I think you will find that as far as your company are concerned , they " see" that you are resting / asleep until 11;58 am and are oblivious to the realities of possibly ( small children / wife / partner working) being awake and up and around since 6.00 am. Correct it is not.'
It is your responsibility to be rested; whether you have been awake all night with kids is irrelevant to the company - it is your responsibility to report unfit for duty at that point.
I agree with many of the above posters as well; are you sitting on tenterhooks at the door from 6am onwards?