awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.ValuedField (goal)
Goal:
K : Type u_1
inst : Field K
Γ₀ : Type u_2
inst_1 : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
⊢ Eq.{u_2 + 1} (α := Γ₀)
    ((Valuation.toMonoidWithZeroHom extensionValuation : (a : hat K) → Γ₀)
        (↑(Exists.choose (p := fun (a : K) ↦
              Eq.{u_2 + 1} (α := ValueGroup₀ (F := Valuation K Γ₀) v)
                (((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) :
                      (a : K) → ValueGroup₀ v)
                    a :
                  ValueGroup₀ (F := Valuation K Γ₀) v)
                (1 : ValueGroup₀ (F := Valuation K Γ₀) v))
              ⋯) :
          hat K) :
      Γ₀)
    (1 : Γ₀)
have simp : Eq (α := Γ₀)
  ((embedding : (a : ValueGroup₀ v) → Γ₀)
      ((Valuation.restrict v : (a : K) → ValueGroup₀ (↑v : K →*₀ Γ₀))
          (Exists.choose (p := fun (a : K) ↦
            Eq.{u_2 + 1} (α := ValueGroup₀ (F := Valuation K Γ₀) v)
              (((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) :
                    (a : K) → ValueGroup₀ v)
                  a :
                ValueGroup₀ (F := Valuation K Γ₀) v)
              (1 : ValueGroup₀ (F := Valuation K Γ₀) v))
            sorry) :
        ValueGroup₀ (F := K →*₀ Γ₀) (↑v : K →*₀ Γ₀)) :
    Γ₀)
  ((embedding : (a : ValueGroup₀ v) → Γ₀) 1 : Γ₀)
Tactic: simp only [Valuation.toMonoidWithZeroHom_coe_eq_coe,
      extensionValuation_toFun, extension_extends, ← map_one (embedding (f := hv.v))]
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.ValuedField (goal)
Goal:
K : Type u_1
inst : Field K
Γ₀ : Type u_2
inst_1 : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
⊢ Eq (α := Γ₀)
    ((embedding : (a : ValueGroup₀ v) → Γ₀)
        ((Valuation.restrict v : (a : K) → ValueGroup₀ (↑v : K →*₀ Γ₀))
            (Exists.choose (p := fun (a : K) ↦
              Eq.{u_2 + 1} (α := ValueGroup₀ (F := Valuation K Γ₀) v)
                (((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) :
                      (a : K) → ValueGroup₀ v)
                    a :
                  ValueGroup₀ (F := Valuation K Γ₀) v)
                (1 : ValueGroup₀ (F := Valuation K Γ₀) v))
              ⋯) :
          ValueGroup₀ (F := K →*₀ Γ₀) (↑v : K →*₀ Γ₀)) :
      Γ₀)
    ((embedding : (a : ValueGroup₀ v) → Γ₀) 1 : Γ₀)
have rw : Eq.{u_2 + 1} (α := ValueGroup₀ (F := K →*₀ Γ₀) (↑v : K →*₀ Γ₀))
  ((Valuation.restrict v : (a : K) → ValueGroup₀ (↑v : K →*₀ Γ₀))
      (Exists.choose (p := fun (a : K) ↦
        Eq.{u_2 + 1} (α := ValueGroup₀ (F := Valuation K Γ₀) v)
          (((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) :
                (a : K) → ValueGroup₀ v)
              a :
            ValueGroup₀ (F := Valuation K Γ₀) v)
          (1 : ValueGroup₀ (F := Valuation K Γ₀) v))
        sorry) :
    ValueGroup₀ (F := K →*₀ Γ₀) (↑v : K →*₀ Γ₀))
  (1 : ValueGroup₀ (F := Valuation K Γ₀) v)
Tactic: rw [embedding_strictMono.injective.eq_iff]
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.ValuedField (goal)
Goal:
K : Type u_1
inst : Field K
Γ₀ : Type u_2
inst_1 : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
a b : ValueGroup₀ (F := Valuation K Γ₀) v
x : K :=
  Exists.choose (p := fun (a_1 : K) ↦
    ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v)
        a_1 =
      a)
    ⋯
hx_def :
  x =
    Exists.choose (p := fun (a_1 : K) ↦
      ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v)
          a_1 =
        a)
      ⋯
hx :
  ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v) x = a
y : K :=
  Exists.choose (p := fun (a : K) ↦
    ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v) a =
      b)
    ⋯
hy_def :
  y =
    Exists.choose (p := fun (a : K) ↦
      ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v)
          a =
        b)
      ⋯
hy :
  ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v) y = b
xy : K :=
  Exists.choose (p := fun (a_1 : K) ↦
    ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v)
        a_1 =
      a * b)
    ⋯
hxy_def :
  xy =
    Exists.choose (p := fun (a_1 : K) ↦
      ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v)
          a_1 =
        a * b)
      ⋯
hxy :
  ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v) xy =
    a * b
hx0 : ¬x = 0
hy0 : ¬y = 0
⊢ Units.mk0 ((v : (a : K) → Γ₀) xy) ⋯ =
    HMul.hMul (α := Γ₀ˣ) (β := Γ₀ˣ) (Units.mk0 ((v : (a : K) → Γ₀) x) ⋯ : Γ₀ˣ)
      (Units.mk0 ((v : (a : K) → Γ₀) y) ⋯ : Γ₀ˣ)
have neg : Units.mk0 ((v : (a : K) → Γ₀) xy) sorry =
  Units.mk0 (HMul.hMul (α := Γ₀) (β := Γ₀) ((v : (a : K) → Γ₀) x : Γ₀) ((v : (a : K) → Γ₀) y : Γ₀)) sorry
Tactic: rw [← Units.mk0_mul]
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.ValuedField (goal)
Goal:
K : Type u_1
inst : Field K
Γ₀ : Type u_2
inst_1 : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
a b : ValueGroup₀ (F := Valuation K Γ₀) v
x : K :=
  Exists.choose (p := fun (a_1 : K) ↦
    ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v)
        a_1 =
      a)
    ⋯
hx_def :
  x =
    Exists.choose (p := fun (a_1 : K) ↦
      ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v)
          a_1 =
        a)
      ⋯
hx :
  ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v) x = a
y : K :=
  Exists.choose (p := fun (a : K) ↦
    ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v) a =
      b)
    ⋯
hy_def :
  y =
    Exists.choose (p := fun (a : K) ↦
      ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v)
          a =
        b)
      ⋯
hy :
  ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v) y = b
xy : K :=
  Exists.choose (p := fun (a_1 : K) ↦
    ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v)
        a_1 =
      a * b)
    ⋯
hxy_def :
  xy =
    Exists.choose (p := fun (a_1 : K) ↦
      ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v)
          a_1 =
        a * b)
      ⋯
hxy :
  ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v) xy =
    a * b
hx0 : ¬x = 0
hy0 : ¬y = 0
neg :
  Units.mk0 ((v : (a : K) → Γ₀) xy) ⋯ =
    Units.mk0 (HMul.hMul (α := Γ₀) (β := Γ₀) ((v : (a : K) → Γ₀) x : Γ₀) ((v : (a : K) → Γ₀) y : Γ₀)) ⋯
⊢ Units.mk0 ((v : (a : K) → Γ₀) xy) ⋯ =
    HMul.hMul (α := Γ₀ˣ) (β := Γ₀ˣ) (Units.mk0 ((v : (a : K) → Γ₀) x) ⋯ : Γ₀ˣ)
      (Units.mk0 ((v : (a : K) → Γ₀) y) ⋯ : Γ₀ˣ)
have neg₁ : HMul.hMul (α := Γ₀) (β := Γ₀) ((v : (a : K) → Γ₀) x : Γ₀) ((v : (a : K) → Γ₀) y : Γ₀) ≠ 0
Tactic: rw [← Units.mk0_mul]
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.ValuedField (goal)
Goal:
K : Type u_1
inst : Field K
Γ₀ : Type u_2
inst_1 : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
a b : ValueGroup₀ (F := Valuation K Γ₀) v
x : K :=
  Exists.choose (p := fun (a_1 : K) ↦
    ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v)
        a_1 =
      a)
    ⋯
hx_def :
  x =
    Exists.choose (p := fun (a_1 : K) ↦
      ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v)
          a_1 =
        a)
      ⋯
hx :
  ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v) x = a
y : K :=
  Exists.choose (p := fun (a : K) ↦
    ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v) a =
      b)
    ⋯
hy_def :
  y =
    Exists.choose (p := fun (a : K) ↦
      ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v)
          a =
        b)
      ⋯
hy :
  ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v) y = b
xy : K :=
  Exists.choose (p := fun (a_1 : K) ↦
    ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v)
        a_1 =
      a * b)
    ⋯
hxy_def :
  xy =
    Exists.choose (p := fun (a_1 : K) ↦
      ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v)
          a_1 =
        a * b)
      ⋯
hxy :
  ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v) xy =
    a * b
hx0 : ¬x = 0
hy0 : ¬y = 0
⊢ Units.mk0 ((v : (a : K) → Γ₀) xy) ⋯ =
    Units.mk0 (HMul.hMul (α := Γ₀) (β := Γ₀) ((v : (a : K) → Γ₀) x : Γ₀) ((v : (a : K) → Γ₀) y : Γ₀)) ⋯
have neg : (↑(Units.mk0 ((v : (a : K) → Γ₀) xy) sorry) : Γ₀) =
  (↑(Units.mk0 (HMul.hMul (α := Γ₀) (β := Γ₀) ((v : (a : K) → Γ₀) x : Γ₀) ((v : (a : K) → Γ₀) y : Γ₀)) sorry) : Γ₀)
Tactic: ext
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.ValuedField (goal)
Goal:
K : Type u_1
inst : Field K
Γ₀ : Type u_2
inst_1 : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
a b : ValueGroup₀ (F := Valuation K Γ₀) v
x : K :=
  Exists.choose (p := fun (a_1 : K) ↦
    ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v)
        a_1 =
      a)
    ⋯
hx_def :
  x =
    Exists.choose (p := fun (a_1 : K) ↦
      ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v)
          a_1 =
        a)
      ⋯
hx :
  ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v) x = a
y : K :=
  Exists.choose (p := fun (a : K) ↦
    ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v) a =
      b)
    ⋯
hy_def :
  y =
    Exists.choose (p := fun (a : K) ↦
      ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v)
          a =
        b)
      ⋯
hy :
  ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v) y = b
xy : K :=
  Exists.choose (p := fun (a_1 : K) ↦
    ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v)
        a_1 =
      a * b)
    ⋯
hxy_def :
  xy =
    Exists.choose (p := fun (a_1 : K) ↦
      ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v)
          a_1 =
        a * b)
      ⋯
hxy :
  ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v) xy =
    a * b
hx0 : ¬x = 0
hy0 : ¬y = 0
⊢ (↑(Units.mk0 ((v : (a : K) → Γ₀) xy) ⋯) : Γ₀) =
    (↑(Units.mk0 (HMul.hMul (α := Γ₀) (β := Γ₀) ((v : (a : K) → Γ₀) x : Γ₀) ((v : (a : K) → Γ₀) y : Γ₀)) ⋯) : Γ₀)
have neg : (v : (a : K) → Γ₀) xy =
  (↑(Units.mk0 (HMul.hMul (α := Γ₀) (β := Γ₀) ((v : (a : K) → Γ₀) x : Γ₀) ((v : (a : K) → Γ₀) y : Γ₀)) sorry) : Γ₀)
Tactic: simp only [Units.val_mk0]
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.ValuedField (goal)
Goal:
K : Type u_1
inst : Field K
Γ₀ : Type u_2
inst_1 : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
a b : ValueGroup₀ (F := Valuation K Γ₀) v
x : K :=
  Exists.choose (p := fun (a_1 : K) ↦
    ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v)
        a_1 =
      a)
    ⋯
hx_def :
  x =
    Exists.choose (p := fun (a_1 : K) ↦
      ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v)
          a_1 =
        a)
      ⋯
hx :
  ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v) x = a
y : K :=
  Exists.choose (p := fun (a : K) ↦
    ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v) a =
      b)
    ⋯
hy_def :
  y =
    Exists.choose (p := fun (a : K) ↦
      ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v)
          a =
        b)
      ⋯
hy :
  ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v) y = b
xy : K :=
  Exists.choose (p := fun (a_1 : K) ↦
    ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v)
        a_1 =
      a * b)
    ⋯
hxy_def :
  xy =
    Exists.choose (p := fun (a_1 : K) ↦
      ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v)
          a_1 =
        a * b)
      ⋯
hxy :
  ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v) xy =
    a * b
hx0 : ¬x = 0
hy0 : ¬y = 0
⊢ (v : (a : K) → Γ₀) xy =
    (↑(Units.mk0 (HMul.hMul (α := Γ₀) (β := Γ₀) ((v : (a : K) → Γ₀) x : Γ₀) ((v : (a : K) → Γ₀) y : Γ₀)) ⋯) : Γ₀)
have neg : (Valuation.restrict v : (a : K) → ValueGroup₀ (↑v : K →*₀ Γ₀)) xy =
  HMul.hMul (α := ValueGroup₀ (F := K →*₀ Γ₀) (↑v : K →*₀ Γ₀)) (β := ValueGroup₀ (F := K →*₀ Γ₀) (↑v : K →*₀ Γ₀))
    ((Valuation.restrict v : (a : K) → ValueGroup₀ (↑v : K →*₀ Γ₀)) x : ValueGroup₀ (F := K →*₀ Γ₀) (↑v : K →*₀ Γ₀))
    ((Valuation.restrict v : (a : K) → ValueGroup₀ (↑v : K →*₀ Γ₀)) y : ValueGroup₀ (F := K →*₀ Γ₀) (↑v : K →*₀ Γ₀))
Tactic: rw [Units.val_mk0, ← map_mul, ← v.restrict_inj, map_mul]
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.ValuedField (goal)
Goal:
K : Type u_1
inst : Field K
Γ₀ : Type u_2
inst_1 : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
a b : ValueGroup₀ (F := Valuation K Γ₀) v
x : K :=
  Exists.choose (p := fun (a_1 : K) ↦
    ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v)
        a_1 =
      a)
    ⋯
hx_def :
  x =
    Exists.choose (p := fun (a_1 : K) ↦
      ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v)
          a_1 =
        a)
      ⋯
hx :
  ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v) x = a
y : K :=
  Exists.choose (p := fun (a : K) ↦
    ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v) a =
      b)
    ⋯
hy_def :
  y =
    Exists.choose (p := fun (a : K) ↦
      ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v)
          a =
        b)
      ⋯
hy :
  ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v) y = b
xy : K :=
  Exists.choose (p := fun (a_1 : K) ↦
    ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v)
        a_1 =
      a * b)
    ⋯
hxy_def :
  xy =
    Exists.choose (p := fun (a_1 : K) ↦
      ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v)
          a_1 =
        a * b)
      ⋯
hxy :
  ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v) xy =
    a * b
hx0 : ¬x = 0
hy0 : ¬y = 0
⊢ ¬Eq.{u_2 + 1} (α := Γ₀) ((extensionValuation : (a : hat K) → Γ₀) (↑xy : hat K) : Γ₀) (0 : Γ₀)
have neg : ¬HMul.hMul (α := ValueGroup₀ (F := Valuation K Γ₀) v) (β := ValueGroup₀ (F := Valuation K Γ₀) v)
      (((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v)
          x :
        ValueGroup₀ (F := Valuation K Γ₀) v)
      (((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v)
          y :
        ValueGroup₀ (F := Valuation K Γ₀) v) =
    0
Tactic: rw [extensionValuation_apply_coe, ← restrict₀_eq_zero_iff, hxy, ← hx, ← hy]
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.ValuedField (goal)
Goal:
K : Type u_1
inst : Field K
Γ₀ : Type u_2
inst_1 : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
⊢ ValueGroup₀ (F := Valuation K Γ₀) v ≃* ValueGroup₀ (F := Valuation.{u_1, u_2} (hat K) Γ₀) extensionValuation
have refine_1 : Function.Injective (α := ValueGroup₀ (F := Valuation K Γ₀) v) (β :=
  ValueGroup₀ (F := Valuation.{u_1, u_2} (hat K) Γ₀) extensionValuation)
  (⇑valueGroup₀_hom_extensionValuation : (a : ValueGroup₀ v) → ValueGroup₀ extensionValuation)
Tactic: refine MulEquiv.ofBijective (valueGroup₀_hom_extensionValuation (hv := hv)) ⟨?_, ?_⟩
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.ValuedField (goal)
Goal:
K : Type u_1
inst : Field K
Γ₀ : Type u_2
inst_1 : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
refine_1 :
  Function.Injective (α := ValueGroup₀ (F := Valuation K Γ₀) v) (β :=
    ValueGroup₀ (F := Valuation.{u_1, u_2} (hat K) Γ₀) extensionValuation)
    (⇑valueGroup₀_hom_extensionValuation : (a : ValueGroup₀ v) → ValueGroup₀ extensionValuation)
⊢ ValueGroup₀ (F := Valuation K Γ₀) v ≃* ValueGroup₀ (F := Valuation.{u_1, u_2} (hat K) Γ₀) extensionValuation
have refine_2 : Function.Surjective (α := ValueGroup₀ (F := Valuation K Γ₀) v) (β :=
  ValueGroup₀ (F := Valuation.{u_1, u_2} (hat K) Γ₀) extensionValuation)
  (⇑valueGroup₀_hom_extensionValuation : (a : ValueGroup₀ v) → ValueGroup₀ extensionValuation)
Tactic: refine MulEquiv.ofBijective (valueGroup₀_hom_extensionValuation (hv := hv)) ⟨?_, ?_⟩
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.ValuedField (hypothesis)
Goal:
K : Type u_1
inst : Field K
Γ₀ : Type u_2
inst_1 : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
a b : ValueGroup₀ (F := Valuation K Γ₀) v
hab :
  Eq (α := ValueGroup₀ (F := Valuation.{u_1, u_2} (hat K) Γ₀) extensionValuation)
    ((valueGroup₀_hom_extensionValuation : (a : ValueGroup₀ v) → ValueGroup₀ extensionValuation) a :
      ValueGroup₀ (F := Valuation.{u_1, u_2} (hat K) Γ₀) extensionValuation)
    ((valueGroup₀_hom_extensionValuation : (a : ValueGroup₀ v) → ValueGroup₀ extensionValuation) b :
      ValueGroup₀ (F := Valuation.{u_1, u_2} (hat K) Γ₀) extensionValuation)
⊢ a = b
have hx_def : let x :=
  Exists.choose (p := fun (a_1 : K) ↦
    ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v)
        a_1 =
      a)
    sorry;
x =
  Exists.choose (p := fun (a_1 : K) ↦
    ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v)
        a_1 =
      a)
    sorry
Tactic: set x := (restrict₀_surjective hv.v a).choose with hx_def
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.ValuedField (hypothesis)
Goal:
K : Type u_1
inst : Field K
Γ₀ : Type u_2
inst_1 : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
a b : ValueGroup₀ (F := Valuation K Γ₀) v
hab :
  Eq (α := ValueGroup₀ (F := Valuation.{u_1, u_2} (hat K) Γ₀) extensionValuation)
    ((valueGroup₀_hom_extensionValuation : (a : ValueGroup₀ v) → ValueGroup₀ extensionValuation) a :
      ValueGroup₀ (F := Valuation.{u_1, u_2} (hat K) Γ₀) extensionValuation)
    ((valueGroup₀_hom_extensionValuation : (a : ValueGroup₀ v) → ValueGroup₀ extensionValuation) b :
      ValueGroup₀ (F := Valuation.{u_1, u_2} (hat K) Γ₀) extensionValuation)
x : K :=
  Exists.choose (p := fun (a_1 : K) ↦
    ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v)
        a_1 =
      a)
    ⋯
hx_def :
  x =
    Exists.choose (p := fun (a_1 : K) ↦
      ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v)
          a_1 =
        a)
      ⋯
hx : ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v)
    (Exists.choose (p := fun (a_1 : K) ↦
      ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v)
          a_1 =
        a)
      ⋯) =
  a :=
  ⋯
⊢ a = b
have hy_def : let y :=
  Exists.choose (p := fun (a : K) ↦
    ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v) a =
      b)
    sorry;
y =
  Exists.choose (p := fun (a : K) ↦
    ((restrict₀ v : MonoidWithZeroHom.{u_1, u_2} K (ValueGroup₀ (F := Valuation K Γ₀) v)) : (a : K) → ValueGroup₀ v) a =
      b)
    sorry
Tactic: set y := (restrict₀_surjective hv.v b).choose with hy_def