Termination w.r.t. Q proof of TRS/AProVEAAECC-ring.trs

Termination w.r.t. Q of the following Term Rewriting System could not be shown:

Q restricted rewrite system:
The TRS R consists of the following rules:

fstsplit₂(0, x) -> nil
fstsplit₂(s₁(n), nil) -> nil
fstsplit₂(s₁(n), cons₂(h, t)) -> cons₂(h, fstsplit₂(n, t))
sndsplit₂(0, x) -> x
sndsplit₂(s₁(n), nil) -> nil
sndsplit₂(s₁(n), cons₂(h, t)) -> sndsplit₂(n, t)
empty₁(nil) -> true
empty₁(cons₂(h, t)) -> false
leq₂(0, m) -> true
leq₂(s₁(n), 0) -> false
leq₂(s₁(n), s₁(m)) -> leq₂(n, m)
length₁(nil) -> 0
length₁(cons₂(h, t)) -> s₁(length₁(t))
app₂(nil, x) -> x
app₂(cons₂(h, t), x) -> cons₂(h, app₂(t, x))
map_f₂(pid, nil) -> nil
map_f₂(pid, cons₂(h, t)) -> app₂(f₂(pid, h), map_f₂(pid, t))
head₁(cons₂(h, t)) -> h
tail₁(cons₂(h, t)) -> t
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_1₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_1)))
if_1₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(sndsplit₂(m, st_1), cons₂(fstsplit₂(m, st_1), in_2), st_2, in_3, st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_2₇(st_1, in_2, st_2, in_3, st_3, m, leq₂(m, length₁(st_2)))
if_2₇(st_1, in_2, st_2, in_3, st_3, m, true) -> if_3₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_2)))
if_3₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, sndsplit₂(m, st_2), cons₂(fstsplit₂(m, st_2), in_3), st_3, m)
if_2₇(st_1, in_2, st_2, in_3, st_3, m, false) -> if_4₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2))))
if_4₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, tail₁(in_2), sndsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2)), cons₂(fstsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2)), in_3), st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_5₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(map_f₂(two, head₁(in_2))))
if_5₇(st_1, in_2, st_2, in_3, st_3, m, true) -> ring₆(st_1, tail₁(in_2), st_2, in_3, st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_6₇(st_1, in_2, st_2, in_3, st_3, m, leq₂(m, length₁(st_3)))
if_6₇(st_1, in_2, st_2, in_3, st_3, m, true) -> if_7₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_3)))
if_7₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, st_2, in_3, sndsplit₂(m, st_3), m)
if_6₇(st_1, in_2, st_2, in_3, st_3, m, false) -> if_8₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, app₂(map_f₂(three, head₁(in_3)), st_3))))
if_8₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, st_2, tail₁(in_3), sndsplit₂(m, app₂(map_f₂(three, head₁(in_3)), st_3)), m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_9₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(map_f₂(three, head₁(in_3))))
if_9₇(st_1, in_2, st_2, in_3, st_3, m, true) -> ring₆(st_1, in_2, st_2, tail₁(in_3), st_3, m)

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

Q restricted rewrite system:
The TRS R consists of the following rules:

fstsplit₂(0, x) -> nil
fstsplit₂(s₁(n), nil) -> nil
fstsplit₂(s₁(n), cons₂(h, t)) -> cons₂(h, fstsplit₂(n, t))
sndsplit₂(0, x) -> x
sndsplit₂(s₁(n), nil) -> nil
sndsplit₂(s₁(n), cons₂(h, t)) -> sndsplit₂(n, t)
empty₁(nil) -> true
empty₁(cons₂(h, t)) -> false
leq₂(0, m) -> true
leq₂(s₁(n), 0) -> false
leq₂(s₁(n), s₁(m)) -> leq₂(n, m)
length₁(nil) -> 0
length₁(cons₂(h, t)) -> s₁(length₁(t))
app₂(nil, x) -> x
app₂(cons₂(h, t), x) -> cons₂(h, app₂(t, x))
map_f₂(pid, nil) -> nil
map_f₂(pid, cons₂(h, t)) -> app₂(f₂(pid, h), map_f₂(pid, t))
head₁(cons₂(h, t)) -> h
tail₁(cons₂(h, t)) -> t
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_1₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_1)))
if_1₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(sndsplit₂(m, st_1), cons₂(fstsplit₂(m, st_1), in_2), st_2, in_3, st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_2₇(st_1, in_2, st_2, in_3, st_3, m, leq₂(m, length₁(st_2)))
if_2₇(st_1, in_2, st_2, in_3, st_3, m, true) -> if_3₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_2)))
if_3₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, sndsplit₂(m, st_2), cons₂(fstsplit₂(m, st_2), in_3), st_3, m)
if_2₇(st_1, in_2, st_2, in_3, st_3, m, false) -> if_4₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2))))
if_4₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, tail₁(in_2), sndsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2)), cons₂(fstsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2)), in_3), st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_5₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(map_f₂(two, head₁(in_2))))
if_5₇(st_1, in_2, st_2, in_3, st_3, m, true) -> ring₆(st_1, tail₁(in_2), st_2, in_3, st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_6₇(st_1, in_2, st_2, in_3, st_3, m, leq₂(m, length₁(st_3)))
if_6₇(st_1, in_2, st_2, in_3, st_3, m, true) -> if_7₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_3)))
if_7₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, st_2, in_3, sndsplit₂(m, st_3), m)
if_6₇(st_1, in_2, st_2, in_3, st_3, m, false) -> if_8₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, app₂(map_f₂(three, head₁(in_3)), st_3))))
if_8₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, st_2, tail₁(in_3), sndsplit₂(m, app₂(map_f₂(three, head₁(in_3)), st_3)), m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_9₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(map_f₂(three, head₁(in_3))))
if_9₇(st_1, in_2, st_2, in_3, st_3, m, true) -> ring₆(st_1, in_2, st_2, tail₁(in_3), st_3, m)

Q is empty.

Using Dependency Pairs [1,13] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

IF_9₇(st_1, in_2, st_2, in_3, st_3, m, true) -> RING₆(st_1, in_2, st_2, tail₁(in_3), st_3, m)
IF_2₇(st_1, in_2, st_2, in_3, st_3, m, false) -> HEAD₁(in_2)
IF_8₇(st_1, in_2, st_2, in_3, st_3, m, false) -> TAIL₁(in_3)
IF_3₇(st_1, in_2, st_2, in_3, st_3, m, false) -> FSTSPLIT₂(m, st_2)
IF_9₇(st_1, in_2, st_2, in_3, st_3, m, true) -> TAIL₁(in_3)
RING₆(st_1, in_2, st_2, in_3, st_3, m) -> HEAD₁(in_3)
SNDSPLIT₂(s₁(n), cons₂(h, t)) -> SNDSPLIT₂(n, t)
IF_4₇(st_1, in_2, st_2, in_3, st_3, m, false) -> MAP_F₂(two, head₁(in_2))
IF_5₇(st_1, in_2, st_2, in_3, st_3, m, true) -> RING₆(st_1, tail₁(in_2), st_2, in_3, st_3, m)
IF_2₇(st_1, in_2, st_2, in_3, st_3, m, false) -> IF_4₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2))))
IF_8₇(st_1, in_2, st_2, in_3, st_3, m, false) -> SNDSPLIT₂(m, app₂(map_f₂(three, head₁(in_3)), st_3))
IF_2₇(st_1, in_2, st_2, in_3, st_3, m, false) -> APP₂(map_f₂(two, head₁(in_2)), st_2)
IF_7₇(st_1, in_2, st_2, in_3, st_3, m, false) -> RING₆(st_1, in_2, st_2, in_3, sndsplit₂(m, st_3), m)
IF_5₇(st_1, in_2, st_2, in_3, st_3, m, true) -> TAIL₁(in_2)
IF_3₇(st_1, in_2, st_2, in_3, st_3, m, false) -> RING₆(st_1, in_2, sndsplit₂(m, st_2), cons₂(fstsplit₂(m, st_2), in_3), st_3, m)
IF_6₇(st_1, in_2, st_2, in_3, st_3, m, false) -> APP₂(map_f₂(three, head₁(in_3)), st_3)
IF_4₇(st_1, in_2, st_2, in_3, st_3, m, false) -> SNDSPLIT₂(m, app₂(map_f₂(two, head₁(in_2)), st_2))
RING₆(st_1, in_2, st_2, in_3, st_3, m) -> MAP_F₂(two, head₁(in_2))
RING₆(st_1, in_2, st_2, in_3, st_3, m) -> LENGTH₁(st_2)
IF_2₇(st_1, in_2, st_2, in_3, st_3, m, true) -> EMPTY₁(fstsplit₂(m, st_2))
APP₂(cons₂(h, t), x) -> APP₂(t, x)
RING₆(st_1, in_2, st_2, in_3, st_3, m) -> EMPTY₁(map_f₂(two, head₁(in_2)))
LENGTH₁(cons₂(h, t)) -> LENGTH₁(t)
IF_1₇(st_1, in_2, st_2, in_3, st_3, m, false) -> FSTSPLIT₂(m, st_1)
RING₆(st_1, in_2, st_2, in_3, st_3, m) -> IF_5₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(map_f₂(two, head₁(in_2))))
IF_8₇(st_1, in_2, st_2, in_3, st_3, m, false) -> HEAD₁(in_3)
IF_6₇(st_1, in_2, st_2, in_3, st_3, m, true) -> EMPTY₁(fstsplit₂(m, st_3))
IF_6₇(st_1, in_2, st_2, in_3, st_3, m, false) -> IF_8₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, app₂(map_f₂(three, head₁(in_3)), st_3))))
IF_4₇(st_1, in_2, st_2, in_3, st_3, m, false) -> FSTSPLIT₂(m, app₂(map_f₂(two, head₁(in_2)), st_2))
IF_6₇(st_1, in_2, st_2, in_3, st_3, m, false) -> EMPTY₁(fstsplit₂(m, app₂(map_f₂(three, head₁(in_3)), st_3)))
IF_2₇(st_1, in_2, st_2, in_3, st_3, m, false) -> MAP_F₂(two, head₁(in_2))
IF_8₇(st_1, in_2, st_2, in_3, st_3, m, false) -> RING₆(st_1, in_2, st_2, tail₁(in_3), sndsplit₂(m, app₂(map_f₂(three, head₁(in_3)), st_3)), m)
IF_8₇(st_1, in_2, st_2, in_3, st_3, m, false) -> APP₂(map_f₂(three, head₁(in_3)), st_3)
IF_6₇(st_1, in_2, st_2, in_3, st_3, m, false) -> HEAD₁(in_3)
IF_2₇(st_1, in_2, st_2, in_3, st_3, m, true) -> IF_3₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_2)))
MAP_F₂(pid, cons₂(h, t)) -> MAP_F₂(pid, t)
IF_4₇(st_1, in_2, st_2, in_3, st_3, m, false) -> RING₆(st_1, tail₁(in_2), sndsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2)), cons₂(fstsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2)), in_3), st_3, m)
RING₆(st_1, in_2, st_2, in_3, st_3, m) -> IF_6₇(st_1, in_2, st_2, in_3, st_3, m, leq₂(m, length₁(st_3)))
RING₆(st_1, in_2, st_2, in_3, st_3, m) -> IF_9₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(map_f₂(three, head₁(in_3))))
IF_3₇(st_1, in_2, st_2, in_3, st_3, m, false) -> SNDSPLIT₂(m, st_2)
IF_6₇(st_1, in_2, st_2, in_3, st_3, m, false) -> MAP_F₂(three, head₁(in_3))
IF_2₇(st_1, in_2, st_2, in_3, st_3, m, false) -> FSTSPLIT₂(m, app₂(map_f₂(two, head₁(in_2)), st_2))
RING₆(st_1, in_2, st_2, in_3, st_3, m) -> LEQ₂(m, length₁(st_3))
IF_7₇(st_1, in_2, st_2, in_3, st_3, m, false) -> SNDSPLIT₂(m, st_3)
MAP_F₂(pid, cons₂(h, t)) -> APP₂(f₂(pid, h), map_f₂(pid, t))
IF_4₇(st_1, in_2, st_2, in_3, st_3, m, false) -> HEAD₁(in_2)
RING₆(st_1, in_2, st_2, in_3, st_3, m) -> LEQ₂(m, length₁(st_2))
IF_1₇(st_1, in_2, st_2, in_3, st_3, m, false) -> RING₆(sndsplit₂(m, st_1), cons₂(fstsplit₂(m, st_1), in_2), st_2, in_3, st_3, m)
IF_2₇(st_1, in_2, st_2, in_3, st_3, m, true) -> FSTSPLIT₂(m, st_2)
IF_6₇(st_1, in_2, st_2, in_3, st_3, m, true) -> FSTSPLIT₂(m, st_3)
FSTSPLIT₂(s₁(n), cons₂(h, t)) -> FSTSPLIT₂(n, t)
RING₆(st_1, in_2, st_2, in_3, st_3, m) -> LENGTH₁(st_3)
RING₆(st_1, in_2, st_2, in_3, st_3, m) -> MAP_F₂(three, head₁(in_3))
RING₆(st_1, in_2, st_2, in_3, st_3, m) -> EMPTY₁(map_f₂(three, head₁(in_3)))
RING₆(st_1, in_2, st_2, in_3, st_3, m) -> EMPTY₁(fstsplit₂(m, st_1))
IF_4₇(st_1, in_2, st_2, in_3, st_3, m, false) -> TAIL₁(in_2)
RING₆(st_1, in_2, st_2, in_3, st_3, m) -> HEAD₁(in_2)
RING₆(st_1, in_2, st_2, in_3, st_3, m) -> IF_2₇(st_1, in_2, st_2, in_3, st_3, m, leq₂(m, length₁(st_2)))
IF_1₇(st_1, in_2, st_2, in_3, st_3, m, false) -> SNDSPLIT₂(m, st_1)
IF_8₇(st_1, in_2, st_2, in_3, st_3, m, false) -> MAP_F₂(three, head₁(in_3))
IF_2₇(st_1, in_2, st_2, in_3, st_3, m, false) -> EMPTY₁(fstsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2)))
RING₆(st_1, in_2, st_2, in_3, st_3, m) -> IF_1₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_1)))
IF_6₇(st_1, in_2, st_2, in_3, st_3, m, false) -> FSTSPLIT₂(m, app₂(map_f₂(three, head₁(in_3)), st_3))
LEQ₂(s₁(n), s₁(m)) -> LEQ₂(n, m)
RING₆(st_1, in_2, st_2, in_3, st_3, m) -> FSTSPLIT₂(m, st_1)
IF_4₇(st_1, in_2, st_2, in_3, st_3, m, false) -> APP₂(map_f₂(two, head₁(in_2)), st_2)
IF_6₇(st_1, in_2, st_2, in_3, st_3, m, true) -> IF_7₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_3)))

The TRS R consists of the following rules:

fstsplit₂(0, x) -> nil
fstsplit₂(s₁(n), nil) -> nil
fstsplit₂(s₁(n), cons₂(h, t)) -> cons₂(h, fstsplit₂(n, t))
sndsplit₂(0, x) -> x
sndsplit₂(s₁(n), nil) -> nil
sndsplit₂(s₁(n), cons₂(h, t)) -> sndsplit₂(n, t)
empty₁(nil) -> true
empty₁(cons₂(h, t)) -> false
leq₂(0, m) -> true
leq₂(s₁(n), 0) -> false
leq₂(s₁(n), s₁(m)) -> leq₂(n, m)
length₁(nil) -> 0
length₁(cons₂(h, t)) -> s₁(length₁(t))
app₂(nil, x) -> x
app₂(cons₂(h, t), x) -> cons₂(h, app₂(t, x))
map_f₂(pid, nil) -> nil
map_f₂(pid, cons₂(h, t)) -> app₂(f₂(pid, h), map_f₂(pid, t))
head₁(cons₂(h, t)) -> h
tail₁(cons₂(h, t)) -> t
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_1₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_1)))
if_1₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(sndsplit₂(m, st_1), cons₂(fstsplit₂(m, st_1), in_2), st_2, in_3, st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_2₇(st_1, in_2, st_2, in_3, st_3, m, leq₂(m, length₁(st_2)))
if_2₇(st_1, in_2, st_2, in_3, st_3, m, true) -> if_3₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_2)))
if_3₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, sndsplit₂(m, st_2), cons₂(fstsplit₂(m, st_2), in_3), st_3, m)
if_2₇(st_1, in_2, st_2, in_3, st_3, m, false) -> if_4₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2))))
if_4₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, tail₁(in_2), sndsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2)), cons₂(fstsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2)), in_3), st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_5₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(map_f₂(two, head₁(in_2))))
if_5₇(st_1, in_2, st_2, in_3, st_3, m, true) -> ring₆(st_1, tail₁(in_2), st_2, in_3, st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_6₇(st_1, in_2, st_2, in_3, st_3, m, leq₂(m, length₁(st_3)))
if_6₇(st_1, in_2, st_2, in_3, st_3, m, true) -> if_7₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_3)))
if_7₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, st_2, in_3, sndsplit₂(m, st_3), m)
if_6₇(st_1, in_2, st_2, in_3, st_3, m, false) -> if_8₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, app₂(map_f₂(three, head₁(in_3)), st_3))))
if_8₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, st_2, tail₁(in_3), sndsplit₂(m, app₂(map_f₂(three, head₁(in_3)), st_3)), m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_9₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(map_f₂(three, head₁(in_3))))
if_9₇(st_1, in_2, st_2, in_3, st_3, m, true) -> ring₆(st_1, in_2, st_2, tail₁(in_3), st_3, m)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

IF_9₇(st_1, in_2, st_2, in_3, st_3, m, true) -> RING₆(st_1, in_2, st_2, tail₁(in_3), st_3, m)
IF_2₇(st_1, in_2, st_2, in_3, st_3, m, false) -> HEAD₁(in_2)
IF_8₇(st_1, in_2, st_2, in_3, st_3, m, false) -> TAIL₁(in_3)
IF_3₇(st_1, in_2, st_2, in_3, st_3, m, false) -> FSTSPLIT₂(m, st_2)
IF_9₇(st_1, in_2, st_2, in_3, st_3, m, true) -> TAIL₁(in_3)
RING₆(st_1, in_2, st_2, in_3, st_3, m) -> HEAD₁(in_3)
SNDSPLIT₂(s₁(n), cons₂(h, t)) -> SNDSPLIT₂(n, t)
IF_4₇(st_1, in_2, st_2, in_3, st_3, m, false) -> MAP_F₂(two, head₁(in_2))
IF_5₇(st_1, in_2, st_2, in_3, st_3, m, true) -> RING₆(st_1, tail₁(in_2), st_2, in_3, st_3, m)
IF_2₇(st_1, in_2, st_2, in_3, st_3, m, false) -> IF_4₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2))))
IF_8₇(st_1, in_2, st_2, in_3, st_3, m, false) -> SNDSPLIT₂(m, app₂(map_f₂(three, head₁(in_3)), st_3))
IF_2₇(st_1, in_2, st_2, in_3, st_3, m, false) -> APP₂(map_f₂(two, head₁(in_2)), st_2)
IF_7₇(st_1, in_2, st_2, in_3, st_3, m, false) -> RING₆(st_1, in_2, st_2, in_3, sndsplit₂(m, st_3), m)
IF_5₇(st_1, in_2, st_2, in_3, st_3, m, true) -> TAIL₁(in_2)
IF_3₇(st_1, in_2, st_2, in_3, st_3, m, false) -> RING₆(st_1, in_2, sndsplit₂(m, st_2), cons₂(fstsplit₂(m, st_2), in_3), st_3, m)
IF_6₇(st_1, in_2, st_2, in_3, st_3, m, false) -> APP₂(map_f₂(three, head₁(in_3)), st_3)
IF_4₇(st_1, in_2, st_2, in_3, st_3, m, false) -> SNDSPLIT₂(m, app₂(map_f₂(two, head₁(in_2)), st_2))
RING₆(st_1, in_2, st_2, in_3, st_3, m) -> MAP_F₂(two, head₁(in_2))
RING₆(st_1, in_2, st_2, in_3, st_3, m) -> LENGTH₁(st_2)
IF_2₇(st_1, in_2, st_2, in_3, st_3, m, true) -> EMPTY₁(fstsplit₂(m, st_2))
APP₂(cons₂(h, t), x) -> APP₂(t, x)
RING₆(st_1, in_2, st_2, in_3, st_3, m) -> EMPTY₁(map_f₂(two, head₁(in_2)))
LENGTH₁(cons₂(h, t)) -> LENGTH₁(t)
IF_1₇(st_1, in_2, st_2, in_3, st_3, m, false) -> FSTSPLIT₂(m, st_1)
RING₆(st_1, in_2, st_2, in_3, st_3, m) -> IF_5₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(map_f₂(two, head₁(in_2))))
IF_8₇(st_1, in_2, st_2, in_3, st_3, m, false) -> HEAD₁(in_3)
IF_6₇(st_1, in_2, st_2, in_3, st_3, m, true) -> EMPTY₁(fstsplit₂(m, st_3))
IF_6₇(st_1, in_2, st_2, in_3, st_3, m, false) -> IF_8₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, app₂(map_f₂(three, head₁(in_3)), st_3))))
IF_4₇(st_1, in_2, st_2, in_3, st_3, m, false) -> FSTSPLIT₂(m, app₂(map_f₂(two, head₁(in_2)), st_2))
IF_6₇(st_1, in_2, st_2, in_3, st_3, m, false) -> EMPTY₁(fstsplit₂(m, app₂(map_f₂(three, head₁(in_3)), st_3)))
IF_2₇(st_1, in_2, st_2, in_3, st_3, m, false) -> MAP_F₂(two, head₁(in_2))
IF_8₇(st_1, in_2, st_2, in_3, st_3, m, false) -> RING₆(st_1, in_2, st_2, tail₁(in_3), sndsplit₂(m, app₂(map_f₂(three, head₁(in_3)), st_3)), m)
IF_8₇(st_1, in_2, st_2, in_3, st_3, m, false) -> APP₂(map_f₂(three, head₁(in_3)), st_3)
IF_6₇(st_1, in_2, st_2, in_3, st_3, m, false) -> HEAD₁(in_3)
IF_2₇(st_1, in_2, st_2, in_3, st_3, m, true) -> IF_3₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_2)))
MAP_F₂(pid, cons₂(h, t)) -> MAP_F₂(pid, t)
IF_4₇(st_1, in_2, st_2, in_3, st_3, m, false) -> RING₆(st_1, tail₁(in_2), sndsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2)), cons₂(fstsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2)), in_3), st_3, m)
RING₆(st_1, in_2, st_2, in_3, st_3, m) -> IF_6₇(st_1, in_2, st_2, in_3, st_3, m, leq₂(m, length₁(st_3)))
RING₆(st_1, in_2, st_2, in_3, st_3, m) -> IF_9₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(map_f₂(three, head₁(in_3))))
IF_3₇(st_1, in_2, st_2, in_3, st_3, m, false) -> SNDSPLIT₂(m, st_2)
IF_6₇(st_1, in_2, st_2, in_3, st_3, m, false) -> MAP_F₂(three, head₁(in_3))
IF_2₇(st_1, in_2, st_2, in_3, st_3, m, false) -> FSTSPLIT₂(m, app₂(map_f₂(two, head₁(in_2)), st_2))
RING₆(st_1, in_2, st_2, in_3, st_3, m) -> LEQ₂(m, length₁(st_3))
IF_7₇(st_1, in_2, st_2, in_3, st_3, m, false) -> SNDSPLIT₂(m, st_3)
MAP_F₂(pid, cons₂(h, t)) -> APP₂(f₂(pid, h), map_f₂(pid, t))
IF_4₇(st_1, in_2, st_2, in_3, st_3, m, false) -> HEAD₁(in_2)
RING₆(st_1, in_2, st_2, in_3, st_3, m) -> LEQ₂(m, length₁(st_2))
IF_1₇(st_1, in_2, st_2, in_3, st_3, m, false) -> RING₆(sndsplit₂(m, st_1), cons₂(fstsplit₂(m, st_1), in_2), st_2, in_3, st_3, m)
IF_2₇(st_1, in_2, st_2, in_3, st_3, m, true) -> FSTSPLIT₂(m, st_2)
IF_6₇(st_1, in_2, st_2, in_3, st_3, m, true) -> FSTSPLIT₂(m, st_3)
FSTSPLIT₂(s₁(n), cons₂(h, t)) -> FSTSPLIT₂(n, t)
RING₆(st_1, in_2, st_2, in_3, st_3, m) -> LENGTH₁(st_3)
RING₆(st_1, in_2, st_2, in_3, st_3, m) -> MAP_F₂(three, head₁(in_3))
RING₆(st_1, in_2, st_2, in_3, st_3, m) -> EMPTY₁(map_f₂(three, head₁(in_3)))
RING₆(st_1, in_2, st_2, in_3, st_3, m) -> EMPTY₁(fstsplit₂(m, st_1))
IF_4₇(st_1, in_2, st_2, in_3, st_3, m, false) -> TAIL₁(in_2)
RING₆(st_1, in_2, st_2, in_3, st_3, m) -> HEAD₁(in_2)
RING₆(st_1, in_2, st_2, in_3, st_3, m) -> IF_2₇(st_1, in_2, st_2, in_3, st_3, m, leq₂(m, length₁(st_2)))
IF_1₇(st_1, in_2, st_2, in_3, st_3, m, false) -> SNDSPLIT₂(m, st_1)
IF_8₇(st_1, in_2, st_2, in_3, st_3, m, false) -> MAP_F₂(three, head₁(in_3))
IF_2₇(st_1, in_2, st_2, in_3, st_3, m, false) -> EMPTY₁(fstsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2)))
RING₆(st_1, in_2, st_2, in_3, st_3, m) -> IF_1₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_1)))
IF_6₇(st_1, in_2, st_2, in_3, st_3, m, false) -> FSTSPLIT₂(m, app₂(map_f₂(three, head₁(in_3)), st_3))
LEQ₂(s₁(n), s₁(m)) -> LEQ₂(n, m)
RING₆(st_1, in_2, st_2, in_3, st_3, m) -> FSTSPLIT₂(m, st_1)
IF_4₇(st_1, in_2, st_2, in_3, st_3, m, false) -> APP₂(map_f₂(two, head₁(in_2)), st_2)
IF_6₇(st_1, in_2, st_2, in_3, st_3, m, true) -> IF_7₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_3)))

The TRS R consists of the following rules:

fstsplit₂(0, x) -> nil
fstsplit₂(s₁(n), nil) -> nil
fstsplit₂(s₁(n), cons₂(h, t)) -> cons₂(h, fstsplit₂(n, t))
sndsplit₂(0, x) -> x
sndsplit₂(s₁(n), nil) -> nil
sndsplit₂(s₁(n), cons₂(h, t)) -> sndsplit₂(n, t)
empty₁(nil) -> true
empty₁(cons₂(h, t)) -> false
leq₂(0, m) -> true
leq₂(s₁(n), 0) -> false
leq₂(s₁(n), s₁(m)) -> leq₂(n, m)
length₁(nil) -> 0
length₁(cons₂(h, t)) -> s₁(length₁(t))
app₂(nil, x) -> x
app₂(cons₂(h, t), x) -> cons₂(h, app₂(t, x))
map_f₂(pid, nil) -> nil
map_f₂(pid, cons₂(h, t)) -> app₂(f₂(pid, h), map_f₂(pid, t))
head₁(cons₂(h, t)) -> h
tail₁(cons₂(h, t)) -> t
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_1₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_1)))
if_1₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(sndsplit₂(m, st_1), cons₂(fstsplit₂(m, st_1), in_2), st_2, in_3, st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_2₇(st_1, in_2, st_2, in_3, st_3, m, leq₂(m, length₁(st_2)))
if_2₇(st_1, in_2, st_2, in_3, st_3, m, true) -> if_3₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_2)))
if_3₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, sndsplit₂(m, st_2), cons₂(fstsplit₂(m, st_2), in_3), st_3, m)
if_2₇(st_1, in_2, st_2, in_3, st_3, m, false) -> if_4₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2))))
if_4₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, tail₁(in_2), sndsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2)), cons₂(fstsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2)), in_3), st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_5₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(map_f₂(two, head₁(in_2))))
if_5₇(st_1, in_2, st_2, in_3, st_3, m, true) -> ring₆(st_1, tail₁(in_2), st_2, in_3, st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_6₇(st_1, in_2, st_2, in_3, st_3, m, leq₂(m, length₁(st_3)))
if_6₇(st_1, in_2, st_2, in_3, st_3, m, true) -> if_7₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_3)))
if_7₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, st_2, in_3, sndsplit₂(m, st_3), m)
if_6₇(st_1, in_2, st_2, in_3, st_3, m, false) -> if_8₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, app₂(map_f₂(three, head₁(in_3)), st_3))))
if_8₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, st_2, tail₁(in_3), sndsplit₂(m, app₂(map_f₂(three, head₁(in_3)), st_3)), m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_9₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(map_f₂(three, head₁(in_3))))
if_9₇(st_1, in_2, st_2, in_3, st_3, m, true) -> ring₆(st_1, in_2, st_2, tail₁(in_3), st_3, m)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 7 SCCs with 45 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

APP₂(cons₂(h, t), x) -> APP₂(t, x)

The TRS R consists of the following rules:

fstsplit₂(0, x) -> nil
fstsplit₂(s₁(n), nil) -> nil
fstsplit₂(s₁(n), cons₂(h, t)) -> cons₂(h, fstsplit₂(n, t))
sndsplit₂(0, x) -> x
sndsplit₂(s₁(n), nil) -> nil
sndsplit₂(s₁(n), cons₂(h, t)) -> sndsplit₂(n, t)
empty₁(nil) -> true
empty₁(cons₂(h, t)) -> false
leq₂(0, m) -> true
leq₂(s₁(n), 0) -> false
leq₂(s₁(n), s₁(m)) -> leq₂(n, m)
length₁(nil) -> 0
length₁(cons₂(h, t)) -> s₁(length₁(t))
app₂(nil, x) -> x
app₂(cons₂(h, t), x) -> cons₂(h, app₂(t, x))
map_f₂(pid, nil) -> nil
map_f₂(pid, cons₂(h, t)) -> app₂(f₂(pid, h), map_f₂(pid, t))
head₁(cons₂(h, t)) -> h
tail₁(cons₂(h, t)) -> t
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_1₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_1)))
if_1₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(sndsplit₂(m, st_1), cons₂(fstsplit₂(m, st_1), in_2), st_2, in_3, st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_2₇(st_1, in_2, st_2, in_3, st_3, m, leq₂(m, length₁(st_2)))
if_2₇(st_1, in_2, st_2, in_3, st_3, m, true) -> if_3₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_2)))
if_3₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, sndsplit₂(m, st_2), cons₂(fstsplit₂(m, st_2), in_3), st_3, m)
if_2₇(st_1, in_2, st_2, in_3, st_3, m, false) -> if_4₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2))))
if_4₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, tail₁(in_2), sndsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2)), cons₂(fstsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2)), in_3), st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_5₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(map_f₂(two, head₁(in_2))))
if_5₇(st_1, in_2, st_2, in_3, st_3, m, true) -> ring₆(st_1, tail₁(in_2), st_2, in_3, st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_6₇(st_1, in_2, st_2, in_3, st_3, m, leq₂(m, length₁(st_3)))
if_6₇(st_1, in_2, st_2, in_3, st_3, m, true) -> if_7₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_3)))
if_7₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, st_2, in_3, sndsplit₂(m, st_3), m)
if_6₇(st_1, in_2, st_2, in_3, st_3, m, false) -> if_8₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, app₂(map_f₂(three, head₁(in_3)), st_3))))
if_8₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, st_2, tail₁(in_3), sndsplit₂(m, app₂(map_f₂(three, head₁(in_3)), st_3)), m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_9₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(map_f₂(three, head₁(in_3))))
if_9₇(st_1, in_2, st_2, in_3, st_3, m, true) -> ring₆(st_1, in_2, st_2, tail₁(in_3), st_3, m)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

APP₂(cons₂(h, t), x) -> APP₂(t, x)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(APP₂(x₁, x₂)) = 2·x₁ + 2·x₁·x₂
POL(cons₂(x₁, x₂)) = 1 + 2·x₂

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
P is empty.
The TRS R consists of the following rules:

fstsplit₂(0, x) -> nil
fstsplit₂(s₁(n), nil) -> nil
fstsplit₂(s₁(n), cons₂(h, t)) -> cons₂(h, fstsplit₂(n, t))
sndsplit₂(0, x) -> x
sndsplit₂(s₁(n), nil) -> nil
sndsplit₂(s₁(n), cons₂(h, t)) -> sndsplit₂(n, t)
empty₁(nil) -> true
empty₁(cons₂(h, t)) -> false
leq₂(0, m) -> true
leq₂(s₁(n), 0) -> false
leq₂(s₁(n), s₁(m)) -> leq₂(n, m)
length₁(nil) -> 0
length₁(cons₂(h, t)) -> s₁(length₁(t))
app₂(nil, x) -> x
app₂(cons₂(h, t), x) -> cons₂(h, app₂(t, x))
map_f₂(pid, nil) -> nil
map_f₂(pid, cons₂(h, t)) -> app₂(f₂(pid, h), map_f₂(pid, t))
head₁(cons₂(h, t)) -> h
tail₁(cons₂(h, t)) -> t
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_1₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_1)))
if_1₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(sndsplit₂(m, st_1), cons₂(fstsplit₂(m, st_1), in_2), st_2, in_3, st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_2₇(st_1, in_2, st_2, in_3, st_3, m, leq₂(m, length₁(st_2)))
if_2₇(st_1, in_2, st_2, in_3, st_3, m, true) -> if_3₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_2)))
if_3₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, sndsplit₂(m, st_2), cons₂(fstsplit₂(m, st_2), in_3), st_3, m)
if_2₇(st_1, in_2, st_2, in_3, st_3, m, false) -> if_4₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2))))
if_4₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, tail₁(in_2), sndsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2)), cons₂(fstsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2)), in_3), st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_5₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(map_f₂(two, head₁(in_2))))
if_5₇(st_1, in_2, st_2, in_3, st_3, m, true) -> ring₆(st_1, tail₁(in_2), st_2, in_3, st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_6₇(st_1, in_2, st_2, in_3, st_3, m, leq₂(m, length₁(st_3)))
if_6₇(st_1, in_2, st_2, in_3, st_3, m, true) -> if_7₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_3)))
if_7₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, st_2, in_3, sndsplit₂(m, st_3), m)
if_6₇(st_1, in_2, st_2, in_3, st_3, m, false) -> if_8₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, app₂(map_f₂(three, head₁(in_3)), st_3))))
if_8₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, st_2, tail₁(in_3), sndsplit₂(m, app₂(map_f₂(three, head₁(in_3)), st_3)), m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_9₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(map_f₂(three, head₁(in_3))))
if_9₇(st_1, in_2, st_2, in_3, st_3, m, true) -> ring₆(st_1, in_2, st_2, tail₁(in_3), st_3, m)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

MAP_F₂(pid, cons₂(h, t)) -> MAP_F₂(pid, t)

The TRS R consists of the following rules:

fstsplit₂(0, x) -> nil
fstsplit₂(s₁(n), nil) -> nil
fstsplit₂(s₁(n), cons₂(h, t)) -> cons₂(h, fstsplit₂(n, t))
sndsplit₂(0, x) -> x
sndsplit₂(s₁(n), nil) -> nil
sndsplit₂(s₁(n), cons₂(h, t)) -> sndsplit₂(n, t)
empty₁(nil) -> true
empty₁(cons₂(h, t)) -> false
leq₂(0, m) -> true
leq₂(s₁(n), 0) -> false
leq₂(s₁(n), s₁(m)) -> leq₂(n, m)
length₁(nil) -> 0
length₁(cons₂(h, t)) -> s₁(length₁(t))
app₂(nil, x) -> x
app₂(cons₂(h, t), x) -> cons₂(h, app₂(t, x))
map_f₂(pid, nil) -> nil
map_f₂(pid, cons₂(h, t)) -> app₂(f₂(pid, h), map_f₂(pid, t))
head₁(cons₂(h, t)) -> h
tail₁(cons₂(h, t)) -> t
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_1₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_1)))
if_1₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(sndsplit₂(m, st_1), cons₂(fstsplit₂(m, st_1), in_2), st_2, in_3, st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_2₇(st_1, in_2, st_2, in_3, st_3, m, leq₂(m, length₁(st_2)))
if_2₇(st_1, in_2, st_2, in_3, st_3, m, true) -> if_3₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_2)))
if_3₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, sndsplit₂(m, st_2), cons₂(fstsplit₂(m, st_2), in_3), st_3, m)
if_2₇(st_1, in_2, st_2, in_3, st_3, m, false) -> if_4₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2))))
if_4₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, tail₁(in_2), sndsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2)), cons₂(fstsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2)), in_3), st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_5₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(map_f₂(two, head₁(in_2))))
if_5₇(st_1, in_2, st_2, in_3, st_3, m, true) -> ring₆(st_1, tail₁(in_2), st_2, in_3, st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_6₇(st_1, in_2, st_2, in_3, st_3, m, leq₂(m, length₁(st_3)))
if_6₇(st_1, in_2, st_2, in_3, st_3, m, true) -> if_7₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_3)))
if_7₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, st_2, in_3, sndsplit₂(m, st_3), m)
if_6₇(st_1, in_2, st_2, in_3, st_3, m, false) -> if_8₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, app₂(map_f₂(three, head₁(in_3)), st_3))))
if_8₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, st_2, tail₁(in_3), sndsplit₂(m, app₂(map_f₂(three, head₁(in_3)), st_3)), m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_9₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(map_f₂(three, head₁(in_3))))
if_9₇(st_1, in_2, st_2, in_3, st_3, m, true) -> ring₆(st_1, in_2, st_2, tail₁(in_3), st_3, m)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

MAP_F₂(pid, cons₂(h, t)) -> MAP_F₂(pid, t)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(MAP_F₂(x₁, x₂)) = 2·x₁·x₂ + 2·x₂
POL(cons₂(x₁, x₂)) = 1 + 2·x₂

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
P is empty.
The TRS R consists of the following rules:

fstsplit₂(0, x) -> nil
fstsplit₂(s₁(n), nil) -> nil
fstsplit₂(s₁(n), cons₂(h, t)) -> cons₂(h, fstsplit₂(n, t))
sndsplit₂(0, x) -> x
sndsplit₂(s₁(n), nil) -> nil
sndsplit₂(s₁(n), cons₂(h, t)) -> sndsplit₂(n, t)
empty₁(nil) -> true
empty₁(cons₂(h, t)) -> false
leq₂(0, m) -> true
leq₂(s₁(n), 0) -> false
leq₂(s₁(n), s₁(m)) -> leq₂(n, m)
length₁(nil) -> 0
length₁(cons₂(h, t)) -> s₁(length₁(t))
app₂(nil, x) -> x
app₂(cons₂(h, t), x) -> cons₂(h, app₂(t, x))
map_f₂(pid, nil) -> nil
map_f₂(pid, cons₂(h, t)) -> app₂(f₂(pid, h), map_f₂(pid, t))
head₁(cons₂(h, t)) -> h
tail₁(cons₂(h, t)) -> t
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_1₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_1)))
if_1₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(sndsplit₂(m, st_1), cons₂(fstsplit₂(m, st_1), in_2), st_2, in_3, st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_2₇(st_1, in_2, st_2, in_3, st_3, m, leq₂(m, length₁(st_2)))
if_2₇(st_1, in_2, st_2, in_3, st_3, m, true) -> if_3₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_2)))
if_3₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, sndsplit₂(m, st_2), cons₂(fstsplit₂(m, st_2), in_3), st_3, m)
if_2₇(st_1, in_2, st_2, in_3, st_3, m, false) -> if_4₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2))))
if_4₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, tail₁(in_2), sndsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2)), cons₂(fstsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2)), in_3), st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_5₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(map_f₂(two, head₁(in_2))))
if_5₇(st_1, in_2, st_2, in_3, st_3, m, true) -> ring₆(st_1, tail₁(in_2), st_2, in_3, st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_6₇(st_1, in_2, st_2, in_3, st_3, m, leq₂(m, length₁(st_3)))
if_6₇(st_1, in_2, st_2, in_3, st_3, m, true) -> if_7₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_3)))
if_7₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, st_2, in_3, sndsplit₂(m, st_3), m)
if_6₇(st_1, in_2, st_2, in_3, st_3, m, false) -> if_8₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, app₂(map_f₂(three, head₁(in_3)), st_3))))
if_8₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, st_2, tail₁(in_3), sndsplit₂(m, app₂(map_f₂(three, head₁(in_3)), st_3)), m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_9₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(map_f₂(three, head₁(in_3))))
if_9₇(st_1, in_2, st_2, in_3, st_3, m, true) -> ring₆(st_1, in_2, st_2, tail₁(in_3), st_3, m)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

LENGTH₁(cons₂(h, t)) -> LENGTH₁(t)

The TRS R consists of the following rules:

fstsplit₂(0, x) -> nil
fstsplit₂(s₁(n), nil) -> nil
fstsplit₂(s₁(n), cons₂(h, t)) -> cons₂(h, fstsplit₂(n, t))
sndsplit₂(0, x) -> x
sndsplit₂(s₁(n), nil) -> nil
sndsplit₂(s₁(n), cons₂(h, t)) -> sndsplit₂(n, t)
empty₁(nil) -> true
empty₁(cons₂(h, t)) -> false
leq₂(0, m) -> true
leq₂(s₁(n), 0) -> false
leq₂(s₁(n), s₁(m)) -> leq₂(n, m)
length₁(nil) -> 0
length₁(cons₂(h, t)) -> s₁(length₁(t))
app₂(nil, x) -> x
app₂(cons₂(h, t), x) -> cons₂(h, app₂(t, x))
map_f₂(pid, nil) -> nil
map_f₂(pid, cons₂(h, t)) -> app₂(f₂(pid, h), map_f₂(pid, t))
head₁(cons₂(h, t)) -> h
tail₁(cons₂(h, t)) -> t
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_1₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_1)))
if_1₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(sndsplit₂(m, st_1), cons₂(fstsplit₂(m, st_1), in_2), st_2, in_3, st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_2₇(st_1, in_2, st_2, in_3, st_3, m, leq₂(m, length₁(st_2)))
if_2₇(st_1, in_2, st_2, in_3, st_3, m, true) -> if_3₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_2)))
if_3₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, sndsplit₂(m, st_2), cons₂(fstsplit₂(m, st_2), in_3), st_3, m)
if_2₇(st_1, in_2, st_2, in_3, st_3, m, false) -> if_4₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2))))
if_4₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, tail₁(in_2), sndsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2)), cons₂(fstsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2)), in_3), st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_5₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(map_f₂(two, head₁(in_2))))
if_5₇(st_1, in_2, st_2, in_3, st_3, m, true) -> ring₆(st_1, tail₁(in_2), st_2, in_3, st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_6₇(st_1, in_2, st_2, in_3, st_3, m, leq₂(m, length₁(st_3)))
if_6₇(st_1, in_2, st_2, in_3, st_3, m, true) -> if_7₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_3)))
if_7₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, st_2, in_3, sndsplit₂(m, st_3), m)
if_6₇(st_1, in_2, st_2, in_3, st_3, m, false) -> if_8₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, app₂(map_f₂(three, head₁(in_3)), st_3))))
if_8₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, st_2, tail₁(in_3), sndsplit₂(m, app₂(map_f₂(three, head₁(in_3)), st_3)), m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_9₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(map_f₂(three, head₁(in_3))))
if_9₇(st_1, in_2, st_2, in_3, st_3, m, true) -> ring₆(st_1, in_2, st_2, tail₁(in_3), st_3, m)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

LENGTH₁(cons₂(h, t)) -> LENGTH₁(t)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(LENGTH₁(x₁)) = 2·x₁ + 2·x₁²
POL(cons₂(x₁, x₂)) = 2 + x₂

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
P is empty.
The TRS R consists of the following rules:

fstsplit₂(0, x) -> nil
fstsplit₂(s₁(n), nil) -> nil
fstsplit₂(s₁(n), cons₂(h, t)) -> cons₂(h, fstsplit₂(n, t))
sndsplit₂(0, x) -> x
sndsplit₂(s₁(n), nil) -> nil
sndsplit₂(s₁(n), cons₂(h, t)) -> sndsplit₂(n, t)
empty₁(nil) -> true
empty₁(cons₂(h, t)) -> false
leq₂(0, m) -> true
leq₂(s₁(n), 0) -> false
leq₂(s₁(n), s₁(m)) -> leq₂(n, m)
length₁(nil) -> 0
length₁(cons₂(h, t)) -> s₁(length₁(t))
app₂(nil, x) -> x
app₂(cons₂(h, t), x) -> cons₂(h, app₂(t, x))
map_f₂(pid, nil) -> nil
map_f₂(pid, cons₂(h, t)) -> app₂(f₂(pid, h), map_f₂(pid, t))
head₁(cons₂(h, t)) -> h
tail₁(cons₂(h, t)) -> t
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_1₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_1)))
if_1₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(sndsplit₂(m, st_1), cons₂(fstsplit₂(m, st_1), in_2), st_2, in_3, st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_2₇(st_1, in_2, st_2, in_3, st_3, m, leq₂(m, length₁(st_2)))
if_2₇(st_1, in_2, st_2, in_3, st_3, m, true) -> if_3₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_2)))
if_3₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, sndsplit₂(m, st_2), cons₂(fstsplit₂(m, st_2), in_3), st_3, m)
if_2₇(st_1, in_2, st_2, in_3, st_3, m, false) -> if_4₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2))))
if_4₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, tail₁(in_2), sndsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2)), cons₂(fstsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2)), in_3), st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_5₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(map_f₂(two, head₁(in_2))))
if_5₇(st_1, in_2, st_2, in_3, st_3, m, true) -> ring₆(st_1, tail₁(in_2), st_2, in_3, st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_6₇(st_1, in_2, st_2, in_3, st_3, m, leq₂(m, length₁(st_3)))
if_6₇(st_1, in_2, st_2, in_3, st_3, m, true) -> if_7₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_3)))
if_7₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, st_2, in_3, sndsplit₂(m, st_3), m)
if_6₇(st_1, in_2, st_2, in_3, st_3, m, false) -> if_8₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, app₂(map_f₂(three, head₁(in_3)), st_3))))
if_8₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, st_2, tail₁(in_3), sndsplit₂(m, app₂(map_f₂(three, head₁(in_3)), st_3)), m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_9₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(map_f₂(three, head₁(in_3))))
if_9₇(st_1, in_2, st_2, in_3, st_3, m, true) -> ring₆(st_1, in_2, st_2, tail₁(in_3), st_3, m)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

LEQ₂(s₁(n), s₁(m)) -> LEQ₂(n, m)

The TRS R consists of the following rules:

fstsplit₂(0, x) -> nil
fstsplit₂(s₁(n), nil) -> nil
fstsplit₂(s₁(n), cons₂(h, t)) -> cons₂(h, fstsplit₂(n, t))
sndsplit₂(0, x) -> x
sndsplit₂(s₁(n), nil) -> nil
sndsplit₂(s₁(n), cons₂(h, t)) -> sndsplit₂(n, t)
empty₁(nil) -> true
empty₁(cons₂(h, t)) -> false
leq₂(0, m) -> true
leq₂(s₁(n), 0) -> false
leq₂(s₁(n), s₁(m)) -> leq₂(n, m)
length₁(nil) -> 0
length₁(cons₂(h, t)) -> s₁(length₁(t))
app₂(nil, x) -> x
app₂(cons₂(h, t), x) -> cons₂(h, app₂(t, x))
map_f₂(pid, nil) -> nil
map_f₂(pid, cons₂(h, t)) -> app₂(f₂(pid, h), map_f₂(pid, t))
head₁(cons₂(h, t)) -> h
tail₁(cons₂(h, t)) -> t
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_1₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_1)))
if_1₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(sndsplit₂(m, st_1), cons₂(fstsplit₂(m, st_1), in_2), st_2, in_3, st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_2₇(st_1, in_2, st_2, in_3, st_3, m, leq₂(m, length₁(st_2)))
if_2₇(st_1, in_2, st_2, in_3, st_3, m, true) -> if_3₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_2)))
if_3₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, sndsplit₂(m, st_2), cons₂(fstsplit₂(m, st_2), in_3), st_3, m)
if_2₇(st_1, in_2, st_2, in_3, st_3, m, false) -> if_4₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2))))
if_4₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, tail₁(in_2), sndsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2)), cons₂(fstsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2)), in_3), st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_5₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(map_f₂(two, head₁(in_2))))
if_5₇(st_1, in_2, st_2, in_3, st_3, m, true) -> ring₆(st_1, tail₁(in_2), st_2, in_3, st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_6₇(st_1, in_2, st_2, in_3, st_3, m, leq₂(m, length₁(st_3)))
if_6₇(st_1, in_2, st_2, in_3, st_3, m, true) -> if_7₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_3)))
if_7₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, st_2, in_3, sndsplit₂(m, st_3), m)
if_6₇(st_1, in_2, st_2, in_3, st_3, m, false) -> if_8₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, app₂(map_f₂(three, head₁(in_3)), st_3))))
if_8₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, st_2, tail₁(in_3), sndsplit₂(m, app₂(map_f₂(three, head₁(in_3)), st_3)), m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_9₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(map_f₂(three, head₁(in_3))))
if_9₇(st_1, in_2, st_2, in_3, st_3, m, true) -> ring₆(st_1, in_2, st_2, tail₁(in_3), st_3, m)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

LEQ₂(s₁(n), s₁(m)) -> LEQ₂(n, m)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(LEQ₂(x₁, x₂)) = 2·x₁·x₂
POL(s₁(x₁)) = 2 + x₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
P is empty.
The TRS R consists of the following rules:

fstsplit₂(0, x) -> nil
fstsplit₂(s₁(n), nil) -> nil
fstsplit₂(s₁(n), cons₂(h, t)) -> cons₂(h, fstsplit₂(n, t))
sndsplit₂(0, x) -> x
sndsplit₂(s₁(n), nil) -> nil
sndsplit₂(s₁(n), cons₂(h, t)) -> sndsplit₂(n, t)
empty₁(nil) -> true
empty₁(cons₂(h, t)) -> false
leq₂(0, m) -> true
leq₂(s₁(n), 0) -> false
leq₂(s₁(n), s₁(m)) -> leq₂(n, m)
length₁(nil) -> 0
length₁(cons₂(h, t)) -> s₁(length₁(t))
app₂(nil, x) -> x
app₂(cons₂(h, t), x) -> cons₂(h, app₂(t, x))
map_f₂(pid, nil) -> nil
map_f₂(pid, cons₂(h, t)) -> app₂(f₂(pid, h), map_f₂(pid, t))
head₁(cons₂(h, t)) -> h
tail₁(cons₂(h, t)) -> t
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_1₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_1)))
if_1₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(sndsplit₂(m, st_1), cons₂(fstsplit₂(m, st_1), in_2), st_2, in_3, st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_2₇(st_1, in_2, st_2, in_3, st_3, m, leq₂(m, length₁(st_2)))
if_2₇(st_1, in_2, st_2, in_3, st_3, m, true) -> if_3₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_2)))
if_3₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, sndsplit₂(m, st_2), cons₂(fstsplit₂(m, st_2), in_3), st_3, m)
if_2₇(st_1, in_2, st_2, in_3, st_3, m, false) -> if_4₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2))))
if_4₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, tail₁(in_2), sndsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2)), cons₂(fstsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2)), in_3), st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_5₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(map_f₂(two, head₁(in_2))))
if_5₇(st_1, in_2, st_2, in_3, st_3, m, true) -> ring₆(st_1, tail₁(in_2), st_2, in_3, st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_6₇(st_1, in_2, st_2, in_3, st_3, m, leq₂(m, length₁(st_3)))
if_6₇(st_1, in_2, st_2, in_3, st_3, m, true) -> if_7₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_3)))
if_7₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, st_2, in_3, sndsplit₂(m, st_3), m)
if_6₇(st_1, in_2, st_2, in_3, st_3, m, false) -> if_8₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, app₂(map_f₂(three, head₁(in_3)), st_3))))
if_8₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, st_2, tail₁(in_3), sndsplit₂(m, app₂(map_f₂(three, head₁(in_3)), st_3)), m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_9₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(map_f₂(three, head₁(in_3))))
if_9₇(st_1, in_2, st_2, in_3, st_3, m, true) -> ring₆(st_1, in_2, st_2, tail₁(in_3), st_3, m)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

SNDSPLIT₂(s₁(n), cons₂(h, t)) -> SNDSPLIT₂(n, t)

The TRS R consists of the following rules:

fstsplit₂(0, x) -> nil
fstsplit₂(s₁(n), nil) -> nil
fstsplit₂(s₁(n), cons₂(h, t)) -> cons₂(h, fstsplit₂(n, t))
sndsplit₂(0, x) -> x
sndsplit₂(s₁(n), nil) -> nil
sndsplit₂(s₁(n), cons₂(h, t)) -> sndsplit₂(n, t)
empty₁(nil) -> true
empty₁(cons₂(h, t)) -> false
leq₂(0, m) -> true
leq₂(s₁(n), 0) -> false
leq₂(s₁(n), s₁(m)) -> leq₂(n, m)
length₁(nil) -> 0
length₁(cons₂(h, t)) -> s₁(length₁(t))
app₂(nil, x) -> x
app₂(cons₂(h, t), x) -> cons₂(h, app₂(t, x))
map_f₂(pid, nil) -> nil
map_f₂(pid, cons₂(h, t)) -> app₂(f₂(pid, h), map_f₂(pid, t))
head₁(cons₂(h, t)) -> h
tail₁(cons₂(h, t)) -> t
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_1₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_1)))
if_1₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(sndsplit₂(m, st_1), cons₂(fstsplit₂(m, st_1), in_2), st_2, in_3, st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_2₇(st_1, in_2, st_2, in_3, st_3, m, leq₂(m, length₁(st_2)))
if_2₇(st_1, in_2, st_2, in_3, st_3, m, true) -> if_3₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_2)))
if_3₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, sndsplit₂(m, st_2), cons₂(fstsplit₂(m, st_2), in_3), st_3, m)
if_2₇(st_1, in_2, st_2, in_3, st_3, m, false) -> if_4₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2))))
if_4₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, tail₁(in_2), sndsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2)), cons₂(fstsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2)), in_3), st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_5₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(map_f₂(two, head₁(in_2))))
if_5₇(st_1, in_2, st_2, in_3, st_3, m, true) -> ring₆(st_1, tail₁(in_2), st_2, in_3, st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_6₇(st_1, in_2, st_2, in_3, st_3, m, leq₂(m, length₁(st_3)))
if_6₇(st_1, in_2, st_2, in_3, st_3, m, true) -> if_7₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_3)))
if_7₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, st_2, in_3, sndsplit₂(m, st_3), m)
if_6₇(st_1, in_2, st_2, in_3, st_3, m, false) -> if_8₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, app₂(map_f₂(three, head₁(in_3)), st_3))))
if_8₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, st_2, tail₁(in_3), sndsplit₂(m, app₂(map_f₂(three, head₁(in_3)), st_3)), m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_9₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(map_f₂(three, head₁(in_3))))
if_9₇(st_1, in_2, st_2, in_3, st_3, m, true) -> ring₆(st_1, in_2, st_2, tail₁(in_3), st_3, m)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

SNDSPLIT₂(s₁(n), cons₂(h, t)) -> SNDSPLIT₂(n, t)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(SNDSPLIT₂(x₁, x₂)) = x₁ + 2·x₁·x₂
POL(cons₂(x₁, x₂)) = x₂
POL(s₁(x₁)) = 2 + x₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

Q DP problem:
P is empty.
The TRS R consists of the following rules:

fstsplit₂(0, x) -> nil
fstsplit₂(s₁(n), nil) -> nil
fstsplit₂(s₁(n), cons₂(h, t)) -> cons₂(h, fstsplit₂(n, t))
sndsplit₂(0, x) -> x
sndsplit₂(s₁(n), nil) -> nil
sndsplit₂(s₁(n), cons₂(h, t)) -> sndsplit₂(n, t)
empty₁(nil) -> true
empty₁(cons₂(h, t)) -> false
leq₂(0, m) -> true
leq₂(s₁(n), 0) -> false
leq₂(s₁(n), s₁(m)) -> leq₂(n, m)
length₁(nil) -> 0
length₁(cons₂(h, t)) -> s₁(length₁(t))
app₂(nil, x) -> x
app₂(cons₂(h, t), x) -> cons₂(h, app₂(t, x))
map_f₂(pid, nil) -> nil
map_f₂(pid, cons₂(h, t)) -> app₂(f₂(pid, h), map_f₂(pid, t))
head₁(cons₂(h, t)) -> h
tail₁(cons₂(h, t)) -> t
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_1₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_1)))
if_1₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(sndsplit₂(m, st_1), cons₂(fstsplit₂(m, st_1), in_2), st_2, in_3, st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_2₇(st_1, in_2, st_2, in_3, st_3, m, leq₂(m, length₁(st_2)))
if_2₇(st_1, in_2, st_2, in_3, st_3, m, true) -> if_3₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_2)))
if_3₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, sndsplit₂(m, st_2), cons₂(fstsplit₂(m, st_2), in_3), st_3, m)
if_2₇(st_1, in_2, st_2, in_3, st_3, m, false) -> if_4₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2))))
if_4₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, tail₁(in_2), sndsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2)), cons₂(fstsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2)), in_3), st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_5₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(map_f₂(two, head₁(in_2))))
if_5₇(st_1, in_2, st_2, in_3, st_3, m, true) -> ring₆(st_1, tail₁(in_2), st_2, in_3, st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_6₇(st_1, in_2, st_2, in_3, st_3, m, leq₂(m, length₁(st_3)))
if_6₇(st_1, in_2, st_2, in_3, st_3, m, true) -> if_7₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_3)))
if_7₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, st_2, in_3, sndsplit₂(m, st_3), m)
if_6₇(st_1, in_2, st_2, in_3, st_3, m, false) -> if_8₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, app₂(map_f₂(three, head₁(in_3)), st_3))))
if_8₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, st_2, tail₁(in_3), sndsplit₂(m, app₂(map_f₂(three, head₁(in_3)), st_3)), m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_9₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(map_f₂(three, head₁(in_3))))
if_9₇(st_1, in_2, st_2, in_3, st_3, m, true) -> ring₆(st_1, in_2, st_2, tail₁(in_3), st_3, m)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

FSTSPLIT₂(s₁(n), cons₂(h, t)) -> FSTSPLIT₂(n, t)

The TRS R consists of the following rules:

fstsplit₂(0, x) -> nil
fstsplit₂(s₁(n), nil) -> nil
fstsplit₂(s₁(n), cons₂(h, t)) -> cons₂(h, fstsplit₂(n, t))
sndsplit₂(0, x) -> x
sndsplit₂(s₁(n), nil) -> nil
sndsplit₂(s₁(n), cons₂(h, t)) -> sndsplit₂(n, t)
empty₁(nil) -> true
empty₁(cons₂(h, t)) -> false
leq₂(0, m) -> true
leq₂(s₁(n), 0) -> false
leq₂(s₁(n), s₁(m)) -> leq₂(n, m)
length₁(nil) -> 0
length₁(cons₂(h, t)) -> s₁(length₁(t))
app₂(nil, x) -> x
app₂(cons₂(h, t), x) -> cons₂(h, app₂(t, x))
map_f₂(pid, nil) -> nil
map_f₂(pid, cons₂(h, t)) -> app₂(f₂(pid, h), map_f₂(pid, t))
head₁(cons₂(h, t)) -> h
tail₁(cons₂(h, t)) -> t
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_1₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_1)))
if_1₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(sndsplit₂(m, st_1), cons₂(fstsplit₂(m, st_1), in_2), st_2, in_3, st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_2₇(st_1, in_2, st_2, in_3, st_3, m, leq₂(m, length₁(st_2)))
if_2₇(st_1, in_2, st_2, in_3, st_3, m, true) -> if_3₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_2)))
if_3₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, sndsplit₂(m, st_2), cons₂(fstsplit₂(m, st_2), in_3), st_3, m)
if_2₇(st_1, in_2, st_2, in_3, st_3, m, false) -> if_4₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2))))
if_4₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, tail₁(in_2), sndsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2)), cons₂(fstsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2)), in_3), st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_5₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(map_f₂(two, head₁(in_2))))
if_5₇(st_1, in_2, st_2, in_3, st_3, m, true) -> ring₆(st_1, tail₁(in_2), st_2, in_3, st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_6₇(st_1, in_2, st_2, in_3, st_3, m, leq₂(m, length₁(st_3)))
if_6₇(st_1, in_2, st_2, in_3, st_3, m, true) -> if_7₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_3)))
if_7₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, st_2, in_3, sndsplit₂(m, st_3), m)
if_6₇(st_1, in_2, st_2, in_3, st_3, m, false) -> if_8₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, app₂(map_f₂(three, head₁(in_3)), st_3))))
if_8₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, st_2, tail₁(in_3), sndsplit₂(m, app₂(map_f₂(three, head₁(in_3)), st_3)), m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_9₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(map_f₂(three, head₁(in_3))))
if_9₇(st_1, in_2, st_2, in_3, st_3, m, true) -> ring₆(st_1, in_2, st_2, tail₁(in_3), st_3, m)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

FSTSPLIT₂(s₁(n), cons₂(h, t)) -> FSTSPLIT₂(n, t)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(FSTSPLIT₂(x₁, x₂)) = x₁ + 2·x₁·x₂
POL(cons₂(x₁, x₂)) = x₂
POL(s₁(x₁)) = 2 + x₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ PisEmptyProof

          ↳ QDP

Q DP problem:
P is empty.
The TRS R consists of the following rules:

fstsplit₂(0, x) -> nil
fstsplit₂(s₁(n), nil) -> nil
fstsplit₂(s₁(n), cons₂(h, t)) -> cons₂(h, fstsplit₂(n, t))
sndsplit₂(0, x) -> x
sndsplit₂(s₁(n), nil) -> nil
sndsplit₂(s₁(n), cons₂(h, t)) -> sndsplit₂(n, t)
empty₁(nil) -> true
empty₁(cons₂(h, t)) -> false
leq₂(0, m) -> true
leq₂(s₁(n), 0) -> false
leq₂(s₁(n), s₁(m)) -> leq₂(n, m)
length₁(nil) -> 0
length₁(cons₂(h, t)) -> s₁(length₁(t))
app₂(nil, x) -> x
app₂(cons₂(h, t), x) -> cons₂(h, app₂(t, x))
map_f₂(pid, nil) -> nil
map_f₂(pid, cons₂(h, t)) -> app₂(f₂(pid, h), map_f₂(pid, t))
head₁(cons₂(h, t)) -> h
tail₁(cons₂(h, t)) -> t
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_1₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_1)))
if_1₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(sndsplit₂(m, st_1), cons₂(fstsplit₂(m, st_1), in_2), st_2, in_3, st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_2₇(st_1, in_2, st_2, in_3, st_3, m, leq₂(m, length₁(st_2)))
if_2₇(st_1, in_2, st_2, in_3, st_3, m, true) -> if_3₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_2)))
if_3₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, sndsplit₂(m, st_2), cons₂(fstsplit₂(m, st_2), in_3), st_3, m)
if_2₇(st_1, in_2, st_2, in_3, st_3, m, false) -> if_4₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2))))
if_4₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, tail₁(in_2), sndsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2)), cons₂(fstsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2)), in_3), st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_5₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(map_f₂(two, head₁(in_2))))
if_5₇(st_1, in_2, st_2, in_3, st_3, m, true) -> ring₆(st_1, tail₁(in_2), st_2, in_3, st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_6₇(st_1, in_2, st_2, in_3, st_3, m, leq₂(m, length₁(st_3)))
if_6₇(st_1, in_2, st_2, in_3, st_3, m, true) -> if_7₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_3)))
if_7₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, st_2, in_3, sndsplit₂(m, st_3), m)
if_6₇(st_1, in_2, st_2, in_3, st_3, m, false) -> if_8₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, app₂(map_f₂(three, head₁(in_3)), st_3))))
if_8₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, st_2, tail₁(in_3), sndsplit₂(m, app₂(map_f₂(three, head₁(in_3)), st_3)), m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_9₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(map_f₂(three, head₁(in_3))))
if_9₇(st_1, in_2, st_2, in_3, st_3, m, true) -> ring₆(st_1, in_2, st_2, tail₁(in_3), st_3, m)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

IF_9₇(st_1, in_2, st_2, in_3, st_3, m, true) -> RING₆(st_1, in_2, st_2, tail₁(in_3), st_3, m)
RING₆(st_1, in_2, st_2, in_3, st_3, m) -> IF_2₇(st_1, in_2, st_2, in_3, st_3, m, leq₂(m, length₁(st_2)))
RING₆(st_1, in_2, st_2, in_3, st_3, m) -> IF_5₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(map_f₂(two, head₁(in_2))))
IF_5₇(st_1, in_2, st_2, in_3, st_3, m, true) -> RING₆(st_1, tail₁(in_2), st_2, in_3, st_3, m)
IF_6₇(st_1, in_2, st_2, in_3, st_3, m, false) -> IF_8₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, app₂(map_f₂(three, head₁(in_3)), st_3))))
IF_2₇(st_1, in_2, st_2, in_3, st_3, m, false) -> IF_4₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2))))
IF_7₇(st_1, in_2, st_2, in_3, st_3, m, false) -> RING₆(st_1, in_2, st_2, in_3, sndsplit₂(m, st_3), m)
RING₆(st_1, in_2, st_2, in_3, st_3, m) -> IF_1₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_1)))
IF_3₇(st_1, in_2, st_2, in_3, st_3, m, false) -> RING₆(st_1, in_2, sndsplit₂(m, st_2), cons₂(fstsplit₂(m, st_2), in_3), st_3, m)
IF_1₇(st_1, in_2, st_2, in_3, st_3, m, false) -> RING₆(sndsplit₂(m, st_1), cons₂(fstsplit₂(m, st_1), in_2), st_2, in_3, st_3, m)
IF_8₇(st_1, in_2, st_2, in_3, st_3, m, false) -> RING₆(st_1, in_2, st_2, tail₁(in_3), sndsplit₂(m, app₂(map_f₂(three, head₁(in_3)), st_3)), m)
IF_2₇(st_1, in_2, st_2, in_3, st_3, m, true) -> IF_3₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_2)))
IF_4₇(st_1, in_2, st_2, in_3, st_3, m, false) -> RING₆(st_1, tail₁(in_2), sndsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2)), cons₂(fstsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2)), in_3), st_3, m)
RING₆(st_1, in_2, st_2, in_3, st_3, m) -> IF_6₇(st_1, in_2, st_2, in_3, st_3, m, leq₂(m, length₁(st_3)))
RING₆(st_1, in_2, st_2, in_3, st_3, m) -> IF_9₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(map_f₂(three, head₁(in_3))))
IF_6₇(st_1, in_2, st_2, in_3, st_3, m, true) -> IF_7₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_3)))

The TRS R consists of the following rules:

fstsplit₂(0, x) -> nil
fstsplit₂(s₁(n), nil) -> nil
fstsplit₂(s₁(n), cons₂(h, t)) -> cons₂(h, fstsplit₂(n, t))
sndsplit₂(0, x) -> x
sndsplit₂(s₁(n), nil) -> nil
sndsplit₂(s₁(n), cons₂(h, t)) -> sndsplit₂(n, t)
empty₁(nil) -> true
empty₁(cons₂(h, t)) -> false
leq₂(0, m) -> true
leq₂(s₁(n), 0) -> false
leq₂(s₁(n), s₁(m)) -> leq₂(n, m)
length₁(nil) -> 0
length₁(cons₂(h, t)) -> s₁(length₁(t))
app₂(nil, x) -> x
app₂(cons₂(h, t), x) -> cons₂(h, app₂(t, x))
map_f₂(pid, nil) -> nil
map_f₂(pid, cons₂(h, t)) -> app₂(f₂(pid, h), map_f₂(pid, t))
head₁(cons₂(h, t)) -> h
tail₁(cons₂(h, t)) -> t
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_1₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_1)))
if_1₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(sndsplit₂(m, st_1), cons₂(fstsplit₂(m, st_1), in_2), st_2, in_3, st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_2₇(st_1, in_2, st_2, in_3, st_3, m, leq₂(m, length₁(st_2)))
if_2₇(st_1, in_2, st_2, in_3, st_3, m, true) -> if_3₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_2)))
if_3₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, sndsplit₂(m, st_2), cons₂(fstsplit₂(m, st_2), in_3), st_3, m)
if_2₇(st_1, in_2, st_2, in_3, st_3, m, false) -> if_4₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2))))
if_4₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, tail₁(in_2), sndsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2)), cons₂(fstsplit₂(m, app₂(map_f₂(two, head₁(in_2)), st_2)), in_3), st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_5₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(map_f₂(two, head₁(in_2))))
if_5₇(st_1, in_2, st_2, in_3, st_3, m, true) -> ring₆(st_1, tail₁(in_2), st_2, in_3, st_3, m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_6₇(st_1, in_2, st_2, in_3, st_3, m, leq₂(m, length₁(st_3)))
if_6₇(st_1, in_2, st_2, in_3, st_3, m, true) -> if_7₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, st_3)))
if_7₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, st_2, in_3, sndsplit₂(m, st_3), m)
if_6₇(st_1, in_2, st_2, in_3, st_3, m, false) -> if_8₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(fstsplit₂(m, app₂(map_f₂(three, head₁(in_3)), st_3))))
if_8₇(st_1, in_2, st_2, in_3, st_3, m, false) -> ring₆(st_1, in_2, st_2, tail₁(in_3), sndsplit₂(m, app₂(map_f₂(three, head₁(in_3)), st_3)), m)
ring₆(st_1, in_2, st_2, in_3, st_3, m) -> if_9₇(st_1, in_2, st_2, in_3, st_3, m, empty₁(map_f₂(three, head₁(in_3))))
if_9₇(st_1, in_2, st_2, in_3, st_3, m, true) -> ring₆(st_1, in_2, st_2, tail₁(in_3), st_3, m)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.