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refactor(Foundations,AffineDisperser): move monic_of_ne_zero to BinaryField (#949) #970

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Merged
SamuelSchlesinger merged 1 commit into from
Oct 25, 2026
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refactor(Foundations,AffineDisperser): move monic_of_ne_zero to BinaryField (#949) #970

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4 changes: 0 additions & 4 deletions Complexity/Computability/Complexity/AffineDisperser/BinaryPolynomial.lean
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Expand Up @@ -320,10 +320,6 @@ theorem modBits_all_false_iff (l f : List Bool) :
← length_modBits l f]
exact degree_polyOfBits_lt _

/-- Every nonzero binary polynomial is monic. -/
theorem monic_of_ne_zero {q : (ZMod 2)[X]} (hq : q ≠ 0) : q.Monic :=
(by decide : ∀ c : ZMod 2, c ≠ 0 → c = 1) _ (leadingCoeff_ne_zero.2 hq)

private theorem natDegree_polyOfCode_eq_log (m : ℕ) : (polyOfCode m).natDegree = Nat.log 2 m := by
rcases eq_or_ne m 0 with rfl | hm
· simp
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8 changes: 6 additions & 2 deletions Complexity/Foundations/FieldTheory/Finite/BinaryField.lean
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Expand Up @@ -54,6 +54,7 @@ need `w ≠ 0`.

## Main results

* `Cslib.monic_of_ne_zero`: every nonzero binary polynomial is monic.
* `Cslib.irreducible_leastIrreducible`, `Cslib.monic_leastIrreducible`,
`Cslib.natDegree_leastIrreducible`: `leastIrreducible w` is monic and irreducible, of degree `w`
when `w ≠ 0`.
Expand Down Expand Up @@ -211,10 +212,13 @@ theorem exists_code_leastIrreducible {p : ℕ} (hp : p ≠ 0) :
· exact hlog ▸ Nat.pow_log_le_self 2 hm0
· exact hlog ▸ Nat.lt_pow_succ_log_self one_lt_two m

/-- Every nonzero binary polynomial is monic. -/
theorem monic_of_ne_zero {q : (ZMod 2)[X]} (hq : q ≠ 0) : q.Monic :=
(by decide : ∀ c : ZMod 2, c ≠ 0 → c = 1) _ (leadingCoeff_ne_zero.2 hq)

/-- `leastIrreducible w` is monic for every degree `w`, as every nonzero binary polynomial is. -/
theorem monic_leastIrreducible (w : ℕ) : (leastIrreducible w).Monic :=
(by decide : ∀ c : ZMod 2, c ≠ 0 → c = 1) _
(leadingCoeff_ne_zero.2 (irreducible_leastIrreducible w).ne_zero)
monic_of_ne_zero (irreducible_leastIrreducible w).ne_zero

/-- **The explicit binary field of degree `w`**: binary polynomials modulo the irreducible
polynomial of degree `w` with the least code. -/
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