awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.Coprime.Ideal (goal)
Goal:
ι : Type u_1
R : Type u_2
inst : CommSemiring R
I : ι → Ideal R
this : DecidableEq ι
a : ι
t : Finset ι
hat : a ∉ t
h : t.Nonempty
ih :
  (∃ (μ : (i : ι) → ↥(⨅ j ∈ t, ⨅ (_ : j ≠ i), I j)), Eq.{u_2 + 1} (α := R) (∑ i ∈ t, (↑(μ i) : R)) 1) ↔
    (↑t : Set ι).Pairwise fun (i j : ι) ↦ I i ⊔ I j = ⊤
μ : (i : ι) → ↥(⨅ j ∈ Finset.cons a t hat, ⨅ (_ : j ≠ i), I j)
hμ_1 : (↑(μ a) : R) + ∑ x ∈ t, (↑(μ x) : R) = 1
hμ :
  Eq.{u_2 + 1} (α := R)
    (∑ x ∈ t,
      HAdd.hAdd (α := R) (Pi.single (M := fun (a : ι) ↦ R) (Exists.choose (p := Membership.mem t) h) (↑(μ a) : R) x : R)
        (↑(μ x) : R))
    1
i : ι
h_1 : i ∈ t
⊢ (↑(HAdd.hAdd (α := (j : ι) → ↥(⨅ j_1 ∈ t, ⨅ (_ : j_1 ≠ j), I j_1)) (β := (i : ι) → ↥(⨅ j ∈ t, ⨅ (_ : j ≠ i), I j))
          (Pi.single (M := fun (i : ι) ↦ ↥(⨅ j ∈ t, ⨅ (_ : j ≠ i), I j)) (Exists.choose (p := Membership.mem t) h)
            (Subtype.mk (p := fun (x : R) ↦
              Membership.mem (γ := Submodule R R) (⨅ j ∈ t, ⨅ (_ : j ≠ Exists.choose (p := Membership.mem t) h), I j) x)
              (↑(μ a) : R) ⋯))
          (fun (i : ι) ↦
            Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submodule R R) (⨅ j ∈ t, ⨅ (_ : j ≠ i), I j) x)
              (↑(μ i) : R) ⋯)
          i) :
      R) =
    HAdd.hAdd (α := R) (Pi.single (M := fun (a : ι) ↦ R) (Exists.choose (p := Membership.mem t) h) (↑(μ a) : R) i : R)
      (↑(μ i) : R)
have h₁ : (↑(Pi.single (M := fun (i : ι) ↦ ↥(⨅ j ∈ t, ⨅ (_ : j ≠ i), I j)) (Exists.choose (p := Membership.mem t) h)
          (Subtype.mk (p := fun (x : R) ↦
              Membership.mem (γ := Submodule R R) (⨅ j ∈ t, ⨅ (_ : j ≠ Exists.choose (p := Membership.mem t) h), I j) x)
              (↑(μ a) : R) sorry :
            ↥(⨅ j ∈ t, ⨅ (_ : j ≠ Exists.choose (p := Membership.mem t) h), I j))
          i) :
      R) +
    (↑(Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submodule R R) (⨅ j ∈ t, ⨅ (_ : j ≠ i), I j) x) (↑(μ i) : R)
          sorry) :
      R) =
  HAdd.hAdd (α := R) (Pi.single (M := fun (a : ι) ↦ R) (Exists.choose (p := Membership.mem t) h) (↑(μ a) : R) i : R)
    (↑(μ i) : R)
Tactic: rw [Pi.add_apply, Submodule.coe_add, Submodule.coe_mk]
-- Lean 4 proof step from Mathlib.RingTheory.Coprime.Ideal (goal)
Goal:
ι : Type u_1
R : Type u_2
inst : CommSemiring R
I : ι → Ideal R
this : DecidableEq ι
a : ι
t : Finset ι
hat : a ∉ t
h : t.Nonempty
ih :
  (∃ (μ : (i : ι) → ↥(⨅ j ∈ t, ⨅ (_ : j ≠ i), I j)), Eq.{u_2 + 1} (α := R) (∑ i ∈ t, (↑(μ i) : R)) 1) ↔
    (↑t : Set ι).Pairwise fun (i j : ι) ↦ I i ⊔ I j = ⊤
μ : (i : ι) → ↥(⨅ j ∈ Finset.cons a t hat, ⨅ (_ : j ≠ i), I j)
hμ_1 : (↑(μ a) : R) + ∑ x ∈ t, (↑(μ x) : R) = 1
hμ :
  Eq.{u_2 + 1} (α := R)
    (∑ x ∈ t,
      HAdd.hAdd (α := R) (Pi.single (M := fun (a : ι) ↦ R) (Exists.choose (p := Membership.mem t) h) (↑(μ a) : R) x : R)
        (↑(μ x) : R))
    1
i : ι
h_1 : i ∈ t
⊢ (↑(Pi.single (M := fun (i : ι) ↦ ↥(⨅ j ∈ t, ⨅ (_ : j ≠ i), I j)) (Exists.choose (p := Membership.mem t) h)
