/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 283 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), 43 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 74 ms]
        (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 0 ms]
        (18) typed CpxTrs
        (19) RewriteLemmaProof [LOWER BOUND(ID), 10 ms]
        (20) typed CpxTrs
        (21) RewriteLemmaProof [LOWER BOUND(ID), 36 ms]
        (22) proven lower bound
        (23) LowerBoundPropagationProof [FINISHED, 0 ms]
        (24) 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:

   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   app(nil, y) -> y
   app(add(n, x), y) -> add(n, app(x, y))
   low(n, nil) -> nil
   low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
   if_low(true, n, add(m, x)) -> add(m, low(n, x))
   if_low(false, n, add(m, x)) -> low(n, x)
   high(n, nil) -> nil
   high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
   if_high(true, n, add(m, x)) -> high(n, x)
   if_high(false, n, add(m, x)) -> add(m, high(n, x))
   quicksort(nil) -> nil
   quicksort(add(n, x)) -> app(quicksort(low(n, x)), add(n, quicksort(high(n, x))))

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:

   minus(x, 0') -> x
   minus(s(x), s(y)) -> minus(x, y)
   quot(0', s(y)) -> 0'
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   le(0', y) -> true
   le(s(x), 0') -> false
   le(s(x), s(y)) -> le(x, y)
   app(nil, y) -> y
   app(add(n, x), y) -> add(n, app(x, y))
   low(n, nil) -> nil
   low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
   if_low(true, n, add(m, x)) -> add(m, low(n, x))
   if_low(false, n, add(m, x)) -> low(n, x)
   high(n, nil) -> nil
   high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
   if_high(true, n, add(m, x)) -> high(n, x)
   if_high(false, n, add(m, x)) -> add(m, high(n, x))
   quicksort(nil) -> nil
   quicksort(add(n, x)) -> app(quicksort(low(n, x)), add(n, quicksort(high(n, x))))

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

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

(4)
Obligation:
TRS:
Rules:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
low(n, nil) -> nil
low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
if_low(true, n, add(m, x)) -> add(m, low(n, x))
if_low(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
if_high(true, n, add(m, x)) -> high(n, x)
if_high(false, n, add(m, x)) -> add(m, high(n, x))
quicksort(nil) -> nil
quicksort(add(n, x)) -> app(quicksort(low(n, x)), add(n, quicksort(high(n, x))))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
le :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
app :: nil:add -> nil:add -> nil:add
nil :: nil:add
add :: 0':s -> nil:add -> nil:add
low :: 0':s -> nil:add -> nil:add
if_low :: true:false -> 0':s -> nil:add -> nil:add
high :: 0':s -> nil:add -> nil:add
if_high :: true:false -> 0':s -> nil:add -> nil:add
quicksort :: nil:add -> nil:add
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
hole_nil:add3_0 :: nil:add
gen_0':s4_0 :: Nat -> 0':s
gen_nil:add5_0 :: Nat -> nil:add

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

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

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

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

(6)
Obligation:
TRS:
Rules:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
low(n, nil) -> nil
low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
if_low(true, n, add(m, x)) -> add(m, low(n, x))
if_low(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
if_high(true, n, add(m, x)) -> high(n, x)
if_high(false, n, add(m, x)) -> add(m, high(n, x))
quicksort(nil) -> nil
quicksort(add(n, x)) -> app(quicksort(low(n, x)), add(n, quicksort(high(n, x))))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
le :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
app :: nil:add -> nil:add -> nil:add
nil :: nil:add
add :: 0':s -> nil:add -> nil:add
low :: 0':s -> nil:add -> nil:add
if_low :: true:false -> 0':s -> nil:add -> nil:add
high :: 0':s -> nil:add -> nil:add
if_high :: true:false -> 0':s -> nil:add -> nil:add
quicksort :: nil:add -> nil:add
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
hole_nil:add3_0 :: nil:add
gen_0':s4_0 :: Nat -> 0':s
gen_nil:add5_0 :: Nat -> nil:add


