/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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), 242 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), 58 ms]
        (14) typed CpxTrs


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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:

   div(x, s(y)) -> d(x, s(y), 0)
   d(x, s(y), z) -> cond(ge(x, z), x, y, z)
   cond(true, x, y, z) -> s(d(x, s(y), plus(s(y), z)))
   cond(false, x, y, z) -> 0
   ge(u, 0) -> true
   ge(0, s(v)) -> false
   ge(s(u), s(v)) -> ge(u, v)
   plus(n, 0) -> n
   plus(n, s(m)) -> s(plus(n, m))

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:

   div(x, s(y)) -> d(x, s(y), 0')
   d(x, s(y), z) -> cond(ge(x, z), x, y, z)
   cond(true, x, y, z) -> s(d(x, s(y), plus(s(y), z)))
   cond(false, x, y, z) -> 0'
   ge(u, 0') -> true
   ge(0', s(v)) -> false
   ge(s(u), s(v)) -> ge(u, v)
   plus(n, 0') -> n
   plus(n, s(m)) -> s(plus(n, m))

S is empty.
Rewrite Strategy: FULL
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(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(4)
Obligation:
TRS:
Rules:
div(x, s(y)) -> d(x, s(y), 0')
d(x, s(y), z) -> cond(ge(x, z), x, y, z)
cond(true, x, y, z) -> s(d(x, s(y), plus(s(y), z)))
cond(false, x, y, z) -> 0'
ge(u, 0') -> true
ge(0', s(v)) -> false
ge(s(u), s(v)) -> ge(u, v)
plus(n, 0') -> n
plus(n, s(m)) -> s(plus(n, m))

Types:
div :: s:0' -> s:0' -> s:0'
s :: s:0' -> s:0'
d :: s:0' -> s:0' -> s:0' -> s:0'
0' :: s:0'
cond :: true:false -> s:0' -> s:0' -> s:0' -> s:0'
ge :: s:0' -> s:0' -> true:false
true :: true:false
plus :: s:0' -> s:0' -> s:0'
false :: true:false
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat -> s:0'

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

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

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(6)
Obligation:
TRS:
Rules:
div(x, s(y)) -> d(x, s(y), 0')
d(x, s(y), z) -> cond(ge(x, z), x, y, z)
cond(true, x, y, z) -> s(d(x, s(y), plus(s(y), z)))
cond(false, x, y, z) -> 0'
ge(u, 0') -> true
ge(0', s(v)) -> false
ge(s(u), s(v)) -> ge(u, v)
plus(n, 0') -> n
plus(n, s(m)) -> s(plus(n, m))

Types:
div :: s:0' -> s:0' -> s:0'
s :: s:0' -> s:0'
d :: s:0' -> s:0' -> s:0' -> s:0'
0' :: s:0'
cond :: true:false -> s:0' -> s:0' -> s:0' -> s:0'
ge :: s:0' -> s:0' -> true:false
true :: true:false
plus :: s:0' -> s:0' -> s:0'
false :: true:false
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat -> s:0'


Generator Equations:
gen_s:0'3_0(0) <=> 0'
gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))


The following defined symbols remain to be analysed:
ge, d, plus

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

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(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ge(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) -> true, rt in Omega(1 + n5_0)

Induction Base:
ge(gen_s:0'3_0(0), gen_s:0'3_0(0)) ->_R^Omega(1)
true

Induction Step:
ge(gen_s:0'3_0(+(n5_0, 1)), gen_s:0'3_0(+(n5_0, 1))) ->_R^Omega(1)
ge(gen_s:0'3_0(n5_0), gen_s:0'3_0(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:
div(x, s(y)) -> d(x, s(y), 0')
d(x, s(y), z) -> cond(ge(x, z), x, y, z)
cond(true, x, y, z) -> s(d(x, s(y), plus(s(y), z)))
cond(false, x, y, z) -> 0'
ge(u, 0') -> true
ge(0', s(v)) -> false
ge(s(u), s(v)) -> ge(u, v)
plus(n, 0') -> n
plus(n, s(m)) -> s(plus(n, m))

Types:
div :: s:0' -> s:0' -> s:0'
s :: s:0' -> s:0'
d :: s:0' -> s:0' -> s:0' -> s:0'
0' :: s:0'
cond :: true:false -> s:0' -> s:0' -> s:0' -> s:0'
ge :: s:0' -> s:0' -> true:false
true :: true:false
plus :: s:0' -> s:0' -> s:0'
false :: true:false
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat -> s:0'


Generator Equations:
gen_s:0'3_0(0) <=> 0'
gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))


The following defined symbols remain to be analysed:
ge, d, plus

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

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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:
div(x, s(y)) -> d(x, s(y), 0')
d(x, s(y), z) -> cond(ge(x, z), x, y, z)
cond(true, x, y, z) -> s(d(x, s(y), plus(s(y), z)))
cond(false, x, y, z) -> 0'
ge(u, 0') -> true
ge(0', s(v)) -> false
ge(s(u), s(v)) -> ge(u, v)
plus(n, 0') -> n
plus(n, s(m)) -> s(plus(n, m))

Types:
div :: s:0' -> s:0' -> s:0'
s :: s:0' -> s:0'
d :: s:0' -> s:0' -> s:0' -> s:0'
0' :: s:0'
cond :: true:false -> s:0' -> s:0' -> s:0' -> s:0'
ge :: s:0' -> s:0' -> true:false
true :: true:false
plus :: s:0' -> s:0' -> s:0'
false :: true:false
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat -> s:0'


Lemmas:
ge(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) -> true, rt in Omega(1 + n5_0)


Generator Equations:
gen_s:0'3_0(0) <=> 0'
gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))


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

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

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(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_s:0'3_0(a), gen_s:0'3_0(n245_0)) -> gen_s:0'3_0(+(n245_0, a)), rt in Omega(1 + n245_0)

Induction Base:
plus(gen_s:0'3_0(a), gen_s:0'3_0(0)) ->_R^Omega(1)
gen_s:0'3_0(a)

Induction Step:
plus(gen_s:0'3_0(a), gen_s:0'3_0(+(n245_0, 1))) ->_R^Omega(1)
s(plus(gen_s:0'3_0(a), gen_s:0'3_0(n245_0))) ->_IH
s(gen_s:0'3_0(+(a, c246_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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(14)
Obligation:
TRS:
Rules:
div(x, s(y)) -> d(x, s(y), 0')
d(x, s(y), z) -> cond(ge(x, z), x, y, z)
cond(true, x, y, z) -> s(d(x, s(y), plus(s(y), z)))
cond(false, x, y, z) -> 0'
ge(u, 0') -> true
ge(0', s(v)) -> false
ge(s(u), s(v)) -> ge(u, v)
plus(n, 0') -> n
plus(n, s(m)) -> s(plus(n, m))

Types:
div :: s:0' -> s:0' -> s:0'
s :: s:0' -> s:0'
d :: s:0' -> s:0' -> s:0' -> s:0'
0' :: s:0'
cond :: true:false -> s:0' -> s:0' -> s:0' -> s:0'
ge :: s:0' -> s:0' -> true:false
true :: true:false
plus :: s:0' -> s:0' -> s:0'
false :: true:false
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat -> s:0'


Lemmas:
ge(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) -> true, rt in Omega(1 + n5_0)
plus(gen_s:0'3_0(a), gen_s:0'3_0(n245_0)) -> gen_s:0'3_0(+(n245_0, a)), rt in Omega(1 + n245_0)


Generator Equations:
gen_s:0'3_0(0) <=> 0'
gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))


The following defined symbols remain to be analysed:
d