/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (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, 31 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:

   odd(Cons(x, xs)) -> even(xs)
   odd(Nil) -> False
   even(Cons(x, xs)) -> odd(xs)
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   even(Nil) -> True
   evenodd(x) -> even(x)

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:

   odd(Cons(x, xs)) -> even(xs)
   odd(Nil) -> False
   even(Cons(x, xs)) -> odd(xs)
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   even(Nil) -> True
   evenodd(x) -> even(x)

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 1. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2, 3, 4]
transitions: 
Cons0(0, 0) -> 0
Nil0() -> 0
False0() -> 0
True0() -> 0
odd0(0) -> 1
even0(0) -> 2
notEmpty0(0) -> 3
evenodd0(0) -> 4
even1(0) -> 1
False1() -> 1
odd1(0) -> 2
True1() -> 3
False1() -> 3
True1() -> 2
even1(0) -> 4
even1(0) -> 2
False1() -> 2
odd1(0) -> 1
odd1(0) -> 4
True1() -> 1
True1() -> 4
False1() -> 4

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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:

   odd(Cons(x, xs)) -> even(xs)
   odd(Nil) -> False
   even(Cons(x, xs)) -> odd(xs)
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   even(Nil) -> True
   evenodd(x) -> even(x)

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

even(Cons(x, Cons(x1_0, xs2_0))) ->^+ even(xs2_0)

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

The pumping substitution is [xs2_0 / Cons(x, Cons(x1_0, xs2_0))].

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


The TRS R consists of the following rules:

   odd(Cons(x, xs)) -> even(xs)
   odd(Nil) -> False
   even(Cons(x, xs)) -> odd(xs)
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   even(Nil) -> True
   evenodd(x) -> even(x)

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:

   odd(Cons(x, xs)) -> even(xs)
   odd(Nil) -> False
   even(Cons(x, xs)) -> odd(xs)
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   even(Nil) -> True
   evenodd(x) -> even(x)

S is empty.
Rewrite Strategy: INNERMOST