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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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

   g(c(x, s(y))) -> g(c(s(x), y))
   f(c(s(x), y)) -> f(c(x, s(y)))
   f(f(x)) -> f(d(f(x)))
   f(x) -> x

S is empty.
Rewrite Strategy: FULL
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(1) DependencyGraphProof (UPPER BOUND(ID))
The following rules are not reachable from basic terms in the dependency graph and can be removed:

f(f(x)) -> f(d(f(x)))

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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:

   g(c(x, s(y))) -> g(c(s(x), y))
   f(c(s(x), y)) -> f(c(x, s(y)))
   f(x) -> x

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

   g(c(x, s(y))) -> g(c(s(x), y))
   f(c(s(x), y)) -> f(c(x, s(y)))
   f(x) -> x

S is empty.
Rewrite Strategy: FULL
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(5) 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 1. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2]
transitions: 
c0(0, 0) -> 0
s0(0) -> 0
g0(0) -> 1
f0(0) -> 2
s1(0) -> 4
c1(4, 0) -> 3
g1(3) -> 1
s1(0) -> 6
c1(0, 6) -> 5
f1(5) -> 2
s1(4) -> 4
s1(6) -> 6
0 -> 2
5 -> 2

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(6)
BOUNDS(1, n^1)

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(7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(8)
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:

   g(c(x, s(y))) -> g(c(s(x), y))
   f(c(s(x), y)) -> f(c(x, s(y)))
   f(f(x)) -> f(d(f(x)))
   f(x) -> x

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

The rewrite sequence

g(c(x, s(y))) ->^+ g(c(s(x), y))

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

The pumping substitution is [y / s(y)].

The result substitution is [x / s(x)].




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(10)
Complex Obligation (BEST)

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(11)
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:

   g(c(x, s(y))) -> g(c(s(x), y))
   f(c(s(x), y)) -> f(c(x, s(y)))
   f(f(x)) -> f(d(f(x)))
   f(x) -> x

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

(12) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(13)
BOUNDS(n^1, INF)

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(14)
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:

   g(c(x, s(y))) -> g(c(s(x), y))
   f(c(s(x), y)) -> f(c(x, s(y)))
   f(f(x)) -> f(d(f(x)))
   f(x) -> x

S is empty.
Rewrite Strategy: FULL