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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 270 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), 86 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 54 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:

   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   double(0) -> 0
   double(s(x)) -> s(s(double(x)))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   plus(s(x), y) -> plus(x, s(y))
   plus(s(x), y) -> s(plus(minus(x, y), double(y)))
   plus(s(plus(x, y)), z) -> s(plus(plus(x, y), z))

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:

   minus(x, 0') -> x
   minus(s(x), s(y)) -> minus(x, y)
   double(0') -> 0'
   double(s(x)) -> s(s(double(x)))
   plus(0', y) -> y
   plus(s(x), y) -> s(plus(x, y))
   plus(s(x), y) -> plus(x, s(y))
   plus(s(x), y) -> s(plus(minus(x, y), double(y)))
   plus(s(plus(x, y)), z) -> s(plus(plus(x, y), z))

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:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
plus(s(x), y) -> plus(x, s(y))
plus(s(x), y) -> s(plus(minus(x, y), double(y)))
plus(s(plus(x, y)), z) -> s(plus(plus(x, y), z))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
double :: 0':s -> 0':s
plus :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s

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

They will be analysed ascendingly in the following order:
minus < plus
double < plus

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(6)
Obligation:
TRS:
Rules:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
plus(s(x), y) -> plus(x, s(y))
plus(s(x), y) -> s(plus(minus(x, y), double(y)))
plus(s(plus(x, y)), z) -> s(plus(plus(x, y), z))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
double :: 0':s -> 0':s
plus :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
minus, double, plus

They will be analysed ascendingly in the following order:
minus < plus
double < plus

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(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) -> gen_0':s2_0(0), rt in Omega(1 + n4_0)

Induction Base:
minus(gen_0':s2_0(0), gen_0':s2_0(0)) ->_R^Omega(1)
gen_0':s2_0(0)

Induction Step:
minus(gen_0':s2_0(+(n4_0, 1)), gen_0':s2_0(+(n4_0, 1))) ->_R^Omega(1)
minus(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) ->_IH
gen_0':s2_0(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:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
plus(s(x), y) -> plus(x, s(y))
plus(s(x), y) -> s(plus(minus(x, y), double(y)))
plus(s(plus(x, y)), z) -> s(plus(plus(x, y), z))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
double :: 0':s -> 0':s
plus :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
minus, double, plus

They will be analysed ascendingly in the following order:
minus < plus
double < plus

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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:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
plus(s(x), y) -> plus(x, s(y))
plus(s(x), y) -> s(plus(minus(x, y), double(y)))
plus(s(plus(x, y)), z) -> s(plus(plus(x, y), z))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
double :: 0':s -> 0':s
plus :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Lemmas:
minus(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) -> gen_0':s2_0(0), rt in Omega(1 + n4_0)


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
double, plus

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

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(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
double(gen_0':s2_0(n240_0)) -> gen_0':s2_0(*(2, n240_0)), rt in Omega(1 + n240_0)

Induction Base:
double(gen_0':s2_0(0)) ->_R^Omega(1)
0'

Induction Step:
double(gen_0':s2_0(+(n240_0, 1))) ->_R^Omega(1)
s(s(double(gen_0':s2_0(n240_0)))) ->_IH
s(s(gen_0':s2_0(*(2, c241_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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(14)
Obligation:
TRS:
Rules:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
plus(s(x), y) -> plus(x, s(y))
plus(s(x), y) -> s(plus(minus(x, y), double(y)))
plus(s(plus(x, y)), z) -> s(plus(plus(x, y), z))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
double :: 0':s -> 0':s
plus :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Lemmas:
minus(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) -> gen_0':s2_0(0), rt in Omega(1 + n4_0)
double(gen_0':s2_0(n240_0)) -> gen_0':s2_0(*(2, n240_0)), rt in Omega(1 + n240_0)


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
plus
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(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_0':s2_0(n484_0), gen_0':s2_0(b)) -> gen_0':s2_0(+(n484_0, b)), rt in Omega(1 + n484_0)

Induction Base:
plus(gen_0':s2_0(0), gen_0':s2_0(b)) ->_R^Omega(1)
gen_0':s2_0(b)

Induction Step:
plus(gen_0':s2_0(+(n484_0, 1)), gen_0':s2_0(b)) ->_R^Omega(1)
s(plus(gen_0':s2_0(n484_0), gen_0':s2_0(b))) ->_IH
s(gen_0':s2_0(+(b, c485_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)