l3lab/ntp-mathlib huggingface dataset:
theorem mk_dvdNotUnit_mk_iff {a b : α} :
    DvdNotUnit (Associates.mk a) (Associates.mk b) ↔ DvdNotUnit a b := by
  rw [DvdNotUnit, DvdNotUnit, mk_ne_zero]
  apply and_congr_right; intro
  constructor
  · contrapose!
    

Proof state:
case h.mp
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CommMonoidWithZero α
a b : α
a✝ : a ≠ 0
⊢ (∀ (x : α), ¬IsUnit x → b ≠ a * x) → ∀ (x : Associates α), ¬IsUnit x → Associates.mk b ≠ Associates.mk a * x

Next tactic:
rw [forall_associated]
theorem mk_dvdNotUnit_mk_iff {a b : α} :
    DvdNotUnit (Associates.mk a) (Associates.mk b) ↔ DvdNotUnit a b := by
  rw [DvdNotUnit, DvdNotUnit, mk_ne_zero]
  apply and_congr_right; intro
  constructor
  · contrapose!
    rw [forall_associated]
    

Proof state:
case h.mp
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CommMonoidWithZero α
a b : α
a✝ : a ≠ 0
⊢ (∀ (x : α), ¬IsUnit x → b ≠ a * x) →
    ∀ (a_2 : α), ¬IsUnit (Associates.mk a_2) → Associates.mk b ≠ Associates.mk a * Associates.mk a_2

Next tactic:
intro h x hx hbax
theorem mk_dvdNotUnit_mk_iff {a b : α} :
    DvdNotUnit (Associates.mk a) (Associates.mk b) ↔ DvdNotUnit a b := by
  rw [DvdNotUnit, DvdNotUnit, mk_ne_zero]
  apply and_congr_right; intro
  constructor
  · contrapose!
    rw [forall_associated]
    intro h x hx hbax
    

Proof state:
case h.mp
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CommMonoidWithZero α
a b : α
a✝ : a ≠ 0
h : ∀ (x : α), ¬IsUnit x → b ≠ a * x
x : α
hx : ¬IsUnit (Associates.mk x)
hbax : Associates.mk b = Associates.mk a * Associates.mk x
⊢ False

Next tactic:
rw [mk_mul_mk, mk_eq_mk_iff_associated] at hbax
theorem mk_dvdNotUnit_mk_iff {a b : α} :
    DvdNotUnit (Associates.mk a) (Associates.mk b) ↔ DvdNotUnit a b := by
  rw [DvdNotUnit, DvdNotUnit, mk_ne_zero]
  apply and_congr_right; intro
  constructor
  · contrapose!
    rw [forall_associated]
    intro h x hx hbax
    rw [mk_mul_mk, mk_eq_mk_iff_associated] at hbax
    

Proof state:
case h.mp
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CommMonoidWithZero α
a b : α
a✝ : a ≠ 0
h : ∀ (x : α), ¬IsUnit x → b ≠ a * x
x : α
hx : ¬IsUnit (Associates.mk x)
hbax : b ~ᵤ a * x
⊢ False

Next tactic:
cases' hbax with u hu
theorem mk_dvdNotUnit_mk_iff {a b : α} :
    DvdNotUnit (Associates.mk a) (Associates.mk b) ↔ DvdNotUnit a b := by
  rw [DvdNotUnit, DvdNotUnit, mk_ne_zero]
  apply and_congr_right; intro
  constructor
  · contrapose!
    rw [forall_associated]
    intro h x hx hbax
    rw [mk_mul_mk, mk_eq_mk_iff_associated] at hbax
    cases' hbax with u hu
    

Proof state:
case h.mp.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CommMonoidWithZero α
a b : α
a✝ : a ≠ 0
h : ∀ (x : α), ¬IsUnit x → b ≠ a * x
x : α
hx : ¬IsUnit (Associates.mk x)
u : αˣ
hu : b * ↑u = a * x
⊢ False

