The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

max(nil) → 0
max(cons(x, nil)) → x
max(cons(x, cons(y, xs))) → if1(ge(x, y), x, y, xs)
if1(true, x, y, xs) → max(cons(x, xs))
if1(false, x, y, xs) → max(cons(y, xs))
del(x, nil) → nil
del(x, cons(y, xs)) → if2(eq(x, y), x, y, xs)
if2(true, x, y, xs) → xs
if2(false, x, y, xs) → cons(y, del(x, xs))
eq(0, 0) → true
eq(0, s(y)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)
sort(xs) → if3(empty(xs), xs)
if3(true, xs) → nil
if3(false, xs) → sort(del(max(xs), xs))
empty(nil) → true
empty(cons(x, xs)) → false
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
max(cons(x, cons(0, xs))) →⁺ max(cons(x, xs))
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [xs / cons(0, xs)].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)

(3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(4) Obligation:

Runtime Complexity Relative TRS:
The TRS R consists of the following rules:

max(nil) → 0'
max(cons(x, nil)) → x
max(cons(x, cons(y, xs))) → if1(ge(x, y), x, y, xs)
if1(true, x, y, xs) → max(cons(x, xs))
if1(false, x, y, xs) → max(cons(y, xs))
del(x, nil) → nil
del(x, cons(y, xs)) → if2(eq(x, y), x, y, xs)
if2(true, x, y, xs) → xs
if2(false, x, y, xs) → cons(y, del(x, xs))
eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
sort(xs) → if3(empty(xs), xs)
if3(true, xs) → nil
if3(false, xs) → sort(del(max(xs), xs))
empty(nil) → true
empty(cons(x, xs)) → false
ge(x, 0') → true
ge(0', s(x)) → false
ge(s(x), s(y)) → ge(x, y)

S is empty.
Rewrite Strategy: FULL

(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(6) Obligation:

TRS:
Rules:
max(nil) → 0'
max(cons(x, nil)) → x
max(cons(x, cons(y, xs))) → if1(ge(x, y), x, y, xs)
if1(true, x, y, xs) → max(cons(x, xs))
if1(false, x, y, xs) → max(cons(y, xs))
del(x, nil) → nil
del(x, cons(y, xs)) → if2(eq(x, y), x, y, xs)
if2(true, x, y, xs) → xs
if2(false, x, y, xs) → cons(y, del(x, xs))
eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
sort(xs) → if3(empty(xs), xs)
if3(true, xs) → nil
if3(false, xs) → sort(del(max(xs), xs))
empty(nil) → true
empty(cons(x, xs)) → false
ge(x, 0') → true
ge(0', s(x)) → false
ge(s(x), s(y)) → ge(x, y)

Types:
max :: nil:cons → 0':s
nil :: nil:cons
0' :: 0':s
cons :: 0':s → nil:cons → nil:cons
if1 :: true:false → 0':s → 0':s → nil:cons → 0':s
ge :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
del :: 0':s → nil:cons → nil:cons
if2 :: true:false → 0':s → 0':s → nil:cons → nil:cons
eq :: 0':s → 0':s → true:false
s :: 0':s → 0':s
sort :: nil:cons → nil:cons
if3 :: true:false → nil:cons → nil:cons
empty :: nil:cons → true:false
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat → 0':s
gen_nil:cons5_0 :: Nat → nil:cons

(7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
max, ge, del, eq, sort

They will be analysed ascendingly in the following order:
ge < max
max < sort
eq < del
del < sort

(8) Obligation:

TRS:
Rules:
max(nil) → 0'
max(cons(x, nil)) → x
max(cons(x, cons(y, xs))) → if1(ge(x, y), x, y, xs)
if1(true, x, y, xs) → max(cons(x, xs))
if1(false, x, y, xs) → max(cons(y, xs))
del(x, nil) → nil
del(x, cons(y, xs)) → if2(eq(x, y), x, y, xs)
if2(true, x, y, xs) → xs
if2(false, x, y, xs) → cons(y, del(x, xs))
eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
sort(xs) → if3(empty(xs), xs)
if3(true, xs) → nil
if3(false, xs) → sort(del(max(xs), xs))
empty(nil) → true
empty(cons(x, xs)) → false
ge(x, 0') → true
ge(0', s(x)) → false
ge(s(x), s(y)) → ge(x, y)

Types:
max :: nil:cons → 0':s
nil :: nil:cons
0' :: 0':s
cons :: 0':s → nil:cons → nil:cons
if1 :: true:false → 0':s → 0':s → nil:cons → 0':s
ge :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
del :: 0':s → nil:cons → nil:cons
if2 :: true:false → 0':s → 0':s → nil:cons → nil:cons
eq :: 0':s → 0':s → true:false
s :: 0':s → 0':s
sort :: nil:cons → nil:cons
if3 :: true:false → nil:cons → nil:cons
empty :: nil:cons → true:false
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
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:
ge, max, del, eq, sort

They will be analysed ascendingly in the following order:
ge < max
max < sort
eq < del
del < sort

(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
ge(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)

Induction Base:
ge(gen_0':s4_0(0), gen_0':s4_0(0)) →_R^Ω(1)
true

Induction Step:
ge(gen_0':s4_0(+(n7_0, 1)), gen_0':s4_0(+(n7_0, 1))) →_R^Ω(1)
ge(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) →_IH
true

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(10) Complex Obligation (BEST)

(11) Obligation:

TRS:
Rules:
max(nil) → 0'
max(cons(x, nil)) → x
max(cons(x, cons(y, xs))) → if1(ge(x, y), x, y, xs)
if1(true, x, y, xs) → max(cons(x, xs))
if1(false, x, y, xs) → max(cons(y, xs))
del(x, nil) → nil
del(x, cons(y, xs)) → if2(eq(x, y), x, y, xs)
if2(true, x, y, xs) → xs
if2(false, x, y, xs) → cons(y, del(x, xs))
eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
sort(xs) → if3(empty(xs), xs)
if3(true, xs) → nil
if3(false, xs) → sort(del(max(xs), xs))
empty(nil) → true
empty(cons(x, xs)) → false
ge(x, 0') → true
ge(0', s(x)) → false
ge(s(x), s(y)) → ge(x, y)

Types:
max :: nil:cons → 0':s
nil :: nil:cons
0' :: 0':s
cons :: 0':s → nil:cons → nil:cons
if1 :: true:false → 0':s → 0':s → nil:cons → 0':s
ge :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
del :: 0':s → nil:cons → nil:cons
if2 :: true:false → 0':s → 0':s → nil:cons → nil:cons
eq :: 0':s → 0':s → true:false
s :: 0':s → 0':s
sort :: nil:cons → nil:cons
if3 :: true:false → nil:cons → nil:cons
empty :: nil:cons → true:false
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat → 0':s
gen_nil:cons5_0 :: Nat → nil:cons

Lemmas:
ge(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)

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:
max, del, eq, sort

They will be analysed ascendingly in the following order:
max < sort
eq < del
del < sort

(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
max(gen_nil:cons5_0(+(1, n348_0))) → gen_0':s4_0(0), rt ∈ Ω(1 + n348₀)

Induction Base:
max(gen_nil:cons5_0(+(1, 0))) →_R^Ω(1)
0'

Induction Step:
max(gen_nil:cons5_0(+(1, +(n348_0, 1)))) →_R^Ω(1)
if1(ge(0', 0'), 0', 0', gen_nil:cons5_0(n348_0)) →_L^Ω(1)
if1(true, 0', 0', gen_nil:cons5_0(n348_0)) →_R^Ω(1)
max(cons(0', gen_nil:cons5_0(n348_0))) →_IH
gen_0':s4_0(0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(13) Complex Obligation (BEST)

(14) Obligation:

TRS:
Rules:
max(nil) → 0'
max(cons(x, nil)) → x
max(cons(x, cons(y, xs))) → if1(ge(x, y), x, y, xs)
if1(true, x, y, xs) → max(cons(x, xs))
if1(false, x, y, xs) → max(cons(y, xs))
del(x, nil) → nil
del(x, cons(y, xs)) → if2(eq(x, y), x, y, xs)
if2(true, x, y, xs) → xs
if2(false, x, y, xs) → cons(y, del(x, xs))
eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
sort(xs) → if3(empty(xs), xs)
if3(true, xs) → nil
if3(false, xs) → sort(del(max(xs), xs))
empty(nil) → true
empty(cons(x, xs)) → false
ge(x, 0') → true
ge(0', s(x)) → false
ge(s(x), s(y)) → ge(x, y)

Types:
max :: nil:cons → 0':s
nil :: nil:cons
0' :: 0':s
cons :: 0':s → nil:cons → nil:cons
if1 :: true:false → 0':s → 0':s → nil:cons → 0':s
ge :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
del :: 0':s → nil:cons → nil:cons
if2 :: true:false → 0':s → 0':s → nil:cons → nil:cons
eq :: 0':s → 0':s → true:false
s :: 0':s → 0':s
sort :: nil:cons → nil:cons
if3 :: true:false → nil:cons → nil:cons
empty :: nil:cons → true:false
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat → 0':s
gen_nil:cons5_0 :: Nat → nil:cons

Lemmas:
ge(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)
max(gen_nil:cons5_0(+(1, n348_0))) → gen_0':s4_0(0), rt ∈ Ω(1 + n348₀)

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:
eq, del, sort

They will be analysed ascendingly in the following order:
eq < del
del < sort

(15) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
eq(gen_0':s4_0(n797_0), gen_0':s4_0(n797_0)) → true, rt ∈ Ω(1 + n797₀)

Induction Base:
eq(gen_0':s4_0(0), gen_0':s4_0(0)) →_R^Ω(1)
true

Induction Step:
eq(gen_0':s4_0(+(n797_0, 1)), gen_0':s4_0(+(n797_0, 1))) →_R^Ω(1)
eq(gen_0':s4_0(n797_0), gen_0':s4_0(n797_0)) →_IH
true

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(16) Complex Obligation (BEST)

(17) Obligation:

TRS:
Rules:
max(nil) → 0'
max(cons(x, nil)) → x
max(cons(x, cons(y, xs))) → if1(ge(x, y), x, y, xs)
if1(true, x, y, xs) → max(cons(x, xs))
if1(false, x, y, xs) → max(cons(y, xs))
del(x, nil) → nil
del(x, cons(y, xs)) → if2(eq(x, y), x, y, xs)
if2(true, x, y, xs) → xs
if2(false, x, y, xs) → cons(y, del(x, xs))
eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
sort(xs) → if3(empty(xs), xs)
if3(true, xs) → nil
if3(false, xs) → sort(del(max(xs), xs))
empty(nil) → true
empty(cons(x, xs)) → false
ge(x, 0') → true
ge(0', s(x)) → false
ge(s(x), s(y)) → ge(x, y)

Types:
max :: nil:cons → 0':s
nil :: nil:cons
0' :: 0':s
cons :: 0':s → nil:cons → nil:cons
if1 :: true:false → 0':s → 0':s → nil:cons → 0':s
ge :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
del :: 0':s → nil:cons → nil:cons
if2 :: true:false → 0':s → 0':s → nil:cons → nil:cons
eq :: 0':s → 0':s → true:false
s :: 0':s → 0':s
sort :: nil:cons → nil:cons
if3 :: true:false → nil:cons → nil:cons
empty :: nil:cons → true:false
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat → 0':s
gen_nil:cons5_0 :: Nat → nil:cons

Lemmas:
ge(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)
max(gen_nil:cons5_0(+(1, n348_0))) → gen_0':s4_0(0), rt ∈ Ω(1 + n348₀)
eq(gen_0':s4_0(n797_0), gen_0':s4_0(n797_0)) → true, rt ∈ Ω(1 + n797₀)

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:
del, sort

They will be analysed ascendingly in the following order:
del < sort

(18) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol del.

(19) Obligation:

TRS:
Rules:
max(nil) → 0'
max(cons(x, nil)) → x
max(cons(x, cons(y, xs))) → if1(ge(x, y), x, y, xs)
if1(true, x, y, xs) → max(cons(x, xs))
if1(false, x, y, xs) → max(cons(y, xs))
del(x, nil) → nil
del(x, cons(y, xs)) → if2(eq(x, y), x, y, xs)
if2(true, x, y, xs) → xs
if2(false, x, y, xs) → cons(y, del(x, xs))
eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
sort(xs) → if3(empty(xs), xs)
if3(true, xs) → nil
if3(false, xs) → sort(del(max(xs), xs))
empty(nil) → true
empty(cons(x, xs)) → false
ge(x, 0') → true
ge(0', s(x)) → false
ge(s(x), s(y)) → ge(x, y)

Types:
max :: nil:cons → 0':s
nil :: nil:cons
0' :: 0':s
cons :: 0':s → nil:cons → nil:cons
if1 :: true:false → 0':s → 0':s → nil:cons → 0':s
ge :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
del :: 0':s → nil:cons → nil:cons
if2 :: true:false → 0':s → 0':s → nil:cons → nil:cons
eq :: 0':s → 0':s → true:false
s :: 0':s → 0':s
sort :: nil:cons → nil:cons
if3 :: true:false → nil:cons → nil:cons
empty :: nil:cons → true:false
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat → 0':s
gen_nil:cons5_0 :: Nat → nil:cons

Lemmas:
ge(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)
max(gen_nil:cons5_0(+(1, n348_0))) → gen_0':s4_0(0), rt ∈ Ω(1 + n348₀)
eq(gen_0':s4_0(n797_0), gen_0':s4_0(n797_0)) → true, rt ∈ Ω(1 + n797₀)

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:
sort

(20) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
sort(gen_nil:cons5_0(n1546_0)) → gen_nil:cons5_0(0), rt ∈ Ω(1 + n1546₀ + n1546₀²)

Induction Base:
sort(gen_nil:cons5_0(0)) →_R^Ω(1)
if3(empty(gen_nil:cons5_0(0)), gen_nil:cons5_0(0)) →_R^Ω(1)
if3(true, gen_nil:cons5_0(0)) →_R^Ω(1)
nil

Induction Step:
sort(gen_nil:cons5_0(+(n1546_0, 1))) →_R^Ω(1)
if3(empty(gen_nil:cons5_0(+(n1546_0, 1))), gen_nil:cons5_0(+(n1546_0, 1))) →_R^Ω(1)
if3(false, gen_nil:cons5_0(+(1, n1546_0))) →_R^Ω(1)
sort(del(max(gen_nil:cons5_0(+(1, n1546_0))), gen_nil:cons5_0(+(1, n1546_0)))) →_L^{Ω(1 + n1546₀)}
sort(del(gen_0':s4_0(0), gen_nil:cons5_0(+(1, n1546_0)))) →_R^Ω(1)
sort(if2(eq(gen_0':s4_0(0), 0'), gen_0':s4_0(0), 0', gen_nil:cons5_0(n1546_0))) →_L^Ω(1)
sort(if2(true, gen_0':s4_0(0), 0', gen_nil:cons5_0(n1546_0))) →_R^Ω(1)
sort(gen_nil:cons5_0(n1546_0)) →_IH
gen_nil:cons5_0(0)

We have rt ∈ Ω(n²) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n²).

(21) Complex Obligation (BEST)

(22) Obligation:

TRS:
Rules:
max(nil) → 0'
max(cons(x, nil)) → x
max(cons(x, cons(y, xs))) → if1(ge(x, y), x, y, xs)
if1(true, x, y, xs) → max(cons(x, xs))
if1(false, x, y, xs) → max(cons(y, xs))
del(x, nil) → nil
del(x, cons(y, xs)) → if2(eq(x, y), x, y, xs)
if2(true, x, y, xs) → xs
if2(false, x, y, xs) → cons(y, del(x, xs))
eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
sort(xs) → if3(empty(xs), xs)
if3(true, xs) → nil
if3(false, xs) → sort(del(max(xs), xs))
empty(nil) → true
empty(cons(x, xs)) → false
ge(x, 0') → true
ge(0', s(x)) → false
ge(s(x), s(y)) → ge(x, y)

Types:
max :: nil:cons → 0':s
nil :: nil:cons
0' :: 0':s
cons :: 0':s → nil:cons → nil:cons
if1 :: true:false → 0':s → 0':s → nil:cons → 0':s
ge :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
del :: 0':s → nil:cons → nil:cons
if2 :: true:false → 0':s → 0':s → nil:cons → nil:cons
eq :: 0':s → 0':s → true:false
s :: 0':s → 0':s
sort :: nil:cons → nil:cons
if3 :: true:false → nil:cons → nil:cons
empty :: nil:cons → true:false
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat → 0':s
gen_nil:cons5_0 :: Nat → nil:cons

Lemmas:
ge(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)
max(gen_nil:cons5_0(+(1, n348_0))) → gen_0':s4_0(0), rt ∈ Ω(1 + n348₀)
eq(gen_0':s4_0(n797_0), gen_0':s4_0(n797_0)) → true, rt ∈ Ω(1 + n797₀)
sort(gen_nil:cons5_0(n1546_0)) → gen_nil:cons5_0(0), rt ∈ Ω(1 + n1546₀ + n1546₀²)

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))

No more defined symbols left to analyse.

(23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
sort(gen_nil:cons5_0(n1546_0)) → gen_nil:cons5_0(0), rt ∈ Ω(1 + n1546₀ + n1546₀²)

(24) BOUNDS(n^2, INF)

(25) Obligation:

TRS:
Rules:
max(nil) → 0'
max(cons(x, nil)) → x
max(cons(x, cons(y, xs))) → if1(ge(x, y), x, y, xs)
if1(true, x, y, xs) → max(cons(x, xs))
if1(false, x, y, xs) → max(cons(y, xs))
del(x, nil) → nil
del(x, cons(y, xs)) → if2(eq(x, y), x, y, xs)
if2(true, x, y, xs) → xs
if2(false, x, y, xs) → cons(y, del(x, xs))
eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
sort(xs) → if3(empty(xs), xs)
if3(true, xs) → nil
if3(false, xs) → sort(del(max(xs), xs))
empty(nil) → true
empty(cons(x, xs)) → false
ge(x, 0') → true
ge(0', s(x)) → false
ge(s(x), s(y)) → ge(x, y)

Types:
max :: nil:cons → 0':s
nil :: nil:cons
0' :: 0':s
cons :: 0':s → nil:cons → nil:cons
if1 :: true:false → 0':s → 0':s → nil:cons → 0':s
ge :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
del :: 0':s → nil:cons → nil:cons
if2 :: true:false → 0':s → 0':s → nil:cons → nil:cons
eq :: 0':s → 0':s → true:false
s :: 0':s → 0':s
sort :: nil:cons → nil:cons
if3 :: true:false → nil:cons → nil:cons
empty :: nil:cons → true:false
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat → 0':s
gen_nil:cons5_0 :: Nat → nil:cons

Lemmas:
ge(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)
max(gen_nil:cons5_0(+(1, n348_0))) → gen_0':s4_0(0), rt ∈ Ω(1 + n348₀)
eq(gen_0':s4_0(n797_0), gen_0':s4_0(n797_0)) → true, rt ∈ Ω(1 + n797₀)
sort(gen_nil:cons5_0(n1546_0)) → gen_nil:cons5_0(0), rt ∈ Ω(1 + n1546₀ + n1546₀²)

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))

No more defined symbols left to analyse.

(26) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
sort(gen_nil:cons5_0(n1546_0)) → gen_nil:cons5_0(0), rt ∈ Ω(1 + n1546₀ + n1546₀²)

(27) BOUNDS(n^2, INF)

(28) Obligation:

TRS:
Rules:
max(nil) → 0'
max(cons(x, nil)) → x
max(cons(x, cons(y, xs))) → if1(ge(x, y), x, y, xs)
if1(true, x, y, xs) → max(cons(x, xs))
if1(false, x, y, xs) → max(cons(y, xs))
del(x, nil) → nil
del(x, cons(y, xs)) → if2(eq(x, y), x, y, xs)
if2(true, x, y, xs) → xs
if2(false, x, y, xs) → cons(y, del(x, xs))
eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
sort(xs) → if3(empty(xs), xs)
if3(true, xs) → nil
if3(false, xs) → sort(del(max(xs), xs))
empty(nil) → true
empty(cons(x, xs)) → false
ge(x, 0') → true
ge(0', s(x)) → false
ge(s(x), s(y)) → ge(x, y)

Types:
max :: nil:cons → 0':s
nil :: nil:cons
0' :: 0':s
cons :: 0':s → nil:cons → nil:cons
if1 :: true:false → 0':s → 0':s → nil:cons → 0':s
ge :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
del :: 0':s → nil:cons → nil:cons
if2 :: true:false → 0':s → 0':s → nil:cons → nil:cons
eq :: 0':s → 0':s → true:false
s :: 0':s → 0':s
sort :: nil:cons → nil:cons
if3 :: true:false → nil:cons → nil:cons
empty :: nil:cons → true:false
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat → 0':s
gen_nil:cons5_0 :: Nat → nil:cons

Lemmas:
ge(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)
max(gen_nil:cons5_0(+(1, n348_0))) → gen_0':s4_0(0), rt ∈ Ω(1 + n348₀)
eq(gen_0':s4_0(n797_0), gen_0':s4_0(n797_0)) → true, rt ∈ Ω(1 + n797₀)

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))

No more defined symbols left to analyse.

(29) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
ge(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)

(30) BOUNDS(n^1, INF)

(31) Obligation:

TRS:
Rules:
max(nil) → 0'
max(cons(x, nil)) → x
max(cons(x, cons(y, xs))) → if1(ge(x, y), x, y, xs)
if1(true, x, y, xs) → max(cons(x, xs))
if1(false, x, y, xs) → max(cons(y, xs))
del(x, nil) → nil
del(x, cons(y, xs)) → if2(eq(x, y), x, y, xs)
if2(true, x, y, xs) → xs
if2(false, x, y, xs) → cons(y, del(x, xs))
eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
sort(xs) → if3(empty(xs), xs)
if3(true, xs) → nil
if3(false, xs) → sort(del(max(xs), xs))
empty(nil) → true
empty(cons(x, xs)) → false
ge(x, 0') → true
ge(0', s(x)) → false
ge(s(x), s(y)) → ge(x, y)

Types:
max :: nil:cons → 0':s
nil :: nil:cons
0' :: 0':s
cons :: 0':s → nil:cons → nil:cons
if1 :: true:false → 0':s → 0':s → nil:cons → 0':s
ge :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
del :: 0':s → nil:cons → nil:cons
if2 :: true:false → 0':s → 0':s → nil:cons → nil:cons
eq :: 0':s → 0':s → true:false
s :: 0':s → 0':s
sort :: nil:cons → nil:cons
if3 :: true:false → nil:cons → nil:cons
empty :: nil:cons → true:false
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat → 0':s
gen_nil:cons5_0 :: Nat → nil:cons

Lemmas:
ge(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)
max(gen_nil:cons5_0(+(1, n348_0))) → gen_0':s4_0(0), rt ∈ Ω(1 + n348₀)

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))

No more defined symbols left to analyse.

(32) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
ge(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)

(33) BOUNDS(n^1, INF)

(34) Obligation:

TRS:
Rules:
max(nil) → 0'
max(cons(x, nil)) → x
max(cons(x, cons(y, xs))) → if1(ge(x, y), x, y, xs)
if1(true, x, y, xs) → max(cons(x, xs))
if1(false, x, y, xs) → max(cons(y, xs))
del(x, nil) → nil
del(x, cons(y, xs)) → if2(eq(x, y), x, y, xs)
if2(true, x, y, xs) → xs
if2(false, x, y, xs) → cons(y, del(x, xs))
eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
sort(xs) → if3(empty(xs), xs)
if3(true, xs) → nil
if3(false, xs) → sort(del(max(xs), xs))
empty(nil) → true
empty(cons(x, xs)) → false
ge(x, 0') → true
ge(0', s(x)) → false
ge(s(x), s(y)) → ge(x, y)

Types:
max :: nil:cons → 0':s
nil :: nil:cons
0' :: 0':s
cons :: 0':s → nil:cons → nil:cons
if1 :: true:false → 0':s → 0':s → nil:cons → 0':s
ge :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
del :: 0':s → nil:cons → nil:cons
if2 :: true:false → 0':s → 0':s → nil:cons → nil:cons
eq :: 0':s → 0':s → true:false
s :: 0':s → 0':s
sort :: nil:cons → nil:cons
if3 :: true:false → nil:cons → nil:cons
empty :: nil:cons → true:false
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat → 0':s
gen_nil:cons5_0 :: Nat → nil:cons

Lemmas:
ge(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)

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))

No more defined symbols left to analyse.

(35) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
ge(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n7₀)

(36) BOUNDS(n^1, INF)