WORST_CASE(Omega(n^1),O(n^1))
proof of input_AMeysv58RM.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-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, 10 ms]
    (4) BOUNDS(1, n^1)
(5) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CdtProblem
    (7) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxRelTRS
    (9) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxRelTRS
    (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) typed CpxTrs
    (13) OrderProof [LOWER BOUND(ID), 0 ms]
    (14) typed CpxTrs
    (15) RewriteLemmaProof [LOWER BOUND(ID), 215 ms]
    (16) typed CpxTrs
    (17) RewriteLemmaProof [LOWER BOUND(ID), 1391 ms]
    (18) BEST
        (19) proven lower bound
            (20) LowerBoundPropagationProof [FINISHED, 0 ms]
            (21) BOUNDS(n^1, INF)
        (22) typed CpxTrs


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(0)
Obligation:
The Runtime Complexity (parallel-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: PARALLEL_INNERMOST
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(1) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
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(2)
Obligation:
The Runtime Complexity (parallel-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: PARALLEL_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.

"[5, 6, 7]
{(5,6,[odd_1|0, even_1|0, even_1|1, 0|1, odd_1|1]), (5,7,[S_1|1]), (6,6,[S_1|0, 0|0]), (7,6,[0|1])}"
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(4)
BOUNDS(1, n^1)

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(5) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
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(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   odd(S(z0)) -> even(z0)
   odd(0) -> 0
   even(S(z0)) -> odd(z0)
   even(0) -> S(0)
Tuples:
   ODD(S(z0)) -> c(EVEN(z0))
   ODD(0) -> c1
   EVEN(S(z0)) -> c2(ODD(z0))
   EVEN(0) -> c3
S tuples:
   ODD(S(z0)) -> c(EVEN(z0))
   ODD(0) -> c1
   EVEN(S(z0)) -> c2(ODD(z0))
   EVEN(0) -> c3
K tuples:none
Defined Rule Symbols:   odd_1, even_1

Defined Pair Symbols:   ODD_1, EVEN_1

Compound Symbols:   c_1, c1, c2_1, c3


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(7) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
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(8)
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:

   ODD(S(z0)) -> c(EVEN(z0))
   ODD(0) -> c1
   EVEN(S(z0)) -> c2(ODD(z0))
   EVEN(0) -> c3

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

   odd(S(z0)) -> even(z0)
   odd(0) -> 0
   even(S(z0)) -> odd(z0)
   even(0) -> S(0)

Rewrite Strategy: INNERMOST
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(9) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
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(10)
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:

   ODD(S(z0)) -> c(EVEN(z0))
   ODD(0') -> c1
   EVEN(S(z0)) -> c2(ODD(z0))
   EVEN(0') -> c3

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

   odd(S(z0)) -> even(z0)
   odd(0') -> 0'
   even(S(z0)) -> odd(z0)
   even(0') -> S(0')

Rewrite Strategy: INNERMOST
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(11) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Inferred types.
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(12)
Obligation:
Innermost TRS:
Rules:
ODD(S(z0)) -> c(EVEN(z0))
ODD(0') -> c1
EVEN(S(z0)) -> c2(ODD(z0))
EVEN(0') -> c3
odd(S(z0)) -> even(z0)
odd(0') -> 0'
even(S(z0)) -> odd(z0)
even(0') -> S(0')

Types:
ODD :: S:0' -> c:c1
S :: S:0' -> S:0'
c :: c2:c3 -> c:c1
EVEN :: S:0' -> c2:c3
0' :: S:0'
c1 :: c:c1
c2 :: c:c1 -> c2:c3
c3 :: c2:c3
odd :: S:0' -> S:0'
even :: S:0' -> S:0'
hole_c:c11_4 :: c:c1
hole_S:0'2_4 :: S:0'
hole_c2:c33_4 :: c2:c3
gen_S:0'4_4 :: Nat -> S:0'

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

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

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(14)
Obligation:
Innermost TRS:
Rules:
ODD(S(z0)) -> c(EVEN(z0))
ODD(0') -> c1
EVEN(S(z0)) -> c2(ODD(z0))
EVEN(0') -> c3
odd(S(z0)) -> even(z0)
odd(0') -> 0'
even(S(z0)) -> odd(z0)
even(0') -> S(0')

Types:
ODD :: S:0' -> c:c1
S :: S:0' -> S:0'
c :: c2:c3 -> c:c1
EVEN :: S:0' -> c2:c3
0' :: S:0'
c1 :: c:c1
c2 :: c:c1 -> c2:c3
c3 :: c2:c3
odd :: S:0' -> S:0'
even :: S:0' -> S:0'
hole_c:c11_4 :: c:c1
hole_S:0'2_4 :: S:0'
hole_c2:c33_4 :: c2:c3
gen_S:0'4_4 :: Nat -> S:0'


Generator Equations:
gen_S:0'4_4(0) <=> 0'
gen_S:0'4_4(+(x, 1)) <=> S(gen_S:0'4_4(x))


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

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

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(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
even(gen_S:0'4_4(*(2, n6_4))) -> gen_S:0'4_4(1), rt in Omega(0)

Induction Base:
even(gen_S:0'4_4(*(2, 0))) ->_R^Omega(0)
S(0')

Induction Step:
even(gen_S:0'4_4(*(2, +(n6_4, 1)))) ->_R^Omega(0)
odd(gen_S:0'4_4(+(1, *(2, n6_4)))) ->_R^Omega(0)
even(gen_S:0'4_4(*(2, n6_4))) ->_IH
gen_S:0'4_4(1)

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
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(16)
Obligation:
Innermost TRS:
Rules:
ODD(S(z0)) -> c(EVEN(z0))
ODD(0') -> c1
EVEN(S(z0)) -> c2(ODD(z0))
EVEN(0') -> c3
odd(S(z0)) -> even(z0)
odd(0') -> 0'
even(S(z0)) -> odd(z0)
even(0') -> S(0')

Types:
ODD :: S:0' -> c:c1
S :: S:0' -> S:0'
c :: c2:c3 -> c:c1
EVEN :: S:0' -> c2:c3
0' :: S:0'
c1 :: c:c1
c2 :: c:c1 -> c2:c3
c3 :: c2:c3
odd :: S:0' -> S:0'
even :: S:0' -> S:0'
hole_c:c11_4 :: c:c1
hole_S:0'2_4 :: S:0'
hole_c2:c33_4 :: c2:c3
gen_S:0'4_4 :: Nat -> S:0'


Lemmas:
even(gen_S:0'4_4(*(2, n6_4))) -> gen_S:0'4_4(1), rt in Omega(0)


Generator Equations:
gen_S:0'4_4(0) <=> 0'
gen_S:0'4_4(+(x, 1)) <=> S(gen_S:0'4_4(x))


The following defined symbols remain to be analysed:
odd, ODD, EVEN

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

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(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
EVEN(gen_S:0'4_4(+(1, *(2, n169_4)))) -> *5_4, rt in Omega(n169_4)

Induction Base:
EVEN(gen_S:0'4_4(+(1, *(2, 0))))

Induction Step:
EVEN(gen_S:0'4_4(+(1, *(2, +(n169_4, 1))))) ->_R^Omega(1)
c2(ODD(gen_S:0'4_4(+(2, *(2, n169_4))))) ->_R^Omega(1)
c2(c(EVEN(gen_S:0'4_4(+(1, *(2, n169_4)))))) ->_IH
c2(c(*5_4))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(18)
Complex Obligation (BEST)

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(19)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
ODD(S(z0)) -> c(EVEN(z0))
ODD(0') -> c1
EVEN(S(z0)) -> c2(ODD(z0))
EVEN(0') -> c3
odd(S(z0)) -> even(z0)
odd(0') -> 0'
even(S(z0)) -> odd(z0)
even(0') -> S(0')

Types:
ODD :: S:0' -> c:c1
S :: S:0' -> S:0'
c :: c2:c3 -> c:c1
EVEN :: S:0' -> c2:c3
0' :: S:0'
c1 :: c:c1
c2 :: c:c1 -> c2:c3
c3 :: c2:c3
odd :: S:0' -> S:0'
even :: S:0' -> S:0'
hole_c:c11_4 :: c:c1
hole_S:0'2_4 :: S:0'
hole_c2:c33_4 :: c2:c3
gen_S:0'4_4 :: Nat -> S:0'


Lemmas:
even(gen_S:0'4_4(*(2, n6_4))) -> gen_S:0'4_4(1), rt in Omega(0)


Generator Equations:
gen_S:0'4_4(0) <=> 0'
gen_S:0'4_4(+(x, 1)) <=> S(gen_S:0'4_4(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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(20) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(21)
BOUNDS(n^1, INF)

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(22)
Obligation:
Innermost TRS:
Rules:
ODD(S(z0)) -> c(EVEN(z0))
ODD(0') -> c1
EVEN(S(z0)) -> c2(ODD(z0))
EVEN(0') -> c3
odd(S(z0)) -> even(z0)
odd(0') -> 0'
even(S(z0)) -> odd(z0)
even(0') -> S(0')

Types:
ODD :: S:0' -> c:c1
S :: S:0' -> S:0'
c :: c2:c3 -> c:c1
EVEN :: S:0' -> c2:c3
0' :: S:0'
c1 :: c:c1
c2 :: c:c1 -> c2:c3
c3 :: c2:c3
odd :: S:0' -> S:0'
even :: S:0' -> S:0'
hole_c:c11_4 :: c:c1
hole_S:0'2_4 :: S:0'
hole_c2:c33_4 :: c2:c3
gen_S:0'4_4 :: Nat -> S:0'


Lemmas:
even(gen_S:0'4_4(*(2, n6_4))) -> gen_S:0'4_4(1), rt in Omega(0)
EVEN(gen_S:0'4_4(+(1, *(2, n169_4)))) -> *5_4, rt in Omega(n169_4)


Generator Equations:
gen_S:0'4_4(0) <=> 0'
gen_S:0'4_4(+(x, 1)) <=> S(gen_S:0'4_4(x))


The following defined symbols remain to be analysed:
ODD

They will be analysed ascendingly in the following order:
ODD = EVEN