awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.WithVal (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
hp0' : LT.lt (α := ℚ) 0 (↑p : ℚ)
hp0 : LT.lt (α := ℝ) 0 (↑p : ℝ)⁻¹
hp1' : LT.lt (α := ℚ) 1 (↑p : ℚ)
hp1 : LT.lt (α := ℝ) (↑p : ℝ)⁻¹ 1
γ :
  (MonoidWithZeroHom.ValueGroup₀ (F := Valuation (WithVal (Rat.padicValuation p)) (WithZero (Multiplicative ℤ)))
      Valued.v)ˣ
x y : WithVal (Rat.padicValuation p)
x' : ℚ := (WithVal.equiv (Rat.padicValuation p) : (a : WithVal (Rat.padicValuation p)) → ℚ) x
hx : x' = (WithVal.equiv (Rat.padicValuation p) : (a : WithVal (Rat.padicValuation p)) → ℚ) x
y' : ℚ := (WithVal.equiv (Rat.padicValuation p) : (a : WithVal (Rat.padicValuation p)) → ℚ) y
hy : y' = (WithVal.equiv (Rat.padicValuation p) : (a : WithVal (Rat.padicValuation p)) → ℚ) y
h :
  (if x' - y' = 0 then 0 else (HPow.hPow (α := ℚ) (↑p : ℚ) (padicValRat p (x' - y')))⁻¹) ≤
    (HPow.hPow (α := ℚ) (↑p : ℚ)
        (HMul.hMul (α := WithZero (Multiplicative ℤ)) (β := WithZero (Multiplicative ℤ))
              ((embedding : (a : MonoidWithZeroHom.ValueGroup₀ Valued.v) → WithZero (Multiplicative ℤ))
                  (↑γ : MonoidWithZeroHom.ValueGroup₀ Valued.v) :
                WithZero (Multiplicative ℤ))
              (exp (-1))).log.natAbs)⁻¹
H : x' - y' = 0
⊢ LT.lt (α := WithZero (Multiplicative ℤ)) (0 : WithZero (Multiplicative ℤ))
    ((embedding : (a : MonoidWithZeroHom.ValueGroup₀ Valued.v) → WithZero (Multiplicative ℤ))
        (↑γ : MonoidWithZeroHom.ValueGroup₀ Valued.v) :
      WithZero (Multiplicative ℤ))
have pos : LT.lt (α := WithZero (Multiplicative ℤ)) (0 : WithZero (Multiplicative ℤ))
  ((embedding : (a : MonoidWithZeroHom.ValueGroup₀ Valued.v) → WithZero (Multiplicative ℤ))
      (↑γ : MonoidWithZeroHom.ValueGroup₀ Valued.v) :
    WithZero (Multiplicative ℤ))
Tactic: simp only [exp_neg]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.WithVal (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
hp0' : LT.lt (α := ℚ) 0 (↑p : ℚ)
hp0 : LT.lt (α := ℝ) 0 (↑p : ℝ)⁻¹
hp1' : LT.lt (α := ℚ) 1 (↑p : ℚ)
hp1 : LT.lt (α := ℝ) (↑p : ℝ)⁻¹ 1
γ :
  (MonoidWithZeroHom.ValueGroup₀ (F := Valuation (WithVal (Rat.padicValuation p)) (WithZero (Multiplicative ℤ)))
      Valued.v)ˣ
x y : WithVal (Rat.padicValuation p)
x' : ℚ := (WithVal.equiv (Rat.padicValuation p) : (a : WithVal (Rat.padicValuation p)) → ℚ) x
hx : x' = (WithVal.equiv (Rat.padicValuation p) : (a : WithVal (Rat.padicValuation p)) → ℚ) x
y' : ℚ := (WithVal.equiv (Rat.padicValuation p) : (a : WithVal (Rat.padicValuation p)) → ℚ) y
hy : y' = (WithVal.equiv (Rat.padicValuation p) : (a : WithVal (Rat.padicValuation p)) → ℚ) y
h :
  (if x' - y' = 0 then 0 else (HPow.hPow (α := ℚ) (↑p : ℚ) (padicValRat p (x' - y')))⁻¹) ≤
    (HPow.hPow (α := ℚ) (↑p : ℚ)
        (HMul.hMul (α := WithZero (Multiplicative ℤ)) (β := WithZero (Multiplicative ℤ))
              ((embedding : (a : MonoidWithZeroHom.ValueGroup₀ Valued.v) → WithZero (Multiplicative ℤ))
                  (↑γ : MonoidWithZeroHom.ValueGroup₀ Valued.v) :
                WithZero (Multiplicative ℤ))