            (Subtype.mk (p := fun (x : R) ↦
                Membership.mem (γ := Submodule R R) (⨅ j ∈ t, ⨅ (_ : j ≠ Exists.choose (p := Membership.mem t) h), I j)
                  x)
                (↑(μ a) : R) ⋯ :
              ↥(⨅ j ∈ t, ⨅ (_ : j ≠ Exists.choose (p := Membership.mem t) h), I j))
            i) :
        R) +
      (↑(Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submodule R R) (⨅ j ∈ t, ⨅ (_ : j ≠ i), I j) x)
            (↑(μ i) : R) ⋯) :
        R) =
    HAdd.hAdd (α := R) (Pi.single (M := fun (a : ι) ↦ R) (Exists.choose (p := Membership.mem t) h) (↑(μ a) : R) i : R)
      (↑(μ i) : R)
have pos : ∀ (hi : i = Exists.choose (p := Membership.mem t) h),
  (↑(Pi.single (M := fun (i : ι) ↦ ↥(⨅ j ∈ t, ⨅ (_ : j ≠ i), I j)) (Exists.choose (p := Membership.mem t) h)
            (Subtype.mk (p := fun (x : R) ↦
                Membership.mem (γ := Submodule R R) (⨅ j ∈ t, ⨅ (_ : j ≠ Exists.choose (p := Membership.mem t) h), I j)
                  x)
                (↑(μ a) : R) sorry :
              ↥(⨅ j ∈ t, ⨅ (_ : j ≠ Exists.choose (p := Membership.mem t) h), I j))
            i) :
        R) +
      (↑(Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submodule R R) (⨅ j ∈ t, ⨅ (_ : j ≠ i), I j) x)
            (↑(μ i) : R) sorry) :
        R) =
    HAdd.hAdd (α := R) (Pi.single (M := fun (a : ι) ↦ R) (Exists.choose (p := Membership.mem t) h) (↑(μ a) : R) i : R)
      (↑(μ i) : R)
Tactic: by_cases hi : i = h.choose
-- Lean 4 proof step from Mathlib.RingTheory.Coprime.Ideal (goal)
Goal:
ι : Type u_1
R : Type u_2
inst : CommSemiring R
I : ι → Ideal R
this : DecidableEq ι
a : ι
t : Finset ι
hat : a ∉ t
h : t.Nonempty
ih :
  (∃ (μ : (i : ι) → ↥(⨅ j ∈ t, ⨅ (_ : j ≠ i), I j)), Eq.{u_2 + 1} (α := R) (∑ i ∈ t, (↑(μ i) : R)) 1) ↔
    (↑t : Set ι).Pairwise fun (i j : ι) ↦ I i ⊔ I j = ⊤
μ : (i : ι) → ↥(⨅ j ∈ Finset.cons a t hat, ⨅ (_ : j ≠ i), I j)
hμ_1 : (↑(μ a) : R) + ∑ x ∈ t, (↑(μ x) : R) = 1
hμ :
  Eq.{u_2 + 1} (α := R)
    (∑ x ∈ t,
      HAdd.hAdd (α := R) (Pi.single (M := fun (a : ι) ↦ R) (Exists.choose (p := Membership.mem t) h) (↑(μ a) : R) x : R)
        (↑(μ x) : R))
    1
i : ι
h_1 : i ∈ t
pos :
  ∀ (hi : i = Exists.choose (p := Membership.mem t) h),
    (↑(Pi.single (M := fun (i : ι) ↦ ↥(⨅ j ∈ t, ⨅ (_ : j ≠ i), I j)) (Exists.choose (p := Membership.mem t) h)
              (Subtype.mk (p := fun (x : R) ↦
                  Membership.mem (γ := Submodule R R)
                    (⨅ j ∈ t, ⨅ (_ : j ≠ Exists.choose (p := Membership.mem t) h), I j) x)
                  (↑(μ a) : R) ⋯ :
                ↥(⨅ j ∈ t, ⨅ (_ : j ≠ Exists.choose (p := Membership.mem t) h), I j))
              i) :
          R) +
        (↑(Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submodule R R) (⨅ j ∈ t, ⨅ (_ : j ≠ i), I j) x)
              (↑(μ i) : R) ⋯) :
          R) =
      HAdd.hAdd (α := R) (Pi.single (M := fun (a : ι) ↦ R) (Exists.choose (p := Membership.mem t) h) (↑(μ a) : R) i : R)
        (↑(μ i) : R)
⊢ (↑(Pi.single (M := fun (i : ι) ↦ ↥(⨅ j ∈ t, ⨅ (_ : j ≠ i), I j)) (Exists.choose (p := Membership.mem t) h)
            (Subtype.mk (p := fun (x : R) ↦
                Membership.mem (γ := Submodule R R) (⨅ j ∈ t, ⨅ (_ : j ≠ Exists.choose (p := Membership.mem t) h), I j)
                  x)
                (↑(μ a) : R) ⋯ :
              ↥(⨅ j ∈ t, ⨅ (_ : j ≠ Exists.choose (p := Membership.mem t) h), I j))
            i) :
        R) +
      (↑(Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submodule R R) (⨅ j ∈ t, ⨅ (_ : j ≠ i), I j) x)
            (↑(μ i) : R) ⋯) :
        R) =
    HAdd.hAdd (α := R) (Pi.single (M := fun (a : ι) ↦ R) (Exists.choose (p := Membership.mem t) h) (↑(μ a) : R) i : R)
      (↑(μ i) : R)
have neg : ∀ (hi : ¬i = Exists.choose (p := Membership.mem t) h),
  (↑(Pi.single (M := fun (i : ι) ↦ ↥(⨅ j ∈ t, ⨅ (_ : j ≠ i), I j)) (Exists.choose (p := Membership.mem t) h)
            (Subtype.mk (p := fun (x : R) ↦
                Membership.mem (γ := Submodule R R) (⨅ j ∈ t, ⨅ (_ : j ≠ Exists.choose (p := Membership.mem t) h), I j)
                  x)
                (↑(μ a) : R) sorry :
              ↥(⨅ j ∈ t, ⨅ (_ : j ≠ Exists.choose (p := Membership.mem t) h), I j))
            i) :
        R) +
      (↑(Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submodule R R) (⨅ j ∈ t, ⨅ (_ : j ≠ i), I j) x)
            (↑(μ i) : R) sorry) :
        R) =
    HAdd.hAdd (α := R) (Pi.single (M := fun (a : ι) ↦ R) (Exists.choose (p := Membership.mem t) h) (↑(μ a) : R) i : R)
      (↑(μ i) : R)
Tactic: by_cases hi : i = h.choose
-- Lean 4 proof step from Mathlib.RingTheory.Coprime.Ideal (goal)
Goal:
ι : Type u_1
R : Type u_2
inst : CommSemiring R
I : ι → Ideal R
this : DecidableEq ι
a : ι
t : Finset ι
hat : a ∉ t
h : t.Nonempty
ih :
  (∃ (μ : (i : ι) → ↥(⨅ j ∈ t, ⨅ (_ : j ≠ i), I j)), Eq.{u_2 + 1} (α := R) (∑ i ∈ t, (↑(μ i) : R)) 1) ↔
    (↑t : Set ι).Pairwise fun (i j : ι) ↦ I i ⊔ I j = ⊤
μ : (i : ι) → ↥(⨅ j ∈ Finset.cons a t hat, ⨅ (_ : j ≠ i), I j)
hμ_1 : (↑(μ a) : R) + ∑ x ∈ t, (↑(μ x) : R) = 1
hμ :
  Eq.{u_2 + 1} (α := R)
    (∑ x ∈ t,
      HAdd.hAdd (α := R) (Pi.single (M := fun (a : ι) ↦ R) (Exists.choose (p := Membership.mem t) h) (↑(μ a) : R) x : R)
        (↑(μ x) : R))
    1
i : ι
h_1 : i ∈ t
hi : i = Exists.choose (p := Membership.mem t) h
⊢ (↑(Pi.single (M := fun (i : ι) ↦ ↥(⨅ j ∈ t, ⨅ (_ : j ≠ i), I j)) (Exists.choose (p := Membership.mem t) h)
            (Subtype.mk (p := fun (x : R) ↦
                Membership.mem (γ := Submodule R R) (⨅ j ∈ t, ⨅ (_ : j ≠ Exists.choose (p := Membership.mem t) h), I j)
                  x)
                (↑(μ a) : R) ⋯ :
              ↥(⨅ j ∈ t, ⨅ (_ : j ≠ Exists.choose (p := Membership.mem t) h), I j))
            i) :
        R) +
      (↑(Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submodule R R) (⨅ j ∈ t, ⨅ (_ : j ≠ i), I j) x)
            (↑(μ i) : R) ⋯) :
        R) =
    HAdd.hAdd (α := R) (Pi.single (M := fun (a : ι) ↦ R) (Exists.choose (p := Membership.mem t) h) (↑(μ a) : R) i : R)
      (↑(μ i) : R)
