Section 11.1: fix two unprovable statements#505
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- The intersection example near `BoundedInterval.inter_eq` claimed
`Ioo 2 4 ∩ Icc 4 6 = Icc 4 4`, which is false as sets
(`Ioo 2 4 ∩ Icc 4 6 = ∅`). Change to `Icc 2 4 ∩ Icc 4 6`, matching
the intended `Icc 4 4` RHS.
- Example 11.1.17 stated `(P' ⊔ P).intervals = {Icc 1 2, Ioo 2 3, Icc 3 4, ∅}`
as a `Finset BoundedInterval` equality, which is unprovable because
`BoundedInterval.inter` is `Classical.choose`-based and does not commit
to a canonical constructor (e.g. nothing forces
`(Icc 1 2 ∩ Ico 1 3 : BoundedInterval) = Icc 1 2`, they are only
equal after mapping toSet).
Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
rkirov
commented
May 23, 2026
| example : ∃ P P' : Partition (Icc 1 4), | ||
| P.intervals = {Ico 1 3, Icc 3 4} ∧ | ||
| P'.intervals = {Icc 1 2, Ioc 2 4} ∧ | ||
| (P' ⊔ P).intervals.image toSet = |
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This one is subtle due to not having extensional equality on BoundedIntervals, which means we can't ever prove even simple A ∩ B = C for BoundedIntervals without going to Sets. I thought about some larger changes to the setup (e.g. constructing computable intersection method for BoundedIntervals), but the easiest way was just change this to confirm equality with Sets.
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The intersection example near
BoundedInterval.inter_eqclaimedIoo 2 4 ∩ Icc 4 6 = Icc 4 4, which is false as sets (Ioo 2 4 ∩ Icc 4 6 = ∅). Change toIcc 2 4 ∩ Icc 4 6, matching the intendedIcc 4 4RHS.Example 11.1.17 stated
(P' ⊔ P).intervals = {Icc 1 2, Ioo 2 3, Icc 3 4, ∅}as aFinset BoundedIntervalequality, which is unprovable becauseBoundedInterval.interisClassical.choose-based and does not commit to a canonical constructor (e.g. nothing forces(Icc 1 2 ∩ Ico 1 3 : BoundedInterval) = Icc 1 2). Restate the third conjunct at the set level, via the coercion image, with a comment explaining why.