/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 715 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_in(0, n) -> ack_out(s(n))
   ack_in(s(m), 0) -> u11(ack_in(m, s(0)))
   u11(ack_out(n)) -> ack_out(n)
   ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
   u21(ack_out(n), m) -> u22(ack_in(m, n))
   u22(ack_out(n)) -> ack_out(n)

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_in(0', n) -> ack_out(s(n))
   ack_in(s(m), 0') -> u11(ack_in(m, s(0')))
   u11(ack_out(n)) -> ack_out(n)
   ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
   u21(ack_out(n), m) -> u22(ack_in(m, n))
   u22(ack_out(n)) -> ack_out(n)

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_in(0', n) -> ack_out(s(n))
ack_in(s(m), 0') -> u11(ack_in(m, s(0')))
u11(ack_out(n)) -> ack_out(n)
ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
u21(ack_out(n), m) -> u22(ack_in(m, n))
u22(ack_out(n)) -> ack_out(n)

Types:
ack_in :: 0':s -> 0':s -> ack_out
0' :: 0':s
ack_out :: 0':s -> ack_out
s :: 0':s -> 0':s
u11 :: ack_out -> ack_out
u21 :: ack_out -> 0':s -> ack_out
u22 :: ack_out -> ack_out
hole_ack_out1_0 :: ack_out
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:
ack_in, u21

They will be analysed ascendingly in the following order:
ack_in = u21

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(6)
Obligation:
TRS:
Rules:
ack_in(0', n) -> ack_out(s(n))
ack_in(s(m), 0') -> u11(ack_in(m, s(0')))
u11(ack_out(n)) -> ack_out(n)
ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
u21(ack_out(n), m) -> u22(ack_in(m, n))
u22(ack_out(n)) -> ack_out(n)

Types:
ack_in :: 0':s -> 0':s -> ack_out
0' :: 0':s
ack_out :: 0':s -> ack_out
s :: 0':s -> 0':s
u11 :: ack_out -> ack_out
u21 :: ack_out -> 0':s -> ack_out
u22 :: ack_out -> ack_out
hole_ack_out1_0 :: ack_out
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:
u21, ack_in

They will be analysed ascendingly in the following order:
ack_in = u21

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(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ack_in(gen_0':s3_0(1), gen_0':s3_0(+(1, n4108_0))) -> *4_0, rt in Omega(n4108_0)

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

Induction Step:
ack_in(gen_0':s3_0(1), gen_0':s3_0(+(1, +(n4108_0, 1)))) ->_R^Omega(1)
u21(ack_in(s(gen_0':s3_0(0)), gen_0':s3_0(+(1, n4108_0))), gen_0':s3_0(0)) ->_IH
u21(*4_0, gen_0':s3_0(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_in(0', n) -> ack_out(s(n))
ack_in(s(m), 0') -> u11(ack_in(m, s(0')))
u11(ack_out(n)) -> ack_out(n)
ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
u21(ack_out(n), m) -> u22(ack_in(m, n))
u22(ack_out(n)) -> ack_out(n)

Types:
ack_in :: 0':s -> 0':s -> ack_out
0' :: 0':s
ack_out :: 0':s -> ack_out
s :: 0':s -> 0':s
u11 :: ack_out -> ack_out
u21 :: ack_out -> 0':s -> ack_out
u22 :: ack_out -> ack_out
hole_ack_out1_0 :: ack_out
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:
ack_in

They will be analysed ascendingly in the following order:
ack_in = u21

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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_in(0', n) -> ack_out(s(n))
ack_in(s(m), 0') -> u11(ack_in(m, s(0')))
u11(ack_out(n)) -> ack_out(n)
ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
u21(ack_out(n), m) -> u22(ack_in(m, n))
u22(ack_out(n)) -> ack_out(n)

Types:
ack_in :: 0':s -> 0':s -> ack_out
0' :: 0':s
ack_out :: 0':s -> ack_out
s :: 0':s -> 0':s
u11 :: ack_out -> ack_out
u21 :: ack_out -> 0':s -> ack_out
u22 :: ack_out -> ack_out
hole_ack_out1_0 :: ack_out
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
ack_in(gen_0':s3_0(1), gen_0':s3_0(+(1, n4108_0))) -> *4_0, rt in Omega(n4108_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:
u21

They will be analysed ascendingly in the following order:
ack_in = u21