/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 234 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) 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:

   cond(true, x) -> cond(and(even(x), gr(x, 0)), p(x))
   and(x, false) -> false
   and(false, x) -> false
   and(true, true) -> true
   even(0) -> true
   even(s(0)) -> false
   even(s(s(x))) -> even(x)
   gr(0, x) -> false
   gr(s(x), 0) -> true
   gr(s(x), s(y)) -> gr(x, y)
   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.
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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:

   cond(true, x) -> cond(and(even(x), gr(x, 0')), p(x))
   and(x, false) -> false
   and(false, x) -> false
   and(true, true) -> true
   even(0') -> true
   even(s(0')) -> false
   even(s(s(x))) -> even(x)
   gr(0', x) -> false
   gr(s(x), 0') -> true
   gr(s(x), s(y)) -> gr(x, y)
   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.
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(4)
Obligation:
TRS:
Rules:
cond(true, x) -> cond(and(even(x), gr(x, 0')), p(x))
and(x, false) -> false
and(false, x) -> false
and(true, true) -> true
even(0') -> true
even(s(0')) -> false
even(s(s(x))) -> even(x)
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
p(0') -> 0'
p(s(x)) -> x

Types:
cond :: true:false -> 0':s:y -> cond
true :: true:false
and :: true:false -> true:false -> true:false
even :: 0':s:y -> true:false
gr :: 0':s:y -> 0':s:y -> true:false
0' :: 0':s:y
p :: 0':s:y -> 0':s:y
false :: true:false
s :: 0':s:y -> 0':s:y
y :: 0':s:y
hole_cond1_0 :: cond
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:
cond, even, gr

They will be analysed ascendingly in the following order:
even < cond
gr < cond

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(6)
Obligation:
TRS:
Rules:
cond(true, x) -> cond(and(even(x), gr(x, 0')), p(x))
and(x, false) -> false
and(false, x) -> false
and(true, true) -> true
even(0') -> true
even(s(0')) -> false
even(s(s(x))) -> even(x)
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
p(0') -> 0'
p(s(x)) -> x

Types:
cond :: true:false -> 0':s:y -> cond
true :: true:false
and :: true:false -> true:false -> true:false
even :: 0':s:y -> true:false
gr :: 0':s:y -> 0':s:y -> true:false
0' :: 0':s:y
p :: 0':s:y -> 0':s:y
false :: true:false
s :: 0':s:y -> 0':s:y
y :: 0':s:y
hole_cond1_0 :: cond
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, cond, gr

They will be analysed ascendingly in the following order:
even < cond
gr < cond

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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:
cond(true, x) -> cond(and(even(x), gr(x, 0')), p(x))
and(x, false) -> false
and(false, x) -> false
and(true, true) -> true
even(0') -> true
even(s(0')) -> false
even(s(s(x))) -> even(x)
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
p(0') -> 0'
p(s(x)) -> x

Types:
cond :: true:false -> 0':s:y -> cond
true :: true:false
and :: true:false -> true:false -> true:false
even :: 0':s:y -> true:false
gr :: 0':s:y -> 0':s:y -> true:false
0' :: 0':s:y
p :: 0':s:y -> 0':s:y
false :: true:false
s :: 0':s:y -> 0':s:y
y :: 0':s:y
hole_cond1_0 :: cond
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, cond, gr

They will be analysed ascendingly in the following order:
even < cond
gr < cond

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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:
cond(true, x) -> cond(and(even(x), gr(x, 0')), p(x))
and(x, false) -> false
and(false, x) -> false
and(true, true) -> true
even(0') -> true
even(s(0')) -> false
even(s(s(x))) -> even(x)
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
p(0') -> 0'
p(s(x)) -> x

Types:
cond :: true:false -> 0':s:y -> cond
true :: true:false
and :: true:false -> true:false -> true:false
even :: 0':s:y -> true:false
gr :: 0':s:y -> 0':s:y -> true:false
0' :: 0':s:y
p :: 0':s:y -> 0':s:y
false :: true:false
s :: 0':s:y -> 0':s:y
y :: 0':s:y
hole_cond1_0 :: cond
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:
gr, cond

They will be analysed ascendingly in the following order:
gr < cond