-- Mathlib/RingTheory/Polynomial/GaussLemma.lean
theorem IsPrimitive.irreducible_iff_irreducible_map_fraction_map {p : R[X]} (hp : p.IsPrimitive) :
    Irreducible p ↔ Irreducible (p.map (algebraMap R K))

Proof state:
case intro.mk.intro.mk.intro.inr
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ≠ 0
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
hcd0 : ¬C (c * d) = 0
u : Rˣ
h1 : C (c * d) * (C ↑u * (integerNormalization R⁰ b).primPart * (integerNormalization R⁰ a).primPart) = C (c * d) * p
hu : c * d * ↑u = (integerNormalization R⁰ a).content * (integerNormalization R⁰ b).content
h0 : a ≠ 0 ∧ b ≠ 0
h : IsUnit (integerNormalization R⁰ a).primPart
⊢ IsUnit a ∨ IsUnit b

Tactic:
left

Result:
case intro.mk.intro.mk.intro.inr.h
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ≠ 0
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
hcd0 : ¬C (c * d) = 0
u : Rˣ
h1 : C (c * d) * (C ↑u * (integerNormalization R⁰ b).primPart * (integerNormalization R⁰ a).primPart) = C (c * d) * p
hu : c * d * ↑u = (integerNormalization R⁰ a).content * (integerNormalization R⁰ b).content
h0 : a ≠ 0 ∧ b ≠ 0
h : IsUnit (integerNormalization R⁰ a).primPart
⊢ IsUnit a

Proof state:
case intro.mk.intro.mk.intro
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ≠ 0
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
hcd0 : ¬C (c * d) = 0
h1 : C c * C d * p = integerNormalization R⁰ a * integerNormalization R⁰ b
u : Rˣ
hu : c * d * ↑u = (integerNormalization R⁰ a).content * (integerNormalization R⁰ b).content
⊢ IsUnit a ∨ IsUnit b

Tactic:
rw [← RingHom.map_mul, eq_comm, (integerNormalization R⁰ a).eq_C_content_mul_primPart,
  (integerNormalization R⁰ b).eq_C_content_mul_primPart, mul_assoc, mul_comm _ (C _ * _), ←
  mul_assoc, ← mul_assoc, ← RingHom.map_mul, ← hu, RingHom.map_mul, mul_assoc, mul_assoc, ←
  mul_assoc (C (u : R))] at h1

Result:
case intro.mk.intro.mk.intro
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ≠ 0
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
hcd0 : ¬C (c * d) = 0
u : Rˣ
h1 : C (c * d) * (C ↑u * (integerNormalization R⁰ b).primPart * (integerNormalization R⁰ a).primPart) = C (c * d) * p
hu : c * d * ↑u = (integerNormalization R⁰ a).content * (integerNormalization R⁰ b).content
⊢ IsUnit a ∨ IsUnit b

Proof state:
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
⊢ IsUnit a ∨ IsUnit b

Tactic:
obtain ⟨⟨c, c0⟩, hc⟩ := integerNormalization_map_to_map R⁰ a

Result:
case intro.mk
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ∈ R⁰
hc : map (algebraMap R K) (integerNormalization R⁰ a) = ↑⟨c, c0⟩ • a
⊢ IsUnit a ∨ IsUnit b

Proof state:
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ≠ 0
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
hcd0 : ¬C (c * d) = 0
u : Rˣ
h1 : C (c * d) * (C ↑u * (integerNormalization R⁰ b).primPart * (integerNormalization R⁰ a).primPart) = C (c * d) * p
hu : c * d * ↑u = (integerNormalization R⁰ a).content * (integerNormalization R⁰ b).content
⊢ a ≠ 0 ∧ b ≠ 0

Tactic:
rw [Ne, Ne, ← not_or, ← mul_eq_zero, ← hab]

Result:
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ≠ 0
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
hcd0 : ¬C (c * d) = 0
u : Rˣ
h1 : C (c * d) * (C ↑u * (integerNormalization R⁰ b).primPart * (integerNormalization R⁰ a).primPart) = C (c * d) * p
hu : c * d * ↑u = (integerNormalization R⁰ a).content * (integerNormalization R⁰ b).content
⊢ ¬map (algebraMap R K) p = 0

Proof state:
case intro.mk.intro.mk
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ≠ 0
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
hcd0 : ¬C (c * d) = 0
h1 : C c * C d * p = integerNormalization R⁰ a * integerNormalization R⁰ b
⊢ IsUnit a ∨ IsUnit b

Tactic:
obtain ⟨u, hu⟩ :
  Associated (c * d)
    (content (integerNormalization R⁰ a) * content (integerNormalization R⁰ b)) := by
  rw [← dvd_dvd_iff_associated, ← normalize_eq_normalize_iff, normalize.map_mul,
    normalize.map_mul, normalize_content, normalize_content, ←
    mul_one (normalize c * normalize d), ← hp.content_eq_one, ← content_C, ← content_C, ←
    content_mul, ← content_mul, ← content_mul, h1]

Result:
case intro.mk.intro.mk.intro
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ≠ 0
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
hcd0 : ¬C (c * d) = 0
h1 : C c * C d * p = integerNormalization R⁰ a * integerNormalization R⁰ b
u : Rˣ
hu : c * d * ↑u = (integerNormalization R⁰ a).content * (integerNormalization R⁰ b).content
⊢ IsUnit a ∨ IsUnit b