mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Polynomial.Degree.IsMonicOfDegree

Goal state before:
pp: case inr
R : Type u_1
inst✝ : Semiring R
p q : R[X]
m n : ℕ
hp : p.IsMonicOfDegree m
hpq : (p * q).IsMonicOfDegree (m + n)
H : Nontrivial R
⊢ q.IsMonicOfDegree n

Tactic:
have h₂ : q.Monic := hp.monic.of_mul_monic_left hpq.monic

Goal state after:
case inr
R : Type u_1
inst✝ : Semiring R
p q : R[X]
m n : ℕ
hp : p.IsMonicOfDegree m
hpq : (p * q).IsMonicOfDegree (m + n)
H : Nontrivial R
h₂ : q.Monic
⊢ q.IsMonicOfDegree n
Module: Mathlib.Algebra.Polynomial.Degree.IsMonicOfDegree

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
a b r : R[X]
m n : ℕ
hmn : n ≤ m
ha : a.IsMonicOfDegree m
hb : b.IsMonicOfDegree n
hr : r.natDegree < m
h : b ∣ a + r
q : R[X]
hq : a + r = q * b
⊢ (a + r).IsMonicOfDegree m

Tactic:
exact ha.add_right hr

Goal state after:
Module: Mathlib.Algebra.Polynomial.Mirror

Goal state before:
pp: case pos
R : Type u_1
inst✝ : Semiring R
n : ℕ
a : R
ha : a = 0
⊢ mirror 0 = 0

Tactic:
mirror_zero

Goal state after:
case pos
R : Type u_1
inst✝ : Semiring R
n : ℕ
a : R
ha : a = 0
⊢ 0 = 0
Module: Mathlib.Algebra.Polynomial.Mirror

Goal state before:
pp: case pos
R : Type u_1
inst✝ : Semiring R
p : R[X]
hp : p = 0
⊢ (mirror 0).natDegree = natDegree 0

Tactic:
mirror_zero

Goal state after:
case pos
R : Type u_1
inst✝ : Semiring R
p : R[X]
hp : p = 0
⊢ natDegree 0 = natDegree 0
Module: Mathlib.Algebra.Polynomial.Mirror

Goal state before:
pp: case pos
R : Type u_1
inst✝ : Semiring R
p : R[X]
hp : p = 0
⊢ (mirror 0).natTrailingDegree = natTrailingDegree 0

Tactic:
mirror_zero

Goal state after:
case pos
R : Type u_1
inst✝ : Semiring R
p : R[X]
hp : p = 0
⊢ natTrailingDegree 0 = natTrailingDegree 0
Module: Mathlib.Algebra.Polynomial.Mirror

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
p : R[X]
⊢ p.mirror.natTrailingDegree = p.natTrailingDegree

Tactic:
by
  by_cases hp : p = 0
  · rw [hp, mirror_zero]
  · rw [mirror, natTrailingDegree_mul_X_pow ((mt reverse_eq_zero.mp) hp),
      natTrailingDegree_reverse, zero_add]

Goal state after:
Module: Mathlib.Algebra.Polynomial.Mirror

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
p : R[X]
h : p.mirror = 0
⊢ mirror 0 = 0

Tactic:
mirror_zero

Goal state after:
R : Type u_1
inst✝ : Semiring R
p : R[X]
h : p.mirror = 0
⊢ 0 = 0
Module: Mathlib.Algebra.Polynomial.Mirror

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
p : R[X]
h : p = 0
⊢ mirror 0 = 0

Tactic:
mirror_zero

Goal state after:
R : Type u_1
inst✝ : Semiring R
p : R[X]
h : p = 0
⊢ 0 = 0
Module: Mathlib.Algebra.Polynomial.Mirror

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
p q : R[X]
inst✝ : NoZeroDivisors R
⊢ (p * q).mirror = p.mirror * q.mirror

Tactic:
by_cases hp : p = 0
  · rw [hp, zero_mul, mirror_zero, zero_mul]
  by_cases hq : q = 0
  · rw [hq, mul_zero, mirror_zero, mul_zero]
  rw [mirror, mirror, mirror, reverse_mul_of_domain, natTrailingDegree_mul hp hq, pow_add]
  rw [mul_assoc, ← mul_assoc q.reverse, ← X_pow_mul (p := reverse q)]
  repeat' rw [mul_assoc]

Goal state after:
Module: Mathlib.Algebra.Polynomial.Homogenize

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
p : R[X]
n : ℕ
m : Fin 2 →₀ ℕ
⊢ MvPolynomial.coeff m (p.homogenize n) = if m 0 + m 1 = n then p.coeff (m 0) else 0

Tactic:
by
  induction p using Polynomial.induction_on' with
  | add p q ihp ihq =>
    simp [*, ite_add_ite]
  | monomial k c =>
    rcases le_or_gt k n with hkn | hnk
    · rw [homogenize_monomial hkn, coeff_monomial, MvPolynomial.coeff_monomial]
      have : (fun₀ | 0 => m 0 | 1 => m 1) = m := by ext i; fin_cases i <;> simp
      aesop
    · aesop (add simp homogenize_monomial_of_lt) (add simp coeff_monomial)

Goal state after:
Module: Mathlib.Algebra.Polynomial.PartialFractions

Goal state before:
pp: case succ.refine_2
R : Type u_1
inst✝¹ : CommRing R
inst✝ : Nontrivial R
f g : R[X]
hg : g.Monic
n : ℕ
q : R[X]
r : Fin n → R[X]
hr : ∀ (i : Fin n), (r i).degree < g.degree
hf : f = q * g ^ n + ∑ i, r i * g ^ ↑i
⊢ q * g ^ n + ∑ i, r i * g ^ ↑i =
    q %ₘ g * g ^ n + (q /ₘ g * g ^ (n + 1) + ∑ i, Fin.snoc r (q %ₘ g) i.castSucc * g ^ ↑i.castSucc)

Tactic:
← add_assoc,

Goal state after:
case succ.refine_2
R : Type u_1
inst✝¹ : CommRing R
inst✝ : Nontrivial R
f g : R[X]
hg : g.Monic
n : ℕ
q : R[X]
r : Fin n → R[X]
hr : ∀ (i : Fin n), (r i).degree < g.degree
hf : f = q * g ^ n + ∑ i, r i * g ^ ↑i
⊢ q * g ^ n + ∑ i, r i * g ^ ↑i =
    q %ₘ g * g ^ n + q /ₘ g * g ^ (n + 1) + ∑ i, Fin.snoc r (q %ₘ g) i.castSucc * g ^ ↑i.castSucc
Module: Mathlib.Algebra.Polynomial.Homogenize

Goal state before:
pp: K : Type u_1
inst✝ : Semifield K
p : K[X]
n : ℕ
hn : p.natDegree ≤ n
x : Fin 2 → K
hx : x 1 ≠ 0
k : ℕ
hk : k ∈ range n.succ
⊢ p.coeff k * (x 0 ^ k * (x 1 ^ n * (x 1 ^ k)⁻¹)) = p.coeff k * (x 0 / x 1) ^ k * x 1 ^ n

pp: case ha
K : Type u_1
inst✝ : Semifield K
p : K[X]
n : ℕ
hn : p.natDegree ≤ n
x : Fin 2 → K
hx : x 1 ≠ 0
k : ℕ
hk : k ∈ range n.succ
⊢ x 1 ≠ 0

pp: case h
K : Type u_1
inst✝ : Semifield K
p : K[X]
n : ℕ
hn : p.natDegree ≤ n
x : Fin 2 → K
hx : x 1 ≠ 0
k : ℕ
hk : k ∈ range n.succ
⊢ k ≤ n

pp: K : Type u_1
inst✝ : Semifield K
p : K[X]
n : ℕ
hn : p.natDegree ≤ n
x : Fin 2 → K
hx : x 1 ≠ 0
k : ℕ
hk : k ∈ range n.succ
⊢ x 1 ^ 0 = 1

pp: K : Type u_1
inst✝ : Semifield K
p : K[X]
n : ℕ
hn : p.natDegree ≤ n
x : Fin 2 → K
hx : x 1 ≠ 0
k : ℕ
hk : k ∈ range n.succ
⊢ x 0 ^ 0 = 1

pp: case h_zero
K : Type u_1
inst✝ : Semifield K
p : K[X]
n : ℕ
hn : p.natDegree ≤ n
x : Fin 2 → K
hx : x 1 ≠ 0
k : ℕ
hk : k ∈ range n.succ
⊢ ∀ (a : Fin 2), x a ^ 0 = 1

pp: case h_add
K : Type u_1
inst✝ : Semifield K
p : K[X]
n : ℕ
hn : p.natDegree ≤ n
x : Fin 2 → K
hx : x 1 ≠ 0
k : ℕ
hk : k ∈ range n.succ
⊢ ∀ (a : Fin 2) (b₁ b₂ : ℕ), x a ^ (b₁ + b₂) = x a ^ b₁ * x a ^ b₂

pp: case h
K : Type u_1
inst✝ : Semifield K
p : K[X]
n : ℕ
hn : p.natDegree ≤ n
x : Fin 2 → K
hx : x 1 ≠ 0
k : ℕ
hk : k ∈ range n.succ
⊢ (fun₀ | 0 => k) 1 = 0

Tactic:
· ring

Goal state after:
case ha
K : Type u_1
inst✝ : Semifield K
p : K[X]
n : ℕ
hn : p.natDegree ≤ n
x : Fin 2 → K
hx : x 1 ≠ 0
k : ℕ
hk : k ∈ range n.succ
⊢ x 1 ≠ 0

case h
K : Type u_1
inst✝ : Semifield K
p : K[X]
n : ℕ
hn : p.natDegree ≤ n
x : Fin 2 → K
hx : x 1 ≠ 0
k : ℕ
hk : k ∈ range n.succ
⊢ k ≤ n

K : Type u_1
inst✝ : Semifield K
p : K[X]
n : ℕ
hn : p.natDegree ≤ n
x : Fin 2 → K
hx : x 1 ≠ 0
k : ℕ
hk : k ∈ range n.succ
⊢ x 1 ^ 0 = 1

