WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).

(0) CpxTRS
(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
    (4) CpxTRS
    (5) CpxTrsMatchBoundsTAProof [FINISHED, 934 ms]
    (6) BOUNDS(1, n^1)
(7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(8) TRS for Loop Detection
    (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (10) BEST
        (11) proven lower bound
            (12) LowerBoundPropagationProof [FINISHED, 0 ms]
            (13) BOUNDS(n^1, INF)
        (14) TRS for Loop Detection


----------------------------------------

(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
   active(__(X, nil)) -> mark(X)
   active(__(nil, X)) -> mark(X)
   active(U11(tt, V)) -> mark(U12(isNeList(V)))
   active(U12(tt)) -> mark(tt)
   active(U21(tt, V1, V2)) -> mark(U22(isList(V1), V2))
   active(U22(tt, V2)) -> mark(U23(isList(V2)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isQid(V)))
   active(U32(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isList(V1), V2))
   active(U42(tt, V2)) -> mark(U43(isNeList(V2)))
   active(U43(tt)) -> mark(tt)
   active(U51(tt, V1, V2)) -> mark(U52(isNeList(V1), V2))
   active(U52(tt, V2)) -> mark(U53(isList(V2)))
   active(U53(tt)) -> mark(tt)
   active(U61(tt, V)) -> mark(U62(isQid(V)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, V)) -> mark(U72(isNePal(V)))
   active(U72(tt)) -> mark(tt)
   active(and(tt, X)) -> mark(X)
   active(isList(V)) -> mark(U11(isPalListKind(V), V))
   active(isList(nil)) -> mark(tt)
   active(isList(__(V1, V2))) -> mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
   active(isNeList(V)) -> mark(U31(isPalListKind(V), V))
   active(isNeList(__(V1, V2))) -> mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
   active(isNeList(__(V1, V2))) -> mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
   active(isNePal(V)) -> mark(U61(isPalListKind(V), V))
   active(isNePal(__(I, __(P, I)))) -> mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
   active(isPal(V)) -> mark(U71(isPalListKind(V), V))
   active(isPal(nil)) -> mark(tt)
   active(isPalListKind(a)) -> mark(tt)
   active(isPalListKind(e)) -> mark(tt)
   active(isPalListKind(i)) -> mark(tt)
   active(isPalListKind(nil)) -> mark(tt)
   active(isPalListKind(o)) -> mark(tt)
   active(isPalListKind(u)) -> mark(tt)
   active(isPalListKind(__(V1, V2))) -> mark(and(isPalListKind(V1), isPalListKind(V2)))
   active(isQid(a)) -> mark(tt)
   active(isQid(e)) -> mark(tt)
   active(isQid(i)) -> mark(tt)
   active(isQid(o)) -> mark(tt)
   active(isQid(u)) -> mark(tt)
   active(__(X1, X2)) -> __(active(X1), X2)
   active(__(X1, X2)) -> __(X1, active(X2))
   active(U11(X1, X2)) -> U11(active(X1), X2)
   active(U12(X)) -> U12(active(X))
   active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
   active(U22(X1, X2)) -> U22(active(X1), X2)
   active(U23(X)) -> U23(active(X))
   active(U31(X1, X2)) -> U31(active(X1), X2)
   active(U32(X)) -> U32(active(X))
   active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
   active(U42(X1, X2)) -> U42(active(X1), X2)
   active(U43(X)) -> U43(active(X))
   active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
   active(U52(X1, X2)) -> U52(active(X1), X2)
   active(U53(X)) -> U53(active(X))
   active(U61(X1, X2)) -> U61(active(X1), X2)
   active(U62(X)) -> U62(active(X))
   active(U71(X1, X2)) -> U71(active(X1), X2)
   active(U72(X)) -> U72(active(X))
   active(and(X1, X2)) -> and(active(X1), X2)
