l3lab/ntp-mathlib huggingface dataset:
/-- We can write the quotient of an ideal over a PID as a product of quotients by principal ideals.
-/
noncomputable def quotientEquivPiSpan (I : Ideal S) (b : Basis ι R S) (hI : I ≠ ⊥) :
    (S ⧸ I) ≃ₗ[R] ∀ i, R ⧸ span ({I.smithCoeffs b hI i} : Set R) := by
  haveI := Fintype.ofFinite ι
  -- Choose `e : S ≃ₗ I` and a basis `b'` for `S` that turns the map
  -- `f := ((Submodule.subtype I).restrictScalars R).comp e` into a diagonal matrix:
  -- there is an `a : ι → ℤ` such that `f (b' i) = a i • b' i`.
  let a := I.smithCoeffs b hI
  let b' := I.ringBasis b hI
  let ab := I.selfBasis b hI
  have ab_eq := I.selfBasis_def b hI
  have mem_I_iff : ∀ x, x ∈ I ↔ ∀ i, a i ∣ b'.repr x i := by
    intro x
    -- Porting note: these lines used to be `simp_rw [ab.mem_ideal_iff', ab_eq]`
    rw [ab.mem_ideal_iff']
    simp_rw [ab_eq]
    have : ∀ (c : ι → R) (i), b'.repr (∑ j : ι, c j • a j • b' j) i = a i * c i := by
      intro c i
      simp only [← MulAction.mul_smul, b'.repr_sum_self, mul_comm]
    constructor
    · rintro ⟨c, rfl⟩ i
      exact ⟨c i, this c i⟩
    · rintro ha
      choose c hc using ha
      exact ⟨c, b'.ext_elem fun i => Eq.trans (hc i) (this c i).symm⟩
  -- Now we map everything through the linear equiv `S ≃ₗ (ι → R)`,
  -- which maps `I` to `I' := Π i, a i ℤ`.
  let I' : Submodule R (ι → R) := Submodule.pi Set.univ fun i => span ({a i} : Set R)
  have : Submodule.map (b'.equivFun : S →ₗ[R] ι → R) (I.restrictScalars R) = I' := by
    ext x
    simp only [Submodule.mem_map, Submodule.mem_pi, mem_span_singleton, Set.mem_univ,
      Submodule.restrictScalars_mem, mem_I_iff, smul_eq_mul, forall_true_left, LinearEquiv.coe_coe,
      Basis.equivFun_apply]
    constructor
    · rintro ⟨y, hy, rfl⟩ i
      exact hy i
    · 

Proof state:
case h.mpr
ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
inst✝³ : IsDomain R
inst✝² : IsPrincipalIdealRing R
inst✝¹ : IsDomain S
inst✝ : Finite ι
I : Ideal S
b : Basis ι R S
hI : I ≠ ⊥
this : Fintype ι
a : ι → R := smithCoeffs b I hI
b' : Basis ι R S := ringBasis b I hI
ab : Basis ι R ↥I := selfBasis b I hI
ab_eq : ∀ (i : ι), ↑((selfBasis b I hI) i) = smithCoeffs b I hI i • (ringBasis b I hI) i
mem_I_iff : ∀ (x : S), x ∈ I ↔ ∀ (i : ι), a i ∣ (b'.repr x) i
I' : Submodule R (ι → R) := Submodule.pi Set.univ fun i => span {a i}
x : ι → R
⊢ (∀ (i : ι), smithCoeffs b I hI i ∣ x i) →
    ∃ y, (∀ (i : ι), smithCoeffs b I hI i ∣ ((ringBasis b I hI).repr y) i) ∧ ⇑((ringBasis b I hI).repr y) = x

Next tactic:
rintro hdvd
/-- We can write the quotient of an ideal over a PID as a product of quotients by principal ideals.
-/
noncomputable def quotientEquivPiSpan (I : Ideal S) (b : Basis ι R S) (hI : I ≠ ⊥) :
    (S ⧸ I) ≃ₗ[R] ∀ i, R ⧸ span ({I.smithCoeffs b hI i} : Set R) := by
  haveI := Fintype.ofFinite ι
  -- Choose `e : S ≃ₗ I` and a basis `b'` for `S` that turns the map
  -- `f := ((Submodule.subtype I).restrictScalars R).comp e` into a diagonal matrix:
  -- there is an `a : ι → ℤ` such that `f (b' i) = a i • b' i`.
  let a := I.smithCoeffs b hI
  let b' := I.ringBasis b hI
  let ab := I.selfBasis b hI
  have ab_eq := I.selfBasis_def b hI
  have mem_I_iff : ∀ x, x ∈ I ↔ ∀ i, a i ∣ b'.repr x i := by
    intro x
    -- Porting note: these lines used to be `simp_rw [ab.mem_ideal_iff', ab_eq]`
    rw [ab.mem_ideal_iff']
    simp_rw [ab_eq]
    have : ∀ (c : ι → R) (i), b'.repr (∑ j : ι, c j • a j • b' j) i = a i * c i := by
      intro c i
      simp only [← MulAction.mul_smul, b'.repr_sum_self, mul_comm]
    constructor
    · rintro ⟨c, rfl⟩ i
      exact ⟨c i, this c i⟩
    · rintro ha
      choose c hc using ha
      exact ⟨c, b'.ext_elem fun i => Eq.trans (hc i) (this c i).symm⟩
  -- Now we map everything through the linear equiv `S ≃ₗ (ι → R)`,
  -- which maps `I` to `I' := Π i, a i ℤ`.
  let I' : Submodule R (ι → R) := Submodule.pi Set.univ fun i => span ({a i} : Set R)
  have : Submodule.map (b'.equivFun : S →ₗ[R] ι → R) (I.restrictScalars R) = I' := by
    ext x
    simp only [Submodule.mem_map, Submodule.mem_pi, mem_span_singleton, Set.mem_univ,
      Submodule.restrictScalars_mem, mem_I_iff, smul_eq_mul, forall_true_left, LinearEquiv.coe_coe,
      Basis.equivFun_apply]
    constructor
    · rintro ⟨y, hy, rfl⟩ i
      exact hy i
    · rintro hdvd
      

Proof state:
case h.mpr
ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
inst✝³ : IsDomain R
inst✝² : IsPrincipalIdealRing R
inst✝¹ : IsDomain S
inst✝ : Finite ι
I : Ideal S
b : Basis ι R S
hI : I ≠ ⊥
this : Fintype ι
a : ι → R := smithCoeffs b I hI
b' : Basis ι R S := ringBasis b I hI
ab : Basis ι R ↥I := selfBasis b I hI
ab_eq : ∀ (i : ι), ↑((selfBasis b I hI) i) = smithCoeffs b I hI i • (ringBasis b I hI) i
mem_I_iff : ∀ (x : S), x ∈ I ↔ ∀ (i : ι), a i ∣ (b'.repr x) i
I' : Submodule R (ι → R) := Submodule.pi Set.univ fun i => span {a i}
x : ι → R
hdvd : ∀ (i : ι), smithCoeffs b I hI i ∣ x i
⊢ ∃ y, (∀ (i : ι), smithCoeffs b I hI i ∣ ((ringBasis b I hI).repr y) i) ∧ ⇑((ringBasis b I hI).repr y) = x

Next tactic:
refine' ⟨∑ i, x i • b' i, fun i => _, _⟩
/-- We can write the quotient of an ideal over a PID as a product of quotients by principal ideals.
-/
noncomputable def quotientEquivPiSpan (I : Ideal S) (b : Basis ι R S) (hI : I ≠ ⊥) :
    (S ⧸ I) ≃ₗ[R] ∀ i, R ⧸ span ({I.smithCoeffs b hI i} : Set R) := by
  haveI := Fintype.ofFinite ι
  -- Choose `e : S ≃ₗ I` and a basis `b'` for `S` that turns the map
  -- `f := ((Submodule.subtype I).restrictScalars R).comp e` into a diagonal matrix:
  -- there is an `a : ι → ℤ` such that `f (b' i) = a i • b' i`.
  let a := I.smithCoeffs b hI
  let b' := I.ringBasis b hI
  let ab := I.selfBasis b hI
  have ab_eq := I.selfBasis_def b hI
  have mem_I_iff : ∀ x, x ∈ I ↔ ∀ i, a i ∣ b'.repr x i := by
    intro x
    -- Porting note: these lines used to be `simp_rw [ab.mem_ideal_iff', ab_eq]`
    rw [ab.mem_ideal_iff']
    simp_rw [ab_eq]
    have : ∀ (c : ι → R) (i), b'.repr (∑ j : ι, c j • a j • b' j) i = a i * c i := by
      intro c i
      simp only [← MulAction.mul_smul, b'.repr_sum_self, mul_comm]
    constructor
    · rintro ⟨c, rfl⟩ i
      exact ⟨c i, this c i⟩
    · rintro ha
      choose c hc using ha
      exact ⟨c, b'.ext_elem fun i => Eq.trans (hc i) (this c i).symm⟩
  -- Now we map everything through the linear equiv `S ≃ₗ (ι → R)`,
  -- which maps `I` to `I' := Π i, a i ℤ`.
  let I' : Submodule R (ι → R) := Submodule.pi Set.univ fun i => span ({a i} : Set R)
  have : Submodule.map (b'.equivFun : S →ₗ[R] ι → R) (I.restrictScalars R) = I' := by
    ext x
    simp only [Submodule.mem_map, Submodule.mem_pi, mem_span_singleton, Set.mem_univ,
      Submodule.restrictScalars_mem, mem_I_iff, smul_eq_mul, forall_true_left, LinearEquiv.coe_coe,
      Basis.equivFun_apply]
    constructor
    · rintro ⟨y, hy, rfl⟩ i
      exact hy i
    · rintro hdvd
      refine' ⟨∑ i, x i • b' i, fun i => _, _⟩ <;> 

Proof state:
case h.mpr.refine'_1
ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
inst✝³ : IsDomain R
inst✝² : IsPrincipalIdealRing R
inst✝¹ : IsDomain S
inst✝ : Finite ι
I : Ideal S
b : Basis ι R S
hI : I ≠ ⊥
this : Fintype ι
a : ι → R := smithCoeffs b I hI
b' : Basis ι R S := ringBasis b I hI
ab : Basis ι R ↥I := selfBasis b I hI
ab_eq : ∀ (i : ι), ↑((selfBasis b I hI) i) = smithCoeffs b I hI i • (ringBasis b I hI) i
mem_I_iff : ∀ (x : S), x ∈ I ↔ ∀ (i : ι), a i ∣ (b'.repr x) i
I' : Submodule R (ι → R) := Submodule.pi Set.univ fun i => span {a i}
x : ι → R
hdvd : ∀ (i : ι), smithCoeffs b I hI i ∣ x i
i : ι
⊢ smithCoeffs b I hI i ∣ ((ringBasis b I hI).repr (∑ i : ι, x i • b' i)) i

Next tactic:
rw [b'.repr_sum_self]
/-- We can write the quotient of an ideal over a PID as a product of quotients by principal ideals.
-/
noncomputable def quotientEquivPiSpan (I : Ideal S) (b : Basis ι R S) (hI : I ≠ ⊥) :
    (S ⧸ I) ≃ₗ[R] ∀ i, R ⧸ span ({I.smithCoeffs b hI i} : Set R) := by
  haveI := Fintype.ofFinite ι
  -- Choose `e : S ≃ₗ I` and a basis `b'` for `S` that turns the map
  -- `f := ((Submodule.subtype I).restrictScalars R).comp e` into a diagonal matrix:
  -- there is an `a : ι → ℤ` such that `f (b' i) = a i • b' i`.
  let a := I.smithCoeffs b hI
  let b' := I.ringBasis b hI
  let ab := I.selfBasis b hI
  have ab_eq := I.selfBasis_def b hI
  have mem_I_iff : ∀ x, x ∈ I ↔ ∀ i, a i ∣ b'.repr x i := by
    intro x
    -- Porting note: these lines used to be `simp_rw [ab.mem_ideal_iff', ab_eq]`
    rw [ab.mem_ideal_iff']
    simp_rw [ab_eq]
    have : ∀ (c : ι → R) (i), b'.repr (∑ j : ι, c j • a j • b' j) i = a i * c i := by
      intro c i
      simp only [← MulAction.mul_smul, b'.repr_sum_self, mul_comm]
    constructor
    · rintro ⟨c, rfl⟩ i
      exact ⟨c i, this c i⟩
    · rintro ha
      choose c hc using ha
      exact ⟨c, b'.ext_elem fun i => Eq.trans (hc i) (this c i).symm⟩
  -- Now we map everything through the linear equiv `S ≃ₗ (ι → R)`,
  -- which maps `I` to `I' := Π i, a i ℤ`.
  let I' : Submodule R (ι → R) := Submodule.pi Set.univ fun i => span ({a i} : Set R)
  have : Submodule.map (b'.equivFun : S →ₗ[R] ι → R) (I.restrictScalars R) = I' := by
    ext x
    simp only [Submodule.mem_map, Submodule.mem_pi, mem_span_singleton, Set.mem_univ,
      Submodule.restrictScalars_mem, mem_I_iff, smul_eq_mul, forall_true_left, LinearEquiv.coe_coe,
      Basis.equivFun_apply]
    constructor
    · rintro ⟨y, hy, rfl⟩ i
      exact hy i
    · rintro hdvd
      refine' ⟨∑ i, x i • b' i, fun i => _, _⟩ <;> 

