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

0 QTRS
↳1 DependencyPairsProof (⇔)
↳2 QDP
↳3 DependencyGraphProof (⇔)
↳4 AND
  ↳5 QDP
    ↳6 QDPOrderProof (⇔)
    ↳7 QDP
    ↳8 PisEmptyProof (⇔)
    ↳9 TRUE
  ↳10 QDP
    ↳11 QDPOrderProof (⇔)
    ↳12 QDP
    ↳13 PisEmptyProof (⇔)
    ↳14 TRUE
  ↳15 QDP
    ↳16 QDPOrderProof (⇔)
    ↳17 QDP
    ↳18 PisEmptyProof (⇔)
    ↳19 TRUE
  ↳20 QDP
    ↳21 QDPOrderProof (⇔)
    ↳22 QDP
    ↳23 QDPOrderProof (⇔)
    ↳24 QDP
    ↳25 PisEmptyProof (⇔)
    ↳26 TRUE
  ↳27 QDP
    ↳28 QDPOrderProof (⇔)
    ↳29 QDP
    ↳30 QDPOrderProof (⇔)
    ↳31 QDP
    ↳32 PisEmptyProof (⇔)
    ↳33 TRUE
  ↳34 QDP
    ↳35 QDPOrderProof (⇔)
    ↳36 QDP
    ↳37 QDPOrderProof (⇔)
    ↳38 QDP
    ↳39 PisEmptyProof (⇔)
    ↳40 TRUE
  ↳41 QDP
    ↳42 QDPOrderProof (⇔)
    ↳43 QDP
    ↳44 QDPOrderProof (⇔)
    ↳45 QDP
    ↳46 PisEmptyProof (⇔)
    ↳47 TRUE
  ↳48 QDP
    ↳49 QDPOrderProof (⇔)
    ↳50 QDP
    ↳51 QDPOrderProof (⇔)
    ↳52 QDP
    ↳53 PisEmptyProof (⇔)
    ↳54 TRUE
  ↳55 QDP
    ↳56 QDPOrderProof (⇔)
    ↳57 QDP
    ↳58 QDPOrderProof (⇔)
    ↳59 QDP
    ↳60 PisEmptyProof (⇔)
    ↳61 TRUE
  ↳62 QDP
    ↳63 QDPOrderProof (⇔)
    ↳64 QDP
    ↳65 QDPOrderProof (⇔)
    ↳66 QDP
    ↳67 PisEmptyProof (⇔)
    ↳68 TRUE
  ↳69 QDP
    ↳70 QDPOrderProof (⇔)
    ↳71 QDP
    ↳72 QDPOrderProof (⇔)
    ↳73 QDP
    ↳74 PisEmptyProof (⇔)
    ↳75 TRUE
  ↳76 QDP
    ↳77 QDPOrderProof (⇔)
    ↳78 QDP
    ↳79 QDPOrderProof (⇔)
    ↳80 QDP
    ↳81 PisEmptyProof (⇔)
    ↳82 TRUE
  ↳83 QDP
    ↳84 QDPOrderProof (⇔)
    ↳85 QDP
    ↳86 QDPOrderProof (⇔)
    ↳87 QDP
    ↳88 PisEmptyProof (⇔)
    ↳89 TRUE
  ↳90 QDP
    ↳91 QDPOrderProof (⇔)
    ↳92 QDP
    ↳93 QDPOrderProof (⇔)
    ↳94 QDP
    ↳95 PisEmptyProof (⇔)
    ↳96 TRUE
  ↳97 QDP
    ↳98 QDPOrderProof (⇔)
    ↳99 QDP
    ↳100 QDPOrderProof (⇔)
    ↳101 QDP
    ↳102 PisEmptyProof (⇔)
    ↳103 TRUE
  ↳104 QDP
    ↳105 QDPOrderProof (⇔)
    ↳106 QDP
    ↳107 QDPOrderProof (⇔)
    ↳108 QDP
    ↳109 QDPOrderProof (⇔)
    ↳110 QDP
    ↳111 QDPOrderProof (⇔)
    ↳112 QDP
    ↳113 QDPOrderProof (⇔)
    ↳114 QDP
    ↳115 QDPOrderProof (⇔)
    ↳116 QDP
    ↳117 QDPOrderProof (⇔)
    ↳118 QDP
    ↳119 QDPOrderProof (⇔)
    ↳120 QDP
    ↳121 QDPOrderProof (⇔)
    ↳122 QDP
    ↳123 QDPOrderProof (⇔)
    ↳124 QDP
    ↳125 PisEmptyProof (⇔)
    ↳126 TRUE
  ↳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 QDPOrderProof (⇔)
    ↳141 QDP
    ↳142 QDPOrderProof (⇔)
    ↳143 QDP
    ↳144 QDPOrderProof (⇔)
    ↳145 QDP
    ↳146 PisEmptyProof (⇔)
    ↳147 TRUE
  ↳148 QDP

(0) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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(zeros) → CONS(0, zeros)
ACTIVE(U41(tt, V2)) → U421(isNatIList(V2))
ACTIVE(U41(tt, V2)) → ISNATILIST(V2)
ACTIVE(U51(tt, V2)) → U521(isNatList(V2))
ACTIVE(U51(tt, V2)) → ISNATLIST(V2)
ACTIVE(U61(tt, L, N)) → U621(isNat(N), L)
ACTIVE(U61(tt, L, N)) → ISNAT(N)
ACTIVE(U62(tt, L)) → S(length(L))
ACTIVE(U62(tt, L)) → LENGTH(L)
ACTIVE(isNat(length(V1))) → U111(isNatList(V1))
ACTIVE(isNat(length(V1))) → ISNATLIST(V1)
ACTIVE(isNat(s(V1))) → U211(isNat(V1))
ACTIVE(isNat(s(V1))) → ISNAT(V1)
ACTIVE(isNatIList(V)) → U311(isNatList(V))
ACTIVE(isNatIList(V)) → ISNATLIST(V)
ACTIVE(isNatIList(cons(V1, V2))) → U411(isNat(V1), V2)
ACTIVE(isNatIList(cons(V1, V2))) → ISNAT(V1)
ACTIVE(isNatList(cons(V1, V2))) → U511(isNat(V1), V2)
ACTIVE(isNatList(cons(V1, V2))) → ISNAT(V1)
ACTIVE(length(cons(N, L))) → U611(isNatList(L), L, N)
ACTIVE(length(cons(N, L))) → ISNATLIST(L)
ACTIVE(cons(X1, X2)) → CONS(active(X1), X2)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(U11(X)) → U111(active(X))
ACTIVE(U11(X)) → ACTIVE(X)
ACTIVE(U21(X)) → U211(active(X))
ACTIVE(U21(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(X1, X2, X3)) → U611(active(X1), X2, X3)
ACTIVE(U61(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U62(X1, X2)) → U621(active(X1), X2)
ACTIVE(U62(X1, X2)) → ACTIVE(X1)
ACTIVE(s(X)) → S(active(X))
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(length(X)) → LENGTH(active(X))
ACTIVE(length(X)) → ACTIVE(X)
CONS(mark(X1), X2) → CONS(X1, X2)
U111(mark(X)) → U111(X)
U211(mark(X)) → U211(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(X1), X2, X3) → U611(X1, X2, X3)
U621(mark(X1), X2) → U621(X1, X2)
S(mark(X)) → S(X)
LENGTH(mark(X)) → LENGTH(X)
PROPER(cons(X1, X2)) → CONS(proper(X1), proper(X2))
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(U11(X)) → U111(proper(X))
PROPER(U11(X)) → PROPER(X)
PROPER(U21(X)) → U211(proper(X))
PROPER(U21(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(isNatIList(X)) → ISNATILIST(proper(X))
PROPER(isNatIList(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(isNatList(X)) → ISNATLIST(proper(X))
PROPER(isNatList(X)) → PROPER(X)
PROPER(U61(X1, X2, X3)) → U611(proper(X1), proper(X2), proper(X3))
PROPER(U61(X1, X2, X3)) → PROPER(X1)
PROPER(U61(X1, X2, X3)) → PROPER(X2)
PROPER(U61(X1, X2, X3)) → PROPER(X3)
PROPER(U62(X1, X2)) → U621(proper(X1), proper(X2))
PROPER(U62(X1, X2)) → PROPER(X1)
PROPER(U62(X1, X2)) → PROPER(X2)
PROPER(isNat(X)) → ISNAT(proper(X))
PROPER(isNat(X)) → PROPER(X)
PROPER(s(X)) → S(proper(X))
PROPER(s(X)) → PROPER(X)
PROPER(length(X)) → LENGTH(proper(X))
PROPER(length(X)) → PROPER(X)
CONS(ok(X1), ok(X2)) → CONS(X1, X2)
U111(ok(X)) → U111(X)
U211(ok(X)) → U211(X)
U311(ok(X)) → U311(X)
U411(ok(X1), ok(X2)) → U411(X1, X2)
U421(ok(X)) → U421(X)
ISNATILIST(ok(X)) → ISNATILIST(X)
U511(ok(X1), ok(X2)) → U511(X1, X2)
U521(ok(X)) → U521(X)
ISNATLIST(ok(X)) → ISNATLIST(X)
U611(ok(X1), ok(X2), ok(X3)) → U611(X1, X2, X3)
U621(ok(X1), ok(X2)) → U621(X1, X2)
ISNAT(ok(X)) → ISNAT(X)
S(ok(X)) → S(X)
LENGTH(ok(X)) → LENGTH(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(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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 18 SCCs with 50 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

ISNAT(ok(X)) → ISNAT(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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.


