l3lab/ntp-mathlib huggingface dataset:
@[to_additive]
theorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :
    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by
  induction i with
  | zero =>
    simp
  | succ i IH =>
    by_cases h : i < n
    · 

Proof state:
case pos
α : Type u_1
β : Type u_2
inst✝ : CommMonoid α
n : ℕ
f : Fin n → α
i : ℕ
IH : prod (take i (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < i) univ, f j
h : i < n
⊢ prod (take (Nat.succ i) (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < Nat.succ i) univ, f j

Next tactic:
have : i < length (ofFn f) := by rwa [length_ofFn f]
@[to_additive]
theorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :
    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by
  induction i with
  | zero =>
    simp
  | succ i IH =>
    by_cases h : i < n
    · have : i < length (ofFn f) := by 

Proof state:
α : Type u_1
β : Type u_2
inst✝ : CommMonoid α
n : ℕ
f : Fin n → α
i : ℕ
IH : prod (take i (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < i) univ, f j
h : i < n
⊢ i < length (ofFn f)

Next tactic:
rwa [length_ofFn f]
@[to_additive]
theorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :
    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by
  induction i with
  | zero =>
    simp
  | succ i IH =>
    by_cases h : i < n
    · have : i < length (ofFn f) := by rwa [length_ofFn f]
      

Proof state:
case pos
α : Type u_1
β : Type u_2
inst✝ : CommMonoid α
n : ℕ
f : Fin n → α
i : ℕ
IH : prod (take i (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < i) univ, f j
h : i < n
this : i < length (ofFn f)
⊢ prod (take (Nat.succ i) (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < Nat.succ i) univ, f j

Next tactic:
rw [prod_take_succ _ _ this]
@[to_additive]
theorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :
    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by
  induction i with
  | zero =>
    simp
  | succ i IH =>
    by_cases h : i < n
    · have : i < length (ofFn f) := by rwa [length_ofFn f]
      rw [prod_take_succ _ _ this]
      

Proof state:
case pos
α : Type u_1
β : Type u_2
inst✝ : CommMonoid α
n : ℕ
f : Fin n → α
i : ℕ
IH : prod (take i (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < i) univ, f j
h : i < n
this : i < length (ofFn f)
⊢ prod (take i (ofFn f)) * get (ofFn f) { val := i, isLt := this } =
    ∏ j in Finset.filter (fun j => ↑j < Nat.succ i) univ, f j

Next tactic:
have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =
          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by
        ext ⟨_, _⟩
        simp [Nat.lt_succ_iff_lt_or_eq]
@[to_additive]
theorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :
    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by
  induction i with
  | zero =>
    simp
  | succ i IH =>
    by_cases h : i < n
    · have : i < length (ofFn f) := by rwa [length_ofFn f]
      rw [prod_take_succ _ _ this]
      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =
          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by
        

Proof state:
α : Type u_1
β : Type u_2
inst✝ : CommMonoid α
n : ℕ
f : Fin n → α
i : ℕ
IH : prod (take i (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < i) univ, f j
h : i < n
this : i < length (ofFn f)
⊢ Finset.filter (fun j => ↑j < i + 1) univ = Finset.filter (fun j => ↑j < i) univ ∪ {{ val := i, isLt := h }}

Next tactic:
ext ⟨_, _⟩
@[to_additive]
theorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :
    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by
  induction i with
  | zero =>
    simp
  | succ i IH =>
    by_cases h : i < n
    · have : i < length (ofFn f) := by rwa [length_ofFn f]
      rw [prod_take_succ _ _ this]
      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =
          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by
        ext ⟨_, _⟩
        

Proof state:
case a.mk
α : Type u_1
β : Type u_2
inst✝ : CommMonoid α
n : ℕ
f : Fin n → α
i : ℕ
IH : prod (take i (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < i) univ, f j
h : i < n
this : i < length (ofFn f)
val✝ : ℕ
isLt✝ : val✝ < n
⊢ { val := val✝, isLt := isLt✝ } ∈ Finset.filter (fun j => ↑j < i + 1) univ ↔
    { val := val✝, isLt := isLt✝ } ∈ Finset.filter (fun j => ↑j < i) univ ∪ {{ val := i, isLt := h }}

Next tactic:
simp [Nat.lt_succ_iff_lt_or_eq]
@[to_additive]
theorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :
    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by
  induction i with
  | zero =>
    simp
  | succ i IH =>
    by_cases h : i < n
    · have : i < length (ofFn f) := by rwa [length_ofFn f]
      rw [prod_take_succ _ _ this]
      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =
          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by
        ext ⟨_, _⟩
        simp [Nat.lt_succ_iff_lt_or_eq]
      

Proof state:
case pos
α : Type u_1
β : Type u_2
inst✝ : CommMonoid α
n : ℕ
f : Fin n → α
i : ℕ
IH : prod (take i (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < i) univ, f j
h : i < n
this : i < length (ofFn f)
A : Finset.filter (fun j => ↑j < i + 1) univ = Finset.filter (fun j => ↑j < i) univ ∪ {{ val := i, isLt := h }}
⊢ prod (take i (ofFn f)) * get (ofFn f) { val := i, isLt := this } =
    ∏ j in Finset.filter (fun j => ↑j < Nat.succ i) univ, f j

Next tactic:
have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)
          (singleton (⟨i, h⟩ : Fin n)) := by simp
@[to_additive]
theorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :
    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by
  induction i with
  | zero =>
    simp
  | succ i IH =>
    by_cases h : i < n
    · have : i < length (ofFn f) := by rwa [length_ofFn f]
      rw [prod_take_succ _ _ this]
      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =
          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by
        ext ⟨_, _⟩
        simp [Nat.lt_succ_iff_lt_or_eq]
      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)
          (singleton (⟨i, h⟩ : Fin n)) := by 

Proof state:
α : Type u_1
β : Type u_2
inst✝ : CommMonoid α
n : ℕ
f : Fin n → α
i : ℕ
IH : prod (take i (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < i) univ, f j
h : i < n
this : i < length (ofFn f)
A : Finset.filter (fun j => ↑j < i + 1) univ = Finset.filter (fun j => ↑j < i) univ ∪ {{ val := i, isLt := h }}
⊢ _root_.Disjoint (Finset.filter (fun j => ↑j < i) univ) {{ val := i, isLt := h }}

Next tactic:
simp
@[to_additive]
theorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :
    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by
  induction i with
  | zero =>
    simp
  | succ i IH =>
    by_cases h : i < n
    · have : i < length (ofFn f) := by rwa [length_ofFn f]
      rw [prod_take_succ _ _ this]
      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =
          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by
        ext ⟨_, _⟩
        simp [Nat.lt_succ_iff_lt_or_eq]
      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)
          (singleton (⟨i, h⟩ : Fin n)) := by simp
      

Proof state:
case pos
α : Type u_1
β : Type u_2
inst✝ : CommMonoid α
n : ℕ
f : Fin n → α
i : ℕ
IH : prod (take i (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < i) univ, f j
h : i < n
this : i < length (ofFn f)
A : Finset.filter (fun j => ↑j < i + 1) univ = Finset.filter (fun j => ↑j < i) univ ∪ {{ val := i, isLt := h }}
B : _root_.Disjoint (Finset.filter (fun j => ↑j < i) univ) {{ val := i, isLt := h }}
⊢ prod (take i (ofFn f)) * get (ofFn f) { val := i, isLt := this } =
    ∏ j in Finset.filter (fun j => ↑j < Nat.succ i) univ, f j

Next tactic:
rw [A, Finset.prod_union B, IH]
@[to_additive]
theorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :
    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by
  induction i with
  | zero =>
    simp
  | succ i IH =>
    by_cases h : i < n
    · have : i < length (ofFn f) := by rwa [length_ofFn f]
      rw [prod_take_succ _ _ this]
      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =
          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by
        ext ⟨_, _⟩
        simp [Nat.lt_succ_iff_lt_or_eq]
      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)
          (singleton (⟨i, h⟩ : Fin n)) := by simp
      rw [A, Finset.prod_union B, IH]
      

Proof state:
case pos
α : Type u_1
β : Type u_2
inst✝ : CommMonoid α
n : ℕ
f : Fin n → α
i : ℕ
IH : prod (take i (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < i) univ, f j
h : i < n
this : i < length (ofFn f)
A : Finset.filter (fun j => ↑j < i + 1) univ = Finset.filter (fun j => ↑j < i) univ ∪ {{ val := i, isLt := h }}
B : _root_.Disjoint (Finset.filter (fun j => ↑j < i) univ) {{ val := i, isLt := h }}
⊢ (∏ j in Finset.filter (fun j => ↑j < i) univ, f j) * get (ofFn f) { val := i, isLt := this } =
    (∏ x in Finset.filter (fun j => ↑j < i) univ, f x) * ∏ x in {{ val := i, isLt := h }}, f x

Next tactic:
simp
@[to_additive]
theorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :
    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by
  induction i with
  | zero =>
    simp
  | succ i IH =>
    by_cases h : i < n
    · have : i < length (ofFn f) := by rwa [length_ofFn f]
      rw [prod_take_succ _ _ this]
      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =
          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by
        ext ⟨_, _⟩
        simp [Nat.lt_succ_iff_lt_or_eq]
      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)
          (singleton (⟨i, h⟩ : Fin n)) := by simp
      rw [A, Finset.prod_union B, IH]
      simp
    · 

Proof state:
case neg
α : Type u_1
β : Type u_2
inst✝ : CommMonoid α
n : ℕ
f : Fin n → α
i : ℕ
IH : prod (take i (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < i) univ, f j
h : ¬i < n
⊢ prod (take (Nat.succ i) (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < Nat.succ i) univ, f j

Next tactic:
have A : (ofFn f).take i = (ofFn f).take i.succ := by
        rw [← length_ofFn f] at h
        have : length (ofFn f) ≤ i := not_lt.mp h
        rw [take_all_of_le this, take_all_of_le (le_trans this (Nat.le_succ _))]
@[to_additive]
theorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :
    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by
  induction i with
  | zero =>
    simp
  | succ i IH =>
    by_cases h : i < n
    · have : i < length (ofFn f) := by rwa [length_ofFn f]
      rw [prod_take_succ _ _ this]
      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =
          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by
        ext ⟨_, _⟩
        simp [Nat.lt_succ_iff_lt_or_eq]
      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)
          (singleton (⟨i, h⟩ : Fin n)) := by simp
      rw [A, Finset.prod_union B, IH]
      simp
    · have A : (ofFn f).take i = (ofFn f).take i.succ := by
        

Proof state:
α : Type u_1
β : Type u_2
inst✝ : CommMonoid α
n : ℕ
f : Fin n → α
i : ℕ
IH : prod (take i (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < i) univ, f j
h : ¬i < n
⊢ take i (ofFn f) = take (Nat.succ i) (ofFn f)

Next tactic:
rw [← length_ofFn f] at h
@[to_additive]
theorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :
    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by
  induction i with
  | zero =>
    simp
  | succ i IH =>
    by_cases h : i < n
    · have : i < length (ofFn f) := by rwa [length_ofFn f]
      rw [prod_take_succ _ _ this]
      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =
          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by
        ext ⟨_, _⟩
        simp [Nat.lt_succ_iff_lt_or_eq]
      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)
          (singleton (⟨i, h⟩ : Fin n)) := by simp
      rw [A, Finset.prod_union B, IH]
      simp
    · have A : (ofFn f).take i = (ofFn f).take i.succ := by
        rw [← length_ofFn f] at h
        

Proof state:
α : Type u_1
β : Type u_2
inst✝ : CommMonoid α
n : ℕ
f : Fin n → α
i : ℕ
IH : prod (take i (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < i) univ, f j
h : ¬i < length (ofFn f)
⊢ take i (ofFn f) = take (Nat.succ i) (ofFn f)

Next tactic:
have : length (ofFn f) ≤ i := not_lt.mp h
@[to_additive]
theorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :
    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by
  induction i with
  | zero =>
    simp
  | succ i IH =>
    by_cases h : i < n
    · have : i < length (ofFn f) := by rwa [length_ofFn f]
      rw [prod_take_succ _ _ this]
      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =
          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by
        ext ⟨_, _⟩
        simp [Nat.lt_succ_iff_lt_or_eq]
      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)
          (singleton (⟨i, h⟩ : Fin n)) := by simp
      rw [A, Finset.prod_union B, IH]
      simp
    · have A : (ofFn f).take i = (ofFn f).take i.succ := by
        rw [← length_ofFn f] at h
        have : length (ofFn f) ≤ i := not_lt.mp h
        