              (exp (-1))).log.natAbs)⁻¹
H : ¬x' - y' = 0
⊢ exp (-padicValRat p (x' - y')) <
    (embedding : (a : MonoidWithZeroHom.ValueGroup₀ Valued.v) → WithZero (Multiplicative ℤ))
      (↑γ : MonoidWithZeroHom.ValueGroup₀ Valued.v)
have neg : -padicValRat p (x' - y') <
  log
    ((embedding : (a : MonoidWithZeroHom.ValueGroup₀ Valued.v) → WithZero (Multiplicative ℤ))
      (↑γ : MonoidWithZeroHom.ValueGroup₀ Valued.v))
Tactic: rw [← lt_log_iff_exp_lt (embedding_unit_ne_zero _)]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.WithVal (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
⊢ IsDenseInducing (α := WithVal (Rat.padicValuation p)) (β := ℚ_[p])
    (⇑(RingHom.comp (Rat.castHom ℚ_[p]) (WithVal.equiv (Rat.padicValuation p)).toRingHom) :
      (a : WithVal (Rat.padicValuation p)) → ℚ_[p])
have refine : DenseRange (X := ℚ_[p]) (α := WithVal (Rat.padicValuation p))
  (⇑(RingHom.comp (Rat.castHom ℚ_[p]) (WithVal.equiv (Rat.padicValuation p)).toRingHom) :
    (a : WithVal (Rat.padicValuation p)) → ℚ_[p])
Tactic: refine Padic.isUniformInducing_cast_withVal.isDenseInducing ?_
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.WithVal (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
y : (Rat.padicValuation p).Completion
⊢ IsDenseInducing.extend ⋯ coe'
      ((extensionHom (RingHom.comp (Rat.castHom ℚ_[p]) (WithVal.equiv (Rat.padicValuation p)).toRingHom) ⋯ :
          (a : Completion (WithVal (Rat.padicValuation p))) → ℚ_[p])
        y) =
    y
have hp : IsClosed (X := Completion (WithVal (Rat.padicValuation p)))
  {a : Completion (WithVal (Rat.padicValuation p)) |
    IsDenseInducing.extend sorry coe'
        ((extensionHom (RingHom.comp (Rat.castHom ℚ_[p]) (WithVal.equiv (Rat.padicValuation p)).toRingHom) sorry :
            (a : Completion (WithVal (Rat.padicValuation p))) → ℚ_[p])
          a) =
      a}
Tactic: induction y using induction_on
Cleared hypotheses: y
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.WithVal (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
y : (Rat.padicValuation p).Completion
hp :
  IsClosed (X := Completion (WithVal (Rat.padicValuation p)))
    {a : Completion (WithVal (Rat.padicValuation p)) |
      IsDenseInducing.extend ⋯ coe'
          ((extensionHom (RingHom.comp (Rat.castHom ℚ_[p]) (WithVal.equiv (Rat.padicValuation p)).toRingHom) ⋯ :
              (a : Completion (WithVal (Rat.padicValuation p))) → ℚ_[p])
            a) =
        a}
⊢ IsDenseInducing.extend ⋯ coe'
      ((extensionHom (RingHom.comp (Rat.castHom ℚ_[p]) (WithVal.equiv (Rat.padicValuation p)).toRingHom) ⋯ :
          (a : Completion (WithVal (Rat.padicValuation p))) → ℚ_[p])
        y) =
    y
have ih : ∀ (a : WithVal (Rat.padicValuation p)),
  Eq (α := (Rat.padicValuation p).Completion)
    (IsDenseInducing.extend sorry coe'
      ((extensionHom (RingHom.comp (Rat.castHom ℚ_[p]) (WithVal.equiv (Rat.padicValuation p)).toRingHom) sorry :
          (a : Completion (WithVal (Rat.padicValuation p))) → ℚ_[p])
        (↑a : Completion (WithVal (Rat.padicValuation p)))))
