WORST_CASE(?,O(n^1))
proof of input_VAcOnkfPQw.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) CpxTrsMatchBoundsTAProof [FINISHED, 139 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:

   not(true) -> false
   not(false) -> true
   evenodd(x, 0) -> not(evenodd(x, s(0)))
   evenodd(0, s(0)) -> false
   evenodd(s(x), s(0)) -> evenodd(x, 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:

   not(true) -> false
   not(false) -> true
   evenodd(x, 0) -> not(evenodd(x, s(0)))
   evenodd(0, s(0)) -> false
   evenodd(s(x), s(0)) -> evenodd(x, 0)

S is empty.
Rewrite Strategy: PARALLEL_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 3. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2]
transitions: 
true0() -> 0
false0() -> 0
00() -> 0
s0(0) -> 0
not0(0) -> 1
evenodd0(0, 0) -> 2
false1() -> 1
true1() -> 1
01() -> 5
s1(5) -> 4
evenodd1(0, 4) -> 3
not1(3) -> 2
false1() -> 2
01() -> 6
evenodd1(0, 6) -> 2
02() -> 9
s2(9) -> 8
evenodd2(0, 8) -> 7
not2(7) -> 2
false1() -> 3
evenodd1(0, 6) -> 3
true2() -> 2
not2(7) -> 3
false1() -> 7
evenodd1(0, 6) -> 7
not2(7) -> 7
true2() -> 3
true2() -> 7
false2() -> 2
false3() -> 2
false3() -> 3
false3() -> 7
true3() -> 2
true3() -> 3
true3() -> 7

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(4)
BOUNDS(1, n^1)