > isPal₁ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
e > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > [U21₂, U22₁] > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
e > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > U41₂ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
e > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > isPal₁ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
i > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > [U21₂, U22₁] > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
i > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > U41₂ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
i > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > isPal₁ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
o > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > [U21₂, U22₁] > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
o > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > U41₂ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
o > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > isPal₁ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
u > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > [U21₂, U22₁] > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
u > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > U41₂ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
u > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > isPal₁ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]

Status:

mark₁: [1]
active₁: [1]
__₂: [2,1]
nil: []
U11₁: [1]
tt: []
U21₂: [1,2]
U22₁: [1]
U41₂: [2,1]
isNeList₁: [1]
U51₂: [1,2]
U52₁: [1]
U71₂: [1,2]
isPal₁: [1]
U81₁: [1]
isQid₁: [1]
isNePal₁: [1]
a: []
e: []
i: []
o: []
u: []
top: []

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(34) Obligation:

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

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(35) PisEmptyProof (EQUIVALENT transformation)

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

(36) TRUE

(37) Obligation:

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

U72¹(ok(X)) → U72¹(X)
U72¹(mark(X)) → U72¹(X)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(38) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U72¹(ok(X)) → U72¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U72¹(x1) = U72¹(x1)
ok(x1) = ok(x1)
mark(x1) = x1
active(x1) = active(x1)
__(x1, x2) = __(x1, x2)
nil = nil
U11(x1) = U11(x1)
tt = tt
U21(x1, x2) = U21(x1, x2)
U22(x1) = U22(x1)
isList(x1) = x1
U31(x1) = x1
U41(x1, x2) = x2
U42(x1) = U42(x1)
isNeList(x1) = x1
U51(x1, x2) = U51(x1, x2)
U52(x1) = U52(x1)
U61(x1) = U61(x1)
U71(x1, x2) = U71(x1, x2)
U72(x1) = U72(x1)
isPal(x1) = isPal(x1)
U81(x1) = U81(x1)
isQid(x1) = x1
isNePal(x1) = isNePal(x1)
a = a
e = e
i = i
o = o
u = u
proper(x1) = proper(x1)
top(x1) = top

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

i > [nil, tt, U21₂, U22₁, U42₁, U52₁, U81₁] > [U72^1₁, ok₁, U61₁, U72₁, isPal₁, a, e]
top > active₁ > [_{_₂}, U51₂, isNePal₁, u, proper₁] > U11₁ > [nil, tt, U21₂, U22₁, U42₁, U52₁, U81₁] > [U72^1₁, ok₁, U61₁, U72₁, isPal₁, a, e]
top > active₁ > [_{_₂}, U51₂, isNePal₁, u, proper₁] > U71₂ > [U72^1₁, ok₁, U61₁, U72₁, isPal₁, a, e]
top > active₁ > [_{_₂}, U51₂, isNePal₁, u, proper₁] > o > [nil, tt, U21₂, U22₁, U42₁, U52₁, U81₁] > [U72^1₁, ok₁, U61₁, U72₁, isPal₁, a, e]

Status:

U72^1₁: [1]
ok₁: [1]
active₁: [1]
__₂: [2,1]
nil: []
U11₁: [1]
tt: []
U21₂: [1,2]
U22₁: [1]
U42₁: [1]
U51₂: [1,2]
U52₁: [1]
U61₁: [1]
U71₂: [1,2]
U72₁: [1]
isPal₁: [1]
U81₁: [1]
isNePal₁: [1]
a: []
e: []
i: []
o: []
u: []
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(39) Obligation:

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

U72¹(mark(X)) → U72¹(X)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(40) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U72¹(mark(X)) → U72¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U72¹(x1) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
__(x1, x2) = __(x1, x2)
nil = nil
U11(x1) = U11(x1)
tt = tt
U21(x1, x2) = U21(x1, x2)
U22(x1) = U22(x1)
isList(x1) = x1
U31(x1) = x1
U41(x1, x2) = U41(x1, x2)
U42(x1) = x1
isNeList(x1) = isNeList(x1)
U51(x1, x2) = U51(x1, x2)
U52(x1) = U52(x1)
U61(x1) = x1
U71(x1, x2) = U71(x1, x2)
U72(x1) = x1
isPal(x1) = isPal(x1)
U81(x1) = U81(x1)
isQid(x1) = isQid(x1)
isNePal(x1) = isNePal(x1)
a = a
e = e
i = i
o = o
u = u
proper(x1) = x1
ok(x1) = x1
top(x1) = top

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

a > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > [U21₂, U22₁] > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
a > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > U41₂ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
a > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > isPal₁ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
e > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > [U21₂, U22₁] > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
e > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > U41₂ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
e > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > isPal₁ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
i > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > [U21₂, U22₁] > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
i > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > U41₂ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
i > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > isPal₁ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
o > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > [U21₂, U22₁] > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
o > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > U41₂ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
o > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > isPal₁ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
u > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > [U21₂, U22₁] > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
u > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > U41₂ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
u > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > isPal₁ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]

Status:

mark₁: [1]
active₁: [1]
__₂: [2,1]
nil: []
U11₁: [1]
tt: []
U21₂: [1,2]
U22₁: [1]
U41₂: [2,1]
isNeList₁: [1]
U51₂: [1,2]
U52₁: [1]
U71₂: [1,2]
isPal₁: [1]
U81₁: [1]
isQid₁: [1]
isNePal₁: [1]
a: []
e: []
i: []
o: []
u: []
top: []

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(41) Obligation:

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

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(42) PisEmptyProof (EQUIVALENT transformation)

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

(43) TRUE

(44) Obligation:

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

U71¹(ok(X1), ok(X2)) → U71¹(X1, X2)
U71¹(mark(X1), X2) → U71¹(X1, X2)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(45) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U71¹(mark(X1), X2) → U71¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U71¹(x1, x2) = U71¹(x1, x2)
ok(x1) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
__(x1, x2) = __(x1, x2)
nil = nil
U11(x1) = U11(x1)
tt = tt
U21(x1, x2) = U21(x1, x2)
U22(x1) = U22(x1)
isList(x1) = isList(x1)
U31(x1) = U31(x1)
U41(x1, x2) = U41(x1, x2)
U42(x1) = U42(x1)
isNeList(x1) = isNeList(x1)
U51(x1, x2) = U51(x1, x2)
U52(x1) = U52(x1)
U61(x1) = U61(x1)
U71(x1, x2) = x1
U72(x1) = x1
isPal(x1) = isPal
U81(x1) = U81(x1)
isQid(x1) = isQid
isNePal(x1) = isNePal
a = a
e = e
i = i
o = o
u = u
proper(x1) = proper(x1)
top(x1) = top

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

U71^1₂ > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > [active₁, U21₂] > [nil, tt, isPal, U81₁] > U42₁ > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > [active₁, U21₂] > [nil, tt, isPal, U81₁] > isNeList₁ > [_{_₂}, isList₁] > U41₂ > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > [active₁, U21₂] > [nil, tt, isPal, U81₁] > isNePal > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > [active₁, U21₂] > U11₁ > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > [active₁, U21₂] > U31₁ > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > [active₁, U21₂] > U51₂ > [_{_₂}, isList₁] > U41₂ > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > [active₁, U21₂] > isQid > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > a > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > e > [nil, tt, isPal, U81₁] > U42₁ > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > e > [nil, tt, isPal, U81₁] > isNeList₁ > [_{_₂}, isList₁] > U41₂ > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > e > [nil, tt, isPal, U81₁] > isNePal > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > i > [nil, tt, isPal, U81₁] > U42₁ > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > i > [nil, tt, isPal, U81₁] > isNeList₁ > [_{_₂}, isList₁] > U41₂ > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > i > [nil, tt, isPal, U81₁] > isNePal > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > o > [mark₁, U22₁, U52₁, U61₁, u, top]

Status:

U71^1₂: [2,1]
mark₁: [1]
active₁: [1]
__₂: [1,2]
nil: []
U11₁: [1]
tt: []
U21₂: [2,1]
U22₁: [1]
isList₁: [1]
U31₁: [1]
U41₂: [2,1]
U42₁: [1]
isNeList₁: [1]
U51₂: [2,1]
U52₁: [1]
U61₁: [1]
isPal: []
U81₁: [1]
isQid: []
isNePal: []
a: []
e: []
i: []
o: []
u: []
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(46) Obligation:

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

U71¹(ok(X1), ok(X2)) → U71¹(X1, X2)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(47) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U71¹(ok(X1), ok(X2)) → U71¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U71¹(x1, x2) = x2
ok(x1) = ok(x1)
active(x1) = x1
__(x1, x2) = x1
mark(x1) = mark
nil = nil
U11(x1) = U11(x1)
tt = tt
U21(x1, x2) = U21(x1)
U22(x1) = U22(x1)
isList(x1) = isList(x1)
U31(x1) = x1
U41(x1, x2) = U41(x2)
U42(x1) = U42(x1)
isNeList(x1) = isNeList(x1)
U51(x1, x2) = x1
U52(x1) = x1
U61(x1) = x1
U71(x1, x2) = x2
U72(x1) = U72(x1)
isPal(x1) = x1
U81(x1) = x1
isQid(x1) = isQid(x1)
isNePal(x1) = isNePal(x1)
a = a
e = e
i = i
o = o
u = u
proper(x1) = proper(x1)
top(x1) = top

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

nil > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > a > mark
nil > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > e > mark
nil > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > i > mark
nil > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > top > mark
o > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > a > mark
o > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > e > mark
o > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > i > mark
o > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > top > mark
u > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > a > mark
u > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > e > mark
u > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > i > mark
u > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > top > mark

Status:

ok₁: [1]
mark: []
nil: []
U11₁: [1]
tt: []
U21₁: [1]
U22₁: [1]
isList₁: [1]
U41₁: [1]
U42₁: [1]
isNeList₁: [1]
U72₁: [1]
isQid₁: [1]
isNePal₁: [1]
a: []
e: []
i: []
o: []
u: []
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(48) Obligation:

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

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(49) PisEmptyProof (EQUIVALENT transformation)

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

(50) TRUE

(51) Obligation:

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

U61¹(ok(X)) → U61¹(X)
U61¹(mark(X)) → U61¹(X)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(52) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U61¹(ok(X)) → U61¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U61¹(x1) = U61¹(x1)
ok(x1) = ok(x1)
mark(x1) = x1
active(x1) = active(x1)
__(x1, x2) = __(x1, x2)
nil = nil
U11(x1) = U11(x1)
tt = tt
U21(x1, x2) = U21(x1, x2)
U22(x1) = U22(x1)
isList(x1) = x1
U31(x1) = x1
U41(x1, x2) = x2
U42(x1) = U42(x1)
isNeList(x1) = x1
U51(x1, x2) = U51(x1, x2)
U52(x1) = U52(x1)
U61(x1) = U61(x1)
U71(x1, x2) = U71(x1, x2)
U72(x1) = U72(x1)
isPal(x1) = isPal(x1)
U81(x1) = U81(x1)
isQid(x1) = x1
isNePal(x1) = isNePal(x1)
a = a
e = e
i = i
o = o
u = u
proper(x1) = proper(x1)
top(x1) = top

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

i > [nil, tt, U21₂, U22₁, U42₁, U52₁, U81₁] > [U61^1₁, ok₁, U61₁, U72₁, isPal₁, a, e]
top > active₁ > [_{_₂}, U51₂, isNePal₁, u, proper₁] > U11₁ > [nil, tt, U21₂, U22₁, U42₁, U52₁, U81₁] > [U61^1₁, ok₁, U61₁, U72₁, isPal₁, a, e]
top > active₁ > [_{_₂}, U51₂, isNePal₁, u, proper₁] > U71₂ > [U61^1₁, ok₁, U61₁, U72₁, isPal₁, a, e]
top > active₁ > [_{_₂}, U51₂, isNePal₁, u, proper₁] > o > [nil, tt, U21₂, U22₁, U42₁, U52₁, U81₁] > [U61^1₁, ok₁, U61₁, U72₁, isPal₁, a, e]

Status:

