l3lab/ntp-mathlib huggingface dataset:
instance (R : Type*) [CommRing R] [hR : HenselianLocalRing R] : HenselianRing R (maximalIdeal R)
    where
  jac := by
    rw [Ideal.jacobson, le_sInf_iff]
    rintro I ⟨-, hI⟩
    exact (eq_maximalIdeal hI).ge
  is_henselian := by
    intro f hf a₀ h₁ h₂
    refine' HenselianLocalRing.is_henselian f hf a₀ h₁ _
    

Proof state:
R : Type u_1
inst✝ : CommRing R
hR : HenselianLocalRing R
f : R[X]
hf : Monic f
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ maximalIdeal R
h₂ : IsUnit ((Ideal.Quotient.mk (maximalIdeal R)) (Polynomial.eval a₀ (derivative f)))
⊢ IsUnit (Polynomial.eval a₀ (derivative f))

Next tactic:
contrapose! h₂
instance (R : Type*) [CommRing R] [hR : HenselianLocalRing R] : HenselianRing R (maximalIdeal R)
    where
  jac := by
    rw [Ideal.jacobson, le_sInf_iff]
    rintro I ⟨-, hI⟩
    exact (eq_maximalIdeal hI).ge
  is_henselian := by
    intro f hf a₀ h₁ h₂
    refine' HenselianLocalRing.is_henselian f hf a₀ h₁ _
    contrapose! h₂
    

Proof state:
R : Type u_1
inst✝ : CommRing R
hR : HenselianLocalRing R
f : R[X]
hf : Monic f
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ maximalIdeal R
h₂ : ¬IsUnit (Polynomial.eval a₀ (derivative f))
⊢ ¬IsUnit ((Ideal.Quotient.mk (maximalIdeal R)) (Polynomial.eval a₀ (derivative f)))

Next tactic:
rw [← mem_nonunits_iff, ← LocalRing.mem_maximalIdeal, ← Ideal.Quotient.eq_zero_iff_mem] at h₂
instance (R : Type*) [CommRing R] [hR : HenselianLocalRing R] : HenselianRing R (maximalIdeal R)
    where
  jac := by
    rw [Ideal.jacobson, le_sInf_iff]
    rintro I ⟨-, hI⟩
    exact (eq_maximalIdeal hI).ge
  is_henselian := by
    intro f hf a₀ h₁ h₂
    refine' HenselianLocalRing.is_henselian f hf a₀ h₁ _
    contrapose! h₂
    rw [← mem_nonunits_iff, ← LocalRing.mem_maximalIdeal, ← Ideal.Quotient.eq_zero_iff_mem] at h₂
    

Proof state:
R : Type u_1
inst✝ : CommRing R
hR : HenselianLocalRing R
f : R[X]
hf : Monic f
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ maximalIdeal R
h₂ : (Ideal.Quotient.mk (maximalIdeal R)) (Polynomial.eval a₀ (derivative f)) = 0
⊢ ¬IsUnit ((Ideal.Quotient.mk (maximalIdeal R)) (Polynomial.eval a₀ (derivative f)))

Next tactic:
rw [h₂]
instance (R : Type*) [CommRing R] [hR : HenselianLocalRing R] : HenselianRing R (maximalIdeal R)
    where
  jac := by
    rw [Ideal.jacobson, le_sInf_iff]
    rintro I ⟨-, hI⟩
    exact (eq_maximalIdeal hI).ge
  is_henselian := by
    intro f hf a₀ h₁ h₂
    refine' HenselianLocalRing.is_henselian f hf a₀ h₁ _
    contrapose! h₂
    rw [← mem_nonunits_iff, ← LocalRing.mem_maximalIdeal, ← Ideal.Quotient.eq_zero_iff_mem] at h₂
    rw [h₂]
    

Proof state:
R : Type u_1
inst✝ : CommRing R
hR : HenselianLocalRing R
f : R[X]
hf : Monic f
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ maximalIdeal R
h₂ : (Ideal.Quotient.mk (maximalIdeal R)) (Polynomial.eval a₀ (derivative f)) = 0
⊢ ¬IsUnit 0

Next tactic:
exact not_isUnit_zero
/-- A ring `R` that is `I`-adically complete is Henselian at `I`. -/
instance (priority := 100) IsAdicComplete.henselianRing (R : Type*) [CommRing R] (I : Ideal R)
    [IsAdicComplete I R] : HenselianRing R I where
  jac := IsAdicComplete.le_jacobson_bot _
  is_henselian := by
    

