awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.RingTheory.Ideal.Quotient.Nilpotent
Goal:
P : ⦃S : Type u_1⦄ → [inst : CommRing S] → Ideal S → Prop
h₁ : ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I : Ideal S), HPow.hPow (β := ℕ) I 2 = ⊥ → P I
h₂ :
  ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I J : Ideal S),
    I ≤ J → P I → P (S := S ⧸ I) (map (F := S →+* S ⧸ I) (Quotient.mk I) J) → P J
S : Type u_1
inst : CommRing S
I : Ideal S
hI' : ¬I = ⊥
H : ∀ m < HAdd.hAdd (α := ℕ) 0 1, ∀ {S : Type u_1} [inst : CommRing S] (I : Ideal S) (hI : I ^ m = ⊥), P I
hI : I = ⊥
⊢ P I

Tactic:
exact (hI' hI).elim

Premises used: False.elim
-- Module: Mathlib.RingTheory.Ideal.Quotient.Nilpotent
Goal:
P : ⦃S : Type u_1⦄ → [inst : CommRing S] → Ideal S → Prop
h₁ : ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I : Ideal S), HPow.hPow (β := ℕ) I 2 = ⊥ → P I
h₂ :
  ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I J : Ideal S),
    I ≤ J → P I → P (S := S ⧸ I) (map (F := S →+* S ⧸ I) (Quotient.mk I) J) → P J
S : Type u_1
inst : CommRing S
I : Ideal S
hI' : ¬I = ⊥
n : ℕ
H : ∀ m < n + 1 + 1, ∀ {S : Type u_1} [inst : CommRing S] (I : Ideal S) (hI : I ^ m = ⊥), P I
hI : I ^ (n + 1 + 1) = ⊥
neg : P (HPow.hPow (β := ℕ) I 2)
neg₁ :
  P (S := S ⧸ HPow.hPow (β := ℕ) I 2)
    (map (F := S →+* S ⧸ HPow.hPow (β := ℕ) I 2) (Quotient.mk (HPow.hPow (β := ℕ) I 2)) I)
⊢ P I

Tactic:
apply h₂ (I ^ 2) _ (Ideal.pow_le_self two_ne_zero)

Premises used: Ideal.pow_le_self, two_ne_zero, HPow.hPow, OfNat.ofNat
-- Module: Mathlib.RingTheory.Ideal.Quotient.Nilpotent
Goal:
P : ⦃S : Type u_1⦄ → [inst : CommRing S] → Ideal S → Prop
h₁ : ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I : Ideal S), HPow.hPow (β := ℕ) I 2 = ⊥ → P I
h₂ :
  ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I J : Ideal S),
    I ≤ J → P I → P (S := S ⧸ I) (map (F := S →+* S ⧸ I) (Quotient.mk I) J) → P J
S : Type u_1
inst : CommRing S
I : Ideal S
hI' : ¬I = ⊥
n : ℕ
H : ∀ m < n + 1 + 1, ∀ {S : Type u_1} [inst : CommRing S] (I : Ideal S) (hI : I ^ m = ⊥), P I
hI : I ^ (n + 1 + 1) = ⊥
neg : HPow.hPow (β := ℕ) I 2 ^ n.succ = ⊥
apply : n.succ < n + 1 + 1
⊢ P (HPow.hPow (β := ℕ) I 2)

Tactic:
apply H n.succ _ (I ^ 2)

Premises used: HPow.hPow, OfNat.ofNat, Nat.succ
-- Module: Mathlib.RingTheory.Ideal.Quotient.Nilpotent
Goal:
P : ⦃S : Type u_1⦄ → [inst : CommRing S] → Ideal S → Prop
h₁ : ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I : Ideal S), HPow.hPow (β := ℕ) I 2 = ⊥ → P I
h₂ :
  ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I J : Ideal S),
    I ≤ J → P I → P (S := S ⧸ I) (map (F := S →+* S ⧸ I) (Quotient.mk I) J) → P J
S : Type u_1
inst : CommRing S
I : Ideal S
hI' : ¬I = ⊥
n : ℕ
H : ∀ m < n + 1 + 1, ∀ {S : Type u_1} [inst : CommRing S] (I : Ideal S) (hI : I ^ m = ⊥), P I
hI : I ^ (n + 1 + 1) = ⊥
neg : I ^ (2 * (n + 1)) ≤ I ^ (n + 1 + 1)
⊢ HPow.hPow (β := ℕ) I 2 ^ n.succ = ⊥

Tactic:
rw [← pow_mul, eq_bot_iff, ← hI, Nat.succ_eq_add_one]

Premises used: Nat.succ_eq_add_one, pow_mul, eq_bot_iff
-- Module: Mathlib.RingTheory.Ideal.Quotient.Nilpotent
Goal:
P : ⦃S : Type u_1⦄ → [inst : CommRing S] → Ideal S → Prop
h₁ : ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I : Ideal S), HPow.hPow (β := ℕ) I 2 = ⊥ → P I
h₂ :
  ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I J : Ideal S),
    I ≤ J → P I → P (S := S ⧸ I) (map (F := S →+* S ⧸ I) (Quotient.mk I) J) → P J
S : Type u_1
inst : CommRing S
I : Ideal S
hI' : ¬I = ⊥
n : ℕ
H : ∀ m < n + 1 + 1, ∀ {S : Type u_1} [inst : CommRing S] (I : Ideal S) (hI : I ^ m = ⊥), P I
hI : I ^ (n + 1 + 1) = ⊥
⊢ n + 1 + 1 ≤ 2 * (n + 1)