   __(mark(X1), X2) -> mark(__(X1, X2))
   __(X1, mark(X2)) -> mark(__(X1, X2))
   U11(mark(X1), X2) -> mark(U11(X1, X2))
   U12(mark(X)) -> mark(U12(X))
   U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
   U22(mark(X1), X2) -> mark(U22(X1, X2))
   U23(mark(X)) -> mark(U23(X))
   U31(mark(X1), X2) -> mark(U31(X1, X2))
   U32(mark(X)) -> mark(U32(X))
   U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
   U42(mark(X1), X2) -> mark(U42(X1, X2))
   U43(mark(X)) -> mark(U43(X))
   U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
   U52(mark(X1), X2) -> mark(U52(X1, X2))
   U53(mark(X)) -> mark(U53(X))
   U61(mark(X1), X2) -> mark(U61(X1, X2))
   U62(mark(X)) -> mark(U62(X))
   U71(mark(X1), X2) -> mark(U71(X1, X2))
   U72(mark(X)) -> mark(U72(X))
   and(mark(X1), X2) -> mark(and(X1, X2))
   proper(__(X1, X2)) -> __(proper(X1), proper(X2))
   proper(nil) -> ok(nil)
   proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
   proper(tt) -> ok(tt)
   proper(U12(X)) -> U12(proper(X))
   proper(isNeList(X)) -> isNeList(proper(X))
   proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
   proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
   proper(isList(X)) -> isList(proper(X))
   proper(U23(X)) -> U23(proper(X))
   proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
   proper(U32(X)) -> U32(proper(X))
   proper(isQid(X)) -> isQid(proper(X))
   proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
   proper(U42(X1, X2)) -> U42(proper(X1), proper(X2))
   proper(U43(X)) -> U43(proper(X))
   proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
   proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
   proper(U53(X)) -> U53(proper(X))
   proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
   proper(U62(X)) -> U62(proper(X))
   proper(U71(X1, X2)) -> U71(proper(X1), proper(X2))
   proper(U72(X)) -> U72(proper(X))
   proper(isNePal(X)) -> isNePal(proper(X))
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(isPalListKind(X)) -> isPalListKind(proper(X))
   proper(isPal(X)) -> isPal(proper(X))
   proper(a) -> ok(a)
   proper(e) -> ok(e)
   proper(i) -> ok(i)
   proper(o) -> ok(o)
   proper(u) -> ok(u)
   __(ok(X1), ok(X2)) -> ok(__(X1, X2))
   U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
   U12(ok(X)) -> ok(U12(X))
   isNeList(ok(X)) -> ok(isNeList(X))
   U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
   U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
   isList(ok(X)) -> ok(isList(X))
   U23(ok(X)) -> ok(U23(X))
   U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
   U32(ok(X)) -> ok(U32(X))
   isQid(ok(X)) -> ok(isQid(X))
   U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
   U42(ok(X1), ok(X2)) -> ok(U42(X1, X2))
   U43(ok(X)) -> ok(U43(X))
   U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
   U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
   U53(ok(X)) -> ok(U53(X))
   U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
   U62(ok(X)) -> ok(U62(X))
   U71(ok(X1), ok(X2)) -> ok(U71(X1, X2))
   U72(ok(X)) -> ok(U72(X))
   isNePal(ok(X)) -> ok(isNePal(X))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isPalListKind(ok(X)) -> ok(isPalListKind(X))
   isPal(ok(X)) -> ok(isPal(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
The following defined symbols can occur below the 0th argument of top: proper, active
The following defined symbols can occur below the 0th argument of proper: proper, active
The following defined symbols can occur below the 0th argument of active: proper, active

