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Mathlib/Algebra/Polynomial/CancelLeads.lean

Lines changed: 1 addition & 1 deletion
Original file line numberDiff line numberDiff line change
@@ -62,7 +62,7 @@ theorem natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm
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apply lt_of_le_of_ne
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· compute_degree!
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rwa [Nat.sub_add_cancel]
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· contrapose! h0
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· contrapose h0
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rw [← leadingCoeff_eq_zero, leadingCoeff, h0, mul_assoc, X_pow_mul, ← tsub_add_cancel_of_le h,
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add_comm _ p.natDegree]
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simp only [coeff_mul_X_pow, coeff_neg, coeff_C_mul, add_tsub_cancel_left, coeff_add]

Mathlib/Algebra/Polynomial/FieldDivision.lean

Lines changed: 6 additions & 6 deletions
Original file line numberDiff line numberDiff line change
@@ -186,7 +186,7 @@ theorem isRoot_of_isRoot_of_dvd_derivative_mul [CharZero R] {f g : R[X]} (hf0 :
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(hfd : f ∣ f.derivative * g) {a : R} (haf : f.IsRoot a) : g.IsRoot a := by
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rcases hfd with ⟨r, hr⟩
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have hdf0 : derivative f ≠ 0 := by
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contrapose! haf
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contrapose haf
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rw [eq_C_of_derivative_eq_zero haf] at hf0 ⊢
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exact not_isRoot_C _ _ <| C_ne_zero.mp hf0
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by_contra hg
@@ -423,13 +423,13 @@ lemma natDegree_mod_lt [Field k] (p : k[X]) {q : k[X]} (hq : q.natDegree ≠ 0)
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(p % q).natDegree < q.natDegree := by
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have hq' : q.leadingCoeff ≠ 0 := by
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rw [leadingCoeff_ne_zero]
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contrapose! hq
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contrapose hq
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simp [hq]
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rw [mod_def]
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refine (natDegree_modByMonic_lt p ?_ ?_).trans_le ?_
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· refine monic_mul_C_of_leadingCoeff_mul_eq_one ?_
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rw [mul_inv_eq_one₀ hq']
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· contrapose! hq
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· contrapose hq
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rw [← natDegree_mul_C_eq_of_mul_eq_one ((inv_mul_eq_one₀ hq').mpr rfl)]
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simp [hq]
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· exact natDegree_mul_C_le q q.leadingCoeff⁻¹
@@ -654,7 +654,7 @@ theorem isCoprime_of_is_root_of_eval_derivative_ne_zero {K : Type*} [Field K] (f
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refine Or.resolve_left
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(EuclideanDomain.dvd_or_coprime (X - C a) (f /ₘ (X - C a))
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(irreducible_of_degree_eq_one (Polynomial.degree_X_sub_C a))) ?_
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contrapose! hf' with h
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contrapose hf' with h
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have : X - C a ∣ derivative f := X_sub_C_dvd_derivative_of_X_sub_C_dvd_divByMonic f h
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rw [← modByMonic_eq_zero_iff_dvd (monic_X_sub_C _), modByMonic_X_sub_C_eq_C_eval] at this
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rwa [← C_inj, C_0]
@@ -669,7 +669,7 @@ theorem irreducible_iff_degree_lt (p : R[X]) (hp0 : p ≠ 0) (hpu : ¬ IsUnit p)
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rw [← irreducible_mul_leadingCoeff_inv,
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(monic_mul_leadingCoeff_inv hp0).irreducible_iff_degree_lt]
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· simp [hp0, natDegree_mul_leadingCoeff_inv]
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· contrapose! hpu
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· contrapose hpu
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exact .of_mul_eq_one _ hpu
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/-- To check a polynomial `p` over a field is irreducible, it suffices to check there are no
@@ -680,7 +680,7 @@ See also: `Polynomial.Monic.irreducible_iff_natDegree'`.
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theorem irreducible_iff_lt_natDegree_lt {p : R[X]} (hp0 : p ≠ 0) (hpu : ¬ IsUnit p) :
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Irreducible p ↔ ∀ q, Monic q → natDegree q ∈ Finset.Ioc 0 (natDegree p / 2) → ¬ q ∣ p := by
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have : p * C (leadingCoeff p)⁻¹ ≠ 1 := by
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contrapose! hpu
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contrapose hpu
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exact .of_mul_eq_one _ hpu
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rw [← irreducible_mul_leadingCoeff_inv,
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(monic_mul_leadingCoeff_inv hp0).irreducible_iff_lt_natDegree_lt this,

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