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))
process(store, m) → if1(store, m, leq(m, length(store)))
if1(store, m, true) → if2(store, m, empty(fstsplit(m, store)))
if1(store, m, false) → if3(store, m, empty(fstsplit(m, app(map_f(self, nil), store))))
if2(store, m, false) → process(app(map_f(self, nil), sndsplit(m, store)), m)
if3(store, m, false) → process(sndsplit(m, app(map_f(self, nil), store)), m)
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))
process'(store, m) → if1'(store, m, leq'(m, length'(store)))
if1'(store, m, true') → if2'(store, m, empty'(fstsplit'(m, store)))
if1'(store, m, false') → if3'(store, m, empty'(fstsplit'(m, app'(map_f'(self', nil'), store))))
if2'(store, m, false') → process'(app'(map_f'(self', nil'), sndsplit'(m, store)), m)
if3'(store, m, false') → process'(sndsplit'(m, app'(map_f'(self', nil'), store)), m)
Sliced the following arguments:
cons'/0
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'(t)) → cons'(fstsplit'(n, t))
sndsplit'(0', x) → x
sndsplit'(s'(n), nil') → nil'
sndsplit'(s'(n), cons'(t)) → sndsplit'(n, t)
empty'(nil') → true'
empty'(cons'(t)) → false'
leq'(0', m) → true'
leq'(s'(n), 0') → false'
leq'(s'(n), s'(m)) → leq'(n, m)
length'(nil') → 0'
length'(cons'(t)) → s'(length'(t))
app'(nil', x) → x
app'(cons'(t), x) → cons'(app'(t, x))
map_f'(nil') → nil'
map_f'(cons'(t)) → app'(f', map_f'(t))
process'(store, m) → if1'(store, m, leq'(m, length'(store)))
if1'(store, m, true') → if2'(store, m, empty'(fstsplit'(m, store)))
if1'(store, m, false') → if3'(store, m, empty'(fstsplit'(m, app'(map_f'(nil'), store))))
if2'(store, m, false') → process'(app'(map_f'(nil'), sndsplit'(m, store)), m)
if3'(store, m, false') → process'(sndsplit'(m, app'(map_f'(nil'), store)), m)
Infered types.
Rules:
fstsplit'(0', x) → nil'
fstsplit'(s'(n), nil') → nil'
fstsplit'(s'(n), cons'(t)) → cons'(fstsplit'(n, t))
sndsplit'(0', x) → x
sndsplit'(s'(n), nil') → nil'
sndsplit'(s'(n), cons'(t)) → sndsplit'(n, t)
empty'(nil') → true'
empty'(cons'(t)) → false'
leq'(0', m) → true'
leq'(s'(n), 0') → false'
leq'(s'(n), s'(m)) → leq'(n, m)
length'(nil') → 0'
length'(cons'(t)) → s'(length'(t))
app'(nil', x) → x
app'(cons'(t), x) → cons'(app'(t, x))
map_f'(nil') → nil'
map_f'(cons'(t)) → app'(f', map_f'(t))
process'(store, m) → if1'(store, m, leq'(m, length'(store)))
if1'(store, m, true') → if2'(store, m, empty'(fstsplit'(m, store)))
if1'(store, m, false') → if3'(store, m, empty'(fstsplit'(m, app'(map_f'(nil'), store))))
if2'(store, m, false') → process'(app'(map_f'(nil'), sndsplit'(m, store)), m)
if3'(store, m, false') → process'(sndsplit'(m, app'(map_f'(nil'), store)), 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'
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'
process' :: nil':cons':f' → 0':s' → process':if1':if2':if3'
if1' :: nil':cons':f' → 0':s' → true':false' → process':if1':if2':if3'
if2' :: nil':cons':f' → 0':s' → true':false' → process':if1':if2':if3'
if3' :: nil':cons':f' → 0':s' → true':false' → process':if1':if2':if3'
_hole_nil':cons':f'1 :: nil':cons':f'
_hole_0':s'2 :: 0':s'
_hole_true':false'3 :: true':false'
_hole_process':if1':if2':if3'4 :: process':if1':if2':if3'
_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', process'
They will be analysed ascendingly in the following order:
fstsplit' < process'
sndsplit' < process'
leq' < process'
length' < process'
app' < map_f'
app' < process'
map_f' < process'
Rules:
fstsplit'(0', x) → nil'
fstsplit'(s'(n), nil') → nil'
fstsplit'(s'(n), cons'(t)) → cons'(fstsplit'(n, t))
sndsplit'(0', x) → x
sndsplit'(s'(n), nil') → nil'
sndsplit'(s'(n), cons'(t)) → sndsplit'(n, t)
empty'(nil') → true'
empty'(cons'(t)) → false'
leq'(0', m) → true'
leq'(s'(n), 0') → false'
leq'(s'(n), s'(m)) → leq'(n, m)
length'(nil') → 0'
length'(cons'(t)) → s'(length'(t))
app'(nil', x) → x
app'(cons'(t), x) → cons'(app'(t, x))
map_f'(nil') → nil'
map_f'(cons'(t)) → app'(f', map_f'(t))
process'(store, m) → if1'(store, m, leq'(m, length'(store)))
if1'(store, m, true') → if2'(store, m, empty'(fstsplit'(m, store)))
if1'(store, m, false') → if3'(store, m, empty'(fstsplit'(m, app'(map_f'(nil'), store))))
if2'(store, m, false') → process'(app'(map_f'(nil'), sndsplit'(m, store)), m)
if3'(store, m, false') → process'(sndsplit'(m, app'(map_f'(nil'), store)), 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'
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'
process' :: nil':cons':f' → 0':s' → process':if1':if2':if3'
if1' :: nil':cons':f' → 0':s' → true':false' → process':if1':if2':if3'
if2' :: nil':cons':f' → 0':s' → true':false' → process':if1':if2':if3'
if3' :: nil':cons':f' → 0':s' → true':false' → process':if1':if2':if3'
_hole_nil':cons':f'1 :: nil':cons':f'
_hole_0':s'2 :: 0':s'
_hole_true':false'3 :: true':false'
_hole_process':if1':if2':if3'4 :: process':if1':if2':if3'
_gen_nil':cons':f'5 :: Nat → nil':cons':f'
_gen_0':s'6 :: Nat → 0':s'
Generator Equations:
_gen_nil':cons':f'5(0) ⇔ nil'
_gen_nil':cons':f'5(+(x, 1)) ⇔ cons'(_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', process'
They will be analysed ascendingly in the following order:
fstsplit' < process'
sndsplit' < process'
leq' < process'
length' < process'
app' < map_f'
app' < process'
map_f' < process'
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)) →RΩ(1)
nil'
Induction Step:
fstsplit'(_gen_0':s'6(+(_$n9, 1)), _gen_nil':cons':f'5(+(_$n9, 1))) →RΩ(1)
cons'(fstsplit'(_gen_0':s'6(_$n9), _gen_nil':cons':f'5(_$n9))) →IH
cons'(_gen_nil':cons':f'5(_$n9))
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
fstsplit'(0', x) → nil'
fstsplit'(s'(n), nil') → nil'
fstsplit'(s'(n), cons'(t)) → cons'(fstsplit'(n, t))
sndsplit'(0', x) → x
sndsplit'(s'(n), nil') → nil'
sndsplit'(s'(n), cons'(t)) → sndsplit'(n, t)
empty'(nil') → true'
empty'(cons'(t)) → false'
leq'(0', m) → true'
leq'(s'(n), 0') → false'
leq'(s'(n), s'(m)) → leq'(n, m)
length'(nil') → 0'
length'(cons'(t)) → s'(length'(t))
app'(nil', x) → x
app'(cons'(t), x) → cons'(app'(t, x))
map_f'(nil') → nil'
map_f'(cons'(t)) → app'(f', map_f'(t))
process'(store, m) → if1'(store, m, leq'(m, length'(store)))
if1'(store, m, true') → if2'(store, m, empty'(fstsplit'(m, store)))
if1'(store, m, false') → if3'(store, m, empty'(fstsplit'(m, app'(map_f'(nil'), store))))
if2'(store, m, false') → process'(app'(map_f'(nil'), sndsplit'(m, store)), m)
if3'(store, m, false') → process'(sndsplit'(m, app'(map_f'(nil'), store)), 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'
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'
process' :: nil':cons':f' → 0':s' → process':if1':if2':if3'
if1' :: nil':cons':f' → 0':s' → true':false' → process':if1':if2':if3'
if2' :: nil':cons':f' → 0':s' → true':false' → process':if1':if2':if3'
if3' :: nil':cons':f' → 0':s' → true':false' → process':if1':if2':if3'
_hole_nil':cons':f'1 :: nil':cons':f'
_hole_0':s'2 :: 0':s'
_hole_true':false'3 :: true':false'
_hole_process':if1':if2':if3'4 :: process':if1':if2':if3'
_gen_nil':cons':f'5 :: Nat → nil':cons':f'
_gen_0':s'6 :: Nat → 0':s'
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'(_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', process'
They will be analysed ascendingly in the following order:
sndsplit' < process'
leq' < process'
length' < process'
app' < map_f'
app' < process'
map_f' < process'
Proved the following rewrite lemma:
sndsplit'(_gen_0':s'6(_n1089), _gen_nil':cons':f'5(_n1089)) → _gen_nil':cons':f'5(0), rt ∈ Ω(1 + n1089)
Induction Base:
sndsplit'(_gen_0':s'6(0), _gen_nil':cons':f'5(0)) →RΩ(1)
_gen_nil':cons':f'5(0)
Induction Step:
sndsplit'(_gen_0':s'6(+(_$n1090, 1)), _gen_nil':cons':f'5(+(_$n1090, 1))) →RΩ(1)
sndsplit'(_gen_0':s'6(_$n1090), _gen_nil':cons':f'5(_$n1090)) →IH
_gen_nil':cons':f'5(0)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
fstsplit'(0', x) → nil'
fstsplit'(s'(n), nil') → nil'
fstsplit'(s'(n), cons'(t)) → cons'(fstsplit'(n, t))
sndsplit'(0', x) → x
sndsplit'(s'(n), nil') → nil'
sndsplit'(s'(n), cons'(t)) → sndsplit'(n, t)
empty'(nil') → true'
empty'(cons'(t)) → false'
leq'(0', m) → true'
leq'(s'(n), 0') → false'
leq'(s'(n), s'(m)) → leq'(n, m)
length'(nil') → 0'
length'(cons'(t)) → s'(length'(t))
app'(nil', x) → x
app'(cons'(t), x) → cons'(app'(t, x))
map_f'(nil') → nil'
map_f'(cons'(t)) → app'(f', map_f'(t))
process'(store, m) → if1'(store, m, leq'(m, length'(store)))
if1'(store, m, true') → if2'(store, m, empty'(fstsplit'(m, store)))
if1'(store, m, false') → if3'(store, m, empty'(fstsplit'(m, app'(map_f'(nil'), store))))
if2'(store, m, false') → process'(app'(map_f'(nil'), sndsplit'(m, store)), m)
if3'(store, m, false') → process'(sndsplit'(m, app'(map_f'(nil'), store)), 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'
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'
process' :: nil':cons':f' → 0':s' → process':if1':if2':if3'
if1' :: nil':cons':f' → 0':s' → true':false' → process':if1':if2':if3'
if2' :: nil':cons':f' → 0':s' → true':false' → process':if1':if2':if3'
if3' :: nil':cons':f' → 0':s' → true':false' → process':if1':if2':if3'
_hole_nil':cons':f'1 :: nil':cons':f'
_hole_0':s'2 :: 0':s'
_hole_true':false'3 :: true':false'
_hole_process':if1':if2':if3'4 :: process':if1':if2':if3'
_gen_nil':cons':f'5 :: Nat → nil':cons':f'
_gen_0':s'6 :: Nat → 0':s'
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(_n1089), _gen_nil':cons':f'5(_n1089)) → _gen_nil':cons':f'5(0), rt ∈ Ω(1 + n1089)
Generator Equations:
_gen_nil':cons':f'5(0) ⇔ nil'
_gen_nil':cons':f'5(+(x, 1)) ⇔ cons'(_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', process'
They will be analysed ascendingly in the following order:
leq' < process'
length' < process'
app' < map_f'
app' < process'
map_f' < process'
Proved the following rewrite lemma:
leq'(_gen_0':s'6(_n2312), _gen_0':s'6(_n2312)) → true', rt ∈ Ω(1 + n2312)
Induction Base:
leq'(_gen_0':s'6(0), _gen_0':s'6(0)) →RΩ(1)
true'
Induction Step:
leq'(_gen_0':s'6(+(_$n2313, 1)), _gen_0':s'6(+(_$n2313, 1))) →RΩ(1)
leq'(_gen_0':s'6(_$n2313), _gen_0':s'6(_$n2313)) →IH
true'
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
fstsplit'(0', x) → nil'
fstsplit'(s'(n), nil') → nil'
fstsplit'(s'(n), cons'(t)) → cons'(fstsplit'(n, t))
sndsplit'(0', x) → x
sndsplit'(s'(n), nil') → nil'
sndsplit'(s'(n), cons'(t)) → sndsplit'(n, t)
empty'(nil') → true'
empty'(cons'(t)) → false'
leq'(0', m) → true'
leq'(s'(n), 0') → false'
leq'(s'(n), s'(m)) → leq'(n, m)
length'(nil') → 0'
length'(cons'(t)) → s'(length'(t))
app'(nil', x) → x
app'(cons'(t), x) → cons'(app'(t, x))
map_f'(nil') → nil'
map_f'(cons'(t)) → app'(f', map_f'(t))
process'(store, m) → if1'(store, m, leq'(m, length'(store)))
if1'(store, m, true') → if2'(store, m, empty'(fstsplit'(m, store)))
if1'(store, m, false') → if3'(store, m, empty'(fstsplit'(m, app'(map_f'(nil'), store))))
if2'(store, m, false') → process'(app'(map_f'(nil'), sndsplit'(m, store)), m)
if3'(store, m, false') → process'(sndsplit'(m, app'(map_f'(nil'), store)), 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'
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'
process' :: nil':cons':f' → 0':s' → process':if1':if2':if3'
if1' :: nil':cons':f' → 0':s' → true':false' → process':if1':if2':if3'
if2' :: nil':cons':f' → 0':s' → true':false' → process':if1':if2':if3'
if3' :: nil':cons':f' → 0':s' → true':false' → process':if1':if2':if3'
_hole_nil':cons':f'1 :: nil':cons':f'
_hole_0':s'2 :: 0':s'
_hole_true':false'3 :: true':false'
_hole_process':if1':if2':if3'4 :: process':if1':if2':if3'
_gen_nil':cons':f'5 :: Nat → nil':cons':f'
_gen_0':s'6 :: Nat → 0':s'
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(_n1089), _gen_nil':cons':f'5(_n1089)) → _gen_nil':cons':f'5(0), rt ∈ Ω(1 + n1089)
leq'(_gen_0':s'6(_n2312), _gen_0':s'6(_n2312)) → true', rt ∈ Ω(1 + n2312)
Generator Equations:
_gen_nil':cons':f'5(0) ⇔ nil'
_gen_nil':cons':f'5(+(x, 1)) ⇔ cons'(_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', process'
They will be analysed ascendingly in the following order:
length' < process'
app' < map_f'
app' < process'
map_f' < process'
Proved the following rewrite lemma:
length'(_gen_nil':cons':f'5(_n3213)) → _gen_0':s'6(_n3213), rt ∈ Ω(1 + n3213)
Induction Base:
length'(_gen_nil':cons':f'5(0)) →RΩ(1)
0'
Induction Step:
length'(_gen_nil':cons':f'5(+(_$n3214, 1))) →RΩ(1)
s'(length'(_gen_nil':cons':f'5(_$n3214))) →IH
s'(_gen_0':s'6(_$n3214))
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
fstsplit'(0', x) → nil'
fstsplit'(s'(n), nil') → nil'
fstsplit'(s'(n), cons'(t)) → cons'(fstsplit'(n, t))
sndsplit'(0', x) → x
sndsplit'(s'(n), nil') → nil'
sndsplit'(s'(n), cons'(t)) → sndsplit'(n, t)
empty'(nil') → true'
empty'(cons'(t)) → false'
leq'(0', m) → true'
leq'(s'(n), 0') → false'
leq'(s'(n), s'(m)) → leq'(n, m)
length'(nil') → 0'
length'(cons'(t)) → s'(length'(t))
app'(nil', x) → x
app'(cons'(t), x) → cons'(app'(t, x))
map_f'(nil') → nil'
map_f'(cons'(t)) → app'(f', map_f'(t))
process'(store, m) → if1'(store, m, leq'(m, length'(store)))
if1'(store, m, true') → if2'(store, m, empty'(fstsplit'(m, store)))
if1'(store, m, false') → if3'(store, m, empty'(fstsplit'(m, app'(map_f'(nil'), store))))
if2'(store, m, false') → process'(app'(map_f'(nil'), sndsplit'(m, store)), m)
if3'(store, m, false') → process'(sndsplit'(m, app'(map_f'(nil'), store)), 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'
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'
process' :: nil':cons':f' → 0':s' → process':if1':if2':if3'
if1' :: nil':cons':f' → 0':s' → true':false' → process':if1':if2':if3'
if2' :: nil':cons':f' → 0':s' → true':false' → process':if1':if2':if3'
if3' :: nil':cons':f' → 0':s' → true':false' → process':if1':if2':if3'
_hole_nil':cons':f'1 :: nil':cons':f'
_hole_0':s'2 :: 0':s'
_hole_true':false'3 :: true':false'
_hole_process':if1':if2':if3'4 :: process':if1':if2':if3'
_gen_nil':cons':f'5 :: Nat → nil':cons':f'
_gen_0':s'6 :: Nat → 0':s'
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(_n1089), _gen_nil':cons':f'5(_n1089)) → _gen_nil':cons':f'5(0), rt ∈ Ω(1 + n1089)
leq'(_gen_0':s'6(_n2312), _gen_0':s'6(_n2312)) → true', rt ∈ Ω(1 + n2312)
length'(_gen_nil':cons':f'5(_n3213)) → _gen_0':s'6(_n3213), rt ∈ Ω(1 + n3213)
Generator Equations:
_gen_nil':cons':f'5(0) ⇔ nil'
_gen_nil':cons':f'5(+(x, 1)) ⇔ cons'(_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', process'
They will be analysed ascendingly in the following order:
app' < map_f'
app' < process'
map_f' < process'
Proved the following rewrite lemma:
app'(_gen_nil':cons':f'5(_n3919), _gen_nil':cons':f'5(b)) → _gen_nil':cons':f'5(+(_n3919, b)), rt ∈ Ω(1 + n3919)
Induction Base:
app'(_gen_nil':cons':f'5(0), _gen_nil':cons':f'5(b)) →RΩ(1)
_gen_nil':cons':f'5(b)
Induction Step:
app'(_gen_nil':cons':f'5(+(_$n3920, 1)), _gen_nil':cons':f'5(_b4088)) →RΩ(1)
cons'(app'(_gen_nil':cons':f'5(_$n3920), _gen_nil':cons':f'5(_b4088))) →IH
cons'(_gen_nil':cons':f'5(+(_$n3920, _b4088)))
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
fstsplit'(0', x) → nil'
fstsplit'(s'(n), nil') → nil'
fstsplit'(s'(n), cons'(t)) → cons'(fstsplit'(n, t))
sndsplit'(0', x) → x
sndsplit'(s'(n), nil') → nil'
sndsplit'(s'(n), cons'(t)) → sndsplit'(n, t)
empty'(nil') → true'
empty'(cons'(t)) → false'
leq'(0', m) → true'
leq'(s'(n), 0') → false'
leq'(s'(n), s'(m)) → leq'(n, m)
length'(nil') → 0'
length'(cons'(t)) → s'(length'(t))
app'(nil', x) → x
app'(cons'(t), x) → cons'(app'(t, x))
map_f'(nil') → nil'
map_f'(cons'(t)) → app'(f', map_f'(t))
process'(store, m) → if1'(store, m, leq'(m, length'(store)))
if1'(store, m, true') → if2'(store, m, empty'(fstsplit'(m, store)))
if1'(store, m, false') → if3'(store, m, empty'(fstsplit'(m, app'(map_f'(nil'), store))))
if2'(store, m, false') → process'(app'(map_f'(nil'), sndsplit'(m, store)), m)
if3'(store, m, false') → process'(sndsplit'(m, app'(map_f'(nil'), store)), 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'
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'
process' :: nil':cons':f' → 0':s' → process':if1':if2':if3'
if1' :: nil':cons':f' → 0':s' → true':false' → process':if1':if2':if3'
if2' :: nil':cons':f' → 0':s' → true':false' → process':if1':if2':if3'
if3' :: nil':cons':f' → 0':s' → true':false' → process':if1':if2':if3'
_hole_nil':cons':f'1 :: nil':cons':f'
_hole_0':s'2 :: 0':s'
_hole_true':false'3 :: true':false'
_hole_process':if1':if2':if3'4 :: process':if1':if2':if3'
_gen_nil':cons':f'5 :: Nat → nil':cons':f'
_gen_0':s'6 :: Nat → 0':s'
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(_n1089), _gen_nil':cons':f'5(_n1089)) → _gen_nil':cons':f'5(0), rt ∈ Ω(1 + n1089)
leq'(_gen_0':s'6(_n2312), _gen_0':s'6(_n2312)) → true', rt ∈ Ω(1 + n2312)
length'(_gen_nil':cons':f'5(_n3213)) → _gen_0':s'6(_n3213), rt ∈ Ω(1 + n3213)
app'(_gen_nil':cons':f'5(_n3919), _gen_nil':cons':f'5(b)) → _gen_nil':cons':f'5(+(_n3919, b)), rt ∈ Ω(1 + n3919)
Generator Equations:
_gen_nil':cons':f'5(0) ⇔ nil'
_gen_nil':cons':f'5(+(x, 1)) ⇔ cons'(_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', process'
They will be analysed ascendingly in the following order:
map_f' < process'
Proved the following rewrite lemma:
map_f'(_gen_nil':cons':f'5(+(1, _n5210))) → _*7, rt ∈ Ω(n5210)
Induction Base:
map_f'(_gen_nil':cons':f'5(+(1, 0)))
Induction Step:
map_f'(_gen_nil':cons':f'5(+(1, +(_$n5211, 1)))) →RΩ(1)
app'(f', map_f'(_gen_nil':cons':f'5(+(1, _$n5211)))) →IH
app'(f', _*7)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
fstsplit'(0', x) → nil'
fstsplit'(s'(n), nil') → nil'
fstsplit'(s'(n), cons'(t)) → cons'(fstsplit'(n, t))
sndsplit'(0', x) → x
sndsplit'(s'(n), nil') → nil'
sndsplit'(s'(n), cons'(t)) → sndsplit'(n, t)
empty'(nil') → true'
empty'(cons'(t)) → false'
leq'(0', m) → true'
leq'(s'(n), 0') → false'
leq'(s'(n), s'(m)) → leq'(n, m)
length'(nil') → 0'
length'(cons'(t)) → s'(length'(t))
app'(nil', x) → x
app'(cons'(t), x) → cons'(app'(t, x))
map_f'(nil') → nil'
map_f'(cons'(t)) → app'(f', map_f'(t))
process'(store, m) → if1'(store, m, leq'(m, length'(store)))
if1'(store, m, true') → if2'(store, m, empty'(fstsplit'(m, store)))
if1'(store, m, false') → if3'(store, m, empty'(fstsplit'(m, app'(map_f'(nil'), store))))
if2'(store, m, false') → process'(app'(map_f'(nil'), sndsplit'(m, store)), m)
if3'(store, m, false') → process'(sndsplit'(m, app'(map_f'(nil'), store)), 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'
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'
process' :: nil':cons':f' → 0':s' → process':if1':if2':if3'
if1' :: nil':cons':f' → 0':s' → true':false' → process':if1':if2':if3'
if2' :: nil':cons':f' → 0':s' → true':false' → process':if1':if2':if3'
if3' :: nil':cons':f' → 0':s' → true':false' → process':if1':if2':if3'
_hole_nil':cons':f'1 :: nil':cons':f'
_hole_0':s'2 :: 0':s'
_hole_true':false'3 :: true':false'
_hole_process':if1':if2':if3'4 :: process':if1':if2':if3'
_gen_nil':cons':f'5 :: Nat → nil':cons':f'
_gen_0':s'6 :: Nat → 0':s'
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(_n1089), _gen_nil':cons':f'5(_n1089)) → _gen_nil':cons':f'5(0), rt ∈ Ω(1 + n1089)
leq'(_gen_0':s'6(_n2312), _gen_0':s'6(_n2312)) → true', rt ∈ Ω(1 + n2312)
length'(_gen_nil':cons':f'5(_n3213)) → _gen_0':s'6(_n3213), rt ∈ Ω(1 + n3213)
app'(_gen_nil':cons':f'5(_n3919), _gen_nil':cons':f'5(b)) → _gen_nil':cons':f'5(+(_n3919, b)), rt ∈ Ω(1 + n3919)
map_f'(_gen_nil':cons':f'5(+(1, _n5210))) → _*7, rt ∈ Ω(n5210)
Generator Equations:
_gen_nil':cons':f'5(0) ⇔ nil'
_gen_nil':cons':f'5(+(x, 1)) ⇔ cons'(_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:
process'
Could not prove a rewrite lemma for the defined symbol process'.
Rules:
fstsplit'(0', x) → nil'
fstsplit'(s'(n), nil') → nil'
fstsplit'(s'(n), cons'(t)) → cons'(fstsplit'(n, t))
sndsplit'(0', x) → x
sndsplit'(s'(n), nil') → nil'
sndsplit'(s'(n), cons'(t)) → sndsplit'(n, t)
empty'(nil') → true'
empty'(cons'(t)) → false'
leq'(0', m) → true'
leq'(s'(n), 0') → false'
leq'(s'(n), s'(m)) → leq'(n, m)
length'(nil') → 0'
length'(cons'(t)) → s'(length'(t))
app'(nil', x) → x
app'(cons'(t), x) → cons'(app'(t, x))
map_f'(nil') → nil'
map_f'(cons'(t)) → app'(f', map_f'(t))
process'(store, m) → if1'(store, m, leq'(m, length'(store)))
if1'(store, m, true') → if2'(store, m, empty'(fstsplit'(m, store)))
if1'(store, m, false') → if3'(store, m, empty'(fstsplit'(m, app'(map_f'(nil'), store))))
if2'(store, m, false') → process'(app'(map_f'(nil'), sndsplit'(m, store)), m)
if3'(store, m, false') → process'(sndsplit'(m, app'(map_f'(nil'), store)), 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'
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'
process' :: nil':cons':f' → 0':s' → process':if1':if2':if3'
if1' :: nil':cons':f' → 0':s' → true':false' → process':if1':if2':if3'
if2' :: nil':cons':f' → 0':s' → true':false' → process':if1':if2':if3'
if3' :: nil':cons':f' → 0':s' → true':false' → process':if1':if2':if3'
_hole_nil':cons':f'1 :: nil':cons':f'
_hole_0':s'2 :: 0':s'
_hole_true':false'3 :: true':false'
_hole_process':if1':if2':if3'4 :: process':if1':if2':if3'
_gen_nil':cons':f'5 :: Nat → nil':cons':f'
_gen_0':s'6 :: Nat → 0':s'
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(_n1089), _gen_nil':cons':f'5(_n1089)) → _gen_nil':cons':f'5(0), rt ∈ Ω(1 + n1089)
leq'(_gen_0':s'6(_n2312), _gen_0':s'6(_n2312)) → true', rt ∈ Ω(1 + n2312)
length'(_gen_nil':cons':f'5(_n3213)) → _gen_0':s'6(_n3213), rt ∈ Ω(1 + n3213)
app'(_gen_nil':cons':f'5(_n3919), _gen_nil':cons':f'5(b)) → _gen_nil':cons':f'5(+(_n3919, b)), rt ∈ Ω(1 + n3919)
map_f'(_gen_nil':cons':f'5(+(1, _n5210))) → _*7, rt ∈ Ω(n5210)
Generator Equations:
_gen_nil':cons':f'5(0) ⇔ nil'
_gen_nil':cons':f'5(+(x, 1)) ⇔ cons'(_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)