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
active(__(X, nil)) -> mark(X)
active(__(nil, X)) -> mark(X)
active(U11(tt, V)) -> mark(U12(isNeList(V)))
active(U12(tt)) -> mark(tt)
active(U21(tt, V1, V2)) -> mark(U22(isList(V1), V2))
active(U22(tt, V2)) -> mark(U23(isList(V2)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isQid(V)))
active(U32(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isList(V1), V2))
active(U42(tt, V2)) -> mark(U43(isNeList(V2)))
active(U43(tt)) -> mark(tt)
active(U51(tt, V1, V2)) -> mark(U52(isNeList(V1), V2))
active(U52(tt, V2)) -> mark(U53(isList(V2)))
active(U53(tt)) -> mark(tt)
active(U61(tt, V)) -> mark(U62(isQid(V)))
active(U62(tt)) -> mark(tt)
active(U71(tt, V)) -> mark(U72(isNePal(V)))
active(U72(tt)) -> mark(tt)
active(and(tt, X)) -> mark(X)
active(isList(V)) -> mark(U11(isPalListKind(V), V))
active(isList(nil)) -> mark(tt)
active(isList(__(V1, V2))) -> mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(V)) -> mark(U31(isPalListKind(V), V))
active(isNeList(__(V1, V2))) -> mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNeList(__(V1, V2))) -> mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
active(isNePal(V)) -> mark(U61(isPalListKind(V), V))
active(isNePal(__(I, __(P, I)))) -> mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
active(isPal(V)) -> mark(U71(isPalListKind(V), V))
active(isPal(nil)) -> mark(tt)
active(isPalListKind(a)) -> mark(tt)
active(isPalListKind(e)) -> mark(tt)
active(isPalListKind(i)) -> mark(tt)
active(isPalListKind(nil)) -> mark(tt)
active(isPalListKind(o)) -> mark(tt)
active(isPalListKind(u)) -> mark(tt)
active(isPalListKind(__(V1, V2))) -> mark(and(isPalListKind(V1), isPalListKind(V2)))
active(isQid(a)) -> mark(tt)
active(isQid(e)) -> mark(tt)
active(isQid(i)) -> mark(tt)
active(isQid(o)) -> mark(tt)
active(isQid(u)) -> mark(tt)
active(__(X1, X2)) -> __(active(X1), X2)
active(__(X1, X2)) -> __(X1, active(X2))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X)) -> U12(active(X))
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X)) -> U32(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2)) -> U42(active(X1), X2)
active(U43(X)) -> U43(active(X))
active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
active(U52(X1, X2)) -> U52(active(X1), X2)
active(U53(X)) -> U53(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X1, X2)) -> U71(active(X1), X2)
active(U72(X)) -> U72(active(X))
active(and(X1, X2)) -> and(active(X1), X2)
proper(__(X1, X2)) -> __(proper(X1), proper(X2))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X)) -> U12(proper(X))
proper(isNeList(X)) -> isNeList(proper(X))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(isList(X)) -> isList(proper(X))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X)) -> U32(proper(X))
proper(isQid(X)) -> isQid(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) -> U42(proper(X1), proper(X2))
proper(U43(X)) -> U43(proper(X))
proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
proper(U53(X)) -> U53(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X1, X2)) -> U71(proper(X1), proper(X2))
proper(U72(X)) -> U72(proper(X))
proper(isNePal(X)) -> isNePal(proper(X))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(isPalListKind(X)) -> isPalListKind(proper(X))
proper(isPal(X)) -> isPal(proper(X))

