/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 286 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), 37 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 468 ms]
        (16) 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:

   tower(x) -> f(a, x, s(0))
   f(a, 0, y) -> y
   f(a, s(x), y) -> f(b, y, s(x))
   f(b, y, x) -> f(a, half(x), exp(y))
   exp(0) -> s(0)
   exp(s(x)) -> double(exp(x))
   double(0) -> 0
   double(s(x)) -> s(s(double(x)))
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(x))) -> s(half(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:

   tower(x) -> f(a, x, s(0'))
   f(a, 0', y) -> y
   f(a, s(x), y) -> f(b, y, s(x))
   f(b, y, x) -> f(a, half(x), exp(y))
   exp(0') -> s(0')
   exp(s(x)) -> double(exp(x))
   double(0') -> 0'
   double(s(x)) -> s(s(double(x)))
   half(0') -> double(0')
   half(s(0')) -> half(0')
   half(s(s(x))) -> s(half(x))

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

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

(4)
Obligation:
TRS:
Rules:
tower(x) -> f(a, x, s(0'))
f(a, 0', y) -> y
f(a, s(x), y) -> f(b, y, s(x))
f(b, y, x) -> f(a, half(x), exp(y))
exp(0') -> s(0')
exp(s(x)) -> double(exp(x))
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
half(0') -> double(0')
half(s(0')) -> half(0')
half(s(s(x))) -> s(half(x))

Types:
tower :: 0':s -> 0':s
f :: a:b -> 0':s -> 0':s -> 0':s
a :: a:b
s :: 0':s -> 0':s
0' :: 0':s
b :: a:b
half :: 0':s -> 0':s
exp :: 0':s -> 0':s
double :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_a:b2_0 :: a:b
gen_0':s3_0 :: Nat -> 0':s

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(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
f, half, exp, double

They will be analysed ascendingly in the following order:
half < f
exp < f
double < half
double < exp

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(6)
Obligation:
TRS:
Rules:
tower(x) -> f(a, x, s(0'))
f(a, 0', y) -> y
f(a, s(x), y) -> f(b, y, s(x))
f(b, y, x) -> f(a, half(x), exp(y))
exp(0') -> s(0')
exp(s(x)) -> double(exp(x))
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
half(0') -> double(0')
half(s(0')) -> half(0')
half(s(s(x))) -> s(half(x))

Types:
tower :: 0':s -> 0':s
f :: a:b -> 0':s -> 0':s -> 0':s
a :: a:b
s :: 0':s -> 0':s
0' :: 0':s
b :: a:b
half :: 0':s -> 0':s
exp :: 0':s -> 0':s
double :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_a:b2_0 :: a:b
gen_0':s3_0 :: Nat -> 0':s


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
double, f, half, exp

They will be analysed ascendingly in the following order:
half < f
exp < f
double < half
double < exp

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(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
double(gen_0':s3_0(n5_0)) -> gen_0':s3_0(*(2, n5_0)), rt in Omega(1 + n5_0)

Induction Base:
double(gen_0':s3_0(0)) ->_R^Omega(1)
0'

Induction Step:
double(gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1)
s(s(double(gen_0':s3_0(n5_0)))) ->_IH
s(s(gen_0':s3_0(*(2, c6_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:

TRS:
Rules:
tower(x) -> f(a, x, s(0'))
f(a, 0', y) -> y
f(a, s(x), y) -> f(b, y, s(x))
f(b, y, x) -> f(a, half(x), exp(y))
exp(0') -> s(0')
exp(s(x)) -> double(exp(x))
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
half(0') -> double(0')
half(s(0')) -> half(0')
half(s(s(x))) -> s(half(x))

Types:
tower :: 0':s -> 0':s
f :: a:b -> 0':s -> 0':s -> 0':s
a :: a:b
s :: 0':s -> 0':s
0' :: 0':s
b :: a:b
half :: 0':s -> 0':s
exp :: 0':s -> 0':s
double :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_a:b2_0 :: a:b
gen_0':s3_0 :: Nat -> 0':s


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
double, f, half, exp

They will be analysed ascendingly in the following order:
half < f
exp < f
double < half
double < exp

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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:
tower(x) -> f(a, x, s(0'))
f(a, 0', y) -> y
f(a, s(x), y) -> f(b, y, s(x))
f(b, y, x) -> f(a, half(x), exp(y))
exp(0') -> s(0')
exp(s(x)) -> double(exp(x))
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
half(0') -> double(0')
half(s(0')) -> half(0')
half(s(s(x))) -> s(half(x))

Types:
tower :: 0':s -> 0':s
f :: a:b -> 0':s -> 0':s -> 0':s
a :: a:b
s :: 0':s -> 0':s
0' :: 0':s
b :: a:b
half :: 0':s -> 0':s
exp :: 0':s -> 0':s
double :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_a:b2_0 :: a:b
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
double(gen_0':s3_0(n5_0)) -> gen_0':s3_0(*(2, n5_0)), rt in Omega(1 + n5_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
half, f, exp

They will be analysed ascendingly in the following order:
half < f
exp < f

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(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
half(gen_0':s3_0(*(2, n253_0))) -> gen_0':s3_0(n253_0), rt in Omega(1 + n253_0)

Induction Base:
half(gen_0':s3_0(*(2, 0))) ->_R^Omega(1)
double(0') ->_L^Omega(1)
gen_0':s3_0(*(2, 0))

Induction Step:
half(gen_0':s3_0(*(2, +(n253_0, 1)))) ->_R^Omega(1)
s(half(gen_0':s3_0(*(2, n253_0)))) ->_IH
s(gen_0':s3_0(c254_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:
tower(x) -> f(a, x, s(0'))
f(a, 0', y) -> y
f(a, s(x), y) -> f(b, y, s(x))
f(b, y, x) -> f(a, half(x), exp(y))
exp(0') -> s(0')
exp(s(x)) -> double(exp(x))
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
half(0') -> double(0')
half(s(0')) -> half(0')
half(s(s(x))) -> s(half(x))

Types:
tower :: 0':s -> 0':s
f :: a:b -> 0':s -> 0':s -> 0':s
a :: a:b
s :: 0':s -> 0':s
0' :: 0':s
b :: a:b
half :: 0':s -> 0':s
exp :: 0':s -> 0':s
double :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_a:b2_0 :: a:b
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
double(gen_0':s3_0(n5_0)) -> gen_0':s3_0(*(2, n5_0)), rt in Omega(1 + n5_0)
half(gen_0':s3_0(*(2, n253_0))) -> gen_0':s3_0(n253_0), rt in Omega(1 + n253_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
exp, f

They will be analysed ascendingly in the following order:
exp < f

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(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
exp(gen_0':s3_0(+(1, n586_0))) -> *4_0, rt in Omega(n586_0)

Induction Base:
exp(gen_0':s3_0(+(1, 0)))

Induction Step:
exp(gen_0':s3_0(+(1, +(n586_0, 1)))) ->_R^Omega(1)
double(exp(gen_0':s3_0(+(1, n586_0)))) ->_IH
double(*4_0)

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

(16)
Obligation:
TRS:
Rules:
tower(x) -> f(a, x, s(0'))
f(a, 0', y) -> y
f(a, s(x), y) -> f(b, y, s(x))
f(b, y, x) -> f(a, half(x), exp(y))
exp(0') -> s(0')
exp(s(x)) -> double(exp(x))
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
half(0') -> double(0')
half(s(0')) -> half(0')
half(s(s(x))) -> s(half(x))

Types:
tower :: 0':s -> 0':s
f :: a:b -> 0':s -> 0':s -> 0':s
a :: a:b
s :: 0':s -> 0':s
0' :: 0':s
b :: a:b
half :: 0':s -> 0':s
exp :: 0':s -> 0':s
double :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_a:b2_0 :: a:b
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
double(gen_0':s3_0(n5_0)) -> gen_0':s3_0(*(2, n5_0)), rt in Omega(1 + n5_0)
half(gen_0':s3_0(*(2, n253_0))) -> gen_0':s3_0(n253_0), rt in Omega(1 + n253_0)
exp(gen_0':s3_0(+(1, n586_0))) -> *4_0, rt in Omega(n586_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
f