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


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


WORST_CASE(Omega(n^2), ?)
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^2, INF).

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) typed CpxTrs
(5) OrderProof [LOWER BOUND(ID), 0 ms]
(6) typed CpxTrs
(7) RewriteLemmaProof [LOWER BOUND(ID), 253 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) typed CpxTrs
        (13) RewriteLemmaProof [LOWER BOUND(ID), 44 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 10 ms]
        (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 5 ms]
        (18) typed CpxTrs
        (19) RewriteLemmaProof [LOWER BOUND(ID), 17 ms]
        (20) proven lower bound
        (21) LowerBoundPropagationProof [FINISHED, 0 ms]
        (22) BOUNDS(n^2, INF)


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

(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, 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: FULL
----------------------------------------

(1) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(2)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, 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: FULL
----------------------------------------

(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(4)
Obligation:
TRS:
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))))

Types:
le :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
app :: nil:cons -> nil:cons -> nil:cons
nil :: nil:cons
cons :: 0':s -> nil:cons -> nil:cons
low :: 0':s -> nil:cons -> nil:cons
iflow :: true:false -> 0':s -> nil:cons -> nil:cons
high :: 0':s -> nil:cons -> nil:cons
ifhigh :: true:false -> 0':s -> nil:cons -> nil:cons
quicksort :: nil:cons -> nil:cons
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_nil:cons3_0 :: nil:cons
gen_0':s4_0 :: Nat -> 0':s
gen_nil:cons5_0 :: Nat -> nil:cons

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
le, app, low, high, quicksort

They will be analysed ascendingly in the following order:
le < low
le < high
app < quicksort
low < quicksort
high < quicksort

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

(6)
Obligation:
TRS:
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))))

Types:
le :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
app :: nil:cons -> nil:cons -> nil:cons
nil :: nil:cons
cons :: 0':s -> nil:cons -> nil:cons
low :: 0':s -> nil:cons -> nil:cons
iflow :: true:false -> 0':s -> nil:cons -> nil:cons
high :: 0':s -> nil:cons -> nil:cons
ifhigh :: true:false -> 0':s -> nil:cons -> nil:cons
quicksort :: nil:cons -> nil:cons
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_nil:cons3_0 :: nil:cons
gen_0':s4_0 :: Nat -> 0':s
gen_nil:cons5_0 :: Nat -> nil:cons


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
gen_nil:cons5_0(0) <=> nil
gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x))


The following defined symbols remain to be analysed:
le, app, low, high, quicksort

They will be analysed ascendingly in the following order:
le < low
le < high
app < quicksort
low < quicksort
high < quicksort

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0)

Induction Base:
le(gen_0':s4_0(0), gen_0':s4_0(0)) ->_R^Omega(1)
true

Induction Step:
le(gen_0':s4_0(+(n7_0, 1)), gen_0':s4_0(+(n7_0, 1))) ->_R^Omega(1)
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) ->_IH
true

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(8)
Complex Obligation (BEST)

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

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

TRS:
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))))

Types:
le :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
app :: nil:cons -> nil:cons -> nil:cons
nil :: nil:cons
cons :: 0':s -> nil:cons -> nil:cons
low :: 0':s -> nil:cons -> nil:cons
iflow :: true:false -> 0':s -> nil:cons -> nil:cons
high :: 0':s -> nil:cons -> nil:cons
ifhigh :: true:false -> 0':s -> nil:cons -> nil:cons
quicksort :: nil:cons -> nil:cons
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_nil:cons3_0 :: nil:cons
gen_0':s4_0 :: Nat -> 0':s
gen_nil:cons5_0 :: Nat -> nil:cons


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
gen_nil:cons5_0(0) <=> nil
gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x))


The following defined symbols remain to be analysed:
le, app, low, high, quicksort

They will be analysed ascendingly in the following order:
le < low
le < high
app < quicksort
low < quicksort
high < quicksort

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
TRS:
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))))

Types:
le :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
app :: nil:cons -> nil:cons -> nil:cons
nil :: nil:cons
cons :: 0':s -> nil:cons -> nil:cons
low :: 0':s -> nil:cons -> nil:cons
iflow :: true:false -> 0':s -> nil:cons -> nil:cons
high :: 0':s -> nil:cons -> nil:cons
ifhigh :: true:false -> 0':s -> nil:cons -> nil:cons
quicksort :: nil:cons -> nil:cons
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_nil:cons3_0 :: nil:cons
gen_0':s4_0 :: Nat -> 0':s
gen_nil:cons5_0 :: Nat -> nil:cons


Lemmas:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0)


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
gen_nil:cons5_0(0) <=> nil
gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x))


