/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 250 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), 51 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:

   lt(0, s(x)) -> true
   lt(x, 0) -> false
   lt(s(x), s(y)) -> lt(x, y)
   logarithm(x) -> ifa(lt(0, x), x)
   ifa(true, x) -> help(x, 1)
   ifa(false, x) -> logZeroError
   help(x, y) -> ifb(lt(y, x), x, y)
   ifb(true, x, y) -> help(half(x), s(y))
   ifb(false, x, y) -> y
   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(x))) -> s(half(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:

   lt(0', s(x)) -> true
   lt(x, 0') -> false
   lt(s(x), s(y)) -> lt(x, y)
   logarithm(x) -> ifa(lt(0', x), x)
   ifa(true, x) -> help(x, 1')
   ifa(false, x) -> logZeroError
   help(x, y) -> ifb(lt(y, x), x, y)
   ifb(true, x, y) -> help(half(x), s(y))
   ifb(false, x, y) -> y
   half(0') -> 0'
   half(s(0')) -> 0'
   half(s(s(x))) -> s(half(x))

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

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

(4)
Obligation:
TRS:
Rules:
lt(0', s(x)) -> true
lt(x, 0') -> false
lt(s(x), s(y)) -> lt(x, y)
logarithm(x) -> ifa(lt(0', x), x)
ifa(true, x) -> help(x, 1')
ifa(false, x) -> logZeroError
help(x, y) -> ifb(lt(y, x), x, y)
ifb(true, x, y) -> help(half(x), s(y))
ifb(false, x, y) -> y
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))

Types:
lt :: 0':s:1':logZeroError -> 0':s:1':logZeroError -> true:false
0' :: 0':s:1':logZeroError
s :: 0':s:1':logZeroError -> 0':s:1':logZeroError
true :: true:false
false :: true:false
logarithm :: 0':s:1':logZeroError -> 0':s:1':logZeroError
ifa :: true:false -> 0':s:1':logZeroError -> 0':s:1':logZeroError
help :: 0':s:1':logZeroError -> 0':s:1':logZeroError -> 0':s:1':logZeroError
1' :: 0':s:1':logZeroError
logZeroError :: 0':s:1':logZeroError
ifb :: true:false -> 0':s:1':logZeroError -> 0':s:1':logZeroError -> 0':s:1':logZeroError
half :: 0':s:1':logZeroError -> 0':s:1':logZeroError
hole_true:false1_0 :: true:false
hole_0':s:1':logZeroError2_0 :: 0':s:1':logZeroError
gen_0':s:1':logZeroError3_0 :: Nat -> 0':s:1':logZeroError

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

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

They will be analysed ascendingly in the following order:
lt < help
half < help

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

(6)
Obligation:
TRS:
Rules:
lt(0', s(x)) -> true
lt(x, 0') -> false
lt(s(x), s(y)) -> lt(x, y)
logarithm(x) -> ifa(lt(0', x), x)
ifa(true, x) -> help(x, 1')
ifa(false, x) -> logZeroError
help(x, y) -> ifb(lt(y, x), x, y)
ifb(true, x, y) -> help(half(x), s(y))
ifb(false, x, y) -> y
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))

Types:
lt :: 0':s:1':logZeroError -> 0':s:1':logZeroError -> true:false
0' :: 0':s:1':logZeroError
s :: 0':s:1':logZeroError -> 0':s:1':logZeroError
true :: true:false
false :: true:false
logarithm :: 0':s:1':logZeroError -> 0':s:1':logZeroError
ifa :: true:false -> 0':s:1':logZeroError -> 0':s:1':logZeroError
help :: 0':s:1':logZeroError -> 0':s:1':logZeroError -> 0':s:1':logZeroError
1' :: 0':s:1':logZeroError
logZeroError :: 0':s:1':logZeroError
ifb :: true:false -> 0':s:1':logZeroError -> 0':s:1':logZeroError -> 0':s:1':logZeroError
half :: 0':s:1':logZeroError -> 0':s:1':logZeroError
hole_true:false1_0 :: true:false
hole_0':s:1':logZeroError2_0 :: 0':s:1':logZeroError
gen_0':s:1':logZeroError3_0 :: Nat -> 0':s:1':logZeroError


Generator Equations:
gen_0':s:1':logZeroError3_0(0) <=> 0'
gen_0':s:1':logZeroError3_0(+(x, 1)) <=> s(gen_0':s:1':logZeroError3_0(x))


The following defined symbols remain to be analysed:
lt, help, half

They will be analysed ascendingly in the following order:
lt < help
half < help

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
lt(gen_0':s:1':logZeroError3_0(n5_0), gen_0':s:1':logZeroError3_0(+(1, n5_0))) -> true, rt in Omega(1 + n5_0)

Induction Base:
lt(gen_0':s:1':logZeroError3_0(0), gen_0':s:1':logZeroError3_0(+(1, 0))) ->_R^Omega(1)
true

Induction Step:
lt(gen_0':s:1':logZeroError3_0(+(n5_0, 1)), gen_0':s:1':logZeroError3_0(+(1, +(n5_0, 1)))) ->_R^Omega(1)
lt(gen_0':s:1':logZeroError3_0(n5_0), gen_0':s:1':logZeroError3_0(+(1, n5_0))) ->_IH
true

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:
lt(0', s(x)) -> true
lt(x, 0') -> false
lt(s(x), s(y)) -> lt(x, y)
logarithm(x) -> ifa(lt(0', x), x)
ifa(true, x) -> help(x, 1')
ifa(false, x) -> logZeroError
help(x, y) -> ifb(lt(y, x), x, y)
ifb(true, x, y) -> help(half(x), s(y))
ifb(false, x, y) -> y
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))

Types:
lt :: 0':s:1':logZeroError -> 0':s:1':logZeroError -> true:false
0' :: 0':s:1':logZeroError
s :: 0':s:1':logZeroError -> 0':s:1':logZeroError
true :: true:false
false :: true:false
logarithm :: 0':s:1':logZeroError -> 0':s:1':logZeroError
ifa :: true:false -> 0':s:1':logZeroError -> 0':s:1':logZeroError
help :: 0':s:1':logZeroError -> 0':s:1':logZeroError -> 0':s:1':logZeroError
1' :: 0':s:1':logZeroError
logZeroError :: 0':s:1':logZeroError
ifb :: true:false -> 0':s:1':logZeroError -> 0':s:1':logZeroError -> 0':s:1':logZeroError
half :: 0':s:1':logZeroError -> 0':s:1':logZeroError
hole_true:false1_0 :: true:false
hole_0':s:1':logZeroError2_0 :: 0':s:1':logZeroError
gen_0':s:1':logZeroError3_0 :: Nat -> 0':s:1':logZeroError


Generator Equations:
gen_0':s:1':logZeroError3_0(0) <=> 0'
gen_0':s:1':logZeroError3_0(+(x, 1)) <=> s(gen_0':s:1':logZeroError3_0(x))


The following defined symbols remain to be analysed:
lt, help, half

They will be analysed ascendingly in the following order:
lt < help
half < help

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
TRS:
Rules:
lt(0', s(x)) -> true
lt(x, 0') -> false
lt(s(x), s(y)) -> lt(x, y)
logarithm(x) -> ifa(lt(0', x), x)
ifa(true, x) -> help(x, 1')
ifa(false, x) -> logZeroError
help(x, y) -> ifb(lt(y, x), x, y)
ifb(true, x, y) -> help(half(x), s(y))
ifb(false, x, y) -> y
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))

Types:
lt :: 0':s:1':logZeroError -> 0':s:1':logZeroError -> true:false
0' :: 0':s:1':logZeroError
s :: 0':s:1':logZeroError -> 0':s:1':logZeroError
true :: true:false
false :: true:false
logarithm :: 0':s:1':logZeroError -> 0':s:1':logZeroError
ifa :: true:false -> 0':s:1':logZeroError -> 0':s:1':logZeroError
help :: 0':s:1':logZeroError -> 0':s:1':logZeroError -> 0':s:1':logZeroError
1' :: 0':s:1':logZeroError
logZeroError :: 0':s:1':logZeroError
ifb :: true:false -> 0':s:1':logZeroError -> 0':s:1':logZeroError -> 0':s:1':logZeroError
half :: 0':s:1':logZeroError -> 0':s:1':logZeroError
hole_true:false1_0 :: true:false
hole_0':s:1':logZeroError2_0 :: 0':s:1':logZeroError
gen_0':s:1':logZeroError3_0 :: Nat -> 0':s:1':logZeroError


Lemmas:
lt(gen_0':s:1':logZeroError3_0(n5_0), gen_0':s:1':logZeroError3_0(+(1, n5_0))) -> true, rt in Omega(1 + n5_0)


Generator Equations:
gen_0':s:1':logZeroError3_0(0) <=> 0'
gen_0':s:1':logZeroError3_0(+(x, 1)) <=> s(gen_0':s:1':logZeroError3_0(x))


The following defined symbols remain to be analysed:
half, help

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

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
half(gen_0':s:1':logZeroError3_0(*(2, n263_0))) -> gen_0':s:1':logZeroError3_0(n263_0), rt in Omega(1 + n263_0)

Induction Base:
half(gen_0':s:1':logZeroError3_0(*(2, 0))) ->_R^Omega(1)
0'

Induction Step:
half(gen_0':s:1':logZeroError3_0(*(2, +(n263_0, 1)))) ->_R^Omega(1)
s(half(gen_0':s:1':logZeroError3_0(*(2, n263_0)))) ->_IH
s(gen_0':s:1':logZeroError3_0(c264_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:
lt(0', s(x)) -> true
lt(x, 0') -> false
lt(s(x), s(y)) -> lt(x, y)
logarithm(x) -> ifa(lt(0', x), x)
ifa(true, x) -> help(x, 1')
ifa(false, x) -> logZeroError
help(x, y) -> ifb(lt(y, x), x, y)
ifb(true, x, y) -> help(half(x), s(y))
ifb(false, x, y) -> y
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))

Types:
lt :: 0':s:1':logZeroError -> 0':s:1':logZeroError -> true:false
0' :: 0':s:1':logZeroError
s :: 0':s:1':logZeroError -> 0':s:1':logZeroError
true :: true:false
false :: true:false
logarithm :: 0':s:1':logZeroError -> 0':s:1':logZeroError
ifa :: true:false -> 0':s:1':logZeroError -> 0':s:1':logZeroError
help :: 0':s:1':logZeroError -> 0':s:1':logZeroError -> 0':s:1':logZeroError
1' :: 0':s:1':logZeroError
logZeroError :: 0':s:1':logZeroError
ifb :: true:false -> 0':s:1':logZeroError -> 0':s:1':logZeroError -> 0':s:1':logZeroError
half :: 0':s:1':logZeroError -> 0':s:1':logZeroError
hole_true:false1_0 :: true:false
hole_0':s:1':logZeroError2_0 :: 0':s:1':logZeroError
gen_0':s:1':logZeroError3_0 :: Nat -> 0':s:1':logZeroError


Lemmas:
lt(gen_0':s:1':logZeroError3_0(n5_0), gen_0':s:1':logZeroError3_0(+(1, n5_0))) -> true, rt in Omega(1 + n5_0)
half(gen_0':s:1':logZeroError3_0(*(2, n263_0))) -> gen_0':s:1':logZeroError3_0(n263_0), rt in Omega(1 + n263_0)


Generator Equations:
gen_0':s:1':logZeroError3_0(0) <=> 0'
gen_0':s:1':logZeroError3_0(+(x, 1)) <=> s(gen_0':s:1':logZeroError3_0(x))


The following defined symbols remain to be analysed:
help