l3lab/ntp-mathlib huggingface dataset:
theorem Monic.irreducible_iff_natDegree' (hp : p.Monic) : Irreducible p ↔ p ≠ 1 ∧
    ∀ f g : R[X], f.Monic → g.Monic → f * g = p → g.natDegree ∉ Ioc 0 (p.natDegree / 2) := by
  simp_rw [hp.irreducible_iff_natDegree, mem_Ioc, Nat.le_div_iff_mul_le zero_lt_two, mul_two]
  apply and_congr_right'
  constructor <;> intro h f g hf hg he <;> subst he
  · 

Proof state:
case h.mp
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
q f g : R[X]
hf : Monic f
hg : Monic g
hp : Monic (f * g)
h : ∀ (f_1 g_1 : R[X]), Monic f_1 → Monic g_1 → f_1 * g_1 = f * g → natDegree f_1 = 0 ∨ natDegree g_1 = 0
⊢ ¬(0 < natDegree g ∧ natDegree g + natDegree g ≤ natDegree (f * g))

Next tactic:
rw [hf.natDegree_mul hg, add_le_add_iff_right]
theorem Monic.irreducible_iff_natDegree' (hp : p.Monic) : Irreducible p ↔ p ≠ 1 ∧
    ∀ f g : R[X], f.Monic → g.Monic → f * g = p → g.natDegree ∉ Ioc 0 (p.natDegree / 2) := by
  simp_rw [hp.irreducible_iff_natDegree, mem_Ioc, Nat.le_div_iff_mul_le zero_lt_two, mul_two]
  apply and_congr_right'
  constructor <;> intro h f g hf hg he <;> subst he
  · rw [hf.natDegree_mul hg, add_le_add_iff_right]
    

Proof state:
case h.mp
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
q f g : R[X]
hf : Monic f
hg : Monic g
hp : Monic (f * g)
h : ∀ (f_1 g_1 : R[X]), Monic f_1 → Monic g_1 → f_1 * g_1 = f * g → natDegree f_1 = 0 ∨ natDegree g_1 = 0
⊢ ¬(0 < natDegree g ∧ natDegree g ≤ natDegree f)

Next tactic:
exact fun ha => (h f g hf hg rfl).elim (ha.1.trans_le ha.2).ne' ha.1.ne'
theorem Monic.irreducible_iff_natDegree' (hp : p.Monic) : Irreducible p ↔ p ≠ 1 ∧
    ∀ f g : R[X], f.Monic → g.Monic → f * g = p → g.natDegree ∉ Ioc 0 (p.natDegree / 2) := by
  simp_rw [hp.irreducible_iff_natDegree, mem_Ioc, Nat.le_div_iff_mul_le zero_lt_two, mul_two]
  apply and_congr_right'
  constructor <;> intro h f g hf hg he <;> subst he
  · rw [hf.natDegree_mul hg, add_le_add_iff_right]
    exact fun ha => (h f g hf hg rfl).elim (ha.1.trans_le ha.2).ne' ha.1.ne'
  · 

Proof state:
case h.mpr
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
q f g : R[X]
hf : Monic f
hg : Monic g
hp : Monic (f * g)
h :
  ∀ (f_1 g_1 : R[X]),
    Monic f_1 → Monic g_1 → f_1 * g_1 = f * g → ¬(0 < natDegree g_1 ∧ natDegree g_1 + natDegree g_1 ≤ natDegree (f * g))
⊢ natDegree f = 0 ∨ natDegree g = 0

Next tactic:
simp_rw [hf.natDegree_mul hg, pos_iff_ne_zero] at h
theorem Monic.irreducible_iff_natDegree' (hp : p.Monic) : Irreducible p ↔ p ≠ 1 ∧
    ∀ f g : R[X], f.Monic → g.Monic → f * g = p → g.natDegree ∉ Ioc 0 (p.natDegree / 2) := by
  simp_rw [hp.irreducible_iff_natDegree, mem_Ioc, Nat.le_div_iff_mul_le zero_lt_two, mul_two]
  apply and_congr_right'
  constructor <;> intro h f g hf hg he <;> subst he
  · rw [hf.natDegree_mul hg, add_le_add_iff_right]
    exact fun ha => (h f g hf hg rfl).elim (ha.1.trans_le ha.2).ne' ha.1.ne'
  · simp_rw [hf.natDegree_mul hg, pos_iff_ne_zero] at h
    

Proof state:
case h.mpr
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
q f g : R[X]
hf : Monic f
hg : Monic g
hp : Monic (f * g)
h :
  ∀ (f_1 g_1 : R[X]),
    Monic f_1 →
      Monic g_1 → f_1 * g_1 = f * g → ¬(natDegree g_1 ≠ 0 ∧ natDegree g_1 + natDegree g_1 ≤ natDegree f + natDegree g)
⊢ natDegree f = 0 ∨ natDegree g = 0

Next tactic:
contrapose! h
theorem Monic.irreducible_iff_natDegree' (hp : p.Monic) : Irreducible p ↔ p ≠ 1 ∧
    ∀ f g : R[X], f.Monic → g.Monic → f * g = p → g.natDegree ∉ Ioc 0 (p.natDegree / 2) := by
  simp_rw [hp.irreducible_iff_natDegree, mem_Ioc, Nat.le_div_iff_mul_le zero_lt_two, mul_two]
  apply and_congr_right'
  constructor <;> intro h f g hf hg he <;> subst he
  · rw [hf.natDegree_mul hg, add_le_add_iff_right]
    exact fun ha => (h f g hf hg rfl).elim (ha.1.trans_le ha.2).ne' ha.1.ne'
  · simp_rw [hf.natDegree_mul hg, pos_iff_ne_zero] at h
    contrapose! h
    

Proof state:
case h.mpr
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
q f g : R[X]
hf : Monic f
hg : Monic g
hp : Monic (f * g)
h : natDegree f ≠ 0 ∧ natDegree g ≠ 0
⊢ ∃ f_1 g_1,
    Monic f_1 ∧
      Monic g_1 ∧ f_1 * g_1 = f * g ∧ natDegree g_1 ≠ 0 ∧ natDegree g_1 + natDegree g_1 ≤ natDegree f + natDegree g

Next tactic:
obtain hl | hl := le_total f.natDegree g.natDegree
theorem Monic.irreducible_iff_natDegree' (hp : p.Monic) : Irreducible p ↔ p ≠ 1 ∧
    ∀ f g : R[X], f.Monic → g.Monic → f * g = p → g.natDegree ∉ Ioc 0 (p.natDegree / 2) := by
  simp_rw [hp.irreducible_iff_natDegree, mem_Ioc, Nat.le_div_iff_mul_le zero_lt_two, mul_two]
  apply and_congr_right'
  constructor <;> intro h f g hf hg he <;> subst he
  · rw [hf.natDegree_mul hg, add_le_add_iff_right]
    exact fun ha => (h f g hf hg rfl).elim (ha.1.trans_le ha.2).ne' ha.1.ne'
  · simp_rw [hf.natDegree_mul hg, pos_iff_ne_zero] at h
    contrapose! h
    obtain hl | hl := le_total f.natDegree g.natDegree
    · 

Proof state:
case h.mpr.inl
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
q f g : R[X]
hf : Monic f
hg : Monic g
hp : Monic (f * g)
h : natDegree f ≠ 0 ∧ natDegree g ≠ 0
hl : natDegree f ≤ natDegree g
⊢ ∃ f_1 g_1,
    Monic f_1 ∧
      Monic g_1 ∧ f_1 * g_1 = f * g ∧ natDegree g_1 ≠ 0 ∧ natDegree g_1 + natDegree g_1 ≤ natDegree f + natDegree g

Next tactic:
exact ⟨g, f, hg, hf, mul_comm g f, h.1, add_le_add_left hl _⟩
theorem Monic.irreducible_iff_natDegree' (hp : p.Monic) : Irreducible p ↔ p ≠ 1 ∧
    ∀ f g : R[X], f.Monic → g.Monic → f * g = p → g.natDegree ∉ Ioc 0 (p.natDegree / 2) := by
  simp_rw [hp.irreducible_iff_natDegree, mem_Ioc, Nat.le_div_iff_mul_le zero_lt_two, mul_two]
  apply and_congr_right'
  constructor <;> intro h f g hf hg he <;> subst he
  · rw [hf.natDegree_mul hg, add_le_add_iff_right]
    exact fun ha => (h f g hf hg rfl).elim (ha.1.trans_le ha.2).ne' ha.1.ne'
  · simp_rw [hf.natDegree_mul hg, pos_iff_ne_zero] at h
    contrapose! h
    obtain hl | hl := le_total f.natDegree g.natDegree
    · exact ⟨g, f, hg, hf, mul_comm g f, h.1, add_le_add_left hl _⟩
    · 

Proof state:
case h.mpr.inr
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
q f g : R[X]
hf : Monic f
hg : Monic g
hp : Monic (f * g)
h : natDegree f ≠ 0 ∧ natDegree g ≠ 0
hl : natDegree g ≤ natDegree f
⊢ ∃ f_1 g_1,
    Monic f_1 ∧
      Monic g_1 ∧ f_1 * g_1 = f * g ∧ natDegree g_1 ≠ 0 ∧ natDegree g_1 + natDegree g_1 ≤ natDegree f + natDegree g

Next tactic:
exact ⟨f, g, hf, hg, rfl, h.2, add_le_add_right hl _⟩
theorem Monic.not_irreducible_iff_exists_add_mul_eq_coeff (hm : p.Monic) (hnd : p.natDegree = 2) :
    ¬Irreducible p ↔ ∃ c₁ c₂, p.coeff 0 = c₁ * c₂ ∧ p.coeff 1 = c₁ + c₂ := by
  

Proof state:
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
p q : R[X]
hm : Monic p
hnd : natDegree p = 2
⊢ ¬Irreducible p ↔ ∃ c₁ c₂, coeff p 0 = c₁ * c₂ ∧ coeff p 1 = c₁ + c₂

Next tactic:
cases subsingleton_or_nontrivial R
theorem Monic.not_irreducible_iff_exists_add_mul_eq_coeff (hm : p.Monic) (hnd : p.natDegree = 2) :
    ¬Irreducible p ↔ ∃ c₁ c₂, p.coeff 0 = c₁ * c₂ ∧ p.coeff 1 = c₁ + c₂ := by
  cases subsingleton_or_nontrivial R
  · 

Proof state:
case inl
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
p q : R[X]
hm : Monic p
hnd : natDegree p = 2
h✝ : Subsingleton R
⊢ ¬Irreducible p ↔ ∃ c₁ c₂, coeff p 0 = c₁ * c₂ ∧ coeff p 1 = c₁ + c₂

Next tactic:
simp [natDegree_of_subsingleton] at hnd
theorem Monic.not_irreducible_iff_exists_add_mul_eq_coeff (hm : p.Monic) (hnd : p.natDegree = 2) :
    ¬Irreducible p ↔ ∃ c₁ c₂, p.coeff 0 = c₁ * c₂ ∧ p.coeff 1 = c₁ + c₂ := by
  cases subsingleton_or_nontrivial R
  · simp [natDegree_of_subsingleton] at hnd
  

Proof state:
case inr
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
p q : R[X]
hm : Monic p
hnd : natDegree p = 2
h✝ : Nontrivial R
⊢ ¬Irreducible p ↔ ∃ c₁ c₂, coeff p 0 = c₁ * c₂ ∧ coeff p 1 = c₁ + c₂

Next tactic:
rw [hm.irreducible_iff_natDegree', and_iff_right, hnd]
theorem Monic.not_irreducible_iff_exists_add_mul_eq_coeff (hm : p.Monic) (hnd : p.natDegree = 2) :
    ¬Irreducible p ↔ ∃ c₁ c₂, p.coeff 0 = c₁ * c₂ ∧ p.coeff 1 = c₁ + c₂ := by
  cases subsingleton_or_nontrivial R
  · simp [natDegree_of_subsingleton] at hnd
  rw [hm.irreducible_iff_natDegree', and_iff_right, hnd]
  

Proof state:
case inr
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
p q : R[X]
hm : Monic p
hnd : natDegree p = 2
h✝ : Nontrivial R
⊢ (¬∀ (f g : R[X]), Monic f → Monic g → f * g = p → natDegree g ∉ Ioc 0 (2 / 2)) ↔
    ∃ c₁ c₂, coeff p 0 = c₁ * c₂ ∧ coeff p 1 = c₁ + c₂
case inr
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
p q : R[X]
hm : Monic p
hnd : natDegree p = 2
h✝ : Nontrivial R
⊢ p ≠ 1

Next tactic:
push_neg
theorem Monic.not_irreducible_iff_exists_add_mul_eq_coeff (hm : p.Monic) (hnd : p.natDegree = 2) :
    ¬Irreducible p ↔ ∃ c₁ c₂, p.coeff 0 = c₁ * c₂ ∧ p.coeff 1 = c₁ + c₂ := by
  cases subsingleton_or_nontrivial R
  · simp [natDegree_of_subsingleton] at hnd
  rw [hm.irreducible_iff_natDegree', and_iff_right, hnd]
  push_neg; 

