/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 93 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(a, a) -> f(a, b)
   f(a, b) -> f(s(a), c)
   f(s(X), c) -> f(X, c)
   f(c, c) -> f(a, a)

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(a, a) -> f(a, b)
   f(a, b) -> f(s(a), c)
   f(s(X), c) -> f(X, c)
   f(c, c) -> f(a, a)

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]
transitions: 
a0() -> 0
b0() -> 0
s0(0) -> 0
c0() -> 0
f0(0, 0) -> 1
a1() -> 2
b1() -> 3
f1(2, 3) -> 1
a1() -> 5
s1(5) -> 4
c1() -> 6
f1(4, 6) -> 1
f1(0, 6) -> 1
a1() -> 7
f1(2, 7) -> 1
a2() -> 8
b2() -> 9
f2(8, 9) -> 1
a2() -> 11
s2(11) -> 10
c2() -> 12
f2(10, 12) -> 1
f2(5, 12) -> 1
a3() -> 14
s3(14) -> 13
c3() -> 15
f3(13, 15) -> 1
f3(11, 15) -> 1
c4() -> 16
f4(14, 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(a, a) -> f(a, b)
   f(a, b) -> f(s(a), c)
   f(s(X), c) -> f(X, c)
   f(c, c) -> f(a, a)

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(s(X), c) ->^+ f(X, c)

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 [ ].




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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(a, a) -> f(a, b)
   f(a, b) -> f(s(a), c)
   f(s(X), c) -> f(X, c)
   f(c, c) -> f(a, a)

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(a, a) -> f(a, b)
   f(a, b) -> f(s(a), c)
   f(s(X), c) -> f(X, c)
   f(c, c) -> f(a, a)

S is empty.
Rewrite Strategy: FULL