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


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WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^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), 316 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) typed CpxTrs


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


The TRS R consists of the following rules:

   g(x, 0) -> 0
   g(d, s(x)) -> s(s(g(d, x)))
   g(h, s(0)) -> 0
   g(h, s(s(x))) -> s(g(h, x))
   double(x) -> g(d, x)
   half(x) -> g(h, x)
   f(s(x), y) -> f(half(s(x)), double(y))
   f(s(0), y) -> y
   id(x) -> f(x, s(0))

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.
----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   g(x, 0') -> 0'
   g(d, s(x)) -> s(s(g(d, x)))
   g(h, s(0')) -> 0'
   g(h, s(s(x))) -> s(g(h, x))
   double(x) -> g(d, x)
   half(x) -> g(h, x)
   f(s(x), y) -> f(half(s(x)), double(y))
   f(s(0'), y) -> y
   id(x) -> f(x, s(0'))

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:
g(x, 0') -> 0'
g(d, s(x)) -> s(s(g(d, x)))
g(h, s(0')) -> 0'
g(h, s(s(x))) -> s(g(h, x))
double(x) -> g(d, x)
half(x) -> g(h, x)
f(s(x), y) -> f(half(s(x)), double(y))
f(s(0'), y) -> y
id(x) -> f(x, s(0'))

Types:
g :: d:h -> 0':s -> 0':s
0' :: 0':s
d :: d:h
s :: 0':s -> 0':s
h :: d:h
double :: 0':s -> 0':s
half :: 0':s -> 0':s
f :: 0':s -> 0':s -> 0':s
id :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_d:h2_0 :: d:h
gen_0':s3_0 :: Nat -> 0':s

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(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
g, f
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(6)
Obligation:
Innermost TRS:
Rules:
g(x, 0') -> 0'
g(d, s(x)) -> s(s(g(d, x)))
g(h, s(0')) -> 0'
g(h, s(s(x))) -> s(g(h, x))
double(x) -> g(d, x)
half(x) -> g(h, x)
f(s(x), y) -> f(half(s(x)), double(y))
f(s(0'), y) -> y
id(x) -> f(x, s(0'))

Types:
g :: d:h -> 0':s -> 0':s
0' :: 0':s
d :: d:h
s :: 0':s -> 0':s
h :: d:h
double :: 0':s -> 0':s
half :: 0':s -> 0':s
f :: 0':s -> 0':s -> 0':s
id :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_d:h2_0 :: d:h
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:
g, f
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(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
g(h, gen_0':s3_0(*(2, n5_0))) -> gen_0':s3_0(n5_0), rt in Omega(1 + n5_0)

Induction Base:
g(h, gen_0':s3_0(*(2, 0))) ->_R^Omega(1)
0'

Induction Step:
g(h, gen_0':s3_0(*(2, +(n5_0, 1)))) ->_R^Omega(1)
s(g(h, gen_0':s3_0(*(2, 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:

Innermost TRS:
Rules:
g(x, 0') -> 0'
g(d, s(x)) -> s(s(g(d, x)))
g(h, s(0')) -> 0'
g(h, s(s(x))) -> s(g(h, x))
double(x) -> g(d, x)
half(x) -> g(h, x)
f(s(x), y) -> f(half(s(x)), double(y))
f(s(0'), y) -> y
id(x) -> f(x, s(0'))

Types:
g :: d:h -> 0':s -> 0':s
0' :: 0':s
d :: d:h
s :: 0':s -> 0':s
h :: d:h
double :: 0':s -> 0':s
half :: 0':s -> 0':s
f :: 0':s -> 0':s -> 0':s
id :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_d:h2_0 :: d:h
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:
g, f
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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:
g(x, 0') -> 0'
g(d, s(x)) -> s(s(g(d, x)))
g(h, s(0')) -> 0'
g(h, s(s(x))) -> s(g(h, x))
double(x) -> g(d, x)
half(x) -> g(h, x)
f(s(x), y) -> f(half(s(x)), double(y))
f(s(0'), y) -> y
id(x) -> f(x, s(0'))

Types:
g :: d:h -> 0':s -> 0':s
0' :: 0':s
d :: d:h
s :: 0':s -> 0':s
h :: d:h
double :: 0':s -> 0':s
half :: 0':s -> 0':s
f :: 0':s -> 0':s -> 0':s
id :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_d:h2_0 :: d:h
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
g(h, gen_0':s3_0(*(2, 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:
f