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
hf : f.natDegree ≤ m
n : ℕ
succ : HPow.hPow (α := R) (-1) (m + 1 + n + m) = HPow.hPow (α := R) (-1) (n + 1)
succ₁ : f.sylvester g (m + 1) (n + 1) (Fin.last (m + 1 + n)) (Fin.castAdd (n + 1) (Fin.last m)) = g.coeff (n + 1)
⊢ HPow.hPow (α := R) (-1) (m + 1 + n + m) *
        f.sylvester g (m + 1) (n + 1) (Fin.last (m + 1 + n)) (Fin.castAdd (n + 1) (Fin.last m)) *
      Matrix.det (n := Fin (m + (n + 1)))
        (Matrix.submatrix (f.sylvester g (m + 1) (n + 1))
          (Fin.castSucc ∘ (⇑(finCongr ⋯) : (a : Fin (m + (n + 1))) → Fin (m + 1 + n)))
          (Fin.succAbove (Fin.castAdd (n + 1) (Fin.last m)) ∘
            (⇑(finCongr ⋯) : (a : Fin (m + (n + 1))) → Fin (m + 1 + n)))) =
    HPow.hPow (α := R) (-1) (n + 1) * g.coeff (n + 1) * Matrix.det (n := Fin (m + (n + 1))) (f.sylvester g m (n + 1))
have succ₂ : Eq (α := Matrix (Fin (m + (n + 1))) (Fin (m + (n + 1))) R)
  (Matrix.submatrix (f.sylvester g (m + 1) (n + 1))
    (Fin.castSucc ∘ (⇑(finCongr sorry) : (a : Fin (m + (n + 1))) → Fin (m + 1 + n)))
    (Fin.succAbove (Fin.castAdd (n + 1) (Fin.last m)) ∘ (⇑(finCongr sorry) : (a : Fin (m + (n + 1))) → Fin (m + 1 + n))))
  (f.sylvester g m (n + 1))
Tactic: congr 2
-- 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
hf : f.natDegree ≤ m
n : ℕ
⊢ HPow.hPow (α := R) (-1) (m + 1 + n + m) = HPow.hPow (α := R) (-1) (n + 1)
have trans : HPow.hPow (α := R) (-1) (m + 1 + n + m) = HPow.hPow (α := R) (-1) (2 * m + (n + 1))
Tactic: trans (-1) ^ (2 * m + (n + 1))
-- 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
hf : f.natDegree ≤ m
n : ℕ
trans : HPow.hPow (α := R) (-1) (m + 1 + n + m) = HPow.hPow (α := R) (-1) (2 * m + (n + 1))
⊢ HPow.hPow (α := R) (-1) (m + 1 + n + m) = HPow.hPow (α := R) (-1) (n + 1)
have trans₁ : HPow.hPow (α := R) (-1) (2 * m + (n + 1)) = HPow.hPow (α := R) (-1) (n + 1)
Tactic: trans (-1) ^ (2 * m + (n + 1))
-- 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
hf : f.natDegree ≤ m
n : ℕ
⊢ HPow.hPow (α := R) (-1) (m + 1 + n + m) = HPow.hPow (α := R) (-1) (2 * m + (n + 1))
have e_a : m + 1 + n + m = 2 * m + (n + 1)
Tactic: congr 1
-- 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
hf : f.natDegree ≤ m
n : ℕ
⊢ f.sylvester g (m + 1) (n + 1) (Fin.last (m + 1 + n)) (Fin.castAdd (n + 1) (Fin.last m)) = g.coeff (n + 1)
have succ : (if m ≤ m + 1 + n ∧ m + 1 + n ≤ m + (n + 1) then g.coeff (m + 1 + n - m) else 0) = g.coeff (n + 1)
Tactic: simp only [sylvester, Set.mem_Icc, Matrix.of_apply, Fin.val_last, Fin.addCases_left]
-- 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
hf : f.natDegree ≤ m
n : ℕ
⊢ (if m ≤ m + 1 + n ∧ m + 1 + n ≤ m + (n + 1) then g.coeff (m + 1 + n - m) else 0) = g.coeff (n + 1)
have succ : g.coeff (m + 1 + n - m) = g.coeff (n + 1)
