awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Group.Finset.Basic (goal)
Goal:
ι : Type u_1
M : Type u_4
s : Finset ι
inst : CommMonoid M
f : ι → M
g : (a : ι) → a ∈ s → ι
hg₁ : ∀ (a : ι) (ha : a ∈ s), f a * f (g a ha) = 1
hg₃ : ∀ (a : ι) (ha : a ∈ s), f a ≠ 1 → g a ha ≠ a
g_mem : ∀ (a : ι) (ha : a ∈ s), g a ha ∈ s
hg₄ : ∀ (a : ι) (ha : a ∈ s), g (g a ha) ⋯ = a
⊢ Eq.{u_4 + 1} (α := M) (∏ x ∈ s, f x) 1
have H : ∀ (s : Finset ι)
  (ih :
    ∀ t ⊂ s,
      ∀ (g : (a : ι) → a ∈ t → ι) (hg₁ : ∀ (a : ι) (ha : a ∈ t), f a * f (g a ha) = 1)
        (hg₃ : ∀ (a : ι) (ha : a ∈ t), f a ≠ 1 → g a ha ≠ a) (g_mem : ∀ (a : ι) (ha : a ∈ t), g a ha ∈ t)
        (hg₄ : ∀ (a : ι) (ha : a ∈ t), g (g a ha) sorry = a), Eq.{u_4 + 1} (α := M) (∏ x ∈ t, f x) 1)
  (g : (a : ι) → a ∈ s → ι) (hg₁ : ∀ (a : ι) (ha : a ∈ s), f a * f (g a ha) = 1)
  (hg₃ : ∀ (a : ι) (ha : a ∈ s), f a ≠ 1 → g a ha ≠ a) (g_mem : ∀ (a : ι) (ha : a ∈ s), g a ha ∈ s)
  (hg₄ : ∀ (a : ι) (ha : a ∈ s), g (g a ha) sorry = a), Eq.{u_4 + 1} (α := M) (∏ x ∈ s, f x) 1
Tactic: induction s using Finset.strongInduction with | H s ih => ?_
Cleared hypotheses: g, hg₁, hg₃, g_mem, hg₄
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Group.Finset.Basic (goal)
Goal:
ι : Type u_1
M : Type u_4
s_1 : Finset ι
inst : CommMonoid M
f : ι → M
s : Finset ι
ih :
  ∀ t ⊂ s,
    ∀ (g : (a : ι) → a ∈ t → ι) (hg₁ : ∀ (a : ι) (ha : a ∈ t), f a * f (g a ha) = 1)
      (hg₃ : ∀ (a : ι) (ha : a ∈ t), f a ≠ 1 → g a ha ≠ a) (g_mem : ∀ (a : ι) (ha : a ∈ t), g a ha ∈ t)
      (hg₄ : ∀ (a : ι) (ha : a ∈ t), g (g a ha) ⋯ = a), Eq.{u_4 + 1} (α := M) (∏ x ∈ t, f x) 1
g : (a : ι) → a ∈ s → ι
hg₁ : ∀ (a : ι) (ha : a ∈ s), f a * f (g a ha) = 1
hg₃ : ∀ (a : ι) (ha : a ∈ s), f a ≠ 1 → g a ha ≠ a
g_mem : ∀ (a : ι) (ha : a ∈ s), g a ha ∈ s
hg₄ : ∀ (a : ι) (ha : a ∈ s), g (g a ha) ⋯ = a
⊢ Eq.{u_4 + 1} (α := M) (∏ x ∈ s, f x) 1
have H : ∀
  (ih :
    ∀ t ⊂ ∅,
      ∀ (g : (a : ι) → a ∈ t → ι) (hg₁ : ∀ (a : ι) (ha : a ∈ t), f a * f (g a ha) = 1)
        (hg₃ : ∀ (a : ι) (ha : a ∈ t), f a ≠ 1 → g a ha ≠ a) (g_mem : ∀ (a : ι) (ha : a ∈ t), g a ha ∈ t)
        (hg₄ : ∀ (a : ι) (ha : a ∈ t), g (g a ha) sorry = a), Eq.{u_4 + 1} (α := M) (∏ x ∈ t, f x) 1)
  (g : (a : ι) → Membership.mem (γ := Finset ι) ∅ a → ι)
  (hg₁ : ∀ (a : ι) (ha : Membership.mem (γ := Finset ι) ∅ a), f a * f (g a ha) = 1)
  (hg₃ : ∀ (a : ι) (ha : Membership.mem (γ := Finset ι) ∅ a), f a ≠ 1 → g a ha ≠ a)
  (g_mem : ∀ (a : ι) (ha : Membership.mem (γ := Finset ι) ∅ a), Membership.mem (γ := Finset ι) ∅ (g a ha))
  (hg₄ : ∀ (a : ι) (ha : Membership.mem (γ := Finset ι) ∅ a), g (g a ha) sorry = a), Eq.{u_4 + 1} (α := M) (∏ x ∈ ∅, f x) 1
Tactic: obtain rfl | ⟨x, hx⟩ := s.eq_empty_or_nonempty
Cleared hypotheses: s, ih, g, hg₁, hg₃, g_mem, hg₄
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Group.Finset.Basic (goal)
Goal:
ι : Type u_1
M : Type u_4
s_1 : Finset ι
inst : CommMonoid M
f : ι → M
s : Finset ι
ih :
  ∀ t ⊂ s,
    ∀ (g : (a : ι) → a ∈ t → ι) (hg₁ : ∀ (a : ι) (ha : a ∈ t), f a * f (g a ha) = 1)
      (hg₃ : ∀ (a : ι) (ha : a ∈ t), f a ≠ 1 → g a ha ≠ a) (g_mem : ∀ (a : ι) (ha : a ∈ t), g a ha ∈ t)
      (hg₄ : ∀ (a : ι) (ha : a ∈ t), g (g a ha) ⋯ = a), Eq.{u_4 + 1} (α := M) (∏ x ∈ t, f x) 1
g : (a : ι) → a ∈ s → ι
hg₁ : ∀ (a : ι) (ha : a ∈ s), f a * f (g a ha) = 1
hg₃ : ∀ (a : ι) (ha : a ∈ s), f a ≠ 1 → g a ha ≠ a
g_mem : ∀ (a : ι) (ha : a ∈ s), g a ha ∈ s
hg₄ : ∀ (a : ι) (ha : a ∈ s), g (g a ha) ⋯ = a
H :
  ∀
    (ih :
      ∀ t ⊂ ∅,
        ∀ (g : (a : ι) → a ∈ t → ι) (hg₁ : ∀ (a : ι) (ha : a ∈ t), f a * f (g a ha) = 1)
          (hg₃ : ∀ (a : ι) (ha : a ∈ t), f a ≠ 1 → g a ha ≠ a) (g_mem : ∀ (a : ι) (ha : a ∈ t), g a ha ∈ t)
          (hg₄ : ∀ (a : ι) (ha : a ∈ t), g (g a ha) ⋯ = a), Eq.{u_4 + 1} (α := M) (∏ x ∈ t, f x) 1)
    (g : (a : ι) → Membership.mem (γ := Finset ι) ∅ a → ι)
    (hg₁ : ∀ (a : ι) (ha : Membership.mem (γ := Finset ι) ∅ a), f a * f (g a ha) = 1)
    (hg₃ : ∀ (a : ι) (ha : Membership.mem (γ := Finset ι) ∅ a), f a ≠ 1 → g a ha ≠ a)
    (g_mem : ∀ (a : ι) (ha : Membership.mem (γ := Finset ι) ∅ a), Membership.mem (γ := Finset ι) ∅ (g a ha))
    (hg₄ : ∀ (a : ι) (ha : Membership.mem (γ := Finset ι) ∅ a), g (g a ha) ⋯ = a),
    Eq.{u_4 + 1} (α := M) (∏ x ∈ ∅, f x) 1
⊢ Eq.{u_4 + 1} (α := M) (∏ x ∈ s, f x) 1
have H₁ : ∀ (x : ι) (hx : x ∈ s), Eq.{u_4 + 1} (α := M) (∏ x ∈ s, f x) 1
Tactic: obtain rfl | ⟨x, hx⟩ := s.eq_empty_or_nonempty
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Group.Finset.Basic (hypothesis)
Goal:
ι : Type u_1
M : Type u_4
s_1 : Finset ι
inst : CommMonoid M
f : ι → M
s : Finset ι
ih :
  ∀ t ⊂ s,
    ∀ (g : (a : ι) → a ∈ t → ι) (hg₁ : ∀ (a : ι) (ha : a ∈ t), f a * f (g a ha) = 1)
      (hg₃ : ∀ (a : ι) (ha : a ∈ t), f a ≠ 1 → g a ha ≠ a) (g_mem : ∀ (a : ι) (ha : a ∈ t), g a ha ∈ t)
      (hg₄ : ∀ (a : ι) (ha : a ∈ t), g (g a ha) ⋯ = a), Eq.{u_4 + 1} (α := M) (∏ x ∈ t, f x) 1
g : (a : ι) → a ∈ s → ι
hg₁ : ∀ (a : ι) (ha : a ∈ s), f a * f (g a ha) = 1
hg₃ : ∀ (a : ι) (ha : a ∈ s), f a ≠ 1 → g a ha ≠ a
g_mem : ∀ (a : ι) (ha : a ∈ s), g a ha ∈ s
hg₄ : ∀ (a : ι) (ha : a ∈ s), g (g a ha) ⋯ = a
x : ι
hx : x ∈ s
⊢ Eq.{u_4 + 1} (α := M) (∏ x ∈ s, f x) 1
have this : {x, g x hx} ⊆ s
Tactic: have : {x, g x hx} ⊆ s := by simp [insert_subset_iff, hx, g_mem]
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Group.Finset.Basic (goal)
Goal:
ι : Type u_1
M : Type u_4
s_1 : Finset ι
inst : CommMonoid M
f : ι → M
s : Finset ι
ih :
  ∀ t ⊂ s,
    ∀ (g : (a : ι) → a ∈ t → ι) (hg₁ : ∀ (a : ι) (ha : a ∈ t), f a * f (g a ha) = 1)
      (hg₃ : ∀ (a : ι) (ha : a ∈ t), f a ≠ 1 → g a ha ≠ a) (g_mem : ∀ (a : ι) (ha : a ∈ t), g a ha ∈ t)
      (hg₄ : ∀ (a : ι) (ha : a ∈ t), g (g a ha) ⋯ = a), Eq.{u_4 + 1} (α := M) (∏ x ∈ t, f x) 1
g : (a : ι) → a ∈ s → ι
hg₁ : ∀ (a : ι) (ha : a ∈ s), f a * f (g a ha) = 1
hg₃ : ∀ (a : ι) (ha : a ∈ s), f a ≠ 1 → g a ha ≠ a
g_mem : ∀ (a : ι) (ha : a ∈ s), g a ha ∈ s
hg₄ : ∀ (a : ι) (ha : a ∈ s), g (g a ha) ⋯ = a
x : ι
hx : x ∈ s
this : {x, g x hx} ⊆ s
h : Eq.{u_4 + 1} (α := M) (∏ x ∈ s \ {x, g x hx}, f x) 1
