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


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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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 246 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), 74 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), 24 ms]
                (20) typed CpxTrs
                (21) RewriteLemmaProof [LOWER BOUND(ID), 2328 ms]
                (22) 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:

   plus(x, 0) -> x
   plus(x, s(y)) -> s(plus(x, y))
   times(0, y) -> 0
   times(x, 0) -> 0
   times(s(x), y) -> plus(times(x, y), y)
   p(s(s(x))) -> s(p(s(x)))
   p(s(0)) -> 0
   fac(s(x)) -> times(fac(p(s(x))), s(x))

S is empty.
Rewrite Strategy: FULL
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(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:

   plus(x, 0') -> x
   plus(x, s(y)) -> s(plus(x, y))
   times(0', y) -> 0'
   times(x, 0') -> 0'
   times(s(x), y) -> plus(times(x, y), y)
   p(s(s(x))) -> s(p(s(x)))
   p(s(0')) -> 0'
   fac(s(x)) -> times(fac(p(s(x))), s(x))

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

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

(4)
Obligation:
TRS:
Rules:
plus(x, 0') -> x
plus(x, s(y)) -> s(plus(x, y))
times(0', y) -> 0'
times(x, 0') -> 0'
times(s(x), y) -> plus(times(x, y), y)
p(s(s(x))) -> s(p(s(x)))
p(s(0')) -> 0'
fac(s(x)) -> times(fac(p(s(x))), s(x))

Types:
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
fac :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
plus, times, p, fac

They will be analysed ascendingly in the following order:
plus < times
times < fac
p < fac

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

(6)
Obligation:
TRS:
Rules:
plus(x, 0') -> x
plus(x, s(y)) -> s(plus(x, y))
times(0', y) -> 0'
times(x, 0') -> 0'
times(s(x), y) -> plus(times(x, y), y)
p(s(s(x))) -> s(p(s(x)))
p(s(0')) -> 0'
fac(s(x)) -> times(fac(p(s(x))), s(x))

Types:
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
fac :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
plus, times, p, fac

They will be analysed ascendingly in the following order:
plus < times
times < fac
p < fac

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_0':s2_0(a), gen_0':s2_0(n4_0)) -> gen_0':s2_0(+(n4_0, a)), rt in Omega(1 + n4_0)

Induction Base:
plus(gen_0':s2_0(a), gen_0':s2_0(0)) ->_R^Omega(1)
gen_0':s2_0(a)

Induction Step:
plus(gen_0':s2_0(a), gen_0':s2_0(+(n4_0, 1))) ->_R^Omega(1)
s(plus(gen_0':s2_0(a), gen_0':s2_0(n4_0))) ->_IH
s(gen_0':s2_0(+(a, c5_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:
plus(x, 0') -> x
plus(x, s(y)) -> s(plus(x, y))
times(0', y) -> 0'
times(x, 0') -> 0'
times(s(x), y) -> plus(times(x, y), y)
p(s(s(x))) -> s(p(s(x)))
p(s(0')) -> 0'
fac(s(x)) -> times(fac(p(s(x))), s(x))

Types:
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
fac :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
plus, times, p, fac

They will be analysed ascendingly in the following order:
plus < times
times < fac
p < fac

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
TRS:
Rules:
plus(x, 0') -> x
plus(x, s(y)) -> s(plus(x, y))
times(0', y) -> 0'
times(x, 0') -> 0'
times(s(x), y) -> plus(times(x, y), y)
p(s(s(x))) -> s(p(s(x)))
p(s(0')) -> 0'
fac(s(x)) -> times(fac(p(s(x))), s(x))

Types:
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
fac :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Lemmas:
plus(gen_0':s2_0(a), gen_0':s2_0(n4_0)) -> gen_0':s2_0(+(n4_0, a)), rt in Omega(1 + n4_0)


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
times, p, fac

They will be analysed ascendingly in the following order:
times < fac
p < fac

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
times(gen_0':s2_0(n475_0), gen_0':s2_0(b)) -> gen_0':s2_0(*(n475_0, b)), rt in Omega(1 + b*n475_0 + n475_0)

Induction Base:
times(gen_0':s2_0(0), gen_0':s2_0(b)) ->_R^Omega(1)
0'

Induction Step:
times(gen_0':s2_0(+(n475_0, 1)), gen_0':s2_0(b)) ->_R^Omega(1)
plus(times(gen_0':s2_0(n475_0), gen_0':s2_0(b)), gen_0':s2_0(b)) ->_IH
plus(gen_0':s2_0(*(c476_0, b)), gen_0':s2_0(b)) ->_L^Omega(1 + b)
gen_0':s2_0(+(b, *(n475_0, b)))

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:
plus(x, 0') -> x
plus(x, s(y)) -> s(plus(x, y))
times(0', y) -> 0'
times(x, 0') -> 0'
times(s(x), y) -> plus(times(x, y), y)
p(s(s(x))) -> s(p(s(x)))
p(s(0')) -> 0'
fac(s(x)) -> times(fac(p(s(x))), s(x))

Types:
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
fac :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Lemmas:
plus(gen_0':s2_0(a), gen_0':s2_0(n4_0)) -> gen_0':s2_0(+(n4_0, a)), rt in Omega(1 + n4_0)


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
times, p, fac

They will be analysed ascendingly in the following order:
times < fac
p < fac

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

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

(17)
BOUNDS(n^2, INF)

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

(18)
Obligation:
TRS:
Rules:
plus(x, 0') -> x
plus(x, s(y)) -> s(plus(x, y))
times(0', y) -> 0'
times(x, 0') -> 0'
times(s(x), y) -> plus(times(x, y), y)
p(s(s(x))) -> s(p(s(x)))
p(s(0')) -> 0'
fac(s(x)) -> times(fac(p(s(x))), s(x))

Types:
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
fac :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Lemmas:
plus(gen_0':s2_0(a), gen_0':s2_0(n4_0)) -> gen_0':s2_0(+(n4_0, a)), rt in Omega(1 + n4_0)
times(gen_0':s2_0(n475_0), gen_0':s2_0(b)) -> gen_0':s2_0(*(n475_0, b)), rt in Omega(1 + b*n475_0 + n475_0)


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
p, fac

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

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
p(gen_0':s2_0(+(1, n1092_0))) -> gen_0':s2_0(n1092_0), rt in Omega(1 + n1092_0)

Induction Base:
p(gen_0':s2_0(+(1, 0))) ->_R^Omega(1)
0'

Induction Step:
p(gen_0':s2_0(+(1, +(n1092_0, 1)))) ->_R^Omega(1)
s(p(s(gen_0':s2_0(n1092_0)))) ->_IH
s(gen_0':s2_0(c1093_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:
plus(x, 0') -> x
plus(x, s(y)) -> s(plus(x, y))
times(0', y) -> 0'
times(x, 0') -> 0'
times(s(x), y) -> plus(times(x, y), y)
p(s(s(x))) -> s(p(s(x)))
p(s(0')) -> 0'
fac(s(x)) -> times(fac(p(s(x))), s(x))

Types:
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
fac :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Lemmas:
plus(gen_0':s2_0(a), gen_0':s2_0(n4_0)) -> gen_0':s2_0(+(n4_0, a)), rt in Omega(1 + n4_0)
times(gen_0':s2_0(n475_0), gen_0':s2_0(b)) -> gen_0':s2_0(*(n475_0, b)), rt in Omega(1 + b*n475_0 + n475_0)
p(gen_0':s2_0(+(1, n1092_0))) -> gen_0':s2_0(n1092_0), rt in Omega(1 + n1092_0)


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
fac
----------------------------------------

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
fac(gen_0':s2_0(+(1, n1276_0))) -> *3_0, rt in Omega(n1276_0 + n1276_0^2)

Induction Base:
fac(gen_0':s2_0(+(1, 0)))

Induction Step:
fac(gen_0':s2_0(+(1, +(n1276_0, 1)))) ->_R^Omega(1)
times(fac(p(s(gen_0':s2_0(+(1, n1276_0))))), s(gen_0':s2_0(+(1, n1276_0)))) ->_L^Omega(2 + n1276_0)
times(fac(gen_0':s2_0(+(1, n1276_0))), s(gen_0':s2_0(+(1, n1276_0)))) ->_IH
times(*3_0, s(gen_0':s2_0(+(1, n1276_0))))

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

(22)
BOUNDS(1, INF)