    (↑a : Completion (WithVal (Rat.padicValuation p)))
Tactic: induction y using induction_on
Cleared hypotheses: y
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.WithVal (hypothesis)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
⊢ IsClosed (X := Completion (WithVal (Rat.padicValuation p)))
    {a : Completion (WithVal (Rat.padicValuation p)) |
      IsDenseInducing.extend ⋯ coe'
          ((extensionHom (RingHom.comp (Rat.castHom ℚ_[p]) (WithVal.equiv (Rat.padicValuation p)).toRingHom) ⋯ :
              (a : Completion (WithVal (Rat.padicValuation p))) → ℚ_[p])
            a) =
        a}
have pf : IsOrderedMonoid (Multiplicative ℤ)
Tactic: generalize_proofs _ _ _ H
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.WithVal (hypothesis)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
pf : IsOrderedMonoid (Multiplicative ℤ)
⊢ IsClosed (X := Completion (WithVal (Rat.padicValuation p)))
    {a : Completion (WithVal (Rat.padicValuation p)) |
      IsDenseInducing.extend ⋯ coe'
          ((extensionHom (RingHom.comp (Rat.castHom ℚ_[p]) (WithVal.equiv (Rat.padicValuation p)).toRingHom) ⋯ :
              (a : Completion (WithVal (Rat.padicValuation p))) → ℚ_[p])
            a) =
        a}
have pf₁ : CharZero ℚ_[p]
Tactic: generalize_proofs _ _ _ H
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.WithVal (hypothesis)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
pf : IsOrderedMonoid (Multiplicative ℤ)
pf₁ : CharZero ℚ_[p]
⊢ IsClosed (X := Completion (WithVal (Rat.padicValuation p)))
    {a : Completion (WithVal (Rat.padicValuation p)) |
      IsDenseInducing.extend ⋯ coe'
          ((extensionHom (RingHom.comp (Rat.castHom ℚ_[p]) (WithVal.equiv (Rat.padicValuation p)).toRingHom) ⋯ :
              (a : Completion (WithVal (Rat.padicValuation p))) → ℚ_[p])
            a) =
        a}
have pf₂ : IsDenseInducing (α := WithVal (Rat.padicValuation p)) (β := ℚ_[p])
  (⇑(RingHom.comp (Rat.castHom ℚ_[p]) (WithVal.equiv (Rat.padicValuation p)).toRingHom) :
    (a : WithVal (Rat.padicValuation p)) → ℚ_[p])
Tactic: generalize_proofs _ _ _ H
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.WithVal (hypothesis)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
pf : IsOrderedMonoid (Multiplicative ℤ)
pf₁ : CharZero ℚ_[p]
pf₂ :
  IsDenseInducing (α := WithVal (Rat.padicValuation p)) (β := ℚ_[p])
    (⇑(RingHom.comp (Rat.castHom ℚ_[p]) (WithVal.equiv (Rat.padicValuation p)).toRingHom) :
      (a : WithVal (Rat.padicValuation p)) → ℚ_[p])
⊢ IsClosed (X := Completion (WithVal (Rat.padicValuation p)))
    {a : Completion (WithVal (Rat.padicValuation p)) |
      IsDenseInducing.extend ⋯ coe'
          ((extensionHom (RingHom.comp (Rat.castHom ℚ_[p]) (WithVal.equiv (Rat.padicValuation p)).toRingHom) ⋯ :
              (a : Completion (WithVal (Rat.padicValuation p))) → ℚ_[p])
            a) =
        a}
have H : IsTopologicalRing (WithVal (Rat.padicValuation p))
Tactic: generalize_proofs _ _ _ H
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.WithVal (hypothesis)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
