exists_sum_eq_one_iff_pairwise_coprime

English

If a and b are coprime naturals, then a.cast b.cast are coprime in any commutative ring.

Русский

Если a и b взаимно просты как натуральные, то a и b в кольце R взаимно просты.

LaTeX

$$IsCoprime (a : R) (b : R)$$

Lean4

theorem exists_sum_eq_one_iff_pairwise_coprime [DecidableEq I] (h : t.Nonempty) :
    (∃ μ : I → R, (∑ i ∈ t, μ i * ∏ j ∈ t \ { i }, s j) = 1) ↔ Pairwise (IsCoprime on fun i : t ↦ s i) := by
  induction h using Finset.Nonempty.cons_induction with
  | singleton => simp [exists_apply_eq, Pairwise, Function.onFun]
  | cons a t hat h ih =>
    rw [pairwise_cons']
    have mem : ∀ x ∈ t, a ∈ insert a t \ { x } := fun x hx ↦
      by
      rw [mem_sdiff, mem_singleton]
      exact ⟨mem_insert_self _ _, fun ha ↦ hat (ha ▸ hx)⟩
    constructor
    · rintro ⟨μ, hμ⟩
      rw [sum_cons, cons_eq_insert, sdiff_singleton_eq_erase, erase_insert hat] at hμ
      refine ⟨ih.mp ⟨Pi.single h.choose (μ a * s h.choose) + μ * fun _ ↦ s a, ?_⟩, fun b hb ↦ ?_⟩
      · rw [prod_eq_mul_prod_diff_singleton h.choose_spec, ← mul_assoc, ← @if_pos _ _ h.choose_spec R (_ * _) 0, ←
          sum_pi_single', ← sum_add_distrib] at hμ
        rw [← hμ, sum_congr rfl]
        intro x hx
        convert add_mul (R := R) _ _ _ using 2
        · by_cases hx : x = h.choose
          · rw [hx, Pi.single_eq_same, Pi.single_eq_same]
          · rw [Pi.single_eq_of_ne hx, Pi.single_eq_of_ne hx, zero_mul]
        · convert (mul_assoc _ _ _).symm
          rw [prod_eq_prod_diff_singleton_mul (mem x hx), mul_comm, sdiff_sdiff_comm, sdiff_singleton_eq_erase a,
            erase_insert hat]
      · have : IsCoprime (s b) (s a) := ⟨μ a * ∏ i ∈ t \ { b }, s i, ∑ i ∈ t, μ i * ∏ j ∈ t \ { i }, s j, ?_⟩
        · exact ⟨this.symm, this⟩
        rw [mul_assoc, ← prod_eq_prod_diff_singleton_mul hb, sum_mul, ← hμ, sum_congr rfl]
        intro x hx
        rw [mul_assoc]
        congr
        rw [prod_eq_prod_diff_singleton_mul (mem x hx) _]
        congr 2
        rw [sdiff_sdiff_comm, sdiff_singleton_eq_erase a, erase_insert hat]
    · rintro ⟨hs, Hb⟩
      obtain ⟨μ, hμ⟩ := ih.mpr hs
      obtain ⟨u, v, huv⟩ := IsCoprime.prod_left fun b hb ↦ (Hb b hb).right
      use fun i ↦ if i = a then u else v * μ i
      have hμ' : (∑ i ∈ t, v * ((μ i * ∏ j ∈ t \ { i }, s j) * s a)) = v * s a := by
        rw [← mul_sum, ← sum_mul, hμ, one_mul]
      rw [sum_cons, cons_eq_insert, sdiff_singleton_eq_erase, erase_insert hat]
      simp only [↓reduceIte, ite_mul]
      rw [← huv, ← hμ', sum_congr rfl]
      intro x hx
      rw [mul_assoc, if_neg fun ha : x = a ↦ hat (ha.casesOn hx)]
      rw [mul_assoc]
      congr
      rw [prod_eq_prod_diff_singleton_mul (mem x hx) _]
      congr 2
      rw [sdiff_sdiff_comm, sdiff_singleton_eq_erase a, erase_insert hat]

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