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
gen_nil:add5_0(0) <=> nil
gen_nil:add5_0(+(x, 1)) <=> add(0', gen_nil:add5_0(x))


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

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

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> gen_0':s4_0(0), rt in Omega(1 + n7_0)

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

Induction Step:
minus(gen_0':s4_0(+(n7_0, 1)), gen_0':s4_0(+(n7_0, 1))) ->_R^Omega(1)
minus(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) ->_IH
gen_0':s4_0(0)

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:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
low(n, nil) -> nil
low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
if_low(true, n, add(m, x)) -> add(m, low(n, x))
if_low(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
if_high(true, n, add(m, x)) -> high(n, x)
if_high(false, n, add(m, x)) -> add(m, high(n, x))
quicksort(nil) -> nil
quicksort(add(n, x)) -> app(quicksort(low(n, x)), add(n, quicksort(high(n, x))))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
le :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
app :: nil:add -> nil:add -> nil:add
nil :: nil:add
add :: 0':s -> nil:add -> nil:add
low :: 0':s -> nil:add -> nil:add
if_low :: true:false -> 0':s -> nil:add -> nil:add
high :: 0':s -> nil:add -> nil:add
if_high :: true:false -> 0':s -> nil:add -> nil:add
quicksort :: nil:add -> nil:add
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
hole_nil:add3_0 :: nil:add
gen_0':s4_0 :: Nat -> 0':s
gen_nil:add5_0 :: Nat -> nil:add


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
gen_nil:add5_0(0) <=> nil
gen_nil:add5_0(+(x, 1)) <=> add(0', gen_nil:add5_0(x))


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

They will be analysed ascendingly in the following order:
minus < quot
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:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
low(n, nil) -> nil
low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
if_low(true, n, add(m, x)) -> add(m, low(n, x))
if_low(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
if_high(true, n, add(m, x)) -> high(n, x)
if_high(false, n, add(m, x)) -> add(m, high(n, x))
quicksort(nil) -> nil
quicksort(add(n, x)) -> app(quicksort(low(n, x)), add(n, quicksort(high(n, x))))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
le :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
app :: nil:add -> nil:add -> nil:add
nil :: nil:add
add :: 0':s -> nil:add -> nil:add
low :: 0':s -> nil:add -> nil:add
if_low :: true:false -> 0':s -> nil:add -> nil:add
high :: 0':s -> nil:add -> nil:add
if_high :: true:false -> 0':s -> nil:add -> nil:add
quicksort :: nil:add -> nil:add
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
hole_nil:add3_0 :: nil:add
gen_0':s4_0 :: Nat -> 0':s
gen_nil:add5_0 :: Nat -> nil:add


Lemmas:
minus(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> gen_0':s4_0(0), 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:add5_0(0) <=> nil
gen_nil:add5_0(+(x, 1)) <=> add(0', gen_nil:add5_0(x))


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

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

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
le(gen_0':s4_0(n507_0), gen_0':s4_0(n507_0)) -> true, rt in Omega(1 + n507_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(+(n507_0, 1)), gen_0':s4_0(+(n507_0, 1))) ->_R^Omega(1)
le(gen_0':s4_0(n507_0), gen_0':s4_0(n507_0)) ->_IH
true

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

(14)
Obligation:
TRS:
Rules:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
low(n, nil) -> nil
low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
if_low(true, n, add(m, x)) -> add(m, low(n, x))
if_low(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
if_high(true, n, add(m, x)) -> high(n, x)
if_high(false, n, add(m, x)) -> add(m, high(n, x))
quicksort(nil) -> nil
quicksort(add(n, x)) -> app(quicksort(low(n, x)), add(n, quicksort(high(n, x))))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
le :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
app :: nil:add -> nil:add -> nil:add
nil :: nil:add
add :: 0':s -> nil:add -> nil:add
low :: 0':s -> nil:add -> nil:add
if_low :: true:false -> 0':s -> nil:add -> nil:add
high :: 0':s -> nil:add -> nil:add
if_high :: true:false -> 0':s -> nil:add -> nil:add
quicksort :: nil:add -> nil:add
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
hole_nil:add3_0 :: nil:add
gen_0':s4_0 :: Nat -> 0':s
gen_nil:add5_0 :: Nat -> nil:add


Lemmas:
minus(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> gen_0':s4_0(0), rt in Omega(1 + n7_0)
le(gen_0':s4_0(n507_0), gen_0':s4_0(n507_0)) -> true, rt in Omega(1 + n507_0)


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
gen_nil:add5_0(0) <=> nil
gen_nil:add5_0(+(x, 1)) <=> add(0', gen_nil:add5_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

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
app(gen_nil:add5_0(n818_0), gen_nil:add5_0(b)) -> gen_nil:add5_0(+(n818_0, b)), rt in Omega(1 + n818_0)

Induction Base:
app(gen_nil:add5_0(0), gen_nil:add5_0(b)) ->_R^Omega(1)
gen_nil:add5_0(b)

Induction Step:
app(gen_nil:add5_0(+(n818_0, 1)), gen_nil:add5_0(b)) ->_R^Omega(1)
add(0', app(gen_nil:add5_0(n818_0), gen_nil:add5_0(b))) ->_IH
add(0', gen_nil:add5_0(+(b, c819_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:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
low(n, nil) -> nil
low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
if_low(true, n, add(m, x)) -> add(m, low(n, x))
if_low(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
if_high(true, n, add(m, x)) -> high(n, x)
if_high(false, n, add(m, x)) -> add(m, high(n, x))
quicksort(nil) -> nil
quicksort(add(n, x)) -> app(quicksort(low(n, x)), add(n, quicksort(high(n, x))))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
le :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
app :: nil:add -> nil:add -> nil:add
nil :: nil:add
add :: 0':s -> nil:add -> nil:add
low :: 0':s -> nil:add -> nil:add
if_low :: true:false -> 0':s -> nil:add -> nil:add
high :: 0':s -> nil:add -> nil:add
if_high :: true:false -> 0':s -> nil:add -> nil:add
quicksort :: nil:add -> nil:add
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
hole_nil:add3_0 :: nil:add
gen_0':s4_0 :: Nat -> 0':s
gen_nil:add5_0 :: Nat -> nil:add


Lemmas:
minus(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> gen_0':s4_0(0), rt in Omega(1 + n7_0)
le(gen_0':s4_0(n507_0), gen_0':s4_0(n507_0)) -> true, rt in Omega(1 + n507_0)
app(gen_nil:add5_0(n818_0), gen_nil:add5_0(b)) -> gen_nil:add5_0(+(n818_0, b)), rt in Omega(1 + n818_0)


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
gen_nil:add5_0(0) <=> nil
gen_nil:add5_0(+(x, 1)) <=> add(0', gen_nil:add5_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

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
low(gen_0':s4_0(0), gen_nil:add5_0(n1743_0)) -> gen_nil:add5_0(n1743_0), rt in Omega(1 + n1743_0)

Induction Base:
low(gen_0':s4_0(0), gen_nil:add5_0(0)) ->_R^Omega(1)
nil

Induction Step:
low(gen_0':s4_0(0), gen_nil:add5_0(+(n1743_0, 1))) ->_R^Omega(1)
if_low(le(0', gen_0':s4_0(0)), gen_0':s4_0(0), add(0', gen_nil:add5_0(n1743_0))) ->_L^Omega(1)
if_low(true, gen_0':s4_0(0), add(0', gen_nil:add5_0(n1743_0))) ->_R^Omega(1)
add(0', low(gen_0':s4_0(0), gen_nil:add5_0(n1743_0))) ->_IH
add(0', gen_nil:add5_0(c1744_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:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
low(n, nil) -> nil
low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
if_low(true, n, add(m, x)) -> add(m, low(n, x))
if_low(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
if_high(true, n, add(m, x)) -> high(n, x)
if_high(false, n, add(m, x)) -> add(m, high(n, x))
quicksort(nil) -> nil
quicksort(add(n, x)) -> app(quicksort(low(n, x)), add(n, quicksort(high(n, x))))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
le :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
app :: nil:add -> nil:add -> nil:add
nil :: nil:add
add :: 0':s -> nil:add -> nil:add
low :: 0':s -> nil:add -> nil:add
if_low :: true:false -> 0':s -> nil:add -> nil:add
high :: 0':s -> nil:add -> nil:add
if_high :: true:false -> 0':s -> nil:add -> nil:add
quicksort :: nil:add -> nil:add
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
hole_nil:add3_0 :: nil:add
gen_0':s4_0 :: Nat -> 0':s
gen_nil:add5_0 :: Nat -> nil:add


Lemmas:
minus(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> gen_0':s4_0(0), rt in Omega(1 + n7_0)
le(gen_0':s4_0(n507_0), gen_0':s4_0(n507_0)) -> true, rt in Omega(1 + n507_0)
app(gen_nil:add5_0(n818_0), gen_nil:add5_0(b)) -> gen_nil:add5_0(+(n818_0, b)), rt in Omega(1 + n818_0)
low(gen_0':s4_0(0), gen_nil:add5_0(n1743_0)) -> gen_nil:add5_0(n1743_0), rt in Omega(1 + n1743_0)


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
gen_nil:add5_0(0) <=> nil
gen_nil:add5_0(+(x, 1)) <=> add(0', gen_nil:add5_0(x))


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

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

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
high(gen_0':s4_0(0), gen_nil:add5_0(n2335_0)) -> gen_nil:add5_0(0), rt in Omega(1 + n2335_0)

Induction Base:
high(gen_0':s4_0(0), gen_nil:add5_0(0)) ->_R^Omega(1)
nil

Induction Step:
high(gen_0':s4_0(0), gen_nil:add5_0(+(n2335_0, 1))) ->_R^Omega(1)
if_high(le(0', gen_0':s4_0(0)), gen_0':s4_0(0), add(0', gen_nil:add5_0(n2335_0))) ->_L^Omega(1)
if_high(true, gen_0':s4_0(0), add(0', gen_nil:add5_0(n2335_0))) ->_R^Omega(1)
high(gen_0':s4_0(0), gen_nil:add5_0(n2335_0)) ->_IH
gen_nil:add5_0(0)

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

(20)
Obligation:
TRS:
Rules:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
low(n, nil) -> nil
low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
if_low(true, n, add(m, x)) -> add(m, low(n, x))
if_low(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
if_high(true, n, add(m, x)) -> high(n, x)
if_high(false, n, add(m, x)) -> add(m, high(n, x))
quicksort(nil) -> nil
quicksort(add(n, x)) -> app(quicksort(low(n, x)), add(n, quicksort(high(n, x))))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
le :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
app :: nil:add -> nil:add -> nil:add
nil :: nil:add
add :: 0':s -> nil:add -> nil:add
low :: 0':s -> nil:add -> nil:add
if_low :: true:false -> 0':s -> nil:add -> nil:add
high :: 0':s -> nil:add -> nil:add
if_high :: true:false -> 0':s -> nil:add -> nil:add
quicksort :: nil:add -> nil:add
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
hole_nil:add3_0 :: nil:add
gen_0':s4_0 :: Nat -> 0':s
gen_nil:add5_0 :: Nat -> nil:add


Lemmas:
minus(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> gen_0':s4_0(0), rt in Omega(1 + n7_0)
le(gen_0':s4_0(n507_0), gen_0':s4_0(n507_0)) -> true, rt in Omega(1 + n507_0)
app(gen_nil:add5_0(n818_0), gen_nil:add5_0(b)) -> gen_nil:add5_0(+(n818_0, b)), rt in Omega(1 + n818_0)
low(gen_0':s4_0(0), gen_nil:add5_0(n1743_0)) -> gen_nil:add5_0(n1743_0), rt in Omega(1 + n1743_0)
high(gen_0':s4_0(0), gen_nil:add5_0(n2335_0)) -> gen_nil:add5_0(0), rt in Omega(1 + n2335_0)


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
gen_nil:add5_0(0) <=> nil
gen_nil:add5_0(+(x, 1)) <=> add(0', gen_nil:add5_0(x))


The following defined symbols remain to be analysed:
quicksort
----------------------------------------

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
quicksort(gen_nil:add5_0(n2923_0)) -> gen_nil:add5_0(n2923_0), rt in Omega(1 + n2923_0 + n2923_0^2)

Induction Base:
quicksort(gen_nil:add5_0(0)) ->_R^Omega(1)
nil

Induction Step:
quicksort(gen_nil:add5_0(+(n2923_0, 1))) ->_R^Omega(1)
app(quicksort(low(0', gen_nil:add5_0(n2923_0))), add(0', quicksort(high(0', gen_nil:add5_0(n2923_0))))) ->_L^Omega(1 + n2923_0)
app(quicksort(gen_nil:add5_0(n2923_0)), add(0', quicksort(high(0', gen_nil:add5_0(n2923_0))))) ->_IH
app(gen_nil:add5_0(c2924_0), add(0', quicksort(high(0', gen_nil:add5_0(n2923_0))))) ->_L^Omega(1 + n2923_0)
app(gen_nil:add5_0(n2923_0), add(0', quicksort(gen_nil:add5_0(0)))) ->_R^Omega(1)
app(gen_nil:add5_0(n2923_0), add(0', nil)) ->_L^Omega(1 + n2923_0)
gen_nil:add5_0(+(n2923_0, +(0, 1)))

We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
----------------------------------------

(22)
Obligation:
Proved the lower bound n^2 for the following obligation:

TRS:
Rules:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
low(n, nil) -> nil
low(n, add(m, x)) -> if_low(le(m, n), n, add(m, x))
if_low(true, n, add(m, x)) -> add(m, low(n, x))
if_low(false, n, add(m, x)) -> low(n, x)
high(n, nil) -> nil
high(n, add(m, x)) -> if_high(le(m, n), n, add(m, x))
if_high(true, n, add(m, x)) -> high(n, x)
if_high(false, n, add(m, x)) -> add(m, high(n, x))
quicksort(nil) -> nil
quicksort(add(n, x)) -> app(quicksort(low(n, x)), add(n, quicksort(high(n, x))))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
le :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
app :: nil:add -> nil:add -> nil:add
nil :: nil:add
add :: 0':s -> nil:add -> nil:add
low :: 0':s -> nil:add -> nil:add
if_low :: true:false -> 0':s -> nil:add -> nil:add
high :: 0':s -> nil:add -> nil:add
if_high :: true:false -> 0':s -> nil:add -> nil:add
quicksort :: nil:add -> nil:add
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
hole_nil:add3_0 :: nil:add
gen_0':s4_0 :: Nat -> 0':s
gen_nil:add5_0 :: Nat -> nil:add


Lemmas:
minus(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> gen_0':s4_0(0), rt in Omega(1 + n7_0)
le(gen_0':s4_0(n507_0), gen_0':s4_0(n507_0)) -> true, rt in Omega(1 + n507_0)
app(gen_nil:add5_0(n818_0), gen_nil:add5_0(b)) -> gen_nil:add5_0(+(n818_0, b)), rt in Omega(1 + n818_0)
low(gen_0':s4_0(0), gen_nil:add5_0(n1743_0)) -> gen_nil:add5_0(n1743_0), rt in Omega(1 + n1743_0)
high(gen_0':s4_0(0), gen_nil:add5_0(n2335_0)) -> gen_nil:add5_0(0), rt in Omega(1 + n2335_0)


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
gen_nil:add5_0(0) <=> nil
gen_nil:add5_0(+(x, 1)) <=> add(0', gen_nil:add5_0(x))


The following defined symbols remain to be analysed:
quicksort
----------------------------------------

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

(24)
BOUNDS(n^2, INF)