/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 281 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), 651 ms]
        (14) 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:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   pred(s(x)) -> x
   minus(x, 0) -> x
   minus(x, s(y)) -> pred(minus(x, y))
   gcd(0, y) -> y
   gcd(s(x), 0) -> s(x)
   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
   if_gcd(true, x, y) -> gcd(minus(x, y), y)
   if_gcd(false, x, y) -> gcd(minus(y, 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.
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(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:

   le(0', y) -> true
   le(s(x), 0') -> false
   le(s(x), s(y)) -> le(x, y)
   pred(s(x)) -> x
   minus(x, 0') -> x
   minus(x, s(y)) -> pred(minus(x, y))
   gcd(0', y) -> y
   gcd(s(x), 0') -> s(x)
   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
   if_gcd(true, x, y) -> gcd(minus(x, y), y)
   if_gcd(false, x, y) -> gcd(minus(y, x), x)

S is empty.
Rewrite Strategy: FULL
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(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(4)
Obligation:
TRS:
Rules:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
pred(s(x)) -> x
minus(x, 0') -> x
minus(x, s(y)) -> pred(minus(x, y))
gcd(0', y) -> y
gcd(s(x), 0') -> s(x)
gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
if_gcd(true, x, y) -> gcd(minus(x, y), y)
if_gcd(false, x, y) -> gcd(minus(y, x), x)

Types:
le :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
pred :: 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
gcd :: 0':s -> 0':s -> 0':s
if_gcd :: true:false -> 0':s -> 0':s -> 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat -> 0':s

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

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

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(6)
Obligation:
TRS:
Rules:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
pred(s(x)) -> x
minus(x, 0') -> x
minus(x, s(y)) -> pred(minus(x, y))
gcd(0', y) -> y
gcd(s(x), 0') -> s(x)
gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
if_gcd(true, x, y) -> gcd(minus(x, y), y)
if_gcd(false, x, y) -> gcd(minus(y, x), x)

Types:
le :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
pred :: 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
gcd :: 0':s -> 0':s -> 0':s
if_gcd :: true:false -> 0':s -> 0':s -> 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
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:
le, minus, gcd

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

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

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

Induction Step:
le(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1)
le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) ->_IH
true

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:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
pred(s(x)) -> x
minus(x, 0') -> x
minus(x, s(y)) -> pred(minus(x, y))
gcd(0', y) -> y
gcd(s(x), 0') -> s(x)
gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
if_gcd(true, x, y) -> gcd(minus(x, y), y)
if_gcd(false, x, y) -> gcd(minus(y, x), x)

Types:
le :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
pred :: 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
gcd :: 0':s -> 0':s -> 0':s
if_gcd :: true:false -> 0':s -> 0':s -> 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
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:
le, minus, gcd

They will be analysed ascendingly in the following order:
le < gcd
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:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
pred(s(x)) -> x
minus(x, 0') -> x
minus(x, s(y)) -> pred(minus(x, y))
gcd(0', y) -> y
gcd(s(x), 0') -> s(x)
gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
if_gcd(true, x, y) -> gcd(minus(x, y), y)
if_gcd(false, x, y) -> gcd(minus(y, x), x)

Types:
le :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
pred :: 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
gcd :: 0':s -> 0':s -> 0':s
if_gcd :: true:false -> 0':s -> 0':s -> 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, 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:
minus, gcd

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

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(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_0':s3_0(a), gen_0':s3_0(+(1, n258_0))) -> *4_0, rt in Omega(n258_0)

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

Induction Step:
minus(gen_0':s3_0(a), gen_0':s3_0(+(1, +(n258_0, 1)))) ->_R^Omega(1)
pred(minus(gen_0':s3_0(a), gen_0':s3_0(+(1, n258_0)))) ->_IH
pred(*4_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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(14)
Obligation:
TRS:
Rules:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
pred(s(x)) -> x
minus(x, 0') -> x
minus(x, s(y)) -> pred(minus(x, y))
gcd(0', y) -> y
gcd(s(x), 0') -> s(x)
gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
if_gcd(true, x, y) -> gcd(minus(x, y), y)
if_gcd(false, x, y) -> gcd(minus(y, x), x)

Types:
le :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
pred :: 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
gcd :: 0':s -> 0':s -> 0':s
if_gcd :: true:false -> 0':s -> 0':s -> 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, rt in Omega(1 + n5_0)
minus(gen_0':s3_0(a), gen_0':s3_0(+(1, n258_0))) -> *4_0, rt in Omega(n258_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:
gcd