K : Type u_1
inst✝ : Semifield K
p : K[X]
n : ℕ
hn : p.natDegree ≤ n
x : Fin 2 → K
hx : x 1 ≠ 0
k : ℕ
hk : k ∈ range n.succ
⊢ x 0 ^ 0 = 1

case h_zero
K : Type u_1
inst✝ : Semifield K
p : K[X]
n : ℕ
hn : p.natDegree ≤ n
x : Fin 2 → K
hx : x 1 ≠ 0
k : ℕ
hk : k ∈ range n.succ
⊢ ∀ (a : Fin 2), x a ^ 0 = 1

case h_add
K : Type u_1
inst✝ : Semifield K
p : K[X]
n : ℕ
hn : p.natDegree ≤ n
x : Fin 2 → K
hx : x 1 ≠ 0
k : ℕ
hk : k ∈ range n.succ
⊢ ∀ (a : Fin 2) (b₁ b₂ : ℕ), x a ^ (b₁ + b₂) = x a ^ b₁ * x a ^ b₂

case h
K : Type u_1
inst✝ : Semifield K
p : K[X]
n : ℕ
hn : p.natDegree ≤ n
x : Fin 2 → K
hx : x 1 ≠ 0
k : ℕ
hk : k ∈ range n.succ
⊢ (fun₀ | 0 => k) 1 = 0
Module: Mathlib.Algebra.Polynomial.PartialFractions

Goal state before:
pp: case succ.snoc.snoc
R : Type u_1
inst✝ : CommRing R
g : R[X]
hg : g.Monic
n : ℕ
ih :
  ∀ {q₁ q₂ : R[X]} {r₁ r₂ : Fin n → R[X]},
    (∀ (i : Fin n), (r₁ i).degree < g.degree) →
      (∀ (i : Fin n), (r₂ i).degree < g.degree) →
        q₁ * g ^ n + ∑ i, r₁ i * g ^ ↑i = q₂ * g ^ n + ∑ i, r₂ i * g ^ ↑i → q₁ = q₂ ∧ r₁ = r₂
q₁ q₂ : R[X]
rs₁ : Fin n → R[X]
r₁ : R[X]
hr₁ : ∀ (i : Fin (n + 1)), (Fin.snoc rs₁ r₁ i).degree < g.degree
rs₂ : Fin n → R[X]
r₂ : R[X]
hr₂ : ∀ (i : Fin (n + 1)), (Fin.snoc rs₂ r₂ i).degree < g.degree
hf : r₁ * g ^ n + (q₁ * g * g ^ n + ∑ x, rs₁ x * g ^ ↑x) = r₂ * g ^ n + (q₂ * g * g ^ n + ∑ x, rs₂ x * g ^ ↑x)
⊢ q₁ = q₂ ∧ Fin.snoc rs₁ r₁ = Fin.snoc rs₂ r₂

Tactic:
← add_assoc,

Goal state after:
case succ.snoc.snoc
R : Type u_1
inst✝ : CommRing R
g : R[X]
hg : g.Monic
n : ℕ
ih :
  ∀ {q₁ q₂ : R[X]} {r₁ r₂ : Fin n → R[X]},
    (∀ (i : Fin n), (r₁ i).degree < g.degree) →
      (∀ (i : Fin n), (r₂ i).degree < g.degree) →
        q₁ * g ^ n + ∑ i, r₁ i * g ^ ↑i = q₂ * g ^ n + ∑ i, r₂ i * g ^ ↑i → q₁ = q₂ ∧ r₁ = r₂
q₁ q₂ : R[X]
rs₁ : Fin n → R[X]
r₁ : R[X]
hr₁ : ∀ (i : Fin (n + 1)), (Fin.snoc rs₁ r₁ i).degree < g.degree
rs₂ : Fin n → R[X]
r₂ : R[X]
hr₂ : ∀ (i : Fin (n + 1)), (Fin.snoc rs₂ r₂ i).degree < g.degree
hf : r₁ * g ^ n + q₁ * g * g ^ n + ∑ x, rs₁ x * g ^ ↑x = r₂ * g ^ n + (q₂ * g * g ^ n + ∑ x, rs₂ x * g ^ ↑x)
⊢ q₁ = q₂ ∧ Fin.snoc rs₁ r₁ = Fin.snoc rs₂ r₂
Module: Mathlib.Algebra.Polynomial.PartialFractions

Goal state before:
pp: case succ.snoc.snoc
R : Type u_1
inst✝ : CommRing R
g : R[X]
hg : g.Monic
n : ℕ
ih :
  ∀ {q₁ q₂ : R[X]} {r₁ r₂ : Fin n → R[X]},
    (∀ (i : Fin n), (r₁ i).degree < g.degree) →
      (∀ (i : Fin n), (r₂ i).degree < g.degree) →
        q₁ * g ^ n + ∑ i, r₁ i * g ^ ↑i = q₂ * g ^ n + ∑ i, r₂ i * g ^ ↑i → q₁ = q₂ ∧ r₁ = r₂
q₁ q₂ : R[X]
rs₁ : Fin n → R[X]
r₁ : R[X]
hr₁ : ∀ (i : Fin (n + 1)), (Fin.snoc rs₁ r₁ i).degree < g.degree
rs₂ : Fin n → R[X]
r₂ : R[X]
hr₂ : ∀ (i : Fin (n + 1)), (Fin.snoc rs₂ r₂ i).degree < g.degree
hf : r₁ * g ^ n + q₁ * g * g ^ n + ∑ x, rs₁ x * g ^ ↑x = r₂ * g ^ n + (q₂ * g * g ^ n + ∑ x, rs₂ x * g ^ ↑x)
⊢ q₁ = q₂ ∧ Fin.snoc rs₁ r₁ = Fin.snoc rs₂ r₂