Tactic:
lia
-- Module: Mathlib.RingTheory.Ideal.Quotient.Nilpotent
Goal:
P : ⦃S : Type u_1⦄ → [inst : CommRing S] → Ideal S → Prop
h₁ : ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I : Ideal S), HPow.hPow (β := ℕ) I 2 = ⊥ → P I
h₂ :
  ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I J : Ideal S),
    I ≤ J → P I → P (S := S ⧸ I) (map (F := S →+* S ⧸ I) (Quotient.mk I) J) → P J
S : Type u_1
inst : CommRing S
I : Ideal S
hI' : ¬I = ⊥
n : ℕ
H : ∀ m < n + 1 + 1, ∀ {S : Type u_1} [inst : CommRing S] (I : Ideal S) (hI : I ^ m = ⊥), P I
hI : I ^ (n + 1 + 1) = ⊥
⊢ I ^ (2 * (n + 1)) ≤ I ^ (n + 1 + 1)

Tactic:
apply Ideal.pow_le_pow_right (by lia)

Premises used: Ideal.pow_le_pow_right
-- Module: Mathlib.RingTheory.Ideal.Quotient.Nilpotent
Goal:
P : ⦃S : Type u_1⦄ → [inst : CommRing S] → Ideal S → Prop
h₁ : ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I : Ideal S), HPow.hPow (β := ℕ) I 2 = ⊥ → P I
h₂ :
  ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I J : Ideal S),
    I ≤ J → P I → P (S := S ⧸ I) (map (F := S →+* S ⧸ I) (Quotient.mk I) J) → P J
S : Type u_1
inst : CommRing S
I : Ideal S
hI' : ¬I = ⊥
n : ℕ
H : ∀ m < n + 1 + 1, ∀ {S : Type u_1} [inst : CommRing S] (I : Ideal S) (hI : I ^ m = ⊥), P I
hI : I ^ (n + 1 + 1) = ⊥
⊢ n.succ < n + 1 + 1

Tactic:
n.succ.lt_succ_self

Premises used: Nat.lt_succ_self, Nat.succ
-- Module: Mathlib.RingTheory.Ideal.Quotient.Nilpotent
Goal:
P : ⦃S : Type u_1⦄ → [inst : CommRing S] → Ideal S → Prop
h₁ : ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I : Ideal S), HPow.hPow (β := ℕ) I 2 = ⊥ → P I
h₂ :
  ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I J : Ideal S),
    I ≤ J → P I → P (S := S ⧸ I) (map (F := S →+* S ⧸ I) (Quotient.mk I) J) → P J
S : Type u_1
inst : CommRing S
I : Ideal S
hI' : ¬I = ⊥
n : ℕ
H : ∀ m < n + 1 + 1, ∀ {S : Type u_1} [inst : CommRing S] (I : Ideal S) (hI : I ^ m = ⊥), P I
hI : I ^ (n + 1 + 1) = ⊥
⊢ n.succ < n + 1 + 1

Tactic:
exact n.succ.lt_succ_self

Premises used: Nat.lt_succ_self, Nat.succ
-- Module: Mathlib.RingTheory.Ideal.Quotient.Nilpotent
Goal:
P : ⦃S : Type u_1⦄ → [inst : CommRing S] → Ideal S → Prop
h₁ : ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I : Ideal S), HPow.hPow (β := ℕ) I 2 = ⊥ → P I
h₂ :
  ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I J : Ideal S),
    I ≤ J → P I → P (S := S ⧸ I) (map (F := S →+* S ⧸ I) (Quotient.mk I) J) → P J
S : Type u_1
inst : CommRing S
I : Ideal S
hI' : ¬I = ⊥
n : ℕ
H : ∀ m < n + 1 + 1, ∀ {S : Type u_1} [inst : CommRing S] (I : Ideal S) (hI : I ^ m = ⊥), P I
hI : I ^ (n + 1 + 1) = ⊥
neg :
  HPow.hPow (α := Ideal (S ⧸ HPow.hPow (β := ℕ) I 2)) (β := ℕ)
      (map (F := S →+* S ⧸ HPow.hPow (β := ℕ) I 2) (Quotient.mk (HPow.hPow (β := ℕ) I 2)) I) 2 =
    ⊥
⊢ P (S := S ⧸ HPow.hPow (β := ℕ) I 2)
    (map (F := S →+* S ⧸ HPow.hPow (β := ℕ) I 2) (Quotient.mk (HPow.hPow (β := ℕ) I 2)) I)

Tactic:
apply h₁
-- Module: Mathlib.RingTheory.Ideal.Quotient.Nilpotent
Goal:
P : ⦃S : Type u_1⦄ → [inst : CommRing S] → Ideal S → Prop
h₁ : ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I : Ideal S), HPow.hPow (β := ℕ) I 2 = ⊥ → P I
h₂ :
  ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I J : Ideal S),
    I ≤ J → P I → P (S := S ⧸ I) (map (F := S →+* S ⧸ I) (Quotient.mk I) J) → P J
S : Type u_1
inst : CommRing S
I : Ideal S
hI' : ¬I = ⊥
n : ℕ
H : ∀ m < n + 1 + 1, ∀ {S : Type u_1} [inst : CommRing S] (I : Ideal S) (hI : I ^ m = ⊥), P I
hI : I ^ (n + 1 + 1) = ⊥
neg :
  Eq (α := Ideal (S ⧸ HPow.hPow (β := ℕ) I 2))
    (map (F := S →+* S ⧸ HPow.hPow (β := ℕ) I 2) (Quotient.mk (HPow.hPow (β := ℕ) I 2)) (HPow.hPow (β := ℕ) I 2)) ⊥
neg₁ neg₂ : Eq.{u_1 + 1} (α := Ideal (S ⧸ HPow.hPow (β := ℕ) I 2)) ⊥ ⊥
⊢ HPow.hPow (α := Ideal (S ⧸ HPow.hPow (β := ℕ) I 2)) (β := ℕ)
      (map (F := S →+* S ⧸ HPow.hPow (β := ℕ) I 2) (Quotient.mk (HPow.hPow (β := ℕ) I 2)) I) 2 =
    ⊥

Tactic:
rw [← Ideal.map_pow, Ideal.map_quotient_self]

Premises used: Ideal.map_quotient_self, Ideal.map_pow
-- Module: Mathlib.RingTheory.Ideal.Quotient.Nilpotent
Goal:
R : Type u_2
inst : CommRing R
I : Ideal R
hI : IsNilpotent I
x : R
⊢ IsUnit (M := R ⧸ I) ((Ideal.Quotient.mk I : (a : R) → R ⧸ I) x : R ⧸ I) ↔ IsUnit x

Tactic:
⟨?_, fun h => h.map <| Ideal.Quotient.mk I⟩

Premises used: Ideal.Quotient.mk, Iff.intro, IsUnit.map
-- Module: Mathlib.RingTheory.Ideal.Quotient.Nilpotent
Goal:
R : Type u_2
inst : CommRing R
I : Ideal R
hI : IsNilpotent I
x : R
refine : IsUnit (M := R ⧸ I) ((Ideal.Quotient.mk I : (a : R) → R ⧸ I) x : R ⧸ I) → IsUnit x
⊢ IsUnit (M := R ⧸ I) ((Ideal.Quotient.mk I : (a : R) → R ⧸ I) x : R ⧸ I) ↔ IsUnit x

Tactic:
refine ⟨?_, fun h => h.map <| Ideal.Quotient.mk I⟩
-- Module: Mathlib.RingTheory.Ideal.Quotient.Nilpotent
Goal:
R : Type u_2
inst : CommRing R
I : Ideal R
hI : IsNilpotent I
x : R
revert : ∀ {x : R}, IsUnit (M := R ⧸ I) ((Ideal.Quotient.mk I : (a : R) → R ⧸ I) x : R ⧸ I) → IsUnit x
⊢ IsUnit (M := R ⧸ I) ((Ideal.Quotient.mk I : (a : R) → R ⧸ I) x : R ⧸ I) → IsUnit x

Tactic:
revert x
-- Module: Mathlib.RingTheory.Ideal.Quotient.Nilpotent
Goal:
R : Type u_2
inst : CommRing R
I : Ideal R
hI : IsNilpotent I
h₁ :
  ∀ ⦃S : Type u_2⦄ [inst : CommRing S] (I : Ideal S),
    HPow.hPow (β := ℕ) I 2 = ⊥ →
      ∀ {x : S}, IsUnit (M := S ⧸ I) ((Ideal.Quotient.mk I : (a : S) → S ⧸ I) x : S ⧸ I) → IsUnit x
h₂ :
  ∀ ⦃S : Type u_2⦄ [inst : CommRing S] (I J : Ideal S),
    I ≤ J →
      (∀ {x : S}, IsUnit (M := S ⧸ I) ((Ideal.Quotient.mk I : (a : S) → S ⧸ I) x : S ⧸ I) → IsUnit x) →
        (∀ {x : S ⧸ I},
            IsUnit (M :=
                HasQuotient.Quotient (S ⧸ I) (B := Ideal (S ⧸ I))
                  (Ideal.map (F := S →+* S ⧸ I) (Ideal.Quotient.mk I) J))
                ((Ideal.Quotient.mk (Ideal.map (F := S →+* S ⧸ I) (Ideal.Quotient.mk I) J) :
                      (a : S ⧸ I) → (S ⧸ I) ⧸ Ideal.map (Ideal.Quotient.mk I) J)
                    x :
                  HasQuotient.Quotient (S ⧸ I) (B := Ideal (S ⧸ I))
                    (Ideal.map (F := S →+* S ⧸ I) (Ideal.Quotient.mk I) J)) →
              IsUnit x) →
          ∀ {x : S}, IsUnit (M := S ⧸ J) ((Ideal.Quotient.mk J : (a : S) → S ⧸ J) x : S ⧸ J) → IsUnit x
⊢ ∀ {x : R}, IsUnit (M := R ⧸ I) ((Ideal.Quotient.mk I : (a : R) → R ⧸ I) x : R ⧸ I) → IsUnit x

Tactic:
apply Ideal.IsNilpotent.induction_on (S := R) I hI

Premises used: Ideal.IsNilpotent.induction_on
-- Module: Mathlib.RingTheory.Ideal.Quotient.Nilpotent
Goal:
R : Type u_2
inst : CommRing R
S : Type u_2
inst_1 : CommRing S
I J : Ideal S
e : I ≤ J
h₁ : ∀ {x : S}, IsUnit (M := S ⧸ I) ((Ideal.Quotient.mk I : (a : S) → S ⧸ I) x : S ⧸ I) → IsUnit x
h₂ :
  ∀ {x : S ⧸ I},
    IsUnit (M :=
        HasQuotient.Quotient (S ⧸ I) (B := Ideal (S ⧸ I)) (Ideal.map (F := S →+* S ⧸ I) (Ideal.Quotient.mk I) J))
        ((Ideal.Quotient.mk (Ideal.map (F := S →+* S ⧸ I) (Ideal.Quotient.mk I) J) :
              (a : S ⧸ I) → (S ⧸ I) ⧸ Ideal.map (Ideal.Quotient.mk I) J)
            x :
          HasQuotient.Quotient (S ⧸ I) (B := Ideal (S ⧸ I)) (Ideal.map (F := S →+* S ⧸ I) (Ideal.Quotient.mk I) J)) →
      IsUnit x
x : S
h₃ : IsUnit (M := S ⧸ J) ((Ideal.Quotient.mk J : (a : S) → S ⧸ J) x : S ⧸ J)
h₄ : IsUnit (M := S ⧸ I) ((Ideal.Quotient.mk I : (a : S) → S ⧸ I) x : S ⧸ I)
⊢ IsUnit x

Tactic:
apply h₁
-- Module: Mathlib.RingTheory.Ideal.Quotient.Nilpotent
Goal:
R : Type u_2
inst : CommRing R
S : Type u_2
inst_1 : CommRing S
I J : Ideal S
e : I ≤ J
h₁ : ∀ {x : S}, IsUnit (M := S ⧸ I) ((Ideal.Quotient.mk I : (a : S) → S ⧸ I) x : S ⧸ I) → IsUnit x
h₂ :
  ∀ {x : S ⧸ I},
    IsUnit (M :=
        HasQuotient.Quotient (S ⧸ I) (B := Ideal (S ⧸ I)) (Ideal.map (F := S →+* S ⧸ I) (Ideal.Quotient.mk I) J))
        ((Ideal.Quotient.mk (Ideal.map (F := S →+* S ⧸ I) (Ideal.Quotient.mk I) J) :
              (a : S ⧸ I) → (S ⧸ I) ⧸ Ideal.map (Ideal.Quotient.mk I) J)
            x :
          HasQuotient.Quotient (S ⧸ I) (B := Ideal (S ⧸ I)) (Ideal.map (F := S →+* S ⧸ I) (Ideal.Quotient.mk I) J)) →
      IsUnit x
x : S
h₃ : IsUnit (M := S ⧸ J) ((Ideal.Quotient.mk J : (a : S) → S ⧸ J) x : S ⧸ J)
h₄ :
  IsUnit (M := HasQuotient.Quotient (S ⧸ I) (B := Ideal (S ⧸ I)) (Ideal.map (F := S →+* S ⧸ I) (Ideal.Quotient.mk I) J))
    ((Ideal.Quotient.mk (Ideal.map (F := S →+* S ⧸ I) (Ideal.Quotient.mk I) J) :
          (a : S ⧸ I) → (S ⧸ I) ⧸ Ideal.map (Ideal.Quotient.mk I) J)
        ((Ideal.Quotient.mk I : (a : S) → S ⧸ I) x) :
      HasQuotient.Quotient (S ⧸ I) (B := Ideal (S ⧸ I)) (Ideal.map (F := S →+* S ⧸ I) (Ideal.Quotient.mk I) J))
⊢ IsUnit (M := S ⧸ I) ((Ideal.Quotient.mk I : (a : S) → S ⧸ I) x : S ⧸ I)

Tactic:
apply h₂
-- Module: Mathlib.RingTheory.Ideal.Quotient.Nilpotent
Goal:
R : Type u_2
inst : CommRing R
S : Type u_2
inst_1 : CommRing S
I J : Ideal S
e : I ≤ J
h₁ : ∀ {x : S}, IsUnit (M := S ⧸ I) ((Ideal.Quotient.mk I : (a : S) → S ⧸ I) x : S ⧸ I) → IsUnit x
h₂ :
  ∀ {x : S ⧸ I},
    IsUnit (M :=
        HasQuotient.Quotient (S ⧸ I) (B := Ideal (S ⧸ I)) (Ideal.map (F := S →+* S ⧸ I) (Ideal.Quotient.mk I) J))
        ((Ideal.Quotient.mk (Ideal.map (F := S →+* S ⧸ I) (Ideal.Quotient.mk I) J) :
              (a : S ⧸ I) → (S ⧸ I) ⧸ Ideal.map (Ideal.Quotient.mk I) J)
            x :
          HasQuotient.Quotient (S ⧸ I) (B := Ideal (S ⧸ I)) (Ideal.map (F := S →+* S ⧸ I) (Ideal.Quotient.mk I) J)) →
      IsUnit x
x : S
h₃ : IsUnit (M := S ⧸ J) ((Ideal.Quotient.mk J : (a : S) → S ⧸ J) x : S ⧸ J)
⊢ IsUnit (M := HasQuotient.Quotient (S ⧸ I) (B := Ideal (S ⧸ I)) (Ideal.map (F := S →+* S ⧸ I) (Ideal.Quotient.mk I) J))
    ((Ideal.Quotient.mk (Ideal.map (F := S →+* S ⧸ I) (Ideal.Quotient.mk I) J) :
          (a : S ⧸ I) → (S ⧸ I) ⧸ Ideal.map (Ideal.Quotient.mk I) J)
        ((Ideal.Quotient.mk I : (a : S) → S ⧸ I) x) :
      HasQuotient.Quotient (S ⧸ I) (B := Ideal (S ⧸ I)) (Ideal.map (F := S →+* S ⧸ I) (Ideal.Quotient.mk I) J))

