/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 65 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(eq(0, 0)) -> mark(true)
   active(eq(s(X), s(Y))) -> mark(eq(X, Y))
   active(eq(X, Y)) -> mark(false)
   active(inf(X)) -> mark(cons(X, inf(s(X))))
   active(take(0, X)) -> mark(nil)
   active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L)))
   active(length(nil)) -> mark(0)
   active(length(cons(X, L))) -> mark(s(length(L)))
   active(inf(X)) -> inf(active(X))
   active(take(X1, X2)) -> take(active(X1), X2)
   active(take(X1, X2)) -> take(X1, active(X2))
   active(length(X)) -> length(active(X))
   inf(mark(X)) -> mark(inf(X))
   take(mark(X1), X2) -> mark(take(X1, X2))
   take(X1, mark(X2)) -> mark(take(X1, X2))
   length(mark(X)) -> mark(length(X))
   proper(eq(X1, X2)) -> eq(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(true) -> ok(true)
   proper(s(X)) -> s(proper(X))
   proper(false) -> ok(false)
   proper(inf(X)) -> inf(proper(X))
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
   proper(nil) -> ok(nil)
   proper(length(X)) -> length(proper(X))
   eq(ok(X1), ok(X2)) -> ok(eq(X1, X2))
   s(ok(X)) -> ok(s(X))
   inf(ok(X)) -> ok(inf(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
   length(ok(X)) -> ok(length(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 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(eq(0, 0)) -> mark(true)
active(eq(s(X), s(Y))) -> mark(eq(X, Y))
active(eq(X, Y)) -> mark(false)
active(inf(X)) -> mark(cons(X, inf(s(X))))
active(take(0, X)) -> mark(nil)
active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L)))
active(length(nil)) -> mark(0)
active(length(cons(X, L))) -> mark(s(length(L)))
active(inf(X)) -> inf(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(length(X)) -> length(active(X))
proper(eq(X1, X2)) -> eq(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(inf(X)) -> inf(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(length(X)) -> length(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:

   inf(mark(X)) -> mark(inf(X))
   take(mark(X1), X2) -> mark(take(X1, X2))
   take(X1, mark(X2)) -> mark(take(X1, X2))
   length(mark(X)) -> mark(length(X))
   proper(0) -> ok(0)
   proper(true) -> ok(true)
   proper(false) -> ok(false)
   proper(nil) -> ok(nil)
   eq(ok(X1), ok(X2)) -> ok(eq(X1, X2))
   s(ok(X)) -> ok(s(X))
   inf(ok(X)) -> ok(inf(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
   length(ok(X)) -> ok(length(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:

   inf(mark(X)) -> mark(inf(X))
   take(mark(X1), X2) -> mark(take(X1, X2))
   take(X1, mark(X2)) -> mark(take(X1, X2))
   length(mark(X)) -> mark(length(X))
   proper(0) -> ok(0)
   proper(true) -> ok(true)
   proper(false) -> ok(false)
   proper(nil) -> ok(nil)
   eq(ok(X1), ok(X2)) -> ok(eq(X1, X2))
   s(ok(X)) -> ok(s(X))
   inf(ok(X)) -> ok(inf(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
   length(ok(X)) -> ok(length(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 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]
transitions: 
mark0(0) -> 0
00() -> 0
ok0(0) -> 0
true0() -> 0
false0() -> 0
nil0() -> 0
active0(0) -> 0
inf0(0) -> 1
take0(0, 0) -> 2
length0(0) -> 3
proper0(0) -> 4
eq0(0, 0) -> 5
s0(0) -> 6
cons0(0, 0) -> 7
top0(0) -> 8
inf1(0) -> 9
mark1(9) -> 1
take1(0, 0) -> 10
mark1(10) -> 2
length1(0) -> 11
mark1(11) -> 3
01() -> 12
ok1(12) -> 4
true1() -> 13
ok1(13) -> 4
false1() -> 14
ok1(14) -> 4
nil1() -> 15
ok1(15) -> 4
eq1(0, 0) -> 16
ok1(16) -> 5
s1(0) -> 17
ok1(17) -> 6
inf1(0) -> 18
ok1(18) -> 1
cons1(0, 0) -> 19
ok1(19) -> 7
take1(0, 0) -> 20
ok1(20) -> 2
length1(0) -> 21
ok1(21) -> 3
proper1(0) -> 22
top1(22) -> 8
active1(0) -> 23
top1(23) -> 8
mark1(9) -> 9
mark1(9) -> 18
mark1(10) -> 10
mark1(10) -> 20
mark1(11) -> 11
mark1(11) -> 21
ok1(12) -> 22
ok1(13) -> 22
ok1(14) -> 22
ok1(15) -> 22
ok1(16) -> 16
ok1(17) -> 17
ok1(18) -> 9
ok1(18) -> 18
ok1(19) -> 19
ok1(20) -> 10
ok1(20) -> 20
ok1(21) -> 11
ok1(21) -> 21
active2(12) -> 24
top2(24) -> 8
active2(13) -> 24
active2(14) -> 24
active2(15) -> 24

