awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Nilpotent (goal)
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, ∀ {S : Type u_1} [inst : CommRing S] (I : Ideal S) (hI : I ^ m = ⊥), P I
hI : I ^ (n + 1) = ⊥
neg :
  ∀ (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 ^ HAdd.hAdd (α := ℕ) 0 1 = ⊥), P I
⊢ P I
have neg₁ : ∀ (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) = ⊥), P I
Tactic: rcases n with - | n
Cleared hypotheses: n, H, hI
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Nilpotent (hypothesis)
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 ^ HAdd.hAdd (α := ℕ) 0 1 = ⊥
⊢ P I
have hI₁ : I = ⊥
Tactic: rw [pow_one] at hI
Cleared hypotheses: hI
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Nilpotent (goal)
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) = ⊥
⊢ P I
have neg : P (HPow.hPow (β := ℕ) I 2)
Tactic: apply h₂ (I ^ 2) _ (Ideal.pow_le_self two_ne_zero)
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Nilpotent (goal)
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)
⊢ P I
have 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)
Tactic: apply h₂ (I ^ 2) _ (Ideal.pow_le_self two_ne_zero)
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Nilpotent (goal)
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) = ⊥
⊢ P (HPow.hPow (β := ℕ) I 2)
have neg : HPow.hPow (β := ℕ) I 2 ^ n.succ = ⊥
Tactic: apply H n.succ _ (I ^ 2)
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Nilpotent (goal)
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 = ⊥
⊢ P (HPow.hPow (β := ℕ) I 2)
have apply : n.succ < n + 1 + 1
Tactic: apply H n.succ _ (I ^ 2)
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Nilpotent (goal)
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) = ⊥
⊢ HPow.hPow (β := ℕ) I 2 ^ n.succ = ⊥
have neg : I ^ (2 * (n + 1)) ≤ I ^ (n + 1 + 1)
Tactic: rw [← pow_mul, eq_bot_iff, ← hI, Nat.succ_eq_add_one]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Nilpotent (goal)
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) = ⊥
⊢ 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)
have 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 =
  ⊥
Tactic: apply h₁
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Nilpotent (goal)
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) = ⊥
⊢ 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 =
    ⊥
have 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)) ⊥
Tactic: rw [← Ideal.map_pow, Ideal.map_quotient_self]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Nilpotent (goal)
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)) ⊥
⊢ 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 =
    ⊥
have neg₁ : Eq.{u_1 + 1} (α := Ideal (S ⧸ HPow.hPow (β := ℕ) I 2)) ⊥ ⊥
Tactic: rw [← Ideal.map_pow, Ideal.map_quotient_self]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Nilpotent (goal)
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
have refine : 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⟩
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Nilpotent (goal)
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
have revert : ∀ {x : R}, IsUnit (M := R ⧸ I) ((Ideal.Quotient.mk I : (a : R) → R ⧸ I) x : R ⧸ I) → IsUnit x
Tactic: revert x
Cleared hypotheses: x
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Nilpotent (goal)
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
have 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
Tactic: apply Ideal.IsNilpotent.induction_on (S := R) I hI
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Nilpotent (goal)
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
⊢ ∀ {x : R}, IsUnit (M := R ⧸ I) ((Ideal.Quotient.mk I : (a : R) → R ⧸ I) x : R ⧸ I) → IsUnit x
have 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
Tactic: apply Ideal.IsNilpotent.induction_on (S := R) I hI
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Nilpotent (goal)
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 x
have h₄ : IsUnit (M := S ⧸ I) ((Ideal.Quotient.mk I : (a : S) → S ⧸ I) x : S ⧸ I)
Tactic: apply h₁
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Nilpotent (goal)
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 := S ⧸ I) ((Ideal.Quotient.mk I : (a : S) → S ⧸ I) x : S ⧸ I)
have 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))
Tactic: apply h₂
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Nilpotent (goal)
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)
⊢ IsUnit x
have h₁ : ∀ (y : S) (hy : (Ideal.Quotient.mk I : (a : S) → S ⧸ I) y = (↑H.unit⁻¹ : S ⧸ I)), IsUnit x
Tactic: obtain ⟨y, hy⟩ := Ideal.Quotient.mk_surjective (↑H.unit⁻¹ : S ⧸ I)
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Nilpotent (goal)
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)
have rw : Eq (α := S ⧸ I) ((Ideal.Quotient.mk I : (a : S) → S ⧸ I) (x * y) : S ⧸ I) (1 : S ⧸ I)
Tactic: rw [map_one, map_mul, hy, IsUnit.mul_val_inv]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Nilpotent (goal)
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)
