WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) CpxTrsMatchBoundsProof [FINISHED, 0 ms]
    (4) BOUNDS(1, n^1)
(5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxTRS
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) typed CpxTrs
    (9) OrderProof [LOWER BOUND(ID), 0 ms]
    (10) typed CpxTrs
    (11) RewriteLemmaProof [LOWER BOUND(ID), 186 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 CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   odd(S(x)) -> even(x)
   even(S(x)) -> odd(x)
   odd(0) -> 0
   even(0) -> S(0)

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(S(x)) -> even(x)
   even(S(x)) -> odd(x)
   odd(0) -> 0
   even(0) -> S(0)

S is empty.
Rewrite Strategy: INNERMOST
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(3) CpxTrsMatchBoundsProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 1. 
The certificate found is represented by the following graph.

"[3, 4, 5]
{(3,4,[odd_1|0, even_1|0, even_1|1, 0|1, odd_1|1]), (3,5,[S_1|1]), (4,4,[S_1|0, 0|0]), (5,4,[0|1])}"
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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(S(x)) -> even(x)
   even(S(x)) -> odd(x)
   odd(0') -> 0'
   even(0') -> S(0')

S is empty.
Rewrite Strategy: INNERMOST
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(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(8)
Obligation:
Innermost TRS:
Rules:
odd(S(x)) -> even(x)
even(S(x)) -> odd(x)
odd(0') -> 0'
even(0') -> S(0')

Types:
odd :: S:0' -> S:0'
S :: S:0' -> S:0'
even :: S:0' -> S:0'
0' :: S:0'
hole_S:0'1_0 :: S:0'
gen_S:0'2_0 :: Nat -> S:0'

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(9) 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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(10)
Obligation:
Innermost TRS:
Rules:
odd(S(x)) -> even(x)
even(S(x)) -> odd(x)
odd(0') -> 0'
even(0') -> S(0')

Types:
odd :: S:0' -> S:0'
S :: S:0' -> S:0'
even :: S:0' -> S:0'
0' :: S:0'
hole_S:0'1_0 :: S:0'
gen_S:0'2_0 :: Nat -> S:0'


Generator Equations:
gen_S:0'2_0(0) <=> 0'
gen_S:0'2_0(+(x, 1)) <=> S(gen_S:0'2_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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(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
even(gen_S:0'2_0(*(2, n4_0))) -> gen_S:0'2_0(1), rt in Omega(1 + n4_0)

Induction Base:
even(gen_S:0'2_0(*(2, 0))) ->_R^Omega(1)
S(0')

Induction Step:
even(gen_S:0'2_0(*(2, +(n4_0, 1)))) ->_R^Omega(1)
odd(gen_S:0'2_0(+(1, *(2, n4_0)))) ->_R^Omega(1)
even(gen_S:0'2_0(*(2, n4_0))) ->_IH
gen_S:0'2_0(1)

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:
odd(S(x)) -> even(x)
even(S(x)) -> odd(x)
odd(0') -> 0'
even(0') -> S(0')

Types:
odd :: S:0' -> S:0'
S :: S:0' -> S:0'
even :: S:0' -> S:0'
0' :: S:0'
hole_S:0'1_0 :: S:0'
gen_S:0'2_0 :: Nat -> S:0'


Generator Equations:
gen_S:0'2_0(0) <=> 0'
gen_S:0'2_0(+(x, 1)) <=> S(gen_S:0'2_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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(14) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(15)
BOUNDS(n^1, INF)

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(16)
Obligation:
Innermost TRS:
Rules:
odd(S(x)) -> even(x)
even(S(x)) -> odd(x)
odd(0') -> 0'
even(0') -> S(0')

Types:
odd :: S:0' -> S:0'
S :: S:0' -> S:0'
even :: S:0' -> S:0'
0' :: S:0'
hole_S:0'1_0 :: S:0'
gen_S:0'2_0 :: Nat -> S:0'


Lemmas:
even(gen_S:0'2_0(*(2, n4_0))) -> gen_S:0'2_0(1), rt in Omega(1 + n4_0)


Generator Equations:
gen_S:0'2_0(0) <=> 0'
gen_S:0'2_0(+(x, 1)) <=> S(gen_S:0'2_0(x))


The following defined symbols remain to be analysed:
odd

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