l3lab/ntp-mathlib huggingface dataset:
noncomputable instance linearOrderedCommGroupWithZero :
    LinearOrderedCommGroupWithZero (ValueGroup A K) :=
  { linearOrder .. with
    mul_assoc := by rintro ⟨a⟩ ⟨b⟩ ⟨c⟩; apply Quotient.sound'; rw [mul_assoc]; apply Setoid.refl'
    one_mul := by rintro ⟨a⟩; apply Quotient.sound'; rw [one_mul]; apply Setoid.refl'
    mul_one := by rintro ⟨a⟩; apply Quotient.sound'; rw [mul_one]; apply Setoid.refl'
    mul_comm := by rintro ⟨a⟩ ⟨b⟩; 

Proof state:
case mk.mk
A : Type u
inst✝⁵ : CommRing A
K : Type v
inst✝⁴ : Field K
inst✝³ : Algebra A K
inst✝² : IsDomain A
inst✝¹ : ValuationRing A
inst✝ : IsFractionRing A K
src✝ : LinearOrder (ValueGroup A K) := linearOrder A K
a✝ : ValueGroup A K
a : K
b✝ : ValueGroup A K
b : K
⊢ Quot.mk Setoid.r a * Quot.mk Setoid.r b = Quot.mk Setoid.r b * Quot.mk Setoid.r a

Next tactic:
apply Quotient.sound'
noncomputable instance linearOrderedCommGroupWithZero :
    LinearOrderedCommGroupWithZero (ValueGroup A K) :=
  { linearOrder .. with
    mul_assoc := by rintro ⟨a⟩ ⟨b⟩ ⟨c⟩; apply Quotient.sound'; rw [mul_assoc]; apply Setoid.refl'
    one_mul := by rintro ⟨a⟩; apply Quotient.sound'; rw [one_mul]; apply Setoid.refl'
    mul_one := by rintro ⟨a⟩; apply Quotient.sound'; rw [mul_one]; apply Setoid.refl'
    mul_comm := by rintro ⟨a⟩ ⟨b⟩; apply Quotient.sound'; 

Proof state:
case mk.mk.a
A : Type u
inst✝⁵ : CommRing A
K : Type v
inst✝⁴ : Field K
inst✝³ : Algebra A K
inst✝² : IsDomain A
inst✝¹ : ValuationRing A
inst✝ : IsFractionRing A K
src✝ : LinearOrder (ValueGroup A K) := linearOrder A K
a✝ : ValueGroup A K
a : K
b✝ : ValueGroup A K
b : K
⊢ Setoid.r (a * b) (b * a)

Next tactic:
rw [mul_comm]
noncomputable instance linearOrderedCommGroupWithZero :
    LinearOrderedCommGroupWithZero (ValueGroup A K) :=
  { linearOrder .. with
    mul_assoc := by rintro ⟨a⟩ ⟨b⟩ ⟨c⟩; apply Quotient.sound'; rw [mul_assoc]; apply Setoid.refl'
    one_mul := by rintro ⟨a⟩; apply Quotient.sound'; rw [one_mul]; apply Setoid.refl'
    mul_one := by rintro ⟨a⟩; apply Quotient.sound'; rw [mul_one]; apply Setoid.refl'
    mul_comm := by rintro ⟨a⟩ ⟨b⟩; apply Quotient.sound'; rw [mul_comm]; 

Proof state:
case mk.mk.a
A : Type u
inst✝⁵ : CommRing A
K : Type v
inst✝⁴ : Field K
inst✝³ : Algebra A K
inst✝² : IsDomain A
inst✝¹ : ValuationRing A
inst✝ : IsFractionRing A K
src✝ : LinearOrder (ValueGroup A K) := linearOrder A K
a✝ : ValueGroup A K
a : K
b✝ : ValueGroup A K
b : K
⊢ Setoid.r (b * a) (b * a)

Next tactic:
apply Setoid.refl'
noncomputable instance linearOrderedCommGroupWithZero :
    LinearOrderedCommGroupWithZero (ValueGroup A K) :=
  { linearOrder .. with
    mul_assoc := by rintro ⟨a⟩ ⟨b⟩ ⟨c⟩; apply Quotient.sound'; rw [mul_assoc]; apply Setoid.refl'
    one_mul := by rintro ⟨a⟩; apply Quotient.sound'; rw [one_mul]; apply Setoid.refl'
    mul_one := by rintro ⟨a⟩; apply Quotient.sound'; rw [mul_one]; apply Setoid.refl'
    mul_comm := by rintro ⟨a⟩ ⟨b⟩; apply Quotient.sound'; rw [mul_comm]; apply Setoid.refl'
    mul_le_mul_left := by
      

Proof state:
A : Type u
inst✝⁵ : CommRing A
K : Type v
inst✝⁴ : Field K
inst✝³ : Algebra A K
inst✝² : IsDomain A
inst✝¹ : ValuationRing A
inst✝ : IsFractionRing A K
src✝ : LinearOrder (ValueGroup A K) := linearOrder A K
⊢ ∀ (a b : ValueGroup A K), a ≤ b → ∀ (c : ValueGroup A K), c * a ≤ c * b

Next tactic:
rintro ⟨a⟩ ⟨b⟩ ⟨c, rfl⟩ ⟨d⟩
noncomputable instance linearOrderedCommGroupWithZero :
    LinearOrderedCommGroupWithZero (ValueGroup A K) :=
  { linearOrder .. with
    mul_assoc := by rintro ⟨a⟩ ⟨b⟩ ⟨c⟩; apply Quotient.sound'; rw [mul_assoc]; apply Setoid.refl'
    one_mul := by rintro ⟨a⟩; apply Quotient.sound'; rw [one_mul]; apply Setoid.refl'
    mul_one := by rintro ⟨a⟩; apply Quotient.sound'; rw [mul_one]; apply Setoid.refl'
    mul_comm := by rintro ⟨a⟩ ⟨b⟩; apply Quotient.sound'; rw [mul_comm]; apply Setoid.refl'
    mul_le_mul_left := by
      rintro ⟨a⟩ ⟨b⟩ ⟨c, rfl⟩ ⟨d⟩
      

Proof state:
case mk.mk.intro.mk
A : Type u
inst✝⁵ : CommRing A
K : Type v
inst✝⁴ : Field K
inst✝³ : Algebra A K
inst✝² : IsDomain A
inst✝¹ : ValuationRing A
inst✝ : IsFractionRing A K
src✝ : LinearOrder (ValueGroup A K) := linearOrder A K
a✝ b✝ : ValueGroup A K
b : K
c : A
c✝ : ValueGroup A K
d : K
⊢ Quot.mk Setoid.r d * Quot.mk Setoid.r (c • b) ≤ Quot.mk Setoid.r d * Quot.mk Setoid.r b

Next tactic:
use c
noncomputable instance linearOrderedCommGroupWithZero :
    LinearOrderedCommGroupWithZero (ValueGroup A K) :=
  { linearOrder .. with
    mul_assoc := by rintro ⟨a⟩ ⟨b⟩ ⟨c⟩; apply Quotient.sound'; rw [mul_assoc]; apply Setoid.refl'
    one_mul := by rintro ⟨a⟩; apply Quotient.sound'; rw [one_mul]; apply Setoid.refl'
    mul_one := by rintro ⟨a⟩; apply Quotient.sound'; rw [mul_one]; apply Setoid.refl'
    mul_comm := by rintro ⟨a⟩ ⟨b⟩; apply Quotient.sound'; rw [mul_comm]; apply Setoid.refl'
    mul_le_mul_left := by
      rintro ⟨a⟩ ⟨b⟩ ⟨c, rfl⟩ ⟨d⟩
      use c; 

Proof state:
case h
A : Type u
inst✝⁵ : CommRing A
K : Type v
inst✝⁴ : Field K
inst✝³ : Algebra A K
inst✝² : IsDomain A
inst✝¹ : ValuationRing A
inst✝ : IsFractionRing A K
src✝ : LinearOrder (ValueGroup A K) := linearOrder A K
a✝ b✝ : ValueGroup A K
b : K
c : A
c✝ : ValueGroup A K
d : K
⊢ c • (d * b) = d * c • b

Next tactic:
simp only [Algebra.smul_def]
noncomputable instance linearOrderedCommGroupWithZero :
    LinearOrderedCommGroupWithZero (ValueGroup A K) :=
  { linearOrder .. with
    mul_assoc := by rintro ⟨a⟩ ⟨b⟩ ⟨c⟩; apply Quotient.sound'; rw [mul_assoc]; apply Setoid.refl'
    one_mul := by rintro ⟨a⟩; apply Quotient.sound'; rw [one_mul]; apply Setoid.refl'
    mul_one := by rintro ⟨a⟩; apply Quotient.sound'; rw [mul_one]; apply Setoid.refl'
    mul_comm := by rintro ⟨a⟩ ⟨b⟩; apply Quotient.sound'; rw [mul_comm]; apply Setoid.refl'
    mul_le_mul_left := by
      rintro ⟨a⟩ ⟨b⟩ ⟨c, rfl⟩ ⟨d⟩
      use c; simp only [Algebra.smul_def]; 