Tactic: rw [if_pos (by 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
hf : f.natDegree ≤ m
n : ℕ
⊢ Eq (α := Matrix (Fin (m + (n + 1))) (Fin (m + (n + 1))) R)
    (Matrix.submatrix (f.sylvester g (m + 1) (n + 1))
      (Fin.castSucc ∘ (⇑(finCongr ⋯) : (a : Fin (m + (n + 1))) → Fin (m + 1 + n)))
      (Fin.succAbove (Fin.castAdd (n + 1) (Fin.last m)) ∘ (⇑(finCongr ⋯) : (a : Fin (m + (n + 1))) → Fin (m + 1 + n))))
    (f.sylvester g m (n + 1))
have succ : ∀ (i j : Fin (m + (n + 1))),
  Matrix.submatrix (f.sylvester g (m + 1) (n + 1))
      (Fin.castSucc ∘ (⇑(finCongr sorry) : (a : Fin (m + (n + 1))) → Fin (m + 1 + n)))
      (Fin.succAbove (Fin.castAdd (n + 1) (Fin.last m)) ∘ (⇑(finCongr sorry) : (a : Fin (m + (n + 1))) → Fin (m + 1 + n))) i
      j =
    f.sylvester g m (n + 1) i j
Tactic: ext i j
-- 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
hf : f.natDegree ≤ m
n : ℕ
i j : Fin (m + (n + 1))
⊢ Matrix.submatrix (f.sylvester g (m + 1) (n + 1))
      (Fin.castSucc ∘ (⇑(finCongr ⋯) : (a : Fin (m + (n + 1))) → Fin (m + 1 + n)))
      (Fin.succAbove (Fin.castAdd (n + 1) (Fin.last m)) ∘ (⇑(finCongr ⋯) : (a : Fin (m + (n + 1))) → Fin (m + 1 + n))) i
      j =
    f.sylvester g m (n + 1) i j
have succ : Eq.{u_1 + 1} (α := R)
  (Fin.addCases (motive := fun (x : Fin (m + 1 + (n + 1))) ↦ R)
      (fun (j₁ : Fin (m + 1)) ↦
        if (↑j₁ : ℕ) ≤ (↑i : ℕ) ∧ (↑i : ℕ) ≤ (↑j₁ : ℕ) + (n + 1) then g.coeff ((↑i : ℕ) - (↑j₁ : ℕ)) else 0)
      (fun (j₁ : Fin (n + 1)) ↦
        if (↑j₁ : ℕ) ≤ (↑i : ℕ) ∧ (↑i : ℕ) ≤ (↑j₁ : ℕ) + (m + 1) then f.coeff ((↑i : ℕ) - (↑j₁ : ℕ)) else 0)
      (Fin.succAbove (Fin.castAdd (n + 1) (Fin.last m)) (Fin.cast sorry j)) :
    R)
  (Fin.addCases (motive := fun (x : Fin (m + (n + 1))) ↦ R)
      (fun (j₁ : Fin m) ↦
        if (↑j₁ : ℕ) ≤ (↑i : ℕ) ∧ (↑i : ℕ) ≤ (↑j₁ : ℕ) + (n + 1) then g.coeff ((↑i : ℕ) - (↑j₁ : ℕ)) else 0)
      (fun (j₁ : Fin (n + 1)) ↦
        if (↑j₁ : ℕ) ≤ (↑i : ℕ) ∧ (↑i : ℕ) ≤ (↑j₁ : ℕ) + m then f.coeff ((↑i : ℕ) - (↑j₁ : ℕ)) else 0)
      j :
    R)
Tactic: simp only [sylvester, Set.mem_Icc, Matrix.submatrix_apply, Function.comp_apply,
        finCongr_apply, Matrix.of_apply, Fin.val_castSucc, Fin.val_cast]
-- 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
hf : f.natDegree ≤ m
n : ℕ
i : Fin (m + (n + 1))
j : Fin m
⊢ Fin.succAbove (Fin.castAdd (n + 1) (Fin.last m)) (Fin.cast ⋯ (Fin.castAdd (n + 1) j)) = Fin.castAdd (n + 1) j.castSucc
have h : (↑(Fin.succAbove (Fin.castAdd (n + 1) (Fin.last m)) (Fin.cast sorry (Fin.castAdd (n + 1) j))) : ℕ) =
  (↑(Fin.castAdd (n + 1) j.castSucc : Fin (m + 1 + (n + 1))) : ℕ)
Tactic: ext
-- 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
hf : f.natDegree ≤ m
n : ℕ
i : Fin (m + (n + 1))
j : Fin m
⊢ Eq.{u_1 + 1} (α := R)
    (Fin.addCases (motive := fun (x : Fin (m + 1 + (n + 1))) ↦ R)
        (fun (j₁ : Fin (m + 1)) ↦
          if (↑j₁ : ℕ) ≤ (↑i : ℕ) ∧ (↑i : ℕ) ≤ (↑j₁ : ℕ) + (n + 1) then g.coeff ((↑i : ℕ) - (↑j₁ : ℕ)) else 0)
        (fun (j₁ : Fin (n + 1)) ↦
          if (↑j₁ : ℕ) ≤ (↑i : ℕ) ∧ (↑i : ℕ) ≤ (↑j₁ : ℕ) + (m + 1) then f.coeff ((↑i : ℕ) - (↑j₁ : ℕ)) else 0)
        (Fin.succAbove (Fin.castAdd (n + 1) (Fin.last m)) (Fin.cast (n := m + (n + 1)) ⋯ (Fin.castAdd (n + 1) j))) :
      R)
    (Fin.addCases (motive := fun (x : Fin (m + (n + 1))) ↦ R)
        (fun (j₁ : Fin m) ↦
          if (↑j₁ : ℕ) ≤ (↑i : ℕ) ∧ (↑i : ℕ) ≤ (↑j₁ : ℕ) + (n + 1) then g.coeff ((↑i : ℕ) - (↑j₁ : ℕ)) else 0)
        (fun (j₁ : Fin (n + 1)) ↦
          if (↑j₁ : ℕ) ≤ (↑i : ℕ) ∧ (↑i : ℕ) ≤ (↑j₁ : ℕ) + m then f.coeff ((↑i : ℕ) - (↑j₁ : ℕ)) else 0)
        (Fin.castAdd (n + 1) j) :
      R)
have this : Fin.succAbove (Fin.castAdd (n + 1) (Fin.last m)) (Fin.cast sorry (Fin.castAdd (n + 1) j)) = Fin.castAdd (n + 1) j.castSucc
Tactic: have : ((Fin.last m).castAdd (n + 1)).succAbove ((j.castAdd (n + 1)).cast (by grind)) =
          j.castSucc.castAdd (n + 1) := by ext; simp [Fin.succAbove, Fin.lt_def]
-- 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
hf : f.natDegree ≤ m
n : ℕ
i : Fin (m + (n + 1))
j : Fin (n + 1)
⊢ Fin.succAbove (Fin.castAdd (n + 1) (Fin.last m)) (Fin.cast ⋯ (Fin.natAdd m j)) = Fin.natAdd (m + 1) j
have h : (↑(Fin.succAbove (Fin.castAdd (n + 1) (Fin.last m)) (Fin.cast sorry (Fin.natAdd m j))) : ℕ) = (↑(Fin.natAdd (m + 1) j) : ℕ)
Tactic: ext
-- 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
hf : f.natDegree ≤ m
n : ℕ
i : Fin (m + (n + 1))
j : Fin (n + 1)
⊢ Eq.{u_1 + 1} (α := R)
    (Fin.addCases (motive := fun (x : Fin (m + 1 + (n + 1))) ↦ R)
        (fun (j₁ : Fin (m + 1)) ↦
          if (↑j₁ : ℕ) ≤ (↑i : ℕ) ∧ (↑i : ℕ) ≤ (↑j₁ : ℕ) + (n + 1) then g.coeff ((↑i : ℕ) - (↑j₁ : ℕ)) else 0)
        (fun (j₁ : Fin (n + 1)) ↦
          if (↑j₁ : ℕ) ≤ (↑i : ℕ) ∧ (↑i : ℕ) ≤ (↑j₁ : ℕ) + (m + 1) then f.coeff ((↑i : ℕ) - (↑j₁ : ℕ)) else 0)
        (Fin.succAbove (Fin.castAdd (n + 1) (Fin.last m)) (Fin.cast ⋯ (Fin.natAdd m j))) :
      R)
    (Fin.addCases (motive := fun (x : Fin (m + (n + 1))) ↦ R)
        (fun (j₁ : Fin m) ↦
          if (↑j₁ : ℕ) ≤ (↑i : ℕ) ∧ (↑i : ℕ) ≤ (↑j₁ : ℕ) + (n + 1) then g.coeff ((↑i : ℕ) - (↑j₁ : ℕ)) else 0)
        (fun (j₁ : Fin (n + 1)) ↦
          if (↑j₁ : ℕ) ≤ (↑i : ℕ) ∧ (↑i : ℕ) ≤ (↑j₁ : ℕ) + m then f.coeff ((↑i : ℕ) - (↑j₁ : ℕ)) else 0)
        (Fin.natAdd m j) :
      R)
have this : Fin.succAbove (Fin.castAdd (n + 1) (Fin.last m)) (Fin.cast sorry (Fin.natAdd m j)) = Fin.natAdd (m + 1) j
Tactic: have : ((Fin.last m).castAdd (n + 1)).succAbove ((j.natAdd m).cast (by grind)) =
          j.natAdd _ := by ext; simp [Fin.succAbove, Fin.lt_def, add_right_comm]
-- 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
hf : f.natDegree ≤ m
n : ℕ
i : Fin (m + (n + 1))
j : Fin (n + 1)
this : Fin.succAbove (Fin.castAdd (n + 1) (Fin.last m)) (Fin.cast ⋯ (Fin.natAdd m j)) = Fin.natAdd (m + 1) j
⊢ Eq.{u_1 + 1} (α := R)
    (Fin.addCases (motive := fun (x : Fin (m + 1 + (n + 1))) ↦ R)
        (fun (j₁ : Fin (m + 1)) ↦
          if (↑j₁ : ℕ) ≤ (↑i : ℕ) ∧ (↑i : ℕ) ≤ (↑j₁ : ℕ) + (n + 1) then g.coeff ((↑i : ℕ) - (↑j₁ : ℕ)) else 0)
        (fun (j₁ : Fin (n + 1)) ↦
          if (↑j₁ : ℕ) ≤ (↑i : ℕ) ∧ (↑i : ℕ) ≤ (↑j₁ : ℕ) + (m + 1) then f.coeff ((↑i : ℕ) - (↑j₁ : ℕ)) else 0)
        (Fin.succAbove (Fin.castAdd (n + 1) (Fin.last m)) (Fin.cast ⋯ (Fin.natAdd m j))) :
      R)
    (Fin.addCases (motive := fun (x : Fin (m + (n + 1))) ↦ R)
        (fun (j₁ : Fin m) ↦
          if (↑j₁ : ℕ) ≤ (↑i : ℕ) ∧ (↑i : ℕ) ≤ (↑j₁ : ℕ) + (n + 1) then g.coeff ((↑i : ℕ) - (↑j₁ : ℕ)) else 0)
        (fun (j₁ : Fin (n + 1)) ↦
          if (↑j₁ : ℕ) ≤ (↑i : ℕ) ∧ (↑i : ℕ) ≤ (↑j₁ : ℕ) + m then f.coeff ((↑i : ℕ) - (↑j₁ : ℕ)) else 0)
        (Fin.natAdd m j) :
      R)
have succ : (if (↑j : ℕ) ≤ (↑i : ℕ) then if (↑i : ℕ) ≤ (↑j : ℕ) + (m + 1) then f.coeff ((↑i : ℕ) - (↑j : ℕ)) else 0 else 0) =
  if (↑j : ℕ) ≤ (↑i : ℕ) then if (↑i : ℕ) ≤ (↑j : ℕ) + m then f.coeff ((↑i : ℕ) - (↑j : ℕ)) else 0 else 0
Tactic: simp only [ite_and, this, Fin.addCases_right]
-- 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
hf : f.natDegree ≤ m
n : ℕ
i : Fin (m + (n + 1))
j : Fin (n + 1)
this : Fin.succAbove (Fin.castAdd (n + 1) (Fin.last m)) (Fin.cast ⋯ (Fin.natAdd m j)) = Fin.natAdd (m + 1) j
⊢ (if (↑j : ℕ) ≤ (↑i : ℕ) then if (↑i : ℕ) ≤ (↑j : ℕ) + (m + 1) then f.coeff ((↑i : ℕ) - (↑j : ℕ)) else 0 else 0) =
    if (↑j : ℕ) ≤ (↑i : ℕ) then if (↑i : ℕ) ≤ (↑j : ℕ) + m then f.coeff ((↑i : ℕ) - (↑j : ℕ)) else 0 else 0
have pos : ∀ (h₁ : (↑j : ℕ) ≤ (↑i : ℕ)) (h₂ : (↑i : ℕ) ≤ (↑j : ℕ) + (m + 1)) (h₃ : (↑i : ℕ) ≤ (↑j : ℕ) + m),
  f.coeff ((↑i : ℕ) - (↑j : ℕ)) = f.coeff ((↑i : ℕ) - (↑j : ℕ))
Tactic: split_ifs with h₁ h₂ h₃ h₃
-- 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
hf : f.natDegree ≤ m
n : ℕ
i : Fin (m + (n + 1))
j : Fin (n + 1)
this : Fin.succAbove (Fin.castAdd (n + 1) (Fin.last m)) (Fin.cast ⋯ (Fin.natAdd m j)) = Fin.natAdd (m + 1) j
pos :
  ∀ (h₁ : (↑j : ℕ) ≤ (↑i : ℕ)) (h₂ : (↑i : ℕ) ≤ (↑j : ℕ) + (m + 1)) (h₃ : (↑i : ℕ) ≤ (↑j : ℕ) + m),
    f.coeff ((↑i : ℕ) - (↑j : ℕ)) = f.coeff ((↑i : ℕ) - (↑j : ℕ))
⊢ (if (↑j : ℕ) ≤ (↑i : ℕ) then if (↑i : ℕ) ≤ (↑j : ℕ) + (m + 1) then f.coeff ((↑i : ℕ) - (↑j : ℕ)) else 0 else 0) =
    if (↑j : ℕ) ≤ (↑i : ℕ) then if (↑i : ℕ) ≤ (↑j : ℕ) + m then f.coeff ((↑i : ℕ) - (↑j : ℕ)) else 0 else 0