U61^1₁: [1]
ok₁: [1]
active₁: [1]
__₂: [2,1]
nil: []
U11₁: [1]
tt: []
U21₂: [1,2]
U22₁: [1]
U42₁: [1]
U51₂: [1,2]
U52₁: [1]
U61₁: [1]
U71₂: [1,2]
U72₁: [1]
isPal₁: [1]
U81₁: [1]
isNePal₁: [1]
a: []
e: []
i: []
o: []
u: []
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(53) Obligation:

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

U61¹(mark(X)) → U61¹(X)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(54) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U61¹(mark(X)) → U61¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U61¹(x1) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
__(x1, x2) = __(x1, x2)
nil = nil
U11(x1) = U11(x1)
tt = tt
U21(x1, x2) = U21(x1, x2)
U22(x1) = U22(x1)
isList(x1) = x1
U31(x1) = x1
U41(x1, x2) = U41(x1, x2)
U42(x1) = x1
isNeList(x1) = isNeList(x1)
U51(x1, x2) = U51(x1, x2)
U52(x1) = U52(x1)
U61(x1) = x1
U71(x1, x2) = U71(x1, x2)
U72(x1) = x1
isPal(x1) = isPal(x1)
U81(x1) = U81(x1)
isQid(x1) = isQid(x1)
isNePal(x1) = isNePal(x1)
a = a
e = e
i = i
o = o
u = u
proper(x1) = x1
ok(x1) = x1
top(x1) = top

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

a > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > [U21₂, U22₁] > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
a > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > U41₂ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
a > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > isPal₁ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
e > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > [U21₂, U22₁] > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
e > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > U41₂ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
e > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > isPal₁ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
i > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > [U21₂, U22₁] > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
i > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > U41₂ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
i > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > isPal₁ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
o > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > [U21₂, U22₁] > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
o > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > U41₂ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
o > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > isPal₁ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
u > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > [U21₂, U22₁] > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
u > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > U41₂ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
u > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > isPal₁ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]

Status:

mark₁: [1]
active₁: [1]
__₂: [2,1]
nil: []
U11₁: [1]
tt: []
U21₂: [1,2]
U22₁: [1]
U41₂: [2,1]
isNeList₁: [1]
U51₂: [1,2]
U52₁: [1]
U71₂: [1,2]
isPal₁: [1]
U81₁: [1]
isQid₁: [1]
isNePal₁: [1]
a: []
e: []
i: []
o: []
u: []
top: []

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(55) Obligation:

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

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(56) PisEmptyProof (EQUIVALENT transformation)

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

(57) TRUE

(58) Obligation:

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

U52¹(ok(X)) → U52¹(X)
U52¹(mark(X)) → U52¹(X)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(59) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U52¹(ok(X)) → U52¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U52¹(x1) = U52¹(x1)
ok(x1) = ok(x1)
mark(x1) = x1
active(x1) = active(x1)
__(x1, x2) = __(x1, x2)
nil = nil
U11(x1) = U11(x1)
tt = tt
U21(x1, x2) = U21(x1, x2)
U22(x1) = U22(x1)
isList(x1) = x1
U31(x1) = x1
U41(x1, x2) = x2
U42(x1) = U42(x1)
isNeList(x1) = x1
U51(x1, x2) = U51(x1, x2)
U52(x1) = U52(x1)
U61(x1) = U61(x1)
U71(x1, x2) = U71(x1, x2)
U72(x1) = U72(x1)
isPal(x1) = isPal(x1)
U81(x1) = U81(x1)
isQid(x1) = x1
isNePal(x1) = isNePal(x1)
a = a
e = e
i = i
o = o
u = u
proper(x1) = proper(x1)
top(x1) = top

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

i > [nil, tt, U21₂, U22₁, U42₁, U52₁, U81₁] > [U52^1₁, ok₁, U61₁, U72₁, isPal₁, a, e]
top > active₁ > [_{_₂}, U51₂, isNePal₁, u, proper₁] > U11₁ > [nil, tt, U21₂, U22₁, U42₁, U52₁, U81₁] > [U52^1₁, ok₁, U61₁, U72₁, isPal₁, a, e]
top > active₁ > [_{_₂}, U51₂, isNePal₁, u, proper₁] > U71₂ > [U52^1₁, ok₁, U61₁, U72₁, isPal₁, a, e]
top > active₁ > [_{_₂}, U51₂, isNePal₁, u, proper₁] > o > [nil, tt, U21₂, U22₁, U42₁, U52₁, U81₁] > [U52^1₁, ok₁, U61₁, U72₁, isPal₁, a, e]

Status:

U52^1₁: [1]
ok₁: [1]
active₁: [1]
__₂: [2,1]
nil: []
U11₁: [1]
tt: []
U21₂: [1,2]
U22₁: [1]
U42₁: [1]
U51₂: [1,2]
U52₁: [1]
U61₁: [1]
U71₂: [1,2]
U72₁: [1]
isPal₁: [1]
U81₁: [1]
isNePal₁: [1]
a: []
e: []
i: []
o: []
u: []
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(60) Obligation:

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

U52¹(mark(X)) → U52¹(X)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(61) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U52¹(mark(X)) → U52¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U52¹(x1) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
__(x1, x2) = __(x1, x2)
nil = nil
U11(x1) = U11(x1)
tt = tt
U21(x1, x2) = U21(x1, x2)
U22(x1) = U22(x1)
isList(x1) = x1
U31(x1) = x1
U41(x1, x2) = U41(x1, x2)
U42(x1) = x1
isNeList(x1) = isNeList(x1)
U51(x1, x2) = U51(x1, x2)
U52(x1) = U52(x1)
U61(x1) = x1
U71(x1, x2) = U71(x1, x2)
U72(x1) = x1
isPal(x1) = isPal(x1)
U81(x1) = U81(x1)
isQid(x1) = isQid(x1)
isNePal(x1) = isNePal(x1)
a = a
e = e
i = i
o = o
u = u
proper(x1) = x1
ok(x1) = x1
top(x1) = top

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

a > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > [U21₂, U22₁] > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
a > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > U41₂ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
a > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > isPal₁ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
e > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > [U21₂, U22₁] > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
e > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > U41₂ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
e > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > isPal₁ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
i > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > [U21₂, U22₁] > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
i > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > U41₂ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
i > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > isPal₁ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
o > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > [U21₂, U22₁] > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
o > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > U41₂ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
o > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > isPal₁ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
u > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > [U21₂, U22₁] > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
u > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > U41₂ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
u > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > isPal₁ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]

Status:

mark₁: [1]
active₁: [1]
__₂: [2,1]
nil: []
U11₁: [1]
tt: []
U21₂: [1,2]
U22₁: [1]
U41₂: [2,1]
isNeList₁: [1]
U51₂: [1,2]
U52₁: [1]
U71₂: [1,2]
isPal₁: [1]
U81₁: [1]
isQid₁: [1]
isNePal₁: [1]
a: []
e: []
i: []
o: []
u: []
top: []

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(62) Obligation:

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

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(63) PisEmptyProof (EQUIVALENT transformation)

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

(64) TRUE

(65) Obligation:

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

U51¹(ok(X1), ok(X2)) → U51¹(X1, X2)
U51¹(mark(X1), X2) → U51¹(X1, X2)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(66) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U51¹(mark(X1), X2) → U51¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U51¹(x1, x2) = U51¹(x1, x2)
ok(x1) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
__(x1, x2) = __(x1, x2)
nil = nil
U11(x1) = U11(x1)
tt = tt
U21(x1, x2) = U21(x1, x2)
U22(x1) = U22(x1)
isList(x1) = isList(x1)
U31(x1) = U31(x1)
U41(x1, x2) = U41(x1, x2)
U42(x1) = U42(x1)
isNeList(x1) = isNeList(x1)
U51(x1, x2) = U51(x1, x2)
U52(x1) = U52(x1)
U61(x1) = U61(x1)
U71(x1, x2) = x1
U72(x1) = x1
isPal(x1) = isPal
U81(x1) = U81(x1)
isQid(x1) = isQid
isNePal(x1) = isNePal
a = a
e = e
i = i
o = o
u = u
proper(x1) = proper(x1)
top(x1) = top

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

U51^1₂ > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > [active₁, U21₂] > [nil, tt, isPal, U81₁] > U42₁ > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > [active₁, U21₂] > [nil, tt, isPal, U81₁] > isNeList₁ > [_{_₂}, isList₁] > U41₂ > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > [active₁, U21₂] > [nil, tt, isPal, U81₁] > isNePal > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > [active₁, U21₂] > U11₁ > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > [active₁, U21₂] > U31₁ > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > [active₁, U21₂] > U51₂ > [_{_₂}, isList₁] > U41₂ > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > [active₁, U21₂] > isQid > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > a > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > e > [nil, tt, isPal, U81₁] > U42₁ > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > e > [nil, tt, isPal, U81₁] > isNeList₁ > [_{_₂}, isList₁] > U41₂ > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > e > [nil, tt, isPal, U81₁] > isNePal > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > i > [nil, tt, isPal, U81₁] > U42₁ > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > i > [nil, tt, isPal, U81₁] > isNeList₁ > [_{_₂}, isList₁] > U41₂ > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > i > [nil, tt, isPal, U81₁] > isNePal > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > o > [mark₁, U22₁, U52₁, U61₁, u, top]

Status:

U51^1₂: [2,1]
mark₁: [1]
active₁: [1]
__₂: [1,2]
nil: []
U11₁: [1]
tt: []
U21₂: [2,1]
U22₁: [1]
isList₁: [1]
U31₁: [1]
U41₂: [2,1]
U42₁: [1]
isNeList₁: [1]
U51₂: [2,1]
U52₁: [1]
U61₁: [1]
isPal: []
U81₁: [1]
isQid: []
isNePal: []
a: []
e: []
i: []
o: []
u: []
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(67) Obligation:

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

U51¹(ok(X1), ok(X2)) → U51¹(X1, X2)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(68) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U51¹(ok(X1), ok(X2)) → U51¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U51¹(x1, x2) = x2
ok(x1) = ok(x1)
active(x1) = x1
__(x1, x2) = x1
mark(x1) = mark
nil = nil
U11(x1) = U11(x1)
tt = tt
U21(x1, x2) = U21(x1)
U22(x1) = U22(x1)
isList(x1) = isList(x1)
U31(x1) = x1
U41(x1, x2) = U41(x2)
U42(x1) = U42(x1)
isNeList(x1) = isNeList(x1)
U51(x1, x2) = x1
U52(x1) = x1
U61(x1) = x1
U71(x1, x2) = x2
U72(x1) = U72(x1)
isPal(x1) = x1
U81(x1) = x1
isQid(x1) = isQid(x1)
isNePal(x1) = isNePal(x1)
a = a
e = e
i = i
o = o
u = u
proper(x1) = proper(x1)
top(x1) = top

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

nil > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > a > mark
nil > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > e > mark
nil > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > i > mark
nil > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > top > mark
o > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > a > mark
o > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > e > mark
o > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > i > mark
o > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > top > mark
u > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > a > mark
u > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > e > mark
u > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > i > mark
u > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > top > mark

Status:

ok₁: [1]
mark: []
nil: []
U11₁: [1]
tt: []
U21₁: [1]
U22₁: [1]
isList₁: [1]
U41₁: [1]
U42₁: [1]
isNeList₁: [1]
U72₁: [1]
isQid₁: [1]
isNePal₁: [1]
a: []
e: []
i: []
o: []
u: []
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(69) Obligation:

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

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(70) PisEmptyProof (EQUIVALENT transformation)

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

(71) TRUE

(72) Obligation:

