-- Mathlib/Algebra/GroupWithZero/Associated.lean
theorem associated_of_dvd_dvd [CancelMonoidWithZero M] {a b : M} (hab : a ∣ b) (hba : b ∣ a) :
    a ~ᵤ b

Proof state:
case intro
M : Type u_1
inst✝ : CancelMonoidWithZero M
a c : M
hba : a * c ∣ a
⊢ a ~ᵤ a * c

Tactic:
rcases hba with ⟨d, a_eq⟩

Result:
case intro.intro
M : Type u_1
inst✝ : CancelMonoidWithZero M
a c d : M
a_eq : a = a * c * d
⊢ a ~ᵤ a * c

Proof state:
M : Type u_1
inst✝ : CancelMonoidWithZero M
a c d : M
a_eq : a = 0
ha0 : ¬a = 0
con : a * c = 0
⊢ False

Tactic:
apply ha0 a_eq

Result:
no goals

Proof state:
case neg
M : Type u_1
inst✝ : CancelMonoidWithZero M
a c d : M
a_eq : a = a * c * d
ha0 : ¬a = 0
hac0 : a * c ≠ 0
this✝ : a * (c * d) = a * 1
hcd : c * d = 1
this : a * c * (d * c) = a * c * 1
⊢ a ~ᵤ a * c

Tactic:
have hdc : d * c = 1 := mul_left_cancel₀ hac0 this

Result:
case neg
M : Type u_1
inst✝ : CancelMonoidWithZero M
a c d : M
a_eq : a = a * c * d
ha0 : ¬a = 0
hac0 : a * c ≠ 0
this✝ : a * (c * d) = a * 1
hcd : c * d = 1
this : a * c * (d * c) = a * c * 1
hdc : d * c = 1
⊢ a ~ᵤ a * c

Proof state:
case neg
M : Type u_1
inst✝ : CancelMonoidWithZero M
a c d : M
a_eq : a = a * c * d
ha0 : ¬a = 0
hac0 : a * c ≠ 0
this✝ : a * (c * d) = a * 1
hcd : c * d = 1
this : a * c * (d * c) = a * c * 1
hdc : d * c = 1
⊢ a ~ᵤ a * c

Tactic:
exact ⟨⟨c, d, hcd, hdc⟩, rfl⟩

Result:
no goals

Proof state:
M : Type u_1
inst✝ : CancelMonoidWithZero M
a c d : M
a_eq : a = a * c * d
ha0 : ¬a = 0
con : a * c = 0
⊢ False

Tactic:
rw [con, zero_mul] at a_eq

Result:
M : Type u_1
inst✝ : CancelMonoidWithZero M
a c d : M
a_eq : a = 0
ha0 : ¬a = 0
con : a * c = 0
⊢ False

Proof state:
case pos
M : Type u_1
inst✝ : CancelMonoidWithZero M
a c d : M
a_eq : a = a * c * d
ha0 : a = 0
⊢ a ~ᵤ a * c

Tactic:
simp_all

Result:
no goals

Proof state:
M : Type u_1
inst✝ : CancelMonoidWithZero M
a c d : M
a_eq : a = a * c * d
ha0 : ¬a = 0
hac0 : a * c ≠ 0
this : a * (c * d) = a * 1
hcd : c * d = 1
⊢ a * c * (d * c) = a * c * 1

Tactic:
rw [← mul_assoc, ← a_eq, mul_one]

Result:
no goals

Proof state:
case neg
M : Type u_1
inst✝ : CancelMonoidWithZero M
a c d : M
a_eq : a = a * c * d
ha0 : ¬a = 0
hac0 : a * c ≠ 0
this : a * (c * d) = a * 1
⊢ a ~ᵤ a * c

Tactic:
have hcd : c * d = 1 := mul_left_cancel₀ ha0 this

Result:
case neg
M : Type u_1
inst✝ : CancelMonoidWithZero M
a c d : M
a_eq : a = a * c * d
ha0 : ¬a = 0
hac0 : a * c ≠ 0
this : a * (c * d) = a * 1
hcd : c * d = 1
⊢ a ~ᵤ a * c

Proof state:
case neg
M : Type u_1
inst✝ : CancelMonoidWithZero M
a c d : M
a_eq : a = a * c * d
ha0 : ¬a = 0
⊢ a ~ᵤ a * c

Tactic:
have hac0 : a * c ≠ 0 := by
  intro con
  rw [con, zero_mul] at a_eq
  apply ha0 a_eq

Result:
case neg
M : Type u_1
inst✝ : CancelMonoidWithZero M
a c d : M
a_eq : a = a * c * d
ha0 : ¬a = 0
hac0 : a * c ≠ 0
⊢ a ~ᵤ a * c

Proof state:
case neg
M : Type u_1
inst✝ : CancelMonoidWithZero M
a c d : M
a_eq : a = a * c * d
ha0 : ¬a = 0
hac0 : a * c ≠ 0
⊢ a ~ᵤ a * c

Tactic:
have : a * (c * d) = a * 1 := by rw [← mul_assoc, ← a_eq, mul_one]

Result:
case neg
M : Type u_1
inst✝ : CancelMonoidWithZero M
a c d : M
a_eq : a = a * c * d
ha0 : ¬a = 0
hac0 : a * c ≠ 0
this : a * (c * d) = a * 1
⊢ a ~ᵤ a * c

Proof state:
case neg
M : Type u_1
inst✝ : CancelMonoidWithZero M
a c d : M
a_eq : a = a * c * d
ha0 : ¬a = 0
hac0 : a * c ≠ 0
this : a * (c * d) = a * 1
hcd : c * d = 1
⊢ a ~ᵤ a * c

Tactic:
have : a * c * (d * c) = a * c * 1 := by rw [← mul_assoc, ← a_eq, mul_one]

Result:
case neg
M : Type u_1
inst✝ : CancelMonoidWithZero M
a c d : M
a_eq : a = a * c * d
ha0 : ¬a = 0
hac0 : a * c ≠ 0
this✝ : a * (c * d) = a * 1
hcd : c * d = 1
this : a * c * (d * c) = a * c * 1
⊢ a ~ᵤ a * c

Proof state:
M : Type u_1
inst✝ : CancelMonoidWithZero M
a c d : M
a_eq : a = a * c * d
ha0 : ¬a = 0
hac0 : a * c ≠ 0
⊢ a * (c * d) = a * 1

Tactic:
rw [← mul_assoc, ← a_eq, mul_one]

Result:
no goals

Proof state:
M : Type u_1
inst✝ : CancelMonoidWithZero M
a c d : M
a_eq : a = a * c * d
ha0 : ¬a = 0
⊢ a * c ≠ 0

Tactic:
intro con

Result:
M : Type u_1
inst✝ : CancelMonoidWithZero M
a c d : M
a_eq : a = a * c * d
ha0 : ¬a = 0
con : a * c = 0
⊢ False

Proof state:
case intro.intro
M : Type u_1
inst✝ : CancelMonoidWithZero M
a c d : M
a_eq : a = a * c * d
⊢ a ~ᵤ a * c

Tactic:
by_cases ha0 : a = 0

Result:
case pos
M : Type u_1
inst✝ : CancelMonoidWithZero M
a c d : M
a_eq : a = a * c * d
ha0 : a = 0
⊢ a ~ᵤ a * c

case neg
M : Type u_1
inst✝ : CancelMonoidWithZero M
a c d : M
a_eq : a = a * c * d
ha0 : ¬a = 0
⊢ a ~ᵤ a * c

Proof state:
M : Type u_1
inst✝ : CancelMonoidWithZero M
a b : M
hab : a ∣ b
hba : b ∣ a
⊢ a ~ᵤ b

Tactic:
rcases hab with ⟨c, rfl⟩

Result:
case intro
M : Type u_1
inst✝ : CancelMonoidWithZero M
a c : M
hba : a * c ∣ a
⊢ a ~ᵤ a * c