⊢ Eq.{u_4 + 1} (α := M) (∏ x ∈ s, f x) 1
have rw : Eq.{u_4 + 1} (α := M) (∏ x ∈ {x, g x hx}, f x) 1
Tactic: rw [← prod_sdiff this, h, one_mul]
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Group.Finset.Basic (hypothesis)
Goal:
ι : Type u_1
M : Type u_4
s_1 : Finset ι
inst : CommMonoid M
f : ι → M
s : Finset ι
ih :
  ∀ t ⊂ s,
    ∀ (g : (a : ι) → a ∈ t → ι) (hg₁ : ∀ (a : ι) (ha : a ∈ t), f a * f (g a ha) = 1)
      (hg₃ : ∀ (a : ι) (ha : a ∈ t), f a ≠ 1 → g a ha ≠ a) (g_mem : ∀ (a : ι) (ha : a ∈ t), g a ha ∈ t)
      (hg₄ : ∀ (a : ι) (ha : a ∈ t), g (g a ha) ⋯ = a), Eq.{u_4 + 1} (α := M) (∏ x ∈ t, f x) 1
g : (a : ι) → a ∈ s → ι
hg₁ : ∀ (a : ι) (ha : a ∈ s), f a * f (g a ha) = 1
hg₃ : ∀ (a : ι) (ha : a ∈ s), f a ≠ 1 → g a ha ≠ a
g_mem : ∀ (a : ι) (ha : a ∈ s), g a ha ∈ s
hg₄ : ∀ (a : ι) (ha : a ∈ s), g (g a ha) ⋯ = a
x : ι
hx : x ∈ s
this : {x, g x hx} ⊆ s
h : Eq.{u_4 + 1} (α := M) (∏ x ∈ s \ {x, g x hx}, f x) 1
⊢ Eq.{u_4 + 1} (α := M) (∏ x ∈ {x, g x hx}, f x) 1
have x₁ : g x hx = x ∨ g x hx ≠ x
Tactic: cases eq_or_ne (g x hx) x with
    | inl hx' => simpa [hx'] using hg₃ x hx
    | inr hx' => grind
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Group.Finset.Basic (goal)
Goal:
ι : Type u_1
M : Type u_4
s_1 : Finset ι
inst : CommMonoid M
f : ι → M
s : Finset ι
ih :
  ∀ t ⊂ s,
    ∀ (g : (a : ι) → a ∈ t → ι) (hg₁ : ∀ (a : ι) (ha : a ∈ t), f a * f (g a ha) = 1)
      (hg₃ : ∀ (a : ι) (ha : a ∈ t), f a ≠ 1 → g a ha ≠ a) (g_mem : ∀ (a : ι) (ha : a ∈ t), g a ha ∈ t)
      (hg₄ : ∀ (a : ι) (ha : a ∈ t), g (g a ha) ⋯ = a), Eq.{u_4 + 1} (α := M) (∏ x ∈ t, f x) 1
g : (a : ι) → a ∈ s → ι
hg₁ : ∀ (a : ι) (ha : a ∈ s), f a * f (g a ha) = 1
hg₃ : ∀ (a : ι) (ha : a ∈ s), f a ≠ 1 → g a ha ≠ a
g_mem : ∀ (a : ι) (ha : a ∈ s), g a ha ∈ s
hg₄ : ∀ (a : ι) (ha : a ∈ s), g (g a ha) ⋯ = a
x : ι
hx : x ∈ s
this : {x, g x hx} ⊆ s
⊢ Eq.{u_4 + 1} (α := M) (∏ x ∈ s, f x) 1
have H : Eq.{u_4 + 1} (α := M) (∏ x ∈ s \ {x, g x hx}, f x) 1
Tactic: suffices h : ∏ x ∈ s \ {x, g x hx}, f x = 1 by
    rw [← prod_sdiff this, h, one_mul]
    cases eq_or_ne (g x hx) x with
    | inl hx' => simpa [hx'] using hg₃ x hx
    | inr hx' => grind
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Group.Finset.Basic (goal)
Goal:
ι : Type u_1
M : Type u_4
s_1 : Finset ι
inst : CommMonoid M
f : ι → M
s : Finset ι
ih :
  ∀ t ⊂ s,
    ∀ (g : (a : ι) → a ∈ t → ι) (hg₁ : ∀ (a : ι) (ha : a ∈ t), f a * f (g a ha) = 1)
      (hg₃ : ∀ (a : ι) (ha : a ∈ t), f a ≠ 1 → g a ha ≠ a) (g_mem : ∀ (a : ι) (ha : a ∈ t), g a ha ∈ t)
      (hg₄ : ∀ (a : ι) (ha : a ∈ t), g (g a ha) ⋯ = a), Eq.{u_4 + 1} (α := M) (∏ x ∈ t, f x) 1
g : (a : ι) → a ∈ s → ι
hg₁ : ∀ (a : ι) (ha : a ∈ s), f a * f (g a ha) = 1
hg₃ : ∀ (a : ι) (ha : a ∈ s), f a ≠ 1 → g a ha ≠ a
g_mem : ∀ (a : ι) (ha : a ∈ s), g a ha ∈ s
hg₄ : ∀ (a : ι) (ha : a ∈ s), g (g a ha) ⋯ = a
x : ι
hx : x ∈ s
this : {x, g x hx} ⊆ s
⊢ Eq.{u_4 + 1} (α := M) (∏ x ∈ s \ {x, g x hx}, f x) 1
have H : ∀ (a : ι) (ha : a ∈ s \ {x, g x hx}), g a sorry ∈ s \ {x, g x hx}
Tactic: suffices h₃ : ∀ a (ha : a ∈ s \ {x, g x hx}), g a (sdiff_subset ha) ∈ s \ {x, g x hx} from
    ih (s \ {x, g x hx}) (ssubset_iff.2 ⟨x, by simp [insert_subset_iff, hx]⟩) _
      (by simp [hg₁]) (fun _ _ => hg₃ _ _) h₃ (fun _ _ => hg₄ _ _)
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Group.Finset.Basic (goal)
Goal:
ι : Type u_1
M : Type u_4
s : Finset ι
inst : CommMonoid M
f : ι → M
g : (a : ι) → a ∈ s → ι
hg₁ : ∀ (a : ι) (ha : a ∈ s), f a * f (g a ha) = 1
hg₃ : ∀ (a : ι) (ha : a ∈ s), f a ≠ 1 → g a ha ≠ a
g_mem : ∀ (a : ι) (ha : a ∈ s), g a ha ∈ s
hg₄ : ∀ (a : ι) (ha : a ∈ s), g (g a ha) ⋯ = a
⊢ Eq.{u_4 + 1} (α := M) (∏ x ∈ s, f x) 1
have H : ∀ (s : Finset ι)
  (ih :
    ∀ t ⊂ s,
      ∀ (g : (a : ι) → a ∈ t → ι) (hg₁ : ∀ (a : ι) (ha : a ∈ t), f a * f (g a ha) = 1)
        (hg₃ : ∀ (a : ι) (ha : a ∈ t), f a ≠ 1 → g a ha ≠ a) (g_mem : ∀ (a : ι) (ha : a ∈ t), g a ha ∈ t)
        (hg₄ : ∀ (a : ι) (ha : a ∈ t), g (g a ha) sorry = a), Eq.{u_4 + 1} (α := M) (∏ x ∈ t, f x) 1)
  (g : (a : ι) → a ∈ s → ι) (hg₁ : ∀ (a : ι) (ha : a ∈ s), f a * f (g a ha) = 1)
  (hg₃ : ∀ (a : ι) (ha : a ∈ s), f a ≠ 1 → g a ha ≠ a) (g_mem : ∀ (a : ι) (ha : a ∈ s), g a ha ∈ s)
  (hg₄ : ∀ (a : ι) (ha : a ∈ s), g (g a ha) sorry = a), Eq.{u_4 + 1} (α := M) (∏ x ∈ s, f x) 1
Tactic: classical
  induction s using Finset.strongInduction with | H s ih => ?_
  obtain rfl | ⟨x, hx⟩ := s.eq_empty_or_nonempty
  · simp
  have : {x, g x hx} ⊆ s := by simp [insert_subset_iff, hx, g_mem]
  suffices h : ∏ x ∈ s \ {x, g x hx}, f x = 1 by
    rw [← prod_sdiff this, h, one_mul]
    cases eq_or_ne (g x hx) x with
    | inl hx' => simpa [hx'] using hg₃ x hx
    | inr hx' => grind
  suffices h₃ : ∀ a (ha : a ∈ s \ {x, g x hx}), g a (sdiff_subset ha) ∈ s \ {x, g x hx} from
    ih (s \ {x, g x hx}) (ssubset_iff.2 ⟨x, by simp [insert_subset_iff, hx]⟩) _
      (by simp [hg₁]) (fun _ _ => hg₃ _ _) h₃ (fun _ _ => hg₄ _ _)
  grind
Cleared hypotheses: g, hg₁, hg₃, g_mem, hg₄
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Group.Finset.Basic (goal)
Goal:
ι : Type u_1
M : Type u_4
s : Finset ι
inst : CommMonoid M
f : ι → M
g : (a : ι) → a ∈ s → ι
hg₁ : ∀ (a : ι) (ha : a ∈ s), f a * f (g a ha) = 1
hg₃ : ∀ (a : ι) (ha : a ∈ s), f a ≠ 1 → g a ha ≠ a
g_mem : ∀ (a : ι) (ha : a ∈ s), g a ha ∈ s
hg₄ : ∀ (a : ι) (ha : a ∈ s), g (g a ha) ⋯ = a
H H₁ :
  ∀ (s : Finset ι)
    (ih :
      ∀ t ⊂ s,
        ∀ (g : (a : ι) → a ∈ t → ι) (hg₁ : ∀ (a : ι) (ha : a ∈ t), f a * f (g a ha) = 1)
          (hg₃ : ∀ (a : ι) (ha : a ∈ t), f a ≠ 1 → g a ha ≠ a) (g_mem : ∀ (a : ι) (ha : a ∈ t), g a ha ∈ t)
          (hg₄ : ∀ (a : ι) (ha : a ∈ t), g (g a ha) ⋯ = a), Eq.{u_4 + 1} (α := M) (∏ x ∈ t, f x) 1)
    (g : (a : ι) → a ∈ s → ι) (hg₁ : ∀ (a : ι) (ha : a ∈ s), f a * f (g a ha) = 1)
    (hg₃ : ∀ (a : ι) (ha : a ∈ s), f a ≠ 1 → g a ha ≠ a) (g_mem : ∀ (a : ι) (ha : a ∈ s), g a ha ∈ s)
    (hg₄ : ∀ (a : ι) (ha : a ∈ s), g (g a ha) ⋯ = a), Eq.{u_4 + 1} (α := M) (∏ x ∈ s, f x) 1
⊢ Eq.{u_4 + 1} (α := M) (∏ x ∈ s, f x) 1
have H₂ : ∀
  (ih :
    ∀ t ⊂ ∅,
      ∀ (g : (a : ι) → a ∈ t → ι) (hg₁ : ∀ (a : ι) (ha : a ∈ t), f a * f (g a ha) = 1)
        (hg₃ : ∀ (a : ι) (ha : a ∈ t), f a ≠ 1 → g a ha ≠ a) (g_mem : ∀ (a : ι) (ha : a ∈ t), g a ha ∈ t)
        (hg₄ : ∀ (a : ι) (ha : a ∈ t), g (g a ha) sorry = a), Eq.{u_4 + 1} (α := M) (∏ x ∈ t, f x) 1)
  (g : (a : ι) → Membership.mem (γ := Finset ι) ∅ a → ι)
  (hg₁ : ∀ (a : ι) (ha : Membership.mem (γ := Finset ι) ∅ a), f a * f (g a ha) = 1)
  (hg₃ : ∀ (a : ι) (ha : Membership.mem (γ := Finset ι) ∅ a), f a ≠ 1 → g a ha ≠ a)
  (g_mem : ∀ (a : ι) (ha : Membership.mem (γ := Finset ι) ∅ a), Membership.mem (γ := Finset ι) ∅ (g a ha))
  (hg₄ : ∀ (a : ι) (ha : Membership.mem (γ := Finset ι) ∅ a), g (g a ha) sorry = a), Eq.{u_4 + 1} (α := M) (∏ x ∈ ∅, f x) 1
Tactic: classical
  induction s using Finset.strongInduction with | H s ih => ?_
  obtain rfl | ⟨x, hx⟩ := s.eq_empty_or_nonempty
  · simp
  have : {x, g x hx} ⊆ s := by simp [insert_subset_iff, hx, g_mem]
  suffices h : ∏ x ∈ s \ {x, g x hx}, f x = 1 by
    rw [← prod_sdiff this, h, one_mul]
    cases eq_or_ne (g x hx) x with
    | inl hx' => simpa [hx'] using hg₃ x hx
    | inr hx' => grind
  suffices h₃ : ∀ a (ha : a ∈ s \ {x, g x hx}), g a (sdiff_subset ha) ∈ s \ {x, g x hx} from
    ih (s \ {x, g x hx}) (ssubset_iff.2 ⟨x, by simp [insert_subset_iff, hx]⟩) _
      (by simp [hg₁]) (fun _ _ => hg₃ _ _) h₃ (fun _ _ => hg₄ _ _)
  grind
Cleared hypotheses: g, hg₁, hg₃, g_mem, hg₄
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Group.Finset.Basic (goal)
Goal:
ι : Type u_1
M : Type u_4
s : Finset ι
inst : CommMonoid M
f : ι → M
g : (a : ι) → a ∈ s → ι
hg₁ : ∀ (a : ι) (ha : a ∈ s), f a * f (g a ha) = 1
hg₃ : ∀ (a : ι) (ha : a ∈ s), f a ≠ 1 → g a ha ≠ a
g_mem : ∀ (a : ι) (ha : a ∈ s), g a ha ∈ s
hg₄ : ∀ (a : ι) (ha : a ∈ s), g (g a ha) ⋯ = a
H H₁ :
  ∀ (s : Finset ι)
    (ih :
      ∀ t ⊂ s,
        ∀ (g : (a : ι) → a ∈ t → ι) (hg₁ : ∀ (a : ι) (ha : a ∈ t), f a * f (g a ha) = 1)
          (hg₃ : ∀ (a : ι) (ha : a ∈ t), f a ≠ 1 → g a ha ≠ a) (g_mem : ∀ (a : ι) (ha : a ∈ t), g a ha ∈ t)
          (hg₄ : ∀ (a : ι) (ha : a ∈ t), g (g a ha) ⋯ = a), Eq.{u_4 + 1} (α := M) (∏ x ∈ t, f x) 1)
    (g : (a : ι) → a ∈ s → ι) (hg₁ : ∀ (a : ι) (ha : a ∈ s), f a * f (g a ha) = 1)
    (hg₃ : ∀ (a : ι) (ha : a ∈ s), f a ≠ 1 → g a ha ≠ a) (g_mem : ∀ (a : ι) (ha : a ∈ s), g a ha ∈ s)
    (hg₄ : ∀ (a : ι) (ha : a ∈ s), g (g a ha) ⋯ = a), Eq.{u_4 + 1} (α := M) (∏ x ∈ s, f x) 1
H₂ :
  ∀
    (ih :
      ∀ t ⊂ ∅,
        ∀ (g : (a : ι) → a ∈ t → ι) (hg₁ : ∀ (a : ι) (ha : a ∈ t), f a * f (g a ha) = 1)
          (hg₃ : ∀ (a : ι) (ha : a ∈ t), f a ≠ 1 → g a ha ≠ a) (g_mem : ∀ (a : ι) (ha : a ∈ t), g a ha ∈ t)
          (hg₄ : ∀ (a : ι) (ha : a ∈ t), g (g a ha) ⋯ = a), Eq.{u_4 + 1} (α := M) (∏ x ∈ t, f x) 1)
    (g : (a : ι) → Membership.mem (γ := Finset ι) ∅ a → ι)
    (hg₁ : ∀ (a : ι) (ha : Membership.mem (γ := Finset ι) ∅ a), f a * f (g a ha) = 1)
    (hg₃ : ∀ (a : ι) (ha : Membership.mem (γ := Finset ι) ∅ a), f a ≠ 1 → g a ha ≠ a)
    (g_mem : ∀ (a : ι) (ha : Membership.mem (γ := Finset ι) ∅ a), Membership.mem (γ := Finset ι) ∅ (g a ha))
    (hg₄ : ∀ (a : ι) (ha : Membership.mem (γ := Finset ι) ∅ a), g (g a ha) ⋯ = a),
    Eq.{u_4 + 1} (α := M) (∏ x ∈ ∅, f x) 1
⊢ Eq.{u_4 + 1} (α := M) (∏ x ∈ s, f x) 1
have H₃ : ∀ (s : Finset ι)
  (ih :
    ∀ t ⊂ s,
      ∀ (g : (a : ι) → a ∈ t → ι) (hg₁ : ∀ (a : ι) (ha : a ∈ t), f a * f (g a ha) = 1)
        (hg₃ : ∀ (a : ι) (ha : a ∈ t), f a ≠ 1 → g a ha ≠ a) (g_mem : ∀ (a : ι) (ha : a ∈ t), g a ha ∈ t)
        (hg₄ : ∀ (a : ι) (ha : a ∈ t), g (g a ha) sorry = a), Eq.{u_4 + 1} (α := M) (∏ x ∈ t, f x) 1)
  (g : (a : ι) → a ∈ s → ι) (hg₁ : ∀ (a : ι) (ha : a ∈ s), f a * f (g a ha) = 1)
  (hg₃ : ∀ (a : ι) (ha : a ∈ s), f a ≠ 1 → g a ha ≠ a) (g_mem : ∀ (a : ι) (ha : a ∈ s), g a ha ∈ s)
  (hg₄ : ∀ (a : ι) (ha : a ∈ s), g (g a ha) sorry = a) (x : ι) (hx : x ∈ s), Eq.{u_4 + 1} (α := M) (∏ x ∈ s, f x) 1
Tactic: classical
  induction s using Finset.strongInduction with | H s ih => ?_
  obtain rfl | ⟨x, hx⟩ := s.eq_empty_or_nonempty
  · simp
  have : {x, g x hx} ⊆ s := by simp [insert_subset_iff, hx, g_mem]
  suffices h : ∏ x ∈ s \ {x, g x hx}, f x = 1 by
    rw [← prod_sdiff this, h, one_mul]
    cases eq_or_ne (g x hx) x with
    | inl hx' => simpa [hx'] using hg₃ x hx
    | inr hx' => grind
  suffices h₃ : ∀ a (ha : a ∈ s \ {x, g x hx}), g a (sdiff_subset ha) ∈ s \ {x, g x hx} from
    ih (s \ {x, g x hx}) (ssubset_iff.2 ⟨x, by simp [insert_subset_iff, hx]⟩) _
      (by simp [hg₁]) (fun _ _ => hg₃ _ _) h₃ (fun _ _ => hg₄ _ _)
  grind
Cleared hypotheses: g, hg₁, hg₃, g_mem, hg₄
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Group.Finset.Basic (goal)
Goal:
ι : Type u_1
κ : Type u_2
M : Type u_4
s : Finset ι
inst : CommMonoid M
inst_1 : DecidableEq κ
f : κ → M
g : ι → κ
⊢ Eq.{u_4 + 1} (α := M) (∏ a ∈ s, f (g a)) (∏ b ∈ image g s, f b ^ #({a ∈ s | g a = b}))
have simp_rw : Eq.{u_4 + 1} (α := M) (∏ a ∈ s, f (g a)) (∏ x ∈ image g s, ∏ _x ∈ s with g _x = x, f x)
Tactic: simp_rw [← prod_const, prod_fiberwise_of_maps_to' fun _ ↦ mem_image_of_mem _]
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Group.Finset.Basic (goal)
Goal:
ι : Type u_1
M : Type u_4
s : Finset ι
inst : CommMonoid M
f : ι → M
R : Setoid ι
inst_1 : DecidableRel (α := ι) (⇑R : ι → ι → Prop)
⊢ Eq.{u_4 + 1} (α := M) (∏ x ∈ s, f x) (∏ xbar ∈ image (Quotient.mk R) s, ∏ y ∈ s with ⟦y⟧ = xbar, f y)
have refine : ∀ (x : ι) (_hx : x ∈ s), Eq.{u_4 + 1} (α := M) (∏ y ∈ s with ⟦y⟧ = ⟦x⟧, f y) (∏ j ∈ s with ⟦j⟧ = ⟦x⟧, f j)
Tactic: refine (Finset.prod_image' f fun x _hx => ?_).symm
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Group.Finset.Basic (goal)
Goal:
ι : Type u_1
M : Type u_4
s : Finset ι
inst : CommMonoid M
f : ι → M
R : Setoid ι
inst_1 : DecidableRel (α := ι) (⇑R : ι → ι → Prop)
h : ∀ x ∈ s, Eq.{u_4 + 1} (α := M) (∏ a ∈ s with (R : ι → ι → Prop) a x, f a) 1
⊢ Eq.{u_4 + 1} (α := M) (∏ x ∈ s, f x) 1
have rw : ∀ x ∈ image (Quotient.mk R) s, Eq.{u_4 + 1} (α := M) (∏ y ∈ s with ⟦y⟧ = x, f y) 1
Tactic: rw [prod_partition R, ← Finset.prod_eq_one]
