/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 751 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


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(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(zeros) -> mark(cons(0, zeros))
   active(and(tt, X)) -> mark(X)
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(s(length(L)))
   active(take(0, IL)) -> mark(nil)
   active(take(s(M), cons(N, IL))) -> mark(cons(N, take(M, IL)))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(and(X1, X2)) -> and(active(X1), X2)
   active(length(X)) -> length(active(X))
   active(s(X)) -> s(active(X))
   active(take(X1, X2)) -> take(active(X1), X2)
   active(take(X1, X2)) -> take(X1, active(X2))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   and(mark(X1), X2) -> mark(and(X1, X2))
   length(mark(X)) -> mark(length(X))
   s(mark(X)) -> mark(s(X))
   take(mark(X1), X2) -> mark(take(X1, X2))
   take(X1, mark(X2)) -> mark(take(X1, X2))
   proper(zeros) -> ok(zeros)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(tt) -> ok(tt)
   proper(length(X)) -> length(proper(X))
   proper(nil) -> ok(nil)
   proper(s(X)) -> s(proper(X))
   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   length(ok(X)) -> ok(length(X))
   s(ok(X)) -> ok(s(X))
   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
   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 cons: active, proper, cons
The following defined symbols can occur below the 1th argument of cons: active, proper, cons
The following defined symbols can occur below the 0th argument of top: active, proper, cons
The following defined symbols can occur below the 0th argument of proper: active, proper, cons
The following defined symbols can occur below the 0th argument of active: active, proper, cons

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
active(and(tt, X)) -> mark(X)
active(length(nil)) -> mark(0)
active(length(cons(N, L))) -> mark(s(length(L)))
active(take(0, IL)) -> mark(nil)
active(take(s(M), cons(N, IL))) -> mark(cons(N, take(M, IL)))
active(and(X1, X2)) -> and(active(X1), X2)
active(length(X)) -> length(active(X))
active(s(X)) -> s(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(length(X)) -> length(proper(X))
proper(s(X)) -> s(proper(X))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))

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

(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:

   active(zeros) -> mark(cons(0, zeros))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   and(mark(X1), X2) -> mark(and(X1, X2))
   length(mark(X)) -> mark(length(X))
   s(mark(X)) -> mark(s(X))
   take(mark(X1), X2) -> mark(take(X1, X2))
   take(X1, mark(X2)) -> mark(take(X1, X2))
   proper(zeros) -> ok(zeros)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(tt) -> ok(tt)
   proper(nil) -> ok(nil)
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   length(ok(X)) -> ok(length(X))
   s(ok(X)) -> ok(s(X))
   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
   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:

   active(zeros) -> mark(cons(0, zeros))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   and(mark(X1), X2) -> mark(and(X1, X2))
   length(mark(X)) -> mark(length(X))
   s(mark(X)) -> mark(s(X))
   take(mark(X1), X2) -> mark(take(X1, X2))
   take(X1, mark(X2)) -> mark(take(X1, X2))
   proper(zeros) -> ok(zeros)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(tt) -> ok(tt)
   proper(nil) -> ok(nil)
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   length(ok(X)) -> ok(length(X))
   s(ok(X)) -> ok(s(X))
   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL
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(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 5. 

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]
transitions: 
zeros0() -> 0
mark0(0) -> 0
00() -> 0
ok0(0) -> 0
tt0() -> 0
nil0() -> 0
active0(0) -> 1
cons0(0, 0) -> 2
and0(0, 0) -> 3
length0(0) -> 4
s0(0) -> 5
take0(0, 0) -> 6
proper0(0) -> 7
top0(0) -> 8
01() -> 10
zeros1() -> 11
cons1(10, 11) -> 9
mark1(9) -> 1
cons1(0, 0) -> 12
mark1(12) -> 2
and1(0, 0) -> 13
mark1(13) -> 3
length1(0) -> 14
mark1(14) -> 4
s1(0) -> 15
mark1(15) -> 5
take1(0, 0) -> 16
mark1(16) -> 6
zeros1() -> 17
ok1(17) -> 7
01() -> 18
ok1(18) -> 7
tt1() -> 19
ok1(19) -> 7
nil1() -> 20
ok1(20) -> 7
cons1(0, 0) -> 21
ok1(21) -> 2
and1(0, 0) -> 22
ok1(22) -> 3
length1(0) -> 23
ok1(23) -> 4
s1(0) -> 24
ok1(24) -> 5
take1(0, 0) -> 25
ok1(25) -> 6
proper1(0) -> 26
top1(26) -> 8
active1(0) -> 27
top1(27) -> 8
mark1(9) -> 27
mark1(12) -> 12
mark1(12) -> 21
mark1(13) -> 13
mark1(13) -> 22
mark1(14) -> 14
mark1(14) -> 23
mark1(15) -> 15
mark1(15) -> 24
mark1(16) -> 16
mark1(16) -> 25
ok1(17) -> 26
ok1(18) -> 26
ok1(19) -> 26
ok1(20) -> 26
ok1(21) -> 12
ok1(21) -> 21
ok1(22) -> 13
ok1(22) -> 22
ok1(23) -> 14
ok1(23) -> 23
ok1(24) -> 15
ok1(24) -> 24
ok1(25) -> 16
ok1(25) -> 25
proper2(9) -> 28
top2(28) -> 8
active2(17) -> 29
top2(29) -> 8
active2(18) -> 29
active2(19) -> 29
active2(20) -> 29
02() -> 31
zeros2() -> 32
cons2(31, 32) -> 30
mark2(30) -> 29
proper2(10) -> 33
proper2(11) -> 34
cons2(33, 34) -> 28
zeros2() -> 35
ok2(35) -> 34
02() -> 36
ok2(36) -> 33
proper3(30) -> 37
top3(37) -> 8
proper3(31) -> 38
proper3(32) -> 39
cons3(38, 39) -> 37
cons3(36, 35) -> 40
ok3(40) -> 28
zeros3() -> 41
ok3(41) -> 39
03() -> 42
ok3(42) -> 38
active3(40) -> 43
top3(43) -> 8
cons4(42, 41) -> 44
ok4(44) -> 37
active4(36) -> 45
cons4(45, 35) -> 43
active4(44) -> 46
top4(46) -> 8
active5(42) -> 47
cons5(47, 41) -> 46

