awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.SimpleRing.Field (goal)
Goal:
A : Type u_1
inst : Ring A
inst_1 : IsSimpleRing A
x : A
hx1_1 : Membership.mem (γ := Subring A) (Subring.center A) x
hx1 : ∀ (g : A), g * x = x * g
hx2 : x ≠ 0
a c : A
⊢ x * (a * c) = a * (x * c)
have rw : x * a * c = a * (x * c)
Tactic: rw [← mul_assoc, ← hx1, mul_assoc]
-- Lean 4 proof step from Mathlib.RingTheory.SimpleRing.Field (goal)
Goal:
A : Type u_1
inst : Ring A
inst_1 : IsSimpleRing A
x : A
hx1_1 : Membership.mem (γ := Subring A) (Subring.center A) x
hx1 : ∀ (g : A), g * x = x * g
hx2 : x ≠ 0
a c : A
rw : x * a * c = a * (x * c)
⊢ x * (a * c) = a * (x * c)
have rw₁ : a * x * c = a * (x * c)
Tactic: rw [← mul_assoc, ← hx1, mul_assoc]
-- Lean 4 proof step from Mathlib.RingTheory.SimpleRing.Field (goal)
Goal:
A : Type u_1
inst : Ring A
inst_1 : IsSimpleRing A
x : A
hx1_1 : Membership.mem (γ := Subring A) (Subring.center A) x
hx1 : ∀ (g : A), g * x = x * g
hx2 : x ≠ 0
a c : A
rw : x * a * c = a * (x * c)
rw₁ : a * x * c = a * (x * c)
⊢ x * (a * c) = a * (x * c)
have rw₂ : a * (x * c) = a * (x * c)
Tactic: rw [← mul_assoc, ← hx1, mul_assoc]
-- Lean 4 proof step from Mathlib.RingTheory.SimpleRing.Field (goal)
Goal:
A : Type u_1
inst : Ring A
inst_1 : IsSimpleRing A
x : A
hx1_1 : Membership.mem (γ := Subring A) (Subring.center A) x
hx1 : ∀ (g : A), g * x = x * g
hx2 : x ≠ 0
b a : A
⊢ x * (a * b) = x * a * b
have rw : x * a * b = x * a * b
Tactic: rw [← mul_assoc, ← hx1, mul_assoc]
-- Lean 4 proof step from Mathlib.RingTheory.SimpleRing.Field (goal)
Goal:
A : Type u_1
inst : Ring A
inst_1 : IsSimpleRing A
x : A
hx1_1 : Membership.mem (γ := Subring A) (Subring.center A) x
hx1 : ∀ (g : A), g * x = x * g
hx2 : x ≠ 0
b a : A
rw : x * a * b = x * a * b
⊢ x * (a * b) = x * a * b
have rw₁ : a * x * b = a * x * b
Tactic: rw [← mul_assoc, ← hx1, mul_assoc]
-- Lean 4 proof step from Mathlib.RingTheory.SimpleRing.Field (goal)
Goal:
A : Type u_1
inst : Ring A
inst_1 : IsSimpleRing A
x : A
hx1_1 : Membership.mem (γ := Subring A) (Subring.center A) x
hx1 : ∀ (g : A), g * x = x * g
hx2 : x ≠ 0
b a : A
rw : x * a * b = x * a * b
rw₁ : a * x * b = a * x * b
⊢ x * (a * b) = x * a * b
have rw₂ : a * (x * b) = a * (x * b)
Tactic: rw [← mul_assoc, ← hx1, mul_assoc]
-- Lean 4 proof step from Mathlib.RingTheory.SimpleRing.Field (hypothesis)
Goal:
A : Type u_1
inst : Ring A
inst_1 : IsSimpleRing A
x : A
hx1_1 : Membership.mem (γ := Subring A) (Subring.center A) x
hx1 : ∀ (g : A), g * x = x * g
hx2 : x ≠ 0
I : TwoSidedIdeal A := mk' (Set.range fun (x_1 : A) ↦ x * x_1) ⋯ ⋯ ⋯ ⋯ ⋯
⊢ ∃ (b : ↥(Subring.center A)),
    Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := Subring A) (Subring.center A) x) x hx1_1 * b = 1
have mem : 1 ∈ I
Tactic: have mem : 1 ∈ I := one_mem_of_ne_zero_mem I hx2 (by simpa [I, mem_mk'] using ⟨1, by simp⟩)
-- Lean 4 proof step from Mathlib.RingTheory.SimpleRing.Field (hypothesis)
Goal:
A : Type u_1
inst : Ring A
inst_1 : IsSimpleRing A
x : A
