/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), ?)
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(0) CpxTRS
(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(2) TRS for Loop Detection
(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
(4) BEST
    (5) proven lower bound
        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
        (7) BOUNDS(n^1, INF)
    (8) 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, INF).


The TRS R consists of the following rules:

   cond_scanr_f_z_xs_1(Cons(0, x11), 0) -> Cons(0, Cons(0, x11))
   cond_scanr_f_z_xs_1(Cons(S(x2), x11), 0) -> Cons(S(x2), Cons(S(x2), x11))
   cond_scanr_f_z_xs_1(Cons(0, x11), M(x2)) -> Cons(0, Cons(0, x11))
   cond_scanr_f_z_xs_1(Cons(S(x40), x23), M(0)) -> Cons(S(x40), Cons(S(x40), x23))
   cond_scanr_f_z_xs_1(Cons(S(x8), x23), M(S(x4))) -> Cons(minus#2(x8, x4), Cons(S(x8), x23))
   cond_scanr_f_z_xs_1(Cons(0, x11), S(x2)) -> Cons(S(x2), Cons(0, x11))
   cond_scanr_f_z_xs_1(Cons(S(x2), x11), S(x4)) -> Cons(plus#2(S(x4), S(x2)), Cons(S(x2), x11))
   scanr#3(Nil) -> Cons(0, Nil)
   scanr#3(Cons(x4, x2)) -> cond_scanr_f_z_xs_1(scanr#3(x2), x4)
   foldl#3(x2, Nil) -> x2
   foldl#3(x6, Cons(x4, x2)) -> foldl#3(max#2(x6, x4), x2)
   minus#2(0, x16) -> 0
   minus#2(S(x20), 0) -> S(x20)
   minus#2(S(x4), S(x2)) -> minus#2(x4, x2)
   plus#2(0, S(x2)) -> S(x2)
   plus#2(S(x4), S(x2)) -> S(plus#2(x4, S(x2)))
   max#2(0, x8) -> x8
   max#2(S(x12), 0) -> S(x12)
   max#2(S(x4), S(x2)) -> S(max#2(x4, x2))
   main(x1) -> foldl#3(0, scanr#3(x1))

S is empty.
Rewrite Strategy: INNERMOST
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(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(2)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   cond_scanr_f_z_xs_1(Cons(0, x11), 0) -> Cons(0, Cons(0, x11))
   cond_scanr_f_z_xs_1(Cons(S(x2), x11), 0) -> Cons(S(x2), Cons(S(x2), x11))
   cond_scanr_f_z_xs_1(Cons(0, x11), M(x2)) -> Cons(0, Cons(0, x11))
   cond_scanr_f_z_xs_1(Cons(S(x40), x23), M(0)) -> Cons(S(x40), Cons(S(x40), x23))
   cond_scanr_f_z_xs_1(Cons(S(x8), x23), M(S(x4))) -> Cons(minus#2(x8, x4), Cons(S(x8), x23))
   cond_scanr_f_z_xs_1(Cons(0, x11), S(x2)) -> Cons(S(x2), Cons(0, x11))
   cond_scanr_f_z_xs_1(Cons(S(x2), x11), S(x4)) -> Cons(plus#2(S(x4), S(x2)), Cons(S(x2), x11))
   scanr#3(Nil) -> Cons(0, Nil)
   scanr#3(Cons(x4, x2)) -> cond_scanr_f_z_xs_1(scanr#3(x2), x4)
   foldl#3(x2, Nil) -> x2
   foldl#3(x6, Cons(x4, x2)) -> foldl#3(max#2(x6, x4), x2)
   minus#2(0, x16) -> 0
   minus#2(S(x20), 0) -> S(x20)
   minus#2(S(x4), S(x2)) -> minus#2(x4, x2)
   plus#2(0, S(x2)) -> S(x2)
   plus#2(S(x4), S(x2)) -> S(plus#2(x4, S(x2)))
   max#2(0, x8) -> x8
   max#2(S(x12), 0) -> S(x12)
   max#2(S(x4), S(x2)) -> S(max#2(x4, x2))
   main(x1) -> foldl#3(0, scanr#3(x1))

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

The rewrite sequence

minus#2(S(x4), S(x2)) ->^+ minus#2(x4, x2)

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

The pumping substitution is [x4 / S(x4), x2 / S(x2)].

