/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) 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), 281 ms]
(10) BEST
    (11) proven lower bound
        (12) LowerBoundPropagationProof [FINISHED, 0 ms]
        (13) BOUNDS(n^1, INF)
    (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 82 ms]
        (16) BOUNDS(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:

   int(0, 0) -> .(0, nil)
   int(0, s(y)) -> .(0, int(s(0), s(y)))
   int(s(x), 0) -> nil
   int(s(x), s(y)) -> int_list(int(x, y))
   int_list(nil) -> nil
   int_list(.(x, y)) -> .(s(x), int_list(y))

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:

   int(0', 0') -> .(0', nil)
   int(0', s(y)) -> .(0', int(s(0'), s(y)))
   int(s(x), 0') -> nil
   int(s(x), s(y)) -> int_list(int(x, y))
   int_list(nil) -> nil
   int_list(.(x, y)) -> .(s(x), int_list(y))

S is empty.
Rewrite Strategy: FULL
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(3) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
./0

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(4)
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:

   int(0', 0') -> .(nil)
   int(0', s(y)) -> .(int(s(0'), s(y)))
   int(s(x), 0') -> nil
   int(s(x), s(y)) -> int_list(int(x, y))
   int_list(nil) -> nil
   int_list(.(y)) -> .(int_list(y))

S is empty.
Rewrite Strategy: FULL
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(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(6)
Obligation:
TRS:
Rules:
int(0', 0') -> .(nil)
int(0', s(y)) -> .(int(s(0'), s(y)))
int(s(x), 0') -> nil
int(s(x), s(y)) -> int_list(int(x, y))
int_list(nil) -> nil
int_list(.(y)) -> .(int_list(y))

Types:
int :: 0':s -> 0':s -> nil:.
0' :: 0':s
. :: nil:. -> nil:.
nil :: nil:.
s :: 0':s -> 0':s
int_list :: nil:. -> nil:.
hole_nil:.1_0 :: nil:.
hole_0':s2_0 :: 0':s
gen_nil:.3_0 :: Nat -> nil:.
gen_0':s4_0 :: Nat -> 0':s

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

They will be analysed ascendingly in the following order:
int_list < int

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(8)
Obligation:
TRS:
Rules:
int(0', 0') -> .(nil)
int(0', s(y)) -> .(int(s(0'), s(y)))
int(s(x), 0') -> nil
int(s(x), s(y)) -> int_list(int(x, y))
int_list(nil) -> nil
int_list(.(y)) -> .(int_list(y))

Types:
int :: 0':s -> 0':s -> nil:.
0' :: 0':s
. :: nil:. -> nil:.
nil :: nil:.
s :: 0':s -> 0':s
int_list :: nil:. -> nil:.
hole_nil:.1_0 :: nil:.
hole_0':s2_0 :: 0':s
gen_nil:.3_0 :: Nat -> nil:.
gen_0':s4_0 :: Nat -> 0':s


Generator Equations:
gen_nil:.3_0(0) <=> nil
gen_nil:.3_0(+(x, 1)) <=> .(gen_nil:.3_0(x))
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


The following defined symbols remain to be analysed:
int_list, int

They will be analysed ascendingly in the following order:
int_list < int

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(9) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
int_list(gen_nil:.3_0(n6_0)) -> gen_nil:.3_0(n6_0), rt in Omega(1 + n6_0)

Induction Base:
int_list(gen_nil:.3_0(0)) ->_R^Omega(1)
nil

Induction Step:
int_list(gen_nil:.3_0(+(n6_0, 1))) ->_R^Omega(1)
.(int_list(gen_nil:.3_0(n6_0))) ->_IH
.(gen_nil:.3_0(c7_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:

TRS:
Rules:
int(0', 0') -> .(nil)
int(0', s(y)) -> .(int(s(0'), s(y)))
int(s(x), 0') -> nil
int(s(x), s(y)) -> int_list(int(x, y))
int_list(nil) -> nil
int_list(.(y)) -> .(int_list(y))

Types:
int :: 0':s -> 0':s -> nil:.
0' :: 0':s
. :: nil:. -> nil:.
nil :: nil:.
s :: 0':s -> 0':s
int_list :: nil:. -> nil:.
hole_nil:.1_0 :: nil:.
hole_0':s2_0 :: 0':s
gen_nil:.3_0 :: Nat -> nil:.
gen_0':s4_0 :: Nat -> 0':s


Generator Equations:
gen_nil:.3_0(0) <=> nil
gen_nil:.3_0(+(x, 1)) <=> .(gen_nil:.3_0(x))
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


The following defined symbols remain to be analysed:
int_list, int

They will be analysed ascendingly in the following order:
int_list < int

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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:
TRS:
Rules:
int(0', 0') -> .(nil)
int(0', s(y)) -> .(int(s(0'), s(y)))
int(s(x), 0') -> nil
int(s(x), s(y)) -> int_list(int(x, y))
int_list(nil) -> nil
int_list(.(y)) -> .(int_list(y))

Types:
int :: 0':s -> 0':s -> nil:.
0' :: 0':s
. :: nil:. -> nil:.
nil :: nil:.
s :: 0':s -> 0':s
int_list :: nil:. -> nil:.
hole_nil:.1_0 :: nil:.
hole_0':s2_0 :: 0':s
gen_nil:.3_0 :: Nat -> nil:.
gen_0':s4_0 :: Nat -> 0':s


Lemmas:
int_list(gen_nil:.3_0(n6_0)) -> gen_nil:.3_0(n6_0), rt in Omega(1 + n6_0)


Generator Equations:
gen_nil:.3_0(0) <=> nil
gen_nil:.3_0(+(x, 1)) <=> .(gen_nil:.3_0(x))
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


The following defined symbols remain to be analysed:
int
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(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
int(gen_0':s4_0(n184_0), gen_0':s4_0(n184_0)) -> gen_nil:.3_0(1), rt in Omega(1 + n184_0)

Induction Base:
int(gen_0':s4_0(0), gen_0':s4_0(0)) ->_R^Omega(1)
.(nil)

Induction Step:
int(gen_0':s4_0(+(n184_0, 1)), gen_0':s4_0(+(n184_0, 1))) ->_R^Omega(1)
int_list(int(gen_0':s4_0(n184_0), gen_0':s4_0(n184_0))) ->_IH
int_list(gen_nil:.3_0(1)) ->_L^Omega(2)
gen_nil:.3_0(1)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(16)
BOUNDS(1, INF)