/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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


WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 86 ms]
    (6) BOUNDS(1, n^1)
(7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CpxTRS
    (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) typed CpxTrs
    (11) OrderProof [LOWER BOUND(ID), 0 ms]
    (12) typed CpxTrs
    (13) RewriteLemmaProof [LOWER BOUND(ID), 438 ms]
    (14) BEST
        (15) proven lower bound
            (16) LowerBoundPropagationProof [FINISHED, 0 ms]
            (17) BOUNDS(n^1, INF)
        (18) typed CpxTrs
            (19) RewriteLemmaProof [LOWER BOUND(ID), 153 ms]
            (20) typed CpxTrs


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

(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(and(tt, X)) -> mark(X)
   active(isList(V)) -> mark(isNeList(V))
   active(isList(nil)) -> mark(tt)
   active(isList(__(V1, V2))) -> mark(and(isList(V1), isList(V2)))
   active(isNeList(V)) -> mark(isQid(V))
   active(isNeList(__(V1, V2))) -> mark(and(isList(V1), isNeList(V2)))
   active(isNeList(__(V1, V2))) -> mark(and(isNeList(V1), isList(V2)))
   active(isNePal(V)) -> mark(isQid(V))
   active(isNePal(__(I, __(P, I)))) -> mark(and(isQid(I), isPal(P)))
   active(isPal(V)) -> mark(isNePal(V))
   active(isPal(nil)) -> mark(tt)
   active(isQid(a)) -> mark(tt)
   active(isQid(e)) -> mark(tt)
   active(isQid(i)) -> mark(tt)
   active(isQid(o)) -> mark(tt)
   active(isQid(u)) -> mark(tt)
   active(__(X1, X2)) -> __(active(X1), X2)
   active(__(X1, X2)) -> __(X1, active(X2))
   active(and(X1, X2)) -> and(active(X1), X2)
   __(mark(X1), X2) -> mark(__(X1, X2))
   __(X1, mark(X2)) -> mark(__(X1, X2))
   and(mark(X1), X2) -> mark(and(X1, X2))
   proper(__(X1, X2)) -> __(proper(X1), proper(X2))
   proper(nil) -> ok(nil)
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(tt) -> ok(tt)
   proper(isList(X)) -> isList(proper(X))
   proper(isNeList(X)) -> isNeList(proper(X))
   proper(isQid(X)) -> isQid(proper(X))
   proper(isNePal(X)) -> isNePal(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))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isList(ok(X)) -> ok(isList(X))
   isNeList(ok(X)) -> ok(isNeList(X))
   isQid(ok(X)) -> ok(isQid(X))
   isNePal(ok(X)) -> ok(isNePal(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(and(tt, X)) -> mark(X)
active(isList(V)) -> mark(isNeList(V))
active(isList(nil)) -> mark(tt)
active(isList(__(V1, V2))) -> mark(and(isList(V1), isList(V2)))
active(isNeList(V)) -> mark(isQid(V))