have pos : (↑(Pi.single (M := fun (i : ι) ↦ ↥(⨅ j ∈ t, ⨅ (_ : j ≠ i), I j)) (Exists.choose (p := Membership.mem t) h)
          (Subtype.mk (p := fun (x : R) ↦
              Membership.mem (γ := Submodule R R) (⨅ j ∈ t, ⨅ (_ : j ≠ Exists.choose (p := Membership.mem t) h), I j) x)
              (↑(μ a) : R) sorry :
            ↥(⨅ j ∈ t, ⨅ (_ : j ≠ Exists.choose (p := Membership.mem t) h), I j))
          (Exists.choose (p := Membership.mem t) h)) :
      R) +
    (↑(Subtype.mk (p := fun (x : R) ↦
          Membership.mem (γ := Submodule R R) (⨅ j ∈ t, ⨅ (_ : j ≠ Exists.choose (p := Membership.mem t) h), I j) x)
          (↑(μ (Exists.choose (p := Membership.mem t) h)) : R) sorry) :
      R) =
  HAdd.hAdd (α := R)
    (Pi.single (M := fun (a : ι) ↦ R) (Exists.choose (p := Membership.mem t) h) (↑(μ a) : R)
        (Exists.choose (p := Membership.mem t) h) :
      R)
    (↑(μ (Exists.choose (p := Membership.mem t) h)) : R)
Tactic: rw [hi, Pi.single_eq_same, Pi.single_eq_same, Submodule.coe_mk]
-- Lean 4 proof step from Mathlib.RingTheory.Coprime.Ideal (goal)
Goal:
ι : Type u_1
R : Type u_2
inst : CommSemiring R
I : ι → Ideal R
this : DecidableEq ι
a : ι
t : Finset ι
hat : a ∉ t
h : t.Nonempty
ih :
  (∃ (μ : (i : ι) → ↥(⨅ j ∈ t, ⨅ (_ : j ≠ i), I j)), Eq.{u_2 + 1} (α := R) (∑ i ∈ t, (↑(μ i) : R)) 1) ↔
    (↑t : Set ι).Pairwise fun (i j : ι) ↦ I i ⊔ I j = ⊤
μ : (i : ι) → ↥(⨅ j ∈ Finset.cons a t hat, ⨅ (_ : j ≠ i), I j)
hμ_1 : (↑(μ a) : R) + ∑ x ∈ t, (↑(μ x) : R) = 1
hμ :
  Eq.{u_2 + 1} (α := R)
    (∑ x ∈ t,
      HAdd.hAdd (α := R) (Pi.single (M := fun (a : ι) ↦ R) (Exists.choose (p := Membership.mem t) h) (↑(μ a) : R) x : R)
        (↑(μ x) : R))
    1
i : ι
h_1 : i ∈ t
hi : i = Exists.choose (p := Membership.mem t) h
pos :
  (↑(Pi.single (M := fun (i : ι) ↦ ↥(⨅ j ∈ t, ⨅ (_ : j ≠ i), I j)) (Exists.choose (p := Membership.mem t) h)
            (Subtype.mk (p := fun (x : R) ↦
                Membership.mem (γ := Submodule R R) (⨅ j ∈ t, ⨅ (_ : j ≠ Exists.choose (p := Membership.mem t) h), I j)
                  x)
                (↑(μ a) : R) ⋯ :
              ↥(⨅ j ∈ t, ⨅ (_ : j ≠ Exists.choose (p := Membership.mem t) h), I j))
            (Exists.choose (p := Membership.mem t) h)) :
        R) +
      (↑(Subtype.mk (p := fun (x : R) ↦
            Membership.mem (γ := Submodule R R) (⨅ j ∈ t, ⨅ (_ : j ≠ Exists.choose (p := Membership.mem t) h), I j) x)
            (↑(μ (Exists.choose (p := Membership.mem t) h)) : R) ⋯) :
        R) =
    HAdd.hAdd (α := R)
      (Pi.single (M := fun (a : ι) ↦ R) (Exists.choose (p := Membership.mem t) h) (↑(μ a) : R)
          (Exists.choose (p := Membership.mem t) h) :
        R)
      (↑(μ (Exists.choose (p := Membership.mem t) h)) : R)
⊢ (↑(Pi.single (M := fun (i : ι) ↦ ↥(⨅ j ∈ t, ⨅ (_ : j ≠ i), I j)) (Exists.choose (p := Membership.mem t) h)
            (Subtype.mk (p := fun (x : R) ↦
                Membership.mem (γ := Submodule R R) (⨅ j ∈ t, ⨅ (_ : j ≠ Exists.choose (p := Membership.mem t) h), I j)
                  x)
                (↑(μ a) : R) ⋯ :
