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


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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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 222 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(from(X)) -> mark(cons(X, from(s(X))))
   active(2ndspos(0, Z)) -> mark(rnil)
   active(2ndspos(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
   active(2ndsneg(0, Z)) -> mark(rnil)
   active(2ndsneg(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z)))
   active(pi(X)) -> mark(2ndspos(X, from(0)))
   active(plus(0, Y)) -> mark(Y)
   active(plus(s(X), Y)) -> mark(s(plus(X, Y)))
   active(times(0, Y)) -> mark(0)
   active(times(s(X), Y)) -> mark(plus(Y, times(X, Y)))
   active(square(X)) -> mark(times(X, X))
   active(s(X)) -> s(active(X))
   active(posrecip(X)) -> posrecip(active(X))
   active(negrecip(X)) -> negrecip(active(X))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(rcons(X1, X2)) -> rcons(active(X1), X2)
   active(rcons(X1, X2)) -> rcons(X1, active(X2))
   active(from(X)) -> from(active(X))
   active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2)
   active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2))
   active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2)
   active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2))
   active(pi(X)) -> pi(active(X))
   active(plus(X1, X2)) -> plus(active(X1), X2)
   active(plus(X1, X2)) -> plus(X1, active(X2))
   active(times(X1, X2)) -> times(active(X1), X2)
   active(times(X1, X2)) -> times(X1, active(X2))
   active(square(X)) -> square(active(X))
   s(mark(X)) -> mark(s(X))
   posrecip(mark(X)) -> mark(posrecip(X))
   negrecip(mark(X)) -> mark(negrecip(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   rcons(mark(X1), X2) -> mark(rcons(X1, X2))
   rcons(X1, mark(X2)) -> mark(rcons(X1, X2))
   from(mark(X)) -> mark(from(X))
   2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2))
   2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2))
   2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2))
   2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2))
   pi(mark(X)) -> mark(pi(X))
   plus(mark(X1), X2) -> mark(plus(X1, X2))
   plus(X1, mark(X2)) -> mark(plus(X1, X2))
   times(mark(X1), X2) -> mark(times(X1, X2))
   times(X1, mark(X2)) -> mark(times(X1, X2))
   square(mark(X)) -> mark(square(X))
   proper(0) -> ok(0)
   proper(s(X)) -> s(proper(X))
   proper(posrecip(X)) -> posrecip(proper(X))
   proper(negrecip(X)) -> negrecip(proper(X))
   proper(nil) -> ok(nil)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(rnil) -> ok(rnil)
   proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2))
   proper(from(X)) -> from(proper(X))
   proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2))
   proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2))
   proper(pi(X)) -> pi(proper(X))
   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
   proper(times(X1, X2)) -> times(proper(X1), proper(X2))
   proper(square(X)) -> square(proper(X))
   s(ok(X)) -> ok(s(X))
   posrecip(ok(X)) -> ok(posrecip(X))
   negrecip(ok(X)) -> ok(negrecip(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2))
   from(ok(X)) -> ok(from(X))
   2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2))
   2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2))
   pi(ok(X)) -> ok(pi(X))
   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
   times(ok(X1), ok(X2)) -> ok(times(X1, X2))
   square(ok(X)) -> ok(square(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(from(X)) -> mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) -> mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) -> mark(rnil)
active(2ndsneg(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) -> mark(2ndspos(X, from(0)))
active(plus(0, Y)) -> mark(Y)
active(plus(s(X), Y)) -> mark(s(plus(X, Y)))
active(times(0, Y)) -> mark(0)
active(times(s(X), Y)) -> mark(plus(Y, times(X, Y)))