hx1_1 : Membership.mem (γ := Subring A) (Subring.center A) x
hx1 : ∀ (g : A), g * x = x * g
hx2 : x ≠ 0
I : TwoSidedIdeal A := mk' (Set.range fun (x_1 : A) ↦ x * x_1) ⋯ ⋯ ⋯ ⋯ ⋯
mem : 1 ∈ I
⊢ ∃ (b : ↥(Subring.center A)),
    Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := Subring A) (Subring.center A) x) x hx1_1 * b = 1
have mem₁ : ∃ (y : A), x * y = 1
Tactic: simp only [TwoSidedIdeal.mem_mk', Set.mem_range, I] at mem
Cleared hypotheses: mem
-- Lean 4 proof step from Mathlib.RingTheory.SimpleRing.Field (goal)
Goal:
A : Type u_1
inst : Ring A
inst_1 : IsSimpleRing A
x : A
hx1_1 : Membership.mem (γ := Subring A) (Subring.center A) x
hx1 : ∀ (g : A), g * x = x * g
hx2 : x ≠ 0
I : TwoSidedIdeal A := mk' (Set.range fun (x_1 : A) ↦ x * x_1) ⋯ ⋯ ⋯ ⋯ ⋯
mem : ∃ (y : A), x * y = 1
⊢ ∃ (b : ↥(Subring.center A)),
    Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := Subring A) (Subring.center A) x) x hx1_1 * b = 1
have obtain : ∀ (y : A) (hy : x * y = 1),
  ∃ (b : ↥(Subring.center A)),
    Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := Subring A) (Subring.center A) x) x hx1_1 * b = 1
Tactic: obtain ⟨y, hy⟩ := mem
Cleared hypotheses: mem
-- Lean 4 proof step from Mathlib.RingTheory.SimpleRing.Field (goal)
Goal:
A : Type u_1
inst : Ring A
inst_1 : IsSimpleRing A
x : A
hx1_1 : Membership.mem (γ := Subring A) (Subring.center A) x
hx1 : ∀ (g : A), g * x = x * g
hx2 : x ≠ 0
I : TwoSidedIdeal A := mk' (Set.range fun (x_1 : A) ↦ x * x_1) ⋯ ⋯ ⋯ ⋯ ⋯
y : A
hy : x * y = 1
⊢ Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := Subring A) (Subring.center A) x) x hx1_1 *
      Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := Subring A) (Subring.center A) x) y ⋯ =
    1
have a : (↑(Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := Subring A) (Subring.center A) x) x hx1_1 *
        Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := Subring A) (Subring.center A) x) y sorry) :
    A) =
  (↑1 : A)
Tactic: ext
-- Lean 4 proof step from Mathlib.RingTheory.SimpleRing.Field (goal)
Goal:
A : Type u_1
inst : Ring A
inst_1 : IsSimpleRing A
x : A
hx1_1 : Membership.mem (γ := Subring A) (Subring.center A) x
hx1 : ∀ (g : A), g * x = x * g
hx2 : x ≠ 0
I : TwoSidedIdeal A := mk' (Set.range fun (x_1 : A) ↦ x * x_1) ⋯ ⋯ ⋯ ⋯ ⋯
y : A
hy : x * y = 1
⊢ ∃ (b : ↥(Subring.center A)),
    Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := Subring A) (Subring.center A) x) x hx1_1 * b = 1
have refine : ∀ (a : A), a * y = y * a
Tactic: refine ⟨⟨y, Subring.mem_center_iff.2 fun a ↦ ?_⟩, by ext; exact hy⟩
-- Lean 4 proof step from Mathlib.RingTheory.SimpleRing.Field (goal)
Goal:
A : Type u_1
inst : Ring A
inst_1 : IsSimpleRing A
x : A
hx1_1 : Membership.mem (γ := Subring A) (Subring.center A) x
hx1 : ∀ (g : A), g * x = x * g
hx2 : x ≠ 0
I : TwoSidedIdeal A := mk' (Set.range fun (x_1 : A) ↦ x * x_1) ⋯ ⋯ ⋯ ⋯ ⋯
y : A
hy : x * y = 1
a : A
⊢ a * y = x * y * a * y
have rw : a * y = 1 * a * y
Tactic: rw [hy, one_mul]
-- Lean 4 proof step from Mathlib.RingTheory.SimpleRing.Field (goal)
Goal:
A : Type u_1
inst : Ring A
inst_1 : IsSimpleRing A
x : A
hx1_1 : Membership.mem (γ := Subring A) (Subring.center A) x
hx1 : ∀ (g : A), g * x = x * g
hx2 : x ≠ 0
I : TwoSidedIdeal A := mk' (Set.range fun (x_1 : A) ↦ x * x_1) ⋯ ⋯ ⋯ ⋯ ⋯
y : A
hy : x * y = 1
a : A
rw : a * y = 1 * a * y
⊢ a * y = x * y * a * y
have rw₁ : a * y = a * y
Tactic: rw [hy, one_mul]
-- Lean 4 proof step from Mathlib.RingTheory.SimpleRing.Field (goal)
Goal:
A : Type u_1
inst : Ring A
inst_1 : IsSimpleRing A
x : A
hx1_1 : Membership.mem (γ := Subring A) (Subring.center A) x
hx1 : ∀ (g : A), g * x = x * g
hx2 : x ≠ 0
I : TwoSidedIdeal A := mk' (Set.range fun (x_1 : A) ↦ x * x_1) ⋯ ⋯ ⋯ ⋯ ⋯
y : A
hy : x * y = 1
a : A
⊢ x * y * a * y = y * x * a * y
have rw : x * y * a * y = x * y * a * y
Tactic: rw [hx1]
-- Lean 4 proof step from Mathlib.RingTheory.SimpleRing.Field (goal)
Goal:
A : Type u_1
inst : Ring A
inst_1 : IsSimpleRing A
x : A
hx1_1 : Membership.mem (γ := Subring A) (Subring.center A) x
hx1 : ∀ (g : A), g * x = x * g
hx2 : x ≠ 0
I : TwoSidedIdeal A := mk' (Set.range fun (x_1 : A) ↦ x * x_1) ⋯ ⋯ ⋯ ⋯ ⋯
y : A
hy : x * y = 1
a : A
⊢ y * x * a * y = y * (x * a) * y
have rw : y * (x * a) * y = y * (x * a) * y
Tactic: rw [mul_assoc y x a]
-- Lean 4 proof step from Mathlib.RingTheory.SimpleRing.Field (goal)
Goal:
A : Type u_1
inst : Ring A
inst_1 : IsSimpleRing A
x : A
hx1_1 : Membership.mem (γ := Subring A) (Subring.center A) x
hx1 : ∀ (g : A), g * x = x * g
hx2 : x ≠ 0
I : TwoSidedIdeal A := mk' (Set.range fun (x_1 : A) ↦ x * x_1) ⋯ ⋯ ⋯ ⋯ ⋯
y : A
hy : x * y = 1
a : A
⊢ y * (x * a) * y = y * (a * x) * y
have rw : y * (x * a) * y = y * (x * a) * y
Tactic: rw [hx1]
-- Lean 4 proof step from Mathlib.RingTheory.SimpleRing.Field (goal)
Goal:
A : Type u_1
inst : Ring A
inst_1 : IsSimpleRing A
x : A
hx1_1 : Membership.mem (γ := Subring A) (Subring.center A) x
hx1 : ∀ (g : A), g * x = x * g
hx2 : x ≠ 0
I : TwoSidedIdeal A := mk' (Set.range fun (x_1 : A) ↦ x * x_1) ⋯ ⋯ ⋯ ⋯ ⋯
y : A
hy : x * y = 1
a : A
⊢ y * (a * x) * y = y * (a * x * y)
have rw : y * (a * x * y) = y * (a * x * y)
Tactic: rw [mul_assoc]
-- Lean 4 proof step from Mathlib.RingTheory.SimpleRing.Field (goal)
Goal:
A : Type u_1
inst : Ring A
inst_1 : IsSimpleRing A
x : A
hx1_1 : Membership.mem (γ := Subring A) (Subring.center A) x
hx1 : ∀ (g : A), g * x = x * g
hx2 : x ≠ 0
I : TwoSidedIdeal A := mk' (Set.range fun (x_1 : A) ↦ x * x_1) ⋯ ⋯ ⋯ ⋯ ⋯
y : A
hy : x * y = 1
a : A
⊢ y * (a * x * y) = y * (a * (x * y))
have rw : y * (a * (x * y)) = y * (a * (x * y))
Tactic: rw [mul_assoc a x y]
-- Lean 4 proof step from Mathlib.RingTheory.SimpleRing.Field (goal)
Goal:
A : Type u_1
inst : Ring A
inst_1 : IsSimpleRing A
x : A
hx1_1 : Membership.mem (γ := Subring A) (Subring.center A) x
hx1 : ∀ (g : A), g * x = x * g
hx2 : x ≠ 0
I : TwoSidedIdeal A := mk' (Set.range fun (x_1 : A) ↦ x * x_1) ⋯ ⋯ ⋯ ⋯ ⋯
y : A
hy : x * y = 1
a : A
⊢ y * (a * (x * y)) = y * a
have rw : y * (a * 1) = y * a
Tactic: rw [hy, mul_one]