Proof state:
case h.mpr.refine'_2
ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
inst✝³ : IsDomain R
inst✝² : IsPrincipalIdealRing R
inst✝¹ : IsDomain S
inst✝ : Finite ι
I : Ideal S
b : Basis ι R S
hI : I ≠ ⊥
this : Fintype ι
a : ι → R := smithCoeffs b I hI
b' : Basis ι R S := ringBasis b I hI
ab : Basis ι R ↥I := selfBasis b I hI
ab_eq : ∀ (i : ι), ↑((selfBasis b I hI) i) = smithCoeffs b I hI i • (ringBasis b I hI) i
mem_I_iff : ∀ (x : S), x ∈ I ↔ ∀ (i : ι), a i ∣ (b'.repr x) i
I' : Submodule R (ι → R) := Submodule.pi Set.univ fun i => span {a i}
x : ι → R
hdvd : ∀ (i : ι), smithCoeffs b I hI i ∣ x i
⊢ ⇑((ringBasis b I hI).repr (∑ i : ι, x i • b' i)) = x

Next tactic:
rw [b'.repr_sum_self]
/-- We can write the quotient of an ideal over a PID as a product of quotients by principal ideals.
-/
noncomputable def quotientEquivPiSpan (I : Ideal S) (b : Basis ι R S) (hI : I ≠ ⊥) :
    (S ⧸ I) ≃ₗ[R] ∀ i, R ⧸ span ({I.smithCoeffs b hI i} : Set R) := by
  haveI := Fintype.ofFinite ι
  -- Choose `e : S ≃ₗ I` and a basis `b'` for `S` that turns the map
  -- `f := ((Submodule.subtype I).restrictScalars R).comp e` into a diagonal matrix:
  -- there is an `a : ι → ℤ` such that `f (b' i) = a i • b' i`.
  let a := I.smithCoeffs b hI
  let b' := I.ringBasis b hI
  let ab := I.selfBasis b hI
  have ab_eq := I.selfBasis_def b hI
  have mem_I_iff : ∀ x, x ∈ I ↔ ∀ i, a i ∣ b'.repr x i := by
    intro x
    -- Porting note: these lines used to be `simp_rw [ab.mem_ideal_iff', ab_eq]`
    rw [ab.mem_ideal_iff']
    simp_rw [ab_eq]
    have : ∀ (c : ι → R) (i), b'.repr (∑ j : ι, c j • a j • b' j) i = a i * c i := by
      intro c i
      simp only [← MulAction.mul_smul, b'.repr_sum_self, mul_comm]
    constructor
    · rintro ⟨c, rfl⟩ i
      exact ⟨c i, this c i⟩
    · rintro ha
      choose c hc using ha
      exact ⟨c, b'.ext_elem fun i => Eq.trans (hc i) (this c i).symm⟩
  -- Now we map everything through the linear equiv `S ≃ₗ (ι → R)`,
  -- which maps `I` to `I' := Π i, a i ℤ`.
  let I' : Submodule R (ι → R) := Submodule.pi Set.univ fun i => span ({a i} : Set R)
  have : Submodule.map (b'.equivFun : S →ₗ[R] ι → R) (I.restrictScalars R) = I' := by
    ext x
    simp only [Submodule.mem_map, Submodule.mem_pi, mem_span_singleton, Set.mem_univ,
      Submodule.restrictScalars_mem, mem_I_iff, smul_eq_mul, forall_true_left, LinearEquiv.coe_coe,
      Basis.equivFun_apply]
    constructor
    · rintro ⟨y, hy, rfl⟩ i
      exact hy i
    · rintro hdvd
      refine' ⟨∑ i, x i • b' i, fun i => _, _⟩ <;> rw [b'.repr_sum_self]
      · 

Proof state:
case h.mpr.refine'_1
ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
inst✝³ : IsDomain R
inst✝² : IsPrincipalIdealRing R
inst✝¹ : IsDomain S
inst✝ : Finite ι
I : Ideal S
b : Basis ι R S
hI : I ≠ ⊥
this : Fintype ι
a : ι → R := smithCoeffs b I hI
b' : Basis ι R S := ringBasis b I hI
ab : Basis ι R ↥I := selfBasis b I hI
ab_eq : ∀ (i : ι), ↑((selfBasis b I hI) i) = smithCoeffs b I hI i • (ringBasis b I hI) i
mem_I_iff : ∀ (x : S), x ∈ I ↔ ∀ (i : ι), a i ∣ (b'.repr x) i
I' : Submodule R (ι → R) := Submodule.pi Set.univ fun i => span {a i}
x : ι → R
hdvd : ∀ (i : ι), smithCoeffs b I hI i ∣ x i
i : ι
⊢ smithCoeffs b I hI i ∣ x i

Next tactic:
exact hdvd i
/-- We can write the quotient of an ideal over a PID as a product of quotients by principal ideals.
-/
noncomputable def quotientEquivPiSpan (I : Ideal S) (b : Basis ι R S) (hI : I ≠ ⊥) :
    (S ⧸ I) ≃ₗ[R] ∀ i, R ⧸ span ({I.smithCoeffs b hI i} : Set R) := by
  haveI := Fintype.ofFinite ι
  -- Choose `e : S ≃ₗ I` and a basis `b'` for `S` that turns the map
  -- `f := ((Submodule.subtype I).restrictScalars R).comp e` into a diagonal matrix:
  -- there is an `a : ι → ℤ` such that `f (b' i) = a i • b' i`.
  let a := I.smithCoeffs b hI
  let b' := I.ringBasis b hI
  let ab := I.selfBasis b hI
  have ab_eq := I.selfBasis_def b hI
  have mem_I_iff : ∀ x, x ∈ I ↔ ∀ i, a i ∣ b'.repr x i := by
    intro x
    -- Porting note: these lines used to be `simp_rw [ab.mem_ideal_iff', ab_eq]`
    rw [ab.mem_ideal_iff']
    simp_rw [ab_eq]
    have : ∀ (c : ι → R) (i), b'.repr (∑ j : ι, c j • a j • b' j) i = a i * c i := by
      intro c i
      simp only [← MulAction.mul_smul, b'.repr_sum_self, mul_comm]
    constructor
    · rintro ⟨c, rfl⟩ i
      exact ⟨c i, this c i⟩
    · rintro ha
      choose c hc using ha
      exact ⟨c, b'.ext_elem fun i => Eq.trans (hc i) (this c i).symm⟩
  -- Now we map everything through the linear equiv `S ≃ₗ (ι → R)`,
  -- which maps `I` to `I' := Π i, a i ℤ`.
  let I' : Submodule R (ι → R) := Submodule.pi Set.univ fun i => span ({a i} : Set R)
  have : Submodule.map (b'.equivFun : S →ₗ[R] ι → R) (I.restrictScalars R) = I' := by
    ext x
    simp only [Submodule.mem_map, Submodule.mem_pi, mem_span_singleton, Set.mem_univ,
      Submodule.restrictScalars_mem, mem_I_iff, smul_eq_mul, forall_true_left, LinearEquiv.coe_coe,
      Basis.equivFun_apply]
    constructor
    · rintro ⟨y, hy, rfl⟩ i
      exact hy i
    · rintro hdvd
      refine' ⟨∑ i, x i • b' i, fun i => _, _⟩ <;> rw [b'.repr_sum_self]
      · exact hdvd i
  

Proof state:
ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
inst✝³ : IsDomain R
inst✝² : IsPrincipalIdealRing R
inst✝¹ : IsDomain S
inst✝ : Finite ι
I : Ideal S
b : Basis ι R S
hI : I ≠ ⊥
this✝ : Fintype ι
a : ι → R := smithCoeffs b I hI
b' : Basis ι R S := ringBasis b I hI
ab : Basis ι R ↥I := selfBasis b I hI
ab_eq : ∀ (i : ι), ↑((selfBasis b I hI) i) = smithCoeffs b I hI i • (ringBasis b I hI) i
mem_I_iff : ∀ (x : S), x ∈ I ↔ ∀ (i : ι), a i ∣ (b'.repr x) i
I' : Submodule R (ι → R) := Submodule.pi Set.univ fun i => span {a i}
this : Submodule.map (↑(Basis.equivFun b')) (Submodule.restrictScalars R I) = I'
⊢ (S ⧸ I) ≃ₗ[R] (i : ι) → R ⧸ span {smithCoeffs b I hI i}

Next tactic:
refine' ((Submodule.Quotient.restrictScalarsEquiv R I).restrictScalars R).symm.trans
    (σ₁₂ := RingHom.id R) (σ₃₂ := RingHom.id R) _
/-- We can write the quotient of an ideal over a PID as a product of quotients by principal ideals.
-/
noncomputable def quotientEquivPiSpan (I : Ideal S) (b : Basis ι R S) (hI : I ≠ ⊥) :
    (S ⧸ I) ≃ₗ[R] ∀ i, R ⧸ span ({I.smithCoeffs b hI i} : Set R) := by
  haveI := Fintype.ofFinite ι
  -- Choose `e : S ≃ₗ I` and a basis `b'` for `S` that turns the map
  -- `f := ((Submodule.subtype I).restrictScalars R).comp e` into a diagonal matrix:
  -- there is an `a : ι → ℤ` such that `f (b' i) = a i • b' i`.
  let a := I.smithCoeffs b hI
  let b' := I.ringBasis b hI
  let ab := I.selfBasis b hI
  have ab_eq := I.selfBasis_def b hI
  have mem_I_iff : ∀ x, x ∈ I ↔ ∀ i, a i ∣ b'.repr x i := by
    intro x
    -- Porting note: these lines used to be `simp_rw [ab.mem_ideal_iff', ab_eq]`
    rw [ab.mem_ideal_iff']
    simp_rw [ab_eq]
    have : ∀ (c : ι → R) (i), b'.repr (∑ j : ι, c j • a j • b' j) i = a i * c i := by
      intro c i
      simp only [← MulAction.mul_smul, b'.repr_sum_self, mul_comm]
    constructor
    · rintro ⟨c, rfl⟩ i
      exact ⟨c i, this c i⟩
    · rintro ha
      choose c hc using ha
      exact ⟨c, b'.ext_elem fun i => Eq.trans (hc i) (this c i).symm⟩
  -- Now we map everything through the linear equiv `S ≃ₗ (ι → R)`,
  -- which maps `I` to `I' := Π i, a i ℤ`.
  let I' : Submodule R (ι → R) := Submodule.pi Set.univ fun i => span ({a i} : Set R)
  have : Submodule.map (b'.equivFun : S →ₗ[R] ι → R) (I.restrictScalars R) = I' := by
    ext x
    simp only [Submodule.mem_map, Submodule.mem_pi, mem_span_singleton, Set.mem_univ,
      Submodule.restrictScalars_mem, mem_I_iff, smul_eq_mul, forall_true_left, LinearEquiv.coe_coe,
      Basis.equivFun_apply]
    constructor
    · rintro ⟨y, hy, rfl⟩ i
      exact hy i
    · rintro hdvd
      refine' ⟨∑ i, x i • b' i, fun i => _, _⟩ <;> rw [b'.repr_sum_self]
      · exact hdvd i
  refine' ((Submodule.Quotient.restrictScalarsEquiv R I).restrictScalars R).symm.trans
    (σ₁₂ := RingHom.id R) (σ₃₂ := RingHom.id R) _
  · 

Proof state:
case refine'_1
ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
inst✝³ : IsDomain R
inst✝² : IsPrincipalIdealRing R
inst✝¹ : IsDomain S
inst✝ : Finite ι
I : Ideal S
b : Basis ι R S
hI : I ≠ ⊥
this✝ : Fintype ι
a : ι → R := smithCoeffs b I hI
b' : Basis ι R S := ringBasis b I hI
ab : Basis ι R ↥I := selfBasis b I hI
ab_eq : ∀ (i : ι), ↑((selfBasis b I hI) i) = smithCoeffs b I hI i • (ringBasis b I hI) i
mem_I_iff : ∀ (x : S), x ∈ I ↔ ∀ (i : ι), a i ∣ (b'.repr x) i
I' : Submodule R (ι → R) := Submodule.pi Set.univ fun i => span {a i}
this : Submodule.map (↑(Basis.equivFun b')) (Submodule.restrictScalars R I) = I'
⊢ RingHomInvPair (RingHom.id R) (RingHom.id R)

Next tactic:
infer_instance
/-- We can write the quotient of an ideal over a PID as a product of quotients by principal ideals.
-/
noncomputable def quotientEquivPiSpan (I : Ideal S) (b : Basis ι R S) (hI : I ≠ ⊥) :
    (S ⧸ I) ≃ₗ[R] ∀ i, R ⧸ span ({I.smithCoeffs b hI i} : Set R) := by
  haveI := Fintype.ofFinite ι
  -- Choose `e : S ≃ₗ I` and a basis `b'` for `S` that turns the map
  -- `f := ((Submodule.subtype I).restrictScalars R).comp e` into a diagonal matrix:
  -- there is an `a : ι → ℤ` such that `f (b' i) = a i • b' i`.
  let a := I.smithCoeffs b hI
  let b' := I.ringBasis b hI
  let ab := I.selfBasis b hI
  have ab_eq := I.selfBasis_def b hI
  have mem_I_iff : ∀ x, x ∈ I ↔ ∀ i, a i ∣ b'.repr x i := by
    intro x
    -- Porting note: these lines used to be `simp_rw [ab.mem_ideal_iff', ab_eq]`
    rw [ab.mem_ideal_iff']
    simp_rw [ab_eq]
    have : ∀ (c : ι → R) (i), b'.repr (∑ j : ι, c j • a j • b' j) i = a i * c i := by
      intro c i
      simp only [← MulAction.mul_smul, b'.repr_sum_self, mul_comm]
    constructor
    · rintro ⟨c, rfl⟩ i
      exact ⟨c i, this c i⟩
    · rintro ha
      choose c hc using ha
      exact ⟨c, b'.ext_elem fun i => Eq.trans (hc i) (this c i).symm⟩
  -- Now we map everything through the linear equiv `S ≃ₗ (ι → R)`,
  -- which maps `I` to `I' := Π i, a i ℤ`.
  let I' : Submodule R (ι → R) := Submodule.pi Set.univ fun i => span ({a i} : Set R)
  have : Submodule.map (b'.equivFun : S →ₗ[R] ι → R) (I.restrictScalars R) = I' := by
    ext x
    simp only [Submodule.mem_map, Submodule.mem_pi, mem_span_singleton, Set.mem_univ,
      Submodule.restrictScalars_mem, mem_I_iff, smul_eq_mul, forall_true_left, LinearEquiv.coe_coe,
      Basis.equivFun_apply]
    constructor
    · rintro ⟨y, hy, rfl⟩ i
      exact hy i
    · rintro hdvd
      refine' ⟨∑ i, x i • b' i, fun i => _, _⟩ <;> rw [b'.repr_sum_self]
      · exact hdvd i
  refine' ((Submodule.Quotient.restrictScalarsEquiv R I).restrictScalars R).symm.trans
    (σ₁₂ := RingHom.id R) (σ₃₂ := RingHom.id R) _
  · infer_instance
  · 

Proof state:
case refine'_2
ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
inst✝³ : IsDomain R
inst✝² : IsPrincipalIdealRing R
inst✝¹ : IsDomain S
inst✝ : Finite ι
I : Ideal S
b : Basis ι R S
hI : I ≠ ⊥
this✝ : Fintype ι
a : ι → R := smithCoeffs b I hI
b' : Basis ι R S := ringBasis b I hI
ab : Basis ι R ↥I := selfBasis b I hI
ab_eq : ∀ (i : ι), ↑((selfBasis b I hI) i) = smithCoeffs b I hI i • (ringBasis b I hI) i
mem_I_iff : ∀ (x : S), x ∈ I ↔ ∀ (i : ι), a i ∣ (b'.repr x) i
I' : Submodule R (ι → R) := Submodule.pi Set.univ fun i => span {a i}
this : Submodule.map (↑(Basis.equivFun b')) (Submodule.restrictScalars R I) = I'
⊢ RingHomInvPair (RingHom.id R) (RingHom.id R)

Next tactic:
infer_instance
/-- We can write the quotient of an ideal over a PID as a product of quotients by principal ideals.
-/
noncomputable def quotientEquivPiSpan (I : Ideal S) (b : Basis ι R S) (hI : I ≠ ⊥) :
    (S ⧸ I) ≃ₗ[R] ∀ i, R ⧸ span ({I.smithCoeffs b hI i} : Set R) := by
  haveI := Fintype.ofFinite ι
  -- Choose `e : S ≃ₗ I` and a basis `b'` for `S` that turns the map
  -- `f := ((Submodule.subtype I).restrictScalars R).comp e` into a diagonal matrix:
  -- there is an `a : ι → ℤ` such that `f (b' i) = a i • b' i`.
  let a := I.smithCoeffs b hI
  let b' := I.ringBasis b hI
  let ab := I.selfBasis b hI
  have ab_eq := I.selfBasis_def b hI
  have mem_I_iff : ∀ x, x ∈ I ↔ ∀ i, a i ∣ b'.repr x i := by
    intro x
    -- Porting note: these lines used to be `simp_rw [ab.mem_ideal_iff', ab_eq]`
    rw [ab.mem_ideal_iff']
    simp_rw [ab_eq]
    have : ∀ (c : ι → R) (i), b'.repr (∑ j : ι, c j • a j • b' j) i = a i * c i := by
      intro c i
      simp only [← MulAction.mul_smul, b'.repr_sum_self, mul_comm]
    constructor
    · rintro ⟨c, rfl⟩ i
      exact ⟨c i, this c i⟩
    · rintro ha
      choose c hc using ha
      exact ⟨c, b'.ext_elem fun i => Eq.trans (hc i) (this c i).symm⟩
  -- Now we map everything through the linear equiv `S ≃ₗ (ι → R)`,
  -- which maps `I` to `I' := Π i, a i ℤ`.
  let I' : Submodule R (ι → R) := Submodule.pi Set.univ fun i => span ({a i} : Set R)
  have : Submodule.map (b'.equivFun : S →ₗ[R] ι → R) (I.restrictScalars R) = I' := by
    ext x
    simp only [Submodule.mem_map, Submodule.mem_pi, mem_span_singleton, Set.mem_univ,
      Submodule.restrictScalars_mem, mem_I_iff, smul_eq_mul, forall_true_left, LinearEquiv.coe_coe,
      Basis.equivFun_apply]
    constructor
    · rintro ⟨y, hy, rfl⟩ i
      exact hy i
    · rintro hdvd
      refine' ⟨∑ i, x i • b' i, fun i => _, _⟩ <;> rw [b'.repr_sum_self]
      · exact hdvd i
  refine' ((Submodule.Quotient.restrictScalarsEquiv R I).restrictScalars R).symm.trans
    (σ₁₂ := RingHom.id R) (σ₃₂ := RingHom.id R) _
  · infer_instance
  · infer_instance
  

Proof state:
case refine'_3
ι : Type u_1
R : Type u_2
S : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
inst✝³ : IsDomain R
inst✝² : IsPrincipalIdealRing R
inst✝¹ : IsDomain S
inst✝ : Finite ι
I : Ideal S
b : Basis ι R S
hI : I ≠ ⊥
this✝ : Fintype ι
a : ι → R := smithCoeffs b I hI
b' : Basis ι R S := ringBasis b I hI
ab : Basis ι R ↥I := selfBasis b I hI
ab_eq : ∀ (i : ι), ↑((selfBasis b I hI) i) = smithCoeffs b I hI i • (ringBasis b I hI) i
mem_I_iff : ∀ (x : S), x ∈ I ↔ ∀ (i : ι), a i ∣ (b'.repr x) i
I' : Submodule R (ι → R) := Submodule.pi Set.univ fun i => span {a i}
this : Submodule.map (↑(Basis.equivFun b')) (Submodule.restrictScalars R I) = I'
⊢ (S ⧸ Submodule.restrictScalars R I) ≃ₗ[R] (i : ι) → R ⧸ span {smithCoeffs b I hI i}

Next tactic:
refine' (Submodule.Quotient.equiv (I.restrictScalars R) I' b'.equivFun this).trans
    (σ₁₂ := RingHom.id R) (σ₃₂ := RingHom.id R) _