Tactic:
h₃.map
        ((DoubleQuot.quotQuotEquivQuotSup I J).trans
              (Ideal.quotEquivOfEq (sup_eq_right.mpr e))).symm.toRingHom

Premises used: Iff.mpr, Ideal.quotEquivOfEq, Quotient.liftOn', AddCon.liftOn, Submodule.Quotient.mk, sup_eq_right, RingEquiv.toRingHom, IsUnit.map, RingEquiv.trans, DoubleQuot.quotQuotEquivQuotSup, RingEquiv.symm
-- Module: Mathlib.RingTheory.Ideal.Quotient.Nilpotent
Goal:
R : Type u_2
inst : CommRing R
S : Type u_2
inst_1 : CommRing S
I J : Ideal S
e : I ≤ J
h₁ : ∀ {x : S}, IsUnit (M := S ⧸ I) ((Ideal.Quotient.mk I : (a : S) → S ⧸ I) x : S ⧸ I) → IsUnit x
h₂ :
  ∀ {x : S ⧸ I},
    IsUnit (M :=
        HasQuotient.Quotient (S ⧸ I) (B := Ideal (S ⧸ I)) (Ideal.map (F := S →+* S ⧸ I) (Ideal.Quotient.mk I) J))
        ((Ideal.Quotient.mk (Ideal.map (F := S →+* S ⧸ I) (Ideal.Quotient.mk I) J) :
              (a : S ⧸ I) → (S ⧸ I) ⧸ Ideal.map (Ideal.Quotient.mk I) J)
            x :
          HasQuotient.Quotient (S ⧸ I) (B := Ideal (S ⧸ I)) (Ideal.map (F := S →+* S ⧸ I) (Ideal.Quotient.mk I) J)) →
      IsUnit x
x : S
h₃ : IsUnit (M := S ⧸ J) ((Ideal.Quotient.mk J : (a : S) → S ⧸ J) x : S ⧸ J)
⊢ IsUnit (M := HasQuotient.Quotient (S ⧸ I) (B := Ideal (S ⧸ I)) (Ideal.map (F := S →+* S ⧸ I) (Ideal.Quotient.mk I) J))
    ((Ideal.Quotient.mk (Ideal.map (F := S →+* S ⧸ I) (Ideal.Quotient.mk I) J) :
          (a : S ⧸ I) → (S ⧸ I) ⧸ Ideal.map (Ideal.Quotient.mk I) J)
        ((Ideal.Quotient.mk I : (a : S) → S ⧸ I) x) :
      HasQuotient.Quotient (S ⧸ I) (B := Ideal (S ⧸ I)) (Ideal.map (F := S →+* S ⧸ I) (Ideal.Quotient.mk I) J))

Tactic:
exact
      h₃.map
        ((DoubleQuot.quotQuotEquivQuotSup I J).trans
              (Ideal.quotEquivOfEq (sup_eq_right.mpr e))).symm.toRingHom

Premises used: Iff.mpr, Ideal.quotEquivOfEq, sup_eq_right, RingEquiv.toRingHom, IsUnit.map, RingEquiv.trans, DoubleQuot.quotQuotEquivQuotSup, RingEquiv.symm
-- Module: Mathlib.RingTheory.Ideal.Quotient.Nilpotent
Goal:
R : Type u_2
inst : CommRing R
S : Type u_2
inst_1 : CommRing S
I : Ideal S
e : HPow.hPow (β := ℕ) I 2 = ⊥
x : S
H : IsUnit (M := S ⧸ I) ((Ideal.Quotient.mk I : (a : S) → S ⧸ I) x : S ⧸ I)
h₁ : ∀ (y : S) (hy : (Ideal.Quotient.mk I : (a : S) → S ⧸ I) y = (↑H.unit⁻¹ : S ⧸ I)), IsUnit x
⊢ IsUnit x

Tactic:
obtain ⟨y, hy⟩ := Ideal.Quotient.mk_surjective (↑H.unit⁻¹ : S ⧸ I)

Premises used: Units.val, Ideal.Quotient.mk_surjective, Inv.inv, IsUnit.unit
-- Module: Mathlib.RingTheory.Ideal.Quotient.Nilpotent
Goal:
R : Type u_2
inst : CommRing R
S : Type u_2
inst_1 : CommRing S
I : Ideal S
e : HPow.hPow (β := ℕ) I 2 = ⊥
x : S
H : IsUnit (M := S ⧸ I) ((Ideal.Quotient.mk I : (a : S) → S ⧸ I) x : S ⧸ I)
y : S
hy : (Ideal.Quotient.mk I : (a : S) → S ⧸ I) y = (↑H.unit⁻¹ : S ⧸ I)
rw : Eq (α := S ⧸ I) ((Ideal.Quotient.mk I : (a : S) → S ⧸ I) (x * y) : S ⧸ I) (1 : S ⧸ I)
rw₁ :
  HMul.hMul (α := S ⧸ I) (β := S ⧸ I) ((Ideal.Quotient.mk I : (a : S) → S ⧸ I) x : S ⧸ I)
      ((Ideal.Quotient.mk I : (a : S) → S ⧸ I) y : S ⧸ I) =
    1
rw₂ : HMul.hMul (α := S ⧸ I) ((Ideal.Quotient.mk I : (a : S) → S ⧸ I) x : S ⧸ I) (↑H.unit⁻¹ : S ⧸ I) = 1
rw₃ rw₄ : Eq.{u_2 + 1} (α := S ⧸ I) 1 1
⊢ Eq (α := S ⧸ I) ((Ideal.Quotient.mk I : (a : S) → S ⧸ I) (x * y) : S ⧸ I)
    ((Ideal.Quotient.mk I : (a : S) → S ⧸ I) 1 : S ⧸ I)

Tactic:
rw [map_one, map_mul, hy, IsUnit.mul_val_inv]

Premises used: IsUnit.mul_val_inv, map_mul, map_one
-- Module: Mathlib.RingTheory.Ideal.Quotient.Nilpotent
Goal:
R : Type u_2
inst : CommRing R
S : Type u_2
inst_1 : CommRing S
I : Ideal S
e : HPow.hPow (β := ℕ) I 2 = ⊥
x : S
H : IsUnit (M := S ⧸ I) ((Ideal.Quotient.mk I : (a : S) → S ⧸ I) x : S ⧸ I)
y : S
hy : (Ideal.Quotient.mk I : (a : S) → S ⧸ I) y = (↑H.unit⁻¹ : S ⧸ I)
⊢ Eq (α := S ⧸ I) ((Ideal.Quotient.mk I : (a : S) → S ⧸ I) (x * y) : S ⧸ I)
    ((Ideal.Quotient.mk I : (a : S) → S ⧸ I) 1 : S ⧸ I)

Tactic:
have : Ideal.Quotient.mk I (x * y) = Ideal.Quotient.mk I 1 := by
      rw [map_one, map_mul, hy, IsUnit.mul_val_inv]
-- Module: Mathlib.RingTheory.Ideal.Quotient.Nilpotent
Goal:
R : Type u_2
inst : CommRing R
S : Type u_2
inst_1 : CommRing S
I : Ideal S
e : HPow.hPow (β := ℕ) I 2 = ⊥
x : S
H : IsUnit (M := S ⧸ I) ((Ideal.Quotient.mk I : (a : S) → S ⧸ I) x : S ⧸ I)
y : S
hy : (Ideal.Quotient.mk I : (a : S) → S ⧸ I) y = (↑H.unit⁻¹ : S ⧸ I)
this :
  Eq (α := S ⧸ I) ((Ideal.Quotient.mk I : (a : S) → S ⧸ I) (x * y) : S ⧸ I)
    ((Ideal.Quotient.mk I : (a : S) → S ⧸ I) 1 : S ⧸ I)
⊢ x * y - 1 ∈ I

Tactic:
rw [Ideal.Quotient.eq] at this

Premises used: Ideal.Quotient.eq
-- Module: Mathlib.RingTheory.Ideal.Quotient.Nilpotent
Goal:
R : Type u_2
inst : CommRing R
S : Type u_2
inst_1 : CommRing S
I : Ideal S
e : HPow.hPow (β := ℕ) I 2 = ⊥
x : S
H : IsUnit (M := S ⧸ I) ((Ideal.Quotient.mk I : (a : S) → S ⧸ I) x : S ⧸ I)
y : S
hy : (Ideal.Quotient.mk I : (a : S) → S ⧸ I) y = (↑H.unit⁻¹ : S ⧸ I)
this : x * y - 1 ∈ I
rw : HPow.hPow (β := ℕ) (x * y - 1) 2 ∈ HPow.hPow (β := ℕ) I 2
⊢ HPow.hPow (β := ℕ) (x * y - 1) 2 = 0

Tactic:
rw [← Ideal.mem_bot, ← e]

Premises used: Ideal.mem_bot
-- Module: Mathlib.RingTheory.Ideal.Quotient.Nilpotent
Goal:
R : Type u_2
inst : CommRing R
S : Type u_2
inst_1 : CommRing S
I : Ideal S
e : HPow.hPow (β := ℕ) I 2 = ⊥
x : S
H : IsUnit (M := S ⧸ I) ((Ideal.Quotient.mk I : (a : S) → S ⧸ I) x : S ⧸ I)
y : S
hy : (Ideal.Quotient.mk I : (a : S) → S ⧸ I) y = (↑H.unit⁻¹ : S ⧸ I)
this : x * y - 1 ∈ I
⊢ HPow.hPow (β := ℕ) (x * y - 1) 2 ∈ HPow.hPow (β := ℕ) I 2