ISNAT(ok(X)) → ISNAT(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNAT(x1)  =  ISNAT(x1)
ok(x1)  =  ok(x1)
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  x1
cons(x1, x2)  =  cons(x1, x2)
0  =  0
U11(x1)  =  U11(x1)
tt  =  tt
U21(x1)  =  x1
U31(x1)  =  x1
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  x1
isNatIList(x1)  =  x1
U51(x1, x2)  =  x2
U52(x1)  =  U52(x1)
isNatList(x1)  =  isNatList(x1)
U61(x1, x2, x3)  =  x2
U62(x1, x2)  =  x2
isNat(x1)  =  x1
s(x1)  =  x1
length(x1)  =  x1
nil  =  nil
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
active1 > [isNatList1, proper1] > zeros > cons2 > U412 > [ok1, U111]
active1 > [isNatList1, proper1] > zeros > 0 > [ok1, U111]
active1 > [isNatList1, proper1] > tt > U521 > [ok1, U111]
nil > 0 > [ok1, U111]
nil > tt > U521 > [ok1, U111]
top > [isNatList1, proper1] > zeros > cons2 > U412 > [ok1, U111]
top > [isNatList1, proper1] > zeros > 0 > [ok1, U111]
top > [isNatList1, proper1] > tt > U521 > [ok1, U111]


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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:

ISNATLIST(ok(X)) → ISNATLIST(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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.


ISNATLIST(ok(X)) → ISNATLIST(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNATLIST(x1)  =  ISNATLIST(x1)
ok(x1)  =  ok(x1)
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  x1
cons(x1, x2)  =  cons(x1, x2)
0  =  0
U11(x1)  =  U11(x1)
tt  =  tt
U21(x1)  =  x1
U31(x1)  =  x1
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  x1
isNatIList(x1)  =  x1
U51(x1, x2)  =  x2
U52(x1)  =  U52(x1)
isNatList(x1)  =  isNatList(x1)
U61(x1, x2, x3)  =  x2
U62(x1, x2)  =  x2
isNat(x1)  =  x1
s(x1)  =  x1
length(x1)  =  x1
nil  =  nil
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
active1 > [isNatList1, proper1] > zeros > cons2 > U412 > [ok1, U111]
active1 > [isNatList1, proper1] > zeros > 0 > [ok1, U111]
active1 > [isNatList1, proper1] > tt > U521 > [ok1, U111]
nil > 0 > [ok1, U111]
nil > tt > U521 > [ok1, U111]
top > [isNatList1, proper1] > zeros > cons2 > U412 > [ok1, U111]
top > [isNatList1, proper1] > zeros > 0 > [ok1, U111]
top > [isNatList1, proper1] > tt > U521 > [ok1, U111]


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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:

ISNATILIST(ok(X)) → ISNATILIST(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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.


ISNATILIST(ok(X)) → ISNATILIST(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNATILIST(x1)  =  ISNATILIST(x1)
ok(x1)  =  ok(x1)
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  x1
cons(x1, x2)  =  cons(x1, x2)
0  =  0
U11(x1)  =  U11(x1)
tt  =  tt
U21(x1)  =  x1
U31(x1)  =  x1
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  x1
isNatIList(x1)  =  x1
U51(x1, x2)  =  x2
U52(x1)  =  U52(x1)
isNatList(x1)  =  isNatList(x1)
U61(x1, x2, x3)  =  x2
U62(x1, x2)  =  x2
isNat(x1)  =  x1
s(x1)  =  x1
length(x1)  =  x1
nil  =  nil
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
active1 > [isNatList1, proper1] > zeros > cons2 > U412 > [ok1, U111]
active1 > [isNatList1, proper1] > zeros > 0 > [ok1, U111]
active1 > [isNatList1, proper1] > tt > U521 > [ok1, U111]
nil > 0 > [ok1, U111]
nil > tt > U521 > [ok1, U111]
top > [isNatList1, proper1] > zeros > cons2 > U412 > [ok1, U111]
top > [isNatList1, proper1] > zeros > 0 > [ok1, U111]
top > [isNatList1, proper1] > tt > U521 > [ok1, U111]


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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:

LENGTH(ok(X)) → LENGTH(X)
LENGTH(mark(X)) → LENGTH(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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.


LENGTH(mark(X)) → LENGTH(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
LENGTH(x1)  =  LENGTH(x1)
ok(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1, x2)
0  =  0
U11(x1)  =  x1
tt  =  tt
U21(x1)  =  U21(x1)
U31(x1)  =  x1
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  U42(x1)
isNatIList(x1)  =  x1
U51(x1, x2)  =  U51(x1, x2)
U52(x1)  =  x1
isNatList(x1)  =  x1
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2)  =  U62(x1, x2)
isNat(x1)  =  isNat(x1)
s(x1)  =  x1
length(x1)  =  x1
nil  =  nil
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
LENGTH1 > mark1
zeros > [0, tt, nil] > [active1, U421, top] > [cons2, U613] > U412 > mark1
zeros > [0, tt, nil] > [active1, U421, top] > U512 > mark1
zeros > [0, tt, nil] > [active1, U421, top] > U622 > mark1
zeros > [0, tt, nil] > [active1, U421, top] > isNat1 > U211 > mark1
proper1 > [0, tt, nil] > [active1, U421, top] > [cons2, U613] > U412 > mark1
proper1 > [0, tt, nil] > [active1, U421, top] > U512 > mark1
proper1 > [0, tt, nil] > [active1, U421, top] > U622 > mark1
proper1 > [0, tt, nil] > [active1, U421, top] > isNat1 > U211 > mark1


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(22) Obligation:

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

LENGTH(ok(X)) → LENGTH(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(23) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


LENGTH(ok(X)) → LENGTH(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
LENGTH(x1)  =  LENGTH(x1)
ok(x1)  =  ok(x1)
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  x1
cons(x1, x2)  =  cons(x1, x2)
0  =  0
U11(x1)  =  U11(x1)
tt  =  tt
U21(x1)  =  x1
U31(x1)  =  x1
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  x1
isNatIList(x1)  =  x1
U51(x1, x2)  =  x2
U52(x1)  =  U52(x1)
isNatList(x1)  =  isNatList(x1)
U61(x1, x2, x3)  =  x2
U62(x1, x2)  =  x2
isNat(x1)  =  x1
s(x1)  =  x1
length(x1)  =  x1
nil  =  nil
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
active1 > [isNatList1, proper1] > zeros > cons2 > U412 > [ok1, U111]
active1 > [isNatList1, proper1] > zeros > 0 > [ok1, U111]
active1 > [isNatList1, proper1] > tt > U521 > [ok1, U111]
nil > 0 > [ok1, U111]
nil > tt > U521 > [ok1, U111]
top > [isNatList1, proper1] > zeros > cons2 > U412 > [ok1, U111]
top > [isNatList1, proper1] > zeros > 0 > [ok1, U111]
top > [isNatList1, proper1] > tt > U521 > [ok1, U111]


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(24) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(25) PisEmptyProof (EQUIVALENT transformation)

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

(26) TRUE

(27) Obligation:

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

S(ok(X)) → S(X)
S(mark(X)) → S(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(28) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


S(mark(X)) → S(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
S(x1)  =  S(x1)
ok(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1, x2)
0  =  0
U11(x1)  =  x1
tt  =  tt
U21(x1)  =  U21(x1)
U31(x1)  =  x1
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  U42(x1)
isNatIList(x1)  =  x1
U51(x1, x2)  =  U51(x1, x2)
U52(x1)  =  x1
isNatList(x1)  =  x1
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2)  =  U62(x1, x2)
isNat(x1)  =  isNat(x1)
s(x1)  =  x1
length(x1)  =  x1
nil  =  nil
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
S1 > mark1
zeros > [0, tt, nil] > [active1, U421, top] > [cons2, U613] > U412 > mark1
zeros > [0, tt, nil] > [active1, U421, top] > U512 > mark1
zeros > [0, tt, nil] > [active1, U421, top] > U622 > mark1
zeros > [0, tt, nil] > [active1, U421, top] > isNat1 > U211 > mark1
proper1 > [0, tt, nil] > [active1, U421, top] > [cons2, U613] > U412 > mark1
proper1 > [0, tt, nil] > [active1, U421, top] > U512 > mark1
proper1 > [0, tt, nil] > [active1, U421, top] > U622 > mark1
proper1 > [0, tt, nil] > [active1, U421, top] > isNat1 > U211 > mark1


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(29) Obligation:

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

S(ok(X)) → S(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(30) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


S(ok(X)) → S(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
S(x1)  =  S(x1)
ok(x1)  =  ok(x1)
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  x1
cons(x1, x2)  =  cons(x1, x2)
0  =  0
U11(x1)  =  U11(x1)
tt  =  tt
U21(x1)  =  x1
U31(x1)  =  x1
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  x1
isNatIList(x1)  =  x1
U51(x1, x2)  =  x2
U52(x1)  =  U52(x1)
isNatList(x1)  =  isNatList(x1)
U61(x1, x2, x3)  =  x2
U62(x1, x2)  =  x2
isNat(x1)  =  x1
s(x1)  =  x1
length(x1)  =  x1
nil  =  nil
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
active1 > [isNatList1, proper1] > zeros > cons2 > U412 > [ok1, U111]
active1 > [isNatList1, proper1] > zeros > 0 > [ok1, U111]
active1 > [isNatList1, proper1] > tt > U521 > [ok1, U111]
nil > 0 > [ok1, U111]
nil > tt > U521 > [ok1, U111]
top > [isNatList1, proper1] > zeros > cons2 > U412 > [ok1, U111]
top > [isNatList1, proper1] > zeros > 0 > [ok1, U111]
top > [isNatList1, proper1] > tt > U521 > [ok1, U111]


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(31) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(32) PisEmptyProof (EQUIVALENT transformation)

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

(33) TRUE

(34) Obligation:

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

U621(ok(X1), ok(X2)) → U621(X1, X2)
U621(mark(X1), X2) → U621(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U621(mark(X1), X2) → U621(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U621(x1, x2)  =  U621(x1, x2)
ok(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1, x2)
0  =  0
U11(x1)  =  U11(x1)
tt  =  tt
U21(x1)  =  x1
U31(x1)  =  x1
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  x1
isNatIList(x1)  =  x1
U51(x1, x2)  =  U51(x1, x2)
U52(x1)  =  U52(x1)
isNatList(x1)  =  x1
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2)  =  U62(x1, x2)
isNat(x1)  =  x1
s(x1)  =  x1
length(x1)  =  x1
nil  =  nil
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
U62^12 > mark1
[zeros, 0, nil, proper1] > [active1, U613] > cons2 > mark1
[zeros, 0, nil, proper1] > [active1, U613] > U111 > mark1
[zeros, 0, nil, proper1] > [active1, U613] > U412 > mark1
[zeros, 0, nil, proper1] > [active1, U613] > U512 > U521 > mark1
[zeros, 0, nil, proper1] > [active1, U613] > U622 > mark1
[zeros, 0, nil, proper1] > tt > U521 > mark1
[zeros, 0, nil, proper1] > tt > U622 > mark1
top > mark1


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(36) Obligation:

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

U621(ok(X1), ok(X2)) → U621(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(37) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U621(ok(X1), ok(X2)) → U621(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U621(x1, x2)  =  x2
ok(x1)  =  ok(x1)
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  x1
cons(x1, x2)  =  x2
0  =  0
U11(x1)  =  U11(x1)
tt  =  tt
U21(x1)  =  U21(x1)
U31(x1)  =  x1
U41(x1, x2)  =  U41(x2)
U42(x1)  =  x1
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51(x2)
U52(x1)  =  x1
isNatList(x1)  =  x1
U61(x1, x2, x3)  =  U61(x1, x2)
U62(x1, x2)  =  U62(x2)
isNat(x1)  =  isNat(x1)
s(x1)  =  x1
length(x1)  =  x1
nil  =  nil
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
[active1, U111, proper1] > zeros > 0 > ok1
[active1, U111, proper1] > tt > isNatIList1 > U411 > ok1
[active1, U111, proper1] > tt > isNatIList1 > [U612, isNat1] > U211 > ok1
[active1, U111, proper1] > tt > U621 > ok1
[active1, U111, proper1] > U511 > ok1
[active1, U111, proper1] > nil


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(38) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(39) PisEmptyProof (EQUIVALENT transformation)

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

(40) TRUE

(41) Obligation:

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

U611(ok(X1), ok(X2), ok(X3)) → U611(X1, X2, X3)
U611(mark(X1), X2, X3) → U611(X1, X2, X3)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U611(ok(X1), ok(X2), ok(X3)) → U611(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U611(x1, x2, x3)  =  U611(x1, x2, x3)
ok(x1)  =  ok(x1)
mark(x1)  =  x1
active(x1)  =  x1
zeros  =  zeros
cons(x1, x2)  =  x2
0  =  0
U11(x1)  =  x1
tt  =  tt
U21(x1)  =  x1
U31(x1)  =  x1
U41(x1, x2)  =  x2
U42(x1)  =  x1
isNatIList(x1)  =  x1
U51(x1, x2)  =  x2
U52(x1)  =  x1
isNatList(x1)  =  x1
U61(x1, x2, x3)  =  x2
U62(x1, x2)  =  x2
isNat(x1)  =  x1
s(x1)  =  x1
length(x1)  =  x1
nil  =  nil
proper(x1)  =  proper
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
proper > [0, nil] > [ok1, zeros] > U61^13
proper > [0, nil] > [ok1, zeros] > tt


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(43) Obligation:

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

U611(mark(X1), X2, X3) → U611(X1, X2, X3)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(44) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U611(mark(X1), X2, X3) → U611(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U611(x1, x2, x3)  =  U611(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1, x2)
0  =  0
U11(x1)  =  x1
tt  =  tt
U21(x1)  =  U21(x1)
U31(x1)  =  U31(x1)
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  U42(x1)
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  x1
U52(x1)  =  x1
isNatList(x1)  =  isNatList
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2)  =  U62(x1, x2)
isNat(x1)  =  isNat
s(x1)  =  x1
length(x1)  =  length(x1)
nil  =  nil
proper(x1)  =  x1
ok(x1)  =  ok
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
U61^12 > mark1
[active1, U421, isNat] > [tt, U211] > [zeros, cons2, isNatIList1, isNatList, U613] > [0, U311, ok] > U622 > mark1
[active1, U421, isNat] > U412 > [zeros, cons2, isNatIList1, isNatList, U613] > [0, U311, ok] > U622 > mark1
[active1, U421, isNat] > length1 > [zeros, cons2, isNatIList1, isNatList, U613] > [0, U311, ok] > U622 > mark1
nil > [0, U311, ok] > U622 > mark1
top > mark1


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(45) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(46) PisEmptyProof (EQUIVALENT transformation)

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

(47) TRUE

(48) 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(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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) 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)  =  U521(x1)
ok(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1, x2)
0  =  0
U11(x1)  =  x1
tt  =  tt
U21(x1)  =  U21(x1)
U31(x1)  =  x1
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  U42(x1)
isNatIList(x1)  =  x1
U51(x1, x2)  =  U51(x1, x2)
U52(x1)  =  x1
isNatList(x1)  =  x1
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2)  =  U62(x1, x2)
isNat(x1)  =  isNat(x1)
s(x1)  =  x1
length(x1)  =  x1
nil  =  nil
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
U52^11 > mark1
zeros > [0, tt, nil] > [active1, U421, top] > [cons2, U613] > U412 > mark1
zeros > [0, tt, nil] > [active1, U421, top] > U512 > mark1
zeros > [0, tt, nil] > [active1, U421, top] > U622 > mark1
zeros > [0, tt, nil] > [active1, U421, top] > isNat1 > U211 > mark1
proper1 > [0, tt, nil] > [active1, U421, top] > [cons2, U613] > U412 > mark1
proper1 > [0, tt, nil] > [active1, U421, top] > U512 > mark1
proper1 > [0, tt, nil] > [active1, U421, top] > U622 > mark1
proper1 > [0, tt, nil] > [active1, U421, top] > isNat1 > U211 > mark1


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(50) Obligation:

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

U521(ok(X)) → U521(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(51) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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)
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  x1
cons(x1, x2)  =  cons(x1, x2)
0  =  0
U11(x1)  =  U11(x1)
tt  =  tt
U21(x1)  =  x1
U31(x1)  =  x1
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  x1
isNatIList(x1)  =  x1
U51(x1, x2)  =  x2
U52(x1)  =  U52(x1)
isNatList(x1)  =  isNatList(x1)
U61(x1, x2, x3)  =  x2
U62(x1, x2)  =  x2
isNat(x1)  =  x1
s(x1)  =  x1
length(x1)  =  x1
nil  =  nil
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
active1 > [isNatList1, proper1] > zeros > cons2 > U412 > [ok1, U111]
active1 > [isNatList1, proper1] > zeros > 0 > [ok1, U111]
active1 > [isNatList1, proper1] > tt > U521 > [ok1, U111]
nil > 0 > [ok1, U111]
nil > tt > U521 > [ok1, U111]
top > [isNatList1, proper1] > zeros > cons2 > U412 > [ok1, U111]
top > [isNatList1, proper1] > zeros > 0 > [ok1, U111]
top > [isNatList1, proper1] > tt > U521 > [ok1, U111]


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(52) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(53) PisEmptyProof (EQUIVALENT transformation)

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

(54) TRUE

(55) 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(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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) 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)  =  U511(x1, x2)
ok(x1)  =  ok(x1)
mark(x1)  =  x1
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  x2
0  =  0
U11(x1)  =  x1
tt  =  tt
U21(x1)  =  x1
U31(x1)  =  x1
U41(x1, x2)  =  x2
U42(x1)  =  U42(x1)
isNatIList(x1)  =  x1
U51(x1, x2)  =  U51(x2)
U52(x1)  =  U52(x1)
isNatList(x1)  =  isNatList(x1)
U61(x1, x2, x3)  =  U61(x2)
U62(x1, x2)  =  x2
isNat(x1)  =  isNat(x1)
s(x1)  =  s(x1)
length(x1)  =  x1
nil  =  nil
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
U51^12 > [ok1, U421]
[zeros, 0, tt, nil] > [active1, U511, U611, isNat1] > U521 > [ok1, U421]
[zeros, 0, tt, nil] > [active1, U511, U611, isNat1] > isNatList1 > [ok1, U421]
[zeros, 0, tt, nil] > [active1, U511, U611, isNat1] > s1 > [ok1, U421]
proper1 > [active1, U511, U611, isNat1] > U521 > [ok1, U421]
proper1 > [active1, U511, U611, isNat1] > isNatList1 > [ok1, U421]
proper1 > [active1, U511, U611, isNat1] > s1 > [ok1, U421]
top > [ok1, U421]


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(57) Obligation:

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

U511(mark(X1), X2) → U511(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(58) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1, x2)
0  =  0
U11(x1)  =  x1
tt  =  tt
U21(x1)  =  U21(x1)
U31(x1)  =  U31(x1)
U41(x1, x2)  =  x1
U42(x1)  =  x1
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  x1
U52(x1)  =  x1
isNatList(x1)  =  isNatList
U61(x1, x2, x3)  =  U61(x1, x2)
U62(x1, x2)  =  U62(x1, x2)
isNat(x1)  =  isNat
s(x1)  =  x1
length(x1)  =  x1
nil  =  nil
proper(x1)  =  x1
ok(x1)  =  x1
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
zeros > [cons2, U612] > mark1
zeros > [cons2, U612] > isNatList > tt
zeros > 0 > mark1
zeros > 0 > tt
nil > mark1
nil > tt
top > active1 > [cons2, U612] > mark1
top > active1 > [cons2, U612] > isNatList > tt
top > active1 > 0 > mark1
top > active1 > 0 > tt
top > active1 > U311 > mark1
top > active1 > U311 > tt
top > active1 > isNatIList > mark1
top > active1 > isNatIList > isNatList > tt
top > active1 > U622 > mark1
top > active1 > isNat > U211 > mark1
top > active1 > isNat > U211 > tt
top > active1 > isNat > isNatList > tt


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(59) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(60) PisEmptyProof (EQUIVALENT transformation)

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

(61) TRUE

(62) 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(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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) 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)  =  U421(x1)
ok(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1, x2)
0  =  0
U11(x1)  =  x1
tt  =  tt
U21(x1)  =  U21(x1)
U31(x1)  =  x1
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  U42(x1)
isNatIList(x1)  =  x1
U51(x1, x2)  =  U51(x1, x2)
U52(x1)  =  x1
isNatList(x1)  =  x1
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2)  =  U62(x1, x2)
isNat(x1)  =  isNat(x1)
s(x1)  =  x1
length(x1)  =  x1
nil  =  nil
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
U42^11 > mark1
zeros > [0, tt, nil] > [active1, U421, top] > [cons2, U613] > U412 > mark1
zeros > [0, tt, nil] > [active1, U421, top] > U512 > mark1
zeros > [0, tt, nil] > [active1, U421, top] > U622 > mark1
zeros > [0, tt, nil] > [active1, U421, top] > isNat1 > U211 > mark1
proper1 > [0, tt, nil] > [active1, U421, top] > [cons2, U613] > U412 > mark1
proper1 > [0, tt, nil] > [active1, U421, top] > U512 > mark1
proper1 > [0, tt, nil] > [active1, U421, top] > U622 > mark1
proper1 > [0, tt, nil] > [active1, U421, top] > isNat1 > U211 > mark1


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(64) Obligation:

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

U421(ok(X)) → U421(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(65) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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)
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  x1
cons(x1, x2)  =  cons(x1, x2)
0  =  0
U11(x1)  =  U11(x1)
tt  =  tt
U21(x1)  =  x1
U31(x1)  =  x1
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  x1
isNatIList(x1)  =  x1
U51(x1, x2)  =  x2
U52(x1)  =  U52(x1)
isNatList(x1)  =  isNatList(x1)
U61(x1, x2, x3)  =  x2
U62(x1, x2)  =  x2
isNat(x1)  =  x1
s(x1)  =  x1
length(x1)  =  x1
nil  =  nil
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
active1 > [isNatList1, proper1] > zeros > cons2 > U412 > [ok1, U111]
active1 > [isNatList1, proper1] > zeros > 0 > [ok1, U111]
active1 > [isNatList1, proper1] > tt > U521 > [ok1, U111]
nil > 0 > [ok1, U111]
nil > tt > U521 > [ok1, U111]
top > [isNatList1, proper1] > zeros > cons2 > U412 > [ok1, U111]
top > [isNatList1, proper1] > zeros > 0 > [ok1, U111]
top > [isNatList1, proper1] > tt > U521 > [ok1, U111]


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(66) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(67) PisEmptyProof (EQUIVALENT transformation)

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

(68) TRUE

(69) 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(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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) 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)  =  U411(x1)
ok(x1)  =  ok(x1)
mark(x1)  =  x1
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  x2
0  =  0
U11(x1)  =  U11(x1)
tt  =  tt
U21(x1)  =  x1
U31(x1)  =  U31(x1)
U41(x1, x2)  =  U41(x2)
U42(x1)  =  x1
isNatIList(x1)  =  x1
U51(x1, x2)  =  U51(x2)
U52(x1)  =  x1
isNatList(x1)  =  x1
U61(x1, x2, x3)  =  U61(x1, x2)
U62(x1, x2)  =  x2
isNat(x1)  =  isNat(x1)
s(x1)  =  x1
length(x1)  =  length(x1)
nil  =  nil
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
proper1 > [active1, tt, U311, U612, length1] > zeros
proper1 > [active1, tt, U311, U612, length1] > 0 > [ok1, U411, U511]
proper1 > [active1, tt, U311, U612, length1] > [U111, isNat1] > [ok1, U411, U511]
proper1 > nil
top > [active1, tt, U311, U612, length1] > zeros
top > [active1, tt, U311, U612, length1] > 0 > [ok1, U411, U511]
top > [active1, tt, U311, U612, length1] > [U111, isNat1] > [ok1, U411, U511]


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(71) Obligation:

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

U411(mark(X1), X2) → U411(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(72) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1, x2)
0  =  0
U11(x1)  =  U11(x1)
tt  =  tt
U21(x1)  =  x1
U31(x1)  =  U31(x1)
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  U42(x1)
isNatIList(x1)  =  x1
U51(x1, x2)  =  x1
U52(x1)  =  x1
isNatList(x1)  =  isNatList
U61(x1, x2, x3)  =  U61(x1, x2)
U62(x1, x2)  =  U62(x1, x2)
isNat(x1)  =  isNat
s(x1)  =  x1
length(x1)  =  length(x1)
nil  =  nil
proper(x1)  =  x1
ok(x1)  =  x1
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
[active1, 0, tt, isNatList, isNat] > zeros > [U41^12, mark1, U311]
[active1, 0, tt, isNatList, isNat] > cons2 > U612 > [U41^12, mark1, U311]
[active1, 0, tt, isNatList, isNat] > U111 > [U41^12, mark1, U311]
[active1, 0, tt, isNatList, isNat] > [U412, U421] > [U41^12, mark1, U311]
[active1, 0, tt, isNatList, isNat] > U622 > length1 > [U41^12, mark1, U311]


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(73) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(74) PisEmptyProof (EQUIVALENT transformation)

