/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 294 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), 29 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 55 ms]
        (16) 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))
   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   inc(s(x)) -> s(inc(x))
   inc(0) -> s(0)
   logarithm(x) -> logIter(x, 0)
   logIter(x, y) -> if(le(s(0), x), le(s(s(0)), x), half(x), inc(y))
   if(false, b, x, y) -> logZeroError
   if(true, false, x, s(y)) -> y
   if(true, true, x, y) -> logIter(x, y)
   f -> g
   f -> h

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))
   le(0', y) -> true
   le(s(x), 0') -> false
   le(s(x), s(y)) -> le(x, y)
   inc(s(x)) -> s(inc(x))
   inc(0') -> s(0')
   logarithm(x) -> logIter(x, 0')
   logIter(x, y) -> if(le(s(0'), x), le(s(s(0')), x), half(x), inc(y))
   if(false, b, x, y) -> logZeroError
   if(true, false, x, s(y)) -> y
   if(true, true, x, y) -> logIter(x, y)
   f -> g
   f -> h

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))
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
inc(s(x)) -> s(inc(x))
inc(0') -> s(0')
logarithm(x) -> logIter(x, 0')
logIter(x, y) -> if(le(s(0'), x), le(s(s(0')), x), half(x), inc(y))
if(false, b, x, y) -> logZeroError
if(true, false, x, s(y)) -> y
if(true, true, x, y) -> logIter(x, y)
f -> g
f -> h

Types:
half :: 0':s:logZeroError -> 0':s:logZeroError
0' :: 0':s:logZeroError
s :: 0':s:logZeroError -> 0':s:logZeroError
le :: 0':s:logZeroError -> 0':s:logZeroError -> true:false
true :: true:false
false :: true:false
inc :: 0':s:logZeroError -> 0':s:logZeroError
logarithm :: 0':s:logZeroError -> 0':s:logZeroError
logIter :: 0':s:logZeroError -> 0':s:logZeroError -> 0':s:logZeroError
if :: true:false -> true:false -> 0':s:logZeroError -> 0':s:logZeroError -> 0':s:logZeroError
logZeroError :: 0':s:logZeroError
f :: g:h
g :: g:h
h :: g:h
hole_0':s:logZeroError1_0 :: 0':s:logZeroError
hole_true:false2_0 :: true:false
hole_g:h3_0 :: g:h
gen_0':s:logZeroError4_0 :: Nat -> 0':s:logZeroError

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
half, le, inc, logIter

They will be analysed ascendingly in the following order:
half < logIter
le < logIter
inc < logIter

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

(6)
Obligation:
TRS:
Rules:
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
inc(s(x)) -> s(inc(x))
inc(0') -> s(0')
logarithm(x) -> logIter(x, 0')
logIter(x, y) -> if(le(s(0'), x), le(s(s(0')), x), half(x), inc(y))
if(false, b, x, y) -> logZeroError
if(true, false, x, s(y)) -> y
if(true, true, x, y) -> logIter(x, y)
f -> g
f -> h

Types:
half :: 0':s:logZeroError -> 0':s:logZeroError
0' :: 0':s:logZeroError
s :: 0':s:logZeroError -> 0':s:logZeroError
le :: 0':s:logZeroError -> 0':s:logZeroError -> true:false
true :: true:false
false :: true:false
inc :: 0':s:logZeroError -> 0':s:logZeroError
logarithm :: 0':s:logZeroError -> 0':s:logZeroError
logIter :: 0':s:logZeroError -> 0':s:logZeroError -> 0':s:logZeroError
if :: true:false -> true:false -> 0':s:logZeroError -> 0':s:logZeroError -> 0':s:logZeroError
logZeroError :: 0':s:logZeroError
f :: g:h
g :: g:h
h :: g:h
hole_0':s:logZeroError1_0 :: 0':s:logZeroError
hole_true:false2_0 :: true:false
hole_g:h3_0 :: g:h
gen_0':s:logZeroError4_0 :: Nat -> 0':s:logZeroError


Generator Equations:
gen_0':s:logZeroError4_0(0) <=> 0'
gen_0':s:logZeroError4_0(+(x, 1)) <=> s(gen_0':s:logZeroError4_0(x))


The following defined symbols remain to be analysed:
half, le, inc, logIter

They will be analysed ascendingly in the following order:
half < logIter
le < logIter
inc < logIter

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
half(gen_0':s:logZeroError4_0(*(2, n6_0))) -> gen_0':s:logZeroError4_0(n6_0), rt in Omega(1 + n6_0)

Induction Base:
half(gen_0':s:logZeroError4_0(*(2, 0))) ->_R^Omega(1)
0'

Induction Step:
half(gen_0':s:logZeroError4_0(*(2, +(n6_0, 1)))) ->_R^Omega(1)
s(half(gen_0':s:logZeroError4_0(*(2, n6_0)))) ->_IH
s(gen_0':s:logZeroError4_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))
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
inc(s(x)) -> s(inc(x))
inc(0') -> s(0')
logarithm(x) -> logIter(x, 0')
logIter(x, y) -> if(le(s(0'), x), le(s(s(0')), x), half(x), inc(y))
if(false, b, x, y) -> logZeroError
if(true, false, x, s(y)) -> y
if(true, true, x, y) -> logIter(x, y)
f -> g
f -> h

Types:
half :: 0':s:logZeroError -> 0':s:logZeroError
0' :: 0':s:logZeroError
s :: 0':s:logZeroError -> 0':s:logZeroError
le :: 0':s:logZeroError -> 0':s:logZeroError -> true:false
true :: true:false
false :: true:false
inc :: 0':s:logZeroError -> 0':s:logZeroError
logarithm :: 0':s:logZeroError -> 0':s:logZeroError
logIter :: 0':s:logZeroError -> 0':s:logZeroError -> 0':s:logZeroError
if :: true:false -> true:false -> 0':s:logZeroError -> 0':s:logZeroError -> 0':s:logZeroError
logZeroError :: 0':s:logZeroError
f :: g:h
g :: g:h
h :: g:h
hole_0':s:logZeroError1_0 :: 0':s:logZeroError
hole_true:false2_0 :: true:false
hole_g:h3_0 :: g:h
gen_0':s:logZeroError4_0 :: Nat -> 0':s:logZeroError


Generator Equations:
gen_0':s:logZeroError4_0(0) <=> 0'
gen_0':s:logZeroError4_0(+(x, 1)) <=> s(gen_0':s:logZeroError4_0(x))


The following defined symbols remain to be analysed:
half, le, inc, logIter

They will be analysed ascendingly in the following order:
half < logIter
le < logIter
inc < logIter

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

(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))
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
inc(s(x)) -> s(inc(x))
inc(0') -> s(0')
logarithm(x) -> logIter(x, 0')
logIter(x, y) -> if(le(s(0'), x), le(s(s(0')), x), half(x), inc(y))
if(false, b, x, y) -> logZeroError
if(true, false, x, s(y)) -> y
if(true, true, x, y) -> logIter(x, y)
f -> g
f -> h

Types:
half :: 0':s:logZeroError -> 0':s:logZeroError
0' :: 0':s:logZeroError
s :: 0':s:logZeroError -> 0':s:logZeroError
le :: 0':s:logZeroError -> 0':s:logZeroError -> true:false
true :: true:false
false :: true:false
inc :: 0':s:logZeroError -> 0':s:logZeroError
logarithm :: 0':s:logZeroError -> 0':s:logZeroError
logIter :: 0':s:logZeroError -> 0':s:logZeroError -> 0':s:logZeroError
if :: true:false -> true:false -> 0':s:logZeroError -> 0':s:logZeroError -> 0':s:logZeroError
logZeroError :: 0':s:logZeroError
f :: g:h
g :: g:h
h :: g:h
hole_0':s:logZeroError1_0 :: 0':s:logZeroError
hole_true:false2_0 :: true:false
hole_g:h3_0 :: g:h
gen_0':s:logZeroError4_0 :: Nat -> 0':s:logZeroError


Lemmas:
half(gen_0':s:logZeroError4_0(*(2, n6_0))) -> gen_0':s:logZeroError4_0(n6_0), rt in Omega(1 + n6_0)


Generator Equations:
gen_0':s:logZeroError4_0(0) <=> 0'
gen_0':s:logZeroError4_0(+(x, 1)) <=> s(gen_0':s:logZeroError4_0(x))


The following defined symbols remain to be analysed:
le, inc, logIter

They will be analysed ascendingly in the following order:
le < logIter
inc < logIter

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
le(gen_0':s:logZeroError4_0(n320_0), gen_0':s:logZeroError4_0(n320_0)) -> true, rt in Omega(1 + n320_0)

Induction Base:
le(gen_0':s:logZeroError4_0(0), gen_0':s:logZeroError4_0(0)) ->_R^Omega(1)
true

Induction Step:
le(gen_0':s:logZeroError4_0(+(n320_0, 1)), gen_0':s:logZeroError4_0(+(n320_0, 1))) ->_R^Omega(1)
le(gen_0':s:logZeroError4_0(n320_0), gen_0':s:logZeroError4_0(n320_0)) ->_IH
true

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))
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
inc(s(x)) -> s(inc(x))
inc(0') -> s(0')
logarithm(x) -> logIter(x, 0')
logIter(x, y) -> if(le(s(0'), x), le(s(s(0')), x), half(x), inc(y))
if(false, b, x, y) -> logZeroError
if(true, false, x, s(y)) -> y
if(true, true, x, y) -> logIter(x, y)
f -> g
f -> h

Types:
half :: 0':s:logZeroError -> 0':s:logZeroError
0' :: 0':s:logZeroError
s :: 0':s:logZeroError -> 0':s:logZeroError
le :: 0':s:logZeroError -> 0':s:logZeroError -> true:false
true :: true:false
false :: true:false
inc :: 0':s:logZeroError -> 0':s:logZeroError
logarithm :: 0':s:logZeroError -> 0':s:logZeroError
logIter :: 0':s:logZeroError -> 0':s:logZeroError -> 0':s:logZeroError
if :: true:false -> true:false -> 0':s:logZeroError -> 0':s:logZeroError -> 0':s:logZeroError
logZeroError :: 0':s:logZeroError
f :: g:h
g :: g:h
h :: g:h
hole_0':s:logZeroError1_0 :: 0':s:logZeroError
hole_true:false2_0 :: true:false
hole_g:h3_0 :: g:h
gen_0':s:logZeroError4_0 :: Nat -> 0':s:logZeroError


Lemmas:
half(gen_0':s:logZeroError4_0(*(2, n6_0))) -> gen_0':s:logZeroError4_0(n6_0), rt in Omega(1 + n6_0)
le(gen_0':s:logZeroError4_0(n320_0), gen_0':s:logZeroError4_0(n320_0)) -> true, rt in Omega(1 + n320_0)


Generator Equations:
gen_0':s:logZeroError4_0(0) <=> 0'
gen_0':s:logZeroError4_0(+(x, 1)) <=> s(gen_0':s:logZeroError4_0(x))


The following defined symbols remain to be analysed:
inc, logIter

They will be analysed ascendingly in the following order:
inc < logIter

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
inc(gen_0':s:logZeroError4_0(n603_0)) -> gen_0':s:logZeroError4_0(+(1, n603_0)), rt in Omega(1 + n603_0)

Induction Base:
inc(gen_0':s:logZeroError4_0(0)) ->_R^Omega(1)
s(0')

Induction Step:
inc(gen_0':s:logZeroError4_0(+(n603_0, 1))) ->_R^Omega(1)
s(inc(gen_0':s:logZeroError4_0(n603_0))) ->_IH
s(gen_0':s:logZeroError4_0(+(1, c604_0)))

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

(16)
Obligation:
TRS:
Rules:
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
inc(s(x)) -> s(inc(x))
inc(0') -> s(0')
logarithm(x) -> logIter(x, 0')
logIter(x, y) -> if(le(s(0'), x), le(s(s(0')), x), half(x), inc(y))
if(false, b, x, y) -> logZeroError
if(true, false, x, s(y)) -> y
if(true, true, x, y) -> logIter(x, y)
f -> g
f -> h

Types:
half :: 0':s:logZeroError -> 0':s:logZeroError
0' :: 0':s:logZeroError
s :: 0':s:logZeroError -> 0':s:logZeroError
le :: 0':s:logZeroError -> 0':s:logZeroError -> true:false
true :: true:false
false :: true:false
inc :: 0':s:logZeroError -> 0':s:logZeroError
logarithm :: 0':s:logZeroError -> 0':s:logZeroError
logIter :: 0':s:logZeroError -> 0':s:logZeroError -> 0':s:logZeroError
if :: true:false -> true:false -> 0':s:logZeroError -> 0':s:logZeroError -> 0':s:logZeroError
logZeroError :: 0':s:logZeroError
f :: g:h
g :: g:h
h :: g:h
hole_0':s:logZeroError1_0 :: 0':s:logZeroError
hole_true:false2_0 :: true:false
hole_g:h3_0 :: g:h
gen_0':s:logZeroError4_0 :: Nat -> 0':s:logZeroError


Lemmas:
half(gen_0':s:logZeroError4_0(*(2, n6_0))) -> gen_0':s:logZeroError4_0(n6_0), rt in Omega(1 + n6_0)
le(gen_0':s:logZeroError4_0(n320_0), gen_0':s:logZeroError4_0(n320_0)) -> true, rt in Omega(1 + n320_0)
inc(gen_0':s:logZeroError4_0(n603_0)) -> gen_0':s:logZeroError4_0(+(1, n603_0)), rt in Omega(1 + n603_0)


Generator Equations:
gen_0':s:logZeroError4_0(0) <=> 0'
gen_0':s:logZeroError4_0(+(x, 1)) <=> s(gen_0':s:logZeroError4_0(x))


The following defined symbols remain to be analysed:
logIter