/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 573 ms]
(8) proven lower bound
(9) LowerBoundPropagationProof [FINISHED, 0 ms]
(10) BOUNDS(n^1, INF)


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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:

   f(x, a(b(y))) -> f(a(b(b(x))), y)
   f(a(x), y) -> f(x, a(y))
   f(b(x), y) -> f(x, b(y))

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:

   f(x, a(b(y))) -> f(a(b(b(x))), y)
   f(a(x), y) -> f(x, a(y))
   f(b(x), y) -> f(x, b(y))

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:
f(x, a(b(y))) -> f(a(b(b(x))), y)
f(a(x), y) -> f(x, a(y))
f(b(x), y) -> f(x, b(y))

Types:
f :: b:a -> b:a -> f
a :: b:a -> b:a
b :: b:a -> b:a
hole_f1_0 :: f
hole_b:a2_0 :: b:a
gen_b:a3_0 :: Nat -> b:a

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(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
f
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(6)
Obligation:
TRS:
Rules:
f(x, a(b(y))) -> f(a(b(b(x))), y)
f(a(x), y) -> f(x, a(y))
f(b(x), y) -> f(x, b(y))

Types:
f :: b:a -> b:a -> f
a :: b:a -> b:a
b :: b:a -> b:a
hole_f1_0 :: f
hole_b:a2_0 :: b:a
gen_b:a3_0 :: Nat -> b:a


Generator Equations:
gen_b:a3_0(0) <=> hole_b:a2_0
gen_b:a3_0(+(x, 1)) <=> a(gen_b:a3_0(x))


The following defined symbols remain to be analysed:
f
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(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
f(gen_b:a3_0(+(1, n5_0)), gen_b:a3_0(b)) -> *4_0, rt in Omega(n5_0)

Induction Base:
f(gen_b:a3_0(+(1, 0)), gen_b:a3_0(b))

Induction Step:
f(gen_b:a3_0(+(1, +(n5_0, 1))), gen_b:a3_0(b)) ->_R^Omega(1)
f(gen_b:a3_0(+(1, n5_0)), a(gen_b:a3_0(b))) ->_IH
*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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(8)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
f(x, a(b(y))) -> f(a(b(b(x))), y)
f(a(x), y) -> f(x, a(y))
f(b(x), y) -> f(x, b(y))

Types:
f :: b:a -> b:a -> f
a :: b:a -> b:a
b :: b:a -> b:a
hole_f1_0 :: f
hole_b:a2_0 :: b:a
gen_b:a3_0 :: Nat -> b:a


Generator Equations:
gen_b:a3_0(0) <=> hole_b:a2_0
gen_b:a3_0(+(x, 1)) <=> a(gen_b:a3_0(x))


The following defined symbols remain to be analysed:
f
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(9) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(10)
BOUNDS(n^1, INF)