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

U42¹(ok(X)) → U42¹(X)
U42¹(mark(X)) → U42¹(X)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(73) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U42¹(ok(X)) → U42¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U42¹(x1) = U42¹(x1)
ok(x1) = ok(x1)
mark(x1) = x1
active(x1) = active(x1)
__(x1, x2) = __(x1, x2)
nil = nil
U11(x1) = U11(x1)
tt = tt
U21(x1, x2) = U21(x1, x2)
U22(x1) = U22(x1)
isList(x1) = x1
U31(x1) = x1
U41(x1, x2) = x2
U42(x1) = U42(x1)
isNeList(x1) = x1
U51(x1, x2) = U51(x1, x2)
U52(x1) = U52(x1)
U61(x1) = U61(x1)
U71(x1, x2) = U71(x1, x2)
U72(x1) = U72(x1)
isPal(x1) = isPal(x1)
U81(x1) = U81(x1)
isQid(x1) = x1
isNePal(x1) = isNePal(x1)
a = a
e = e
i = i
o = o
u = u
proper(x1) = proper(x1)
top(x1) = top

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

i > [nil, tt, U21₂, U22₁, U42₁, U52₁, U81₁] > [U42^1₁, ok₁, U61₁, U72₁, isPal₁, a, e]
top > active₁ > [_{_₂}, U51₂, isNePal₁, u, proper₁] > U11₁ > [nil, tt, U21₂, U22₁, U42₁, U52₁, U81₁] > [U42^1₁, ok₁, U61₁, U72₁, isPal₁, a, e]
top > active₁ > [_{_₂}, U51₂, isNePal₁, u, proper₁] > U71₂ > [U42^1₁, ok₁, U61₁, U72₁, isPal₁, a, e]
top > active₁ > [_{_₂}, U51₂, isNePal₁, u, proper₁] > o > [nil, tt, U21₂, U22₁, U42₁, U52₁, U81₁] > [U42^1₁, ok₁, U61₁, U72₁, isPal₁, a, e]

Status:

U42^1₁: [1]
ok₁: [1]
active₁: [1]
__₂: [2,1]
nil: []
U11₁: [1]
tt: []
U21₂: [1,2]
U22₁: [1]
U42₁: [1]
U51₂: [1,2]
U52₁: [1]
U61₁: [1]
U71₂: [1,2]
U72₁: [1]
isPal₁: [1]
U81₁: [1]
isNePal₁: [1]
a: []
e: []
i: []
o: []
u: []
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(74) Obligation:

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

U42¹(mark(X)) → U42¹(X)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(75) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U42¹(mark(X)) → U42¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U42¹(x1) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
__(x1, x2) = __(x1, x2)
nil = nil
U11(x1) = U11(x1)
tt = tt
U21(x1, x2) = U21(x1, x2)
U22(x1) = U22(x1)
isList(x1) = x1
U31(x1) = x1
U41(x1, x2) = U41(x1, x2)
U42(x1) = x1
isNeList(x1) = isNeList(x1)
U51(x1, x2) = U51(x1, x2)
U52(x1) = U52(x1)
U61(x1) = x1
U71(x1, x2) = U71(x1, x2)
U72(x1) = x1
isPal(x1) = isPal(x1)
U81(x1) = U81(x1)
isQid(x1) = isQid(x1)
isNePal(x1) = isNePal(x1)
a = a
e = e
i = i
o = o
u = u
proper(x1) = x1
ok(x1) = x1
top(x1) = top

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

a > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > [U21₂, U22₁] > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
a > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > U41₂ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
a > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > isPal₁ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
e > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > [U21₂, U22₁] > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
e > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > U41₂ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
e > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > isPal₁ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
i > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > [U21₂, U22₁] > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
i > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > U41₂ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
i > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > isPal₁ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
o > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > [U21₂, U22₁] > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
o > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > U41₂ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
o > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > isPal₁ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
u > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > [U21₂, U22₁] > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
u > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > U41₂ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
u > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > isPal₁ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]

Status:

mark₁: [1]
active₁: [1]
__₂: [2,1]
nil: []
U11₁: [1]
tt: []
U21₂: [1,2]
U22₁: [1]
U41₂: [2,1]
isNeList₁: [1]
U51₂: [1,2]
U52₁: [1]
U71₂: [1,2]
isPal₁: [1]
U81₁: [1]
isQid₁: [1]
isNePal₁: [1]
a: []
e: []
i: []
o: []
u: []
top: []

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(76) Obligation:

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

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(77) PisEmptyProof (EQUIVALENT transformation)

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

(78) TRUE

(79) Obligation:

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

U41¹(ok(X1), ok(X2)) → U41¹(X1, X2)
U41¹(mark(X1), X2) → U41¹(X1, X2)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(80) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U41¹(mark(X1), X2) → U41¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U41¹(x1, x2) = U41¹(x1, x2)
ok(x1) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
__(x1, x2) = __(x1, x2)
nil = nil
U11(x1) = U11(x1)
tt = tt
U21(x1, x2) = U21(x1, x2)
U22(x1) = U22(x1)
isList(x1) = isList(x1)
U31(x1) = U31(x1)
U41(x1, x2) = U41(x1, x2)
U42(x1) = U42(x1)
isNeList(x1) = isNeList(x1)
U51(x1, x2) = U51(x1, x2)
U52(x1) = U52(x1)
U61(x1) = U61(x1)
U71(x1, x2) = x1
U72(x1) = x1
isPal(x1) = isPal
U81(x1) = U81(x1)
isQid(x1) = isQid
isNePal(x1) = isNePal
a = a
e = e
i = i
o = o
u = u
proper(x1) = proper(x1)
top(x1) = top

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

U41^1₂ > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > [active₁, U21₂] > [nil, tt, isPal, U81₁] > U42₁ > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > [active₁, U21₂] > [nil, tt, isPal, U81₁] > isNeList₁ > [_{_₂}, isList₁] > U41₂ > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > [active₁, U21₂] > [nil, tt, isPal, U81₁] > isNePal > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > [active₁, U21₂] > U11₁ > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > [active₁, U21₂] > U31₁ > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > [active₁, U21₂] > U51₂ > [_{_₂}, isList₁] > U41₂ > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > [active₁, U21₂] > isQid > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > a > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > e > [nil, tt, isPal, U81₁] > U42₁ > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > e > [nil, tt, isPal, U81₁] > isNeList₁ > [_{_₂}, isList₁] > U41₂ > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > e > [nil, tt, isPal, U81₁] > isNePal > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > i > [nil, tt, isPal, U81₁] > U42₁ > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > i > [nil, tt, isPal, U81₁] > isNeList₁ > [_{_₂}, isList₁] > U41₂ > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > i > [nil, tt, isPal, U81₁] > isNePal > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > o > [mark₁, U22₁, U52₁, U61₁, u, top]

Status:

U41^1₂: [2,1]
mark₁: [1]
active₁: [1]
__₂: [1,2]
nil: []
U11₁: [1]
tt: []
U21₂: [2,1]
U22₁: [1]
isList₁: [1]
U31₁: [1]
U41₂: [2,1]
U42₁: [1]
isNeList₁: [1]
U51₂: [2,1]
U52₁: [1]
U61₁: [1]
isPal: []
U81₁: [1]
isQid: []
isNePal: []
a: []
e: []
i: []
o: []
u: []
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(81) Obligation:

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

U41¹(ok(X1), ok(X2)) → U41¹(X1, X2)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(82) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U41¹(ok(X1), ok(X2)) → U41¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U41¹(x1, x2) = x2
ok(x1) = ok(x1)
active(x1) = x1
__(x1, x2) = x1
mark(x1) = mark
nil = nil
U11(x1) = U11(x1)
tt = tt
U21(x1, x2) = U21(x1)
U22(x1) = U22(x1)
isList(x1) = isList(x1)
U31(x1) = x1
U41(x1, x2) = U41(x2)
U42(x1) = U42(x1)
isNeList(x1) = isNeList(x1)
U51(x1, x2) = x1
U52(x1) = x1
U61(x1) = x1
U71(x1, x2) = x2
U72(x1) = U72(x1)
isPal(x1) = x1
U81(x1) = x1
isQid(x1) = isQid(x1)
isNePal(x1) = isNePal(x1)
a = a
e = e
i = i
o = o
u = u
proper(x1) = proper(x1)
top(x1) = top

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

nil > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > a > mark
nil > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > e > mark
nil > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > i > mark
nil > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > top > mark
o > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > a > mark
o > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > e > mark
o > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > i > mark
o > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > top > mark
u > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > a > mark
u > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > e > mark
u > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > i > mark
u > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > top > mark

Status:

ok₁: [1]
mark: []
nil: []
U11₁: [1]
tt: []
U21₁: [1]
U22₁: [1]
isList₁: [1]
U41₁: [1]
U42₁: [1]
isNeList₁: [1]
U72₁: [1]
isQid₁: [1]
isNePal₁: [1]
a: []
e: []
i: []
o: []
u: []
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(83) Obligation:

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

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(84) PisEmptyProof (EQUIVALENT transformation)

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

(85) TRUE

(86) Obligation:

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

U31¹(ok(X)) → U31¹(X)
U31¹(mark(X)) → U31¹(X)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

U31¹(ok(X)) → U31¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U31¹(x1) = U31¹(x1)
ok(x1) = ok(x1)
mark(x1) = x1
active(x1) = active(x1)
__(x1, x2) = __(x1, x2)
nil = nil
U11(x1) = U11(x1)
tt = tt
U21(x1, x2) = U21(x1, x2)
U22(x1) = U22(x1)
isList(x1) = x1
U31(x1) = x1
U41(x1, x2) = x2
U42(x1) = U42(x1)
isNeList(x1) = x1
U51(x1, x2) = U51(x1, x2)
U52(x1) = U52(x1)
U61(x1) = U61(x1)
U71(x1, x2) = U71(x1, x2)
U72(x1) = U72(x1)
isPal(x1) = isPal(x1)
U81(x1) = U81(x1)
isQid(x1) = x1
isNePal(x1) = isNePal(x1)
a = a
e = e
i = i
o = o
u = u
proper(x1) = proper(x1)
top(x1) = top

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

i > [nil, tt, U21₂, U22₁, U42₁, U52₁, U81₁] > [U31^1₁, ok₁, U61₁, U72₁, isPal₁, a, e]
top > active₁ > [_{_₂}, U51₂, isNePal₁, u, proper₁] > U11₁ > [nil, tt, U21₂, U22₁, U42₁, U52₁, U81₁] > [U31^1₁, ok₁, U61₁, U72₁, isPal₁, a, e]
top > active₁ > [_{_₂}, U51₂, isNePal₁, u, proper₁] > U71₂ > [U31^1₁, ok₁, U61₁, U72₁, isPal₁, a, e]
top > active₁ > [_{_₂}, U51₂, isNePal₁, u, proper₁] > o > [nil, tt, U21₂, U22₁, U42₁, U52₁, U81₁] > [U31^1₁, ok₁, U61₁, U72₁, isPal₁, a, e]

Status:

U31^1₁: [1]
ok₁: [1]
active₁: [1]
__₂: [2,1]
nil: []
U11₁: [1]
tt: []
U21₂: [1,2]
U22₁: [1]
U42₁: [1]
U51₂: [1,2]
U52₁: [1]
U61₁: [1]
U71₂: [1,2]
U72₁: [1]
isPal₁: [1]
U81₁: [1]
isNePal₁: [1]
a: []
e: []
i: []
o: []
u: []
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(88) Obligation:

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

U31¹(mark(X)) → U31¹(X)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U31¹(mark(X)) → U31¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U31¹(x1) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
__(x1, x2) = __(x1, x2)
nil = nil
U11(x1) = U11(x1)
tt = tt
U21(x1, x2) = U21(x1, x2)
U22(x1) = U22(x1)
isList(x1) = x1
U31(x1) = x1
U41(x1, x2) = U41(x1, x2)
U42(x1) = x1
isNeList(x1) = isNeList(x1)
U51(x1, x2) = U51(x1, x2)
U52(x1) = U52(x1)
U61(x1) = x1
U71(x1, x2) = U71(x1, x2)
U72(x1) = x1
isPal(x1) = isPal(x1)
U81(x1) = U81(x1)
isQid(x1) = isQid(x1)
isNePal(x1) = isNePal(x1)
a = a
e = e
i = i
o = o
u = u
proper(x1) = x1
ok(x1) = x1
top(x1) = top