Proof state:
α : Type u_1
β : Type u_2
inst✝ : CommMonoid α
n : ℕ
f : Fin n → α
i : ℕ
IH : prod (take i (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < i) univ, f j
h : ¬i < length (ofFn f)
this : length (ofFn f) ≤ i
⊢ take i (ofFn f) = take (Nat.succ i) (ofFn f)

Next tactic:
rw [take_all_of_le this, take_all_of_le (le_trans this (Nat.le_succ _))]
@[to_additive]
theorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :
    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by
  induction i with
  | zero =>
    simp
  | succ i IH =>
    by_cases h : i < n
    · have : i < length (ofFn f) := by rwa [length_ofFn f]
      rw [prod_take_succ _ _ this]
      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =
          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by
        ext ⟨_, _⟩
        simp [Nat.lt_succ_iff_lt_or_eq]
      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)
          (singleton (⟨i, h⟩ : Fin n)) := by simp
      rw [A, Finset.prod_union B, IH]
      simp
    · have A : (ofFn f).take i = (ofFn f).take i.succ := by
        rw [← length_ofFn f] at h
        have : length (ofFn f) ≤ i := not_lt.mp h
        rw [take_all_of_le this, take_all_of_le (le_trans this (Nat.le_succ _))]
      

Proof state:
case neg
α : Type u_1
β : Type u_2
inst✝ : CommMonoid α
n : ℕ
f : Fin n → α
i : ℕ
IH : prod (take i (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < i) univ, f j
h : ¬i < n
A : take i (ofFn f) = take (Nat.succ i) (ofFn f)
⊢ prod (take (Nat.succ i) (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < Nat.succ i) univ, f j

Next tactic:
have B : ∀ j : Fin n, ((j : ℕ) < i.succ) = ((j : ℕ) < i) := by
        intro j
        have : (j : ℕ) < i := lt_of_lt_of_le j.2 (not_lt.mp h)
        simp [this, lt_trans this (Nat.lt_succ_self _)]
@[to_additive]
theorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :
    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by
  induction i with
  | zero =>
    simp
  | succ i IH =>
    by_cases h : i < n
    · have : i < length (ofFn f) := by rwa [length_ofFn f]
      rw [prod_take_succ _ _ this]
      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =
          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by
        ext ⟨_, _⟩
        simp [Nat.lt_succ_iff_lt_or_eq]
      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)
          (singleton (⟨i, h⟩ : Fin n)) := by simp
      rw [A, Finset.prod_union B, IH]
      simp
    · have A : (ofFn f).take i = (ofFn f).take i.succ := by
        rw [← length_ofFn f] at h
        have : length (ofFn f) ≤ i := not_lt.mp h
        rw [take_all_of_le this, take_all_of_le (le_trans this (Nat.le_succ _))]
      have B : ∀ j : Fin n, ((j : ℕ) < i.succ) = ((j : ℕ) < i) := by
        

Proof state:
α : Type u_1
β : Type u_2
inst✝ : CommMonoid α
n : ℕ
f : Fin n → α
i : ℕ
IH : prod (take i (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < i) univ, f j
h : ¬i < n
A : take i (ofFn f) = take (Nat.succ i) (ofFn f)
⊢ ∀ (j : Fin n), (↑j < Nat.succ i) = (↑j < i)

Next tactic:
intro j
@[to_additive]
theorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :
    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by
  induction i with
  | zero =>
    simp
  | succ i IH =>
    by_cases h : i < n
    · have : i < length (ofFn f) := by rwa [length_ofFn f]
      rw [prod_take_succ _ _ this]
      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =
          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by
        ext ⟨_, _⟩
        simp [Nat.lt_succ_iff_lt_or_eq]
      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)
          (singleton (⟨i, h⟩ : Fin n)) := by simp
      rw [A, Finset.prod_union B, IH]
      simp
    · have A : (ofFn f).take i = (ofFn f).take i.succ := by
        rw [← length_ofFn f] at h
        have : length (ofFn f) ≤ i := not_lt.mp h
        rw [take_all_of_le this, take_all_of_le (le_trans this (Nat.le_succ _))]
      have B : ∀ j : Fin n, ((j : ℕ) < i.succ) = ((j : ℕ) < i) := by
        intro j
        