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

(75) TRUE

(76) 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(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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) 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)  =  U311(x1)
ok(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1, x2)
0  =  0
U11(x1)  =  x1
tt  =  tt
U21(x1)  =  U21(x1)
U31(x1)  =  x1
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  U42(x1)
isNatIList(x1)  =  x1
U51(x1, x2)  =  U51(x1, x2)
U52(x1)  =  x1
isNatList(x1)  =  x1
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2)  =  U62(x1, x2)
isNat(x1)  =  isNat(x1)
s(x1)  =  x1
length(x1)  =  x1
nil  =  nil
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
U31^11 > mark1
zeros > [0, tt, nil] > [active1, U421, top] > [cons2, U613] > U412 > mark1
zeros > [0, tt, nil] > [active1, U421, top] > U512 > mark1
zeros > [0, tt, nil] > [active1, U421, top] > U622 > mark1
zeros > [0, tt, nil] > [active1, U421, top] > isNat1 > U211 > mark1
proper1 > [0, tt, nil] > [active1, U421, top] > [cons2, U613] > U412 > mark1
proper1 > [0, tt, nil] > [active1, U421, top] > U512 > mark1
proper1 > [0, tt, nil] > [active1, U421, top] > U622 > mark1
proper1 > [0, tt, nil] > [active1, U421, top] > isNat1 > U211 > mark1


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(78) Obligation:

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

U311(ok(X)) → U311(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(79) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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)
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  x1
cons(x1, x2)  =  cons(x1, x2)
0  =  0
U11(x1)  =  U11(x1)
tt  =  tt
U21(x1)  =  x1
U31(x1)  =  x1
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  x1
isNatIList(x1)  =  x1
U51(x1, x2)  =  x2
U52(x1)  =  U52(x1)
isNatList(x1)  =  isNatList(x1)
U61(x1, x2, x3)  =  x2
U62(x1, x2)  =  x2
isNat(x1)  =  x1
s(x1)  =  x1
length(x1)  =  x1
nil  =  nil
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
active1 > [isNatList1, proper1] > zeros > cons2 > U412 > [ok1, U111]
active1 > [isNatList1, proper1] > zeros > 0 > [ok1, U111]
active1 > [isNatList1, proper1] > tt > U521 > [ok1, U111]
nil > 0 > [ok1, U111]
nil > tt > U521 > [ok1, U111]
top > [isNatList1, proper1] > zeros > cons2 > U412 > [ok1, U111]
top > [isNatList1, proper1] > zeros > 0 > [ok1, U111]
top > [isNatList1, proper1] > tt > U521 > [ok1, U111]


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(80) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(81) PisEmptyProof (EQUIVALENT transformation)

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

(82) TRUE

(83) Obligation:

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

U211(ok(X)) → U211(X)
U211(mark(X)) → U211(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U211(mark(X)) → U211(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U211(x1)  =  U211(x1)
ok(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1, x2)
0  =  0
U11(x1)  =  x1
tt  =  tt
U21(x1)  =  U21(x1)
U31(x1)  =  x1
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  U42(x1)
isNatIList(x1)  =  x1
U51(x1, x2)  =  U51(x1, x2)
U52(x1)  =  x1
isNatList(x1)  =  x1
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2)  =  U62(x1, x2)
isNat(x1)  =  isNat(x1)
s(x1)  =  x1
length(x1)  =  x1
nil  =  nil
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
U21^11 > mark1
zeros > [0, tt, nil] > [active1, U421, top] > [cons2, U613] > U412 > mark1
zeros > [0, tt, nil] > [active1, U421, top] > U512 > mark1
zeros > [0, tt, nil] > [active1, U421, top] > U622 > mark1
zeros > [0, tt, nil] > [active1, U421, top] > isNat1 > U211 > mark1
proper1 > [0, tt, nil] > [active1, U421, top] > [cons2, U613] > U412 > mark1
proper1 > [0, tt, nil] > [active1, U421, top] > U512 > mark1
proper1 > [0, tt, nil] > [active1, U421, top] > U622 > mark1
proper1 > [0, tt, nil] > [active1, U421, top] > isNat1 > U211 > mark1


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(85) Obligation:

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

U211(ok(X)) → U211(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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.

(86) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U211(ok(X)) → U211(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U211(x1)  =  U211(x1)
ok(x1)  =  ok(x1)
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  x1
cons(x1, x2)  =  cons(x1, x2)
0  =  0
U11(x1)  =  U11(x1)
tt  =  tt
U21(x1)  =  x1
U31(x1)  =  x1
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  x1
isNatIList(x1)  =  x1
U51(x1, x2)  =  x2
U52(x1)  =  U52(x1)
isNatList(x1)  =  isNatList(x1)
U61(x1, x2, x3)  =  x2
U62(x1, x2)  =  x2
isNat(x1)  =  x1
s(x1)  =  x1
length(x1)  =  x1
nil  =  nil
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
active1 > [isNatList1, proper1] > zeros > cons2 > U412 > [ok1, U111]
active1 > [isNatList1, proper1] > zeros > 0 > [ok1, U111]
active1 > [isNatList1, proper1] > tt > U521 > [ok1, U111]
nil > 0 > [ok1, U111]
nil > tt > U521 > [ok1, U111]
top > [isNatList1, proper1] > zeros > cons2 > U412 > [ok1, U111]
top > [isNatList1, proper1] > zeros > 0 > [ok1, U111]
top > [isNatList1, proper1] > tt > U521 > [ok1, U111]


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(87) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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.

(88) PisEmptyProof (EQUIVALENT transformation)

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

(89) TRUE

(90) 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(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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) 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)  =  U111(x1)
ok(x1)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1, x2)
0  =  0
U11(x1)  =  x1
tt  =  tt
U21(x1)  =  U21(x1)
U31(x1)  =  x1
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  U42(x1)
isNatIList(x1)  =  x1
U51(x1, x2)  =  U51(x1, x2)
U52(x1)  =  x1
isNatList(x1)  =  x1
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2)  =  U62(x1, x2)
isNat(x1)  =  isNat(x1)
s(x1)  =  x1
length(x1)  =  x1
nil  =  nil
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
U11^11 > mark1
zeros > [0, tt, nil] > [active1, U421, top] > [cons2, U613] > U412 > mark1
zeros > [0, tt, nil] > [active1, U421, top] > U512 > mark1
zeros > [0, tt, nil] > [active1, U421, top] > U622 > mark1
zeros > [0, tt, nil] > [active1, U421, top] > isNat1 > U211 > mark1
proper1 > [0, tt, nil] > [active1, U421, top] > [cons2, U613] > U412 > mark1
proper1 > [0, tt, nil] > [active1, U421, top] > U512 > mark1
proper1 > [0, tt, nil] > [active1, U421, top] > U622 > mark1
proper1 > [0, tt, nil] > [active1, U421, top] > isNat1 > U211 > mark1


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(92) Obligation:

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

U111(ok(X)) → U111(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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.

(93) 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)
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  x1
cons(x1, x2)  =  cons(x1, x2)
0  =  0
U11(x1)  =  U11(x1)
tt  =  tt
U21(x1)  =  x1
U31(x1)  =  x1
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  x1
isNatIList(x1)  =  x1
U51(x1, x2)  =  x2
U52(x1)  =  U52(x1)
isNatList(x1)  =  isNatList(x1)
U61(x1, x2, x3)  =  x2
U62(x1, x2)  =  x2
isNat(x1)  =  x1
s(x1)  =  x1
length(x1)  =  x1
nil  =  nil
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
active1 > [isNatList1, proper1] > zeros > cons2 > U412 > [ok1, U111]
active1 > [isNatList1, proper1] > zeros > 0 > [ok1, U111]
active1 > [isNatList1, proper1] > tt > U521 > [ok1, U111]
nil > 0 > [ok1, U111]
nil > tt > U521 > [ok1, U111]
top > [isNatList1, proper1] > zeros > cons2 > U412 > [ok1, U111]
top > [isNatList1, proper1] > zeros > 0 > [ok1, U111]
top > [isNatList1, proper1] > tt > U521 > [ok1, U111]


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(94) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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.

(95) PisEmptyProof (EQUIVALENT transformation)

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

(96) TRUE

(97) Obligation:

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

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

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(98) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


CONS(ok(X1), ok(X2)) → CONS(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS(x1, x2)  =  CONS(x1)
ok(x1)  =  ok(x1)
mark(x1)  =  x1
active(x1)  =  x1
zeros  =  zeros
cons(x1, x2)  =  x2
0  =  0
U11(x1)  =  U11(x1)
tt  =  tt
U21(x1)  =  x1
U31(x1)  =  x1
U41(x1, x2)  =  x2
U42(x1)  =  x1
isNatIList(x1)  =  x1
U51(x1, x2)  =  x2
U52(x1)  =  x1
isNatList(x1)  =  x1
U61(x1, x2, x3)  =  U61(x2)
U62(x1, x2)  =  U62(x2)
isNat(x1)  =  isNat(x1)
s(x1)  =  x1
length(x1)  =  length(x1)
nil  =  nil
proper(x1)  =  proper(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
CONS1 > ok1
zeros > tt > ok1
nil > 0 > tt > ok1
top > [U611, U621, length1, proper1] > U111 > tt > ok1
top > [U611, U621, length1, proper1] > isNat1 > tt > ok1


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(99) Obligation:

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

CONS(mark(X1), X2) → CONS(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(100) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


CONS(mark(X1), X2) → CONS(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS(x1, x2)  =  CONS(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1, x2)
0  =  0
U11(x1)  =  U11(x1)
tt  =  tt
U21(x1)  =  x1
U31(x1)  =  x1
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  x1
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  x1
U52(x1)  =  x1
isNatList(x1)  =  isNatList
U61(x1, x2, x3)  =  U61(x1, x2)
U62(x1, x2)  =  U62(x1, x2)
isNat(x1)  =  isNat
s(x1)  =  x1
length(x1)  =  length(x1)
nil  =  nil
proper(x1)  =  x1
ok(x1)  =  x1
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
[active1, 0, tt, isNatList, isNat] > zeros > mark1
[active1, 0, tt, isNatList, isNat] > U111 > mark1
[active1, 0, tt, isNatList, isNat] > isNatIList1 > [cons2, U412] > U612 > mark1
[active1, 0, tt, isNatList, isNat] > U622 > length1 > mark1


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(101) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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.

