-- Mathlib/LinearAlgebra/Matrix/Transvection.lean
theorem mul_listTransvecRow_last_row (hM : M (inr unit) (inr unit) ≠ 0) (i : Fin r) :
    (M * (listTransvecRow M).prod) (inr unit) (inl i) = 0

Proof state:
case pos
𝕜 : Type u_3
inst✝ : Field 𝕜
r : ℕ
M : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
hM : M (inr ()) (inr ()) ≠ 0
i : Fin r
n : ℕ
IH : n ≤ r → (M * (List.take n (listTransvecRow M)).prod) (inr ()) (inl i) = if n ≤ ↑i then M (inr ()) (inl i) else 0
hk : n + 1 ≤ r
hnr : n < r
n' : Fin r := ⟨n, hnr⟩
A : (listTransvecRow M)[n]? = some (transvection (inr ()) (inl n') (-M (inr ()) (inl n') / M (inr ()) (inr ())))
h : n' = i
hni : n = ↑i
⊢ (M * (List.take n (listTransvecRow M)).prod *
        transvection (inr ()) (inl n') (-M (inr ()) (inl n') / M (inr ()) (inr ())))
      (inr ()) (inl i) =
    if n + 1 ≤ ↑i then M (inr ()) (inl i) else 0

Tactic:
have : ¬n.succ ≤ i := by simp only [← hni, n.lt_succ_self, not_le]

Result:
case pos
𝕜 : Type u_3
inst✝ : Field 𝕜
r : ℕ
M : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
hM : M (inr ()) (inr ()) ≠ 0
i : Fin r
n : ℕ
IH : n ≤ r → (M * (List.take n (listTransvecRow M)).prod) (inr ()) (inl i) = if n ≤ ↑i then M (inr ()) (inl i) else 0
hk : n + 1 ≤ r
hnr : n < r
n' : Fin r := ⟨n, hnr⟩
A : (listTransvecRow M)[n]? = some (transvection (inr ()) (inl n') (-M (inr ()) (inl n') / M (inr ()) (inr ())))
h : n' = i
hni : n = ↑i
this : ¬n.succ ≤ ↑i
⊢ (M * (List.take n (listTransvecRow M)).prod *
        transvection (inr ()) (inl n') (-M (inr ()) (inl n') / M (inr ()) (inr ())))
      (inr ()) (inl i) =
    if n + 1 ≤ ↑i then M (inr ()) (inl i) else 0

Proof state:
case mk
𝕜 : Type u_3
inst✝ : Field 𝕜
r : ℕ
M : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
hM : M (inr ()) (inr ()) ≠ 0
n : ℕ
hk : n + 1 ≤ r
hnr : n < r
n' : Fin r := ⟨n, hnr⟩
A : (listTransvecRow M)[n]? = some (transvection (inr ()) (inl n') (-M (inr ()) (inl n') / M (inr ()) (inr ())))
val✝ : ℕ
isLt✝ : val✝ < r
IH :
  n ≤ r →
    (M * (List.take n (listTransvecRow M)).prod) (inr ()) (inl ⟨val✝, isLt✝⟩) =
      if n ≤ ↑⟨val✝, isLt✝⟩ then M (inr ()) (inl ⟨val✝, isLt✝⟩) else 0
h : n' = ⟨val✝, isLt✝⟩
⊢ n = ↑⟨val✝, isLt✝⟩

Tactic:
simp only [n', Fin.mk_eq_mk] at h

Result:
case mk
𝕜 : Type u_3
inst✝ : Field 𝕜
r : ℕ
M : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
hM : M (inr ()) (inr ()) ≠ 0
n : ℕ
hk : n + 1 ≤ r
hnr : n < r
n' : Fin r := ⟨n, hnr⟩
A : (listTransvecRow M)[n]? = some (transvection (inr ()) (inl n') (-M (inr ()) (inl n') / M (inr ()) (inr ())))
val✝ : ℕ
isLt✝ : val✝ < r
IH :
  n ≤ r →
    (M * (List.take n (listTransvecRow M)).prod) (inr ()) (inl ⟨val✝, isLt✝⟩) =
      if n ≤ ↑⟨val✝, isLt✝⟩ then M (inr ()) (inl ⟨val✝, isLt✝⟩) else 0
h : n = val✝
⊢ n = ↑⟨val✝, isLt✝⟩

Proof state:
𝕜 : Type u_3
inst✝ : Field 𝕜
r : ℕ
M : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
hM : M (inr ()) (inr ()) ≠ 0
i : Fin r
n : ℕ
IH : n ≤ r → (M * (List.take n (listTransvecRow M)).prod) (inr ()) (inl i) = if n ≤ ↑i then M (inr ()) (inl i) else 0
hk : n + 1 ≤ r
hnr : n < r
n' : Fin r := ⟨n, hnr⟩
⊢ (listTransvecRow M)[n]? = some (transvection (inr ()) (inl n') (-M (inr ()) (inl n') / M (inr ()) (inr ())))

Tactic:
simp only [n', listTransvecRow, List.ofFnNthVal, hnr, dif_pos, List.getElem?_ofFn]

Result:
no goals

Proof state:
case neg
𝕜 : Type u_3
inst✝ : Field 𝕜
r : ℕ
M : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
hM : M (inr ()) (inr ()) ≠ 0
i : Fin r
n : ℕ
IH : n ≤ r → (M * (List.take n (listTransvecRow M)).prod) (inr ()) (inl i) = if n ≤ ↑i then M (inr ()) (inl i) else 0
hk : n + 1 ≤ r
hnr : n < r
n' : Fin r := ⟨n, hnr⟩
A : (listTransvecRow M)[n]? = some (transvection (inr ()) (inl n') (-M (inr ()) (inl n') / M (inr ()) (inr ())))
h : ¬n' = i
hni : n ≠ ↑i
⊢ (if n ≤ ↑i then M (inr ()) (inl i) else 0) = if n + 1 ≤ ↑i then M (inr ()) (inl i) else 0

Tactic:
rcases le_or_lt (n + 1) i with (hi | hi)

Result:
case neg.inl
𝕜 : Type u_3
inst✝ : Field 𝕜
r : ℕ
M : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
hM : M (inr ()) (inr ()) ≠ 0
i : Fin r
n : ℕ
IH : n ≤ r → (M * (List.take n (listTransvecRow M)).prod) (inr ()) (inl i) = if n ≤ ↑i then M (inr ()) (inl i) else 0
hk : n + 1 ≤ r
hnr : n < r
n' : Fin r := ⟨n, hnr⟩
A : (listTransvecRow M)[n]? = some (transvection (inr ()) (inl n') (-M (inr ()) (inl n') / M (inr ()) (inr ())))
h : ¬n' = i
hni : n ≠ ↑i
hi : n + 1 ≤ ↑i
⊢ (if n ≤ ↑i then M (inr ()) (inl i) else 0) = if n + 1 ≤ ↑i then M (inr ()) (inl i) else 0

case neg.inr
𝕜 : Type u_3
inst✝ : Field 𝕜
r : ℕ
M : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
hM : M (inr ()) (inr ()) ≠ 0
i : Fin r
n : ℕ
IH : n ≤ r → (M * (List.take n (listTransvecRow M)).prod) (inr ()) (inl i) = if n ≤ ↑i then M (inr ()) (inl i) else 0
hk : n + 1 ≤ r
hnr : n < r
n' : Fin r := ⟨n, hnr⟩
A : (listTransvecRow M)[n]? = some (transvection (inr ()) (inl n') (-M (inr ()) (inl n') / M (inr ()) (inr ())))
h : ¬n' = i
hni : n ≠ ↑i
hi : ↑i < n + 1
⊢ (if n ≤ ↑i then M (inr ()) (inl i) else 0) = if n + 1 ≤ ↑i then M (inr ()) (inl i) else 0

Proof state:
𝕜 : Type u_3
inst✝ : Field 𝕜
r : ℕ
M : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
hM : M (inr ()) (inr ()) ≠ 0
i : Fin r
⊢ ∀ k ≤ r, (M * (List.take k (listTransvecRow M)).prod) (inr ()) (inl i) = if k ≤ ↑i then M (inr ()) (inl i) else 0

Tactic:
intro k hk

Result:
𝕜 : Type u_3
inst✝ : Field 𝕜
r : ℕ
M : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
hM : M (inr ()) (inr ()) ≠ 0
i : Fin r
k : ℕ
hk : k ≤ r
⊢ (M * (List.take k (listTransvecRow M)).prod) (inr ()) (inl i) = if k ≤ ↑i then M (inr ()) (inl i) else 0

Proof state:
𝕜 : Type u_3
inst✝ : Field 𝕜
r : ℕ
M : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
hM : M (inr ()) (inr ()) ≠ 0
i : Fin r
n : ℕ
IH : n ≤ r → (M * (List.take n (listTransvecRow M)).prod) (inr ()) (inl i) = if n ≤ ↑i then M (inr ()) (inl i) else 0
hk : n + 1 ≤ r
hnr : n < r
n' : Fin r := ⟨n, hnr⟩
A : (listTransvecRow M)[n]? = some (transvection (inr ()) (inl n') (-M (inr ()) (inl n') / M (inr ()) (inr ())))
h : ¬n' = i
⊢ n ≠ ↑i

Tactic:
rintro rfl

Result:
𝕜 : Type u_3
inst✝ : Field 𝕜
r : ℕ
M : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
hM : M (inr ()) (inr ()) ≠ 0
i : Fin r
IH :
  ↑i ≤ r → (M * (List.take (↑i) (listTransvecRow M)).prod) (inr ()) (inl i) = if ↑i ≤ ↑i then M (inr ()) (inl i) else 0
hk : ↑i + 1 ≤ r
hnr : ↑i < r
n' : Fin r := ⟨↑i, hnr⟩
A : (listTransvecRow M)[↑i]? = some (transvection (inr ()) (inl n') (-M (inr ()) (inl n') / M (inr ()) (inr ())))
h : ¬n' = i
⊢ False

Proof state:
𝕜 : Type u_3
inst✝ : Field 𝕜
r : ℕ
M : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
hM : M (inr ()) (inr ()) ≠ 0
i : Fin r
H : ∀ k ≤ r, (M * (List.take k (listTransvecRow M)).prod) (inr ()) (inl i) = if k ≤ ↑i then M (inr ()) (inl i) else 0
⊢ (listTransvecRow M).length = r

Tactic:
simp [listTransvecRow]

Result:
no goals

Proof state:
case succ
𝕜 : Type u_3
inst✝ : Field 𝕜
r : ℕ
M : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
hM : M (inr ()) (inr ()) ≠ 0
i : Fin r
n : ℕ
IH : n ≤ r → (M * (List.take n (listTransvecRow M)).prod) (inr ()) (inl i) = if n ≤ ↑i then M (inr ()) (inl i) else 0
hk : n + 1 ≤ r
hnr : n < r
⊢ (M * (List.take (n + 1) (listTransvecRow M)).prod) (inr ()) (inl i) = if n + 1 ≤ ↑i then M (inr ()) (inl i) else 0

Tactic:
let n' : Fin r := ⟨n, hnr⟩

Result:
case succ
𝕜 : Type u_3
inst✝ : Field 𝕜
r : ℕ
M : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
hM : M (inr ()) (inr ()) ≠ 0
i : Fin r
n : ℕ
IH : n ≤ r → (M * (List.take n (listTransvecRow M)).prod) (inr ()) (inl i) = if n ≤ ↑i then M (inr ()) (inl i) else 0
hk : n + 1 ≤ r
hnr : n < r
n' : Fin r := ⟨n, hnr⟩
⊢ (M * (List.take (n + 1) (listTransvecRow M)).prod) (inr ()) (inl i) = if n + 1 ≤ ↑i then M (inr ()) (inl i) else 0