-- Lean 4 proof step from Mathlib.RingTheory.SimpleRing.Field (goal)
Goal:
A : Type u_1
inst : Ring A
inst_1 : IsSimpleRing A
x : A
hx1_1 : Membership.mem (γ := Subring A) (Subring.center A) x
hx1 : ∀ (g : A), g * x = x * g
hx2 : x ≠ 0
I : TwoSidedIdeal A := mk' (Set.range fun (x_1 : A) ↦ x * x_1) ⋯ ⋯ ⋯ ⋯ ⋯
y : A
hy : x * y = 1
a : A
rw : y * (a * 1) = y * a
⊢ y * (a * (x * y)) = y * a
have rw₁ : y * a = y * a
Tactic: rw [hy, mul_one]
-- Lean 4 proof step from Mathlib.RingTheory.SimpleRing.Field (goal)
Goal:
A : Type u_1
inst : Ring A
inst_1 : IsSimpleRing A
x : A
hx1_1 : Membership.mem (γ := Subring A) (Subring.center A) x
hx1 : ∀ (g : A), g * x = x * g
hx2 : x ≠ 0
I : TwoSidedIdeal A := mk' (Set.range fun (x_1 : A) ↦ x * x_1) ⋯ ⋯ ⋯ ⋯ ⋯
y : A
hy : x * y = 1
a : A
⊢ a * y = y * a
have calc : a * y = x * y * a * y
Tactic: calc a * y
      _ = (x * y) * a * y := by rw [hy, one_mul]
      _ = (y * x) * a * y := by rw [hx1]
      _ = y * (x * a) * y := by rw [mul_assoc y x a]
      _ = y * (a * x) * y := by rw [hx1]
      _ = y * ((a * x) * y) := by rw [mul_assoc]
      _ = y * (a * (x * y)) := by rw [mul_assoc a x y]
      _ = y * a := by rw [hy, mul_one]
-- Lean 4 proof step from Mathlib.RingTheory.SimpleRing.Field (goal)
Goal:
A : Type u_1
inst : Ring A
inst_1 : IsSimpleRing A
x : A
hx1_1 : Membership.mem (γ := Subring A) (Subring.center A) x
hx1 : ∀ (g : A), g * x = x * g
hx2 : x ≠ 0
I : TwoSidedIdeal A := mk' (Set.range fun (x_1 : A) ↦ x * x_1) ⋯ ⋯ ⋯ ⋯ ⋯
y : A
hy : x * y = 1
a : A
calc : a * y = x * y * a * y
⊢ a * y = y * a
have calc₁ : x * y * a * y = y * x * a * y
Tactic: calc a * y
      _ = (x * y) * a * y := by rw [hy, one_mul]
      _ = (y * x) * a * y := by rw [hx1]
      _ = y * (x * a) * y := by rw [mul_assoc y x a]
      _ = y * (a * x) * y := by rw [hx1]
      _ = y * ((a * x) * y) := by rw [mul_assoc]
      _ = y * (a * (x * y)) := by rw [mul_assoc a x y]
      _ = y * a := by rw [hy, mul_one]
-- Lean 4 proof step from Mathlib.RingTheory.SimpleRing.Field (goal)
Goal:
A : Type u_1
inst : Ring A
inst_1 : IsSimpleRing A
x : A
hx1_1 : Membership.mem (γ := Subring A) (Subring.center A) x
hx1 : ∀ (g : A), g * x = x * g
hx2 : x ≠ 0
I : TwoSidedIdeal A := mk' (Set.range fun (x_1 : A) ↦ x * x_1) ⋯ ⋯ ⋯ ⋯ ⋯
y : A
hy : x * y = 1
a : A
calc : a * y = x * y * a * y
calc₁ : x * y * a * y = y * x * a * y
⊢ a * y = y * a
have calc₂ : y * x * a * y = y * (x * a) * y
Tactic: calc a * y
      _ = (x * y) * a * y := by rw [hy, one_mul]
      _ = (y * x) * a * y := by rw [hx1]
      _ = y * (x * a) * y := by rw [mul_assoc y x a]
      _ = y * (a * x) * y := by rw [hx1]
      _ = y * ((a * x) * y) := by rw [mul_assoc]
      _ = y * (a * (x * y)) := by rw [mul_assoc a x y]
      _ = y * a := by rw [hy, mul_one]
-- Lean 4 proof step from Mathlib.RingTheory.SimpleRing.Field (goal)
Goal:
A : Type u_1
inst : Ring A
inst_1 : IsSimpleRing A
x : A
hx1_1 : Membership.mem (γ := Subring A) (Subring.center A) x
hx1 : ∀ (g : A), g * x = x * g
hx2 : x ≠ 0
I : TwoSidedIdeal A := mk' (Set.range fun (x_1 : A) ↦ x * x_1) ⋯ ⋯ ⋯ ⋯ ⋯
y : A
hy : x * y = 1
a : A
calc : a * y = x * y * a * y
calc₁ : x * y * a * y = y * x * a * y
calc₂ : y * x * a * y = y * (x * a) * y
⊢ a * y = y * a
have calc₃ : y * (x * a) * y = y * (a * x) * y
Tactic: calc a * y
      _ = (x * y) * a * y := by rw [hy, one_mul]
      _ = (y * x) * a * y := by rw [hx1]
      _ = y * (x * a) * y := by rw [mul_assoc y x a]
      _ = y * (a * x) * y := by rw [hx1]
      _ = y * ((a * x) * y) := by rw [mul_assoc]
      _ = y * (a * (x * y)) := by rw [mul_assoc a x y]
      _ = y * a := by rw [hy, mul_one]
-- Lean 4 proof step from Mathlib.RingTheory.SimpleRing.Field (goal)
Goal:
A : Type u_1
inst : Ring A
inst_1 : IsSimpleRing A
x : A
hx1_1 : Membership.mem (γ := Subring A) (Subring.center A) x
hx1 : ∀ (g : A), g * x = x * g
hx2 : x ≠ 0
I : TwoSidedIdeal A := mk' (Set.range fun (x_1 : A) ↦ x * x_1) ⋯ ⋯ ⋯ ⋯ ⋯
y : A
hy : x * y = 1
a : A
calc : a * y = x * y * a * y
calc₁ : x * y * a * y = y * x * a * y
calc₂ : y * x * a * y = y * (x * a) * y
calc₃ : y * (x * a) * y = y * (a * x) * y
⊢ a * y = y * a
have calc₄ : y * (a * x) * y = y * (a * x * y)
Tactic: calc a * y
      _ = (x * y) * a * y := by rw [hy, one_mul]
      _ = (y * x) * a * y := by rw [hx1]
      _ = y * (x * a) * y := by rw [mul_assoc y x a]
      _ = y * (a * x) * y := by rw [hx1]
      _ = y * ((a * x) * y) := by rw [mul_assoc]
      _ = y * (a * (x * y)) := by rw [mul_assoc a x y]
      _ = y * a := by rw [hy, mul_one]
-- Lean 4 proof step from Mathlib.RingTheory.SimpleRing.Field (goal)
Goal:
A : Type u_1
inst : Ring A
inst_1 : IsSimpleRing A
x : A
hx1_1 : Membership.mem (γ := Subring A) (Subring.center A) x
hx1 : ∀ (g : A), g * x = x * g
hx2 : x ≠ 0
I : TwoSidedIdeal A := mk' (Set.range fun (x_1 : A) ↦ x * x_1) ⋯ ⋯ ⋯ ⋯ ⋯
y : A
hy : x * y = 1
a : A
calc : a * y = x * y * a * y
calc₁ : x * y * a * y = y * x * a * y
calc₂ : y * x * a * y = y * (x * a) * y
calc₃ : y * (x * a) * y = y * (a * x) * y
calc₄ : y * (a * x) * y = y * (a * x * y)
⊢ a * y = y * a
have calc₅ : y * (a * x * y) = y * (a * (x * y))
Tactic: calc a * y
      _ = (x * y) * a * y := by rw [hy, one_mul]
      _ = (y * x) * a * y := by rw [hx1]
      _ = y * (x * a) * y := by rw [mul_assoc y x a]
      _ = y * (a * x) * y := by rw [hx1]
      _ = y * ((a * x) * y) := by rw [mul_assoc]
      _ = y * (a * (x * y)) := by rw [mul_assoc a x y]
      _ = y * a := by rw [hy, mul_one]
-- Lean 4 proof step from Mathlib.RingTheory.SimpleRing.Field (goal)
Goal:
A : Type u_1
inst : Ring A
inst_1 : IsSimpleRing A
x : A
hx1_1 : Membership.mem (γ := Subring A) (Subring.center A) x
hx1 : ∀ (g : A), g * x = x * g
hx2 : x ≠ 0
I : TwoSidedIdeal A := mk' (Set.range fun (x_1 : A) ↦ x * x_1) ⋯ ⋯ ⋯ ⋯ ⋯
y : A
hy : x * y = 1
a : A
calc : a * y = x * y * a * y
calc₁ : x * y * a * y = y * x * a * y
calc₂ : y * x * a * y = y * (x * a) * y
calc₃ : y * (x * a) * y = y * (a * x) * y
calc₄ : y * (a * x) * y = y * (a * x * y)
calc₅ : y * (a * x * y) = y * (a * (x * y))
⊢ a * y = y * a
have calc₆ : y * (a * (x * y)) = y * a
Tactic: calc a * y
      _ = (x * y) * a * y := by rw [hy, one_mul]
      _ = (y * x) * a * y := by rw [hx1]
      _ = y * (x * a) * y := by rw [mul_assoc y x a]
      _ = y * (a * x) * y := by rw [hx1]
      _ = y * ((a * x) * y) := by rw [mul_assoc]
      _ = y * (a * (x * y)) := by rw [mul_assoc a x y]
      _ = y * a := by rw [hy, mul_one]
-- Lean 4 proof step from Mathlib.RingTheory.SimpleRing.Field (goal)
Goal:
A : Type u_1
inst : CommRing A
h : IsSimpleRing A
⊢ IsField.{u_1} ↥⊤
have rw : IsField.{u_1} ↥(Subring.center A)
Tactic: rw [← Subring.center_eq_top A]
-- Lean 4 proof step from Mathlib.RingTheory.Smooth.AdicCompletion (goal)
Goal:
R : Type u_1
A : Type u_2
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
S : Type u_3
inst_3 : CommRing S
inst_4 : Algebra R S
I : Ideal S
f : AlgHom.{u_1, u_2, u_3} R A (S ⧸ I)
inst_5 : FormallySmooth R A
m : ℕ
T : Type u_3 := S ⧸ I ^ (m + 1 + 1)
J : Ideal T := Ideal.map (F := S →ₐ[R] S ⧸ I ^ (m + 1 + 1)) (Ideal.Quotient.mkₐ R (I ^ (m + 1 + 1))) (I ^ (m + 1))
q : AlgHom.{u_1, u_2, u_3} R A (T ⧸ J) :=
  AlgHom.comp
    (↑(AlgEquiv.symm (DoubleQuot.quotQuotEquivQuotOfLEₐ R ⋯)) :
      S ⧸ I ^ (m + 1) →ₐ[R] (S ⧸ I ^ (m + 1).succ) ⧸ Ideal.map (Ideal.Quotient.mkₐ R (I ^ (m + 1).succ)) (I ^ (m + 1)))
    (liftAdicCompletionAux (m + 1))
⊢ J ^ (m + 1 + 1) = 0
have rw : Eq (α := Ideal T)
  (Ideal.map (F := S →ₐ[R] S ⧸ I ^ (m + 1 + 1)) (Ideal.Quotient.mkₐ R (I ^ (m + 1 + 1))) (I ^ ((m + 1) * (m + 1 + 1))))
  ⊥
Tactic: rw [← Ideal.map_pow, Submodule.zero_eq_bot, ← pow_mul]
-- Lean 4 proof step from Mathlib.RingTheory.Smooth.AdicCompletion (goal)
Goal:
R : Type u_1
A : Type u_2
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
S : Type u_3
inst_3 : CommRing S
inst_4 : Algebra R S
I : Ideal S
f : AlgHom.{u_1, u_2, u_3} R A (S ⧸ I)
inst_5 : FormallySmooth R A
⊢ Eq (α := AlgHom.{u_1, u_2, u_3} R A (S ⧸ HPow.hPow (β := ℕ) I 0))
    (AlgHom.comp (Ideal.Quotient.factorₐ R ⋯) (liftAdicCompletionAux I f (HAdd.hAdd (α := ℕ) 0 1)))
    (liftAdicCompletionAux I f 0)
have zero : ∀ (x : A),
  Eq.{u_3 + 1} (α := S ⧸ HPow.hPow (β := ℕ) I 0)
    (((AlgHom.comp (Ideal.Quotient.factorₐ R sorry) (liftAdicCompletionAux I f (HAdd.hAdd (α := ℕ) 0 1)) :
            AlgHom.{u_1, u_2, u_3} R A (S ⧸ HPow.hPow (β := ℕ) I 0)) :
          (a : A) → S ⧸ I ^ 0)
        x :
      S ⧸ HPow.hPow (β := ℕ) I 0)
    (((liftAdicCompletionAux I f 0 : AlgHom.{u_1, u_2, u_3} R A (S ⧸ HPow.hPow (β := ℕ) I 0)) : (a : A) → S ⧸ I ^ 0) x :
      S ⧸ HPow.hPow (β := ℕ) I 0)
Tactic: ext
-- Lean 4 proof step from Mathlib.RingTheory.Smooth.AdicCompletion (goal)
Goal:
R : Type u_1
A : Type u_2
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
S : Type u_3
inst_3 : CommRing S
inst_4 : Algebra R S
I : Ideal S
f : AlgHom.{u_1, u_2, u_3} R A (S ⧸ I)
inst_5 : FormallySmooth R A
x : A
⊢ Eq.{u_3 + 1} (α := S ⧸ HPow.hPow (β := ℕ) I 0)
    (((AlgHom.comp (Ideal.Quotient.factorₐ R ⋯) (liftAdicCompletionAux I f (HAdd.hAdd (α := ℕ) 0 1)) :
