WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) 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), 1884 ms]
(8) proven lower bound
(9) LowerBoundPropagationProof [FINISHED, 0 ms]
(10) 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:

   f(x, 0, 0) -> s(x)
   f(0, y, 0) -> s(y)
   f(0, 0, z) -> s(z)
   f(s(0), y, z) -> f(0, s(y), s(z))
   f(s(x), s(y), 0) -> f(x, y, s(0))
   f(s(x), 0, s(z)) -> f(x, s(0), z)
   f(0, s(0), s(0)) -> s(s(0))
   f(s(x), s(y), s(z)) -> f(x, y, f(s(x), s(y), z))
   f(0, s(s(y)), s(0)) -> f(0, y, s(0))
   f(0, s(0), s(s(z))) -> f(0, s(0), z)
   f(0, s(s(y)), s(s(z))) -> f(0, y, f(0, s(s(y)), s(z)))

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:

   f(x, 0', 0') -> s(x)
   f(0', y, 0') -> s(y)
   f(0', 0', z) -> s(z)
   f(s(0'), y, z) -> f(0', s(y), s(z))
   f(s(x), s(y), 0') -> f(x, y, s(0'))
   f(s(x), 0', s(z)) -> f(x, s(0'), z)
   f(0', s(0'), s(0')) -> s(s(0'))
   f(s(x), s(y), s(z)) -> f(x, y, f(s(x), s(y), z))
   f(0', s(s(y)), s(0')) -> f(0', y, s(0'))
   f(0', s(0'), s(s(z))) -> f(0', s(0'), z)
   f(0', s(s(y)), s(s(z))) -> f(0', y, f(0', s(s(y)), s(z)))

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

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

Types:
f :: 0':s -> 0':s -> 0':s -> 0':s
0' :: 0':s
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
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(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
f(gen_0':s2_0(0), gen_0':s2_0(+(1, *(2, n4_0))), gen_0':s2_0(1)) -> gen_0':s2_0(2), rt in Omega(1 + n4_0)

Induction Base:
f(gen_0':s2_0(0), gen_0':s2_0(+(1, *(2, 0))), gen_0':s2_0(1)) ->_R^Omega(1)
s(s(0'))

Induction Step:
f(gen_0':s2_0(0), gen_0':s2_0(+(1, *(2, +(n4_0, 1)))), gen_0':s2_0(1)) ->_R^Omega(1)
f(0', gen_0':s2_0(+(1, *(2, n4_0))), s(0')) ->_IH
gen_0':s2_0(2)

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:

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

Types:
f :: 0':s -> 0':s -> 0':s -> 0':s
0' :: 0':s
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
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(9) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(10)
BOUNDS(n^1, INF)