Proof state:
case inr
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
p q : R[X]
hm : Monic p
hnd : natDegree p = 2
h✝ : Nontrivial R
⊢ (∃ f g, Monic f ∧ Monic g ∧ f * g = p ∧ natDegree g ∈ Ioc 0 (2 / 2)) ↔
    ∃ c₁ c₂, coeff p 0 = c₁ * c₂ ∧ coeff p 1 = c₁ + c₂
case inr
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
p q : R[X]
hm : Monic p
hnd : natDegree p = 2
h✝ : Nontrivial R
⊢ p ≠ 1

Next tactic:
constructor
theorem Monic.not_irreducible_iff_exists_add_mul_eq_coeff (hm : p.Monic) (hnd : p.natDegree = 2) :
    ¬Irreducible p ↔ ∃ c₁ c₂, p.coeff 0 = c₁ * c₂ ∧ p.coeff 1 = c₁ + c₂ := by
  cases subsingleton_or_nontrivial R
  · simp [natDegree_of_subsingleton] at hnd
  rw [hm.irreducible_iff_natDegree', and_iff_right, hnd]
  push_neg; constructor
  · 

Proof state:
case inr.mp
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
p q : R[X]
hm : Monic p
hnd : natDegree p = 2
h✝ : Nontrivial R
⊢ (∃ f g, Monic f ∧ Monic g ∧ f * g = p ∧ natDegree g ∈ Ioc 0 (2 / 2)) →
    ∃ c₁ c₂, coeff p 0 = c₁ * c₂ ∧ coeff p 1 = c₁ + c₂

Next tactic:
rintro ⟨a, b, ha, hb, rfl, hdb⟩
theorem Monic.not_irreducible_iff_exists_add_mul_eq_coeff (hm : p.Monic) (hnd : p.natDegree = 2) :
    ¬Irreducible p ↔ ∃ c₁ c₂, p.coeff 0 = c₁ * c₂ ∧ p.coeff 1 = c₁ + c₂ := by
  cases subsingleton_or_nontrivial R
  · simp [natDegree_of_subsingleton] at hnd
  rw [hm.irreducible_iff_natDegree', and_iff_right, hnd]
  push_neg; constructor
  · rintro ⟨a, b, ha, hb, rfl, hdb⟩
    

Proof state:
case inr.mp.intro.intro.intro.intro.intro
R : Type u
S : Type v
T : Type w
a✝ b✝ : R
n : ℕ
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
q : R[X]
h✝ : Nontrivial R
a b : R[X]
ha : Monic a
hb : Monic b
hdb : natDegree b ∈ Ioc 0 (2 / 2)
hm : Monic (a * b)
hnd : natDegree (a * b) = 2
⊢ ∃ c₁ c₂, coeff (a * b) 0 = c₁ * c₂ ∧ coeff (a * b) 1 = c₁ + c₂

Next tactic:
simp only [zero_lt_two, Nat.div_self, ge_iff_le,
      Nat.Ioc_succ_singleton, zero_add, mem_singleton] at hdb
theorem Monic.not_irreducible_iff_exists_add_mul_eq_coeff (hm : p.Monic) (hnd : p.natDegree = 2) :
    ¬Irreducible p ↔ ∃ c₁ c₂, p.coeff 0 = c₁ * c₂ ∧ p.coeff 1 = c₁ + c₂ := by
  cases subsingleton_or_nontrivial R
  · simp [natDegree_of_subsingleton] at hnd
  rw [hm.irreducible_iff_natDegree', and_iff_right, hnd]
  push_neg; constructor
  · rintro ⟨a, b, ha, hb, rfl, hdb⟩
    simp only [zero_lt_two, Nat.div_self, ge_iff_le,
      Nat.Ioc_succ_singleton, zero_add, mem_singleton] at hdb
    

Proof state:
case inr.mp.intro.intro.intro.intro.intro
R : Type u
S : Type v
T : Type w
a✝ b✝ : R
n : ℕ
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
q : R[X]
h✝ : Nontrivial R
a b : R[X]
ha : Monic a
hb : Monic b
hm : Monic (a * b)
hnd : natDegree (a * b) = 2
hdb : natDegree b = 1
⊢ ∃ c₁ c₂, coeff (a * b) 0 = c₁ * c₂ ∧ coeff (a * b) 1 = c₁ + c₂

