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


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


WORST_CASE(Omega(n^1), ?)
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^1, 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), 265 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), 58 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 26 ms]
        (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 38 ms]
        (18) typed CpxTrs
        (19) RewriteLemmaProof [LOWER BOUND(ID), 38 ms]
        (20) BOUNDS(1, INF)


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

(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:

   +(0, y) -> y
   +(s(x), y) -> s(+(x, y))
   ++(nil, ys) -> ys
   ++(:(x, xs), ys) -> :(x, ++(xs, ys))
   sum(:(x, nil)) -> :(x, nil)
   sum(:(x, :(y, xs))) -> sum(:(+(x, y), xs))
   sum(++(xs, :(x, :(y, ys)))) -> sum(++(xs, sum(:(x, :(y, ys)))))
   -(x, 0) -> x
   -(0, s(y)) -> 0
   -(s(x), s(y)) -> -(x, y)
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(-(x, y), s(y)))
   length(nil) -> 0
   length(:(x, xs)) -> s(length(xs))
   hd(:(x, xs)) -> x
   avg(xs) -> quot(hd(sum(xs)), length(xs))

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:

   +'(0', y) -> y
   +'(s(x), y) -> s(+'(x, y))
   ++(nil, ys) -> ys
   ++(:(x, xs), ys) -> :(x, ++(xs, ys))
   sum(:(x, nil)) -> :(x, nil)
   sum(:(x, :(y, xs))) -> sum(:(+'(x, y), xs))
   sum(++(xs, :(x, :(y, ys)))) -> sum(++(xs, sum(:(x, :(y, ys)))))
   -(x, 0') -> x
   -(0', s(y)) -> 0'
   -(s(x), s(y)) -> -(x, y)
   quot(0', s(y)) -> 0'
   quot(s(x), s(y)) -> s(quot(-(x, y), s(y)))
   length(nil) -> 0'
   length(:(x, xs)) -> s(length(xs))
   hd(:(x, xs)) -> x
   avg(xs) -> quot(hd(sum(xs)), length(xs))

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

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

(4)
Obligation:
TRS:
Rules:
+'(0', y) -> y
+'(s(x), y) -> s(+'(x, y))
++(nil, ys) -> ys
++(:(x, xs), ys) -> :(x, ++(xs, ys))
sum(:(x, nil)) -> :(x, nil)
sum(:(x, :(y, xs))) -> sum(:(+'(x, y), xs))
sum(++(xs, :(x, :(y, ys)))) -> sum(++(xs, sum(:(x, :(y, ys)))))
-(x, 0') -> x
-(0', s(y)) -> 0'
-(s(x), s(y)) -> -(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(-(x, y), s(y)))
length(nil) -> 0'
length(:(x, xs)) -> s(length(xs))
hd(:(x, xs)) -> x
avg(xs) -> quot(hd(sum(xs)), length(xs))

Types:
+' :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
++ :: nil:: -> nil:: -> nil::
nil :: nil::
: :: 0':s -> nil:: -> nil::
sum :: nil:: -> nil::
- :: 0':s -> 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
length :: nil:: -> 0':s
hd :: nil:: -> 0':s
avg :: nil:: -> 0':s
hole_0':s1_0 :: 0':s
hole_nil::2_0 :: nil::
gen_0':s3_0 :: Nat -> 0':s
gen_nil::4_0 :: Nat -> nil::

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
+', ++, sum, -, quot, length

They will be analysed ascendingly in the following order:
+' < sum
++ < sum
- < quot

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

(6)
Obligation:
TRS:
Rules:
+'(0', y) -> y
+'(s(x), y) -> s(+'(x, y))
++(nil, ys) -> ys
++(:(x, xs), ys) -> :(x, ++(xs, ys))
sum(:(x, nil)) -> :(x, nil)
sum(:(x, :(y, xs))) -> sum(:(+'(x, y), xs))
sum(++(xs, :(x, :(y, ys)))) -> sum(++(xs, sum(:(x, :(y, ys)))))
-(x, 0') -> x
-(0', s(y)) -> 0'
-(s(x), s(y)) -> -(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(-(x, y), s(y)))
length(nil) -> 0'
length(:(x, xs)) -> s(length(xs))
hd(:(x, xs)) -> x
avg(xs) -> quot(hd(sum(xs)), length(xs))

Types:
+' :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
++ :: nil:: -> nil:: -> nil::
nil :: nil::
: :: 0':s -> nil:: -> nil::
sum :: nil:: -> nil::
- :: 0':s -> 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
length :: nil:: -> 0':s
hd :: nil:: -> 0':s
avg :: nil:: -> 0':s
hole_0':s1_0 :: 0':s
hole_nil::2_0 :: nil::
gen_0':s3_0 :: Nat -> 0':s
gen_nil::4_0 :: Nat -> nil::


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
gen_nil::4_0(0) <=> nil
gen_nil::4_0(+(x, 1)) <=> :(0', gen_nil::4_0(x))


The following defined symbols remain to be analysed:
+', ++, sum, -, quot, length

They will be analysed ascendingly in the following order:
+' < sum
++ < sum
- < quot

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

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

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

Induction Step:
+'(gen_0':s3_0(+(n6_0, 1)), gen_0':s3_0(b)) ->_R^Omega(1)
s(+'(gen_0':s3_0(n6_0), gen_0':s3_0(b))) ->_IH
s(gen_0':s3_0(+(b, 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:
+'(0', y) -> y
+'(s(x), y) -> s(+'(x, y))
++(nil, ys) -> ys
++(:(x, xs), ys) -> :(x, ++(xs, ys))
sum(:(x, nil)) -> :(x, nil)
sum(:(x, :(y, xs))) -> sum(:(+'(x, y), xs))
sum(++(xs, :(x, :(y, ys)))) -> sum(++(xs, sum(:(x, :(y, ys)))))
-(x, 0') -> x
-(0', s(y)) -> 0'
-(s(x), s(y)) -> -(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(-(x, y), s(y)))
length(nil) -> 0'
length(:(x, xs)) -> s(length(xs))
hd(:(x, xs)) -> x
avg(xs) -> quot(hd(sum(xs)), length(xs))

Types:
+' :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
++ :: nil:: -> nil:: -> nil::
nil :: nil::
: :: 0':s -> nil:: -> nil::
sum :: nil:: -> nil::
- :: 0':s -> 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
length :: nil:: -> 0':s
hd :: nil:: -> 0':s
avg :: nil:: -> 0':s
hole_0':s1_0 :: 0':s
hole_nil::2_0 :: nil::
gen_0':s3_0 :: Nat -> 0':s
gen_nil::4_0 :: Nat -> nil::


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
gen_nil::4_0(0) <=> nil
gen_nil::4_0(+(x, 1)) <=> :(0', gen_nil::4_0(x))


The following defined symbols remain to be analysed:
+', ++, sum, -, quot, length

They will be analysed ascendingly in the following order:
+' < sum
++ < sum
- < quot

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
TRS:
Rules:
+'(0', y) -> y
+'(s(x), y) -> s(+'(x, y))
++(nil, ys) -> ys
++(:(x, xs), ys) -> :(x, ++(xs, ys))
sum(:(x, nil)) -> :(x, nil)
sum(:(x, :(y, xs))) -> sum(:(+'(x, y), xs))
sum(++(xs, :(x, :(y, ys)))) -> sum(++(xs, sum(:(x, :(y, ys)))))
-(x, 0') -> x
-(0', s(y)) -> 0'
-(s(x), s(y)) -> -(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(-(x, y), s(y)))
length(nil) -> 0'
length(:(x, xs)) -> s(length(xs))
hd(:(x, xs)) -> x
avg(xs) -> quot(hd(sum(xs)), length(xs))

Types:
+' :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
++ :: nil:: -> nil:: -> nil::
nil :: nil::
: :: 0':s -> nil:: -> nil::
sum :: nil:: -> nil::
- :: 0':s -> 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
length :: nil:: -> 0':s
hd :: nil:: -> 0':s
avg :: nil:: -> 0':s
hole_0':s1_0 :: 0':s
hole_nil::2_0 :: nil::
gen_0':s3_0 :: Nat -> 0':s
gen_nil::4_0 :: Nat -> nil::


Lemmas:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n6_0, b)), 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::4_0(0) <=> nil
gen_nil::4_0(+(x, 1)) <=> :(0', gen_nil::4_0(x))


The following defined symbols remain to be analysed:
++, sum, -, quot, length

They will be analysed ascendingly in the following order:
++ < sum
- < quot

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
++(gen_nil::4_0(n723_0), gen_nil::4_0(b)) -> gen_nil::4_0(+(n723_0, b)), rt in Omega(1 + n723_0)

Induction Base:
++(gen_nil::4_0(0), gen_nil::4_0(b)) ->_R^Omega(1)
gen_nil::4_0(b)

Induction Step:
++(gen_nil::4_0(+(n723_0, 1)), gen_nil::4_0(b)) ->_R^Omega(1)
:(0', ++(gen_nil::4_0(n723_0), gen_nil::4_0(b))) ->_IH
:(0', gen_nil::4_0(+(b, c724_0)))

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

(14)
Obligation:
TRS:
Rules:
+'(0', y) -> y
+'(s(x), y) -> s(+'(x, y))
++(nil, ys) -> ys
++(:(x, xs), ys) -> :(x, ++(xs, ys))
sum(:(x, nil)) -> :(x, nil)
sum(:(x, :(y, xs))) -> sum(:(+'(x, y), xs))
sum(++(xs, :(x, :(y, ys)))) -> sum(++(xs, sum(:(x, :(y, ys)))))
-(x, 0') -> x
-(0', s(y)) -> 0'
-(s(x), s(y)) -> -(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(-(x, y), s(y)))
length(nil) -> 0'
length(:(x, xs)) -> s(length(xs))
hd(:(x, xs)) -> x
avg(xs) -> quot(hd(sum(xs)), length(xs))

Types:
+' :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
++ :: nil:: -> nil:: -> nil::
nil :: nil::
: :: 0':s -> nil:: -> nil::
sum :: nil:: -> nil::
- :: 0':s -> 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
length :: nil:: -> 0':s
hd :: nil:: -> 0':s
avg :: nil:: -> 0':s
hole_0':s1_0 :: 0':s
hole_nil::2_0 :: nil::
gen_0':s3_0 :: Nat -> 0':s
gen_nil::4_0 :: Nat -> nil::


Lemmas:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n6_0, b)), rt in Omega(1 + n6_0)
++(gen_nil::4_0(n723_0), gen_nil::4_0(b)) -> gen_nil::4_0(+(n723_0, b)), rt in Omega(1 + n723_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
gen_nil::4_0(0) <=> nil
gen_nil::4_0(+(x, 1)) <=> :(0', gen_nil::4_0(x))


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

They will be analysed ascendingly in the following order:
- < quot

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
sum(gen_nil::4_0(+(1, n1576_0))) -> gen_nil::4_0(1), rt in Omega(1 + n1576_0)

Induction Base:
sum(gen_nil::4_0(+(1, 0))) ->_R^Omega(1)
:(0', nil)

Induction Step:
sum(gen_nil::4_0(+(1, +(n1576_0, 1)))) ->_R^Omega(1)
sum(:(+'(0', 0'), gen_nil::4_0(n1576_0))) ->_L^Omega(1)
sum(:(gen_0':s3_0(+(0, 0)), gen_nil::4_0(n1576_0))) ->_IH
gen_nil::4_0(1)

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

(16)
Obligation:
TRS:
Rules:
+'(0', y) -> y
+'(s(x), y) -> s(+'(x, y))
++(nil, ys) -> ys
++(:(x, xs), ys) -> :(x, ++(xs, ys))
sum(:(x, nil)) -> :(x, nil)
sum(:(x, :(y, xs))) -> sum(:(+'(x, y), xs))
sum(++(xs, :(x, :(y, ys)))) -> sum(++(xs, sum(:(x, :(y, ys)))))
-(x, 0') -> x
-(0', s(y)) -> 0'
-(s(x), s(y)) -> -(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(-(x, y), s(y)))
length(nil) -> 0'
length(:(x, xs)) -> s(length(xs))
hd(:(x, xs)) -> x
avg(xs) -> quot(hd(sum(xs)), length(xs))

Types:
+' :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
++ :: nil:: -> nil:: -> nil::
nil :: nil::
: :: 0':s -> nil:: -> nil::
sum :: nil:: -> nil::
- :: 0':s -> 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
length :: nil:: -> 0':s
hd :: nil:: -> 0':s
avg :: nil:: -> 0':s
hole_0':s1_0 :: 0':s
hole_nil::2_0 :: nil::
gen_0':s3_0 :: Nat -> 0':s
gen_nil::4_0 :: Nat -> nil::


Lemmas:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n6_0, b)), rt in Omega(1 + n6_0)
++(gen_nil::4_0(n723_0), gen_nil::4_0(b)) -> gen_nil::4_0(+(n723_0, b)), rt in Omega(1 + n723_0)
sum(gen_nil::4_0(+(1, n1576_0))) -> gen_nil::4_0(1), rt in Omega(1 + n1576_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
gen_nil::4_0(0) <=> nil
gen_nil::4_0(+(x, 1)) <=> :(0', gen_nil::4_0(x))


The following defined symbols remain to be analysed:
-, quot, length

They will be analysed ascendingly in the following order:
- < quot

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
-(gen_0':s3_0(n1997_0), gen_0':s3_0(n1997_0)) -> gen_0':s3_0(0), rt in Omega(1 + n1997_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(+(n1997_0, 1)), gen_0':s3_0(+(n1997_0, 1))) ->_R^Omega(1)
-(gen_0':s3_0(n1997_0), gen_0':s3_0(n1997_0)) ->_IH
gen_0':s3_0(0)

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

(18)
Obligation:
TRS:
Rules:
+'(0', y) -> y
+'(s(x), y) -> s(+'(x, y))
++(nil, ys) -> ys
++(:(x, xs), ys) -> :(x, ++(xs, ys))
sum(:(x, nil)) -> :(x, nil)
sum(:(x, :(y, xs))) -> sum(:(+'(x, y), xs))
sum(++(xs, :(x, :(y, ys)))) -> sum(++(xs, sum(:(x, :(y, ys)))))
-(x, 0') -> x
-(0', s(y)) -> 0'
-(s(x), s(y)) -> -(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(-(x, y), s(y)))
length(nil) -> 0'
length(:(x, xs)) -> s(length(xs))
hd(:(x, xs)) -> x
avg(xs) -> quot(hd(sum(xs)), length(xs))

Types:
+' :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
++ :: nil:: -> nil:: -> nil::
nil :: nil::
: :: 0':s -> nil:: -> nil::
sum :: nil:: -> nil::
- :: 0':s -> 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
length :: nil:: -> 0':s
hd :: nil:: -> 0':s
avg :: nil:: -> 0':s
hole_0':s1_0 :: 0':s
hole_nil::2_0 :: nil::
gen_0':s3_0 :: Nat -> 0':s
gen_nil::4_0 :: Nat -> nil::


Lemmas:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n6_0, b)), rt in Omega(1 + n6_0)
++(gen_nil::4_0(n723_0), gen_nil::4_0(b)) -> gen_nil::4_0(+(n723_0, b)), rt in Omega(1 + n723_0)
sum(gen_nil::4_0(+(1, n1576_0))) -> gen_nil::4_0(1), rt in Omega(1 + n1576_0)
-(gen_0':s3_0(n1997_0), gen_0':s3_0(n1997_0)) -> gen_0':s3_0(0), rt in Omega(1 + n1997_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
gen_nil::4_0(0) <=> nil
gen_nil::4_0(+(x, 1)) <=> :(0', gen_nil::4_0(x))


The following defined symbols remain to be analysed:
quot, length
----------------------------------------

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
length(gen_nil::4_0(n2624_0)) -> gen_0':s3_0(n2624_0), rt in Omega(1 + n2624_0)

Induction Base:
length(gen_nil::4_0(0)) ->_R^Omega(1)
0'

Induction Step:
length(gen_nil::4_0(+(n2624_0, 1))) ->_R^Omega(1)
s(length(gen_nil::4_0(n2624_0))) ->_IH
s(gen_0':s3_0(c2625_0))

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

(20)
BOUNDS(1, INF)