active(isNeList(__(V1, V2))) -> mark(and(isList(V1), isNeList(V2)))
active(isNeList(__(V1, V2))) -> mark(and(isNeList(V1), isList(V2)))
active(isNePal(V)) -> mark(isQid(V))
active(isNePal(__(I, __(P, I)))) -> mark(and(isQid(I), isPal(P)))
active(isPal(V)) -> mark(isNePal(V))
active(isPal(nil)) -> mark(tt)
active(isQid(a)) -> mark(tt)
active(isQid(e)) -> mark(tt)
active(isQid(i)) -> mark(tt)
active(isQid(o)) -> mark(tt)
active(isQid(u)) -> mark(tt)
active(__(X1, X2)) -> __(active(X1), X2)
active(__(X1, X2)) -> __(X1, active(X2))
active(and(X1, X2)) -> and(active(X1), X2)
proper(__(X1, X2)) -> __(proper(X1), proper(X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(isList(X)) -> isList(proper(X))
proper(isNeList(X)) -> isNeList(proper(X))
proper(isQid(X)) -> isQid(proper(X))
proper(isNePal(X)) -> isNePal(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))
   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))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isList(ok(X)) -> ok(isList(X))
   isNeList(ok(X)) -> ok(isNeList(X))
   isQid(ok(X)) -> ok(isQid(X))
   isNePal(ok(X)) -> ok(isNePal(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))
   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))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isList(ok(X)) -> ok(isList(X))
   isNeList(ok(X)) -> ok(isNeList(X))
   isQid(ok(X)) -> ok(isQid(X))
   isNePal(ok(X)) -> ok(isNePal(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]
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
and0(0, 0) -> 2
proper0(0) -> 3
isList0(0) -> 4
isNeList0(0) -> 5
isQid0(0) -> 6
isNePal0(0) -> 7
isPal0(0) -> 8
top0(0) -> 9
__1(0, 0) -> 10
mark1(10) -> 1
and1(0, 0) -> 11
mark1(11) -> 2
nil1() -> 12
ok1(12) -> 3
tt1() -> 13
ok1(13) -> 3
a1() -> 14
ok1(14) -> 3
e1() -> 15
ok1(15) -> 3
i1() -> 16
ok1(16) -> 3
o1() -> 17
ok1(17) -> 3
u1() -> 18
ok1(18) -> 3
__1(0, 0) -> 19
ok1(19) -> 1
and1(0, 0) -> 20
ok1(20) -> 2
isList1(0) -> 21
ok1(21) -> 4
isNeList1(0) -> 22
ok1(22) -> 5
isQid1(0) -> 23
ok1(23) -> 6
isNePal1(0) -> 24
ok1(24) -> 7
isPal1(0) -> 25
ok1(25) -> 8
proper1(0) -> 26
top1(26) -> 9
active1(0) -> 27
top1(27) -> 9
mark1(10) -> 10
mark1(10) -> 19
mark1(11) -> 11
mark1(11) -> 20
ok1(12) -> 26
ok1(13) -> 26
ok1(14) -> 26
ok1(15) -> 26
ok1(16) -> 26
ok1(17) -> 26
ok1(18) -> 26
ok1(19) -> 10
ok1(19) -> 19
ok1(20) -> 11
ok1(20) -> 20
ok1(21) -> 21
ok1(22) -> 22
ok1(23) -> 23
ok1(24) -> 24
ok1(25) -> 25
active2(12) -> 28
top2(28) -> 9
active2(13) -> 28
active2(14) -> 28
active2(15) -> 28
active2(16) -> 28
active2(17) -> 28
active2(18) -> 28

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

(6)
BOUNDS(1, n^1)

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

(7) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(8)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


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(and(tt, X)) -> mark(X)
   active(isList(V)) -> mark(isNeList(V))
   active(isList(nil)) -> mark(tt)
   active(isList(__(V1, V2))) -> mark(and(isList(V1), isList(V2)))
   active(isNeList(V)) -> mark(isQid(V))
   active(isNeList(__(V1, V2))) -> mark(and(isList(V1), isNeList(V2)))
   active(isNeList(__(V1, V2))) -> mark(and(isNeList(V1), isList(V2)))
   active(isNePal(V)) -> mark(isQid(V))
   active(isNePal(__(I, __(P, I)))) -> mark(and(isQid(I), isPal(P)))
   active(isPal(V)) -> mark(isNePal(V))
   active(isPal(nil)) -> mark(tt)
   active(isQid(a)) -> mark(tt)
   active(isQid(e)) -> mark(tt)
   active(isQid(i)) -> mark(tt)
   active(isQid(o)) -> mark(tt)
   active(isQid(u)) -> mark(tt)
   active(__(X1, X2)) -> __(active(X1), X2)
   active(__(X1, X2)) -> __(X1, active(X2))
   active(and(X1, X2)) -> and(active(X1), X2)
   __(mark(X1), X2) -> mark(__(X1, X2))
   __(X1, mark(X2)) -> mark(__(X1, X2))
   and(mark(X1), X2) -> mark(and(X1, X2))
   proper(__(X1, X2)) -> __(proper(X1), proper(X2))
   proper(nil) -> ok(nil)
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(tt) -> ok(tt)
   proper(isList(X)) -> isList(proper(X))
   proper(isNeList(X)) -> isNeList(proper(X))
   proper(isQid(X)) -> isQid(proper(X))
   proper(isNePal(X)) -> isNePal(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))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isList(ok(X)) -> ok(isList(X))
   isNeList(ok(X)) -> ok(isNeList(X))
   isQid(ok(X)) -> ok(isQid(X))
   isNePal(ok(X)) -> ok(isNePal(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) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(10)
Obligation:
TRS:
Rules:
active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
active(__(X, nil)) -> mark(X)
active(__(nil, X)) -> mark(X)
active(and(tt, X)) -> mark(X)
active(isList(V)) -> mark(isNeList(V))
active(isList(nil)) -> mark(tt)
active(isList(__(V1, V2))) -> mark(and(isList(V1), isList(V2)))
active(isNeList(V)) -> mark(isQid(V))
active(isNeList(__(V1, V2))) -> mark(and(isList(V1), isNeList(V2)))
active(isNeList(__(V1, V2))) -> mark(and(isNeList(V1), isList(V2)))
active(isNePal(V)) -> mark(isQid(V))
active(isNePal(__(I, __(P, I)))) -> mark(and(isQid(I), isPal(P)))
active(isPal(V)) -> mark(isNePal(V))
active(isPal(nil)) -> mark(tt)
active(isQid(a)) -> mark(tt)
active(isQid(e)) -> mark(tt)
active(isQid(i)) -> mark(tt)
active(isQid(o)) -> mark(tt)
active(isQid(u)) -> mark(tt)
active(__(X1, X2)) -> __(active(X1), X2)
active(__(X1, X2)) -> __(X1, active(X2))
active(and(X1, X2)) -> and(active(X1), X2)
__(mark(X1), X2) -> mark(__(X1, X2))
__(X1, mark(X2)) -> mark(__(X1, X2))
and(mark(X1), X2) -> mark(and(X1, X2))
proper(__(X1, X2)) -> __(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(isList(X)) -> isList(proper(X))
proper(isNeList(X)) -> isNeList(proper(X))
proper(isQid(X)) -> isQid(proper(X))
proper(isNePal(X)) -> isNePal(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))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
isList(ok(X)) -> ok(isList(X))
isNeList(ok(X)) -> ok(isNeList(X))
isQid(ok(X)) -> ok(isQid(X))
isNePal(ok(X)) -> ok(isNePal(X))
isPal(ok(X)) -> ok(isPal(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
__ :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
mark :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
nil :: mark:nil:tt:a:e:i:o:u:ok
and :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
tt :: mark:nil:tt:a:e:i:o:u:ok
isList :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
isNeList :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
isQid :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
isNePal :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
isPal :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
a :: mark:nil:tt:a:e:i:o:u:ok
e :: mark:nil:tt:a:e:i:o:u:ok
i :: mark:nil:tt:a:e:i:o:u:ok
o :: mark:nil:tt:a:e:i:o:u:ok
u :: mark:nil:tt:a:e:i:o:u:ok
proper :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
ok :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
top :: mark:nil:tt:a:e:i:o:u:ok -> top
hole_mark:nil:tt:a:e:i:o:u:ok1_0 :: mark:nil:tt:a:e:i:o:u:ok
hole_top2_0 :: top
gen_mark:nil:tt:a:e:i:o:u:ok3_0 :: Nat -> mark:nil:tt:a:e:i:o:u:ok

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

(11) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
active, __, isNeList, and, isList, isQid, isPal, isNePal, proper, top