              ↥(⨅ j ∈ t, ⨅ (_ : j ≠ Exists.choose (p := Membership.mem t) h), I j))
            i) :
        R) +
      (↑(Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submodule R R) (⨅ j ∈ t, ⨅ (_ : j ≠ i), I j) x)
            (↑(μ i) : R) ⋯) :
        R) =
    HAdd.hAdd (α := R) (Pi.single (M := fun (a : ι) ↦ R) (Exists.choose (p := Membership.mem t) h) (↑(μ a) : R) i : R)
      (↑(μ i) : R)
have pos₁ : (↑(Subtype.mk (p := fun (x : R) ↦
          Membership.mem (γ := Submodule R R) (⨅ j ∈ t, ⨅ (_ : j ≠ Exists.choose (p := Membership.mem t) h), I j) x)
          (↑(μ a) : R) sorry) :
      R) +
    (↑(Subtype.mk (p := fun (x : R) ↦
          Membership.mem (γ := Submodule R R) (⨅ j ∈ t, ⨅ (_ : j ≠ Exists.choose (p := Membership.mem t) h), I j) x)
          (↑(μ (Exists.choose (p := Membership.mem t) h)) : R) sorry) :
      R) =
  HAdd.hAdd (α := R)
    (Pi.single (M := fun (a : ι) ↦ R) (Exists.choose (p := Membership.mem t) h) (↑(μ a) : R)
        (Exists.choose (p := Membership.mem t) h) :
      R)
    (↑(μ (Exists.choose (p := Membership.mem t) h)) : R)
Tactic: rw [hi, Pi.single_eq_same, Pi.single_eq_same, Submodule.coe_mk]
-- Lean 4 proof step from Mathlib.RingTheory.Coprime.Ideal (goal)
Goal:
ι : Type u_1
R : Type u_2
inst : CommSemiring R
I : ι → Ideal R
this : DecidableEq ι
a : ι
t : Finset ι
hat : a ∉ t
h : t.Nonempty
ih :
  (∃ (μ : (i : ι) → ↥(⨅ j ∈ t, ⨅ (_ : j ≠ i), I j)), Eq.{u_2 + 1} (α := R) (∑ i ∈ t, (↑(μ i) : R)) 1) ↔
    (↑t : Set ι).Pairwise fun (i j : ι) ↦ I i ⊔ I j = ⊤
μ : (i : ι) → ↥(⨅ j ∈ Finset.cons a t hat, ⨅ (_ : j ≠ i), I j)
hμ_1 : (↑(μ a) : R) + ∑ x ∈ t, (↑(μ x) : R) = 1
hμ :
  Eq.{u_2 + 1} (α := R)
    (∑ x ∈ t,
      HAdd.hAdd (α := R) (Pi.single (M := fun (a : ι) ↦ R) (Exists.choose (p := Membership.mem t) h) (↑(μ a) : R) x : R)
        (↑(μ x) : R))
    1
i : ι
h_1 : i ∈ t
hi : ¬i = Exists.choose (p := Membership.mem t) h
⊢ (↑(Pi.single (M := fun (i : ι) ↦ ↥(⨅ j ∈ t, ⨅ (_ : j ≠ i), I j)) (Exists.choose (p := Membership.mem t) h)
            (Subtype.mk (p := fun (x : R) ↦
                Membership.mem (γ := Submodule R R) (⨅ j ∈ t, ⨅ (_ : j ≠ Exists.choose (p := Membership.mem t) h), I j)
                  x)
                (↑(μ a) : R) ⋯ :
              ↥(⨅ j ∈ t, ⨅ (_ : j ≠ Exists.choose (p := Membership.mem t) h), I j))
            i) :
        R) +
      (↑(Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submodule R R) (⨅ j ∈ t, ⨅ (_ : j ≠ i), I j) x)
            (↑(μ i) : R) ⋯) :
        R) =
    HAdd.hAdd (α := R) (Pi.single (M := fun (a : ι) ↦ R) (Exists.choose (p := Membership.mem t) h) (↑(μ a) : R) i : R)
      (↑(μ i) : R)
have neg : (↑0 : R) +
    (↑(Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submodule R R) (⨅ j ∈ t, ⨅ (_ : j ≠ i), I j) x) (↑(μ i) : R)
          sorry) :
      R) =
  HAdd.hAdd (α := R) (Pi.single (M := fun (a : ι) ↦ R) (Exists.choose (p := Membership.mem t) h) (↑(μ a) : R) i : R)
    (↑(μ i) : R)
Tactic: rw [Pi.single_eq_of_ne hi, Pi.single_eq_of_ne hi, Submodule.coe_zero]
-- Lean 4 proof step from Mathlib.RingTheory.Coprime.Ideal (goal)
