WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxTRS
(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) CpxTrsMatchBoundsTAProof [FINISHED, 0 ms]
    (4) BOUNDS(1, n^1)
(5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxTRS
    (7) SlicingProof [LOWER BOUND(ID), 0 ms]
    (8) CpxTRS
    (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) typed CpxTrs
    (11) OrderProof [LOWER BOUND(ID), 0 ms]
    (12) typed CpxTrs
    (13) RewriteLemmaProof [LOWER BOUND(ID), 139 ms]
    (14) BEST
        (15) proven lower bound
            (16) LowerBoundPropagationProof [FINISHED, 0 ms]
            (17) BOUNDS(n^1, INF)
        (18) typed CpxTrs


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


The TRS R consists of the following rules:

   odd(Cons(x, xs)) -> even(xs)
   odd(Nil) -> False
   even(Cons(x, xs)) -> odd(xs)
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   even(Nil) -> True
   evenodd(x) -> even(x)

S is empty.
Rewrite Strategy: INNERMOST
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(1) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
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(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   odd(Cons(x, xs)) -> even(xs)
   odd(Nil) -> False
   even(Cons(x, xs)) -> odd(xs)
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   even(Nil) -> True
   evenodd(x) -> even(x)

S is empty.
Rewrite Strategy: INNERMOST
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(3) CpxTrsMatchBoundsTAProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 1. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2, 3, 4]
transitions: 
Cons0(0, 0) -> 0
Nil0() -> 0
False0() -> 0
True0() -> 0
odd0(0) -> 1
even0(0) -> 2
notEmpty0(0) -> 3
evenodd0(0) -> 4
even1(0) -> 1
False1() -> 1
odd1(0) -> 2
True1() -> 3
False1() -> 3
True1() -> 2
even1(0) -> 4
even1(0) -> 2
False1() -> 2
odd1(0) -> 1
odd1(0) -> 4
True1() -> 1
True1() -> 4
False1() -> 4

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(4)
BOUNDS(1, n^1)

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(5) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
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(6)
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:

   odd(Cons(x, xs)) -> even(xs)
   odd(Nil) -> False
   even(Cons(x, xs)) -> odd(xs)
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   even(Nil) -> True
   evenodd(x) -> even(x)

S is empty.
Rewrite Strategy: INNERMOST
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(7) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
Cons/0

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(8)
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:

   odd(Cons(xs)) -> even(xs)
   odd(Nil) -> False
   even(Cons(xs)) -> odd(xs)
   notEmpty(Cons(xs)) -> True
   notEmpty(Nil) -> False
   even(Nil) -> True
   evenodd(x) -> even(x)

S is empty.
Rewrite Strategy: INNERMOST
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(9) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(10)
Obligation:
Innermost TRS:
Rules:
odd(Cons(xs)) -> even(xs)
odd(Nil) -> False
even(Cons(xs)) -> odd(xs)
notEmpty(Cons(xs)) -> True
notEmpty(Nil) -> False
even(Nil) -> True
evenodd(x) -> even(x)

Types:
odd :: Cons:Nil -> False:True
Cons :: Cons:Nil -> Cons:Nil
even :: Cons:Nil -> False:True
Nil :: Cons:Nil
False :: False:True
notEmpty :: Cons:Nil -> False:True
True :: False:True
evenodd :: Cons:Nil -> False:True
hole_False:True1_0 :: False:True
hole_Cons:Nil2_0 :: Cons:Nil
gen_Cons:Nil3_0 :: Nat -> Cons:Nil

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(11) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
odd, even

They will be analysed ascendingly in the following order:
odd = even

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(12)
Obligation:
Innermost TRS:
Rules:
odd(Cons(xs)) -> even(xs)
odd(Nil) -> False
even(Cons(xs)) -> odd(xs)
notEmpty(Cons(xs)) -> True
notEmpty(Nil) -> False
even(Nil) -> True
evenodd(x) -> even(x)

Types:
odd :: Cons:Nil -> False:True
Cons :: Cons:Nil -> Cons:Nil
even :: Cons:Nil -> False:True
Nil :: Cons:Nil
False :: False:True
notEmpty :: Cons:Nil -> False:True
True :: False:True
evenodd :: Cons:Nil -> False:True
hole_False:True1_0 :: False:True
hole_Cons:Nil2_0 :: Cons:Nil
gen_Cons:Nil3_0 :: Nat -> Cons:Nil


Generator Equations:
gen_Cons:Nil3_0(0) <=> Nil
gen_Cons:Nil3_0(+(x, 1)) <=> Cons(gen_Cons:Nil3_0(x))


The following defined symbols remain to be analysed:
even, odd

They will be analysed ascendingly in the following order:
odd = even

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(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
even(gen_Cons:Nil3_0(*(2, n5_0))) -> True, rt in Omega(1 + n5_0)

Induction Base:
even(gen_Cons:Nil3_0(*(2, 0))) ->_R^Omega(1)
True

Induction Step:
even(gen_Cons:Nil3_0(*(2, +(n5_0, 1)))) ->_R^Omega(1)
odd(gen_Cons:Nil3_0(+(1, *(2, n5_0)))) ->_R^Omega(1)
even(gen_Cons:Nil3_0(*(2, n5_0))) ->_IH
True

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)
Complex Obligation (BEST)

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(15)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
odd(Cons(xs)) -> even(xs)
odd(Nil) -> False
even(Cons(xs)) -> odd(xs)
notEmpty(Cons(xs)) -> True
notEmpty(Nil) -> False
even(Nil) -> True
evenodd(x) -> even(x)

Types:
odd :: Cons:Nil -> False:True
Cons :: Cons:Nil -> Cons:Nil
even :: Cons:Nil -> False:True
Nil :: Cons:Nil
False :: False:True
notEmpty :: Cons:Nil -> False:True
True :: False:True
evenodd :: Cons:Nil -> False:True
hole_False:True1_0 :: False:True
hole_Cons:Nil2_0 :: Cons:Nil
gen_Cons:Nil3_0 :: Nat -> Cons:Nil


Generator Equations:
gen_Cons:Nil3_0(0) <=> Nil
gen_Cons:Nil3_0(+(x, 1)) <=> Cons(gen_Cons:Nil3_0(x))


The following defined symbols remain to be analysed:
even, odd

They will be analysed ascendingly in the following order:
odd = even

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(16) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(17)
BOUNDS(n^1, INF)

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(18)
Obligation:
Innermost TRS:
Rules:
odd(Cons(xs)) -> even(xs)
odd(Nil) -> False
even(Cons(xs)) -> odd(xs)
notEmpty(Cons(xs)) -> True
notEmpty(Nil) -> False
even(Nil) -> True
evenodd(x) -> even(x)

Types:
odd :: Cons:Nil -> False:True
Cons :: Cons:Nil -> Cons:Nil
even :: Cons:Nil -> False:True
Nil :: Cons:Nil
False :: False:True
notEmpty :: Cons:Nil -> False:True
True :: False:True
evenodd :: Cons:Nil -> False:True
hole_False:True1_0 :: False:True
hole_Cons:Nil2_0 :: Cons:Nil
gen_Cons:Nil3_0 :: Nat -> Cons:Nil


Lemmas:
even(gen_Cons:Nil3_0(*(2, n5_0))) -> True, rt in Omega(1 + n5_0)


Generator Equations:
gen_Cons:Nil3_0(0) <=> Nil
gen_Cons:Nil3_0(+(x, 1)) <=> Cons(gen_Cons:Nil3_0(x))


The following defined symbols remain to be analysed:
odd

They will be analysed ascendingly in the following order:
odd = even