Proof state:
α : Type u_1
β : Type u_2
inst✝ : CommMonoid α
n : ℕ
f : Fin n → α
i : ℕ
IH : prod (take i (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < i) univ, f j
h : ¬i < n
A : take i (ofFn f) = take (Nat.succ i) (ofFn f)
j : Fin n
⊢ (↑j < Nat.succ i) = (↑j < i)

Next tactic:
have : (j : ℕ) < i := lt_of_lt_of_le j.2 (not_lt.mp h)
@[to_additive]
theorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :
    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by
  induction i with
  | zero =>
    simp
  | succ i IH =>
    by_cases h : i < n
    · have : i < length (ofFn f) := by rwa [length_ofFn f]
      rw [prod_take_succ _ _ this]
      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =
          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by
        ext ⟨_, _⟩
        simp [Nat.lt_succ_iff_lt_or_eq]
      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)
          (singleton (⟨i, h⟩ : Fin n)) := by simp
      rw [A, Finset.prod_union B, IH]
      simp
    · have A : (ofFn f).take i = (ofFn f).take i.succ := by
        rw [← length_ofFn f] at h
        have : length (ofFn f) ≤ i := not_lt.mp h
        rw [take_all_of_le this, take_all_of_le (le_trans this (Nat.le_succ _))]
      have B : ∀ j : Fin n, ((j : ℕ) < i.succ) = ((j : ℕ) < i) := by
        intro j
        have : (j : ℕ) < i := lt_of_lt_of_le j.2 (not_lt.mp h)
        

Proof state:
α : Type u_1
β : Type u_2
inst✝ : CommMonoid α
n : ℕ
f : Fin n → α
i : ℕ
IH : prod (take i (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < i) univ, f j
h : ¬i < n
A : take i (ofFn f) = take (Nat.succ i) (ofFn f)
j : Fin n
this : ↑j < i
⊢ (↑j < Nat.succ i) = (↑j < i)

Next tactic:
simp [this, lt_trans this (Nat.lt_succ_self _)]
@[to_additive]
theorem prod_take_ofFn {n : ℕ} (f : Fin n → α) (i : ℕ) :
    ((ofFn f).take i).prod = ∏ j in Finset.univ.filter fun j : Fin n => j.val < i, f j := by
  induction i with
  | zero =>
    simp
  | succ i IH =>
    by_cases h : i < n
    · have : i < length (ofFn f) := by rwa [length_ofFn f]
      rw [prod_take_succ _ _ this]
      have A : ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =
          ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) ∪ {(⟨i, h⟩ : Fin n)} := by
        ext ⟨_, _⟩
        simp [Nat.lt_succ_iff_lt_or_eq]
      have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ)
          (singleton (⟨i, h⟩ : Fin n)) := by simp
      rw [A, Finset.prod_union B, IH]
      simp
    · have A : (ofFn f).take i = (ofFn f).take i.succ := by
        rw [← length_ofFn f] at h
        have : length (ofFn f) ≤ i := not_lt.mp h
        rw [take_all_of_le this, take_all_of_le (le_trans this (Nat.le_succ _))]
      have B : ∀ j : Fin n, ((j : ℕ) < i.succ) = ((j : ℕ) < i) := by
        intro j
        have : (j : ℕ) < i := lt_of_lt_of_le j.2 (not_lt.mp h)
        simp [this, lt_trans this (Nat.lt_succ_self _)]
      

Proof state:
case neg
α : Type u_1
β : Type u_2
inst✝ : CommMonoid α
n : ℕ
f : Fin n → α
i : ℕ
IH : prod (take i (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < i) univ, f j
h : ¬i < n
A : take i (ofFn f) = take (Nat.succ i) (ofFn f)
B : ∀ (j : Fin n), (↑j < Nat.succ i) = (↑j < i)
⊢ prod (take (Nat.succ i) (ofFn f)) = ∏ j in Finset.filter (fun j => ↑j < Nat.succ i) univ, f j

Next tactic:
simp [← A, B, IH]
@[to_additive]
theorem prod_ofFn {n : ℕ} {f : Fin n → α} : (ofFn f).prod = ∏ i, f i := by
  

Proof state:
α : Type u_1
β : Type u_2
inst✝ : CommMonoid α
n : ℕ
f : Fin n → α
⊢ prod (ofFn f) = ∏ i : Fin n, f i

Next tactic:
convert prod_take_ofFn f n
@[to_additive]
theorem prod_ofFn {n : ℕ} {f : Fin n → α} : (ofFn f).prod = ∏ i, f i := by
  convert prod_take_ofFn f n
  · 

Proof state:
case h.e'_2.h.e'_4
α : Type u_1
β : Type u_2
inst✝ : CommMonoid α
n : ℕ
f : Fin n → α
⊢ ofFn f = take n (ofFn f)

Next tactic:
rw [take_all_of_le (le_of_eq (length_ofFn f))]
@[to_additive]
theorem prod_ofFn {n : ℕ} {f : Fin n → α} : (ofFn f).prod = ∏ i, f i := by
  convert prod_take_ofFn f n
  · rw [take_all_of_le (le_of_eq (length_ofFn f))]
  · 

Proof state:
case h.e'_3.h
α : Type u_1
β : Type u_2
inst✝ : CommMonoid α
n : ℕ
f : Fin n → α
⊢ univ = Finset.filter (fun j => ↑j < n) univ

Next tactic:
simp
@[to_additive]
theorem alternatingProd_eq_finset_prod {G : Type*} [CommGroup G] :
    ∀ (L : List G), alternatingProd L = ∏ i : Fin L.length, L.get i ^ (-1 : ℤ) ^ (i : ℕ)
  | [] => by
    

Proof state:
α : Type u_1
β : Type u_2
G : Type u_3
inst✝ : CommGroup G
⊢ alternatingProd [] = ∏ i : Fin (length []), get [] i ^ (-1) ^ ↑i

Next tactic:
rw [alternatingProd, Finset.prod_eq_one]
@[to_additive]
theorem alternatingProd_eq_finset_prod {G : Type*} [CommGroup G] :
    ∀ (L : List G), alternatingProd L = ∏ i : Fin L.length, L.get i ^ (-1 : ℤ) ^ (i : ℕ)
  | [] => by
    rw [alternatingProd, Finset.prod_eq_one]
    

Proof state:
α : Type u_1
β : Type u_2
G : Type u_3
inst✝ : CommGroup G
⊢ ∀ x ∈ univ, get [] x ^ (-1) ^ ↑x = 1

Next tactic:
rintro ⟨i, ⟨⟩⟩
@[to_additive]
theorem alternatingProd_eq_finset_prod {G : Type*} [CommGroup G] :
    ∀ (L : List G), alternatingProd L = ∏ i : Fin L.length, L.get i ^ (-1 : ℤ) ^ (i : ℕ)
  | [] => by
    rw [alternatingProd, Finset.prod_eq_one]
    rintro ⟨i, ⟨⟩⟩
  | g::[] => by
    

Proof state:
α : Type u_1
β : Type u_2
G : Type u_3
inst✝ : CommGroup G
g : G
⊢ alternatingProd [g] = ∏ i : Fin (length [g]), get [g] i ^ (-1) ^ ↑i

Next tactic:
show g = ∏ i : Fin 1, [g].get i ^ (-1 : ℤ) ^ (i : ℕ)
@[to_additive]
theorem alternatingProd_eq_finset_prod {G : Type*} [CommGroup G] :
    ∀ (L : List G), alternatingProd L = ∏ i : Fin L.length, L.get i ^ (-1 : ℤ) ^ (i : ℕ)
  | [] => by
    rw [alternatingProd, Finset.prod_eq_one]
    rintro ⟨i, ⟨⟩⟩
  | g::[] => by
    show g = ∏ i : Fin 1, [g].get i ^ (-1 : ℤ) ^ (i : ℕ)
    

Proof state:
α : Type u_1
β : Type u_2
G : Type u_3
inst✝ : CommGroup G
g : G
⊢ g = ∏ i : Fin 1, get [g] i ^ (-1) ^ ↑i

Next tactic:
rw [Fin.prod_univ_succ]