/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 271 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), 46 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 156 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:

   top(free(x)) -> top(check(new(x)))
   new(free(x)) -> free(new(x))
   old(free(x)) -> free(old(x))
   new(serve) -> free(serve)
   old(serve) -> free(serve)
   check(free(x)) -> free(check(x))
   check(new(x)) -> new(check(x))
   check(old(x)) -> old(check(x))
   check(old(x)) -> old(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:

   top(free(x)) -> top(check(new(x)))
   new(free(x)) -> free(new(x))
   old(free(x)) -> free(old(x))
   new(serve) -> free(serve)
   old(serve) -> free(serve)
   check(free(x)) -> free(check(x))
   check(new(x)) -> new(check(x))
   check(old(x)) -> old(check(x))
   check(old(x)) -> old(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:
top(free(x)) -> top(check(new(x)))
new(free(x)) -> free(new(x))
old(free(x)) -> free(old(x))
new(serve) -> free(serve)
old(serve) -> free(serve)
check(free(x)) -> free(check(x))
check(new(x)) -> new(check(x))
check(old(x)) -> old(check(x))
check(old(x)) -> old(x)

Types:
top :: free:serve -> top
free :: free:serve -> free:serve
check :: free:serve -> free:serve
new :: free:serve -> free:serve
old :: free:serve -> free:serve
serve :: free:serve
hole_top1_0 :: top
hole_free:serve2_0 :: free:serve
gen_free:serve3_0 :: Nat -> free:serve

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(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
top, check, new, old

They will be analysed ascendingly in the following order:
check < top
new < top
new < check
old < check

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(6)
Obligation:
TRS:
Rules:
top(free(x)) -> top(check(new(x)))
new(free(x)) -> free(new(x))
old(free(x)) -> free(old(x))
new(serve) -> free(serve)
old(serve) -> free(serve)
check(free(x)) -> free(check(x))
check(new(x)) -> new(check(x))
check(old(x)) -> old(check(x))
check(old(x)) -> old(x)

Types:
top :: free:serve -> top
free :: free:serve -> free:serve
check :: free:serve -> free:serve
new :: free:serve -> free:serve
old :: free:serve -> free:serve
serve :: free:serve
hole_top1_0 :: top
hole_free:serve2_0 :: free:serve
gen_free:serve3_0 :: Nat -> free:serve


Generator Equations:
gen_free:serve3_0(0) <=> serve
gen_free:serve3_0(+(x, 1)) <=> free(gen_free:serve3_0(x))


The following defined symbols remain to be analysed:
new, top, check, old

They will be analysed ascendingly in the following order:
check < top
new < top
new < check
old < check

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(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
new(gen_free:serve3_0(n5_0)) -> gen_free:serve3_0(+(1, n5_0)), rt in Omega(1 + n5_0)

Induction Base:
new(gen_free:serve3_0(0)) ->_R^Omega(1)
free(serve)

Induction Step:
new(gen_free:serve3_0(+(n5_0, 1))) ->_R^Omega(1)
free(new(gen_free:serve3_0(n5_0))) ->_IH
free(gen_free:serve3_0(+(1, 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:
top(free(x)) -> top(check(new(x)))
new(free(x)) -> free(new(x))
old(free(x)) -> free(old(x))
new(serve) -> free(serve)
old(serve) -> free(serve)
check(free(x)) -> free(check(x))
check(new(x)) -> new(check(x))
check(old(x)) -> old(check(x))
check(old(x)) -> old(x)

Types:
top :: free:serve -> top
free :: free:serve -> free:serve
check :: free:serve -> free:serve
new :: free:serve -> free:serve
old :: free:serve -> free:serve
serve :: free:serve
hole_top1_0 :: top
hole_free:serve2_0 :: free:serve
gen_free:serve3_0 :: Nat -> free:serve


Generator Equations:
gen_free:serve3_0(0) <=> serve
gen_free:serve3_0(+(x, 1)) <=> free(gen_free:serve3_0(x))


The following defined symbols remain to be analysed:
new, top, check, old

They will be analysed ascendingly in the following order:
check < top
new < top
new < check
old < check

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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:
top(free(x)) -> top(check(new(x)))
new(free(x)) -> free(new(x))
old(free(x)) -> free(old(x))
new(serve) -> free(serve)
old(serve) -> free(serve)
check(free(x)) -> free(check(x))
check(new(x)) -> new(check(x))
check(old(x)) -> old(check(x))
check(old(x)) -> old(x)

Types:
top :: free:serve -> top
free :: free:serve -> free:serve
check :: free:serve -> free:serve
new :: free:serve -> free:serve
old :: free:serve -> free:serve
serve :: free:serve
hole_top1_0 :: top
hole_free:serve2_0 :: free:serve
gen_free:serve3_0 :: Nat -> free:serve


Lemmas:
new(gen_free:serve3_0(n5_0)) -> gen_free:serve3_0(+(1, n5_0)), rt in Omega(1 + n5_0)


Generator Equations:
gen_free:serve3_0(0) <=> serve
gen_free:serve3_0(+(x, 1)) <=> free(gen_free:serve3_0(x))


The following defined symbols remain to be analysed:
old, top, check

They will be analysed ascendingly in the following order:
check < top
old < check

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(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
old(gen_free:serve3_0(n194_0)) -> gen_free:serve3_0(+(1, n194_0)), rt in Omega(1 + n194_0)

Induction Base:
old(gen_free:serve3_0(0)) ->_R^Omega(1)
free(serve)

Induction Step:
old(gen_free:serve3_0(+(n194_0, 1))) ->_R^Omega(1)
free(old(gen_free:serve3_0(n194_0))) ->_IH
free(gen_free:serve3_0(+(1, c195_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:
top(free(x)) -> top(check(new(x)))
new(free(x)) -> free(new(x))
old(free(x)) -> free(old(x))
new(serve) -> free(serve)
old(serve) -> free(serve)
check(free(x)) -> free(check(x))
check(new(x)) -> new(check(x))
check(old(x)) -> old(check(x))
check(old(x)) -> old(x)

Types:
top :: free:serve -> top
free :: free:serve -> free:serve
check :: free:serve -> free:serve
new :: free:serve -> free:serve
old :: free:serve -> free:serve
serve :: free:serve
hole_top1_0 :: top
hole_free:serve2_0 :: free:serve
gen_free:serve3_0 :: Nat -> free:serve


Lemmas:
new(gen_free:serve3_0(n5_0)) -> gen_free:serve3_0(+(1, n5_0)), rt in Omega(1 + n5_0)
old(gen_free:serve3_0(n194_0)) -> gen_free:serve3_0(+(1, n194_0)), rt in Omega(1 + n194_0)


Generator Equations:
gen_free:serve3_0(0) <=> serve
gen_free:serve3_0(+(x, 1)) <=> free(gen_free:serve3_0(x))


The following defined symbols remain to be analysed:
check, top

They will be analysed ascendingly in the following order:
check < top

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(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
check(gen_free:serve3_0(+(1, n387_0))) -> *4_0, rt in Omega(n387_0)

Induction Base:
check(gen_free:serve3_0(+(1, 0)))

Induction Step:
check(gen_free:serve3_0(+(1, +(n387_0, 1)))) ->_R^Omega(1)
free(check(gen_free:serve3_0(+(1, n387_0)))) ->_IH
free(*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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(16)
Obligation:
TRS:
Rules:
top(free(x)) -> top(check(new(x)))
new(free(x)) -> free(new(x))
old(free(x)) -> free(old(x))
new(serve) -> free(serve)
old(serve) -> free(serve)
check(free(x)) -> free(check(x))
check(new(x)) -> new(check(x))
check(old(x)) -> old(check(x))
check(old(x)) -> old(x)

Types:
top :: free:serve -> top
free :: free:serve -> free:serve
check :: free:serve -> free:serve
new :: free:serve -> free:serve
old :: free:serve -> free:serve
serve :: free:serve
hole_top1_0 :: top
hole_free:serve2_0 :: free:serve
gen_free:serve3_0 :: Nat -> free:serve


Lemmas:
new(gen_free:serve3_0(n5_0)) -> gen_free:serve3_0(+(1, n5_0)), rt in Omega(1 + n5_0)
old(gen_free:serve3_0(n194_0)) -> gen_free:serve3_0(+(1, n194_0)), rt in Omega(1 + n194_0)
check(gen_free:serve3_0(+(1, n387_0))) -> *4_0, rt in Omega(n387_0)


Generator Equations:
gen_free:serve3_0(0) <=> serve
gen_free:serve3_0(+(x, 1)) <=> free(gen_free:serve3_0(x))


The following defined symbols remain to be analysed:
top