/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


--------------------------------------------------------------------------------


WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 495 ms]
(4) CpxRelTRS
(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


----------------------------------------

(0)
Obligation:
The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   le(0, Y) -> true
   le(s(X), 0) -> false
   le(s(X), s(Y)) -> le(X, Y)
   app(nil, Y) -> Y
   app(cons(N, L), Y) -> cons(N, app(L, Y))
   low(N, nil) -> nil
   low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
   iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
   iflow(false, N, cons(M, L)) -> low(N, L)
   high(N, nil) -> nil
   high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
   ifhigh(true, N, cons(M, L)) -> high(N, L)
   ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
   quicksort(nil) -> nil
   quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

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

(1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID))
The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem:

   encArg(0) -> 0
   encArg(true) -> true
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(false) -> false
   encArg(nil) -> nil
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2))
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encArg(cons_low(x_1, x_2)) -> low(encArg(x_1), encArg(x_2))
   encArg(cons_iflow(x_1, x_2, x_3)) -> iflow(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_high(x_1, x_2)) -> high(encArg(x_1), encArg(x_2))
   encArg(cons_ifhigh(x_1, x_2, x_3)) -> ifhigh(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_quicksort(x_1)) -> quicksort(encArg(x_1))
   encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_low(x_1, x_2) -> low(encArg(x_1), encArg(x_2))
   encode_iflow(x_1, x_2, x_3) -> iflow(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_high(x_1, x_2) -> high(encArg(x_1), encArg(x_2))
   encode_ifhigh(x_1, x_2, x_3) -> ifhigh(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_quicksort(x_1) -> quicksort(encArg(x_1))


----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   le(0, Y) -> true
   le(s(X), 0) -> false
   le(s(X), s(Y)) -> le(X, Y)
   app(nil, Y) -> Y
   app(cons(N, L), Y) -> cons(N, app(L, Y))
   low(N, nil) -> nil
   low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
   iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
   iflow(false, N, cons(M, L)) -> low(N, L)
   high(N, nil) -> nil
   high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
   ifhigh(true, N, cons(M, L)) -> high(N, L)
   ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
   quicksort(nil) -> nil
   quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

The (relative) TRS S consists of the following rules:

   encArg(0) -> 0
   encArg(true) -> true
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(false) -> false
   encArg(nil) -> nil
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2))
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encArg(cons_low(x_1, x_2)) -> low(encArg(x_1), encArg(x_2))
   encArg(cons_iflow(x_1, x_2, x_3)) -> iflow(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_high(x_1, x_2)) -> high(encArg(x_1), encArg(x_2))
   encArg(cons_ifhigh(x_1, x_2, x_3)) -> ifhigh(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_quicksort(x_1)) -> quicksort(encArg(x_1))
   encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_low(x_1, x_2) -> low(encArg(x_1), encArg(x_2))
   encode_iflow(x_1, x_2, x_3) -> iflow(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_high(x_1, x_2) -> high(encArg(x_1), encArg(x_2))
   encode_ifhigh(x_1, x_2, x_3) -> ifhigh(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_quicksort(x_1) -> quicksort(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   le(0, Y) -> true
   le(s(X), 0) -> false
   le(s(X), s(Y)) -> le(X, Y)
   app(nil, Y) -> Y
   app(cons(N, L), Y) -> cons(N, app(L, Y))
   low(N, nil) -> nil
   low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
   iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
   iflow(false, N, cons(M, L)) -> low(N, L)
   high(N, nil) -> nil
   high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
   ifhigh(true, N, cons(M, L)) -> high(N, L)
   ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
   quicksort(nil) -> nil
   quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

The (relative) TRS S consists of the following rules:

   encArg(0) -> 0
   encArg(true) -> true
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(false) -> false
   encArg(nil) -> nil
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2))
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encArg(cons_low(x_1, x_2)) -> low(encArg(x_1), encArg(x_2))
   encArg(cons_iflow(x_1, x_2, x_3)) -> iflow(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_high(x_1, x_2)) -> high(encArg(x_1), encArg(x_2))
   encArg(cons_ifhigh(x_1, x_2, x_3)) -> ifhigh(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_quicksort(x_1)) -> quicksort(encArg(x_1))
   encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_low(x_1, x_2) -> low(encArg(x_1), encArg(x_2))
   encode_iflow(x_1, x_2, x_3) -> iflow(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_high(x_1, x_2) -> high(encArg(x_1), encArg(x_2))
   encode_ifhigh(x_1, x_2, x_3) -> ifhigh(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_quicksort(x_1) -> quicksort(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

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

(6)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   le(0, Y) -> true
   le(s(X), 0) -> false
   le(s(X), s(Y)) -> le(X, Y)
   app(nil, Y) -> Y
   app(cons(N, L), Y) -> cons(N, app(L, Y))
   low(N, nil) -> nil
   low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
   iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
   iflow(false, N, cons(M, L)) -> low(N, L)
   high(N, nil) -> nil
   high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
   ifhigh(true, N, cons(M, L)) -> high(N, L)
   ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
   quicksort(nil) -> nil
   quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

