awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Resultant.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
f g : R[X]
m n : ℕ
r : R
M₁ : Matrix (Fin (m + n)) (Fin (m + n)) R := f.sylvester (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) r : R[X]) g) m n
M₂ : Matrix (Fin (m + n)) (Fin (m + n)) R := f.sylvester g m n
M : (i : ℕ) → Matrix (Fin (m + n)) (Fin (m + n)) R :=
  fun (i : ℕ) ↦
    (Matrix.of : (a : Fin (m + n) → Fin (m + n) → R) → Matrix (Fin (m + n)) (Fin (m + n)) R) fun (j₁ j₂ : Fin (m + n)) ↦
      if (↑j₂ : ℕ) < i then M₁ j₁ j₂ else M₂ j₁ j₂
i : ℕ
IH : ∀ (hi : i ≤ m), (M i).det = r ^ i * M₂.det
hi : i + 1 ≤ m
this :
  (M i).updateCol (⟨i, ⋯⟩ : Fin (m + n))
      (HSMul.hSMul (β := (j : Fin (m + n)) → R) r fun (j : Fin (m + n)) ↦ M i j (⟨i, ⋯⟩ : Fin (m + n))) =
    M (i + 1)
⊢ (M (i + 1)).det = r ^ (i + 1) * M₂.det
have rw : (M (i + 1)).det = r * r ^ i * M₂.det
Tactic: rw [pow_succ', mul_assoc, ← IH (by lia), ← this, Matrix.det_updateCol_smul,
          Matrix.updateCol_eq_self]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Resultant.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
f g : R[X]
m n : ℕ
r : R
M₁ : Matrix (Fin (m + n)) (Fin (m + n)) R := f.sylvester (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) r : R[X]) g) m n
M₂ : Matrix (Fin (m + n)) (Fin (m + n)) R := f.sylvester g m n
M : (i : ℕ) → Matrix (Fin (m + n)) (Fin (m + n)) R :=
  fun (i : ℕ) ↦
    (Matrix.of : (a : Fin (m + n) → Fin (m + n) → R) → Matrix (Fin (m + n)) (Fin (m + n)) R) fun (j₁ j₂ : Fin (m + n)) ↦
      if (↑j₂ : ℕ) < i then M₁ j₁ j₂ else M₂ j₁ j₂
i : ℕ
IH : ∀ (hi : i ≤ m), (M i).det = r ^ i * M₂.det
hi : i + 1 ≤ m
this :
  (M i).updateCol (⟨i, ⋯⟩ : Fin (m + n))
      (HSMul.hSMul (β := (j : Fin (m + n)) → R) r fun (j : Fin (m + n)) ↦ M i j (⟨i, ⋯⟩ : Fin (m + n))) =
    M (i + 1)
rw : (M (i + 1)).det = r * r ^ i * M₂.det
⊢ (M (i + 1)).det = r ^ (i + 1) * M₂.det
have rw₁ : (M (i + 1)).det = r * (r ^ i * M₂.det)
Tactic: rw [pow_succ', mul_assoc, ← IH (by lia), ← this, Matrix.det_updateCol_smul,
          Matrix.updateCol_eq_self]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Resultant.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
f g : R[X]
m n : ℕ
r : R
M₁ : Matrix (Fin (m + n)) (Fin (m + n)) R := f.sylvester (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) r : R[X]) g) m n
M₂ : Matrix (Fin (m + n)) (Fin (m + n)) R := f.sylvester g m n
M : (i : ℕ) → Matrix (Fin (m + n)) (Fin (m + n)) R :=
  fun (i : ℕ) ↦
    (Matrix.of : (a : Fin (m + n) → Fin (m + n) → R) → Matrix (Fin (m + n)) (Fin (m + n)) R) fun (j₁ j₂ : Fin (m + n)) ↦
      if (↑j₂ : ℕ) < i then M₁ j₁ j₂ else M₂ j₁ j₂
i : ℕ
IH : ∀ (hi : i ≤ m), (M i).det = r ^ i * M₂.det
hi : i + 1 ≤ m
this :
  (M i).updateCol (⟨i, ⋯⟩ : Fin (m + n))
      (HSMul.hSMul (β := (j : Fin (m + n)) → R) r fun (j : Fin (m + n)) ↦ M i j (⟨i, ⋯⟩ : Fin (m + n))) =
    M (i + 1)
rw : (M (i + 1)).det = r * r ^ i * M₂.det
rw₁ : (M (i + 1)).det = r * (r ^ i * M₂.det)
⊢ (M (i + 1)).det = r ^ (i + 1) * M₂.det
have rw₂ : (M (i + 1)).det = r * (M i).det
Tactic: rw [pow_succ', mul_assoc, ← IH (by lia), ← this, Matrix.det_updateCol_smul,
          Matrix.updateCol_eq_self]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Resultant.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
f g : R[X]
m n : ℕ
r : R
M₁ : Matrix (Fin (m + n)) (Fin (m + n)) R := f.sylvester (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) r : R[X]) g) m n
M₂ : Matrix (Fin (m + n)) (Fin (m + n)) R := f.sylvester g m n
M : (i : ℕ) → Matrix (Fin (m + n)) (Fin (m + n)) R :=
  fun (i : ℕ) ↦
    (Matrix.of : (a : Fin (m + n) → Fin (m + n) → R) → Matrix (Fin (m + n)) (Fin (m + n)) R) fun (j₁ j₂ : Fin (m + n)) ↦
      if (↑j₂ : ℕ) < i then M₁ j₁ j₂ else M₂ j₁ j₂
i : ℕ
IH : ∀ (hi : i ≤ m), (M i).det = r ^ i * M₂.det
hi : i + 1 ≤ m
this :
  (M i).updateCol (⟨i, ⋯⟩ : Fin (m + n))
      (HSMul.hSMul (β := (j : Fin (m + n)) → R) r fun (j : Fin (m + n)) ↦ M i j (⟨i, ⋯⟩ : Fin (m + n))) =
    M (i + 1)
rw : (M (i + 1)).det = r * r ^ i * M₂.det
rw₁ : (M (i + 1)).det = r * (r ^ i * M₂.det)
rw₂ : (M (i + 1)).det = r * (M i).det
⊢ (M (i + 1)).det = r ^ (i + 1) * M₂.det
have rw₃ : ((M i).updateCol (⟨i, sorry⟩ : Fin (m + n))
      (HSMul.hSMul (β := (j : Fin (m + n)) → R) r fun (j : Fin (m + n)) ↦ M i j (⟨i, sorry⟩ : Fin (m + n)))).det =
  r * (M i).det
Tactic: rw [pow_succ', mul_assoc, ← IH (by lia), ← this, Matrix.det_updateCol_smul,
          Matrix.updateCol_eq_self]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Resultant.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
f g : R[X]
m n : ℕ
r : R
M₁ : Matrix (Fin (m + n)) (Fin (m + n)) R := f.sylvester (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) r : R[X]) g) m n
M₂ : Matrix (Fin (m + n)) (Fin (m + n)) R := f.sylvester g m n
M : (i : ℕ) → Matrix (Fin (m + n)) (Fin (m + n)) R :=
  fun (i : ℕ) ↦
    (Matrix.of : (a : Fin (m + n) → Fin (m + n) → R) → Matrix (Fin (m + n)) (Fin (m + n)) R) fun (j₁ j₂ : Fin (m + n)) ↦
      if (↑j₂ : ℕ) < i then M₁ j₁ j₂ else M₂ j₁ j₂
i : ℕ
IH : ∀ (hi : i ≤ m), (M i).det = r ^ i * M₂.det
hi : i + 1 ≤ m
this :
  (M i).updateCol (⟨i, ⋯⟩ : Fin (m + n))
      (HSMul.hSMul (β := (j : Fin (m + n)) → R) r fun (j : Fin (m + n)) ↦ M i j (⟨i, ⋯⟩ : Fin (m + n))) =
    M (i + 1)
rw : (M (i + 1)).det = r * r ^ i * M₂.det
rw₁ : (M (i + 1)).det = r * (r ^ i * M₂.det)
rw₂ : (M (i + 1)).det = r * (M i).det
rw₃ :
  ((M i).updateCol (⟨i, ⋯⟩ : Fin (m + n))
        (HSMul.hSMul (β := (j : Fin (m + n)) → R) r fun (j : Fin (m + n)) ↦ M i j (⟨i, ⋯⟩ : Fin (m + n)))).det =
    r * (M i).det
⊢ (M (i + 1)).det = r ^ (i + 1) * M₂.det
have rw₄ : r * ((M i).updateCol (⟨i, sorry⟩ : Fin (m + n)) fun (j : Fin (m + n)) ↦ M i j (⟨i, sorry⟩ : Fin (m + n))).det = r * (M i).det
Tactic: rw [pow_succ', mul_assoc, ← IH (by lia), ← this, Matrix.det_updateCol_smul,
          Matrix.updateCol_eq_self]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Resultant.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
f g : R[X]
m n : ℕ
r : R
