awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- 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
p : ι → Prop
inst_1 : DecidablePred p
⊢ Eq.{u_4 + 1} (α := M) (∏ x ∈ Finset.subtype p s, f (↑x : ι)) (∏ x ∈ s with p x, f x)
have this : Eq.{u_4 + 1} (α := M) (∏ x ∈ map (Embedding.subtype p) (Finset.subtype p s), f x)
  (∏ x ∈ Finset.subtype p s, f ((Embedding.subtype p : (a : Subtype p) → ι) x))
Tactic: have := prod_map (s.subtype p) (Function.Embedding.subtype _) f
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Group.Finset.Basic (goal)
Goal:
ι : Type u_1
M : Type u_4
inst : CommMonoid M
p : ι → Prop
s : Finset { x : ι // p x }
f : { x : ι // p x } → M
g : ι → M
h : ∀ x ∈ s, g (↑x : ι) = f x
⊢ Eq.{u_4 + 1} (α := M) (∏ x ∈ map (Embedding.subtype fun (x : ι) ↦ p x) s, g x) (∏ x ∈ s, f x)
have rw : Eq.{u_4 + 1} (α := M) (∏ x ∈ s, g ((Embedding.subtype fun (x : ι) ↦ p x : (a : { x : ι // p x }) → ι) x)) (∏ x ∈ s, f x)
Tactic: rw [Finset.prod_map]
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Group.Finset.Basic (goal)
Goal:
ι : Type u_1
M : Type u_4
inst : CommMonoid M
p : ι → Prop
F : Fintype (Subtype p)
s : Finset ι
h : ∀ (x : ι), x ∈ s ↔ p x
f : ι → M
⊢ Eq.{u_4 + 1} (α := M) (∏ a ∈ s, f a) (∏ a : Subtype p, f (↑a : ι))
have obtain : ∀ {F : Fintype ↥s} (h : ∀ (x : ι), x ∈ s ↔ (fun (x : ι) ↦ x ∈ s) x),
  Eq.{u_4 + 1} (α := M) (∏ a ∈ s, f a) (∏ a : ↥s, f (↑a : ι))
Tactic: obtain rfl : p = (· ∈ s) := by simp [h]
Cleared hypotheses: p, 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
s : Finset ι
f : ι → M
F : Fintype ↥s
h : ∀ (x : ι), x ∈ s ↔ (fun (x : ι) ↦ x ∈ s) x
⊢ Eq.{u_4 + 1} (α := M) (∏ a ∈ s, f a) (∏ a : ↥s, f (↑a : ι))
have rw : Eq.{u_4 + 1} (α := M) (∏ i : ↥s, f (↑i : ι)) (∏ a : ↥s, f (↑a : ι))
Tactic: rw [← prod_coe_sort]
-- 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 : Fintype ι
s : Set ι
inst_2 : DecidablePred fun (x : ι) ↦ x ∈ s
f : ι → M
g : (↑s : Type u_1) → M
w : ∀ (x : ι) (hx : x ∈ s), f x = g (Subtype.mk (p := fun (x : ι) ↦ x ∈ s) x hx)
w' : ∀ x ∉ s, f x = 1
⊢ Eq.{u_4 + 1} (α := M) (∏ i : ι, f i) (∏ i : (↑s : Type u_1), g i)
have rw : Eq.{u_4 + 1} (α := M) (∏ a : { x : ι // x ∈ s }, f (↑a : ι)) (∏ i : (↑s : Type u_1), g i)
Tactic: rw [← prod_subset s.toFinset.subset_univ (by simpa), prod_subtype (p := (· ∈ s)) _ (by simp)]
-- 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 : Fintype ι
s : Set ι
inst_2 : DecidablePred fun (x : ι) ↦ x ∈ s
f : ι → M
g : (↑s : Type u_1) → M
w : ∀ (x : ι) (hx : x ∈ s), f x = g (Subtype.mk (p := fun (x : ι) ↦ x ∈ s) x hx)
w' : ∀ x ∉ s, f x = 1
rw : Eq.{u_4 + 1} (α := M) (∏ a : { x : ι // x ∈ s }, f (↑a : ι)) (∏ i : (↑s : Type u_1), g i)
⊢ Eq.{u_4 + 1} (α := M) (∏ i : ι, f i) (∏ i : (↑s : Type u_1), g i)
have rw₁ : Fintype { x : ι // x ∈ s }
Tactic: rw [← prod_subset s.toFinset.subset_univ (by simpa), prod_subtype (p := (· ∈ s)) _ (by simp)]
-- 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 : ↥s → M
⊢ Eq.{u_4 + 1} (α := M) (∏ x : ↥s, f x) (∏ x ∈ s, extend (Subtype.val (p := fun (x : ι) ↦ x ∈ s)) f 1 x)
have rw : Eq.{u_4 + 1} (α := M) (∏ x ∈ s.attach, f x)