Proof state:
R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
inst✝ : IsAdicComplete I R
⊢ ∀ (f : R[X]),
    Monic f →
      ∀ (a₀ : R),
        Polynomial.eval a₀ f ∈ I →
          IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f))) → ∃ a, IsRoot f a ∧ a - a₀ ∈ I

Next tactic:
intro f _ a₀ h₁ h₂
/-- A ring `R` that is `I`-adically complete is Henselian at `I`. -/
instance (priority := 100) IsAdicComplete.henselianRing (R : Type*) [CommRing R] (I : Ideal R)
    [IsAdicComplete I R] : HenselianRing R I where
  jac := IsAdicComplete.le_jacobson_bot _
  is_henselian := by
    intro f _ a₀ h₁ h₂
    classical
      

Proof state:
R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
inst✝ : IsAdicComplete I R
f : R[X]
x✝ : Monic f
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ I
h₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))
⊢ ∃ a, IsRoot f a ∧ a - a₀ ∈ I

Next tactic:
let f' := derivative f
/-- A ring `R` that is `I`-adically complete is Henselian at `I`. -/
instance (priority := 100) IsAdicComplete.henselianRing (R : Type*) [CommRing R] (I : Ideal R)
    [IsAdicComplete I R] : HenselianRing R I where
  jac := IsAdicComplete.le_jacobson_bot _
  is_henselian := by
    intro f _ a₀ h₁ h₂
    classical
      let f' := derivative f
      -- we define a sequence `c n` by starting at `a₀` and then continually
      -- applying the function sending `b` to `b - f(b)/f'(b)` (Newton's method).
      -- Note that `f'.eval b` is a unit, because `b` has the same residue as `a₀` modulo `I`.
      

Proof state:
R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
inst✝ : IsAdicComplete I R
f : R[X]
x✝ : Monic f
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ I
h₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))
f' : R[X] := derivative f
⊢ ∃ a, IsRoot f a ∧ a - a₀ ∈ I

Next tactic:
let c : ℕ → R := fun n => Nat.recOn n a₀ fun _ b => b - f.eval b * Ring.inverse (f'.eval b)
/-- A ring `R` that is `I`-adically complete is Henselian at `I`. -/
instance (priority := 100) IsAdicComplete.henselianRing (R : Type*) [CommRing R] (I : Ideal R)
    [IsAdicComplete I R] : HenselianRing R I where
  jac := IsAdicComplete.le_jacobson_bot _
  is_henselian := by
    intro f _ a₀ h₁ h₂
    classical
      let f' := derivative f
      -- we define a sequence `c n` by starting at `a₀` and then continually
      -- applying the function sending `b` to `b - f(b)/f'(b)` (Newton's method).
      -- Note that `f'.eval b` is a unit, because `b` has the same residue as `a₀` modulo `I`.
      let c : ℕ → R := fun n => Nat.recOn n a₀ fun _ b => b - f.eval b * Ring.inverse (f'.eval b)
      

Proof state:
R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
inst✝ : IsAdicComplete I R
f : R[X]
x✝ : Monic f
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ I
h₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))
f' : R[X] := derivative f
c : ℕ → R := fun n => Nat.recOn n a₀ fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')
⊢ ∃ a, IsRoot f a ∧ a - a₀ ∈ I

Next tactic:
have hc : ∀ n, c (n + 1) = c n - f.eval (c n) * Ring.inverse (f'.eval (c n)) := by
        intro n
        simp only [Nat.rec_add_one]
/-- A ring `R` that is `I`-adically complete is Henselian at `I`. -/
instance (priority := 100) IsAdicComplete.henselianRing (R : Type*) [CommRing R] (I : Ideal R)
    [IsAdicComplete I R] : HenselianRing R I where
  jac := IsAdicComplete.le_jacobson_bot _
  is_henselian := by
    intro f _ a₀ h₁ h₂
    classical
      let f' := derivative f
      -- we define a sequence `c n` by starting at `a₀` and then continually
      -- applying the function sending `b` to `b - f(b)/f'(b)` (Newton's method).
      -- Note that `f'.eval b` is a unit, because `b` has the same residue as `a₀` modulo `I`.
      let c : ℕ → R := fun n => Nat.recOn n a₀ fun _ b => b - f.eval b * Ring.inverse (f'.eval b)
      have hc : ∀ n, c (n + 1) = c n - f.eval (c n) * Ring.inverse (f'.eval (c n)) := by
        

Proof state:
R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
inst✝ : IsAdicComplete I R
f : R[X]
x✝ : Monic f
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ I
h₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))
f' : R[X] := derivative f
c : ℕ → R := fun n => Nat.recOn n a₀ fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')
⊢ ∀ (n : ℕ), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')

Next tactic:
intro n
/-- A ring `R` that is `I`-adically complete is Henselian at `I`. -/
instance (priority := 100) IsAdicComplete.henselianRing (R : Type*) [CommRing R] (I : Ideal R)
    [IsAdicComplete I R] : HenselianRing R I where
  jac := IsAdicComplete.le_jacobson_bot _
  is_henselian := by
    intro f _ a₀ h₁ h₂
    classical
      let f' := derivative f
      -- we define a sequence `c n` by starting at `a₀` and then continually
      -- applying the function sending `b` to `b - f(b)/f'(b)` (Newton's method).
      -- Note that `f'.eval b` is a unit, because `b` has the same residue as `a₀` modulo `I`.
      let c : ℕ → R := fun n => Nat.recOn n a₀ fun _ b => b - f.eval b * Ring.inverse (f'.eval b)
      have hc : ∀ n, c (n + 1) = c n - f.eval (c n) * Ring.inverse (f'.eval (c n)) := by
        intro n
        

Proof state:
R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
inst✝ : IsAdicComplete I R
f : R[X]
x✝ : Monic f
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ I
h₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))
f' : R[X] := derivative f
c : ℕ → R := fun n => Nat.recOn n a₀ fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')
n : ℕ
⊢ c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')

Next tactic:
simp only [Nat.rec_add_one]
/-- A ring `R` that is `I`-adically complete is Henselian at `I`. -/
instance (priority := 100) IsAdicComplete.henselianRing (R : Type*) [CommRing R] (I : Ideal R)
    [IsAdicComplete I R] : HenselianRing R I where
  jac := IsAdicComplete.le_jacobson_bot _
  is_henselian := by
    intro f _ a₀ h₁ h₂
    classical
      let f' := derivative f
      -- we define a sequence `c n` by starting at `a₀` and then continually
      -- applying the function sending `b` to `b - f(b)/f'(b)` (Newton's method).
      -- Note that `f'.eval b` is a unit, because `b` has the same residue as `a₀` modulo `I`.
      let c : ℕ → R := fun n => Nat.recOn n a₀ fun _ b => b - f.eval b * Ring.inverse (f'.eval b)
      have hc : ∀ n, c (n + 1) = c n - f.eval (c n) * Ring.inverse (f'.eval (c n)) := by
        intro n
        simp only [Nat.rec_add_one]
      -- we now spend some time determining properties of the sequence `c : ℕ → R`
      -- `hc_mod`: for every `n`, we have `c n ≡ a₀ [SMOD I]`
      -- `hf'c`  : for every `n`, `f'.eval (c n)` is a unit
      -- `hfcI`  : for every `n`, `f.eval (c n)` is contained in `I ^ (n+1)`
      

Proof state:
R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
inst✝ : IsAdicComplete I R
f : R[X]
x✝ : Monic f
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ I
h₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))
f' : R[X] := derivative f
c : ℕ → R := fun n => Nat.recOn n a₀ fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')
hc : ∀ (n : ℕ), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')
⊢ ∃ a, IsRoot f a ∧ a - a₀ ∈ I

Next tactic:
have hc_mod : ∀ n, c n ≡ a₀ [SMOD I] := by
        intro n
        induction' n with n ih
        · rfl
        rw [Nat.succ_eq_add_one, hc, sub_eq_add_neg, ← add_zero a₀]
        refine' ih.add _
        rw [SModEq.zero, Ideal.neg_mem_iff]
        refine' I.mul_mem_right _ _
        rw [← SModEq.zero] at h₁ ⊢
        exact (ih.eval f).trans h₁
/-- A ring `R` that is `I`-adically complete is Henselian at `I`. -/
instance (priority := 100) IsAdicComplete.henselianRing (R : Type*) [CommRing R] (I : Ideal R)
    [IsAdicComplete I R] : HenselianRing R I where
  jac := IsAdicComplete.le_jacobson_bot _
  is_henselian := by
    intro f _ a₀ h₁ h₂
    classical
      let f' := derivative f
      -- we define a sequence `c n` by starting at `a₀` and then continually
      -- applying the function sending `b` to `b - f(b)/f'(b)` (Newton's method).
      -- Note that `f'.eval b` is a unit, because `b` has the same residue as `a₀` modulo `I`.
      let c : ℕ → R := fun n => Nat.recOn n a₀ fun _ b => b - f.eval b * Ring.inverse (f'.eval b)
      have hc : ∀ n, c (n + 1) = c n - f.eval (c n) * Ring.inverse (f'.eval (c n)) := by
        intro n
        simp only [Nat.rec_add_one]
      -- we now spend some time determining properties of the sequence `c : ℕ → R`
      -- `hc_mod`: for every `n`, we have `c n ≡ a₀ [SMOD I]`
      -- `hf'c`  : for every `n`, `f'.eval (c n)` is a unit
      -- `hfcI`  : for every `n`, `f.eval (c n)` is contained in `I ^ (n+1)`
      have hc_mod : ∀ n, c n ≡ a₀ [SMOD I] := by
        

Proof state:
R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
inst✝ : IsAdicComplete I R
f : R[X]
x✝ : Monic f
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ I
h₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))
f' : R[X] := derivative f
c : ℕ → R := fun n => Nat.recOn n a₀ fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')
hc : ∀ (n : ℕ), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')
⊢ ∀ (n : ℕ), c n ≡ a₀ [SMOD I]

Next tactic:
intro n
/-- A ring `R` that is `I`-adically complete is Henselian at `I`. -/
instance (priority := 100) IsAdicComplete.henselianRing (R : Type*) [CommRing R] (I : Ideal R)
    [IsAdicComplete I R] : HenselianRing R I where
  jac := IsAdicComplete.le_jacobson_bot _
  is_henselian := by
    intro f _ a₀ h₁ h₂
    classical
      let f' := derivative f
      -- we define a sequence `c n` by starting at `a₀` and then continually
      -- applying the function sending `b` to `b - f(b)/f'(b)` (Newton's method).
      -- Note that `f'.eval b` is a unit, because `b` has the same residue as `a₀` modulo `I`.
      let c : ℕ → R := fun n => Nat.recOn n a₀ fun _ b => b - f.eval b * Ring.inverse (f'.eval b)
      have hc : ∀ n, c (n + 1) = c n - f.eval (c n) * Ring.inverse (f'.eval (c n)) := by
        intro n
        simp only [Nat.rec_add_one]
      -- we now spend some time determining properties of the sequence `c : ℕ → R`
      -- `hc_mod`: for every `n`, we have `c n ≡ a₀ [SMOD I]`
      -- `hf'c`  : for every `n`, `f'.eval (c n)` is a unit
      -- `hfcI`  : for every `n`, `f.eval (c n)` is contained in `I ^ (n+1)`
      have hc_mod : ∀ n, c n ≡ a₀ [SMOD I] := by
        intro n
        

Proof state:
R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
inst✝ : IsAdicComplete I R
f : R[X]
x✝ : Monic f
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ I
h₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))
f' : R[X] := derivative f
c : ℕ → R := fun n => Nat.recOn n a₀ fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')
hc : ∀ (n : ℕ), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')
n : ℕ
⊢ c n ≡ a₀ [SMOD I]

Next tactic:
induction' n with n ih
/-- A ring `R` that is `I`-adically complete is Henselian at `I`. -/
instance (priority := 100) IsAdicComplete.henselianRing (R : Type*) [CommRing R] (I : Ideal R)
    [IsAdicComplete I R] : HenselianRing R I where
  jac := IsAdicComplete.le_jacobson_bot _
  is_henselian := by
    intro f _ a₀ h₁ h₂
    classical
      let f' := derivative f
      -- we define a sequence `c n` by starting at `a₀` and then continually
      -- applying the function sending `b` to `b - f(b)/f'(b)` (Newton's method).
      -- Note that `f'.eval b` is a unit, because `b` has the same residue as `a₀` modulo `I`.
      let c : ℕ → R := fun n => Nat.recOn n a₀ fun _ b => b - f.eval b * Ring.inverse (f'.eval b)
      have hc : ∀ n, c (n + 1) = c n - f.eval (c n) * Ring.inverse (f'.eval (c n)) := by
        intro n
        simp only [Nat.rec_add_one]
      -- we now spend some time determining properties of the sequence `c : ℕ → R`
      -- `hc_mod`: for every `n`, we have `c n ≡ a₀ [SMOD I]`
      -- `hf'c`  : for every `n`, `f'.eval (c n)` is a unit
      -- `hfcI`  : for every `n`, `f.eval (c n)` is contained in `I ^ (n+1)`
      have hc_mod : ∀ n, c n ≡ a₀ [SMOD I] := by
        intro n
        induction' n with n ih
        · 

Proof state:
case zero
R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
inst✝ : IsAdicComplete I R
f : R[X]
x✝ : Monic f
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ I
h₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))
f' : R[X] := derivative f
c : ℕ → R := fun n => Nat.recOn n a₀ fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')
hc : ∀ (n : ℕ), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')
⊢ c Nat.zero ≡ a₀ [SMOD I]

Next tactic:
rfl
/-- A ring `R` that is `I`-adically complete is Henselian at `I`. -/
instance (priority := 100) IsAdicComplete.henselianRing (R : Type*) [CommRing R] (I : Ideal R)
    [IsAdicComplete I R] : HenselianRing R I where
  jac := IsAdicComplete.le_jacobson_bot _
  is_henselian := by
    intro f _ a₀ h₁ h₂
    classical
      let f' := derivative f
      -- we define a sequence `c n` by starting at `a₀` and then continually
      -- applying the function sending `b` to `b - f(b)/f'(b)` (Newton's method).
      -- Note that `f'.eval b` is a unit, because `b` has the same residue as `a₀` modulo `I`.
      let c : ℕ → R := fun n => Nat.recOn n a₀ fun _ b => b - f.eval b * Ring.inverse (f'.eval b)
      have hc : ∀ n, c (n + 1) = c n - f.eval (c n) * Ring.inverse (f'.eval (c n)) := by
        intro n
        simp only [Nat.rec_add_one]
      -- we now spend some time determining properties of the sequence `c : ℕ → R`
      -- `hc_mod`: for every `n`, we have `c n ≡ a₀ [SMOD I]`
      -- `hf'c`  : for every `n`, `f'.eval (c n)` is a unit
      -- `hfcI`  : for every `n`, `f.eval (c n)` is contained in `I ^ (n+1)`
      have hc_mod : ∀ n, c n ≡ a₀ [SMOD I] := by
        intro n
        induction' n with n ih
        · rfl
        

Proof state:
case succ
R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
inst✝ : IsAdicComplete I R
f : R[X]
x✝ : Monic f
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ I
h₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))
f' : R[X] := derivative f
c : ℕ → R := fun n => Nat.recOn n a₀ fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')
hc : ∀ (n : ℕ), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')
n : ℕ
ih : c n ≡ a₀ [SMOD I]
⊢ c (Nat.succ n) ≡ a₀ [SMOD I]

Next tactic:
rw [Nat.succ_eq_add_one, hc, sub_eq_add_neg, ← add_zero a₀]
/-- A ring `R` that is `I`-adically complete is Henselian at `I`. -/
instance (priority := 100) IsAdicComplete.henselianRing (R : Type*) [CommRing R] (I : Ideal R)
    [IsAdicComplete I R] : HenselianRing R I where
  jac := IsAdicComplete.le_jacobson_bot _
  is_henselian := by
    intro f _ a₀ h₁ h₂
    classical
      let f' := derivative f
      -- we define a sequence `c n` by starting at `a₀` and then continually
      -- applying the function sending `b` to `b - f(b)/f'(b)` (Newton's method).
      -- Note that `f'.eval b` is a unit, because `b` has the same residue as `a₀` modulo `I`.
      let c : ℕ → R := fun n => Nat.recOn n a₀ fun _ b => b - f.eval b * Ring.inverse (f'.eval b)
      have hc : ∀ n, c (n + 1) = c n - f.eval (c n) * Ring.inverse (f'.eval (c n)) := by
        intro n
        simp only [Nat.rec_add_one]
      -- we now spend some time determining properties of the sequence `c : ℕ → R`
      -- `hc_mod`: for every `n`, we have `c n ≡ a₀ [SMOD I]`
      -- `hf'c`  : for every `n`, `f'.eval (c n)` is a unit
      -- `hfcI`  : for every `n`, `f.eval (c n)` is contained in `I ^ (n+1)`
      have hc_mod : ∀ n, c n ≡ a₀ [SMOD I] := by
        intro n
        induction' n with n ih
        · rfl
        rw [Nat.succ_eq_add_one, hc, sub_eq_add_neg, ← add_zero a₀]
        

Proof state:
case succ
R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
inst✝ : IsAdicComplete I R
f : R[X]
x✝ : Monic f
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ I
h₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))
f' : R[X] := derivative f
c : ℕ → R := fun n => Nat.recOn n a₀ fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')
hc : ∀ (n : ℕ), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')
n : ℕ
ih : c n ≡ a₀ [SMOD I]
⊢ c n + -(Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')) ≡ a₀ + 0 [SMOD I]

Next tactic:
refine' ih.add _
/-- A ring `R` that is `I`-adically complete is Henselian at `I`. -/
instance (priority := 100) IsAdicComplete.henselianRing (R : Type*) [CommRing R] (I : Ideal R)
    [IsAdicComplete I R] : HenselianRing R I where
  jac := IsAdicComplete.le_jacobson_bot _
  is_henselian := by
    intro f _ a₀ h₁ h₂
    classical
      let f' := derivative f
      -- we define a sequence `c n` by starting at `a₀` and then continually
      -- applying the function sending `b` to `b - f(b)/f'(b)` (Newton's method).
      -- Note that `f'.eval b` is a unit, because `b` has the same residue as `a₀` modulo `I`.
      let c : ℕ → R := fun n => Nat.recOn n a₀ fun _ b => b - f.eval b * Ring.inverse (f'.eval b)
      have hc : ∀ n, c (n + 1) = c n - f.eval (c n) * Ring.inverse (f'.eval (c n)) := by
        intro n
        simp only [Nat.rec_add_one]
      -- we now spend some time determining properties of the sequence `c : ℕ → R`
      -- `hc_mod`: for every `n`, we have `c n ≡ a₀ [SMOD I]`
      -- `hf'c`  : for every `n`, `f'.eval (c n)` is a unit
      -- `hfcI`  : for every `n`, `f.eval (c n)` is contained in `I ^ (n+1)`
      have hc_mod : ∀ n, c n ≡ a₀ [SMOD I] := by
        intro n
        induction' n with n ih
        · rfl
        rw [Nat.succ_eq_add_one, hc, sub_eq_add_neg, ← add_zero a₀]
        refine' ih.add _
        

Proof state:
case succ
R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
inst✝ : IsAdicComplete I R
f : R[X]
x✝ : Monic f
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ I
h₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))
f' : R[X] := derivative f
c : ℕ → R := fun n => Nat.recOn n a₀ fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')
hc : ∀ (n : ℕ), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')
n : ℕ
ih : c n ≡ a₀ [SMOD I]
⊢ -(Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')) ≡ 0 [SMOD I]

Next tactic:
rw [SModEq.zero, Ideal.neg_mem_iff]
/-- A ring `R` that is `I`-adically complete is Henselian at `I`. -/
instance (priority := 100) IsAdicComplete.henselianRing (R : Type*) [CommRing R] (I : Ideal R)
    [IsAdicComplete I R] : HenselianRing R I where
  jac := IsAdicComplete.le_jacobson_bot _
  is_henselian := by
    intro f _ a₀ h₁ h₂
    classical
      let f' := derivative f
      -- we define a sequence `c n` by starting at `a₀` and then continually
      -- applying the function sending `b` to `b - f(b)/f'(b)` (Newton's method).
      -- Note that `f'.eval b` is a unit, because `b` has the same residue as `a₀` modulo `I`.
      let c : ℕ → R := fun n => Nat.recOn n a₀ fun _ b => b - f.eval b * Ring.inverse (f'.eval b)
      have hc : ∀ n, c (n + 1) = c n - f.eval (c n) * Ring.inverse (f'.eval (c n)) := by
        intro n
        simp only [Nat.rec_add_one]
      -- we now spend some time determining properties of the sequence `c : ℕ → R`
      -- `hc_mod`: for every `n`, we have `c n ≡ a₀ [SMOD I]`
      -- `hf'c`  : for every `n`, `f'.eval (c n)` is a unit
      -- `hfcI`  : for every `n`, `f.eval (c n)` is contained in `I ^ (n+1)`
      have hc_mod : ∀ n, c n ≡ a₀ [SMOD I] := by
        intro n
        induction' n with n ih
        · rfl
        rw [Nat.succ_eq_add_one, hc, sub_eq_add_neg, ← add_zero a₀]
        refine' ih.add _
        rw [SModEq.zero, Ideal.neg_mem_iff]
        

Proof state:
case succ
R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
inst✝ : IsAdicComplete I R
f : R[X]
x✝ : Monic f
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ I
h₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))
f' : R[X] := derivative f
c : ℕ → R := fun n => Nat.recOn n a₀ fun x b => b - Polynomial.eval b f * Ring.inverse (Polynomial.eval b f')
hc : ∀ (n : ℕ), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')
n : ℕ
ih : c n ≡ a₀ [SMOD I]
⊢ Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f') ∈ I

Next tactic:
refine' I.mul_mem_right _ _