The result substitution is [ ].




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(4)
Complex Obligation (BEST)

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(5)
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, INF).


The TRS R consists of the following rules:

   cond_scanr_f_z_xs_1(Cons(0, x11), 0) -> Cons(0, Cons(0, x11))
   cond_scanr_f_z_xs_1(Cons(S(x2), x11), 0) -> Cons(S(x2), Cons(S(x2), x11))
   cond_scanr_f_z_xs_1(Cons(0, x11), M(x2)) -> Cons(0, Cons(0, x11))
   cond_scanr_f_z_xs_1(Cons(S(x40), x23), M(0)) -> Cons(S(x40), Cons(S(x40), x23))
   cond_scanr_f_z_xs_1(Cons(S(x8), x23), M(S(x4))) -> Cons(minus#2(x8, x4), Cons(S(x8), x23))
   cond_scanr_f_z_xs_1(Cons(0, x11), S(x2)) -> Cons(S(x2), Cons(0, x11))
   cond_scanr_f_z_xs_1(Cons(S(x2), x11), S(x4)) -> Cons(plus#2(S(x4), S(x2)), Cons(S(x2), x11))
   scanr#3(Nil) -> Cons(0, Nil)
   scanr#3(Cons(x4, x2)) -> cond_scanr_f_z_xs_1(scanr#3(x2), x4)
   foldl#3(x2, Nil) -> x2
   foldl#3(x6, Cons(x4, x2)) -> foldl#3(max#2(x6, x4), x2)
   minus#2(0, x16) -> 0
   minus#2(S(x20), 0) -> S(x20)
   minus#2(S(x4), S(x2)) -> minus#2(x4, x2)
   plus#2(0, S(x2)) -> S(x2)
   plus#2(S(x4), S(x2)) -> S(plus#2(x4, S(x2)))
   max#2(0, x8) -> x8
   max#2(S(x12), 0) -> S(x12)
   max#2(S(x4), S(x2)) -> S(max#2(x4, x2))
   main(x1) -> foldl#3(0, scanr#3(x1))

S is empty.
Rewrite Strategy: INNERMOST
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(6) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(7)
BOUNDS(n^1, INF)

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(8)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   cond_scanr_f_z_xs_1(Cons(0, x11), 0) -> Cons(0, Cons(0, x11))
   cond_scanr_f_z_xs_1(Cons(S(x2), x11), 0) -> Cons(S(x2), Cons(S(x2), x11))
   cond_scanr_f_z_xs_1(Cons(0, x11), M(x2)) -> Cons(0, Cons(0, x11))
   cond_scanr_f_z_xs_1(Cons(S(x40), x23), M(0)) -> Cons(S(x40), Cons(S(x40), x23))
   cond_scanr_f_z_xs_1(Cons(S(x8), x23), M(S(x4))) -> Cons(minus#2(x8, x4), Cons(S(x8), x23))
   cond_scanr_f_z_xs_1(Cons(0, x11), S(x2)) -> Cons(S(x2), Cons(0, x11))
   cond_scanr_f_z_xs_1(Cons(S(x2), x11), S(x4)) -> Cons(plus#2(S(x4), S(x2)), Cons(S(x2), x11))
   scanr#3(Nil) -> Cons(0, Nil)
   scanr#3(Cons(x4, x2)) -> cond_scanr_f_z_xs_1(scanr#3(x2), x4)
   foldl#3(x2, Nil) -> x2
   foldl#3(x6, Cons(x4, x2)) -> foldl#3(max#2(x6, x4), x2)
   minus#2(0, x16) -> 0
   minus#2(S(x20), 0) -> S(x20)
   minus#2(S(x4), S(x2)) -> minus#2(x4, x2)
   plus#2(0, S(x2)) -> S(x2)
   plus#2(S(x4), S(x2)) -> S(plus#2(x4, S(x2)))
   max#2(0, x8) -> x8
   max#2(S(x12), 0) -> S(x12)
   max#2(S(x4), S(x2)) -> S(max#2(x4, x2))
   main(x1) -> foldl#3(0, scanr#3(x1))

S is empty.
Rewrite Strategy: INNERMOST