active(square(X)) -> mark(times(X, X))
active(s(X)) -> s(active(X))
active(posrecip(X)) -> posrecip(active(X))
active(negrecip(X)) -> negrecip(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(rcons(X1, X2)) -> rcons(active(X1), X2)
active(rcons(X1, X2)) -> rcons(X1, active(X2))
active(from(X)) -> from(active(X))
active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2))
active(pi(X)) -> pi(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(times(X1, X2)) -> times(active(X1), X2)
active(times(X1, X2)) -> times(X1, active(X2))
active(square(X)) -> square(active(X))
proper(s(X)) -> s(proper(X))
proper(posrecip(X)) -> posrecip(proper(X))
proper(negrecip(X)) -> negrecip(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) -> pi(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(times(X1, X2)) -> times(proper(X1), proper(X2))
proper(square(X)) -> square(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:

   s(mark(X)) -> mark(s(X))
   posrecip(mark(X)) -> mark(posrecip(X))
   negrecip(mark(X)) -> mark(negrecip(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   rcons(mark(X1), X2) -> mark(rcons(X1, X2))
   rcons(X1, mark(X2)) -> mark(rcons(X1, X2))
   from(mark(X)) -> mark(from(X))
   2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2))
   2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2))
   2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2))
   2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2))
   pi(mark(X)) -> mark(pi(X))
   plus(mark(X1), X2) -> mark(plus(X1, X2))
   plus(X1, mark(X2)) -> mark(plus(X1, X2))
   times(mark(X1), X2) -> mark(times(X1, X2))
   times(X1, mark(X2)) -> mark(times(X1, X2))
   square(mark(X)) -> mark(square(X))
   proper(0) -> ok(0)
   proper(nil) -> ok(nil)
   proper(rnil) -> ok(rnil)
   s(ok(X)) -> ok(s(X))
   posrecip(ok(X)) -> ok(posrecip(X))
   negrecip(ok(X)) -> ok(negrecip(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2))
   from(ok(X)) -> ok(from(X))
   2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2))
   2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2))
   pi(ok(X)) -> ok(pi(X))
   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
   times(ok(X1), ok(X2)) -> ok(times(X1, X2))
   square(ok(X)) -> ok(square(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:

   s(mark(X)) -> mark(s(X))
   posrecip(mark(X)) -> mark(posrecip(X))
   negrecip(mark(X)) -> mark(negrecip(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   rcons(mark(X1), X2) -> mark(rcons(X1, X2))
   rcons(X1, mark(X2)) -> mark(rcons(X1, X2))
   from(mark(X)) -> mark(from(X))
   2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2))
   2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2))
   2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2))
   2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2))
   pi(mark(X)) -> mark(pi(X))
   plus(mark(X1), X2) -> mark(plus(X1, X2))
   plus(X1, mark(X2)) -> mark(plus(X1, X2))
   times(mark(X1), X2) -> mark(times(X1, X2))
   times(X1, mark(X2)) -> mark(times(X1, X2))
   square(mark(X)) -> mark(square(X))
   proper(0) -> ok(0)
   proper(nil) -> ok(nil)
   proper(rnil) -> ok(rnil)
   s(ok(X)) -> ok(s(X))
   posrecip(ok(X)) -> ok(posrecip(X))
   negrecip(ok(X)) -> ok(negrecip(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2))
   from(ok(X)) -> ok(from(X))
   2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2))
   2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2))
   pi(ok(X)) -> ok(pi(X))
   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
   times(ok(X1), ok(X2)) -> ok(times(X1, X2))
   square(ok(X)) -> ok(square(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]
transitions: 
mark0(0) -> 0
00() -> 0
ok0(0) -> 0
nil0() -> 0
rnil0() -> 0
active0(0) -> 0
s0(0) -> 1
posrecip0(0) -> 2
negrecip0(0) -> 3
cons0(0, 0) -> 4
rcons0(0, 0) -> 5
from0(0) -> 6
2ndspos0(0, 0) -> 7
2ndsneg0(0, 0) -> 8
pi0(0) -> 9
plus0(0, 0) -> 10
times0(0, 0) -> 11
square0(0) -> 12
proper0(0) -> 13
top0(0) -> 14
s1(0) -> 15
mark1(15) -> 1
posrecip1(0) -> 16
mark1(16) -> 2
negrecip1(0) -> 17
mark1(17) -> 3
cons1(0, 0) -> 18
mark1(18) -> 4
rcons1(0, 0) -> 19
mark1(19) -> 5
from1(0) -> 20
mark1(20) -> 6
2ndspos1(0, 0) -> 21
mark1(21) -> 7
2ndsneg1(0, 0) -> 22
mark1(22) -> 8
pi1(0) -> 23
mark1(23) -> 9
plus1(0, 0) -> 24
mark1(24) -> 10
times1(0, 0) -> 25
mark1(25) -> 11
square1(0) -> 26
mark1(26) -> 12
01() -> 27
ok1(27) -> 13
nil1() -> 28
ok1(28) -> 13
rnil1() -> 29
ok1(29) -> 13
s1(0) -> 30
ok1(30) -> 1
posrecip1(0) -> 31
ok1(31) -> 2
negrecip1(0) -> 32
ok1(32) -> 3
cons1(0, 0) -> 33
ok1(33) -> 4
rcons1(0, 0) -> 34
ok1(34) -> 5
from1(0) -> 35
ok1(35) -> 6
2ndspos1(0, 0) -> 36
ok1(36) -> 7
2ndsneg1(0, 0) -> 37
ok1(37) -> 8
pi1(0) -> 38
ok1(38) -> 9
plus1(0, 0) -> 39
ok1(39) -> 10
times1(0, 0) -> 40
ok1(40) -> 11
square1(0) -> 41
ok1(41) -> 12
proper1(0) -> 42
top1(42) -> 14
active1(0) -> 43
top1(43) -> 14
mark1(15) -> 15
mark1(15) -> 30
mark1(16) -> 16
mark1(16) -> 31
mark1(17) -> 17
mark1(17) -> 32
mark1(18) -> 18
mark1(18) -> 33
mark1(19) -> 19
mark1(19) -> 34
mark1(20) -> 20
mark1(20) -> 35
mark1(21) -> 21
mark1(21) -> 36
mark1(22) -> 22
mark1(22) -> 37
mark1(23) -> 23
mark1(23) -> 38
mark1(24) -> 24
mark1(24) -> 39
mark1(25) -> 25
mark1(25) -> 40
mark1(26) -> 26
mark1(26) -> 41
ok1(27) -> 42
ok1(28) -> 42
ok1(29) -> 42
ok1(30) -> 15
ok1(30) -> 30
ok1(31) -> 16
ok1(31) -> 31
ok1(32) -> 17
ok1(32) -> 32
ok1(33) -> 18
ok1(33) -> 33
ok1(34) -> 19
ok1(34) -> 34
ok1(35) -> 20
ok1(35) -> 35
ok1(36) -> 21
ok1(36) -> 36
ok1(37) -> 22
ok1(37) -> 37
ok1(38) -> 23
ok1(38) -> 38
ok1(39) -> 24
ok1(39) -> 39
ok1(40) -> 25
ok1(40) -> 40
ok1(41) -> 26
ok1(41) -> 41
active2(27) -> 44
top2(44) -> 14
active2(28) -> 44
active2(29) -> 44

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

(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(from(X)) -> mark(cons(X, from(s(X))))
   active(2ndspos(0, Z)) -> mark(rnil)
   active(2ndspos(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
   active(2ndsneg(0, Z)) -> mark(rnil)
   active(2ndsneg(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z)))
   active(pi(X)) -> mark(2ndspos(X, from(0)))
   active(plus(0, Y)) -> mark(Y)
   active(plus(s(X), Y)) -> mark(s(plus(X, Y)))
   active(times(0, Y)) -> mark(0)
   active(times(s(X), Y)) -> mark(plus(Y, times(X, Y)))
   active(square(X)) -> mark(times(X, X))
   active(s(X)) -> s(active(X))
   active(posrecip(X)) -> posrecip(active(X))
   active(negrecip(X)) -> negrecip(active(X))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(rcons(X1, X2)) -> rcons(active(X1), X2)
   active(rcons(X1, X2)) -> rcons(X1, active(X2))
   active(from(X)) -> from(active(X))
   active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2)
   active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2))
   active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2)
   active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2))
   active(pi(X)) -> pi(active(X))
   active(plus(X1, X2)) -> plus(active(X1), X2)
   active(plus(X1, X2)) -> plus(X1, active(X2))
   active(times(X1, X2)) -> times(active(X1), X2)
   active(times(X1, X2)) -> times(X1, active(X2))
   active(square(X)) -> square(active(X))
   s(mark(X)) -> mark(s(X))
   posrecip(mark(X)) -> mark(posrecip(X))
   negrecip(mark(X)) -> mark(negrecip(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   rcons(mark(X1), X2) -> mark(rcons(X1, X2))
   rcons(X1, mark(X2)) -> mark(rcons(X1, X2))
   from(mark(X)) -> mark(from(X))
   2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2))
   2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2))
   2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2))
   2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2))
   pi(mark(X)) -> mark(pi(X))
   plus(mark(X1), X2) -> mark(plus(X1, X2))
   plus(X1, mark(X2)) -> mark(plus(X1, X2))
   times(mark(X1), X2) -> mark(times(X1, X2))
   times(X1, mark(X2)) -> mark(times(X1, X2))
   square(mark(X)) -> mark(square(X))
   proper(0) -> ok(0)
   proper(s(X)) -> s(proper(X))
   proper(posrecip(X)) -> posrecip(proper(X))
   proper(negrecip(X)) -> negrecip(proper(X))
   proper(nil) -> ok(nil)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(rnil) -> ok(rnil)
   proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2))
   proper(from(X)) -> from(proper(X))
   proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2))
   proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2))
   proper(pi(X)) -> pi(proper(X))
   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
   proper(times(X1, X2)) -> times(proper(X1), proper(X2))
   proper(square(X)) -> square(proper(X))
   s(ok(X)) -> ok(s(X))
   posrecip(ok(X)) -> ok(posrecip(X))
   negrecip(ok(X)) -> ok(negrecip(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2))
   from(ok(X)) -> ok(from(X))
   2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2))
   2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2))
   pi(ok(X)) -> ok(pi(X))
   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
   times(ok(X1), ok(X2)) -> ok(times(X1, X2))
   square(ok(X)) -> ok(square(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

times(ok(X1), ok(X2)) ->^+ ok(times(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(from(X)) -> mark(cons(X, from(s(X))))
   active(2ndspos(0, Z)) -> mark(rnil)
   active(2ndspos(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
   active(2ndsneg(0, Z)) -> mark(rnil)
   active(2ndsneg(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z)))
   active(pi(X)) -> mark(2ndspos(X, from(0)))
   active(plus(0, Y)) -> mark(Y)
   active(plus(s(X), Y)) -> mark(s(plus(X, Y)))
   active(times(0, Y)) -> mark(0)
   active(times(s(X), Y)) -> mark(plus(Y, times(X, Y)))
   active(square(X)) -> mark(times(X, X))
   active(s(X)) -> s(active(X))
   active(posrecip(X)) -> posrecip(active(X))
   active(negrecip(X)) -> negrecip(active(X))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(rcons(X1, X2)) -> rcons(active(X1), X2)
   active(rcons(X1, X2)) -> rcons(X1, active(X2))
   active(from(X)) -> from(active(X))
   active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2)
   active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2))
   active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2)
   active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2))
   active(pi(X)) -> pi(active(X))
   active(plus(X1, X2)) -> plus(active(X1), X2)
   active(plus(X1, X2)) -> plus(X1, active(X2))
   active(times(X1, X2)) -> times(active(X1), X2)
   active(times(X1, X2)) -> times(X1, active(X2))
   active(square(X)) -> square(active(X))
   s(mark(X)) -> mark(s(X))
   posrecip(mark(X)) -> mark(posrecip(X))
   negrecip(mark(X)) -> mark(negrecip(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   rcons(mark(X1), X2) -> mark(rcons(X1, X2))
   rcons(X1, mark(X2)) -> mark(rcons(X1, X2))
   from(mark(X)) -> mark(from(X))
   2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2))
   2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2))
   2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2))
   2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2))
   pi(mark(X)) -> mark(pi(X))
   plus(mark(X1), X2) -> mark(plus(X1, X2))
   plus(X1, mark(X2)) -> mark(plus(X1, X2))
   times(mark(X1), X2) -> mark(times(X1, X2))
   times(X1, mark(X2)) -> mark(times(X1, X2))
   square(mark(X)) -> mark(square(X))
   proper(0) -> ok(0)
   proper(s(X)) -> s(proper(X))
   proper(posrecip(X)) -> posrecip(proper(X))
   proper(negrecip(X)) -> negrecip(proper(X))
   proper(nil) -> ok(nil)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(rnil) -> ok(rnil)
   proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2))
   proper(from(X)) -> from(proper(X))
   proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2))
   proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2))
   proper(pi(X)) -> pi(proper(X))