⊢ 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)
have 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
Tactic: rw [map_one, map_mul, hy, IsUnit.mul_val_inv]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Nilpotent (goal)
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
⊢ 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)
have rw₂ : HMul.hMul (α := S ⧸ I) ((Ideal.Quotient.mk I : (a : S) → S ⧸ I) x : S ⧸ I) (↑H.unit⁻¹ : S ⧸ I) = 1
Tactic: rw [map_one, map_mul, hy, IsUnit.mul_val_inv]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Nilpotent (goal)
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
⊢ 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)
have rw₃ : Eq.{u_2 + 1} (α := S ⧸ I) 1 1
Tactic: rw [map_one, map_mul, hy, IsUnit.mul_val_inv]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Nilpotent (hypothesis)
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)
⊢ IsUnit x
have 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)
Tactic: have : Ideal.Quotient.mk I (x * y) = Ideal.Quotient.mk I 1 := by
      rw [map_one, map_mul, hy, IsUnit.mul_val_inv]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Nilpotent (hypothesis)
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)
⊢ IsUnit x
have this₁ : x * y - 1 ∈ I
Tactic: rw [Ideal.Quotient.eq] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Nilpotent (goal)
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
have rw : HPow.hPow (β := ℕ) (x * y - 1) 2 ∈ HPow.hPow (β := ℕ) I 2
Tactic: rw [← Ideal.mem_bot, ← e]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Nilpotent (hypothesis)
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
⊢ IsUnit x
have this₁ : 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 _
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Nilpotent (goal)
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
have rw : 1 - x * (y * (2 - x * y)) = HPow.hPow (β := ℕ) (x * y - 1) 2
Tactic: rw [eq_comm, ← sub_eq_zero, ← this]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Nilpotent (hypothesis)
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
⊢ IsUnit x
have this₁ : x * (y * (2 - x * y)) = 1
Tactic: have : x * (y * (2 - x * y)) = 1 := by
      rw [eq_comm, ← sub_eq_zero, ← this]
      ring
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Nilpotent (goal)
Goal:
S : Type u_1
inst : CommRing S
I : Ideal S
x : S
n : ℕ
hn : n ≠ 0
⊢ 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)
have 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)
Tactic: rw [← IsNilpotent.isUnit_quotient_mk_iff (I := Ideal.map (Ideal.Quotient.mk (I ^ n)) I)]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Nilpotent (goal)
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)
⊢ 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)
have hI : IsNilpotent (R := Ideal (S ⧸ I ^ n)) (map (F := S →+* S ⧸ I ^ n) (mk (I ^ n)) I)
Tactic: rw [← IsNilpotent.isUnit_quotient_mk_iff (I := Ideal.map (Ideal.Quotient.mk (I ^ n)) I)]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Nilpotent (goal)
Goal:
S : Type u_1
inst : CommRing S
I : Ideal S
x : S
n : ℕ
hn : n ≠ 0
⊢ 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)
have rw : IsUnit (M := S ⧸ I)
    ((DoubleQuot.quotQuotEquivQuotOfLE sorry : (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)
Tactic: rw [← isUnit_map_iff (DoubleQuot.quotQuotEquivQuotOfLE (Ideal.pow_le_self hn))]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Nilpotent (goal)
Goal:
S : Type u_1
inst : CommRing S
I : Ideal S
x : S
n : ℕ
hn : n ≠ 0
⊢ IsNilpotent (R := Ideal (S ⧸ I ^ n)) (map (F := S →+* S ⧸ I ^ n) (mk (I ^ n)) I)
have h : HPow.hPow (α := Ideal (S ⧸ I ^ n)) (map (F := S →+* S ⧸ I ^ n) (mk (I ^ n)) I) n = 0
Tactic: use n
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Nilpotent (goal)
Goal:
S : Type u_1
inst : CommRing S
I : Ideal S
inst_1 : I.IsMaximal
n : ℕ
hn : n ≠ 0
x : S
this : Field (S ⧸ I) := Quotient.field I
⊢ IsUnit (M := S ⧸ I ^ n) ((mk (I ^ n) : (a : S) → S ⧸ I ^ n) x : S ⧸ I ^ n) ↔ x ∉ I
have rw : Ne (α := S ⧸ I) ((mk I : (a : S) → S ⧸ I) x : S ⧸ I) (0 : S ⧸ I) ↔ x ∉ I
Tactic: rw [isUnit_mk_pow_iff_isUnit_mk I hn, isUnit_iff_ne_zero]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Nilpotent (goal)
Goal:
S : Type u_1
inst : CommRing S
I : Ideal S
inst_1 : I.IsMaximal
n : ℕ
x : S
hx : x ∉ I
⊢ IsUnit (M := S ⧸ I ^ n) ((mk (I ^ n) : (a : S) → S ⧸ I ^ n) x : S ⧸ I ^ n)
have pos : ∀ (hn : n = 0), IsUnit (M := S ⧸ I ^ n) ((mk (I ^ n) : (a : S) → S ⧸ I ^ n) x : S ⧸ I ^ n)
Tactic: by_cases! hn : n = 0
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Quotient.Nilpotent (goal)
Goal:
S : Type u_1
inst : CommRing S
I : Ideal S
inst_1 : I.IsMaximal
n : ℕ
x : S
hx : x ∉ I
pos : ∀ (hn : n = 0), IsUnit (M := S ⧸ I ^ n) ((mk (I ^ n) : (a : S) → S ⧸ I ^ n) x : S ⧸ I ^ n)
⊢ IsUnit (M := S ⧸ I ^ n) ((mk (I ^ n) : (a : S) → S ⧸ I ^ n) x : S ⧸ I ^ n)
have neg : ∀ (hn : n ≠ 0), IsUnit (M := S ⧸ I ^ n) ((mk (I ^ n) : (a : S) → S ⧸ I ^ n) x : S ⧸ I ^ n)
Tactic: by_cases! hn : n = 0