/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^1), ?)
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, 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), 347 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), 92 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 136 ms]
        (16) typed CpxTrs


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(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   min(x, 0) -> 0
   min(0, y) -> 0
   min(s(x), s(y)) -> s(min(x, y))
   max(x, 0) -> x
   max(0, y) -> y
   max(s(x), s(y)) -> s(max(x, y))
   minus(x, 0) -> x
   minus(s(x), s(y)) -> s(minus(x, any(y)))
   gcd(s(x), s(y)) -> gcd(minus(max(x, y), min(x, y)), s(min(x, y)))
   any(s(x)) -> s(s(any(x)))
   any(x) -> 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^1, INF).


The TRS R consists of the following rules:

   min(x, 0') -> 0'
   min(0', y) -> 0'
   min(s(x), s(y)) -> s(min(x, y))
   max(x, 0') -> x
   max(0', y) -> y
   max(s(x), s(y)) -> s(max(x, y))
   minus(x, 0') -> x
   minus(s(x), s(y)) -> s(minus(x, any(y)))
   gcd(s(x), s(y)) -> gcd(minus(max(x, y), min(x, y)), s(min(x, y)))
   any(s(x)) -> s(s(any(x)))
   any(x) -> x

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

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

(4)
Obligation:
TRS:
Rules:
min(x, 0') -> 0'
min(0', y) -> 0'
min(s(x), s(y)) -> s(min(x, y))
max(x, 0') -> x
max(0', y) -> y
max(s(x), s(y)) -> s(max(x, y))
minus(x, 0') -> x
minus(s(x), s(y)) -> s(minus(x, any(y)))
gcd(s(x), s(y)) -> gcd(minus(max(x, y), min(x, y)), s(min(x, y)))
any(s(x)) -> s(s(any(x)))
any(x) -> x

Types:
min :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
max :: 0':s -> 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
any :: 0':s -> 0':s
gcd :: 0':s -> 0':s -> gcd
hole_0':s1_0 :: 0':s
hole_gcd2_0 :: gcd
gen_0':s3_0 :: Nat -> 0':s

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
min, max, minus, any, gcd

They will be analysed ascendingly in the following order:
min < gcd
max < gcd
any < minus
minus < gcd

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

(6)
Obligation:
TRS:
Rules:
min(x, 0') -> 0'
min(0', y) -> 0'
min(s(x), s(y)) -> s(min(x, y))
max(x, 0') -> x
max(0', y) -> y
max(s(x), s(y)) -> s(max(x, y))
minus(x, 0') -> x
minus(s(x), s(y)) -> s(minus(x, any(y)))
gcd(s(x), s(y)) -> gcd(minus(max(x, y), min(x, y)), s(min(x, y)))
any(s(x)) -> s(s(any(x)))
any(x) -> x

Types:
min :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
max :: 0':s -> 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
any :: 0':s -> 0':s
gcd :: 0':s -> 0':s -> gcd
hole_0':s1_0 :: 0':s
hole_gcd2_0 :: gcd
gen_0':s3_0 :: Nat -> 0':s


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
min, max, minus, any, gcd

They will be analysed ascendingly in the following order:
min < gcd
max < gcd
any < minus
minus < gcd

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
min(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(n5_0), rt in Omega(1 + n5_0)

Induction Base:
min(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
0'

Induction Step:
min(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1)
s(min(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0))) ->_IH
s(gen_0':s3_0(c6_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:

TRS:
Rules:
min(x, 0') -> 0'
min(0', y) -> 0'
min(s(x), s(y)) -> s(min(x, y))
max(x, 0') -> x
max(0', y) -> y
max(s(x), s(y)) -> s(max(x, y))
minus(x, 0') -> x
minus(s(x), s(y)) -> s(minus(x, any(y)))
gcd(s(x), s(y)) -> gcd(minus(max(x, y), min(x, y)), s(min(x, y)))
any(s(x)) -> s(s(any(x)))
any(x) -> x

Types:
min :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
max :: 0':s -> 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
any :: 0':s -> 0':s
gcd :: 0':s -> 0':s -> gcd
hole_0':s1_0 :: 0':s
hole_gcd2_0 :: gcd
gen_0':s3_0 :: Nat -> 0':s


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
min, max, minus, any, gcd

They will be analysed ascendingly in the following order:
min < gcd
max < gcd
any < minus
minus < gcd

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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:
TRS:
Rules:
min(x, 0') -> 0'
min(0', y) -> 0'
min(s(x), s(y)) -> s(min(x, y))
max(x, 0') -> x
max(0', y) -> y
max(s(x), s(y)) -> s(max(x, y))
minus(x, 0') -> x
minus(s(x), s(y)) -> s(minus(x, any(y)))
gcd(s(x), s(y)) -> gcd(minus(max(x, y), min(x, y)), s(min(x, y)))
any(s(x)) -> s(s(any(x)))
any(x) -> x

Types:
min :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
max :: 0':s -> 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
any :: 0':s -> 0':s
gcd :: 0':s -> 0':s -> gcd
hole_0':s1_0 :: 0':s
hole_gcd2_0 :: gcd
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
min(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(n5_0), rt in Omega(1 + n5_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
max, minus, any, gcd

They will be analysed ascendingly in the following order:
max < gcd
any < minus
minus < gcd

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(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
max(gen_0':s3_0(n301_0), gen_0':s3_0(n301_0)) -> gen_0':s3_0(n301_0), rt in Omega(1 + n301_0)

Induction Base:
max(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
gen_0':s3_0(0)

Induction Step:
max(gen_0':s3_0(+(n301_0, 1)), gen_0':s3_0(+(n301_0, 1))) ->_R^Omega(1)
s(max(gen_0':s3_0(n301_0), gen_0':s3_0(n301_0))) ->_IH
s(gen_0':s3_0(c302_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:
min(x, 0') -> 0'
min(0', y) -> 0'
min(s(x), s(y)) -> s(min(x, y))
max(x, 0') -> x
max(0', y) -> y
max(s(x), s(y)) -> s(max(x, y))
minus(x, 0') -> x
minus(s(x), s(y)) -> s(minus(x, any(y)))
gcd(s(x), s(y)) -> gcd(minus(max(x, y), min(x, y)), s(min(x, y)))
any(s(x)) -> s(s(any(x)))
any(x) -> x

Types:
min :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
max :: 0':s -> 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
any :: 0':s -> 0':s
gcd :: 0':s -> 0':s -> gcd
hole_0':s1_0 :: 0':s
hole_gcd2_0 :: gcd
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
min(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(n5_0), rt in Omega(1 + n5_0)
max(gen_0':s3_0(n301_0), gen_0':s3_0(n301_0)) -> gen_0':s3_0(n301_0), rt in Omega(1 + n301_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
any, minus, gcd

They will be analysed ascendingly in the following order:
any < minus
minus < gcd

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(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
any(gen_0':s3_0(+(1, n677_0))) -> *4_0, rt in Omega(n677_0)

Induction Base:
any(gen_0':s3_0(+(1, 0)))

Induction Step:
any(gen_0':s3_0(+(1, +(n677_0, 1)))) ->_R^Omega(1)
s(s(any(gen_0':s3_0(+(1, n677_0))))) ->_IH
s(s(*4_0))

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

(16)
Obligation:
TRS:
Rules:
min(x, 0') -> 0'
min(0', y) -> 0'
min(s(x), s(y)) -> s(min(x, y))
max(x, 0') -> x
max(0', y) -> y
max(s(x), s(y)) -> s(max(x, y))
minus(x, 0') -> x
minus(s(x), s(y)) -> s(minus(x, any(y)))
gcd(s(x), s(y)) -> gcd(minus(max(x, y), min(x, y)), s(min(x, y)))
any(s(x)) -> s(s(any(x)))
any(x) -> x

Types:
min :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
max :: 0':s -> 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
any :: 0':s -> 0':s
gcd :: 0':s -> 0':s -> gcd
hole_0':s1_0 :: 0':s
hole_gcd2_0 :: gcd
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
min(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(n5_0), rt in Omega(1 + n5_0)
max(gen_0':s3_0(n301_0), gen_0':s3_0(n301_0)) -> gen_0':s3_0(n301_0), rt in Omega(1 + n301_0)
any(gen_0':s3_0(+(1, n677_0))) -> *4_0, rt in Omega(n677_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
minus, gcd

They will be analysed ascendingly in the following order:
minus < gcd