They will be analysed ascendingly in the following order:
__ < active
isNeList < active
and < active
isList < active
isQid < active
isPal < active
isNePal < active
active < top
__ < proper
isNeList < proper
and < proper
isList < proper
isQid < proper
isPal < proper
isNePal < proper
proper < top

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

(12)
Obligation:
TRS:
Rules:
active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
active(__(X, nil)) -> mark(X)
active(__(nil, X)) -> mark(X)
active(and(tt, X)) -> mark(X)
active(isList(V)) -> mark(isNeList(V))
active(isList(nil)) -> mark(tt)
active(isList(__(V1, V2))) -> mark(and(isList(V1), isList(V2)))
active(isNeList(V)) -> mark(isQid(V))
active(isNeList(__(V1, V2))) -> mark(and(isList(V1), isNeList(V2)))
active(isNeList(__(V1, V2))) -> mark(and(isNeList(V1), isList(V2)))
active(isNePal(V)) -> mark(isQid(V))
active(isNePal(__(I, __(P, I)))) -> mark(and(isQid(I), isPal(P)))
active(isPal(V)) -> mark(isNePal(V))
active(isPal(nil)) -> mark(tt)
active(isQid(a)) -> mark(tt)
active(isQid(e)) -> mark(tt)
active(isQid(i)) -> mark(tt)
active(isQid(o)) -> mark(tt)
active(isQid(u)) -> mark(tt)
active(__(X1, X2)) -> __(active(X1), X2)
active(__(X1, X2)) -> __(X1, active(X2))
active(and(X1, X2)) -> and(active(X1), X2)
__(mark(X1), X2) -> mark(__(X1, X2))
__(X1, mark(X2)) -> mark(__(X1, X2))
and(mark(X1), X2) -> mark(and(X1, X2))
proper(__(X1, X2)) -> __(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(isList(X)) -> isList(proper(X))
proper(isNeList(X)) -> isNeList(proper(X))
proper(isQid(X)) -> isQid(proper(X))
proper(isNePal(X)) -> isNePal(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))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
isList(ok(X)) -> ok(isList(X))
isNeList(ok(X)) -> ok(isNeList(X))
isQid(ok(X)) -> ok(isQid(X))
isNePal(ok(X)) -> ok(isNePal(X))
isPal(ok(X)) -> ok(isPal(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
__ :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
mark :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
nil :: mark:nil:tt:a:e:i:o:u:ok
and :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
tt :: mark:nil:tt:a:e:i:o:u:ok
isList :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
isNeList :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
isQid :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
isNePal :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
isPal :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
a :: mark:nil:tt:a:e:i:o:u:ok
e :: mark:nil:tt:a:e:i:o:u:ok
i :: mark:nil:tt:a:e:i:o:u:ok
o :: mark:nil:tt:a:e:i:o:u:ok
u :: mark:nil:tt:a:e:i:o:u:ok
proper :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
ok :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
top :: mark:nil:tt:a:e:i:o:u:ok -> top
hole_mark:nil:tt:a:e:i:o:u:ok1_0 :: mark:nil:tt:a:e:i:o:u:ok
hole_top2_0 :: top
gen_mark:nil:tt:a:e:i:o:u:ok3_0 :: Nat -> mark:nil:tt:a:e:i:o:u:ok


Generator Equations:
gen_mark:nil:tt:a:e:i:o:u:ok3_0(0) <=> nil
gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(x, 1)) <=> mark(gen_mark:nil:tt:a:e:i:o:u:ok3_0(x))


The following defined symbols remain to be analysed:
__, active, isNeList, and, isList, isQid, isPal, isNePal, proper, top

They will be analysed ascendingly in the following order:
__ < active
isNeList < active
and < active
isList < active
isQid < active
isPal < active
isNePal < active
active < top
__ < proper
isNeList < proper
and < proper
isList < proper
isQid < proper
isPal < proper
isNePal < proper
proper < top