The following defined symbols remain to be analysed:
app, low, high, quicksort

They will be analysed ascendingly in the following order:
app < quicksort
low < quicksort
high < quicksort

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
app(gen_nil:cons5_0(n288_0), gen_nil:cons5_0(b)) -> gen_nil:cons5_0(+(n288_0, b)), rt in Omega(1 + n288_0)

Induction Base:
app(gen_nil:cons5_0(0), gen_nil:cons5_0(b)) ->_R^Omega(1)
gen_nil:cons5_0(b)

Induction Step:
app(gen_nil:cons5_0(+(n288_0, 1)), gen_nil:cons5_0(b)) ->_R^Omega(1)
cons(0', app(gen_nil:cons5_0(n288_0), gen_nil:cons5_0(b))) ->_IH
cons(0', gen_nil:cons5_0(+(b, c289_0)))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(14)
Obligation:
TRS:
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))))

Types:
le :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
app :: nil:cons -> nil:cons -> nil:cons
nil :: nil:cons
cons :: 0':s -> nil:cons -> nil:cons
low :: 0':s -> nil:cons -> nil:cons
iflow :: true:false -> 0':s -> nil:cons -> nil:cons
high :: 0':s -> nil:cons -> nil:cons
ifhigh :: true:false -> 0':s -> nil:cons -> nil:cons
quicksort :: nil:cons -> nil:cons
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_nil:cons3_0 :: nil:cons
gen_0':s4_0 :: Nat -> 0':s
gen_nil:cons5_0 :: Nat -> nil:cons


Lemmas:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0)
app(gen_nil:cons5_0(n288_0), gen_nil:cons5_0(b)) -> gen_nil:cons5_0(+(n288_0, b)), rt in Omega(1 + n288_0)


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
gen_nil:cons5_0(0) <=> nil
gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x))


The following defined symbols remain to be analysed:
low, high, quicksort

They will be analysed ascendingly in the following order:
low < quicksort
high < quicksort

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
low(gen_0':s4_0(0), gen_nil:cons5_0(n1123_0)) -> gen_nil:cons5_0(n1123_0), rt in Omega(1 + n1123_0)

Induction Base:
low(gen_0':s4_0(0), gen_nil:cons5_0(0)) ->_R^Omega(1)
nil

Induction Step:
low(gen_0':s4_0(0), gen_nil:cons5_0(+(n1123_0, 1))) ->_R^Omega(1)
iflow(le(0', gen_0':s4_0(0)), gen_0':s4_0(0), cons(0', gen_nil:cons5_0(n1123_0))) ->_L^Omega(1)
iflow(true, gen_0':s4_0(0), cons(0', gen_nil:cons5_0(n1123_0))) ->_R^Omega(1)
cons(0', low(gen_0':s4_0(0), gen_nil:cons5_0(n1123_0))) ->_IH
cons(0', gen_nil:cons5_0(c1124_0))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(16)
Obligation:
TRS:
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))))

Types:
le :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
app :: nil:cons -> nil:cons -> nil:cons
nil :: nil:cons
cons :: 0':s -> nil:cons -> nil:cons
low :: 0':s -> nil:cons -> nil:cons
iflow :: true:false -> 0':s -> nil:cons -> nil:cons
high :: 0':s -> nil:cons -> nil:cons
ifhigh :: true:false -> 0':s -> nil:cons -> nil:cons
quicksort :: nil:cons -> nil:cons
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_nil:cons3_0 :: nil:cons
gen_0':s4_0 :: Nat -> 0':s
gen_nil:cons5_0 :: Nat -> nil:cons


Lemmas:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0)
app(gen_nil:cons5_0(n288_0), gen_nil:cons5_0(b)) -> gen_nil:cons5_0(+(n288_0, b)), rt in Omega(1 + n288_0)
low(gen_0':s4_0(0), gen_nil:cons5_0(n1123_0)) -> gen_nil:cons5_0(n1123_0), rt in Omega(1 + n1123_0)


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
gen_nil:cons5_0(0) <=> nil
gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x))


The following defined symbols remain to be analysed:
high, quicksort

They will be analysed ascendingly in the following order:
high < quicksort

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
high(gen_0':s4_0(0), gen_nil:cons5_0(n1630_0)) -> gen_nil:cons5_0(0), rt in Omega(1 + n1630_0)