Proof state:
case h
A : Type u
inst✝⁵ : CommRing A
K : Type v
inst✝⁴ : Field K
inst✝³ : Algebra A K
inst✝² : IsDomain A
inst✝¹ : ValuationRing A
inst✝ : IsFractionRing A K
src✝ : LinearOrder (ValueGroup A K) := linearOrder A K
a✝ b✝ : ValueGroup A K
b : K
c : A
c✝ : ValueGroup A K
d : K
⊢ (algebraMap A K) c * (d * b) = d * ((algebraMap A K) c * b)

Next tactic:
ring
noncomputable instance linearOrderedCommGroupWithZero :
    LinearOrderedCommGroupWithZero (ValueGroup A K) :=
  { linearOrder .. with
    mul_assoc := by rintro ⟨a⟩ ⟨b⟩ ⟨c⟩; apply Quotient.sound'; rw [mul_assoc]; apply Setoid.refl'
    one_mul := by rintro ⟨a⟩; apply Quotient.sound'; rw [one_mul]; apply Setoid.refl'
    mul_one := by rintro ⟨a⟩; apply Quotient.sound'; rw [mul_one]; apply Setoid.refl'
    mul_comm := by rintro ⟨a⟩ ⟨b⟩; apply Quotient.sound'; rw [mul_comm]; apply Setoid.refl'
    mul_le_mul_left := by
      rintro ⟨a⟩ ⟨b⟩ ⟨c, rfl⟩ ⟨d⟩
      use c; simp only [Algebra.smul_def]; ring
    zero_mul := by 

Proof state:
A : Type u
inst✝⁵ : CommRing A
K : Type v
inst✝⁴ : Field K
inst✝³ : Algebra A K
inst✝² : IsDomain A
inst✝¹ : ValuationRing A
inst✝ : IsFractionRing A K
src✝ : LinearOrder (ValueGroup A K) := linearOrder A K
⊢ ∀ (a : ValueGroup A K), 0 * a = 0

Next tactic:
rintro ⟨a⟩
noncomputable instance linearOrderedCommGroupWithZero :
    LinearOrderedCommGroupWithZero (ValueGroup A K) :=
  { linearOrder .. with
    mul_assoc := by rintro ⟨a⟩ ⟨b⟩ ⟨c⟩; apply Quotient.sound'; rw [mul_assoc]; apply Setoid.refl'
    one_mul := by rintro ⟨a⟩; apply Quotient.sound'; rw [one_mul]; apply Setoid.refl'
    mul_one := by rintro ⟨a⟩; apply Quotient.sound'; rw [mul_one]; apply Setoid.refl'
    mul_comm := by rintro ⟨a⟩ ⟨b⟩; apply Quotient.sound'; rw [mul_comm]; apply Setoid.refl'
    mul_le_mul_left := by
      rintro ⟨a⟩ ⟨b⟩ ⟨c, rfl⟩ ⟨d⟩
      use c; simp only [Algebra.smul_def]; ring
    zero_mul := by rintro ⟨a⟩; 

Proof state:
case mk
A : Type u
inst✝⁵ : CommRing A
K : Type v
inst✝⁴ : Field K
inst✝³ : Algebra A K
inst✝² : IsDomain A
inst✝¹ : ValuationRing A
inst✝ : IsFractionRing A K
src✝ : LinearOrder (ValueGroup A K) := linearOrder A K
a✝ : ValueGroup A K
a : K
⊢ 0 * Quot.mk Setoid.r a = 0

Next tactic:
apply Quotient.sound'
noncomputable instance linearOrderedCommGroupWithZero :
    LinearOrderedCommGroupWithZero (ValueGroup A K) :=
  { linearOrder .. with
    mul_assoc := by rintro ⟨a⟩ ⟨b⟩ ⟨c⟩; apply Quotient.sound'; rw [mul_assoc]; apply Setoid.refl'
    one_mul := by rintro ⟨a⟩; apply Quotient.sound'; rw [one_mul]; apply Setoid.refl'
    mul_one := by rintro ⟨a⟩; apply Quotient.sound'; rw [mul_one]; apply Setoid.refl'
    mul_comm := by rintro ⟨a⟩ ⟨b⟩; apply Quotient.sound'; rw [mul_comm]; apply Setoid.refl'
    mul_le_mul_left := by
      rintro ⟨a⟩ ⟨b⟩ ⟨c, rfl⟩ ⟨d⟩
      use c; simp only [Algebra.smul_def]; ring
    zero_mul := by rintro ⟨a⟩; apply Quotient.sound'; 

