/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) SlicingProof [LOWER BOUND(ID), 0 ms]
(4) CpxTRS
(5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) typed CpxTrs
(7) OrderProof [LOWER BOUND(ID), 0 ms]
(8) typed CpxTrs
(9) RewriteLemmaProof [LOWER BOUND(ID), 260 ms]
(10) BEST
    (11) proven lower bound
        (12) LowerBoundPropagationProof [FINISHED, 0 ms]
        (13) BOUNDS(n^1, INF)
    (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 368 ms]
        (16) BOUNDS(1, INF)


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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:

   numbers -> d(0)
   d(x) -> if(le(x, nr), x)
   if(true, x) -> cons(x, d(s(x)))
   if(false, x) -> nil
   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   nr -> ack(s(s(s(s(s(s(0)))))), 0)
   ack(0, x) -> s(x)
   ack(s(x), 0) -> ack(x, s(0))
   ack(s(x), s(y)) -> ack(x, ack(s(x), y))

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:

   numbers -> d(0')
   d(x) -> if(le(x, nr), x)
   if(true, x) -> cons(x, d(s(x)))
   if(false, x) -> nil
   le(0', y) -> true
   le(s(x), 0') -> false
   le(s(x), s(y)) -> le(x, y)
   nr -> ack(s(s(s(s(s(s(0')))))), 0')
   ack(0', x) -> s(x)
   ack(s(x), 0') -> ack(x, s(0'))
   ack(s(x), s(y)) -> ack(x, ack(s(x), y))

S is empty.
Rewrite Strategy: FULL
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(3) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
cons/0

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(4)
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:

   numbers -> d(0')
   d(x) -> if(le(x, nr), x)
   if(true, x) -> cons(d(s(x)))
   if(false, x) -> nil
   le(0', y) -> true
   le(s(x), 0') -> false
   le(s(x), s(y)) -> le(x, y)
   nr -> ack(s(s(s(s(s(s(0')))))), 0')
   ack(0', x) -> s(x)
   ack(s(x), 0') -> ack(x, s(0'))
   ack(s(x), s(y)) -> ack(x, ack(s(x), y))

S is empty.
Rewrite Strategy: FULL
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(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(6)
Obligation:
TRS:
Rules:
numbers -> d(0')
d(x) -> if(le(x, nr), x)
if(true, x) -> cons(d(s(x)))
if(false, x) -> nil
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
nr -> ack(s(s(s(s(s(s(0')))))), 0')
ack(0', x) -> s(x)
ack(s(x), 0') -> ack(x, s(0'))
ack(s(x), s(y)) -> ack(x, ack(s(x), y))

Types:
numbers :: cons:nil
d :: 0':s -> cons:nil
0' :: 0':s
if :: true:false -> 0':s -> cons:nil
le :: 0':s -> 0':s -> true:false
nr :: 0':s
true :: true:false
cons :: cons:nil -> cons:nil
s :: 0':s -> 0':s
false :: true:false
nil :: cons:nil
ack :: 0':s -> 0':s -> 0':s
hole_cons:nil1_0 :: cons:nil
hole_0':s2_0 :: 0':s
hole_true:false3_0 :: true:false
gen_cons:nil4_0 :: Nat -> cons:nil
gen_0':s5_0 :: Nat -> 0':s

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(7) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
d, le, ack

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

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(8)
Obligation:
TRS:
Rules:
numbers -> d(0')
d(x) -> if(le(x, nr), x)
if(true, x) -> cons(d(s(x)))
if(false, x) -> nil
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
nr -> ack(s(s(s(s(s(s(0')))))), 0')
ack(0', x) -> s(x)
ack(s(x), 0') -> ack(x, s(0'))
ack(s(x), s(y)) -> ack(x, ack(s(x), y))

Types:
numbers :: cons:nil
d :: 0':s -> cons:nil
0' :: 0':s
if :: true:false -> 0':s -> cons:nil
le :: 0':s -> 0':s -> true:false
nr :: 0':s
true :: true:false
cons :: cons:nil -> cons:nil
s :: 0':s -> 0':s
false :: true:false
nil :: cons:nil
ack :: 0':s -> 0':s -> 0':s
hole_cons:nil1_0 :: cons:nil
hole_0':s2_0 :: 0':s
hole_true:false3_0 :: true:false
gen_cons:nil4_0 :: Nat -> cons:nil
gen_0':s5_0 :: Nat -> 0':s


Generator Equations:
gen_cons:nil4_0(0) <=> nil
gen_cons:nil4_0(+(x, 1)) <=> cons(gen_cons:nil4_0(x))
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))


The following defined symbols remain to be analysed:
le, d, ack

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

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(9) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
le(gen_0':s5_0(n7_0), gen_0':s5_0(n7_0)) -> true, rt in Omega(1 + n7_0)

Induction Base:
le(gen_0':s5_0(0), gen_0':s5_0(0)) ->_R^Omega(1)
true

Induction Step:
le(gen_0':s5_0(+(n7_0, 1)), gen_0':s5_0(+(n7_0, 1))) ->_R^Omega(1)
le(gen_0':s5_0(n7_0), gen_0':s5_0(n7_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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(10)
Complex Obligation (BEST)

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(11)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
numbers -> d(0')
d(x) -> if(le(x, nr), x)
if(true, x) -> cons(d(s(x)))
if(false, x) -> nil
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
nr -> ack(s(s(s(s(s(s(0')))))), 0')
ack(0', x) -> s(x)
ack(s(x), 0') -> ack(x, s(0'))
ack(s(x), s(y)) -> ack(x, ack(s(x), y))

Types:
numbers :: cons:nil
d :: 0':s -> cons:nil
0' :: 0':s
if :: true:false -> 0':s -> cons:nil
le :: 0':s -> 0':s -> true:false
nr :: 0':s
true :: true:false
cons :: cons:nil -> cons:nil
s :: 0':s -> 0':s
false :: true:false
nil :: cons:nil
ack :: 0':s -> 0':s -> 0':s
hole_cons:nil1_0 :: cons:nil
hole_0':s2_0 :: 0':s
hole_true:false3_0 :: true:false
gen_cons:nil4_0 :: Nat -> cons:nil
gen_0':s5_0 :: Nat -> 0':s


Generator Equations:
gen_cons:nil4_0(0) <=> nil
gen_cons:nil4_0(+(x, 1)) <=> cons(gen_cons:nil4_0(x))
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))


The following defined symbols remain to be analysed:
le, d, ack

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

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(12) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(13)
BOUNDS(n^1, INF)

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(14)
Obligation:
TRS:
Rules:
numbers -> d(0')
d(x) -> if(le(x, nr), x)
if(true, x) -> cons(d(s(x)))
if(false, x) -> nil
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
nr -> ack(s(s(s(s(s(s(0')))))), 0')
ack(0', x) -> s(x)
ack(s(x), 0') -> ack(x, s(0'))
ack(s(x), s(y)) -> ack(x, ack(s(x), y))

Types:
numbers :: cons:nil
d :: 0':s -> cons:nil
0' :: 0':s
if :: true:false -> 0':s -> cons:nil
le :: 0':s -> 0':s -> true:false
nr :: 0':s
true :: true:false
cons :: cons:nil -> cons:nil
s :: 0':s -> 0':s
false :: true:false
nil :: cons:nil
ack :: 0':s -> 0':s -> 0':s
hole_cons:nil1_0 :: cons:nil
hole_0':s2_0 :: 0':s
hole_true:false3_0 :: true:false
gen_cons:nil4_0 :: Nat -> cons:nil
gen_0':s5_0 :: Nat -> 0':s


Lemmas:
le(gen_0':s5_0(n7_0), gen_0':s5_0(n7_0)) -> true, rt in Omega(1 + n7_0)


Generator Equations:
gen_cons:nil4_0(0) <=> nil
gen_cons:nil4_0(+(x, 1)) <=> cons(gen_cons:nil4_0(x))
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))


The following defined symbols remain to be analysed:
d, ack
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(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ack(gen_0':s5_0(1), gen_0':s5_0(+(1, n910_0))) -> *6_0, rt in Omega(n910_0)

Induction Base:
ack(gen_0':s5_0(1), gen_0':s5_0(+(1, 0)))

Induction Step:
ack(gen_0':s5_0(1), gen_0':s5_0(+(1, +(n910_0, 1)))) ->_R^Omega(1)
ack(gen_0':s5_0(0), ack(s(gen_0':s5_0(0)), gen_0':s5_0(+(1, n910_0)))) ->_IH
ack(gen_0':s5_0(0), *6_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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(16)
BOUNDS(1, INF)