(102) PisEmptyProof (EQUIVALENT transformation)

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

(103) TRUE

(104) Obligation:

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

PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(U11(X)) → PROPER(X)
PROPER(U21(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(isNatIList(X)) → PROPER(X)
PROPER(U51(X1, X2)) → PROPER(X1)
PROPER(U51(X1, X2)) → PROPER(X2)
PROPER(U52(X)) → PROPER(X)
PROPER(isNatList(X)) → PROPER(X)
PROPER(U61(X1, X2, X3)) → PROPER(X1)
PROPER(U61(X1, X2, X3)) → PROPER(X2)
PROPER(U61(X1, X2, X3)) → PROPER(X3)
PROPER(U62(X1, X2)) → PROPER(X1)
PROPER(U62(X1, X2)) → PROPER(X2)
PROPER(isNat(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
PROPER(length(X)) → PROPER(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(cons(X1, X2)) → PROPER(X1)
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(U61(X1, X2, X3)) → PROPER(X1)
PROPER(U61(X1, X2, X3)) → PROPER(X2)
PROPER(U61(X1, X2, X3)) → PROPER(X3)
PROPER(U62(X1, X2)) → PROPER(X1)
PROPER(U62(X1, X2)) → PROPER(X2)
PROPER(isNat(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)
cons(x1, x2)  =  cons(x1, x2)
U11(x1)  =  x1
U21(x1)  =  x1
U31(x1)  =  x1
U41(x1, x2)  =  U41(x1, x2)
U42(x1)  =  x1
isNatIList(x1)  =  x1
U51(x1, x2)  =  U51(x1, x2)
U52(x1)  =  x1
isNatList(x1)  =  x1
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2)  =  U62(x1, x2)
isNat(x1)  =  isNat(x1)
s(x1)  =  x1
length(x1)  =  x1
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  x1
0  =  0
tt  =  tt
nil  =  nil
proper(x1)  =  proper(x1)
ok(x1)  =  ok
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
[active1, top] > U622 > ok > [cons2, U512, U613] > U412
[active1, top] > U622 > ok > isNat1 > tt
[active1, top] > zeros
[active1, top] > 0
proper1 > U622 > ok > [cons2, U512, U613] > U412
proper1 > U622 > ok > isNat1 > tt
proper1 > zeros
proper1 > 0
proper1 > nil


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(106) Obligation:

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

PROPER(U11(X)) → PROPER(X)
PROPER(U21(X)) → PROPER(X)
PROPER(U31(X)) → PROPER(X)
PROPER(U42(X)) → PROPER(X)
PROPER(isNatIList(X)) → PROPER(X)
PROPER(U52(X)) → PROPER(X)
PROPER(isNatList(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
PROPER(length(X)) → PROPER(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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.

(107) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PROPER(U21(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)  =  x1
U21(x1)  =  U21(x1)
U31(x1)  =  x1
U42(x1)  =  x1
isNatIList(x1)  =  x1
U52(x1)  =  x1
isNatList(x1)  =  x1
s(x1)  =  x1
length(x1)  =  x1
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  x1
cons(x1, x2)  =  cons(x1, x2)
0  =  0
tt  =  tt
U41(x1, x2)  =  U41(x1, x2)
U51(x1, x2)  =  U51(x1, x2)
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2)  =  U62(x1, x2)
isNat(x1)  =  isNat(x1)
nil  =  nil
proper(x1)  =  x1
ok(x1)  =  x1
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
[U211, active1] > zeros
[U211, active1] > [cons2, U512, U613] > U622
[U211, active1] > 0
[U211, active1] > tt > U622
[U211, active1] > tt > isNat1
[U211, active1] > U412
nil > 0


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(108) Obligation:

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

PROPER(U11(X)) → PROPER(X)
PROPER(U31(X)) → PROPER(X)
PROPER(U42(X)) → PROPER(X)
PROPER(isNatIList(X)) → PROPER(X)
PROPER(U52(X)) → PROPER(X)
PROPER(isNatList(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
PROPER(length(X)) → PROPER(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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.

(109) 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)
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)
U31(x1)  =  x1
U42(x1)  =  x1
isNatIList(x1)  =  x1
U52(x1)  =  x1
isNatList(x1)  =  x1
s(x1)  =  x1
length(x1)  =  x1
active(x1)  =  x1
zeros  =  zeros
mark(x1)  =  mark
cons(x1, x2)  =  cons
0  =  0
tt  =  tt
U21(x1)  =  U21
U41(x1, x2)  =  x2
U51(x1, x2)  =  x2
U61(x1, x2, x3)  =  x3
U62(x1, x2)  =  x2
isNat(x1)  =  isNat(x1)
nil  =  nil
proper(x1)  =  proper(x1)
ok(x1)  =  ok
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
PROPER1 > [mark, isNat1, nil]
[tt, U21] > [U111, proper1] > zeros > cons > ok > [mark, isNat1, nil]
[tt, U21] > [U111, proper1] > 0 > ok > [mark, isNat1, nil]
top > [mark, isNat1, nil]


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(110) Obligation:

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

PROPER(U31(X)) → PROPER(X)
PROPER(U42(X)) → PROPER(X)
PROPER(isNatIList(X)) → PROPER(X)
PROPER(U52(X)) → PROPER(X)
PROPER(isNatList(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
PROPER(length(X)) → PROPER(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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.

(111) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PROPER(isNatIList(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
isNatIList(x1)  =  isNatIList(x1)
U52(x1)  =  x1
isNatList(x1)  =  x1
s(x1)  =  x1
length(x1)  =  x1
active(x1)  =  x1
zeros  =  zeros
mark(x1)  =  x1
cons(x1, x2)  =  x2
0  =  0
U11(x1)  =  x1
tt  =  tt
U21(x1)  =  x1
U41(x1, x2)  =  U41(x2)
U51(x1, x2)  =  x2
U61(x1, x2, x3)  =  x2
U62(x1, x2)  =  x2
isNat(x1)  =  x1
nil  =  nil
proper(x1)  =  x1
ok(x1)  =  x1
top(x1)  =  top(x1)

Lexicographic Path Order [LPO].
Precedence:
[PROPER1, isNatIList1, U411] > tt
[0, nil] > tt


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(112) 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(isNatList(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
PROPER(length(X)) → PROPER(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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.

(113) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PROPER(isNatList(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
isNatList(x1)  =  isNatList(x1)
s(x1)  =  x1
length(x1)  =  x1
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  x1
cons(x1, x2)  =  cons(x1, x2)
0  =  0
U11(x1)  =  x1
tt  =  tt
U21(x1)  =  x1
U41(x1, x2)  =  x2
isNatIList(x1)  =  x1
U51(x1, x2)  =  x2
U61(x1, x2, x3)  =  U61(x2, x3)
U62(x1, x2)  =  x2
isNat(x1)  =  isNat(x1)
nil  =  nil
proper(x1)  =  x1
ok(x1)  =  x1
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
isNat1 > [isNatList1, active1, tt] > cons2
isNat1 > [isNatList1, active1, tt] > U612
nil > [zeros, 0] > [isNatList1, active1, tt] > cons2
nil > [zeros, 0] > [isNatList1, active1, tt] > U612
top > [isNatList1, active1, tt] > cons2
top > [isNatList1, active1, tt] > U612


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(114) Obligation:

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

PROPER(U31(X)) → PROPER(X)
PROPER(U42(X)) → PROPER(X)
PROPER(U52(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
PROPER(length(X)) → PROPER(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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.


PROPER(s(X)) → PROPER(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
U31(x1)  =  x1
U42(x1)  =  x1
U52(x1)  =  x1
s(x1)  =  s(x1)
length(x1)  =  x1
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  mark
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21(x1)
U41(x1, x2)  =  U41(x1, x2)
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51(x1)
isNatList(x1)  =  isNatList
U61(x1, x2, x3)  =  x1
U62(x1, x2)  =  x1
isNat(x1)  =  x1
nil  =  nil
proper(x1)  =  x1
ok(x1)  =  x1
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
PROPER1 > isNatList
[active1, U412] > zeros > 0 > isNatList
[active1, U412] > isNatIList1 > tt > s1 > U211 > [mark, cons, U11] > U511 > isNatList
nil > isNatList
top > isNatList


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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:

PROPER(U31(X)) → PROPER(X)
PROPER(U42(X)) → PROPER(X)
PROPER(U52(X)) → PROPER(X)
PROPER(length(X)) → PROPER(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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.


PROPER(length(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
length(x1)  =  length(x1)
active(x1)  =  x1
zeros  =  zeros
mark(x1)  =  mark
cons(x1, x2)  =  x2
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U41(x1, x2)  =  U41(x1, x2)
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
isNatList(x1)  =  x1
U61(x1, x2, x3)  =  x1
U62(x1, x2)  =  U62(x1)
isNat(x1)  =  isNat
s(x1)  =  s
nil  =  nil
proper(x1)  =  x1
ok(x1)  =  x1
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
PROPER1 > [mark, s]
zeros > 0 > [mark, s]
[U11, tt, isNat, nil] > length1 > 0 > [mark, s]
[U11, tt, isNat, nil] > U21 > [mark, s]
[U11, tt, isNat, nil] > isNatIList1 > [mark, s]
[U11, tt, isNat, nil] > U621 > [mark, s]
U412 > isNatIList1 > [mark, s]
U51 > [mark, s]
top > [mark, s]


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(118) Obligation:

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

PROPER(U31(X)) → PROPER(X)
PROPER(U42(X)) → PROPER(X)
PROPER(U52(X)) → PROPER(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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) 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)
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  mark(x1)
cons(x1, x2)  =  cons(x1, x2)
0  =  0
U11(x1)  =  x1
tt  =  tt
U21(x1)  =  x1
U41(x1, x2)  =  U41(x1)
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51(x1)
isNatList(x1)  =  isNatList
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2)  =  U62(x1, x2)
isNat(x1)  =  isNat
s(x1)  =  x1
length(x1)  =  length(x1)
nil  =  nil
proper(x1)  =  x1
ok(x1)  =  ok
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
[active1, length1] > U521 > PROPER1
[active1, length1] > U521 > [cons2, tt, U613, U622] > [mark1, U411, U511] > ok
[active1, length1] > zeros > [cons2, tt, U613, U622] > [mark1, U411, U511] > ok
[active1, length1] > zeros > 0 > ok
[active1, length1] > [isNatIList, isNatList, isNat] > [cons2, tt, U613, U622] > [mark1, U411, U511] > ok
nil > 0 > ok


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(120) Obligation:

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

PROPER(U31(X)) → PROPER(X)
PROPER(U42(X)) → PROPER(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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.

(121) 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)
U42(x1)  =  x1
active(x1)  =  x1
zeros  =  zeros
mark(x1)  =  x1
cons(x1, x2)  =  x2
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U41(x1, x2)  =  U41(x2)
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51(x2)
U52(x1)  =  U52
isNatList(x1)  =  isNatList(x1)
U61(x1, x2, x3)  =  U61(x1, x2)
U62(x1, x2)  =  U62(x2)
isNat(x1)  =  x1
s(x1)  =  s
length(x1)  =  length(x1)
nil  =  nil
proper(x1)  =  proper(x1)
ok(x1)  =  ok
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
[U11, length1, proper1] > zeros > 0 > [U52, ok] > tt
[U11, length1, proper1] > [U411, isNatIList1] > U311 > [U52, ok] > tt
[U11, length1, proper1] > [U411, isNatIList1] > isNatList1 > U511 > [U52, ok] > tt
[U11, length1, proper1] > U612 > [U621, s] > U21 > [U52, ok] > tt
[U11, length1, proper1] > nil


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(122) Obligation:

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

PROPER(U42(X)) → PROPER(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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.

(123) 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)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
U42(x1)  =  U42(x1)
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  mark
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  x1
tt  =  tt
U21(x1)  =  x1
U31(x1)  =  U31
U41(x1, x2)  =  U41
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51(x2)
U52(x1)  =  U52
isNatList(x1)  =  x1
U61(x1, x2, x3)  =  U61(x2, x3)
U62(x1, x2)  =  U62(x1, x2)
isNat(x1)  =  x1
s(x1)  =  s
length(x1)  =  length
nil  =  nil
proper(x1)  =  x1
ok(x1)  =  ok
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
PROPER1 > [0, nil, ok]
length > [U421, active1, zeros, cons, U31, U52, U612, top] > [mark, U41, U511, s] > isNatIList > [0, nil, ok]
length > [U421, active1, zeros, cons, U31, U52, U612, top] > [mark, U41, U511, s] > U622 > [0, nil, ok]
length > [U421, active1, zeros, cons, U31, U52, U612, top] > tt > isNatIList > [0, nil, ok]
length > [U421, active1, zeros, cons, U31, U52, U612, top] > tt > U622 > [0, nil, ok]


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(124) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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.

(125) PisEmptyProof (EQUIVALENT transformation)

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

(126) TRUE

(127) Obligation:

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

ACTIVE(U11(X)) → ACTIVE(X)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(U21(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(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U62(X1, X2)) → ACTIVE(X1)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(length(X)) → ACTIVE(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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.


ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(U61(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U62(X1, X2)) → ACTIVE(X1)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
U11(x1)  =  x1
cons(x1, x2)  =  cons(x1, x2)
U21(x1)  =  x1
U31(x1)  =  x1
U41(x1, x2)  =  x1
U42(x1)  =  x1
U51(x1, x2)  =  x1
U52(x1)  =  x1
U61(x1, x2, x3)  =  U61(x1, x2)
U62(x1, x2)  =  U62(x1, x2)
s(x1)  =  x1
length(x1)  =  x1
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  x1
0  =  0
tt  =  tt
isNatIList(x1)  =  isNatIList
isNatList(x1)  =  isNatList
isNat(x1)  =  isNat
nil  =  nil
proper(x1)  =  x1
ok(x1)  =  x1
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
[active1, tt, isNat] > cons2 > U612 > U622
[active1, tt, isNat] > cons2 > isNatList
[active1, tt, isNat] > zeros
[active1, tt, isNat] > 0
[active1, tt, isNat] > isNatIList


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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:

ACTIVE(U11(X)) → ACTIVE(X)
ACTIVE(U21(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(s(X)) → ACTIVE(X)
ACTIVE(length(X)) → ACTIVE(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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.


ACTIVE(U11(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)
U11(x1)  =  U11(x1)
U21(x1)  =  x1
U31(x1)  =  x1
U41(x1, x2)  =  x1
U42(x1)  =  x1
U51(x1, x2)  =  x1
U52(x1)  =  x1
s(x1)  =  x1
length(x1)  =  x1
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  mark(x1)
cons(x1, x2)  =  cons(x1, x2)
0  =  0
tt  =  tt
isNatIList(x1)  =  isNatIList
isNatList(x1)  =  isNatList
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2)  =  U62(x1, x2)
isNat(x1)  =  isNat
nil  =  nil
proper(x1)  =  x1
ok(x1)  =  ok
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
zeros > [active1, tt, isNatIList, U622] > [U111, isNat] > [0, isNatList, U613, ok] > mark1
zeros > [active1, tt, isNatIList, U622] > cons2 > [0, isNatList, U613, ok] > mark1
nil > [0, isNatList, U613, ok] > mark1


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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:

ACTIVE(U21(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(s(X)) → ACTIVE(X)
ACTIVE(length(X)) → ACTIVE(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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.


ACTIVE(U52(X)) → ACTIVE(X)
ACTIVE(s(X)) → ACTIVE(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
U21(x1)  =  x1
U31(x1)  =  x1
U41(x1, x2)  =  x1
U42(x1)  =  x1
U51(x1, x2)  =  x1
U52(x1)  =  U52(x1)
s(x1)  =  s(x1)
length(x1)  =  x1
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  x1
cons(x1, x2)  =  cons(x2)
0  =  0
U11(x1)  =  U11
tt  =  tt
isNatIList(x1)  =  isNatIList
isNatList(x1)  =  isNatList
U61(x1, x2, x3)  =  x2
U62(x1, x2)  =  x2
isNat(x1)  =  isNat
nil  =  nil
proper(x1)  =  x1
ok(x1)  =  ok
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
ACTIVE1 > U521
nil > [U11, ok] > s1 > U521
top > [active1, cons1, 0, isNatList, isNat] > zeros > [U11, ok] > s1 > U521
top > [active1, cons1, 0, isNatList, isNat] > tt > isNatIList > [U11, ok] > s1 > U521


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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:

ACTIVE(U21(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(length(X)) → ACTIVE(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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.


ACTIVE(U31(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)
U21(x1)  =  x1
U31(x1)  =  U31(x1)
U41(x1, x2)  =  x1
U42(x1)  =  x1
U51(x1, x2)  =  x1
length(x1)  =  x1
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  mark
cons(x1, x2)  =  cons(x1, x2)
0  =  0
U11(x1)  =  x1
tt  =  tt
isNatIList(x1)  =  isNatIList
U52(x1)  =  x1
isNatList(x1)  =  isNatList(x1)
U61(x1, x2, x3)  =  x1
U62(x1, x2)  =  U62(x1, x2)
isNat(x1)  =  isNat
s(x1)  =  s(x1)
nil  =  nil
proper(x1)  =  proper(x1)
ok(x1)  =  x1
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
ACTIVE1 > isNatList1
[0, nil] > [mark, tt] > cons2 > isNatList1
[0, nil] > [mark, tt] > isNatIList > isNatList1
[0, nil] > [mark, tt] > s1 > isNatList1
[0, nil] > [mark, tt] > top > isNatList1
proper1 > [active1, U622, isNat] > U311 > [mark, tt] > cons2 > isNatList1
proper1 > [active1, U622, isNat] > U311 > [mark, tt] > isNatIList > isNatList1
proper1 > [active1, U622, isNat] > U311 > [mark, tt] > s1 > isNatList1
proper1 > [active1, U622, isNat] > U311 > [mark, tt] > top > isNatList1
proper1 > [active1, U622, isNat] > zeros > [mark, tt] > cons2 > isNatList1
proper1 > [active1, U622, isNat] > zeros > [mark, tt] > isNatIList > isNatList1
proper1 > [active1, U622, isNat] > zeros > [mark, tt] > s1 > isNatList1
proper1 > [active1, U622, isNat] > zeros > [mark, tt] > top > isNatList1


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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:

ACTIVE(U21(X)) → ACTIVE(X)
ACTIVE(U41(X1, X2)) → ACTIVE(X1)
ACTIVE(U42(X)) → ACTIVE(X)
ACTIVE(U51(X1, X2)) → ACTIVE(X1)
ACTIVE(length(X)) → ACTIVE(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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.


ACTIVE(U51(X1, X2)) → ACTIVE(X1)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
U21(x1)  =  x1
U41(x1, x2)  =  x1
U42(x1)  =  x1
U51(x1, x2)  =  U51(x1, x2)
length(x1)  =  x1
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  mark
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1)  =  U11
tt  =  tt
U31(x1)  =  U31
isNatIList(x1)  =  x1
U52(x1)  =  U52
isNatList(x1)  =  x1
U61(x1, x2, x3)  =  x1
U62(x1, x2)  =  U62
isNat(x1)  =  x1
s(x1)  =  x1
nil  =  nil
proper(x1)  =  proper(x1)
ok(x1)  =  x1
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
proper1 > 0 > [mark, U31]
proper1 > U11 > [active1, cons1] > [U512, U52] > ACTIVE1
proper1 > U11 > [active1, cons1] > [U512, U52] > [mark, U31]
proper1 > U11 > [active1, cons1] > zeros
proper1 > U11 > [active1, cons1] > U62 > [mark, U31]
proper1 > tt > [U512, U52] > ACTIVE1
proper1 > tt > [U512, U52] > [mark, U31]
proper1 > tt > U62 > [mark, U31]
proper1 > nil


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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:

ACTIVE(U21(X)) → ACTIVE(X)
ACTIVE(U41(X1, X2)) → ACTIVE(X1)
ACTIVE(U42(X)) → ACTIVE(X)
ACTIVE(length(X)) → ACTIVE(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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.


ACTIVE(U21(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)
U21(x1)  =  U21(x1)
U41(x1, x2)  =  x1
U42(x1)  =  x1
length(x1)  =  x1
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  mark
cons(x1, x2)  =  cons(x2)
0  =  0
U11(x1)  =  U11(x1)
tt  =  tt
U31(x1)  =  U31(x1)
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51(x2)
U52(x1)  =  x1
isNatList(x1)  =  isNatList(x1)
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2)  =  U62(x1)
isNat(x1)  =  x1
s(x1)  =  s(x1)
nil  =  nil
proper(x1)  =  proper(x1)
ok(x1)  =  ok(x1)
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
ACTIVE1 > [U211, zeros, U111, ok1]
[U511, proper1] > [active1, isNatIList1, U621] > 0 > [mark, cons1, U311, U613] > isNatList1 > [U211, zeros, U111, ok1]
[U511, proper1] > [active1, isNatIList1, U621] > tt > s1 > [mark, cons1, U311, U613] > isNatList1 > [U211, zeros, U111, ok1]
[U511, proper1] > nil > [U211, zeros, U111, ok1]
top > [active1, isNatIList1, U621] > 0 > [mark, cons1, U311, U613] > isNatList1 > [U211, zeros, U111, ok1]
top > [active1, isNatIList1, U621] > tt > s1 > [mark, cons1, U311, U613] > isNatList1 > [U211, zeros, U111, ok1]


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(139) Obligation:

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

ACTIVE(U41(X1, X2)) → ACTIVE(X1)
ACTIVE(U42(X)) → ACTIVE(X)
ACTIVE(length(X)) → ACTIVE(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(length(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)
U41(x1, x2)  =  x1
U42(x1)  =  x1
length(x1)  =  length(x1)
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  mark(x1)
cons(x1, x2)  =  cons(x1, x2)
0  =  0
U11(x1)  =  x1
tt  =  tt
U21(x1)  =  x1
U31(x1)  =  U31(x1)
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  x1
U52(x1)  =  x1
isNatList(x1)  =  isNatList
U61(x1, x2, x3)  =  U61(x1, x2, x3)
U62(x1, x2)  =  U62(x1, x2)
isNat(x1)  =  isNat
s(x1)  =  s(x1)
nil  =  nil
proper(x1)  =  proper(x1)
ok(x1)  =  x1
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
ACTIVE1 > [tt, isNatList, nil]
proper1 > [length1, active1, 0, U311, isNatIList, U613, U622, isNat] > zeros > cons2 > mark1 > [tt, isNatList, nil]
proper1 > [length1, active1, 0, U311, isNatIList, U613, U622, isNat] > s1 > mark1 > [tt, isNatList, nil]
top > [length1, active1, 0, U311, isNatIList, U613, U622, isNat] > zeros > cons2 > mark1 > [tt, isNatList, nil]
top > [length1, active1, 0, U311, isNatIList, U613, U622, isNat] > s1 > mark1 > [tt, isNatList, nil]


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(141) Obligation:

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

ACTIVE(U41(X1, X2)) → ACTIVE(X1)
ACTIVE(U42(X)) → ACTIVE(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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.

(142) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(U41(X1, X2)) → ACTIVE(X1)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
U41(x1, x2)  =  U41(x1)
U42(x1)  =  x1
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  x1
cons(x1, x2)  =  x1
0  =  0
U11(x1)  =  U11
tt  =  tt
U21(x1)  =  U21
U31(x1)  =  U31
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  x1
U52(x1)  =  x1
isNatList(x1)  =  isNatList
U61(x1, x2, x3)  =  U61(x3)
U62(x1, x2)  =  x1
isNat(x1)  =  isNat
s(x1)  =  s
length(x1)  =  x1
nil  =  nil
proper(x1)  =  proper(x1)
ok(x1)  =  x1
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
[active1, 0, tt, nil] > s > [U411, isNatIList, proper1] > zeros
[active1, 0, tt, nil] > s > [U411, isNatIList, proper1] > U21
[active1, 0, tt, nil] > s > [U411, isNatIList, proper1] > U31
[active1, 0, tt, nil] > s > [U411, isNatIList, proper1] > isNatList > [U11, isNat]
[active1, 0, tt, nil] > s > [U411, isNatIList, proper1] > U611 > [U11, isNat]


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(143) Obligation:

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

ACTIVE(U42(X)) → ACTIVE(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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.

(144) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(U42(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)
U42(x1)  =  U42(x1)
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  mark
cons(x1, x2)  =  cons
0  =  0
U11(x1)  =  x1
tt  =  tt
U21(x1)  =  x1
U31(x1)  =  U31
U41(x1, x2)  =  U41
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51(x2)
U52(x1)  =  U52
isNatList(x1)  =  x1
U61(x1, x2, x3)  =  U61(x2, x3)
U62(x1, x2)  =  U62(x1, x2)
isNat(x1)  =  x1
s(x1)  =  s
length(x1)  =  length
nil  =  nil
proper(x1)  =  x1
ok(x1)  =  ok
top(x1)  =  top

Lexicographic Path Order [LPO].
Precedence:
ACTIVE1 > [0, nil, ok]
length > [U421, active1, zeros, cons, U31, U52, U612, top] > [mark, U41, U511, s] > isNatIList > [0, nil, ok]
length > [U421, active1, zeros, cons, U31, U52, U612, top] > [mark, U41, U511, s] > U622 > [0, nil, ok]
length > [U421, active1, zeros, cons, U31, U52, U612, top] > tt > isNatIList > [0, nil, ok]
length > [U421, active1, zeros, cons, U31, U52, U612, top] > tt > U622 > [0, nil, ok]


The following usable rules [FROCOS05] were oriented:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(145) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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.

(146) PisEmptyProof (EQUIVALENT transformation)

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

(147) TRUE

(148) 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(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(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(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(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(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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.