/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) SlicingProof [LOWER BOUND(ID), 0 ms]
(4) CpxTRS
(5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) typed CpxTrs
(7) OrderProof [LOWER BOUND(ID), 0 ms]
(8) typed CpxTrs
(9) RewriteLemmaProof [LOWER BOUND(ID), 549 ms]
(10) proven lower bound
(11) LowerBoundPropagationProof [FINISHED, 0 ms]
(12) BOUNDS(n^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:

   ack(Cons(x, xs), Nil) -> ack(xs, Cons(Nil, Nil))
   ack(Cons(x', xs'), Cons(x, xs)) -> ack(xs', ack(Cons(x', xs'), xs))
   ack(Nil, n) -> Cons(Cons(Nil, Nil), n)
   goal(m, n) -> ack(m, n)

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:

   ack(Cons(x, xs), Nil) -> ack(xs, Cons(Nil, Nil))
   ack(Cons(x', xs'), Cons(x, xs)) -> ack(xs', ack(Cons(x', xs'), xs))
   ack(Nil, n) -> Cons(Cons(Nil, Nil), n)
   goal(m, n) -> ack(m, n)

S is empty.
Rewrite Strategy: INNERMOST
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(3) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
Cons/0

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(4)
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:

   ack(Cons(xs), Nil) -> ack(xs, Cons(Nil))
   ack(Cons(xs'), Cons(xs)) -> ack(xs', ack(Cons(xs'), xs))
   ack(Nil, n) -> Cons(n)
   goal(m, n) -> ack(m, n)

S is empty.
Rewrite Strategy: INNERMOST
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(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(6)
Obligation:
Innermost TRS:
Rules:
ack(Cons(xs), Nil) -> ack(xs, Cons(Nil))
ack(Cons(xs'), Cons(xs)) -> ack(xs', ack(Cons(xs'), xs))
ack(Nil, n) -> Cons(n)
goal(m, n) -> ack(m, n)

Types:
ack :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat -> Cons:Nil

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(7) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
ack
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(8)
Obligation:
Innermost TRS:
Rules:
ack(Cons(xs), Nil) -> ack(xs, Cons(Nil))
ack(Cons(xs'), Cons(xs)) -> ack(xs', ack(Cons(xs'), xs))
ack(Nil, n) -> Cons(n)
goal(m, n) -> ack(m, n)

Types:
ack :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat -> Cons:Nil


Generator Equations:
gen_Cons:Nil2_0(0) <=> Nil
gen_Cons:Nil2_0(+(x, 1)) <=> Cons(gen_Cons:Nil2_0(x))


The following defined symbols remain to be analysed:
ack
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(9) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ack(gen_Cons:Nil2_0(1), gen_Cons:Nil2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0)

Induction Base:
ack(gen_Cons:Nil2_0(1), gen_Cons:Nil2_0(+(1, 0)))

Induction Step:
ack(gen_Cons:Nil2_0(1), gen_Cons:Nil2_0(+(1, +(n4_0, 1)))) ->_R^Omega(1)
ack(gen_Cons:Nil2_0(0), ack(Cons(gen_Cons:Nil2_0(0)), gen_Cons:Nil2_0(+(1, n4_0)))) ->_IH
ack(gen_Cons:Nil2_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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(10)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
ack(Cons(xs), Nil) -> ack(xs, Cons(Nil))
ack(Cons(xs'), Cons(xs)) -> ack(xs', ack(Cons(xs'), xs))
ack(Nil, n) -> Cons(n)
goal(m, n) -> ack(m, n)

Types:
ack :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat -> Cons:Nil


Generator Equations:
gen_Cons:Nil2_0(0) <=> Nil
gen_Cons:Nil2_0(+(x, 1)) <=> Cons(gen_Cons:Nil2_0(x))


The following defined symbols remain to be analysed:
ack
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(11) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(12)
BOUNDS(n^1, INF)