/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) 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), 273 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) typed CpxTrs
        (13) RewriteLemmaProof [LOWER BOUND(ID), 9 ms]
        (14) typed CpxTrs


----------------------------------------

(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:

   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(x))) -> s(half(x))
   lastbit(0) -> 0
   lastbit(s(0)) -> s(0)
   lastbit(s(s(x))) -> lastbit(x)
   conv(0) -> cons(nil, 0)
   conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x)))

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(1) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(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:

   half(0') -> 0'
   half(s(0')) -> 0'
   half(s(s(x))) -> s(half(x))
   lastbit(0') -> 0'
   lastbit(s(0')) -> s(0')
   lastbit(s(s(x))) -> lastbit(x)
   conv(0') -> cons(nil, 0')
   conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x)))

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(4)
Obligation:
TRS:
Rules:
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
lastbit(0') -> 0'
lastbit(s(0')) -> s(0')
lastbit(s(s(x))) -> lastbit(x)
conv(0') -> cons(nil, 0')
conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x)))

Types:
half :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
lastbit :: 0':s -> 0':s
conv :: 0':s -> nil:cons
cons :: nil:cons -> 0':s -> nil:cons
nil :: nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat -> 0':s
gen_nil:cons4_0 :: Nat -> nil:cons

----------------------------------------

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
half, lastbit, conv

They will be analysed ascendingly in the following order:
half < conv
lastbit < conv

----------------------------------------

(6)
Obligation:
TRS:
Rules:
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
lastbit(0') -> 0'
lastbit(s(0')) -> s(0')
lastbit(s(s(x))) -> lastbit(x)
conv(0') -> cons(nil, 0')
conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x)))

Types:
half :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
lastbit :: 0':s -> 0':s
conv :: 0':s -> nil:cons
cons :: nil:cons -> 0':s -> nil:cons
nil :: nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat -> 0':s
gen_nil:cons4_0 :: Nat -> nil:cons


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
gen_nil:cons4_0(0) <=> nil
gen_nil:cons4_0(+(x, 1)) <=> cons(gen_nil:cons4_0(x), 0')


The following defined symbols remain to be analysed:
half, lastbit, conv

They will be analysed ascendingly in the following order:
half < conv
lastbit < conv

----------------------------------------

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
half(gen_0':s3_0(*(2, n6_0))) -> gen_0':s3_0(n6_0), rt in Omega(1 + n6_0)

Induction Base:
half(gen_0':s3_0(*(2, 0))) ->_R^Omega(1)
0'

Induction Step:
half(gen_0':s3_0(*(2, +(n6_0, 1)))) ->_R^Omega(1)
s(half(gen_0':s3_0(*(2, n6_0)))) ->_IH
s(gen_0':s3_0(c7_0))

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

(8)
Complex Obligation (BEST)

----------------------------------------

(9)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
lastbit(0') -> 0'
lastbit(s(0')) -> s(0')
lastbit(s(s(x))) -> lastbit(x)
conv(0') -> cons(nil, 0')
conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x)))

Types:
half :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
lastbit :: 0':s -> 0':s
conv :: 0':s -> nil:cons
cons :: nil:cons -> 0':s -> nil:cons
nil :: nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat -> 0':s
gen_nil:cons4_0 :: Nat -> nil:cons


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
gen_nil:cons4_0(0) <=> nil
gen_nil:cons4_0(+(x, 1)) <=> cons(gen_nil:cons4_0(x), 0')


The following defined symbols remain to be analysed:
half, lastbit, conv

They will be analysed ascendingly in the following order:
half < conv
lastbit < conv

----------------------------------------

(10) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(11)
BOUNDS(n^1, INF)

----------------------------------------

(12)
Obligation:
TRS:
Rules:
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
lastbit(0') -> 0'
lastbit(s(0')) -> s(0')
lastbit(s(s(x))) -> lastbit(x)
conv(0') -> cons(nil, 0')
conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x)))

Types:
half :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
lastbit :: 0':s -> 0':s
conv :: 0':s -> nil:cons
cons :: nil:cons -> 0':s -> nil:cons
nil :: nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat -> 0':s
gen_nil:cons4_0 :: Nat -> nil:cons


Lemmas:
half(gen_0':s3_0(*(2, n6_0))) -> gen_0':s3_0(n6_0), rt in Omega(1 + n6_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
gen_nil:cons4_0(0) <=> nil
gen_nil:cons4_0(+(x, 1)) <=> cons(gen_nil:cons4_0(x), 0')


The following defined symbols remain to be analysed:
lastbit, conv

They will be analysed ascendingly in the following order:
lastbit < conv

----------------------------------------

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
lastbit(gen_0':s3_0(*(2, n300_0))) -> gen_0':s3_0(0), rt in Omega(1 + n300_0)

Induction Base:
lastbit(gen_0':s3_0(*(2, 0))) ->_R^Omega(1)
0'

Induction Step:
lastbit(gen_0':s3_0(*(2, +(n300_0, 1)))) ->_R^Omega(1)
lastbit(gen_0':s3_0(*(2, n300_0))) ->_IH
gen_0':s3_0(0)

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

(14)
Obligation:
TRS:
Rules:
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
lastbit(0') -> 0'
lastbit(s(0')) -> s(0')
lastbit(s(s(x))) -> lastbit(x)
conv(0') -> cons(nil, 0')
conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x)))

Types:
half :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
lastbit :: 0':s -> 0':s
conv :: 0':s -> nil:cons
cons :: nil:cons -> 0':s -> nil:cons
nil :: nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat -> 0':s
gen_nil:cons4_0 :: Nat -> nil:cons


Lemmas:
half(gen_0':s3_0(*(2, n6_0))) -> gen_0':s3_0(n6_0), rt in Omega(1 + n6_0)
lastbit(gen_0':s3_0(*(2, n300_0))) -> gen_0':s3_0(0), rt in Omega(1 + n300_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
gen_nil:cons4_0(0) <=> nil
gen_nil:cons4_0(+(x, 1)) <=> cons(gen_nil:cons4_0(x), 0')


The following defined symbols remain to be analysed:
conv