l3lab/ntp-mathlib huggingface dataset:
theorem mem_span_C_X_sub_C_X_sub_C_iff_eval_eval_eq_zero {b : R[X]} {P : R[X][X]} :
    P ∈ Ideal.span {C (X - C a), X - C b} ↔ (P.eval b).eval a = 0 := by
  rw [Ideal.mem_span_pair]
  constructor <;> 

Proof state:
case mpr
R : Type u
S : Type v
T : Type w
A : Type z
a b✝ : R
n : ℕ
inst✝ : CommRing R
p q b : R[X]
P : R[X][X]
⊢ eval a (eval b P) = 0 → ∃ a_2 b_1, a_2 * C (X - C a) + b_1 * (X - C b) = P

Next tactic:
intro h
theorem mem_span_C_X_sub_C_X_sub_C_iff_eval_eval_eq_zero {b : R[X]} {P : R[X][X]} :
    P ∈ Ideal.span {C (X - C a), X - C b} ↔ (P.eval b).eval a = 0 := by
  rw [Ideal.mem_span_pair]
  constructor <;> intro h
  · 

Proof state:
case mp
R : Type u
S : Type v
T : Type w
A : Type z
a b✝ : R
n : ℕ
inst✝ : CommRing R
p q b : R[X]
P : R[X][X]
h : ∃ a_1 b_1, a_1 * C (X - C a) + b_1 * (X - C b) = P
⊢ eval a (eval b P) = 0

Next tactic:
rcases h with ⟨_, _, rfl⟩
theorem mem_span_C_X_sub_C_X_sub_C_iff_eval_eval_eq_zero {b : R[X]} {P : R[X][X]} :
    P ∈ Ideal.span {C (X - C a), X - C b} ↔ (P.eval b).eval a = 0 := by
  rw [Ideal.mem_span_pair]
  constructor <;> intro h
  · rcases h with ⟨_, _, rfl⟩
    

Proof state:
case mp.intro.intro
R : Type u
S : Type v
T : Type w
A : Type z
a b✝ : R
n : ℕ
inst✝ : CommRing R
p q b : R[X]
w✝¹ w✝ : R[X][X]
⊢ eval a (eval b (w✝¹ * C (X - C a) + w✝ * (X - C b))) = 0

Next tactic:
simp only [eval_C, eval_X, eval_add, eval_sub, eval_mul, add_zero, mul_zero, sub_self]
theorem mem_span_C_X_sub_C_X_sub_C_iff_eval_eval_eq_zero {b : R[X]} {P : R[X][X]} :
    P ∈ Ideal.span {C (X - C a), X - C b} ↔ (P.eval b).eval a = 0 := by
  rw [Ideal.mem_span_pair]
  constructor <;> intro h
  · rcases h with ⟨_, _, rfl⟩
    simp only [eval_C, eval_X, eval_add, eval_sub, eval_mul, add_zero, mul_zero, sub_self]
  · 

Proof state:
case mpr
R : Type u
S : Type v
T : Type w
A : Type z
a b✝ : R
n : ℕ
inst✝ : CommRing R
p q b : R[X]
P : R[X][X]
h : eval a (eval b P) = 0
⊢ ∃ a_1 b_1, a_1 * C (X - C a) + b_1 * (X - C b) = P

Next tactic:
rcases dvd_iff_isRoot.mpr h with ⟨p, hp⟩
theorem mem_span_C_X_sub_C_X_sub_C_iff_eval_eval_eq_zero {b : R[X]} {P : R[X][X]} :
    P ∈ Ideal.span {C (X - C a), X - C b} ↔ (P.eval b).eval a = 0 := by
  rw [Ideal.mem_span_pair]
  constructor <;> intro h
  · rcases h with ⟨_, _, rfl⟩
    simp only [eval_C, eval_X, eval_add, eval_sub, eval_mul, add_zero, mul_zero, sub_self]
  · rcases dvd_iff_isRoot.mpr h with ⟨p, hp⟩
    

Proof state:
case mpr.intro
R : Type u
S : Type v
T : Type w
A : Type z
a b✝ : R
n : ℕ
inst✝ : CommRing R
p✝ q b : R[X]
P : R[X][X]
h : eval a (eval b P) = 0
p : R[X]
hp : eval b P = (X - C a) * p
⊢ ∃ a_1 b_1, a_1 * C (X - C a) + b_1 * (X - C b) = P

