/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) CpxTrsMatchBoundsTAProof [FINISHED, 385 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:

   f(x, a(b(c(y)))) -> f(b(c(a(b(x)))), y)
   f(a(x), y) -> f(x, a(y))
   f(b(x), y) -> f(x, b(y))
   f(c(x), y) -> f(x, c(y))

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:

   f(x, a(b(c(y)))) -> f(b(c(a(b(x)))), y)
   f(a(x), y) -> f(x, a(y))
   f(b(x), y) -> f(x, b(y))
   f(c(x), y) -> f(x, c(y))

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 3. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1]
transitions: 
a0(0) -> 0
b0(0) -> 0
c0(0) -> 0
f0(0, 0) -> 1
b1(0) -> 5
a1(5) -> 4
c1(4) -> 3
b1(3) -> 2
f1(2, 0) -> 1
a1(0) -> 6
f1(0, 6) -> 1
b1(0) -> 7
f1(0, 7) -> 1
c1(0) -> 8
f1(0, 8) -> 1
b1(2) -> 5
a1(6) -> 6
a1(7) -> 6
a1(8) -> 6
b1(6) -> 7
b1(7) -> 7
b1(8) -> 7
b2(0) -> 9
f2(3, 9) -> 1
c1(6) -> 8
c1(7) -> 8
c1(8) -> 8
b2(0) -> 13
a2(13) -> 12
c2(12) -> 11
b2(11) -> 10
f2(10, 0) -> 1
f2(10, 6) -> 1
f2(10, 7) -> 1
f2(10, 8) -> 1
c2(9) -> 14
f2(4, 14) -> 1
b1(10) -> 5
b2(10) -> 13
a2(14) -> 15
f2(5, 15) -> 1
b3(0) -> 16
f3(11, 16) -> 1
b3(6) -> 16
b3(7) -> 16
b3(8) -> 16
b2(15) -> 9
f2(0, 9) -> 1
f2(2, 9) -> 1
f2(10, 9) -> 1
c3(16) -> 17
f3(12, 17) -> 1
a1(9) -> 6
b1(9) -> 7
b2(9) -> 9
b3(9) -> 16
c1(9) -> 8
a3(17) -> 18
f3(13, 18) -> 1
b3(18) -> 16
f3(0, 16) -> 1
f3(10, 16) -> 1
a1(16) -> 6
b1(16) -> 7
b3(16) -> 16
c1(16) -> 8
f2(10, 16) -> 1

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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:

   f(x, a(b(c(y)))) -> f(b(c(a(b(x)))), y)
   f(a(x), y) -> f(x, a(y))
   f(b(x), y) -> f(x, b(y))
   f(c(x), y) -> f(x, c(y))

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

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

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

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

The result substitution is [y / c(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 (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   f(x, a(b(c(y)))) -> f(b(c(a(b(x)))), y)
   f(a(x), y) -> f(x, a(y))
   f(b(x), y) -> f(x, b(y))
   f(c(x), y) -> f(x, c(y))

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:

   f(x, a(b(c(y)))) -> f(b(c(a(b(x)))), y)
   f(a(x), y) -> f(x, a(y))
   f(b(x), y) -> f(x, b(y))
   f(c(x), y) -> f(x, c(y))

S is empty.
Rewrite Strategy: FULL