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

eq(0, 0) → true
eq(0, s(x)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
app(nil, y) → y
tail(nil) → nil
null(nil) → true
rm(n, nil) → nil
if_rm(true, n, add(m, x)) → rm(n, x)
minsort(x) → mins(x, nil, nil)
mins(x, y, z) → if(null(x), x, y, z)
if(true, x, y, z) → z
if(false, x, y, z) → if2(eq(head(x), min(x)), x, y, z)

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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

eq'(0', 0') → true'
eq'(0', s'(x)) → false'
eq'(s'(x), 0') → false'
eq'(s'(x), s'(y)) → eq'(x, y)
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
app'(nil', y) → y
tail'(nil') → nil'
null'(nil') → true'
rm'(n, nil') → nil'
if_rm'(true', n, add'(m, x)) → rm'(n, x)
minsort'(x) → mins'(x, nil', nil')
mins'(x, y, z) → if'(null'(x), x, y, z)
if'(true', x, y, z) → z
if'(false', x, y, z) → if2'(eq'(head'(x), min'(x)), x, y, z)

Rewrite Strategy: INNERMOST

Infered types.

Rules:
eq'(0', 0') → true'
eq'(0', s'(x)) → false'
eq'(s'(x), 0') → false'
eq'(s'(x), s'(y)) → eq'(x, y)
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
app'(nil', y) → y
tail'(nil') → nil'
null'(nil') → true'
rm'(n, nil') → nil'
if_rm'(true', n, add'(m, x)) → rm'(n, x)
minsort'(x) → mins'(x, nil', nil')
mins'(x, y, z) → if'(null'(x), x, y, z)
if'(true', x, y, z) → z
if'(false', x, y, z) → if2'(eq'(head'(x), min'(x)), x, y, z)

Types:
eq' :: 0':s' → 0':s' → true':false'
0' :: 0':s'
true' :: true':false'
s' :: 0':s' → 0':s'
false' :: true':false'
le' :: 0':s' → 0':s' → true':false'
if_min' :: true':false' → nil':add' → 0':s'
_hole_true':false'1 :: true':false'
_hole_0':s'2 :: 0':s'
_gen_0':s'4 :: Nat → 0':s'

Heuristically decided to analyse the following defined symbols:
eq', le', app', min', rm', mins'

They will be analysed ascendingly in the following order:
eq' < rm'
eq' < mins'
le' < min'
app' < mins'
min' < mins'
rm' < mins'

Rules:
eq'(0', 0') → true'
eq'(0', s'(x)) → false'
eq'(s'(x), 0') → false'
eq'(s'(x), s'(y)) → eq'(x, y)
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
app'(nil', y) → y
tail'(nil') → nil'
null'(nil') → true'
rm'(n, nil') → nil'
if_rm'(true', n, add'(m, x)) → rm'(n, x)
minsort'(x) → mins'(x, nil', nil')
mins'(x, y, z) → if'(null'(x), x, y, z)
if'(true', x, y, z) → z
if'(false', x, y, z) → if2'(eq'(head'(x), min'(x)), x, y, z)

Types:
eq' :: 0':s' → 0':s' → true':false'
0' :: 0':s'
true' :: true':false'
s' :: 0':s' → 0':s'
false' :: true':false'
le' :: 0':s' → 0':s' → true':false'
if_min' :: true':false' → nil':add' → 0':s'
_hole_true':false'1 :: true':false'
_hole_0':s'2 :: 0':s'
_gen_0':s'4 :: Nat → 0':s'

Generator Equations:
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))

The following defined symbols remain to be analysed:
eq', le', app', min', rm', mins'

They will be analysed ascendingly in the following order:
eq' < rm'
eq' < mins'
le' < min'
app' < mins'
min' < mins'
rm' < mins'

Proved the following rewrite lemma:
eq'(_gen_0':s'4(_n7), _gen_0':s'4(_n7)) → true', rt ∈ Ω(1 + n7)

Induction Base:
eq'(_gen_0':s'4(0), _gen_0':s'4(0)) →RΩ(1)
true'

Induction Step:
eq'(_gen_0':s'4(+(_\$n8, 1)), _gen_0':s'4(+(_\$n8, 1))) →RΩ(1)
eq'(_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:
eq'(0', 0') → true'
eq'(0', s'(x)) → false'
eq'(s'(x), 0') → false'
eq'(s'(x), s'(y)) → eq'(x, y)
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
app'(nil', y) → y
tail'(nil') → nil'
null'(nil') → true'
rm'(n, nil') → nil'
if_rm'(true', n, add'(m, x)) → rm'(n, x)
minsort'(x) → mins'(x, nil', nil')
mins'(x, y, z) → if'(null'(x), x, y, z)
if'(true', x, y, z) → z
if'(false', x, y, z) → if2'(eq'(head'(x), min'(x)), x, y, z)

Types:
eq' :: 0':s' → 0':s' → true':false'
0' :: 0':s'
true' :: true':false'
s' :: 0':s' → 0':s'
false' :: true':false'
le' :: 0':s' → 0':s' → true':false'
if_min' :: true':false' → nil':add' → 0':s'
_hole_true':false'1 :: true':false'
_hole_0':s'2 :: 0':s'
_gen_0':s'4 :: Nat → 0':s'

Lemmas:
eq'(_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))

The following defined symbols remain to be analysed:
le', app', min', rm', mins'

They will be analysed ascendingly in the following order:
le' < min'
app' < mins'
min' < mins'
rm' < mins'

Proved the following rewrite lemma:
le'(_gen_0':s'4(_n1563), _gen_0':s'4(_n1563)) → true', rt ∈ Ω(1 + n1563)

Induction Base:
le'(_gen_0':s'4(0), _gen_0':s'4(0)) →RΩ(1)
true'

Induction Step:
le'(_gen_0':s'4(+(_\$n1564, 1)), _gen_0':s'4(+(_\$n1564, 1))) →RΩ(1)
le'(_gen_0':s'4(_\$n1564), _gen_0':s'4(_\$n1564)) →IH
true'

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

Rules:
eq'(0', 0') → true'
eq'(0', s'(x)) → false'
eq'(s'(x), 0') → false'
eq'(s'(x), s'(y)) → eq'(x, y)
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
app'(nil', y) → y
tail'(nil') → nil'
null'(nil') → true'
rm'(n, nil') → nil'
if_rm'(true', n, add'(m, x)) → rm'(n, x)
minsort'(x) → mins'(x, nil', nil')
mins'(x, y, z) → if'(null'(x), x, y, z)
if'(true', x, y, z) → z
if'(false', x, y, z) → if2'(eq'(head'(x), min'(x)), x, y, z)

Types:
eq' :: 0':s' → 0':s' → true':false'
0' :: 0':s'
true' :: true':false'
s' :: 0':s' → 0':s'
false' :: true':false'
le' :: 0':s' → 0':s' → true':false'
if_min' :: true':false' → nil':add' → 0':s'
_hole_true':false'1 :: true':false'
_hole_0':s'2 :: 0':s'
_gen_0':s'4 :: Nat → 0':s'

Lemmas:
eq'(_gen_0':s'4(_n7), _gen_0':s'4(_n7)) → true', rt ∈ Ω(1 + n7)
le'(_gen_0':s'4(_n1563), _gen_0':s'4(_n1563)) → true', rt ∈ Ω(1 + n1563)

Generator Equations:
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))

The following defined symbols remain to be analysed:
app', min', rm', mins'

They will be analysed ascendingly in the following order:
app' < mins'
min' < mins'
rm' < mins'

Proved the following rewrite lemma:

Induction Base:

Induction Step:

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

Rules:
eq'(0', 0') → true'
eq'(0', s'(x)) → false'
eq'(s'(x), 0') → false'
eq'(s'(x), s'(y)) → eq'(x, y)
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
app'(nil', y) → y
tail'(nil') → nil'
null'(nil') → true'
rm'(n, nil') → nil'
if_rm'(true', n, add'(m, x)) → rm'(n, x)
minsort'(x) → mins'(x, nil', nil')
mins'(x, y, z) → if'(null'(x), x, y, z)
if'(true', x, y, z) → z
if'(false', x, y, z) → if2'(eq'(head'(x), min'(x)), x, y, z)

Types:
eq' :: 0':s' → 0':s' → true':false'
0' :: 0':s'
true' :: true':false'
s' :: 0':s' → 0':s'
false' :: true':false'
le' :: 0':s' → 0':s' → true':false'
if_min' :: true':false' → nil':add' → 0':s'
_hole_true':false'1 :: true':false'
_hole_0':s'2 :: 0':s'
_gen_0':s'4 :: Nat → 0':s'

Lemmas:
eq'(_gen_0':s'4(_n7), _gen_0':s'4(_n7)) → true', rt ∈ Ω(1 + n7)
le'(_gen_0':s'4(_n1563), _gen_0':s'4(_n1563)) → true', rt ∈ Ω(1 + n1563)

Generator Equations:
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))

The following defined symbols remain to be analysed:
min', rm', mins'

They will be analysed ascendingly in the following order:
min' < mins'
rm' < mins'

Proved the following rewrite lemma:
min'(_gen_nil':add'5(+(1, _n5110))) → _gen_0':s'4(0), rt ∈ Ω(1 + n5110)

Induction Base:
0'

Induction Step:
_gen_0':s'4(0)

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

Rules:
eq'(0', 0') → true'
eq'(0', s'(x)) → false'
eq'(s'(x), 0') → false'
eq'(s'(x), s'(y)) → eq'(x, y)
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
app'(nil', y) → y
tail'(nil') → nil'
null'(nil') → true'
rm'(n, nil') → nil'
if_rm'(true', n, add'(m, x)) → rm'(n, x)
minsort'(x) → mins'(x, nil', nil')
mins'(x, y, z) → if'(null'(x), x, y, z)
if'(true', x, y, z) → z
if'(false', x, y, z) → if2'(eq'(head'(x), min'(x)), x, y, z)

Types:
eq' :: 0':s' → 0':s' → true':false'
0' :: 0':s'
true' :: true':false'
s' :: 0':s' → 0':s'
false' :: true':false'
le' :: 0':s' → 0':s' → true':false'
if_min' :: true':false' → nil':add' → 0':s'
_hole_true':false'1 :: true':false'
_hole_0':s'2 :: 0':s'
_gen_0':s'4 :: Nat → 0':s'

Lemmas:
eq'(_gen_0':s'4(_n7), _gen_0':s'4(_n7)) → true', rt ∈ Ω(1 + n7)
le'(_gen_0':s'4(_n1563), _gen_0':s'4(_n1563)) → true', rt ∈ Ω(1 + n1563)
min'(_gen_nil':add'5(+(1, _n5110))) → _gen_0':s'4(0), rt ∈ Ω(1 + n5110)

Generator Equations:
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))

The following defined symbols remain to be analysed:
rm', mins'

They will be analysed ascendingly in the following order:
rm' < mins'

Proved the following rewrite lemma:

Induction Base:
nil'

Induction Step:

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

Rules:
eq'(0', 0') → true'
eq'(0', s'(x)) → false'
eq'(s'(x), 0') → false'
eq'(s'(x), s'(y)) → eq'(x, y)
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
app'(nil', y) → y
tail'(nil') → nil'
null'(nil') → true'
rm'(n, nil') → nil'
if_rm'(true', n, add'(m, x)) → rm'(n, x)
minsort'(x) → mins'(x, nil', nil')
mins'(x, y, z) → if'(null'(x), x, y, z)
if'(true', x, y, z) → z
if'(false', x, y, z) → if2'(eq'(head'(x), min'(x)), x, y, z)

Types:
eq' :: 0':s' → 0':s' → true':false'
0' :: 0':s'
true' :: true':false'
s' :: 0':s' → 0':s'
false' :: true':false'
le' :: 0':s' → 0':s' → true':false'
if_min' :: true':false' → nil':add' → 0':s'
_hole_true':false'1 :: true':false'
_hole_0':s'2 :: 0':s'
_gen_0':s'4 :: Nat → 0':s'

Lemmas:
eq'(_gen_0':s'4(_n7), _gen_0':s'4(_n7)) → true', rt ∈ Ω(1 + n7)
le'(_gen_0':s'4(_n1563), _gen_0':s'4(_n1563)) → true', rt ∈ Ω(1 + n1563)
min'(_gen_nil':add'5(+(1, _n5110))) → _gen_0':s'4(0), rt ∈ Ω(1 + n5110)

Generator Equations:
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))

The following defined symbols remain to be analysed:
mins'

Could not prove a rewrite lemma for the defined symbol mins'.

Rules:
eq'(0', 0') → true'
eq'(0', s'(x)) → false'
eq'(s'(x), 0') → false'
eq'(s'(x), s'(y)) → eq'(x, y)
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
app'(nil', y) → y
tail'(nil') → nil'
null'(nil') → true'
rm'(n, nil') → nil'
if_rm'(true', n, add'(m, x)) → rm'(n, x)
minsort'(x) → mins'(x, nil', nil')
mins'(x, y, z) → if'(null'(x), x, y, z)
if'(true', x, y, z) → z
if'(false', x, y, z) → if2'(eq'(head'(x), min'(x)), x, y, z)

Types:
eq' :: 0':s' → 0':s' → true':false'
0' :: 0':s'
true' :: true':false'
s' :: 0':s' → 0':s'
false' :: true':false'
le' :: 0':s' → 0':s' → true':false'
if_min' :: true':false' → nil':add' → 0':s'
_hole_true':false'1 :: true':false'
_hole_0':s'2 :: 0':s'
_gen_0':s'4 :: Nat → 0':s'

Lemmas:
eq'(_gen_0':s'4(_n7), _gen_0':s'4(_n7)) → true', rt ∈ Ω(1 + n7)
le'(_gen_0':s'4(_n1563), _gen_0':s'4(_n1563)) → true', rt ∈ Ω(1 + n1563)
min'(_gen_nil':add'5(+(1, _n5110))) → _gen_0':s'4(0), rt ∈ Ω(1 + n5110)

Generator Equations:
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
eq'(_gen_0':s'4(_n7), _gen_0':s'4(_n7)) → true', rt ∈ Ω(1 + n7)