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
min(nil) → 0
minIter(add(n, x), y, m) → if_min(le(n, m), x, y, m)
if_min(true, x, y, m) → m
if_min(false, x, y, m) → minIter(x, y, m)
tail(nil) → nil
null(nil) → true
rm(n, nil) → nil
if_rm(true, n, add(m, x)) → rm(n, x)
minsort(nil, nil) → nil

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
min'(nil') → 0'
minIter'(add'(n, x), y, m) → if_min'(le'(n, m), x, y, m)
if_min'(true', x, y, m) → m
if_min'(false', x, y, m) → minIter'(x, y, m)
tail'(nil') → nil'
null'(nil') → true'
rm'(n, nil') → nil'
if_rm'(true', n, add'(m, x)) → rm'(n, x)
minsort'(nil', nil') → nil'

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
min'(nil') → 0'
minIter'(add'(n, x), y, m) → if_min'(le'(n, m), x, y, m)
if_min'(true', x, y, m) → m
if_min'(false', x, y, m) → minIter'(x, y, m)
tail'(nil') → nil'
null'(nil') → true'
rm'(n, nil') → nil'
if_rm'(true', n, add'(m, x)) → rm'(n, x)
minsort'(nil', nil') → nil'

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'
_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', minIter', rm', minsort'

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

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
min'(nil') → 0'
minIter'(add'(n, x), y, m) → if_min'(le'(n, m), x, y, m)
if_min'(true', x, y, m) → m
if_min'(false', x, y, m) → minIter'(x, y, m)
tail'(nil') → nil'
null'(nil') → true'
rm'(n, nil') → nil'
if_rm'(true', n, add'(m, x)) → rm'(n, x)
minsort'(nil', nil') → nil'

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'
_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', minIter', rm', minsort'

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

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
min'(nil') → 0'
minIter'(add'(n, x), y, m) → if_min'(le'(n, m), x, y, m)
if_min'(true', x, y, m) → m
if_min'(false', x, y, m) → minIter'(x, y, m)
tail'(nil') → nil'
null'(nil') → true'
rm'(n, nil') → nil'
if_rm'(true', n, add'(m, x)) → rm'(n, x)
minsort'(nil', nil') → nil'

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'
_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', minIter', rm', minsort'

They will be analysed ascendingly in the following order:
le' < minIter'
app' < minsort'
rm' < minsort'

Proved the following rewrite lemma:
le'(_gen_0':s'4(_n1533), _gen_0':s'4(_n1533)) → true', rt ∈ Ω(1 + n1533)

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

Induction Step:
le'(_gen_0':s'4(+(_\$n1534, 1)), _gen_0':s'4(+(_\$n1534, 1))) →RΩ(1)
le'(_gen_0':s'4(_\$n1534), _gen_0':s'4(_\$n1534)) →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
min'(nil') → 0'
minIter'(add'(n, x), y, m) → if_min'(le'(n, m), x, y, m)
if_min'(true', x, y, m) → m
if_min'(false', x, y, m) → minIter'(x, y, m)
tail'(nil') → nil'
null'(nil') → true'
rm'(n, nil') → nil'
if_rm'(true', n, add'(m, x)) → rm'(n, x)
minsort'(nil', nil') → nil'

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'
_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(_n1533), _gen_0':s'4(_n1533)) → true', rt ∈ Ω(1 + n1533)

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', minIter', rm', minsort'

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

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
min'(nil') → 0'
minIter'(add'(n, x), y, m) → if_min'(le'(n, m), x, y, m)
if_min'(true', x, y, m) → m
if_min'(false', x, y, m) → minIter'(x, y, m)
tail'(nil') → nil'
null'(nil') → true'
rm'(n, nil') → nil'
if_rm'(true', n, add'(m, x)) → rm'(n, x)
minsort'(nil', nil') → nil'

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'
_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(_n1533), _gen_0':s'4(_n1533)) → true', rt ∈ Ω(1 + n1533)

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:
minIter', rm', minsort'

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

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

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
min'(nil') → 0'
minIter'(add'(n, x), y, m) → if_min'(le'(n, m), x, y, m)
if_min'(true', x, y, m) → m
if_min'(false', x, y, m) → minIter'(x, y, m)
tail'(nil') → nil'
null'(nil') → true'
rm'(n, nil') → nil'
if_rm'(true', n, add'(m, x)) → rm'(n, x)
minsort'(nil', nil') → nil'

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'
_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(_n1533), _gen_0':s'4(_n1533)) → true', rt ∈ Ω(1 + n1533)

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', minsort'

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

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
min'(nil') → 0'
minIter'(add'(n, x), y, m) → if_min'(le'(n, m), x, y, m)
if_min'(true', x, y, m) → m
if_min'(false', x, y, m) → minIter'(x, y, m)
tail'(nil') → nil'
null'(nil') → true'
rm'(n, nil') → nil'
if_rm'(true', n, add'(m, x)) → rm'(n, x)
minsort'(nil', nil') → nil'

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'
_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(_n1533), _gen_0':s'4(_n1533)) → true', rt ∈ Ω(1 + n1533)

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:
minsort'

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

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
min'(nil') → 0'
minIter'(add'(n, x), y, m) → if_min'(le'(n, m), x, y, m)
if_min'(true', x, y, m) → m
if_min'(false', x, y, m) → minIter'(x, y, m)
tail'(nil') → nil'
null'(nil') → true'
rm'(n, nil') → nil'
if_rm'(true', n, add'(m, x)) → rm'(n, x)
minsort'(nil', nil') → nil'

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'
_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(_n1533), _gen_0':s'4(_n1533)) → true', rt ∈ Ω(1 + n1533)