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


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


WORST_CASE(Omega(n^3), ?)
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^3, 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), 270 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), 29 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), 745 ms]
                (20) typed CpxTrs
                (21) RewriteLemmaProof [LOWER BOUND(ID), 6428 ms]
                (22) proven lower bound
                (23) LowerBoundPropagationProof [FINISHED, 0 ms]
                (24) BOUNDS(n^3, INF)


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

(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^3, INF).


The TRS R consists of the following rules:

   fac(0) -> 1
   fac(s(x)) -> *(s(x), fac(x))
   floop(0, y) -> y
   floop(s(x), y) -> floop(x, *(s(x), y))
   *(x, 0) -> 0
   *(x, s(y)) -> +(*(x, y), x)
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))
   1 -> s(0)
   fac(0) -> s(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^3, INF).


The TRS R consists of the following rules:

   fac(0') -> 1'
   fac(s(x)) -> *'(s(x), fac(x))
   floop(0', y) -> y
   floop(s(x), y) -> floop(x, *'(s(x), y))
   *'(x, 0') -> 0'
   *'(x, s(y)) -> +'(*'(x, y), x)
   +'(x, 0') -> x
   +'(x, s(y)) -> s(+'(x, y))
   1' -> s(0')
   fac(0') -> s(0')

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

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

(4)
Obligation:
TRS:
Rules:
fac(0') -> 1'
fac(s(x)) -> *'(s(x), fac(x))
floop(0', y) -> y
floop(s(x), y) -> floop(x, *'(s(x), y))
*'(x, 0') -> 0'
*'(x, s(y)) -> +'(*'(x, y), x)
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
1' -> s(0')
fac(0') -> s(0')

Types:
fac :: 0':s -> 0':s
0' :: 0':s
1' :: 0':s
s :: 0':s -> 0':s
*' :: 0':s -> 0':s -> 0':s
floop :: 0':s -> 0':s -> 0':s
+' :: 0':s -> 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:
fac, *', floop, +'

They will be analysed ascendingly in the following order:
*' < fac
*' < floop
+' < *'

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

(6)
Obligation:
TRS:
Rules:
fac(0') -> 1'
fac(s(x)) -> *'(s(x), fac(x))
floop(0', y) -> y
floop(s(x), y) -> floop(x, *'(s(x), y))
*'(x, 0') -> 0'
*'(x, s(y)) -> +'(*'(x, y), x)
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
1' -> s(0')
fac(0') -> s(0')

Types:
fac :: 0':s -> 0':s
0' :: 0':s
1' :: 0':s
s :: 0':s -> 0':s
*' :: 0':s -> 0':s -> 0':s
floop :: 0':s -> 0':s -> 0':s
+' :: 0':s -> 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:
+', fac, *', floop

They will be analysed ascendingly in the following order:
*' < fac
*' < floop
+' < *'

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
+'(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:
+'(gen_0':s2_0(a), gen_0':s2_0(0)) ->_R^Omega(1)
gen_0':s2_0(a)

Induction Step:
+'(gen_0':s2_0(a), gen_0':s2_0(+(n4_0, 1))) ->_R^Omega(1)
s(+'(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:
fac(0') -> 1'
fac(s(x)) -> *'(s(x), fac(x))
floop(0', y) -> y
floop(s(x), y) -> floop(x, *'(s(x), y))
*'(x, 0') -> 0'
*'(x, s(y)) -> +'(*'(x, y), x)
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
1' -> s(0')
fac(0') -> s(0')

Types:
fac :: 0':s -> 0':s
0' :: 0':s
1' :: 0':s
s :: 0':s -> 0':s
*' :: 0':s -> 0':s -> 0':s
floop :: 0':s -> 0':s -> 0':s
+' :: 0':s -> 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:
+', fac, *', floop

They will be analysed ascendingly in the following order:
*' < fac
*' < floop
+' < *'

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
TRS:
Rules:
fac(0') -> 1'
fac(s(x)) -> *'(s(x), fac(x))
floop(0', y) -> y
floop(s(x), y) -> floop(x, *'(s(x), y))
*'(x, 0') -> 0'
*'(x, s(y)) -> +'(*'(x, y), x)
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
1' -> s(0')
fac(0') -> s(0')

Types:
fac :: 0':s -> 0':s
0' :: 0':s
1' :: 0':s
s :: 0':s -> 0':s
*' :: 0':s -> 0':s -> 0':s
floop :: 0':s -> 0':s -> 0':s
+' :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Lemmas:
+'(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:
*', fac, floop

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

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
*'(gen_0':s2_0(a), gen_0':s2_0(n511_0)) -> gen_0':s2_0(*(n511_0, a)), rt in Omega(1 + a*n511_0 + n511_0)

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

Induction Step:
*'(gen_0':s2_0(a), gen_0':s2_0(+(n511_0, 1))) ->_R^Omega(1)
+'(*'(gen_0':s2_0(a), gen_0':s2_0(n511_0)), gen_0':s2_0(a)) ->_IH
+'(gen_0':s2_0(*(c512_0, a)), gen_0':s2_0(a)) ->_L^Omega(1 + a)
gen_0':s2_0(+(a, *(n511_0, a)))

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:
fac(0') -> 1'
fac(s(x)) -> *'(s(x), fac(x))
floop(0', y) -> y
floop(s(x), y) -> floop(x, *'(s(x), y))
*'(x, 0') -> 0'
*'(x, s(y)) -> +'(*'(x, y), x)
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
1' -> s(0')
fac(0') -> s(0')

Types:
fac :: 0':s -> 0':s
0' :: 0':s
1' :: 0':s
s :: 0':s -> 0':s
*' :: 0':s -> 0':s -> 0':s
floop :: 0':s -> 0':s -> 0':s
+' :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Lemmas:
+'(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:
*', fac, floop

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

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

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

(17)
BOUNDS(n^2, INF)

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

(18)
Obligation:
TRS:
Rules:
fac(0') -> 1'
fac(s(x)) -> *'(s(x), fac(x))
floop(0', y) -> y
floop(s(x), y) -> floop(x, *'(s(x), y))
*'(x, 0') -> 0'
*'(x, s(y)) -> +'(*'(x, y), x)
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
1' -> s(0')
fac(0') -> s(0')

Types:
fac :: 0':s -> 0':s
0' :: 0':s
1' :: 0':s
s :: 0':s -> 0':s
*' :: 0':s -> 0':s -> 0':s
floop :: 0':s -> 0':s -> 0':s
+' :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Lemmas:
+'(gen_0':s2_0(a), gen_0':s2_0(n4_0)) -> gen_0':s2_0(+(n4_0, a)), rt in Omega(1 + n4_0)
*'(gen_0':s2_0(a), gen_0':s2_0(n511_0)) -> gen_0':s2_0(*(n511_0, a)), rt in Omega(1 + a*n511_0 + n511_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, floop
----------------------------------------

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
fac(gen_0':s2_0(+(1, n1127_0))) -> *3_0, rt in Omega(n1127_0)

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

Induction Step:
fac(gen_0':s2_0(+(1, +(n1127_0, 1)))) ->_R^Omega(1)
*'(s(gen_0':s2_0(+(1, n1127_0))), fac(gen_0':s2_0(+(1, n1127_0)))) ->_IH
*'(s(gen_0':s2_0(+(1, n1127_0))), *3_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:
fac(0') -> 1'
fac(s(x)) -> *'(s(x), fac(x))
floop(0', y) -> y
floop(s(x), y) -> floop(x, *'(s(x), y))
*'(x, 0') -> 0'
*'(x, s(y)) -> +'(*'(x, y), x)
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
1' -> s(0')
fac(0') -> s(0')

Types:
fac :: 0':s -> 0':s
0' :: 0':s
1' :: 0':s
s :: 0':s -> 0':s
*' :: 0':s -> 0':s -> 0':s
floop :: 0':s -> 0':s -> 0':s
+' :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Lemmas:
+'(gen_0':s2_0(a), gen_0':s2_0(n4_0)) -> gen_0':s2_0(+(n4_0, a)), rt in Omega(1 + n4_0)
*'(gen_0':s2_0(a), gen_0':s2_0(n511_0)) -> gen_0':s2_0(*(n511_0, a)), rt in Omega(1 + a*n511_0 + n511_0)
fac(gen_0':s2_0(+(1, n1127_0))) -> *3_0, rt in Omega(n1127_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:
floop
----------------------------------------

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
floop(gen_0':s2_0(n2194_0), gen_0':s2_0(b)) -> *3_0, rt in Omega(b*n2194_0 + b*n2194_0^2 + n2194_0)

Induction Base:
floop(gen_0':s2_0(0), gen_0':s2_0(b))

Induction Step:
floop(gen_0':s2_0(+(n2194_0, 1)), gen_0':s2_0(b)) ->_R^Omega(1)
floop(gen_0':s2_0(n2194_0), *'(s(gen_0':s2_0(n2194_0)), gen_0':s2_0(b))) ->_L^Omega(1 + 2*b + b*n2194_0)
floop(gen_0':s2_0(n2194_0), gen_0':s2_0(*(b, +(n2194_0, 1)))) ->_IH
*3_0

We have rt in Omega(n^3) and sz in O(n). Thus, we have irc_R in Omega(n^3).
----------------------------------------

(22)
Obligation:
Proved the lower bound n^3 for the following obligation:

TRS:
Rules:
fac(0') -> 1'
fac(s(x)) -> *'(s(x), fac(x))
floop(0', y) -> y
floop(s(x), y) -> floop(x, *'(s(x), y))
*'(x, 0') -> 0'
*'(x, s(y)) -> +'(*'(x, y), x)
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
1' -> s(0')
fac(0') -> s(0')

Types:
fac :: 0':s -> 0':s
0' :: 0':s
1' :: 0':s
s :: 0':s -> 0':s
*' :: 0':s -> 0':s -> 0':s
floop :: 0':s -> 0':s -> 0':s
+' :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Lemmas:
+'(gen_0':s2_0(a), gen_0':s2_0(n4_0)) -> gen_0':s2_0(+(n4_0, a)), rt in Omega(1 + n4_0)
*'(gen_0':s2_0(a), gen_0':s2_0(n511_0)) -> gen_0':s2_0(*(n511_0, a)), rt in Omega(1 + a*n511_0 + n511_0)
fac(gen_0':s2_0(+(1, n1127_0))) -> *3_0, rt in Omega(n1127_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:
floop
----------------------------------------

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

(24)
BOUNDS(n^3, INF)