/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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 (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), 441 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), 249 ms]
        (14) BOUNDS(1, INF)


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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:

   plus(s(s(x)), y) -> s(plus(x, s(y)))
   plus(x, s(s(y))) -> s(plus(s(x), y))
   plus(s(0), y) -> s(y)
   plus(0, y) -> y
   ack(0, y) -> s(y)
   ack(s(x), 0) -> ack(x, s(0))
   ack(s(x), s(y)) -> ack(x, plus(y, ack(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^1, INF).


The TRS R consists of the following rules:

   plus(s(s(x)), y) -> s(plus(x, s(y)))
   plus(x, s(s(y))) -> s(plus(s(x), y))
   plus(s(0'), y) -> s(y)
   plus(0', y) -> y
   ack(0', y) -> s(y)
   ack(s(x), 0') -> ack(x, s(0'))
   ack(s(x), s(y)) -> ack(x, plus(y, ack(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:
plus(s(s(x)), y) -> s(plus(x, s(y)))
plus(x, s(s(y))) -> s(plus(s(x), y))
plus(s(0'), y) -> s(y)
plus(0', y) -> y
ack(0', y) -> s(y)
ack(s(x), 0') -> ack(x, s(0'))
ack(s(x), s(y)) -> ack(x, plus(y, ack(s(x), y)))

Types:
plus :: s:0' -> s:0' -> s:0'
s :: s:0' -> s:0'
0' :: s:0'
ack :: 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:
plus, ack

They will be analysed ascendingly in the following order:
plus < ack

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(6)
Obligation:
Innermost TRS:
Rules:
plus(s(s(x)), y) -> s(plus(x, s(y)))
plus(x, s(s(y))) -> s(plus(s(x), y))
plus(s(0'), y) -> s(y)
plus(0', y) -> y
ack(0', y) -> s(y)
ack(s(x), 0') -> ack(x, s(0'))
ack(s(x), s(y)) -> ack(x, plus(y, ack(s(x), y)))

Types:
plus :: s:0' -> s:0' -> s:0'
s :: s:0' -> s:0'
0' :: s:0'
ack :: 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:
plus, ack

They will be analysed ascendingly in the following order:
plus < ack

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

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

Induction Step:
plus(gen_s:0'2_0(+(1, *(2, +(n4_0, 1)))), gen_s:0'2_0(b)) ->_R^Omega(1)
s(plus(gen_s:0'2_0(+(1, *(2, n4_0))), s(gen_s:0'2_0(b)))) ->_IH
s(gen_s:0'2_0(+(+(1, +(b, 1)), *(2, 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:
plus(s(s(x)), y) -> s(plus(x, s(y)))
plus(x, s(s(y))) -> s(plus(s(x), y))
plus(s(0'), y) -> s(y)
plus(0', y) -> y
ack(0', y) -> s(y)
ack(s(x), 0') -> ack(x, s(0'))
ack(s(x), s(y)) -> ack(x, plus(y, ack(s(x), y)))

Types:
plus :: s:0' -> s:0' -> s:0'
s :: s:0' -> s:0'
0' :: s:0'
ack :: 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:
plus, ack

They will be analysed ascendingly in the following order:
plus < ack

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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:
plus(s(s(x)), y) -> s(plus(x, s(y)))
plus(x, s(s(y))) -> s(plus(s(x), y))
plus(s(0'), y) -> s(y)
plus(0', y) -> y
ack(0', y) -> s(y)
ack(s(x), 0') -> ack(x, s(0'))
ack(s(x), s(y)) -> ack(x, plus(y, ack(s(x), y)))

Types:
plus :: s:0' -> s:0' -> s:0'
s :: s:0' -> s:0'
0' :: s:0'
ack :: s:0' -> s:0' -> s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat -> s:0'


Lemmas:
plus(gen_s:0'2_0(+(1, *(2, n4_0))), gen_s:0'2_0(b)) -> gen_s:0'2_0(+(+(1, *(2, n4_0)), b)), 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:
ack
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(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ack(gen_s:0'2_0(1), gen_s:0'2_0(+(1, n860_0))) -> *3_0, rt in Omega(n860_0)

Induction Base:
ack(gen_s:0'2_0(1), gen_s:0'2_0(+(1, 0)))

Induction Step:
ack(gen_s:0'2_0(1), gen_s:0'2_0(+(1, +(n860_0, 1)))) ->_R^Omega(1)
ack(gen_s:0'2_0(0), plus(gen_s:0'2_0(+(1, n860_0)), ack(s(gen_s:0'2_0(0)), gen_s:0'2_0(+(1, n860_0))))) ->_IH
ack(gen_s:0'2_0(0), plus(gen_s:0'2_0(+(1, n860_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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(14)
BOUNDS(1, INF)