            AlgHom.{u_1, u_2, u_3} R A (S ⧸ HPow.hPow (β := ℕ) I 0)) :
          (a : A) → S ⧸ I ^ 0)
        x :
      S ⧸ HPow.hPow (β := ℕ) I 0)
    (((liftAdicCompletionAux I f 0 : AlgHom.{u_1, u_2, u_3} R A (S ⧸ HPow.hPow (β := ℕ) I 0)) : (a : A) → S ⧸ I ^ 0) x :
      S ⧸ HPow.hPow (β := ℕ) I 0)
have zero : Eq.{u_3 + 1} (α := S ⧸ HPow.hPow (β := ℕ) I 0) 0 1
Tactic: apply eq_of_zero_eq_one
-- Lean 4 proof step from Mathlib.RingTheory.Smooth.AdicCompletion (goal)
Goal:
R : Type u_1
A : Type u_2
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
S : Type u_3
inst_3 : CommRing S
inst_4 : Algebra R S
I : Ideal S
f : AlgHom.{u_1, u_2, u_3} R A (S ⧸ I)
inst_5 : FormallySmooth R A
m : ℕ
⊢ Eq (α := AlgHom.{u_1, u_2, u_3} R A (S ⧸ I ^ (m + 1)))
    (AlgHom.comp (Ideal.Quotient.factorₐ R ⋯) (liftAdicCompletionAux I f (m + 1 + 1)))
    (liftAdicCompletionAux I f (m + 1))
have succ : AlgHom.comp
    (AlgHom.comp
      (↑(DoubleQuot.quotQuotEquivQuotOfLEₐ R sorry) :
        (S ⧸ I ^ (m + 1).succ) ⧸ Ideal.map (Ideal.Quotient.mkₐ R (I ^ (m + 1).succ)) (I ^ (m + 1)) →ₐ[R]
          S ⧸ I ^ (m + 1))
      (Ideal.Quotient.mkₐ R
        (Ideal.map (F := S →ₐ[R] S ⧸ I ^ (m + 1).succ) (Ideal.Quotient.mkₐ R (I ^ (m + 1).succ)) (I ^ (m + 1)))))
    (lift (Ideal.map (F := S →ₐ[R] S ⧸ I ^ (m + 1 + 1)) (Ideal.Quotient.mkₐ R (I ^ (m + 1 + 1))) (I ^ (m + 1))) sorry
      (AlgHom.comp
        (↑(AlgEquiv.symm (DoubleQuot.quotQuotEquivQuotOfLEₐ R sorry)) :
          S ⧸ I ^ (m + 1) →ₐ[R]
            (S ⧸ I ^ (m + 1).succ) ⧸ Ideal.map (Ideal.Quotient.mkₐ R (I ^ (m + 1).succ)) (I ^ (m + 1)))
        (liftAdicCompletionAux I f (m + 1)))) =
  liftAdicCompletionAux I f (m + 1)
Tactic: rw [liftAdicCompletionAux, ← DoubleQuot.quotQuotEquivQuotOfLEₐ_comp_mkₐ]
-- Lean 4 proof step from Mathlib.RingTheory.Smooth.AdicCompletion (goal)
Goal:
R : Type u_1
A : Type u_2
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
S : Type u_3
inst_3 : CommRing S
inst_4 : Algebra R S
I : Ideal S
f : AlgHom.{u_1, u_2, u_3} R A (S ⧸ I)
inst_5 : FormallySmooth R A
m : ℕ
⊢ AlgHom.comp
      (AlgHom.comp
        (↑(DoubleQuot.quotQuotEquivQuotOfLEₐ R ⋯) :
          (S ⧸ I ^ (m + 1).succ) ⧸ Ideal.map (Ideal.Quotient.mkₐ R (I ^ (m + 1).succ)) (I ^ (m + 1)) →ₐ[R]
            S ⧸ I ^ (m + 1))
        (Ideal.Quotient.mkₐ R
          (Ideal.map (F := S →ₐ[R] S ⧸ I ^ (m + 1).succ) (Ideal.Quotient.mkₐ R (I ^ (m + 1).succ)) (I ^ (m + 1)))))
      (lift (Ideal.map (F := S →ₐ[R] S ⧸ I ^ (m + 1 + 1)) (Ideal.Quotient.mkₐ R (I ^ (m + 1 + 1))) (I ^ (m + 1))) ⋯
        (AlgHom.comp
          (↑(AlgEquiv.symm (DoubleQuot.quotQuotEquivQuotOfLEₐ R ⋯)) :
            S ⧸ I ^ (m + 1) →ₐ[R]
              (S ⧸ I ^ (m + 1).succ) ⧸ Ideal.map (Ideal.Quotient.mkₐ R (I ^ (m + 1).succ)) (I ^ (m + 1)))
          (liftAdicCompletionAux I f (m + 1)))) =
    liftAdicCompletionAux I f (m + 1)
have succ : ∀ (x : A),
  Eq.{u_3 + 1} (α := S ⧸ I ^ (m + 1))
    ((AlgHom.comp
            (AlgHom.comp
