/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (innermost) 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, 35 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 (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   g(S(x), y) -> g(x, S(y))
   f(y, S(x)) -> f(S(y), x)
   g(0, x2) -> x2
   f(x1, 0) -> g(x1, 0)

S is empty.
Rewrite Strategy: INNERMOST
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(1) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
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(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   g(S(x), y) -> g(x, S(y))
   f(y, S(x)) -> f(S(y), x)
   g(0, x2) -> x2
   f(x1, 0) -> g(x1, 0)

S is empty.
Rewrite Strategy: INNERMOST
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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 2. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2]
transitions: 
S0(0) -> 0
00() -> 0
g0(0, 0) -> 1
f0(0, 0) -> 2
S1(0) -> 3
g1(0, 3) -> 1
S1(0) -> 4
f1(4, 0) -> 2
01() -> 5
g1(0, 5) -> 2
S1(3) -> 3
S1(5) -> 3
g1(0, 3) -> 2
S1(4) -> 4
g1(4, 5) -> 2
S2(5) -> 6
g2(0, 6) -> 2
g2(4, 6) -> 2
S1(6) -> 3
S2(6) -> 6
0 -> 1
3 -> 1
3 -> 2
5 -> 2
6 -> 2

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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 (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   g(S(x), y) -> g(x, S(y))
   f(y, S(x)) -> f(S(y), x)
   g(0, x2) -> x2
   f(x1, 0) -> g(x1, 0)

S is empty.
Rewrite Strategy: INNERMOST
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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

f(y, S(x)) ->^+ f(S(y), x)

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

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

The result substitution is [y / S(y)].




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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 (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   g(S(x), y) -> g(x, S(y))
   f(y, S(x)) -> f(S(y), x)
   g(0, x2) -> x2
   f(x1, 0) -> g(x1, 0)

S is empty.
Rewrite Strategy: INNERMOST
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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 (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   g(S(x), y) -> g(x, S(y))
   f(y, S(x)) -> f(S(y), x)
   g(0, x2) -> x2
   f(x1, 0) -> g(x1, 0)

S is empty.
Rewrite Strategy: INNERMOST