/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


--------------------------------------------------------------------------------


WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) SlicingProof [LOWER BOUND(ID), 0 ms]
(4) CpxTRS
(5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) typed CpxTrs
(7) OrderProof [LOWER BOUND(ID), 0 ms]
(8) typed CpxTrs
(9) RewriteLemmaProof [LOWER BOUND(ID), 334 ms]
(10) BEST
    (11) proven lower bound
        (12) LowerBoundPropagationProof [FINISHED, 0 ms]
        (13) BOUNDS(n^1, INF)
    (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 45 ms]
        (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 0 ms]
        (18) typed CpxTrs
        (19) RewriteLemmaProof [LOWER BOUND(ID), 2 ms]
        (20) typed CpxTrs
        (21) RewriteLemmaProof [LOWER BOUND(ID), 17 ms]
        (22) typed CpxTrs
        (23) RewriteLemmaProof [LOWER BOUND(ID), 2156 ms]
        (24) typed CpxTrs


----------------------------------------

(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


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)

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(1) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(2)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


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)

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(3) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
cons/0
map_f/0
f/0
f/1

----------------------------------------

(4)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


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)

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(6)
Obligation:
TRS:
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:f1_0 :: nil:cons:f
hole_0':s2_0 :: 0':s
hole_true:false3_0 :: true:false
hole_process:if1:if2:if34_0 :: process:if1:if2:if3
gen_nil:cons:f5_0 :: Nat -> nil:cons:f
gen_0':s6_0 :: Nat -> 0':s

----------------------------------------

(7) OrderProof (LOWER BOUND(ID))
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

----------------------------------------

(8)
Obligation:
TRS:
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:f1_0 :: nil:cons:f
hole_0':s2_0 :: 0':s
hole_true:false3_0 :: true:false
hole_process:if1:if2:if34_0 :: process:if1:if2:if3
gen_nil:cons:f5_0 :: Nat -> nil:cons:f
gen_0':s6_0 :: Nat -> 0':s


Generator Equations:
gen_nil:cons:f5_0(0) <=> nil
gen_nil:cons:f5_0(+(x, 1)) <=> cons(gen_nil:cons:f5_0(x))
gen_0':s6_0(0) <=> 0'
gen_0':s6_0(+(x, 1)) <=> s(gen_0':s6_0(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

----------------------------------------

(9) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
fstsplit(gen_0':s6_0(n8_0), gen_nil:cons:f5_0(n8_0)) -> gen_nil:cons:f5_0(n8_0), rt in Omega(1 + n8_0)

Induction Base:
fstsplit(gen_0':s6_0(0), gen_nil:cons:f5_0(0)) ->_R^Omega(1)
nil

Induction Step:
fstsplit(gen_0':s6_0(+(n8_0, 1)), gen_nil:cons:f5_0(+(n8_0, 1))) ->_R^Omega(1)
cons(fstsplit(gen_0':s6_0(n8_0), gen_nil:cons:f5_0(n8_0))) ->_IH
cons(gen_nil:cons:f5_0(c9_0))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(10)
Complex Obligation (BEST)

----------------------------------------

(11)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
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:f1_0 :: nil:cons:f
hole_0':s2_0 :: 0':s
hole_true:false3_0 :: true:false
hole_process:if1:if2:if34_0 :: process:if1:if2:if3
gen_nil:cons:f5_0 :: Nat -> nil:cons:f
gen_0':s6_0 :: Nat -> 0':s


Generator Equations:
gen_nil:cons:f5_0(0) <=> nil
gen_nil:cons:f5_0(+(x, 1)) <=> cons(gen_nil:cons:f5_0(x))
gen_0':s6_0(0) <=> 0'
gen_0':s6_0(+(x, 1)) <=> s(gen_0':s6_0(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

----------------------------------------

(12) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(13)
BOUNDS(n^1, INF)

----------------------------------------

(14)
Obligation:
TRS:
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:f1_0 :: nil:cons:f
hole_0':s2_0 :: 0':s
hole_true:false3_0 :: true:false
hole_process:if1:if2:if34_0 :: process:if1:if2:if3
gen_nil:cons:f5_0 :: Nat -> nil:cons:f
gen_0':s6_0 :: Nat -> 0':s


Lemmas:
fstsplit(gen_0':s6_0(n8_0), gen_nil:cons:f5_0(n8_0)) -> gen_nil:cons:f5_0(n8_0), rt in Omega(1 + n8_0)


Generator Equations:
gen_nil:cons:f5_0(0) <=> nil
gen_nil:cons:f5_0(+(x, 1)) <=> cons(gen_nil:cons:f5_0(x))
gen_0':s6_0(0) <=> 0'
gen_0':s6_0(+(x, 1)) <=> s(gen_0':s6_0(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

----------------------------------------

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
sndsplit(gen_0':s6_0(n502_0), gen_nil:cons:f5_0(n502_0)) -> gen_nil:cons:f5_0(0), rt in Omega(1 + n502_0)

Induction Base:
sndsplit(gen_0':s6_0(0), gen_nil:cons:f5_0(0)) ->_R^Omega(1)
gen_nil:cons:f5_0(0)

Induction Step:
sndsplit(gen_0':s6_0(+(n502_0, 1)), gen_nil:cons:f5_0(+(n502_0, 1))) ->_R^Omega(1)
sndsplit(gen_0':s6_0(n502_0), gen_nil:cons:f5_0(n502_0)) ->_IH
gen_nil:cons:f5_0(0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(16)
Obligation:
TRS:
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:f1_0 :: nil:cons:f
hole_0':s2_0 :: 0':s
hole_true:false3_0 :: true:false
hole_process:if1:if2:if34_0 :: process:if1:if2:if3
gen_nil:cons:f5_0 :: Nat -> nil:cons:f
gen_0':s6_0 :: Nat -> 0':s


Lemmas:
fstsplit(gen_0':s6_0(n8_0), gen_nil:cons:f5_0(n8_0)) -> gen_nil:cons:f5_0(n8_0), rt in Omega(1 + n8_0)
sndsplit(gen_0':s6_0(n502_0), gen_nil:cons:f5_0(n502_0)) -> gen_nil:cons:f5_0(0), rt in Omega(1 + n502_0)


Generator Equations:
gen_nil:cons:f5_0(0) <=> nil
gen_nil:cons:f5_0(+(x, 1)) <=> cons(gen_nil:cons:f5_0(x))
gen_0':s6_0(0) <=> 0'
gen_0':s6_0(+(x, 1)) <=> s(gen_0':s6_0(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

----------------------------------------

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
leq(gen_0':s6_0(n1054_0), gen_0':s6_0(n1054_0)) -> true, rt in Omega(1 + n1054_0)

Induction Base:
leq(gen_0':s6_0(0), gen_0':s6_0(0)) ->_R^Omega(1)
true

Induction Step:
leq(gen_0':s6_0(+(n1054_0, 1)), gen_0':s6_0(+(n1054_0, 1))) ->_R^Omega(1)
leq(gen_0':s6_0(n1054_0), gen_0':s6_0(n1054_0)) ->_IH
true

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(18)
Obligation:
TRS:
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:f1_0 :: nil:cons:f
hole_0':s2_0 :: 0':s
hole_true:false3_0 :: true:false
hole_process:if1:if2:if34_0 :: process:if1:if2:if3
gen_nil:cons:f5_0 :: Nat -> nil:cons:f
gen_0':s6_0 :: Nat -> 0':s


Lemmas:
fstsplit(gen_0':s6_0(n8_0), gen_nil:cons:f5_0(n8_0)) -> gen_nil:cons:f5_0(n8_0), rt in Omega(1 + n8_0)
sndsplit(gen_0':s6_0(n502_0), gen_nil:cons:f5_0(n502_0)) -> gen_nil:cons:f5_0(0), rt in Omega(1 + n502_0)
leq(gen_0':s6_0(n1054_0), gen_0':s6_0(n1054_0)) -> true, rt in Omega(1 + n1054_0)


Generator Equations:
gen_nil:cons:f5_0(0) <=> nil
gen_nil:cons:f5_0(+(x, 1)) <=> cons(gen_nil:cons:f5_0(x))
gen_0':s6_0(0) <=> 0'
gen_0':s6_0(+(x, 1)) <=> s(gen_0':s6_0(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

----------------------------------------

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
length(gen_nil:cons:f5_0(n1389_0)) -> gen_0':s6_0(n1389_0), rt in Omega(1 + n1389_0)

Induction Base:
length(gen_nil:cons:f5_0(0)) ->_R^Omega(1)
0'

Induction Step:
length(gen_nil:cons:f5_0(+(n1389_0, 1))) ->_R^Omega(1)
s(length(gen_nil:cons:f5_0(n1389_0))) ->_IH
s(gen_0':s6_0(c1390_0))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(20)
Obligation:
TRS:
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:f1_0 :: nil:cons:f
hole_0':s2_0 :: 0':s
hole_true:false3_0 :: true:false
hole_process:if1:if2:if34_0 :: process:if1:if2:if3
gen_nil:cons:f5_0 :: Nat -> nil:cons:f
gen_0':s6_0 :: Nat -> 0':s


Lemmas:
fstsplit(gen_0':s6_0(n8_0), gen_nil:cons:f5_0(n8_0)) -> gen_nil:cons:f5_0(n8_0), rt in Omega(1 + n8_0)
sndsplit(gen_0':s6_0(n502_0), gen_nil:cons:f5_0(n502_0)) -> gen_nil:cons:f5_0(0), rt in Omega(1 + n502_0)
leq(gen_0':s6_0(n1054_0), gen_0':s6_0(n1054_0)) -> true, rt in Omega(1 + n1054_0)
length(gen_nil:cons:f5_0(n1389_0)) -> gen_0':s6_0(n1389_0), rt in Omega(1 + n1389_0)


Generator Equations:
gen_nil:cons:f5_0(0) <=> nil
gen_nil:cons:f5_0(+(x, 1)) <=> cons(gen_nil:cons:f5_0(x))
gen_0':s6_0(0) <=> 0'
gen_0':s6_0(+(x, 1)) <=> s(gen_0':s6_0(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

----------------------------------------

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
app(gen_nil:cons:f5_0(n1677_0), gen_nil:cons:f5_0(b)) -> gen_nil:cons:f5_0(+(n1677_0, b)), rt in Omega(1 + n1677_0)

Induction Base:
app(gen_nil:cons:f5_0(0), gen_nil:cons:f5_0(b)) ->_R^Omega(1)
gen_nil:cons:f5_0(b)

Induction Step:
app(gen_nil:cons:f5_0(+(n1677_0, 1)), gen_nil:cons:f5_0(b)) ->_R^Omega(1)
cons(app(gen_nil:cons:f5_0(n1677_0), gen_nil:cons:f5_0(b))) ->_IH
cons(gen_nil:cons:f5_0(+(b, c1678_0)))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(22)
Obligation:
TRS:
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:f1_0 :: nil:cons:f
hole_0':s2_0 :: 0':s
hole_true:false3_0 :: true:false
hole_process:if1:if2:if34_0 :: process:if1:if2:if3
gen_nil:cons:f5_0 :: Nat -> nil:cons:f
gen_0':s6_0 :: Nat -> 0':s


Lemmas:
fstsplit(gen_0':s6_0(n8_0), gen_nil:cons:f5_0(n8_0)) -> gen_nil:cons:f5_0(n8_0), rt in Omega(1 + n8_0)
sndsplit(gen_0':s6_0(n502_0), gen_nil:cons:f5_0(n502_0)) -> gen_nil:cons:f5_0(0), rt in Omega(1 + n502_0)
leq(gen_0':s6_0(n1054_0), gen_0':s6_0(n1054_0)) -> true, rt in Omega(1 + n1054_0)
length(gen_nil:cons:f5_0(n1389_0)) -> gen_0':s6_0(n1389_0), rt in Omega(1 + n1389_0)
app(gen_nil:cons:f5_0(n1677_0), gen_nil:cons:f5_0(b)) -> gen_nil:cons:f5_0(+(n1677_0, b)), rt in Omega(1 + n1677_0)


Generator Equations:
gen_nil:cons:f5_0(0) <=> nil
gen_nil:cons:f5_0(+(x, 1)) <=> cons(gen_nil:cons:f5_0(x))
gen_0':s6_0(0) <=> 0'
gen_0':s6_0(+(x, 1)) <=> s(gen_0':s6_0(x))


The following defined symbols remain to be analysed:
map_f, process

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

----------------------------------------

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
map_f(gen_nil:cons:f5_0(+(1, n2574_0))) -> *7_0, rt in Omega(n2574_0)

Induction Base:
map_f(gen_nil:cons:f5_0(+(1, 0)))

Induction Step:
map_f(gen_nil:cons:f5_0(+(1, +(n2574_0, 1)))) ->_R^Omega(1)
app(f, map_f(gen_nil:cons:f5_0(+(1, n2574_0)))) ->_IH
app(f, *7_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(24)
Obligation:
TRS:
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:f1_0 :: nil:cons:f
hole_0':s2_0 :: 0':s
hole_true:false3_0 :: true:false
hole_process:if1:if2:if34_0 :: process:if1:if2:if3
gen_nil:cons:f5_0 :: Nat -> nil:cons:f
gen_0':s6_0 :: Nat -> 0':s


Lemmas:
fstsplit(gen_0':s6_0(n8_0), gen_nil:cons:f5_0(n8_0)) -> gen_nil:cons:f5_0(n8_0), rt in Omega(1 + n8_0)
sndsplit(gen_0':s6_0(n502_0), gen_nil:cons:f5_0(n502_0)) -> gen_nil:cons:f5_0(0), rt in Omega(1 + n502_0)
leq(gen_0':s6_0(n1054_0), gen_0':s6_0(n1054_0)) -> true, rt in Omega(1 + n1054_0)
length(gen_nil:cons:f5_0(n1389_0)) -> gen_0':s6_0(n1389_0), rt in Omega(1 + n1389_0)
app(gen_nil:cons:f5_0(n1677_0), gen_nil:cons:f5_0(b)) -> gen_nil:cons:f5_0(+(n1677_0, b)), rt in Omega(1 + n1677_0)
map_f(gen_nil:cons:f5_0(+(1, n2574_0))) -> *7_0, rt in Omega(n2574_0)


Generator Equations:
gen_nil:cons:f5_0(0) <=> nil
gen_nil:cons:f5_0(+(x, 1)) <=> cons(gen_nil:cons:f5_0(x))
gen_0':s6_0(0) <=> 0'
gen_0':s6_0(+(x, 1)) <=> s(gen_0':s6_0(x))


The following defined symbols remain to be analysed:
process