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

a > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > [U21₂, U22₁] > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
a > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > U41₂ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
a > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > isPal₁ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
e > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > [U21₂, U22₁] > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
e > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > U41₂ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
e > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > isPal₁ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
i > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > [U21₂, U22₁] > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
i > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > U41₂ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
i > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > isPal₁ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
o > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > [U21₂, U22₁] > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
o > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > U41₂ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
o > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > isPal₁ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
u > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > [U21₂, U22₁] > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
u > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > U41₂ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
u > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > isPal₁ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]

Status:

mark₁: [1]
active₁: [1]
__₂: [2,1]
nil: []
U11₁: [1]
tt: []
U21₂: [1,2]
U22₁: [1]
U41₂: [2,1]
isNeList₁: [1]
U51₂: [1,2]
U52₁: [1]
U71₂: [1,2]
isPal₁: [1]
U81₁: [1]
isQid₁: [1]
isNePal₁: [1]
a: []
e: []
i: []
o: []
u: []
top: []

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(90) Obligation:

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

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(91) PisEmptyProof (EQUIVALENT transformation)

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

(92) TRUE

(93) Obligation:

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

U22¹(ok(X)) → U22¹(X)
U22¹(mark(X)) → U22¹(X)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

U22¹(ok(X)) → U22¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U22¹(x1) = U22¹(x1)
ok(x1) = ok(x1)
mark(x1) = x1
active(x1) = active(x1)
__(x1, x2) = __(x1, x2)
nil = nil
U11(x1) = U11(x1)
tt = tt
U21(x1, x2) = U21(x1, x2)
U22(x1) = U22(x1)
isList(x1) = x1
U31(x1) = x1
U41(x1, x2) = x2
U42(x1) = U42(x1)
isNeList(x1) = x1
U51(x1, x2) = U51(x1, x2)
U52(x1) = U52(x1)
U61(x1) = U61(x1)
U71(x1, x2) = U71(x1, x2)
U72(x1) = U72(x1)
isPal(x1) = isPal(x1)
U81(x1) = U81(x1)
isQid(x1) = x1
isNePal(x1) = isNePal(x1)
a = a
e = e
i = i
o = o
u = u
proper(x1) = proper(x1)
top(x1) = top

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

i > [nil, tt, U21₂, U22₁, U42₁, U52₁, U81₁] > [U22^1₁, ok₁, U61₁, U72₁, isPal₁, a, e]
top > active₁ > [_{_₂}, U51₂, isNePal₁, u, proper₁] > U11₁ > [nil, tt, U21₂, U22₁, U42₁, U52₁, U81₁] > [U22^1₁, ok₁, U61₁, U72₁, isPal₁, a, e]
top > active₁ > [_{_₂}, U51₂, isNePal₁, u, proper₁] > U71₂ > [U22^1₁, ok₁, U61₁, U72₁, isPal₁, a, e]
top > active₁ > [_{_₂}, U51₂, isNePal₁, u, proper₁] > o > [nil, tt, U21₂, U22₁, U42₁, U52₁, U81₁] > [U22^1₁, ok₁, U61₁, U72₁, isPal₁, a, e]

Status:

U22^1₁: [1]
ok₁: [1]
active₁: [1]
__₂: [2,1]
nil: []
U11₁: [1]
tt: []
U21₂: [1,2]
U22₁: [1]
U42₁: [1]
U51₂: [1,2]
U52₁: [1]
U61₁: [1]
U71₂: [1,2]
U72₁: [1]
isPal₁: [1]
U81₁: [1]
isNePal₁: [1]
a: []
e: []
i: []
o: []
u: []
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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:

U22¹(mark(X)) → U22¹(X)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

U22¹(mark(X)) → U22¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U22¹(x1) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
__(x1, x2) = __(x1, x2)
nil = nil
U11(x1) = U11(x1)
tt = tt
U21(x1, x2) = U21(x1, x2)
U22(x1) = U22(x1)
isList(x1) = x1
U31(x1) = x1
U41(x1, x2) = U41(x1, x2)
U42(x1) = x1
isNeList(x1) = isNeList(x1)
U51(x1, x2) = U51(x1, x2)
U52(x1) = U52(x1)
U61(x1) = x1
U71(x1, x2) = U71(x1, x2)
U72(x1) = x1
isPal(x1) = isPal(x1)
U81(x1) = U81(x1)
isQid(x1) = isQid(x1)
isNePal(x1) = isNePal(x1)
a = a
e = e
i = i
o = o
u = u
proper(x1) = x1
ok(x1) = x1
top(x1) = top

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

a > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > [U21₂, U22₁] > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
a > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > U41₂ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
a > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > isPal₁ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
e > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > [U21₂, U22₁] > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
e > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > U41₂ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
e > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > isPal₁ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
i > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > [U21₂, U22₁] > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
i > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > U41₂ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
i > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > isPal₁ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
o > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > [U21₂, U22₁] > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
o > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > U41₂ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
o > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > isPal₁ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
u > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > [U21₂, U22₁] > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
u > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > U41₂ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
u > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > isPal₁ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]

Status:

mark₁: [1]
active₁: [1]
__₂: [2,1]
nil: []
U11₁: [1]
tt: []
U21₂: [1,2]
U22₁: [1]
U41₂: [2,1]
isNeList₁: [1]
U51₂: [1,2]
U52₁: [1]
U71₂: [1,2]
isPal₁: [1]
U81₁: [1]
isQid₁: [1]
isNePal₁: [1]
a: []
e: []
i: []
o: []
u: []
top: []

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(97) Obligation:

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

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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) PisEmptyProof (EQUIVALENT transformation)

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

(99) TRUE

(100) Obligation:

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

U21¹(ok(X1), ok(X2)) → U21¹(X1, X2)
U21¹(mark(X1), X2) → U21¹(X1, X2)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(101) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U21¹(mark(X1), X2) → U21¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U21¹(x1, x2) = U21¹(x1, x2)
ok(x1) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
__(x1, x2) = __(x1, x2)
nil = nil
U11(x1) = U11(x1)
tt = tt
U21(x1, x2) = U21(x1, x2)
U22(x1) = U22(x1)
isList(x1) = isList(x1)
U31(x1) = U31(x1)
U41(x1, x2) = U41(x1, x2)
U42(x1) = U42(x1)
isNeList(x1) = isNeList(x1)
U51(x1, x2) = U51(x1, x2)
U52(x1) = U52(x1)
U61(x1) = U61(x1)
U71(x1, x2) = x1
U72(x1) = x1
isPal(x1) = isPal
U81(x1) = U81(x1)
isQid(x1) = isQid
isNePal(x1) = isNePal
a = a
e = e
i = i
o = o
u = u
proper(x1) = proper(x1)
top(x1) = top

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

U21^1₂ > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > [active₁, U21₂] > [nil, tt, isPal, U81₁] > U42₁ > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > [active₁, U21₂] > [nil, tt, isPal, U81₁] > isNeList₁ > [_{_₂}, isList₁] > U41₂ > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > [active₁, U21₂] > [nil, tt, isPal, U81₁] > isNePal > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > [active₁, U21₂] > U11₁ > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > [active₁, U21₂] > U31₁ > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > [active₁, U21₂] > U51₂ > [_{_₂}, isList₁] > U41₂ > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > [active₁, U21₂] > isQid > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > a > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > e > [nil, tt, isPal, U81₁] > U42₁ > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > e > [nil, tt, isPal, U81₁] > isNeList₁ > [_{_₂}, isList₁] > U41₂ > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > e > [nil, tt, isPal, U81₁] > isNePal > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > i > [nil, tt, isPal, U81₁] > U42₁ > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > i > [nil, tt, isPal, U81₁] > isNeList₁ > [_{_₂}, isList₁] > U41₂ > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > i > [nil, tt, isPal, U81₁] > isNePal > [mark₁, U22₁, U52₁, U61₁, u, top]
proper₁ > o > [mark₁, U22₁, U52₁, U61₁, u, top]

Status:

U21^1₂: [2,1]
mark₁: [1]
active₁: [1]
__₂: [1,2]
nil: []
U11₁: [1]
tt: []
U21₂: [2,1]
U22₁: [1]
isList₁: [1]
U31₁: [1]
U41₂: [2,1]
U42₁: [1]
isNeList₁: [1]
U51₂: [2,1]
U52₁: [1]
U61₁: [1]
isPal: []
U81₁: [1]
isQid: []
isNePal: []
a: []
e: []
i: []
o: []
u: []
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(102) Obligation:

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

U21¹(ok(X1), ok(X2)) → U21¹(X1, X2)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(103) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U21¹(ok(X1), ok(X2)) → U21¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U21¹(x1, x2) = x2
ok(x1) = ok(x1)
active(x1) = x1
__(x1, x2) = x1
mark(x1) = mark
nil = nil
U11(x1) = U11(x1)
tt = tt
U21(x1, x2) = U21(x1)
U22(x1) = U22(x1)
isList(x1) = isList(x1)
U31(x1) = x1
U41(x1, x2) = U41(x2)
U42(x1) = U42(x1)
isNeList(x1) = isNeList(x1)
U51(x1, x2) = x1
U52(x1) = x1
U61(x1) = x1
U71(x1, x2) = x2
U72(x1) = U72(x1)
isPal(x1) = x1
U81(x1) = x1
isQid(x1) = isQid(x1)
isNePal(x1) = isNePal(x1)
a = a
e = e
i = i
o = o
u = u
proper(x1) = proper(x1)
top(x1) = top

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

nil > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > a > mark
nil > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > e > mark
nil > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > i > mark
nil > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > top > mark
o > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > a > mark
o > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > e > mark
o > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > i > mark
o > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > top > mark
u > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > a > mark
u > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > e > mark
u > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > i > mark
u > tt > [ok₁, U11₁, U21₁, U22₁, isList₁, U41₁, U42₁, isNeList₁, U72₁, isQid₁, isNePal₁, proper₁] > top > mark

Status:

ok₁: [1]
mark: []
nil: []
U11₁: [1]
tt: []
U21₁: [1]
U22₁: [1]
isList₁: [1]
U41₁: [1]
U42₁: [1]
isNeList₁: [1]
U72₁: [1]
isQid₁: [1]
isNePal₁: [1]
a: []
e: []
i: []
o: []
u: []
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(104) Obligation:

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

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(105) PisEmptyProof (EQUIVALENT transformation)

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

(106) TRUE

(107) Obligation:

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

U11¹(ok(X)) → U11¹(X)
U11¹(mark(X)) → U11¹(X)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(108) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11¹(ok(X)) → U11¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U11¹(x1) = U11¹(x1)
ok(x1) = ok(x1)
mark(x1) = x1
active(x1) = active(x1)
__(x1, x2) = __(x1, x2)
nil = nil
U11(x1) = U11(x1)
tt = tt
U21(x1, x2) = U21(x1, x2)
U22(x1) = U22(x1)
isList(x1) = x1
U31(x1) = x1
U41(x1, x2) = x2
U42(x1) = U42(x1)
isNeList(x1) = x1
U51(x1, x2) = U51(x1, x2)
U52(x1) = U52(x1)
U61(x1) = U61(x1)
U71(x1, x2) = U71(x1, x2)
U72(x1) = U72(x1)
isPal(x1) = isPal(x1)
U81(x1) = U81(x1)
isQid(x1) = x1
isNePal(x1) = isNePal(x1)
a = a
e = e
i = i
o = o
u = u
proper(x1) = proper(x1)
top(x1) = top

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

i > [nil, tt, U21₂, U22₁, U42₁, U52₁, U81₁] > [U11^1₁, ok₁, U61₁, U72₁, isPal₁, a, e]
top > active₁ > [_{_₂}, U51₂, isNePal₁, u, proper₁] > U11₁ > [nil, tt, U21₂, U22₁, U42₁, U52₁, U81₁] > [U11^1₁, ok₁, U61₁, U72₁, isPal₁, a, e]
top > active₁ > [_{_₂}, U51₂, isNePal₁, u, proper₁] > U71₂ > [U11^1₁, ok₁, U61₁, U72₁, isPal₁, a, e]
top > active₁ > [_{_₂}, U51₂, isNePal₁, u, proper₁] > o > [nil, tt, U21₂, U22₁, U42₁, U52₁, U81₁] > [U11^1₁, ok₁, U61₁, U72₁, isPal₁, a, e]

