/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 315 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:

   f(a, g(y)) -> g(g(y))
   f(g(x), a) -> f(x, g(a))
   f(g(x), g(y)) -> h(g(y), x, g(y))
   h(g(x), y, z) -> f(y, h(x, y, z))
   h(a, y, z) -> 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(a, g(y)) -> g(g(y))
   f(g(x), a) -> f(x, g(a))
   f(g(x), g(y)) -> h(g(y), x, g(y))
   h(g(x), y, z) -> f(y, h(x, y, z))
   h(a, y, z) -> 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(a, g(y)) -> g(g(y))
f(g(x), a) -> f(x, g(a))
f(g(x), g(y)) -> h(g(y), x, g(y))
h(g(x), y, z) -> f(y, h(x, y, z))
h(a, y, z) -> z

Types:
f :: a:g -> a:g -> a:g
a :: a:g
g :: a:g -> a:g
h :: a:g -> a:g -> a:g -> a:g
hole_a:g1_0 :: a:g
gen_a:g2_0 :: Nat -> a:g

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(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
f, h

They will be analysed ascendingly in the following order:
f = h

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(6)
Obligation:
TRS:
Rules:
f(a, g(y)) -> g(g(y))
f(g(x), a) -> f(x, g(a))
f(g(x), g(y)) -> h(g(y), x, g(y))
h(g(x), y, z) -> f(y, h(x, y, z))
h(a, y, z) -> z

Types:
f :: a:g -> a:g -> a:g
a :: a:g
g :: a:g -> a:g
h :: a:g -> a:g -> a:g -> a:g
hole_a:g1_0 :: a:g
gen_a:g2_0 :: Nat -> a:g


Generator Equations:
gen_a:g2_0(0) <=> a
gen_a:g2_0(+(x, 1)) <=> g(gen_a:g2_0(x))


The following defined symbols remain to be analysed:
h, f

They will be analysed ascendingly in the following order:
f = h

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(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
h(gen_a:g2_0(n4_0), gen_a:g2_0(0), gen_a:g2_0(1)) -> gen_a:g2_0(+(1, n4_0)), rt in Omega(1 + n4_0)

Induction Base:
h(gen_a:g2_0(0), gen_a:g2_0(0), gen_a:g2_0(1)) ->_R^Omega(1)
gen_a:g2_0(1)

Induction Step:
h(gen_a:g2_0(+(n4_0, 1)), gen_a:g2_0(0), gen_a:g2_0(1)) ->_R^Omega(1)
f(gen_a:g2_0(0), h(gen_a:g2_0(n4_0), gen_a:g2_0(0), gen_a:g2_0(1))) ->_IH
f(gen_a:g2_0(0), gen_a:g2_0(+(1, c5_0))) ->_R^Omega(1)
g(g(gen_a:g2_0(n4_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:
f(a, g(y)) -> g(g(y))
f(g(x), a) -> f(x, g(a))
f(g(x), g(y)) -> h(g(y), x, g(y))
h(g(x), y, z) -> f(y, h(x, y, z))
h(a, y, z) -> z

Types:
f :: a:g -> a:g -> a:g
a :: a:g
g :: a:g -> a:g
h :: a:g -> a:g -> a:g -> a:g
hole_a:g1_0 :: a:g
gen_a:g2_0 :: Nat -> a:g


Generator Equations:
gen_a:g2_0(0) <=> a
gen_a:g2_0(+(x, 1)) <=> g(gen_a:g2_0(x))


The following defined symbols remain to be analysed:
h, f

They will be analysed ascendingly in the following order:
f = h

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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:
f(a, g(y)) -> g(g(y))
f(g(x), a) -> f(x, g(a))
f(g(x), g(y)) -> h(g(y), x, g(y))
h(g(x), y, z) -> f(y, h(x, y, z))
h(a, y, z) -> z

Types:
f :: a:g -> a:g -> a:g
a :: a:g
g :: a:g -> a:g
h :: a:g -> a:g -> a:g -> a:g
hole_a:g1_0 :: a:g
gen_a:g2_0 :: Nat -> a:g


Lemmas:
h(gen_a:g2_0(n4_0), gen_a:g2_0(0), gen_a:g2_0(1)) -> gen_a:g2_0(+(1, n4_0)), rt in Omega(1 + n4_0)


Generator Equations:
gen_a:g2_0(0) <=> a
gen_a:g2_0(+(x, 1)) <=> g(gen_a:g2_0(x))


The following defined symbols remain to be analysed:
f

They will be analysed ascendingly in the following order:
f = h