/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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), 444 ms]
(10) BEST
    (11) proven lower bound
        (12) LowerBoundPropagationProof [FINISHED, 0 ms]
        (13) BOUNDS(n^1, INF)
    (14) typed CpxTrs


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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:

   a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y))
   mark(f(X1, X2)) -> a__f(mark(X1), X2)
   mark(g(X)) -> g(mark(X))
   a__f(X1, X2) -> f(X1, X2)

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:

   a__f(g(X), Y) -> a__f(mark(X), f(g(X), Y))
   mark(f(X1, X2)) -> a__f(mark(X1), X2)
   mark(g(X)) -> g(mark(X))
   a__f(X1, X2) -> f(X1, X2)

S is empty.
Rewrite Strategy: INNERMOST
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(3) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
a__f/1
f/1

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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:

   a__f(g(X)) -> a__f(mark(X))
   mark(f(X1)) -> a__f(mark(X1))
   mark(g(X)) -> g(mark(X))
   a__f(X1) -> f(X1)

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:
a__f(g(X)) -> a__f(mark(X))
mark(f(X1)) -> a__f(mark(X1))
mark(g(X)) -> g(mark(X))
a__f(X1) -> f(X1)

Types:
a__f :: g:f -> g:f
g :: g:f -> g:f
mark :: g:f -> g:f
f :: g:f -> g:f
hole_g:f1_0 :: g:f
gen_g:f2_0 :: Nat -> g:f

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(7) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
a__f, mark

They will be analysed ascendingly in the following order:
a__f = mark

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(8)
Obligation:
Innermost TRS:
Rules:
a__f(g(X)) -> a__f(mark(X))
mark(f(X1)) -> a__f(mark(X1))
mark(g(X)) -> g(mark(X))
a__f(X1) -> f(X1)

Types:
a__f :: g:f -> g:f
g :: g:f -> g:f
mark :: g:f -> g:f
f :: g:f -> g:f
hole_g:f1_0 :: g:f
gen_g:f2_0 :: Nat -> g:f


Generator Equations:
gen_g:f2_0(0) <=> hole_g:f1_0
gen_g:f2_0(+(x, 1)) <=> g(gen_g:f2_0(x))


The following defined symbols remain to be analysed:
mark, a__f

They will be analysed ascendingly in the following order:
a__f = mark

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(9) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
mark(gen_g:f2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0)

Induction Base:
mark(gen_g:f2_0(+(1, 0)))

Induction Step:
mark(gen_g:f2_0(+(1, +(n4_0, 1)))) ->_R^Omega(1)
g(mark(gen_g:f2_0(+(1, n4_0)))) ->_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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(10)
Complex Obligation (BEST)

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(11)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
a__f(g(X)) -> a__f(mark(X))
mark(f(X1)) -> a__f(mark(X1))
mark(g(X)) -> g(mark(X))
a__f(X1) -> f(X1)

Types:
a__f :: g:f -> g:f
g :: g:f -> g:f
mark :: g:f -> g:f
f :: g:f -> g:f
hole_g:f1_0 :: g:f
gen_g:f2_0 :: Nat -> g:f


Generator Equations:
gen_g:f2_0(0) <=> hole_g:f1_0
gen_g:f2_0(+(x, 1)) <=> g(gen_g:f2_0(x))


The following defined symbols remain to be analysed:
mark, a__f

They will be analysed ascendingly in the following order:
a__f = mark

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(12) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(13)
BOUNDS(n^1, INF)

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(14)
Obligation:
Innermost TRS:
Rules:
a__f(g(X)) -> a__f(mark(X))
mark(f(X1)) -> a__f(mark(X1))
mark(g(X)) -> g(mark(X))
a__f(X1) -> f(X1)

Types:
a__f :: g:f -> g:f
g :: g:f -> g:f
mark :: g:f -> g:f
f :: g:f -> g:f
hole_g:f1_0 :: g:f
gen_g:f2_0 :: Nat -> g:f


Lemmas:
mark(gen_g:f2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0)


Generator Equations:
gen_g:f2_0(0) <=> hole_g:f1_0
gen_g:f2_0(+(x, 1)) <=> g(gen_g:f2_0(x))


The following defined symbols remain to be analysed:
a__f

They will be analysed ascendingly in the following order:
a__f = mark