have neg : ∀ (h₁ : (↑j : ℕ) ≤ (↑i : ℕ)) (h₂ : (↑i : ℕ) ≤ (↑j : ℕ) + (m + 1)) (h₃ : ¬(↑i : ℕ) ≤ (↑j : ℕ) + m),
  f.coeff ((↑i : ℕ) - (↑j : ℕ)) = 0
Tactic: split_ifs with h₁ h₂ h₃ h₃
-- 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
hf : f.natDegree ≤ m
n : ℕ
i : Fin (m + (n + 1))
j : Fin (n + 1)
this : Fin.succAbove (Fin.castAdd (n + 1) (Fin.last m)) (Fin.cast ⋯ (Fin.natAdd m j)) = Fin.natAdd (m + 1) j
pos :
  ∀ (h₁ : (↑j : ℕ) ≤ (↑i : ℕ)) (h₂ : (↑i : ℕ) ≤ (↑j : ℕ) + (m + 1)) (h₃ : (↑i : ℕ) ≤ (↑j : ℕ) + m),
    f.coeff ((↑i : ℕ) - (↑j : ℕ)) = f.coeff ((↑i : ℕ) - (↑j : ℕ))
neg :
  ∀ (h₁ : (↑j : ℕ) ≤ (↑i : ℕ)) (h₂ : (↑i : ℕ) ≤ (↑j : ℕ) + (m + 1)) (h₃ : ¬(↑i : ℕ) ≤ (↑j : ℕ) + m),
    f.coeff ((↑i : ℕ) - (↑j : ℕ)) = 0
⊢ (if (↑j : ℕ) ≤ (↑i : ℕ) then if (↑i : ℕ) ≤ (↑j : ℕ) + (m + 1) then f.coeff ((↑i : ℕ) - (↑j : ℕ)) else 0 else 0) =
    if (↑j : ℕ) ≤ (↑i : ℕ) then if (↑i : ℕ) ≤ (↑j : ℕ) + m then f.coeff ((↑i : ℕ) - (↑j : ℕ)) else 0 else 0
have pos₁ : ∀ (h₁ : (↑j : ℕ) ≤ (↑i : ℕ)) (h₂ : ¬(↑i : ℕ) ≤ (↑j : ℕ) + (m + 1)) (h₃ : (↑i : ℕ) ≤ (↑j : ℕ) + m),
  0 = f.coeff ((↑i : ℕ) - (↑j : ℕ))
Tactic: split_ifs with h₁ h₂ h₃ h₃
-- 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
hf : f.natDegree ≤ m
n : ℕ
i : Fin (m + (n + 1))
j : Fin (n + 1)
this : Fin.succAbove (Fin.castAdd (n + 1) (Fin.last m)) (Fin.cast ⋯ (Fin.natAdd m j)) = Fin.natAdd (m + 1) j
pos :
  ∀ (h₁ : (↑j : ℕ) ≤ (↑i : ℕ)) (h₂ : (↑i : ℕ) ≤ (↑j : ℕ) + (m + 1)) (h₃ : (↑i : ℕ) ≤ (↑j : ℕ) + m),
    f.coeff ((↑i : ℕ) - (↑j : ℕ)) = f.coeff ((↑i : ℕ) - (↑j : ℕ))
neg :
  ∀ (h₁ : (↑j : ℕ) ≤ (↑i : ℕ)) (h₂ : (↑i : ℕ) ≤ (↑j : ℕ) + (m + 1)) (h₃ : ¬(↑i : ℕ) ≤ (↑j : ℕ) + m),
    f.coeff ((↑i : ℕ) - (↑j : ℕ)) = 0
pos₁ :
  ∀ (h₁ : (↑j : ℕ) ≤ (↑i : ℕ)) (h₂ : ¬(↑i : ℕ) ≤ (↑j : ℕ) + (m + 1)) (h₃ : (↑i : ℕ) ≤ (↑j : ℕ) + m),
    0 = f.coeff ((↑i : ℕ) - (↑j : ℕ))
⊢ (if (↑j : ℕ) ≤ (↑i : ℕ) then if (↑i : ℕ) ≤ (↑j : ℕ) + (m + 1) then f.coeff ((↑i : ℕ) - (↑j : ℕ)) else 0 else 0) =
    if (↑j : ℕ) ≤ (↑i : ℕ) then if (↑i : ℕ) ≤ (↑j : ℕ) + m then f.coeff ((↑i : ℕ) - (↑j : ℕ)) else 0 else 0
have neg₁ : ∀ (h₁ : (↑j : ℕ) ≤ (↑i : ℕ)) (h₂ : ¬(↑i : ℕ) ≤ (↑j : ℕ) + (m + 1)) (h₃ : ¬(↑i : ℕ) ≤ (↑j : ℕ) + m),
  Eq.{u_1 + 1} (α := R) 0 0
Tactic: split_ifs with h₁ h₂ h₃ h₃
-- 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
hf : f.natDegree ≤ m
n : ℕ
i : Fin (m + (n + 1))
j : Fin (n + 1)
this : Fin.succAbove (Fin.castAdd (n + 1) (Fin.last m)) (Fin.cast ⋯ (Fin.natAdd m j)) = Fin.natAdd (m + 1) j
pos :
  ∀ (h₁ : (↑j : ℕ) ≤ (↑i : ℕ)) (h₂ : (↑i : ℕ) ≤ (↑j : ℕ) + (m + 1)) (h₃ : (↑i : ℕ) ≤ (↑j : ℕ) + m),
    f.coeff ((↑i : ℕ) - (↑j : ℕ)) = f.coeff ((↑i : ℕ) - (↑j : ℕ))
neg :
  ∀ (h₁ : (↑j : ℕ) ≤ (↑i : ℕ)) (h₂ : (↑i : ℕ) ≤ (↑j : ℕ) + (m + 1)) (h₃ : ¬(↑i : ℕ) ≤ (↑j : ℕ) + m),
    f.coeff ((↑i : ℕ) - (↑j : ℕ)) = 0
pos₁ :
  ∀ (h₁ : (↑j : ℕ) ≤ (↑i : ℕ)) (h₂ : ¬(↑i : ℕ) ≤ (↑j : ℕ) + (m + 1)) (h₃ : (↑i : ℕ) ≤ (↑j : ℕ) + m),
    0 = f.coeff ((↑i : ℕ) - (↑j : ℕ))
neg₁ :
  ∀ (h₁ : (↑j : ℕ) ≤ (↑i : ℕ)) (h₂ : ¬(↑i : ℕ) ≤ (↑j : ℕ) + (m + 1)) (h₃ : ¬(↑i : ℕ) ≤ (↑j : ℕ) + m),
    Eq.{u_1 + 1} (α := R) 0 0
⊢ (if (↑j : ℕ) ≤ (↑i : ℕ) then if (↑i : ℕ) ≤ (↑j : ℕ) + (m + 1) then f.coeff ((↑i : ℕ) - (↑j : ℕ)) else 0 else 0) =
    if (↑j : ℕ) ≤ (↑i : ℕ) then if (↑i : ℕ) ≤ (↑j : ℕ) + m then f.coeff ((↑i : ℕ) - (↑j : ℕ)) else 0 else 0
have neg₂ : ∀ (h₁ : ¬(↑j : ℕ) ≤ (↑i : ℕ)), Eq.{u_1 + 1} (α := R) 0 0
Tactic: split_ifs with h₁ h₂ h₃ h₃
-- 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
hf : f.natDegree ≤ m
n : ℕ
i j : Fin (m + (n + 1))
⊢ Eq.{u_1 + 1} (α := R)
    (Fin.addCases (motive := fun (x : Fin (m + 1 + (n + 1))) ↦ R)
        (fun (j₁ : Fin (m + 1)) ↦
          if (↑j₁ : ℕ) ≤ (↑i : ℕ) ∧ (↑i : ℕ) ≤ (↑j₁ : ℕ) + (n + 1) then g.coeff ((↑i : ℕ) - (↑j₁ : ℕ)) else 0)
        (fun (j₁ : Fin (n + 1)) ↦
          if (↑j₁ : ℕ) ≤ (↑i : ℕ) ∧ (↑i : ℕ) ≤ (↑j₁ : ℕ) + (m + 1) then f.coeff ((↑i : ℕ) - (↑j₁ : ℕ)) else 0)
        (Fin.succAbove (Fin.castAdd (n + 1) (Fin.last m)) (Fin.cast ⋯ j)) :
      R)
    (Fin.addCases (motive := fun (x : Fin (m + (n + 1))) ↦ R)
        (fun (j₁ : Fin m) ↦
          if (↑j₁ : ℕ) ≤ (↑i : ℕ) ∧ (↑i : ℕ) ≤ (↑j₁ : ℕ) + (n + 1) then g.coeff ((↑i : ℕ) - (↑j₁ : ℕ)) else 0)
        (fun (j₁ : Fin (n + 1)) ↦
          if (↑j₁ : ℕ) ≤ (↑i : ℕ) ∧ (↑i : ℕ) ≤ (↑j₁ : ℕ) + m then f.coeff ((↑i : ℕ) - (↑j₁ : ℕ)) else 0)
        j :
      R)
have succ : ∀ (j : Fin m),
  Eq.{u_1 + 1} (α := R)
    (Fin.addCases (motive := fun (x : Fin (m + 1 + (n + 1))) ↦ R)
        (fun (j₁ : Fin (m + 1)) ↦
          if (↑j₁ : ℕ) ≤ (↑i : ℕ) ∧ (↑i : ℕ) ≤ (↑j₁ : ℕ) + (n + 1) then g.coeff ((↑i : ℕ) - (↑j₁ : ℕ)) else 0)
        (fun (j₁ : Fin (n + 1)) ↦
          if (↑j₁ : ℕ) ≤ (↑i : ℕ) ∧ (↑i : ℕ) ≤ (↑j₁ : ℕ) + (m + 1) then f.coeff ((↑i : ℕ) - (↑j₁ : ℕ)) else 0)
        (Fin.succAbove (Fin.castAdd (n + 1) (Fin.last m)) (Fin.cast (n := m + (n + 1)) sorry (Fin.castAdd (n + 1) j))) :
      R)
    (Fin.addCases (motive := fun (x : Fin (m + (n + 1))) ↦ R)
        (fun (j₁ : Fin m) ↦
          if (↑j₁ : ℕ) ≤ (↑i : ℕ) ∧ (↑i : ℕ) ≤ (↑j₁ : ℕ) + (n + 1) then g.coeff ((↑i : ℕ) - (↑j₁ : ℕ)) else 0)
        (fun (j₁ : Fin (n + 1)) ↦
          if (↑j₁ : ℕ) ≤ (↑i : ℕ) ∧ (↑i : ℕ) ≤ (↑j₁ : ℕ) + m then f.coeff ((↑i : ℕ) - (↑j₁ : ℕ)) else 0)
        (Fin.castAdd (n + 1) j) :
      R)
Tactic: induction j using Fin.addCases with
      | left j =>
        have : ((Fin.last m).castAdd (n + 1)).succAbove ((j.castAdd (n + 1)).cast (by grind)) =
          j.castSucc.castAdd (n + 1) := by ext; simp [Fin.succAbove, Fin.lt_def]
        simp [this]
      | right j =>
        have : ((Fin.last m).castAdd (n + 1)).succAbove ((j.natAdd m).cast (by grind)) =
          j.natAdd _ := by ext; simp [Fin.succAbove, Fin.lt_def, add_right_comm]
        simp only [ite_and, this, Fin.addCases_right]
        split_ifs with h₁ h₂ h₃ h₃ <;> try lia
        exact coeff_eq_zero_of_natDegree_lt (by lia)
Cleared hypotheses: j
-- 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
hf : f.natDegree ≤ m
n : ℕ
b : Fin (m + 1 + (n + 1))
hb : b ≠ Fin.castAdd (n + 1) (Fin.last m)
⊢ HPow.hPow (α := R) (-1) ((↑(Fin.last ((m + 1).add n)) : ℕ) + (↑b : ℕ)) *
        f.sylvester g (m + 1) (n + 1) (Fin.last ((m + 1).add n)) b *
      (Matrix.submatrix (f.sylvester g (m + 1) (n + 1)) (Fin.succAbove (Fin.last ((m + 1).add n))) b.succAbove).det =
    0
have suffices : f.sylvester g (m + 1) (n + 1) (Fin.last (m + 1 + n)) b = 0
Tactic: suffices f.sylvester g (m + 1) (n + 1) (.last (m + 1 + n)) b = 0 by simp [this]
-- 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
hf : f.natDegree ≤ m
n : ℕ
b : Fin (m + 1)
hb :
  Ne (α := Fin ((m + 1).add n).succ) (Fin.castAdd (n + 1) b : Fin (m + 1 + (n + 1))) (Fin.castAdd (n + 1) (Fin.last m))
⊢ f.sylvester g (m + 1) (n + 1) (Fin.last (m + 1 + n)) (Fin.castAdd (n + 1) b) = 0
have hb₁ : ¬(↑b : ℕ) = m
Tactic: simp only [Nat.add_eq, Nat.succ_eq_add_one, ne_eq, Fin.ext_iff, Fin.val_castAdd,
        Fin.val_last] at hb
Cleared hypotheses: hb
-- 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
hf : f.natDegree ≤ m
n : ℕ
b : Fin (m + 1)
hb : ¬(↑b : ℕ) = m
⊢ f.sylvester g (m + 1) (n + 1) (Fin.last (m + 1 + n)) (Fin.castAdd (n + 1) b) = 0
have left : (↑b : ℕ) ≤ m + 1 + n → m + 1 + n ≤ (↑b : ℕ) + (n + 1) → g.coeff (m + 1 + n - (↑b : ℕ)) = 0
Tactic: simp only [sylvester, Set.mem_Icc, Matrix.of_apply, Fin.val_last, Fin.addCases_left,
        ite_eq_right_iff, and_imp]
-- 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
hf : f.natDegree ≤ m
n : ℕ
b : Fin (m + 1 + (n + 1))
hb : b ≠ Fin.castAdd (n + 1) (Fin.last m)
⊢ f.sylvester g (m + 1) (n + 1) (Fin.last (m + 1 + n)) b = 0
have left : ∀ (b : Fin (m + 1))
  (hb :
    Ne (α := Fin ((m + 1).add n).succ) (Fin.castAdd (n + 1) b : Fin (m + 1 + (n + 1)))
      (Fin.castAdd (n + 1) (Fin.last m))),
  f.sylvester g (m + 1) (n + 1) (Fin.last (m + 1 + n)) (Fin.castAdd (n + 1) b) = 0
Tactic: induction b using Fin.addCases with
    | left b =>
      simp only [Nat.add_eq, Nat.succ_eq_add_one, ne_eq, Fin.ext_iff, Fin.val_castAdd,
        Fin.val_last] at hb
      simp only [sylvester, Set.mem_Icc, Matrix.of_apply, Fin.val_last, Fin.addCases_left,
        ite_eq_right_iff, and_imp]
      intros
      lia
    | right i => simpa [sylvester] using fun _ _ ↦ coeff_eq_zero_of_natDegree_lt (by lia)
Cleared hypotheses: b, hb
-- 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
hf : f.natDegree ≤ m
n : ℕ
b : Fin (m + 1 + (n + 1))
hb : b ≠ Fin.castAdd (n + 1) (Fin.last m)
left :
  ∀ (b : Fin (m + 1))
    (hb :
      Ne (α := Fin ((m + 1).add n).succ) (Fin.castAdd (n + 1) b : Fin (m + 1 + (n + 1)))
        (Fin.castAdd (n + 1) (Fin.last m))),
    f.sylvester g (m + 1) (n + 1) (Fin.last (m + 1 + n)) (Fin.castAdd (n + 1) b) = 0
⊢ f.sylvester g (m + 1) (n + 1) (Fin.last (m + 1 + n)) b = 0
have right : ∀ (i : Fin (n + 1)) (hb : Fin.natAdd (m + 1) i ≠ Fin.castAdd (n + 1) (Fin.last m)),
  f.sylvester g (m + 1) (n + 1) (Fin.last (m + 1 + n) : Fin (m + 1 + n + 1)) (Fin.natAdd (m + 1) i) = 0
Tactic: induction b using Fin.addCases with
    | left b =>
      simp only [Nat.add_eq, Nat.succ_eq_add_one, ne_eq, Fin.ext_iff, Fin.val_castAdd,
        Fin.val_last] at hb
      simp only [sylvester, Set.mem_Icc, Matrix.of_apply, Fin.val_last, Fin.addCases_left,
        ite_eq_right_iff, and_imp]
      intros
      lia
    | right i => simpa [sylvester] using fun _ _ ↦ coeff_eq_zero_of_natDegree_lt (by lia)
Cleared hypotheses: b, hb
-- 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 k_1 : ℕ
hf : f.natDegree ≤ m
k : ℕ
IH : f.resultant g (m + k) n = HPow.hPow (α := R) (-1) (n * k) * g.coeff n ^ k * f.resultant g m n
⊢ f.resultant g (m + (k + 1)) n = HPow.hPow (α := R) (-1) (n * (k + 1)) * g.coeff n ^ (k + 1) * f.resultant g m n
have succ : HPow.hPow (α := R) (-1) n * g.coeff n * (HPow.hPow (α := R) (-1) (n * k) * g.coeff n ^ k * f.resultant g m n) =
  HPow.hPow (α := R) (-1) (n * (k + 1)) * g.coeff n ^ (k + 1) * f.resultant g m n
Tactic: simp [← add_assoc, resultant_succ_left_deg, hf.trans, IH]
-- 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 k : ℕ
hf : f.natDegree ≤ m
⊢ f.resultant g (m + k) n = HPow.hPow (α := R) (-1) (n * k) * g.coeff n ^ k * f.resultant g m n
have zero : f.resultant g (m + 0) n = HPow.hPow (α := R) (-1) (n * 0) * HPow.hPow (β := ℕ) (g.coeff n) 0 * f.resultant g m n
Tactic: induction k with
  | zero => simp
  | succ k IH => simp [← add_assoc, resultant_succ_left_deg, hf.trans, IH]; ring
-- 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 k : ℕ
hf : f.natDegree ≤ m
zero : f.resultant g (m + 0) n = HPow.hPow (α := R) (-1) (n * 0) * HPow.hPow (β := ℕ) (g.coeff n) 0 * f.resultant g m n
⊢ f.resultant g (m + k) n = HPow.hPow (α := R) (-1) (n * k) * g.coeff n ^ k * f.resultant g m n
have succ : ∀ (k : ℕ) (IH : f.resultant g (m + k) n = HPow.hPow (α := R) (-1) (n * k) * g.coeff n ^ k * f.resultant g m n),
  f.resultant g (m + (k + 1)) n = HPow.hPow (α := R) (-1) (n * (k + 1)) * g.coeff n ^ (k + 1) * f.resultant g m n
Tactic: induction k with
  | zero => simp
  | succ k IH => simp [← add_assoc, resultant_succ_left_deg, hf.trans, IH]; ring