l3lab/ntp-mathlib huggingface dataset:
/-- 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]
        refine' I.mul_mem_right _ _
        

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 ∈ I

Next tactic:
rw [← SModEq.zero] at 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
        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 ⊢
        

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 ≡ 0 [SMOD 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 ≡ 0 [SMOD I]

Next tactic:
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
        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
      

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')
hc_mod : ∀ (n : ℕ), c n ≡ a₀ [SMOD I]
⊢ ∃ a, IsRoot f a ∧ a - a₀ ∈ I

Next tactic:
have hf'c : ∀ n, IsUnit (f'.eval (c n)) := by
        intro n
        haveI := isLocalRingHom_of_le_jacobson_bot I (IsAdicComplete.le_jacobson_bot I)
        apply isUnit_of_map_unit (Ideal.Quotient.mk I)
        convert h₂ using 1
        exact SModEq.def.mp ((hc_mod n).eval _)
/-- 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]
        refine' I.mul_mem_right _ _
        rw [← SModEq.zero] at h₁ ⊢
        exact (ih.eval f).trans h₁
      have hf'c : ∀ n, IsUnit (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')
hc : ∀ (n : ℕ), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')
hc_mod : ∀ (n : ℕ), c n ≡ a₀ [SMOD I]
⊢ ∀ (n : ℕ), IsUnit (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
        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]
        refine' I.mul_mem_right _ _
        rw [← SModEq.zero] at h₁ ⊢
        exact (ih.eval f).trans h₁
      have hf'c : ∀ n, IsUnit (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')
hc : ∀ (n : ℕ), c (n + 1) = c n - Polynomial.eval (c n) f * Ring.inverse (Polynomial.eval (c n) f')
hc_mod : ∀ (n : ℕ), c n ≡ a₀ [SMOD I]
n : ℕ
⊢ IsUnit (Polynomial.eval (c n) f')

Next tactic:
haveI := isLocalRingHom_of_le_jacobson_bot I (IsAdicComplete.le_jacobson_bot I)
/-- 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]
        refine' I.mul_mem_right _ _
        rw [← SModEq.zero] at h₁ ⊢
        exact (ih.eval f).trans h₁
      have hf'c : ∀ n, IsUnit (f'.eval (c n)) := by
        intro n
        haveI := isLocalRingHom_of_le_jacobson_bot I (IsAdicComplete.le_jacobson_bot 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
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')
hc_mod : ∀ (n : ℕ), c n ≡ a₀ [SMOD I]
n : ℕ
this : IsLocalRingHom (Ideal.Quotient.mk I)
⊢ IsUnit (Polynomial.eval (c n) f')

Next tactic:
apply isUnit_of_map_unit (Ideal.Quotient.mk I)
/-- 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]
        refine' I.mul_mem_right _ _
        rw [← SModEq.zero] at h₁ ⊢
        exact (ih.eval f).trans h₁
      have hf'c : ∀ n, IsUnit (f'.eval (c n)) := by
        intro n
        haveI := isLocalRingHom_of_le_jacobson_bot I (IsAdicComplete.le_jacobson_bot I)
        apply isUnit_of_map_unit (Ideal.Quotient.mk I)
        

Proof state:
case h
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')
hc_mod : ∀ (n : ℕ), c n ≡ a₀ [SMOD I]
n : ℕ
this : IsLocalRingHom (Ideal.Quotient.mk I)
⊢ IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval (c n) f'))

Next tactic:
convert h₂ using 1
/-- 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]
        refine' I.mul_mem_right _ _
        rw [← SModEq.zero] at h₁ ⊢
        exact (ih.eval f).trans h₁
      have hf'c : ∀ n, IsUnit (f'.eval (c n)) := by
        intro n
        haveI := isLocalRingHom_of_le_jacobson_bot I (IsAdicComplete.le_jacobson_bot I)
        apply isUnit_of_map_unit (Ideal.Quotient.mk I)
        convert h₂ using 1
        

Proof state:
case h.e'_3
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')
hc_mod : ∀ (n : ℕ), c n ≡ a₀ [SMOD I]
n : ℕ
this : IsLocalRingHom (Ideal.Quotient.mk I)
⊢ (Ideal.Quotient.mk I) (Polynomial.eval (c n) f') = (Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f))

Next tactic:
exact SModEq.def.mp ((hc_mod n).eval _)
/-- 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]
        refine' I.mul_mem_right _ _
        rw [← SModEq.zero] at h₁ ⊢
        exact (ih.eval f).trans h₁
      have hf'c : ∀ n, IsUnit (f'.eval (c n)) := by
        intro n
        haveI := isLocalRingHom_of_le_jacobson_bot I (IsAdicComplete.le_jacobson_bot I)
        apply isUnit_of_map_unit (Ideal.Quotient.mk I)
        convert h₂ using 1
        exact SModEq.def.mp ((hc_mod n).eval _)
      have hfcI : ∀ n, f.eval (c n) ∈ I ^ (n + 1) := 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')
hc_mod : ∀ (n : ℕ), c n ≡ a₀ [SMOD I]
hf'c : ∀ (n : ℕ), IsUnit (Polynomial.eval (c n) f')
⊢ ∀ (n : ℕ), Polynomial.eval (c n) f ∈ I ^ (n + 1)

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
        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
      have hf'c : ∀ n, IsUnit (f'.eval (c n)) := by
        intro n
        haveI := isLocalRingHom_of_le_jacobson_bot I (IsAdicComplete.le_jacobson_bot I)
        apply isUnit_of_map_unit (Ideal.Quotient.mk I)
        convert h₂ using 1
        exact SModEq.def.mp ((hc_mod n).eval _)
      have hfcI : ∀ n, f.eval (c n) ∈ I ^ (n + 1) := 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')
hc_mod : ∀ (n : ℕ), c n ≡ a₀ [SMOD I]
hf'c : ∀ (n : ℕ), IsUnit (Polynomial.eval (c n) f')
n : ℕ
⊢ Polynomial.eval (c n) f ∈ I ^ (n + 1)

Next tactic:
induction' n with n ih