Next tactic:
rcases @X_sub_C_dvd_sub_C_eval _ b _ P with ⟨q, hq⟩
theorem mem_span_C_X_sub_C_X_sub_C_iff_eval_eval_eq_zero {b : R[X]} {P : R[X][X]} :
    P ∈ Ideal.span {C (X - C a), X - C b} ↔ (P.eval b).eval a = 0 := by
  rw [Ideal.mem_span_pair]
  constructor <;> intro h
  · rcases h with ⟨_, _, rfl⟩
    simp only [eval_C, eval_X, eval_add, eval_sub, eval_mul, add_zero, mul_zero, sub_self]
  · rcases dvd_iff_isRoot.mpr h with ⟨p, hp⟩
    rcases @X_sub_C_dvd_sub_C_eval _ b _ P with ⟨q, hq⟩
    

Proof state:
case mpr.intro.intro
R : Type u
S : Type v
T : Type w
A : Type z
a b✝ : R
n : ℕ
inst✝ : CommRing R
p✝ q✝ b : R[X]
P : R[X][X]
h : eval a (eval b P) = 0
p : R[X]
hp : eval b P = (X - C a) * p
q : R[X][X]
hq : P - C (eval b P) = (X - C b) * q
⊢ ∃ a_1 b_1, a_1 * C (X - C a) + b_1 * (X - C b) = P

Next tactic:
exact ⟨C p, q, by rw [mul_comm, mul_comm q, eq_add_of_sub_eq' hq, hp, C_mul]⟩
theorem mem_span_C_X_sub_C_X_sub_C_iff_eval_eval_eq_zero {b : R[X]} {P : R[X][X]} :
    P ∈ Ideal.span {C (X - C a), X - C b} ↔ (P.eval b).eval a = 0 := by
  rw [Ideal.mem_span_pair]
  constructor <;> intro h
  · rcases h with ⟨_, _, rfl⟩
    simp only [eval_C, eval_X, eval_add, eval_sub, eval_mul, add_zero, mul_zero, sub_self]
  · rcases dvd_iff_isRoot.mpr h with ⟨p, hp⟩
    rcases @X_sub_C_dvd_sub_C_eval _ b _ P with ⟨q, hq⟩
    exact ⟨C p, q, by 

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a b✝ : R
n : ℕ
inst✝ : CommRing R
p✝ q✝ b : R[X]
P : R[X][X]
h : eval a (eval b P) = 0
p : R[X]
hp : eval b P = (X - C a) * p
q : R[X][X]
hq : P - C (eval b P) = (X - C b) * q
⊢ C p * C (X - C a) + q * (X - C b) = P

Next tactic:
rw [mul_comm, mul_comm q, eq_add_of_sub_eq' hq, hp, C_mul]
theorem modByMonic_X (p : R[X]) : p %ₘ X = C (p.eval 0) := by
  

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝ : CommRing R
p✝ q p : R[X]
⊢ p %ₘ X = C (eval 0 p)

Next tactic:
rw [← modByMonic_X_sub_C_eq_C_eval, C_0, sub_zero]
theorem eval₂_modByMonic_eq_self_of_root [CommRing S] {f : R →+* S} {p q : R[X]} (hq : q.Monic)
    {x : S} (hx : q.eval₂ f x = 0) : (p %ₘ q).eval₂ f x = p.eval₂ f x := by
  

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝¹ : CommRing R
p✝ q✝ : R[X]
inst✝ : CommRing S
f : R →+* S
p q : R[X]
hq : Monic q
x : S
hx : eval₂ f x q = 0
⊢ eval₂ f x (p %ₘ q) = eval₂ f x p

Next tactic:
rw [modByMonic_eq_sub_mul_div p hq, eval₂_sub, eval₂_mul, hx, zero_mul, sub_zero]
theorem sub_dvd_eval_sub (a b : R) (p : R[X]) : a - b ∣ p.eval a - p.eval b := by
  

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a✝ b✝ : R
n : ℕ
inst✝ : CommRing R
p✝ q : R[X]
a b : R
p : R[X]
⊢ a - b ∣ eval a p - eval b p

Next tactic:
suffices X - C b ∣ p - C (p.eval b) by
    simpa only [coe_evalRingHom, eval_sub, eval_X, eval_C] using (evalRingHom a).map_dvd this
theorem sub_dvd_eval_sub (a b : R) (p : R[X]) : a - b ∣ p.eval a - p.eval b := by
  suffices X - C b ∣ p - C (p.eval b) by
    

