awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Ring.Nat (goal)
Goal:
ι : Type u_1
M : Type u_2
f : ι → M
s : Finset M
hb : ∀ b ∈ s, Set.Finite.{u_1} (α := ι) {a : ι | f a = b}
s₁ s₂ s₃ : Nat.card.{u_1} (↑(f ⁻¹' (↑s : Set M)) : Type u_1) = ∑ b ∈ s, Nat.card.{u_1} { a : ι // f a = b }
s₄ :
  let t := Set.Finite.toFinset (s := (↑s : Set M) ∩ Set.range f) ⋯;
  Nat.card.{u_1} (↑(f ⁻¹' (↑t : Set M)) : Type u_1) = ∑ b ∈ s, Nat.card.{u_1} { a : ι // f a = b }
s₅ :
  let t := Set.Finite.toFinset (s := (↑s : Set M) ∩ Set.range f) ⋯;
  Nat.card.{u_1} (↑(f ⁻¹' (↑t : Set M)) : Type u_1) = ∑ b ∈ t, Nat.card.{u_1} (↑{a : ι | f a = b} : Type u_1)
⊢ Nat.card.{u_1} (↑(f ⁻¹' (↑s : Set M)) : Type u_1) = ∑ b ∈ s, Nat.card.{u_1} { a : ι // f a = b }
have s₆ : let t := Set.Finite.toFinset (s := (↑s : Set M) ∩ Set.range f) sorry;
∀ (ht : (f ⁻¹' (↑t : Set M)).Finite),
  Nat.card.{u_1} (↑(f ⁻¹' (↑t : Set M)) : Type u_1) = ∑ b ∈ t, Nat.card.{u_1} (↑{a : ι | f a = b} : Type u_1)
Tactic: classical
  -- `t = s ∩ Set.range f` as a `Finset`
  let t := (Set.finite_coe_iff.mp (Finite.Set.finite_inter_of_left ↑s (Set.range f))).toFinset
  rw [show Nat.card (f ⁻¹' s) = Nat.card (f ⁻¹' t) by simp [t]]
  rw [show ∑ b ∈ s, Nat.card {a //f a = b} = ∑ b ∈ t, Nat.card {a | f a = b} by
    exact (Finset.sum_subset (by simp [t]) (by aesop)).symm]
  have ht : Set.Finite (f ⁻¹' t) := Set.Finite.preimage' (finite_toSet t) (by aesop)
  rw [Nat.card_eq_card_finite_toFinset ht, Finset.card_eq_sum_card_image (f := f)]
  refine Finset.sum_congr ?_ fun m hm ↦ ?_
  · simpa [← Finset.coe_inj, t] using Set.image_preimage_eq_inter_range
  · rw [Nat.card_eq_card_finite_toFinset (hb _ (by aesop))]
    suffices {a | f a = m} ⊆ ht.toFinset from
      congr_arg (Finset.card ·) (Finset.ext_iff.mpr fun a ↦ by simpa using fun h ↦ this h)
    intro _ h
    simp_all
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Ring.Nat (goal)
Goal:
ι : Type u_1
M : Type u_2
f : ι → M
s : Finset M
hb : ∀ b ∈ s, Set.Finite.{u_1} (α := ι) {a : ι | f a = b}
s₁ s₂ s₃ : Nat.card.{u_1} (↑(f ⁻¹' (↑s : Set M)) : Type u_1) = ∑ b ∈ s, Nat.card.{u_1} { a : ι // f a = b }
s₄ :
  let t := Set.Finite.toFinset (s := (↑s : Set M) ∩ Set.range f) ⋯;
  Nat.card.{u_1} (↑(f ⁻¹' (↑t : Set M)) : Type u_1) = ∑ b ∈ s, Nat.card.{u_1} { a : ι // f a = b }
s₅ :
  let t := Set.Finite.toFinset (s := (↑s : Set M) ∩ Set.range f) ⋯;
  Nat.card.{u_1} (↑(f ⁻¹' (↑t : Set M)) : Type u_1) = ∑ b ∈ t, Nat.card.{u_1} (↑{a : ι | f a = b} : Type u_1)
s₆ :
  let t := Set.Finite.toFinset (s := (↑s : Set M) ∩ Set.range f) ⋯;
  ∀ (ht : (f ⁻¹' (↑t : Set M)).Finite),
    Nat.card.{u_1} (↑(f ⁻¹' (↑t : Set M)) : Type u_1) = ∑ b ∈ t, Nat.card.{u_1} (↑{a : ι | f a = b} : Type u_1)
⊢ Nat.card.{u_1} (↑(f ⁻¹' (↑s : Set M)) : Type u_1) = ∑ b ∈ s, Nat.card.{u_1} { a : ι // f a = b }
have s₇ : let t := Set.Finite.toFinset (s := (↑s : Set M) ∩ Set.range f) sorry;
∀ (ht : (f ⁻¹' (↑t : Set M)).Finite),
  Eq (α := ℕ)
    (∑ b ∈ image f (ht.toFinset (s := f ⁻¹' (↑t : Set M))), #({a ∈ ht.toFinset (s := f ⁻¹' (↑t : Set M)) | f a = b}))
    (∑ b ∈ t, Nat.card.{u_1} (↑{a : ι | f a = b} : Type u_1))
Tactic: classical
  -- `t = s ∩ Set.range f` as a `Finset`
  let t := (Set.finite_coe_iff.mp (Finite.Set.finite_inter_of_left ↑s (Set.range f))).toFinset
  rw [show Nat.card (f ⁻¹' s) = Nat.card (f ⁻¹' t) by simp [t]]
  rw [show ∑ b ∈ s, Nat.card {a //f a = b} = ∑ b ∈ t, Nat.card {a | f a = b} by
    exact (Finset.sum_subset (by simp [t]) (by aesop)).symm]
  have ht : Set.Finite (f ⁻¹' t) := Set.Finite.preimage' (finite_toSet t) (by aesop)
  rw [Nat.card_eq_card_finite_toFinset ht, Finset.card_eq_sum_card_image (f := f)]
  refine Finset.sum_congr ?_ fun m hm ↦ ?_
  · simpa [← Finset.coe_inj, t] using Set.image_preimage_eq_inter_range
  · rw [Nat.card_eq_card_finite_toFinset (hb _ (by aesop))]
    suffices {a | f a = m} ⊆ ht.toFinset from
      congr_arg (Finset.card ·) (Finset.ext_iff.mpr fun a ↦ by simpa using fun h ↦ this h)
    intro _ h
    simp_all
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Ring.Nat (goal)
Goal:
ι : Type u_1
M : Type u_2
f : ι → M
s : Finset M
hb : ∀ b ∈ s, Set.Finite.{u_1} (α := ι) {a : ι | f a = b}
s₁ s₂ s₃ : Nat.card.{u_1} (↑(f ⁻¹' (↑s : Set M)) : Type u_1) = ∑ b ∈ s, Nat.card.{u_1} { a : ι // f a = b }
s₄ :
  let t := Set.Finite.toFinset (s := (↑s : Set M) ∩ Set.range f) ⋯;
  Nat.card.{u_1} (↑(f ⁻¹' (↑t : Set M)) : Type u_1) = ∑ b ∈ s, Nat.card.{u_1} { a : ι // f a = b }
s₅ :
  let t := Set.Finite.toFinset (s := (↑s : Set M) ∩ Set.range f) ⋯;
  Nat.card.{u_1} (↑(f ⁻¹' (↑t : Set M)) : Type u_1) = ∑ b ∈ t, Nat.card.{u_1} (↑{a : ι | f a = b} : Type u_1)
s₆ :
  let t := Set.Finite.toFinset (s := (↑s : Set M) ∩ Set.range f) ⋯;
  ∀ (ht : (f ⁻¹' (↑t : Set M)).Finite),
    Nat.card.{u_1} (↑(f ⁻¹' (↑t : Set M)) : Type u_1) = ∑ b ∈ t, Nat.card.{u_1} (↑{a : ι | f a = b} : Type u_1)
s₇ :
  let t := Set.Finite.toFinset (s := (↑s : Set M) ∩ Set.range f) ⋯;
  ∀ (ht : (f ⁻¹' (↑t : Set M)).Finite),
    Eq (α := ℕ)
      (∑ b ∈ image f (ht.toFinset (s := f ⁻¹' (↑t : Set M))), #({a ∈ ht.toFinset (s := f ⁻¹' (↑t : Set M)) | f a = b}))
      (∑ b ∈ t, Nat.card.{u_1} (↑{a : ι | f a = b} : Type u_1))
⊢ Nat.card.{u_1} (↑(f ⁻¹' (↑s : Set M)) : Type u_1) = ∑ b ∈ s, Nat.card.{u_1} { a : ι // f a = b }
have refine_1 : let t := Set.Finite.toFinset (s := (↑s : Set M) ∩ Set.range f) sorry;
∀ (ht : (f ⁻¹' (↑t : Set M)).Finite), image f (ht.toFinset (s := f ⁻¹' (↑t : Set M))) = t
Tactic: classical
  -- `t = s ∩ Set.range f` as a `Finset`
  let t := (Set.finite_coe_iff.mp (Finite.Set.finite_inter_of_left ↑s (Set.range f))).toFinset
  rw [show Nat.card (f ⁻¹' s) = Nat.card (f ⁻¹' t) by simp [t]]
  rw [show ∑ b ∈ s, Nat.card {a //f a = b} = ∑ b ∈ t, Nat.card {a | f a = b} by
    exact (Finset.sum_subset (by simp [t]) (by aesop)).symm]
  have ht : Set.Finite (f ⁻¹' t) := Set.Finite.preimage' (finite_toSet t) (by aesop)
  rw [Nat.card_eq_card_finite_toFinset ht, Finset.card_eq_sum_card_image (f := f)]
  refine Finset.sum_congr ?_ fun m hm ↦ ?_
  · simpa [← Finset.coe_inj, t] using Set.image_preimage_eq_inter_range
  · rw [Nat.card_eq_card_finite_toFinset (hb _ (by aesop))]
    suffices {a | f a = m} ⊆ ht.toFinset from
      congr_arg (Finset.card ·) (Finset.ext_iff.mpr fun a ↦ by simpa using fun h ↦ this h)
    intro _ h
    simp_all
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Ring.Nat (goal)
Goal:
ι : Type u_1
M : Type u_2
f : ι → M
s : Finset M
hb : ∀ b ∈ s, Set.Finite.{u_1} (α := ι) {a : ι | f a = b}
s₁ s₂ s₃ : Nat.card.{u_1} (↑(f ⁻¹' (↑s : Set M)) : Type u_1) = ∑ b ∈ s, Nat.card.{u_1} { a : ι // f a = b }
s₄ :
  let t := Set.Finite.toFinset (s := (↑s : Set M) ∩ Set.range f) ⋯;
  Nat.card.{u_1} (↑(f ⁻¹' (↑t : Set M)) : Type u_1) = ∑ b ∈ s, Nat.card.{u_1} { a : ι // f a = b }
s₅ :
  let t := Set.Finite.toFinset (s := (↑s : Set M) ∩ Set.range f) ⋯;
  Nat.card.{u_1} (↑(f ⁻¹' (↑t : Set M)) : Type u_1) = ∑ b ∈ t, Nat.card.{u_1} (↑{a : ι | f a = b} : Type u_1)
s₆ :
  let t := Set.Finite.toFinset (s := (↑s : Set M) ∩ Set.range f) ⋯;
  ∀ (ht : (f ⁻¹' (↑t : Set M)).Finite),
    Nat.card.{u_1} (↑(f ⁻¹' (↑t : Set M)) : Type u_1) = ∑ b ∈ t, Nat.card.{u_1} (↑{a : ι | f a = b} : Type u_1)
s₇ :
  let t := Set.Finite.toFinset (s := (↑s : Set M) ∩ Set.range f) ⋯;
  ∀ (ht : (f ⁻¹' (↑t : Set M)).Finite),
    Eq (α := ℕ)
      (∑ b ∈ image f (ht.toFinset (s := f ⁻¹' (↑t : Set M))), #({a ∈ ht.toFinset (s := f ⁻¹' (↑t : Set M)) | f a = b}))
      (∑ b ∈ t, Nat.card.{u_1} (↑{a : ι | f a = b} : Type u_1))
refine_1 :
  let t := Set.Finite.toFinset (s := (↑s : Set M) ∩ Set.range f) ⋯;
  ∀ (ht : (f ⁻¹' (↑t : Set M)).Finite), image f (ht.toFinset (s := f ⁻¹' (↑t : Set M))) = t
⊢ Nat.card.{u_1} (↑(f ⁻¹' (↑s : Set M)) : Type u_1) = ∑ b ∈ s, Nat.card.{u_1} { a : ι // f a = b }
have refine_2 : let t := Set.Finite.toFinset (s := (↑s : Set M) ∩ Set.range f) sorry;
∀ (ht : (f ⁻¹' (↑t : Set M)).Finite) (m : M) (hm : m ∈ t),
  #({a ∈ ht.toFinset (s := f ⁻¹' (↑t : Set M)) | f a = m}) = Nat.card.{u_1} (↑{a : ι | f a = m} : Type u_1)
Tactic: classical
  -- `t = s ∩ Set.range f` as a `Finset`
  let t := (Set.finite_coe_iff.mp (Finite.Set.finite_inter_of_left ↑s (Set.range f))).toFinset
  rw [show Nat.card (f ⁻¹' s) = Nat.card (f ⁻¹' t) by simp [t]]
  rw [show ∑ b ∈ s, Nat.card {a //f a = b} = ∑ b ∈ t, Nat.card {a | f a = b} by
    exact (Finset.sum_subset (by simp [t]) (by aesop)).symm]
  have ht : Set.Finite (f ⁻¹' t) := Set.Finite.preimage' (finite_toSet t) (by aesop)
  rw [Nat.card_eq_card_finite_toFinset ht, Finset.card_eq_sum_card_image (f := f)]
  refine Finset.sum_congr ?_ fun m hm ↦ ?_
  · simpa [← Finset.coe_inj, t] using Set.image_preimage_eq_inter_range
  · rw [Nat.card_eq_card_finite_toFinset (hb _ (by aesop))]
    suffices {a | f a = m} ⊆ ht.toFinset from
      congr_arg (Finset.card ·) (Finset.ext_iff.mpr fun a ↦ by simpa using fun h ↦ this h)
    intro _ h
    simp_all
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Sym (goal)
Goal:
ι : Type u_1
M : Type u_2
inst : LinearOrder ι
inst_1 : AddCommMonoid M
s : Finset ι
p : Sym2 ι → M
⊢ Eq.{u_2 + 1} (α := M) (∑ i ∈ s.sym2 with ¬i.IsDiag, p i) (∑ i ∈ s.offDiag with i.1 < i.2, p s(i.1, i.2))
have rw : Eq.{u_2 + 1} (α := M) (∑ i ∈ s.sym2 with ¬i.IsDiag, p i) (∑ i ∈ s.offDiag with i.1 ≤ i.2, p s(i.1, i.2))
Tactic: rw [Finset.offDiag_filter_lt_eq_filter_le]
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Sym (goal)
Goal:
ι : Type u_1
M : Type u_2
inst : LinearOrder ι
inst_1 : AddCommMonoid M
s : Finset ι
p : Sym2 ι → M
| ∑ i ∈ s.offDiag with i.1 ≤ i.2, p s(i.1, i.2)
have rw : ∑ x ∈ Finset.subtype (fun (i : ι × ι) ↦ i.1 ≤ i.2) s.offDiag, p s((↑x : ι × ι).1, (↑x : ι × ι).2)
Tactic: rw [← Finset.sum_subtype_eq_sum_filter]
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Sym (goal)
Goal:
ι : Type u_1
M : Type u_2
inst : LinearOrder ι
inst_1 : AddCommMonoid M
s : Finset ι
p : Sym2 ι → M
⊢ Eq.{u_2 + 1} (α := M) (∑ i ∈ s.sym2 with ¬i.IsDiag, p i) (∑ i ∈ s.offDiag with i.1 ≤ i.2, p s(i.1, i.2))
have conv_rhs : Eq.{u_2 + 1} (α := M) (∑ i ∈ s.sym2 with ¬i.IsDiag, p i)
  (∑ x ∈ Finset.subtype (fun (i : ι × ι) ↦ i.1 ≤ i.2) s.offDiag, p s((↑x : ι × ι).1, (↑x : ι × ι).2))
Tactic: conv_rhs => rw [← Finset.sum_subtype_eq_sum_filter]
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Sym (goal)
Goal:
ι : Type u_1
M : Type u_2
inst : LinearOrder ι
inst_1 : AddCommMonoid M
s : Finset ι
p : Sym2 ι → M
⊢ Eq.{u_2 + 1} (α := M) (∑ i ∈ s.sym2 with ¬i.IsDiag, p i)
    (∑ x ∈ Finset.subtype (fun (i : ι × ι) ↦ i.1 ≤ i.2) s.offDiag, p s((↑x : ι × ι).1, (↑x : ι × ι).2))
have refine_1 : ∀ (i : { p : ι × ι // p.1 ≤ p.2 }),
  Membership.mem (γ := Finset { p : ι × ι // p.1 ≤ p.2 }) (Finset.subtype (fun (i : ι × ι) ↦ i.1 ≤ i.2) s.offDiag) i ↔
    Membership.mem (α := Sym2 ι) ({i ∈ s.sym2 | ¬i.IsDiag})
      ((Equiv.symm Sym2.sortEquiv : (a : { p : ι × ι // p.1 ≤ p.2 }) → Sym2 ι) i : Sym2 ι)
Tactic: refine (Finset.sum_equiv Sym2.sortEquiv.symm ?_ ?_).symm
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.Sym (goal)
Goal:
ι : Type u_1
M : Type u_2
inst : LinearOrder ι
inst_1 : AddCommMonoid M
s : Finset ι
p : Sym2 ι → M
refine_1 :
  ∀ (i : { p : ι × ι // p.1 ≤ p.2 }),
    Membership.mem (γ := Finset { p : ι × ι // p.1 ≤ p.2 }) (Finset.subtype (fun (i : ι × ι) ↦ i.1 ≤ i.2) s.offDiag) i ↔
      Membership.mem (α := Sym2 ι) ({i ∈ s.sym2 | ¬i.IsDiag})
        ((Equiv.symm Sym2.sortEquiv : (a : { p : ι × ι // p.1 ≤ p.2 }) → Sym2 ι) i : Sym2 ι)
⊢ Eq.{u_2 + 1} (α := M) (∑ i ∈ s.sym2 with ¬i.IsDiag, p i)
    (∑ x ∈ Finset.subtype (fun (i : ι × ι) ↦ i.1 ≤ i.2) s.offDiag, p s((↑x : ι × ι).1, (↑x : ι × ι).2))
have refine_2 : ∀ (i : { p : ι × ι // p.1 ≤ p.2 }),
  Membership.mem (γ := Finset { p : ι × ι // p.1 ≤ p.2 }) (Finset.subtype (fun (i : ι × ι) ↦ i.1 ≤ i.2) s.offDiag) i →
    p s((↑i : ι × ι).1, (↑i : ι × ι).2) = p ((Equiv.symm Sym2.sortEquiv : (a : { p : ι × ι // p.1 ≤ p.2 }) → Sym2 ι) i)
Tactic: refine (Finset.sum_equiv Sym2.sortEquiv.symm ?_ ?_).symm
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.WithTop (goal)
Goal:
ι : Type u_1
M : Type u_2
inst : AddCommMonoid M
s : Finset ι
f : ι → WithTop M
⊢ Eq.{u_2 + 1} (α := WithTop M) (∑ i ∈ s, f i) ⊤ ↔ ∃ i ∈ s, f i = ⊤
have empty : Eq.{u_2 + 1} (α := WithTop M) (∑ i ∈ ∅, f i) ⊤ ↔ ∃ (i : ι), Membership.mem (γ := Finset ι) ∅ i ∧ f i = ⊤
Tactic: induction s using Finset.cons_induction
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.WithTop (goal)
Goal:
ι : Type u_1
M : Type u_2
inst : AddCommMonoid M
s : Finset ι
f : ι → WithTop M
empty : Eq.{u_2 + 1} (α := WithTop M) (∑ i ∈ ∅, f i) ⊤ ↔ ∃ (i : ι), Membership.mem (γ := Finset ι) ∅ i ∧ f i = ⊤
⊢ Eq.{u_2 + 1} (α := WithTop M) (∑ i ∈ s, f i) ⊤ ↔ ∃ i ∈ s, f i = ⊤
have cons : ∀ (a : ι) (s : Finset ι) (h : a ∉ s),
  (Eq.{u_2 + 1} (α := WithTop M) (∑ i ∈ s, f i) ⊤ ↔ ∃ i ∈ s, f i = ⊤) →
    (Eq.{u_2 + 1} (α := WithTop M) (∑ i ∈ cons a s h, f i) ⊤ ↔ ∃ i ∈ cons a s h, f i = ⊤)
Tactic: induction s using Finset.cons_induction
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.WithTop (goal)
Goal:
ι : Type u_1
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : NoZeroDivisors M₀
inst_2 : Nontrivial M₀
inst_3 : DecidableEq M₀
s : Finset ι
f : ι → WithTop M₀
i : ι
hi : i ∈ s
hi' : f i = ⊤
h : ∀ j ∈ s, f j ≠ 0
⊢ Eq.{u_3 + 1} (α := WithTop M₀) (∏ j ∈ s, f j) ⊤
have rw : (∏ x ∈ s.erase i, f x) * f i = ⊤
Tactic: rw [← prod_erase_mul _ _ hi]
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.WithTop (goal)
Goal:
ι : Type u_1
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : NoZeroDivisors M₀
inst_2 : Nontrivial M₀
inst_3 : DecidableEq M₀
s : Finset ι
f : ι → WithTop M₀
i : ι
hi : i ∈ s
hi' : f i = ⊤
h : ∀ j ∈ s, f j ≠ 0
⊢ Eq.{u_3 + 1} (α := WithTop M₀) (∏ j ∈ s, f j) ⊤
have classical : (∏ x ∈ s.erase i, f x) * f i = ⊤
Tactic: classical rw [← prod_erase_mul _ _ hi]
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.WithTop (goal)
Goal:
ι : Type u_1
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : NoZeroDivisors M₀
inst_2 : Nontrivial M₀
inst_3 : DecidableEq M₀
s : Finset ι
f : ι → WithTop M₀
i : ι
hi : i ∈ s
hi' : f i = ⊤
h : ∀ j ∈ s, f j ≠ 0
⊢ (∏ x ∈ s.erase i, f x) * f i = ⊤
have refine : Ne.{u_3 + 1} (α := WithTop M₀) (∏ x ∈ s.erase i, f x) 0
Tactic: refine WithTop.mul_eq_top_iff.mpr (Or.inl ⟨?_, hi'⟩)
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.WithTop (goal)
Goal:
ι : Type u_1
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : NoZeroDivisors M₀
inst_2 : Nontrivial M₀
inst_3 : DecidableEq M₀
s : Finset ι
f : ι → WithTop M₀
i : ι
hi : i ∈ s
hi' : f i = ⊤
h : ∀ j ∈ s, f j ≠ 0
⊢ Ne.{u_3 + 1} (α := WithTop M₀) (∏ x ∈ s.erase i, f x) 0
have refine : ∀ a ∈ s.erase i, f a ≠ 0
Tactic: refine prod_ne_zero_iff.mpr ?_
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.WithTop (goal)
Goal:
ι : Type u_1
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : NoZeroDivisors M₀
inst_2 : Nontrivial M₀
inst_3 : DecidableEq M₀
s : Finset ι
f : ι → WithTop M₀
i : ι
hi : i ∈ s
h : Eq.{u_3 + 1} (α := WithTop M₀) (∏ j ∈ s, f j) ⊤
h0 : f i = 0
⊢ False
have apply : Eq.{u_3 + 1} (α := WithTop M₀) ⊤ 0
Tactic: apply WithTop.top_ne_zero (α := M₀)
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.WithTop (goal)
Goal:
ι : Type u_1
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : NoZeroDivisors M₀
inst_2 : Nontrivial M₀
inst_3 : DecidableEq M₀
s : Finset ι
f : ι → WithTop M₀
h : Eq.{u_3 + 1} (α := WithTop M₀) (∏ j ∈ s, f j) ⊤
⊢ ∃ i ∈ s, f i = ⊤
have contrapose! : ∀ (h : ∀ (i : ι) (_ : i ∈ s), f i ≠ ⊤), Ne.{u_3 + 1} (α := WithTop M₀) (∏ j ∈ s, f j) ⊤
Tactic: contrapose! h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.WithTop (goal)
Goal:
ι : Type u_1
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : NoZeroDivisors M₀
inst_2 : Nontrivial M₀
inst_3 : DecidableEq M₀
s : Finset ι
f : ι → WithTop M₀
⊢ Eq.{u_3 + 1} (α := WithTop M₀) (∏ j ∈ s, f j) ⊤ ↔ (∃ i ∈ s, f i = ⊤) ∧ ∀ i ∈ s, f i ≠ 0
have mp : Eq.{u_3 + 1} (α := WithTop M₀) (∏ j ∈ s, f j) ⊤ → (∃ i ∈ s, f i = ⊤) ∧ ∀ i ∈ s, f i ≠ 0
Tactic: constructor
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.WithTop (goal)
Goal:
ι : Type u_1
M₀ : Type u_3
inst : CommMonoidWithZero M₀
inst_1 : NoZeroDivisors M₀
inst_2 : Nontrivial M₀
inst_3 : DecidableEq M₀
s : Finset ι
f : ι → WithTop M₀
mp : Eq.{u_3 + 1} (α := WithTop M₀) (∏ j ∈ s, f j) ⊤ → (∃ i ∈ s, f i = ⊤) ∧ ∀ i ∈ s, f i ≠ 0
⊢ Eq.{u_3 + 1} (α := WithTop M₀) (∏ j ∈ s, f j) ⊤ ↔ (∃ i ∈ s, f i = ⊤) ∧ ∀ i ∈ s, f i ≠ 0
have mpr : ((∃ i ∈ s, f i = ⊤) ∧ ∀ i ∈ s, f i ≠ 0) → Eq.{u_3 + 1} (α := WithTop M₀) (∏ j ∈ s, f j) ⊤
Tactic: constructor
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.WithTop (goal)
Goal:
ι : Type u_1
M : Type u_2
inst : AddCommMonoid M
s : Finset ι
f : ι → WithBot M
⊢ Eq.{u_2 + 1} (α := WithBot M) (∑ i ∈ s, f i) ⊥ ↔ ∃ i ∈ s, f i = ⊥
have empty : Eq.{u_2 + 1} (α := WithBot M) (∑ i ∈ ∅, f i) ⊥ ↔ ∃ (i : ι), Membership.mem (γ := Finset ι) ∅ i ∧ f i = ⊥
Tactic: induction s using Finset.cons_induction
-- Lean 4 proof step from Mathlib.Algebra.BigOperators.WithTop (goal)
Goal:
ι : Type u_1
M : Type u_2
inst : AddCommMonoid M
s : Finset ι
f : ι → WithBot M
empty : Eq.{u_2 + 1} (α := WithBot M) (∑ i ∈ ∅, f i) ⊥ ↔ ∃ (i : ι), Membership.mem (γ := Finset ι) ∅ i ∧ f i = ⊥
⊢ Eq.{u_2 + 1} (α := WithBot M) (∑ i ∈ s, f i) ⊥ ↔ ∃ i ∈ s, f i = ⊥
have cons : ∀ (a : ι) (s : Finset ι) (h : a ∉ s),
  (Eq.{u_2 + 1} (α := WithBot M) (∑ i ∈ s, f i) ⊥ ↔ ∃ i ∈ s, f i = ⊥) →
    (Eq.{u_2 + 1} (α := WithBot M) (∑ i ∈ cons a s h, f i) ⊥ ↔ ∃ i ∈ cons a s h, f i = ⊥)
Tactic: induction s using Finset.cons_induction
-- Lean 4 proof step from Mathlib.Algebra.BrauerGroup.Defs (goal)
Goal:
K : Type u
inst : Field K
A : CSA.{u, u_1} K
B : CSA.{u, u_2} K
C : CSA.{u, u_3} K
hAB : IsBrauerEquivalent A B
hBC : IsBrauerEquivalent B C
⊢ IsBrauerEquivalent A C
have obtain : ∀ (n m : ℕ) (hn : n ≠ 0) (hm : m ≠ 0)
  (iso1 :
    AlgEquiv.{u, u_1, u_2} K (Matrix (Fin n) (Fin n) (↑A.toAlgCat : Type u_1))
      (Matrix (Fin m) (Fin m) (↑B.toAlgCat : Type u_2))),
  IsBrauerEquivalent A C
Tactic: obtain ⟨n, m, hn, hm, ⟨iso1⟩⟩ := hAB
Cleared hypotheses: hAB
-- Lean 4 proof step from Mathlib.Algebra.BrauerGroup.Defs (goal)
Goal:
K : Type u
inst : Field K
A : CSA.{u, u_1} K
B : CSA.{u, u_2} K
C : CSA.{u, u_3} K
hBC : IsBrauerEquivalent B C
n m : ℕ
hn : n ≠ 0
hm : m ≠ 0
iso1 :
  AlgEquiv.{u, u_1, u_2} K (Matrix (Fin n) (Fin n) (↑A.toAlgCat : Type u_1))
    (Matrix (Fin m) (Fin m) (↑B.toAlgCat : Type u_2))
⊢ IsBrauerEquivalent A C
have obtain : ∀ (p q : ℕ) (hp : p ≠ 0) (hq : q ≠ 0)
  (iso2 :
    AlgEquiv.{u, u_2, u_3} K (Matrix (Fin p) (Fin p) (↑B.toAlgCat : Type u_2))
      (Matrix (Fin q) (Fin q) (↑C.toAlgCat : Type u_3))),
  IsBrauerEquivalent A C
Tactic: obtain ⟨p, q, hp, hq, ⟨iso2⟩⟩ := hBC
Cleared hypotheses: hBC
-- Lean 4 proof step from Mathlib.Algebra.Category.AlgCat.Limits (goal)
Goal:
R : Type u
inst : CommRing R
J : Type v
inst_1 : Category.{t, v} J
F : J ⥤ AlgCat.{w, u} R
inst_2 :
  Small.{w, max v w}
    (↑(F ⋙
            forget (AlgCat.{w, u} R) (FC := fun (x1 x2 : AlgCat.{w, u} R) ↦ (↑x1 : Type w) →ₐ[R] (↑x2 : Type w)) (CC :=
              carrier)).sections :
      Type (max v w))
x x_1 : J
f : x ⟶ x_1
⊢ Eq.{w + 1} (α :=
    Functor.obj
        (Functor.obj (Functor.const J)
          (of R
            (Types.Small.limitCone
                (F ⋙
                  forget (AlgCat.{w, u} R) (FC := fun (x1 x2 : AlgCat.{w, u} R) ↦ (↑x1 : Type w) →ₐ[R] (↑x2 : Type w))
                    (CC := carrier))).pt))
        x ⟶
      F.obj x_1)
    (CategoryTheory.Functor.map
        (Functor.obj (Functor.const J)
          (of R
            (Types.Small.limitCone
                (F ⋙
                  forget (AlgCat.{w, u} R) (FC := fun (x1 x2 : AlgCat.{w, u} R) ↦ (↑x1 : Type w) →ₐ[R] (↑x2 : Type w))
                    (CC := carrier))).pt))
        f ≫
      ofHom (limitπAlgHom F x_1))
    (ofHom (limitπAlgHom F x) ≫ F.map f)
have hf : ∀
  (x_2 :
    (↑(Functor.obj
          (Functor.obj (Functor.const J)
            (of R
              (Types.Small.limitCone
                  (F ⋙
                    forget (AlgCat.{w, u} R) (FC := fun (x1 x2 : AlgCat.{w, u} R) ↦ (↑x1 : Type w) →ₐ[R] (↑x2 : Type w))
                      (CC := carrier))).pt))
          x) :
      Type w)),
  Eq (α := (↑(F.obj x_1) : Type w))
    ((Hom.hom
            (CategoryTheory.Functor.map
                (Functor.obj (Functor.const J)
                  (of R
                    (Types.Small.limitCone
                        (F ⋙
                          forget (AlgCat.{w, u} R) (FC := fun (x1 x2 : AlgCat.{w, u} R) ↦
                            (↑x1 : Type w) →ₐ[R] (↑x2 : Type w)) (CC := carrier))).pt))
                f ≫
              ofHom (limitπAlgHom F x_1)) :
          (a : (↑(((Functor.const J).obj (of R (Types.Small.limitCone (F ⋙ forget (AlgCat R))).pt)).obj x) : Type w)) →
            (↑(F.obj x_1) : Type w))
        x_2 :
      (↑(F.obj x_1) : Type w))
    ((Hom.hom (ofHom (limitπAlgHom F x) ≫ F.map f) :
          (a : (↑(((Functor.const J).obj (of R (Types.Small.limitCone (F ⋙ forget (AlgCat R))).pt)).obj x) : Type w)) →
            (↑(F.obj x_1) : Type w))
        x_2 :
      (↑(F.obj x_1) : Type w))
Tactic: ext
-- Lean 4 proof step from Mathlib.Algebra.Category.AlgCat.Limits (goal)
Goal:
R : Type u
inst : CommRing R
J : Type v
inst_1 : Category.{t, v} J
F : J ⥤ AlgCat.{w, u} R
inst_2 :
  Small.{w, max v w}
    (↑(F ⋙
            forget (AlgCat.{w, u} R) (FC := fun (x1 x2 : AlgCat.{w, u} R) ↦ (↑x1 : Type w) →ₐ[R] (↑x2 : Type w)) (CC :=
              carrier)).sections :
      Type (max v w))
⊢ IsLimit (limitCone F)
have refine_1 : ∀ (s : Cone F),
  EquivLike.coe (E :=
      (↑(F ⋙
                forget (AlgCat.{w, u} R) (FC := fun (x1 x2 : AlgCat.{w, u} R) ↦ (↑x1 : Type w) →ₐ[R] (↑x2 : Type w))
                  (CC := carrier)).sections :
          Type (max w v)) ≃
        Shrink.{w, max w v}
          (↑(F ⋙
                  forget (AlgCat.{w, u} R) (FC := fun (x1 x2 : AlgCat.{w, u} R) ↦ (↑x1 : Type w) →ₐ[R] (↑x2 : Type w))
                    (CC := carrier)).sections :
            Type (max w v)))
      (equivShrink
        (↑(F ⋙
                forget (AlgCat.{w, u} R) (FC := fun (x1 x2 : AlgCat.{w, u} R) ↦ (↑x1 : Type w) →ₐ[R] (↑x2 : Type w))
                  (CC := carrier)).sections :
          Type (max w v)))
      (Subtype.mk (p :=
        fun
          (x :
            (j : J) →
              (F ⋙
                    forget (AlgCat.{w, u} R) (FC := fun (x1 x2 : AlgCat.{w, u} R) ↦ (↑x1 : Type w) →ₐ[R] (↑x2 : Type w))
                      (CC := carrier)).obj
                j) ↦
        x ∈
          (F ⋙
              forget (AlgCat.{w, u} R) (FC := fun (x1 x2 : AlgCat.{w, u} R) ↦ (↑x1 : Type w) →ₐ[R] (↑x2 : Type w))
                (CC := carrier)).sections)
        (fun (j : J) ↦
          (ConcreteCategory.hom (FC := TypeCat.Fun)
                (NatTrans.app
                  (Cone.π
                    (Functor.mapCone
                      (forget (AlgCat.{w, u} R) (FC := fun (x1 x2 : AlgCat.{w, u} R) ↦
                        (↑x1 : Type w) →ₐ[R] (↑x2 : Type w)) (CC := carrier))
                      s))
                  j) :
              (a : (fun (X : Type w) ↦ X) (((Functor.const J).obj ((forget (AlgCat R)).mapCone s).pt).obj j)) →
                (fun (X : Type w) ↦ X) ((F ⋙ forget (AlgCat R)).obj j))
            1)
        sorry) =
    1
Tactic: refine
    IsLimit.ofFaithful (forget (AlgCat R)) (Types.Small.limitConeIsLimit.{v, w} _)
      (fun s => ofHom
        { toFun := _, map_one' := ?_, map_mul' := ?_, map_zero' := ?_, map_add' := ?_,
          commutes' := ?_ })
      (fun s => rfl)