/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), ?)
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 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 (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   lt(0, s(X)) -> true
   lt(s(X), 0) -> false
   lt(s(X), s(Y)) -> lt(X, Y)
   append(nil, Y) -> Y
   append(add(N, X), Y) -> add(N, append(X, Y))
   split(N, nil) -> pair(nil, nil)
   split(N, add(M, Y)) -> f_1(split(N, Y), N, M, Y)
   f_1(pair(X, Z), N, M, Y) -> f_2(lt(N, M), N, M, Y, X, Z)
   f_2(true, N, M, Y, X, Z) -> pair(X, add(M, Z))
   f_2(false, N, M, Y, X, Z) -> pair(add(M, X), Z)
   qsort(nil) -> nil
   qsort(add(N, X)) -> f_3(split(N, X), N, X)
   f_3(pair(Y, Z), N, X) -> append(qsort(Y), add(X, qsort(Z)))

S is empty.
Rewrite Strategy: FULL
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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 (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   lt(0, s(X)) -> true
   lt(s(X), 0) -> false
   lt(s(X), s(Y)) -> lt(X, Y)
   append(nil, Y) -> Y
   append(add(N, X), Y) -> add(N, append(X, Y))
   split(N, nil) -> pair(nil, nil)
   split(N, add(M, Y)) -> f_1(split(N, Y), N, M, Y)
   f_1(pair(X, Z), N, M, Y) -> f_2(lt(N, M), N, M, Y, X, Z)
   f_2(true, N, M, Y, X, Z) -> pair(X, add(M, Z))
   f_2(false, N, M, Y, X, Z) -> pair(add(M, X), Z)
   qsort(nil) -> nil
   qsort(add(N, X)) -> f_3(split(N, X), N, X)
   f_3(pair(Y, Z), N, X) -> append(qsort(Y), add(X, qsort(Z)))

S is empty.
Rewrite Strategy: FULL
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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

split(N, add(M, Y)) ->^+ f_1(split(N, Y), N, M, Y)

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

The pumping substitution is [Y / add(M, Y)].

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


The TRS R consists of the following rules:

   lt(0, s(X)) -> true
   lt(s(X), 0) -> false
   lt(s(X), s(Y)) -> lt(X, Y)
   append(nil, Y) -> Y
   append(add(N, X), Y) -> add(N, append(X, Y))
   split(N, nil) -> pair(nil, nil)
   split(N, add(M, Y)) -> f_1(split(N, Y), N, M, Y)
   f_1(pair(X, Z), N, M, Y) -> f_2(lt(N, M), N, M, Y, X, Z)
   f_2(true, N, M, Y, X, Z) -> pair(X, add(M, Z))
   f_2(false, N, M, Y, X, Z) -> pair(add(M, X), Z)
   qsort(nil) -> nil
   qsort(add(N, X)) -> f_3(split(N, X), N, X)
   f_3(pair(Y, Z), N, X) -> append(qsort(Y), add(X, qsort(Z)))

S is empty.
Rewrite Strategy: FULL
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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 (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   lt(0, s(X)) -> true
   lt(s(X), 0) -> false
   lt(s(X), s(Y)) -> lt(X, Y)
   append(nil, Y) -> Y
   append(add(N, X), Y) -> add(N, append(X, Y))
   split(N, nil) -> pair(nil, nil)
   split(N, add(M, Y)) -> f_1(split(N, Y), N, M, Y)
   f_1(pair(X, Z), N, M, Y) -> f_2(lt(N, M), N, M, Y, X, Z)
   f_2(true, N, M, Y, X, Z) -> pair(X, add(M, Z))
   f_2(false, N, M, Y, X, Z) -> pair(add(M, X), Z)
   qsort(nil) -> nil
   qsort(add(N, X)) -> f_3(split(N, X), N, X)
   f_3(pair(Y, Z), N, X) -> append(qsort(Y), add(X, qsort(Z)))

S is empty.
Rewrite Strategy: FULL