M₁ : Matrix (Fin (m + n)) (Fin (m + n)) R := f.sylvester (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) r : R[X]) g) m n
M₂ : Matrix (Fin (m + n)) (Fin (m + n)) R := f.sylvester g m n
M : (i : ℕ) → Matrix (Fin (m + n)) (Fin (m + n)) R :=
  fun (i : ℕ) ↦
    (Matrix.of : (a : Fin (m + n) → Fin (m + n) → R) → Matrix (Fin (m + n)) (Fin (m + n)) R) fun (j₁ j₂ : Fin (m + n)) ↦
      if (↑j₂ : ℕ) < i then M₁ j₁ j₂ else M₂ j₁ j₂
i : ℕ
IH : ∀ (hi : i ≤ m), (M i).det = r ^ i * M₂.det
hi : i + 1 ≤ m
this :
  (M i).updateCol (⟨i, ⋯⟩ : Fin (m + n))
      (HSMul.hSMul (β := (j : Fin (m + n)) → R) r fun (j : Fin (m + n)) ↦ M i j (⟨i, ⋯⟩ : Fin (m + n))) =
    M (i + 1)
rw : (M (i + 1)).det = r * r ^ i * M₂.det
rw₁ : (M (i + 1)).det = r * (r ^ i * M₂.det)
rw₂ : (M (i + 1)).det = r * (M i).det
rw₃ :
  ((M i).updateCol (⟨i, ⋯⟩ : Fin (m + n))
        (HSMul.hSMul (β := (j : Fin (m + n)) → R) r fun (j : Fin (m + n)) ↦ M i j (⟨i, ⋯⟩ : Fin (m + n)))).det =
    r * (M i).det
rw₄ :
  r * ((M i).updateCol (⟨i, ⋯⟩ : Fin (m + n)) fun (j : Fin (m + n)) ↦ M i j (⟨i, ⋯⟩ : Fin (m + n))).det = r * (M i).det
⊢ (M (i + 1)).det = r ^ (i + 1) * M₂.det
have rw₅ : r * (M i).det = r * (M i).det
Tactic: rw [pow_succ', mul_assoc, ← IH (by lia), ← this, Matrix.det_updateCol_smul,
          Matrix.updateCol_eq_self]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Resultant.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
f g : R[X]
m n : ℕ
r : R
M₁ : Matrix (Fin (m + n)) (Fin (m + n)) R := f.sylvester (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) r : R[X]) g) m n
M₂ : Matrix (Fin (m + n)) (Fin (m + n)) R := f.sylvester g m n
M : (i : ℕ) → Matrix (Fin (m + n)) (Fin (m + n)) R :=
  fun (i : ℕ) ↦
    (Matrix.of : (a : Fin (m + n) → Fin (m + n) → R) → Matrix (Fin (m + n)) (Fin (m + n)) R) fun (j₁ j₂ : Fin (m + n)) ↦
      if (↑j₂ : ℕ) < i then M₁ j₁ j₂ else M₂ j₁ j₂
i : ℕ
IH : ∀ (hi : i ≤ m), (M i).det = r ^ i * M₂.det
hi : i + 1 ≤ m
⊢ (M (i + 1)).det = r ^ (i + 1) * M₂.det
have succ : (M i).updateCol (⟨i, sorry⟩ : Fin (m + n))
    (HSMul.hSMul (β := (j : Fin (m + n)) → R) r fun (j : Fin (m + n)) ↦ M i j (⟨i, sorry⟩ : Fin (m + n))) =
  M (i + 1)
Tactic: suffices (M i).updateCol ⟨i, by lia⟩ (r • fun j ↦ M i j ⟨i, by lia⟩) = (M (i + 1)) by
        rw [pow_succ', mul_assoc, ← IH (by lia), ← this, Matrix.det_updateCol_smul,
          Matrix.updateCol_eq_self]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Resultant.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
f g : R[X]
m n : ℕ
r : R
M₁ : Matrix (Fin (m + n)) (Fin (m + n)) R := f.sylvester (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) r : R[X]) g) m n
M₂ : Matrix (Fin (m + n)) (Fin (m + n)) R := f.sylvester g m n
M : (i : ℕ) → Matrix (Fin (m + n)) (Fin (m + n)) R :=
  fun (i : ℕ) ↦
    (Matrix.of : (a : Fin (m + n) → Fin (m + n) → R) → Matrix (Fin (m + n)) (Fin (m + n)) R) fun (j₁ j₂ : Fin (m + n)) ↦
      if (↑j₂ : ℕ) < i then M₁ j₁ j₂ else M₂ j₁ j₂
i : ℕ
IH : ∀ (hi : i ≤ m), (M i).det = r ^ i * M₂.det
hi : i + 1 ≤ m
⊢ (M i).updateCol (⟨i, ⋯⟩ : Fin (m + n))
      (HSMul.hSMul (β := (j : Fin (m + n)) → R) r fun (j : Fin (m + n)) ↦ M i j (⟨i, ⋯⟩ : Fin (m + n))) =
    M (i + 1)
have succ : ∀ (j₁ j₂ : Fin (m + n)),
  (M i).updateCol (⟨i, sorry⟩ : Fin (m + n))
      (HSMul.hSMul (β := (j : Fin (m + n)) → R) r fun (j : Fin (m + n)) ↦ M i j (⟨i, sorry⟩ : Fin (m + n))) j₁ j₂ =
    M (i + 1) j₁ j₂
Tactic: ext j₁ j₂
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Resultant.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
f g : R[X]
m n : ℕ
r : R
M₁ : Matrix (Fin (m + n)) (Fin (m + n)) R := f.sylvester (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) r : R[X]) g) m n
M₂ : Matrix (Fin (m + n)) (Fin (m + n)) R := f.sylvester g m n
M : (i : ℕ) → Matrix (Fin (m + n)) (Fin (m + n)) R :=
  fun (i : ℕ) ↦
    (Matrix.of : (a : Fin (m + n) → Fin (m + n) → R) → Matrix (Fin (m + n)) (Fin (m + n)) R) fun (j₁ j₂ : Fin (m + n)) ↦
      if (↑j₂ : ℕ) < i then M₁ j₁ j₂ else M₂ j₁ j₂
i : ℕ
IH : ∀ (hi : i ≤ m), (M i).det = r ^ i * M₂.det
hi : i + 1 ≤ m
j₁ j₂ : Fin (m + n)
⊢ (M i).updateCol (⟨i, ⋯⟩ : Fin (m + n))
      (HSMul.hSMul (β := (j : Fin (m + n)) → R) r fun (j : Fin (m + n)) ↦ M i j (⟨i, ⋯⟩ : Fin (m + n))) j₁ j₂ =
    M (i + 1) j₁ j₂
have succ : (if (↑j₂ : ℕ) = i then r * f.sylvester g m n j₁ (⟨i, sorry⟩ : Fin (m + n))
  else if (↑j₂ : ℕ) < i then M₁ j₁ j₂ else f.sylvester g m n j₁ j₂) =
  if (↑j₂ : ℕ) < i + 1 then M₁ j₁ j₂ else f.sylvester g m n j₁ j₂
Tactic: simp only [Matrix.of_apply, lt_self_iff_false, ↓reduceIte, Matrix.updateCol_apply,
        Fin.ext_iff, Pi.smul_apply, smul_eq_mul, M, M₂]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Resultant.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
f g : R[X]
m n : ℕ
r : R
M₁ : Matrix (Fin (m + n)) (Fin (m + n)) R := f.sylvester (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) r : R[X]) g) m n
M₂ : Matrix (Fin (m + n)) (Fin (m + n)) R := f.sylvester g m n
M : (i : ℕ) → Matrix (Fin (m + n)) (Fin (m + n)) R :=
  fun (i : ℕ) ↦
    (Matrix.of : (a : Fin (m + n) → Fin (m + n) → R) → Matrix (Fin (m + n)) (Fin (m + n)) R) fun (j₁ j₂ : Fin (m + n)) ↦
      if (↑j₂ : ℕ) < i then M₁ j₁ j₂ else M₂ j₁ j₂
i : ℕ
IH : ∀ (hi : i ≤ m), (M i).det = r ^ i * M₂.det
hi : i + 1 ≤ m
j₁ j₂ : Fin (m + n)
⊢ (if (↑j₂ : ℕ) = i then r * f.sylvester g m n j₁ (⟨i, ⋯⟩ : Fin (m + n))
    else if (↑j₂ : ℕ) < i then M₁ j₁ j₂ else f.sylvester g m n j₁ j₂) =
    if (↑j₂ : ℕ) < i + 1 then M₁ j₁ j₂ else f.sylvester g m n j₁ j₂
have succ : ∀ (IH : ∀ (hi : (↑j₂ : ℕ) ≤ m), (M (↑j₂ : ℕ)).det = r ^ (↑j₂ : ℕ) * M₂.det) (hi : (↑j₂ : ℕ) + 1 ≤ m),
  (if (↑j₂ : ℕ) = (↑j₂ : ℕ) then r * f.sylvester g m n j₁ (⟨(↑j₂ : ℕ), sorry⟩ : Fin (m + n))
    else if (↑j₂ : ℕ) < (↑j₂ : ℕ) then M₁ j₁ j₂ else f.sylvester g m n j₁ j₂) =
    if (↑j₂ : ℕ) < (↑j₂ : ℕ) + 1 then M₁ j₁ j₂ else f.sylvester g m n j₁ j₂
Tactic: obtain rfl | h := eq_or_ne ↑j₂ i
Cleared hypotheses: i, IH, hi
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Resultant.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
f g : R[X]
m n : ℕ
r : R
M₁ : Matrix (Fin (m + n)) (Fin (m + n)) R := f.sylvester (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) r : R[X]) g) m n
M₂ : Matrix (Fin (m + n)) (Fin (m + n)) R := f.sylvester g m n
M : (i : ℕ) → Matrix (Fin (m + n)) (Fin (m + n)) R :=
  fun (i : ℕ) ↦
    (Matrix.of : (a : Fin (m + n) → Fin (m + n) → R) → Matrix (Fin (m + n)) (Fin (m + n)) R) fun (j₁ j₂ : Fin (m + n)) ↦
      if (↑j₂ : ℕ) < i then M₁ j₁ j₂ else M₂ j₁ j₂
i : ℕ
IH : ∀ (hi : i ≤ m), (M i).det = r ^ i * M₂.det
hi : i + 1 ≤ m
j₁ j₂ : Fin (m + n)
succ :
  ∀ (IH : ∀ (hi : (↑j₂ : ℕ) ≤ m), (M (↑j₂ : ℕ)).det = r ^ (↑j₂ : ℕ) * M₂.det) (hi : (↑j₂ : ℕ) + 1 ≤ m),
    (if (↑j₂ : ℕ) = (↑j₂ : ℕ) then r * f.sylvester g m n j₁ (⟨(↑j₂ : ℕ), ⋯⟩ : Fin (m + n))
      else if (↑j₂ : ℕ) < (↑j₂ : ℕ) then M₁ j₁ j₂ else f.sylvester g m n j₁ j₂) =
      if (↑j₂ : ℕ) < (↑j₂ : ℕ) + 1 then M₁ j₁ j₂ else f.sylvester g m n j₁ j₂
⊢ (if (↑j₂ : ℕ) = i then r * f.sylvester g m n j₁ (⟨i, ⋯⟩ : Fin (m + n))
    else if (↑j₂ : ℕ) < i then M₁ j₁ j₂ else f.sylvester g m n j₁ j₂) =
    if (↑j₂ : ℕ) < i + 1 then M₁ j₁ j₂ else f.sylvester g m n j₁ j₂
have succ₁ : ∀ (h : (↑j₂ : ℕ) ≠ i),
  (if (↑j₂ : ℕ) = i then r * f.sylvester g m n j₁ (⟨i, sorry⟩ : Fin (m + n))
    else if (↑j₂ : ℕ) < i then M₁ j₁ j₂ else f.sylvester g m n j₁ j₂) =
    if (↑j₂ : ℕ) < i + 1 then M₁ j₁ j₂ else f.sylvester g m n j₁ j₂
Tactic: obtain rfl | h := eq_or_ne ↑j₂ i
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Resultant.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
f g : R[X]
m n : ℕ
r : R
M₁ : Matrix (Fin (m + n)) (Fin (m + n)) R := f.sylvester (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) r : R[X]) g) m n
M₂ : Matrix (Fin (m + n)) (Fin (m + n)) R := f.sylvester g m n
M : (i : ℕ) → Matrix (Fin (m + n)) (Fin (m + n)) R :=
  fun (i : ℕ) ↦
    (Matrix.of : (a : Fin (m + n) → Fin (m + n) → R) → Matrix (Fin (m + n)) (Fin (m + n)) R) fun (j₁ j₂ : Fin (m + n)) ↦
      if (↑j₂ : ℕ) < i then M₁ j₁ j₂ else M₂ j₁ j₂
j₁ j₂ : Fin (m + n)
IH : ∀ (hi : (↑j₂ : ℕ) ≤ m), (M (↑j₂ : ℕ)).det = r ^ (↑j₂ : ℕ) * M₂.det
hi : (↑j₂ : ℕ) + 1 ≤ m
⊢ (if (↑j₂ : ℕ) = (↑j₂ : ℕ) then r * f.sylvester g m n j₁ (⟨(↑j₂ : ℕ), ⋯⟩ : Fin (m + n))
    else if (↑j₂ : ℕ) < (↑j₂ : ℕ) then M₁ j₁ j₂ else f.sylvester g m n j₁ j₂) =
    if (↑j₂ : ℕ) < (↑j₂ : ℕ) + 1 then M₁ j₁ j₂ else f.sylvester g m n j₁ j₂
have succ : HMul.hMul (β := R) r
    (Fin.addCases (motive := fun (x : Fin (m + n)) ↦ R)
        (fun (j₁_1 : Fin m) ↦
          if (↑j₁_1 : ℕ) ≤ (↑j₁ : ℕ) ∧ (↑j₁ : ℕ) ≤ (↑j₁_1 : ℕ) + n then g.coeff ((↑j₁ : ℕ) - (↑j₁_1 : ℕ)) else 0)
        (fun (j₁_1 : Fin n) ↦
          if (↑j₁_1 : ℕ) ≤ (↑j₁ : ℕ) ∧ (↑j₁ : ℕ) ≤ (↑j₁_1 : ℕ) + m then f.coeff ((↑j₁ : ℕ) - (↑j₁_1 : ℕ)) else 0)
        j₂ :
      R) =
  Fin.addCases (motive := fun (x : Fin (m + n)) ↦ R)
    (fun (j₁_1 : Fin m) ↦
      if (↑j₁_1 : ℕ) ≤ (↑j₁ : ℕ) ∧ (↑j₁ : ℕ) ≤ (↑j₁_1 : ℕ) + n then r * g.coeff ((↑j₁ : ℕ) - (↑j₁_1 : ℕ)) else 0)
    (fun (j₁_1 : Fin n) ↦
      if (↑j₁_1 : ℕ) ≤ (↑j₁ : ℕ) ∧ (↑j₁ : ℕ) ≤ (↑j₁_1 : ℕ) + m then f.coeff ((↑j₁ : ℕ) - (↑j₁_1 : ℕ)) else 0)
    j₂
Tactic: simp only [↓reduceIte, sylvester, Set.mem_Icc, Fin.eta, Matrix.of_apply,
          lt_add_iff_pos_right, zero_lt_one, coeff_C_mul, M₁]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Resultant.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
f g : R[X]
m n : ℕ
r : R
M₁ : Matrix (Fin (m + n)) (Fin (m + n)) R := f.sylvester (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) r : R[X]) g) m n
M₂ : Matrix (Fin (m + n)) (Fin (m + n)) R := f.sylvester g m n
M : (i : ℕ) → Matrix (Fin (m + n)) (Fin (m + n)) R :=
  fun (i : ℕ) ↦
    (Matrix.of : (a : Fin (m + n) → Fin (m + n) → R) → Matrix (Fin (m + n)) (Fin (m + n)) R) fun (j₁ j₂ : Fin (m + n)) ↦
      if (↑j₂ : ℕ) < i then M₁ j₁ j₂ else M₂ j₁ j₂
j₁ j₂ : Fin (m + n)
IH : ∀ (hi : (↑j₂ : ℕ) ≤ m), (M (↑j₂ : ℕ)).det = r ^ (↑j₂ : ℕ) * M₂.det
hi : (↑j₂ : ℕ) + 1 ≤ m
⊢ HMul.hMul (β := R) r
      (Fin.addCases (motive := fun (x : Fin (m + n)) ↦ R)
          (fun (j₁_1 : Fin m) ↦
            if (↑j₁_1 : ℕ) ≤ (↑j₁ : ℕ) ∧ (↑j₁ : ℕ) ≤ (↑j₁_1 : ℕ) + n then g.coeff ((↑j₁ : ℕ) - (↑j₁_1 : ℕ)) else 0)
          (fun (j₁_1 : Fin n) ↦
            if (↑j₁_1 : ℕ) ≤ (↑j₁ : ℕ) ∧ (↑j₁ : ℕ) ≤ (↑j₁_1 : ℕ) + m then f.coeff ((↑j₁ : ℕ) - (↑j₁_1 : ℕ)) else 0)
          j₂ :
        R) =
    Fin.addCases (motive := fun (x : Fin (m + n)) ↦ R)
      (fun (j₁_1 : Fin m) ↦
        if (↑j₁_1 : ℕ) ≤ (↑j₁ : ℕ) ∧ (↑j₁ : ℕ) ≤ (↑j₁_1 : ℕ) + n then r * g.coeff ((↑j₁ : ℕ) - (↑j₁_1 : ℕ)) else 0)
      (fun (j₁_1 : Fin n) ↦
        if (↑j₁_1 : ℕ) ≤ (↑j₁ : ℕ) ∧ (↑j₁ : ℕ) ≤ (↑j₁_1 : ℕ) + m then f.coeff ((↑j₁ : ℕ) - (↑j₁_1 : ℕ)) else 0)
      j₂
have succ : ∀ (j₂ : Fin m)
  (IH :
    ∀ (hi : (↑(Fin.castAdd n j₂) : ℕ) ≤ m), (M (↑(Fin.castAdd n j₂) : ℕ)).det = r ^ (↑(Fin.castAdd n j₂) : ℕ) * M₂.det)
  (hi : (↑(Fin.castAdd n j₂) : ℕ) + 1 ≤ m),
  HMul.hMul (β := R) r
      (Fin.addCases (motive := fun (x : Fin (m + n)) ↦ R)
          (fun (j₁_1 : Fin m) ↦
            if (↑j₁_1 : ℕ) ≤ (↑j₁ : ℕ) ∧ (↑j₁ : ℕ) ≤ (↑j₁_1 : ℕ) + n then g.coeff ((↑j₁ : ℕ) - (↑j₁_1 : ℕ)) else 0)
          (fun (j₁_1 : Fin n) ↦
            if (↑j₁_1 : ℕ) ≤ (↑j₁ : ℕ) ∧ (↑j₁ : ℕ) ≤ (↑j₁_1 : ℕ) + m then f.coeff ((↑j₁ : ℕ) - (↑j₁_1 : ℕ)) else 0)
          (Fin.castAdd n j₂) :
        R) =
    Fin.addCases (motive := fun (x : Fin (m + n)) ↦ R)
      (fun (j₁_1 : Fin m) ↦
        if (↑j₁_1 : ℕ) ≤ (↑j₁ : ℕ) ∧ (↑j₁ : ℕ) ≤ (↑j₁_1 : ℕ) + n then r * g.coeff ((↑j₁ : ℕ) - (↑j₁_1 : ℕ)) else 0)
      (fun (j₁_1 : Fin n) ↦
        if (↑j₁_1 : ℕ) ≤ (↑j₁ : ℕ) ∧ (↑j₁ : ℕ) ≤ (↑j₁_1 : ℕ) + m then f.coeff ((↑j₁ : ℕ) - (↑j₁_1 : ℕ)) else 0)
      (Fin.castAdd n j₂)
Tactic: induction j₂ using Fin.addCases with
        | left j₂ => simp
        | right i => simp at hi
Cleared hypotheses: j₂, IH, hi
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Resultant.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
f g : R[X]
m n : ℕ
r : R
M₁ : Matrix (Fin (m + n)) (Fin (m + n)) R := f.sylvester (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) r : R[X]) g) m n
M₂ : Matrix (Fin (m + n)) (Fin (m + n)) R := f.sylvester g m n
M : (i : ℕ) → Matrix (Fin (m + n)) (Fin (m + n)) R :=
  fun (i : ℕ) ↦
    (Matrix.of : (a : Fin (m + n) → Fin (m + n) → R) → Matrix (Fin (m + n)) (Fin (m + n)) R) fun (j₁ j₂ : Fin (m + n)) ↦
      if (↑j₂ : ℕ) < i then M₁ j₁ j₂ else M₂ j₁ j₂
i : ℕ
hi : i ≤ m
⊢ (M i).det = r ^ i * M₂.det
have zero : ∀ (hi : 0 ≤ m), (M 0).det = HPow.hPow (β := ℕ) r 0 * M₂.det
Tactic: induction i with
    | zero => simp [M]; rfl
    | succ i IH =>
      suffices (M i).updateCol ⟨i, by lia⟩ (r • fun j ↦ M i j ⟨i, by lia⟩) = (M (i + 1)) by
        rw [pow_succ', mul_assoc, ← IH (by lia), ← this, Matrix.det_updateCol_smul,
          Matrix.updateCol_eq_self]
      ext j₁ j₂
      simp only [Matrix.of_apply, lt_self_iff_false, ↓reduceIte, Matrix.updateCol_apply,
        Fin.ext_iff, Pi.smul_apply, smul_eq_mul, M, M₂]
      obtain rfl | h := eq_or_ne ↑j₂ i
      · simp only [↓reduceIte, sylvester, Set.mem_Icc, Fin.eta, Matrix.of_apply,
          lt_add_iff_pos_right, zero_lt_one, coeff_C_mul, M₁]
        induction j₂ using Fin.addCases with
        | left j₂ => simp
        | right i => simp at hi
      lia
Cleared hypotheses: i, hi
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Resultant.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
f g : R[X]
m n : ℕ
r : R
M₁ : Matrix (Fin (m + n)) (Fin (m + n)) R := f.sylvester (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) r : R[X]) g) m n
M₂ : Matrix (Fin (m + n)) (Fin (m + n)) R := f.sylvester g m n
M : (i : ℕ) → Matrix (Fin (m + n)) (Fin (m + n)) R :=
  fun (i : ℕ) ↦
    (Matrix.of : (a : Fin (m + n) → Fin (m + n) → R) → Matrix (Fin (m + n)) (Fin (m + n)) R) fun (j₁ j₂ : Fin (m + n)) ↦
      if (↑j₂ : ℕ) < i then M₁ j₁ j₂ else M₂ j₁ j₂
i : ℕ
hi : i ≤ m
zero : ∀ (hi : 0 ≤ m), (M 0).det = HPow.hPow (β := ℕ) r 0 * M₂.det
⊢ (M i).det = r ^ i * M₂.det
have succ : ∀ (i : ℕ) (IH : ∀ (hi : i ≤ m), (M i).det = r ^ i * M₂.det) (hi : i + 1 ≤ m), (M (i + 1)).det = r ^ (i + 1) * M₂.det
Tactic: induction i with
    | zero => simp [M]; rfl
    | succ i IH =>
      suffices (M i).updateCol ⟨i, by lia⟩ (r • fun j ↦ M i j ⟨i, by lia⟩) = (M (i + 1)) by
        rw [pow_succ', mul_assoc, ← IH (by lia), ← this, Matrix.det_updateCol_smul,
          Matrix.updateCol_eq_self]
      ext j₁ j₂
      simp only [Matrix.of_apply, lt_self_iff_false, ↓reduceIte, Matrix.updateCol_apply,
        Fin.ext_iff, Pi.smul_apply, smul_eq_mul, M, M₂]
      obtain rfl | h := eq_or_ne ↑j₂ i
      · simp only [↓reduceIte, sylvester, Set.mem_Icc, Fin.eta, Matrix.of_apply,
          lt_add_iff_pos_right, zero_lt_one, coeff_C_mul, M₁]
        induction j₂ using Fin.addCases with
        | left j₂ => simp
        | right i => simp at hi
      lia
Cleared hypotheses: i, hi
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Resultant.Basic (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
f g : R[X]
m n : ℕ
r : R
M₁ : Matrix (Fin (m + n)) (Fin (m + n)) R := f.sylvester (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) r : R[X]) g) m n
M₂ : Matrix (Fin (m + n)) (Fin (m + n)) R := f.sylvester g m n
M : (i : ℕ) → Matrix (Fin (m + n)) (Fin (m + n)) R :=
  fun (i : ℕ) ↦
    (Matrix.of : (a : Fin (m + n) → Fin (m + n) → R) → Matrix (Fin (m + n)) (Fin (m + n)) R) fun (j₁ j₂ : Fin (m + n)) ↦
      if (↑j₂ : ℕ) < i then M₁ j₁ j₂ else M₂ j₁ j₂
⊢ f.resultant (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) r : R[X]) g) m n = r ^ m * f.resultant g m n
have this : ∀ (i : ℕ) (hi : i ≤ m), (M i).det = r ^ i * M₂.det
Tactic: have (i : ℕ) (hi : i ≤ m) : (M i).det = r ^ i * M₂.det := by
    induction i with
    | zero => simp [M]; rfl
    | succ i IH =>
      suffices (M i).updateCol ⟨i, by lia⟩ (r • fun j ↦ M i j ⟨i, by lia⟩) = (M (i + 1)) by
        rw [pow_succ', mul_assoc, ← IH (by lia), ← this, Matrix.det_updateCol_smul,
          Matrix.updateCol_eq_self]
      ext j₁ j₂
      simp only [Matrix.of_apply, lt_self_iff_false, ↓reduceIte, Matrix.updateCol_apply,
        Fin.ext_iff, Pi.smul_apply, smul_eq_mul, M, M₂]
      obtain rfl | h := eq_or_ne ↑j₂ i
      · simp only [↓reduceIte, sylvester, Set.mem_Icc, Fin.eta, Matrix.of_apply,
          lt_add_iff_pos_right, zero_lt_one, coeff_C_mul, M₁]
        induction j₂ using Fin.addCases with
        | left j₂ => simp
        | right i => simp at hi
      lia
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Resultant.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
f g : R[X]
m n : ℕ
r : R
M₁ : Matrix (Fin (m + n)) (Fin (m + n)) R := f.sylvester (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) r : R[X]) g) m n
M₂ : Matrix (Fin (m + n)) (Fin (m + n)) R := f.sylvester g m n
M : (i : ℕ) → Matrix (Fin (m + n)) (Fin (m + n)) R :=
  fun (i : ℕ) ↦
    (Matrix.of : (a : Fin (m + n) → Fin (m + n) → R) → Matrix (Fin (m + n)) (Fin (m + n)) R) fun (j₁ j₂ : Fin (m + n)) ↦
      if (↑j₂ : ℕ) < i then M₁ j₁ j₂ else M₂ j₁ j₂
this : ∀ (i : ℕ) (hi : i ≤ m), (M i).det = r ^ i * M₂.det
⊢ f.resultant (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) r : R[X]) g) m n = r ^ m * f.resultant g m n
have rw : (f.sylvester (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) r : R[X]) g) m n).det = (M m).det
Tactic: rw [resultant, resultant, ← this m le_rfl]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Resultant.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
f g : R[X]
m n : ℕ
r : R
M₁ : Matrix (Fin (m + n)) (Fin (m + n)) R := f.sylvester (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) r : R[X]) g) m n
M₂ : Matrix (Fin (m + n)) (Fin (m + n)) R := f.sylvester g m n
M : (i : ℕ) → Matrix (Fin (m + n)) (Fin (m + n)) R :=
  fun (i : ℕ) ↦
    (Matrix.of : (a : Fin (m + n) → Fin (m + n) → R) → Matrix (Fin (m + n)) (Fin (m + n)) R) fun (j₁ j₂ : Fin (m + n)) ↦
      if (↑j₂ : ℕ) < i then M₁ j₁ j₂ else M₂ j₁ j₂
this : ∀ (i : ℕ) (hi : i ≤ m), (M i).det = r ^ i * M₂.det
⊢ (f.sylvester (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) r : R[X]) g) m n).det = (M m).det
have e_M : f.sylvester (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) r : R[X]) g) m n = M m
Tactic: congr 1