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a✝ b✝ : R
n : ℕ
inst✝ : CommRing R
p✝ q : R[X]
a b : R
p : R[X]
this : X - C b ∣ p - C (eval b p)
⊢ a - b ∣ eval a p - eval b p

Next tactic:
simpa only [coe_evalRingHom, eval_sub, eval_X, eval_C] using (evalRingHom a).map_dvd this
theorem sub_dvd_eval_sub (a b : R) (p : R[X]) : a - b ∣ p.eval a - p.eval b := by
  suffices X - C b ∣ p - C (p.eval b) by
    simpa only [coe_evalRingHom, eval_sub, eval_X, eval_C] using (evalRingHom a).map_dvd this
  

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a✝ b✝ : R
n : ℕ
inst✝ : CommRing R
p✝ q : R[X]
a b : R
p : R[X]
⊢ X - C b ∣ p - C (eval b p)

Next tactic:
simp [dvd_iff_isRoot]
theorem ker_evalRingHom (x : R) : RingHom.ker (evalRingHom x) = Ideal.span {X - C x} := by
  

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝ : CommRing R
p q : R[X]
x : R
⊢ RingHom.ker (evalRingHom x) = Ideal.span {X - C x}

Next tactic:
ext y
theorem ker_evalRingHom (x : R) : RingHom.ker (evalRingHom x) = Ideal.span {X - C x} := by
  ext y
  

Proof state:
case h
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝ : CommRing R
p q : R[X]
x : R
y : R[X]
⊢ y ∈ RingHom.ker (evalRingHom x) ↔ y ∈ Ideal.span {X - C x}

Next tactic:
simp [Ideal.mem_span_singleton, dvd_iff_isRoot, RingHom.mem_ker]
@[simp]
theorem rootMultiplicity_eq_zero_iff {p : R[X]} {x : R} :
    rootMultiplicity x p = 0 ↔ IsRoot p x → p = 0 := by
  

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝ : CommRing R
p✝ q p : R[X]
x : R
⊢ rootMultiplicity x p = 0 ↔ IsRoot p x → p = 0

Next tactic:
classical
  simp only [rootMultiplicity_eq_multiplicity, dite_eq_left_iff, PartENat.get_eq_iff_eq_coe,
    Nat.cast_zero, multiplicity.multiplicity_eq_zero, dvd_iff_isRoot, not_imp_not]
@[simp]
theorem rootMultiplicity_eq_zero_iff {p : R[X]} {x : R} :
    rootMultiplicity x p = 0 ↔ IsRoot p x → p = 0 := by
  classical
  

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝ : CommRing R
p✝ q p : R[X]
x : R
⊢ rootMultiplicity x p = 0 ↔ IsRoot p x → p = 0

Next tactic:
simp only [rootMultiplicity_eq_multiplicity, dite_eq_left_iff, PartENat.get_eq_iff_eq_coe,
    Nat.cast_zero, multiplicity.multiplicity_eq_zero, dvd_iff_isRoot, not_imp_not]
@[simp]
theorem rootMultiplicity_pos' {p : R[X]} {x : R} : 0 < rootMultiplicity x p ↔ p ≠ 0 ∧ IsRoot p x :=
  by 

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝ : CommRing R
p✝ q p : R[X]
x : R
⊢ 0 < rootMultiplicity x p ↔ p ≠ 0 ∧ IsRoot p x

Next tactic:
rw [pos_iff_ne_zero, Ne.def, rootMultiplicity_eq_zero_iff, not_imp, and_comm]
theorem eval_divByMonic_pow_rootMultiplicity_ne_zero {p : R[X]} (a : R) (hp : p ≠ 0) :
    eval a (p /ₘ (X - C a) ^ rootMultiplicity a p) ≠ 0 := by
  

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a✝ b : R
n : ℕ
inst✝ : CommRing R
p✝ q p : R[X]
a : R
hp : p ≠ 0
⊢ eval a (p /ₘ (X - C a) ^ rootMultiplicity a p) ≠ 0

Next tactic:
classical
  haveI : Nontrivial R := Nontrivial.of_polynomial_ne hp
  rw [Ne.def, ← IsRoot.def, ← dvd_iff_isRoot]
  rintro ⟨q, hq⟩
  have := pow_mul_divByMonic_rootMultiplicity_eq p a
  rw [hq, ← mul_assoc, ← pow_succ', rootMultiplicity_eq_multiplicity, dif_neg hp] at this
  exact
    multiplicity.is_greatest'
      (multiplicity_finite_of_degree_pos_of_monic
        (show (0 : WithBot ℕ) < degree (X - C a) by rw [degree_X_sub_C]; exact by decide)
        (monic_X_sub_C _) hp)
      (Nat.lt_succ_self _) (dvd_of_mul_right_eq _ this)
theorem eval_divByMonic_pow_rootMultiplicity_ne_zero {p : R[X]} (a : R) (hp : p ≠ 0) :
    eval a (p /ₘ (X - C a) ^ rootMultiplicity a p) ≠ 0 := by
  classical
  

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a✝ b : R
n : ℕ
inst✝ : CommRing R
p✝ q p : R[X]
a : R
hp : p ≠ 0
⊢ eval a (p /ₘ (X - C a) ^ rootMultiplicity a p) ≠ 0

Next tactic:
haveI : Nontrivial R := Nontrivial.of_polynomial_ne hp
theorem eval_divByMonic_pow_rootMultiplicity_ne_zero {p : R[X]} (a : R) (hp : p ≠ 0) :
    eval a (p /ₘ (X - C a) ^ rootMultiplicity a p) ≠ 0 := by
  classical
  haveI : Nontrivial R := Nontrivial.of_polynomial_ne hp
  

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a✝ b : R
n : ℕ
inst✝ : CommRing R
p✝ q p : R[X]
a : R
hp : p ≠ 0
this : Nontrivial R
⊢ eval a (p /ₘ (X - C a) ^ rootMultiplicity a p) ≠ 0

Next tactic:
rw [Ne.def, ← IsRoot.def, ← dvd_iff_isRoot]
theorem eval_divByMonic_pow_rootMultiplicity_ne_zero {p : R[X]} (a : R) (hp : p ≠ 0) :
    eval a (p /ₘ (X - C a) ^ rootMultiplicity a p) ≠ 0 := by
  classical
  haveI : Nontrivial R := Nontrivial.of_polynomial_ne hp
  rw [Ne.def, ← IsRoot.def, ← dvd_iff_isRoot]
  

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a✝ b : R
n : ℕ
inst✝ : CommRing R
p✝ q p : R[X]
a : R
hp : p ≠ 0
this : Nontrivial R
⊢ ¬X - C a ∣ p /ₘ (X - C a) ^ rootMultiplicity a p

Next tactic:
rintro ⟨q, hq⟩
theorem eval_divByMonic_pow_rootMultiplicity_ne_zero {p : R[X]} (a : R) (hp : p ≠ 0) :
    eval a (p /ₘ (X - C a) ^ rootMultiplicity a p) ≠ 0 := by
  classical
  haveI : Nontrivial R := Nontrivial.of_polynomial_ne hp
  rw [Ne.def, ← IsRoot.def, ← dvd_iff_isRoot]
  rintro ⟨q, hq⟩
  

Proof state:
case intro
R : Type u
S : Type v
T : Type w
A : Type z
a✝ b : R
n : ℕ
inst✝ : CommRing R
p✝ q✝ p : R[X]
a : R
hp : p ≠ 0
this : Nontrivial R
q : R[X]
hq : p /ₘ (X - C a) ^ rootMultiplicity a p = (X - C a) * q
⊢ False

Next tactic:
have := pow_mul_divByMonic_rootMultiplicity_eq p a
theorem eval_divByMonic_pow_rootMultiplicity_ne_zero {p : R[X]} (a : R) (hp : p ≠ 0) :
    eval a (p /ₘ (X - C a) ^ rootMultiplicity a p) ≠ 0 := by
  classical
  haveI : Nontrivial R := Nontrivial.of_polynomial_ne hp
  rw [Ne.def, ← IsRoot.def, ← dvd_iff_isRoot]
  rintro ⟨q, hq⟩
  have := pow_mul_divByMonic_rootMultiplicity_eq p a
  

Proof state:
case intro
R : Type u
S : Type v
T : Type w
A : Type z
a✝ b : R
n : ℕ
inst✝ : CommRing R
p✝ q✝ p : R[X]
a : R
hp : p ≠ 0
this✝ : Nontrivial R
q : R[X]
hq : p /ₘ (X - C a) ^ rootMultiplicity a p = (X - C a) * q
this : (X - C a) ^ rootMultiplicity a p * (p /ₘ (X - C a) ^ rootMultiplicity a p) = p
⊢ False

