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feat(ReedSolomon): decode-outcome iff for Gao decoding #268

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alexanderlhicks merged 1 commit into from
Jul 12, 2026
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feat(ReedSolomon): decode-outcome iff for Gao decoding #268

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10 changes: 10 additions & 0 deletions CompPoly/Univariate/ReedSolomon/GaoCorrectness.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -310,6 +310,16 @@ theorem decode_none_farness [Field F]
rw [Gao.decode_eq_some k D r hkn msg (Nat.le_of_not_lt hle)] at hnone
exact Option.some_ne_none _ hnone

/-- Outcome characterization: `Gao.decode` returns `none` iff the received word is beyond
the guaranteed radius `⌊(n-k)/2⌋` of every codeword (combines `decode_eq_none` and
`decode_none_farness`). -/
theorem decode_eq_none_iff [Field F]
(k : ℕ) (D : Domain F) (r : Vector F D.n) (hkn : k < D.n) :
Gao.decode k D r = none ↔
∀ msg : Vector F k, D.n - k < 2 * hammingDist r.get (encode D msg).get :=
⟨decode_none_farness k D r hkn,
fun h => decode_eq_none k D r hkn fun msg => by have := h msg; omega⟩

/-- Uniqueness: within the guaranteed-decoding radius `⌊(D.n-k)/2⌋`, two messages
whose codewords lie within that radius of `r` decode to the same polynomial. -/
theorem decode_unique [Field F] (k : ℕ) (D : Domain F) (r : Vector F D.n)
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