Next tactic:
apply h (x * ↑u⁻¹)
theorem mk_dvdNotUnit_mk_iff {a b : α} :
    DvdNotUnit (Associates.mk a) (Associates.mk b) ↔ DvdNotUnit a b := by
  rw [DvdNotUnit, DvdNotUnit, mk_ne_zero]
  apply and_congr_right; intro
  constructor
  · contrapose!
    rw [forall_associated]
    intro h x hx hbax
    rw [mk_mul_mk, mk_eq_mk_iff_associated] at hbax
    cases' hbax with u hu
    apply h (x * ↑u⁻¹)
    · 

Proof state:
case h.mp.intro._
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CommMonoidWithZero α
a b : α
a✝ : a ≠ 0
h : ∀ (x : α), ¬IsUnit x → b ≠ a * x
x : α
hx : ¬IsUnit (Associates.mk x)
u : αˣ
hu : b * ↑u = a * x
⊢ ¬IsUnit (x * ↑u⁻¹)

Next tactic:
rw [isUnit_mk] at hx
theorem mk_dvdNotUnit_mk_iff {a b : α} :
    DvdNotUnit (Associates.mk a) (Associates.mk b) ↔ DvdNotUnit a b := by
  rw [DvdNotUnit, DvdNotUnit, mk_ne_zero]
  apply and_congr_right; intro
  constructor
  · contrapose!
    rw [forall_associated]
    intro h x hx hbax
    rw [mk_mul_mk, mk_eq_mk_iff_associated] at hbax
    cases' hbax with u hu
    apply h (x * ↑u⁻¹)
    · rw [isUnit_mk] at hx
      

Proof state:
case h.mp.intro._
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CommMonoidWithZero α
a b : α
a✝ : a ≠ 0
h : ∀ (x : α), ¬IsUnit x → b ≠ a * x
x : α
hx : ¬IsUnit x
u : αˣ
hu : b * ↑u = a * x
⊢ ¬IsUnit (x * ↑u⁻¹)

Next tactic:
rw [Associated.isUnit_iff]
theorem mk_dvdNotUnit_mk_iff {a b : α} :
    DvdNotUnit (Associates.mk a) (Associates.mk b) ↔ DvdNotUnit a b := by
  rw [DvdNotUnit, DvdNotUnit, mk_ne_zero]
  apply and_congr_right; intro
  constructor
  · contrapose!
    rw [forall_associated]
    intro h x hx hbax
    rw [mk_mul_mk, mk_eq_mk_iff_associated] at hbax
    cases' hbax with u hu
    apply h (x * ↑u⁻¹)
    · rw [isUnit_mk] at hx
      rw [Associated.isUnit_iff]
      

Proof state:
case h.mp.intro._
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CommMonoidWithZero α
a b : α
a✝ : a ≠ 0
h : ∀ (x : α), ¬IsUnit x → b ≠ a * x
x : α
hx : ¬IsUnit x
u : αˣ
hu : b * ↑u = a * x
⊢ ¬IsUnit ?m.231758
case h.mp.intro._
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CommMonoidWithZero α
a b : α
a✝ : a ≠ 0
h : ∀ (x : α), ¬IsUnit x → b ≠ a * x
x : α
hx : ¬IsUnit x
u : αˣ
hu : b * ↑u = a * x
⊢ x * ↑u⁻¹ ~ᵤ ?m.231758
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CommMonoidWithZero α
a b : α
a✝ : a ≠ 0
h : ∀ (x : α), ¬IsUnit x → b ≠ a * x
x : α
hx : ¬IsUnit x
u : αˣ
hu : b * ↑u = a * x
⊢ α

Next tactic:
apply hx
theorem mk_dvdNotUnit_mk_iff {a b : α} :
    DvdNotUnit (Associates.mk a) (Associates.mk b) ↔ DvdNotUnit a b := by
  rw [DvdNotUnit, DvdNotUnit, mk_ne_zero]
  apply and_congr_right; intro
  constructor
  · contrapose!
    rw [forall_associated]
    intro h x hx hbax
    rw [mk_mul_mk, mk_eq_mk_iff_associated] at hbax
    cases' hbax with u hu
    apply h (x * ↑u⁻¹)
    · rw [isUnit_mk] at hx
      rw [Associated.isUnit_iff]
      apply hx
      

Proof state:
case h.mp.intro._
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CommMonoidWithZero α
a b : α
a✝ : a ≠ 0
h : ∀ (x : α), ¬IsUnit x → b ≠ a * x
x : α
hx : ¬IsUnit x
u : αˣ
hu : b * ↑u = a * x
⊢ x * ↑u⁻¹ ~ᵤ x

Next tactic:
use u
theorem mk_dvdNotUnit_mk_iff {a b : α} :
    DvdNotUnit (Associates.mk a) (Associates.mk b) ↔ DvdNotUnit a b := by
  rw [DvdNotUnit, DvdNotUnit, mk_ne_zero]
  apply and_congr_right; intro
  constructor
  · contrapose!
    rw [forall_associated]
    intro h x hx hbax
    rw [mk_mul_mk, mk_eq_mk_iff_associated] at hbax
    cases' hbax with u hu
    apply h (x * ↑u⁻¹)
    · rw [isUnit_mk] at hx
      rw [Associated.isUnit_iff]
      apply hx
      use u
      

Proof state:
case h
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CommMonoidWithZero α
a b : α
a✝ : a ≠ 0
h : ∀ (x : α), ¬IsUnit x → b ≠ a * x
x : α
hx : ¬IsUnit x
u : αˣ
hu : b * ↑u = a * x
⊢ x * ↑u⁻¹ * ↑u = x

Next tactic:
simp
theorem mk_dvdNotUnit_mk_iff {a b : α} :
    DvdNotUnit (Associates.mk a) (Associates.mk b) ↔ DvdNotUnit a b := by
  rw [DvdNotUnit, DvdNotUnit, mk_ne_zero]
  apply and_congr_right; intro
  constructor
  · contrapose!
    rw [forall_associated]
    intro h x hx hbax
    rw [mk_mul_mk, mk_eq_mk_iff_associated] at hbax
    cases' hbax with u hu
    apply h (x * ↑u⁻¹)
    · rw [isUnit_mk] at hx
      rw [Associated.isUnit_iff]
      apply hx
      use u
      simp
    

Proof state:
case h.mp.intro.a
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CommMonoidWithZero α
a b : α
a✝ : a ≠ 0
h : ∀ (x : α), ¬IsUnit x → b ≠ a * x
x : α
hx : ¬IsUnit (Associates.mk x)
u : αˣ
hu : b * ↑u = a * x
⊢ b = a * (x * ↑u⁻¹)

Next tactic:
simp [← mul_assoc, ← hu]
theorem mk_dvdNotUnit_mk_iff {a b : α} :
    DvdNotUnit (Associates.mk a) (Associates.mk b) ↔ DvdNotUnit a b := by
  rw [DvdNotUnit, DvdNotUnit, mk_ne_zero]
  apply and_congr_right; intro
  constructor
  · contrapose!
    rw [forall_associated]
    intro h x hx hbax
    rw [mk_mul_mk, mk_eq_mk_iff_associated] at hbax
    cases' hbax with u hu
    apply h (x * ↑u⁻¹)
    · rw [isUnit_mk] at hx
      rw [Associated.isUnit_iff]
      apply hx
      use u
      simp
    simp [← mul_assoc, ← hu]
  · 

Proof state:
case h.mpr
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CommMonoidWithZero α
a b : α
a✝ : a ≠ 0
⊢ (∃ x, ¬IsUnit x ∧ b = a * x) → ∃ x, ¬IsUnit x ∧ Associates.mk b = Associates.mk a * x

Next tactic:
rintro ⟨x, ⟨hx, rfl⟩⟩
theorem mk_dvdNotUnit_mk_iff {a b : α} :
    DvdNotUnit (Associates.mk a) (Associates.mk b) ↔ DvdNotUnit a b := by
  rw [DvdNotUnit, DvdNotUnit, mk_ne_zero]
  apply and_congr_right; intro
  constructor
  · contrapose!
    rw [forall_associated]
    intro h x hx hbax
    rw [mk_mul_mk, mk_eq_mk_iff_associated] at hbax
    cases' hbax with u hu
    apply h (x * ↑u⁻¹)
    · rw [isUnit_mk] at hx
      rw [Associated.isUnit_iff]
      apply hx
      use u
      simp
    simp [← mul_assoc, ← hu]
  · rintro ⟨x, ⟨hx, rfl⟩⟩
    

Proof state:
case h.mpr.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CommMonoidWithZero α
a : α
a✝ : a ≠ 0
x : α
hx : ¬IsUnit x
⊢ ∃ x_1, ¬IsUnit x_1 ∧ Associates.mk (a * x) = Associates.mk a * x_1

Next tactic:
use Associates.mk x
theorem mk_dvdNotUnit_mk_iff {a b : α} :
    DvdNotUnit (Associates.mk a) (Associates.mk b) ↔ DvdNotUnit a b := by
  rw [DvdNotUnit, DvdNotUnit, mk_ne_zero]
  apply and_congr_right; intro
  constructor
  · contrapose!
    rw [forall_associated]
    intro h x hx hbax
    rw [mk_mul_mk, mk_eq_mk_iff_associated] at hbax
    cases' hbax with u hu
    apply h (x * ↑u⁻¹)
    · rw [isUnit_mk] at hx
      rw [Associated.isUnit_iff]
      apply hx
      use u
      simp
    simp [← mul_assoc, ← hu]
  · rintro ⟨x, ⟨hx, rfl⟩⟩
    use Associates.mk x
    

Proof state:
case h
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CommMonoidWithZero α
a : α
a✝ : a ≠ 0
x : α
hx : ¬IsUnit x
⊢ ¬IsUnit (Associates.mk x) ∧ Associates.mk (a * x) = Associates.mk a * Associates.mk x

Next tactic:
simp [isUnit_mk, mk_mul_mk, hx]
theorem dvdNotUnit_of_lt {a b : Associates α} (hlt : a < b) : DvdNotUnit a b := by
  

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CommMonoidWithZero α
a b : Associates α
hlt : a < b
⊢ DvdNotUnit a b

Next tactic:
constructor
theorem dvdNotUnit_of_lt {a b : Associates α} (hlt : a < b) : DvdNotUnit a b := by
  constructor;
  · 

Proof state:
case left
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CommMonoidWithZero α
a b : Associates α
hlt : a < b
⊢ a ≠ 0

Next tactic:
rintro rfl
theorem dvdNotUnit_of_lt {a b : Associates α} (hlt : a < b) : DvdNotUnit a b := by
  constructor;
  · rintro rfl
    

Proof state:
case left
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CommMonoidWithZero α
b : Associates α
hlt : 0 < b
⊢ False

Next tactic:
apply not_lt_of_le _ hlt
theorem dvdNotUnit_of_lt {a b : Associates α} (hlt : a < b) : DvdNotUnit a b := by
  constructor;
  · rintro rfl
    apply not_lt_of_le _ hlt
    

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CommMonoidWithZero α
b : Associates α
hlt : 0 < b
⊢ b ≤ 0

Next tactic:
apply dvd_zero
theorem dvdNotUnit_of_lt {a b : Associates α} (hlt : a < b) : DvdNotUnit a b := by
  constructor;
  · rintro rfl
    apply not_lt_of_le _ hlt
    apply dvd_zero
  

Proof state:
case right
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CommMonoidWithZero α
a b : Associates α
hlt : a < b
⊢ ∃ x, ¬IsUnit x ∧ b = a * x

Next tactic:
rcases hlt with ⟨⟨x, rfl⟩, ndvd⟩
theorem dvdNotUnit_of_lt {a b : Associates α} (hlt : a < b) : DvdNotUnit a b := by
  constructor;
  · rintro rfl
    apply not_lt_of_le _ hlt
    apply dvd_zero
  rcases hlt with ⟨⟨x, rfl⟩, ndvd⟩
  

Proof state:
case right.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CommMonoidWithZero α
a : Associates α
x : Associates α
ndvd : ¬a * x ∣ a
⊢ ∃ x_1, ¬IsUnit x_1 ∧ a * x = a * x_1

Next tactic:
refine' ⟨x, _, rfl⟩
theorem dvdNotUnit_of_lt {a b : Associates α} (hlt : a < b) : DvdNotUnit a b := by
  constructor;
  · rintro rfl
    apply not_lt_of_le _ hlt
    apply dvd_zero
  rcases hlt with ⟨⟨x, rfl⟩, ndvd⟩
  refine' ⟨x, _, rfl⟩
  

Proof state:
case right.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CommMonoidWithZero α
a : Associates α
x : Associates α
ndvd : ¬a * x ∣ a
⊢ ¬IsUnit x

Next tactic:
contrapose! ndvd
theorem dvdNotUnit_of_lt {a b : Associates α} (hlt : a < b) : DvdNotUnit a b := by
  constructor;
  · rintro rfl
    apply not_lt_of_le _ hlt
    apply dvd_zero
  rcases hlt with ⟨⟨x, rfl⟩, ndvd⟩
  refine' ⟨x, _, rfl⟩
  contrapose! ndvd
  

Proof state:
case right.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CommMonoidWithZero α
a : Associates α
x : Associates α
ndvd : IsUnit x
⊢ a * x ∣ a

Next tactic:
rcases ndvd with ⟨u, rfl⟩
theorem dvdNotUnit_of_lt {a b : Associates α} (hlt : a < b) : DvdNotUnit a b := by
  constructor;
  · rintro rfl
    apply not_lt_of_le _ hlt
    apply dvd_zero
  rcases hlt with ⟨⟨x, rfl⟩, ndvd⟩
  refine' ⟨x, _, rfl⟩
  contrapose! ndvd
  rcases ndvd with ⟨u, rfl⟩
  

Proof state:
case right.intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CommMonoidWithZero α
a : Associates α
u : (Associates α)ˣ
⊢ a * ↑u ∣ a

Next tactic:
simp
theorem irreducible_iff_prime_iff :
    (∀ a : α, Irreducible a ↔ Prime a) ↔ ∀ a : Associates α, Irreducible a ↔ Prime a := by
  

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CommMonoidWithZero α
⊢ (∀ (a : α), Irreducible a ↔ Prime a) ↔ ∀ (a : Associates α), Irreducible a ↔ Prime a

Next tactic:
simp_rw [forall_associated, irreducible_mk, prime_mk]
instance instCancelCommMonoidWithZero : CancelCommMonoidWithZero (Associates α) :=
{ (by 

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
⊢ CommMonoidWithZero (Associates α)

Next tactic:
infer_instance
instance instCancelCommMonoidWithZero : CancelCommMonoidWithZero (Associates α) :=
{ (by infer_instance : CommMonoidWithZero (Associates α)) with
  mul_left_cancel_of_ne_zero := by
    

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
src✝ : CommMonoidWithZero (Associates α) := inferInstance
⊢ ∀ {a b c : Associates α}, a ≠ 0 → a * b = a * c → b = c

Next tactic:
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩ ha h
instance instCancelCommMonoidWithZero : CancelCommMonoidWithZero (Associates α) :=
{ (by infer_instance : CommMonoidWithZero (Associates α)) with
  mul_left_cancel_of_ne_zero := by
    rintro ⟨a⟩ ⟨b⟩ ⟨c⟩ ha h
    

Proof state:
case mk.mk.mk
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
src✝ : CommMonoidWithZero (Associates α) := inferInstance
a✝ : Associates α
a : α
b✝ : Associates α
b : α
c✝ : Associates α
c : α
ha : Quot.mk Setoid.r a ≠ 0
h : Quot.mk Setoid.r a * Quot.mk Setoid.r b = Quot.mk Setoid.r a * Quot.mk Setoid.r c
⊢ Quot.mk Setoid.r b = Quot.mk Setoid.r c

Next tactic:
rcases Quotient.exact' h with ⟨u, hu⟩
instance instCancelCommMonoidWithZero : CancelCommMonoidWithZero (Associates α) :=
{ (by infer_instance : CommMonoidWithZero (Associates α)) with
  mul_left_cancel_of_ne_zero := by
    rintro ⟨a⟩ ⟨b⟩ ⟨c⟩ ha h
    rcases Quotient.exact' h with ⟨u, hu⟩
    

Proof state:
case mk.mk.mk.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
src✝ : CommMonoidWithZero (Associates α) := inferInstance
a✝ : Associates α
a : α
b✝ : Associates α
b : α
c✝ : Associates α
c : α
ha : Quot.mk Setoid.r a ≠ 0
h : Quot.mk Setoid.r a * Quot.mk Setoid.r b = Quot.mk Setoid.r a * Quot.mk Setoid.r c
u : αˣ
hu : a * b * ↑u = a * c
⊢ Quot.mk Setoid.r b = Quot.mk Setoid.r c

Next tactic:
have hu : a * (b * ↑u) = a * c := by rwa [← mul_assoc]
instance instCancelCommMonoidWithZero : CancelCommMonoidWithZero (Associates α) :=
{ (by infer_instance : CommMonoidWithZero (Associates α)) with
  mul_left_cancel_of_ne_zero := by
    rintro ⟨a⟩ ⟨b⟩ ⟨c⟩ ha h
    rcases Quotient.exact' h with ⟨u, hu⟩
    have hu : a * (b * ↑u) = a * c := by 

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
src✝ : CommMonoidWithZero (Associates α) := inferInstance
a✝ : Associates α
a : α
b✝ : Associates α
b : α
c✝ : Associates α
c : α
ha : Quot.mk Setoid.r a ≠ 0
h : Quot.mk Setoid.r a * Quot.mk Setoid.r b = Quot.mk Setoid.r a * Quot.mk Setoid.r c
u : αˣ
hu : a * b * ↑u = a * c
⊢ a * (b * ↑u) = a * c

Next tactic:
rwa [← mul_assoc]
instance instCancelCommMonoidWithZero : CancelCommMonoidWithZero (Associates α) :=
{ (by infer_instance : CommMonoidWithZero (Associates α)) with
  mul_left_cancel_of_ne_zero := by
    rintro ⟨a⟩ ⟨b⟩ ⟨c⟩ ha h
    rcases Quotient.exact' h with ⟨u, hu⟩
    have hu : a * (b * ↑u) = a * c := by rwa [← mul_assoc]
    

Proof state:
case mk.mk.mk.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
src✝ : CommMonoidWithZero (Associates α) := inferInstance
a✝ : Associates α
a : α
b✝ : Associates α
b : α
c✝ : Associates α
c : α
ha : Quot.mk Setoid.r a ≠ 0
h : Quot.mk Setoid.r a * Quot.mk Setoid.r b = Quot.mk Setoid.r a * Quot.mk Setoid.r c
u : αˣ
hu✝ : a * b * ↑u = a * c
hu : a * (b * ↑u) = a * c
⊢ Quot.mk Setoid.r b = Quot.mk Setoid.r c

Next tactic:
exact Quotient.sound' ⟨u, mul_left_cancel₀ (mk_ne_zero.1 ha) hu⟩
instance : NoZeroDivisors (Associates α) := by 

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
⊢ NoZeroDivisors (Associates α)

Next tactic:
infer_instance
theorem le_of_mul_le_mul_left (a b c : Associates α) (ha : a ≠ 0) : a * b ≤ a * c → b ≤ c
  | ⟨d, hd⟩ => ⟨d, mul_left_cancel₀ ha <| by 

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
a b c : Associates α
ha : a ≠ 0
d : Associates α
hd : a * c = a * b * d
⊢ a * c = a * (b * d)

Next tactic:
rwa [← mul_assoc]
theorem one_or_eq_of_le_of_prime : ∀ p m : Associates α, Prime p → m ≤ p → m = 1 ∨ m = p
  | p, m, ⟨hp0, _, h⟩, ⟨d, r⟩ => by
    

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
p m : Associates α
hp0 : p ≠ 0
left✝ : ¬IsUnit p
h : ∀ (a b : Associates α), p ∣ a * b → p ∣ a ∨ p ∣ b
d : Associates α
r : p = m * d
⊢ m = 1 ∨ m = p

Next tactic:
have dvd_rfl' : p ∣ m * d := by rw[r]
theorem one_or_eq_of_le_of_prime : ∀ p m : Associates α, Prime p → m ≤ p → m = 1 ∨ m = p
  | p, m, ⟨hp0, _, h⟩, ⟨d, r⟩ => by
    have dvd_rfl' : p ∣ m * d := by 

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
p m : Associates α
hp0 : p ≠ 0
left✝ : ¬IsUnit p
h : ∀ (a b : Associates α), p ∣ a * b → p ∣ a ∨ p ∣ b
d : Associates α
r : p = m * d
⊢ p ∣ m * d

Next tactic:
rw[r]
theorem one_or_eq_of_le_of_prime : ∀ p m : Associates α, Prime p → m ≤ p → m = 1 ∨ m = p
  | p, m, ⟨hp0, _, h⟩, ⟨d, r⟩ => by
    have dvd_rfl' : p ∣ m * d := by rw[r]
    

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
p m : Associates α
hp0 : p ≠ 0
left✝ : ¬IsUnit p
h : ∀ (a b : Associates α), p ∣ a * b → p ∣ a ∨ p ∣ b
d : Associates α
r : p = m * d
dvd_rfl' : p ∣ m * d
⊢ m = 1 ∨ m = p

Next tactic:
rw [r]
theorem one_or_eq_of_le_of_prime : ∀ p m : Associates α, Prime p → m ≤ p → m = 1 ∨ m = p
  | p, m, ⟨hp0, _, h⟩, ⟨d, r⟩ => by
    have dvd_rfl' : p ∣ m * d := by rw[r]
    rw [r]
    

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
p m : Associates α
hp0 : p ≠ 0
left✝ : ¬IsUnit p
h : ∀ (a b : Associates α), p ∣ a * b → p ∣ a ∨ p ∣ b
d : Associates α
r : p = m * d
dvd_rfl' : p ∣ m * d
⊢ m = 1 ∨ m = m * d

Next tactic:
match h m d dvd_rfl' with
    | Or.inl h' =>
      by_cases h : m = 0
      case pos =>
        simp [h, zero_mul]
      case neg =>
        rw [r] at h'
        have : m * d ≤ m * 1 := by simpa using h'
        have : d ≤ 1 := Associates.le_of_mul_le_mul_left m d 1 ‹m ≠ 0› this
        have : d = 1 := bot_unique this
        simp [this]
    | Or.inr h' =>
        by_cases h : d = 0
        case pos =>
          rw [r] at hp0
          have : m * d = 0 := by rw [h]; simp
          contradiction
        case neg =>
          rw [r] at h'
          have : d * m ≤ d * 1 := by simpa [mul_comm] using h'
          exact Or.inl <| bot_unique <| Associates.le_of_mul_le_mul_left d m 1 ‹d ≠ 0› this
theorem one_or_eq_of_le_of_prime : ∀ p m : Associates α, Prime p → m ≤ p → m = 1 ∨ m = p
  | p, m, ⟨hp0, _, h⟩, ⟨d, r⟩ => by
    have dvd_rfl' : p ∣ m * d := by rw[r]
    rw [r]
    match h m d dvd_rfl' with
    | Or.inl h' =>
      

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
p m : Associates α
hp0 : p ≠ 0
left✝ : ¬IsUnit p
h : ∀ (a b : Associates α), p ∣ a * b → p ∣ a ∨ p ∣ b
d : Associates α
r : p = m * d
dvd_rfl' : p ∣ m * d
h' : p ∣ m
⊢ m = 1 ∨ m = m * d

Next tactic:
by_cases h : m = 0
theorem one_or_eq_of_le_of_prime : ∀ p m : Associates α, Prime p → m ≤ p → m = 1 ∨ m = p
  | p, m, ⟨hp0, _, h⟩, ⟨d, r⟩ => by
    have dvd_rfl' : p ∣ m * d := by rw[r]
    rw [r]
    match h m d dvd_rfl' with
    | Or.inl h' =>
      by_cases h : m = 0
      

Proof state:
case pos
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
p m : Associates α
hp0 : p ≠ 0
left✝ : ¬IsUnit p
h✝ : ∀ (a b : Associates α), p ∣ a * b → p ∣ a ∨ p ∣ b
d : Associates α
r : p = m * d
dvd_rfl' : p ∣ m * d
h' : p ∣ m
h : m = 0
⊢ m = 1 ∨ m = m * d
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
p m : Associates α
hp0 : p ≠ 0
left✝ : ¬IsUnit p
h✝ : ∀ (a b : Associates α), p ∣ a * b → p ∣ a ∨ p ∣ b
d : Associates α
r : p = m * d
dvd_rfl' : p ∣ m * d
h' : p ∣ m
h : ¬m = 0
⊢ m = 1 ∨ m = m * d

Next tactic:
case pos =>
        simp [h, zero_mul]
theorem one_or_eq_of_le_of_prime : ∀ p m : Associates α, Prime p → m ≤ p → m = 1 ∨ m = p
  | p, m, ⟨hp0, _, h⟩, ⟨d, r⟩ => by
    have dvd_rfl' : p ∣ m * d := by rw[r]
    rw [r]
    match h m d dvd_rfl' with
    | Or.inl h' =>
      by_cases h : m = 0
      

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
p m : Associates α
hp0 : p ≠ 0
left✝ : ¬IsUnit p
h✝ : ∀ (a b : Associates α), p ∣ a * b → p ∣ a ∨ p ∣ b
d : Associates α
r : p = m * d
dvd_rfl' : p ∣ m * d
h' : p ∣ m
h : m = 0
⊢ m = 1 ∨ m = m * d

Next tactic:
case pos =>
        simp [h, zero_mul]
theorem one_or_eq_of_le_of_prime : ∀ p m : Associates α, Prime p → m ≤ p → m = 1 ∨ m = p
  | p, m, ⟨hp0, _, h⟩, ⟨d, r⟩ => by
    have dvd_rfl' : p ∣ m * d := by rw[r]
    rw [r]
    match h m d dvd_rfl' with
    | Or.inl h' =>
      by_cases h : m = 0
      case pos =>
        

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
p m : Associates α
hp0 : p ≠ 0
left✝ : ¬IsUnit p
h✝ : ∀ (a b : Associates α), p ∣ a * b → p ∣ a ∨ p ∣ b
d : Associates α
r : p = m * d
dvd_rfl' : p ∣ m * d
h' : p ∣ m
h : m = 0
⊢ m = 1 ∨ m = m * d

Next tactic:
simp [h, zero_mul]
theorem one_or_eq_of_le_of_prime : ∀ p m : Associates α, Prime p → m ≤ p → m = 1 ∨ m = p
  | p, m, ⟨hp0, _, h⟩, ⟨d, r⟩ => by
    have dvd_rfl' : p ∣ m * d := by rw[r]
    rw [r]
    match h m d dvd_rfl' with
    | Or.inl h' =>
      by_cases h : m = 0
      case pos =>
        simp [h, zero_mul]
      

Proof state:
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : CancelCommMonoidWithZero α
p m : Associates α
hp0 : p ≠ 0
left✝ : ¬IsUnit p
h✝ : ∀ (a b : Associates α), p ∣ a * b → p ∣ a ∨ p ∣ b
d : Associates α
r : p = m * d
dvd_rfl' : p ∣ m * d
h' : p ∣ m
h : ¬m = 0
⊢ m = 1 ∨ m = m * d

Next tactic:
case neg =>
        rw [r] at h'
        have : m * d ≤ m * 1 := by simpa using h'
        have : d ≤ 1 := Associates.le_of_mul_le_mul_left m d 1 ‹m ≠ 0› this
        have : d = 1 := bot_unique this
        simp [this]