Goal:
ι : Type u_1
R : Type u_2
inst : CommSemiring R
I : ι → Ideal R
this : DecidableEq ι
a : ι
t : Finset ι
hat : a ∉ t
h : t.Nonempty
ih :
  (∃ (μ : (i : ι) → ↥(⨅ j ∈ t, ⨅ (_ : j ≠ i), I j)), Eq.{u_2 + 1} (α := R) (∑ i ∈ t, (↑(μ i) : R)) 1) ↔
    (↑t : Set ι).Pairwise fun (i j : ι) ↦ I i ⊔ I j = ⊤
μ : (i : ι) → ↥(⨅ j ∈ Finset.cons a t hat, ⨅ (_ : j ≠ i), I j)
hμ_1 : (↑(μ a) : R) + ∑ x ∈ t, (↑(μ x) : R) = 1
hμ :
  Eq.{u_2 + 1} (α := R)
    (∑ x ∈ t,
      HAdd.hAdd (α := R) (Pi.single (M := fun (a : ι) ↦ R) (Exists.choose (p := Membership.mem t) h) (↑(μ a) : R) x : R)
        (↑(μ x) : R))
    1
i : ι
h_1 : i ∈ t
hi : ¬i = Exists.choose (p := Membership.mem t) h
neg :
  (↑0 : R) +
      (↑(Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submodule R R) (⨅ j ∈ t, ⨅ (_ : j ≠ i), I j) x)
            (↑(μ i) : R) ⋯) :
        R) =
    HAdd.hAdd (α := R) (Pi.single (M := fun (a : ι) ↦ R) (Exists.choose (p := Membership.mem t) h) (↑(μ a) : R) i : R)
      (↑(μ i) : R)
⊢ (↑(Pi.single (M := fun (i : ι) ↦ ↥(⨅ j ∈ t, ⨅ (_ : j ≠ i), I j)) (Exists.choose (p := Membership.mem t) h)
            (Subtype.mk (p := fun (x : R) ↦
                Membership.mem (γ := Submodule R R) (⨅ j ∈ t, ⨅ (_ : j ≠ Exists.choose (p := Membership.mem t) h), I j)
                  x)
                (↑(μ a) : R) ⋯ :
              ↥(⨅ j ∈ t, ⨅ (_ : j ≠ Exists.choose (p := Membership.mem t) h), I j))
            i) :
        R) +
      (↑(Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submodule R R) (⨅ j ∈ t, ⨅ (_ : j ≠ i), I j) x)
            (↑(μ i) : R) ⋯) :
        R) =
    HAdd.hAdd (α := R) (Pi.single (M := fun (a : ι) ↦ R) (Exists.choose (p := Membership.mem t) h) (↑(μ a) : R) i : R)
      (↑(μ i) : R)
have neg₁ : (↑0 : R) +
    (↑(Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submodule R R) (⨅ j ∈ t, ⨅ (_ : j ≠ i), I j) x) (↑(μ i) : R)
          sorry) :
      R) =
  0 + (↑(μ i) : R)
Tactic: rw [Pi.single_eq_of_ne hi, Pi.single_eq_of_ne hi, Submodule.coe_zero]
-- Lean 4 proof step from Mathlib.RingTheory.Coprime.Ideal (goal)
Goal:
ι : Type u_1
R : Type u_2
inst : CommSemiring R
I : ι → Ideal R
this : DecidableEq ι
a : ι
t : Finset ι
hat : a ∉ t
h : t.Nonempty
ih :
  (∃ (μ : (i : ι) → ↥(⨅ j ∈ t, ⨅ (_ : j ≠ i), I j)), Eq.{u_2 + 1} (α := R) (∑ i ∈ t, (↑(μ i) : R)) 1) ↔
    (↑t : Set ι).Pairwise fun (i j : ι) ↦ I i ⊔ I j = ⊤
μ : (i : ι) → ↥(⨅ j ∈ Finset.cons a t hat, ⨅ (_ : j ≠ i), I j)
hμ_1 : (↑(μ a) : R) + ∑ x ∈ t, (↑(μ x) : R) = 1
hμ :
  Eq.{u_2 + 1} (α := R)
    (∑ x ∈ t,
      HAdd.hAdd (α := R) (Pi.single (M := fun (a : ι) ↦ R) (Exists.choose (p := Membership.mem t) h) (↑(μ a) : R) x : R)
        (↑(μ x) : R))
    1
i : ι
h_1 : i ∈ t
hi : ¬i = Exists.choose (p := Membership.mem t) h
neg :
  (↑0 : R) +
      (↑(Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submodule R R) (⨅ j ∈ t, ⨅ (_ : j ≠ i), I j) x)
            (↑(μ i) : R) ⋯) :
        R) =
    HAdd.hAdd (α := R) (Pi.single (M := fun (a : ι) ↦ R) (Exists.choose (p := Membership.mem t) h) (↑(μ a) : R) i : R)
      (↑(μ i) : R)
neg₁ :
  (↑0 : R) +
      (↑(Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submodule R R) (⨅ j ∈ t, ⨅ (_ : j ≠ i), I j) x)
            (↑(μ i) : R) ⋯) :
        R) =
    0 + (↑(μ i) : R)
⊢ (↑(Pi.single (M := fun (i : ι) ↦ ↥(⨅ j ∈ t, ⨅ (_ : j ≠ i), I j)) (Exists.choose (p := Membership.mem t) h)
            (Subtype.mk (p := fun (x : R) ↦
                Membership.mem (γ := Submodule R R) (⨅ j ∈ t, ⨅ (_ : j ≠ Exists.choose (p := Membership.mem t) h), I j)
                  x)
                (↑(μ a) : R) ⋯ :
              ↥(⨅ j ∈ t, ⨅ (_ : j ≠ Exists.choose (p := Membership.mem t) h), I j))
            i) :
        R) +
      (↑(Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submodule R R) (⨅ j ∈ t, ⨅ (_ : j ≠ i), I j) x)
            (↑(μ i) : R) ⋯) :
        R) =
    HAdd.hAdd (α := R) (Pi.single (M := fun (a : ι) ↦ R) (Exists.choose (p := Membership.mem t) h) (↑(μ a) : R) i : R)
      (↑(μ i) : R)
have neg₂ : 0 +
    (↑(Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Submodule R R) (⨅ j ∈ t, ⨅ (_ : j ≠ i), I j) x) (↑(μ i) : R)
          sorry) :
      R) =
  0 + (↑(μ i) : R)
Tactic: rw [Pi.single_eq_of_ne hi, Pi.single_eq_of_ne hi, Submodule.coe_zero]
-- Lean 4 proof step from Mathlib.RingTheory.Coprime.Ideal (goal)
Goal:
ι : Type u_1
R : Type u_2
inst : CommSemiring R
I : ι → Ideal R
this : DecidableEq ι
a : ι
t : Finset ι
hat : a ∉ t
h : t.Nonempty
ih :
  (∃ (μ : (i : ι) → ↥(⨅ j ∈ t, ⨅ (_ : j ≠ i), I j)), Eq.{u_2 + 1} (α := R) (∑ i ∈ t, (↑(μ i) : R)) 1) ↔
    (↑t : Set ι).Pairwise fun (i j : ι) ↦ I i ⊔ I j = ⊤
μ : (i : ι) → ↥(⨅ j ∈ Finset.cons a t hat, ⨅ (_ : j ≠ i), I j)
hμ : (↑(μ a) : R) + ∑ x ∈ t, (↑(μ x) : R) = 1
b : ι
hb : b ∈ (↑t : Set ι)
ab : a ≠ b
⊢ I a ⊔ I b = ⊤
have rw : ∃ y ∈ I a, ∃ z ∈ I b, y + z = 1
Tactic: rw [eq_top_iff_one, Submodule.mem_sup]
-- Lean 4 proof step from Mathlib.RingTheory.Coprime.Ideal (goal)
Goal:
ι : Type u_1
R : Type u_2
inst : CommSemiring R
I : ι → Ideal R
this : DecidableEq ι
a : ι
t : Finset ι
hat : a ∉ t
h : t.Nonempty
ih :
  (∃ (μ : (i : ι) → ↥(⨅ j ∈ t, ⨅ (_ : j ≠ i), I j)), Eq.{u_2 + 1} (α := R) (∑ i ∈ t, (↑(μ i) : R)) 1) ↔
    (↑t : Set ι).Pairwise fun (i j : ι) ↦ I i ⊔ I j = ⊤
μ : (i : ι) → ↥(⨅ j ∈ Finset.cons a t hat, ⨅ (_ : j ≠ i), I j)
hμ : (↑(μ a) : R) + ∑ x ∈ t, (↑(μ x) : R) = 1
b : ι
hb : b ∈ (↑t : Set ι)
ab : a ≠ b
⊢ ∃ y ∈ I a, ∃ z ∈ I b, y + z = 1
have rw : ∀ (hμ : ∑ x ∈ t, (↑(μ x) : R) + (↑(μ a) : R) = 1) (b : ι) (hb : b ∈ (↑t : Set ι)) (ab : a ≠ b),
  ∃ y ∈ I a, ∃ z ∈ I b, y + z = 1
Tactic: rw [add_comm] at hμ
Cleared hypotheses: hμ, b, hb, ab
-- Lean 4 proof step from Mathlib.RingTheory.Coprime.Ideal (goal)
Goal:
ι : Type u_1
R : Type u_2
inst : CommSemiring R
I : ι → Ideal R
this : DecidableEq ι
a : ι
t : Finset ι
hat : a ∉ t
h : t.Nonempty
ih :
  (∃ (μ : (i : ι) → ↥(⨅ j ∈ t, ⨅ (_ : j ≠ i), I j)), Eq.{u_2 + 1} (α := R) (∑ i ∈ t, (↑(μ i) : R)) 1) ↔
    (↑t : Set ι).Pairwise fun (i j : ι) ↦ I i ⊔ I j = ⊤
μ : (i : ι) → ↥(⨅ j ∈ Finset.cons a t hat, ⨅ (_ : j ≠ i), I j)