----------------------------------------

(2)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   __(mark(X1), X2) -> mark(__(X1, X2))
   __(X1, mark(X2)) -> mark(__(X1, X2))
   U11(mark(X1), X2) -> mark(U11(X1, X2))
   U12(mark(X)) -> mark(U12(X))
   U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
   U22(mark(X1), X2) -> mark(U22(X1, X2))
   U23(mark(X)) -> mark(U23(X))
   U31(mark(X1), X2) -> mark(U31(X1, X2))
   U32(mark(X)) -> mark(U32(X))
   U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
   U42(mark(X1), X2) -> mark(U42(X1, X2))
   U43(mark(X)) -> mark(U43(X))
   U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
   U52(mark(X1), X2) -> mark(U52(X1, X2))
   U53(mark(X)) -> mark(U53(X))
   U61(mark(X1), X2) -> mark(U61(X1, X2))
   U62(mark(X)) -> mark(U62(X))
   U71(mark(X1), X2) -> mark(U71(X1, X2))
   U72(mark(X)) -> mark(U72(X))
   and(mark(X1), X2) -> mark(and(X1, X2))
   proper(nil) -> ok(nil)
   proper(tt) -> ok(tt)
   proper(a) -> ok(a)
   proper(e) -> ok(e)
   proper(i) -> ok(i)
   proper(o) -> ok(o)
   proper(u) -> ok(u)
   __(ok(X1), ok(X2)) -> ok(__(X1, X2))
   U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
   U12(ok(X)) -> ok(U12(X))
   isNeList(ok(X)) -> ok(isNeList(X))
   U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
   U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
   isList(ok(X)) -> ok(isList(X))
   U23(ok(X)) -> ok(U23(X))
   U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
   U32(ok(X)) -> ok(U32(X))
   isQid(ok(X)) -> ok(isQid(X))
   U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
   U42(ok(X1), ok(X2)) -> ok(U42(X1, X2))
   U43(ok(X)) -> ok(U43(X))
   U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
   U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
   U53(ok(X)) -> ok(U53(X))
   U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
   U62(ok(X)) -> ok(U62(X))
   U71(ok(X1), ok(X2)) -> ok(U71(X1, X2))
   U72(ok(X)) -> ok(U72(X))
   isNePal(ok(X)) -> ok(isNePal(X))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isPalListKind(ok(X)) -> ok(isPalListKind(X))
   isPal(ok(X)) -> ok(isPal(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(3) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(4)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   __(mark(X1), X2) -> mark(__(X1, X2))
   __(X1, mark(X2)) -> mark(__(X1, X2))
   U11(mark(X1), X2) -> mark(U11(X1, X2))
   U12(mark(X)) -> mark(U12(X))
   U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
   U22(mark(X1), X2) -> mark(U22(X1, X2))
   U23(mark(X)) -> mark(U23(X))
   U31(mark(X1), X2) -> mark(U31(X1, X2))
   U32(mark(X)) -> mark(U32(X))
   U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
   U42(mark(X1), X2) -> mark(U42(X1, X2))
   U43(mark(X)) -> mark(U43(X))
   U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
   U52(mark(X1), X2) -> mark(U52(X1, X2))
   U53(mark(X)) -> mark(U53(X))
   U61(mark(X1), X2) -> mark(U61(X1, X2))
   U62(mark(X)) -> mark(U62(X))
   U71(mark(X1), X2) -> mark(U71(X1, X2))
   U72(mark(X)) -> mark(U72(X))
   and(mark(X1), X2) -> mark(and(X1, X2))
   proper(nil) -> ok(nil)
   proper(tt) -> ok(tt)
   proper(a) -> ok(a)
   proper(e) -> ok(e)
   proper(i) -> ok(i)
   proper(o) -> ok(o)
   proper(u) -> ok(u)
   __(ok(X1), ok(X2)) -> ok(__(X1, X2))
   U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
   U12(ok(X)) -> ok(U12(X))
   isNeList(ok(X)) -> ok(isNeList(X))
   U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
   U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
   isList(ok(X)) -> ok(isList(X))
   U23(ok(X)) -> ok(U23(X))
   U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
   U32(ok(X)) -> ok(U32(X))
   isQid(ok(X)) -> ok(isQid(X))
   U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
   U42(ok(X1), ok(X2)) -> ok(U42(X1, X2))
   U43(ok(X)) -> ok(U43(X))
   U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
   U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
   U53(ok(X)) -> ok(U53(X))
   U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
   U62(ok(X)) -> ok(U62(X))
