l3lab/ntp-mathlib huggingface dataset:
/-- Reduction to diagonal form by elementary operations is invariant under reindexing. -/
theorem reindex_exists_list_transvec_mul_mul_list_transvec_eq_diagonal (M : Matrix p p 𝕜)
    (e : p ≃ n)
    (H :
      ∃ (L L' : List (TransvectionStruct n 𝕜)) (D : n → 𝕜),
        (L.map toMatrix).prod * Matrix.reindexAlgEquiv 𝕜 e M * (L'.map toMatrix).prod =
          diagonal D) :
    ∃ (L L' : List (TransvectionStruct p 𝕜)) (D : p → 𝕜),
      (L.map toMatrix).prod * M * (L'.map toMatrix).prod = diagonal D := by
  rcases H with ⟨L₀, L₀', D₀, h₀⟩
  refine' ⟨L₀.map (reindexEquiv e.symm), L₀'.map (reindexEquiv e.symm), D₀ ∘ e, _⟩
  have : M = reindexAlgEquiv 𝕜 e.symm (reindexAlgEquiv 𝕜 e M) := by
    

Proof state:
n : Type u_1
p : Type u_2
R : Type u₂
𝕜 : Type u_3
inst✝⁵ : Field 𝕜
inst✝⁴ : DecidableEq n
inst✝³ : DecidableEq p
inst✝² : CommRing R
r : ℕ
M✝ : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
inst✝¹ : Fintype n
inst✝ : Fintype p
M : Matrix p p 𝕜
e : p ≃ n
L₀ L₀' : List (TransvectionStruct n 𝕜)
D₀ : n → 𝕜
h₀ : List.prod (List.map toMatrix L₀) * (reindexAlgEquiv 𝕜 e) M * List.prod (List.map toMatrix L₀') = diagonal D₀
⊢ M = (reindexAlgEquiv 𝕜 e.symm) ((reindexAlgEquiv 𝕜 e) M)

Next tactic:
simp only [Equiv.symm_symm, submatrix_submatrix, reindex_apply, submatrix_id_id,
      Equiv.symm_comp_self, reindexAlgEquiv_apply]
/-- Reduction to diagonal form by elementary operations is invariant under reindexing. -/
theorem reindex_exists_list_transvec_mul_mul_list_transvec_eq_diagonal (M : Matrix p p 𝕜)
    (e : p ≃ n)
    (H :
      ∃ (L L' : List (TransvectionStruct n 𝕜)) (D : n → 𝕜),
        (L.map toMatrix).prod * Matrix.reindexAlgEquiv 𝕜 e M * (L'.map toMatrix).prod =
          diagonal D) :
    ∃ (L L' : List (TransvectionStruct p 𝕜)) (D : p → 𝕜),
      (L.map toMatrix).prod * M * (L'.map toMatrix).prod = diagonal D := by
  rcases H with ⟨L₀, L₀', D₀, h₀⟩
  refine' ⟨L₀.map (reindexEquiv e.symm), L₀'.map (reindexEquiv e.symm), D₀ ∘ e, _⟩
  have : M = reindexAlgEquiv 𝕜 e.symm (reindexAlgEquiv 𝕜 e M) := by
    simp only [Equiv.symm_symm, submatrix_submatrix, reindex_apply, submatrix_id_id,
      Equiv.symm_comp_self, reindexAlgEquiv_apply]
  

Proof state:
case intro.intro.intro
n : Type u_1
p : Type u_2
R : Type u₂
𝕜 : Type u_3
inst✝⁵ : Field 𝕜
inst✝⁴ : DecidableEq n
inst✝³ : DecidableEq p
inst✝² : CommRing R
r : ℕ
M✝ : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
inst✝¹ : Fintype n
inst✝ : Fintype p
M : Matrix p p 𝕜
e : p ≃ n
L₀ L₀' : List (TransvectionStruct n 𝕜)
D₀ : n → 𝕜
h₀ : List.prod (List.map toMatrix L₀) * (reindexAlgEquiv 𝕜 e) M * List.prod (List.map toMatrix L₀') = diagonal D₀
this : M = (reindexAlgEquiv 𝕜 e.symm) ((reindexAlgEquiv 𝕜 e) M)
⊢ List.prod (List.map toMatrix (List.map (reindexEquiv e.symm) L₀)) * M *
      List.prod (List.map toMatrix (List.map (reindexEquiv e.symm) L₀')) =
    diagonal (D₀ ∘ ⇑e)

Next tactic:
rw [this]
/-- Reduction to diagonal form by elementary operations is invariant under reindexing. -/
theorem reindex_exists_list_transvec_mul_mul_list_transvec_eq_diagonal (M : Matrix p p 𝕜)
    (e : p ≃ n)
    (H :
      ∃ (L L' : List (TransvectionStruct n 𝕜)) (D : n → 𝕜),
        (L.map toMatrix).prod * Matrix.reindexAlgEquiv 𝕜 e M * (L'.map toMatrix).prod =
          diagonal D) :
    ∃ (L L' : List (TransvectionStruct p 𝕜)) (D : p → 𝕜),
      (L.map toMatrix).prod * M * (L'.map toMatrix).prod = diagonal D := by
  rcases H with ⟨L₀, L₀', D₀, h₀⟩
  refine' ⟨L₀.map (reindexEquiv e.symm), L₀'.map (reindexEquiv e.symm), D₀ ∘ e, _⟩
  have : M = reindexAlgEquiv 𝕜 e.symm (reindexAlgEquiv 𝕜 e M) := by
    simp only [Equiv.symm_symm, submatrix_submatrix, reindex_apply, submatrix_id_id,
      Equiv.symm_comp_self, reindexAlgEquiv_apply]
  rw [this]
  

Proof state:
case intro.intro.intro
n : Type u_1
p : Type u_2
R : Type u₂
𝕜 : Type u_3
inst✝⁵ : Field 𝕜
inst✝⁴ : DecidableEq n
inst✝³ : DecidableEq p
inst✝² : CommRing R
r : ℕ
M✝ : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
inst✝¹ : Fintype n
inst✝ : Fintype p
M : Matrix p p 𝕜
e : p ≃ n
L₀ L₀' : List (TransvectionStruct n 𝕜)
D₀ : n → 𝕜
h₀ : List.prod (List.map toMatrix L₀) * (reindexAlgEquiv 𝕜 e) M * List.prod (List.map toMatrix L₀') = diagonal D₀
this : M = (reindexAlgEquiv 𝕜 e.symm) ((reindexAlgEquiv 𝕜 e) M)
⊢ List.prod (List.map toMatrix (List.map (reindexEquiv e.symm) L₀)) *
        (reindexAlgEquiv 𝕜 e.symm) ((reindexAlgEquiv 𝕜 e) M) *
      List.prod (List.map toMatrix (List.map (reindexEquiv e.symm) L₀')) =
    diagonal (D₀ ∘ ⇑e)

Next tactic:
simp only [toMatrix_reindexEquiv_prod, List.map_map, reindexAlgEquiv_apply]
/-- Reduction to diagonal form by elementary operations is invariant under reindexing. -/
theorem reindex_exists_list_transvec_mul_mul_list_transvec_eq_diagonal (M : Matrix p p 𝕜)
    (e : p ≃ n)
    (H :
      ∃ (L L' : List (TransvectionStruct n 𝕜)) (D : n → 𝕜),
        (L.map toMatrix).prod * Matrix.reindexAlgEquiv 𝕜 e M * (L'.map toMatrix).prod =
          diagonal D) :
    ∃ (L L' : List (TransvectionStruct p 𝕜)) (D : p → 𝕜),
      (L.map toMatrix).prod * M * (L'.map toMatrix).prod = diagonal D := by
  rcases H with ⟨L₀, L₀', D₀, h₀⟩
  refine' ⟨L₀.map (reindexEquiv e.symm), L₀'.map (reindexEquiv e.symm), D₀ ∘ e, _⟩
  have : M = reindexAlgEquiv 𝕜 e.symm (reindexAlgEquiv 𝕜 e M) := by
    simp only [Equiv.symm_symm, submatrix_submatrix, reindex_apply, submatrix_id_id,
      Equiv.symm_comp_self, reindexAlgEquiv_apply]
  rw [this]
  simp only [toMatrix_reindexEquiv_prod, List.map_map, reindexAlgEquiv_apply]
  

Proof state:
case intro.intro.intro
n : Type u_1
p : Type u_2
R : Type u₂
𝕜 : Type u_3
inst✝⁵ : Field 𝕜
inst✝⁴ : DecidableEq n
inst✝³ : DecidableEq p
inst✝² : CommRing R
r : ℕ
M✝ : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
inst✝¹ : Fintype n
inst✝ : Fintype p
M : Matrix p p 𝕜
e : p ≃ n
L₀ L₀' : List (TransvectionStruct n 𝕜)
D₀ : n → 𝕜
h₀ : List.prod (List.map toMatrix L₀) * (reindexAlgEquiv 𝕜 e) M * List.prod (List.map toMatrix L₀') = diagonal D₀
this : M = (reindexAlgEquiv 𝕜 e.symm) ((reindexAlgEquiv 𝕜 e) M)
⊢ (reindex e.symm e.symm) (List.prod (List.map toMatrix L₀)) * (reindex e.symm e.symm) ((reindex e e) M) *
      (reindex e.symm e.symm) (List.prod (List.map toMatrix L₀')) =
    diagonal (D₀ ∘ ⇑e)

Next tactic:
simp only [← reindexAlgEquiv_apply, ← reindexAlgEquiv_mul, h₀]
/-- Reduction to diagonal form by elementary operations is invariant under reindexing. -/
theorem reindex_exists_list_transvec_mul_mul_list_transvec_eq_diagonal (M : Matrix p p 𝕜)
    (e : p ≃ n)
    (H :
      ∃ (L L' : List (TransvectionStruct n 𝕜)) (D : n → 𝕜),
        (L.map toMatrix).prod * Matrix.reindexAlgEquiv 𝕜 e M * (L'.map toMatrix).prod =
          diagonal D) :
    ∃ (L L' : List (TransvectionStruct p 𝕜)) (D : p → 𝕜),
      (L.map toMatrix).prod * M * (L'.map toMatrix).prod = diagonal D := by
  rcases H with ⟨L₀, L₀', D₀, h₀⟩
  refine' ⟨L₀.map (reindexEquiv e.symm), L₀'.map (reindexEquiv e.symm), D₀ ∘ e, _⟩
  have : M = reindexAlgEquiv 𝕜 e.symm (reindexAlgEquiv 𝕜 e M) := by
    simp only [Equiv.symm_symm, submatrix_submatrix, reindex_apply, submatrix_id_id,
      Equiv.symm_comp_self, reindexAlgEquiv_apply]
  rw [this]
  simp only [toMatrix_reindexEquiv_prod, List.map_map, reindexAlgEquiv_apply]
  simp only [← reindexAlgEquiv_apply, ← reindexAlgEquiv_mul, h₀]
  

Proof state:
case intro.intro.intro
n : Type u_1
p : Type u_2
R : Type u₂
𝕜 : Type u_3
inst✝⁵ : Field 𝕜
inst✝⁴ : DecidableEq n
inst✝³ : DecidableEq p
inst✝² : CommRing R
r : ℕ
M✝ : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
inst✝¹ : Fintype n
inst✝ : Fintype p
M : Matrix p p 𝕜
e : p ≃ n
L₀ L₀' : List (TransvectionStruct n 𝕜)
D₀ : n → 𝕜
h₀ : List.prod (List.map toMatrix L₀) * (reindexAlgEquiv 𝕜 e) M * List.prod (List.map toMatrix L₀') = diagonal D₀
this : M = (reindexAlgEquiv 𝕜 e.symm) ((reindexAlgEquiv 𝕜 e) M)
⊢ (reindexAlgEquiv 𝕜 e.symm) (diagonal D₀) = diagonal (D₀ ∘ ⇑e)

Next tactic:
simp only [Equiv.symm_symm, reindex_apply, submatrix_diagonal_equiv, reindexAlgEquiv_apply]
/-- Any matrix can be reduced to diagonal form by elementary operations. Formulated here on `Type 0`
because we will make an induction using `Fin r`.
See `exists_list_transvec_mul_mul_list_transvec_eq_diagonal` for the general version (which follows
from this one and reindexing). -/
theorem exists_list_transvec_mul_mul_list_transvec_eq_diagonal_aux (n : Type) [Fintype n]
    [DecidableEq n] (M : Matrix n n 𝕜) :
    ∃ (L L' : List (TransvectionStruct n 𝕜)) (D : n → 𝕜),
      (L.map toMatrix).prod * M * (L'.map toMatrix).prod = diagonal D := by
  

Proof state:
n✝ : Type u_1
p : Type u_2
R : Type u₂
𝕜 : Type u_3
inst✝⁷ : Field 𝕜
inst✝⁶ : DecidableEq n✝
inst✝⁵ : DecidableEq p
inst✝⁴ : CommRing R
r : ℕ
M✝ : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
inst✝³ : Fintype n✝
inst✝² : Fintype p
n : Type
inst✝¹ : Fintype n
inst✝ : DecidableEq n
M : Matrix n n 𝕜
⊢ ∃ L L' D, List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L') = diagonal D

Next tactic:
induction' hn : Fintype.card n with r IH generalizing n M
/-- Any matrix can be reduced to diagonal form by elementary operations. Formulated here on `Type 0`
because we will make an induction using `Fin r`.
See `exists_list_transvec_mul_mul_list_transvec_eq_diagonal` for the general version (which follows
from this one and reindexing). -/
theorem exists_list_transvec_mul_mul_list_transvec_eq_diagonal_aux (n : Type) [Fintype n]
    [DecidableEq n] (M : Matrix n n 𝕜) :
    ∃ (L L' : List (TransvectionStruct n 𝕜)) (D : n → 𝕜),
      (L.map toMatrix).prod * M * (L'.map toMatrix).prod = diagonal D := by
  induction' hn : Fintype.card n with r IH generalizing n M
  · 

Proof state:
case zero
n✝ : Type u_1
p : Type u_2
R : Type u₂
𝕜 : Type u_3
inst✝⁷ : Field 𝕜
inst✝⁶ : DecidableEq n✝
inst✝⁵ : DecidableEq p
inst✝⁴ : CommRing R
r : ℕ
M✝ : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
inst✝³ : Fintype n✝
inst✝² : Fintype p
n : Type
inst✝¹ : Fintype n
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hn : Fintype.card n = Nat.zero
⊢ ∃ L L' D, List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L') = diagonal D

Next tactic:
refine' ⟨List.nil, List.nil, fun _ => 1, _⟩
/-- Any matrix can be reduced to diagonal form by elementary operations. Formulated here on `Type 0`
because we will make an induction using `Fin r`.
See `exists_list_transvec_mul_mul_list_transvec_eq_diagonal` for the general version (which follows
from this one and reindexing). -/
theorem exists_list_transvec_mul_mul_list_transvec_eq_diagonal_aux (n : Type) [Fintype n]
    [DecidableEq n] (M : Matrix n n 𝕜) :
    ∃ (L L' : List (TransvectionStruct n 𝕜)) (D : n → 𝕜),
      (L.map toMatrix).prod * M * (L'.map toMatrix).prod = diagonal D := by
  induction' hn : Fintype.card n with r IH generalizing n M
  · refine' ⟨List.nil, List.nil, fun _ => 1, _⟩
    

Proof state:
case zero
n✝ : Type u_1
p : Type u_2
R : Type u₂
𝕜 : Type u_3
inst✝⁷ : Field 𝕜
inst✝⁶ : DecidableEq n✝
inst✝⁵ : DecidableEq p
inst✝⁴ : CommRing R
r : ℕ
M✝ : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
inst✝³ : Fintype n✝
inst✝² : Fintype p
n : Type
inst✝¹ : Fintype n
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hn : Fintype.card n = Nat.zero
⊢ List.prod (List.map toMatrix []) * M * List.prod (List.map toMatrix []) = diagonal fun x => 1

Next tactic:
ext i j
/-- Any matrix can be reduced to diagonal form by elementary operations. Formulated here on `Type 0`
because we will make an induction using `Fin r`.
See `exists_list_transvec_mul_mul_list_transvec_eq_diagonal` for the general version (which follows
from this one and reindexing). -/
theorem exists_list_transvec_mul_mul_list_transvec_eq_diagonal_aux (n : Type) [Fintype n]
    [DecidableEq n] (M : Matrix n n 𝕜) :
    ∃ (L L' : List (TransvectionStruct n 𝕜)) (D : n → 𝕜),
      (L.map toMatrix).prod * M * (L'.map toMatrix).prod = diagonal D := by
  induction' hn : Fintype.card n with r IH generalizing n M
  · refine' ⟨List.nil, List.nil, fun _ => 1, _⟩
    ext i j
    

Proof state:
case zero.a.h
n✝ : Type u_1
p : Type u_2
R : Type u₂
𝕜 : Type u_3
inst✝⁷ : Field 𝕜
inst✝⁶ : DecidableEq n✝
inst✝⁵ : DecidableEq p
inst✝⁴ : CommRing R
r : ℕ
M✝ : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
inst✝³ : Fintype n✝
inst✝² : Fintype p
n : Type
inst✝¹ : Fintype n
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hn : Fintype.card n = Nat.zero
i j : n
⊢ (List.prod (List.map toMatrix []) * M * List.prod (List.map toMatrix [])) i j = diagonal (fun x => 1) i j

Next tactic:
rw [Fintype.card_eq_zero_iff] at hn
/-- Any matrix can be reduced to diagonal form by elementary operations. Formulated here on `Type 0`
because we will make an induction using `Fin r`.
See `exists_list_transvec_mul_mul_list_transvec_eq_diagonal` for the general version (which follows
from this one and reindexing). -/
theorem exists_list_transvec_mul_mul_list_transvec_eq_diagonal_aux (n : Type) [Fintype n]
    [DecidableEq n] (M : Matrix n n 𝕜) :
    ∃ (L L' : List (TransvectionStruct n 𝕜)) (D : n → 𝕜),
      (L.map toMatrix).prod * M * (L'.map toMatrix).prod = diagonal D := by
  induction' hn : Fintype.card n with r IH generalizing n M
  · refine' ⟨List.nil, List.nil, fun _ => 1, _⟩
    ext i j
    rw [Fintype.card_eq_zero_iff] at hn
    

Proof state:
case zero.a.h
n✝ : Type u_1
p : Type u_2
R : Type u₂
𝕜 : Type u_3
inst✝⁷ : Field 𝕜
inst✝⁶ : DecidableEq n✝
inst✝⁵ : DecidableEq p
inst✝⁴ : CommRing R
r : ℕ
M✝ : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
inst✝³ : Fintype n✝
inst✝² : Fintype p
n : Type
inst✝¹ : Fintype n
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hn : IsEmpty n
i j : n
⊢ (List.prod (List.map toMatrix []) * M * List.prod (List.map toMatrix [])) i j = diagonal (fun x => 1) i j

Next tactic:
exact hn.elim' i
/-- Any matrix can be reduced to diagonal form by elementary operations. Formulated here on `Type 0`
because we will make an induction using `Fin r`.
See `exists_list_transvec_mul_mul_list_transvec_eq_diagonal` for the general version (which follows
from this one and reindexing). -/
theorem exists_list_transvec_mul_mul_list_transvec_eq_diagonal_aux (n : Type) [Fintype n]
    [DecidableEq n] (M : Matrix n n 𝕜) :
    ∃ (L L' : List (TransvectionStruct n 𝕜)) (D : n → 𝕜),
      (L.map toMatrix).prod * M * (L'.map toMatrix).prod = diagonal D := by
  induction' hn : Fintype.card n with r IH generalizing n M
  · refine' ⟨List.nil, List.nil, fun _ => 1, _⟩
    ext i j
    rw [Fintype.card_eq_zero_iff] at hn
    exact hn.elim' i
  · 

Proof state:
case succ
n✝ : Type u_1
p : Type u_2
R : Type u₂
𝕜 : Type u_3
inst✝⁷ : Field 𝕜
inst✝⁶ : DecidableEq n✝
inst✝⁵ : DecidableEq p
inst✝⁴ : CommRing R
r✝ : ℕ
M✝ : Matrix (Fin r✝ ⊕ Unit) (Fin r✝ ⊕ Unit) 𝕜
inst✝³ : Fintype n✝
inst✝² : Fintype p
r : ℕ
IH :
  ∀ (n : Type) [inst : Fintype n] [inst_1 : DecidableEq n] (M : Matrix n n 𝕜),
    Fintype.card n = r → ∃ L L' D, List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L') = diagonal D
n : Type
inst✝¹ : Fintype n
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hn : Fintype.card n = Nat.succ r
⊢ ∃ L L' D, List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L') = diagonal D

Next tactic:
have e : n ≃ Sum (Fin r) Unit := by
      refine' Fintype.equivOfCardEq _
      rw [hn]
      rw [@Fintype.card_sum (Fin r) Unit _ _]
      simp
/-- Any matrix can be reduced to diagonal form by elementary operations. Formulated here on `Type 0`
because we will make an induction using `Fin r`.
See `exists_list_transvec_mul_mul_list_transvec_eq_diagonal` for the general version (which follows
from this one and reindexing). -/
theorem exists_list_transvec_mul_mul_list_transvec_eq_diagonal_aux (n : Type) [Fintype n]
    [DecidableEq n] (M : Matrix n n 𝕜) :
    ∃ (L L' : List (TransvectionStruct n 𝕜)) (D : n → 𝕜),
      (L.map toMatrix).prod * M * (L'.map toMatrix).prod = diagonal D := by
  induction' hn : Fintype.card n with r IH generalizing n M
  · refine' ⟨List.nil, List.nil, fun _ => 1, _⟩
    ext i j
    rw [Fintype.card_eq_zero_iff] at hn
    exact hn.elim' i
  · have e : n ≃ Sum (Fin r) Unit := by
      

Proof state:
n✝ : Type u_1
p : Type u_2
R : Type u₂
𝕜 : Type u_3
inst✝⁷ : Field 𝕜
inst✝⁶ : DecidableEq n✝
inst✝⁵ : DecidableEq p
inst✝⁴ : CommRing R
r✝ : ℕ
M✝ : Matrix (Fin r✝ ⊕ Unit) (Fin r✝ ⊕ Unit) 𝕜
inst✝³ : Fintype n✝
inst✝² : Fintype p
r : ℕ
IH :
  ∀ (n : Type) [inst : Fintype n] [inst_1 : DecidableEq n] (M : Matrix n n 𝕜),
    Fintype.card n = r → ∃ L L' D, List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L') = diagonal D
n : Type
inst✝¹ : Fintype n
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hn : Fintype.card n = Nat.succ r
⊢ n ≃ Fin r ⊕ Unit

Next tactic:
refine' Fintype.equivOfCardEq _
/-- Any matrix can be reduced to diagonal form by elementary operations. Formulated here on `Type 0`
because we will make an induction using `Fin r`.
See `exists_list_transvec_mul_mul_list_transvec_eq_diagonal` for the general version (which follows
from this one and reindexing). -/
theorem exists_list_transvec_mul_mul_list_transvec_eq_diagonal_aux (n : Type) [Fintype n]
    [DecidableEq n] (M : Matrix n n 𝕜) :
    ∃ (L L' : List (TransvectionStruct n 𝕜)) (D : n → 𝕜),
      (L.map toMatrix).prod * M * (L'.map toMatrix).prod = diagonal D := by
  induction' hn : Fintype.card n with r IH generalizing n M
  · refine' ⟨List.nil, List.nil, fun _ => 1, _⟩
    ext i j
    rw [Fintype.card_eq_zero_iff] at hn
    exact hn.elim' i
  · have e : n ≃ Sum (Fin r) Unit := by
      refine' Fintype.equivOfCardEq _
      

Proof state:
n✝ : Type u_1
p : Type u_2
R : Type u₂
𝕜 : Type u_3
inst✝⁷ : Field 𝕜
inst✝⁶ : DecidableEq n✝
inst✝⁵ : DecidableEq p
inst✝⁴ : CommRing R
r✝ : ℕ
M✝ : Matrix (Fin r✝ ⊕ Unit) (Fin r✝ ⊕ Unit) 𝕜
inst✝³ : Fintype n✝
inst✝² : Fintype p
r : ℕ
IH :
  ∀ (n : Type) [inst : Fintype n] [inst_1 : DecidableEq n] (M : Matrix n n 𝕜),
    Fintype.card n = r → ∃ L L' D, List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L') = diagonal D
n : Type
inst✝¹ : Fintype n
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hn : Fintype.card n = Nat.succ r
⊢ Fintype.card n = Fintype.card (Fin r ⊕ Unit)

Next tactic:
rw [hn]
/-- Any matrix can be reduced to diagonal form by elementary operations. Formulated here on `Type 0`
because we will make an induction using `Fin r`.
See `exists_list_transvec_mul_mul_list_transvec_eq_diagonal` for the general version (which follows
from this one and reindexing). -/
theorem exists_list_transvec_mul_mul_list_transvec_eq_diagonal_aux (n : Type) [Fintype n]
    [DecidableEq n] (M : Matrix n n 𝕜) :
    ∃ (L L' : List (TransvectionStruct n 𝕜)) (D : n → 𝕜),
      (L.map toMatrix).prod * M * (L'.map toMatrix).prod = diagonal D := by
  induction' hn : Fintype.card n with r IH generalizing n M
  · refine' ⟨List.nil, List.nil, fun _ => 1, _⟩
    ext i j
    rw [Fintype.card_eq_zero_iff] at hn
    exact hn.elim' i
  · have e : n ≃ Sum (Fin r) Unit := by
      refine' Fintype.equivOfCardEq _
      rw [hn]
      

Proof state:
n✝ : Type u_1
p : Type u_2
R : Type u₂
𝕜 : Type u_3
inst✝⁷ : Field 𝕜
inst✝⁶ : DecidableEq n✝
inst✝⁵ : DecidableEq p
inst✝⁴ : CommRing R
r✝ : ℕ
M✝ : Matrix (Fin r✝ ⊕ Unit) (Fin r✝ ⊕ Unit) 𝕜
inst✝³ : Fintype n✝
inst✝² : Fintype p
r : ℕ
IH :
  ∀ (n : Type) [inst : Fintype n] [inst_1 : DecidableEq n] (M : Matrix n n 𝕜),
    Fintype.card n = r → ∃ L L' D, List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L') = diagonal D
n : Type
inst✝¹ : Fintype n
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hn : Fintype.card n = Nat.succ r
⊢ Nat.succ r = Fintype.card (Fin r ⊕ Unit)

Next tactic:
rw [@Fintype.card_sum (Fin r) Unit _ _]
/-- Any matrix can be reduced to diagonal form by elementary operations. Formulated here on `Type 0`
because we will make an induction using `Fin r`.
See `exists_list_transvec_mul_mul_list_transvec_eq_diagonal` for the general version (which follows
from this one and reindexing). -/
theorem exists_list_transvec_mul_mul_list_transvec_eq_diagonal_aux (n : Type) [Fintype n]
    [DecidableEq n] (M : Matrix n n 𝕜) :
    ∃ (L L' : List (TransvectionStruct n 𝕜)) (D : n → 𝕜),
      (L.map toMatrix).prod * M * (L'.map toMatrix).prod = diagonal D := by
  induction' hn : Fintype.card n with r IH generalizing n M
  · refine' ⟨List.nil, List.nil, fun _ => 1, _⟩
    ext i j
    rw [Fintype.card_eq_zero_iff] at hn
    exact hn.elim' i
  · have e : n ≃ Sum (Fin r) Unit := by
      refine' Fintype.equivOfCardEq _
      rw [hn]
      rw [@Fintype.card_sum (Fin r) Unit _ _]
      

Proof state:
n✝ : Type u_1
p : Type u_2
R : Type u₂
𝕜 : Type u_3
inst✝⁷ : Field 𝕜
inst✝⁶ : DecidableEq n✝
inst✝⁵ : DecidableEq p
inst✝⁴ : CommRing R
r✝ : ℕ
M✝ : Matrix (Fin r✝ ⊕ Unit) (Fin r✝ ⊕ Unit) 𝕜
inst✝³ : Fintype n✝
inst✝² : Fintype p
r : ℕ
IH :
  ∀ (n : Type) [inst : Fintype n] [inst_1 : DecidableEq n] (M : Matrix n n 𝕜),
    Fintype.card n = r → ∃ L L' D, List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L') = diagonal D
n : Type
inst✝¹ : Fintype n
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hn : Fintype.card n = Nat.succ r
⊢ Nat.succ r = Fintype.card (Fin r) + Fintype.card Unit

Next tactic:
simp
/-- Any matrix can be reduced to diagonal form by elementary operations. Formulated here on `Type 0`
because we will make an induction using `Fin r`.
See `exists_list_transvec_mul_mul_list_transvec_eq_diagonal` for the general version (which follows
from this one and reindexing). -/
theorem exists_list_transvec_mul_mul_list_transvec_eq_diagonal_aux (n : Type) [Fintype n]
    [DecidableEq n] (M : Matrix n n 𝕜) :
    ∃ (L L' : List (TransvectionStruct n 𝕜)) (D : n → 𝕜),
      (L.map toMatrix).prod * M * (L'.map toMatrix).prod = diagonal D := by
  induction' hn : Fintype.card n with r IH generalizing n M
  · refine' ⟨List.nil, List.nil, fun _ => 1, _⟩
    ext i j
    rw [Fintype.card_eq_zero_iff] at hn
    exact hn.elim' i
  · have e : n ≃ Sum (Fin r) Unit := by
      refine' Fintype.equivOfCardEq _
      rw [hn]
      rw [@Fintype.card_sum (Fin r) Unit _ _]
      simp
    

Proof state:
case succ
n✝ : Type u_1
p : Type u_2
R : Type u₂
𝕜 : Type u_3
inst✝⁷ : Field 𝕜
inst✝⁶ : DecidableEq n✝
inst✝⁵ : DecidableEq p
inst✝⁴ : CommRing R
r✝ : ℕ
M✝ : Matrix (Fin r✝ ⊕ Unit) (Fin r✝ ⊕ Unit) 𝕜
inst✝³ : Fintype n✝
inst✝² : Fintype p
r : ℕ
IH :
  ∀ (n : Type) [inst : Fintype n] [inst_1 : DecidableEq n] (M : Matrix n n 𝕜),
    Fintype.card n = r → ∃ L L' D, List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L') = diagonal D
n : Type
inst✝¹ : Fintype n
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hn : Fintype.card n = Nat.succ r
e : n ≃ Fin r ⊕ Unit
⊢ ∃ L L' D, List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L') = diagonal D

Next tactic:
apply reindex_exists_list_transvec_mul_mul_list_transvec_eq_diagonal M e
/-- Any matrix can be reduced to diagonal form by elementary operations. Formulated here on `Type 0`
because we will make an induction using `Fin r`.
See `exists_list_transvec_mul_mul_list_transvec_eq_diagonal` for the general version (which follows
from this one and reindexing). -/
theorem exists_list_transvec_mul_mul_list_transvec_eq_diagonal_aux (n : Type) [Fintype n]
    [DecidableEq n] (M : Matrix n n 𝕜) :
    ∃ (L L' : List (TransvectionStruct n 𝕜)) (D : n → 𝕜),
      (L.map toMatrix).prod * M * (L'.map toMatrix).prod = diagonal D := by
  induction' hn : Fintype.card n with r IH generalizing n M
  · refine' ⟨List.nil, List.nil, fun _ => 1, _⟩
    ext i j
    rw [Fintype.card_eq_zero_iff] at hn
    exact hn.elim' i
  · have e : n ≃ Sum (Fin r) Unit := by
      refine' Fintype.equivOfCardEq _
      rw [hn]
      rw [@Fintype.card_sum (Fin r) Unit _ _]
      simp
    apply reindex_exists_list_transvec_mul_mul_list_transvec_eq_diagonal M e
    

Proof state:
case succ
n✝ : Type u_1
p : Type u_2
R : Type u₂
𝕜 : Type u_3
inst✝⁷ : Field 𝕜
inst✝⁶ : DecidableEq n✝
inst✝⁵ : DecidableEq p
inst✝⁴ : CommRing R
r✝ : ℕ
M✝ : Matrix (Fin r✝ ⊕ Unit) (Fin r✝ ⊕ Unit) 𝕜
inst✝³ : Fintype n✝
inst✝² : Fintype p
r : ℕ
IH :
  ∀ (n : Type) [inst : Fintype n] [inst_1 : DecidableEq n] (M : Matrix n n 𝕜),
    Fintype.card n = r → ∃ L L' D, List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L') = diagonal D
n : Type
inst✝¹ : Fintype n
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hn : Fintype.card n = Nat.succ r
e : n ≃ Fin r ⊕ Unit
⊢ ∃ L L' D, List.prod (List.map toMatrix L) * (reindexAlgEquiv 𝕜 e) M * List.prod (List.map toMatrix L') = diagonal D

Next tactic:
apply
      exists_list_transvec_mul_mul_list_transvec_eq_diagonal_induction fun N =>
        IH (Fin r) N (by simp)