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
__(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n5_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) -> *4_0, rt in Omega(n5_0)

Induction Base:
__(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, 0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b))

Induction Step:
__(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, +(n5_0, 1))), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) ->_R^Omega(1)
mark(__(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n5_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(14)
Complex Obligation (BEST)

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

(15)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
active(__(X, nil)) -> mark(X)
active(__(nil, X)) -> mark(X)
active(and(tt, X)) -> mark(X)
active(isList(V)) -> mark(isNeList(V))
active(isList(nil)) -> mark(tt)
active(isList(__(V1, V2))) -> mark(and(isList(V1), isList(V2)))
active(isNeList(V)) -> mark(isQid(V))
active(isNeList(__(V1, V2))) -> mark(and(isList(V1), isNeList(V2)))
active(isNeList(__(V1, V2))) -> mark(and(isNeList(V1), isList(V2)))
active(isNePal(V)) -> mark(isQid(V))
active(isNePal(__(I, __(P, I)))) -> mark(and(isQid(I), isPal(P)))
active(isPal(V)) -> mark(isNePal(V))
active(isPal(nil)) -> mark(tt)
active(isQid(a)) -> mark(tt)
active(isQid(e)) -> mark(tt)
active(isQid(i)) -> mark(tt)
active(isQid(o)) -> mark(tt)
active(isQid(u)) -> mark(tt)
active(__(X1, X2)) -> __(active(X1), X2)
active(__(X1, X2)) -> __(X1, active(X2))
active(and(X1, X2)) -> and(active(X1), X2)
__(mark(X1), X2) -> mark(__(X1, X2))
__(X1, mark(X2)) -> mark(__(X1, X2))
and(mark(X1), X2) -> mark(and(X1, X2))
proper(__(X1, X2)) -> __(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(isList(X)) -> isList(proper(X))
proper(isNeList(X)) -> isNeList(proper(X))
proper(isQid(X)) -> isQid(proper(X))
proper(isNePal(X)) -> isNePal(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))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
isList(ok(X)) -> ok(isList(X))
isNeList(ok(X)) -> ok(isNeList(X))
isQid(ok(X)) -> ok(isQid(X))
isNePal(ok(X)) -> ok(isNePal(X))
isPal(ok(X)) -> ok(isPal(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
__ :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
mark :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
nil :: mark:nil:tt:a:e:i:o:u:ok
and :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
tt :: mark:nil:tt:a:e:i:o:u:ok
isList :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
isNeList :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
isQid :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
isNePal :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
isPal :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
a :: mark:nil:tt:a:e:i:o:u:ok
e :: mark:nil:tt:a:e:i:o:u:ok
i :: mark:nil:tt:a:e:i:o:u:ok
o :: mark:nil:tt:a:e:i:o:u:ok
u :: mark:nil:tt:a:e:i:o:u:ok
proper :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
ok :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
top :: mark:nil:tt:a:e:i:o:u:ok -> top
hole_mark:nil:tt:a:e:i:o:u:ok1_0 :: mark:nil:tt:a:e:i:o:u:ok
hole_top2_0 :: top
gen_mark:nil:tt:a:e:i:o:u:ok3_0 :: Nat -> mark:nil:tt:a:e:i:o:u:ok


Generator Equations:
gen_mark:nil:tt:a:e:i:o:u:ok3_0(0) <=> nil
gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(x, 1)) <=> mark(gen_mark:nil:tt:a:e:i:o:u:ok3_0(x))


The following defined symbols remain to be analysed:
__, active, isNeList, and, isList, isQid, isPal, isNePal, proper, top

They will be analysed ascendingly in the following order:
__ < active
isNeList < active
and < active
isList < active
isQid < active
isPal < active
isNePal < active
active < top
__ < proper
isNeList < proper
and < proper
isList < proper
isQid < proper
isPal < proper
isNePal < proper
proper < top

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

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

(17)
BOUNDS(n^1, INF)

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

(18)
Obligation:
TRS:
Rules:
active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
active(__(X, nil)) -> mark(X)
active(__(nil, X)) -> mark(X)
active(and(tt, X)) -> mark(X)
active(isList(V)) -> mark(isNeList(V))
active(isList(nil)) -> mark(tt)
active(isList(__(V1, V2))) -> mark(and(isList(V1), isList(V2)))
active(isNeList(V)) -> mark(isQid(V))
active(isNeList(__(V1, V2))) -> mark(and(isList(V1), isNeList(V2)))
active(isNeList(__(V1, V2))) -> mark(and(isNeList(V1), isList(V2)))
active(isNePal(V)) -> mark(isQid(V))
active(isNePal(__(I, __(P, I)))) -> mark(and(isQid(I), isPal(P)))
active(isPal(V)) -> mark(isNePal(V))
active(isPal(nil)) -> mark(tt)
active(isQid(a)) -> mark(tt)
active(isQid(e)) -> mark(tt)
active(isQid(i)) -> mark(tt)
active(isQid(o)) -> mark(tt)
active(isQid(u)) -> mark(tt)
active(__(X1, X2)) -> __(active(X1), X2)
active(__(X1, X2)) -> __(X1, active(X2))
active(and(X1, X2)) -> and(active(X1), X2)
__(mark(X1), X2) -> mark(__(X1, X2))
__(X1, mark(X2)) -> mark(__(X1, X2))
and(mark(X1), X2) -> mark(and(X1, X2))
proper(__(X1, X2)) -> __(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(isList(X)) -> isList(proper(X))
proper(isNeList(X)) -> isNeList(proper(X))
proper(isQid(X)) -> isQid(proper(X))
proper(isNePal(X)) -> isNePal(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))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
isList(ok(X)) -> ok(isList(X))
isNeList(ok(X)) -> ok(isNeList(X))
isQid(ok(X)) -> ok(isQid(X))
isNePal(ok(X)) -> ok(isNePal(X))
isPal(ok(X)) -> ok(isPal(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
__ :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
mark :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
nil :: mark:nil:tt:a:e:i:o:u:ok
and :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
tt :: mark:nil:tt:a:e:i:o:u:ok
isList :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
isNeList :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
isQid :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
isNePal :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
isPal :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
a :: mark:nil:tt:a:e:i:o:u:ok
e :: mark:nil:tt:a:e:i:o:u:ok
i :: mark:nil:tt:a:e:i:o:u:ok
o :: mark:nil:tt:a:e:i:o:u:ok
u :: mark:nil:tt:a:e:i:o:u:ok
proper :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
ok :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
top :: mark:nil:tt:a:e:i:o:u:ok -> top
hole_mark:nil:tt:a:e:i:o:u:ok1_0 :: mark:nil:tt:a:e:i:o:u:ok
hole_top2_0 :: top
gen_mark:nil:tt:a:e:i:o:u:ok3_0 :: Nat -> mark:nil:tt:a:e:i:o:u:ok


Lemmas:
__(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n5_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) -> *4_0, rt in Omega(n5_0)


Generator Equations:
gen_mark:nil:tt:a:e:i:o:u:ok3_0(0) <=> nil
gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(x, 1)) <=> mark(gen_mark:nil:tt:a:e:i:o:u:ok3_0(x))


The following defined symbols remain to be analysed:
isNeList, active, and, isList, isQid, isPal, isNePal, proper, top

They will be analysed ascendingly in the following order:
isNeList < active
and < active
isList < active
isQid < active
isPal < active
isNePal < active
active < top
isNeList < proper
and < proper
isList < proper
isQid < proper
isPal < proper
isNePal < proper
proper < top

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
and(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n1320_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) -> *4_0, rt in Omega(n1320_0)

Induction Base:
and(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, 0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b))

Induction Step:
and(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, +(n1320_0, 1))), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) ->_R^Omega(1)
mark(and(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n1320_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(20)
Obligation:
TRS:
Rules:
active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
active(__(X, nil)) -> mark(X)
active(__(nil, X)) -> mark(X)
active(and(tt, X)) -> mark(X)
active(isList(V)) -> mark(isNeList(V))
active(isList(nil)) -> mark(tt)
active(isList(__(V1, V2))) -> mark(and(isList(V1), isList(V2)))
active(isNeList(V)) -> mark(isQid(V))
active(isNeList(__(V1, V2))) -> mark(and(isList(V1), isNeList(V2)))
active(isNeList(__(V1, V2))) -> mark(and(isNeList(V1), isList(V2)))
active(isNePal(V)) -> mark(isQid(V))
active(isNePal(__(I, __(P, I)))) -> mark(and(isQid(I), isPal(P)))
active(isPal(V)) -> mark(isNePal(V))
active(isPal(nil)) -> mark(tt)
active(isQid(a)) -> mark(tt)
active(isQid(e)) -> mark(tt)
active(isQid(i)) -> mark(tt)
active(isQid(o)) -> mark(tt)
active(isQid(u)) -> mark(tt)
active(__(X1, X2)) -> __(active(X1), X2)
active(__(X1, X2)) -> __(X1, active(X2))
active(and(X1, X2)) -> and(active(X1), X2)
__(mark(X1), X2) -> mark(__(X1, X2))
__(X1, mark(X2)) -> mark(__(X1, X2))
and(mark(X1), X2) -> mark(and(X1, X2))
proper(__(X1, X2)) -> __(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(isList(X)) -> isList(proper(X))
proper(isNeList(X)) -> isNeList(proper(X))
proper(isQid(X)) -> isQid(proper(X))
proper(isNePal(X)) -> isNePal(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))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
isList(ok(X)) -> ok(isList(X))
isNeList(ok(X)) -> ok(isNeList(X))
isQid(ok(X)) -> ok(isQid(X))
isNePal(ok(X)) -> ok(isNePal(X))
isPal(ok(X)) -> ok(isPal(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
__ :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
mark :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
nil :: mark:nil:tt:a:e:i:o:u:ok
and :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
tt :: mark:nil:tt:a:e:i:o:u:ok
isList :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
isNeList :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
isQid :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
isNePal :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
isPal :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
a :: mark:nil:tt:a:e:i:o:u:ok
e :: mark:nil:tt:a:e:i:o:u:ok
i :: mark:nil:tt:a:e:i:o:u:ok
o :: mark:nil:tt:a:e:i:o:u:ok
u :: mark:nil:tt:a:e:i:o:u:ok
proper :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
ok :: mark:nil:tt:a:e:i:o:u:ok -> mark:nil:tt:a:e:i:o:u:ok
top :: mark:nil:tt:a:e:i:o:u:ok -> top
hole_mark:nil:tt:a:e:i:o:u:ok1_0 :: mark:nil:tt:a:e:i:o:u:ok
hole_top2_0 :: top
gen_mark:nil:tt:a:e:i:o:u:ok3_0 :: Nat -> mark:nil:tt:a:e:i:o:u:ok


Lemmas:
__(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n5_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
and(gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(1, n1320_0)), gen_mark:nil:tt:a:e:i:o:u:ok3_0(b)) -> *4_0, rt in Omega(n1320_0)


Generator Equations:
gen_mark:nil:tt:a:e:i:o:u:ok3_0(0) <=> nil
gen_mark:nil:tt:a:e:i:o:u:ok3_0(+(x, 1)) <=> mark(gen_mark:nil:tt:a:e:i:o:u:ok3_0(x))


The following defined symbols remain to be analysed:
isList, active, isQid, isPal, isNePal, proper, top

They will be analysed ascendingly in the following order:
isList < active
isQid < active
isPal < active
isNePal < active
active < top
isList < proper
isQid < proper
isPal < proper
isNePal < proper
proper < top