Status:

U11^1₁: [1]
ok₁: [1]
active₁: [1]
__₂: [2,1]
nil: []
U11₁: [1]
tt: []
U21₂: [1,2]
U22₁: [1]
U42₁: [1]
U51₂: [1,2]
U52₁: [1]
U61₁: [1]
U71₂: [1,2]
U72₁: [1]
isPal₁: [1]
U81₁: [1]
isNePal₁: [1]
a: []
e: []
i: []
o: []
u: []
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(109) Obligation:

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

U11¹(mark(X)) → U11¹(X)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(110) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11¹(mark(X)) → U11¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U11¹(x1) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
__(x1, x2) = __(x1, x2)
nil = nil
U11(x1) = U11(x1)
tt = tt
U21(x1, x2) = U21(x1, x2)
U22(x1) = U22(x1)
isList(x1) = x1
U31(x1) = x1
U41(x1, x2) = U41(x1, x2)
U42(x1) = x1
isNeList(x1) = isNeList(x1)
U51(x1, x2) = U51(x1, x2)
U52(x1) = U52(x1)
U61(x1) = x1
U71(x1, x2) = U71(x1, x2)
U72(x1) = x1
isPal(x1) = isPal(x1)
U81(x1) = U81(x1)
isQid(x1) = isQid(x1)
isNePal(x1) = isNePal(x1)
a = a
e = e
i = i
o = o
u = u
proper(x1) = x1
ok(x1) = x1
top(x1) = top

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

a > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > [U21₂, U22₁] > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
a > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > U41₂ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
a > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > isPal₁ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
e > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > [U21₂, U22₁] > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
e > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > U41₂ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
e > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > isPal₁ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
i > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > [U21₂, U22₁] > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
i > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > U41₂ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
i > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > isPal₁ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
o > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > [U21₂, U22₁] > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
o > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > U41₂ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
o > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > isPal₁ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
u > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > [U21₂, U22₁] > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
u > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > __₂ > U41₂ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]
u > [nil, tt] > [active₁, U51₂, U52₁, U71₂, top] > isPal₁ > [mark₁, U11₁, isNeList₁, U81₁, isQid₁, isNePal₁]

Status:

mark₁: [1]
active₁: [1]
__₂: [2,1]
nil: []
U11₁: [1]
tt: []
U21₂: [1,2]
U22₁: [1]
U41₂: [2,1]
isNeList₁: [1]
U51₂: [1,2]
U52₁: [1]
U71₂: [1,2]
isPal₁: [1]
U81₁: [1]
isQid₁: [1]
isNePal₁: [1]
a: []
e: []
i: []
o: []
u: []
top: []

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(111) Obligation:

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

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(112) PisEmptyProof (EQUIVALENT transformation)

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

(113) TRUE

(114) Obligation:

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

__¹(X1, mark(X2)) → __¹(X1, X2)
__¹(mark(X1), X2) → __¹(X1, X2)
__¹(ok(X1), ok(X2)) → __¹(X1, X2)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(115) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

__¹(mark(X1), X2) → __¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
__¹(x1, x2) = x1
mark(x1) = mark(x1)
ok(x1) = x1
active(x1) = active(x1)
__(x1, x2) = __(x1, x2)
nil = nil
U11(x1) = x1
tt = tt
U21(x1, x2) = U21(x1, x2)
U22(x1) = x1
isList(x1) = isList(x1)
U31(x1) = U31(x1)
U41(x1, x2) = U41(x1, x2)
U42(x1) = x1
isNeList(x1) = isNeList(x1)
U51(x1, x2) = U51(x1, x2)
U52(x1) = U52(x1)
U61(x1) = U61(x1)
U71(x1, x2) = U71(x1, x2)
U72(x1) = U72(x1)
isPal(x1) = x1
U81(x1) = x1
isQid(x1) = isQid(x1)
isNePal(x1) = x1
a = a
e = e
i = i
o = o
u = u
proper(x1) = x1
top(x1) = top

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

active₁ > U51₂ > [_{_₂}, tt, U21₂, isList₁, U41₂, isNeList₁, U52₁, U61₁, U71₂] > U31₁ > [mark₁, nil, a, e]
active₁ > U51₂ > [_{_₂}, tt, U21₂, isList₁, U41₂, isNeList₁, U52₁, U61₁, U71₂] > U72₁ > [mark₁, nil, a, e]
active₁ > isQid₁ > [_{_₂}, tt, U21₂, isList₁, U41₂, isNeList₁, U52₁, U61₁, U71₂] > U31₁ > [mark₁, nil, a, e]
active₁ > isQid₁ > [_{_₂}, tt, U21₂, isList₁, U41₂, isNeList₁, U52₁, U61₁, U71₂] > U72₁ > [mark₁, nil, a, e]
o > [mark₁, nil, a, e]
u > [_{_₂}, tt, U21₂, isList₁, U41₂, isNeList₁, U52₁, U61₁, U71₂] > U31₁ > [mark₁, nil, a, e]
u > [_{_₂}, tt, U21₂, isList₁, U41₂, isNeList₁, U52₁, U61₁, U71₂] > U72₁ > [mark₁, nil, a, e]

Status:

mark₁: [1]
active₁: [1]
__₂: [2,1]
nil: []
tt: []
U21₂: [1,2]
isList₁: [1]
U31₁: [1]
U41₂: [1,2]
isNeList₁: [1]
U51₂: [1,2]
U52₁: [1]
U61₁: [1]
U71₂: [1,2]
U72₁: [1]
isQid₁: [1]
a: []
e: []
i: []
o: []
u: []
top: []

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(116) Obligation:

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

__¹(X1, mark(X2)) → __¹(X1, X2)
__¹(ok(X1), ok(X2)) → __¹(X1, X2)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(117) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

__¹(X1, mark(X2)) → __¹(X1, X2)
__¹(ok(X1), ok(X2)) → __¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
__¹(x1, x2) = __¹(x2)
mark(x1) = mark(x1)
ok(x1) = ok(x1)
active(x1) = active(x1)
__(x1, x2) = __(x1, x2)
nil = nil
U11(x1) = U11(x1)
tt = tt
U21(x1, x2) = U21(x1, x2)
U22(x1) = U22(x1)
isList(x1) = isList(x1)
U31(x1) = U31(x1)
U41(x1, x2) = U41(x1, x2)
U42(x1) = U42(x1)
isNeList(x1) = isNeList(x1)
U51(x1, x2) = U51(x1, x2)
U52(x1) = U52(x1)
U61(x1) = U61(x1)
U71(x1, x2) = U71(x1, x2)
U72(x1) = U72(x1)
isPal(x1) = isPal(x1)
U81(x1) = U81(x1)
isQid(x1) = isQid(x1)
isNePal(x1) = isNePal(x1)
a = a
e = e
i = i
o = o
u = u
proper(x1) = proper(x1)
top(x1) = top

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

active₁ > [U11₁, isList₁, U31₁, U41₂, U72₁, proper₁] > nil
active₁ > [U11₁, isList₁, U31₁, U41₂, U72₁, proper₁] > U21₂ > U22₁ > [mark₁, U42₁, U52₁, i, o, u] > [ok₁, tt, isNeList₁]
active₁ > [U11₁, isList₁, U31₁, U41₂, U72₁, proper₁] > isPal₁ > [_{_₂}, U51₂, U61₁, U71₂, isQid₁, isNePal₁] > [mark₁, U42₁, U52₁, i, o, u] > [ok₁, tt, isNeList₁]
active₁ > [U11₁, isList₁, U31₁, U41₂, U72₁, proper₁] > isPal₁ > U81₁ > [mark₁, U42₁, U52₁, i, o, u] > [ok₁, tt, isNeList₁]
active₁ > [U11₁, isList₁, U31₁, U41₂, U72₁, proper₁] > a
e > [mark₁, U42₁, U52₁, i, o, u] > [ok₁, tt, isNeList₁]

Status:

__^1₁: [1]
mark₁: [1]
ok₁: [1]
active₁: [1]
__₂: [1,2]
nil: []
U11₁: [1]
tt: []
U21₂: [1,2]
U22₁: [1]
isList₁: [1]
U31₁: [1]
U41₂: [1,2]
U42₁: [1]
isNeList₁: [1]
U51₂: [1,2]
U52₁: [1]
U61₁: [1]
U71₂: [1,2]
U72₁: [1]
isPal₁: [1]
U81₁: [1]
isQid₁: [1]
isNePal₁: [1]
a: []
e: []
i: []
o: []
u: []
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(118) Obligation:

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

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(119) PisEmptyProof (EQUIVALENT transformation)

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

(120) TRUE

(121) Obligation:

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

PROPER(__(X1, X2)) → PROPER(X2)
PROPER(__(X1, X2)) → PROPER(X1)
PROPER(U11(X)) → PROPER(X)
PROPER(U21(X1, X2)) → PROPER(X1)
PROPER(U21(X1, X2)) → PROPER(X2)
PROPER(U22(X)) → PROPER(X)
PROPER(isList(X)) → PROPER(X)
PROPER(U31(X)) → PROPER(X)
PROPER(U41(X1, X2)) → PROPER(X1)
PROPER(U41(X1, X2)) → PROPER(X2)
PROPER(U42(X)) → PROPER(X)
PROPER(isNeList(X)) → PROPER(X)
PROPER(U51(X1, X2)) → PROPER(X1)
PROPER(U51(X1, X2)) → PROPER(X2)
PROPER(U52(X)) → PROPER(X)
PROPER(U61(X)) → PROPER(X)
PROPER(U71(X1, X2)) → PROPER(X1)
PROPER(U71(X1, X2)) → PROPER(X2)
PROPER(U72(X)) → PROPER(X)
PROPER(isPal(X)) → PROPER(X)
PROPER(U81(X)) → PROPER(X)
PROPER(isQid(X)) → PROPER(X)
PROPER(isNePal(X)) → PROPER(X)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(122) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(__(X1, X2)) → PROPER(X2)
PROPER(__(X1, X2)) → PROPER(X1)
PROPER(U21(X1, X2)) → PROPER(X1)
PROPER(U21(X1, X2)) → PROPER(X2)
PROPER(U41(X1, X2)) → PROPER(X1)
PROPER(U41(X1, X2)) → PROPER(X2)
PROPER(U51(X1, X2)) → PROPER(X1)
PROPER(U51(X1, X2)) → PROPER(X2)
PROPER(U71(X1, X2)) → PROPER(X1)
PROPER(U71(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)
__(x1, x2) = __(x1, x2)
U11(x1) = x1
U21(x1, x2) = U21(x1, x2)
U22(x1) = x1
isList(x1) = x1
U31(x1) = x1
U41(x1, x2) = U41(x1, x2)
U42(x1) = x1
isNeList(x1) = x1
U51(x1, x2) = U51(x1, x2)
U52(x1) = x1
U61(x1) = x1
U71(x1, x2) = U71(x1, x2)
U72(x1) = x1
isPal(x1) = x1
U81(x1) = x1
isQid(x1) = x1
isNePal(x1) = x1
active(x1) = active(x1)
mark(x1) = x1
nil = nil
tt = tt
a = a
e = e
i = i
o = o
u = u
proper(x1) = x1
ok(x1) = x1
top(x1) = top

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

nil > tt
a > tt
e > tt
i > tt
o > tt
u > tt
top > active₁ > [PROPER₁, U21₂]
top > active₁ > __₂ > U51₂
top > active₁ > U41₂
top > active₁ > U71₂
top > active₁ > tt

Status:

PROPER₁: [1]
__₂: [2,1]
U21₂: [2,1]
U41₂: [1,2]
U51₂: [2,1]
U71₂: [2,1]
active₁: [1]
nil: []
tt: []
a: []
e: []
i: []
o: []
u: []
top: []

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(123) Obligation:

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

PROPER(U11(X)) → PROPER(X)
PROPER(U22(X)) → PROPER(X)
PROPER(isList(X)) → PROPER(X)
PROPER(U31(X)) → PROPER(X)
PROPER(U42(X)) → PROPER(X)
PROPER(isNeList(X)) → PROPER(X)
PROPER(U52(X)) → PROPER(X)
PROPER(U61(X)) → PROPER(X)
PROPER(U72(X)) → PROPER(X)
PROPER(isPal(X)) → PROPER(X)
PROPER(U81(X)) → PROPER(X)
PROPER(isQid(X)) → PROPER(X)
PROPER(isNePal(X)) → PROPER(X)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(124) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(U11(X)) → PROPER(X)
PROPER(U22(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1) = PROPER(x1)
U11(x1) = U11(x1)
U22(x1) = U22(x1)
isList(x1) = x1
U31(x1) = x1
U42(x1) = x1
isNeList(x1) = x1
U52(x1) = x1
U61(x1) = x1
U72(x1) = x1
isPal(x1) = x1
U81(x1) = x1
isQid(x1) = x1
isNePal(x1) = x1
active(x1) = active(x1)
__(x1, x2) = __(x1, x2)
mark(x1) = mark(x1)
nil = nil
tt = tt
U21(x1, x2) = U21(x1, x2)
U41(x1, x2) = U41(x1, x2)
U51(x1, x2) = U51(x1, x2)
U71(x1, x2) = U71(x1, x2)
a = a
e = e
i = i
o = o
u = u
proper(x1) = x1
ok(x1) = x1
top(x1) = top

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

active₁ > __₂ > U21₂ > [U11₁, U22₁, mark₁] > [tt, a, i]
active₁ > __₂ > U41₂ > [U11₁, U22₁, mark₁] > [tt, a, i]
active₁ > __₂ > U51₂ > [U11₁, U22₁, mark₁] > [tt, a, i]
active₁ > __₂ > U71₂ > [U11₁, U22₁, mark₁] > [tt, a, i]
nil > [U11₁, U22₁, mark₁] > [tt, a, i]
e > [tt, a, i]
o > [U11₁, U22₁, mark₁] > [tt, a, i]
u > [U11₁, U22₁, mark₁] > [tt, a, i]

Status:

PROPER₁: [1]
U11₁: [1]
U22₁: [1]
active₁: [1]
__₂: [2,1]
mark₁: [1]
nil: []
tt: []
U21₂: [1,2]
U41₂: [1,2]
U51₂: [1,2]
U71₂: [1,2]
a: []
e: []
i: []
o: []
u: []
top: []

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(125) Obligation:

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

PROPER(isList(X)) → PROPER(X)
PROPER(U31(X)) → PROPER(X)
PROPER(U42(X)) → PROPER(X)
PROPER(isNeList(X)) → PROPER(X)
PROPER(U52(X)) → PROPER(X)
PROPER(U61(X)) → PROPER(X)
PROPER(U72(X)) → PROPER(X)
PROPER(isPal(X)) → PROPER(X)
PROPER(U81(X)) → PROPER(X)
PROPER(isQid(X)) → PROPER(X)
PROPER(isNePal(X)) → PROPER(X)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(126) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(isList(X)) → PROPER(X)
PROPER(isNeList(X)) → PROPER(X)
PROPER(isPal(X)) → PROPER(X)
PROPER(U81(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
isList(x1) = isList(x1)
U31(x1) = x1
U42(x1) = x1
isNeList(x1) = isNeList(x1)
U52(x1) = x1
U61(x1) = x1
U72(x1) = x1
isPal(x1) = isPal(x1)
U81(x1) = U81(x1)
isQid(x1) = x1
isNePal(x1) = x1
active(x1) = x1
__(x1, x2) = __(x1, x2)
mark(x1) = x1
nil = nil
U11(x1) = U11
tt = tt
U21(x1, x2) = U21(x1)
U22(x1) = U22
U41(x1, x2) = U41(x2)
U51(x1, x2) = U51(x1, x2)
U71(x1, x2) = U71(x1, x2)
a = a
e = e
i = i
o = o
u = u
proper(x1) = x1
ok(x1) = x1
top(x1) = top(x1)

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

__₂ > [isPal₁, U71₂] > U81₁ > tt
__₂ > U21₁ > isList₁ > [isNeList₁, U41₁]
__₂ > U21₁ > isList₁ > U11 > tt
__₂ > U21₁ > U22 > tt
__₂ > U51₂ > isList₁ > [isNeList₁, U41₁]
__₂ > U51₂ > isList₁ > U11 > tt
nil > tt
a > tt
e > tt
i > tt
o > tt
u > tt

Status:

isList₁: [1]
isNeList₁: [1]
isPal₁: [1]
U81₁: [1]
__₂: [1,2]
nil: []
U11: []
tt: []
U21₁: [1]
U22: []
U41₁: [1]
U51₂: [1,2]
U71₂: [2,1]
a: []
e: []
i: []
o: []
u: []
top₁: [1]

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(127) Obligation:

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

PROPER(U31(X)) → PROPER(X)
PROPER(U42(X)) → PROPER(X)
PROPER(U52(X)) → PROPER(X)
PROPER(U61(X)) → PROPER(X)
PROPER(U72(X)) → PROPER(X)
PROPER(isQid(X)) → PROPER(X)
PROPER(isNePal(X)) → PROPER(X)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(128) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(isNePal(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1) = PROPER(x1)
U31(x1) = x1
U42(x1) = x1
U52(x1) = x1
U61(x1) = x1
U72(x1) = x1
isQid(x1) = x1
isNePal(x1) = isNePal(x1)
active(x1) = x1
__(x1, x2) = __
mark(x1) = mark
nil = nil
U11(x1) = U11
tt = tt
U21(x1, x2) = U21
U22(x1) = U22
isList(x1) = isList
U41(x1, x2) = x1
isNeList(x1) = isNeList
U51(x1, x2) = x1
U71(x1, x2) = U71
isPal(x1) = isPal
U81(x1) = U81
a = a
e = e
i = i
o = o
u = u
proper(x1) = proper(x1)
ok(x1) = x1
top(x1) = top

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

[U11, isList] > [isNePal₁, tt, U21, isPal, proper₁] > PROPER₁
[U11, isList] > [isNePal₁, tt, U21, isPal, proper₁] > [, mark, U22, isNeList, U71, U81, o]
[U11, isList] > [isNePal₁, tt, U21, isPal, proper₁] > nil
a > [isNePal₁, tt, U21, isPal, proper₁] > PROPER₁
a > [isNePal₁, tt, U21, isPal, proper₁] > [, mark, U22, isNeList, U71, U81, o]
a > [isNePal₁, tt, U21, isPal, proper₁] > nil
e > [isNePal₁, tt, U21, isPal, proper₁] > PROPER₁
e > [isNePal₁, tt, U21, isPal, proper₁] > [, mark, U22, isNeList, U71, U81, o]
e > [isNePal₁, tt, U21, isPal, proper₁] > nil
i > [isNePal₁, tt, U21, isPal, proper₁] > PROPER₁
i > [isNePal₁, tt, U21, isPal, proper₁] > [, mark, U22, isNeList, U71, U81, o]
i > [isNePal₁, tt, U21, isPal, proper₁] > nil
u > [isNePal₁, tt, U21, isPal, proper₁] > PROPER₁
u > [isNePal₁, tt, U21, isPal, proper₁] > [, mark, U22, isNeList, U71, U81, o]
u > [isNePal₁, tt, U21, isPal, proper₁] > nil
top > [isNePal₁, tt, U21, isPal, proper₁] > PROPER₁
top > [isNePal₁, tt, U21, isPal, proper₁] > [, mark, U22, isNeList, U71, U81, o]
top > [isNePal₁, tt, U21, isPal, proper₁] > nil

Status:

PROPER₁: [1]
isNePal₁: [1]
_: []
mark: []
nil: []
U11: []
tt: []
U21: []
U22: []
isList: []
isNeList: []
U71: []
isPal: []
U81: []
a: []
e: []
i: []
o: []
u: []
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(129) Obligation:

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

PROPER(U31(X)) → PROPER(X)
PROPER(U42(X)) → PROPER(X)
PROPER(U52(X)) → PROPER(X)
PROPER(U61(X)) → PROPER(X)
PROPER(U72(X)) → PROPER(X)
PROPER(isQid(X)) → PROPER(X)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(130) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(U52(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1) = PROPER(x1)
U31(x1) = x1
U42(x1) = x1
U52(x1) = U52(x1)
U61(x1) = x1
U72(x1) = x1
isQid(x1) = x1
active(x1) = active(x1)
__(x1, x2) = __(x1, x2)
mark(x1) = mark
nil = nil
U11(x1) = U11(x1)
tt = tt
U21(x1, x2) = x2
U22(x1) = U22(x1)
isList(x1) = x1
U41(x1, x2) = U41(x1, x2)
isNeList(x1) = x1
U51(x1, x2) = U51
U71(x1, x2) = x2
isPal(x1) = isPal(x1)
U81(x1) = U81(x1)
isNePal(x1) = isNePal
a = a
e = e
i = i
o = o
u = u
proper(x1) = proper(x1)
ok(x1) = ok
top(x1) = top

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

U51 > proper₁ > isPal₁ > ok > [active₁, _{_₂}, U11₁, tt, U22₁] > U52₁ > [PROPER₁, mark, nil, U81₁, u]
U51 > proper₁ > isPal₁ > ok > [active₁, _{_₂}, U11₁, tt, U22₁] > U41₂ > [PROPER₁, mark, nil, U81₁, u]
U51 > proper₁ > a > [PROPER₁, mark, nil, U81₁, u]
U51 > proper₁ > e > [active₁, _{_₂}, U11₁, tt, U22₁] > U52₁ > [PROPER₁, mark, nil, U81₁, u]
U51 > proper₁ > e > [active₁, _{_₂}, U11₁, tt, U22₁] > U41₂ > [PROPER₁, mark, nil, U81₁, u]
U51 > proper₁ > i > [active₁, _{_₂}, U11₁, tt, U22₁] > U52₁ > [PROPER₁, mark, nil, U81₁, u]
U51 > proper₁ > i > [active₁, _{_₂}, U11₁, tt, U22₁] > U41₂ > [PROPER₁, mark, nil, U81₁, u]
U51 > proper₁ > o > [active₁, _{_₂}, U11₁, tt, U22₁] > U52₁ > [PROPER₁, mark, nil, U81₁, u]
U51 > proper₁ > o > [active₁, _{_₂}, U11₁, tt, U22₁] > U41₂ > [PROPER₁, mark, nil, U81₁, u]
isNePal > proper₁ > isPal₁ > ok > [active₁, _{_₂}, U11₁, tt, U22₁] > U52₁ > [PROPER₁, mark, nil, U81₁, u]
isNePal > proper₁ > isPal₁ > ok > [active₁, _{_₂}, U11₁, tt, U22₁] > U41₂ > [PROPER₁, mark, nil, U81₁, u]
isNePal > proper₁ > a > [PROPER₁, mark, nil, U81₁, u]
isNePal > proper₁ > e > [active₁, _{_₂}, U11₁, tt, U22₁] > U52₁ > [PROPER₁, mark, nil, U81₁, u]
isNePal > proper₁ > e > [active₁, _{_₂}, U11₁, tt, U22₁] > U41₂ > [PROPER₁, mark, nil, U81₁, u]
isNePal > proper₁ > i > [active₁, _{_₂}, U11₁, tt, U22₁] > U52₁ > [PROPER₁, mark, nil, U81₁, u]
isNePal > proper₁ > i > [active₁, _{_₂}, U11₁, tt, U22₁] > U41₂ > [PROPER₁, mark, nil, U81₁, u]
isNePal > proper₁ > o > [active₁, _{_₂}, U11₁, tt, U22₁] > U52₁ > [PROPER₁, mark, nil, U81₁, u]
isNePal > proper₁ > o > [active₁, _{_₂}, U11₁, tt, U22₁] > U41₂ > [PROPER₁, mark, nil, U81₁, u]
top > proper₁ > isPal₁ > ok > [active₁, _{_₂}, U11₁, tt, U22₁] > U52₁ > [PROPER₁, mark, nil, U81₁, u]
top > proper₁ > isPal₁ > ok > [active₁, _{_₂}, U11₁, tt, U22₁] > U41₂ > [PROPER₁, mark, nil, U81₁, u]
top > proper₁ > a > [PROPER₁, mark, nil, U81₁, u]
top > proper₁ > e > [active₁, _{_₂}, U11₁, tt, U22₁] > U52₁ > [PROPER₁, mark, nil, U81₁, u]
top > proper₁ > e > [active₁, _{_₂}, U11₁, tt, U22₁] > U41₂ > [PROPER₁, mark, nil, U81₁, u]
top > proper₁ > i > [active₁, _{_₂}, U11₁, tt, U22₁] > U52₁ > [PROPER₁, mark, nil, U81₁, u]
top > proper₁ > i > [active₁, _{_₂}, U11₁, tt, U22₁] > U41₂ > [PROPER₁, mark, nil, U81₁, u]
top > proper₁ > o > [active₁, _{_₂}, U11₁, tt, U22₁] > U52₁ > [PROPER₁, mark, nil, U81₁, u]
top > proper₁ > o > [active₁, _{_₂}, U11₁, tt, U22₁] > U41₂ > [PROPER₁, mark, nil, U81₁, u]

Status:

PROPER₁: [1]
U52₁: [1]
active₁: [1]
__₂: [1,2]
mark: []
nil: []
U11₁: [1]
tt: []
U22₁: [1]
U41₂: [2,1]
U51: []
isPal₁: [1]
U81₁: [1]
isNePal: []
a: []
e: []
i: []
o: []
u: []
proper₁: [1]
ok: []
top: []

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(131) Obligation:

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

PROPER(U31(X)) → PROPER(X)
PROPER(U42(X)) → PROPER(X)
PROPER(U61(X)) → PROPER(X)
PROPER(U72(X)) → PROPER(X)
PROPER(isQid(X)) → PROPER(X)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(132) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(U72(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1) = PROPER(x1)
U31(x1) = x1
U42(x1) = x1
U61(x1) = x1
U72(x1) = U72(x1)
isQid(x1) = x1
active(x1) = active(x1)
__(x1, x2) = __(x1, x2)
mark(x1) = mark(x1)
nil = nil
U11(x1) = U11(x1)
tt = tt
U21(x1, x2) = U21(x1, x2)
U22(x1) = U22(x1)
isList(x1) = isList(x1)
U41(x1, x2) = U41(x1, x2)
isNeList(x1) = x1
U51(x1, x2) = U51(x1, x2)
U52(x1) = U52(x1)
U71(x1, x2) = U71(x1, x2)
isPal(x1) = isPal(x1)
U81(x1) = U81(x1)
isNePal(x1) = x1
a = a
e = e
i = i
o = o
u = u
proper(x1) = x1
ok(x1) = x1
top(x1) = top

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

[active₁, _{_₂}, U11₁, U51₂, U52₁] > [U21₂, U22₁] > [U72₁, mark₁, U81₁, u] > tt > isPal₁
[active₁, _{_₂}, U11₁, U51₂, U52₁] > [U21₂, U22₁] > isList₁ > tt > isPal₁
[active₁, _{_₂}, U11₁, U51₂, U52₁] > U41₂ > [U72₁, mark₁, U81₁, u] > tt > isPal₁
[active₁, _{_₂}, U11₁, U51₂, U52₁] > U71₂ > [U72₁, mark₁, U81₁, u] > tt > isPal₁
a > tt > isPal₁
e > tt > isPal₁
o > [U72₁, mark₁, U81₁, u] > tt > isPal₁

Status:

PROPER₁: [1]
U72₁: [1]
active₁: [1]
__₂: [1,2]
mark₁: [1]
nil: []
U11₁: [1]
tt: []
U21₂: [1,2]
U22₁: [1]
isList₁: [1]
U41₂: [1,2]
U51₂: [1,2]
U52₁: [1]
U71₂: [1,2]
isPal₁: [1]
U81₁: [1]
a: []
e: []
i: []
o: []
u: []
top: []

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(133) Obligation:

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

PROPER(U31(X)) → PROPER(X)
PROPER(U42(X)) → PROPER(X)
PROPER(U61(X)) → PROPER(X)
PROPER(isQid(X)) → PROPER(X)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(134) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(U42(X)) → PROPER(X)
PROPER(U61(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1) = PROPER(x1)
U31(x1) = x1
U42(x1) = U42(x1)
U61(x1) = U61(x1)
isQid(x1) = x1
active(x1) = x1
__(x1, x2) = __(x1, x2)
mark(x1) = x1
nil = nil
U11(x1) = U11
tt = tt
U21(x1, x2) = U21
U22(x1) = U22
isList(x1) = isList
U41(x1, x2) = U41(x2)
isNeList(x1) = x1
U51(x1, x2) = x1
U52(x1) = U52
U71(x1, x2) = x1
U72(x1) = U72
isPal(x1) = isPal
U81(x1) = U81
isNePal(x1) = isNePal(x1)
a = a
e = e
i = i
o = o
u = u
proper(x1) = x1
ok(x1) = x1
top(x1) = top(x1)

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

__₂ > [U42₁, U41₁]
nil > [U42₁, U41₁]
isList > U11 > [tt, U52, U72, U81, i] > [U42₁, U41₁]
isList > U21 > U22 > [tt, U52, U72, U81, i] > [U42₁, U41₁]
isPal > [tt, U52, U72, U81, i] > [U42₁, U41₁]
isNePal₁ > [PROPER₁, U61₁] > [tt, U52, U72, U81, i] > [U42₁, U41₁]
a > [tt, U52, U72, U81, i] > [U42₁, U41₁]
e > [tt, U52, U72, U81, i] > [U42₁, U41₁]
o > [tt, U52, U72, U81, i] > [U42₁, U41₁]
u > [tt, U52, U72, U81, i] > [U42₁, U41₁]
top₁ > [U42₁, U41₁]

Status:

PROPER₁: [1]
U42₁: [1]
U61₁: [1]
__₂: [1,2]
nil: []
U11: []
tt: []
U21: []
U22: []
isList: []
U41₁: [1]
U52: []
U72: []
isPal: []
U81: []
isNePal₁: [1]
a: []
e: []
i: []
o: []
u: []
top₁: [1]

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(135) Obligation:

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

PROPER(U31(X)) → PROPER(X)
PROPER(isQid(X)) → PROPER(X)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(136) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(isQid(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1) = PROPER(x1)
U31(x1) = x1
isQid(x1) = isQid(x1)
active(x1) = active(x1)
__(x1, x2) = __(x1, x2)
mark(x1) = x1
nil = nil
U11(x1) = U11
tt = tt
U21(x1, x2) = U21
U22(x1) = U22
isList(x1) = x1
U41(x1, x2) = U41(x2)
U42(x1) = U42
isNeList(x1) = x1
U51(x1, x2) = U51(x1, x2)
U52(x1) = U52
U61(x1) = U61
U71(x1, x2) = U71
U72(x1) = U72(x1)
isPal(x1) = isPal
U81(x1) = U81
isNePal(x1) = x1
a = a
e = e
i = i
o = o
u = u
proper(x1) = x1
ok(x1) = x1
top(x1) = top

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

PROPER₁ > U61
a > [nil, tt] > U61
e > [nil, tt] > U61
i > U61
o > U61
u > [nil, tt] > U61
top > [active₁, U71] > isQid₁ > [nil, tt] > U61
top > [active₁, U71] > __₂ > U21 > U61
top > [active₁, U71] > __₂ > U41₁ > U61
top > [active₁, U71] > __₂ > U51₂ > U61
top > [active₁, U71] > U11 > [nil, tt] > U61
top > [active₁, U71] > U22 > [nil, tt] > U61
top > [active₁, U71] > U42 > [nil, tt] > U61
top > [active₁, U71] > U52 > [nil, tt] > U61
top > [active₁, U71] > U72₁ > [nil, tt] > U61
top > [active₁, U71] > isPal > [nil, tt] > U61
top > [active₁, U71] > U81 > [nil, tt] > U61

Status:

PROPER₁: [1]
isQid₁: [1]
active₁: [1]
__₂: [2,1]
nil: []
U11: []
tt: []
U21: []
U22: []
U41₁: [1]
U42: []
U51₂: [1,2]
U52: []
U61: []
U71: []
U72₁: [1]
isPal: []
U81: []
a: []
e: []
i: []
o: []
u: []
top: []

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(137) Obligation:

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

PROPER(U31(X)) → PROPER(X)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(138) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(U31(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1) = PROPER(x1)
U31(x1) = U31(x1)
active(x1) = x1
__(x1, x2) = __
mark(x1) = mark
nil = nil
U11(x1) = U11(x1)
tt = tt
U21(x1, x2) = U21(x1)
U22(x1) = U22(x1)
isList(x1) = isList(x1)
U41(x1, x2) = U41
U42(x1) = U42
isNeList(x1) = isNeList(x1)
U51(x1, x2) = U51
U52(x1) = U52
U61(x1) = U61
U71(x1, x2) = U71
U72(x1) = U72
isPal(x1) = isPal
U81(x1) = U81
isQid(x1) = isQid
isNePal(x1) = isNePal
a = a
e = e
i = i
o = o
u = u
proper(x1) = x1
ok(x1) = x1
top(x1) = top

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

nil > [, mark, U51] > [U11₁, U22₁]
U21₁ > isList₁ > [, mark, U51] > [U11₁, U22₁]
U41 > U42 > [, mark, U51] > [U11₁, U22₁]
U41 > isNeList₁ > U31₁ > PROPER₁ > [U11₁, U22₁]
U41 > isNeList₁ > U31₁ > [, mark, U51] > [U11₁, U22₁]
U41 > isNeList₁ > isList₁ > [, mark, U51] > [U11₁, U22₁]
U41 > isNeList₁ > isQid > [, mark, U51] > [U11₁, U22₁]
U61 > tt > isList₁ > [, mark, U51] > [U11₁, U22₁]
U61 > tt > U42 > [, mark, U51] > [U11₁, U22₁]
U61 > tt > U52 > [, mark, U51] > [U11₁, U22₁]
U61 > tt > U72 > [, mark, U51] > [U11₁, U22₁]
U61 > tt > isPal > U81 > [, mark, U51] > [U11₁, U22₁]
[U71, isNePal] > U72 > [, mark, U51] > [U11₁, U22₁]
[U71, isNePal] > isPal > U81 > [, mark, U51] > [U11₁, U22₁]
[U71, isNePal] > isQid > [, mark, U51] > [U11₁, U22₁]
a > [U11₁, U22₁]
e > [U11₁, U22₁]
i > [U11₁, U22₁]
o > [U11₁, U22₁]
u > [U11₁, U22₁]
top > [U11₁, U22₁]

Status:

PROPER₁: [1]
U31₁: [1]
_: []
mark: []
nil: []
U11₁: [1]
tt: []
U21₁: [1]
U22₁: [1]
isList₁: [1]
U41: []
U42: []
isNeList₁: [1]
U51: []
U52: []
U61: []
U71: []
U72: []
isPal: []
U81: []
isQid: []
isNePal: []
a: []
e: []
i: []
o: []
u: []
top: []

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(139) Obligation:

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

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(140) PisEmptyProof (EQUIVALENT transformation)

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

(141) TRUE

(142) Obligation:

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

ACTIVE(__(X1, X2)) → ACTIVE(X2)
ACTIVE(__(X1, X2)) → ACTIVE(X1)
ACTIVE(U11(X)) → ACTIVE(X)
ACTIVE(U21(X1, X2)) → ACTIVE(X1)
ACTIVE(U22(X)) → ACTIVE(X)
ACTIVE(U31(X)) → ACTIVE(X)
ACTIVE(U41(X1, X2)) → ACTIVE(X1)
ACTIVE(U42(X)) → ACTIVE(X)
ACTIVE(U51(X1, X2)) → ACTIVE(X1)
ACTIVE(U52(X)) → ACTIVE(X)
ACTIVE(U61(X)) → ACTIVE(X)
ACTIVE(U71(X1, X2)) → ACTIVE(X1)
ACTIVE(U72(X)) → ACTIVE(X)
ACTIVE(U81(X)) → ACTIVE(X)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(143) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(__(X1, X2)) → ACTIVE(X2)
ACTIVE(__(X1, X2)) → ACTIVE(X1)
ACTIVE(U11(X)) → ACTIVE(X)
ACTIVE(U21(X1, X2)) → ACTIVE(X1)
ACTIVE(U22(X)) → ACTIVE(X)
ACTIVE(U31(X)) → ACTIVE(X)
ACTIVE(U41(X1, X2)) → ACTIVE(X1)
ACTIVE(U42(X)) → ACTIVE(X)
ACTIVE(U51(X1, X2)) → ACTIVE(X1)
ACTIVE(U52(X)) → ACTIVE(X)
ACTIVE(U61(X)) → ACTIVE(X)
ACTIVE(U71(X1, X2)) → ACTIVE(X1)
ACTIVE(U72(X)) → ACTIVE(X)
ACTIVE(U81(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)
__(x1, x2) = __(x1, x2)
U11(x1) = U11(x1)
U21(x1, x2) = U21(x1, x2)
U22(x1) = U22(x1)
U31(x1) = U31(x1)
U41(x1, x2) = U41(x1, x2)
U42(x1) = U42(x1)
U51(x1, x2) = U51(x1, x2)
U52(x1) = U52(x1)
U61(x1) = U61(x1)
U71(x1, x2) = U71(x1)
U72(x1) = U72(x1)
U81(x1) = U81(x1)
active(x1) = active(x1)
mark(x1) = mark(x1)
nil = nil
tt = tt
isList(x1) = isList(x1)
isNeList(x1) = x1
isPal(x1) = isPal
isQid(x1) = isQid
isNePal(x1) = isNePal
a = a
e = e
i = i
o = o
u = u
proper(x1) = x1
ok(x1) = x1
top(x1) = top

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

e > [U11₁, mark₁, a, i, o, u] > tt
top > [_{_₂}, U21₂, U52₁, active₁, isPal] > U22₁ > [ACTIVE₁, U72₁, U81₁] > [U11₁, mark₁, a, i, o, u] > tt
top > [_{_₂}, U21₂, U52₁, active₁, isPal] > U31₁ > [ACTIVE₁, U72₁, U81₁] > [U11₁, mark₁, a, i, o, u] > tt
top > [_{_₂}, U21₂, U52₁, active₁, isPal] > U41₂ > [ACTIVE₁, U72₁, U81₁] > [U11₁, mark₁, a, i, o, u] > tt
top > [_{_₂}, U21₂, U52₁, active₁, isPal] > U42₁ > [ACTIVE₁, U72₁, U81₁] > [U11₁, mark₁, a, i, o, u] > tt
top > [_{_₂}, U21₂, U52₁, active₁, isPal] > [U51₂, isList₁] > [ACTIVE₁, U72₁, U81₁] > [U11₁, mark₁, a, i, o, u] > tt
top > [_{_₂}, U21₂, U52₁, active₁, isPal] > isNePal > U61₁ > [ACTIVE₁, U72₁, U81₁] > [U11₁, mark₁, a, i, o, u] > tt
top > [_{_₂}, U21₂, U52₁, active₁, isPal] > isNePal > U71₁ > [ACTIVE₁, U72₁, U81₁] > [U11₁, mark₁, a, i, o, u] > tt
top > [_{_₂}, U21₂, U52₁, active₁, isPal] > isNePal > isQid > [U11₁, mark₁, a, i, o, u] > tt

Status:

ACTIVE₁: [1]
__₂: [2,1]
U11₁: [1]
U21₂: [1,2]
U22₁: [1]
U31₁: [1]
U41₂: [2,1]
U42₁: [1]
U51₂: [2,1]
U52₁: [1]
U61₁: [1]
U71₁: [1]
U72₁: [1]
U81₁: [1]
active₁: [1]
mark₁: [1]
nil: []
tt: []
isList₁: [1]
isPal: []
isQid: []
isNePal: []
a: []
e: []
i: []
o: []
u: []
top: []

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(144) Obligation:

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

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(145) PisEmptyProof (EQUIVALENT transformation)

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

(146) TRUE

(147) 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(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(148) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TOP(mark(X)) → TOP(proper(X))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
TOP(x1) = x1
ok(x1) = x1
active(x1) = x1
mark(x1) = mark(x1)
proper(x1) = x1
__(x1, x2) = __(x1, x2)
nil = nil
U11(x1) = U11(x1)
tt = tt
U21(x1, x2) = U21(x1, x2)
U22(x1) = U22(x1)
isList(x1) = isList(x1)
U31(x1) = U31(x1)
U41(x1, x2) = U41(x1, x2)
U42(x1) = U42(x1)
isNeList(x1) = isNeList(x1)
U51(x1, x2) = U51(x1, x2)
U52(x1) = U52(x1)
U61(x1) = U61(x1)
U71(x1, x2) = U71(x1, x2)
U72(x1) = U72(x1)
isPal(x1) = isPal(x1)
U81(x1) = U81(x1)
isQid(x1) = x1
isNePal(x1) = isNePal(x1)
a = a
e = e
i = i
o = o
u = u
top(x1) = top

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

__₂ > U21₂ > isList₁ > [nil, tt, isNeList₁] > U22₁ > [mark₁, U11₁, U31₁, U42₁, U52₁, U61₁]
__₂ > U41₂ > [nil, tt, isNeList₁] > U22₁ > [mark₁, U11₁, U31₁, U42₁, U52₁, U61₁]
__₂ > U51₂ > isList₁ > [nil, tt, isNeList₁] > U22₁ > [mark₁, U11₁, U31₁, U42₁, U52₁, U61₁]
__₂ > [U71₂, isPal₁] > U72₁ > [nil, tt, isNeList₁] > U22₁ > [mark₁, U11₁, U31₁, U42₁, U52₁, U61₁]
__₂ > [U71₂, isPal₁] > U81₁ > [nil, tt, isNeList₁] > U22₁ > [mark₁, U11₁, U31₁, U42₁, U52₁, U61₁]
__₂ > [U71₂, isPal₁] > isNePal₁ > [mark₁, U11₁, U31₁, U42₁, U52₁, U61₁]
a > [nil, tt, isNeList₁] > U22₁ > [mark₁, U11₁, U31₁, U42₁, U52₁, U61₁]
e > [nil, tt, isNeList₁] > U22₁ > [mark₁, U11₁, U31₁, U42₁, U52₁, U61₁]
i > [nil, tt, isNeList₁] > U22₁ > [mark₁, U11₁, U31₁, U42₁, U52₁, U61₁]
o > [nil, tt, isNeList₁] > U22₁ > [mark₁, U11₁, U31₁, U42₁, U52₁, U61₁]
u > [nil, tt, isNeList₁] > U22₁ > [mark₁, U11₁, U31₁, U42₁, U52₁, U61₁]
top > [mark₁, U11₁, U31₁, U42₁, U52₁, U61₁]

Status:

mark₁: [1]
__₂: [1,2]
nil: []
U11₁: [1]
tt: []
U21₂: [2,1]
U22₁: [1]
isList₁: [1]
U31₁: [1]
U41₂: [2,1]
U42₁: [1]
isNeList₁: [1]
U51₂: [1,2]
U52₁: [1]
U61₁: [1]
U71₂: [2,1]
U72₁: [1]
isPal₁: [1]
U81₁: [1]
isNePal₁: [1]
a: []
e: []
i: []
o: []
u: []
top: []

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(149) Obligation:

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

TOP(ok(X)) → TOP(active(X))

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(150) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TOP(ok(X)) → TOP(active(X))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
TOP(x1) = TOP(x1)
ok(x1) = ok(x1)
active(x1) = x1
__(x1, x2) = __(x1, x2)
mark(x1) = mark
nil = nil
U11(x1) = x1
tt = tt
U21(x1, x2) = U21(x1)
U22(x1) = U22(x1)
isList(x1) = isList(x1)
U31(x1) = x1
U41(x1, x2) = U41(x1, x2)
U42(x1) = U42(x1)
isNeList(x1) = isNeList(x1)
U51(x1, x2) = U51(x2)
U52(x1) = x1
U61(x1) = U61(x1)
U71(x1, x2) = x1
U72(x1) = x1
isPal(x1) = isPal(x1)
U81(x1) = x1
isQid(x1) = x1
isNePal(x1) = isNePal(x1)
a = a
e = e
i = i
o = o
u = u
proper(x1) = proper(x1)
top(x1) = top

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

a > [TOP₁, ok₁, U21₁, U22₁, U42₁, isNeList₁, U61₁, isNePal₁] > tt > [mark, u]
top > proper₁ > nil > [TOP₁, ok₁, U21₁, U22₁, U42₁, isNeList₁, U61₁, isNePal₁] > tt > [mark, u]
top > proper₁ > U51₁ > [_{_₂}, isList₁, U41₂] > [TOP₁, ok₁, U21₁, U22₁, U42₁, isNeList₁, U61₁, isNePal₁] > tt > [mark, u]
top > proper₁ > isPal₁ > [TOP₁, ok₁, U21₁, U22₁, U42₁, isNeList₁, U61₁, isNePal₁] > tt > [mark, u]
top > proper₁ > e > [mark, u]
top > proper₁ > i > [TOP₁, ok₁, U21₁, U22₁, U42₁, isNeList₁, U61₁, isNePal₁] > tt > [mark, u]
top > proper₁ > o > [mark, u]

Status:

TOP₁: [1]
ok₁: [1]
__₂: [1,2]
mark: []
nil: []
tt: []
U21₁: [1]
U22₁: [1]
isList₁: [1]
U41₂: [1,2]
U42₁: [1]
isNeList₁: [1]
U51₁: [1]
U61₁: [1]
isPal₁: [1]
isNePal₁: [1]
a: []
e: []
i: []
o: []
u: []
proper₁: [1]
top: []

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(151) Obligation:

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

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(tt)
active(U21(tt, V2)) → mark(U22(isList(V2)))
active(U22(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNeList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt, P)) → mark(U72(isPal(P)))
active(U72(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(isList(V)) → mark(U11(isNeList(V)))
active(isList(nil)) → mark(tt)
active(isList(__(V1, V2))) → mark(U21(isList(V1), V2))
active(isNeList(V)) → mark(U31(isQid(V)))
active(isNeList(__(V1, V2))) → mark(U41(isList(V1), V2))
active(isNeList(__(V1, V2))) → mark(U51(isNeList(V1), V2))
active(isNePal(V)) → mark(U61(isQid(V)))
active(isNePal(__(I, __(P, I)))) → mark(U71(isQid(I), P))
active(isPal(V)) → mark(U81(isNePal(V)))
active(isPal(nil)) → mark(tt)
active(isQid(a)) → mark(tt)
active(isQid(e)) → mark(tt)
active(isQid(i)) → mark(tt)
active(isQid(o)) → mark(tt)
active(isQid(u)) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X)) → U81(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X)) → mark(U81(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isList(X)) → isList(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNeList(X)) → isNeList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(isPal(X)) → isPal(proper(X))
proper(U81(X)) → U81(proper(X))
proper(isQid(X)) → isQid(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
proper(a) → ok(a)
proper(e) → ok(e)
proper(i) → ok(i)
proper(o) → ok(o)
proper(u) → ok(u)
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isList(ok(X)) → ok(isList(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNeList(ok(X)) → ok(isNeList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
isPal(ok(X)) → ok(isPal(X))
U81(ok(X)) → ok(U81(X))
isQid(ok(X)) → ok(isQid(X))
isNePal(ok(X)) → ok(isNePal(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.

(152) PisEmptyProof (EQUIVALENT transformation)

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