The (relative) TRS S consists of the following rules:

   encArg(0) -> 0
   encArg(true) -> true
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(false) -> false
   encArg(nil) -> nil
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2))
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encArg(cons_low(x_1, x_2)) -> low(encArg(x_1), encArg(x_2))
   encArg(cons_iflow(x_1, x_2, x_3)) -> iflow(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_high(x_1, x_2)) -> high(encArg(x_1), encArg(x_2))
   encArg(cons_ifhigh(x_1, x_2, x_3)) -> ifhigh(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_quicksort(x_1)) -> quicksort(encArg(x_1))
   encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_low(x_1, x_2) -> low(encArg(x_1), encArg(x_2))
   encode_iflow(x_1, x_2, x_3) -> iflow(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_high(x_1, x_2) -> high(encArg(x_1), encArg(x_2))
   encode_ifhigh(x_1, x_2, x_3) -> ifhigh(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_quicksort(x_1) -> quicksort(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

(7) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

app(cons(N, L), Y) ->^+ cons(N, app(L, Y))

gives rise to a decreasing loop by considering the right hand sides subterm at position [1].

The pumping substitution is [L / cons(N, L)].

The result substitution is [ ].




----------------------------------------

(8)
Complex Obligation (BEST)

----------------------------------------

(9)
Obligation:
Proved the lower bound n^1 for the following obligation:

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


The TRS R consists of the following rules:

   le(0, Y) -> true
   le(s(X), 0) -> false
   le(s(X), s(Y)) -> le(X, Y)
   app(nil, Y) -> Y
   app(cons(N, L), Y) -> cons(N, app(L, Y))
   low(N, nil) -> nil
   low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
   iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
   iflow(false, N, cons(M, L)) -> low(N, L)
   high(N, nil) -> nil
   high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
   ifhigh(true, N, cons(M, L)) -> high(N, L)
   ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
   quicksort(nil) -> nil
   quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

The (relative) TRS S consists of the following rules:

   encArg(0) -> 0
   encArg(true) -> true
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(false) -> false
   encArg(nil) -> nil
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2))
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encArg(cons_low(x_1, x_2)) -> low(encArg(x_1), encArg(x_2))
   encArg(cons_iflow(x_1, x_2, x_3)) -> iflow(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_high(x_1, x_2)) -> high(encArg(x_1), encArg(x_2))
   encArg(cons_ifhigh(x_1, x_2, x_3)) -> ifhigh(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_quicksort(x_1)) -> quicksort(encArg(x_1))
   encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_low(x_1, x_2) -> low(encArg(x_1), encArg(x_2))
   encode_iflow(x_1, x_2, x_3) -> iflow(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_high(x_1, x_2) -> high(encArg(x_1), encArg(x_2))
   encode_ifhigh(x_1, x_2, x_3) -> ifhigh(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_quicksort(x_1) -> quicksort(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

(10) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(11)
BOUNDS(n^1, INF)

----------------------------------------

(12)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   le(0, Y) -> true
   le(s(X), 0) -> false
   le(s(X), s(Y)) -> le(X, Y)
   app(nil, Y) -> Y
   app(cons(N, L), Y) -> cons(N, app(L, Y))
   low(N, nil) -> nil
   low(N, cons(M, L)) -> iflow(le(M, N), N, cons(M, L))
   iflow(true, N, cons(M, L)) -> cons(M, low(N, L))
   iflow(false, N, cons(M, L)) -> low(N, L)
   high(N, nil) -> nil
   high(N, cons(M, L)) -> ifhigh(le(M, N), N, cons(M, L))
   ifhigh(true, N, cons(M, L)) -> high(N, L)
   ifhigh(false, N, cons(M, L)) -> cons(M, high(N, L))
   quicksort(nil) -> nil
   quicksort(cons(N, L)) -> app(quicksort(low(N, L)), cons(N, quicksort(high(N, L))))

The (relative) TRS S consists of the following rules:

   encArg(0) -> 0
   encArg(true) -> true
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(false) -> false
   encArg(nil) -> nil
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2))
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encArg(cons_low(x_1, x_2)) -> low(encArg(x_1), encArg(x_2))
   encArg(cons_iflow(x_1, x_2, x_3)) -> iflow(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_high(x_1, x_2)) -> high(encArg(x_1), encArg(x_2))
   encArg(cons_ifhigh(x_1, x_2, x_3)) -> ifhigh(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_quicksort(x_1)) -> quicksort(encArg(x_1))
   encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_low(x_1, x_2) -> low(encArg(x_1), encArg(x_2))
   encode_iflow(x_1, x_2, x_3) -> iflow(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_high(x_1, x_2) -> high(encArg(x_1), encArg(x_2))
   encode_ifhigh(x_1, x_2, x_3) -> ifhigh(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_quicksort(x_1) -> quicksort(encArg(x_1))

Rewrite Strategy: INNERMOST