Proof state:
case mk.a
A : Type u
inst✝⁵ : CommRing A
K : Type v
inst✝⁴ : Field K
inst✝³ : Algebra A K
inst✝² : IsDomain A
inst✝¹ : ValuationRing A
inst✝ : IsFractionRing A K
src✝ : LinearOrder (ValueGroup A K) := linearOrder A K
a✝ : ValueGroup A K
a : K
⊢ Setoid.r (0 * a) 0

Next tactic:
rw [zero_mul]
noncomputable instance linearOrderedCommGroupWithZero :
    LinearOrderedCommGroupWithZero (ValueGroup A K) :=
  { linearOrder .. with
    mul_assoc := by rintro ⟨a⟩ ⟨b⟩ ⟨c⟩; apply Quotient.sound'; rw [mul_assoc]; apply Setoid.refl'
    one_mul := by rintro ⟨a⟩; apply Quotient.sound'; rw [one_mul]; apply Setoid.refl'
    mul_one := by rintro ⟨a⟩; apply Quotient.sound'; rw [mul_one]; apply Setoid.refl'
    mul_comm := by rintro ⟨a⟩ ⟨b⟩; apply Quotient.sound'; rw [mul_comm]; apply Setoid.refl'
    mul_le_mul_left := by
      rintro ⟨a⟩ ⟨b⟩ ⟨c, rfl⟩ ⟨d⟩
      use c; simp only [Algebra.smul_def]; ring
    zero_mul := by rintro ⟨a⟩; apply Quotient.sound'; rw [zero_mul]; 

Proof state:
case mk.a
A : Type u
inst✝⁵ : CommRing A
K : Type v
inst✝⁴ : Field K
inst✝³ : Algebra A K
inst✝² : IsDomain A
inst✝¹ : ValuationRing A
inst✝ : IsFractionRing A K
src✝ : LinearOrder (ValueGroup A K) := linearOrder A K
a✝ : ValueGroup A K
a : K
⊢ Setoid.r 0 0

Next tactic:
apply Setoid.refl'
noncomputable instance linearOrderedCommGroupWithZero :
    LinearOrderedCommGroupWithZero (ValueGroup A K) :=
  { linearOrder .. with
    mul_assoc := by rintro ⟨a⟩ ⟨b⟩ ⟨c⟩; apply Quotient.sound'; rw [mul_assoc]; apply Setoid.refl'
    one_mul := by rintro ⟨a⟩; apply Quotient.sound'; rw [one_mul]; apply Setoid.refl'
    mul_one := by rintro ⟨a⟩; apply Quotient.sound'; rw [mul_one]; apply Setoid.refl'
    mul_comm := by rintro ⟨a⟩ ⟨b⟩; apply Quotient.sound'; rw [mul_comm]; apply Setoid.refl'
    mul_le_mul_left := by
      rintro ⟨a⟩ ⟨b⟩ ⟨c, rfl⟩ ⟨d⟩
      use c; simp only [Algebra.smul_def]; ring
    zero_mul := by rintro ⟨a⟩; apply Quotient.sound'; rw [zero_mul]; apply Setoid.refl'
    mul_zero := by 

Proof state:
A : Type u
inst✝⁵ : CommRing A
K : Type v
inst✝⁴ : Field K
inst✝³ : Algebra A K
inst✝² : IsDomain A
inst✝¹ : ValuationRing A
inst✝ : IsFractionRing A K
src✝ : LinearOrder (ValueGroup A K) := linearOrder A K
⊢ ∀ (a : ValueGroup A K), a * 0 = 0

Next tactic:
rintro ⟨a⟩
noncomputable instance linearOrderedCommGroupWithZero :
    LinearOrderedCommGroupWithZero (ValueGroup A K) :=
  { linearOrder .. with
    mul_assoc := by rintro ⟨a⟩ ⟨b⟩ ⟨c⟩; apply Quotient.sound'; rw [mul_assoc]; apply Setoid.refl'
    one_mul := by rintro ⟨a⟩; apply Quotient.sound'; rw [one_mul]; apply Setoid.refl'
    mul_one := by rintro ⟨a⟩; apply Quotient.sound'; rw [mul_one]; apply Setoid.refl'
    mul_comm := by rintro ⟨a⟩ ⟨b⟩; apply Quotient.sound'; rw [mul_comm]; apply Setoid.refl'
    mul_le_mul_left := by
      rintro ⟨a⟩ ⟨b⟩ ⟨c, rfl⟩ ⟨d⟩
      use c; simp only [Algebra.smul_def]; ring
    zero_mul := by rintro ⟨a⟩; apply Quotient.sound'; rw [zero_mul]; apply Setoid.refl'
    mul_zero := by rintro ⟨a⟩; 