pf : IsOrderedMonoid (Multiplicative ℤ)
pf₁ : CharZero ℚ_[p]
pf₂ :
  IsDenseInducing (α := WithVal (Rat.padicValuation p)) (β := ℚ_[p])
    (⇑(RingHom.comp (Rat.castHom ℚ_[p]) (WithVal.equiv (Rat.padicValuation p)).toRingHom) :
      (a : WithVal (Rat.padicValuation p)) → ℚ_[p])
H : IsTopologicalRing (WithVal (Rat.padicValuation p))
⊢ IsClosed (X := Completion (WithVal (Rat.padicValuation p)))
    {a : Completion (WithVal (Rat.padicValuation p)) |
      IsDenseInducing.extend ⋯ coe'
          ((extensionHom (RingHom.comp (Rat.castHom ℚ_[p]) (WithVal.equiv (Rat.padicValuation p)).toRingHom) ⋯ :
              (a : Completion (WithVal (Rat.padicValuation p))) → ℚ_[p])
            a) =
        a}
have pf₃ : IsUniformAddGroup (WithVal (Rat.padicValuation p))
Tactic: generalize_proofs _ _ _ H
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.WithVal (hypothesis)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
pf : IsOrderedMonoid (Multiplicative ℤ)
pf₁ : CharZero ℚ_[p]
pf₂ :
  IsDenseInducing (α := WithVal (Rat.padicValuation p)) (β := ℚ_[p])
    (⇑(RingHom.comp (Rat.castHom ℚ_[p]) (WithVal.equiv (Rat.padicValuation p)).toRingHom) :
      (a : WithVal (Rat.padicValuation p)) → ℚ_[p])
H : IsTopologicalRing (WithVal (Rat.padicValuation p))
pf₃ : IsUniformAddGroup (WithVal (Rat.padicValuation p))
⊢ IsClosed (X := Completion (WithVal (Rat.padicValuation p)))
    {a : Completion (WithVal (Rat.padicValuation p)) |
      IsDenseInducing.extend ⋯ coe'
          ((extensionHom (RingHom.comp (Rat.castHom ℚ_[p]) (WithVal.equiv (Rat.padicValuation p)).toRingHom) ⋯ :
              (a : Completion (WithVal (Rat.padicValuation p))) → ℚ_[p])
            a) =
        a}
have pf₄ : IsUniformAddGroup ℚ_[p]
Tactic: generalize_proofs _ _ _ H
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.WithVal (hypothesis)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
pf : IsOrderedMonoid (Multiplicative ℤ)
pf₁ : CharZero ℚ_[p]
pf₂ :
  IsDenseInducing (α := WithVal (Rat.padicValuation p)) (β := ℚ_[p])
    (⇑(RingHom.comp (Rat.castHom ℚ_[p]) (WithVal.equiv (Rat.padicValuation p)).toRingHom) :
      (a : WithVal (Rat.padicValuation p)) → ℚ_[p])
H : IsTopologicalRing (WithVal (Rat.padicValuation p))
pf₃ : IsUniformAddGroup (WithVal (Rat.padicValuation p))
pf₄ : IsUniformAddGroup ℚ_[p]
⊢ IsClosed (X := Completion (WithVal (Rat.padicValuation p)))
    {a : Completion (WithVal (Rat.padicValuation p)) |
      IsDenseInducing.extend ⋯ coe'
          ((extensionHom (RingHom.comp (Rat.castHom ℚ_[p]) (WithVal.equiv (Rat.padicValuation p)).toRingHom) ⋯ :
              (a : Completion (WithVal (Rat.padicValuation p))) → ℚ_[p])
            a) =
        a}
have pf₅ : IsTopologicalRing ℚ_[p]
Tactic: generalize_proofs _ _ _ H
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.WithVal (hypothesis)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
pf : IsOrderedMonoid (Multiplicative ℤ)
pf₁ : CharZero ℚ_[p]
pf₂ :
  IsDenseInducing (α := WithVal (Rat.padicValuation p)) (β := ℚ_[p])
    (⇑(RingHom.comp (Rat.castHom ℚ_[p]) (WithVal.equiv (Rat.padicValuation p)).toRingHom) :
      (a : WithVal (Rat.padicValuation p)) → ℚ_[p])
H : IsTopologicalRing (WithVal (Rat.padicValuation p))
pf₃ : IsUniformAddGroup (WithVal (Rat.padicValuation p))
pf₄ : IsUniformAddGroup ℚ_[p]
pf₅ : IsTopologicalRing ℚ_[p]
⊢ IsClosed (X := Completion (WithVal (Rat.padicValuation p)))
    {a : Completion (WithVal (Rat.padicValuation p)) |
      IsDenseInducing.extend ⋯ coe'
          ((extensionHom (RingHom.comp (Rat.castHom ℚ_[p]) (WithVal.equiv (Rat.padicValuation p)).toRingHom) ⋯ :
              (a : Completion (WithVal (Rat.padicValuation p))) → ℚ_[p])
            a) =
        a}
have pf₆ : Continuous (X := WithVal (Rat.padicValuation p)) (Y := ℚ_[p])
  (⇑(RingHom.comp (Rat.castHom ℚ_[p]) (WithVal.equiv (Rat.padicValuation p)).toRingHom) :
    (a : WithVal (Rat.padicValuation p)) → ℚ_[p])
Tactic: generalize_proofs _ _ _ H
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.WithVal (hypothesis)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
pf : IsOrderedMonoid (Multiplicative ℤ)
pf₁ : CharZero ℚ_[p]
pf₂ :
  IsDenseInducing (α := WithVal (Rat.padicValuation p)) (β := ℚ_[p])
    (⇑(RingHom.comp (Rat.castHom ℚ_[p]) (WithVal.equiv (Rat.padicValuation p)).toRingHom) :
      (a : WithVal (Rat.padicValuation p)) → ℚ_[p])
H : IsTopologicalRing (WithVal (Rat.padicValuation p))
pf₃ : IsUniformAddGroup (WithVal (Rat.padicValuation p))
pf₄ : IsUniformAddGroup ℚ_[p]
pf₅ : IsTopologicalRing ℚ_[p]
pf₆ :
  Continuous (X := WithVal (Rat.padicValuation p)) (Y := ℚ_[p])
    (⇑(RingHom.comp (Rat.castHom ℚ_[p]) (WithVal.equiv (Rat.padicValuation p)).toRingHom) :
      (a : WithVal (Rat.padicValuation p)) → ℚ_[p])
⊢ IsClosed (X := Completion (WithVal (Rat.padicValuation p)))
    {a : Completion (WithVal (Rat.padicValuation p)) |
      IsDenseInducing.extend ⋯ coe'
          ((extensionHom (RingHom.comp (Rat.castHom ℚ_[p]) (WithVal.equiv (Rat.padicValuation p)).toRingHom) ⋯ :
              (a : Completion (WithVal (Rat.padicValuation p))) → ℚ_[p])
            a) =
        a}
have pf₇ : CompleteSpace ℚ_[p]
Tactic: generalize_proofs _ _ _ H
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.WithVal (hypothesis)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
pf : IsOrderedMonoid (Multiplicative ℤ)
pf₁ : CharZero ℚ_[p]
pf₂ :
  IsDenseInducing (α := WithVal (Rat.padicValuation p)) (β := ℚ_[p])
    (⇑(RingHom.comp (Rat.castHom ℚ_[p]) (WithVal.equiv (Rat.padicValuation p)).toRingHom) :
      (a : WithVal (Rat.padicValuation p)) → ℚ_[p])
H : IsTopologicalRing (WithVal (Rat.padicValuation p))
pf₃ : IsUniformAddGroup (WithVal (Rat.padicValuation p))
pf₄ : IsUniformAddGroup ℚ_[p]
pf₅ : IsTopologicalRing ℚ_[p]
pf₆ :
  Continuous (X := WithVal (Rat.padicValuation p)) (Y := ℚ_[p])
    (⇑(RingHom.comp (Rat.castHom ℚ_[p]) (WithVal.equiv (Rat.padicValuation p)).toRingHom) :
      (a : WithVal (Rat.padicValuation p)) → ℚ_[p])
pf₇ : CompleteSpace ℚ_[p]
⊢ IsClosed (X := Completion (WithVal (Rat.padicValuation p)))
    {a : Completion (WithVal (Rat.padicValuation p)) |
      IsDenseInducing.extend ⋯ coe'
          ((extensionHom (RingHom.comp (Rat.castHom ℚ_[p]) (WithVal.equiv (Rat.padicValuation p)).toRingHom) ⋯ :
              (a : Completion (WithVal (Rat.padicValuation p))) → ℚ_[p])
            a) =
        a}
have pf₈ : T0Space ℚ_[p]
Tactic: generalize_proofs _ _ _ H
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.WithVal (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
pf : IsOrderedMonoid (Multiplicative ℤ)
pf_1 : CharZero ℚ_[p]
pf_2 :
  IsDenseInducing (α := WithVal (Rat.padicValuation p)) (β := ℚ_[p])
    (⇑(RingHom.comp (Rat.castHom ℚ_[p]) (WithVal.equiv (Rat.padicValuation p)).toRingHom) :
      (a : WithVal (Rat.padicValuation p)) → ℚ_[p])
H : IsTopologicalRing (WithVal (Rat.padicValuation p))
pf_3 : IsUniformAddGroup (WithVal (Rat.padicValuation p))
pf_4 : IsUniformAddGroup ℚ_[p]
pf_5 : IsTopologicalRing ℚ_[p]
pf_6 :
  Continuous (X := WithVal (Rat.padicValuation p)) (Y := ℚ_[p])
    (⇑(RingHom.comp (Rat.castHom ℚ_[p]) (WithVal.equiv (Rat.padicValuation p)).toRingHom) :
      (a : WithVal (Rat.padicValuation p)) → ℚ_[p])
pf_7 : CompleteSpace ℚ_[p]
pf_8 : T0Space ℚ_[p]
⊢ IsClosed (X := Completion (WithVal (Rat.padicValuation p)))
    {a : Completion (WithVal (Rat.padicValuation p)) |
      pf_2.extend coe'
          ((extensionHom (RingHom.comp (Rat.castHom ℚ_[p]) (WithVal.equiv (Rat.padicValuation p)).toRingHom) pf_6 :
              (a : Completion (WithVal (Rat.padicValuation p))) → ℚ_[p])
            a) =
        a}
have hp : Continuous (Y := (Rat.padicValuation p).Completion) fun (a : Completion (WithVal (Rat.padicValuation p))) ↦
  pf_2.extend coe'
    ((extensionHom (RingHom.comp (Rat.castHom ℚ_[p]) (WithVal.equiv (Rat.padicValuation p)).toRingHom) pf_6 :
        (a : Completion (WithVal (Rat.padicValuation p))) → ℚ_[p])
      a)
Tactic: refine isClosed_eq ?_ continuous_id
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.WithVal (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
a : WithVal (Rat.padicValuation p)
⊢ Eq (α := (Rat.padicValuation p).Completion)
    (IsDenseInducing.extend ⋯ coe'
      ((extensionHom (RingHom.comp (Rat.castHom ℚ_[p]) (WithVal.equiv (Rat.padicValuation p)).toRingHom) ⋯ :
          (a : Completion (WithVal (Rat.padicValuation p))) → ℚ_[p])
        (↑a : Completion (WithVal (Rat.padicValuation p)))))
    (↑a : Completion (WithVal (Rat.padicValuation p)))
have ih : Eq (α := (Rat.padicValuation p).Completion)
  (IsDenseInducing.extend sorry coe'
    ((RingHom.comp (Rat.castHom ℚ_[p]) (WithVal.equiv (Rat.padicValuation p)).toRingHom :
        (a : WithVal (Rat.padicValuation p)) → ℚ_[p])
      a))
  (↑a : Completion (WithVal (Rat.padicValuation p)))
Tactic: rw [extensionHom_coe]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.WithVal (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
a : WithVal (Rat.padicValuation p)
⊢ Eq (α := (Rat.padicValuation p).Completion)
    (IsDenseInducing.extend ⋯ coe'
      ((RingHom.comp (Rat.castHom ℚ_[p]) (WithVal.equiv (Rat.padicValuation p)).toRingHom :
          (a : WithVal (Rat.padicValuation p)) → ℚ_[p])
        a))
