/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 33 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(nil) -> nil
   f(.(nil, y)) -> .(nil, f(y))
   f(.(.(x, y), z)) -> f(.(x, .(y, z)))
   g(nil) -> nil
   g(.(x, nil)) -> .(g(x), nil)
   g(.(x, .(y, z))) -> g(.(.(x, y), z))

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(nil) -> nil
   f(.(nil, y)) -> .(nil, f(y))
   f(.(.(x, y), z)) -> f(.(x, .(y, z)))
   g(nil) -> nil
   g(.(x, nil)) -> .(g(x), nil)
   g(.(x, .(y, z))) -> g(.(.(x, y), z))

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 1. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2]
transitions: 
nil0() -> 0
.0(0, 0) -> 0
f0(0) -> 1
g0(0) -> 2
nil1() -> 1
nil1() -> 3
f1(0) -> 4
.1(3, 4) -> 1
.1(0, 0) -> 6
.1(0, 6) -> 5
f1(5) -> 1
nil1() -> 2
g1(0) -> 7
nil1() -> 8
.1(7, 8) -> 2
.1(0, 0) -> 10
.1(10, 0) -> 9
g1(9) -> 2
nil1() -> 4
.1(3, 4) -> 4
f1(6) -> 4
f1(5) -> 4
.1(0, 6) -> 6
nil1() -> 7
.1(7, 8) -> 7
g1(10) -> 7
g1(9) -> 7
.1(10, 0) -> 10

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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(nil) -> nil
   f(.(nil, y)) -> .(nil, f(y))
   f(.(.(x, y), z)) -> f(.(x, .(y, z)))
   g(nil) -> nil
   g(.(x, nil)) -> .(g(x), nil)
   g(.(x, .(y, z))) -> g(.(.(x, y), z))

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

g(.(x, nil)) ->^+ .(g(x), nil)

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

The pumping substitution is [x / .(x, nil)].

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(nil) -> nil
   f(.(nil, y)) -> .(nil, f(y))
   f(.(.(x, y), z)) -> f(.(x, .(y, z)))
   g(nil) -> nil
   g(.(x, nil)) -> .(g(x), nil)
   g(.(x, .(y, z))) -> g(.(.(x, y), z))

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(nil) -> nil
   f(.(nil, y)) -> .(nil, f(y))
   f(.(.(x, y), z)) -> f(.(x, .(y, z)))
   g(nil) -> nil
   g(.(x, nil)) -> .(g(x), nil)
   g(.(x, .(y, z))) -> g(.(.(x, y), z))

S is empty.
Rewrite Strategy: FULL