/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 428 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(0, 1, x) -> f(g(x), g(x), x)
   f(g(x), y, z) -> g(f(x, y, z))
   f(x, g(y), z) -> g(f(x, y, z))
   f(x, y, g(z)) -> g(f(x, y, z))

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(0', 1', x) -> f(g(x), g(x), x)
   f(g(x), y, z) -> g(f(x, y, z))
   f(x, g(y), z) -> g(f(x, y, z))
   f(x, y, g(z)) -> g(f(x, y, z))

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(0', 1', x) -> f(g(x), g(x), x)
f(g(x), y, z) -> g(f(x, y, z))
f(x, g(y), z) -> g(f(x, y, z))
f(x, y, g(z)) -> g(f(x, y, z))

Types:
f :: 0':1':g -> 0':1':g -> 0':1':g -> 0':1':g
0' :: 0':1':g
1' :: 0':1':g
g :: 0':1':g -> 0':1':g
hole_0':1':g1_0 :: 0':1':g
gen_0':1':g2_0 :: Nat -> 0':1':g

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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(0', 1', x) -> f(g(x), g(x), x)
f(g(x), y, z) -> g(f(x, y, z))
f(x, g(y), z) -> g(f(x, y, z))
f(x, y, g(z)) -> g(f(x, y, z))

Types:
f :: 0':1':g -> 0':1':g -> 0':1':g -> 0':1':g
0' :: 0':1':g
1' :: 0':1':g
g :: 0':1':g -> 0':1':g
hole_0':1':g1_0 :: 0':1':g
gen_0':1':g2_0 :: Nat -> 0':1':g


Generator Equations:
gen_0':1':g2_0(0) <=> 1'
gen_0':1':g2_0(+(x, 1)) <=> g(gen_0':1':g2_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':1':g2_0(+(1, n4_0)), gen_0':1':g2_0(b), gen_0':1':g2_0(c)) -> *3_0, rt in Omega(n4_0)

Induction Base:
f(gen_0':1':g2_0(+(1, 0)), gen_0':1':g2_0(b), gen_0':1':g2_0(c))

Induction Step:
f(gen_0':1':g2_0(+(1, +(n4_0, 1))), gen_0':1':g2_0(b), gen_0':1':g2_0(c)) ->_R^Omega(1)
g(f(gen_0':1':g2_0(+(1, n4_0)), gen_0':1':g2_0(b), gen_0':1':g2_0(c))) ->_IH
g(*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)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
f(0', 1', x) -> f(g(x), g(x), x)
f(g(x), y, z) -> g(f(x, y, z))
f(x, g(y), z) -> g(f(x, y, z))
f(x, y, g(z)) -> g(f(x, y, z))

Types:
f :: 0':1':g -> 0':1':g -> 0':1':g -> 0':1':g
0' :: 0':1':g
1' :: 0':1':g
g :: 0':1':g -> 0':1':g
hole_0':1':g1_0 :: 0':1':g
gen_0':1':g2_0 :: Nat -> 0':1':g


Generator Equations:
gen_0':1':g2_0(0) <=> 1'
gen_0':1':g2_0(+(x, 1)) <=> g(gen_0':1':g2_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)