   U71(ok(X1), ok(X2)) -> ok(U71(X1, X2))
   U72(ok(X)) -> ok(U72(X))
   isNePal(ok(X)) -> ok(isNePal(X))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isPalListKind(ok(X)) -> ok(isPalListKind(X))
   isPal(ok(X)) -> ok(isPal(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(5) CpxTrsMatchBoundsTAProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 2. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27]
transitions: 
mark0(0) -> 0
nil0() -> 0
ok0(0) -> 0
tt0() -> 0
a0() -> 0
e0() -> 0
i0() -> 0
o0() -> 0
u0() -> 0
active0(0) -> 0
__0(0, 0) -> 1
U110(0, 0) -> 2
U120(0) -> 3
U210(0, 0, 0) -> 4
U220(0, 0) -> 5
U230(0) -> 6
U310(0, 0) -> 7
U320(0) -> 8
U410(0, 0, 0) -> 9
U420(0, 0) -> 10
U430(0) -> 11
U510(0, 0, 0) -> 12
U520(0, 0) -> 13
U530(0) -> 14
U610(0, 0) -> 15
U620(0) -> 16
U710(0, 0) -> 17
U720(0) -> 18
and0(0, 0) -> 19
proper0(0) -> 20
isNeList0(0) -> 21
isList0(0) -> 22
isQid0(0) -> 23
isNePal0(0) -> 24
isPalListKind0(0) -> 25
isPal0(0) -> 26
top0(0) -> 27
__1(0, 0) -> 28
mark1(28) -> 1
U111(0, 0) -> 29
mark1(29) -> 2
U121(0) -> 30
mark1(30) -> 3
U211(0, 0, 0) -> 31
mark1(31) -> 4
U221(0, 0) -> 32
mark1(32) -> 5
U231(0) -> 33
mark1(33) -> 6
U311(0, 0) -> 34
mark1(34) -> 7
U321(0) -> 35
mark1(35) -> 8
U411(0, 0, 0) -> 36
mark1(36) -> 9
U421(0, 0) -> 37
mark1(37) -> 10
U431(0) -> 38
mark1(38) -> 11
U511(0, 0, 0) -> 39
mark1(39) -> 12
U521(0, 0) -> 40
mark1(40) -> 13
U531(0) -> 41
mark1(41) -> 14
U611(0, 0) -> 42
mark1(42) -> 15
U621(0) -> 43
mark1(43) -> 16
U711(0, 0) -> 44
mark1(44) -> 17
U721(0) -> 45
mark1(45) -> 18
and1(0, 0) -> 46
mark1(46) -> 19
nil1() -> 47
ok1(47) -> 20
tt1() -> 48
ok1(48) -> 20
a1() -> 49
ok1(49) -> 20
e1() -> 50
ok1(50) -> 20
i1() -> 51
ok1(51) -> 20
o1() -> 52
ok1(52) -> 20
u1() -> 53
ok1(53) -> 20
__1(0, 0) -> 54
ok1(54) -> 1
U111(0, 0) -> 55
ok1(55) -> 2
U121(0) -> 56
ok1(56) -> 3
isNeList1(0) -> 57
ok1(57) -> 21
U211(0, 0, 0) -> 58
ok1(58) -> 4
U221(0, 0) -> 59
ok1(59) -> 5
isList1(0) -> 60
ok1(60) -> 22
U231(0) -> 61
ok1(61) -> 6
U311(0, 0) -> 62
ok1(62) -> 7
U321(0) -> 63
ok1(63) -> 8
isQid1(0) -> 64
ok1(64) -> 23
U411(0, 0, 0) -> 65
ok1(65) -> 9
U421(0, 0) -> 66
ok1(66) -> 10
U431(0) -> 67
ok1(67) -> 11
U511(0, 0, 0) -> 68
ok1(68) -> 12
U521(0, 0) -> 69
ok1(69) -> 13
U531(0) -> 70
ok1(70) -> 14
U611(0, 0) -> 71
ok1(71) -> 15
U621(0) -> 72
ok1(72) -> 16
U711(0, 0) -> 73
ok1(73) -> 17
U721(0) -> 74
ok1(74) -> 18
isNePal1(0) -> 75
ok1(75) -> 24
and1(0, 0) -> 76
ok1(76) -> 19
isPalListKind1(0) -> 77
ok1(77) -> 25
isPal1(0) -> 78
ok1(78) -> 26
proper1(0) -> 79
top1(79) -> 27
active1(0) -> 80
top1(80) -> 27
mark1(28) -> 28
mark1(28) -> 54
mark1(29) -> 29
mark1(29) -> 55
mark1(30) -> 30
mark1(30) -> 56
mark1(31) -> 31
mark1(31) -> 58
mark1(32) -> 32
mark1(32) -> 59
mark1(33) -> 33
mark1(33) -> 61
mark1(34) -> 34
mark1(34) -> 62
mark1(35) -> 35
mark1(35) -> 63
mark1(36) -> 36
mark1(36) -> 65
mark1(37) -> 37
mark1(37) -> 66
mark1(38) -> 38
mark1(38) -> 67
mark1(39) -> 39
mark1(39) -> 68
mark1(40) -> 40
mark1(40) -> 69
mark1(41) -> 41
mark1(41) -> 70
mark1(42) -> 42
mark1(42) -> 71
mark1(43) -> 43
mark1(43) -> 72
mark1(44) -> 44
mark1(44) -> 73
mark1(45) -> 45
mark1(45) -> 74
mark1(46) -> 46
mark1(46) -> 76
ok1(47) -> 79
ok1(48) -> 79
ok1(49) -> 79
ok1(50) -> 79
ok1(51) -> 79
ok1(52) -> 79
ok1(53) -> 79
ok1(54) -> 28
ok1(54) -> 54
ok1(55) -> 29
ok1(55) -> 55
ok1(56) -> 30
ok1(56) -> 56
ok1(57) -> 57
ok1(58) -> 31
ok1(58) -> 58
ok1(59) -> 32
ok1(59) -> 59
ok1(60) -> 60
ok1(61) -> 33
ok1(61) -> 61
ok1(62) -> 34
ok1(62) -> 62
ok1(63) -> 35
ok1(63) -> 63
ok1(64) -> 64
ok1(65) -> 36
ok1(65) -> 65
ok1(66) -> 37
ok1(66) -> 66
ok1(67) -> 38
ok1(67) -> 67
ok1(68) -> 39
ok1(68) -> 68
ok1(69) -> 40
ok1(69) -> 69
ok1(70) -> 41
ok1(70) -> 70
ok1(71) -> 42
ok1(71) -> 71
ok1(72) -> 43
ok1(72) -> 72
ok1(73) -> 44
ok1(73) -> 73
ok1(74) -> 45
ok1(74) -> 74
ok1(75) -> 75
ok1(76) -> 46
ok1(76) -> 76
ok1(77) -> 77
ok1(78) -> 78
active2(47) -> 81
top2(81) -> 27
active2(48) -> 81
active2(49) -> 81
active2(50) -> 81
active2(51) -> 81
active2(52) -> 81
active2(53) -> 81