Induction Base:
high(gen_0':s4_0(0), gen_nil:cons5_0(0)) ->_R^Omega(1)
nil

Induction Step:
high(gen_0':s4_0(0), gen_nil:cons5_0(+(n1630_0, 1))) ->_R^Omega(1)
ifhigh(le(0', gen_0':s4_0(0)), gen_0':s4_0(0), cons(0', gen_nil:cons5_0(n1630_0))) ->_L^Omega(1)
ifhigh(true, gen_0':s4_0(0), cons(0', gen_nil:cons5_0(n1630_0))) ->_R^Omega(1)
high(gen_0':s4_0(0), gen_nil:cons5_0(n1630_0)) ->_IH
gen_nil:cons5_0(0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(18)
Obligation:
TRS:
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))))

Types:
le :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
app :: nil:cons -> nil:cons -> nil:cons
nil :: nil:cons
cons :: 0':s -> nil:cons -> nil:cons
low :: 0':s -> nil:cons -> nil:cons
iflow :: true:false -> 0':s -> nil:cons -> nil:cons
high :: 0':s -> nil:cons -> nil:cons
ifhigh :: true:false -> 0':s -> nil:cons -> nil:cons
quicksort :: nil:cons -> nil:cons
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_nil:cons3_0 :: nil:cons
gen_0':s4_0 :: Nat -> 0':s
gen_nil:cons5_0 :: Nat -> nil:cons


Lemmas:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0)
app(gen_nil:cons5_0(n288_0), gen_nil:cons5_0(b)) -> gen_nil:cons5_0(+(n288_0, b)), rt in Omega(1 + n288_0)
low(gen_0':s4_0(0), gen_nil:cons5_0(n1123_0)) -> gen_nil:cons5_0(n1123_0), rt in Omega(1 + n1123_0)
high(gen_0':s4_0(0), gen_nil:cons5_0(n1630_0)) -> gen_nil:cons5_0(0), rt in Omega(1 + n1630_0)


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
gen_nil:cons5_0(0) <=> nil
gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x))


The following defined symbols remain to be analysed:
quicksort
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(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
quicksort(gen_nil:cons5_0(n2133_0)) -> gen_nil:cons5_0(n2133_0), rt in Omega(1 + n2133_0 + n2133_0^2)

Induction Base:
quicksort(gen_nil:cons5_0(0)) ->_R^Omega(1)
nil

Induction Step:
quicksort(gen_nil:cons5_0(+(n2133_0, 1))) ->_R^Omega(1)
app(quicksort(low(0', gen_nil:cons5_0(n2133_0))), cons(0', quicksort(high(0', gen_nil:cons5_0(n2133_0))))) ->_L^Omega(1 + n2133_0)
app(quicksort(gen_nil:cons5_0(n2133_0)), cons(0', quicksort(high(0', gen_nil:cons5_0(n2133_0))))) ->_IH
app(gen_nil:cons5_0(c2134_0), cons(0', quicksort(high(0', gen_nil:cons5_0(n2133_0))))) ->_L^Omega(1 + n2133_0)
app(gen_nil:cons5_0(n2133_0), cons(0', quicksort(gen_nil:cons5_0(0)))) ->_R^Omega(1)
app(gen_nil:cons5_0(n2133_0), cons(0', nil)) ->_L^Omega(1 + n2133_0)
gen_nil:cons5_0(+(n2133_0, +(0, 1)))

We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
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(20)
Obligation:
Proved the lower bound n^2 for the following obligation:

TRS:
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))))

Types:
le :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
app :: nil:cons -> nil:cons -> nil:cons
nil :: nil:cons
cons :: 0':s -> nil:cons -> nil:cons
low :: 0':s -> nil:cons -> nil:cons
iflow :: true:false -> 0':s -> nil:cons -> nil:cons
high :: 0':s -> nil:cons -> nil:cons
ifhigh :: true:false -> 0':s -> nil:cons -> nil:cons
quicksort :: nil:cons -> nil:cons
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_nil:cons3_0 :: nil:cons
gen_0':s4_0 :: Nat -> 0':s
gen_nil:cons5_0 :: Nat -> nil:cons


Lemmas:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0)
app(gen_nil:cons5_0(n288_0), gen_nil:cons5_0(b)) -> gen_nil:cons5_0(+(n288_0, b)), rt in Omega(1 + n288_0)
low(gen_0':s4_0(0), gen_nil:cons5_0(n1123_0)) -> gen_nil:cons5_0(n1123_0), rt in Omega(1 + n1123_0)
high(gen_0':s4_0(0), gen_nil:cons5_0(n1630_0)) -> gen_nil:cons5_0(0), rt in Omega(1 + n1630_0)


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
gen_nil:cons5_0(0) <=> nil
gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x))


The following defined symbols remain to be analysed:
quicksort
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(21) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(22)
BOUNDS(n^2, INF)