/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (innermost) 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), 312 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), 80 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), 735 ms]
                (20) BOUNDS(1, INF)


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(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).


The TRS R consists of the following rules:

   p(s(x)) -> x
   fact(0) -> s(0)
   fact(s(x)) -> *(s(x), fact(p(s(x))))
   *(0, y) -> 0
   *(s(x), y) -> +(*(x, y), y)
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))

S is empty.
Rewrite Strategy: INNERMOST
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(1) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
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(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).


The TRS R consists of the following rules:

   p(s(x)) -> x
   fact(0') -> s(0')
   fact(s(x)) -> *'(s(x), fact(p(s(x))))
   *'(0', y) -> 0'
   *'(s(x), y) -> +'(*'(x, y), y)
   +'(x, 0') -> x
   +'(x, s(y)) -> s(+'(x, y))

S is empty.
Rewrite Strategy: INNERMOST
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(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(4)
Obligation:
Innermost TRS:
Rules:
p(s(x)) -> x
fact(0') -> s(0')
fact(s(x)) -> *'(s(x), fact(p(s(x))))
*'(0', y) -> 0'
*'(s(x), y) -> +'(*'(x, y), y)
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))

Types:
p :: s:0' -> s:0'
s :: s:0' -> s:0'
fact :: s:0' -> s:0'
0' :: s:0'
*' :: s:0' -> s:0' -> s:0'
+' :: s:0' -> s:0' -> s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat -> s:0'

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(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
fact, *', +'

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

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(6)
Obligation:
Innermost TRS:
Rules:
p(s(x)) -> x
fact(0') -> s(0')
fact(s(x)) -> *'(s(x), fact(p(s(x))))
*'(0', y) -> 0'
*'(s(x), y) -> +'(*'(x, y), y)
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))

Types:
p :: s:0' -> s:0'
s :: s:0' -> s:0'
fact :: s:0' -> s:0'
0' :: s:0'
*' :: s:0' -> s:0' -> s:0'
+' :: s:0' -> s:0' -> s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat -> s:0'


Generator Equations:
gen_s:0'2_0(0) <=> 0'
gen_s:0'2_0(+(x, 1)) <=> s(gen_s:0'2_0(x))


The following defined symbols remain to be analysed:
+', fact, *'

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

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(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
+'(gen_s:0'2_0(a), gen_s:0'2_0(n4_0)) -> gen_s:0'2_0(+(n4_0, a)), rt in Omega(1 + n4_0)

Induction Base:
+'(gen_s:0'2_0(a), gen_s:0'2_0(0)) ->_R^Omega(1)
gen_s:0'2_0(a)

Induction Step:
+'(gen_s:0'2_0(a), gen_s:0'2_0(+(n4_0, 1))) ->_R^Omega(1)
s(+'(gen_s:0'2_0(a), gen_s:0'2_0(n4_0))) ->_IH
s(gen_s:0'2_0(+(a, c5_0)))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(8)
Complex Obligation (BEST)

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

Innermost TRS:
Rules:
p(s(x)) -> x
fact(0') -> s(0')
fact(s(x)) -> *'(s(x), fact(p(s(x))))
*'(0', y) -> 0'
*'(s(x), y) -> +'(*'(x, y), y)
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))

Types:
p :: s:0' -> s:0'
s :: s:0' -> s:0'
fact :: s:0' -> s:0'
0' :: s:0'
*' :: s:0' -> s:0' -> s:0'
+' :: s:0' -> s:0' -> s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat -> s:0'


Generator Equations:
gen_s:0'2_0(0) <=> 0'
gen_s:0'2_0(+(x, 1)) <=> s(gen_s:0'2_0(x))


The following defined symbols remain to be analysed:
+', fact, *'

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

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(10) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(11)
BOUNDS(n^1, INF)

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(12)
Obligation:
Innermost TRS:
Rules:
p(s(x)) -> x
fact(0') -> s(0')
fact(s(x)) -> *'(s(x), fact(p(s(x))))
*'(0', y) -> 0'
*'(s(x), y) -> +'(*'(x, y), y)
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))

Types:
p :: s:0' -> s:0'
s :: s:0' -> s:0'
fact :: s:0' -> s:0'
0' :: s:0'
*' :: s:0' -> s:0' -> s:0'
+' :: s:0' -> s:0' -> s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat -> s:0'


Lemmas:
+'(gen_s:0'2_0(a), gen_s:0'2_0(n4_0)) -> gen_s:0'2_0(+(n4_0, a)), rt in Omega(1 + n4_0)


Generator Equations:
gen_s:0'2_0(0) <=> 0'
gen_s:0'2_0(+(x, 1)) <=> s(gen_s:0'2_0(x))


The following defined symbols remain to be analysed:
*', fact

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

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(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
*'(gen_s:0'2_0(n513_0), gen_s:0'2_0(b)) -> gen_s:0'2_0(*(n513_0, b)), rt in Omega(1 + b*n513_0 + n513_0)

Induction Base:
*'(gen_s:0'2_0(0), gen_s:0'2_0(b)) ->_R^Omega(1)
0'

Induction Step:
*'(gen_s:0'2_0(+(n513_0, 1)), gen_s:0'2_0(b)) ->_R^Omega(1)
+'(*'(gen_s:0'2_0(n513_0), gen_s:0'2_0(b)), gen_s:0'2_0(b)) ->_IH
+'(gen_s:0'2_0(*(c514_0, b)), gen_s:0'2_0(b)) ->_L^Omega(1 + b)
gen_s:0'2_0(+(b, *(n513_0, b)))

We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
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(14)
Complex Obligation (BEST)

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(15)
Obligation:
Proved the lower bound n^2 for the following obligation:

Innermost TRS:
Rules:
p(s(x)) -> x
fact(0') -> s(0')
fact(s(x)) -> *'(s(x), fact(p(s(x))))
*'(0', y) -> 0'
*'(s(x), y) -> +'(*'(x, y), y)
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))

Types:
p :: s:0' -> s:0'
s :: s:0' -> s:0'
fact :: s:0' -> s:0'
0' :: s:0'
*' :: s:0' -> s:0' -> s:0'
+' :: s:0' -> s:0' -> s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat -> s:0'


Lemmas:
+'(gen_s:0'2_0(a), gen_s:0'2_0(n4_0)) -> gen_s:0'2_0(+(n4_0, a)), rt in Omega(1 + n4_0)


Generator Equations:
gen_s:0'2_0(0) <=> 0'
gen_s:0'2_0(+(x, 1)) <=> s(gen_s:0'2_0(x))


The following defined symbols remain to be analysed:
*', fact

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

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(16) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(17)
BOUNDS(n^2, INF)

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(18)
Obligation:
Innermost TRS:
Rules:
p(s(x)) -> x
fact(0') -> s(0')
fact(s(x)) -> *'(s(x), fact(p(s(x))))
*'(0', y) -> 0'
*'(s(x), y) -> +'(*'(x, y), y)
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))

Types:
p :: s:0' -> s:0'
s :: s:0' -> s:0'
fact :: s:0' -> s:0'
0' :: s:0'
*' :: s:0' -> s:0' -> s:0'
+' :: s:0' -> s:0' -> s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat -> s:0'


Lemmas:
+'(gen_s:0'2_0(a), gen_s:0'2_0(n4_0)) -> gen_s:0'2_0(+(n4_0, a)), rt in Omega(1 + n4_0)
*'(gen_s:0'2_0(n513_0), gen_s:0'2_0(b)) -> gen_s:0'2_0(*(n513_0, b)), rt in Omega(1 + b*n513_0 + n513_0)


Generator Equations:
gen_s:0'2_0(0) <=> 0'
gen_s:0'2_0(+(x, 1)) <=> s(gen_s:0'2_0(x))


The following defined symbols remain to be analysed:
fact
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(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
fact(gen_s:0'2_0(n1126_0)) -> *3_0, rt in Omega(n1126_0)

Induction Base:
fact(gen_s:0'2_0(0))

Induction Step:
fact(gen_s:0'2_0(+(n1126_0, 1))) ->_R^Omega(1)
*'(s(gen_s:0'2_0(n1126_0)), fact(p(s(gen_s:0'2_0(n1126_0))))) ->_R^Omega(1)
*'(s(gen_s:0'2_0(n1126_0)), fact(gen_s:0'2_0(n1126_0))) ->_IH
*'(s(gen_s:0'2_0(n1126_0)), *3_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(20)
BOUNDS(1, INF)