/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 232 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), 49 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:

   times(x, y) -> help(x, y, 0)
   help(x, y, c) -> if(lt(c, y), x, y, c)
   if(true, x, y, c) -> plus(x, help(x, y, s(c)))
   if(false, x, y, c) -> 0
   lt(0, s(x)) -> true
   lt(s(x), 0) -> false
   lt(s(x), s(y)) -> lt(x, y)
   plus(x, 0) -> x
   plus(0, x) -> x
   plus(x, s(y)) -> s(plus(x, y))
   plus(s(x), y) -> s(plus(x, y))

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:

   times(x, y) -> help(x, y, 0')
   help(x, y, c) -> if(lt(c, y), x, y, c)
   if(true, x, y, c) -> plus(x, help(x, y, s(c)))
   if(false, x, y, c) -> 0'
   lt(0', s(x)) -> true
   lt(s(x), 0') -> false
   lt(s(x), s(y)) -> lt(x, y)
   plus(x, 0') -> x
   plus(0', x) -> x
   plus(x, s(y)) -> s(plus(x, y))
   plus(s(x), y) -> s(plus(x, y))

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

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

(4)
Obligation:
TRS:
Rules:
times(x, y) -> help(x, y, 0')
help(x, y, c) -> if(lt(c, y), x, y, c)
if(true, x, y, c) -> plus(x, help(x, y, s(c)))
if(false, x, y, c) -> 0'
lt(0', s(x)) -> true
lt(s(x), 0') -> false
lt(s(x), s(y)) -> lt(x, y)
plus(x, 0') -> x
plus(0', x) -> x
plus(x, s(y)) -> s(plus(x, y))
plus(s(x), y) -> s(plus(x, y))

Types:
times :: 0':s -> 0':s -> 0':s
help :: 0':s -> 0':s -> 0':s -> 0':s
0' :: 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
lt :: 0':s -> 0':s -> true:false
true :: true:false
plus :: 0':s -> 0':s -> 0':s
s :: 0':s -> 0':s
false :: true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s

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

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

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

(6)
Obligation:
TRS:
Rules:
times(x, y) -> help(x, y, 0')
help(x, y, c) -> if(lt(c, y), x, y, c)
if(true, x, y, c) -> plus(x, help(x, y, s(c)))
if(false, x, y, c) -> 0'
lt(0', s(x)) -> true
lt(s(x), 0') -> false
lt(s(x), s(y)) -> lt(x, y)
plus(x, 0') -> x
plus(0', x) -> x
plus(x, s(y)) -> s(plus(x, y))
plus(s(x), y) -> s(plus(x, y))

Types:
times :: 0':s -> 0':s -> 0':s
help :: 0':s -> 0':s -> 0':s -> 0':s
0' :: 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
lt :: 0':s -> 0':s -> true:false
true :: true:false
plus :: 0':s -> 0':s -> 0':s
s :: 0':s -> 0':s
false :: true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


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


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

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

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) -> true, rt in Omega(1 + n5_0)

Induction Base:
lt(gen_0':s3_0(0), gen_0':s3_0(+(1, 0))) ->_R^Omega(1)
true

Induction Step:
lt(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(1, +(n5_0, 1)))) ->_R^Omega(1)
lt(gen_0':s3_0(n5_0), gen_0':s3_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).
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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:
times(x, y) -> help(x, y, 0')
help(x, y, c) -> if(lt(c, y), x, y, c)
if(true, x, y, c) -> plus(x, help(x, y, s(c)))
if(false, x, y, c) -> 0'
lt(0', s(x)) -> true
lt(s(x), 0') -> false
lt(s(x), s(y)) -> lt(x, y)
plus(x, 0') -> x
plus(0', x) -> x
plus(x, s(y)) -> s(plus(x, y))
plus(s(x), y) -> s(plus(x, y))

Types:
times :: 0':s -> 0':s -> 0':s
help :: 0':s -> 0':s -> 0':s -> 0':s
0' :: 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
lt :: 0':s -> 0':s -> true:false
true :: true:false
plus :: 0':s -> 0':s -> 0':s
s :: 0':s -> 0':s
false :: true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


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


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

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

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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:
times(x, y) -> help(x, y, 0')
help(x, y, c) -> if(lt(c, y), x, y, c)
if(true, x, y, c) -> plus(x, help(x, y, s(c)))
if(false, x, y, c) -> 0'
lt(0', s(x)) -> true
lt(s(x), 0') -> false
lt(s(x), s(y)) -> lt(x, y)
plus(x, 0') -> x
plus(0', x) -> x
plus(x, s(y)) -> s(plus(x, y))
plus(s(x), y) -> s(plus(x, y))

Types:
times :: 0':s -> 0':s -> 0':s
help :: 0':s -> 0':s -> 0':s -> 0':s
0' :: 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
lt :: 0':s -> 0':s -> true:false
true :: true:false
plus :: 0':s -> 0':s -> 0':s
s :: 0':s -> 0':s
false :: true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) -> true, rt in Omega(1 + n5_0)


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


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

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

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_0':s3_0(a), gen_0':s3_0(n269_0)) -> gen_0':s3_0(+(n269_0, a)), rt in Omega(1 + n269_0)

Induction Base:
plus(gen_0':s3_0(a), gen_0':s3_0(0)) ->_R^Omega(1)
gen_0':s3_0(a)

Induction Step:
plus(gen_0':s3_0(a), gen_0':s3_0(+(n269_0, 1))) ->_R^Omega(1)
s(plus(gen_0':s3_0(a), gen_0':s3_0(n269_0))) ->_IH
s(gen_0':s3_0(+(a, c270_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:
times(x, y) -> help(x, y, 0')
help(x, y, c) -> if(lt(c, y), x, y, c)
if(true, x, y, c) -> plus(x, help(x, y, s(c)))
if(false, x, y, c) -> 0'
lt(0', s(x)) -> true
lt(s(x), 0') -> false
lt(s(x), s(y)) -> lt(x, y)
plus(x, 0') -> x
plus(0', x) -> x
plus(x, s(y)) -> s(plus(x, y))
plus(s(x), y) -> s(plus(x, y))

Types:
times :: 0':s -> 0':s -> 0':s
help :: 0':s -> 0':s -> 0':s -> 0':s
0' :: 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
lt :: 0':s -> 0':s -> true:false
true :: true:false
plus :: 0':s -> 0':s -> 0':s
s :: 0':s -> 0':s
false :: true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) -> true, rt in Omega(1 + n5_0)
plus(gen_0':s3_0(a), gen_0':s3_0(n269_0)) -> gen_0':s3_0(+(n269_0, a)), rt in Omega(1 + n269_0)


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


The following defined symbols remain to be analysed:
help