/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 264 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), 645 ms]
        (14) 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:

   plus(x, y) -> plusIter(x, y, 0)
   plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z)
   ifPlus(true, x, y, z) -> y
   ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z))
   le(s(x), 0) -> false
   le(0, y) -> true
   le(s(x), s(y)) -> le(x, y)
   sum(xs) -> sumIter(xs, 0)
   sumIter(xs, x) -> ifSum(isempty(xs), xs, x, plus(x, head(xs)))
   ifSum(true, xs, x, y) -> x
   ifSum(false, xs, x, y) -> sumIter(tail(xs), y)
   isempty(nil) -> true
   isempty(cons(x, xs)) -> false
   head(nil) -> error
   head(cons(x, xs)) -> x
   tail(nil) -> nil
   tail(cons(x, xs)) -> xs
   a -> b
   a -> c

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:

   plus(x, y) -> plusIter(x, y, 0')
   plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z)
   ifPlus(true, x, y, z) -> y
   ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z))
   le(s(x), 0') -> false
   le(0', y) -> true
   le(s(x), s(y)) -> le(x, y)
   sum(xs) -> sumIter(xs, 0')
   sumIter(xs, x) -> ifSum(isempty(xs), xs, x, plus(x, head(xs)))
   ifSum(true, xs, x, y) -> x
   ifSum(false, xs, x, y) -> sumIter(tail(xs), y)
   isempty(nil) -> true
   isempty(cons(x, xs)) -> false
   head(nil) -> error
   head(cons(x, xs)) -> x
   tail(nil) -> nil
   tail(cons(x, xs)) -> xs
   a -> b
   a -> c

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

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

(4)
Obligation:
TRS:
Rules:
plus(x, y) -> plusIter(x, y, 0')
plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z)
ifPlus(true, x, y, z) -> y
ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z))
le(s(x), 0') -> false
le(0', y) -> true
le(s(x), s(y)) -> le(x, y)
sum(xs) -> sumIter(xs, 0')
sumIter(xs, x) -> ifSum(isempty(xs), xs, x, plus(x, head(xs)))
ifSum(true, xs, x, y) -> x
ifSum(false, xs, x, y) -> sumIter(tail(xs), y)
isempty(nil) -> true
isempty(cons(x, xs)) -> false
head(nil) -> error
head(cons(x, xs)) -> x
tail(nil) -> nil
tail(cons(x, xs)) -> xs
a -> b
a -> c

Types:
plus :: 0':s:error -> 0':s:error -> 0':s:error
plusIter :: 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error
0' :: 0':s:error
ifPlus :: true:false -> 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error
le :: 0':s:error -> 0':s:error -> true:false
true :: true:false
false :: true:false
s :: 0':s:error -> 0':s:error
sum :: nil:cons -> 0':s:error
sumIter :: nil:cons -> 0':s:error -> 0':s:error
ifSum :: true:false -> nil:cons -> 0':s:error -> 0':s:error -> 0':s:error
isempty :: nil:cons -> true:false
head :: nil:cons -> 0':s:error
tail :: nil:cons -> nil:cons
nil :: nil:cons
cons :: 0':s:error -> nil:cons -> nil:cons
error :: 0':s:error
a :: b:c
b :: b:c
c :: b:c
hole_0':s:error1_0 :: 0':s:error
hole_true:false2_0 :: true:false
hole_nil:cons3_0 :: nil:cons
hole_b:c4_0 :: b:c
gen_0':s:error5_0 :: Nat -> 0':s:error
gen_nil:cons6_0 :: Nat -> nil:cons

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
plusIter, le, sumIter

They will be analysed ascendingly in the following order:
le < plusIter

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

(6)
Obligation:
TRS:
Rules:
plus(x, y) -> plusIter(x, y, 0')
plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z)
ifPlus(true, x, y, z) -> y
ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z))
le(s(x), 0') -> false
le(0', y) -> true
le(s(x), s(y)) -> le(x, y)
sum(xs) -> sumIter(xs, 0')
sumIter(xs, x) -> ifSum(isempty(xs), xs, x, plus(x, head(xs)))
ifSum(true, xs, x, y) -> x
ifSum(false, xs, x, y) -> sumIter(tail(xs), y)
isempty(nil) -> true
isempty(cons(x, xs)) -> false
head(nil) -> error
head(cons(x, xs)) -> x
tail(nil) -> nil
tail(cons(x, xs)) -> xs
a -> b
a -> c

Types:
plus :: 0':s:error -> 0':s:error -> 0':s:error
plusIter :: 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error
0' :: 0':s:error
ifPlus :: true:false -> 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error
le :: 0':s:error -> 0':s:error -> true:false
true :: true:false
false :: true:false
s :: 0':s:error -> 0':s:error
sum :: nil:cons -> 0':s:error
sumIter :: nil:cons -> 0':s:error -> 0':s:error
ifSum :: true:false -> nil:cons -> 0':s:error -> 0':s:error -> 0':s:error
isempty :: nil:cons -> true:false
head :: nil:cons -> 0':s:error
tail :: nil:cons -> nil:cons
nil :: nil:cons
cons :: 0':s:error -> nil:cons -> nil:cons
error :: 0':s:error
a :: b:c
b :: b:c
c :: b:c
hole_0':s:error1_0 :: 0':s:error
hole_true:false2_0 :: true:false
hole_nil:cons3_0 :: nil:cons
hole_b:c4_0 :: b:c
gen_0':s:error5_0 :: Nat -> 0':s:error
gen_nil:cons6_0 :: Nat -> nil:cons


Generator Equations:
gen_0':s:error5_0(0) <=> 0'
gen_0':s:error5_0(+(x, 1)) <=> s(gen_0':s:error5_0(x))
gen_nil:cons6_0(0) <=> nil
gen_nil:cons6_0(+(x, 1)) <=> cons(0', gen_nil:cons6_0(x))


The following defined symbols remain to be analysed:
le, plusIter, sumIter

They will be analysed ascendingly in the following order:
le < plusIter

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
le(gen_0':s:error5_0(+(1, n8_0)), gen_0':s:error5_0(n8_0)) -> false, rt in Omega(1 + n8_0)

Induction Base:
le(gen_0':s:error5_0(+(1, 0)), gen_0':s:error5_0(0)) ->_R^Omega(1)
false

Induction Step:
le(gen_0':s:error5_0(+(1, +(n8_0, 1))), gen_0':s:error5_0(+(n8_0, 1))) ->_R^Omega(1)
le(gen_0':s:error5_0(+(1, n8_0)), gen_0':s:error5_0(n8_0)) ->_IH
false

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:
plus(x, y) -> plusIter(x, y, 0')
plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z)
ifPlus(true, x, y, z) -> y
ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z))
le(s(x), 0') -> false
le(0', y) -> true
le(s(x), s(y)) -> le(x, y)
sum(xs) -> sumIter(xs, 0')
sumIter(xs, x) -> ifSum(isempty(xs), xs, x, plus(x, head(xs)))
ifSum(true, xs, x, y) -> x
ifSum(false, xs, x, y) -> sumIter(tail(xs), y)
isempty(nil) -> true
isempty(cons(x, xs)) -> false
head(nil) -> error
head(cons(x, xs)) -> x
tail(nil) -> nil
tail(cons(x, xs)) -> xs
a -> b
a -> c

Types:
plus :: 0':s:error -> 0':s:error -> 0':s:error
plusIter :: 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error
0' :: 0':s:error
ifPlus :: true:false -> 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error
le :: 0':s:error -> 0':s:error -> true:false
true :: true:false
false :: true:false
s :: 0':s:error -> 0':s:error
sum :: nil:cons -> 0':s:error
sumIter :: nil:cons -> 0':s:error -> 0':s:error
ifSum :: true:false -> nil:cons -> 0':s:error -> 0':s:error -> 0':s:error
isempty :: nil:cons -> true:false
head :: nil:cons -> 0':s:error
tail :: nil:cons -> nil:cons
nil :: nil:cons
cons :: 0':s:error -> nil:cons -> nil:cons
error :: 0':s:error
a :: b:c
b :: b:c
c :: b:c
hole_0':s:error1_0 :: 0':s:error
hole_true:false2_0 :: true:false
hole_nil:cons3_0 :: nil:cons
hole_b:c4_0 :: b:c
gen_0':s:error5_0 :: Nat -> 0':s:error
gen_nil:cons6_0 :: Nat -> nil:cons


Generator Equations:
gen_0':s:error5_0(0) <=> 0'
gen_0':s:error5_0(+(x, 1)) <=> s(gen_0':s:error5_0(x))
gen_nil:cons6_0(0) <=> nil
gen_nil:cons6_0(+(x, 1)) <=> cons(0', gen_nil:cons6_0(x))


The following defined symbols remain to be analysed:
le, plusIter, sumIter

They will be analysed ascendingly in the following order:
le < plusIter

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
TRS:
Rules:
plus(x, y) -> plusIter(x, y, 0')
plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z)
ifPlus(true, x, y, z) -> y
ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z))
le(s(x), 0') -> false
le(0', y) -> true
le(s(x), s(y)) -> le(x, y)
sum(xs) -> sumIter(xs, 0')
sumIter(xs, x) -> ifSum(isempty(xs), xs, x, plus(x, head(xs)))
ifSum(true, xs, x, y) -> x
ifSum(false, xs, x, y) -> sumIter(tail(xs), y)
isempty(nil) -> true
isempty(cons(x, xs)) -> false
head(nil) -> error
head(cons(x, xs)) -> x
tail(nil) -> nil
tail(cons(x, xs)) -> xs
a -> b
a -> c

Types:
plus :: 0':s:error -> 0':s:error -> 0':s:error
plusIter :: 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error
0' :: 0':s:error
ifPlus :: true:false -> 0':s:error -> 0':s:error -> 0':s:error -> 0':s:error
le :: 0':s:error -> 0':s:error -> true:false
true :: true:false
false :: true:false
s :: 0':s:error -> 0':s:error
sum :: nil:cons -> 0':s:error
sumIter :: nil:cons -> 0':s:error -> 0':s:error
ifSum :: true:false -> nil:cons -> 0':s:error -> 0':s:error -> 0':s:error
isempty :: nil:cons -> true:false
head :: nil:cons -> 0':s:error
tail :: nil:cons -> nil:cons
nil :: nil:cons
cons :: 0':s:error -> nil:cons -> nil:cons
error :: 0':s:error
a :: b:c
b :: b:c
c :: b:c
hole_0':s:error1_0 :: 0':s:error
hole_true:false2_0 :: true:false
hole_nil:cons3_0 :: nil:cons
hole_b:c4_0 :: b:c
gen_0':s:error5_0 :: Nat -> 0':s:error
gen_nil:cons6_0 :: Nat -> nil:cons


Lemmas:
le(gen_0':s:error5_0(+(1, n8_0)), gen_0':s:error5_0(n8_0)) -> false, rt in Omega(1 + n8_0)


Generator Equations:
gen_0':s:error5_0(0) <=> 0'
gen_0':s:error5_0(+(x, 1)) <=> s(gen_0':s:error5_0(x))
gen_nil:cons6_0(0) <=> nil
gen_nil:cons6_0(+(x, 1)) <=> cons(0', gen_nil:cons6_0(x))


The following defined symbols remain to be analysed:
plusIter, sumIter
----------------------------------------

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
sumIter(gen_nil:cons6_0(n2686_0), gen_0':s:error5_0(1)) -> *7_0, rt in Omega(n2686_0)

Induction Base:
sumIter(gen_nil:cons6_0(0), gen_0':s:error5_0(1))

Induction Step:
sumIter(gen_nil:cons6_0(+(n2686_0, 1)), gen_0':s:error5_0(1)) ->_R^Omega(1)
ifSum(isempty(gen_nil:cons6_0(+(n2686_0, 1))), gen_nil:cons6_0(+(n2686_0, 1)), gen_0':s:error5_0(1), plus(gen_0':s:error5_0(1), head(gen_nil:cons6_0(+(n2686_0, 1))))) ->_R^Omega(1)
ifSum(false, gen_nil:cons6_0(+(1, n2686_0)), gen_0':s:error5_0(1), plus(gen_0':s:error5_0(1), head(gen_nil:cons6_0(+(1, n2686_0))))) ->_R^Omega(1)
ifSum(false, gen_nil:cons6_0(+(1, n2686_0)), gen_0':s:error5_0(1), plus(gen_0':s:error5_0(1), 0')) ->_R^Omega(1)
ifSum(false, gen_nil:cons6_0(+(1, n2686_0)), gen_0':s:error5_0(1), plusIter(gen_0':s:error5_0(1), 0', 0')) ->_R^Omega(1)
ifSum(false, gen_nil:cons6_0(+(1, n2686_0)), gen_0':s:error5_0(1), ifPlus(le(gen_0':s:error5_0(1), 0'), gen_0':s:error5_0(1), 0', 0')) ->_L^Omega(1)
ifSum(false, gen_nil:cons6_0(+(1, n2686_0)), gen_0':s:error5_0(1), ifPlus(false, gen_0':s:error5_0(1), 0', 0')) ->_R^Omega(1)
ifSum(false, gen_nil:cons6_0(+(1, n2686_0)), gen_0':s:error5_0(1), plusIter(gen_0':s:error5_0(1), s(0'), s(0'))) ->_R^Omega(1)
ifSum(false, gen_nil:cons6_0(+(1, n2686_0)), gen_0':s:error5_0(1), ifPlus(le(gen_0':s:error5_0(1), s(0')), gen_0':s:error5_0(1), s(0'), s(0'))) ->_R^Omega(1)
ifSum(false, gen_nil:cons6_0(+(1, n2686_0)), gen_0':s:error5_0(1), ifPlus(le(gen_0':s:error5_0(0), 0'), gen_0':s:error5_0(1), s(0'), s(0'))) ->_R^Omega(1)
ifSum(false, gen_nil:cons6_0(+(1, n2686_0)), gen_0':s:error5_0(1), ifPlus(true, gen_0':s:error5_0(1), s(0'), s(0'))) ->_R^Omega(1)
ifSum(false, gen_nil:cons6_0(+(1, n2686_0)), gen_0':s:error5_0(1), s(0')) ->_R^Omega(1)
sumIter(tail(gen_nil:cons6_0(+(1, n2686_0))), s(0')) ->_R^Omega(1)
sumIter(gen_nil:cons6_0(n2686_0), s(0')) ->_IH
*7_0

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

(14)
BOUNDS(1, INF)