l3lab/ntp-mathlib huggingface dataset:
instance [Field R] : Field (HahnSeries Γ R) :=
  { inferInstanceAs (IsDomain (HahnSeries Γ R)),
    inferInstanceAs (CommRing (HahnSeries Γ R)) with
    inv := fun x =>
      if x0 : x = 0 then 0
      else
        C (x.coeff x.order)⁻¹ * (single (-x.order)) 1 *
          (SummableFamily.powers _ (unit_aux x (inv_mul_cancel (coeff_order_ne_zero x0)))).hsum
    inv_zero := dif_pos rfl
    mul_inv_cancel := fun x x0 => by
      refine' (congr rfl (dif_neg x0)).trans _
      

Proof state:
Γ : Type u_1
R : Type u_2
inst✝¹ : LinearOrderedAddCommGroup Γ
inst✝ : Field R
src✝¹ : IsDomain (HahnSeries Γ R) := inferInstanceAs (IsDomain (HahnSeries Γ R))
src✝ : CommRing (HahnSeries Γ R) := inferInstanceAs (CommRing (HahnSeries Γ R))
x : HahnSeries Γ R
x0 : x ≠ 0
⊢ x *
      (C (coeff x (order x))⁻¹ * (single (-order x)) 1 *
        SummableFamily.hsum
          (SummableFamily.powers (1 - C (coeff x (order x))⁻¹ * (single (-order x)) 1 * x)
            (_ : 0 < (addVal Γ R) (1 - C (coeff x (order x))⁻¹ * (single (-order x)) 1 * x)))) =
    1

Next tactic:
have h :=
        SummableFamily.one_sub_self_mul_hsum_powers
          (unit_aux x (inv_mul_cancel (coeff_order_ne_zero x0)))
instance [Field R] : Field (HahnSeries Γ R) :=
  { inferInstanceAs (IsDomain (HahnSeries Γ R)),
    inferInstanceAs (CommRing (HahnSeries Γ R)) with
    inv := fun x =>
      if x0 : x = 0 then 0
      else
        C (x.coeff x.order)⁻¹ * (single (-x.order)) 1 *
          (SummableFamily.powers _ (unit_aux x (inv_mul_cancel (coeff_order_ne_zero x0)))).hsum
    inv_zero := dif_pos rfl
    mul_inv_cancel := fun x x0 => by
      refine' (congr rfl (dif_neg x0)).trans _
      have h :=
        SummableFamily.one_sub_self_mul_hsum_powers
          (unit_aux x (inv_mul_cancel (coeff_order_ne_zero x0)))
      

Proof state:
Γ : Type u_1
R : Type u_2
inst✝¹ : LinearOrderedAddCommGroup Γ
inst✝ : Field R
src✝¹ : IsDomain (HahnSeries Γ R) := inferInstanceAs (IsDomain (HahnSeries Γ R))
src✝ : CommRing (HahnSeries Γ R) := inferInstanceAs (CommRing (HahnSeries Γ R))
x : HahnSeries Γ R
x0 : x ≠ 0
h :
  (1 - (1 - C (coeff x (order x))⁻¹ * (single (-order x)) 1 * x)) *
      SummableFamily.hsum
        (SummableFamily.powers (1 - C (coeff x (order x))⁻¹ * (single (-order x)) 1 * x)
          (_ : 0 < (addVal Γ R) (1 - C (coeff x (order x))⁻¹ * (single (-order x)) 1 * x))) =
    1
⊢ x *
      (C (coeff x (order x))⁻¹ * (single (-order x)) 1 *
        SummableFamily.hsum
          (SummableFamily.powers (1 - C (coeff x (order x))⁻¹ * (single (-order x)) 1 * x)
            (_ : 0 < (addVal Γ R) (1 - C (coeff x (order x))⁻¹ * (single (-order x)) 1 * x)))) =
    1

Next tactic:
rw [sub_sub_cancel] at h
instance [Field R] : Field (HahnSeries Γ R) :=
  { inferInstanceAs (IsDomain (HahnSeries Γ R)),
    inferInstanceAs (CommRing (HahnSeries Γ R)) with
    inv := fun x =>
      if x0 : x = 0 then 0
      else
        C (x.coeff x.order)⁻¹ * (single (-x.order)) 1 *
          (SummableFamily.powers _ (unit_aux x (inv_mul_cancel (coeff_order_ne_zero x0)))).hsum
    inv_zero := dif_pos rfl
    mul_inv_cancel := fun x x0 => by
      refine' (congr rfl (dif_neg x0)).trans _
      have h :=
        SummableFamily.one_sub_self_mul_hsum_powers
          (unit_aux x (inv_mul_cancel (coeff_order_ne_zero x0)))
      rw [sub_sub_cancel] at h
      

Proof state:
Γ : Type u_1
R : Type u_2
inst✝¹ : LinearOrderedAddCommGroup Γ
inst✝ : Field R
src✝¹ : IsDomain (HahnSeries Γ R) := inferInstanceAs (IsDomain (HahnSeries Γ R))
src✝ : CommRing (HahnSeries Γ R) := inferInstanceAs (CommRing (HahnSeries Γ R))
x : HahnSeries Γ R
x0 : x ≠ 0
h :
  C (coeff x (order x))⁻¹ * (single (-order x)) 1 * x *
      SummableFamily.hsum
        (SummableFamily.powers (1 - C (coeff x (order x))⁻¹ * (single (-order x)) 1 * x)
          (_ : 0 < (addVal Γ R) (1 - C (coeff x (order x))⁻¹ * (single (-order x)) 1 * x))) =
    1
⊢ x *
      (C (coeff x (order x))⁻¹ * (single (-order x)) 1 *
        SummableFamily.hsum
          (SummableFamily.powers (1 - C (coeff x (order x))⁻¹ * (single (-order x)) 1 * x)
            (_ : 0 < (addVal Γ R) (1 - C (coeff x (order x))⁻¹ * (single (-order x)) 1 * x)))) =
    1

Next tactic:
rw [← mul_assoc, mul_comm x, h]
/-- The directed system of `R`-modules of the form `R/J`, where `J` is an ideal of `R`,
determined by the functor `I`  -/
def ringModIdeals (I : D ⥤ Ideal R) : D ⥤ ModuleCat.{u} R where
  obj t := ModuleCat.of R <| R ⧸ I.obj t
  map w := Submodule.mapQ _ _ LinearMap.id (I.map w).down.down
  -- Porting note: was 'obviously'
  map_comp f g := by 

Proof state:
R : Type u
inst✝¹ : CommRing R
D : Type v
inst✝ : SmallCategory D
I : D ⥤ Ideal R
X✝ Y✝ Z✝ : D
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
⊢ { obj := fun t => ModuleCat.of R (R ⧸ I.obj t),
          map := fun {X Y} w => Submodule.mapQ (I.obj X) (I.obj Y) LinearMap.id (_ : I.obj X ≤ I.obj Y) }.map
      (f ≫ g) =
    { obj := fun t => ModuleCat.of R (R ⧸ I.obj t),
            map := fun {X Y} w => Submodule.mapQ (I.obj X) (I.obj Y) LinearMap.id (_ : I.obj X ≤ I.obj Y) }.map
        f ≫
      { obj := fun t => ModuleCat.of R (R ⧸ I.obj t),
            map := fun {X Y} w => Submodule.mapQ (I.obj X) (I.obj Y) LinearMap.id (_ : I.obj X ≤ I.obj Y) }.map
        g

Next tactic:
apply Submodule.linearMap_qext
/-- The directed system of `R`-modules of the form `R/J`, where `J` is an ideal of `R`,
determined by the functor `I`  -/
def ringModIdeals (I : D ⥤ Ideal R) : D ⥤ ModuleCat.{u} R where
  obj t := ModuleCat.of R <| R ⧸ I.obj t
  map w := Submodule.mapQ _ _ LinearMap.id (I.map w).down.down
  -- Porting note: was 'obviously'
  map_comp f g := by apply Submodule.linearMap_qext; 

Proof state:
case h
R : Type u
inst✝¹ : CommRing R
D : Type v
inst✝ : SmallCategory D
I : D ⥤ Ideal R
X✝ Y✝ Z✝ : D
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
⊢ { obj := fun t => ModuleCat.of R (R ⧸ I.obj t),
            map := fun {X Y} w => Submodule.mapQ (I.obj X) (I.obj Y) LinearMap.id (_ : I.obj X ≤ I.obj Y) }.map
        (f ≫ g) ∘ₗ
      Submodule.mkQ (I.obj X✝) =
    ({ obj := fun t => ModuleCat.of R (R ⧸ I.obj t),
              map := fun {X Y} w => Submodule.mapQ (I.obj X) (I.obj Y) LinearMap.id (_ : I.obj X ≤ I.obj Y) }.map
          f ≫
        { obj := fun t => ModuleCat.of R (R ⧸ I.obj t),
              map := fun {X Y} w => Submodule.mapQ (I.obj X) (I.obj Y) LinearMap.id (_ : I.obj X ≤ I.obj Y) }.map
          g) ∘ₗ
      Submodule.mkQ (I.obj X✝)

Next tactic:
rfl
lemma hasColimitDiagram (I : D ⥤ Ideal R) (i : ℕ) :
    HasColimit (diagram I i) := by
  

Proof state:
R : Type (max u v)
inst✝¹ : CommRing R
D : Type v
inst✝ : SmallCategory D
I : D ⥤ Ideal R
i : ℕ
⊢ HasColimit (diagram I i)

Next tactic:
have : HasColimitsOfShape Dᵒᵖ (AddCommGroupCatMax.{u, v}) := inferInstance
lemma hasColimitDiagram (I : D ⥤ Ideal R) (i : ℕ) :
    HasColimit (diagram I i) := by
  have : HasColimitsOfShape Dᵒᵖ (AddCommGroupCatMax.{u, v}) := inferInstance
  

Proof state:
R : Type (max u v)
inst✝¹ : CommRing R
D : Type v
inst✝ : SmallCategory D
I : D ⥤ Ideal R
i : ℕ
this : HasColimitsOfShape Dᵒᵖ AddCommGroupCatMax
⊢ HasColimit (diagram I i)

Next tactic:
infer_instance
/-- Lifting `idealPowersDiagram J` from a diagram valued in `ideals R` to a diagram
valued in `SelfLERadical J`. -/
def idealPowersToSelfLERadical (J : Ideal R) : ℕᵒᵖ ⥤ SelfLERadical J :=
  FullSubcategory.lift _ (idealPowersDiagram J) fun k => by
    

Proof state:
R : Type u
inst✝ : CommRing R
J : Ideal R
k : ℕᵒᵖ
⊢ J ≤ Ideal.radical ((idealPowersDiagram J).obj k)

Next tactic:
change _ ≤ (J ^ unop k).radical
/-- Lifting `idealPowersDiagram J` from a diagram valued in `ideals R` to a diagram
valued in `SelfLERadical J`. -/
def idealPowersToSelfLERadical (J : Ideal R) : ℕᵒᵖ ⥤ SelfLERadical J :=
  FullSubcategory.lift _ (idealPowersDiagram J) fun k => by
    change _ ≤ (J ^ unop k).radical
    

Proof state:
R : Type u
inst✝ : CommRing R
J : Ideal R
k : ℕᵒᵖ
⊢ J ≤ Ideal.radical (J ^ k.unop)

Next tactic:
cases' unop k with n
/-- Lifting `idealPowersDiagram J` from a diagram valued in `ideals R` to a diagram
valued in `SelfLERadical J`. -/
def idealPowersToSelfLERadical (J : Ideal R) : ℕᵒᵖ ⥤ SelfLERadical J :=
  FullSubcategory.lift _ (idealPowersDiagram J) fun k => by
    change _ ≤ (J ^ unop k).radical
    cases' unop k with n
    · 

Proof state:
case zero
R : Type u
inst✝ : CommRing R
J : Ideal R
k : ℕᵒᵖ
⊢ J ≤ Ideal.radical (J ^ Nat.zero)

Next tactic:
simp [Ideal.radical_top, pow_zero, Ideal.one_eq_top, le_top, Nat.zero_eq]
/-- Lifting `idealPowersDiagram J` from a diagram valued in `ideals R` to a diagram
valued in `SelfLERadical J`. -/
def idealPowersToSelfLERadical (J : Ideal R) : ℕᵒᵖ ⥤ SelfLERadical J :=
  FullSubcategory.lift _ (idealPowersDiagram J) fun k => by
    change _ ≤ (J ^ unop k).radical
    cases' unop k with n
    · simp [Ideal.radical_top, pow_zero, Ideal.one_eq_top, le_top, Nat.zero_eq]
    · 

Proof state:
case succ
R : Type u
inst✝ : CommRing R
J : Ideal R
k : ℕᵒᵖ
n : ℕ
⊢ J ≤ Ideal.radical (J ^ Nat.succ n)

Next tactic:
simp only [J.radical_pow _ n.succ_pos, Ideal.le_radical]
/-- The lemma below essentially says that `idealPowersToSelfLERadical I` is initial in
`selfLERadicalDiagram I`.

PORTING NOTE: This lemma should probably be moved to `Mathlib/RingTheory/Finiteness`
to be near `Ideal.exists_radical_pow_le_of_fg`, which it generalizes. -/
theorem Ideal.exists_pow_le_of_le_radical_of_fG (hIJ : I ≤ J.radical) (hJ : J.radical.FG) :
    ∃ k : ℕ, I ^ k ≤ J := by
  

Proof state:
R : Type u
inst✝ : CommRing R
I J K : Ideal R
hIJ : I ≤ Ideal.radical J
hJ : Ideal.FG (Ideal.radical J)
⊢ ∃ k, I ^ k ≤ J

Next tactic:
obtain ⟨k, hk⟩ := J.exists_radical_pow_le_of_fg hJ
/-- The lemma below essentially says that `idealPowersToSelfLERadical I` is initial in
`selfLERadicalDiagram I`.

PORTING NOTE: This lemma should probably be moved to `Mathlib/RingTheory/Finiteness`
to be near `Ideal.exists_radical_pow_le_of_fg`, which it generalizes. -/
theorem Ideal.exists_pow_le_of_le_radical_of_fG (hIJ : I ≤ J.radical) (hJ : J.radical.FG) :
    ∃ k : ℕ, I ^ k ≤ J := by
  obtain ⟨k, hk⟩ := J.exists_radical_pow_le_of_fg hJ
  

Proof state:
case intro
R : Type u
inst✝ : CommRing R
I J K : Ideal R
hIJ : I ≤ Ideal.radical J
hJ : Ideal.FG (Ideal.radical J)
k : ℕ
hk : Ideal.radical J ^ k ≤ J
⊢ ∃ k, I ^ k ≤ J

Next tactic:
use k
/-- The lemma below essentially says that `idealPowersToSelfLERadical I` is initial in
`selfLERadicalDiagram I`.

PORTING NOTE: This lemma should probably be moved to `Mathlib/RingTheory/Finiteness`
to be near `Ideal.exists_radical_pow_le_of_fg`, which it generalizes. -/
theorem Ideal.exists_pow_le_of_le_radical_of_fG (hIJ : I ≤ J.radical) (hJ : J.radical.FG) :
    ∃ k : ℕ, I ^ k ≤ J := by
  obtain ⟨k, hk⟩ := J.exists_radical_pow_le_of_fg hJ
  use k
  

Proof state:
case h
R : Type u
inst✝ : CommRing R
I J K : Ideal R
hIJ : I ≤ Ideal.radical J
hJ : Ideal.FG (Ideal.radical J)
k : ℕ
hk : Ideal.radical J ^ k ≤ J
⊢ I ^ k ≤ J

Next tactic:
calc
    I ^ k ≤ J.radical ^ k := Ideal.pow_right_mono hIJ _
    _ ≤ J := hk
/-- The diagram of powers of `J` is initial in the diagram of all ideals with
radical containing `J`. This uses noetherianness. -/
instance ideal_powers_initial [hR : IsNoetherian R R] :
    Functor.Initial (idealPowersToSelfLERadical J) where
  out J' := by
    

Proof state:
R : Type u
inst✝ : CommRing R
I J K : Ideal R
hR : IsNoetherian R R
J' : SelfLERadical J
⊢ IsConnected (CostructuredArrow (idealPowersToSelfLERadical J) J')