hμ : ∑ x ∈ t, (↑(μ x) : R) + (↑(μ a) : R) = 1
b : ι
hb : b ∈ (↑t : Set ι)
ab : a ≠ b
⊢ ∃ y ∈ I a, ∃ z ∈ I b, y + z = 1
have refine_1 : ∑ x ∈ t, (↑(μ x) : R) ∈ I a
Tactic: refine ⟨_, ?_, _, ?_, hμ⟩
-- Lean 4 proof step from Mathlib.RingTheory.Coprime.Ideal (goal)
Goal:
ι : Type u_1
R : Type u_2
inst : CommSemiring R
I : ι → Ideal R
this : DecidableEq ι
a : ι
t : Finset ι
hat : a ∉ t
h : t.Nonempty
ih :
  (∃ (μ : (i : ι) → ↥(⨅ j ∈ t, ⨅ (_ : j ≠ i), I j)), Eq.{u_2 + 1} (α := R) (∑ i ∈ t, (↑(μ i) : R)) 1) ↔
    (↑t : Set ι).Pairwise fun (i j : ι) ↦ I i ⊔ I j = ⊤
μ : (i : ι) → ↥(⨅ j ∈ Finset.cons a t hat, ⨅ (_ : j ≠ i), I j)
hμ : ∑ x ∈ t, (↑(μ x) : R) + (↑(μ a) : R) = 1
b : ι
hb : b ∈ (↑t : Set ι)
ab : a ≠ b
refine_1 : ∑ x ∈ t, (↑(μ x) : R) ∈ I a
⊢ ∃ y ∈ I a, ∃ z ∈ I b, y + z = 1
have refine_2 : (↑(μ a) : R) ∈ I b
Tactic: refine ⟨_, ?_, _, ?_, hμ⟩
-- Lean 4 proof step from Mathlib.RingTheory.Coprime.Ideal (goal)
Goal:
ι : Type u_1
R : Type u_2
inst : CommSemiring R
I : ι → Ideal R
this : DecidableEq ι
a : ι
t : Finset ι
hat : a ∉ t
h : t.Nonempty
ih :
  (∃ (μ : (i : ι) → ↥(⨅ j ∈ t, ⨅ (_ : j ≠ i), I j)), Eq.{u_2 + 1} (α := R) (∑ i ∈ t, (↑(μ i) : R)) 1) ↔
    (↑t : Set ι).Pairwise fun (i j : ι) ↦ I i ⊔ I j = ⊤
μ : (i : ι) → ↥(⨅ j ∈ Finset.cons a t hat, ⨅ (_ : j ≠ i), I j)
hμ : ∑ x ∈ t, (↑(μ x) : R) + (↑(μ a) : R) = 1
b : ι
hb : b ∈ (↑t : Set ι)
ab : a ≠ b
⊢ ∑ x ∈ t, (↑(μ x) : R) ∈ I a
have refine_1 : ∀ (x : ι) (hx : x ∈ t), (↑(μ x) : R) ∈ I a
Tactic: refine sum_mem _ fun x hx => ?_
-- Lean 4 proof step from Mathlib.RingTheory.Coprime.Ideal (hypothesis)
Goal:
ι : Type u_1
R : Type u_2
inst : CommSemiring R
I : ι → Ideal R
this : DecidableEq ι
a : ι
t : Finset ι
hat : a ∉ t
h : t.Nonempty
ih :
  (∃ (μ : (i : ι) → ↥(⨅ j ∈ t, ⨅ (_ : j ≠ i), I j)), Eq.{u_2 + 1} (α := R) (∑ i ∈ t, (↑(μ i) : R)) 1) ↔
    (↑t : Set ι).Pairwise fun (i j : ι) ↦ I i ⊔ I j = ⊤
μ : (i : ι) → ↥(⨅ j ∈ Finset.cons a t hat, ⨅ (_ : j ≠ i), I j)
hμ : ∑ x ∈ t, (↑(μ x) : R) + (↑(μ a) : R) = 1
b : ι
hb : b ∈ (↑t : Set ι)
ab : a ≠ b
x : ι
hx : x ∈ t
⊢ (↑(μ x) : R) ∈ I a
have this₁ : Membership.mem (γ := Submodule R R) (⨅ j ∈ Finset.cons a t hat, ⨅ (_ : j ≠ x), I j) (↑(μ x) : R)
Tactic: have := Submodule.coe_mem (μ x)
-- Lean 4 proof step from Mathlib.RingTheory.Coprime.Ideal (hypothesis)
Goal:
ι : Type u_1
R : Type u_2
inst : CommSemiring R
I : ι → Ideal R
this_1 : DecidableEq ι
a : ι
t : Finset ι
hat : a ∉ t
h : t.Nonempty
ih :
  (∃ (μ : (i : ι) → ↥(⨅ j ∈ t, ⨅ (_ : j ≠ i), I j)), Eq.{u_2 + 1} (α := R) (∑ i ∈ t, (↑(μ i) : R)) 1) ↔
    (↑t : Set ι).Pairwise fun (i j : ι) ↦ I i ⊔ I j = ⊤
μ : (i : ι) → ↥(⨅ j ∈ Finset.cons a t hat, ⨅ (_ : j ≠ i), I j)
hμ : ∑ x ∈ t, (↑(μ x) : R) + (↑(μ a) : R) = 1
b : ι
hb : b ∈ (↑t : Set ι)
ab : a ≠ b
x : ι
hx : x ∈ t
this : Membership.mem (γ := Submodule R R) (⨅ j ∈ Finset.cons a t hat, ⨅ (_ : j ≠ x), I j) (↑(μ x) : R)
⊢ (↑(μ x) : R) ∈ I a
have this₁ : ∀ (i : ι) (i_1 : i ∈ Finset.cons a t hat) (i_2 : i ≠ x), (↑(μ x) : R) ∈ I i
Tactic: simp only [mem_iInf] at this
Cleared hypotheses: this