Proof state:
case mk
A : Type u
inst✝⁵ : CommRing A
K : Type v
inst✝⁴ : Field K
inst✝³ : Algebra A K
inst✝² : IsDomain A
inst✝¹ : ValuationRing A
inst✝ : IsFractionRing A K
src✝ : LinearOrder (ValueGroup A K) := linearOrder A K
a✝ : ValueGroup A K
a : K
⊢ Quot.mk Setoid.r a * 0 = 0

Next tactic:
apply Quotient.sound'
noncomputable instance linearOrderedCommGroupWithZero :
    LinearOrderedCommGroupWithZero (ValueGroup A K) :=
  { linearOrder .. with
    mul_assoc := by rintro ⟨a⟩ ⟨b⟩ ⟨c⟩; apply Quotient.sound'; rw [mul_assoc]; apply Setoid.refl'
    one_mul := by rintro ⟨a⟩; apply Quotient.sound'; rw [one_mul]; apply Setoid.refl'
    mul_one := by rintro ⟨a⟩; apply Quotient.sound'; rw [mul_one]; apply Setoid.refl'
    mul_comm := by rintro ⟨a⟩ ⟨b⟩; apply Quotient.sound'; rw [mul_comm]; apply Setoid.refl'
    mul_le_mul_left := by
      rintro ⟨a⟩ ⟨b⟩ ⟨c, rfl⟩ ⟨d⟩
      use c; simp only [Algebra.smul_def]; ring
    zero_mul := by rintro ⟨a⟩; apply Quotient.sound'; rw [zero_mul]; apply Setoid.refl'
    mul_zero := by rintro ⟨a⟩; apply Quotient.sound'; 

Proof state:
case mk.a
A : Type u
inst✝⁵ : CommRing A
K : Type v
inst✝⁴ : Field K
inst✝³ : Algebra A K
inst✝² : IsDomain A
inst✝¹ : ValuationRing A
inst✝ : IsFractionRing A K
src✝ : LinearOrder (ValueGroup A K) := linearOrder A K
a✝ : ValueGroup A K
a : K
⊢ Setoid.r (a * 0) 0

Next tactic:
rw [mul_zero]
noncomputable instance linearOrderedCommGroupWithZero :
    LinearOrderedCommGroupWithZero (ValueGroup A K) :=
  { linearOrder .. with
    mul_assoc := by rintro ⟨a⟩ ⟨b⟩ ⟨c⟩; apply Quotient.sound'; rw [mul_assoc]; apply Setoid.refl'
    one_mul := by rintro ⟨a⟩; apply Quotient.sound'; rw [one_mul]; apply Setoid.refl'
    mul_one := by rintro ⟨a⟩; apply Quotient.sound'; rw [mul_one]; apply Setoid.refl'
    mul_comm := by rintro ⟨a⟩ ⟨b⟩; apply Quotient.sound'; rw [mul_comm]; apply Setoid.refl'
    mul_le_mul_left := by
      rintro ⟨a⟩ ⟨b⟩ ⟨c, rfl⟩ ⟨d⟩
      use c; simp only [Algebra.smul_def]; ring
    zero_mul := by rintro ⟨a⟩; apply Quotient.sound'; rw [zero_mul]; apply Setoid.refl'
    mul_zero := by rintro ⟨a⟩; apply Quotient.sound'; rw [mul_zero]; 

Proof state:
case mk.a
A : Type u
inst✝⁵ : CommRing A
K : Type v
inst✝⁴ : Field K
inst✝³ : Algebra A K
inst✝² : IsDomain A
inst✝¹ : ValuationRing A
inst✝ : IsFractionRing A K
src✝ : LinearOrder (ValueGroup A K) := linearOrder A K
a✝ : ValueGroup A K
a : K
⊢ Setoid.r 0 0

Next tactic:
apply Setoid.refl'
noncomputable instance linearOrderedCommGroupWithZero :
    LinearOrderedCommGroupWithZero (ValueGroup A K) :=
  { linearOrder .. with
    mul_assoc := by rintro ⟨a⟩ ⟨b⟩ ⟨c⟩; apply Quotient.sound'; rw [mul_assoc]; apply Setoid.refl'
    one_mul := by rintro ⟨a⟩; apply Quotient.sound'; rw [one_mul]; apply Setoid.refl'
    mul_one := by rintro ⟨a⟩; apply Quotient.sound'; rw [mul_one]; apply Setoid.refl'
    mul_comm := by rintro ⟨a⟩ ⟨b⟩; apply Quotient.sound'; rw [mul_comm]; apply Setoid.refl'
    mul_le_mul_left := by
      rintro ⟨a⟩ ⟨b⟩ ⟨c, rfl⟩ ⟨d⟩
      use c; simp only [Algebra.smul_def]; ring
    zero_mul := by rintro ⟨a⟩; apply Quotient.sound'; rw [zero_mul]; apply Setoid.refl'
    mul_zero := by rintro ⟨a⟩; apply Quotient.sound'; rw [mul_zero]; apply Setoid.refl'
    zero_le_one := ⟨0, by 

Proof state:
A : Type u
inst✝⁵ : CommRing A
K : Type v
inst✝⁴ : Field K
inst✝³ : Algebra A K
inst✝² : IsDomain A
inst✝¹ : ValuationRing A
inst✝ : IsFractionRing A K
src✝ : LinearOrder (ValueGroup A K) := linearOrder A K
⊢ 0 • 1 = 0

Next tactic:
rw [zero_smul]
noncomputable instance linearOrderedCommGroupWithZero :
    LinearOrderedCommGroupWithZero (ValueGroup A K) :=
  { linearOrder .. with
    mul_assoc := by rintro ⟨a⟩ ⟨b⟩ ⟨c⟩; apply Quotient.sound'; rw [mul_assoc]; apply Setoid.refl'
    one_mul := by rintro ⟨a⟩; apply Quotient.sound'; rw [one_mul]; apply Setoid.refl'
    mul_one := by rintro ⟨a⟩; apply Quotient.sound'; rw [mul_one]; apply Setoid.refl'
    mul_comm := by rintro ⟨a⟩ ⟨b⟩; apply Quotient.sound'; rw [mul_comm]; apply Setoid.refl'
    mul_le_mul_left := by
      rintro ⟨a⟩ ⟨b⟩ ⟨c, rfl⟩ ⟨d⟩
      use c; simp only [Algebra.smul_def]; ring
    zero_mul := by rintro ⟨a⟩; apply Quotient.sound'; rw [zero_mul]; apply Setoid.refl'
    mul_zero := by rintro ⟨a⟩; apply Quotient.sound'; rw [mul_zero]; apply Setoid.refl'
    zero_le_one := ⟨0, by rw [zero_smul]⟩
    exists_pair_ne := by
      

Proof state:
A : Type u
inst✝⁵ : CommRing A
K : Type v
inst✝⁴ : Field K
inst✝³ : Algebra A K
inst✝² : IsDomain A
inst✝¹ : ValuationRing A
inst✝ : IsFractionRing A K
src✝ : LinearOrder (ValueGroup A K) := linearOrder A K
⊢ ∃ x y, x ≠ y

Next tactic:
use 0, 1
noncomputable instance linearOrderedCommGroupWithZero :
    LinearOrderedCommGroupWithZero (ValueGroup A K) :=
  { linearOrder .. with
    mul_assoc := by rintro ⟨a⟩ ⟨b⟩ ⟨c⟩; apply Quotient.sound'; rw [mul_assoc]; apply Setoid.refl'
    one_mul := by rintro ⟨a⟩; apply Quotient.sound'; rw [one_mul]; apply Setoid.refl'
    mul_one := by rintro ⟨a⟩; apply Quotient.sound'; rw [mul_one]; apply Setoid.refl'
    mul_comm := by rintro ⟨a⟩ ⟨b⟩; apply Quotient.sound'; rw [mul_comm]; apply Setoid.refl'
    mul_le_mul_left := by
      rintro ⟨a⟩ ⟨b⟩ ⟨c, rfl⟩ ⟨d⟩
      use c; simp only [Algebra.smul_def]; ring
    zero_mul := by rintro ⟨a⟩; apply Quotient.sound'; rw [zero_mul]; apply Setoid.refl'
    mul_zero := by rintro ⟨a⟩; apply Quotient.sound'; rw [mul_zero]; apply Setoid.refl'
    zero_le_one := ⟨0, by rw [zero_smul]⟩
    exists_pair_ne := by
      use 0, 1
      

Proof state:
case h
A : Type u
inst✝⁵ : CommRing A
K : Type v
inst✝⁴ : Field K
inst✝³ : Algebra A K
inst✝² : IsDomain A
inst✝¹ : ValuationRing A
inst✝ : IsFractionRing A K
src✝ : LinearOrder (ValueGroup A K) := linearOrder A K
⊢ 0 ≠ 1

Next tactic:
intro c
noncomputable instance linearOrderedCommGroupWithZero :
    LinearOrderedCommGroupWithZero (ValueGroup A K) :=
  { linearOrder .. with
    mul_assoc := by rintro ⟨a⟩ ⟨b⟩ ⟨c⟩; apply Quotient.sound'; rw [mul_assoc]; apply Setoid.refl'
    one_mul := by rintro ⟨a⟩; apply Quotient.sound'; rw [one_mul]; apply Setoid.refl'
    mul_one := by rintro ⟨a⟩; apply Quotient.sound'; rw [mul_one]; apply Setoid.refl'
    mul_comm := by rintro ⟨a⟩ ⟨b⟩; apply Quotient.sound'; rw [mul_comm]; apply Setoid.refl'
    mul_le_mul_left := by
      rintro ⟨a⟩ ⟨b⟩ ⟨c, rfl⟩ ⟨d⟩
      use c; simp only [Algebra.smul_def]; ring
    zero_mul := by rintro ⟨a⟩; apply Quotient.sound'; rw [zero_mul]; apply Setoid.refl'
    mul_zero := by rintro ⟨a⟩; apply Quotient.sound'; rw [mul_zero]; apply Setoid.refl'
    zero_le_one := ⟨0, by rw [zero_smul]⟩
    exists_pair_ne := by
      use 0, 1
      intro c; 

Proof state:
case h
A : Type u
inst✝⁵ : CommRing A
K : Type v
inst✝⁴ : Field K
inst✝³ : Algebra A K
inst✝² : IsDomain A
inst✝¹ : ValuationRing A
inst✝ : IsFractionRing A K
src✝ : LinearOrder (ValueGroup A K) := linearOrder A K
c : 0 = 1
⊢ False

Next tactic:
obtain ⟨d, hd⟩ := Quotient.exact' c
noncomputable instance linearOrderedCommGroupWithZero :
    LinearOrderedCommGroupWithZero (ValueGroup A K) :=
  { linearOrder .. with
    mul_assoc := by rintro ⟨a⟩ ⟨b⟩ ⟨c⟩; apply Quotient.sound'; rw [mul_assoc]; apply Setoid.refl'
    one_mul := by rintro ⟨a⟩; apply Quotient.sound'; rw [one_mul]; apply Setoid.refl'
    mul_one := by rintro ⟨a⟩; apply Quotient.sound'; rw [mul_one]; apply Setoid.refl'
    mul_comm := by rintro ⟨a⟩ ⟨b⟩; apply Quotient.sound'; rw [mul_comm]; apply Setoid.refl'
    mul_le_mul_left := by
      rintro ⟨a⟩ ⟨b⟩ ⟨c, rfl⟩ ⟨d⟩
      use c; simp only [Algebra.smul_def]; ring
    zero_mul := by rintro ⟨a⟩; apply Quotient.sound'; rw [zero_mul]; apply Setoid.refl'
    mul_zero := by rintro ⟨a⟩; apply Quotient.sound'; rw [mul_zero]; apply Setoid.refl'
    zero_le_one := ⟨0, by rw [zero_smul]⟩
    exists_pair_ne := by
      use 0, 1
      intro c; obtain ⟨d, hd⟩ := Quotient.exact' c
      

Proof state:
case h.intro
A : Type u
inst✝⁵ : CommRing A
K : Type v
inst✝⁴ : Field K
inst✝³ : Algebra A K
inst✝² : IsDomain A
inst✝¹ : ValuationRing A
inst✝ : IsFractionRing A K
src✝ : LinearOrder (ValueGroup A K) := linearOrder A K
c : 0 = 1
d : Aˣ
hd : (fun m => m • 1) d = 0
⊢ False

Next tactic:
apply_fun fun t => d⁻¹ • t at hd
noncomputable instance linearOrderedCommGroupWithZero :
    LinearOrderedCommGroupWithZero (ValueGroup A K) :=
  { linearOrder .. with
    mul_assoc := by rintro ⟨a⟩ ⟨b⟩ ⟨c⟩; apply Quotient.sound'; rw [mul_assoc]; apply Setoid.refl'
    one_mul := by rintro ⟨a⟩; apply Quotient.sound'; rw [one_mul]; apply Setoid.refl'
    mul_one := by rintro ⟨a⟩; apply Quotient.sound'; rw [mul_one]; apply Setoid.refl'
    mul_comm := by rintro ⟨a⟩ ⟨b⟩; apply Quotient.sound'; rw [mul_comm]; apply Setoid.refl'
    mul_le_mul_left := by
      rintro ⟨a⟩ ⟨b⟩ ⟨c, rfl⟩ ⟨d⟩
      use c; simp only [Algebra.smul_def]; ring
    zero_mul := by rintro ⟨a⟩; apply Quotient.sound'; rw [zero_mul]; apply Setoid.refl'
    mul_zero := by rintro ⟨a⟩; apply Quotient.sound'; rw [mul_zero]; apply Setoid.refl'
    zero_le_one := ⟨0, by rw [zero_smul]⟩
    exists_pair_ne := by
      use 0, 1
      intro c; obtain ⟨d, hd⟩ := Quotient.exact' c
      apply_fun fun t => d⁻¹ • t at hd
      

Proof state:
case h.intro
A : Type u
inst✝⁵ : CommRing A
K : Type v
inst✝⁴ : Field K
inst✝³ : Algebra A K
inst✝² : IsDomain A
inst✝¹ : ValuationRing A
inst✝ : IsFractionRing A K
src✝ : LinearOrder (ValueGroup A K) := linearOrder A K
c : 0 = 1
d : Aˣ
hd : d⁻¹ • (fun m => m • 1) d = d⁻¹ • 0
⊢ False

Next tactic:
simp only [inv_smul_smul, smul_zero, one_ne_zero] at hd
noncomputable instance linearOrderedCommGroupWithZero :
    LinearOrderedCommGroupWithZero (ValueGroup A K) :=
  { linearOrder .. with
    mul_assoc := by rintro ⟨a⟩ ⟨b⟩ ⟨c⟩; apply Quotient.sound'; rw [mul_assoc]; apply Setoid.refl'
    one_mul := by rintro ⟨a⟩; apply Quotient.sound'; rw [one_mul]; apply Setoid.refl'
    mul_one := by rintro ⟨a⟩; apply Quotient.sound'; rw [mul_one]; apply Setoid.refl'
    mul_comm := by rintro ⟨a⟩ ⟨b⟩; apply Quotient.sound'; rw [mul_comm]; apply Setoid.refl'
    mul_le_mul_left := by
      rintro ⟨a⟩ ⟨b⟩ ⟨c, rfl⟩ ⟨d⟩
      use c; simp only [Algebra.smul_def]; ring
    zero_mul := by rintro ⟨a⟩; apply Quotient.sound'; rw [zero_mul]; apply Setoid.refl'
    mul_zero := by rintro ⟨a⟩; apply Quotient.sound'; rw [mul_zero]; apply Setoid.refl'
    zero_le_one := ⟨0, by rw [zero_smul]⟩
    exists_pair_ne := by
      use 0, 1
      intro c; obtain ⟨d, hd⟩ := Quotient.exact' c
      apply_fun fun t => d⁻¹ • t at hd
      simp only [inv_smul_smul, smul_zero, one_ne_zero] at hd
    inv_zero := by 

Proof state:
A : Type u
inst✝⁵ : CommRing A
K : Type v
inst✝⁴ : Field K
inst✝³ : Algebra A K
inst✝² : IsDomain A
inst✝¹ : ValuationRing A
inst✝ : IsFractionRing A K
src✝ : LinearOrder (ValueGroup A K) := linearOrder A K
⊢ 0⁻¹ = 0

Next tactic:
apply Quotient.sound'
noncomputable instance linearOrderedCommGroupWithZero :
    LinearOrderedCommGroupWithZero (ValueGroup A K) :=
  { linearOrder .. with
    mul_assoc := by rintro ⟨a⟩ ⟨b⟩ ⟨c⟩; apply Quotient.sound'; rw [mul_assoc]; apply Setoid.refl'
    one_mul := by rintro ⟨a⟩; apply Quotient.sound'; rw [one_mul]; apply Setoid.refl'
    mul_one := by rintro ⟨a⟩; apply Quotient.sound'; rw [mul_one]; apply Setoid.refl'
    mul_comm := by rintro ⟨a⟩ ⟨b⟩; apply Quotient.sound'; rw [mul_comm]; apply Setoid.refl'
    mul_le_mul_left := by
      rintro ⟨a⟩ ⟨b⟩ ⟨c, rfl⟩ ⟨d⟩
      use c; simp only [Algebra.smul_def]; ring
    zero_mul := by rintro ⟨a⟩; apply Quotient.sound'; rw [zero_mul]; apply Setoid.refl'
    mul_zero := by rintro ⟨a⟩; apply Quotient.sound'; rw [mul_zero]; apply Setoid.refl'
    zero_le_one := ⟨0, by rw [zero_smul]⟩
    exists_pair_ne := by
      use 0, 1
      intro c; obtain ⟨d, hd⟩ := Quotient.exact' c
      apply_fun fun t => d⁻¹ • t at hd
      simp only [inv_smul_smul, smul_zero, one_ne_zero] at hd
    inv_zero := by apply Quotient.sound'; 

Proof state:
case a
A : Type u
inst✝⁵ : CommRing A
K : Type v
inst✝⁴ : Field K
inst✝³ : Algebra A K
inst✝² : IsDomain A
inst✝¹ : ValuationRing A
inst✝ : IsFractionRing A K
src✝ : LinearOrder (ValueGroup A K) := linearOrder A K
⊢ Setoid.r 0⁻¹ 0

Next tactic:
rw [inv_zero]