/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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 (innermost) 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), 564 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), 295 ms]
        (14) typed CpxTrs


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(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   double(x) -> permute(x, x, a)
   permute(x, y, a) -> permute(isZero(x), x, b)
   permute(false, x, b) -> permute(ack(x, x), p(x), c)
   permute(true, x, b) -> 0
   permute(y, x, c) -> s(s(permute(x, y, a)))
   p(0) -> 0
   p(s(x)) -> x
   ack(0, x) -> plus(x, s(0))
   ack(s(x), 0) -> ack(x, s(0))
   ack(s(x), s(y)) -> ack(x, ack(s(x), y))
   plus(0, y) -> y
   plus(s(x), y) -> plus(x, s(y))
   plus(x, s(s(y))) -> s(plus(s(x), y))
   plus(x, s(0)) -> s(x)
   plus(x, 0) -> x
   isZero(0) -> true
   isZero(s(x)) -> false

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

(1) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   double(x) -> permute(x, x, a)
   permute(x, y, a) -> permute(isZero(x), x, b)
   permute(false, x, b) -> permute(ack(x, x), p(x), c)
   permute(true, x, b) -> 0'
   permute(y, x, c) -> s(s(permute(x, y, a)))
   p(0') -> 0'
   p(s(x)) -> x
   ack(0', x) -> plus(x, s(0'))
   ack(s(x), 0') -> ack(x, s(0'))
   ack(s(x), s(y)) -> ack(x, ack(s(x), y))
   plus(0', y) -> y
   plus(s(x), y) -> plus(x, s(y))
   plus(x, s(s(y))) -> s(plus(s(x), y))
   plus(x, s(0')) -> s(x)
   plus(x, 0') -> x
   isZero(0') -> true
   isZero(s(x)) -> false

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

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

(4)
Obligation:
Innermost TRS:
Rules:
double(x) -> permute(x, x, a)
permute(x, y, a) -> permute(isZero(x), x, b)
permute(false, x, b) -> permute(ack(x, x), p(x), c)
permute(true, x, b) -> 0'
permute(y, x, c) -> s(s(permute(x, y, a)))
p(0') -> 0'
p(s(x)) -> x
ack(0', x) -> plus(x, s(0'))
ack(s(x), 0') -> ack(x, s(0'))
ack(s(x), s(y)) -> ack(x, ack(s(x), y))
plus(0', y) -> y
plus(s(x), y) -> plus(x, s(y))
plus(x, s(s(y))) -> s(plus(s(x), y))
plus(x, s(0')) -> s(x)
plus(x, 0') -> x
isZero(0') -> true
isZero(s(x)) -> false

Types:
double :: false:true:0':s -> false:true:0':s
permute :: false:true:0':s -> false:true:0':s -> a:b:c -> false:true:0':s
a :: a:b:c
isZero :: false:true:0':s -> false:true:0':s
b :: a:b:c
false :: false:true:0':s
ack :: false:true:0':s -> false:true:0':s -> false:true:0':s
p :: false:true:0':s -> false:true:0':s
c :: a:b:c
true :: false:true:0':s
0' :: false:true:0':s
s :: false:true:0':s -> false:true:0':s
plus :: false:true:0':s -> false:true:0':s -> false:true:0':s
hole_false:true:0':s1_0 :: false:true:0':s
hole_a:b:c2_0 :: a:b:c
gen_false:true:0':s3_0 :: Nat -> false:true:0':s

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

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

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

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(6)
Obligation:
Innermost TRS:
Rules:
double(x) -> permute(x, x, a)
permute(x, y, a) -> permute(isZero(x), x, b)
permute(false, x, b) -> permute(ack(x, x), p(x), c)
permute(true, x, b) -> 0'
permute(y, x, c) -> s(s(permute(x, y, a)))
p(0') -> 0'
p(s(x)) -> x
ack(0', x) -> plus(x, s(0'))
ack(s(x), 0') -> ack(x, s(0'))
ack(s(x), s(y)) -> ack(x, ack(s(x), y))
plus(0', y) -> y
plus(s(x), y) -> plus(x, s(y))
plus(x, s(s(y))) -> s(plus(s(x), y))
plus(x, s(0')) -> s(x)
plus(x, 0') -> x
isZero(0') -> true
isZero(s(x)) -> false

Types:
double :: false:true:0':s -> false:true:0':s
permute :: false:true:0':s -> false:true:0':s -> a:b:c -> false:true:0':s
a :: a:b:c
isZero :: false:true:0':s -> false:true:0':s
b :: a:b:c
false :: false:true:0':s
ack :: false:true:0':s -> false:true:0':s -> false:true:0':s
p :: false:true:0':s -> false:true:0':s
c :: a:b:c
true :: false:true:0':s
0' :: false:true:0':s
s :: false:true:0':s -> false:true:0':s
plus :: false:true:0':s -> false:true:0':s -> false:true:0':s
hole_false:true:0':s1_0 :: false:true:0':s
hole_a:b:c2_0 :: a:b:c
gen_false:true:0':s3_0 :: Nat -> false:true:0':s


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


The following defined symbols remain to be analysed:
plus, permute, ack

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

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(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_false:true:0':s3_0(a), gen_false:true:0':s3_0(+(2, *(2, n5_0)))) -> *4_0, rt in Omega(n5_0)

Induction Base:
plus(gen_false:true:0':s3_0(a), gen_false:true:0':s3_0(+(2, *(2, 0))))

Induction Step:
plus(gen_false:true:0':s3_0(a), gen_false:true:0':s3_0(+(2, *(2, +(n5_0, 1))))) ->_R^Omega(1)
s(plus(s(gen_false:true:0':s3_0(a)), gen_false:true:0':s3_0(+(2, *(2, n5_0))))) ->_IH
s(*4_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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(8)
Complex Obligation (BEST)

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(9)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
double(x) -> permute(x, x, a)
permute(x, y, a) -> permute(isZero(x), x, b)
permute(false, x, b) -> permute(ack(x, x), p(x), c)
permute(true, x, b) -> 0'
permute(y, x, c) -> s(s(permute(x, y, a)))
p(0') -> 0'
p(s(x)) -> x
ack(0', x) -> plus(x, s(0'))
ack(s(x), 0') -> ack(x, s(0'))
ack(s(x), s(y)) -> ack(x, ack(s(x), y))
plus(0', y) -> y
plus(s(x), y) -> plus(x, s(y))
plus(x, s(s(y))) -> s(plus(s(x), y))
plus(x, s(0')) -> s(x)
plus(x, 0') -> x
isZero(0') -> true
isZero(s(x)) -> false

Types:
double :: false:true:0':s -> false:true:0':s
permute :: false:true:0':s -> false:true:0':s -> a:b:c -> false:true:0':s
a :: a:b:c
isZero :: false:true:0':s -> false:true:0':s
b :: a:b:c
false :: false:true:0':s
ack :: false:true:0':s -> false:true:0':s -> false:true:0':s
p :: false:true:0':s -> false:true:0':s
c :: a:b:c
true :: false:true:0':s
0' :: false:true:0':s
s :: false:true:0':s -> false:true:0':s
plus :: false:true:0':s -> false:true:0':s -> false:true:0':s
hole_false:true:0':s1_0 :: false:true:0':s
hole_a:b:c2_0 :: a:b:c
gen_false:true:0':s3_0 :: Nat -> false:true:0':s


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


The following defined symbols remain to be analysed:
plus, permute, ack

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

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(10) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
Innermost TRS:
Rules:
double(x) -> permute(x, x, a)
permute(x, y, a) -> permute(isZero(x), x, b)
permute(false, x, b) -> permute(ack(x, x), p(x), c)
permute(true, x, b) -> 0'
permute(y, x, c) -> s(s(permute(x, y, a)))
p(0') -> 0'
p(s(x)) -> x
ack(0', x) -> plus(x, s(0'))
ack(s(x), 0') -> ack(x, s(0'))
ack(s(x), s(y)) -> ack(x, ack(s(x), y))
plus(0', y) -> y
plus(s(x), y) -> plus(x, s(y))
plus(x, s(s(y))) -> s(plus(s(x), y))
plus(x, s(0')) -> s(x)
plus(x, 0') -> x
isZero(0') -> true
isZero(s(x)) -> false

Types:
double :: false:true:0':s -> false:true:0':s
permute :: false:true:0':s -> false:true:0':s -> a:b:c -> false:true:0':s
a :: a:b:c
isZero :: false:true:0':s -> false:true:0':s
b :: a:b:c
false :: false:true:0':s
ack :: false:true:0':s -> false:true:0':s -> false:true:0':s
p :: false:true:0':s -> false:true:0':s
c :: a:b:c
true :: false:true:0':s
0' :: false:true:0':s
s :: false:true:0':s -> false:true:0':s
plus :: false:true:0':s -> false:true:0':s -> false:true:0':s
hole_false:true:0':s1_0 :: false:true:0':s
hole_a:b:c2_0 :: a:b:c
gen_false:true:0':s3_0 :: Nat -> false:true:0':s


Lemmas:
plus(gen_false:true:0':s3_0(a), gen_false:true:0':s3_0(+(2, *(2, n5_0)))) -> *4_0, rt in Omega(n5_0)


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


The following defined symbols remain to be analysed:
ack, permute

They will be analysed ascendingly in the following order:
ack < permute

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ack(gen_false:true:0':s3_0(1), gen_false:true:0':s3_0(+(1, n2887_0))) -> *4_0, rt in Omega(n2887_0)

Induction Base:
ack(gen_false:true:0':s3_0(1), gen_false:true:0':s3_0(+(1, 0)))

Induction Step:
ack(gen_false:true:0':s3_0(1), gen_false:true:0':s3_0(+(1, +(n2887_0, 1)))) ->_R^Omega(1)
ack(gen_false:true:0':s3_0(0), ack(s(gen_false:true:0':s3_0(0)), gen_false:true:0':s3_0(+(1, n2887_0)))) ->_IH
ack(gen_false:true:0':s3_0(0), *4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(14)
Obligation:
Innermost TRS:
Rules:
double(x) -> permute(x, x, a)
permute(x, y, a) -> permute(isZero(x), x, b)
permute(false, x, b) -> permute(ack(x, x), p(x), c)
permute(true, x, b) -> 0'
permute(y, x, c) -> s(s(permute(x, y, a)))
p(0') -> 0'
p(s(x)) -> x
ack(0', x) -> plus(x, s(0'))
ack(s(x), 0') -> ack(x, s(0'))
ack(s(x), s(y)) -> ack(x, ack(s(x), y))
plus(0', y) -> y
plus(s(x), y) -> plus(x, s(y))
plus(x, s(s(y))) -> s(plus(s(x), y))
plus(x, s(0')) -> s(x)
plus(x, 0') -> x
isZero(0') -> true
isZero(s(x)) -> false

Types:
double :: false:true:0':s -> false:true:0':s
permute :: false:true:0':s -> false:true:0':s -> a:b:c -> false:true:0':s
a :: a:b:c
isZero :: false:true:0':s -> false:true:0':s
b :: a:b:c
false :: false:true:0':s
ack :: false:true:0':s -> false:true:0':s -> false:true:0':s
p :: false:true:0':s -> false:true:0':s
c :: a:b:c
true :: false:true:0':s
0' :: false:true:0':s
s :: false:true:0':s -> false:true:0':s
plus :: false:true:0':s -> false:true:0':s -> false:true:0':s
hole_false:true:0':s1_0 :: false:true:0':s
hole_a:b:c2_0 :: a:b:c
gen_false:true:0':s3_0 :: Nat -> false:true:0':s


Lemmas:
plus(gen_false:true:0':s3_0(a), gen_false:true:0':s3_0(+(2, *(2, n5_0)))) -> *4_0, rt in Omega(n5_0)
ack(gen_false:true:0':s3_0(1), gen_false:true:0':s3_0(+(1, n2887_0))) -> *4_0, rt in Omega(n2887_0)


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


The following defined symbols remain to be analysed:
permute