Tactic:
← add_assoc,

Goal state after:
case succ.snoc.snoc
R : Type u_1
inst✝ : CommRing R
g : R[X]
hg : g.Monic
n : ℕ
ih :
  ∀ {q₁ q₂ : R[X]} {r₁ r₂ : Fin n → R[X]},
    (∀ (i : Fin n), (r₁ i).degree < g.degree) →
      (∀ (i : Fin n), (r₂ i).degree < g.degree) →
        q₁ * g ^ n + ∑ i, r₁ i * g ^ ↑i = q₂ * g ^ n + ∑ i, r₂ i * g ^ ↑i → q₁ = q₂ ∧ r₁ = r₂
q₁ q₂ : R[X]
rs₁ : Fin n → R[X]
r₁ : R[X]
hr₁ : ∀ (i : Fin (n + 1)), (Fin.snoc rs₁ r₁ i).degree < g.degree
rs₂ : Fin n → R[X]
r₂ : R[X]
hr₂ : ∀ (i : Fin (n + 1)), (Fin.snoc rs₂ r₂ i).degree < g.degree
hf : r₁ * g ^ n + q₁ * g * g ^ n + ∑ x, rs₁ x * g ^ ↑x = r₂ * g ^ n + q₂ * g * g ^ n + ∑ x, rs₂ x * g ^ ↑x
⊢ q₁ = q₂ ∧ Fin.snoc rs₁ r₁ = Fin.snoc rs₂ r₂
Module: Mathlib.Algebra.Polynomial.Homogenize

Goal state before:
pp: R : Type u_1
inst✝¹ : CommSemiring R
M : Type u_2
inst✝ : AddCommMonoid M
p : R[X]
f : R → M
⊢ ((p.homogenize p.natDegree).coeff.sum fun x c ↦ f c) = p.sum fun x c ↦ f c

Tactic:
by
  rw [MvPolynomial.sum_def, sum_def p]
  -- We set up a bijection between the sets indexing the terms on both sides
  -- and show that it maps the terms in the one sum to those in the other.
  refine Finset.sum_nbij' (fun s ↦ s 0) (fun n ↦ fun₀ | 0 => n | 1 => p.natDegree - n)
    (fun s hs ↦ ?_) (fun n hn ↦ ?_) (fun s hs ↦ ?_) (fun n hn ↦ by simp)
    fun s hs ↦ ?_
  · simpa [coeff_homogenize, sum_eq_natDegree_of_mem_support_homogenize p hs] using hs
  · simpa [coeff_homogenize, mem_support_iff.mp hn]
      using Nat.add_sub_of_le <| le_natDegree_of_mem_supp n hn
  · -- speeds up `grind` quite a bit
    grind only [= Finsupp.update_apply, = Finsupp.single_apply,
      sum_eq_natDegree_of_mem_support_homogenize p hs]
  · simp [coeff_homogenize, sum_eq_natDegree_of_mem_support_homogenize p hs]

Goal state after:
Module: Mathlib.Algebra.Polynomial.Smeval

Goal state before:
pp: case monomial
R : Type u_1
inst✝⁴ : Semiring R
p : R[X]
S : Type u_2
inst✝³ : NonAssocSemiring S
inst✝² : Module R S
inst✝¹ : Pow S ℕ
inst✝ : NatPowAssoc S
n : ℕ
a : R
⊢ ((monomial n) a).smeval 0 = ((monomial n) a).coeff 0 • 1

Tactic:
cases n with

Goal state after:
Module: Mathlib.Algebra.Polynomial.PartialFractions

Goal state before:
pp: case cons
R : Type u_1
inst✝² : CommRing R
inst✝¹ : Nontrivial R
ι : Type u_2
inst✝ : DecidableEq ι
f : R[X]
g : ι → R[X]
i : ι
s : Finset ι
hi : i ∉ s
ih :
  (∀ i ∈ s, (g i).Monic) →
    ((↑s).Pairwise fun i j ↦ IsCoprime (g i) (g j)) →
      ∃ q r, (∀ i ∈ s, (r i).degree < (g i).degree) ∧ f = q * ∏ i ∈ s, g i + ∑ i ∈ s, r i * ∏ k ∈ s.erase i, g k
hg : (g i).Monic ∧ ∀ x ∈ s, (g x).Monic
hgg : ((↑s).Pairwise fun i j ↦ IsCoprime (g i) (g j)) ∧ ∀ b ∈ ↑s, i ≠ b → IsCoprime (g i) (g b) ∧ IsCoprime (g b) (g i)
q : R[X]
r : ι → R[X]
hf : f = q * ∏ i ∈ s, g i + ∑ i ∈ s, r i * ∏ k ∈ s.erase i, g k
⊢ ∃ q r,
    (∀ i_1 ∈ Finset.cons i s hi, (r i_1).degree < (g i_1).degree) ∧
      f =
        q * ∏ i ∈ Finset.cons i s hi, g i +
          ∑ i_1 ∈ Finset.cons i s hi, r i_1 * ∏ k ∈ (Finset.cons i s hi).erase i_1, g k