----------------------------------------

(6)
BOUNDS(1, n^1)

----------------------------------------

(7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(8)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
   active(__(X, nil)) -> mark(X)
   active(__(nil, X)) -> mark(X)
   active(U11(tt, V)) -> mark(U12(isNeList(V)))
   active(U12(tt)) -> mark(tt)
   active(U21(tt, V1, V2)) -> mark(U22(isList(V1), V2))
   active(U22(tt, V2)) -> mark(U23(isList(V2)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isQid(V)))
   active(U32(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isList(V1), V2))
   active(U42(tt, V2)) -> mark(U43(isNeList(V2)))
   active(U43(tt)) -> mark(tt)
   active(U51(tt, V1, V2)) -> mark(U52(isNeList(V1), V2))
   active(U52(tt, V2)) -> mark(U53(isList(V2)))
   active(U53(tt)) -> mark(tt)
   active(U61(tt, V)) -> mark(U62(isQid(V)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, V)) -> mark(U72(isNePal(V)))
   active(U72(tt)) -> mark(tt)
   active(and(tt, X)) -> mark(X)
   active(isList(V)) -> mark(U11(isPalListKind(V), V))
   active(isList(nil)) -> mark(tt)
   active(isList(__(V1, V2))) -> mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
   active(isNeList(V)) -> mark(U31(isPalListKind(V), V))
   active(isNeList(__(V1, V2))) -> mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
   active(isNeList(__(V1, V2))) -> mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
   active(isNePal(V)) -> mark(U61(isPalListKind(V), V))
   active(isNePal(__(I, __(P, I)))) -> mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
   active(isPal(V)) -> mark(U71(isPalListKind(V), V))
   active(isPal(nil)) -> mark(tt)
   active(isPalListKind(a)) -> mark(tt)
   active(isPalListKind(e)) -> mark(tt)
   active(isPalListKind(i)) -> mark(tt)
   active(isPalListKind(nil)) -> mark(tt)
   active(isPalListKind(o)) -> mark(tt)
   active(isPalListKind(u)) -> mark(tt)
   active(isPalListKind(__(V1, V2))) -> mark(and(isPalListKind(V1), isPalListKind(V2)))
   active(isQid(a)) -> mark(tt)
   active(isQid(e)) -> mark(tt)
   active(isQid(i)) -> mark(tt)
   active(isQid(o)) -> mark(tt)
   active(isQid(u)) -> mark(tt)
   active(__(X1, X2)) -> __(active(X1), X2)
   active(__(X1, X2)) -> __(X1, active(X2))
   active(U11(X1, X2)) -> U11(active(X1), X2)
   active(U12(X)) -> U12(active(X))
   active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
   active(U22(X1, X2)) -> U22(active(X1), X2)
   active(U23(X)) -> U23(active(X))
   active(U31(X1, X2)) -> U31(active(X1), X2)
   active(U32(X)) -> U32(active(X))
   active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
   active(U42(X1, X2)) -> U42(active(X1), X2)
   active(U43(X)) -> U43(active(X))
   active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
   active(U52(X1, X2)) -> U52(active(X1), X2)
   active(U53(X)) -> U53(active(X))
   active(U61(X1, X2)) -> U61(active(X1), X2)
   active(U62(X)) -> U62(active(X))
   active(U71(X1, X2)) -> U71(active(X1), X2)
   active(U72(X)) -> U72(active(X))
   active(and(X1, X2)) -> and(active(X1), X2)
   __(mark(X1), X2) -> mark(__(X1, X2))
   __(X1, mark(X2)) -> mark(__(X1, X2))
   U11(mark(X1), X2) -> mark(U11(X1, X2))
   U12(mark(X)) -> mark(U12(X))
   U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
   U22(mark(X1), X2) -> mark(U22(X1, X2))
   U23(mark(X)) -> mark(U23(X))
   U31(mark(X1), X2) -> mark(U31(X1, X2))
   U32(mark(X)) -> mark(U32(X))
   U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
   U42(mark(X1), X2) -> mark(U42(X1, X2))
   U43(mark(X)) -> mark(U43(X))
   U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
   U52(mark(X1), X2) -> mark(U52(X1, X2))
   U53(mark(X)) -> mark(U53(X))
   U61(mark(X1), X2) -> mark(U61(X1, X2))
   U62(mark(X)) -> mark(U62(X))
   U71(mark(X1), X2) -> mark(U71(X1, X2))
   U72(mark(X)) -> mark(U72(X))
   and(mark(X1), X2) -> mark(and(X1, X2))
   proper(__(X1, X2)) -> __(proper(X1), proper(X2))
   proper(nil) -> ok(nil)
   proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
   proper(tt) -> ok(tt)
   proper(U12(X)) -> U12(proper(X))
   proper(isNeList(X)) -> isNeList(proper(X))
   proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
   proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
   proper(isList(X)) -> isList(proper(X))
   proper(U23(X)) -> U23(proper(X))
   proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
   proper(U32(X)) -> U32(proper(X))
   proper(isQid(X)) -> isQid(proper(X))
   proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
   proper(U42(X1, X2)) -> U42(proper(X1), proper(X2))
   proper(U43(X)) -> U43(proper(X))
   proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
   proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
   proper(U53(X)) -> U53(proper(X))
   proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
   proper(U62(X)) -> U62(proper(X))
   proper(U71(X1, X2)) -> U71(proper(X1), proper(X2))
   proper(U72(X)) -> U72(proper(X))
   proper(isNePal(X)) -> isNePal(proper(X))
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(isPalListKind(X)) -> isPalListKind(proper(X))
   proper(isPal(X)) -> isPal(proper(X))
   proper(a) -> ok(a)
   proper(e) -> ok(e)
   proper(i) -> ok(i)
   proper(o) -> ok(o)
   proper(u) -> ok(u)
   __(ok(X1), ok(X2)) -> ok(__(X1, X2))
   U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
   U12(ok(X)) -> ok(U12(X))
   isNeList(ok(X)) -> ok(isNeList(X))
   U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
   U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
   isList(ok(X)) -> ok(isList(X))
   U23(ok(X)) -> ok(U23(X))
   U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
   U32(ok(X)) -> ok(U32(X))
   isQid(ok(X)) -> ok(isQid(X))
   U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
   U42(ok(X1), ok(X2)) -> ok(U42(X1, X2))
   U43(ok(X)) -> ok(U43(X))
   U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
   U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
   U53(ok(X)) -> ok(U53(X))
   U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
   U62(ok(X)) -> ok(U62(X))
   U71(ok(X1), ok(X2)) -> ok(U71(X1, X2))
   U72(ok(X)) -> ok(U72(X))
   isNePal(ok(X)) -> ok(isNePal(X))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isPalListKind(ok(X)) -> ok(isPalListKind(X))
   isPal(ok(X)) -> ok(isPal(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(9) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

