/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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), 235 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), 91 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:

   cond1(true, x) -> cond2(even(x), x)
   cond2(true, x) -> cond1(neq(x, 0), div2(x))
   cond2(false, x) -> cond1(neq(x, 0), p(x))
   neq(0, 0) -> false
   neq(0, s(x)) -> true
   neq(s(x), 0) -> true
   neq(s(x), s(y)) -> neq(x, y)
   even(0) -> true
   even(s(0)) -> false
   even(s(s(x))) -> even(x)
   div2(0) -> 0
   div2(s(0)) -> 0
   div2(s(s(x))) -> s(div2(x))
   p(0) -> 0
   p(s(x)) -> x

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.
----------------------------------------

(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:

   cond1(true, x) -> cond2(even(x), x)
   cond2(true, x) -> cond1(neq(x, 0'), div2(x))
   cond2(false, x) -> cond1(neq(x, 0'), p(x))
   neq(0', 0') -> false
   neq(0', s(x)) -> true
   neq(s(x), 0') -> true
   neq(s(x), s(y)) -> neq(x, y)
   even(0') -> true
   even(s(0')) -> false
   even(s(s(x))) -> even(x)
   div2(0') -> 0'
   div2(s(0')) -> 0'
   div2(s(s(x))) -> s(div2(x))
   p(0') -> 0'
   p(s(x)) -> x

S is empty.
Rewrite Strategy: FULL
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(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(4)
Obligation:
TRS:
Rules:
cond1(true, x) -> cond2(even(x), x)
cond2(true, x) -> cond1(neq(x, 0'), div2(x))
cond2(false, x) -> cond1(neq(x, 0'), p(x))
neq(0', 0') -> false
neq(0', s(x)) -> true
neq(s(x), 0') -> true
neq(s(x), s(y)) -> neq(x, y)
even(0') -> true
even(s(0')) -> false
even(s(s(x))) -> even(x)
div2(0') -> 0'
div2(s(0')) -> 0'
div2(s(s(x))) -> s(div2(x))
p(0') -> 0'
p(s(x)) -> x

Types:
cond1 :: true:false -> 0':s:y -> cond1:cond2
true :: true:false
cond2 :: true:false -> 0':s:y -> cond1:cond2
even :: 0':s:y -> true:false
neq :: 0':s:y -> 0':s:y -> true:false
0' :: 0':s:y
div2 :: 0':s:y -> 0':s:y
false :: true:false
p :: 0':s:y -> 0':s:y
s :: 0':s:y -> 0':s:y
y :: 0':s:y
hole_cond1:cond21_0 :: cond1:cond2
hole_true:false2_0 :: true:false
hole_0':s:y3_0 :: 0':s:y
gen_0':s:y4_0 :: Nat -> 0':s:y

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(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
cond1, cond2, even, neq, div2

They will be analysed ascendingly in the following order:
cond1 = cond2
even < cond1
neq < cond2
div2 < cond2

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

(6)
Obligation:
TRS:
Rules:
cond1(true, x) -> cond2(even(x), x)
cond2(true, x) -> cond1(neq(x, 0'), div2(x))
cond2(false, x) -> cond1(neq(x, 0'), p(x))
neq(0', 0') -> false
neq(0', s(x)) -> true
neq(s(x), 0') -> true
neq(s(x), s(y)) -> neq(x, y)
even(0') -> true
even(s(0')) -> false
even(s(s(x))) -> even(x)
div2(0') -> 0'
div2(s(0')) -> 0'
div2(s(s(x))) -> s(div2(x))
p(0') -> 0'
p(s(x)) -> x

Types:
cond1 :: true:false -> 0':s:y -> cond1:cond2
true :: true:false
cond2 :: true:false -> 0':s:y -> cond1:cond2
even :: 0':s:y -> true:false
neq :: 0':s:y -> 0':s:y -> true:false
0' :: 0':s:y
div2 :: 0':s:y -> 0':s:y
false :: true:false
p :: 0':s:y -> 0':s:y
s :: 0':s:y -> 0':s:y
y :: 0':s:y
hole_cond1:cond21_0 :: cond1:cond2
hole_true:false2_0 :: true:false
hole_0':s:y3_0 :: 0':s:y
gen_0':s:y4_0 :: Nat -> 0':s:y


Generator Equations:
gen_0':s:y4_0(0) <=> 0'
gen_0':s:y4_0(+(x, 1)) <=> s(gen_0':s:y4_0(x))


The following defined symbols remain to be analysed:
even, cond1, cond2, neq, div2

They will be analysed ascendingly in the following order:
cond1 = cond2
even < cond1
neq < cond2
div2 < cond2

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(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
even(gen_0':s:y4_0(*(2, n6_0))) -> true, rt in Omega(1 + n6_0)

Induction Base:
even(gen_0':s:y4_0(*(2, 0))) ->_R^Omega(1)
true

Induction Step:
even(gen_0':s:y4_0(*(2, +(n6_0, 1)))) ->_R^Omega(1)
even(gen_0':s:y4_0(*(2, n6_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:
cond1(true, x) -> cond2(even(x), x)
cond2(true, x) -> cond1(neq(x, 0'), div2(x))
cond2(false, x) -> cond1(neq(x, 0'), p(x))
neq(0', 0') -> false
neq(0', s(x)) -> true
neq(s(x), 0') -> true
neq(s(x), s(y)) -> neq(x, y)
even(0') -> true
even(s(0')) -> false
even(s(s(x))) -> even(x)
div2(0') -> 0'
div2(s(0')) -> 0'
div2(s(s(x))) -> s(div2(x))
p(0') -> 0'
p(s(x)) -> x

Types:
cond1 :: true:false -> 0':s:y -> cond1:cond2
true :: true:false
cond2 :: true:false -> 0':s:y -> cond1:cond2
even :: 0':s:y -> true:false
neq :: 0':s:y -> 0':s:y -> true:false
0' :: 0':s:y
div2 :: 0':s:y -> 0':s:y
false :: true:false
p :: 0':s:y -> 0':s:y
s :: 0':s:y -> 0':s:y
y :: 0':s:y
hole_cond1:cond21_0 :: cond1:cond2
hole_true:false2_0 :: true:false
hole_0':s:y3_0 :: 0':s:y
gen_0':s:y4_0 :: Nat -> 0':s:y


Generator Equations:
gen_0':s:y4_0(0) <=> 0'
gen_0':s:y4_0(+(x, 1)) <=> s(gen_0':s:y4_0(x))


The following defined symbols remain to be analysed:
even, cond1, cond2, neq, div2

They will be analysed ascendingly in the following order:
cond1 = cond2
even < cond1
neq < cond2
div2 < cond2

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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:
cond1(true, x) -> cond2(even(x), x)
cond2(true, x) -> cond1(neq(x, 0'), div2(x))
cond2(false, x) -> cond1(neq(x, 0'), p(x))
neq(0', 0') -> false
neq(0', s(x)) -> true
neq(s(x), 0') -> true
neq(s(x), s(y)) -> neq(x, y)
even(0') -> true
even(s(0')) -> false
even(s(s(x))) -> even(x)
div2(0') -> 0'
div2(s(0')) -> 0'
div2(s(s(x))) -> s(div2(x))
p(0') -> 0'
p(s(x)) -> x

Types:
cond1 :: true:false -> 0':s:y -> cond1:cond2
true :: true:false
cond2 :: true:false -> 0':s:y -> cond1:cond2
even :: 0':s:y -> true:false
neq :: 0':s:y -> 0':s:y -> true:false
0' :: 0':s:y
div2 :: 0':s:y -> 0':s:y
false :: true:false
p :: 0':s:y -> 0':s:y
s :: 0':s:y -> 0':s:y
y :: 0':s:y
hole_cond1:cond21_0 :: cond1:cond2
hole_true:false2_0 :: true:false
hole_0':s:y3_0 :: 0':s:y
gen_0':s:y4_0 :: Nat -> 0':s:y


Lemmas:
even(gen_0':s:y4_0(*(2, n6_0))) -> true, rt in Omega(1 + n6_0)


Generator Equations:
gen_0':s:y4_0(0) <=> 0'
gen_0':s:y4_0(+(x, 1)) <=> s(gen_0':s:y4_0(x))


The following defined symbols remain to be analysed:
neq, cond1, cond2, div2

They will be analysed ascendingly in the following order:
cond1 = cond2
neq < cond2
div2 < cond2

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(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
div2(gen_0':s:y4_0(*(2, n193_0))) -> gen_0':s:y4_0(n193_0), rt in Omega(1 + n193_0)

Induction Base:
div2(gen_0':s:y4_0(*(2, 0))) ->_R^Omega(1)
0'

Induction Step:
div2(gen_0':s:y4_0(*(2, +(n193_0, 1)))) ->_R^Omega(1)
s(div2(gen_0':s:y4_0(*(2, n193_0)))) ->_IH
s(gen_0':s:y4_0(c194_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:
cond1(true, x) -> cond2(even(x), x)
cond2(true, x) -> cond1(neq(x, 0'), div2(x))
cond2(false, x) -> cond1(neq(x, 0'), p(x))
neq(0', 0') -> false
neq(0', s(x)) -> true
neq(s(x), 0') -> true
neq(s(x), s(y)) -> neq(x, y)
even(0') -> true
even(s(0')) -> false
even(s(s(x))) -> even(x)
div2(0') -> 0'
div2(s(0')) -> 0'
div2(s(s(x))) -> s(div2(x))
p(0') -> 0'
p(s(x)) -> x

Types:
cond1 :: true:false -> 0':s:y -> cond1:cond2
true :: true:false
cond2 :: true:false -> 0':s:y -> cond1:cond2
even :: 0':s:y -> true:false
neq :: 0':s:y -> 0':s:y -> true:false
0' :: 0':s:y
div2 :: 0':s:y -> 0':s:y
false :: true:false
p :: 0':s:y -> 0':s:y
s :: 0':s:y -> 0':s:y
y :: 0':s:y
hole_cond1:cond21_0 :: cond1:cond2
hole_true:false2_0 :: true:false
hole_0':s:y3_0 :: 0':s:y
gen_0':s:y4_0 :: Nat -> 0':s:y


Lemmas:
even(gen_0':s:y4_0(*(2, n6_0))) -> true, rt in Omega(1 + n6_0)
div2(gen_0':s:y4_0(*(2, n193_0))) -> gen_0':s:y4_0(n193_0), rt in Omega(1 + n193_0)


Generator Equations:
gen_0':s:y4_0(0) <=> 0'
gen_0':s:y4_0(+(x, 1)) <=> s(gen_0':s:y4_0(x))


The following defined symbols remain to be analysed:
cond2, cond1

They will be analysed ascendingly in the following order:
cond1 = cond2