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Group.Finset.Basic (goal)
Goal:
ι : Type u_1
M : Type u_4
s : Finset ι
inst : CommMonoid M
f : ι → M
R : Setoid ι
inst_1 : DecidableRel (α := ι) (⇑R : ι → ι → Prop)
h : ∀ x ∈ s, Eq.{u_4 + 1} (α := M) (∏ a ∈ s with (R : ι → ι → Prop) a x, f a) 1
xbar : Quotient R
xbar_in_s : xbar ∈ image (Quotient.mk R) s
⊢ Eq.{u_4 + 1} (α := M) (∏ y ∈ s with ⟦y⟧ = xbar, f y) 1
have obtain : ∀ (x : ι) (x_in_s : x ∈ s) (xbar_in_s : ⟦x⟧ ∈ image (Quotient.mk R) s),
  Eq.{u_4 + 1} (α := M) (∏ y ∈ s with ⟦y⟧ = ⟦x⟧, f y) 1
Tactic: obtain ⟨x, x_in_s, rfl⟩ := mem_image.mp xbar_in_s
Cleared hypotheses: xbar, xbar_in_s
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Group.Finset.Basic (hypothesis)
Goal:
ι : Type u_1
M : Type u_4
s : Finset ι
inst : CommMonoid M
f : ι → M
R : Setoid ι
inst_1 : DecidableRel (α := ι) (⇑R : ι → ι → Prop)
h : ∀ x ∈ s, Eq.{u_4 + 1} (α := M) (∏ a ∈ s with (R : ι → ι → Prop) a x, f a) 1
x : ι
x_in_s : x ∈ s
xbar_in_s : ⟦x⟧ ∈ image (Quotient.mk R) s
⊢ Eq.{u_4 + 1} (α := M) (∏ y ∈ s with ⟦y⟧ = ⟦x⟧, f y) 1
have h₁ : ∀ x ∈ s, Eq.{u_4 + 1} (α := M) (∏ x ∈ s with ⟦x⟧ = ⟦x⟧, f x) 1
Tactic: simp only [← Quotient.eq] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Group.Finset.Basic (goal)
Goal:
ι : Type u_1
M : Type u_4
inst : CommMonoid M
s : Finset ι
hc : #s ≤ 1
f : ι → M
b : M
h : ∏ x ∈ s, f x = b
x : ι
hx : x ∈ s
⊢ f x = b
have pos : ∀ (hc0 : #s = 0), f x = b
Tactic: by_cases hc0 : #s = 0
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Group.Finset.Basic (goal)
Goal:
ι : Type u_1
M : Type u_4
inst : CommMonoid M
s : Finset ι
hc : #s ≤ 1
f : ι → M
b : M
h : ∏ x ∈ s, f x = b
x : ι
hx : x ∈ s
pos : ∀ (hc0 : #s = 0), f x = b
⊢ f x = b
have neg : ∀ (hc0 : ¬#s = 0), f x = b
Tactic: by_cases hc0 : #s = 0
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Group.Finset.Basic (hypothesis)
Goal:
ι : Type u_1
M : Type u_4
inst : CommMonoid M
s : Finset ι
hc : #s ≤ 1
f : ι → M
b : M
h : ∏ x ∈ s, f x = b
x : ι
hx : x ∈ s
hc0 : ¬#s = 0
⊢ f x = b
have h1 : #s = 1
Tactic: have h1 : #s = 1 := le_antisymm hc (Nat.one_le_of_lt (Nat.pos_of_ne_zero hc0))
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Group.Finset.Basic (hypothesis)
Goal:
ι : Type u_1
M : Type u_4
inst : CommMonoid M
s : Finset ι
hc : #s ≤ 1
f : ι → M
b : M
h : ∏ x ∈ s, f x = b
x : ι
hx : x ∈ s
hc0 : ¬#s = 0
h1 : #s = 1
⊢ f x = b
have h1₁ : ∃ (a : ι), s = {a}
Tactic: rw [card_eq_one] at h1
Cleared hypotheses: h1
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Group.Finset.Basic (goal)
Goal:
ι : Type u_1
M : Type u_4
inst : CommMonoid M
inst_1 : DecidableEq ι
s : Finset ι
f : ι → M
a : ι
h : a ∈ s
⊢ f a * ∏ x ∈ s.erase a, f x = ∏ x ∈ s, f x
have rw : Eq.{u_4 + 1} (α := M) (∏ x ∈ insert a (s.erase a), f x) (∏ x ∈ s, f x)
Tactic: rw [← prod_insert (notMem_erase a s), insert_erase h]
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Group.Finset.Basic (goal)
Goal:
ι : Type u_1
M : Type u_4
inst : CommMonoid M
inst_1 : DecidableEq ι
