/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (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), 800 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 (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   ack(0, y) -> s(y)
   ack(s(x), 0) -> ack(x, s(0))
   ack(s(x), s(y)) -> ack(x, ack(s(x), y))
   f(s(x), y) -> f(x, s(x))
   f(x, s(y)) -> f(y, x)
   f(x, y) -> ack(x, y)
   ack(s(x), y) -> f(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:

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

Types:
ack :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
f :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s

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(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
ack, f

They will be analysed ascendingly in the following order:
ack = f

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

Types:
ack :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
f :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
f, ack

They will be analysed ascendingly in the following order:
ack = f

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(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ack(gen_0':s2_0(1), gen_0':s2_0(+(1, n1529_0))) -> *3_0, rt in Omega(n1529_0)

Induction Base:
ack(gen_0':s2_0(1), gen_0':s2_0(+(1, 0)))

Induction Step:
ack(gen_0':s2_0(1), gen_0':s2_0(+(1, +(n1529_0, 1)))) ->_R^Omega(1)
ack(gen_0':s2_0(0), ack(s(gen_0':s2_0(0)), gen_0':s2_0(+(1, n1529_0)))) ->_IH
ack(gen_0':s2_0(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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(8)
Complex Obligation (BEST)

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(9)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
ack(0', y) -> s(y)
ack(s(x), 0') -> ack(x, s(0'))
ack(s(x), s(y)) -> ack(x, ack(s(x), y))
f(s(x), y) -> f(x, s(x))
f(x, s(y)) -> f(y, x)
f(x, y) -> ack(x, y)
ack(s(x), y) -> f(x, x)

Types:
ack :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
f :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
ack

They will be analysed ascendingly in the following order:
ack = 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:
TRS:
Rules:
ack(0', y) -> s(y)
ack(s(x), 0') -> ack(x, s(0'))
ack(s(x), s(y)) -> ack(x, ack(s(x), y))
f(s(x), y) -> f(x, s(x))
f(x, s(y)) -> f(y, x)
f(x, y) -> ack(x, y)
ack(s(x), y) -> f(x, x)

Types:
ack :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
f :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Lemmas:
ack(gen_0':s2_0(1), gen_0':s2_0(+(1, n1529_0))) -> *3_0, rt in Omega(n1529_0)


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
f

They will be analysed ascendingly in the following order:
ack = f