Tactic:
have hjs {j : ι} (hj : j ∈ s) : i ≠ j := fun hij => hi (hij ▸ hj)

Goal state after:
case cons
R : Type u_1
inst✝² : CommRing R
inst✝¹ : Nontrivial R
ι : Type u_2
inst✝ : DecidableEq ι
f : R[X]
g : ι → R[X]
i : ι
s : Finset ι
hi : i ∉ s
ih :
  (∀ i ∈ s, (g i).Monic) →
    ((↑s).Pairwise fun i j ↦ IsCoprime (g i) (g j)) →
      ∃ q r, (∀ i ∈ s, (r i).degree < (g i).degree) ∧ f = q * ∏ i ∈ s, g i + ∑ i ∈ s, r i * ∏ k ∈ s.erase i, g k
hg : (g i).Monic ∧ ∀ x ∈ s, (g x).Monic
hgg : ((↑s).Pairwise fun i j ↦ IsCoprime (g i) (g j)) ∧ ∀ b ∈ ↑s, i ≠ b → IsCoprime (g i) (g b) ∧ IsCoprime (g b) (g i)
q : R[X]
r : ι → R[X]
hf : f = q * ∏ i ∈ s, g i + ∑ i ∈ s, r i * ∏ k ∈ s.erase i, g k
hjs : ∀ {j : ι}, j ∈ s → i ≠ j
⊢ ∃ q r,
    (∀ i_1 ∈ Finset.cons i s hi, (r i_1).degree < (g i_1).degree) ∧
      f =
        q * ∏ i ∈ Finset.cons i s hi, g i +
          ∑ i_1 ∈ Finset.cons i s hi, r i_1 * ∏ k ∈ (Finset.cons i s hi).erase i_1, g k
Module: Mathlib.Algebra.Polynomial.Smeval

Goal state before:
pp: case monomial
R : Type u_1
inst✝⁴ : Semiring R
p q : R[X]
S : Type u_2
inst✝³ : Semiring S
inst✝² : Module R S
inst✝¹ : IsScalarTower R S S
inst✝ : SMulCommClass R S S
x y : S
hc : Commute x y
n : ℕ
a : R
⊢ Commute (a • x ^ n) (q.smeval y)

Tactic:
refine Commute.smul_left ?_ a

Goal state after:
case monomial
R : Type u_1
inst✝⁴ : Semiring R
p q : R[X]
S : Type u_2
inst✝³ : Semiring S
inst✝² : Module R S
inst✝¹ : IsScalarTower R S S
inst✝ : SMulCommClass R S S
x y : S
hc : Commute x y
n : ℕ
a : R
⊢ Commute (x ^ n) (q.smeval y)
Module: Mathlib.Algebra.Polynomial.UnitTrinomial

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
k m n : ℕ
u v w : R
hkm : k < m
hmn : m < n
hu : u ≠ 0
hw : w ≠ 0
⊢ C w * X ^ k + (C v * X ^ (n - m + k) + C u * X ^ n) = trinomial k (n - m + k) n w v u

Tactic:
← add_assoc,

Goal state after:
R : Type u_1
inst✝ : Semiring R
k m n : ℕ
u v w : R
hkm : k < m
hmn : m < n
hu : u ≠ 0
hw : w ≠ 0
⊢ C w * X ^ k + C v * X ^ (n - m + k) + C u * X ^ n = trinomial k (n - m + k) n w v u
Module: Mathlib.Algebra.Polynomial.PartialFractions

Goal state before:
pp: case cons.refine_2
R : Type u_1
inst✝² : CommRing R
inst✝¹ : Nontrivial R
ι : Type u_2
inst✝ : DecidableEq ι
f : R[X]
g : ι → R[X]
i : ι
s : Finset ι
hi : i ∉ s
ih :
  (∀ i ∈ s, (g i).Monic) →
    ((↑s).Pairwise fun i j ↦ IsCoprime (g i) (g j)) →
      ∃ q r, (∀ i ∈ s, (r i).degree < (g i).degree) ∧ f = q * ∏ i ∈ s, g i + ∑ i ∈ s, r i * ∏ k ∈ s.erase i, g k
hg : (g i).Monic ∧ ∀ x ∈ s, (g x).Monic
hgg : ((↑s).Pairwise fun i j ↦ IsCoprime (g i) (g j)) ∧ ∀ b ∈ ↑s, i ≠ b → IsCoprime (g i) (g b) ∧ IsCoprime (g b) (g i)
q : R[X]
r : ι → R[X]
hf : f = q * ∏ i ∈ s, g i + ∑ i ∈ s, r i * ∏ k ∈ s.erase i, g k
hjs : ∀ {j : ι}, j ∈ s → i ≠ j
a b : ι → R[X]
hab : ∀ j ∈ s, a j * g i + b j * g j = 1
j : ι
hj : j ∈ s
⊢ r j * ∏ k ∈ s.erase j, g k =
    r j * b j %ₘ g i * ∏ x ∈ s, g x +
      ((r j * b j /ₘ g i + r j * a j /ₘ g j) * (g i * ∏ x ∈ s, g x) + r j * a j %ₘ g j * (g i * ∏ x ∈ s.erase j, g x))

Tactic:
← Finset.mul_prod_erase s g hj

Goal state after:
case cons.refine_2
R : Type u_1
inst✝² : CommRing R
inst✝¹ : Nontrivial R
ι : Type u_2
inst✝ : DecidableEq ι
f : R[X]
g : ι → R[X]
i : ι
s : Finset ι
hi : i ∉ s
ih :
  (∀ i ∈ s, (g i).Monic) →
    ((↑s).Pairwise fun i j ↦ IsCoprime (g i) (g j)) →
      ∃ q r, (∀ i ∈ s, (r i).degree < (g i).degree) ∧ f = q * ∏ i ∈ s, g i + ∑ i ∈ s, r i * ∏ k ∈ s.erase i, g k
hg : (g i).Monic ∧ ∀ x ∈ s, (g x).Monic
hgg : ((↑s).Pairwise fun i j ↦ IsCoprime (g i) (g j)) ∧ ∀ b ∈ ↑s, i ≠ b → IsCoprime (g i) (g b) ∧ IsCoprime (g b) (g i)
q : R[X]
r : ι → R[X]
hf : f = q * ∏ i ∈ s, g i + ∑ i ∈ s, r i * ∏ k ∈ s.erase i, g k
hjs : ∀ {j : ι}, j ∈ s → i ≠ j
a b : ι → R[X]
hab : ∀ j ∈ s, a j * g i + b j * g j = 1
j : ι
hj : j ∈ s
⊢ r j * ∏ k ∈ s.erase j, g k =
    r j * b j %ₘ g i * (g j * ∏ x ∈ s.erase j, g x) +
      ((r j * b j /ₘ g i + r j * a j /ₘ g j) * (g i * (g j * ∏ x ∈ s.erase j, g x)) +
        r j * a j %ₘ g j * (g i * ∏ x ∈ s.erase j, g x))
Module: Mathlib.Algebra.Polynomial.PartialFractions

Goal state before:
pp: case cons.refine_2
R : Type u_1
inst✝² : CommRing R
inst✝¹ : Nontrivial R
ι : Type u_2
inst✝ : DecidableEq ι
f : R[X]
g : ι → R[X]
i : ι
s : Finset ι
hi : i ∉ s
ih :
  (∀ i ∈ s, (g i).Monic) →
    ((↑s).Pairwise fun i j ↦ IsCoprime (g i) (g j)) →
      ∃ q r, (∀ i ∈ s, (r i).degree < (g i).degree) ∧ f = q * ∏ i ∈ s, g i + ∑ i ∈ s, r i * ∏ k ∈ s.erase i, g k
hg : (g i).Monic ∧ ∀ x ∈ s, (g x).Monic
hgg : ((↑s).Pairwise fun i j ↦ IsCoprime (g i) (g j)) ∧ ∀ b ∈ ↑s, i ≠ b → IsCoprime (g i) (g b) ∧ IsCoprime (g b) (g i)
q : R[X]
r : ι → R[X]
hf : f = q * ∏ i ∈ s, g i + ∑ i ∈ s, r i * ∏ k ∈ s.erase i, g k
hjs : ∀ {j : ι}, j ∈ s → i ≠ j
a b : ι → R[X]
hab : ∀ j ∈ s, a j * g i + b j * g j = 1
j : ι
hj : j ∈ s
⊢ r j = r j * b j %ₘ g i * g j + (r j * b j /ₘ g i * g i * g j + r j * a j /ₘ g j * g i * g j + r j * a j %ₘ g j * g i)

Tactic:
← add_assoc,

Goal state after:
case cons.refine_2
R : Type u_1
inst✝² : CommRing R
inst✝¹ : Nontrivial R
ι : Type u_2
inst✝ : DecidableEq ι
f : R[X]
g : ι → R[X]
i : ι
s : Finset ι
hi : i ∉ s
ih :
  (∀ i ∈ s, (g i).Monic) →
    ((↑s).Pairwise fun i j ↦ IsCoprime (g i) (g j)) →
      ∃ q r, (∀ i ∈ s, (r i).degree < (g i).degree) ∧ f = q * ∏ i ∈ s, g i + ∑ i ∈ s, r i * ∏ k ∈ s.erase i, g k
hg : (g i).Monic ∧ ∀ x ∈ s, (g x).Monic
hgg : ((↑s).Pairwise fun i j ↦ IsCoprime (g i) (g j)) ∧ ∀ b ∈ ↑s, i ≠ b → IsCoprime (g i) (g b) ∧ IsCoprime (g b) (g i)
q : R[X]
r : ι → R[X]
hf : f = q * ∏ i ∈ s, g i + ∑ i ∈ s, r i * ∏ k ∈ s.erase i, g k
hjs : ∀ {j : ι}, j ∈ s → i ≠ j
a b : ι → R[X]
hab : ∀ j ∈ s, a j * g i + b j * g j = 1
j : ι
hj : j ∈ s
⊢ r j = r j * b j %ₘ g i * g j + (r j * b j /ₘ g i * g i * g j + r j * a j /ₘ g j * g i * g j) + r j * a j %ₘ g j * g i
Module: Mathlib.Algebra.Polynomial.PartialFractions

Goal state before:
pp: case cons.refine_2
R : Type u_1
inst✝² : CommRing R
inst✝¹ : Nontrivial R
ι : Type u_2
inst✝ : DecidableEq ι
f : R[X]
g : ι → R[X]
i : ι
s : Finset ι
hi : i ∉ s
ih :
  (∀ i ∈ s, (g i).Monic) →
    ((↑s).Pairwise fun i j ↦ IsCoprime (g i) (g j)) →
      ∃ q r, (∀ i ∈ s, (r i).degree < (g i).degree) ∧ f = q * ∏ i ∈ s, g i + ∑ i ∈ s, r i * ∏ k ∈ s.erase i, g k
hg : (g i).Monic ∧ ∀ x ∈ s, (g x).Monic
hgg : ((↑s).Pairwise fun i j ↦ IsCoprime (g i) (g j)) ∧ ∀ b ∈ ↑s, i ≠ b → IsCoprime (g i) (g b) ∧ IsCoprime (g b) (g i)
q : R[X]
r : ι → R[X]
hf : f = q * ∏ i ∈ s, g i + ∑ i ∈ s, r i * ∏ k ∈ s.erase i, g k
hjs : ∀ {j : ι}, j ∈ s → i ≠ j
a b : ι → R[X]
hab : ∀ j ∈ s, a j * g i + b j * g j = 1
j : ι
hj : j ∈ s
⊢ r j = r j * b j %ₘ g i * g j + (r j * b j /ₘ g i * g i * g j + r j * a j /ₘ g j * g i * g j) + r j * a j %ₘ g j * g i

Tactic:
← add_assoc,

Goal state after:
case cons.refine_2
R : Type u_1
inst✝² : CommRing R
inst✝¹ : Nontrivial R
ι : Type u_2
inst✝ : DecidableEq ι
f : R[X]
g : ι → R[X]
i : ι
s : Finset ι
hi : i ∉ s
ih :
  (∀ i ∈ s, (g i).Monic) →
    ((↑s).Pairwise fun i j ↦ IsCoprime (g i) (g j)) →
      ∃ q r, (∀ i ∈ s, (r i).degree < (g i).degree) ∧ f = q * ∏ i ∈ s, g i + ∑ i ∈ s, r i * ∏ k ∈ s.erase i, g k
hg : (g i).Monic ∧ ∀ x ∈ s, (g x).Monic
hgg : ((↑s).Pairwise fun i j ↦ IsCoprime (g i) (g j)) ∧ ∀ b ∈ ↑s, i ≠ b → IsCoprime (g i) (g b) ∧ IsCoprime (g b) (g i)
q : R[X]
r : ι → R[X]
hf : f = q * ∏ i ∈ s, g i + ∑ i ∈ s, r i * ∏ k ∈ s.erase i, g k
hjs : ∀ {j : ι}, j ∈ s → i ≠ j
a b : ι → R[X]
hab : ∀ j ∈ s, a j * g i + b j * g j = 1
j : ι
hj : j ∈ s
⊢ r j = r j * b j %ₘ g i * g j + r j * b j /ₘ g i * g i * g j + r j * a j /ₘ g j * g i * g j + r j * a j %ₘ g j * g i
Module: Mathlib.Algebra.Polynomial.PartialFractions

Goal state before:
pp: case cons
R : Type u_1
inst✝¹ : CommRing R
ι : Type u_2
inst✝ : DecidableEq ι
g : ι → R[X]
q₁ q₂ : R[X]
r₁ r₂ : ι → R[X]
i : ι
s : Finset ι
hi : i ∉ s
ih :
  (∀ i ∈ s, (g i).Monic) →
    ((↑s).Pairwise fun i j ↦ IsCoprime (g i) (g j)) →
      (∀ i ∈ s, (r₁ i).degree < (g i).degree) →
        (∀ i ∈ s, (r₂ i).degree < (g i).degree) →
          q₁ * ∏ i ∈ s, g i + ∑ i ∈ s, r₁ i * ∏ k ∈ s.erase i, g k =
              q₂ * ∏ i ∈ s, g i + ∑ i ∈ s, r₂ i * ∏ k ∈ s.erase i, g k →
            q₁ = q₂ ∧ ∀ i ∈ s, r₁ i = r₂ i
hg : (g i).Monic ∧ ∀ x ∈ s, (g x).Monic
hgg : (↑(Finset.cons i s hi)).Pairwise fun i j ↦ IsCoprime (g i) (g j)
hr₁ : (r₁ i).degree < (g i).degree ∧ ∀ x ∈ s, (r₁ x).degree < (g x).degree
hr₂ : (r₂ i).degree < (g i).degree ∧ ∀ x ∈ s, (r₂ x).degree < (g x).degree
hf :
  q₁ * (g i * ∏ x ∈ s, g x) + (r₁ i * ∏ k ∈ s, g k + ∑ x ∈ s, r₁ x * ∏ k ∈ (Finset.cons i s hi).erase x, g k) =
    q₂ * (g i * ∏ x ∈ s, g x) + (r₂ i * ∏ k ∈ s, g k + ∑ x ∈ s, r₂ x * ∏ k ∈ (Finset.cons i s hi).erase x, g k)
cop : IsCoprime (g i) (∏ i ∈ s, g i)
⊢ q₁ = q₂ ∧ ∀ i_1 ∈ Finset.cons i s hi, r₁ i_1 = r₂ i_1

Tactic:
have hjs {j : ι} (hj : j ∈ s) : i ≠ j := fun hij => hi (hij ▸ hj)

Goal state after:
case cons
R : Type u_1
inst✝¹ : CommRing R
ι : Type u_2
inst✝ : DecidableEq ι
g : ι → R[X]
q₁ q₂ : R[X]
r₁ r₂ : ι → R[X]
i : ι
s : Finset ι
hi : i ∉ s
ih :
  (∀ i ∈ s, (g i).Monic) →
    ((↑s).Pairwise fun i j ↦ IsCoprime (g i) (g j)) →
      (∀ i ∈ s, (r₁ i).degree < (g i).degree) →
        (∀ i ∈ s, (r₂ i).degree < (g i).degree) →
          q₁ * ∏ i ∈ s, g i + ∑ i ∈ s, r₁ i * ∏ k ∈ s.erase i, g k =
              q₂ * ∏ i ∈ s, g i + ∑ i ∈ s, r₂ i * ∏ k ∈ s.erase i, g k →
            q₁ = q₂ ∧ ∀ i ∈ s, r₁ i = r₂ i
hg : (g i).Monic ∧ ∀ x ∈ s, (g x).Monic
hgg : (↑(Finset.cons i s hi)).Pairwise fun i j ↦ IsCoprime (g i) (g j)
hr₁ : (r₁ i).degree < (g i).degree ∧ ∀ x ∈ s, (r₁ x).degree < (g x).degree
hr₂ : (r₂ i).degree < (g i).degree ∧ ∀ x ∈ s, (r₂ x).degree < (g x).degree
hf :
  q₁ * (g i * ∏ x ∈ s, g x) + (r₁ i * ∏ k ∈ s, g k + ∑ x ∈ s, r₁ x * ∏ k ∈ (Finset.cons i s hi).erase x, g k) =
    q₂ * (g i * ∏ x ∈ s, g x) + (r₂ i * ∏ k ∈ s, g k + ∑ x ∈ s, r₂ x * ∏ k ∈ (Finset.cons i s hi).erase x, g k)
cop : IsCoprime (g i) (∏ i ∈ s, g i)
hjs : ∀ {j : ι}, j ∈ s → i ≠ j
⊢ q₁ = q₂ ∧ ∀ i_1 ∈ Finset.cons i s hi, r₁ i_1 = r₂ i_1
Module: Mathlib.Algebra.Polynomial.RuleOfSigns

Goal state before:
pp: R : Type u_1
inst✝² : Ring R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
P : R[X]
η : R
hη : 0 < η
hP₀ : 0 < P.leadingCoeff
hc : P.nextCoeff < 0
d : ℕ
hd : P.natDegree = d + 1
hndxP : ((X - C η) * P).natDegree = P.natDegree + 1
hndxeP : ((X - C η) * P.eraseLead).natDegree = P.natDegree
hQ : ((X - C η) * P).nextCoeff = P.coeff d - η * P.coeff (d + 1)
⊢ 0 < η * P.coeff (d + 1)

Tactic:
exact hd ▸ mul_pos hη hP₀

Goal state after:
Module: Mathlib.Algebra.Polynomial.UnitTrinomial

Goal state before:
pp: case h
p : ℤ[X]
k m n : ℕ
hkm : k < m
hmn : m < n
u v w : ℤˣ
hp : p = trinomial k m n ↑u ↑v ↑w
key : n - m + k < n
⊢ m + (n - m + k) ∉ Set.Ioo (k + n) (n + n)

Tactic:
← add_assoc,

Goal state after:
case h
p : ℤ[X]
k m n : ℕ
hkm : k < m
hmn : m < n
u v w : ℤˣ
hp : p = trinomial k m n ↑u ↑v ↑w
key : n - m + k < n
⊢ m + (n - m) + k ∉ Set.Ioo (k + n) (n + n)
Module: Mathlib.Algebra.Polynomial.UnitTrinomial

Goal state before:
pp: p q : ℤ[X]
k m m' n : ℕ
hkm : k < m
hmn : m < n
hkm' : k < m'
hmn' : m' < n
u v w : ℤˣ
hp : p = trinomial k m n ↑u ↑v ↑w
hq : q = trinomial k m' n ↑u ↑v ↑w
h : p * p.mirror = q * q.mirror
⊢ q = p ∨ q = p.mirror

Tactic:
by
  let f (p : ℤ[X]) : ℤ[X] := ⟨.ofCoeff <| .filter (· ∈ Set.Ioo (k + n) (n + n)) p.toFinsupp.coeff⟩
  replace h := congr_arg f h
  replace h := (irreducible_aux1 hkm hmn u v w hp).trans h
  replace h := h.trans (irreducible_aux1 hkm' hmn' u v w hq).symm
  rw [(isUnit_C.mpr v.isUnit).mul_right_inj] at h
  rw [binomial_eq_binomial u.ne_zero w.ne_zero] at h
  simp only [add_left_inj, Units.val_inj] at h
  rcases h with (⟨rfl, -⟩ | ⟨rfl, rfl, h⟩ | ⟨-, hm, hm'⟩)
  · exact Or.inl (hq.trans hp.symm)
  · refine Or.inr ?_
    rw [← trinomial_mirror hkm' hmn' u.ne_zero u.ne_zero, eq_comm, mirror_eq_iff] at hp
    exact hq.trans hp
  · grind

Goal state after: