WORST_CASE(Omega(n^1),?)
proof of /export/starexec/sandbox/benchmark/theBenchmark.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


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

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 200 ms]
(2) CpxRelTRS
(3) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CpxRelTRS
(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), 285 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 90 ms]
(12) BEST
    (13) proven lower bound
        (14) LowerBoundPropagationProof [FINISHED, 0 ms]
        (15) BOUNDS(n^1, INF)
    (16) typed CpxTrs


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


The TRS R consists of the following rules:

   mergesort(Cons(x', Cons(x, xs))) -> splitmerge(Cons(x', Cons(x, xs)), Nil, Nil)
   mergesort(Cons(x, Nil)) -> Cons(x, Nil)
   merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](<=(x', x), Cons(x', xs'), Cons(x, xs))
   merge(Cons(x, xs), Nil) -> Cons(x, xs)
   splitmerge(Cons(x, xs), xs1, xs2) -> splitmerge(xs, Cons(x, xs2), xs1)
   splitmerge(Nil, xs1, xs2) -> merge(mergesort(xs1), mergesort(xs2))
   mergesort(Nil) -> Nil
   merge(Nil, xs2) -> xs2
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   goal(xs) -> mergesort(xs)

The (relative) TRS S consists of the following rules:

   <=(S(x), S(y)) -> <=(x, y)
   <=(0, y) -> True
   <=(S(x), 0) -> False
   merge[Ite](False, xs1, Cons(x, xs)) -> Cons(x, merge(xs1, xs))
   merge[Ite](True, Cons(x, xs), xs2) -> Cons(x, merge(xs, xs2))

Rewrite Strategy: INNERMOST
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(1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
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(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   mergesort(Cons(x', Cons(x, xs))) -> splitmerge(Cons(x', Cons(x, xs)), Nil, Nil)
   mergesort(Cons(x, Nil)) -> Cons(x, Nil)
   merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](<=(x', x), Cons(x', xs'), Cons(x, xs))
   merge(Cons(x, xs), Nil) -> Cons(x, xs)
   splitmerge(Cons(x, xs), xs1, xs2) -> splitmerge(xs, Cons(x, xs2), xs1)
   splitmerge(Nil, xs1, xs2) -> merge(mergesort(xs1), mergesort(xs2))
   mergesort(Nil) -> Nil
   merge(Nil, xs2) -> xs2
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   goal(xs) -> mergesort(xs)

The (relative) TRS S consists of the following rules:

   <=(S(x), S(y)) -> <=(x, y)
   <=(0, y) -> True
   <=(S(x), 0) -> False
   merge[Ite](False, xs1, Cons(x, xs)) -> Cons(x, merge(xs1, xs))
   merge[Ite](True, Cons(x, xs), xs2) -> Cons(x, merge(xs, xs2))

Rewrite Strategy: INNERMOST
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(3) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   mergesort(Cons(x', Cons(x, xs))) -> splitmerge(Cons(x', Cons(x, xs)), Nil, Nil)
   mergesort(Cons(x, Nil)) -> Cons(x, Nil)
   merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](<=(x', x), Cons(x', xs'), Cons(x, xs))
   merge(Cons(x, xs), Nil) -> Cons(x, xs)
   splitmerge(Cons(x, xs), xs1, xs2) -> splitmerge(xs, Cons(x, xs2), xs1)
   splitmerge(Nil, xs1, xs2) -> merge(mergesort(xs1), mergesort(xs2))
   mergesort(Nil) -> Nil
   merge(Nil, xs2) -> xs2
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   goal(xs) -> mergesort(xs)

The (relative) TRS S consists of the following rules:

   <=(S(x), S(y)) -> <=(x, y)
   <=(0', y) -> True
   <=(S(x), 0') -> False
   merge[Ite](False, xs1, Cons(x, xs)) -> Cons(x, merge(xs1, xs))
   merge[Ite](True, Cons(x, xs), xs2) -> Cons(x, merge(xs, xs2))

Rewrite Strategy: INNERMOST
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(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(6)
Obligation:
Innermost TRS:
Rules:
mergesort(Cons(x', Cons(x, xs))) -> splitmerge(Cons(x', Cons(x, xs)), Nil, Nil)
mergesort(Cons(x, Nil)) -> Cons(x, Nil)
merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](<=(x', x), Cons(x', xs'), Cons(x, xs))
merge(Cons(x, xs), Nil) -> Cons(x, xs)
splitmerge(Cons(x, xs), xs1, xs2) -> splitmerge(xs, Cons(x, xs2), xs1)
splitmerge(Nil, xs1, xs2) -> merge(mergesort(xs1), mergesort(xs2))
mergesort(Nil) -> Nil
merge(Nil, xs2) -> xs2
notEmpty(Cons(x, xs)) -> True
notEmpty(Nil) -> False
goal(xs) -> mergesort(xs)
<=(S(x), S(y)) -> <=(x, y)
<=(0', y) -> True
<=(S(x), 0') -> False
merge[Ite](False, xs1, Cons(x, xs)) -> Cons(x, merge(xs1, xs))
merge[Ite](True, Cons(x, xs), xs2) -> Cons(x, merge(xs, xs2))

Types:
mergesort :: Cons:Nil -> Cons:Nil
Cons :: S:0' -> Cons:Nil -> Cons:Nil
splitmerge :: Cons:Nil -> Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
merge :: Cons:Nil -> Cons:Nil -> Cons:Nil
merge[Ite] :: True:False -> Cons:Nil -> Cons:Nil -> Cons:Nil
<= :: S:0' -> S:0' -> True:False
notEmpty :: Cons:Nil -> True:False
True :: True:False
False :: True:False
goal :: Cons:Nil -> Cons:Nil
S :: S:0' -> S:0'
0' :: S:0'
hole_Cons:Nil1_0 :: Cons:Nil
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat -> Cons:Nil
gen_S:0'5_0 :: Nat -> S:0'

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(7) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
mergesort, splitmerge, merge, <=

They will be analysed ascendingly in the following order:
mergesort = splitmerge
merge < splitmerge
<= < merge

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(8)
Obligation:
Innermost TRS:
Rules:
mergesort(Cons(x', Cons(x, xs))) -> splitmerge(Cons(x', Cons(x, xs)), Nil, Nil)
mergesort(Cons(x, Nil)) -> Cons(x, Nil)
merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](<=(x', x), Cons(x', xs'), Cons(x, xs))
merge(Cons(x, xs), Nil) -> Cons(x, xs)
splitmerge(Cons(x, xs), xs1, xs2) -> splitmerge(xs, Cons(x, xs2), xs1)
splitmerge(Nil, xs1, xs2) -> merge(mergesort(xs1), mergesort(xs2))
mergesort(Nil) -> Nil
merge(Nil, xs2) -> xs2
notEmpty(Cons(x, xs)) -> True
notEmpty(Nil) -> False
goal(xs) -> mergesort(xs)
<=(S(x), S(y)) -> <=(x, y)
<=(0', y) -> True
<=(S(x), 0') -> False
merge[Ite](False, xs1, Cons(x, xs)) -> Cons(x, merge(xs1, xs))
merge[Ite](True, Cons(x, xs), xs2) -> Cons(x, merge(xs, xs2))

Types:
mergesort :: Cons:Nil -> Cons:Nil
Cons :: S:0' -> Cons:Nil -> Cons:Nil
splitmerge :: Cons:Nil -> Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
merge :: Cons:Nil -> Cons:Nil -> Cons:Nil
merge[Ite] :: True:False -> Cons:Nil -> Cons:Nil -> Cons:Nil
<= :: S:0' -> S:0' -> True:False
notEmpty :: Cons:Nil -> True:False
True :: True:False
False :: True:False
goal :: Cons:Nil -> Cons:Nil
S :: S:0' -> S:0'
0' :: S:0'
hole_Cons:Nil1_0 :: Cons:Nil
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat -> Cons:Nil
gen_S:0'5_0 :: Nat -> S:0'


Generator Equations:
gen_Cons:Nil4_0(0) <=> Nil
gen_Cons:Nil4_0(+(x, 1)) <=> Cons(0', gen_Cons:Nil4_0(x))
gen_S:0'5_0(0) <=> 0'
gen_S:0'5_0(+(x, 1)) <=> S(gen_S:0'5_0(x))


The following defined symbols remain to be analysed:
<=, mergesort, splitmerge, merge

They will be analysed ascendingly in the following order:
mergesort = splitmerge
merge < splitmerge
<= < merge

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(9) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
<=(gen_S:0'5_0(n7_0), gen_S:0'5_0(n7_0)) -> True, rt in Omega(0)

Induction Base:
<=(gen_S:0'5_0(0), gen_S:0'5_0(0)) ->_R^Omega(0)
True

Induction Step:
<=(gen_S:0'5_0(+(n7_0, 1)), gen_S:0'5_0(+(n7_0, 1))) ->_R^Omega(0)
<=(gen_S:0'5_0(n7_0), gen_S:0'5_0(n7_0)) ->_IH
True

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
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(10)
Obligation:
Innermost TRS:
Rules:
mergesort(Cons(x', Cons(x, xs))) -> splitmerge(Cons(x', Cons(x, xs)), Nil, Nil)
mergesort(Cons(x, Nil)) -> Cons(x, Nil)
merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](<=(x', x), Cons(x', xs'), Cons(x, xs))
merge(Cons(x, xs), Nil) -> Cons(x, xs)
splitmerge(Cons(x, xs), xs1, xs2) -> splitmerge(xs, Cons(x, xs2), xs1)
splitmerge(Nil, xs1, xs2) -> merge(mergesort(xs1), mergesort(xs2))
mergesort(Nil) -> Nil
merge(Nil, xs2) -> xs2
notEmpty(Cons(x, xs)) -> True
notEmpty(Nil) -> False
goal(xs) -> mergesort(xs)
<=(S(x), S(y)) -> <=(x, y)
<=(0', y) -> True
<=(S(x), 0') -> False
merge[Ite](False, xs1, Cons(x, xs)) -> Cons(x, merge(xs1, xs))
merge[Ite](True, Cons(x, xs), xs2) -> Cons(x, merge(xs, xs2))

Types:
mergesort :: Cons:Nil -> Cons:Nil
Cons :: S:0' -> Cons:Nil -> Cons:Nil
splitmerge :: Cons:Nil -> Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
merge :: Cons:Nil -> Cons:Nil -> Cons:Nil
merge[Ite] :: True:False -> Cons:Nil -> Cons:Nil -> Cons:Nil
<= :: S:0' -> S:0' -> True:False
notEmpty :: Cons:Nil -> True:False
True :: True:False
False :: True:False
goal :: Cons:Nil -> Cons:Nil
S :: S:0' -> S:0'
0' :: S:0'
hole_Cons:Nil1_0 :: Cons:Nil
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat -> Cons:Nil
gen_S:0'5_0 :: Nat -> S:0'


Lemmas:
<=(gen_S:0'5_0(n7_0), gen_S:0'5_0(n7_0)) -> True, rt in Omega(0)


Generator Equations:
gen_Cons:Nil4_0(0) <=> Nil
gen_Cons:Nil4_0(+(x, 1)) <=> Cons(0', gen_Cons:Nil4_0(x))
gen_S:0'5_0(0) <=> 0'
gen_S:0'5_0(+(x, 1)) <=> S(gen_S:0'5_0(x))


The following defined symbols remain to be analysed:
merge, mergesort, splitmerge

They will be analysed ascendingly in the following order:
mergesort = splitmerge
merge < splitmerge

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(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
merge(gen_Cons:Nil4_0(n258_0), gen_Cons:Nil4_0(1)) -> gen_Cons:Nil4_0(+(1, n258_0)), rt in Omega(1 + n258_0)

Induction Base:
merge(gen_Cons:Nil4_0(0), gen_Cons:Nil4_0(1)) ->_R^Omega(1)
gen_Cons:Nil4_0(1)

Induction Step:
merge(gen_Cons:Nil4_0(+(n258_0, 1)), gen_Cons:Nil4_0(1)) ->_R^Omega(1)
merge[Ite](<=(0', 0'), Cons(0', gen_Cons:Nil4_0(n258_0)), Cons(0', gen_Cons:Nil4_0(0))) ->_L^Omega(0)
merge[Ite](True, Cons(0', gen_Cons:Nil4_0(n258_0)), Cons(0', gen_Cons:Nil4_0(0))) ->_R^Omega(0)
Cons(0', merge(gen_Cons:Nil4_0(n258_0), Cons(0', gen_Cons:Nil4_0(0)))) ->_IH
Cons(0', gen_Cons:Nil4_0(+(1, c259_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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(12)
Complex Obligation (BEST)

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(13)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
mergesort(Cons(x', Cons(x, xs))) -> splitmerge(Cons(x', Cons(x, xs)), Nil, Nil)
mergesort(Cons(x, Nil)) -> Cons(x, Nil)
merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](<=(x', x), Cons(x', xs'), Cons(x, xs))
merge(Cons(x, xs), Nil) -> Cons(x, xs)
splitmerge(Cons(x, xs), xs1, xs2) -> splitmerge(xs, Cons(x, xs2), xs1)
splitmerge(Nil, xs1, xs2) -> merge(mergesort(xs1), mergesort(xs2))
mergesort(Nil) -> Nil
merge(Nil, xs2) -> xs2
notEmpty(Cons(x, xs)) -> True
notEmpty(Nil) -> False
goal(xs) -> mergesort(xs)
<=(S(x), S(y)) -> <=(x, y)
<=(0', y) -> True
<=(S(x), 0') -> False
merge[Ite](False, xs1, Cons(x, xs)) -> Cons(x, merge(xs1, xs))
merge[Ite](True, Cons(x, xs), xs2) -> Cons(x, merge(xs, xs2))

Types:
mergesort :: Cons:Nil -> Cons:Nil
Cons :: S:0' -> Cons:Nil -> Cons:Nil
splitmerge :: Cons:Nil -> Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
merge :: Cons:Nil -> Cons:Nil -> Cons:Nil
merge[Ite] :: True:False -> Cons:Nil -> Cons:Nil -> Cons:Nil
<= :: S:0' -> S:0' -> True:False
notEmpty :: Cons:Nil -> True:False
True :: True:False
False :: True:False
goal :: Cons:Nil -> Cons:Nil
S :: S:0' -> S:0'
0' :: S:0'
hole_Cons:Nil1_0 :: Cons:Nil
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat -> Cons:Nil
gen_S:0'5_0 :: Nat -> S:0'


Lemmas:
<=(gen_S:0'5_0(n7_0), gen_S:0'5_0(n7_0)) -> True, rt in Omega(0)


Generator Equations:
gen_Cons:Nil4_0(0) <=> Nil
gen_Cons:Nil4_0(+(x, 1)) <=> Cons(0', gen_Cons:Nil4_0(x))
gen_S:0'5_0(0) <=> 0'
gen_S:0'5_0(+(x, 1)) <=> S(gen_S:0'5_0(x))


The following defined symbols remain to be analysed:
merge, mergesort, splitmerge

They will be analysed ascendingly in the following order:
mergesort = splitmerge
merge < splitmerge

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

(15)
BOUNDS(n^1, INF)

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(16)
Obligation:
Innermost TRS:
Rules:
mergesort(Cons(x', Cons(x, xs))) -> splitmerge(Cons(x', Cons(x, xs)), Nil, Nil)
mergesort(Cons(x, Nil)) -> Cons(x, Nil)
merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](<=(x', x), Cons(x', xs'), Cons(x, xs))
merge(Cons(x, xs), Nil) -> Cons(x, xs)
splitmerge(Cons(x, xs), xs1, xs2) -> splitmerge(xs, Cons(x, xs2), xs1)
splitmerge(Nil, xs1, xs2) -> merge(mergesort(xs1), mergesort(xs2))
mergesort(Nil) -> Nil
merge(Nil, xs2) -> xs2
notEmpty(Cons(x, xs)) -> True
notEmpty(Nil) -> False
goal(xs) -> mergesort(xs)
<=(S(x), S(y)) -> <=(x, y)
<=(0', y) -> True
<=(S(x), 0') -> False
merge[Ite](False, xs1, Cons(x, xs)) -> Cons(x, merge(xs1, xs))
merge[Ite](True, Cons(x, xs), xs2) -> Cons(x, merge(xs, xs2))

Types:
mergesort :: Cons:Nil -> Cons:Nil
Cons :: S:0' -> Cons:Nil -> Cons:Nil
splitmerge :: Cons:Nil -> Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
merge :: Cons:Nil -> Cons:Nil -> Cons:Nil
merge[Ite] :: True:False -> Cons:Nil -> Cons:Nil -> Cons:Nil
<= :: S:0' -> S:0' -> True:False
notEmpty :: Cons:Nil -> True:False
True :: True:False
False :: True:False
goal :: Cons:Nil -> Cons:Nil
S :: S:0' -> S:0'
0' :: S:0'
hole_Cons:Nil1_0 :: Cons:Nil
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat -> Cons:Nil
gen_S:0'5_0 :: Nat -> S:0'


Lemmas:
<=(gen_S:0'5_0(n7_0), gen_S:0'5_0(n7_0)) -> True, rt in Omega(0)
merge(gen_Cons:Nil4_0(n258_0), gen_Cons:Nil4_0(1)) -> gen_Cons:Nil4_0(+(1, n258_0)), rt in Omega(1 + n258_0)


Generator Equations:
gen_Cons:Nil4_0(0) <=> Nil
gen_Cons:Nil4_0(+(x, 1)) <=> Cons(0', gen_Cons:Nil4_0(x))
gen_S:0'5_0(0) <=> 0'
gen_S:0'5_0(+(x, 1)) <=> S(gen_S:0'5_0(x))


The following defined symbols remain to be analysed:
splitmerge, mergesort

They will be analysed ascendingly in the following order:
mergesort = splitmerge