  (∏ x ∈ s.attach, extend (Subtype.val (p := fun (x : ι) ↦ x ∈ s)) f 1 (↑x : ι))
Tactic: rw [univ_eq_attach, ← Finset.prod_attach s]
-- 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 : ↥s → M
⊢ Eq.{u_4 + 1} (α := M) (∏ x ∈ s.attach, f x)
    (∏ x ∈ s.attach, extend (Subtype.val (p := fun (x : ι) ↦ x ∈ s)) f 1 (↑x : ι))
have e_f : ∀ (x : ι) (hx : x ∈ s),
  f (Subtype.mk (p := fun (x : ι) ↦ x ∈ s) x hx) =
    extend (Subtype.val (p := fun (x : ι) ↦ x ∈ s)) f 1 (↑(Subtype.mk (p := fun (x : ι) ↦ x ∈ s) x hx) : ι)
Tactic: congr with ⟨x, hx⟩
-- 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 : ↥s → M
x : ι
hx : x ∈ s
⊢ f (Subtype.mk (p := fun (x : ι) ↦ x ∈ s) x hx) =
    extend (Subtype.val (p := fun (x : ι) ↦ x ∈ s)) f 1 (↑(Subtype.mk (p := fun (x : ι) ↦ x ∈ s) x hx) : ι)
have e_f : f (Subtype.mk (p := fun (x : ι) ↦ x ∈ s) x hx) = f (Subtype.mk (p := fun (x : ι) ↦ x ∈ s) x hx)
Tactic: rw [Subtype.val_injective.extend_apply]
-- 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
inst : CommMonoid M
s : Finset ι
t : Finset κ
f : ι → M
g : κ → M
i : (a : ι) → a ∈ s → f a ≠ 1 → κ
hi : ∀ (a : ι) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t
i_inj :
  ∀ (a₁ : ι) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (a₂ : ι) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1),
    i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂
i_surj : ∀ b ∈ t, g b ≠ 1 → ∃ (a : ι) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ = b
h : ∀ (a : ι) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)
⊢ Eq.{u_4 + 1} (α := M) (∏ x ∈ s, f x) (∏ x ∈ s with f x ≠ 1, f x)
have rw : Eq.{u_4 + 1} (α := M) (∏ x ∈ s, f x) (∏ x ∈ s, f x)
Tactic: rw [prod_filter_ne_one]
-- 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
inst : CommMonoid M
s : Finset ι
t : Finset κ
f : ι → M
g : κ → M
i : (a : ι) → a ∈ s → f a ≠ 1 → κ
hi : ∀ (a : ι) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t
i_inj :
  ∀ (a₁ : ι) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (a₂ : ι) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1),
    i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂
i_surj : ∀ b ∈ t, g b ≠ 1 → ∃ (a : ι) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ = b
h : ∀ (a : ι) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)
⊢ Eq.{u_4 + 1} (α := M) (∏ x ∈ s, f x) (∏ x ∈ t, g x)
have calc_1 : ∀ (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}), (fun (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}) ↦ i a sorry sorry) a ha ∈ {x ∈ t | g x ≠ 1}
Tactic: calc
    ∏ x ∈ s, f x = ∏ x ∈ s with f x ≠ 1, f x := by rw [prod_filter_ne_one]
    _ = ∏ x ∈ t with g x ≠ 1, g x :=
      prod_bij (fun a ha => i a (mem_filter.mp ha).1 <| by simpa using (mem_filter.mp ha).2)
        ?_ ?_ ?_ ?_
    _ = ∏ x ∈ t, g x := prod_filter_ne_one _
-- 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
inst : CommMonoid M
s : Finset ι
t : Finset κ
f : ι → M
g : κ → M
i : (a : ι) → a ∈ s → f a ≠ 1 → κ
hi : ∀ (a : ι) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t
i_inj :
  ∀ (a₁ : ι) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (a₂ : ι) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1),
    i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂
i_surj : ∀ b ∈ t, g b ≠ 1 → ∃ (a : ι) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ = b
h : ∀ (a : ι) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)
calc_1 :
  ∀ (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}), (fun (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}) ↦ i a ⋯ ⋯) a ha ∈ {x ∈ t | g x ≠ 1}
⊢ Eq.{u_4 + 1} (α := M) (∏ x ∈ s, f x) (∏ x ∈ t, g x)
have calc_2 : ∀ (a₁ : ι) (ha₁ : a₁ ∈ {x ∈ s | f x ≠ 1}) (a₂ : ι) (ha₂ : a₂ ∈ {x ∈ s | f x ≠ 1}),
  Eq.{u_2 + 1} (α := κ) ((fun (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}) ↦ i a sorry sorry) a₁ ha₁)
      ((fun (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}) ↦ i a sorry sorry) a₂ ha₂) →
    a₁ = a₂
Tactic: calc
    ∏ x ∈ s, f x = ∏ x ∈ s with f x ≠ 1, f x := by rw [prod_filter_ne_one]
    _ = ∏ x ∈ t with g x ≠ 1, g x :=
      prod_bij (fun a ha => i a (mem_filter.mp ha).1 <| by simpa using (mem_filter.mp ha).2)
        ?_ ?_ ?_ ?_
    _ = ∏ x ∈ t, g x := prod_filter_ne_one _
-- 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
inst : CommMonoid M
s : Finset ι
t : Finset κ
f : ι → M
g : κ → M
i : (a : ι) → a ∈ s → f a ≠ 1 → κ
hi : ∀ (a : ι) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t
i_inj :
  ∀ (a₁ : ι) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (a₂ : ι) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1),
    i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂
i_surj : ∀ b ∈ t, g b ≠ 1 → ∃ (a : ι) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ = b
h : ∀ (a : ι) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)
calc_1 :
  ∀ (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}), (fun (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}) ↦ i a ⋯ ⋯) a ha ∈ {x ∈ t | g x ≠ 1}
calc_2 :
  ∀ (a₁ : ι) (ha₁ : a₁ ∈ {x ∈ s | f x ≠ 1}) (a₂ : ι) (ha₂ : a₂ ∈ {x ∈ s | f x ≠ 1}),
    Eq.{u_2 + 1} (α := κ) ((fun (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}) ↦ i a ⋯ ⋯) a₁ ha₁)
        ((fun (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}) ↦ i a ⋯ ⋯) a₂ ha₂) →
      a₁ = a₂
⊢ Eq.{u_4 + 1} (α := M) (∏ x ∈ s, f x) (∏ x ∈ t, g x)
have calc_3 : ∀ b ∈ {x ∈ t | g x ≠ 1},
  ∃ (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}), (fun (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}) ↦ i a sorry sorry) a ha = b
Tactic: calc
    ∏ x ∈ s, f x = ∏ x ∈ s with f x ≠ 1, f x := by rw [prod_filter_ne_one]
    _ = ∏ x ∈ t with g x ≠ 1, g x :=
      prod_bij (fun a ha => i a (mem_filter.mp ha).1 <| by simpa using (mem_filter.mp ha).2)
        ?_ ?_ ?_ ?_
    _ = ∏ x ∈ t, g x := prod_filter_ne_one _
-- 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
inst : CommMonoid M
s : Finset ι
t : Finset κ
f : ι → M
g : κ → M
i : (a : ι) → a ∈ s → f a ≠ 1 → κ
hi : ∀ (a : ι) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t
i_inj :
  ∀ (a₁ : ι) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (a₂ : ι) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1),
    i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂
i_surj : ∀ b ∈ t, g b ≠ 1 → ∃ (a : ι) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ = b
h : ∀ (a : ι) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)
calc_1 :
  ∀ (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}), (fun (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}) ↦ i a ⋯ ⋯) a ha ∈ {x ∈ t | g x ≠ 1}
calc_2 :
  ∀ (a₁ : ι) (ha₁ : a₁ ∈ {x ∈ s | f x ≠ 1}) (a₂ : ι) (ha₂ : a₂ ∈ {x ∈ s | f x ≠ 1}),
    Eq.{u_2 + 1} (α := κ) ((fun (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}) ↦ i a ⋯ ⋯) a₁ ha₁)
        ((fun (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}) ↦ i a ⋯ ⋯) a₂ ha₂) →
      a₁ = a₂
calc_3 :
  ∀ b ∈ {x ∈ t | g x ≠ 1},
    ∃ (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}), (fun (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}) ↦ i a ⋯ ⋯) a ha = b
⊢ Eq.{u_4 + 1} (α := M) (∏ x ∈ s, f x) (∏ x ∈ t, g x)
have calc_4 : ∀ (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}), f a = g ((fun (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}) ↦ i a sorry sorry) a ha)
Tactic: calc
    ∏ x ∈ s, f x = ∏ x ∈ s with f x ≠ 1, f x := by rw [prod_filter_ne_one]
    _ = ∏ x ∈ t with g x ≠ 1, g x :=
      prod_bij (fun a ha => i a (mem_filter.mp ha).1 <| by simpa using (mem_filter.mp ha).2)
        ?_ ?_ ?_ ?_
    _ = ∏ x ∈ t, g x := prod_filter_ne_one _
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Group.Finset.Basic (hypothesis)
Goal:
ι : Type u_1
κ : Type u_2
M : Type u_4
inst : CommMonoid M
s : Finset ι
t : Finset κ
f : ι → M
g : κ → M
i : (a : ι) → a ∈ s → f a ≠ 1 → κ
hi : ∀ (a : ι) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t
i_inj :
  ∀ (a₁ : ι) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (a₂ : ι) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1),
    i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂
i_surj : ∀ b ∈ t, g b ≠ 1 → ∃ (a : ι) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ = b
h : ∀ (a : ι) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)
b : κ
hb : b ∈ {x ∈ t | g x ≠ 1}
⊢ ∃ (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}), (fun (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}) ↦ i a ⋯ ⋯) a ha = b
have h₁ : b ∈ t
Tactic: refine (mem_filter.mp hb).elim fun h₁ h₂ ↦ ?_
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Group.Finset.Basic (hypothesis)
Goal:
ι : Type u_1
κ : Type u_2
M : Type u_4
inst : CommMonoid M
s : Finset ι
t : Finset κ
f : ι → M
g : κ → M
i : (a : ι) → a ∈ s → f a ≠ 1 → κ
hi : ∀ (a : ι) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t
i_inj :
  ∀ (a₁ : ι) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (a₂ : ι) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1),
    i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂
i_surj : ∀ b ∈ t, g b ≠ 1 → ∃ (a : ι) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ = b
h : ∀ (a : ι) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)
b : κ
hb : b ∈ {x ∈ t | g x ≠ 1}
h₁ : b ∈ t
⊢ ∃ (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}), (fun (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}) ↦ i a ⋯ ⋯) a ha = b
have h₂ : g b ≠ 1
Tactic: refine (mem_filter.mp hb).elim fun h₁ h₂ ↦ ?_
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Group.Finset.Basic (hypothesis)
Goal:
ι : Type u_1
κ : Type u_2
M : Type u_4
inst : CommMonoid M
s : Finset ι
t : Finset κ
f : ι → M
g : κ → M
i : (a : ι) → a ∈ s → f a ≠ 1 → κ
hi : ∀ (a : ι) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t
i_inj :
  ∀ (a₁ : ι) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (a₂ : ι) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1),
    i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂
i_surj : ∀ b ∈ t, g b ≠ 1 → ∃ (a : ι) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ = b
h : ∀ (a : ι) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)
b : κ
hb : b ∈ {x ∈ t | g x ≠ 1}
h₁ : b ∈ t
h₂ : g b ≠ 1
H : g b = 1
⊢ False
have h₃ : Ne.{u_4 + 1} (α := M) 1 1
Tactic: rw [H] at h₂
Cleared hypotheses: h₂, H
-- 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
inst : CommMonoid M
s : Finset ι
t : Finset κ
f : ι → M
g : κ → M
i : (a : ι) → a ∈ s → f a ≠ 1 → κ
hi : ∀ (a : ι) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t
i_inj :
  ∀ (a₁ : ι) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (a₂ : ι) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1),
    i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂
i_surj : ∀ b ∈ t, g b ≠ 1 → ∃ (a : ι) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ = b
h : ∀ (a : ι) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)
b : κ
hb : b ∈ {x ∈ t | g x ≠ 1}
h₁ : b ∈ t
h₂ : g b ≠ 1
⊢ ∃ (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}), (fun (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}) ↦ i a ⋯ ⋯) a ha = b
have calc_3 : ∀ (a : ι) (ha₁ : a ∈ s) (ha₂ : f a ≠ 1) (eq : i a ha₁ ha₂ = b),
  ∃ (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}), (fun (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}) ↦ i a sorry sorry) a ha = b
Tactic: obtain ⟨a, ha₁, ha₂, eq⟩ := i_surj b h₁ fun H ↦ by rw [H] at h₂; simp at h₂
-- 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
inst : CommMonoid M
s : Finset ι
t : Finset κ
f : ι → M
g : κ → M
i : (a : ι) → a ∈ s → f a ≠ 1 → κ
hi : ∀ (a : ι) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t
i_inj :
  ∀ (a₁ : ι) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (a₂ : ι) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1),
    i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂
i_surj : ∀ b ∈ t, g b ≠ 1 → ∃ (a : ι) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ = b
h : ∀ (a : ι) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)
⊢ Eq.{u_4 + 1} (α := M) (∏ x ∈ s, f x) (∏ x ∈ t, g x)
have x : Eq.{u_4 + 1} (α := M) (∏ x ∈ s, f x) (∏ x ∈ s with f x ≠ 1, f x)
Tactic: classical
  calc
    ∏ x ∈ s, f x = ∏ x ∈ s with f x ≠ 1, f x := by rw [prod_filter_ne_one]
    _ = ∏ x ∈ t with g x ≠ 1, g x :=
      prod_bij (fun a ha => i a (mem_filter.mp ha).1 <| by simpa using (mem_filter.mp ha).2)
        ?_ ?_ ?_ ?_
    _ = ∏ x ∈ t, g x := prod_filter_ne_one _
  · grind
  · solve_by_elim
  · intro b hb
    refine (mem_filter.mp hb).elim fun h₁ h₂ ↦ ?_
    obtain ⟨a, ha₁, ha₂, eq⟩ := i_surj b h₁ fun H ↦ by rw [H] at h₂; simp at h₂
    exact ⟨a, mem_filter.mpr ⟨ha₁, ha₂⟩, eq⟩
  · solve_by_elim
-- 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
inst : CommMonoid M
s : Finset ι
t : Finset κ
f : ι → M
g : κ → M
i : (a : ι) → a ∈ s → f a ≠ 1 → κ
hi : ∀ (a : ι) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t
i_inj :
  ∀ (a₁ : ι) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (a₂ : ι) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1),
    i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂
i_surj : ∀ b ∈ t, g b ≠ 1 → ∃ (a : ι) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ = b
h : ∀ (a : ι) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)
x : Eq.{u_4 + 1} (α := M) (∏ x ∈ s, f x) (∏ x ∈ s with f x ≠ 1, f x)
⊢ Eq.{u_4 + 1} (α := M) (∏ x ∈ s, f x) (∏ x ∈ t, g x)
have calc_1 : ∀ (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}), (fun (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}) ↦ i a sorry sorry) a ha ∈ {x ∈ t | g x ≠ 1}
Tactic: classical
  calc
    ∏ x ∈ s, f x = ∏ x ∈ s with f x ≠ 1, f x := by rw [prod_filter_ne_one]
    _ = ∏ x ∈ t with g x ≠ 1, g x :=
      prod_bij (fun a ha => i a (mem_filter.mp ha).1 <| by simpa using (mem_filter.mp ha).2)
        ?_ ?_ ?_ ?_
    _ = ∏ x ∈ t, g x := prod_filter_ne_one _
  · grind
  · solve_by_elim
  · intro b hb
    refine (mem_filter.mp hb).elim fun h₁ h₂ ↦ ?_
    obtain ⟨a, ha₁, ha₂, eq⟩ := i_surj b h₁ fun H ↦ by rw [H] at h₂; simp at h₂
    exact ⟨a, mem_filter.mpr ⟨ha₁, ha₂⟩, eq⟩
  · solve_by_elim
-- 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
inst : CommMonoid M
s : Finset ι
t : Finset κ
f : ι → M
g : κ → M
i : (a : ι) → a ∈ s → f a ≠ 1 → κ
hi : ∀ (a : ι) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t
i_inj :
  ∀ (a₁ : ι) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (a₂ : ι) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1),
    i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂
i_surj : ∀ b ∈ t, g b ≠ 1 → ∃ (a : ι) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ = b
h : ∀ (a : ι) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)
x : Eq.{u_4 + 1} (α := M) (∏ x ∈ s, f x) (∏ x ∈ s with f x ≠ 1, f x)
calc_1 :
  ∀ (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}), (fun (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}) ↦ i a ⋯ ⋯) a ha ∈ {x ∈ t | g x ≠ 1}
⊢ Eq.{u_4 + 1} (α := M) (∏ x ∈ s, f x) (∏ x ∈ t, g x)
have calc_2 : ∀ (a₁ : ι) (ha₁ : a₁ ∈ {x ∈ s | f x ≠ 1}) (a₂ : ι) (ha₂ : a₂ ∈ {x ∈ s | f x ≠ 1}),
  Eq.{u_2 + 1} (α := κ) ((fun (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}) ↦ i a sorry sorry) a₁ ha₁)
      ((fun (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}) ↦ i a sorry sorry) a₂ ha₂) →
    a₁ = a₂
Tactic: classical
  calc
    ∏ x ∈ s, f x = ∏ x ∈ s with f x ≠ 1, f x := by rw [prod_filter_ne_one]
    _ = ∏ x ∈ t with g x ≠ 1, g x :=
      prod_bij (fun a ha => i a (mem_filter.mp ha).1 <| by simpa using (mem_filter.mp ha).2)
        ?_ ?_ ?_ ?_
    _ = ∏ x ∈ t, g x := prod_filter_ne_one _
  · grind
  · solve_by_elim
  · intro b hb
    refine (mem_filter.mp hb).elim fun h₁ h₂ ↦ ?_
    obtain ⟨a, ha₁, ha₂, eq⟩ := i_surj b h₁ fun H ↦ by rw [H] at h₂; simp at h₂
    exact ⟨a, mem_filter.mpr ⟨ha₁, ha₂⟩, eq⟩
  · solve_by_elim
-- 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
inst : CommMonoid M
s : Finset ι
t : Finset κ
f : ι → M
g : κ → M
i : (a : ι) → a ∈ s → f a ≠ 1 → κ
hi : ∀ (a : ι) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t
i_inj :
  ∀ (a₁ : ι) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (a₂ : ι) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1),
    i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂
i_surj : ∀ b ∈ t, g b ≠ 1 → ∃ (a : ι) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ = b
h : ∀ (a : ι) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)
x : Eq.{u_4 + 1} (α := M) (∏ x ∈ s, f x) (∏ x ∈ s with f x ≠ 1, f x)
calc_1 :
  ∀ (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}), (fun (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}) ↦ i a ⋯ ⋯) a ha ∈ {x ∈ t | g x ≠ 1}
calc_2 :
  ∀ (a₁ : ι) (ha₁ : a₁ ∈ {x ∈ s | f x ≠ 1}) (a₂ : ι) (ha₂ : a₂ ∈ {x ∈ s | f x ≠ 1}),
    Eq.{u_2 + 1} (α := κ) ((fun (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}) ↦ i a ⋯ ⋯) a₁ ha₁)
        ((fun (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}) ↦ i a ⋯ ⋯) a₂ ha₂) →
      a₁ = a₂
⊢ Eq.{u_4 + 1} (α := M) (∏ x ∈ s, f x) (∏ x ∈ t, g x)
have calc_3 : ∀ b ∈ {x ∈ t | g x ≠ 1},
  ∃ (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}), (fun (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}) ↦ i a sorry sorry) a ha = b
Tactic: classical
  calc
    ∏ x ∈ s, f x = ∏ x ∈ s with f x ≠ 1, f x := by rw [prod_filter_ne_one]
    _ = ∏ x ∈ t with g x ≠ 1, g x :=
      prod_bij (fun a ha => i a (mem_filter.mp ha).1 <| by simpa using (mem_filter.mp ha).2)
        ?_ ?_ ?_ ?_
    _ = ∏ x ∈ t, g x := prod_filter_ne_one _
  · grind
  · solve_by_elim
  · intro b hb
    refine (mem_filter.mp hb).elim fun h₁ h₂ ↦ ?_
    obtain ⟨a, ha₁, ha₂, eq⟩ := i_surj b h₁ fun H ↦ by rw [H] at h₂; simp at h₂
    exact ⟨a, mem_filter.mpr ⟨ha₁, ha₂⟩, eq⟩
  · solve_by_elim
-- 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
inst : CommMonoid M
s : Finset ι
t : Finset κ
f : ι → M
g : κ → M
i : (a : ι) → a ∈ s → f a ≠ 1 → κ
hi : ∀ (a : ι) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t
i_inj :
  ∀ (a₁ : ι) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (a₂ : ι) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1),
    i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂
i_surj : ∀ b ∈ t, g b ≠ 1 → ∃ (a : ι) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ = b
h : ∀ (a : ι) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)
x : Eq.{u_4 + 1} (α := M) (∏ x ∈ s, f x) (∏ x ∈ s with f x ≠ 1, f x)
calc_1 :
  ∀ (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}), (fun (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}) ↦ i a ⋯ ⋯) a ha ∈ {x ∈ t | g x ≠ 1}
calc_2 :
  ∀ (a₁ : ι) (ha₁ : a₁ ∈ {x ∈ s | f x ≠ 1}) (a₂ : ι) (ha₂ : a₂ ∈ {x ∈ s | f x ≠ 1}),
    Eq.{u_2 + 1} (α := κ) ((fun (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}) ↦ i a ⋯ ⋯) a₁ ha₁)
        ((fun (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}) ↦ i a ⋯ ⋯) a₂ ha₂) →
      a₁ = a₂
calc_3 :
  ∀ b ∈ {x ∈ t | g x ≠ 1},
    ∃ (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}), (fun (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}) ↦ i a ⋯ ⋯) a ha = b
⊢ Eq.{u_4 + 1} (α := M) (∏ x ∈ s, f x) (∏ x ∈ t, g x)
have calc_4 : ∀ (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}), f a = g ((fun (a : ι) (ha : a ∈ {x ∈ s | f x ≠ 1}) ↦ i a sorry sorry) a ha)
Tactic: classical
  calc
    ∏ x ∈ s, f x = ∏ x ∈ s with f x ≠ 1, f x := by rw [prod_filter_ne_one]
    _ = ∏ x ∈ t with g x ≠ 1, g x :=
      prod_bij (fun a ha => i a (mem_filter.mp ha).1 <| by simpa using (mem_filter.mp ha).2)
        ?_ ?_ ?_ ?_
    _ = ∏ x ∈ t, g x := prod_filter_ne_one _
  · grind
  · solve_by_elim
  · intro b hb
    refine (mem_filter.mp hb).elim fun h₁ h₂ ↦ ?_
    obtain ⟨a, ha₁, ha₂, eq⟩ := i_surj b h₁ fun H ↦ by rw [H] at h₂; simp at h₂
    exact ⟨a, mem_filter.mpr ⟨ha₁, ha₂⟩, eq⟩
  · solve_by_elim
-- 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
h : Ne.{u_4 + 1} (α := M) (∏ x ∈ s, f x) 1
⊢ ∃ a ∈ s, f a ≠ 1
have contrapose! : ∀ (h : ∀ (a : ι) (_ : a ∈ s), f a = 1), Eq.{u_4 + 1} (α := M) (∏ x ∈ s, f x) 1
Tactic: contrapose! h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Group.Finset.Basic (goal)
Goal:
M : Type u_4
inst : CommMonoid M
f : ℕ → M
n : ℕ
⊢ ∏ x ∈ range (n + 1), f x = f n * ∏ x ∈ range n, f x
have rw : ∏ x ∈ insert n (range n), f x = f n * ∏ x ∈ range n, f x
Tactic: rw [range_add_one, prod_insert notMem_range_self]
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Group.Finset.Basic (goal)
Goal:
M : Type u_4
inst : CommMonoid M
f : ℕ → M
n : ℕ
rw : ∏ x ∈ insert n (range n), f x = f n * ∏ x ∈ range n, f x
⊢ ∏ x ∈ range (n + 1), f x = f n * ∏ x ∈ range n, f x
have rw₁ : f n * ∏ x ∈ range n, f x = f n * ∏ x ∈ range n, f x
Tactic: rw [range_add_one, prod_insert notMem_range_self]
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Group.Finset.Basic (goal)
Goal:
M : Type u_4
inst : CommMonoid M
f : ℕ → M
n : ℕ
⊢ ∏ k ∈ range (n + 1 + 1), f k = (∏ k ∈ range (n + 1), f (k + 1)) * f 0
have rw : ∏ k ∈ range (n + 1 + 1), f k = (∏ x ∈ range n, f (x + 1)) * f (n + 1) * f 0
Tactic: rw [prod_range_succ _ n, mul_right_comm, ← prod_range_succ' _ n, prod_range_succ]
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Group.Finset.Basic (goal)
Goal:
M : Type u_4
inst : CommMonoid M
f : ℕ → M
n : ℕ
rw : ∏ k ∈ range (n + 1 + 1), f k = (∏ x ∈ range n, f (x + 1)) * f (n + 1) * f 0
⊢ ∏ k ∈ range (n + 1 + 1), f k = (∏ k ∈ range (n + 1), f (k + 1)) * f 0
have rw₁ : ∏ k ∈ range (n + 1 + 1), f k = (∏ x ∈ range n, f (x + 1)) * f 0 * f (n + 1)
Tactic: rw [prod_range_succ _ n, mul_right_comm, ← prod_range_succ' _ n, prod_range_succ]
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Group.Finset.Basic (goal)
Goal:
M : Type u_4
inst : CommMonoid M
f : ℕ → M
n : ℕ
rw : ∏ k ∈ range (n + 1 + 1), f k = (∏ x ∈ range n, f (x + 1)) * f (n + 1) * f 0
rw₁ : ∏ k ∈ range (n + 1 + 1), f k = (∏ x ∈ range n, f (x + 1)) * f 0 * f (n + 1)
⊢ ∏ k ∈ range (n + 1 + 1), f k = (∏ k ∈ range (n + 1), f (k + 1)) * f 0
have rw₂ : ∏ k ∈ range (n + 1 + 1), f k = (∏ k ∈ range (n + 1), f k) * f (n + 1)
Tactic: rw [prod_range_succ _ n, mul_right_comm, ← prod_range_succ' _ n, prod_range_succ]