    (↑a : Completion (WithVal (Rat.padicValuation p)))
have ih : Continuous (X := WithVal (Rat.padicValuation p)) (Y := (Rat.padicValuation p).Completion) coe'
Tactic: apply IsDenseInducing.extend_eq
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.WithVal (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
y : ℚ_[p]
⊢ (extensionHom (RingHom.comp (Rat.castHom ℚ_[p]) (WithVal.equiv (Rat.padicValuation p)).toRingHom) ⋯ :
        (a : Completion (WithVal (Rat.padicValuation p))) → ℚ_[p])
      (IsDenseInducing.extend ⋯ coe' y) =
    y
have hp : IsClosed (X := ℚ_[p])
  {x : ℚ_[p] |
    (extensionHom (RingHom.comp (Rat.castHom ℚ_[p]) (WithVal.equiv (Rat.padicValuation p)).toRingHom) sorry :
          (a : Completion (WithVal (Rat.padicValuation p))) → ℚ_[p])
        (IsDenseInducing.extend sorry coe' x) =
      x}
Tactic: induction y using isClosed_property (Padic.denseRange_ratCast p)
Cleared hypotheses: y
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.WithVal (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
y : ℚ_[p]
hp :
  IsClosed (X := ℚ_[p])
    {x : ℚ_[p] |
      (extensionHom (RingHom.comp (Rat.castHom ℚ_[p]) (WithVal.equiv (Rat.padicValuation p)).toRingHom) ⋯ :
            (a : Completion (WithVal (Rat.padicValuation p))) → ℚ_[p])
          (IsDenseInducing.extend ⋯ coe' x) =
        x}
⊢ (extensionHom (RingHom.comp (Rat.castHom ℚ_[p]) (WithVal.equiv (Rat.padicValuation p)).toRingHom) ⋯ :
        (a : Completion (WithVal (Rat.padicValuation p))) → ℚ_[p])
      (IsDenseInducing.extend ⋯ coe' y) =
    y
have h : ∀ (a : ℚ),
  Eq (α := ℚ_[p])
    ((extensionHom (RingHom.comp (Rat.castHom ℚ_[p]) (WithVal.equiv (Rat.padicValuation p)).toRingHom) sorry :
          (a : Completion (WithVal (Rat.padicValuation p))) → ℚ_[p])
        (IsDenseInducing.extend sorry coe' (↑a : ℚ_[p])) :
      ℚ_[p])
    ((↑a : ℚ_[p]) : ℚ_[p])
Tactic: induction y using isClosed_property (Padic.denseRange_ratCast p)
Cleared hypotheses: y
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.WithVal (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
⊢ IsClosed (X := ℚ_[p])
    {x : ℚ_[p] |
      (extensionHom (RingHom.comp (Rat.castHom ℚ_[p]) (WithVal.equiv (Rat.padicValuation p)).toRingHom) ⋯ :
            (a : Completion (WithVal (Rat.padicValuation p))) → ℚ_[p])
          (IsDenseInducing.extend ⋯ coe' x) =
        x}
have hp : Continuous (Y := ℚ_[p]) fun (x : ℚ_[p]) ↦
  (extensionHom (RingHom.comp (Rat.castHom ℚ_[p]) (WithVal.equiv (Rat.padicValuation p)).toRingHom) sorry :
      (a : Completion (WithVal (Rat.padicValuation p))) → ℚ_[p])
    (IsDenseInducing.extend sorry coe' x)
Tactic: refine isClosed_eq ?_ continuous_id
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.WithVal (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
⊢ Continuous (Y := ℚ_[p]) fun (x : ℚ_[p]) ↦
    (extensionHom (RingHom.comp (Rat.castHom ℚ_[p]) (WithVal.equiv (Rat.padicValuation p)).toRingHom) ⋯ :
        (a : Completion (WithVal (Rat.padicValuation p))) → ℚ_[p])
      (IsDenseInducing.extend ⋯ coe' x)
have hp : Continuous (X := ℚ_[p]) (Y := Completion (WithVal (Rat.padicValuation p))) (IsDenseInducing.extend sorry coe')
Tactic: refine continuous_extension.comp ?_
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.WithVal (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
a q : ℚ
⊢ Eq (α := Completion (WithVal (Rat.padicValuation p))) (IsDenseInducing.extend ⋯ coe' (↑q : ℚ_[p]))
    (↑(((WithVal.equiv (Rat.padicValuation p)).symm : (a : ℚ) → WithVal (Rat.padicValuation p)) q) :
      Completion (WithVal (Rat.padicValuation p)))
have hf : Continuous (X := WithVal (Rat.padicValuation p)) (Y := Completion (WithVal (Rat.padicValuation p))) coe'
Tactic: apply IsDenseInducing.extend_eq
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.WithVal (hypothesis)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
a : ℚ
⊢ Eq (α := ℚ_[p])
    ((extensionHom (RingHom.comp (Rat.castHom ℚ_[p]) (WithVal.equiv (Rat.padicValuation p)).toRingHom) ⋯ :
          (a : Completion (WithVal (Rat.padicValuation p))) → ℚ_[p])
        (IsDenseInducing.extend ⋯ coe' (↑a : ℚ_[p])) :
      ℚ_[p])
    ((↑a : ℚ_[p]) : ℚ_[p])
have this : ∀ (q : ℚ),
  Eq (α := Completion (WithVal (Rat.padicValuation p))) (IsDenseInducing.extend sorry coe' (↑q : ℚ_[p]))
    (↑(((WithVal.equiv (Rat.padicValuation p)).symm : (a : ℚ) → WithVal (Rat.padicValuation p)) q) :
      Completion (WithVal (Rat.padicValuation p)))
Tactic: have : ∀ q : ℚ, Padic.isDenseInducing_cast_withVal.extend coe' q = coe'
        ((WithVal.equiv (Rat.padicValuation p)).symm q) := by
        intro q
        apply IsDenseInducing.extend_eq
        exact continuous_coe _
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.WithVal (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
a : ℚ
this :
  ∀ (q : ℚ),
    Eq (α := Completion (WithVal (Rat.padicValuation p))) (IsDenseInducing.extend ⋯ coe' (↑q : ℚ_[p]))
      (↑(((WithVal.equiv (Rat.padicValuation p)).symm : (a : ℚ) → WithVal (Rat.padicValuation p)) q) :
        Completion (WithVal (Rat.padicValuation p)))
⊢ Eq (α := ℚ_[p])
    ((extensionHom (RingHom.comp (Rat.castHom ℚ_[p]) (WithVal.equiv (Rat.padicValuation p)).toRingHom) ⋯ :
          (a : Completion (WithVal (Rat.padicValuation p))) → ℚ_[p])
        (IsDenseInducing.extend ⋯ coe' (↑a : ℚ_[p])) :
      ℚ_[p])
    ((↑a : ℚ_[p]) : ℚ_[p])
have h : Eq (α := ℚ_[p])
  ((RingHom.comp (Rat.castHom ℚ_[p]) (WithVal.equiv (Rat.padicValuation p)).toRingHom :
        (a : WithVal (Rat.padicValuation p)) → ℚ_[p])
      (((WithVal.equiv (Rat.padicValuation p)).symm : (a : ℚ) → WithVal (Rat.padicValuation p)) a) :
    ℚ_[p])
  ((↑a : ℚ_[p]) : ℚ_[p])
Tactic: rw [this, extensionHom_coe]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.WithVal (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
x : ℚ
⊢ norm (E := ℚ_[p]) (↑x : ℚ_[p]) ≤ 1 ↔
    LE.le (α := WithZero (Multiplicative ℤ))
      ((Rat.padicValuation p : (a : ℚ) → WithZero (Multiplicative ℤ)) x : WithZero (Multiplicative ℤ))
      (1 : WithZero (Multiplicative ℤ))
have rw : norm (E := ℚ_[p]) (↑x : ℚ_[p]) ≤ 1 ↔ ¬p ∣ x.den
Tactic: rw [Rat.padicValuation_le_one_iff]