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


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


WORST_CASE(Omega(n^2), ?)
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^2, INF).

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) typed CpxTrs
(5) OrderProof [LOWER BOUND(ID), 0 ms]
(6) typed CpxTrs
(7) RewriteLemmaProof [LOWER BOUND(ID), 353 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) typed CpxTrs
        (13) RewriteLemmaProof [LOWER BOUND(ID), 4183 ms]
        (14) BEST
            (15) proven lower bound
                (16) LowerBoundPropagationProof [FINISHED, 0 ms]
                (17) BOUNDS(n^2, INF)
            (18) typed CpxTrs
                (19) RewriteLemmaProof [LOWER BOUND(ID), 4 ms]
                (20) typed CpxTrs
                (21) RewriteLemmaProof [LOWER BOUND(ID), 0 ms]
                (22) typed CpxTrs
                (23) RewriteLemmaProof [LOWER BOUND(ID), 401 ms]
                (24) BOUNDS(1, INF)


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

(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:

   +(x, 0) -> x
   +(0, x) -> x
   +(s(x), s(y)) -> s(s(+(x, y)))
   +(+(x, y), z) -> +(x, +(y, z))
   *(x, 0) -> 0
   *(0, x) -> 0
   *(s(x), s(y)) -> s(+(*(x, y), +(x, y)))
   *(*(x, y), z) -> *(x, *(y, z))
   app(nil, l) -> l
   app(cons(x, l1), l2) -> cons(x, app(l1, l2))
   sum(nil) -> 0
   sum(cons(x, l)) -> +(x, sum(l))
   sum(app(l1, l2)) -> +(sum(l1), sum(l2))
   prod(nil) -> s(0)
   prod(cons(x, l)) -> *(x, prod(l))
   prod(app(l1, l2)) -> *(prod(l1), prod(l2))

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:

   +'(x, 0') -> x
   +'(0', x) -> x
   +'(s(x), s(y)) -> s(s(+'(x, y)))
   +'(+'(x, y), z) -> +'(x, +'(y, z))
   *'(x, 0') -> 0'
   *'(0', x) -> 0'
   *'(s(x), s(y)) -> s(+'(*'(x, y), +'(x, y)))
   *'(*'(x, y), z) -> *'(x, *'(y, z))
   app(nil, l) -> l
   app(cons(x, l1), l2) -> cons(x, app(l1, l2))
   sum(nil) -> 0'
   sum(cons(x, l)) -> +'(x, sum(l))
   sum(app(l1, l2)) -> +'(sum(l1), sum(l2))
   prod(nil) -> s(0')
   prod(cons(x, l)) -> *'(x, prod(l))
   prod(app(l1, l2)) -> *'(prod(l1), prod(l2))

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

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

(4)
Obligation:
TRS:
Rules:
+'(x, 0') -> x
+'(0', x) -> x
+'(s(x), s(y)) -> s(s(+'(x, y)))
+'(+'(x, y), z) -> +'(x, +'(y, z))
*'(x, 0') -> 0'
*'(0', x) -> 0'
*'(s(x), s(y)) -> s(+'(*'(x, y), +'(x, y)))
*'(*'(x, y), z) -> *'(x, *'(y, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0'
sum(cons(x, l)) -> +'(x, sum(l))
sum(app(l1, l2)) -> +'(sum(l1), sum(l2))
prod(nil) -> s(0')
prod(cons(x, l)) -> *'(x, prod(l))
prod(app(l1, l2)) -> *'(prod(l1), prod(l2))

Types:
+' :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
*' :: 0':s -> 0':s -> 0':s
app :: nil:cons -> nil:cons -> nil:cons
nil :: nil:cons
cons :: 0':s -> nil:cons -> nil:cons
sum :: nil:cons -> 0':s
prod :: nil:cons -> 0':s
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat -> 0':s
gen_nil:cons4_0 :: Nat -> nil:cons

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
+', *', app, sum, prod

They will be analysed ascendingly in the following order:
+' < *'
+' < sum
*' < prod

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

(6)
Obligation:
TRS:
Rules:
+'(x, 0') -> x
+'(0', x) -> x
+'(s(x), s(y)) -> s(s(+'(x, y)))
+'(+'(x, y), z) -> +'(x, +'(y, z))
*'(x, 0') -> 0'
*'(0', x) -> 0'
*'(s(x), s(y)) -> s(+'(*'(x, y), +'(x, y)))
*'(*'(x, y), z) -> *'(x, *'(y, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0'
sum(cons(x, l)) -> +'(x, sum(l))
sum(app(l1, l2)) -> +'(sum(l1), sum(l2))
prod(nil) -> s(0')
prod(cons(x, l)) -> *'(x, prod(l))
prod(app(l1, l2)) -> *'(prod(l1), prod(l2))

Types:
+' :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
*' :: 0':s -> 0':s -> 0':s
app :: nil:cons -> nil:cons -> nil:cons
nil :: nil:cons
cons :: 0':s -> nil:cons -> nil:cons
sum :: nil:cons -> 0':s
prod :: nil:cons -> 0':s
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat -> 0':s
gen_nil:cons4_0 :: Nat -> nil:cons


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
gen_nil:cons4_0(0) <=> nil
gen_nil:cons4_0(+(x, 1)) <=> cons(0', gen_nil:cons4_0(x))


The following defined symbols remain to be analysed:
+', *', app, sum, prod

They will be analysed ascendingly in the following order:
+' < *'
+' < sum
*' < prod

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(n6_0)) -> gen_0':s3_0(*(2, n6_0)), rt in Omega(1 + n6_0)

Induction Base:
+'(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
gen_0':s3_0(0)

Induction Step:
+'(gen_0':s3_0(+(n6_0, 1)), gen_0':s3_0(+(n6_0, 1))) ->_R^Omega(1)
s(s(+'(gen_0':s3_0(n6_0), gen_0':s3_0(n6_0)))) ->_IH
s(s(gen_0':s3_0(*(2, c7_0))))

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

(8)
Complex Obligation (BEST)

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

(9)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
+'(x, 0') -> x
+'(0', x) -> x
+'(s(x), s(y)) -> s(s(+'(x, y)))
+'(+'(x, y), z) -> +'(x, +'(y, z))
*'(x, 0') -> 0'
*'(0', x) -> 0'
*'(s(x), s(y)) -> s(+'(*'(x, y), +'(x, y)))
*'(*'(x, y), z) -> *'(x, *'(y, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0'
sum(cons(x, l)) -> +'(x, sum(l))
sum(app(l1, l2)) -> +'(sum(l1), sum(l2))
prod(nil) -> s(0')
prod(cons(x, l)) -> *'(x, prod(l))
prod(app(l1, l2)) -> *'(prod(l1), prod(l2))

Types:
+' :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
*' :: 0':s -> 0':s -> 0':s
app :: nil:cons -> nil:cons -> nil:cons
nil :: nil:cons
cons :: 0':s -> nil:cons -> nil:cons
sum :: nil:cons -> 0':s
prod :: nil:cons -> 0':s
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat -> 0':s
gen_nil:cons4_0 :: Nat -> nil:cons


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
gen_nil:cons4_0(0) <=> nil
gen_nil:cons4_0(+(x, 1)) <=> cons(0', gen_nil:cons4_0(x))


The following defined symbols remain to be analysed:
+', *', app, sum, prod

They will be analysed ascendingly in the following order:
+' < *'
+' < sum
*' < prod

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
TRS:
Rules:
+'(x, 0') -> x
+'(0', x) -> x
+'(s(x), s(y)) -> s(s(+'(x, y)))
+'(+'(x, y), z) -> +'(x, +'(y, z))
*'(x, 0') -> 0'
*'(0', x) -> 0'
*'(s(x), s(y)) -> s(+'(*'(x, y), +'(x, y)))
*'(*'(x, y), z) -> *'(x, *'(y, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0'
sum(cons(x, l)) -> +'(x, sum(l))
sum(app(l1, l2)) -> +'(sum(l1), sum(l2))
prod(nil) -> s(0')
prod(cons(x, l)) -> *'(x, prod(l))
prod(app(l1, l2)) -> *'(prod(l1), prod(l2))

Types:
+' :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
*' :: 0':s -> 0':s -> 0':s
app :: nil:cons -> nil:cons -> nil:cons
nil :: nil:cons
cons :: 0':s -> nil:cons -> nil:cons
sum :: nil:cons -> 0':s
prod :: nil:cons -> 0':s
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat -> 0':s
gen_nil:cons4_0 :: Nat -> nil:cons


Lemmas:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(n6_0)) -> gen_0':s3_0(*(2, n6_0)), rt in Omega(1 + n6_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
gen_nil:cons4_0(0) <=> nil
gen_nil:cons4_0(+(x, 1)) <=> cons(0', gen_nil:cons4_0(x))


The following defined symbols remain to be analysed:
*', app, sum, prod

They will be analysed ascendingly in the following order:
*' < prod

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
*'(gen_0':s3_0(n719_0), gen_0':s3_0(n719_0)) -> *5_0, rt in Omega(n719_0 + n719_0^2)

Induction Base:
*'(gen_0':s3_0(0), gen_0':s3_0(0))

Induction Step:
*'(gen_0':s3_0(+(n719_0, 1)), gen_0':s3_0(+(n719_0, 1))) ->_R^Omega(1)
s(+'(*'(gen_0':s3_0(n719_0), gen_0':s3_0(n719_0)), +'(gen_0':s3_0(n719_0), gen_0':s3_0(n719_0)))) ->_IH
s(+'(*5_0, +'(gen_0':s3_0(n719_0), gen_0':s3_0(n719_0)))) ->_L^Omega(1 + n719_0)
s(+'(*5_0, gen_0':s3_0(*(2, n719_0))))

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

(14)
Complex Obligation (BEST)

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

(15)
Obligation:
Proved the lower bound n^2 for the following obligation:

TRS:
Rules:
+'(x, 0') -> x
+'(0', x) -> x
+'(s(x), s(y)) -> s(s(+'(x, y)))
+'(+'(x, y), z) -> +'(x, +'(y, z))
*'(x, 0') -> 0'
*'(0', x) -> 0'
*'(s(x), s(y)) -> s(+'(*'(x, y), +'(x, y)))
*'(*'(x, y), z) -> *'(x, *'(y, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0'
sum(cons(x, l)) -> +'(x, sum(l))
sum(app(l1, l2)) -> +'(sum(l1), sum(l2))
prod(nil) -> s(0')
prod(cons(x, l)) -> *'(x, prod(l))
prod(app(l1, l2)) -> *'(prod(l1), prod(l2))

Types:
+' :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
*' :: 0':s -> 0':s -> 0':s
app :: nil:cons -> nil:cons -> nil:cons
nil :: nil:cons
cons :: 0':s -> nil:cons -> nil:cons
sum :: nil:cons -> 0':s
prod :: nil:cons -> 0':s
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat -> 0':s
gen_nil:cons4_0 :: Nat -> nil:cons


Lemmas:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(n6_0)) -> gen_0':s3_0(*(2, n6_0)), rt in Omega(1 + n6_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
gen_nil:cons4_0(0) <=> nil
gen_nil:cons4_0(+(x, 1)) <=> cons(0', gen_nil:cons4_0(x))


The following defined symbols remain to be analysed:
*', app, sum, prod

They will be analysed ascendingly in the following order:
*' < prod

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

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

(17)
BOUNDS(n^2, INF)

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

(18)
Obligation:
TRS:
Rules:
+'(x, 0') -> x
+'(0', x) -> x
+'(s(x), s(y)) -> s(s(+'(x, y)))
+'(+'(x, y), z) -> +'(x, +'(y, z))
*'(x, 0') -> 0'
*'(0', x) -> 0'
*'(s(x), s(y)) -> s(+'(*'(x, y), +'(x, y)))
*'(*'(x, y), z) -> *'(x, *'(y, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0'
sum(cons(x, l)) -> +'(x, sum(l))
sum(app(l1, l2)) -> +'(sum(l1), sum(l2))
prod(nil) -> s(0')
prod(cons(x, l)) -> *'(x, prod(l))
prod(app(l1, l2)) -> *'(prod(l1), prod(l2))

Types:
+' :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
*' :: 0':s -> 0':s -> 0':s
app :: nil:cons -> nil:cons -> nil:cons
nil :: nil:cons
cons :: 0':s -> nil:cons -> nil:cons
sum :: nil:cons -> 0':s
prod :: nil:cons -> 0':s
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat -> 0':s
gen_nil:cons4_0 :: Nat -> nil:cons


Lemmas:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(n6_0)) -> gen_0':s3_0(*(2, n6_0)), rt in Omega(1 + n6_0)
*'(gen_0':s3_0(n719_0), gen_0':s3_0(n719_0)) -> *5_0, rt in Omega(n719_0 + n719_0^2)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
gen_nil:cons4_0(0) <=> nil
gen_nil:cons4_0(+(x, 1)) <=> cons(0', gen_nil:cons4_0(x))


The following defined symbols remain to be analysed:
app, sum, prod
----------------------------------------

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
app(gen_nil:cons4_0(n13211_0), gen_nil:cons4_0(b)) -> gen_nil:cons4_0(+(n13211_0, b)), rt in Omega(1 + n13211_0)

Induction Base:
app(gen_nil:cons4_0(0), gen_nil:cons4_0(b)) ->_R^Omega(1)
gen_nil:cons4_0(b)

Induction Step:
app(gen_nil:cons4_0(+(n13211_0, 1)), gen_nil:cons4_0(b)) ->_R^Omega(1)
cons(0', app(gen_nil:cons4_0(n13211_0), gen_nil:cons4_0(b))) ->_IH
cons(0', gen_nil:cons4_0(+(b, c13212_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:
+'(x, 0') -> x
+'(0', x) -> x
+'(s(x), s(y)) -> s(s(+'(x, y)))
+'(+'(x, y), z) -> +'(x, +'(y, z))
*'(x, 0') -> 0'
*'(0', x) -> 0'
*'(s(x), s(y)) -> s(+'(*'(x, y), +'(x, y)))
*'(*'(x, y), z) -> *'(x, *'(y, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0'
sum(cons(x, l)) -> +'(x, sum(l))
sum(app(l1, l2)) -> +'(sum(l1), sum(l2))
prod(nil) -> s(0')
prod(cons(x, l)) -> *'(x, prod(l))
prod(app(l1, l2)) -> *'(prod(l1), prod(l2))

Types:
+' :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
*' :: 0':s -> 0':s -> 0':s
app :: nil:cons -> nil:cons -> nil:cons
nil :: nil:cons
cons :: 0':s -> nil:cons -> nil:cons
sum :: nil:cons -> 0':s
prod :: nil:cons -> 0':s
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat -> 0':s
gen_nil:cons4_0 :: Nat -> nil:cons


Lemmas:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(n6_0)) -> gen_0':s3_0(*(2, n6_0)), rt in Omega(1 + n6_0)
*'(gen_0':s3_0(n719_0), gen_0':s3_0(n719_0)) -> *5_0, rt in Omega(n719_0 + n719_0^2)
app(gen_nil:cons4_0(n13211_0), gen_nil:cons4_0(b)) -> gen_nil:cons4_0(+(n13211_0, b)), rt in Omega(1 + n13211_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
gen_nil:cons4_0(0) <=> nil
gen_nil:cons4_0(+(x, 1)) <=> cons(0', gen_nil:cons4_0(x))


The following defined symbols remain to be analysed:
sum, prod
----------------------------------------

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
sum(gen_nil:cons4_0(n14223_0)) -> gen_0':s3_0(0), rt in Omega(1 + n14223_0)

Induction Base:
sum(gen_nil:cons4_0(0)) ->_R^Omega(1)
0'

Induction Step:
sum(gen_nil:cons4_0(+(n14223_0, 1))) ->_R^Omega(1)
+'(0', sum(gen_nil:cons4_0(n14223_0))) ->_IH
+'(0', gen_0':s3_0(0)) ->_L^Omega(1)
gen_0':s3_0(*(2, 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:
+'(x, 0') -> x
+'(0', x) -> x
+'(s(x), s(y)) -> s(s(+'(x, y)))
+'(+'(x, y), z) -> +'(x, +'(y, z))
*'(x, 0') -> 0'
*'(0', x) -> 0'
*'(s(x), s(y)) -> s(+'(*'(x, y), +'(x, y)))
*'(*'(x, y), z) -> *'(x, *'(y, z))
app(nil, l) -> l
app(cons(x, l1), l2) -> cons(x, app(l1, l2))
sum(nil) -> 0'
sum(cons(x, l)) -> +'(x, sum(l))
sum(app(l1, l2)) -> +'(sum(l1), sum(l2))
prod(nil) -> s(0')
prod(cons(x, l)) -> *'(x, prod(l))
prod(app(l1, l2)) -> *'(prod(l1), prod(l2))

Types:
+' :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
*' :: 0':s -> 0':s -> 0':s
app :: nil:cons -> nil:cons -> nil:cons
nil :: nil:cons
cons :: 0':s -> nil:cons -> nil:cons
sum :: nil:cons -> 0':s
prod :: nil:cons -> 0':s
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat -> 0':s
gen_nil:cons4_0 :: Nat -> nil:cons


Lemmas:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(n6_0)) -> gen_0':s3_0(*(2, n6_0)), rt in Omega(1 + n6_0)
*'(gen_0':s3_0(n719_0), gen_0':s3_0(n719_0)) -> *5_0, rt in Omega(n719_0 + n719_0^2)
app(gen_nil:cons4_0(n13211_0), gen_nil:cons4_0(b)) -> gen_nil:cons4_0(+(n13211_0, b)), rt in Omega(1 + n13211_0)
sum(gen_nil:cons4_0(n14223_0)) -> gen_0':s3_0(0), rt in Omega(1 + n14223_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
gen_nil:cons4_0(0) <=> nil
gen_nil:cons4_0(+(x, 1)) <=> cons(0', gen_nil:cons4_0(x))


The following defined symbols remain to be analysed:
prod
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(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
prod(gen_nil:cons4_0(n14816_0)) -> *5_0, rt in Omega(n14816_0)

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

Induction Step:
prod(gen_nil:cons4_0(+(n14816_0, 1))) ->_R^Omega(1)
*'(0', prod(gen_nil:cons4_0(n14816_0))) ->_IH
*'(0', *5_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(24)
BOUNDS(1, INF)