Next tactic:
have hda := hnd
theorem Monic.not_irreducible_iff_exists_add_mul_eq_coeff (hm : p.Monic) (hnd : p.natDegree = 2) :
    ¬Irreducible p ↔ ∃ c₁ c₂, p.coeff 0 = c₁ * c₂ ∧ p.coeff 1 = c₁ + c₂ := by
  cases subsingleton_or_nontrivial R
  · simp [natDegree_of_subsingleton] at hnd
  rw [hm.irreducible_iff_natDegree', and_iff_right, hnd]
  push_neg; constructor
  · rintro ⟨a, b, ha, hb, rfl, hdb⟩
    simp only [zero_lt_two, Nat.div_self, ge_iff_le,
      Nat.Ioc_succ_singleton, zero_add, mem_singleton] at hdb
    have hda := hnd
    

Proof state:
case inr.mp.intro.intro.intro.intro.intro
R : Type u
S : Type v
T : Type w
a✝ b✝ : R
n : ℕ
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
q : R[X]
h✝ : Nontrivial R
a b : R[X]
ha : Monic a
hb : Monic b
hm : Monic (a * b)
hnd : natDegree (a * b) = 2
hdb : natDegree b = 1
hda : natDegree (a * b) = 2
⊢ ∃ c₁ c₂, coeff (a * b) 0 = c₁ * c₂ ∧ coeff (a * b) 1 = c₁ + c₂

Next tactic:
rw [ha.natDegree_mul hb, hdb] at hda
theorem Monic.not_irreducible_iff_exists_add_mul_eq_coeff (hm : p.Monic) (hnd : p.natDegree = 2) :
    ¬Irreducible p ↔ ∃ c₁ c₂, p.coeff 0 = c₁ * c₂ ∧ p.coeff 1 = c₁ + c₂ := by
  cases subsingleton_or_nontrivial R
  · simp [natDegree_of_subsingleton] at hnd
  rw [hm.irreducible_iff_natDegree', and_iff_right, hnd]
  push_neg; constructor
  · rintro ⟨a, b, ha, hb, rfl, hdb⟩
    simp only [zero_lt_two, Nat.div_self, ge_iff_le,
      Nat.Ioc_succ_singleton, zero_add, mem_singleton] at hdb
    have hda := hnd
    rw [ha.natDegree_mul hb, hdb] at hda
    

Proof state:
case inr.mp.intro.intro.intro.intro.intro
R : Type u
S : Type v
T : Type w
a✝ b✝ : R
n : ℕ
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
q : R[X]
h✝ : Nontrivial R
a b : R[X]
ha : Monic a
hb : Monic b
hm : Monic (a * b)
hnd : natDegree (a * b) = 2
hdb : natDegree b = 1
hda : natDegree a + 1 = 2
⊢ ∃ c₁ c₂, coeff (a * b) 0 = c₁ * c₂ ∧ coeff (a * b) 1 = c₁ + c₂

Next tactic:
use a.coeff 0, b.coeff 0, mul_coeff_zero a b
theorem Monic.not_irreducible_iff_exists_add_mul_eq_coeff (hm : p.Monic) (hnd : p.natDegree = 2) :
    ¬Irreducible p ↔ ∃ c₁ c₂, p.coeff 0 = c₁ * c₂ ∧ p.coeff 1 = c₁ + c₂ := by
  cases subsingleton_or_nontrivial R
  · simp [natDegree_of_subsingleton] at hnd
  rw [hm.irreducible_iff_natDegree', and_iff_right, hnd]
  push_neg; constructor
  · rintro ⟨a, b, ha, hb, rfl, hdb⟩
    simp only [zero_lt_two, Nat.div_self, ge_iff_le,
      Nat.Ioc_succ_singleton, zero_add, mem_singleton] at hdb
    have hda := hnd
    rw [ha.natDegree_mul hb, hdb] at hda
    use a.coeff 0, b.coeff 0, mul_coeff_zero a b
    

Proof state:
case right
R : Type u
S : Type v
T : Type w
a✝ b✝ : R
n : ℕ
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
q : R[X]
h✝ : Nontrivial R
a b : R[X]
ha : Monic a
hb : Monic b
hm : Monic (a * b)
hnd : natDegree (a * b) = 2
hdb : natDegree b = 1
hda : natDegree a + 1 = 2
⊢ coeff (a * b) 1 = coeff a 0 + coeff b 0

Next tactic:
simpa only [nextCoeff, hnd, add_right_cancel hda, hdb] using ha.nextCoeff_mul hb
theorem Monic.not_irreducible_iff_exists_add_mul_eq_coeff (hm : p.Monic) (hnd : p.natDegree = 2) :
    ¬Irreducible p ↔ ∃ c₁ c₂, p.coeff 0 = c₁ * c₂ ∧ p.coeff 1 = c₁ + c₂ := by
  cases subsingleton_or_nontrivial R
  · simp [natDegree_of_subsingleton] at hnd
  rw [hm.irreducible_iff_natDegree', and_iff_right, hnd]
  push_neg; constructor
  · rintro ⟨a, b, ha, hb, rfl, hdb⟩
    simp only [zero_lt_two, Nat.div_self, ge_iff_le,
      Nat.Ioc_succ_singleton, zero_add, mem_singleton] at hdb
    have hda := hnd
    rw [ha.natDegree_mul hb, hdb] at hda
    use a.coeff 0, b.coeff 0, mul_coeff_zero a b
    simpa only [nextCoeff, hnd, add_right_cancel hda, hdb] using ha.nextCoeff_mul hb
  · 

Proof state:
case inr.mpr
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
p q : R[X]
hm : Monic p
hnd : natDegree p = 2
h✝ : Nontrivial R
⊢ (∃ c₁ c₂, coeff p 0 = c₁ * c₂ ∧ coeff p 1 = c₁ + c₂) →
    ∃ f g, Monic f ∧ Monic g ∧ f * g = p ∧ natDegree g ∈ Ioc 0 (2 / 2)

Next tactic:
rintro ⟨c₁, c₂, hmul, hadd⟩
theorem Monic.not_irreducible_iff_exists_add_mul_eq_coeff (hm : p.Monic) (hnd : p.natDegree = 2) :
    ¬Irreducible p ↔ ∃ c₁ c₂, p.coeff 0 = c₁ * c₂ ∧ p.coeff 1 = c₁ + c₂ := by
  cases subsingleton_or_nontrivial R
  · simp [natDegree_of_subsingleton] at hnd
  rw [hm.irreducible_iff_natDegree', and_iff_right, hnd]
  push_neg; constructor
  · rintro ⟨a, b, ha, hb, rfl, hdb⟩
    simp only [zero_lt_two, Nat.div_self, ge_iff_le,
      Nat.Ioc_succ_singleton, zero_add, mem_singleton] at hdb
    have hda := hnd
    rw [ha.natDegree_mul hb, hdb] at hda
    use a.coeff 0, b.coeff 0, mul_coeff_zero a b
    simpa only [nextCoeff, hnd, add_right_cancel hda, hdb] using ha.nextCoeff_mul hb
  · rintro ⟨c₁, c₂, hmul, hadd⟩
    

Proof state:
case inr.mpr.intro.intro.intro
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
p q : R[X]
hm : Monic p
hnd : natDegree p = 2
h✝ : Nontrivial R
c₁ c₂ : R
hmul : coeff p 0 = c₁ * c₂
hadd : coeff p 1 = c₁ + c₂
⊢ ∃ f g, Monic f ∧ Monic g ∧ f * g = p ∧ natDegree g ∈ Ioc 0 (2 / 2)

Next tactic:
refine
      ⟨X + C c₁, X + C c₂, monic_X_add_C _, monic_X_add_C _, ?_, ?_ ⟩
theorem Monic.not_irreducible_iff_exists_add_mul_eq_coeff (hm : p.Monic) (hnd : p.natDegree = 2) :
    ¬Irreducible p ↔ ∃ c₁ c₂, p.coeff 0 = c₁ * c₂ ∧ p.coeff 1 = c₁ + c₂ := by
  cases subsingleton_or_nontrivial R
  · simp [natDegree_of_subsingleton] at hnd
  rw [hm.irreducible_iff_natDegree', and_iff_right, hnd]
  push_neg; constructor
  · rintro ⟨a, b, ha, hb, rfl, hdb⟩
    simp only [zero_lt_two, Nat.div_self, ge_iff_le,
      Nat.Ioc_succ_singleton, zero_add, mem_singleton] at hdb
    have hda := hnd
    rw [ha.natDegree_mul hb, hdb] at hda
    use a.coeff 0, b.coeff 0, mul_coeff_zero a b
    simpa only [nextCoeff, hnd, add_right_cancel hda, hdb] using ha.nextCoeff_mul hb
  · rintro ⟨c₁, c₂, hmul, hadd⟩
    refine
      ⟨X + C c₁, X + C c₂, monic_X_add_C _, monic_X_add_C _, ?_, ?_ ⟩
    · 

Proof state:
case inr.mpr.intro.intro.intro.refine_1
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
p q : R[X]
hm : Monic p
hnd : natDegree p = 2
h✝ : Nontrivial R
c₁ c₂ : R
hmul : coeff p 0 = c₁ * c₂
hadd : coeff p 1 = c₁ + c₂
⊢ (X + C c₁) * (X + C c₂) = p

Next tactic:
rw [p.as_sum_range_C_mul_X_pow, hnd, Finset.sum_range_succ, Finset.sum_range_succ,
        Finset.sum_range_one, ← hnd, hm.coeff_natDegree, hnd, hmul, hadd, C_mul, C_add, C_1]
theorem Monic.not_irreducible_iff_exists_add_mul_eq_coeff (hm : p.Monic) (hnd : p.natDegree = 2) :
    ¬Irreducible p ↔ ∃ c₁ c₂, p.coeff 0 = c₁ * c₂ ∧ p.coeff 1 = c₁ + c₂ := by
  cases subsingleton_or_nontrivial R
  · simp [natDegree_of_subsingleton] at hnd
  rw [hm.irreducible_iff_natDegree', and_iff_right, hnd]
  push_neg; constructor
  · rintro ⟨a, b, ha, hb, rfl, hdb⟩
    simp only [zero_lt_two, Nat.div_self, ge_iff_le,
      Nat.Ioc_succ_singleton, zero_add, mem_singleton] at hdb
    have hda := hnd
    rw [ha.natDegree_mul hb, hdb] at hda
    use a.coeff 0, b.coeff 0, mul_coeff_zero a b
    simpa only [nextCoeff, hnd, add_right_cancel hda, hdb] using ha.nextCoeff_mul hb
  · rintro ⟨c₁, c₂, hmul, hadd⟩
    refine
      ⟨X + C c₁, X + C c₂, monic_X_add_C _, monic_X_add_C _, ?_, ?_ ⟩
    · rw [p.as_sum_range_C_mul_X_pow, hnd, Finset.sum_range_succ, Finset.sum_range_succ,
        Finset.sum_range_one, ← hnd, hm.coeff_natDegree, hnd, hmul, hadd, C_mul, C_add, C_1]
      

Proof state:
case inr.mpr.intro.intro.intro.refine_1
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
p q : R[X]
hm : Monic p
hnd : natDegree p = 2
h✝ : Nontrivial R
c₁ c₂ : R
hmul : coeff p 0 = c₁ * c₂
hadd : coeff p 1 = c₁ + c₂
⊢ (X + C c₁) * (X + C c₂) = C c₁ * C c₂ * X ^ 0 + (C c₁ + C c₂) * X ^ 1 + 1 * X ^ 2

Next tactic:
ring
theorem Monic.not_irreducible_iff_exists_add_mul_eq_coeff (hm : p.Monic) (hnd : p.natDegree = 2) :
    ¬Irreducible p ↔ ∃ c₁ c₂, p.coeff 0 = c₁ * c₂ ∧ p.coeff 1 = c₁ + c₂ := by
  cases subsingleton_or_nontrivial R
  · simp [natDegree_of_subsingleton] at hnd
  rw [hm.irreducible_iff_natDegree', and_iff_right, hnd]
  push_neg; constructor
  · rintro ⟨a, b, ha, hb, rfl, hdb⟩
    simp only [zero_lt_two, Nat.div_self, ge_iff_le,
      Nat.Ioc_succ_singleton, zero_add, mem_singleton] at hdb
    have hda := hnd
    rw [ha.natDegree_mul hb, hdb] at hda
    use a.coeff 0, b.coeff 0, mul_coeff_zero a b
    simpa only [nextCoeff, hnd, add_right_cancel hda, hdb] using ha.nextCoeff_mul hb
  · rintro ⟨c₁, c₂, hmul, hadd⟩
    refine
      ⟨X + C c₁, X + C c₂, monic_X_add_C _, monic_X_add_C _, ?_, ?_ ⟩
    · rw [p.as_sum_range_C_mul_X_pow, hnd, Finset.sum_range_succ, Finset.sum_range_succ,
        Finset.sum_range_one, ← hnd, hm.coeff_natDegree, hnd, hmul, hadd, C_mul, C_add, C_1]
      ring
    · 

Proof state:
case inr.mpr.intro.intro.intro.refine_2
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
p q : R[X]
hm : Monic p
hnd : natDegree p = 2
h✝ : Nontrivial R
c₁ c₂ : R
hmul : coeff p 0 = c₁ * c₂
hadd : coeff p 1 = c₁ + c₂
⊢ natDegree (X + C c₂) ∈ Ioc 0 (2 / 2)

Next tactic:
rw [mem_Ioc, natDegree_X_add_C _]
theorem Monic.not_irreducible_iff_exists_add_mul_eq_coeff (hm : p.Monic) (hnd : p.natDegree = 2) :
    ¬Irreducible p ↔ ∃ c₁ c₂, p.coeff 0 = c₁ * c₂ ∧ p.coeff 1 = c₁ + c₂ := by
  cases subsingleton_or_nontrivial R
  · simp [natDegree_of_subsingleton] at hnd
  rw [hm.irreducible_iff_natDegree', and_iff_right, hnd]
  push_neg; constructor
  · rintro ⟨a, b, ha, hb, rfl, hdb⟩
    simp only [zero_lt_two, Nat.div_self, ge_iff_le,
      Nat.Ioc_succ_singleton, zero_add, mem_singleton] at hdb
    have hda := hnd
    rw [ha.natDegree_mul hb, hdb] at hda
    use a.coeff 0, b.coeff 0, mul_coeff_zero a b
    simpa only [nextCoeff, hnd, add_right_cancel hda, hdb] using ha.nextCoeff_mul hb
  · rintro ⟨c₁, c₂, hmul, hadd⟩
    refine
      ⟨X + C c₁, X + C c₂, monic_X_add_C _, monic_X_add_C _, ?_, ?_ ⟩
    · rw [p.as_sum_range_C_mul_X_pow, hnd, Finset.sum_range_succ, Finset.sum_range_succ,
        Finset.sum_range_one, ← hnd, hm.coeff_natDegree, hnd, hmul, hadd, C_mul, C_add, C_1]
      ring
    · rw [mem_Ioc, natDegree_X_add_C _]
      

Proof state:
case inr.mpr.intro.intro.intro.refine_2
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
p q : R[X]
hm : Monic p
hnd : natDegree p = 2
h✝ : Nontrivial R
c₁ c₂ : R
hmul : coeff p 0 = c₁ * c₂
hadd : coeff p 1 = c₁ + c₂
⊢ 0 < 1 ∧ 1 ≤ 2 / 2

Next tactic:
simp
theorem Monic.not_irreducible_iff_exists_add_mul_eq_coeff (hm : p.Monic) (hnd : p.natDegree = 2) :
    ¬Irreducible p ↔ ∃ c₁ c₂, p.coeff 0 = c₁ * c₂ ∧ p.coeff 1 = c₁ + c₂ := by
  cases subsingleton_or_nontrivial R
  · simp [natDegree_of_subsingleton] at hnd
  rw [hm.irreducible_iff_natDegree', and_iff_right, hnd]
  push_neg; constructor
  · rintro ⟨a, b, ha, hb, rfl, hdb⟩
    simp only [zero_lt_two, Nat.div_self, ge_iff_le,
      Nat.Ioc_succ_singleton, zero_add, mem_singleton] at hdb
    have hda := hnd
    rw [ha.natDegree_mul hb, hdb] at hda
    use a.coeff 0, b.coeff 0, mul_coeff_zero a b
    simpa only [nextCoeff, hnd, add_right_cancel hda, hdb] using ha.nextCoeff_mul hb
  · rintro ⟨c₁, c₂, hmul, hadd⟩
    refine
      ⟨X + C c₁, X + C c₂, monic_X_add_C _, monic_X_add_C _, ?_, ?_ ⟩
    · rw [p.as_sum_range_C_mul_X_pow, hnd, Finset.sum_range_succ, Finset.sum_range_succ,
        Finset.sum_range_one, ← hnd, hm.coeff_natDegree, hnd, hmul, hadd, C_mul, C_add, C_1]
      ring
    · rw [mem_Ioc, natDegree_X_add_C _]
      simp
  · 

Proof state:
case inr
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
p q : R[X]
hm : Monic p
hnd : natDegree p = 2
h✝ : Nontrivial R
⊢ p ≠ 1

Next tactic:
rintro rfl