Next tactic:
rw [hq, ← mul_assoc, ← pow_succ', rootMultiplicity_eq_multiplicity, dif_neg hp] at this
theorem eval_divByMonic_pow_rootMultiplicity_ne_zero {p : R[X]} (a : R) (hp : p ≠ 0) :
    eval a (p /ₘ (X - C a) ^ rootMultiplicity a p) ≠ 0 := by
  classical
  haveI : Nontrivial R := Nontrivial.of_polynomial_ne hp
  rw [Ne.def, ← IsRoot.def, ← dvd_iff_isRoot]
  rintro ⟨q, hq⟩
  have := pow_mul_divByMonic_rootMultiplicity_eq p a
  rw [hq, ← mul_assoc, ← pow_succ', rootMultiplicity_eq_multiplicity, dif_neg hp] at this
  

Proof state:
case intro
R : Type u
S : Type v
T : Type w
A : Type z
a✝ b : R
n : ℕ
inst✝ : CommRing R
p✝ q✝ p : R[X]
a : R
hp : p ≠ 0
this✝ : Nontrivial R
q : R[X]
hq : p /ₘ (X - C a) ^ rootMultiplicity a p = (X - C a) * q
this : (X - C a) ^ (Part.get (multiplicity (X - C a) p) (_ : multiplicity.Finite (X - C a) p) + 1) * q = p
⊢ False

Next tactic:
exact
    multiplicity.is_greatest'
      (multiplicity_finite_of_degree_pos_of_monic
        (show (0 : WithBot ℕ) < degree (X - C a) by rw [degree_X_sub_C]; exact by decide)
        (monic_X_sub_C _) hp)
      (Nat.lt_succ_self _) (dvd_of_mul_right_eq _ this)
theorem eval_divByMonic_pow_rootMultiplicity_ne_zero {p : R[X]} (a : R) (hp : p ≠ 0) :
    eval a (p /ₘ (X - C a) ^ rootMultiplicity a p) ≠ 0 := by
  classical
  haveI : Nontrivial R := Nontrivial.of_polynomial_ne hp
  rw [Ne.def, ← IsRoot.def, ← dvd_iff_isRoot]
  rintro ⟨q, hq⟩
  have := pow_mul_divByMonic_rootMultiplicity_eq p a
  rw [hq, ← mul_assoc, ← pow_succ', rootMultiplicity_eq_multiplicity, dif_neg hp] at this
  exact
    multiplicity.is_greatest'
      (multiplicity_finite_of_degree_pos_of_monic
        (show (0 : WithBot ℕ) < degree (X - C a) by 

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a✝ b : R
n : ℕ
inst✝ : CommRing R
p✝ q✝ p : R[X]
a : R
hp : p ≠ 0
this✝ : Nontrivial R
q : R[X]
hq : p /ₘ (X - C a) ^ rootMultiplicity a p = (X - C a) * q
this : (X - C a) ^ (Part.get (multiplicity (X - C a) p) (_ : multiplicity.Finite (X - C a) p) + 1) * q = p
⊢ 0 < degree (X - C a)

Next tactic:
rw [degree_X_sub_C]
theorem eval_divByMonic_pow_rootMultiplicity_ne_zero {p : R[X]} (a : R) (hp : p ≠ 0) :
    eval a (p /ₘ (X - C a) ^ rootMultiplicity a p) ≠ 0 := by
  classical
  haveI : Nontrivial R := Nontrivial.of_polynomial_ne hp
  rw [Ne.def, ← IsRoot.def, ← dvd_iff_isRoot]
  rintro ⟨q, hq⟩
  have := pow_mul_divByMonic_rootMultiplicity_eq p a
  rw [hq, ← mul_assoc, ← pow_succ', rootMultiplicity_eq_multiplicity, dif_neg hp] at this
  exact
    multiplicity.is_greatest'
      (multiplicity_finite_of_degree_pos_of_monic
        (show (0 : WithBot ℕ) < degree (X - C a) by rw [degree_X_sub_C]; 

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a✝ b : R
n : ℕ
inst✝ : CommRing R
p✝ q✝ p : R[X]
a : R
hp : p ≠ 0
this✝ : Nontrivial R
q : R[X]
hq : p /ₘ (X - C a) ^ rootMultiplicity a p = (X - C a) * q
this : (X - C a) ^ (Part.get (multiplicity (X - C a) p) (_ : multiplicity.Finite (X - C a) p) + 1) * q = p
⊢ 0 < 1

Next tactic:
exact by decide
theorem eval_divByMonic_pow_rootMultiplicity_ne_zero {p : R[X]} (a : R) (hp : p ≠ 0) :
    eval a (p /ₘ (X - C a) ^ rootMultiplicity a p) ≠ 0 := by
  classical
  haveI : Nontrivial R := Nontrivial.of_polynomial_ne hp
  rw [Ne.def, ← IsRoot.def, ← dvd_iff_isRoot]
  rintro ⟨q, hq⟩
  have := pow_mul_divByMonic_rootMultiplicity_eq p a
  rw [hq, ← mul_assoc, ← pow_succ', rootMultiplicity_eq_multiplicity, dif_neg hp] at this
  exact
    multiplicity.is_greatest'
      (multiplicity_finite_of_degree_pos_of_monic
        (show (0 : WithBot ℕ) < degree (X - C a) by rw [degree_X_sub_C]; exact by 

Proof state:
R : Type u
S : Type v
T : Type w
A : Type z
a✝ b : R
n : ℕ
inst✝ : CommRing R
p✝ q✝ p : R[X]
a : R
hp : p ≠ 0
this✝ : Nontrivial R
q : R[X]
hq : p /ₘ (X - C a) ^ rootMultiplicity a p = (X - C a) * q
this : (X - C a) ^ (Part.get (multiplicity (X - C a) p) (_ : multiplicity.Finite (X - C a) p) + 1) * q = p
⊢ 0 < 1

Next tactic:
decide
theorem Associates.isAtom_iff {p : Associates M} (h₁ : p ≠ 0) : IsAtom p ↔ Irreducible p :=
  ⟨fun hp =>
    ⟨by 

Proof state:
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
p : Associates M
h₁ : p ≠ 0
hp : IsAtom p
⊢ ¬IsUnit p

Next tactic:
simpa only [Associates.isUnit_iff_eq_one] using hp.1
theorem Associates.isAtom_iff {p : Associates M} (h₁ : p ≠ 0) : IsAtom p ↔ Irreducible p :=
  ⟨fun hp =>
    ⟨by simpa only [Associates.isUnit_iff_eq_one] using hp.1, fun a b h =>
      (hp.le_iff.mp ⟨_, h⟩).casesOn (fun ha => Or.inl (a.isUnit_iff_eq_one.mpr ha)) fun ha =>
        Or.inr
          (show IsUnit b by
            

Proof state:
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
p : Associates M
h₁ : p ≠ 0
hp : IsAtom p
a b : Associates M
h : p = a * b
ha : a = p
⊢ IsUnit b

Next tactic:
rw [ha] at h
theorem Associates.isAtom_iff {p : Associates M} (h₁ : p ≠ 0) : IsAtom p ↔ Irreducible p :=
  ⟨fun hp =>
    ⟨by simpa only [Associates.isUnit_iff_eq_one] using hp.1, fun a b h =>
      (hp.le_iff.mp ⟨_, h⟩).casesOn (fun ha => Or.inl (a.isUnit_iff_eq_one.mpr ha)) fun ha =>
        Or.inr
          (show IsUnit b by
            rw [ha] at h
            

Proof state:
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
p : Associates M
h₁ : p ≠ 0
hp : IsAtom p
a b : Associates M
h : p = p * b
ha : a = p
⊢ IsUnit b

Next tactic:
apply isUnit_of_associated_mul (show Associated (p * b) p by conv_rhs => rw [h]) h₁
theorem Associates.isAtom_iff {p : Associates M} (h₁ : p ≠ 0) : IsAtom p ↔ Irreducible p :=
  ⟨fun hp =>
    ⟨by simpa only [Associates.isUnit_iff_eq_one] using hp.1, fun a b h =>
      (hp.le_iff.mp ⟨_, h⟩).casesOn (fun ha => Or.inl (a.isUnit_iff_eq_one.mpr ha)) fun ha =>
        Or.inr
          (show IsUnit b by
            rw [ha] at h
            apply isUnit_of_associated_mul (show Associated (p * b) p by 

Proof state:
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
p : Associates M
h₁ : p ≠ 0
hp : IsAtom p
a b : Associates M
h : p = p * b
ha : a = p
⊢ Associated (p * b) p

Next tactic:
conv_rhs => rw [h]
theorem Associates.isAtom_iff {p : Associates M} (h₁ : p ≠ 0) : IsAtom p ↔ Irreducible p :=
  ⟨fun hp =>
    ⟨by simpa only [Associates.isUnit_iff_eq_one] using hp.1, fun a b h =>
      (hp.le_iff.mp ⟨_, h⟩).casesOn (fun ha => Or.inl (a.isUnit_iff_eq_one.mpr ha)) fun ha =>
        Or.inr
          (show IsUnit b by
            rw [ha] at h
            apply isUnit_of_associated_mul (show Associated (p * b) p by conv_rhs => 

Proof state:
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
p : Associates M
h₁ : p ≠ 0
hp : IsAtom p
a b : Associates M
h : p = p * b
ha : a = p
| p

Next tactic:
rw [h]
theorem Associates.isAtom_iff {p : Associates M} (h₁ : p ≠ 0) : IsAtom p ↔ Irreducible p :=
  ⟨fun hp =>
    ⟨by simpa only [Associates.isUnit_iff_eq_one] using hp.1, fun a b h =>
      (hp.le_iff.mp ⟨_, h⟩).casesOn (fun ha => Or.inl (a.isUnit_iff_eq_one.mpr ha)) fun ha =>
        Or.inr
          (show IsUnit b by
            rw [ha] at h
            apply isUnit_of_associated_mul (show Associated (p * b) p by conv_rhs => rw [h]) h₁)⟩,
    fun hp =>
    ⟨by 

Proof state:
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
p : Associates M
h₁ : p ≠ 0
hp : Irreducible p
⊢ p ≠ ⊥

Next tactic:
simpa only [Associates.isUnit_iff_eq_one, Associates.bot_eq_one] using hp.1
theorem Associates.isAtom_iff {p : Associates M} (h₁ : p ≠ 0) : IsAtom p ↔ Irreducible p :=
  ⟨fun hp =>
    ⟨by simpa only [Associates.isUnit_iff_eq_one] using hp.1, fun a b h =>
      (hp.le_iff.mp ⟨_, h⟩).casesOn (fun ha => Or.inl (a.isUnit_iff_eq_one.mpr ha)) fun ha =>
        Or.inr
          (show IsUnit b by
            rw [ha] at h
            apply isUnit_of_associated_mul (show Associated (p * b) p by conv_rhs => rw [h]) h₁)⟩,
    fun hp =>
    ⟨by simpa only [Associates.isUnit_iff_eq_one, Associates.bot_eq_one] using hp.1,
      fun b ⟨⟨a, hab⟩, hb⟩ =>
      (hp.isUnit_or_isUnit hab).casesOn
        (fun hb => show b = ⊥ by 

Proof state:
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
p : Associates M
h₁ : p ≠ 0
hp : Irreducible p
b : Associates M
x✝ : b < p
a : Associates M
hab : p = b * a
hb✝ : ¬p ∣ b
hb : IsUnit b
⊢ b = ⊥

Next tactic:
rwa [Associates.isUnit_iff_eq_one, ← Associates.bot_eq_one] at hb
theorem Associates.isAtom_iff {p : Associates M} (h₁ : p ≠ 0) : IsAtom p ↔ Irreducible p :=
  ⟨fun hp =>
    ⟨by simpa only [Associates.isUnit_iff_eq_one] using hp.1, fun a b h =>
      (hp.le_iff.mp ⟨_, h⟩).casesOn (fun ha => Or.inl (a.isUnit_iff_eq_one.mpr ha)) fun ha =>
        Or.inr
          (show IsUnit b by
            rw [ha] at h
            apply isUnit_of_associated_mul (show Associated (p * b) p by conv_rhs => rw [h]) h₁)⟩,
    fun hp =>
    ⟨by simpa only [Associates.isUnit_iff_eq_one, Associates.bot_eq_one] using hp.1,
      fun b ⟨⟨a, hab⟩, hb⟩ =>
      (hp.isUnit_or_isUnit hab).casesOn
        (fun hb => show b = ⊥ by rwa [Associates.isUnit_iff_eq_one, ← Associates.bot_eq_one] at hb)
        fun ha =>
        absurd
          (show p ∣ b from
            ⟨(ha.unit⁻¹ : Units _), by 

Proof state:
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
p : Associates M
h₁ : p ≠ 0
hp : Irreducible p
b : Associates M
x✝ : b < p
a : Associates M
hab : p = b * a
hb : ¬p ∣ b
ha : IsUnit a
⊢ b = p * ↑(IsUnit.unit ha)⁻¹

Next tactic:
rw [hab, mul_assoc, IsUnit.mul_val_inv ha, mul_one]
theorem exists_chain_of_prime_pow {p : Associates M} {n : ℕ} (hn : n ≠ 0) (hp : Prime p) :
    ∃ c : Fin (n + 1) → Associates M,
      c 1 = p ∧ StrictMono c ∧ ∀ {r : Associates M}, r ≤ p ^ n ↔ ∃ i, r = c i := by
  

Proof state:
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
p : Associates M
n : ℕ
hn : n ≠ 0
hp : Prime p
⊢ ∃ c, c 1 = p ∧ StrictMono c ∧ ∀ {r : Associates M}, r ≤ p ^ n ↔ ∃ i, r = c i

Next tactic:
refine' ⟨fun i => p ^ (i : ℕ), _, fun n m h => _, @fun y => ⟨fun h => _, _⟩⟩
theorem exists_chain_of_prime_pow {p : Associates M} {n : ℕ} (hn : n ≠ 0) (hp : Prime p) :
    ∃ c : Fin (n + 1) → Associates M,
      c 1 = p ∧ StrictMono c ∧ ∀ {r : Associates M}, r ≤ p ^ n ↔ ∃ i, r = c i := by
  refine' ⟨fun i => p ^ (i : ℕ), _, fun n m h => _, @fun y => ⟨fun h => _, _⟩⟩
  · 

Proof state:
case refine'_1
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
p : Associates M
n : ℕ
hn : n ≠ 0
hp : Prime p
⊢ (fun i => p ^ ↑i) 1 = p

Next tactic:
dsimp only
theorem exists_chain_of_prime_pow {p : Associates M} {n : ℕ} (hn : n ≠ 0) (hp : Prime p) :
    ∃ c : Fin (n + 1) → Associates M,
      c 1 = p ∧ StrictMono c ∧ ∀ {r : Associates M}, r ≤ p ^ n ↔ ∃ i, r = c i := by
  refine' ⟨fun i => p ^ (i : ℕ), _, fun n m h => _, @fun y => ⟨fun h => _, _⟩⟩
  · dsimp only
    

Proof state:
case refine'_1
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
p : Associates M
n : ℕ
hn : n ≠ 0
hp : Prime p
⊢ p ^ ↑1 = p

Next tactic:
rw [Fin.val_one', Nat.mod_eq_of_lt, pow_one]
theorem exists_chain_of_prime_pow {p : Associates M} {n : ℕ} (hn : n ≠ 0) (hp : Prime p) :
    ∃ c : Fin (n + 1) → Associates M,
      c 1 = p ∧ StrictMono c ∧ ∀ {r : Associates M}, r ≤ p ^ n ↔ ∃ i, r = c i := by
  refine' ⟨fun i => p ^ (i : ℕ), _, fun n m h => _, @fun y => ⟨fun h => _, _⟩⟩
  · dsimp only
    rw [Fin.val_one', Nat.mod_eq_of_lt, pow_one]
    

Proof state:
case refine'_1
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
p : Associates M
n : ℕ
hn : n ≠ 0
hp : Prime p
⊢ 1 < n + 1

Next tactic:
exact Nat.lt_succ_of_le (Nat.one_le_iff_ne_zero.mpr hn)
theorem exists_chain_of_prime_pow {p : Associates M} {n : ℕ} (hn : n ≠ 0) (hp : Prime p) :
    ∃ c : Fin (n + 1) → Associates M,
      c 1 = p ∧ StrictMono c ∧ ∀ {r : Associates M}, r ≤ p ^ n ↔ ∃ i, r = c i := by
  refine' ⟨fun i => p ^ (i : ℕ), _, fun n m h => _, @fun y => ⟨fun h => _, _⟩⟩
  · dsimp only
    rw [Fin.val_one', Nat.mod_eq_of_lt, pow_one]
    exact Nat.lt_succ_of_le (Nat.one_le_iff_ne_zero.mpr hn)
  · 

Proof state:
case refine'_2
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
p : Associates M
n✝ : ℕ
hn : n✝ ≠ 0
hp : Prime p
n m : Fin (n✝ + 1)
h : n < m
⊢ (fun i => p ^ ↑i) n < (fun i => p ^ ↑i) m

Next tactic:
exact Associates.dvdNotUnit_iff_lt.mp
        ⟨pow_ne_zero n hp.ne_zero, p ^ (m - n : ℕ),
          not_isUnit_of_not_isUnit_dvd hp.not_unit (dvd_pow dvd_rfl (Nat.sub_pos_of_lt h).ne'),
          (pow_mul_pow_sub p h.le).symm⟩