Next tactic:
apply (config := {allowSynthFailures := true }) zigzag_isConnected
/-- The diagram of powers of `J` is initial in the diagram of all ideals with
radical containing `J`. This uses noetherianness. -/
instance ideal_powers_initial [hR : IsNoetherian R R] :
    Functor.Initial (idealPowersToSelfLERadical J) where
  out J' := by
    apply (config := {allowSynthFailures := true }) zigzag_isConnected
    · 

Proof state:
case inst
R : Type u
inst✝ : CommRing R
I J K : Ideal R
hR : IsNoetherian R R
J' : SelfLERadical J
⊢ Nonempty (CostructuredArrow (idealPowersToSelfLERadical J) J')

Next tactic:
obtain ⟨k, hk⟩ := Ideal.exists_pow_le_of_le_radical_of_fG J'.2 (isNoetherian_def.mp hR _)
/-- The diagram of powers of `J` is initial in the diagram of all ideals with
radical containing `J`. This uses noetherianness. -/
instance ideal_powers_initial [hR : IsNoetherian R R] :
    Functor.Initial (idealPowersToSelfLERadical J) where
  out J' := by
    apply (config := {allowSynthFailures := true }) zigzag_isConnected
    · obtain ⟨k, hk⟩ := Ideal.exists_pow_le_of_le_radical_of_fG J'.2 (isNoetherian_def.mp hR _)
      

Proof state:
case inst.intro
R : Type u
inst✝ : CommRing R
I J K : Ideal R
hR : IsNoetherian R R
J' : SelfLERadical J
k : ℕ
hk : J ^ k ≤ J'.obj
⊢ Nonempty (CostructuredArrow (idealPowersToSelfLERadical J) J')

Next tactic:
exact ⟨CostructuredArrow.mk (⟨⟨hk⟩⟩ : (idealPowersToSelfLERadical J).obj (op k) ⟶ J')⟩
/-- The diagram of powers of `J` is initial in the diagram of all ideals with
radical containing `J`. This uses noetherianness. -/
instance ideal_powers_initial [hR : IsNoetherian R R] :
    Functor.Initial (idealPowersToSelfLERadical J) where
  out J' := by
    apply (config := {allowSynthFailures := true }) zigzag_isConnected
    · obtain ⟨k, hk⟩ := Ideal.exists_pow_le_of_le_radical_of_fG J'.2 (isNoetherian_def.mp hR _)
      exact ⟨CostructuredArrow.mk (⟨⟨hk⟩⟩ : (idealPowersToSelfLERadical J).obj (op k) ⟶ J')⟩
    · 

Proof state:
case h
R : Type u
inst✝ : CommRing R
I J K : Ideal R
hR : IsNoetherian R R
J' : SelfLERadical J
⊢ ∀ (j₁ j₂ : CostructuredArrow (idealPowersToSelfLERadical J) J'), Zigzag j₁ j₂

Next tactic:
intro j1 j2
/-- The diagram of powers of `J` is initial in the diagram of all ideals with
radical containing `J`. This uses noetherianness. -/
instance ideal_powers_initial [hR : IsNoetherian R R] :
    Functor.Initial (idealPowersToSelfLERadical J) where
  out J' := by
    apply (config := {allowSynthFailures := true }) zigzag_isConnected
    · obtain ⟨k, hk⟩ := Ideal.exists_pow_le_of_le_radical_of_fG J'.2 (isNoetherian_def.mp hR _)
      exact ⟨CostructuredArrow.mk (⟨⟨hk⟩⟩ : (idealPowersToSelfLERadical J).obj (op k) ⟶ J')⟩
    · intro j1 j2
      

Proof state:
case h
R : Type u
inst✝ : CommRing R
I J K : Ideal R
hR : IsNoetherian R R
J' : SelfLERadical J
j1 j2 : CostructuredArrow (idealPowersToSelfLERadical J) J'
⊢ Zigzag j1 j2

Next tactic:
apply Relation.ReflTransGen.single
/-- The diagram of powers of `J` is initial in the diagram of all ideals with
radical containing `J`. This uses noetherianness. -/
instance ideal_powers_initial [hR : IsNoetherian R R] :
    Functor.Initial (idealPowersToSelfLERadical J) where
  out J' := by
    apply (config := {allowSynthFailures := true }) zigzag_isConnected
    · obtain ⟨k, hk⟩ := Ideal.exists_pow_le_of_le_radical_of_fG J'.2 (isNoetherian_def.mp hR _)
      exact ⟨CostructuredArrow.mk (⟨⟨hk⟩⟩ : (idealPowersToSelfLERadical J).obj (op k) ⟶ J')⟩
    · intro j1 j2
      apply Relation.ReflTransGen.single
      -- The inclusions `J^n1 ≤ J'` and `J^n2 ≤ J'` always form a triangle, based on
      -- which exponent is larger.
      

Proof state:
case h.hab
R : Type u
inst✝ : CommRing R
I J K : Ideal R
hR : IsNoetherian R R
J' : SelfLERadical J
j1 j2 : CostructuredArrow (idealPowersToSelfLERadical J) J'
⊢ Zag j1 j2

Next tactic:
rcases le_total (unop j1.left) (unop j2.left) with h | h
/-- The diagram of powers of `J` is initial in the diagram of all ideals with
radical containing `J`. This uses noetherianness. -/
instance ideal_powers_initial [hR : IsNoetherian R R] :
    Functor.Initial (idealPowersToSelfLERadical J) where
  out J' := by
    apply (config := {allowSynthFailures := true }) zigzag_isConnected
    · obtain ⟨k, hk⟩ := Ideal.exists_pow_le_of_le_radical_of_fG J'.2 (isNoetherian_def.mp hR _)
      exact ⟨CostructuredArrow.mk (⟨⟨hk⟩⟩ : (idealPowersToSelfLERadical J).obj (op k) ⟶ J')⟩
    · intro j1 j2
      apply Relation.ReflTransGen.single
      -- The inclusions `J^n1 ≤ J'` and `J^n2 ≤ J'` always form a triangle, based on
      -- which exponent is larger.
      rcases le_total (unop j1.left) (unop j2.left) with h | h
      

Proof state:
case h.hab.inl
R : Type u
inst✝ : CommRing R
I J K : Ideal R
hR : IsNoetherian R R
J' : SelfLERadical J
j1 j2 : CostructuredArrow (idealPowersToSelfLERadical J) J'
h : j1.left.unop ≤ j2.left.unop
⊢ Zag j1 j2
case h.hab.inr
R : Type u
inst✝ : CommRing R
I J K : Ideal R
hR : IsNoetherian R R
J' : SelfLERadical J
j1 j2 : CostructuredArrow (idealPowersToSelfLERadical J) J'
h : j2.left.unop ≤ j1.left.unop
⊢ Zag j1 j2

Next tactic:
right
/-- The diagram of powers of `J` is initial in the diagram of all ideals with
radical containing `J`. This uses noetherianness. -/
instance ideal_powers_initial [hR : IsNoetherian R R] :
    Functor.Initial (idealPowersToSelfLERadical J) where
  out J' := by
    apply (config := {allowSynthFailures := true }) zigzag_isConnected
    · obtain ⟨k, hk⟩ := Ideal.exists_pow_le_of_le_radical_of_fG J'.2 (isNoetherian_def.mp hR _)
      exact ⟨CostructuredArrow.mk (⟨⟨hk⟩⟩ : (idealPowersToSelfLERadical J).obj (op k) ⟶ J')⟩
    · intro j1 j2
      apply Relation.ReflTransGen.single
      -- The inclusions `J^n1 ≤ J'` and `J^n2 ≤ J'` always form a triangle, based on
      -- which exponent is larger.
      rcases le_total (unop j1.left) (unop j2.left) with h | h
      right; 

Proof state:
case h.hab.inl.h
R : Type u
inst✝ : CommRing R
I J K : Ideal R
hR : IsNoetherian R R
J' : SelfLERadical J
j1 j2 : CostructuredArrow (idealPowersToSelfLERadical J) J'
h : j1.left.unop ≤ j2.left.unop
⊢ Nonempty (j2 ⟶ j1)
case h.hab.inr
R : Type u
inst✝ : CommRing R
I J K : Ideal R
hR : IsNoetherian R R
J' : SelfLERadical J
j1 j2 : CostructuredArrow (idealPowersToSelfLERadical J) J'
h : j2.left.unop ≤ j1.left.unop
⊢ Zag j1 j2

Next tactic:
exact ⟨CostructuredArrow.homMk (homOfLE h).op (AsTrue.get trivial)⟩
/-- The diagram of powers of `J` is initial in the diagram of all ideals with
radical containing `J`. This uses noetherianness. -/
instance ideal_powers_initial [hR : IsNoetherian R R] :
    Functor.Initial (idealPowersToSelfLERadical J) where
  out J' := by
    apply (config := {allowSynthFailures := true }) zigzag_isConnected
    · obtain ⟨k, hk⟩ := Ideal.exists_pow_le_of_le_radical_of_fG J'.2 (isNoetherian_def.mp hR _)
      exact ⟨CostructuredArrow.mk (⟨⟨hk⟩⟩ : (idealPowersToSelfLERadical J).obj (op k) ⟶ J')⟩
    · intro j1 j2
      apply Relation.ReflTransGen.single
      -- The inclusions `J^n1 ≤ J'` and `J^n2 ≤ J'` always form a triangle, based on
      -- which exponent is larger.
      rcases le_total (unop j1.left) (unop j2.left) with h | h
      right; exact ⟨CostructuredArrow.homMk (homOfLE h).op (AsTrue.get trivial)⟩
      

Proof state:
case h.hab.inr
R : Type u
inst✝ : CommRing R
I J K : Ideal R
hR : IsNoetherian R R
J' : SelfLERadical J
j1 j2 : CostructuredArrow (idealPowersToSelfLERadical J) J'
h : j2.left.unop ≤ j1.left.unop
⊢ Zag j1 j2

Next tactic:
left
/-- The diagram of powers of `J` is initial in the diagram of all ideals with
radical containing `J`. This uses noetherianness. -/
instance ideal_powers_initial [hR : IsNoetherian R R] :
    Functor.Initial (idealPowersToSelfLERadical J) where
  out J' := by
    apply (config := {allowSynthFailures := true }) zigzag_isConnected
    · obtain ⟨k, hk⟩ := Ideal.exists_pow_le_of_le_radical_of_fG J'.2 (isNoetherian_def.mp hR _)
      exact ⟨CostructuredArrow.mk (⟨⟨hk⟩⟩ : (idealPowersToSelfLERadical J).obj (op k) ⟶ J')⟩
    · intro j1 j2
      apply Relation.ReflTransGen.single
      -- The inclusions `J^n1 ≤ J'` and `J^n2 ≤ J'` always form a triangle, based on
      -- which exponent is larger.
      rcases le_total (unop j1.left) (unop j2.left) with h | h
      right; exact ⟨CostructuredArrow.homMk (homOfLE h).op (AsTrue.get trivial)⟩
      left; 

Proof state:
case h.hab.inr.h
R : Type u
inst✝ : CommRing R
I J K : Ideal R
hR : IsNoetherian R R
J' : SelfLERadical J
j1 j2 : CostructuredArrow (idealPowersToSelfLERadical J) J'
h : j2.left.unop ≤ j1.left.unop
⊢ Nonempty (j1 ⟶ j2)

Next tactic:
exact ⟨CostructuredArrow.homMk (homOfLE h).op (AsTrue.get trivial)⟩
/-- Casting from the full subcategory of ideals with radical containing `J` to the full
subcategory of ideals with radical containing `K`. -/
def SelfLERadical.cast (hJK : J.radical = K.radical) : SelfLERadical J ⥤ SelfLERadical K :=
  FullSubcategory.map fun L hL => by
    

Proof state:
R : Type u
inst✝ : CommRing R
I J K : Ideal R
hJK : Ideal.radical J = Ideal.radical K
L : Ideal R
hL : J ≤ Ideal.radical L
⊢ K ≤ Ideal.radical L

Next tactic:
rw [← Ideal.radical_le_radical_iff] at hL ⊢
/-- Casting from the full subcategory of ideals with radical containing `J` to the full
subcategory of ideals with radical containing `K`. -/
def SelfLERadical.cast (hJK : J.radical = K.radical) : SelfLERadical J ⥤ SelfLERadical K :=
  FullSubcategory.map fun L hL => by
    rw [← Ideal.radical_le_radical_iff] at hL ⊢
    

Proof state:
R : Type u
inst✝ : CommRing R
I J K : Ideal R
hJK : Ideal.radical J = Ideal.radical K
L : Ideal R
hL : Ideal.radical J ≤ Ideal.radical L
⊢ Ideal.radical K ≤ Ideal.radical L

Next tactic:
exact hJK.symm.trans_le hL
@[simp]
lemma N₂_map_isoΓ₀_hom_app_f (X : ChainComplex C ℕ) :
    (N₂.map (isoΓ₀.hom.app X)).f = PInfty := by
  

Proof state:
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
inst✝¹ : IsIdempotentComplete C
inst✝ : HasFiniteCoproducts C
X : ChainComplex C ℕ
⊢ (N₂.map (isoΓ₀.hom.app X)).f = PInfty

Next tactic:
ext
@[simp]
lemma N₂_map_isoΓ₀_hom_app_f (X : ChainComplex C ℕ) :
    (N₂.map (isoΓ₀.hom.app X)).f = PInfty := by
  ext
  

Proof state:
case h
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
inst✝¹ : IsIdempotentComplete C
inst✝ : HasFiniteCoproducts C
X : ChainComplex C ℕ
i✝ : ℕ
⊢ HomologicalComplex.Hom.f (N₂.map (isoΓ₀.hom.app X)).f i✝ = HomologicalComplex.Hom.f PInfty i✝

Next tactic:
apply comp_id
/-- The natural isomorphism `NΓ' satisfies the compatibility that is needed
for the construction of our counit isomorphism `η` -/
theorem hη :
    Compatibility.τ₀ =
      Compatibility.τ₁ isoN₁ isoΓ₀
        (N₁Γ₀ : Γ ⋙ N₁ ≅ (toKaroubiEquivalence (ChainComplex C ℕ)).functor) := by
  

Proof state:
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
inst✝¹ : IsIdempotentComplete C
inst✝ : HasFiniteCoproducts C
⊢ Compatibility.τ₀ = Compatibility.τ₁ isoN₁ isoΓ₀ N₁Γ₀

Next tactic:
ext K : 3
/-- The natural isomorphism `NΓ' satisfies the compatibility that is needed
for the construction of our counit isomorphism `η` -/
theorem hη :
    Compatibility.τ₀ =
      Compatibility.τ₁ isoN₁ isoΓ₀
        (N₁Γ₀ : Γ ⋙ N₁ ≅ (toKaroubiEquivalence (ChainComplex C ℕ)).functor) := by
  ext K : 3
  

Proof state:
case w.w.h
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
inst✝¹ : IsIdempotentComplete C
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
⊢ Compatibility.τ₀.hom.app K = (Compatibility.τ₁ isoN₁ isoΓ₀ N₁Γ₀).hom.app K

Next tactic:
simp only [Compatibility.τ₀_hom_app, Compatibility.τ₁_hom_app]
/-- The natural isomorphism `NΓ' satisfies the compatibility that is needed
for the construction of our counit isomorphism `η` -/
theorem hη :
    Compatibility.τ₀ =
      Compatibility.τ₁ isoN₁ isoΓ₀
        (N₁Γ₀ : Γ ⋙ N₁ ≅ (toKaroubiEquivalence (ChainComplex C ℕ)).functor) := by
  ext K : 3
  simp only [Compatibility.τ₀_hom_app, Compatibility.τ₁_hom_app]
  

Proof state:
case w.w.h
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
inst✝¹ : IsIdempotentComplete C
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
⊢ Preadditive.DoldKan.equivalence.counitIso.hom.app ((toKaroubiEquivalence (ChainComplex C ℕ)).functor.obj K) =
    Preadditive.DoldKan.equivalence.functor.map (isoΓ₀.hom.app K) ≫ isoN₁.hom.app (Γ.obj K) ≫ N₁Γ₀.hom.app K

Next tactic:
refine' (N₂Γ₂_compatible_with_N₁Γ₀ K).trans (by simp )