   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
   proper(times(X1, X2)) -> times(proper(X1), proper(X2))
   proper(square(X)) -> square(proper(X))
   s(ok(X)) -> ok(s(X))
   posrecip(ok(X)) -> ok(posrecip(X))
   negrecip(ok(X)) -> ok(negrecip(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2))
   from(ok(X)) -> ok(from(X))
   2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2))
   2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2))
   pi(ok(X)) -> ok(pi(X))
   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
   times(ok(X1), ok(X2)) -> ok(times(X1, X2))
   square(ok(X)) -> ok(square(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(from(X)) -> mark(cons(X, from(s(X))))
   active(2ndspos(0, Z)) -> mark(rnil)
   active(2ndspos(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
   active(2ndsneg(0, Z)) -> mark(rnil)
   active(2ndsneg(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z)))
   active(pi(X)) -> mark(2ndspos(X, from(0)))
   active(plus(0, Y)) -> mark(Y)
   active(plus(s(X), Y)) -> mark(s(plus(X, Y)))
   active(times(0, Y)) -> mark(0)
   active(times(s(X), Y)) -> mark(plus(Y, times(X, Y)))
   active(square(X)) -> mark(times(X, X))
   active(s(X)) -> s(active(X))
   active(posrecip(X)) -> posrecip(active(X))
   active(negrecip(X)) -> negrecip(active(X))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(rcons(X1, X2)) -> rcons(active(X1), X2)
   active(rcons(X1, X2)) -> rcons(X1, active(X2))
   active(from(X)) -> from(active(X))
   active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2)
   active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2))
   active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2)
   active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2))
   active(pi(X)) -> pi(active(X))
   active(plus(X1, X2)) -> plus(active(X1), X2)
   active(plus(X1, X2)) -> plus(X1, active(X2))
   active(times(X1, X2)) -> times(active(X1), X2)
   active(times(X1, X2)) -> times(X1, active(X2))
   active(square(X)) -> square(active(X))
   s(mark(X)) -> mark(s(X))
   posrecip(mark(X)) -> mark(posrecip(X))
   negrecip(mark(X)) -> mark(negrecip(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   rcons(mark(X1), X2) -> mark(rcons(X1, X2))
   rcons(X1, mark(X2)) -> mark(rcons(X1, X2))
   from(mark(X)) -> mark(from(X))
   2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2))
   2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2))
   2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2))
   2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2))
   pi(mark(X)) -> mark(pi(X))
   plus(mark(X1), X2) -> mark(plus(X1, X2))
   plus(X1, mark(X2)) -> mark(plus(X1, X2))
   times(mark(X1), X2) -> mark(times(X1, X2))
   times(X1, mark(X2)) -> mark(times(X1, X2))
   square(mark(X)) -> mark(square(X))
   proper(0) -> ok(0)
   proper(s(X)) -> s(proper(X))
   proper(posrecip(X)) -> posrecip(proper(X))
   proper(negrecip(X)) -> negrecip(proper(X))
   proper(nil) -> ok(nil)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(rnil) -> ok(rnil)
   proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2))
   proper(from(X)) -> from(proper(X))
   proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2))
   proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2))
   proper(pi(X)) -> pi(proper(X))
   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
   proper(times(X1, X2)) -> times(proper(X1), proper(X2))
   proper(square(X)) -> square(proper(X))
   s(ok(X)) -> ok(s(X))
   posrecip(ok(X)) -> ok(posrecip(X))
   negrecip(ok(X)) -> ok(negrecip(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2))
   from(ok(X)) -> ok(from(X))
   2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2))
   2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2))
   pi(ok(X)) -> ok(pi(X))
   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
   times(ok(X1), ok(X2)) -> ok(times(X1, X2))
   square(ok(X)) -> ok(square(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL