WORST_CASE(?,O(n^1))
proof of input_nhlZi6fxRf.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
(3) CpxTrsMatchBoundsProof [FINISHED, 0 ms]
(4) BOUNDS(1, n^1)


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(0)
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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(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.

"[1, 2, 3]
{(1,2,[odd_1|0, even_1|0, even_1|1, 0|1, odd_1|1]), (1,3,[S_1|1]), (2,2,[S_1|0, 0|0]), (3,2,[0|1])}"
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(4)
BOUNDS(1, n^1)