/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 253 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:

   minus(x, y) -> cond(min(x, y), x, y)
   cond(y, x, y) -> s(minus(x, s(y)))
   min(0, v) -> 0
   min(u, 0) -> 0
   min(s(u), s(v)) -> s(min(u, v))

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:

   minus(x, y) -> cond(min(x, y), x, y)
   cond(y, x, y) -> s(minus(x, s(y)))
   min(0', v) -> 0'
   min(u, 0') -> 0'
   min(s(u), s(v)) -> s(min(u, v))

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:
minus(x, y) -> cond(min(x, y), x, y)
cond(y, x, y) -> s(minus(x, s(y)))
min(0', v) -> 0'
min(u, 0') -> 0'
min(s(u), s(v)) -> s(min(u, v))

Types:
minus :: s:0' -> s:0' -> s:0'
cond :: s:0' -> s:0' -> s:0' -> s:0'
min :: s:0' -> s:0' -> s:0'
s :: s:0' -> s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat -> s:0'

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

They will be analysed ascendingly in the following order:
min < minus

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(6)
Obligation:
TRS:
Rules:
minus(x, y) -> cond(min(x, y), x, y)
cond(y, x, y) -> s(minus(x, s(y)))
min(0', v) -> 0'
min(u, 0') -> 0'
min(s(u), s(v)) -> s(min(u, v))

Types:
minus :: s:0' -> s:0' -> s:0'
cond :: s:0' -> s:0' -> s:0' -> s:0'
min :: s:0' -> s:0' -> s:0'
s :: s:0' -> s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat -> s:0'


Generator Equations:
gen_s:0'2_0(0) <=> 0'
gen_s:0'2_0(+(x, 1)) <=> s(gen_s:0'2_0(x))


The following defined symbols remain to be analysed:
min, minus

They will be analysed ascendingly in the following order:
min < minus

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(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
min(gen_s:0'2_0(n4_0), gen_s:0'2_0(n4_0)) -> gen_s:0'2_0(n4_0), rt in Omega(1 + n4_0)

Induction Base:
min(gen_s:0'2_0(0), gen_s:0'2_0(0)) ->_R^Omega(1)
0'

Induction Step:
min(gen_s:0'2_0(+(n4_0, 1)), gen_s:0'2_0(+(n4_0, 1))) ->_R^Omega(1)
s(min(gen_s:0'2_0(n4_0), gen_s:0'2_0(n4_0))) ->_IH
s(gen_s:0'2_0(c5_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:
minus(x, y) -> cond(min(x, y), x, y)
cond(y, x, y) -> s(minus(x, s(y)))
min(0', v) -> 0'
min(u, 0') -> 0'
min(s(u), s(v)) -> s(min(u, v))

Types:
minus :: s:0' -> s:0' -> s:0'
cond :: s:0' -> s:0' -> s:0' -> s:0'
min :: s:0' -> s:0' -> s:0'
s :: s:0' -> s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat -> s:0'


Generator Equations:
gen_s:0'2_0(0) <=> 0'
gen_s:0'2_0(+(x, 1)) <=> s(gen_s:0'2_0(x))


The following defined symbols remain to be analysed:
min, minus

They will be analysed ascendingly in the following order:
min < minus

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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:
minus(x, y) -> cond(min(x, y), x, y)
cond(y, x, y) -> s(minus(x, s(y)))
min(0', v) -> 0'
min(u, 0') -> 0'
min(s(u), s(v)) -> s(min(u, v))

Types:
minus :: s:0' -> s:0' -> s:0'
cond :: s:0' -> s:0' -> s:0' -> s:0'
min :: s:0' -> s:0' -> s:0'
s :: s:0' -> s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat -> s:0'


Lemmas:
min(gen_s:0'2_0(n4_0), gen_s:0'2_0(n4_0)) -> gen_s:0'2_0(n4_0), rt in Omega(1 + n4_0)


Generator Equations:
gen_s:0'2_0(0) <=> 0'
gen_s:0'2_0(+(x, 1)) <=> s(gen_s:0'2_0(x))


The following defined symbols remain to be analysed:
minus