/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) DependencyGraphProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) NestedDefinedSymbolProof [UPPER BOUND(ID), 3 ms]
    (4) CpxTRS
    (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
    (6) CpxTRS
    (7) CpxTrsMatchBoundsProof [FINISHED, 0 ms]
    (8) BOUNDS(1, n^1)
(9) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(10) TRS for Loop Detection
    (11) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (12) BEST
        (13) proven lower bound
            (14) LowerBoundPropagationProof [FINISHED, 0 ms]
            (15) BOUNDS(n^1, INF)
        (16) 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:

   rec(rec(x)) -> sent(rec(x))
   rec(sent(x)) -> sent(rec(x))
   rec(no(x)) -> sent(rec(x))
   rec(bot) -> up(sent(bot))
   rec(up(x)) -> up(rec(x))
   sent(up(x)) -> up(sent(x))
   no(up(x)) -> up(no(x))
   top(rec(up(x))) -> top(check(rec(x)))
   top(sent(up(x))) -> top(check(rec(x)))
   top(no(up(x))) -> top(check(rec(x)))
   check(up(x)) -> up(check(x))
   check(sent(x)) -> sent(check(x))
   check(rec(x)) -> rec(check(x))
   check(no(x)) -> no(check(x))
   check(no(x)) -> no(x)

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(1) DependencyGraphProof (UPPER BOUND(ID))
The following rules are not reachable from basic terms in the dependency graph and can be removed:

top(rec(up(x))) -> top(check(rec(x)))
top(sent(up(x))) -> top(check(rec(x)))
top(no(up(x))) -> top(check(rec(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:

   rec(rec(x)) -> sent(rec(x))
   rec(sent(x)) -> sent(rec(x))
   rec(no(x)) -> sent(rec(x))
   rec(bot) -> up(sent(bot))
   rec(up(x)) -> up(rec(x))
   sent(up(x)) -> up(sent(x))
   no(up(x)) -> up(no(x))
   check(up(x)) -> up(check(x))
   check(sent(x)) -> sent(check(x))
   check(rec(x)) -> rec(check(x))
   check(no(x)) -> no(check(x))
   check(no(x)) -> no(x)

S is empty.
Rewrite Strategy: FULL
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(3) NestedDefinedSymbolProof (UPPER BOUND(ID))
The TRS does not nest defined symbols.
Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
rec(rec(x)) -> sent(rec(x))
rec(sent(x)) -> sent(rec(x))
rec(no(x)) -> sent(rec(x))
check(sent(x)) -> sent(check(x))
check(rec(x)) -> rec(check(x))
check(no(x)) -> no(check(x))
check(no(x)) -> no(x)

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

(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:

   rec(bot) -> up(sent(bot))
   rec(up(x)) -> up(rec(x))
   sent(up(x)) -> up(sent(x))
   no(up(x)) -> up(no(x))
   check(up(x)) -> up(check(x))

S is empty.
Rewrite Strategy: FULL
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(5) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
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(6)
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:

   rec(bot) -> up(sent(bot))
   rec(up(x)) -> up(rec(x))
   sent(up(x)) -> up(sent(x))
   no(up(x)) -> up(no(x))
   check(up(x)) -> up(check(x))

S is empty.
Rewrite Strategy: FULL
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(7) CpxTrsMatchBoundsProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 1. 
The certificate found is represented by the following graph.

"[20, 21, 22, 23, 24, 25, 26, 27]
{(20,21,[rec_1|0, sent_1|0, no_1|0, check_1|0]), (20,22,[up_1|1]), (20,24,[up_1|1]), (20,25,[up_1|1]), (20,26,[up_1|1]), (20,27,[up_1|1]), (21,21,[bot|0, up_1|0]), (22,23,[sent_1|1]), (23,21,[bot|1]), (24,21,[rec_1|1]), (24,22,[up_1|1]), (24,24,[up_1|1]), (25,21,[sent_1|1]), (25,25,[up_1|1]), (26,21,[no_1|1]), (26,26,[up_1|1]), (27,21,[check_1|1]), (27,27,[up_1|1])}"
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(8)
BOUNDS(1, n^1)

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(9) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(10)
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:

   rec(rec(x)) -> sent(rec(x))
   rec(sent(x)) -> sent(rec(x))
   rec(no(x)) -> sent(rec(x))
   rec(bot) -> up(sent(bot))
   rec(up(x)) -> up(rec(x))
   sent(up(x)) -> up(sent(x))
   no(up(x)) -> up(no(x))
   top(rec(up(x))) -> top(check(rec(x)))
   top(sent(up(x))) -> top(check(rec(x)))
   top(no(up(x))) -> top(check(rec(x)))
   check(up(x)) -> up(check(x))
   check(sent(x)) -> sent(check(x))
   check(rec(x)) -> rec(check(x))
   check(no(x)) -> no(check(x))
   check(no(x)) -> no(x)

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

The rewrite sequence

rec(up(x)) ->^+ up(rec(x))

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

The pumping substitution is [x / up(x)].

The result substitution is [ ].




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(12)
Complex Obligation (BEST)

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(13)
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:

   rec(rec(x)) -> sent(rec(x))
   rec(sent(x)) -> sent(rec(x))
   rec(no(x)) -> sent(rec(x))
   rec(bot) -> up(sent(bot))
   rec(up(x)) -> up(rec(x))
   sent(up(x)) -> up(sent(x))
   no(up(x)) -> up(no(x))
   top(rec(up(x))) -> top(check(rec(x)))
   top(sent(up(x))) -> top(check(rec(x)))
   top(no(up(x))) -> top(check(rec(x)))
   check(up(x)) -> up(check(x))
   check(sent(x)) -> sent(check(x))
   check(rec(x)) -> rec(check(x))
   check(no(x)) -> no(check(x))
   check(no(x)) -> no(x)

S is empty.
Rewrite Strategy: FULL
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(14) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(15)
BOUNDS(n^1, INF)

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(16)
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:

   rec(rec(x)) -> sent(rec(x))
   rec(sent(x)) -> sent(rec(x))
   rec(no(x)) -> sent(rec(x))
   rec(bot) -> up(sent(bot))
   rec(up(x)) -> up(rec(x))
   sent(up(x)) -> up(sent(x))
   no(up(x)) -> up(no(x))
   top(rec(up(x))) -> top(check(rec(x)))
   top(sent(up(x))) -> top(check(rec(x)))
   top(no(up(x))) -> top(check(rec(x)))
   check(up(x)) -> up(check(x))
   check(sent(x)) -> sent(check(x))
   check(rec(x)) -> rec(check(x))
   check(no(x)) -> no(check(x))
   check(no(x)) -> no(x)

S is empty.
Rewrite Strategy: FULL