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Now, I'm showing that functions called
lowerandupperexist. Recall that they are currently given axiomatically, but to show equivalence to the circuit, we need a more computational definition. This will be a (long) line of technical lemmas:minSubnorm, smaller thanmaxNorm.minSubnorm, then you are zero. If you are larger thanmaxNorm, then you are infinity.ExtRat, there will either be aPackedFloatsmaller than it, or itis equal to a packed float (similar for upper)PackedFloats, there is a finite number ofPackedFloats between them (from the gap theorem). This one is a bit tricky, I'm not quite sure what the "good" statement of this theorem is. I may just exhibit that thePackedFloats are finite, and can be put into an ordered list for the non-NaN fragment, which immediately gives you the ordering theorems. On paper, stuff that follows from finiteness is "immediate", but I'm not sure what the best way to express this in a proof assistant. Maybe @Léo Stefanesco , @Henrik Böving , @Abdalrhman M Mohamed have thoughts.