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RS-membership lemmas for evalOnPoints (☺️ very cute ☺️) #660

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Merged
alexanderlhicks merged 2 commits into from
Jul 21, 2026
Merged

RS-membership lemmas for evalOnPoints (☺️ very cute ☺️) #660

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075de67
Add some cute little lemmas
ElijahVlasov Jul 21, 2026
9bed5e6
Merge branch 'main' into ElijahVlasov/cute-little-lemmas
alexanderlhicks Jul 21, 2026
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14 changes: 12 additions & 2 deletions ArkLib/Data/CodingTheory/ReedSolomon.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -185,8 +185,6 @@ lemma mem_code_iff_exists_polynomial {n : ℕ} {α : ι ↪ F} {f : ι → F} :
[Polynomial.degreeLT,
Polynomial.degree_lt_iff_coeff_zero])



lemma mem_code_iff_exists_polynomial_of_ne_zero {n : ℕ} [ne : NeZero n] {α : ι ↪ F} {f : ι → F} :
f ∈ code α n ↔ ∃ p : Polynomial F, p.natDegree < n ∧ f = evalOnPoints α p := by
rw [mem_code_iff_exists_polynomial]
Expand All @@ -200,6 +198,18 @@ lemma mem_code_iff_exists_polynomial_of_ne_zero {n : ℕ} [ne : NeZero n] {α :
(add simp [Polynomial.natDegree_lt_iff_degree_lt])
(add safe (by omega))

/-- `evalOnPoints α p` belongs to an RS-code of degree `n`,
if `p.degree < n`. -/
lemma evalOnPoints_mem_code_of_degree_lt {α : ι ↪ F} {p : F[X]} (h_deg : p.degree < n) :
evalOnPoints α p ∈ code α n :=
mem_code_of_polynomial_of_degree_lt_of_eval p h_deg (by simp [evalOnPoints])

/-- `evalOnPoints α p` belongs to an RS-code of degree `n`,
if `p.natDegree < n`. -/
lemma evalOnPoints_mem_code_of_natDegree_lt {α : ι ↪ F} {p : F[X]} (h_deg : p.natDegree < n) :
evalOnPoints α p ∈ code α n :=
mem_code_of_polynomial_of_natDegree_lt_of_eval p h_deg (by simp [evalOnPoints])

/-- **Monotonicity of `code` in the degree bound.** If `n ≤ m`, the degree-`n` Reed-Solomon code
is contained in the degree-`m` code over the same domain. -/
@[mono]
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