              (↑(DoubleQuot.quotQuotEquivQuotOfLEₐ R sorry) :
                (S ⧸ I ^ (m + 1).succ) ⧸ Ideal.map (Ideal.Quotient.mkₐ R (I ^ (m + 1).succ)) (I ^ (m + 1)) →ₐ[R]
                  S ⧸ I ^ (m + 1))
              (Ideal.Quotient.mkₐ R
                (Ideal.map (F := S →ₐ[R] S ⧸ I ^ (m + 1).succ) (Ideal.Quotient.mkₐ R (I ^ (m + 1).succ))
                  (I ^ (m + 1)))))
            (lift (Ideal.map (F := S →ₐ[R] S ⧸ I ^ (m + 1 + 1)) (Ideal.Quotient.mkₐ R (I ^ (m + 1 + 1))) (I ^ (m + 1)))
              sorry
              (AlgHom.comp
                (↑(AlgEquiv.symm (DoubleQuot.quotQuotEquivQuotOfLEₐ R sorry)) :
                  S ⧸ I ^ (m + 1) →ₐ[R]
                    (S ⧸ I ^ (m + 1).succ) ⧸ Ideal.map (Ideal.Quotient.mkₐ R (I ^ (m + 1).succ)) (I ^ (m + 1)))
                (liftAdicCompletionAux I f (m + 1)))) :
          (a : A) → S ⧸ I ^ (m + 1))
        x :
      S ⧸ I ^ (m + 1))
    (((liftAdicCompletionAux I f (m + 1) : AlgHom.{u_1, u_2, u_3} R A (S ⧸ I ^ (m + 1))) : (a : A) → S ⧸ I ^ (m + 1))
        x :
      S ⧸ I ^ (m + 1))
Tactic: ext
-- Lean 4 proof step from Mathlib.RingTheory.Smooth.AdicCompletion (goal)
Goal:
R : Type u_1
A : Type u_2
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
S : Type u_3
inst_3 : CommRing S
inst_4 : Algebra R S
I : Ideal S
f : AlgHom.{u_1, u_2, u_3} R A (S ⧸ I)
inst_5 : FormallySmooth R A
m : ℕ
⊢ AlgHom.comp (Ideal.Quotient.factorₐ R ⋯) (liftAdicCompletionAux I f (m + 1)) = liftAdicCompletionAux I f m
have zero : Eq (α := AlgHom.{u_1, u_2, u_3} R A (S ⧸ HPow.hPow (β := ℕ) I 0))
  (AlgHom.comp (Ideal.Quotient.factorₐ R sorry) (liftAdicCompletionAux I f (HAdd.hAdd (α := ℕ) 0 1)))
  (liftAdicCompletionAux I f 0)
Tactic: cases m with
  | zero =>
    ext
    apply eq_of_zero_eq_one
    simp [subsingleton_iff_zero_eq_one]
  | succ m =>
    rw [liftAdicCompletionAux, ← DoubleQuot.quotQuotEquivQuotOfLEₐ_comp_mkₐ]
    ext
    simp
Cleared hypotheses: m
-- Lean 4 proof step from Mathlib.RingTheory.Smooth.AdicCompletion (goal)
Goal:
R : Type u_1
A : Type u_2
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
S : Type u_3
inst_3 : CommRing S
inst_4 : Algebra R S
I : Ideal S
f : AlgHom.{u_1, u_2, u_3} R A (S ⧸ I)
inst_5 : FormallySmooth R A
m : ℕ
zero :
  Eq (α := AlgHom.{u_1, u_2, u_3} R A (S ⧸ HPow.hPow (β := ℕ) I 0))
    (AlgHom.comp (Ideal.Quotient.factorₐ R ⋯) (liftAdicCompletionAux I f (HAdd.hAdd (α := ℕ) 0 1)))
    (liftAdicCompletionAux I f 0)
⊢ AlgHom.comp (Ideal.Quotient.factorₐ R ⋯) (liftAdicCompletionAux I f (m + 1)) = liftAdicCompletionAux I f m
have succ : ∀ (m : ℕ),
  Eq (α := AlgHom.{u_1, u_2, u_3} R A (S ⧸ I ^ (m + 1)))
    (AlgHom.comp (Ideal.Quotient.factorₐ R sorry) (liftAdicCompletionAux I f (m + 1 + 1)))
    (liftAdicCompletionAux I f (m + 1))
Tactic: cases m with
  | zero =>
    ext
    apply eq_of_zero_eq_one
    simp [subsingleton_iff_zero_eq_one]
  | succ m =>
    rw [liftAdicCompletionAux, ← DoubleQuot.quotQuotEquivQuotOfLEₐ_comp_mkₐ]
    ext
    simp
Cleared hypotheses: m