/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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), 291 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), 69 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 115 ms]
        (16) typed CpxTrs


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

(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:

   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   min(x, 0) -> 0
   min(0, y) -> 0
   min(s(x), s(y)) -> s(min(x, y))
   twice(0) -> 0
   twice(s(x)) -> s(s(twice(x)))
   f(s(x), s(y)) -> f(-(y, min(x, y)), s(twice(min(x, y))))
   f(s(x), s(y)) -> f(-(x, min(x, y)), s(twice(min(x, y))))

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:

   -(x, 0') -> x
   -(s(x), s(y)) -> -(x, y)
   min(x, 0') -> 0'
   min(0', y) -> 0'
   min(s(x), s(y)) -> s(min(x, y))
   twice(0') -> 0'
   twice(s(x)) -> s(s(twice(x)))
   f(s(x), s(y)) -> f(-(y, min(x, y)), s(twice(min(x, y))))
   f(s(x), s(y)) -> f(-(x, min(x, y)), s(twice(min(x, y))))

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

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

(4)
Obligation:
Innermost TRS:
Rules:
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
min(x, 0') -> 0'
min(0', y) -> 0'
min(s(x), s(y)) -> s(min(x, y))
twice(0') -> 0'
twice(s(x)) -> s(s(twice(x)))
f(s(x), s(y)) -> f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) -> f(-(x, min(x, y)), s(twice(min(x, y))))

Types:
- :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
min :: 0':s -> 0':s -> 0':s
twice :: 0':s -> 0':s
f :: 0':s -> 0':s -> f
hole_0':s1_0 :: 0':s
hole_f2_0 :: f
gen_0':s3_0 :: Nat -> 0':s

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
-, min, twice, f

They will be analysed ascendingly in the following order:
- < f
min < f
twice < f

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

(6)
Obligation:
Innermost TRS:
Rules:
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
min(x, 0') -> 0'
min(0', y) -> 0'
min(s(x), s(y)) -> s(min(x, y))
twice(0') -> 0'
twice(s(x)) -> s(s(twice(x)))
f(s(x), s(y)) -> f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) -> f(-(x, min(x, y)), s(twice(min(x, y))))

Types:
- :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
min :: 0':s -> 0':s -> 0':s
twice :: 0':s -> 0':s
f :: 0':s -> 0':s -> f
hole_0':s1_0 :: 0':s
hole_f2_0 :: f
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:
-, min, twice, f

They will be analysed ascendingly in the following order:
- < f
min < f
twice < f

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
-(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(0), rt in Omega(1 + n5_0)

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

Induction Step:
-(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1)
-(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) ->_IH
gen_0':s3_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:

Innermost TRS:
Rules:
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
min(x, 0') -> 0'
min(0', y) -> 0'
min(s(x), s(y)) -> s(min(x, y))
twice(0') -> 0'
twice(s(x)) -> s(s(twice(x)))
f(s(x), s(y)) -> f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) -> f(-(x, min(x, y)), s(twice(min(x, y))))

Types:
- :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
min :: 0':s -> 0':s -> 0':s
twice :: 0':s -> 0':s
f :: 0':s -> 0':s -> f
hole_0':s1_0 :: 0':s
hole_f2_0 :: f
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:
-, min, twice, f

They will be analysed ascendingly in the following order:
- < f
min < f
twice < f

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

(10) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
Innermost TRS:
Rules:
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
min(x, 0') -> 0'
min(0', y) -> 0'
min(s(x), s(y)) -> s(min(x, y))
twice(0') -> 0'
twice(s(x)) -> s(s(twice(x)))
f(s(x), s(y)) -> f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) -> f(-(x, min(x, y)), s(twice(min(x, y))))

Types:
- :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
min :: 0':s -> 0':s -> 0':s
twice :: 0':s -> 0':s
f :: 0':s -> 0':s -> f
hole_0':s1_0 :: 0':s
hole_f2_0 :: f
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
-(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(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:
min, twice, f

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

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(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
min(gen_0':s3_0(n297_0), gen_0':s3_0(n297_0)) -> gen_0':s3_0(n297_0), rt in Omega(1 + n297_0)

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

Induction Step:
min(gen_0':s3_0(+(n297_0, 1)), gen_0':s3_0(+(n297_0, 1))) ->_R^Omega(1)
s(min(gen_0':s3_0(n297_0), gen_0':s3_0(n297_0))) ->_IH
s(gen_0':s3_0(c298_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:
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
min(x, 0') -> 0'
min(0', y) -> 0'
min(s(x), s(y)) -> s(min(x, y))
twice(0') -> 0'
twice(s(x)) -> s(s(twice(x)))
f(s(x), s(y)) -> f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) -> f(-(x, min(x, y)), s(twice(min(x, y))))

Types:
- :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
min :: 0':s -> 0':s -> 0':s
twice :: 0':s -> 0':s
f :: 0':s -> 0':s -> f
hole_0':s1_0 :: 0':s
hole_f2_0 :: f
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
-(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(0), rt in Omega(1 + n5_0)
min(gen_0':s3_0(n297_0), gen_0':s3_0(n297_0)) -> gen_0':s3_0(n297_0), rt in Omega(1 + n297_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:
twice, f

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

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
twice(gen_0':s3_0(n635_0)) -> gen_0':s3_0(*(2, n635_0)), rt in Omega(1 + n635_0)

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

Induction Step:
twice(gen_0':s3_0(+(n635_0, 1))) ->_R^Omega(1)
s(s(twice(gen_0':s3_0(n635_0)))) ->_IH
s(s(gen_0':s3_0(*(2, c636_0))))

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

(16)
Obligation:
Innermost TRS:
Rules:
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
min(x, 0') -> 0'
min(0', y) -> 0'
min(s(x), s(y)) -> s(min(x, y))
twice(0') -> 0'
twice(s(x)) -> s(s(twice(x)))
f(s(x), s(y)) -> f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) -> f(-(x, min(x, y)), s(twice(min(x, y))))

Types:
- :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
min :: 0':s -> 0':s -> 0':s
twice :: 0':s -> 0':s
f :: 0':s -> 0':s -> f
hole_0':s1_0 :: 0':s
hole_f2_0 :: f
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
-(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(0), rt in Omega(1 + n5_0)
min(gen_0':s3_0(n297_0), gen_0':s3_0(n297_0)) -> gen_0':s3_0(n297_0), rt in Omega(1 + n297_0)
twice(gen_0':s3_0(n635_0)) -> gen_0':s3_0(*(2, n635_0)), rt in Omega(1 + n635_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