feat: add Dyadic.divAtPrec for round-down dyadic division at given precision#13654
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feat: add Dyadic.divAtPrec for round-down dyadic division at given precision#13654kim-em wants to merge 5 commits into
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…ecision This PR adds `Dyadic.divAtPrec a b prec`, returning the greatest dyadic with precision at most `prec` which is less than or equal to `a/b` (and `0` when `b = 0`). Mirroring the existing `invAtPrec`, the characterising lemmas `divAtPrec_mul_le` and `lt_divAtPrec_add_inc_mul` are also provided. Compared to composing `invAtPrec` with multiplication, `divAtPrec` rounds `a/b` down uniformly (no sign-dependent rounding flip), states the precision at the output rather than requiring callers to back-solve from `‖a‖`, and fuses the bignum division with renormalisation. Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
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Discharges the existing TODOs by adding `eq_invAtPrec` and `eq_divAtPrec`, factored through a public helper `eq_toDyadic_of_precision_le`: any dyadic with precision at most `prec` lying in `[y.toRat, y.toRat + 2 ^ (-prec))` is the floor of `q` at that precision. Adds a (non-public) import of `Init.Data.Int.Order` for `Int.le_antisymm` and `Int.lt_add_one_iff`. Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
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Hoist the `hxr`/`hbr` bridges from `0 < x` to `0 < x.toRat` to the top of each proof and inline them at the use sites, replacing the repeated `have h_pos`, `have h_le`, and inline `Rat.ne_of_gt`/`Rat.le_of_lt` derivations. Also drops the trailing intermediate `have`s in favour of direct `calc` steps. No semantic changes; ~38 fewer lines. Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
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…q_divAtPrec_one` Add `invAtPrec_eq_divAtPrec_one`, the (theorem-level only) equality `invAtPrec x prec = divAtPrec 1 x prec`. The definition of `invAtPrec` is left untouched so its runtime path remains tight. The three characterising lemmas about `invAtPrec` now follow as one-line corollaries of their `divAtPrec` counterparts. Also add two strict bounds relating `a * invAtPrec b prec` to `divAtPrec a b prec` (which are *not* equal in general): each isolates `a * invAtPrec b prec` on one side, with the two pinning the gap to within `2 ^ (-prec)` and `a * 2 ^ (-prec)` respectively. Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
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…ic.Inv The previous commit's docstrings overstated two things. `invAtPrec` is now only propositionally equal to `divAtPrec 1`, not definitionally; and the gap between `a * invAtPrec b prec` and `divAtPrec a b prec` is asymmetric — bounded by `2 ^ (-prec)` on one side and by `a * 2 ^ (-prec)` on the other, not by `a * 2 ^ (-prec)` in both directions. Update the prose to match the theorems. Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
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This PR adds
Dyadic.divAtPrec a b prec, returning the greatest dyadic with precision at mostprecwhich is less than or equal toa/b(and0whenb = 0). Mirroring the existinginvAtPrec, the characterising lemmasdivAtPrec_mul_leandlt_divAtPrec_add_inc_mulare also provided.Compared to composing
invAtPrecwith multiplication,divAtPrecroundsa/bdown uniformly (no sign-dependent rounding flip), states the precision at the output rather than requiring callers to back-solve from‖a‖, and fuses the bignum division with renormalisation.invAtPrecremains the right primitive when the reciprocal is reused.Closes #13653.
🤖 Prepared with Claude Code