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


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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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 605 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(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))
   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))
   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))
   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))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL
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(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(and(X1, X2)) -> and(active(X1), X2)
active(length(X)) -> length(active(X))
active(s(X)) -> s(active(X))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(length(X)) -> length(proper(X))
proper(s(X)) -> s(proper(X))

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(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))
   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))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL
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(3) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
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(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))
   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))
   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]
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
proper0(0) -> 6
top0(0) -> 7
01() -> 9
zeros1() -> 10
cons1(9, 10) -> 8
mark1(8) -> 1
cons1(0, 0) -> 11
mark1(11) -> 2
and1(0, 0) -> 12
mark1(12) -> 3
length1(0) -> 13
mark1(13) -> 4
s1(0) -> 14
mark1(14) -> 5
zeros1() -> 15
ok1(15) -> 6
01() -> 16
ok1(16) -> 6
tt1() -> 17
ok1(17) -> 6
nil1() -> 18
ok1(18) -> 6
cons1(0, 0) -> 19
ok1(19) -> 2
and1(0, 0) -> 20
ok1(20) -> 3
length1(0) -> 21
ok1(21) -> 4
s1(0) -> 22
ok1(22) -> 5
proper1(0) -> 23
top1(23) -> 7
active1(0) -> 24
top1(24) -> 7
mark1(8) -> 24
mark1(11) -> 11
mark1(11) -> 19
mark1(12) -> 12
mark1(12) -> 20
mark1(13) -> 13
mark1(13) -> 21
mark1(14) -> 14
mark1(14) -> 22
ok1(15) -> 23
ok1(16) -> 23
ok1(17) -> 23
ok1(18) -> 23
ok1(19) -> 11
ok1(19) -> 19
ok1(20) -> 12
ok1(20) -> 20
ok1(21) -> 13
ok1(21) -> 21
ok1(22) -> 14
ok1(22) -> 22
proper2(8) -> 25
top2(25) -> 7
active2(15) -> 26
top2(26) -> 7
active2(16) -> 26
active2(17) -> 26
active2(18) -> 26
02() -> 28
zeros2() -> 29
cons2(28, 29) -> 27
mark2(27) -> 26
proper2(9) -> 30
proper2(10) -> 31
cons2(30, 31) -> 25
zeros2() -> 32
ok2(32) -> 31
02() -> 33
ok2(33) -> 30
proper3(27) -> 34
top3(34) -> 7
proper3(28) -> 35
proper3(29) -> 36
cons3(35, 36) -> 34
cons3(33, 32) -> 37
ok3(37) -> 25
zeros3() -> 38
ok3(38) -> 36
03() -> 39
ok3(39) -> 35
active3(37) -> 40
top3(40) -> 7
cons4(39, 38) -> 41
ok4(41) -> 34
active4(33) -> 42
cons4(42, 32) -> 40
active4(41) -> 43
top4(43) -> 7
active5(39) -> 44
cons5(44, 38) -> 43

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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.
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(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(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))
   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))
   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))
   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))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL
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(9) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

s(mark(X)) ->^+ mark(s(X))

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

The pumping substitution is [X / mark(X)].

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(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))
   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))
   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))
   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))
   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.
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(13)
BOUNDS(n^1, INF)

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(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(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))
   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))
   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))
   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))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL