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

fstsplit(0, x) → nil
fstsplit(s(n), nil) → nil
fstsplit(s(n), cons(h, t)) → cons(h, fstsplit(n, t))
sndsplit(0, x) → x
sndsplit(s(n), nil) → nil
sndsplit(s(n), cons(h, t)) → sndsplit(n, t)
empty(nil) → true
empty(cons(h, t)) → false
leq(0, m) → true
leq(s(n), 0) → false
leq(s(n), s(m)) → leq(n, m)
length(nil) → 0
length(cons(h, t)) → s(length(t))
app(nil, x) → x
app(cons(h, t), x) → cons(h, app(t, x))
map_f(pid, nil) → nil
map_f(pid, cons(h, t)) → app(f(pid, h), map_f(pid, t))
head(cons(h, t)) → h
tail(cons(h, t)) → t
ring(st_1, in_2, st_2, in_3, st_3, m) → if_1(st_1, in_2, st_2, in_3, st_3, m, empty(fstsplit(m, st_1)))
if_1(st_1, in_2, st_2, in_3, st_3, m, false) → ring(sndsplit(m, st_1), cons(fstsplit(m, st_1), in_2), st_2, in_3, st_3, m)
ring(st_1, in_2, st_2, in_3, st_3, m) → if_2(st_1, in_2, st_2, in_3, st_3, m, leq(m, length(st_2)))
if_2(st_1, in_2, st_2, in_3, st_3, m, true) → if_3(st_1, in_2, st_2, in_3, st_3, m, empty(fstsplit(m, st_2)))
if_3(st_1, in_2, st_2, in_3, st_3, m, false) → ring(st_1, in_2, sndsplit(m, st_2), cons(fstsplit(m, st_2), in_3), st_3, m)
if_2(st_1, in_2, st_2, in_3, st_3, m, false) → if_4(st_1, in_2, st_2, in_3, st_3, m, empty(fstsplit(m, app(map_f(two, head(in_2)), st_2))))
if_4(st_1, in_2, st_2, in_3, st_3, m, false) → ring(st_1, tail(in_2), sndsplit(m, app(map_f(two, head(in_2)), st_2)), cons(fstsplit(m, app(map_f(two, head(in_2)), st_2)), in_3), st_3, m)
ring(st_1, in_2, st_2, in_3, st_3, m) → if_5(st_1, in_2, st_2, in_3, st_3, m, empty(map_f(two, head(in_2))))
if_5(st_1, in_2, st_2, in_3, st_3, m, true) → ring(st_1, tail(in_2), st_2, in_3, st_3, m)
ring(st_1, in_2, st_2, in_3, st_3, m) → if_6(st_1, in_2, st_2, in_3, st_3, m, leq(m, length(st_3)))
if_6(st_1, in_2, st_2, in_3, st_3, m, true) → if_7(st_1, in_2, st_2, in_3, st_3, m, empty(fstsplit(m, st_3)))
if_7(st_1, in_2, st_2, in_3, st_3, m, false) → ring(st_1, in_2, st_2, in_3, sndsplit(m, st_3), m)
if_6(st_1, in_2, st_2, in_3, st_3, m, false) → if_8(st_1, in_2, st_2, in_3, st_3, m, empty(fstsplit(m, app(map_f(three, head(in_3)), st_3))))
if_8(st_1, in_2, st_2, in_3, st_3, m, false) → ring(st_1, in_2, st_2, tail(in_3), sndsplit(m, app(map_f(three, head(in_3)), st_3)), m)
ring(st_1, in_2, st_2, in_3, st_3, m) → if_9(st_1, in_2, st_2, in_3, st_3, m, empty(map_f(three, head(in_3))))
if_9(st_1, in_2, st_2, in_3, st_3, m, true) → ring(st_1, in_2, st_2, tail(in_3), st_3, m)

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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

fstsplit'(0', x) → nil'
fstsplit'(s'(n), nil') → nil'
fstsplit'(s'(n), cons'(h, t)) → cons'(h, fstsplit'(n, t))
sndsplit'(0', x) → x
sndsplit'(s'(n), nil') → nil'
sndsplit'(s'(n), cons'(h, t)) → sndsplit'(n, t)
empty'(nil') → true'
empty'(cons'(h, t)) → false'
leq'(0', m) → true'
leq'(s'(n), 0') → false'
leq'(s'(n), s'(m)) → leq'(n, m)
length'(nil') → 0'
length'(cons'(h, t)) → s'(length'(t))
app'(nil', x) → x
app'(cons'(h, t), x) → cons'(h, app'(t, x))
map_f'(pid, nil') → nil'
map_f'(pid, cons'(h, t)) → app'(f'(pid, h), map_f'(pid, t))
head'(cons'(h, t)) → h
tail'(cons'(h, t)) → t
ring'(st_1, in_2, st_2, in_3, st_3, m) → if_1'(st_1, in_2, st_2, in_3, st_3, m, empty'(fstsplit'(m, st_1)))
if_1'(st_1, in_2, st_2, in_3, st_3, m, false') → ring'(sndsplit'(m, st_1), cons'(fstsplit'(m, st_1), in_2), st_2, in_3, st_3, m)
ring'(st_1, in_2, st_2, in_3, st_3, m) → if_2'(st_1, in_2, st_2, in_3, st_3, m, leq'(m, length'(st_2)))
if_2'(st_1, in_2, st_2, in_3, st_3, m, true') → if_3'(st_1, in_2, st_2, in_3, st_3, m, empty'(fstsplit'(m, st_2)))
if_3'(st_1, in_2, st_2, in_3, st_3, m, false') → ring'(st_1, in_2, sndsplit'(m, st_2), cons'(fstsplit'(m, st_2), in_3), st_3, m)
if_2'(st_1, in_2, st_2, in_3, st_3, m, false') → if_4'(st_1, in_2, st_2, in_3, st_3, m, empty'(fstsplit'(m, app'(map_f'(two', head'(in_2)), st_2))))
if_4'(st_1, in_2, st_2, in_3, st_3, m, false') → ring'(st_1, tail'(in_2), sndsplit'(m, app'(map_f'(two', head'(in_2)), st_2)), cons'(fstsplit'(m, app'(map_f'(two', head'(in_2)), st_2)), in_3), st_3, m)
ring'(st_1, in_2, st_2, in_3, st_3, m) → if_5'(st_1, in_2, st_2, in_3, st_3, m, empty'(map_f'(two', head'(in_2))))
if_5'(st_1, in_2, st_2, in_3, st_3, m, true') → ring'(st_1, tail'(in_2), st_2, in_3, st_3, m)
ring'(st_1, in_2, st_2, in_3, st_3, m) → if_6'(st_1, in_2, st_2, in_3, st_3, m, leq'(m, length'(st_3)))
if_6'(st_1, in_2, st_2, in_3, st_3, m, true') → if_7'(st_1, in_2, st_2, in_3, st_3, m, empty'(fstsplit'(m, st_3)))
if_7'(st_1, in_2, st_2, in_3, st_3, m, false') → ring'(st_1, in_2, st_2, in_3, sndsplit'(m, st_3), m)
if_6'(st_1, in_2, st_2, in_3, st_3, m, false') → if_8'(st_1, in_2, st_2, in_3, st_3, m, empty'(fstsplit'(m, app'(map_f'(three', head'(in_3)), st_3))))
if_8'(st_1, in_2, st_2, in_3, st_3, m, false') → ring'(st_1, in_2, st_2, tail'(in_3), sndsplit'(m, app'(map_f'(three', head'(in_3)), st_3)), m)
ring'(st_1, in_2, st_2, in_3, st_3, m) → if_9'(st_1, in_2, st_2, in_3, st_3, m, empty'(map_f'(three', head'(in_3))))
if_9'(st_1, in_2, st_2, in_3, st_3, m, true') → ring'(st_1, in_2, st_2, tail'(in_3), st_3, m)

Rewrite Strategy: INNERMOST

Sliced the following arguments:
map_f'/0
f'/0
f'/1

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

fstsplit'(0', x) → nil'
fstsplit'(s'(n), nil') → nil'
fstsplit'(s'(n), cons'(h, t)) → cons'(h, fstsplit'(n, t))
sndsplit'(0', x) → x
sndsplit'(s'(n), nil') → nil'
sndsplit'(s'(n), cons'(h, t)) → sndsplit'(n, t)
empty'(nil') → true'
empty'(cons'(h, t)) → false'
leq'(0', m) → true'
leq'(s'(n), 0') → false'
leq'(s'(n), s'(m)) → leq'(n, m)
length'(nil') → 0'
length'(cons'(h, t)) → s'(length'(t))
app'(nil', x) → x
app'(cons'(h, t), x) → cons'(h, app'(t, x))
map_f'(nil') → nil'
map_f'(cons'(h, t)) → app'(f', map_f'(t))
head'(cons'(h, t)) → h
tail'(cons'(h, t)) → t
ring'(st_1, in_2, st_2, in_3, st_3, m) → if_1'(st_1, in_2, st_2, in_3, st_3, m, empty'(fstsplit'(m, st_1)))
if_1'(st_1, in_2, st_2, in_3, st_3, m, false') → ring'(sndsplit'(m, st_1), cons'(fstsplit'(m, st_1), in_2), st_2, in_3, st_3, m)
ring'(st_1, in_2, st_2, in_3, st_3, m) → if_2'(st_1, in_2, st_2, in_3, st_3, m, leq'(m, length'(st_2)))
if_2'(st_1, in_2, st_2, in_3, st_3, m, true') → if_3'(st_1, in_2, st_2, in_3, st_3, m, empty'(fstsplit'(m, st_2)))
if_3'(st_1, in_2, st_2, in_3, st_3, m, false') → ring'(st_1, in_2, sndsplit'(m, st_2), cons'(fstsplit'(m, st_2), in_3), st_3, m)
if_2'(st_1, in_2, st_2, in_3, st_3, m, false') → if_4'(st_1, in_2, st_2, in_3, st_3, m, empty'(fstsplit'(m, app'(map_f'(head'(in_2)), st_2))))
if_4'(st_1, in_2, st_2, in_3, st_3, m, false') → ring'(st_1, tail'(in_2), sndsplit'(m, app'(map_f'(head'(in_2)), st_2)), cons'(fstsplit'(m, app'(map_f'(head'(in_2)), st_2)), in_3), st_3, m)
ring'(st_1, in_2, st_2, in_3, st_3, m) → if_5'(st_1, in_2, st_2, in_3, st_3, m, empty'(map_f'(head'(in_2))))
if_5'(st_1, in_2, st_2, in_3, st_3, m, true') → ring'(st_1, tail'(in_2), st_2, in_3, st_3, m)
ring'(st_1, in_2, st_2, in_3, st_3, m) → if_6'(st_1, in_2, st_2, in_3, st_3, m, leq'(m, length'(st_3)))
if_6'(st_1, in_2, st_2, in_3, st_3, m, true') → if_7'(st_1, in_2, st_2, in_3, st_3, m, empty'(fstsplit'(m, st_3)))
if_7'(st_1, in_2, st_2, in_3, st_3, m, false') → ring'(st_1, in_2, st_2, in_3, sndsplit'(m, st_3), m)
if_6'(st_1, in_2, st_2, in_3, st_3, m, false') → if_8'(st_1, in_2, st_2, in_3, st_3, m, empty'(fstsplit'(m, app'(map_f'(head'(in_3)), st_3))))
if_8'(st_1, in_2, st_2, in_3, st_3, m, false') → ring'(st_1, in_2, st_2, tail'(in_3), sndsplit'(m, app'(map_f'(head'(in_3)), st_3)), m)
ring'(st_1, in_2, st_2, in_3, st_3, m) → if_9'(st_1, in_2, st_2, in_3, st_3, m, empty'(map_f'(head'(in_3))))
if_9'(st_1, in_2, st_2, in_3, st_3, m, true') → ring'(st_1, in_2, st_2, tail'(in_3), st_3, m)

Rewrite Strategy: INNERMOST

Infered types.

Rules:
fstsplit'(0', x) → nil'
fstsplit'(s'(n), nil') → nil'
fstsplit'(s'(n), cons'(h, t)) → cons'(h, fstsplit'(n, t))
sndsplit'(0', x) → x
sndsplit'(s'(n), nil') → nil'
sndsplit'(s'(n), cons'(h, t)) → sndsplit'(n, t)
empty'(nil') → true'
empty'(cons'(h, t)) → false'
leq'(0', m) → true'
leq'(s'(n), 0') → false'
leq'(s'(n), s'(m)) → leq'(n, m)
length'(nil') → 0'
length'(cons'(h, t)) → s'(length'(t))
app'(nil', x) → x
app'(cons'(h, t), x) → cons'(h, app'(t, x))
map_f'(nil') → nil'
map_f'(cons'(h, t)) → app'(f', map_f'(t))
head'(cons'(h, t)) → h
tail'(cons'(h, t)) → t
ring'(st_1, in_2, st_2, in_3, st_3, m) → if_1'(st_1, in_2, st_2, in_3, st_3, m, empty'(fstsplit'(m, st_1)))
if_1'(st_1, in_2, st_2, in_3, st_3, m, false') → ring'(sndsplit'(m, st_1), cons'(fstsplit'(m, st_1), in_2), st_2, in_3, st_3, m)
ring'(st_1, in_2, st_2, in_3, st_3, m) → if_2'(st_1, in_2, st_2, in_3, st_3, m, leq'(m, length'(st_2)))
if_2'(st_1, in_2, st_2, in_3, st_3, m, true') → if_3'(st_1, in_2, st_2, in_3, st_3, m, empty'(fstsplit'(m, st_2)))
if_3'(st_1, in_2, st_2, in_3, st_3, m, false') → ring'(st_1, in_2, sndsplit'(m, st_2), cons'(fstsplit'(m, st_2), in_3), st_3, m)
if_2'(st_1, in_2, st_2, in_3, st_3, m, false') → if_4'(st_1, in_2, st_2, in_3, st_3, m, empty'(fstsplit'(m, app'(map_f'(head'(in_2)), st_2))))
if_4'(st_1, in_2, st_2, in_3, st_3, m, false') → ring'(st_1, tail'(in_2), sndsplit'(m, app'(map_f'(head'(in_2)), st_2)), cons'(fstsplit'(m, app'(map_f'(head'(in_2)), st_2)), in_3), st_3, m)
ring'(st_1, in_2, st_2, in_3, st_3, m) → if_5'(st_1, in_2, st_2, in_3, st_3, m, empty'(map_f'(head'(in_2))))
if_5'(st_1, in_2, st_2, in_3, st_3, m, true') → ring'(st_1, tail'(in_2), st_2, in_3, st_3, m)
ring'(st_1, in_2, st_2, in_3, st_3, m) → if_6'(st_1, in_2, st_2, in_3, st_3, m, leq'(m, length'(st_3)))
if_6'(st_1, in_2, st_2, in_3, st_3, m, true') → if_7'(st_1, in_2, st_2, in_3, st_3, m, empty'(fstsplit'(m, st_3)))
if_7'(st_1, in_2, st_2, in_3, st_3, m, false') → ring'(st_1, in_2, st_2, in_3, sndsplit'(m, st_3), m)
if_6'(st_1, in_2, st_2, in_3, st_3, m, false') → if_8'(st_1, in_2, st_2, in_3, st_3, m, empty'(fstsplit'(m, app'(map_f'(head'(in_3)), st_3))))
if_8'(st_1, in_2, st_2, in_3, st_3, m, false') → ring'(st_1, in_2, st_2, tail'(in_3), sndsplit'(m, app'(map_f'(head'(in_3)), st_3)), m)
ring'(st_1, in_2, st_2, in_3, st_3, m) → if_9'(st_1, in_2, st_2, in_3, st_3, m, empty'(map_f'(head'(in_3))))
if_9'(st_1, in_2, st_2, in_3, st_3, m, true') → ring'(st_1, in_2, st_2, tail'(in_3), st_3, m)

Types:
fstsplit' :: 0':s' → nil':cons':f' → nil':cons':f'
0' :: 0':s'
nil' :: nil':cons':f'
s' :: 0':s' → 0':s'
cons' :: nil':cons':f' → nil':cons':f' → nil':cons':f'
sndsplit' :: 0':s' → nil':cons':f' → nil':cons':f'
empty' :: nil':cons':f' → true':false'
true' :: true':false'
false' :: true':false'
leq' :: 0':s' → 0':s' → true':false'
length' :: nil':cons':f' → 0':s'
app' :: nil':cons':f' → nil':cons':f' → nil':cons':f'
map_f' :: nil':cons':f' → nil':cons':f'
f' :: nil':cons':f'
head' :: nil':cons':f' → nil':cons':f'
tail' :: nil':cons':f' → nil':cons':f'
ring' :: nil':cons':f' → nil':cons':f' → nil':cons':f' → nil':cons':f' → nil':cons':f' → 0':s' → ring':if_1':if_2':if_3':if_4':if_5':if_6':if_7':if_8':if_9'
if_1' :: nil':cons':f' → nil':cons':f' → nil':cons':f' → nil':cons':f' → nil':cons':f' → 0':s' → true':false' → ring':if_1':if_2':if_3':if_4':if_5':if_6':if_7':if_8':if_9'
if_2' :: nil':cons':f' → nil':cons':f' → nil':cons':f' → nil':cons':f' → nil':cons':f' → 0':s' → true':false' → ring':if_1':if_2':if_3':if_4':if_5':if_6':if_7':if_8':if_9'
if_3' :: nil':cons':f' → nil':cons':f' → nil':cons':f' → nil':cons':f' → nil':cons':f' → 0':s' → true':false' → ring':if_1':if_2':if_3':if_4':if_5':if_6':if_7':if_8':if_9'
if_4' :: nil':cons':f' → nil':cons':f' → nil':cons':f' → nil':cons':f' → nil':cons':f' → 0':s' → true':false' → ring':if_1':if_2':if_3':if_4':if_5':if_6':if_7':if_8':if_9'
if_5' :: nil':cons':f' → nil':cons':f' → nil':cons':f' → nil':cons':f' → nil':cons':f' → 0':s' → true':false' → ring':if_1':if_2':if_3':if_4':if_5':if_6':if_7':if_8':if_9'
if_6' :: nil':cons':f' → nil':cons':f' → nil':cons':f' → nil':cons':f' → nil':cons':f' → 0':s' → true':false' → ring':if_1':if_2':if_3':if_4':if_5':if_6':if_7':if_8':if_9'
if_7' :: nil':cons':f' → nil':cons':f' → nil':cons':f' → nil':cons':f' → nil':cons':f' → 0':s' → true':false' → ring':if_1':if_2':if_3':if_4':if_5':if_6':if_7':if_8':if_9'
if_8' :: nil':cons':f' → nil':cons':f' → nil':cons':f' → nil':cons':f' → nil':cons':f' → 0':s' → true':false' → ring':if_1':if_2':if_3':if_4':if_5':if_6':if_7':if_8':if_9'
if_9' :: nil':cons':f' → nil':cons':f' → nil':cons':f' → nil':cons':f' → nil':cons':f' → 0':s' → true':false' → ring':if_1':if_2':if_3':if_4':if_5':if_6':if_7':if_8':if_9'
_hole_nil':cons':f'1 :: nil':cons':f'
_hole_0':s'2 :: 0':s'
_hole_true':false'3 :: true':false'
_hole_ring':if_1':if_2':if_3':if_4':if_5':if_6':if_7':if_8':if_9'4 :: ring':if_1':if_2':if_3':if_4':if_5':if_6':if_7':if_8':if_9'
_gen_nil':cons':f'5 :: Nat → nil':cons':f'
_gen_0':s'6 :: Nat → 0':s'

Heuristically decided to analyse the following defined symbols:
fstsplit', sndsplit', leq', length', app', map_f', ring'

They will be analysed ascendingly in the following order:
fstsplit' < ring'
sndsplit' < ring'
leq' < ring'
length' < ring'
app' < map_f'
app' < ring'
map_f' < ring'

Generator Equations:
_gen_nil':cons':f'5(0) ⇔ nil'
_gen_nil':cons':f'5(+(x, 1)) ⇔ cons'(nil', _gen_nil':cons':f'5(x))
_gen_0':s'6(0) ⇔ 0'
_gen_0':s'6(+(x, 1)) ⇔ s'(_gen_0':s'6(x))

The following defined symbols remain to be analysed:
fstsplit', sndsplit', leq', length', app', map_f', ring'

They will be analysed ascendingly in the following order:
fstsplit' < ring'
sndsplit' < ring'
leq' < ring'
length' < ring'
app' < map_f'
app' < ring'
map_f' < ring'

Proved the following rewrite lemma:
fstsplit'(_gen_0':s'6(_n8), _gen_nil':cons':f'5(_n8)) → _gen_nil':cons':f'5(_n8), rt ∈ Ω(1 + _n8)

Induction Base:
fstsplit'(_gen_0':s'6(0), _gen_nil':cons':f'5(0)) →_R^Ω(1)
nil'

Induction Step:
fstsplit'(_gen_0':s'6(+(_$n9, 1)), _gen_nil':cons':f'5(+(_$n9, 1))) →_R^Ω(1)
cons'(nil', fstsplit'(_gen_0':s'6(_$n9), _gen_nil':cons':f'5(_$n9))) →_IH
cons'(nil', _gen_nil':cons':f'5(_$n9))

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

Lemmas:
fstsplit'(_gen_0':s'6(_n8), _gen_nil':cons':f'5(_n8)) → _gen_nil':cons':f'5(_n8), rt ∈ Ω(1 + _n8)

Generator Equations:
_gen_nil':cons':f'5(0) ⇔ nil'
_gen_nil':cons':f'5(+(x, 1)) ⇔ cons'(nil', _gen_nil':cons':f'5(x))
_gen_0':s'6(0) ⇔ 0'
_gen_0':s'6(+(x, 1)) ⇔ s'(_gen_0':s'6(x))

The following defined symbols remain to be analysed:
sndsplit', leq', length', app', map_f', ring'

They will be analysed ascendingly in the following order:
sndsplit' < ring'
leq' < ring'
length' < ring'
app' < map_f'
app' < ring'
map_f' < ring'

Proved the following rewrite lemma:
sndsplit'(_gen_0':s'6(_n3664), _gen_nil':cons':f'5(_n3664)) → _gen_nil':cons':f'5(0), rt ∈ Ω(1 + _n3664)

Induction Base:
sndsplit'(_gen_0':s'6(0), _gen_nil':cons':f'5(0)) →_R^Ω(1)
_gen_nil':cons':f'5(0)

Induction Step:
sndsplit'(_gen_0':s'6(+(_$n3665, 1)), _gen_nil':cons':f'5(+(_$n3665, 1))) →_R^Ω(1)
sndsplit'(_gen_0':s'6(_$n3665), _gen_nil':cons':f'5(_$n3665)) →_IH
_gen_nil':cons':f'5(0)

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

Generator Equations:
_gen_nil':cons':f'5(0) ⇔ nil'
_gen_nil':cons':f'5(+(x, 1)) ⇔ cons'(nil', _gen_nil':cons':f'5(x))
_gen_0':s'6(0) ⇔ 0'
_gen_0':s'6(+(x, 1)) ⇔ s'(_gen_0':s'6(x))

The following defined symbols remain to be analysed:
leq', length', app', map_f', ring'

They will be analysed ascendingly in the following order:
leq' < ring'
length' < ring'
app' < map_f'
app' < ring'
map_f' < ring'

Proved the following rewrite lemma:
leq'(_gen_0':s'6(_n7229), _gen_0':s'6(_n7229)) → true', rt ∈ Ω(1 + _n7229)

Induction Base:
leq'(_gen_0':s'6(0), _gen_0':s'6(0)) →_R^Ω(1)
true'

Induction Step:
leq'(_gen_0':s'6(+(_$n7230, 1)), _gen_0':s'6(+(_$n7230, 1))) →_R^Ω(1)
leq'(_gen_0':s'6(_$n7230), _gen_0':s'6(_$n7230)) →_IH
true'

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

Generator Equations:
_gen_nil':cons':f'5(0) ⇔ nil'
_gen_nil':cons':f'5(+(x, 1)) ⇔ cons'(nil', _gen_nil':cons':f'5(x))
_gen_0':s'6(0) ⇔ 0'
_gen_0':s'6(+(x, 1)) ⇔ s'(_gen_0':s'6(x))

The following defined symbols remain to be analysed:
length', app', map_f', ring'

They will be analysed ascendingly in the following order:
length' < ring'
app' < map_f'
app' < ring'
map_f' < ring'

Proved the following rewrite lemma:
length'(_gen_nil':cons':f'5(_n10224)) → _gen_0':s'6(_n10224), rt ∈ Ω(1 + _n10224)

Induction Base:
length'(_gen_nil':cons':f'5(0)) →_R^Ω(1)
0'

Induction Step:
length'(_gen_nil':cons':f'5(+(_$n10225, 1))) →_R^Ω(1)
s'(length'(_gen_nil':cons':f'5(_$n10225))) →_IH
s'(_gen_0':s'6(_$n10225))

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

Generator Equations:
_gen_nil':cons':f'5(0) ⇔ nil'
_gen_nil':cons':f'5(+(x, 1)) ⇔ cons'(nil', _gen_nil':cons':f'5(x))
_gen_0':s'6(0) ⇔ 0'
_gen_0':s'6(+(x, 1)) ⇔ s'(_gen_0':s'6(x))

The following defined symbols remain to be analysed:
app', map_f', ring'

They will be analysed ascendingly in the following order:
app' < map_f'
app' < ring'
map_f' < ring'

Proved the following rewrite lemma:
app'(_gen_nil':cons':f'5(_n12436), _gen_nil':cons':f'5(b)) → _gen_nil':cons':f'5(+(_n12436, b)), rt ∈ Ω(1 + _n12436)

Induction Base:
app'(_gen_nil':cons':f'5(0), _gen_nil':cons':f'5(b)) →_R^Ω(1)
_gen_nil':cons':f'5(b)

Induction Step:
app'(_gen_nil':cons':f'5(+(_$n12437, 1)), _gen_nil':cons':f'5(_b12613)) →_R^Ω(1)
cons'(nil', app'(_gen_nil':cons':f'5(_$n12437), _gen_nil':cons':f'5(_b12613))) →_IH
cons'(nil', _gen_nil':cons':f'5(+(_$n12437, _b12613)))

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

Lemmas:
fstsplit'(_gen_0':s'6(_n8), _gen_nil':cons':f'5(_n8)) → _gen_nil':cons':f'5(_n8), rt ∈ Ω(1 + _n8)
sndsplit'(_gen_0':s'6(_n3664), _gen_nil':cons':f'5(_n3664)) → _gen_nil':cons':f'5(0), rt ∈ Ω(1 + _n3664)
leq'(_gen_0':s'6(_n7229), _gen_0':s'6(_n7229)) → true', rt ∈ Ω(1 + _n7229)
length'(_gen_nil':cons':f'5(_n10224)) → _gen_0':s'6(_n10224), rt ∈ Ω(1 + _n10224)
app'(_gen_nil':cons':f'5(_n12436), _gen_nil':cons':f'5(b)) → _gen_nil':cons':f'5(+(_n12436, b)), rt ∈ Ω(1 + _n12436)

Generator Equations:
_gen_nil':cons':f'5(0) ⇔ nil'
_gen_nil':cons':f'5(+(x, 1)) ⇔ cons'(nil', _gen_nil':cons':f'5(x))
_gen_0':s'6(0) ⇔ 0'
_gen_0':s'6(+(x, 1)) ⇔ s'(_gen_0':s'6(x))

The following defined symbols remain to be analysed:
map_f', ring'

They will be analysed ascendingly in the following order:
map_f' < ring'

Proved the following rewrite lemma:
map_f'(_gen_nil':cons':f'5(+(1, _n16262))) → _*7, rt ∈ Ω(_n16262)

Induction Base:
map_f'(_gen_nil':cons':f'5(+(1, 0)))

Induction Step:
map_f'(_gen_nil':cons':f'5(+(1, +(_$n16263, 1)))) →_R^Ω(1)
app'(f', map_f'(_gen_nil':cons':f'5(+(1, _$n16263)))) →_IH
app'(f', _*7)

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

Generator Equations:
_gen_nil':cons':f'5(0) ⇔ nil'
_gen_nil':cons':f'5(+(x, 1)) ⇔ cons'(nil', _gen_nil':cons':f'5(x))
_gen_0':s'6(0) ⇔ 0'
_gen_0':s'6(+(x, 1)) ⇔ s'(_gen_0':s'6(x))

The following defined symbols remain to be analysed:
ring'

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

Generator Equations:
_gen_nil':cons':f'5(0) ⇔ nil'
_gen_nil':cons':f'5(+(x, 1)) ⇔ cons'(nil', _gen_nil':cons':f'5(x))
_gen_0':s'6(0) ⇔ 0'
_gen_0':s'6(+(x, 1)) ⇔ s'(_gen_0':s'6(x))

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
fstsplit'(_gen_0':s'6(_n8), _gen_nil':cons':f'5(_n8)) → _gen_nil':cons':f'5(_n8), rt ∈ Ω(1 + _n8)