/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) 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), 290 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), 37 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 76 ms]
        (16) proven lower bound
        (17) LowerBoundPropagationProof [FINISHED, 0 ms]
        (18) BOUNDS(n^2, 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:

   isEmpty(cons(x, xs)) -> false
   isEmpty(nil) -> true
   isZero(0) -> true
   isZero(s(x)) -> false
   head(cons(x, xs)) -> x
   tail(cons(x, xs)) -> xs
   tail(nil) -> nil
   p(s(s(x))) -> s(p(s(x)))
   p(s(0)) -> 0
   p(0) -> 0
   inc(s(x)) -> s(inc(x))
   inc(0) -> s(0)
   sumList(xs, y) -> if(isEmpty(xs), isZero(head(xs)), y, tail(xs), cons(p(head(xs)), tail(xs)), inc(y))
   if(true, b, y, xs, ys, x) -> y
   if(false, true, y, xs, ys, x) -> sumList(xs, y)
   if(false, false, y, xs, ys, x) -> sumList(ys, x)
   sum(xs) -> sumList(xs, 0)

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:

   isEmpty(cons(x, xs)) -> false
   isEmpty(nil) -> true
   isZero(0') -> true
   isZero(s(x)) -> false
   head(cons(x, xs)) -> x
   tail(cons(x, xs)) -> xs
   tail(nil) -> nil
   p(s(s(x))) -> s(p(s(x)))
   p(s(0')) -> 0'
   p(0') -> 0'
   inc(s(x)) -> s(inc(x))
   inc(0') -> s(0')
   sumList(xs, y) -> if(isEmpty(xs), isZero(head(xs)), y, tail(xs), cons(p(head(xs)), tail(xs)), inc(y))
   if(true, b, y, xs, ys, x) -> y
   if(false, true, y, xs, ys, x) -> sumList(xs, y)
   if(false, false, y, xs, ys, x) -> sumList(ys, x)
   sum(xs) -> sumList(xs, 0')

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

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

(4)
Obligation:
TRS:
Rules:
isEmpty(cons(x, xs)) -> false
isEmpty(nil) -> true
isZero(0') -> true
isZero(s(x)) -> false
head(cons(x, xs)) -> x
tail(cons(x, xs)) -> xs
tail(nil) -> nil
p(s(s(x))) -> s(p(s(x)))
p(s(0')) -> 0'
p(0') -> 0'
inc(s(x)) -> s(inc(x))
inc(0') -> s(0')
sumList(xs, y) -> if(isEmpty(xs), isZero(head(xs)), y, tail(xs), cons(p(head(xs)), tail(xs)), inc(y))
if(true, b, y, xs, ys, x) -> y
if(false, true, y, xs, ys, x) -> sumList(xs, y)
if(false, false, y, xs, ys, x) -> sumList(ys, x)
sum(xs) -> sumList(xs, 0')

Types:
isEmpty :: cons:nil -> false:true
cons :: 0':s -> cons:nil -> cons:nil
false :: false:true
nil :: cons:nil
true :: false:true
isZero :: 0':s -> false:true
0' :: 0':s
s :: 0':s -> 0':s
head :: cons:nil -> 0':s
tail :: cons:nil -> cons:nil
p :: 0':s -> 0':s
inc :: 0':s -> 0':s
sumList :: cons:nil -> 0':s -> 0':s
if :: false:true -> false:true -> 0':s -> cons:nil -> cons:nil -> 0':s -> 0':s
sum :: cons:nil -> 0':s
hole_false:true1_0 :: false:true
hole_cons:nil2_0 :: cons:nil
hole_0':s3_0 :: 0':s
gen_cons:nil4_0 :: Nat -> cons:nil
gen_0':s5_0 :: Nat -> 0':s

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
p, inc, sumList

They will be analysed ascendingly in the following order:
p < sumList
inc < sumList

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

(6)
Obligation:
TRS:
Rules:
isEmpty(cons(x, xs)) -> false
isEmpty(nil) -> true
isZero(0') -> true
isZero(s(x)) -> false
head(cons(x, xs)) -> x
tail(cons(x, xs)) -> xs
tail(nil) -> nil
p(s(s(x))) -> s(p(s(x)))
p(s(0')) -> 0'
p(0') -> 0'
inc(s(x)) -> s(inc(x))
inc(0') -> s(0')
sumList(xs, y) -> if(isEmpty(xs), isZero(head(xs)), y, tail(xs), cons(p(head(xs)), tail(xs)), inc(y))
if(true, b, y, xs, ys, x) -> y
if(false, true, y, xs, ys, x) -> sumList(xs, y)
if(false, false, y, xs, ys, x) -> sumList(ys, x)
sum(xs) -> sumList(xs, 0')

Types:
isEmpty :: cons:nil -> false:true
cons :: 0':s -> cons:nil -> cons:nil
false :: false:true
nil :: cons:nil
true :: false:true
isZero :: 0':s -> false:true
0' :: 0':s
s :: 0':s -> 0':s
head :: cons:nil -> 0':s
tail :: cons:nil -> cons:nil
p :: 0':s -> 0':s
inc :: 0':s -> 0':s
sumList :: cons:nil -> 0':s -> 0':s
if :: false:true -> false:true -> 0':s -> cons:nil -> cons:nil -> 0':s -> 0':s
sum :: cons:nil -> 0':s
hole_false:true1_0 :: false:true
hole_cons:nil2_0 :: cons:nil
hole_0':s3_0 :: 0':s
gen_cons:nil4_0 :: Nat -> cons:nil
gen_0':s5_0 :: Nat -> 0':s


Generator Equations:
gen_cons:nil4_0(0) <=> nil
gen_cons:nil4_0(+(x, 1)) <=> cons(0', gen_cons:nil4_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, inc, sumList

They will be analysed ascendingly in the following order:
p < sumList
inc < sumList

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

(7) 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).
----------------------------------------

(8)
Complex Obligation (BEST)

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

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

TRS:
Rules:
isEmpty(cons(x, xs)) -> false
isEmpty(nil) -> true
isZero(0') -> true
isZero(s(x)) -> false
head(cons(x, xs)) -> x
tail(cons(x, xs)) -> xs
tail(nil) -> nil
p(s(s(x))) -> s(p(s(x)))
p(s(0')) -> 0'
p(0') -> 0'
inc(s(x)) -> s(inc(x))
inc(0') -> s(0')
sumList(xs, y) -> if(isEmpty(xs), isZero(head(xs)), y, tail(xs), cons(p(head(xs)), tail(xs)), inc(y))
if(true, b, y, xs, ys, x) -> y
if(false, true, y, xs, ys, x) -> sumList(xs, y)
if(false, false, y, xs, ys, x) -> sumList(ys, x)
sum(xs) -> sumList(xs, 0')

Types:
isEmpty :: cons:nil -> false:true
cons :: 0':s -> cons:nil -> cons:nil
false :: false:true
nil :: cons:nil
true :: false:true
isZero :: 0':s -> false:true
0' :: 0':s
s :: 0':s -> 0':s
head :: cons:nil -> 0':s
tail :: cons:nil -> cons:nil
p :: 0':s -> 0':s
inc :: 0':s -> 0':s
sumList :: cons:nil -> 0':s -> 0':s
if :: false:true -> false:true -> 0':s -> cons:nil -> cons:nil -> 0':s -> 0':s
sum :: cons:nil -> 0':s
hole_false:true1_0 :: false:true
hole_cons:nil2_0 :: cons:nil
hole_0':s3_0 :: 0':s
gen_cons:nil4_0 :: Nat -> cons:nil
gen_0':s5_0 :: Nat -> 0':s


Generator Equations:
gen_cons:nil4_0(0) <=> nil
gen_cons:nil4_0(+(x, 1)) <=> cons(0', gen_cons:nil4_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, inc, sumList

They will be analysed ascendingly in the following order:
p < sumList
inc < sumList

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
TRS:
Rules:
isEmpty(cons(x, xs)) -> false
isEmpty(nil) -> true
isZero(0') -> true
isZero(s(x)) -> false
head(cons(x, xs)) -> x
tail(cons(x, xs)) -> xs
tail(nil) -> nil
p(s(s(x))) -> s(p(s(x)))
p(s(0')) -> 0'
p(0') -> 0'
inc(s(x)) -> s(inc(x))
inc(0') -> s(0')
sumList(xs, y) -> if(isEmpty(xs), isZero(head(xs)), y, tail(xs), cons(p(head(xs)), tail(xs)), inc(y))
if(true, b, y, xs, ys, x) -> y
if(false, true, y, xs, ys, x) -> sumList(xs, y)
if(false, false, y, xs, ys, x) -> sumList(ys, x)
sum(xs) -> sumList(xs, 0')

Types:
isEmpty :: cons:nil -> false:true
cons :: 0':s -> cons:nil -> cons:nil
false :: false:true
nil :: cons:nil
true :: false:true
isZero :: 0':s -> false:true
0' :: 0':s
s :: 0':s -> 0':s
head :: cons:nil -> 0':s
tail :: cons:nil -> cons:nil
p :: 0':s -> 0':s
inc :: 0':s -> 0':s
sumList :: cons:nil -> 0':s -> 0':s
if :: false:true -> false:true -> 0':s -> cons:nil -> cons:nil -> 0':s -> 0':s
sum :: cons:nil -> 0':s
hole_false:true1_0 :: false:true
hole_cons:nil2_0 :: cons:nil
hole_0':s3_0 :: 0':s
gen_cons:nil4_0 :: Nat -> cons:nil
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_cons:nil4_0(0) <=> nil
gen_cons:nil4_0(+(x, 1)) <=> cons(0', gen_cons:nil4_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:
inc, sumList

They will be analysed ascendingly in the following order:
inc < sumList

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
inc(gen_0':s5_0(n264_0)) -> gen_0':s5_0(+(1, n264_0)), rt in Omega(1 + n264_0)

Induction Base:
inc(gen_0':s5_0(0)) ->_R^Omega(1)
s(0')

Induction Step:
inc(gen_0':s5_0(+(n264_0, 1))) ->_R^Omega(1)
s(inc(gen_0':s5_0(n264_0))) ->_IH
s(gen_0':s5_0(+(1, c265_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:
isEmpty(cons(x, xs)) -> false
isEmpty(nil) -> true
isZero(0') -> true
isZero(s(x)) -> false
head(cons(x, xs)) -> x
tail(cons(x, xs)) -> xs
tail(nil) -> nil
p(s(s(x))) -> s(p(s(x)))
p(s(0')) -> 0'
p(0') -> 0'
inc(s(x)) -> s(inc(x))
inc(0') -> s(0')
sumList(xs, y) -> if(isEmpty(xs), isZero(head(xs)), y, tail(xs), cons(p(head(xs)), tail(xs)), inc(y))
if(true, b, y, xs, ys, x) -> y
if(false, true, y, xs, ys, x) -> sumList(xs, y)
if(false, false, y, xs, ys, x) -> sumList(ys, x)
sum(xs) -> sumList(xs, 0')

Types:
isEmpty :: cons:nil -> false:true
cons :: 0':s -> cons:nil -> cons:nil
false :: false:true
nil :: cons:nil
true :: false:true
isZero :: 0':s -> false:true
0' :: 0':s
s :: 0':s -> 0':s
head :: cons:nil -> 0':s
tail :: cons:nil -> cons:nil
p :: 0':s -> 0':s
inc :: 0':s -> 0':s
sumList :: cons:nil -> 0':s -> 0':s
if :: false:true -> false:true -> 0':s -> cons:nil -> cons:nil -> 0':s -> 0':s
sum :: cons:nil -> 0':s
hole_false:true1_0 :: false:true
hole_cons:nil2_0 :: cons:nil
hole_0':s3_0 :: 0':s
gen_cons:nil4_0 :: Nat -> cons:nil
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)
inc(gen_0':s5_0(n264_0)) -> gen_0':s5_0(+(1, n264_0)), rt in Omega(1 + n264_0)


Generator Equations:
gen_cons:nil4_0(0) <=> nil
gen_cons:nil4_0(+(x, 1)) <=> cons(0', gen_cons:nil4_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:
sumList
----------------------------------------

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
sumList(gen_cons:nil4_0(n531_0), gen_0':s5_0(b)) -> gen_0':s5_0(b), rt in Omega(1 + b + b*n531_0 + n531_0)

Induction Base:
sumList(gen_cons:nil4_0(0), gen_0':s5_0(b)) ->_R^Omega(1)
if(isEmpty(gen_cons:nil4_0(0)), isZero(head(gen_cons:nil4_0(0))), gen_0':s5_0(b), tail(gen_cons:nil4_0(0)), cons(p(head(gen_cons:nil4_0(0))), tail(gen_cons:nil4_0(0))), inc(gen_0':s5_0(b))) ->_R^Omega(1)
if(true, isZero(head(gen_cons:nil4_0(0))), gen_0':s5_0(b), tail(gen_cons:nil4_0(0)), cons(p(head(gen_cons:nil4_0(0))), tail(gen_cons:nil4_0(0))), inc(gen_0':s5_0(b))) ->_R^Omega(1)
if(true, isZero(head(gen_cons:nil4_0(0))), gen_0':s5_0(b), nil, cons(p(head(gen_cons:nil4_0(0))), tail(gen_cons:nil4_0(0))), inc(gen_0':s5_0(b))) ->_R^Omega(1)
if(true, isZero(head(gen_cons:nil4_0(0))), gen_0':s5_0(b), nil, cons(p(head(gen_cons:nil4_0(0))), nil), inc(gen_0':s5_0(b))) ->_L^Omega(1 + b)
if(true, isZero(head(gen_cons:nil4_0(0))), gen_0':s5_0(b), nil, cons(p(head(gen_cons:nil4_0(0))), nil), gen_0':s5_0(+(1, b))) ->_R^Omega(1)
gen_0':s5_0(b)

Induction Step:
sumList(gen_cons:nil4_0(+(n531_0, 1)), gen_0':s5_0(b)) ->_R^Omega(1)
if(isEmpty(gen_cons:nil4_0(+(n531_0, 1))), isZero(head(gen_cons:nil4_0(+(n531_0, 1)))), gen_0':s5_0(b), tail(gen_cons:nil4_0(+(n531_0, 1))), cons(p(head(gen_cons:nil4_0(+(n531_0, 1)))), tail(gen_cons:nil4_0(+(n531_0, 1)))), inc(gen_0':s5_0(b))) ->_R^Omega(1)
if(false, isZero(head(gen_cons:nil4_0(+(1, n531_0)))), gen_0':s5_0(b), tail(gen_cons:nil4_0(+(1, n531_0))), cons(p(head(gen_cons:nil4_0(+(1, n531_0)))), tail(gen_cons:nil4_0(+(1, n531_0)))), inc(gen_0':s5_0(b))) ->_R^Omega(1)
if(false, isZero(0'), gen_0':s5_0(b), tail(gen_cons:nil4_0(+(1, n531_0))), cons(p(head(gen_cons:nil4_0(+(1, n531_0)))), tail(gen_cons:nil4_0(+(1, n531_0)))), inc(gen_0':s5_0(b))) ->_R^Omega(1)
if(false, true, gen_0':s5_0(b), tail(gen_cons:nil4_0(+(1, n531_0))), cons(p(head(gen_cons:nil4_0(+(1, n531_0)))), tail(gen_cons:nil4_0(+(1, n531_0)))), inc(gen_0':s5_0(b))) ->_R^Omega(1)
if(false, true, gen_0':s5_0(b), gen_cons:nil4_0(n531_0), cons(p(head(gen_cons:nil4_0(+(1, n531_0)))), tail(gen_cons:nil4_0(+(1, n531_0)))), inc(gen_0':s5_0(b))) ->_R^Omega(1)
if(false, true, gen_0':s5_0(b), gen_cons:nil4_0(n531_0), cons(p(0'), tail(gen_cons:nil4_0(+(1, n531_0)))), inc(gen_0':s5_0(b))) ->_R^Omega(1)
if(false, true, gen_0':s5_0(b), gen_cons:nil4_0(n531_0), cons(0', tail(gen_cons:nil4_0(+(1, n531_0)))), inc(gen_0':s5_0(b))) ->_R^Omega(1)
if(false, true, gen_0':s5_0(b), gen_cons:nil4_0(n531_0), cons(0', gen_cons:nil4_0(n531_0)), inc(gen_0':s5_0(b))) ->_L^Omega(1 + b)
if(false, true, gen_0':s5_0(b), gen_cons:nil4_0(n531_0), cons(0', gen_cons:nil4_0(n531_0)), gen_0':s5_0(+(1, b))) ->_R^Omega(1)
sumList(gen_cons:nil4_0(n531_0), gen_0':s5_0(b)) ->_IH
gen_0':s5_0(b)

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

(16)
Obligation:
Proved the lower bound n^2 for the following obligation:

TRS:
Rules:
isEmpty(cons(x, xs)) -> false
isEmpty(nil) -> true
isZero(0') -> true
isZero(s(x)) -> false
head(cons(x, xs)) -> x
tail(cons(x, xs)) -> xs
tail(nil) -> nil
p(s(s(x))) -> s(p(s(x)))
p(s(0')) -> 0'
p(0') -> 0'
inc(s(x)) -> s(inc(x))
inc(0') -> s(0')
sumList(xs, y) -> if(isEmpty(xs), isZero(head(xs)), y, tail(xs), cons(p(head(xs)), tail(xs)), inc(y))
if(true, b, y, xs, ys, x) -> y
if(false, true, y, xs, ys, x) -> sumList(xs, y)
if(false, false, y, xs, ys, x) -> sumList(ys, x)
sum(xs) -> sumList(xs, 0')

Types:
isEmpty :: cons:nil -> false:true
cons :: 0':s -> cons:nil -> cons:nil
false :: false:true
nil :: cons:nil
true :: false:true
isZero :: 0':s -> false:true
0' :: 0':s
s :: 0':s -> 0':s
head :: cons:nil -> 0':s
tail :: cons:nil -> cons:nil
p :: 0':s -> 0':s
inc :: 0':s -> 0':s
sumList :: cons:nil -> 0':s -> 0':s
if :: false:true -> false:true -> 0':s -> cons:nil -> cons:nil -> 0':s -> 0':s
sum :: cons:nil -> 0':s
hole_false:true1_0 :: false:true
hole_cons:nil2_0 :: cons:nil
hole_0':s3_0 :: 0':s
gen_cons:nil4_0 :: Nat -> cons:nil
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)
inc(gen_0':s5_0(n264_0)) -> gen_0':s5_0(+(1, n264_0)), rt in Omega(1 + n264_0)


Generator Equations:
gen_cons:nil4_0(0) <=> nil
gen_cons:nil4_0(+(x, 1)) <=> cons(0', gen_cons:nil4_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:
sumList
----------------------------------------

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

(18)
BOUNDS(n^2, INF)