/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^4), ?)
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^4, 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), 249 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), 23 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), 21 ms]
                (20) BEST
                    (21) proven lower bound
                        (22) LowerBoundPropagationProof [FINISHED, 0 ms]
                        (23) BOUNDS(n^3, INF)
                    (24) typed CpxTrs
                        (25) RewriteLemmaProof [LOWER BOUND(ID), 1113 ms]
                        (26) proven lower bound
                        (27) LowerBoundPropagationProof [FINISHED, 0 ms]
                        (28) BOUNDS(n^4, INF)


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

(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^4, INF).


The TRS R consists of the following rules:

   plus(0, x) -> x
   plus(s(x), y) -> s(plus(p(s(x)), y))
   times(0, y) -> 0
   times(s(x), y) -> plus(y, times(p(s(x)), y))
   p(s(0)) -> 0
   p(s(s(x))) -> s(p(s(x)))
   fac(0, x) -> x
   fac(s(x), y) -> fac(p(s(x)), times(s(x), y))
   factorial(x) -> fac(x, 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^4, INF).


The TRS R consists of the following rules:

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

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

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

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

Types:
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
p :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
fac :: 0':s -> 0':s -> 0':s
factorial :: 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, p, times, fac

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

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

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

Types:
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
p :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
fac :: 0':s -> 0':s -> 0':s
factorial :: 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:
p, plus, times, fac

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

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

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

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

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

Types:
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
p :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
fac :: 0':s -> 0':s -> 0':s
factorial :: 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:
p, plus, times, fac

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

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

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

(11)
BOUNDS(n^1, INF)

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

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

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


Lemmas:
p(gen_0':s2_0(+(1, n4_0))) -> gen_0':s2_0(n4_0), 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:
plus, times, fac

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

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_0':s2_0(n193_0), gen_0':s2_0(b)) -> gen_0':s2_0(+(n193_0, b)), rt in Omega(1 + n193_0 + n193_0^2)

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

Induction Step:
plus(gen_0':s2_0(+(n193_0, 1)), gen_0':s2_0(b)) ->_R^Omega(1)
s(plus(p(s(gen_0':s2_0(n193_0))), gen_0':s2_0(b))) ->_L^Omega(1 + n193_0)
s(plus(gen_0':s2_0(n193_0), gen_0':s2_0(b))) ->_IH
s(gen_0':s2_0(+(b, c194_0)))

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

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


Lemmas:
p(gen_0':s2_0(+(1, n4_0))) -> gen_0':s2_0(n4_0), 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:
plus, times, fac

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

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

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

(17)
BOUNDS(n^2, INF)

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

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

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


Lemmas:
p(gen_0':s2_0(+(1, n4_0))) -> gen_0':s2_0(n4_0), rt in Omega(1 + n4_0)
plus(gen_0':s2_0(n193_0), gen_0':s2_0(b)) -> gen_0':s2_0(+(n193_0, b)), rt in Omega(1 + n193_0 + n193_0^2)


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, fac

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

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
times(gen_0':s2_0(n578_0), gen_0':s2_0(b)) -> gen_0':s2_0(*(n578_0, b)), rt in Omega(1 + b*n578_0 + b^2*n578_0 + n578_0 + n578_0^2)

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

Induction Step:
times(gen_0':s2_0(+(n578_0, 1)), gen_0':s2_0(b)) ->_R^Omega(1)
plus(gen_0':s2_0(b), times(p(s(gen_0':s2_0(n578_0))), gen_0':s2_0(b))) ->_L^Omega(1 + n578_0)
plus(gen_0':s2_0(b), times(gen_0':s2_0(n578_0), gen_0':s2_0(b))) ->_IH
plus(gen_0':s2_0(b), gen_0':s2_0(*(c579_0, b))) ->_L^Omega(1 + b + b^2)
gen_0':s2_0(+(b, *(n578_0, b)))

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

(20)
Complex Obligation (BEST)

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

(21)
Obligation:
Proved the lower bound n^3 for the following obligation:

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

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


Lemmas:
p(gen_0':s2_0(+(1, n4_0))) -> gen_0':s2_0(n4_0), rt in Omega(1 + n4_0)
plus(gen_0':s2_0(n193_0), gen_0':s2_0(b)) -> gen_0':s2_0(+(n193_0, b)), rt in Omega(1 + n193_0 + n193_0^2)


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, fac

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

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

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

(23)
BOUNDS(n^3, INF)

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

(24)
Obligation:
TRS:
Rules:
plus(0', x) -> x
plus(s(x), y) -> s(plus(p(s(x)), y))
times(0', y) -> 0'
times(s(x), y) -> plus(y, times(p(s(x)), y))
p(s(0')) -> 0'
p(s(s(x))) -> s(p(s(x)))
fac(0', x) -> x
fac(s(x), y) -> fac(p(s(x)), times(s(x), y))
factorial(x) -> fac(x, s(0'))

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


Lemmas:
p(gen_0':s2_0(+(1, n4_0))) -> gen_0':s2_0(n4_0), rt in Omega(1 + n4_0)
plus(gen_0':s2_0(n193_0), gen_0':s2_0(b)) -> gen_0':s2_0(+(n193_0, b)), rt in Omega(1 + n193_0 + n193_0^2)
times(gen_0':s2_0(n578_0), gen_0':s2_0(b)) -> gen_0':s2_0(*(n578_0, b)), rt in Omega(1 + b*n578_0 + b^2*n578_0 + n578_0 + n578_0^2)


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

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
fac(gen_0':s2_0(n1122_0), gen_0':s2_0(b)) -> *3_0, rt in Omega(b*n1122_0 + b*n1122_0^2 + b^2*n1122_0 + b^2*n1122_0^2 + n1122_0 + n1122_0^2 + n1122_0^3)

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

Induction Step:
fac(gen_0':s2_0(+(n1122_0, 1)), gen_0':s2_0(b)) ->_R^Omega(1)
fac(p(s(gen_0':s2_0(n1122_0))), times(s(gen_0':s2_0(n1122_0)), gen_0':s2_0(b))) ->_L^Omega(1 + n1122_0)
fac(gen_0':s2_0(n1122_0), times(s(gen_0':s2_0(n1122_0)), gen_0':s2_0(b))) ->_L^Omega(3 + b + b*n1122_0 + b^2 + b^2*n1122_0 + 3*n1122_0 + n1122_0^2)
fac(gen_0':s2_0(n1122_0), gen_0':s2_0(*(+(n1122_0, 1), b))) ->_IH
*3_0

We have rt in Omega(n^4) and sz in O(n). Thus, we have irc_R in Omega(n^4).
----------------------------------------

(26)
Obligation:
Proved the lower bound n^4 for the following obligation:

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

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


Lemmas:
p(gen_0':s2_0(+(1, n4_0))) -> gen_0':s2_0(n4_0), rt in Omega(1 + n4_0)
plus(gen_0':s2_0(n193_0), gen_0':s2_0(b)) -> gen_0':s2_0(+(n193_0, b)), rt in Omega(1 + n193_0 + n193_0^2)
times(gen_0':s2_0(n578_0), gen_0':s2_0(b)) -> gen_0':s2_0(*(n578_0, b)), rt in Omega(1 + b*n578_0 + b^2*n578_0 + n578_0 + n578_0^2)


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

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

(28)
BOUNDS(n^4, INF)