Runtime Complexity TRS:
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)))
app(nil, y) → y
reverse(nil) → nil
shuffle(nil) → nil
concat(leaf, y) → y
concat(cons(u, v), y) → cons(u, concat(v, y))
less_leaves(x, leaf) → false
less_leaves(leaf, cons(w, z)) → true
less_leaves(cons(u, v), cons(w, z)) → less_leaves(concat(u, v), concat(w, z))

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

Runtime Complexity TRS:
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)))
app'(nil', y) → y
reverse'(nil') → nil'
shuffle'(nil') → nil'
concat'(leaf', y) → y
concat'(cons'(u, v), y) → cons'(u, concat'(v, y))
less_leaves'(x, leaf') → false'
less_leaves'(leaf', cons'(w, z)) → true'
less_leaves'(cons'(u, v), cons'(w, z)) → less_leaves'(concat'(u, v), concat'(w, z))

Rewrite Strategy: INNERMOST

Sliced the following arguments:

Runtime Complexity TRS:
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)))
app'(nil', y) → y
reverse'(nil') → nil'
shuffle'(nil') → nil'
concat'(leaf', y) → y
concat'(cons'(u, v), y) → cons'(u, concat'(v, y))
less_leaves'(x, leaf') → false'
less_leaves'(leaf', cons'(w, z)) → true'
less_leaves'(cons'(u, v), cons'(w, z)) → less_leaves'(concat'(u, v), concat'(w, z))

Rewrite Strategy: INNERMOST

Infered types.

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)))
app'(nil', y) → y
reverse'(nil') → nil'
shuffle'(nil') → nil'
concat'(leaf', y) → y
concat'(cons'(u, v), y) → cons'(u, concat'(v, y))
less_leaves'(x, leaf') → false'
less_leaves'(leaf', cons'(w, z)) → true'
less_leaves'(cons'(u, v), cons'(w, z)) → less_leaves'(concat'(u, v), concat'(w, z))

Types:
minus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
quot' :: 0':s' → 0':s' → 0':s'
concat' :: leaf':cons' → leaf':cons' → leaf':cons'
leaf' :: leaf':cons'
cons' :: leaf':cons' → leaf':cons' → leaf':cons'
less_leaves' :: leaf':cons' → leaf':cons' → false':true'
false' :: false':true'
true' :: false':true'
_hole_0':s'1 :: 0':s'
_hole_leaf':cons'3 :: leaf':cons'
_hole_false':true'4 :: false':true'
_gen_0':s'5 :: Nat → 0':s'
_gen_leaf':cons'7 :: Nat → leaf':cons'

Heuristically decided to analyse the following defined symbols:
minus', quot', app', reverse', shuffle', concat', less_leaves'

They will be analysed ascendingly in the following order:
minus' < quot'
app' < reverse'
reverse' < shuffle'
concat' < less_leaves'

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)))
app'(nil', y) → y
reverse'(nil') → nil'
shuffle'(nil') → nil'
concat'(leaf', y) → y
concat'(cons'(u, v), y) → cons'(u, concat'(v, y))
less_leaves'(x, leaf') → false'
less_leaves'(leaf', cons'(w, z)) → true'
less_leaves'(cons'(u, v), cons'(w, z)) → less_leaves'(concat'(u, v), concat'(w, z))

Types:
minus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
quot' :: 0':s' → 0':s' → 0':s'
concat' :: leaf':cons' → leaf':cons' → leaf':cons'
leaf' :: leaf':cons'
cons' :: leaf':cons' → leaf':cons' → leaf':cons'
less_leaves' :: leaf':cons' → leaf':cons' → false':true'
false' :: false':true'
true' :: false':true'
_hole_0':s'1 :: 0':s'
_hole_leaf':cons'3 :: leaf':cons'
_hole_false':true'4 :: false':true'
_gen_0':s'5 :: Nat → 0':s'
_gen_leaf':cons'7 :: Nat → leaf':cons'

Generator Equations:
_gen_0':s'5(0) ⇔ 0'
_gen_0':s'5(+(x, 1)) ⇔ s'(_gen_0':s'5(x))
_gen_leaf':cons'7(0) ⇔ leaf'
_gen_leaf':cons'7(+(x, 1)) ⇔ cons'(leaf', _gen_leaf':cons'7(x))

The following defined symbols remain to be analysed:
minus', quot', app', reverse', shuffle', concat', less_leaves'

They will be analysed ascendingly in the following order:
minus' < quot'
app' < reverse'
reverse' < shuffle'
concat' < less_leaves'

Proved the following rewrite lemma:
minus'(_gen_0':s'5(_n9), _gen_0':s'5(_n9)) → _gen_0':s'5(0), rt ∈ Ω(1 + n9)

Induction Base:
minus'(_gen_0':s'5(0), _gen_0':s'5(0)) →RΩ(1)
_gen_0':s'5(0)

Induction Step:
minus'(_gen_0':s'5(+(_\$n10, 1)), _gen_0':s'5(+(_\$n10, 1))) →RΩ(1)
minus'(_gen_0':s'5(_\$n10), _gen_0':s'5(_\$n10)) →IH
_gen_0':s'5(0)

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

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)))
app'(nil', y) → y
reverse'(nil') → nil'
shuffle'(nil') → nil'
concat'(leaf', y) → y
concat'(cons'(u, v), y) → cons'(u, concat'(v, y))
less_leaves'(x, leaf') → false'
less_leaves'(leaf', cons'(w, z)) → true'
less_leaves'(cons'(u, v), cons'(w, z)) → less_leaves'(concat'(u, v), concat'(w, z))

Types:
minus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
quot' :: 0':s' → 0':s' → 0':s'
concat' :: leaf':cons' → leaf':cons' → leaf':cons'
leaf' :: leaf':cons'
cons' :: leaf':cons' → leaf':cons' → leaf':cons'
less_leaves' :: leaf':cons' → leaf':cons' → false':true'
false' :: false':true'
true' :: false':true'
_hole_0':s'1 :: 0':s'
_hole_leaf':cons'3 :: leaf':cons'
_hole_false':true'4 :: false':true'
_gen_0':s'5 :: Nat → 0':s'
_gen_leaf':cons'7 :: Nat → leaf':cons'

Lemmas:
minus'(_gen_0':s'5(_n9), _gen_0':s'5(_n9)) → _gen_0':s'5(0), rt ∈ Ω(1 + n9)

Generator Equations:
_gen_0':s'5(0) ⇔ 0'
_gen_0':s'5(+(x, 1)) ⇔ s'(_gen_0':s'5(x))
_gen_leaf':cons'7(0) ⇔ leaf'
_gen_leaf':cons'7(+(x, 1)) ⇔ cons'(leaf', _gen_leaf':cons'7(x))

The following defined symbols remain to be analysed:
quot', app', reverse', shuffle', concat', less_leaves'

They will be analysed ascendingly in the following order:
app' < reverse'
reverse' < shuffle'
concat' < less_leaves'

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

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)))
app'(nil', y) → y
reverse'(nil') → nil'
shuffle'(nil') → nil'
concat'(leaf', y) → y
concat'(cons'(u, v), y) → cons'(u, concat'(v, y))
less_leaves'(x, leaf') → false'
less_leaves'(leaf', cons'(w, z)) → true'
less_leaves'(cons'(u, v), cons'(w, z)) → less_leaves'(concat'(u, v), concat'(w, z))

Types:
minus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
quot' :: 0':s' → 0':s' → 0':s'
concat' :: leaf':cons' → leaf':cons' → leaf':cons'
leaf' :: leaf':cons'
cons' :: leaf':cons' → leaf':cons' → leaf':cons'
less_leaves' :: leaf':cons' → leaf':cons' → false':true'
false' :: false':true'
true' :: false':true'
_hole_0':s'1 :: 0':s'
_hole_leaf':cons'3 :: leaf':cons'
_hole_false':true'4 :: false':true'
_gen_0':s'5 :: Nat → 0':s'
_gen_leaf':cons'7 :: Nat → leaf':cons'

Lemmas:
minus'(_gen_0':s'5(_n9), _gen_0':s'5(_n9)) → _gen_0':s'5(0), rt ∈ Ω(1 + n9)

Generator Equations:
_gen_0':s'5(0) ⇔ 0'
_gen_0':s'5(+(x, 1)) ⇔ s'(_gen_0':s'5(x))
_gen_leaf':cons'7(0) ⇔ leaf'
_gen_leaf':cons'7(+(x, 1)) ⇔ cons'(leaf', _gen_leaf':cons'7(x))

The following defined symbols remain to be analysed:
app', reverse', shuffle', concat', less_leaves'

They will be analysed ascendingly in the following order:
app' < reverse'
reverse' < shuffle'
concat' < less_leaves'

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:
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)))
app'(nil', y) → y
reverse'(nil') → nil'
shuffle'(nil') → nil'
concat'(leaf', y) → y
concat'(cons'(u, v), y) → cons'(u, concat'(v, y))
less_leaves'(x, leaf') → false'
less_leaves'(leaf', cons'(w, z)) → true'
less_leaves'(cons'(u, v), cons'(w, z)) → less_leaves'(concat'(u, v), concat'(w, z))

Types:
minus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
quot' :: 0':s' → 0':s' → 0':s'
concat' :: leaf':cons' → leaf':cons' → leaf':cons'
leaf' :: leaf':cons'
cons' :: leaf':cons' → leaf':cons' → leaf':cons'
less_leaves' :: leaf':cons' → leaf':cons' → false':true'
false' :: false':true'
true' :: false':true'
_hole_0':s'1 :: 0':s'
_hole_leaf':cons'3 :: leaf':cons'
_hole_false':true'4 :: false':true'
_gen_0':s'5 :: Nat → 0':s'
_gen_leaf':cons'7 :: Nat → leaf':cons'

Lemmas:
minus'(_gen_0':s'5(_n9), _gen_0':s'5(_n9)) → _gen_0':s'5(0), rt ∈ Ω(1 + n9)

Generator Equations:
_gen_0':s'5(0) ⇔ 0'
_gen_0':s'5(+(x, 1)) ⇔ s'(_gen_0':s'5(x))
_gen_leaf':cons'7(0) ⇔ leaf'
_gen_leaf':cons'7(+(x, 1)) ⇔ cons'(leaf', _gen_leaf':cons'7(x))

The following defined symbols remain to be analysed:
reverse', shuffle', concat', less_leaves'

They will be analysed ascendingly in the following order:
reverse' < shuffle'
concat' < less_leaves'

Proved the following rewrite lemma:

Induction Base:
nil'

Induction Step:

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

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)))
app'(nil', y) → y
reverse'(nil') → nil'
shuffle'(nil') → nil'
concat'(leaf', y) → y
concat'(cons'(u, v), y) → cons'(u, concat'(v, y))
less_leaves'(x, leaf') → false'
less_leaves'(leaf', cons'(w, z)) → true'
less_leaves'(cons'(u, v), cons'(w, z)) → less_leaves'(concat'(u, v), concat'(w, z))

Types:
minus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
quot' :: 0':s' → 0':s' → 0':s'
concat' :: leaf':cons' → leaf':cons' → leaf':cons'
leaf' :: leaf':cons'
cons' :: leaf':cons' → leaf':cons' → leaf':cons'
less_leaves' :: leaf':cons' → leaf':cons' → false':true'
false' :: false':true'
true' :: false':true'
_hole_0':s'1 :: 0':s'
_hole_leaf':cons'3 :: leaf':cons'
_hole_false':true'4 :: false':true'
_gen_0':s'5 :: Nat → 0':s'
_gen_leaf':cons'7 :: Nat → leaf':cons'

Lemmas:
minus'(_gen_0':s'5(_n9), _gen_0':s'5(_n9)) → _gen_0':s'5(0), rt ∈ Ω(1 + n9)

Generator Equations:
_gen_0':s'5(0) ⇔ 0'
_gen_0':s'5(+(x, 1)) ⇔ s'(_gen_0':s'5(x))
_gen_leaf':cons'7(0) ⇔ leaf'
_gen_leaf':cons'7(+(x, 1)) ⇔ cons'(leaf', _gen_leaf':cons'7(x))

The following defined symbols remain to be analysed:
shuffle', concat', less_leaves'

They will be analysed ascendingly in the following order:
concat' < less_leaves'

Proved the following rewrite lemma:
shuffle'(_gen_nil':add'6(_n3044)) → _gen_nil':add'6(_n3044), rt ∈ Ω(1 + n3044 + n30442 + n30443)

Induction Base:
nil'

Induction Step:

We have rt ∈ Ω(n3) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n3).

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)))
app'(nil', y) → y
reverse'(nil') → nil'
shuffle'(nil') → nil'
concat'(leaf', y) → y
concat'(cons'(u, v), y) → cons'(u, concat'(v, y))
less_leaves'(x, leaf') → false'
less_leaves'(leaf', cons'(w, z)) → true'
less_leaves'(cons'(u, v), cons'(w, z)) → less_leaves'(concat'(u, v), concat'(w, z))

Types:
minus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
quot' :: 0':s' → 0':s' → 0':s'
concat' :: leaf':cons' → leaf':cons' → leaf':cons'
leaf' :: leaf':cons'
cons' :: leaf':cons' → leaf':cons' → leaf':cons'
less_leaves' :: leaf':cons' → leaf':cons' → false':true'
false' :: false':true'
true' :: false':true'
_hole_0':s'1 :: 0':s'
_hole_leaf':cons'3 :: leaf':cons'
_hole_false':true'4 :: false':true'
_gen_0':s'5 :: Nat → 0':s'
_gen_leaf':cons'7 :: Nat → leaf':cons'

Lemmas:
minus'(_gen_0':s'5(_n9), _gen_0':s'5(_n9)) → _gen_0':s'5(0), rt ∈ Ω(1 + n9)
shuffle'(_gen_nil':add'6(_n3044)) → _gen_nil':add'6(_n3044), rt ∈ Ω(1 + n3044 + n30442 + n30443)

Generator Equations:
_gen_0':s'5(0) ⇔ 0'
_gen_0':s'5(+(x, 1)) ⇔ s'(_gen_0':s'5(x))
_gen_leaf':cons'7(0) ⇔ leaf'
_gen_leaf':cons'7(+(x, 1)) ⇔ cons'(leaf', _gen_leaf':cons'7(x))

The following defined symbols remain to be analysed:
concat', less_leaves'

They will be analysed ascendingly in the following order:
concat' < less_leaves'

Proved the following rewrite lemma:
concat'(_gen_leaf':cons'7(_n3755), _gen_leaf':cons'7(b)) → _gen_leaf':cons'7(+(_n3755, b)), rt ∈ Ω(1 + n3755)

Induction Base:
concat'(_gen_leaf':cons'7(0), _gen_leaf':cons'7(b)) →RΩ(1)
_gen_leaf':cons'7(b)

Induction Step:
concat'(_gen_leaf':cons'7(+(_\$n3756, 1)), _gen_leaf':cons'7(_b4076)) →RΩ(1)
cons'(leaf', concat'(_gen_leaf':cons'7(_\$n3756), _gen_leaf':cons'7(_b4076))) →IH
cons'(leaf', _gen_leaf':cons'7(+(_\$n3756, _b4076)))

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

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)))
app'(nil', y) → y
reverse'(nil') → nil'
shuffle'(nil') → nil'
concat'(leaf', y) → y
concat'(cons'(u, v), y) → cons'(u, concat'(v, y))
less_leaves'(x, leaf') → false'
less_leaves'(leaf', cons'(w, z)) → true'
less_leaves'(cons'(u, v), cons'(w, z)) → less_leaves'(concat'(u, v), concat'(w, z))

Types:
minus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
quot' :: 0':s' → 0':s' → 0':s'
concat' :: leaf':cons' → leaf':cons' → leaf':cons'
leaf' :: leaf':cons'
cons' :: leaf':cons' → leaf':cons' → leaf':cons'
less_leaves' :: leaf':cons' → leaf':cons' → false':true'
false' :: false':true'
true' :: false':true'
_hole_0':s'1 :: 0':s'
_hole_leaf':cons'3 :: leaf':cons'
_hole_false':true'4 :: false':true'
_gen_0':s'5 :: Nat → 0':s'
_gen_leaf':cons'7 :: Nat → leaf':cons'

Lemmas:
minus'(_gen_0':s'5(_n9), _gen_0':s'5(_n9)) → _gen_0':s'5(0), rt ∈ Ω(1 + n9)
shuffle'(_gen_nil':add'6(_n3044)) → _gen_nil':add'6(_n3044), rt ∈ Ω(1 + n3044 + n30442 + n30443)
concat'(_gen_leaf':cons'7(_n3755), _gen_leaf':cons'7(b)) → _gen_leaf':cons'7(+(_n3755, b)), rt ∈ Ω(1 + n3755)

Generator Equations:
_gen_0':s'5(0) ⇔ 0'
_gen_0':s'5(+(x, 1)) ⇔ s'(_gen_0':s'5(x))
_gen_leaf':cons'7(0) ⇔ leaf'
_gen_leaf':cons'7(+(x, 1)) ⇔ cons'(leaf', _gen_leaf':cons'7(x))

The following defined symbols remain to be analysed:
less_leaves'

Proved the following rewrite lemma:
less_leaves'(_gen_leaf':cons'7(_n5027), _gen_leaf':cons'7(_n5027)) → false', rt ∈ Ω(1 + n5027)

Induction Base:
less_leaves'(_gen_leaf':cons'7(0), _gen_leaf':cons'7(0)) →RΩ(1)
false'

Induction Step:
less_leaves'(_gen_leaf':cons'7(+(_\$n5028, 1)), _gen_leaf':cons'7(+(_\$n5028, 1))) →RΩ(1)
less_leaves'(concat'(leaf', _gen_leaf':cons'7(_\$n5028)), concat'(leaf', _gen_leaf':cons'7(_\$n5028))) →LΩ(1)
less_leaves'(_gen_leaf':cons'7(+(0, _\$n5028)), concat'(leaf', _gen_leaf':cons'7(_\$n5028))) →LΩ(1)
less_leaves'(_gen_leaf':cons'7(_\$n5028), _gen_leaf':cons'7(+(0, _\$n5028))) →IH
false'

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

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)))
app'(nil', y) → y
reverse'(nil') → nil'
shuffle'(nil') → nil'
concat'(leaf', y) → y
concat'(cons'(u, v), y) → cons'(u, concat'(v, y))
less_leaves'(x, leaf') → false'
less_leaves'(leaf', cons'(w, z)) → true'
less_leaves'(cons'(u, v), cons'(w, z)) → less_leaves'(concat'(u, v), concat'(w, z))

Types:
minus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
quot' :: 0':s' → 0':s' → 0':s'
concat' :: leaf':cons' → leaf':cons' → leaf':cons'
leaf' :: leaf':cons'
cons' :: leaf':cons' → leaf':cons' → leaf':cons'
less_leaves' :: leaf':cons' → leaf':cons' → false':true'
false' :: false':true'
true' :: false':true'
_hole_0':s'1 :: 0':s'
_hole_leaf':cons'3 :: leaf':cons'
_hole_false':true'4 :: false':true'
_gen_0':s'5 :: Nat → 0':s'
_gen_leaf':cons'7 :: Nat → leaf':cons'

Lemmas:
minus'(_gen_0':s'5(_n9), _gen_0':s'5(_n9)) → _gen_0':s'5(0), rt ∈ Ω(1 + n9)
shuffle'(_gen_nil':add'6(_n3044)) → _gen_nil':add'6(_n3044), rt ∈ Ω(1 + n3044 + n30442 + n30443)
concat'(_gen_leaf':cons'7(_n3755), _gen_leaf':cons'7(b)) → _gen_leaf':cons'7(+(_n3755, b)), rt ∈ Ω(1 + n3755)
less_leaves'(_gen_leaf':cons'7(_n5027), _gen_leaf':cons'7(_n5027)) → false', rt ∈ Ω(1 + n5027)

Generator Equations:
_gen_0':s'5(0) ⇔ 0'
_gen_0':s'5(+(x, 1)) ⇔ s'(_gen_0':s'5(x))