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(6)
BOUNDS(1, n^1)

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(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(zeros) -> mark(cons(0, zeros))
   active(and(tt, X)) -> mark(X)
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(s(length(L)))
   active(take(0, IL)) -> mark(nil)
   active(take(s(M), cons(N, IL))) -> mark(cons(N, take(M, IL)))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(and(X1, X2)) -> and(active(X1), X2)
   active(length(X)) -> length(active(X))
   active(s(X)) -> s(active(X))
   active(take(X1, X2)) -> take(active(X1), X2)
   active(take(X1, X2)) -> take(X1, active(X2))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   and(mark(X1), X2) -> mark(and(X1, X2))
   length(mark(X)) -> mark(length(X))
   s(mark(X)) -> mark(s(X))
   take(mark(X1), X2) -> mark(take(X1, X2))
   take(X1, mark(X2)) -> mark(take(X1, X2))
   proper(zeros) -> ok(zeros)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(tt) -> ok(tt)
   proper(length(X)) -> length(proper(X))
   proper(nil) -> ok(nil)
   proper(s(X)) -> s(proper(X))
   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   length(ok(X)) -> ok(length(X))
   s(ok(X)) -> ok(s(X))
   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
   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

take(ok(X1), ok(X2)) ->^+ ok(take(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 [ ].




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(10)
Complex Obligation (BEST)

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(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(zeros) -> mark(cons(0, zeros))
   active(and(tt, X)) -> mark(X)
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(s(length(L)))
   active(take(0, IL)) -> mark(nil)
   active(take(s(M), cons(N, IL))) -> mark(cons(N, take(M, IL)))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(and(X1, X2)) -> and(active(X1), X2)
   active(length(X)) -> length(active(X))
   active(s(X)) -> s(active(X))
   active(take(X1, X2)) -> take(active(X1), X2)
   active(take(X1, X2)) -> take(X1, active(X2))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   and(mark(X1), X2) -> mark(and(X1, X2))
   length(mark(X)) -> mark(length(X))
   s(mark(X)) -> mark(s(X))
   take(mark(X1), X2) -> mark(take(X1, X2))
   take(X1, mark(X2)) -> mark(take(X1, X2))
   proper(zeros) -> ok(zeros)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(tt) -> ok(tt)
   proper(length(X)) -> length(proper(X))
   proper(nil) -> ok(nil)
   proper(s(X)) -> s(proper(X))
   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   length(ok(X)) -> ok(length(X))
   s(ok(X)) -> ok(s(X))
   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL
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(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(zeros) -> mark(cons(0, zeros))
   active(and(tt, X)) -> mark(X)
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(s(length(L)))
   active(take(0, IL)) -> mark(nil)
   active(take(s(M), cons(N, IL))) -> mark(cons(N, take(M, IL)))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(and(X1, X2)) -> and(active(X1), X2)
   active(length(X)) -> length(active(X))
   active(s(X)) -> s(active(X))
   active(take(X1, X2)) -> take(active(X1), X2)
   active(take(X1, X2)) -> take(X1, active(X2))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   and(mark(X1), X2) -> mark(and(X1, X2))
   length(mark(X)) -> mark(length(X))
   s(mark(X)) -> mark(s(X))
   take(mark(X1), X2) -> mark(take(X1, X2))
   take(X1, mark(X2)) -> mark(take(X1, X2))
   proper(zeros) -> ok(zeros)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(tt) -> ok(tt)
   proper(length(X)) -> length(proper(X))
   proper(nil) -> ok(nil)
   proper(s(X)) -> s(proper(X))
   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   length(ok(X)) -> ok(length(X))
   s(ok(X)) -> ok(s(X))
   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL