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


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


WORST_CASE(Omega(n^2), ?)
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^2, 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), 255 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), 583 ms]
        (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 226 ms]
        (18) BEST
            (19) proven lower bound
                (20) LowerBoundPropagationProof [FINISHED, 0 ms]
                (21) BOUNDS(n^2, INF)
            (22) typed CpxTrs


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

(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).


The TRS R consists of the following rules:

   empty(nil) -> true
   empty(cons(x, y)) -> false
   tail(nil) -> nil
   tail(cons(x, y)) -> y
   head(cons(x, y)) -> x
   zero(0) -> true
   zero(s(x)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(x))) -> s(p(s(x)))
   intlist(x) -> if_intlist(empty(x), x)
   if_intlist(true, x) -> nil
   if_intlist(false, x) -> cons(s(head(x)), intlist(tail(x)))
   int(x, y) -> if_int(zero(x), zero(y), x, y)
   if_int(true, b, x, y) -> if1(b, x, y)
   if_int(false, b, x, y) -> if2(b, x, y)
   if1(true, x, y) -> cons(0, nil)
   if1(false, x, y) -> cons(0, int(s(0), y))
   if2(true, x, y) -> nil
   if2(false, x, y) -> intlist(int(p(x), p(y)))

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^2, INF).


The TRS R consists of the following rules:

   empty(nil) -> true
   empty(cons(x, y)) -> false
   tail(nil) -> nil
   tail(cons(x, y)) -> y
   head(cons(x, y)) -> x
   zero(0') -> true
   zero(s(x)) -> false
   p(0') -> 0'
   p(s(0')) -> 0'
   p(s(s(x))) -> s(p(s(x)))
   intlist(x) -> if_intlist(empty(x), x)
   if_intlist(true, x) -> nil
   if_intlist(false, x) -> cons(s(head(x)), intlist(tail(x)))
   int(x, y) -> if_int(zero(x), zero(y), x, y)
   if_int(true, b, x, y) -> if1(b, x, y)
   if_int(false, b, x, y) -> if2(b, x, y)
   if1(true, x, y) -> cons(0', nil)
   if1(false, x, y) -> cons(0', int(s(0'), y))
   if2(true, x, y) -> nil
   if2(false, x, y) -> intlist(int(p(x), p(y)))

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

(3) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
if1/1

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

(4)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).


The TRS R consists of the following rules:

   empty(nil) -> true
   empty(cons(x, y)) -> false
   tail(nil) -> nil
   tail(cons(x, y)) -> y
   head(cons(x, y)) -> x
   zero(0') -> true
   zero(s(x)) -> false
   p(0') -> 0'
   p(s(0')) -> 0'
   p(s(s(x))) -> s(p(s(x)))
   intlist(x) -> if_intlist(empty(x), x)
   if_intlist(true, x) -> nil
   if_intlist(false, x) -> cons(s(head(x)), intlist(tail(x)))
   int(x, y) -> if_int(zero(x), zero(y), x, y)
   if_int(true, b, x, y) -> if1(b, y)
   if_int(false, b, x, y) -> if2(b, x, y)
   if1(true, y) -> cons(0', nil)
   if1(false, y) -> cons(0', int(s(0'), y))
   if2(true, x, y) -> nil
   if2(false, x, y) -> intlist(int(p(x), p(y)))

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

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

(6)
Obligation:
TRS:
Rules:
empty(nil) -> true
empty(cons(x, y)) -> false
tail(nil) -> nil
tail(cons(x, y)) -> y
head(cons(x, y)) -> x
zero(0') -> true
zero(s(x)) -> false
p(0') -> 0'
p(s(0')) -> 0'
p(s(s(x))) -> s(p(s(x)))
intlist(x) -> if_intlist(empty(x), x)
if_intlist(true, x) -> nil
if_intlist(false, x) -> cons(s(head(x)), intlist(tail(x)))
int(x, y) -> if_int(zero(x), zero(y), x, y)
if_int(true, b, x, y) -> if1(b, y)
if_int(false, b, x, y) -> if2(b, x, y)
if1(true, y) -> cons(0', nil)
if1(false, y) -> cons(0', int(s(0'), y))
if2(true, x, y) -> nil
if2(false, x, y) -> intlist(int(p(x), p(y)))

Types:
empty :: nil:cons -> true:false
nil :: nil:cons
true :: true:false
cons :: 0':s -> nil:cons -> nil:cons
false :: true:false
tail :: nil:cons -> nil:cons
head :: nil:cons -> 0':s
zero :: 0':s -> true:false
0' :: 0':s
s :: 0':s -> 0':s
p :: 0':s -> 0':s
intlist :: nil:cons -> nil:cons
if_intlist :: true:false -> nil:cons -> nil:cons
int :: 0':s -> 0':s -> nil:cons
if_int :: true:false -> true:false -> 0':s -> 0':s -> nil:cons
if1 :: true:false -> 0':s -> nil:cons
if2 :: true:false -> 0':s -> 0':s -> nil:cons
hole_true:false1_0 :: true:false
hole_nil:cons2_0 :: nil:cons
hole_0':s3_0 :: 0':s
gen_nil:cons4_0 :: Nat -> nil:cons
gen_0':s5_0 :: Nat -> 0':s

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

(7) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
p, intlist, int, if1

They will be analysed ascendingly in the following order:
p < int
intlist < int
int = if1

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

(8)
Obligation:
TRS:
Rules:
empty(nil) -> true
empty(cons(x, y)) -> false
tail(nil) -> nil
tail(cons(x, y)) -> y
head(cons(x, y)) -> x
zero(0') -> true
zero(s(x)) -> false
p(0') -> 0'
p(s(0')) -> 0'
p(s(s(x))) -> s(p(s(x)))
intlist(x) -> if_intlist(empty(x), x)
if_intlist(true, x) -> nil
if_intlist(false, x) -> cons(s(head(x)), intlist(tail(x)))
int(x, y) -> if_int(zero(x), zero(y), x, y)
if_int(true, b, x, y) -> if1(b, y)
if_int(false, b, x, y) -> if2(b, x, y)
if1(true, y) -> cons(0', nil)
if1(false, y) -> cons(0', int(s(0'), y))
if2(true, x, y) -> nil
if2(false, x, y) -> intlist(int(p(x), p(y)))

Types:
empty :: nil:cons -> true:false
nil :: nil:cons
true :: true:false
cons :: 0':s -> nil:cons -> nil:cons
false :: true:false
tail :: nil:cons -> nil:cons
head :: nil:cons -> 0':s
zero :: 0':s -> true:false
0' :: 0':s
s :: 0':s -> 0':s
p :: 0':s -> 0':s
intlist :: nil:cons -> nil:cons
if_intlist :: true:false -> nil:cons -> nil:cons
int :: 0':s -> 0':s -> nil:cons
if_int :: true:false -> true:false -> 0':s -> 0':s -> nil:cons
if1 :: true:false -> 0':s -> nil:cons
if2 :: true:false -> 0':s -> 0':s -> nil:cons
hole_true:false1_0 :: true:false
hole_nil:cons2_0 :: nil:cons
hole_0':s3_0 :: 0':s
gen_nil:cons4_0 :: Nat -> nil:cons
gen_0':s5_0 :: Nat -> 0':s


Generator Equations:
gen_nil:cons4_0(0) <=> nil
gen_nil:cons4_0(+(x, 1)) <=> cons(0', gen_nil:cons4_0(x))
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))


The following defined symbols remain to be analysed:
p, intlist, int, if1

They will be analysed ascendingly in the following order:
p < int
intlist < int
int = if1

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

(9) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
p(gen_0':s5_0(+(1, n7_0))) -> gen_0':s5_0(n7_0), rt in Omega(1 + n7_0)

Induction Base:
p(gen_0':s5_0(+(1, 0))) ->_R^Omega(1)
0'

Induction Step:
p(gen_0':s5_0(+(1, +(n7_0, 1)))) ->_R^Omega(1)
s(p(s(gen_0':s5_0(n7_0)))) ->_IH
s(gen_0':s5_0(c8_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:
empty(nil) -> true
empty(cons(x, y)) -> false
tail(nil) -> nil
tail(cons(x, y)) -> y
head(cons(x, y)) -> x
zero(0') -> true
zero(s(x)) -> false
p(0') -> 0'
p(s(0')) -> 0'
p(s(s(x))) -> s(p(s(x)))
intlist(x) -> if_intlist(empty(x), x)
if_intlist(true, x) -> nil
if_intlist(false, x) -> cons(s(head(x)), intlist(tail(x)))
int(x, y) -> if_int(zero(x), zero(y), x, y)
if_int(true, b, x, y) -> if1(b, y)
if_int(false, b, x, y) -> if2(b, x, y)
if1(true, y) -> cons(0', nil)
if1(false, y) -> cons(0', int(s(0'), y))
if2(true, x, y) -> nil
if2(false, x, y) -> intlist(int(p(x), p(y)))

Types:
empty :: nil:cons -> true:false
nil :: nil:cons
true :: true:false
cons :: 0':s -> nil:cons -> nil:cons
false :: true:false
tail :: nil:cons -> nil:cons
head :: nil:cons -> 0':s
zero :: 0':s -> true:false
0' :: 0':s
s :: 0':s -> 0':s
p :: 0':s -> 0':s
intlist :: nil:cons -> nil:cons
if_intlist :: true:false -> nil:cons -> nil:cons
int :: 0':s -> 0':s -> nil:cons
if_int :: true:false -> true:false -> 0':s -> 0':s -> nil:cons
if1 :: true:false -> 0':s -> nil:cons
if2 :: true:false -> 0':s -> 0':s -> nil:cons
hole_true:false1_0 :: true:false
hole_nil:cons2_0 :: nil:cons
hole_0':s3_0 :: 0':s
gen_nil:cons4_0 :: Nat -> nil:cons
gen_0':s5_0 :: Nat -> 0':s


Generator Equations:
gen_nil:cons4_0(0) <=> nil
gen_nil:cons4_0(+(x, 1)) <=> cons(0', gen_nil:cons4_0(x))
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))


The following defined symbols remain to be analysed:
p, intlist, int, if1

They will be analysed ascendingly in the following order:
p < int
intlist < int
int = if1

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

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

(13)
BOUNDS(n^1, INF)

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

(14)
Obligation:
TRS:
Rules:
empty(nil) -> true
empty(cons(x, y)) -> false
tail(nil) -> nil
tail(cons(x, y)) -> y
head(cons(x, y)) -> x
zero(0') -> true
zero(s(x)) -> false
p(0') -> 0'
p(s(0')) -> 0'
p(s(s(x))) -> s(p(s(x)))
intlist(x) -> if_intlist(empty(x), x)
if_intlist(true, x) -> nil
if_intlist(false, x) -> cons(s(head(x)), intlist(tail(x)))
int(x, y) -> if_int(zero(x), zero(y), x, y)
if_int(true, b, x, y) -> if1(b, y)
if_int(false, b, x, y) -> if2(b, x, y)
if1(true, y) -> cons(0', nil)
if1(false, y) -> cons(0', int(s(0'), y))
if2(true, x, y) -> nil
if2(false, x, y) -> intlist(int(p(x), p(y)))

Types:
empty :: nil:cons -> true:false
nil :: nil:cons
true :: true:false
cons :: 0':s -> nil:cons -> nil:cons
false :: true:false
tail :: nil:cons -> nil:cons
head :: nil:cons -> 0':s
zero :: 0':s -> true:false
0' :: 0':s
s :: 0':s -> 0':s
p :: 0':s -> 0':s
intlist :: nil:cons -> nil:cons
if_intlist :: true:false -> nil:cons -> nil:cons
int :: 0':s -> 0':s -> nil:cons
if_int :: true:false -> true:false -> 0':s -> 0':s -> nil:cons
if1 :: true:false -> 0':s -> nil:cons
if2 :: true:false -> 0':s -> 0':s -> nil:cons
hole_true:false1_0 :: true:false
hole_nil:cons2_0 :: nil:cons
hole_0':s3_0 :: 0':s
gen_nil:cons4_0 :: Nat -> nil:cons
gen_0':s5_0 :: Nat -> 0':s


Lemmas:
p(gen_0':s5_0(+(1, n7_0))) -> gen_0':s5_0(n7_0), rt in Omega(1 + n7_0)


Generator Equations:
gen_nil:cons4_0(0) <=> nil
gen_nil:cons4_0(+(x, 1)) <=> cons(0', gen_nil:cons4_0(x))
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))


The following defined symbols remain to be analysed:
intlist, int, if1

They will be analysed ascendingly in the following order:
intlist < int
int = if1

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
intlist(gen_nil:cons4_0(n312_0)) -> *6_0, rt in Omega(n312_0)

Induction Base:
intlist(gen_nil:cons4_0(0))

Induction Step:
intlist(gen_nil:cons4_0(+(n312_0, 1))) ->_R^Omega(1)
if_intlist(empty(gen_nil:cons4_0(+(n312_0, 1))), gen_nil:cons4_0(+(n312_0, 1))) ->_R^Omega(1)
if_intlist(false, gen_nil:cons4_0(+(1, n312_0))) ->_R^Omega(1)
cons(s(head(gen_nil:cons4_0(+(1, n312_0)))), intlist(tail(gen_nil:cons4_0(+(1, n312_0))))) ->_R^Omega(1)
cons(s(0'), intlist(tail(gen_nil:cons4_0(+(1, n312_0))))) ->_R^Omega(1)
cons(s(0'), intlist(gen_nil:cons4_0(n312_0))) ->_IH
cons(s(0'), *6_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:
empty(nil) -> true
empty(cons(x, y)) -> false
tail(nil) -> nil
tail(cons(x, y)) -> y
head(cons(x, y)) -> x
zero(0') -> true
zero(s(x)) -> false
p(0') -> 0'
p(s(0')) -> 0'
p(s(s(x))) -> s(p(s(x)))
intlist(x) -> if_intlist(empty(x), x)
if_intlist(true, x) -> nil
if_intlist(false, x) -> cons(s(head(x)), intlist(tail(x)))
int(x, y) -> if_int(zero(x), zero(y), x, y)
if_int(true, b, x, y) -> if1(b, y)
if_int(false, b, x, y) -> if2(b, x, y)
if1(true, y) -> cons(0', nil)
if1(false, y) -> cons(0', int(s(0'), y))
if2(true, x, y) -> nil
if2(false, x, y) -> intlist(int(p(x), p(y)))

Types:
empty :: nil:cons -> true:false
nil :: nil:cons
true :: true:false
cons :: 0':s -> nil:cons -> nil:cons
false :: true:false
tail :: nil:cons -> nil:cons
head :: nil:cons -> 0':s
zero :: 0':s -> true:false
0' :: 0':s
s :: 0':s -> 0':s
p :: 0':s -> 0':s
intlist :: nil:cons -> nil:cons
if_intlist :: true:false -> nil:cons -> nil:cons
int :: 0':s -> 0':s -> nil:cons
if_int :: true:false -> true:false -> 0':s -> 0':s -> nil:cons
if1 :: true:false -> 0':s -> nil:cons
if2 :: true:false -> 0':s -> 0':s -> nil:cons
hole_true:false1_0 :: true:false
hole_nil:cons2_0 :: nil:cons
hole_0':s3_0 :: 0':s
gen_nil:cons4_0 :: Nat -> nil:cons
gen_0':s5_0 :: Nat -> 0':s


Lemmas:
p(gen_0':s5_0(+(1, n7_0))) -> gen_0':s5_0(n7_0), rt in Omega(1 + n7_0)
intlist(gen_nil:cons4_0(n312_0)) -> *6_0, rt in Omega(n312_0)


Generator Equations:
gen_nil:cons4_0(0) <=> nil
gen_nil:cons4_0(+(x, 1)) <=> cons(0', gen_nil:cons4_0(x))
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))


The following defined symbols remain to be analysed:
if1, int

They will be analysed ascendingly in the following order:
int = if1

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
int(gen_0':s5_0(+(1, n5442_0)), gen_0':s5_0(n5442_0)) -> gen_nil:cons4_0(0), rt in Omega(1 + n5442_0 + n5442_0^2)

Induction Base:
int(gen_0':s5_0(+(1, 0)), gen_0':s5_0(0)) ->_R^Omega(1)
if_int(zero(gen_0':s5_0(+(1, 0))), zero(gen_0':s5_0(0)), gen_0':s5_0(+(1, 0)), gen_0':s5_0(0)) ->_R^Omega(1)
if_int(false, zero(gen_0':s5_0(0)), gen_0':s5_0(1), gen_0':s5_0(0)) ->_R^Omega(1)
if_int(false, true, gen_0':s5_0(1), gen_0':s5_0(0)) ->_R^Omega(1)
if2(true, gen_0':s5_0(1), gen_0':s5_0(0)) ->_R^Omega(1)
nil

Induction Step:
int(gen_0':s5_0(+(1, +(n5442_0, 1))), gen_0':s5_0(+(n5442_0, 1))) ->_R^Omega(1)
if_int(zero(gen_0':s5_0(+(1, +(n5442_0, 1)))), zero(gen_0':s5_0(+(n5442_0, 1))), gen_0':s5_0(+(1, +(n5442_0, 1))), gen_0':s5_0(+(n5442_0, 1))) ->_R^Omega(1)
if_int(false, zero(gen_0':s5_0(+(1, n5442_0))), gen_0':s5_0(+(2, n5442_0)), gen_0':s5_0(+(1, n5442_0))) ->_R^Omega(1)
if_int(false, false, gen_0':s5_0(+(2, n5442_0)), gen_0':s5_0(+(1, n5442_0))) ->_R^Omega(1)
if2(false, gen_0':s5_0(+(2, n5442_0)), gen_0':s5_0(+(1, n5442_0))) ->_R^Omega(1)
intlist(int(p(gen_0':s5_0(+(2, n5442_0))), p(gen_0':s5_0(+(1, n5442_0))))) ->_L^Omega(2 + n5442_0)
intlist(int(gen_0':s5_0(+(1, n5442_0)), p(gen_0':s5_0(+(1, n5442_0))))) ->_L^Omega(1 + n5442_0)
intlist(int(gen_0':s5_0(+(1, n5442_0)), gen_0':s5_0(n5442_0))) ->_IH
intlist(gen_nil:cons4_0(0)) ->_R^Omega(1)
if_intlist(empty(gen_nil:cons4_0(0)), gen_nil:cons4_0(0)) ->_R^Omega(1)
if_intlist(true, gen_nil:cons4_0(0)) ->_R^Omega(1)
nil

We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
----------------------------------------

(18)
Complex Obligation (BEST)

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

(19)
Obligation:
Proved the lower bound n^2 for the following obligation:

TRS:
Rules:
empty(nil) -> true
empty(cons(x, y)) -> false
tail(nil) -> nil
tail(cons(x, y)) -> y
head(cons(x, y)) -> x
zero(0') -> true
zero(s(x)) -> false
p(0') -> 0'
p(s(0')) -> 0'
p(s(s(x))) -> s(p(s(x)))
intlist(x) -> if_intlist(empty(x), x)
if_intlist(true, x) -> nil
if_intlist(false, x) -> cons(s(head(x)), intlist(tail(x)))
int(x, y) -> if_int(zero(x), zero(y), x, y)
if_int(true, b, x, y) -> if1(b, y)
if_int(false, b, x, y) -> if2(b, x, y)
if1(true, y) -> cons(0', nil)
if1(false, y) -> cons(0', int(s(0'), y))
if2(true, x, y) -> nil
if2(false, x, y) -> intlist(int(p(x), p(y)))

Types:
empty :: nil:cons -> true:false
nil :: nil:cons
true :: true:false
cons :: 0':s -> nil:cons -> nil:cons
false :: true:false
tail :: nil:cons -> nil:cons
head :: nil:cons -> 0':s
zero :: 0':s -> true:false
0' :: 0':s
s :: 0':s -> 0':s
p :: 0':s -> 0':s
intlist :: nil:cons -> nil:cons
if_intlist :: true:false -> nil:cons -> nil:cons
int :: 0':s -> 0':s -> nil:cons
if_int :: true:false -> true:false -> 0':s -> 0':s -> nil:cons
if1 :: true:false -> 0':s -> nil:cons
if2 :: true:false -> 0':s -> 0':s -> nil:cons
hole_true:false1_0 :: true:false
hole_nil:cons2_0 :: nil:cons
hole_0':s3_0 :: 0':s
gen_nil:cons4_0 :: Nat -> nil:cons
gen_0':s5_0 :: Nat -> 0':s


Lemmas:
p(gen_0':s5_0(+(1, n7_0))) -> gen_0':s5_0(n7_0), rt in Omega(1 + n7_0)
intlist(gen_nil:cons4_0(n312_0)) -> *6_0, rt in Omega(n312_0)


Generator Equations:
gen_nil:cons4_0(0) <=> nil
gen_nil:cons4_0(+(x, 1)) <=> cons(0', gen_nil:cons4_0(x))
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))


The following defined symbols remain to be analysed:
int

They will be analysed ascendingly in the following order:
int = if1

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

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

(21)
BOUNDS(n^2, INF)

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

(22)
Obligation:
TRS:
Rules:
empty(nil) -> true
empty(cons(x, y)) -> false
tail(nil) -> nil
tail(cons(x, y)) -> y
head(cons(x, y)) -> x
zero(0') -> true
zero(s(x)) -> false
p(0') -> 0'
p(s(0')) -> 0'
p(s(s(x))) -> s(p(s(x)))
intlist(x) -> if_intlist(empty(x), x)
if_intlist(true, x) -> nil
if_intlist(false, x) -> cons(s(head(x)), intlist(tail(x)))
int(x, y) -> if_int(zero(x), zero(y), x, y)
if_int(true, b, x, y) -> if1(b, y)
if_int(false, b, x, y) -> if2(b, x, y)
if1(true, y) -> cons(0', nil)
if1(false, y) -> cons(0', int(s(0'), y))
if2(true, x, y) -> nil
if2(false, x, y) -> intlist(int(p(x), p(y)))

Types:
empty :: nil:cons -> true:false
nil :: nil:cons
true :: true:false
cons :: 0':s -> nil:cons -> nil:cons
false :: true:false
tail :: nil:cons -> nil:cons
head :: nil:cons -> 0':s
zero :: 0':s -> true:false
0' :: 0':s
s :: 0':s -> 0':s
p :: 0':s -> 0':s
intlist :: nil:cons -> nil:cons
if_intlist :: true:false -> nil:cons -> nil:cons
int :: 0':s -> 0':s -> nil:cons
if_int :: true:false -> true:false -> 0':s -> 0':s -> nil:cons
if1 :: true:false -> 0':s -> nil:cons
if2 :: true:false -> 0':s -> 0':s -> nil:cons
hole_true:false1_0 :: true:false
hole_nil:cons2_0 :: nil:cons
hole_0':s3_0 :: 0':s
gen_nil:cons4_0 :: Nat -> nil:cons
gen_0':s5_0 :: Nat -> 0':s


Lemmas:
p(gen_0':s5_0(+(1, n7_0))) -> gen_0':s5_0(n7_0), rt in Omega(1 + n7_0)
intlist(gen_nil:cons4_0(n312_0)) -> *6_0, rt in Omega(n312_0)
int(gen_0':s5_0(+(1, n5442_0)), gen_0':s5_0(n5442_0)) -> gen_nil:cons4_0(0), rt in Omega(1 + n5442_0 + n5442_0^2)


Generator Equations:
gen_nil:cons4_0(0) <=> nil
gen_nil:cons4_0(+(x, 1)) <=> cons(0', gen_nil:cons4_0(x))
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))


The following defined symbols remain to be analysed:
if1

They will be analysed ascendingly in the following order:
int = if1