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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(eq(0, 0)) -> mark(true)
   active(eq(s(X), s(Y))) -> mark(eq(X, Y))
   active(eq(X, Y)) -> mark(false)
   active(inf(X)) -> mark(cons(X, inf(s(X))))
   active(take(0, X)) -> mark(nil)
   active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L)))
   active(length(nil)) -> mark(0)
   active(length(cons(X, L))) -> mark(s(length(L)))
   active(inf(X)) -> inf(active(X))
   active(take(X1, X2)) -> take(active(X1), X2)
   active(take(X1, X2)) -> take(X1, active(X2))
   active(length(X)) -> length(active(X))
   inf(mark(X)) -> mark(inf(X))
   take(mark(X1), X2) -> mark(take(X1, X2))
   take(X1, mark(X2)) -> mark(take(X1, X2))
   length(mark(X)) -> mark(length(X))
   proper(eq(X1, X2)) -> eq(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(true) -> ok(true)
   proper(s(X)) -> s(proper(X))
   proper(false) -> ok(false)
   proper(inf(X)) -> inf(proper(X))
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
   proper(nil) -> ok(nil)
   proper(length(X)) -> length(proper(X))
   eq(ok(X1), ok(X2)) -> ok(eq(X1, X2))
   s(ok(X)) -> ok(s(X))
   inf(ok(X)) -> ok(inf(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
   length(ok(X)) -> ok(length(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

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(eq(0, 0)) -> mark(true)
   active(eq(s(X), s(Y))) -> mark(eq(X, Y))
   active(eq(X, Y)) -> mark(false)
   active(inf(X)) -> mark(cons(X, inf(s(X))))
   active(take(0, X)) -> mark(nil)
   active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L)))
   active(length(nil)) -> mark(0)
   active(length(cons(X, L))) -> mark(s(length(L)))
   active(inf(X)) -> inf(active(X))
   active(take(X1, X2)) -> take(active(X1), X2)
   active(take(X1, X2)) -> take(X1, active(X2))
   active(length(X)) -> length(active(X))
   inf(mark(X)) -> mark(inf(X))
   take(mark(X1), X2) -> mark(take(X1, X2))
   take(X1, mark(X2)) -> mark(take(X1, X2))
   length(mark(X)) -> mark(length(X))
   proper(eq(X1, X2)) -> eq(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(true) -> ok(true)
   proper(s(X)) -> s(proper(X))
   proper(false) -> ok(false)
   proper(inf(X)) -> inf(proper(X))
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
   proper(nil) -> ok(nil)
   proper(length(X)) -> length(proper(X))
   eq(ok(X1), ok(X2)) -> ok(eq(X1, X2))
   s(ok(X)) -> ok(s(X))
   inf(ok(X)) -> ok(inf(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
   length(ok(X)) -> ok(length(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(eq(0, 0)) -> mark(true)
   active(eq(s(X), s(Y))) -> mark(eq(X, Y))
   active(eq(X, Y)) -> mark(false)
   active(inf(X)) -> mark(cons(X, inf(s(X))))
   active(take(0, X)) -> mark(nil)
   active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L)))
   active(length(nil)) -> mark(0)
   active(length(cons(X, L))) -> mark(s(length(L)))
   active(inf(X)) -> inf(active(X))
   active(take(X1, X2)) -> take(active(X1), X2)
   active(take(X1, X2)) -> take(X1, active(X2))
   active(length(X)) -> length(active(X))
   inf(mark(X)) -> mark(inf(X))
   take(mark(X1), X2) -> mark(take(X1, X2))
   take(X1, mark(X2)) -> mark(take(X1, X2))
   length(mark(X)) -> mark(length(X))
   proper(eq(X1, X2)) -> eq(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(true) -> ok(true)
   proper(s(X)) -> s(proper(X))
   proper(false) -> ok(false)
   proper(inf(X)) -> inf(proper(X))
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
   proper(nil) -> ok(nil)
   proper(length(X)) -> length(proper(X))
   eq(ok(X1), ok(X2)) -> ok(eq(X1, X2))
   s(ok(X)) -> ok(s(X))
   inf(ok(X)) -> ok(inf(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
   length(ok(X)) -> ok(length(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL