awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.LinearAlgebra.FreeModule.PID (goal)
Goal:
R : Type u_1
inst : CommSemiring R
I : Ideal R
inst_1 : Submodule.IsPrincipal I
x : R
hx : x ∈ I
⊢ Membership.mem (γ := Ideal R) (Ideal.span {x}) (generator I) ↔ I ≤ Ideal.span {x}
have conv_rhs : Membership.mem (γ := Ideal R) (Ideal.span {x}) (generator I) ↔ I ≤ Ideal.span {x}
Tactic: conv_rhs => rw [← span_singleton_generator I, Submodule.span_singleton_le_iff_mem]
-- Lean 4 proof step from Mathlib.LinearAlgebra.FreeModule.PID (goal)
Goal:
R : Type u_1
inst : CommSemiring R
I : Ideal R
inst_1 : Submodule.IsPrincipal I
x : R
hx : x ∈ I
conv_rhs : Membership.mem (γ := Ideal R) (Ideal.span {x}) (generator I) ↔ I ≤ Ideal.span {x}
⊢ Membership.mem (γ := Ideal R) (Ideal.span {x}) (generator I) ↔ I ≤ Ideal.span {x}
have conv_rhs₁ : Membership.mem (γ := Ideal R) (Ideal.span {x}) (generator I) ↔
  Membership.mem (γ := Submodule R R) (Ideal.span {x}) (generator I)
Tactic: conv_rhs => rw [← span_singleton_generator I, Submodule.span_singleton_le_iff_mem]
-- Lean 4 proof step from Mathlib.LinearAlgebra.FreeModule.PID (hypothesis)
Goal:
R : Type u_2
inst : CommRing R
M : Type u_3
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : IsPrincipalIdealRing R
N O : Submodule R M
hNO : N ≤ O
ϕ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R
hϕ : ∀ (ψ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R), ¬ϕ.submoduleImage N < ψ.submoduleImage N
inst_4 : (ϕ.submoduleImage N).IsPrincipal
y : M
yN : y ∈ N
ϕy_eq : (ϕ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) = generator (ϕ.submoduleImage N)
ψ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R
a : R := generator (ϕ.submoduleImage N)
d : R :=
  generator
    (span R (insert a (singleton (α := R) ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) : R))))
⊢ generator (ϕ.submoduleImage N) ∣ (ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯)
have d_dvd_left : d ∣ a
Tactic: have d_dvd_left : d ∣ a := (mem_iff_generator_dvd _).mp (subset_span (mem_insert _ _))
-- Lean 4 proof step from Mathlib.LinearAlgebra.FreeModule.PID (hypothesis)
Goal:
R : Type u_2
inst : CommRing R
M : Type u_3
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : IsPrincipalIdealRing R
N O : Submodule R M
hNO : N ≤ O
ϕ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R
hϕ : ∀ (ψ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R), ¬ϕ.submoduleImage N < ψ.submoduleImage N
inst_4 : (ϕ.submoduleImage N).IsPrincipal
y : M
yN : y ∈ N
ϕy_eq : (ϕ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) = generator (ϕ.submoduleImage N)
ψ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R
a : R := generator (ϕ.submoduleImage N)
d : R :=
  generator
    (span R (insert a (singleton (α := R) ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) : R))))
d_dvd_left : d ∣ a
⊢ generator (ϕ.submoduleImage N) ∣ (ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯)
have d_dvd_right : d ∣ (ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y sorry)
Tactic: have d_dvd_right : d ∣ ψ ⟨y, hNO yN⟩ :=
    (mem_iff_generator_dvd _).mp (subset_span (mem_insert_of_mem _ (mem_singleton _)))
-- Lean 4 proof step from Mathlib.LinearAlgebra.FreeModule.PID (goal)
Goal:
R : Type u_2
inst : CommRing R
M : Type u_3
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : IsPrincipalIdealRing R
N O : Submodule R M
hNO : N ≤ O
ϕ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R
hϕ : ∀ (ψ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R), ¬ϕ.submoduleImage N < ψ.submoduleImage N
inst_4 : (ϕ.submoduleImage N).IsPrincipal
y : M
yN : y ∈ N
ϕy_eq : (ϕ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) = generator (ϕ.submoduleImage N)
ψ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R
a : R := generator (ϕ.submoduleImage N)
d : R :=
  generator
    (span R (insert a (singleton (α := R) ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) : R))))
d_dvd_left : d ∣ a
d_dvd_right : d ∣ (ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯)
⊢ generator (ϕ.submoduleImage N) ∣ (ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯)
have refine : generator (ϕ.submoduleImage N) ∣ d
Tactic: refine dvd_trans ?_ d_dvd_right
-- Lean 4 proof step from Mathlib.LinearAlgebra.FreeModule.PID (goal)
Goal:
R : Type u_2
inst : CommRing R
M : Type u_3
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : IsPrincipalIdealRing R
N O : Submodule R M
hNO : N ≤ O
ϕ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R
hϕ : ∀ (ψ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R), ¬ϕ.submoduleImage N < ψ.submoduleImage N
inst_4 : (ϕ.submoduleImage N).IsPrincipal
y : M
yN : y ∈ N
ϕy_eq : (ϕ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) = generator (ϕ.submoduleImage N)
ψ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R
a : R := generator (ϕ.submoduleImage N)
d : R :=
  generator
    (span R (insert a (singleton (α := R) ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) : R))))
d_dvd_left : d ∣ a
d_dvd_right : d ∣ (ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯)
⊢ generator (ϕ.submoduleImage N) ∣ d
have rw : Eq (α := Ideal R)
  (R ∙
    generator
      (span R (insert a (singleton (α := R) ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y sorry) : R)))))
  (R ∙ generator (ϕ.submoduleImage N))
Tactic: rw [dvd_generator_iff, Ideal.span, ←
    span_singleton_generator (Submodule.span R {a, ψ ⟨y, hNO yN⟩})]
-- Lean 4 proof step from Mathlib.LinearAlgebra.FreeModule.PID (goal)
Goal:
R : Type u_2
inst : CommRing R
M : Type u_3
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : IsPrincipalIdealRing R
N O : Submodule R M
hNO : N ≤ O
ϕ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R
hϕ : ∀ (ψ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R), ¬ϕ.submoduleImage N < ψ.submoduleImage N
inst_4 : (ϕ.submoduleImage N).IsPrincipal
y : M
yN : y ∈ N
ϕy_eq : (ϕ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) = generator (ϕ.submoduleImage N)
ψ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R
a : R := generator (ϕ.submoduleImage N)
d : R :=
  generator
    (span R (insert a (singleton (α := R) ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) : R))))
d_dvd_left : d ∣ a
d_dvd_right : d ∣ (ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯)
rw :
  Eq (α := Ideal R)
    (R ∙
      generator
        (span R (insert a (singleton (α := R) ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) : R)))))
    (R ∙ generator (ϕ.submoduleImage N))
⊢ generator (ϕ.submoduleImage N) ∣ d
have rw₁ : Membership.mem (γ := Ideal R)
  (span R (insert a (singleton (α := R) ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y sorry) : R))))
  (generator (ϕ.submoduleImage N))
Tactic: rw [dvd_generator_iff, Ideal.span, ←
    span_singleton_generator (Submodule.span R {a, ψ ⟨y, hNO yN⟩})]
-- Lean 4 proof step from Mathlib.LinearAlgebra.FreeModule.PID (goal)
Goal:
R : Type u_2
inst : CommRing R
M : Type u_3
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : IsPrincipalIdealRing R
N O : Submodule R M
hNO : N ≤ O
ϕ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R
hϕ : ∀ (ψ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R), ¬ϕ.submoduleImage N < ψ.submoduleImage N
inst_4 : (ϕ.submoduleImage N).IsPrincipal
y : M
yN : y ∈ N
ϕy_eq : (ϕ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) = generator (ϕ.submoduleImage N)
ψ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R
a : R := generator (ϕ.submoduleImage N)
d : R :=
  generator
    (span R (insert a (singleton (α := R) ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) : R))))
d_dvd_left : d ∣ a
d_dvd_right : d ∣ (ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯)
⊢ ∃ (r₁ : R) (r₂ : R),
    d = r₁ * a + HMul.hMul (β := R) r₂ ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) : R)
have obtain : ∀ (r₁ r₂' : R)
  (hr₂' :
    Membership.mem (γ := Submodule R R)
      (span R (singleton (α := R) ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y sorry) : R))) r₂')
  (hr₁ :
    generator
        (span R (insert a (singleton (α := R) ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y sorry) : R)))) =
      r₁ • a + r₂'),
  ∃ (r₁ : R) (r₂ : R),
    d = r₁ * a + HMul.hMul (β := R) r₂ ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y sorry) : R)
Tactic: obtain ⟨r₁, r₂', hr₂', hr₁⟩ :=
        mem_span_insert.mp (IsPrincipal.generator_mem (Submodule.span R {a, ψ ⟨y, hNO yN⟩}))
-- Lean 4 proof step from Mathlib.LinearAlgebra.FreeModule.PID (goal)
Goal:
R : Type u_2
inst : CommRing R
M : Type u_3
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : IsPrincipalIdealRing R
N O : Submodule R M
hNO : N ≤ O
ϕ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R
hϕ : ∀ (ψ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R), ¬ϕ.submoduleImage N < ψ.submoduleImage N
inst_4 : (ϕ.submoduleImage N).IsPrincipal
y : M
yN : y ∈ N
ϕy_eq : (ϕ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) = generator (ϕ.submoduleImage N)
ψ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R
a : R := generator (ϕ.submoduleImage N)
d : R :=
  generator
    (span R (insert a (singleton (α := R) ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) : R))))
d_dvd_left : d ∣ a
d_dvd_right : d ∣ (ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯)
r₁ r₂' : R
hr₂' :
  Membership.mem (γ := Submodule R R)
    (span R (singleton (α := R) ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) : R))) r₂'
hr₁ :
  generator
      (span R (insert a (singleton (α := R) ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) : R)))) =
    r₁ • a + r₂'
⊢ ∃ (r₁ : R) (r₂ : R),
    d = r₁ * a + HMul.hMul (β := R) r₂ ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) : R)
have obtain : ∀ (r₂ : R)
  (hr₂' :
    Membership.mem (γ := Submodule R R)
      (span R (singleton (α := R) ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y sorry) : R)))
      (HSMul.hSMul (β := R) r₂ ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y sorry) : R)))
  (hr₁ :
    generator
        (span R (insert a (singleton (α := R) ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y sorry) : R)))) =
      r₁ • a + HSMul.hSMul (β := R) r₂ ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y sorry) : R)),
  ∃ (r₁ : R) (r₂ : R),
    d = r₁ * a + HMul.hMul (β := R) r₂ ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y sorry) : R)
Tactic: obtain ⟨r₂, rfl⟩ := mem_span_singleton.mp hr₂'
Cleared hypotheses: r₂', hr₂', hr₁
-- Lean 4 proof step from Mathlib.LinearAlgebra.FreeModule.PID (goal)
Goal:
R : Type u_2
inst : CommRing R
M : Type u_3
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : IsPrincipalIdealRing R
N O : Submodule R M
hNO : N ≤ O
ϕ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R
hϕ : ∀ (ψ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R), ¬ϕ.submoduleImage N < ψ.submoduleImage N
inst_4 : (ϕ.submoduleImage N).IsPrincipal
y : M
yN : y ∈ N
ϕy_eq : (ϕ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) = generator (ϕ.submoduleImage N)
ψ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R
a : R := generator (ϕ.submoduleImage N)
d : R :=
  generator
    (span R (insert a (singleton (α := R) ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) : R))))
d_dvd_left : d ∣ a
d_dvd_right : d ∣ (ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯)
⊢ Eq (α := Ideal R)
    (R ∙
      generator
        (span R (insert a (singleton (α := R) ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) : R)))))
    (R ∙ generator (ϕ.submoduleImage N))
have obtain : ∀ (r₁ r₂ : R)
  (d_eq : d = r₁ * a + HMul.hMul (β := R) r₂ ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y sorry) : R)),
  Eq (α := Ideal R)
    (R ∙
      generator
        (span R (insert a (singleton (α := R) ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y sorry) : R)))))
    (R ∙ generator (ϕ.submoduleImage N))
Tactic: obtain ⟨r₁, r₂, d_eq⟩ : ∃ r₁ r₂ : R, d = r₁ * a + r₂ * ψ ⟨y, hNO yN⟩ := by
      obtain ⟨r₁, r₂', hr₂', hr₁⟩ :=
        mem_span_insert.mp (IsPrincipal.generator_mem (Submodule.span R {a, ψ ⟨y, hNO yN⟩}))
      obtain ⟨r₂, rfl⟩ := mem_span_singleton.mp hr₂'
      exact ⟨r₁, r₂, hr₁⟩
-- Lean 4 proof step from Mathlib.LinearAlgebra.FreeModule.PID (goal)
Goal:
R : Type u_2
inst : CommRing R
M : Type u_3
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : IsPrincipalIdealRing R
N O : Submodule R M
hNO : N ≤ O
ϕ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R
hϕ : ∀ (ψ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R), ¬ϕ.submoduleImage N < ψ.submoduleImage N
inst_4 : (ϕ.submoduleImage N).IsPrincipal
y : M
yN : y ∈ N
ϕy_eq : (ϕ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) = generator (ϕ.submoduleImage N)
ψ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R
a : R := generator (ϕ.submoduleImage N)
d : R :=
  generator
    (span R (insert a (singleton (α := R) ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) : R))))
d_dvd_left : d ∣ a
d_dvd_right : d ∣ (ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯)
r₁ r₂ : R
d_eq : d = r₁ * a + HMul.hMul (β := R) r₂ ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) : R)
ψ' : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R := r₁ • ϕ + r₂ • ψ
⊢ R ∙ d ≤ ψ'.submoduleImage N
have rw : ∃ (y : M) (yN : y ∈ N), (ψ' : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y sorry) = d
Tactic: rw [span_le, singleton_subset_iff, SetLike.mem_coe, LinearMap.mem_submoduleImage_of_le hNO]
-- Lean 4 proof step from Mathlib.LinearAlgebra.FreeModule.PID (goal)
Goal:
R : Type u_2
inst : CommRing R
M : Type u_3
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : IsPrincipalIdealRing R
N O : Submodule R M
hNO : N ≤ O
ϕ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R
hϕ : ∀ (ψ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R), ¬ϕ.submoduleImage N < ψ.submoduleImage N
inst_4 : (ϕ.submoduleImage N).IsPrincipal
y : M
yN : y ∈ N
ϕy_eq : (ϕ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) = generator (ϕ.submoduleImage N)
ψ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R
a : R := generator (ϕ.submoduleImage N)
d : R :=
  generator
    (span R (insert a (singleton (α := R) ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) : R))))
d_dvd_left : d ∣ a
d_dvd_right : d ∣ (ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯)
r₁ r₂ : R
d_eq : d = r₁ * a + HMul.hMul (β := R) r₂ ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) : R)
ψ' : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R := r₁ • ϕ + r₂ • ψ
⊢ ∃ (y : M) (yN : y ∈ N), (ψ' : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) = d
have refine : (ψ' : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y sorry) = d
Tactic: refine ⟨y, yN, ?_⟩
-- Lean 4 proof step from Mathlib.LinearAlgebra.FreeModule.PID (goal)
Goal:
R : Type u_2
inst : CommRing R
M : Type u_3
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : IsPrincipalIdealRing R
N O : Submodule R M
hNO : N ≤ O
ϕ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R
hϕ : ∀ (ψ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R), ¬ϕ.submoduleImage N < ψ.submoduleImage N
inst_4 : (ϕ.submoduleImage N).IsPrincipal
y : M
yN : y ∈ N
ϕy_eq : (ϕ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) = generator (ϕ.submoduleImage N)
ψ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R
a : R := generator (ϕ.submoduleImage N)
d : R :=
  generator
    (span R (insert a (singleton (α := R) ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) : R))))
d_dvd_left : d ∣ a
d_dvd_right : d ∣ (ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯)
r₁ r₂ : R
d_eq : d = r₁ * a + HMul.hMul (β := R) r₂ ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) : R)
ψ' : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R := r₁ • ϕ + r₂ • ψ
⊢ (ψ' : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) = d
have change : HMul.hMul (β := R) r₁ ((ϕ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y sorry) : R) +
    HMul.hMul (β := R) r₂ ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y sorry) : R) =
  d
Tactic: change r₁ * ϕ ⟨y, hNO yN⟩ + r₂ * ψ ⟨y, hNO yN⟩ = d
-- Lean 4 proof step from Mathlib.LinearAlgebra.FreeModule.PID (goal)
Goal:
R : Type u_2
inst : CommRing R
M : Type u_3
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : IsPrincipalIdealRing R
N O : Submodule R M
hNO : N ≤ O
ϕ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R
hϕ : ∀ (ψ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R), ¬ϕ.submoduleImage N < ψ.submoduleImage N
inst_4 : (ϕ.submoduleImage N).IsPrincipal
y : M
yN : y ∈ N
ϕy_eq : (ϕ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) = generator (ϕ.submoduleImage N)
ψ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R
a : R := generator (ϕ.submoduleImage N)
d : R :=
  generator
    (span R (insert a (singleton (α := R) ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) : R))))
d_dvd_left : d ∣ a
d_dvd_right : d ∣ (ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯)
r₁ r₂ : R
d_eq : d = r₁ * a + HMul.hMul (β := R) r₂ ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) : R)
ψ' : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R := r₁ • ϕ + r₂ • ψ
⊢ HMul.hMul (β := R) r₁ ((ϕ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) : R) +
      HMul.hMul (β := R) r₂ ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) : R) =
    d
have rw : HMul.hMul (β := R) r₁ ((ϕ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y sorry) : R) +
    HMul.hMul (β := R) r₂ ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y sorry) : R) =
  r₁ * a + HMul.hMul (β := R) r₂ ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y sorry) : R)
Tactic: rw [d_eq, ϕy_eq]
-- Lean 4 proof step from Mathlib.LinearAlgebra.FreeModule.PID (goal)
Goal:
R : Type u_2
inst : CommRing R
M : Type u_3
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : IsPrincipalIdealRing R
N O : Submodule R M
hNO : N ≤ O
ϕ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R
hϕ : ∀ (ψ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R), ¬ϕ.submoduleImage N < ψ.submoduleImage N
inst_4 : (ϕ.submoduleImage N).IsPrincipal
y : M
yN : y ∈ N
ϕy_eq : (ϕ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) = generator (ϕ.submoduleImage N)
ψ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R
a : R := generator (ϕ.submoduleImage N)
d : R :=
  generator
    (span R (insert a (singleton (α := R) ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) : R))))
d_dvd_left : d ∣ a
d_dvd_right : d ∣ (ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯)
r₁ r₂ : R
d_eq : d = r₁ * a + HMul.hMul (β := R) r₂ ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) : R)
ψ' : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R := r₁ • ϕ + r₂ • ψ
rw :
  HMul.hMul (β := R) r₁ ((ϕ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) : R) +
      HMul.hMul (β := R) r₂ ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) : R) =
    r₁ * a + HMul.hMul (β := R) r₂ ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) : R)
⊢ HMul.hMul (β := R) r₁ ((ϕ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) : R) +
      HMul.hMul (β := R) r₂ ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) : R) =
    d
have rw₁ : r₁ * generator (ϕ.submoduleImage N) +
    HMul.hMul (β := R) r₂ ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y sorry) : R) =
  r₁ * a + HMul.hMul (β := R) r₂ ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y sorry) : R)
Tactic: rw [d_eq, ϕy_eq]
-- Lean 4 proof step from Mathlib.LinearAlgebra.FreeModule.PID (hypothesis)
Goal:
R : Type u_2
inst : CommRing R
M : Type u_3
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : IsPrincipalIdealRing R
N O : Submodule R M
hNO : N ≤ O
ϕ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R
hϕ : ∀ (ψ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R), ¬ϕ.submoduleImage N < ψ.submoduleImage N
inst_4 : (ϕ.submoduleImage N).IsPrincipal
y : M
yN : y ∈ N
ϕy_eq : (ϕ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) = generator (ϕ.submoduleImage N)
ψ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R
a : R := generator (ϕ.submoduleImage N)
d : R :=
  generator
    (span R (insert a (singleton (α := R) ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) : R))))
d_dvd_left : d ∣ a
d_dvd_right : d ∣ (ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯)
r₁ r₂ : R
d_eq : d = r₁ * a + HMul.hMul (β := R) r₂ ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) : R)
ψ' : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R := r₁ • ϕ + r₂ • ψ
⊢ Eq (α := Ideal R)
    (R ∙
      generator
        (span R (insert a (singleton (α := R) ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) : R)))))
    (R ∙ generator (ϕ.submoduleImage N))
have this : R ∙ d ≤ ψ'.submoduleImage N
Tactic: have : span R {d} ≤ ψ'.submoduleImage N := by
      rw [span_le, singleton_subset_iff, SetLike.mem_coe, LinearMap.mem_submoduleImage_of_le hNO]
      refine ⟨y, yN, ?_⟩
      change r₁ * ϕ ⟨y, hNO yN⟩ + r₂ * ψ ⟨y, hNO yN⟩ = d
      rw [d_eq, ϕy_eq]
-- Lean 4 proof step from Mathlib.LinearAlgebra.FreeModule.PID (goal)
Goal:
R : Type u_2
inst : CommRing R
M : Type u_3
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : IsPrincipalIdealRing R
N O : Submodule R M
hNO : N ≤ O
ϕ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R
hϕ : ∀ (ψ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R), ¬ϕ.submoduleImage N < ψ.submoduleImage N
inst_4 : (ϕ.submoduleImage N).IsPrincipal
y : M
yN : y ∈ N
ϕy_eq : (ϕ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) = generator (ϕ.submoduleImage N)
ψ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R
a : R := generator (ϕ.submoduleImage N)
d : R :=
  generator
    (span R (insert a (singleton (α := R) ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) : R))))
d_dvd_left : d ∣ a
d_dvd_right : d ∣ (ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯)
r₁ r₂ : R
d_eq : d = r₁ * a + HMul.hMul (β := R) r₂ ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) : R)
ψ' : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R := r₁ • ϕ + r₂ • ψ
this : R ∙ d ≤ ψ'.submoduleImage N
⊢ Eq (α := Ideal R)
    (R ∙
      generator
        (span R (insert a (singleton (α := R) ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) : R)))))
    (R ∙ generator (ϕ.submoduleImage N))
have le_antisymm : ψ'.submoduleImage N = R ∙ generator (ϕ.submoduleImage N)
Tactic: refine
      le_antisymm (this.trans (le_of_eq ?_)) (Ideal.span_singleton_le_span_singleton.mpr d_dvd_left)
-- Lean 4 proof step from Mathlib.LinearAlgebra.FreeModule.PID (goal)
Goal:
R : Type u_2
inst : CommRing R
M : Type u_3
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : IsPrincipalIdealRing R
N O : Submodule R M
hNO : N ≤ O
ϕ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R
hϕ : ∀ (ψ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R), ¬ϕ.submoduleImage N < ψ.submoduleImage N
inst_4 : (ϕ.submoduleImage N).IsPrincipal
y : M
yN : y ∈ N
ϕy_eq : (ϕ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) = generator (ϕ.submoduleImage N)
ψ : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R
a : R := generator (ϕ.submoduleImage N)
d : R :=
  generator
    (span R (insert a (singleton (α := R) ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) : R))))
d_dvd_left : d ∣ a
d_dvd_right : d ∣ (ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯)
r₁ r₂ : R
d_eq : d = r₁ * a + HMul.hMul (β := R) r₂ ((ψ : (a : ↥O) → R) (Subtype.mk (p := fun (x : M) ↦ x ∈ O) y ⋯) : R)
ψ' : LinearMap.{u_2, u_2, u_3, u_2} (RingHom.id R) (↥O) R := r₁ • ϕ + r₂ • ψ
this : R ∙ d ≤ ψ'.submoduleImage N
⊢ ψ'.submoduleImage N = R ∙ generator (ϕ.submoduleImage N)
have rw : ψ'.submoduleImage N = ϕ.submoduleImage N
Tactic: rw [span_singleton_generator]