s : Finset ι
f : ι → M
a : ι
h : a ∈ s
rw : Eq.{u_4 + 1} (α := M) (∏ x ∈ insert a (s.erase a), f x) (∏ x ∈ s, f x)
⊢ f a * ∏ x ∈ s.erase a, f x = ∏ x ∈ s, f x
have rw₁ : Eq.{u_4 + 1} (α := M) (∏ x ∈ s, f x) (∏ x ∈ s, f x)
Tactic: rw [← prod_insert (notMem_erase a s), insert_erase h]
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Group.Finset.Basic (goal)
Goal:
ι : Type u_1
M : Type u_4
inst : CommMonoid M
inst_1 : DecidableEq ι
s : Finset ι
f : ι → M
a : ι
h : a ∈ s
⊢ (∏ x ∈ s.erase a, f x) * f a = ∏ x ∈ s, f x
have rw : f a * ∏ x ∈ s.erase a, f x = ∏ x ∈ s, f x
Tactic: rw [mul_comm, mul_prod_erase s f h]
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Group.Finset.Basic (goal)
Goal:
ι : Type u_1
M : Type u_4
inst : CommMonoid M
inst_1 : DecidableEq ι
s : Finset ι
f : ι → M
a : ι
h : a ∈ s
rw : f a * ∏ x ∈ s.erase a, f x = ∏ x ∈ s, f x
⊢ (∏ x ∈ s.erase a, f x) * f a = ∏ x ∈ s, f x
have rw₁ : Eq.{u_4 + 1} (α := M) (∏ x ∈ s, f x) (∏ x ∈ s, f x)
Tactic: rw [mul_comm, mul_prod_erase s f h]
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Group.Finset.Basic (goal)
Goal:
ι : Type u_1
M : Type u_4
inst : CommMonoid M
inst_1 : DecidableEq ι
s : Finset ι
f : ι → M
a : ι
h : f a = 1
⊢ Eq.{u_4 + 1} (α := M) (∏ x ∈ s.erase a, f x) (∏ x ∈ s, f x)
have rw : Eq.{u_4 + 1} (α := M) (∏ x ∈ s \ {a}, f x) (∏ x ∈ s, f x)
Tactic: rw [← sdiff_singleton_eq_erase]
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Group.Finset.Basic (goal)
Goal:
ι : Type u_1
M : Type u_4
inst : CommMonoid M
inst_1 : DecidableEq ι
s : Finset ι
f : ι → M
a : ι
h : f a = 1
⊢ Eq.{u_4 + 1} (α := M) (∏ x ∈ s \ {a}, f x) (∏ x ∈ s, f x)
have refine : ∀ (x : ι) (hx : x ∈ s) (hnx : x ∉ s \ {a}), f x = 1
Tactic: refine prod_subset sdiff_subset fun x hx hnx => ?_
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Group.Finset.Basic (goal)
Goal:
ι : Type u_1
κ : Type u_5
inst : DecidableEq ι
inst_1 : CommMonoid κ
inst_2 : LT κ
inst_3 : MulLeftStrictMono κ
s : Finset ι
d : ι
hd : d ∈ s
f : ι → κ
hdf : 1 < f d
| ∏ m ∈ s, f m
have rw : ∏ m ∈ insert d (s.erase d), f m
Tactic: rw [← Finset.insert_erase hd]
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Group.Finset.Basic (goal)
Goal:
ι : Type u_1
κ : Type u_5
inst : DecidableEq ι
inst_1 : CommMonoid κ
inst_2 : LT κ
inst_3 : MulLeftStrictMono κ
s : Finset ι
d : ι
hd : d ∈ s
f : ι → κ
hdf : 1 < f d
⊢ LT.lt.{u_5} (α := κ) (∏ m ∈ s.erase d, f m) (∏ m ∈ s, f m)
have conv : LT.lt.{u_5} (α := κ) (∏ m ∈ s.erase d, f m) (∏ m ∈ insert d (s.erase d), f m)
Tactic: conv in ∏ m ∈ s, f m => rw [← Finset.insert_erase hd]
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Group.Finset.Basic (goal)
Goal:
ι : Type u_1
κ : Type u_5
inst : DecidableEq ι
inst_1 : CommMonoid κ
inst_2 : LT κ
inst_3 : MulLeftStrictMono κ
s : Finset ι
d : ι
hd : d ∈ s
f : ι → κ
hdf : 1 < f d
⊢ LT.lt.{u_5} (α := κ) (∏ m ∈ s.erase d, f m) (∏ m ∈ insert d (s.erase d), f m)
have rw : ∏ m ∈ s.erase d, f m < f d * ∏ x ∈ s.erase d, f x
Tactic: rw [Finset.prod_insert (Finset.notMem_erase d s)]