Tactic:
Ideal.pow_mem_pow this _

Premises used: Set.Mem, Ideal.pow_mem_pow
-- Module: Mathlib.RingTheory.Ideal.Quotient.Nilpotent
Goal:
R : Type u_2
inst : CommRing R
S : Type u_2
inst_1 : CommRing S
I : Ideal S
e : HPow.hPow (β := ℕ) I 2 = ⊥
x : S
H : IsUnit (M := S ⧸ I) ((Ideal.Quotient.mk I : (a : S) → S ⧸ I) x : S ⧸ I)
y : S
hy : (Ideal.Quotient.mk I : (a : S) → S ⧸ I) y = (↑H.unit⁻¹ : S ⧸ I)
this : x * y - 1 ∈ I
⊢ HPow.hPow (β := ℕ) (x * y - 1) 2 ∈ HPow.hPow (β := ℕ) I 2

Tactic:
exact Ideal.pow_mem_pow this _

Premises used: Ideal.pow_mem_pow
-- Module: Mathlib.RingTheory.Ideal.Quotient.Nilpotent
Goal:
R : Type u_2
inst : CommRing R
S : Type u_2
inst_1 : CommRing S
I : Ideal S
e : HPow.hPow (β := ℕ) I 2 = ⊥
x : S
H : IsUnit (M := S ⧸ I) ((Ideal.Quotient.mk I : (a : S) → S ⧸ I) x : S ⧸ I)
y : S
hy : (Ideal.Quotient.mk I : (a : S) → S ⧸ I) y = (↑H.unit⁻¹ : S ⧸ I)
this : x * y - 1 ∈ I
⊢ HPow.hPow (β := ℕ) (x * y - 1) 2 = 0

Tactic:
have : (x * y - 1) ^ 2 = 0 := by
      rw [← Ideal.mem_bot, ← e]
      exact Ideal.pow_mem_pow this _
-- Module: Mathlib.RingTheory.Ideal.Quotient.Nilpotent
Goal:
R : Type u_2
inst : CommRing R
S : Type u_2
inst_1 : CommRing S
I : Ideal S
e : HPow.hPow (β := ℕ) I 2 = ⊥
x : S
H : IsUnit (M := S ⧸ I) ((Ideal.Quotient.mk I : (a : S) → S ⧸ I) x : S ⧸ I)
y : S
hy : (Ideal.Quotient.mk I : (a : S) → S ⧸ I) y = (↑H.unit⁻¹ : S ⧸ I)
this_1 : x * y - 1 ∈ I
this : HPow.hPow (β := ℕ) (x * y - 1) 2 = 0
rw : 1 - x * (y * (2 - x * y)) = HPow.hPow (β := ℕ) (x * y - 1) 2
⊢ x * (y * (2 - x * y)) = 1

Tactic:
rw [eq_comm, ← sub_eq_zero, ← this]

Premises used: sub_eq_zero, eq_comm
-- Module: Mathlib.RingTheory.Ideal.Quotient.Nilpotent
Goal:
R : Type u_2
inst : CommRing R
S : Type u_2
inst_1 : CommRing S
I : Ideal S
e : HPow.hPow (β := ℕ) I 2 = ⊥
x : S
H : IsUnit (M := S ⧸ I) ((Ideal.Quotient.mk I : (a : S) → S ⧸ I) x : S ⧸ I)
y : S
hy : (Ideal.Quotient.mk I : (a : S) → S ⧸ I) y = (↑H.unit⁻¹ : S ⧸ I)
this_1 : x * y - 1 ∈ I
this : HPow.hPow (β := ℕ) (x * y - 1) 2 = 0
⊢ 1 - x * (y * (2 - x * y)) = HPow.hPow (β := ℕ) (x * y - 1) 2

Tactic:
ring
-- Module: Mathlib.RingTheory.Ideal.Quotient.Nilpotent
Goal:
R : Type u_2
inst : CommRing R
S : Type u_2
inst_1 : CommRing S
I : Ideal S
e : HPow.hPow (β := ℕ) I 2 = ⊥
x : S
H : IsUnit (M := S ⧸ I) ((Ideal.Quotient.mk I : (a : S) → S ⧸ I) x : S ⧸ I)
y : S
hy : (Ideal.Quotient.mk I : (a : S) → S ⧸ I) y = (↑H.unit⁻¹ : S ⧸ I)
this_1 : x * y - 1 ∈ I
this : HPow.hPow (β := ℕ) (x * y - 1) 2 = 0
⊢ x * (y * (2 - x * y)) = 1

Tactic:
have : x * (y * (2 - x * y)) = 1 := by
      rw [eq_comm, ← sub_eq_zero, ← this]
      ring
-- Module: Mathlib.RingTheory.Ideal.Quotient.Nilpotent
Goal:
R : Type u_2
inst : CommRing R
S : Type u_2
inst_1 : CommRing S
I : Ideal S
e : HPow.hPow (β := ℕ) I 2 = ⊥
x : S
H : IsUnit (M := S ⧸ I) ((Ideal.Quotient.mk I : (a : S) → S ⧸ I) x : S ⧸ I)
y : S
hy : (Ideal.Quotient.mk I : (a : S) → S ⧸ I) y = (↑H.unit⁻¹ : S ⧸ I)
this_1 : x * y - 1 ∈ I
this_2 : HPow.hPow (β := ℕ) (x * y - 1) 2 = 0
this : x * (y * (2 - x * y)) = 1
⊢ IsUnit x