U52(ok(X1), ok(X2)) ->^+ ok(U52(X1, X2))

gives rise to a decreasing loop by considering the right hand sides subterm at position [0].

The pumping substitution is [X1 / ok(X1), X2 / ok(X2)].

The result substitution is [ ].




----------------------------------------

(10)
Complex Obligation (BEST)

----------------------------------------

(11)
Obligation:
Proved the lower bound n^1 for the following obligation:

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
   active(__(X, nil)) -> mark(X)
   active(__(nil, X)) -> mark(X)
   active(U11(tt, V)) -> mark(U12(isNeList(V)))
   active(U12(tt)) -> mark(tt)
   active(U21(tt, V1, V2)) -> mark(U22(isList(V1), V2))
   active(U22(tt, V2)) -> mark(U23(isList(V2)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isQid(V)))
   active(U32(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isList(V1), V2))
   active(U42(tt, V2)) -> mark(U43(isNeList(V2)))
   active(U43(tt)) -> mark(tt)
   active(U51(tt, V1, V2)) -> mark(U52(isNeList(V1), V2))
   active(U52(tt, V2)) -> mark(U53(isList(V2)))
   active(U53(tt)) -> mark(tt)
   active(U61(tt, V)) -> mark(U62(isQid(V)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, V)) -> mark(U72(isNePal(V)))
   active(U72(tt)) -> mark(tt)
   active(and(tt, X)) -> mark(X)
   active(isList(V)) -> mark(U11(isPalListKind(V), V))
   active(isList(nil)) -> mark(tt)
   active(isList(__(V1, V2))) -> mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
   active(isNeList(V)) -> mark(U31(isPalListKind(V), V))
   active(isNeList(__(V1, V2))) -> mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
   active(isNeList(__(V1, V2))) -> mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
   active(isNePal(V)) -> mark(U61(isPalListKind(V), V))
   active(isNePal(__(I, __(P, I)))) -> mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
   active(isPal(V)) -> mark(U71(isPalListKind(V), V))
   active(isPal(nil)) -> mark(tt)
   active(isPalListKind(a)) -> mark(tt)
   active(isPalListKind(e)) -> mark(tt)
   active(isPalListKind(i)) -> mark(tt)
   active(isPalListKind(nil)) -> mark(tt)
   active(isPalListKind(o)) -> mark(tt)
   active(isPalListKind(u)) -> mark(tt)
   active(isPalListKind(__(V1, V2))) -> mark(and(isPalListKind(V1), isPalListKind(V2)))
   active(isQid(a)) -> mark(tt)
   active(isQid(e)) -> mark(tt)
   active(isQid(i)) -> mark(tt)
   active(isQid(o)) -> mark(tt)
   active(isQid(u)) -> mark(tt)
   active(__(X1, X2)) -> __(active(X1), X2)
   active(__(X1, X2)) -> __(X1, active(X2))
   active(U11(X1, X2)) -> U11(active(X1), X2)
   active(U12(X)) -> U12(active(X))
   active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
   active(U22(X1, X2)) -> U22(active(X1), X2)
   active(U23(X)) -> U23(active(X))
   active(U31(X1, X2)) -> U31(active(X1), X2)
   active(U32(X)) -> U32(active(X))
   active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
   active(U42(X1, X2)) -> U42(active(X1), X2)
   active(U43(X)) -> U43(active(X))
   active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
   active(U52(X1, X2)) -> U52(active(X1), X2)
   active(U53(X)) -> U53(active(X))
   active(U61(X1, X2)) -> U61(active(X1), X2)
   active(U62(X)) -> U62(active(X))
   active(U71(X1, X2)) -> U71(active(X1), X2)
   active(U72(X)) -> U72(active(X))
   active(and(X1, X2)) -> and(active(X1), X2)
   __(mark(X1), X2) -> mark(__(X1, X2))
   __(X1, mark(X2)) -> mark(__(X1, X2))
   U11(mark(X1), X2) -> mark(U11(X1, X2))
   U12(mark(X)) -> mark(U12(X))
   U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
   U22(mark(X1), X2) -> mark(U22(X1, X2))
   U23(mark(X)) -> mark(U23(X))
   U31(mark(X1), X2) -> mark(U31(X1, X2))
   U32(mark(X)) -> mark(U32(X))
   U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
   U42(mark(X1), X2) -> mark(U42(X1, X2))
   U43(mark(X)) -> mark(U43(X))
   U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
   U52(mark(X1), X2) -> mark(U52(X1, X2))
   U53(mark(X)) -> mark(U53(X))
   U61(mark(X1), X2) -> mark(U61(X1, X2))
   U62(mark(X)) -> mark(U62(X))
   U71(mark(X1), X2) -> mark(U71(X1, X2))
   U72(mark(X)) -> mark(U72(X))
   and(mark(X1), X2) -> mark(and(X1, X2))
   proper(__(X1, X2)) -> __(proper(X1), proper(X2))
   proper(nil) -> ok(nil)
   proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
   proper(tt) -> ok(tt)
   proper(U12(X)) -> U12(proper(X))
   proper(isNeList(X)) -> isNeList(proper(X))
   proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
   proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
   proper(isList(X)) -> isList(proper(X))
   proper(U23(X)) -> U23(proper(X))
   proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
   proper(U32(X)) -> U32(proper(X))
   proper(isQid(X)) -> isQid(proper(X))
   proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
   proper(U42(X1, X2)) -> U42(proper(X1), proper(X2))
   proper(U43(X)) -> U43(proper(X))
   proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
   proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
   proper(U53(X)) -> U53(proper(X))
   proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
   proper(U62(X)) -> U62(proper(X))
   proper(U71(X1, X2)) -> U71(proper(X1), proper(X2))
   proper(U72(X)) -> U72(proper(X))
   proper(isNePal(X)) -> isNePal(proper(X))
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(isPalListKind(X)) -> isPalListKind(proper(X))
   proper(isPal(X)) -> isPal(proper(X))
   proper(a) -> ok(a)
   proper(e) -> ok(e)
   proper(i) -> ok(i)
   proper(o) -> ok(o)
   proper(u) -> ok(u)
   __(ok(X1), ok(X2)) -> ok(__(X1, X2))
   U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
   U12(ok(X)) -> ok(U12(X))
   isNeList(ok(X)) -> ok(isNeList(X))
   U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
   U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
   isList(ok(X)) -> ok(isList(X))
   U23(ok(X)) -> ok(U23(X))
   U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
   U32(ok(X)) -> ok(U32(X))
   isQid(ok(X)) -> ok(isQid(X))
   U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
   U42(ok(X1), ok(X2)) -> ok(U42(X1, X2))
   U43(ok(X)) -> ok(U43(X))
   U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
   U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
   U53(ok(X)) -> ok(U53(X))
   U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
   U62(ok(X)) -> ok(U62(X))
   U71(ok(X1), ok(X2)) -> ok(U71(X1, X2))
   U72(ok(X)) -> ok(U72(X))
   isNePal(ok(X)) -> ok(isNePal(X))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isPalListKind(ok(X)) -> ok(isPalListKind(X))
   isPal(ok(X)) -> ok(isPal(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(12) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(13)
BOUNDS(n^1, INF)

----------------------------------------

(14)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
   active(__(X, nil)) -> mark(X)
   active(__(nil, X)) -> mark(X)
   active(U11(tt, V)) -> mark(U12(isNeList(V)))
   active(U12(tt)) -> mark(tt)
   active(U21(tt, V1, V2)) -> mark(U22(isList(V1), V2))
   active(U22(tt, V2)) -> mark(U23(isList(V2)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isQid(V)))
   active(U32(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isList(V1), V2))
   active(U42(tt, V2)) -> mark(U43(isNeList(V2)))
   active(U43(tt)) -> mark(tt)
   active(U51(tt, V1, V2)) -> mark(U52(isNeList(V1), V2))
   active(U52(tt, V2)) -> mark(U53(isList(V2)))
   active(U53(tt)) -> mark(tt)
   active(U61(tt, V)) -> mark(U62(isQid(V)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt, V)) -> mark(U72(isNePal(V)))
   active(U72(tt)) -> mark(tt)
   active(and(tt, X)) -> mark(X)
   active(isList(V)) -> mark(U11(isPalListKind(V), V))
   active(isList(nil)) -> mark(tt)
   active(isList(__(V1, V2))) -> mark(U21(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
   active(isNeList(V)) -> mark(U31(isPalListKind(V), V))
   active(isNeList(__(V1, V2))) -> mark(U41(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
   active(isNeList(__(V1, V2))) -> mark(U51(and(isPalListKind(V1), isPalListKind(V2)), V1, V2))
   active(isNePal(V)) -> mark(U61(isPalListKind(V), V))
   active(isNePal(__(I, __(P, I)))) -> mark(and(and(isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P))))
   active(isPal(V)) -> mark(U71(isPalListKind(V), V))
   active(isPal(nil)) -> mark(tt)
   active(isPalListKind(a)) -> mark(tt)
   active(isPalListKind(e)) -> mark(tt)
   active(isPalListKind(i)) -> mark(tt)
   active(isPalListKind(nil)) -> mark(tt)
   active(isPalListKind(o)) -> mark(tt)
   active(isPalListKind(u)) -> mark(tt)
   active(isPalListKind(__(V1, V2))) -> mark(and(isPalListKind(V1), isPalListKind(V2)))
   active(isQid(a)) -> mark(tt)
   active(isQid(e)) -> mark(tt)
   active(isQid(i)) -> mark(tt)
   active(isQid(o)) -> mark(tt)
   active(isQid(u)) -> mark(tt)
   active(__(X1, X2)) -> __(active(X1), X2)
   active(__(X1, X2)) -> __(X1, active(X2))
   active(U11(X1, X2)) -> U11(active(X1), X2)
   active(U12(X)) -> U12(active(X))
   active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
   active(U22(X1, X2)) -> U22(active(X1), X2)
   active(U23(X)) -> U23(active(X))
   active(U31(X1, X2)) -> U31(active(X1), X2)
   active(U32(X)) -> U32(active(X))
   active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
   active(U42(X1, X2)) -> U42(active(X1), X2)
   active(U43(X)) -> U43(active(X))
   active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
   active(U52(X1, X2)) -> U52(active(X1), X2)
   active(U53(X)) -> U53(active(X))
   active(U61(X1, X2)) -> U61(active(X1), X2)
   active(U62(X)) -> U62(active(X))
   active(U71(X1, X2)) -> U71(active(X1), X2)
   active(U72(X)) -> U72(active(X))
   active(and(X1, X2)) -> and(active(X1), X2)
   __(mark(X1), X2) -> mark(__(X1, X2))
   __(X1, mark(X2)) -> mark(__(X1, X2))
   U11(mark(X1), X2) -> mark(U11(X1, X2))
   U12(mark(X)) -> mark(U12(X))
   U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
   U22(mark(X1), X2) -> mark(U22(X1, X2))
   U23(mark(X)) -> mark(U23(X))
   U31(mark(X1), X2) -> mark(U31(X1, X2))
   U32(mark(X)) -> mark(U32(X))
   U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
   U42(mark(X1), X2) -> mark(U42(X1, X2))
   U43(mark(X)) -> mark(U43(X))
   U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
   U52(mark(X1), X2) -> mark(U52(X1, X2))
   U53(mark(X)) -> mark(U53(X))
   U61(mark(X1), X2) -> mark(U61(X1, X2))
   U62(mark(X)) -> mark(U62(X))
   U71(mark(X1), X2) -> mark(U71(X1, X2))
   U72(mark(X)) -> mark(U72(X))
   and(mark(X1), X2) -> mark(and(X1, X2))
   proper(__(X1, X2)) -> __(proper(X1), proper(X2))
   proper(nil) -> ok(nil)
   proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
   proper(tt) -> ok(tt)
   proper(U12(X)) -> U12(proper(X))
   proper(isNeList(X)) -> isNeList(proper(X))
   proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
   proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
   proper(isList(X)) -> isList(proper(X))
   proper(U23(X)) -> U23(proper(X))
   proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
   proper(U32(X)) -> U32(proper(X))
   proper(isQid(X)) -> isQid(proper(X))
   proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
   proper(U42(X1, X2)) -> U42(proper(X1), proper(X2))
   proper(U43(X)) -> U43(proper(X))
   proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
   proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
   proper(U53(X)) -> U53(proper(X))
   proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
   proper(U62(X)) -> U62(proper(X))
   proper(U71(X1, X2)) -> U71(proper(X1), proper(X2))
   proper(U72(X)) -> U72(proper(X))
   proper(isNePal(X)) -> isNePal(proper(X))
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(isPalListKind(X)) -> isPalListKind(proper(X))
   proper(isPal(X)) -> isPal(proper(X))
   proper(a) -> ok(a)
   proper(e) -> ok(e)
   proper(i) -> ok(i)
   proper(o) -> ok(o)
   proper(u) -> ok(u)
   __(ok(X1), ok(X2)) -> ok(__(X1, X2))
   U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
   U12(ok(X)) -> ok(U12(X))
   isNeList(ok(X)) -> ok(isNeList(X))
   U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
   U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
   isList(ok(X)) -> ok(isList(X))
   U23(ok(X)) -> ok(U23(X))
   U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
   U32(ok(X)) -> ok(U32(X))
   isQid(ok(X)) -> ok(isQid(X))
   U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
   U42(ok(X1), ok(X2)) -> ok(U42(X1, X2))
   U43(ok(X)) -> ok(U43(X))
   U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
   U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
   U53(ok(X)) -> ok(U53(X))
   U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
   U62(ok(X)) -> ok(U62(X))
   U71(ok(X1), ok(X2)) -> ok(U71(X1, X2))
   U72(ok(X)) -> ok(U72(X))
   isNePal(ok(X)) -> ok(isNePal(X))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isPalListKind(ok(X)) -> ok(isPalListKind(X))
   isPal(ok(X)) -> ok(isPal(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL