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))))
Renamed function symbols to avoid clashes with predefined symbol.
Runtime Complexity TRS:
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))))
Infered types.
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':false'1 :: true':false'
_hole_0':s'2 :: 0':s'
_hole_nil':cons'3 :: nil':cons'
_gen_0':s'4 :: Nat → 0':s'
_gen_nil':cons'5 :: Nat → nil':cons'
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'
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':false'1 :: true':false'
_hole_0':s'2 :: 0':s'
_hole_nil':cons'3 :: nil':cons'
_gen_0':s'4 :: Nat → 0':s'
_gen_nil':cons'5 :: Nat → nil':cons'
Generator Equations:
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))
_gen_nil':cons'5(0) ⇔ nil'
_gen_nil':cons'5(+(x, 1)) ⇔ cons'(0', _gen_nil':cons'5(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'
Proved the following rewrite lemma:
le'(_gen_0':s'4(_n7), _gen_0':s'4(_n7)) → true', rt ∈ Ω(1 + n7)
Induction Base:
le'(_gen_0':s'4(0), _gen_0':s'4(0)) →RΩ(1)
true'
Induction Step:
le'(_gen_0':s'4(+(_$n8, 1)), _gen_0':s'4(+(_$n8, 1))) →RΩ(1)
le'(_gen_0':s'4(_$n8), _gen_0':s'4(_$n8)) →IH
true'
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
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':false'1 :: true':false'
_hole_0':s'2 :: 0':s'
_hole_nil':cons'3 :: nil':cons'
_gen_0':s'4 :: Nat → 0':s'
_gen_nil':cons'5 :: Nat → nil':cons'
Lemmas:
le'(_gen_0':s'4(_n7), _gen_0':s'4(_n7)) → true', rt ∈ Ω(1 + n7)
Generator Equations:
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))
_gen_nil':cons'5(0) ⇔ nil'
_gen_nil':cons'5(+(x, 1)) ⇔ cons'(0', _gen_nil':cons'5(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'
Proved the following rewrite lemma:
app'(_gen_nil':cons'5(_n842), _gen_nil':cons'5(b)) → _gen_nil':cons'5(+(_n842, b)), rt ∈ Ω(1 + n842)
Induction Base:
app'(_gen_nil':cons'5(0), _gen_nil':cons'5(b)) →RΩ(1)
_gen_nil':cons'5(b)
Induction Step:
app'(_gen_nil':cons'5(+(_$n843, 1)), _gen_nil':cons'5(_b1073)) →RΩ(1)
cons'(0', app'(_gen_nil':cons'5(_$n843), _gen_nil':cons'5(_b1073))) →IH
cons'(0', _gen_nil':cons'5(+(_$n843, _b1073)))
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
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':false'1 :: true':false'
_hole_0':s'2 :: 0':s'
_hole_nil':cons'3 :: nil':cons'
_gen_0':s'4 :: Nat → 0':s'
_gen_nil':cons'5 :: Nat → nil':cons'
Lemmas:
le'(_gen_0':s'4(_n7), _gen_0':s'4(_n7)) → true', rt ∈ Ω(1 + n7)
app'(_gen_nil':cons'5(_n842), _gen_nil':cons'5(b)) → _gen_nil':cons'5(+(_n842, b)), rt ∈ Ω(1 + n842)
Generator Equations:
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))
_gen_nil':cons'5(0) ⇔ nil'
_gen_nil':cons'5(+(x, 1)) ⇔ cons'(0', _gen_nil':cons'5(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'
Proved the following rewrite lemma:
low'(_gen_0':s'4(0), _gen_nil':cons'5(_n2145)) → _gen_nil':cons'5(_n2145), rt ∈ Ω(1 + n2145)
Induction Base:
low'(_gen_0':s'4(0), _gen_nil':cons'5(0)) →RΩ(1)
nil'
Induction Step:
low'(_gen_0':s'4(0), _gen_nil':cons'5(+(_$n2146, 1))) →RΩ(1)
iflow'(le'(0', _gen_0':s'4(0)), _gen_0':s'4(0), cons'(0', _gen_nil':cons'5(_$n2146))) →LΩ(1)
iflow'(true', _gen_0':s'4(0), cons'(0', _gen_nil':cons'5(_$n2146))) →RΩ(1)
cons'(0', low'(_gen_0':s'4(0), _gen_nil':cons'5(_$n2146))) →IH
cons'(0', _gen_nil':cons'5(_$n2146))
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
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':false'1 :: true':false'
_hole_0':s'2 :: 0':s'
_hole_nil':cons'3 :: nil':cons'
_gen_0':s'4 :: Nat → 0':s'
_gen_nil':cons'5 :: Nat → nil':cons'
Lemmas:
le'(_gen_0':s'4(_n7), _gen_0':s'4(_n7)) → true', rt ∈ Ω(1 + n7)
app'(_gen_nil':cons'5(_n842), _gen_nil':cons'5(b)) → _gen_nil':cons'5(+(_n842, b)), rt ∈ Ω(1 + n842)
low'(_gen_0':s'4(0), _gen_nil':cons'5(_n2145)) → _gen_nil':cons'5(_n2145), rt ∈ Ω(1 + n2145)
Generator Equations:
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))
_gen_nil':cons'5(0) ⇔ nil'
_gen_nil':cons'5(+(x, 1)) ⇔ cons'(0', _gen_nil':cons'5(x))
The following defined symbols remain to be analysed:
high', quicksort'
They will be analysed ascendingly in the following order:
high' < quicksort'
Proved the following rewrite lemma:
high'(_gen_0':s'4(0), _gen_nil':cons'5(_n4688)) → _gen_nil':cons'5(0), rt ∈ Ω(1 + n4688)
Induction Base:
high'(_gen_0':s'4(0), _gen_nil':cons'5(0)) →RΩ(1)
nil'
Induction Step:
high'(_gen_0':s'4(0), _gen_nil':cons'5(+(_$n4689, 1))) →RΩ(1)
ifhigh'(le'(0', _gen_0':s'4(0)), _gen_0':s'4(0), cons'(0', _gen_nil':cons'5(_$n4689))) →LΩ(1)
ifhigh'(true', _gen_0':s'4(0), cons'(0', _gen_nil':cons'5(_$n4689))) →RΩ(1)
high'(_gen_0':s'4(0), _gen_nil':cons'5(_$n4689)) →IH
_gen_nil':cons'5(0)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
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':false'1 :: true':false'
_hole_0':s'2 :: 0':s'
_hole_nil':cons'3 :: nil':cons'
_gen_0':s'4 :: Nat → 0':s'
_gen_nil':cons'5 :: Nat → nil':cons'
Lemmas:
le'(_gen_0':s'4(_n7), _gen_0':s'4(_n7)) → true', rt ∈ Ω(1 + n7)
app'(_gen_nil':cons'5(_n842), _gen_nil':cons'5(b)) → _gen_nil':cons'5(+(_n842, b)), rt ∈ Ω(1 + n842)
low'(_gen_0':s'4(0), _gen_nil':cons'5(_n2145)) → _gen_nil':cons'5(_n2145), rt ∈ Ω(1 + n2145)
high'(_gen_0':s'4(0), _gen_nil':cons'5(_n4688)) → _gen_nil':cons'5(0), rt ∈ Ω(1 + n4688)
Generator Equations:
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))
_gen_nil':cons'5(0) ⇔ nil'
_gen_nil':cons'5(+(x, 1)) ⇔ cons'(0', _gen_nil':cons'5(x))
The following defined symbols remain to be analysed:
quicksort'
Proved the following rewrite lemma:
quicksort'(_gen_nil':cons'5(_n7087)) → _gen_nil':cons'5(_n7087), rt ∈ Ω(1 + n7087 + n70872)
Induction Base:
quicksort'(_gen_nil':cons'5(0)) →RΩ(1)
nil'
Induction Step:
quicksort'(_gen_nil':cons'5(+(_$n7088, 1))) →RΩ(1)
app'(quicksort'(low'(0', _gen_nil':cons'5(_$n7088))), cons'(0', quicksort'(high'(0', _gen_nil':cons'5(_$n7088))))) →LΩ(1 + $n7088)
app'(quicksort'(_gen_nil':cons'5(_$n7088)), cons'(0', quicksort'(high'(0', _gen_nil':cons'5(_$n7088))))) →IH
app'(_gen_nil':cons'5(_$n7088), cons'(0', quicksort'(high'(0', _gen_nil':cons'5(_$n7088))))) →LΩ(1 + $n7088)
app'(_gen_nil':cons'5(_$n7088), cons'(0', quicksort'(_gen_nil':cons'5(0)))) →RΩ(1)
app'(_gen_nil':cons'5(_$n7088), cons'(0', nil')) →LΩ(1 + $n7088)
_gen_nil':cons'5(+(_$n7088, +(0, 1)))
We have rt ∈ Ω(n2) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n2).
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':false'1 :: true':false'
_hole_0':s'2 :: 0':s'
_hole_nil':cons'3 :: nil':cons'
_gen_0':s'4 :: Nat → 0':s'
_gen_nil':cons'5 :: Nat → nil':cons'
Lemmas:
le'(_gen_0':s'4(_n7), _gen_0':s'4(_n7)) → true', rt ∈ Ω(1 + n7)
app'(_gen_nil':cons'5(_n842), _gen_nil':cons'5(b)) → _gen_nil':cons'5(+(_n842, b)), rt ∈ Ω(1 + n842)
low'(_gen_0':s'4(0), _gen_nil':cons'5(_n2145)) → _gen_nil':cons'5(_n2145), rt ∈ Ω(1 + n2145)
high'(_gen_0':s'4(0), _gen_nil':cons'5(_n4688)) → _gen_nil':cons'5(0), rt ∈ Ω(1 + n4688)
quicksort'(_gen_nil':cons'5(_n7087)) → _gen_nil':cons'5(_n7087), rt ∈ Ω(1 + n7087 + n70872)
Generator Equations:
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))
_gen_nil':cons'5(0) ⇔ nil'
_gen_nil':cons'5(+(x, 1)) ⇔ cons'(0', _gen_nil':cons'5(x))
No more defined symbols left to analyse.
The lowerbound Ω(n2) was proven with the following lemma:
quicksort'(_gen_nil':cons'5(_n7087)) → _gen_nil':cons'5(_n7087), rt ∈ Ω(1 + n7087 + n70872)