Tactic:
.of_mul_eq_one _ this

Premises used: IsUnit.of_mul_eq_one
-- Module: Mathlib.RingTheory.Ideal.Quotient.Nilpotent
Goal:
R : Type u_2
inst : CommRing R
S : Type u_2
inst_1 : CommRing S
I : Ideal S
e : HPow.hPow (β := ℕ) I 2 = ⊥
x : S
H : IsUnit (M := S ⧸ I) ((Ideal.Quotient.mk I : (a : S) → S ⧸ I) x : S ⧸ I)
y : S
hy : (Ideal.Quotient.mk I : (a : S) → S ⧸ I) y = (↑H.unit⁻¹ : S ⧸ I)
this_1 : x * y - 1 ∈ I
this_2 : HPow.hPow (β := ℕ) (x * y - 1) 2 = 0
this : x * (y * (2 - x * y)) = 1
⊢ IsUnit x

Tactic:
exact .of_mul_eq_one _ this
-- Module: Mathlib.RingTheory.Ideal.Quotient.Nilpotent
Goal:
S : Type u_1
inst : CommRing S
I : Ideal S
x : S
n : ℕ
hn : n ≠ 0
rw :
  IsUnit (M := HasQuotient.Quotient (S ⧸ I ^ n) (B := Ideal (S ⧸ I ^ n)) (map (F := S →+* S ⧸ I ^ n) (mk (I ^ n)) I))
      ((mk (map (F := S →+* S ⧸ I ^ n) (mk (I ^ n)) I) : (a : S ⧸ I ^ n) → (S ⧸ I ^ n) ⧸ map (mk (I ^ n)) I)
          ((mk (I ^ n) : (a : S) → S ⧸ I ^ n) x) :
        HasQuotient.Quotient (S ⧸ I ^ n) (B := Ideal (S ⧸ I ^ n)) (map (F := S →+* S ⧸ I ^ n) (mk (I ^ n)) I)) ↔
    IsUnit (M := S ⧸ I) ((mk I : (a : S) → S ⧸ I) x : S ⧸ I)
hI : IsNilpotent (R := Ideal (S ⧸ I ^ n)) (map (F := S →+* S ⧸ I ^ n) (mk (I ^ n)) I)
⊢ IsUnit (M := S ⧸ I ^ n) ((mk (I ^ n) : (a : S) → S ⧸ I ^ n) x : S ⧸ I ^ n) ↔
    IsUnit (M := S ⧸ I) ((mk I : (a : S) → S ⧸ I) x : S ⧸ I)

Tactic:
rw [← IsNilpotent.isUnit_quotient_mk_iff (I := Ideal.map (Ideal.Quotient.mk (I ^ n)) I)]

Premises used: IsNilpotent.isUnit_quotient_mk_iff
-- Module: Mathlib.RingTheory.Ideal.Quotient.Nilpotent
Goal:
S : Type u_1
inst : CommRing S
I : Ideal S
x : S
n : ℕ
hn : n ≠ 0
rw :
  IsUnit (M := S ⧸ I)
      ((DoubleQuot.quotQuotEquivQuotOfLE ⋯ : (a : (S ⧸ I ^ n) ⧸ map (mk (I ^ n)) I) → S ⧸ I)
          ((mk (map (F := S →+* S ⧸ I ^ n) (mk (I ^ n)) I) : (a : S ⧸ I ^ n) → (S ⧸ I ^ n) ⧸ map (mk (I ^ n)) I)
            ((mk (I ^ n) : (a : S) → S ⧸ I ^ n) x)) :
        S ⧸ I) ↔
    IsUnit (M := S ⧸ I) ((mk I : (a : S) → S ⧸ I) x : S ⧸ I)
⊢ IsUnit (M := HasQuotient.Quotient (S ⧸ I ^ n) (B := Ideal (S ⧸ I ^ n)) (map (F := S →+* S ⧸ I ^ n) (mk (I ^ n)) I))
      ((mk (map (F := S →+* S ⧸ I ^ n) (mk (I ^ n)) I) : (a : S ⧸ I ^ n) → (S ⧸ I ^ n) ⧸ map (mk (I ^ n)) I)
          ((mk (I ^ n) : (a : S) → S ⧸ I ^ n) x) :
        HasQuotient.Quotient (S ⧸ I ^ n) (B := Ideal (S ⧸ I ^ n)) (map (F := S →+* S ⧸ I ^ n) (mk (I ^ n)) I)) ↔
    IsUnit (M := S ⧸ I) ((mk I : (a : S) → S ⧸ I) x : S ⧸ I)

Tactic:
rw [← isUnit_map_iff (DoubleQuot.quotQuotEquivQuotOfLE (Ideal.pow_le_self hn))]

Premises used: DoubleQuot.quotQuotEquivQuotOfLE, Ideal.pow_le_self, isUnit_map_iff