/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) CpxTrsMatchBoundsTAProof [FINISHED, 98 ms]
    (4) BOUNDS(1, n^1)
(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(6) TRS for Loop Detection
    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (8) BEST
        (9) proven lower bound
            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
            (11) BOUNDS(n^1, INF)
        (12) 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:

   a__zeros -> cons(0, zeros)
   a__tail(cons(X, XS)) -> mark(XS)
   mark(zeros) -> a__zeros
   mark(tail(X)) -> a__tail(mark(X))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(0) -> 0
   a__zeros -> zeros
   a__tail(X) -> tail(X)

S is empty.
Rewrite Strategy: FULL
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(1) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
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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:

   a__zeros -> cons(0, zeros)
   a__tail(cons(X, XS)) -> mark(XS)
   mark(zeros) -> a__zeros
   mark(tail(X)) -> a__tail(mark(X))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(0) -> 0
   a__zeros -> zeros
   a__tail(X) -> tail(X)

S is empty.
Rewrite Strategy: FULL
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(3) 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 4. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2, 3]
transitions: 
cons0(0, 0) -> 0
00() -> 0
zeros0() -> 0
tail0(0) -> 0
a__zeros0() -> 1
a__tail0(0) -> 2
mark0(0) -> 3
01() -> 4
zeros1() -> 5
cons1(4, 5) -> 1
mark1(0) -> 2
a__zeros1() -> 3
mark1(0) -> 6
a__tail1(6) -> 3
mark1(0) -> 7
cons1(7, 0) -> 3
01() -> 3
zeros1() -> 1
tail1(0) -> 2
02() -> 8
zeros2() -> 9
cons2(8, 9) -> 3
a__zeros1() -> 2
a__zeros1() -> 6
a__zeros1() -> 7
a__tail1(6) -> 2
a__tail1(6) -> 6
a__tail1(6) -> 7
cons1(7, 0) -> 2
cons1(7, 0) -> 6
cons1(7, 0) -> 7
01() -> 2
01() -> 6
01() -> 7
zeros2() -> 3
tail2(6) -> 3
cons2(8, 9) -> 2
cons2(8, 9) -> 6
cons2(8, 9) -> 7
mark2(0) -> 2
mark2(0) -> 3
mark2(0) -> 6
mark2(0) -> 7
zeros2() -> 2
zeros2() -> 6
zeros2() -> 7
tail2(6) -> 2
tail2(6) -> 6
tail2(6) -> 7
mark2(9) -> 2
mark2(9) -> 3
mark2(9) -> 6
mark2(9) -> 7
a__zeros3() -> 2
a__zeros3() -> 3
a__zeros3() -> 6
a__zeros3() -> 7
04() -> 10
zeros4() -> 11
cons4(10, 11) -> 2
cons4(10, 11) -> 3
cons4(10, 11) -> 6
cons4(10, 11) -> 7
zeros4() -> 2
zeros4() -> 3
zeros4() -> 6
zeros4() -> 7
mark2(11) -> 2
mark2(11) -> 3
mark2(11) -> 6
mark2(11) -> 7

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(4)
BOUNDS(1, n^1)

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(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(6)
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:

   a__zeros -> cons(0, zeros)
   a__tail(cons(X, XS)) -> mark(XS)
   mark(zeros) -> a__zeros
   mark(tail(X)) -> a__tail(mark(X))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(0) -> 0
   a__zeros -> zeros
   a__tail(X) -> tail(X)

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

The rewrite sequence

mark(tail(X)) ->^+ a__tail(mark(X))

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

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

The result substitution is [ ].




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(8)
Complex Obligation (BEST)

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(9)
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:

   a__zeros -> cons(0, zeros)
   a__tail(cons(X, XS)) -> mark(XS)
   mark(zeros) -> a__zeros
   mark(tail(X)) -> a__tail(mark(X))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(0) -> 0
   a__zeros -> zeros
   a__tail(X) -> tail(X)

S is empty.
Rewrite Strategy: FULL
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(10) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(11)
BOUNDS(n^1, INF)

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(12)
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:

   a__zeros -> cons(0, zeros)
   a__tail(cons(X, XS)) -> mark(XS)
   mark(zeros) -> a__zeros
   mark(tail(X)) -> a__tail(mark(X))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(0) -> 0
   a__zeros -> zeros
   a__tail(X) -> tail(X)

S is empty.
Rewrite Strategy: FULL