awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x_1 y : ℤ_[p]
x : ℕ → ℤ_[p]
hx : ∀ {m n : ℕ}, m ≤ n → ‖x m - x n‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑m : ℤ))
x' : CauSeq ℤ_[p] norm := Subtype.mk (p := fun (f : ℕ → ℤ_[p]) ↦ IsCauSeq norm f) x ⋯
n : ℕ
this : 0 < HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ))
i : ℕ
hi :
  ∀ j ≥ i,
    ∀ k ≥ j,
      ‖(↑x' : ℕ → ℤ_[p]) k - (↑(const norm x'.lim) : ℕ → ℤ_[p]) j‖ < HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ))
pos : ∀ (hin : i ≤ n), ‖x n - x'.lim‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ))
⊢ ‖x n - x'.lim‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ))
have neg : ∀ (hin : n < i), ‖x n - x'.lim‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ))
Tactic: by_cases! hin : i ≤ n
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (hypothesis)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x_1 y : ℤ_[p]
x : ℕ → ℤ_[p]
hx : ∀ {m n : ℕ}, m ≤ n → ‖x m - x n‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑m : ℤ))
x' : CauSeq ℤ_[p] norm := Subtype.mk (p := fun (f : ℕ → ℤ_[p]) ↦ IsCauSeq norm f) x ⋯
n : ℕ
this : 0 < HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ))
i : ℕ
hi :
  ∀ j ≥ i,
    ∀ k ≥ j,
      ‖(↑x' : ℕ → ℤ_[p]) k - (↑(const norm x'.lim) : ℕ → ℤ_[p]) j‖ < HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ))
hin : n < i
⊢ ‖x n - x'.lim‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ))
have hi₁ : ‖(↑x' : ℕ → ℤ_[p]) i - (↑(const norm x'.lim) : ℕ → ℤ_[p]) i‖ < HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ))
Tactic: specialize hi i le_rfl i le_rfl
Cleared hypotheses: hi
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (hypothesis)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x_1 y : ℤ_[p]
x : ℕ → ℤ_[p]
hx : ∀ {m n : ℕ}, m ≤ n → ‖x m - x n‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑m : ℤ))
x' : CauSeq ℤ_[p] norm := Subtype.mk (p := fun (f : ℕ → ℤ_[p]) ↦ IsCauSeq norm f) x ⋯
n : ℕ
this : 0 < HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ))
i : ℕ
hin : n < i
hi : ‖(↑x' : ℕ → ℤ_[p]) i - (↑(const norm x'.lim) : ℕ → ℤ_[p]) i‖ < HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ))
⊢ ‖x n - x'.lim‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ))
have hx₁ : ‖x n - x i‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ))
Tactic: specialize hx hin.le
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (hypothesis)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x_1 y : ℤ_[p]
x : ℕ → ℤ_[p]
hx_1 : ∀ {m n : ℕ}, m ≤ n → ‖x m - x n‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑m : ℤ))
x' : CauSeq ℤ_[p] norm := Subtype.mk (p := fun (f : ℕ → ℤ_[p]) ↦ IsCauSeq norm f) x ⋯
n : ℕ
this : 0 < HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ))
i : ℕ
hin : n < i
hi : ‖(↑x' : ℕ → ℤ_[p]) i - (↑(const norm x'.lim) : ℕ → ℤ_[p]) i‖ < HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ))
hx : ‖x n - x i‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ))
⊢ ‖x n - x'.lim‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ))
have this₁ : ‖x n - x i + (x i - x'.lim)‖ ≤ max ‖x n - x i‖ ‖x i - x'.lim‖
Tactic: have := nonarchimedean (x n - x i : ℤ_[p]) (x i - x'.lim)
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (hypothesis)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x_1 y : ℤ_[p]
x : ℕ → ℤ_[p]
hx_1 : ∀ {m n : ℕ}, m ≤ n → ‖x m - x n‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑m : ℤ))
x' : CauSeq ℤ_[p] norm := Subtype.mk (p := fun (f : ℕ → ℤ_[p]) ↦ IsCauSeq norm f) x ⋯
n : ℕ
this_1 : 0 < HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ))
i : ℕ
hin : n < i
hi : ‖(↑x' : ℕ → ℤ_[p]) i - (↑(const norm x'.lim) : ℕ → ℤ_[p]) i‖ < HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ))
hx : ‖x n - x i‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ))
this : ‖x n - x i + (x i - x'.lim)‖ ≤ max ‖x n - x i‖ ‖x i - x'.lim‖
⊢ ‖x n - x'.lim‖ ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑n : ℤ))
have this₁ : ‖x n - x'.lim‖ ≤ max ‖x n - x i‖ ‖x i - x'.lim‖
Tactic: rw [sub_add_sub_cancel] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x_1 y : ℤ_[p]
x_2 : ↥(nonZeroDivisors ℤ_[p])
x : ℤ_[p]
hx : Membership.mem (γ := Submonoid ℤ_[p]) (nonZeroDivisors ℤ_[p]) x
⊢ IsUnit (M := ℚ_[p])
    ((algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p])
        (↑(Subtype.mk (p := fun (x : ℤ_[p]) ↦ Membership.mem (γ := Submonoid ℤ_[p]) (nonZeroDivisors ℤ_[p]) x) x hx) :
          ℤ_[p]) :
      ℚ_[p])
have rwa : IsUnit
  (↑(↑(Subtype.mk (p := fun (x : ℤ_[p]) ↦ Membership.mem (γ := Submonoid ℤ_[p]) (nonZeroDivisors ℤ_[p]) x) x hx) :
        ℤ_[p]) :
    ℚ_[p])
Tactic: rwa [algebraMap_apply, isUnit_iff_ne_zero, PadicInt.coe_ne_zero, ←
      mem_nonZeroDivisors_iff_ne_zero]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x_1 y : ℤ_[p]
x_2 : ↥(nonZeroDivisors ℤ_[p])
x : ℤ_[p]
hx : Membership.mem (γ := Submonoid ℤ_[p]) (nonZeroDivisors ℤ_[p]) x
rwa :
  IsUnit
    (↑(↑(Subtype.mk (p := fun (x : ℤ_[p]) ↦ Membership.mem (γ := Submonoid ℤ_[p]) (nonZeroDivisors ℤ_[p]) x) x hx) :
          ℤ_[p]) :
      ℚ_[p])
⊢ IsUnit (M := ℚ_[p])
    ((algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p])
        (↑(Subtype.mk (p := fun (x : ℤ_[p]) ↦ Membership.mem (γ := Submonoid ℤ_[p]) (nonZeroDivisors ℤ_[p]) x) x hx) :
          ℤ_[p]) :
      ℚ_[p])
have rwa₁ : (↑(↑(Subtype.mk (p := fun (x : ℤ_[p]) ↦ Membership.mem (γ := Submonoid ℤ_[p]) (nonZeroDivisors ℤ_[p]) x) x hx) :
        ℤ_[p]) :
    ℚ_[p]) ≠
  0
Tactic: rwa [algebraMap_apply, isUnit_iff_ne_zero, PadicInt.coe_ne_zero, ←
      mem_nonZeroDivisors_iff_ne_zero]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x_1 y : ℤ_[p]
x_2 : ↥(nonZeroDivisors ℤ_[p])
x : ℤ_[p]
hx : Membership.mem (γ := Submonoid ℤ_[p]) (nonZeroDivisors ℤ_[p]) x
rwa :
  IsUnit
    (↑(↑(Subtype.mk (p := fun (x : ℤ_[p]) ↦ Membership.mem (γ := Submonoid ℤ_[p]) (nonZeroDivisors ℤ_[p]) x) x hx) :
          ℤ_[p]) :
      ℚ_[p])
rwa₁ :
  (↑(↑(Subtype.mk (p := fun (x : ℤ_[p]) ↦ Membership.mem (γ := Submonoid ℤ_[p]) (nonZeroDivisors ℤ_[p]) x) x hx) :
          ℤ_[p]) :
      ℚ_[p]) ≠
    0
⊢ IsUnit (M := ℚ_[p])
    ((algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p])
        (↑(Subtype.mk (p := fun (x : ℤ_[p]) ↦ Membership.mem (γ := Submonoid ℤ_[p]) (nonZeroDivisors ℤ_[p]) x) x hx) :
          ℤ_[p]) :
      ℚ_[p])
have rwa₂ : (↑(Subtype.mk (p := fun (x : ℤ_[p]) ↦ Membership.mem (γ := Submonoid ℤ_[p]) (nonZeroDivisors ℤ_[p]) x) x hx) : ℤ_[p]) ≠
  0
Tactic: rwa [algebraMap_apply, isUnit_iff_ne_zero, PadicInt.coe_ne_zero, ←
      mem_nonZeroDivisors_iff_ne_zero]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x_1 y : ℤ_[p]
x_2 : ↥(nonZeroDivisors ℤ_[p])
x : ℤ_[p]
hx : Membership.mem (γ := Submonoid ℤ_[p]) (nonZeroDivisors ℤ_[p]) x
rwa :
  IsUnit
    (↑(↑(Subtype.mk (p := fun (x : ℤ_[p]) ↦ Membership.mem (γ := Submonoid ℤ_[p]) (nonZeroDivisors ℤ_[p]) x) x hx) :
          ℤ_[p]) :
      ℚ_[p])
rwa₁ :
  (↑(↑(Subtype.mk (p := fun (x : ℤ_[p]) ↦ Membership.mem (γ := Submonoid ℤ_[p]) (nonZeroDivisors ℤ_[p]) x) x hx) :
          ℤ_[p]) :
      ℚ_[p]) ≠
    0
rwa₂ :
  (↑(Subtype.mk (p := fun (x : ℤ_[p]) ↦ Membership.mem (γ := Submonoid ℤ_[p]) (nonZeroDivisors ℤ_[p]) x) x hx) :
      ℤ_[p]) ≠
    0
⊢ IsUnit (M := ℚ_[p])
    ((algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p])
        (↑(Subtype.mk (p := fun (x : ℤ_[p]) ↦ Membership.mem (γ := Submonoid ℤ_[p]) (nonZeroDivisors ℤ_[p]) x) x hx) :
          ℤ_[p]) :
      ℚ_[p])
have rwa₃ : Membership.mem (γ := Submonoid ℤ_[p]) (nonZeroDivisors ℤ_[p])
  (↑(Subtype.mk (p := fun (x : ℤ_[p]) ↦ Membership.mem (γ := Submonoid ℤ_[p]) (nonZeroDivisors ℤ_[p]) x) x hx) : ℤ_[p])
Tactic: rwa [algebraMap_apply, isUnit_iff_ne_zero, PadicInt.coe_ne_zero, ←
      mem_nonZeroDivisors_iff_ne_zero]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x_1 y : ℤ_[p]
x : ℚ_[p]
⊢ ∃ (x_2 : ℤ_[p] × ↥(nonZeroDivisors ℤ_[p])),
    HMul.hMul (β := ℚ_[p]) x ((algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) (↑x_2.2 : ℤ_[p]) : ℚ_[p]) =
      (algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) x_2.1
have pos : ∀ (hx : ‖x‖ ≤ 1),
  ∃ (x_2 : ℤ_[p] × ↥(nonZeroDivisors ℤ_[p])),
    HMul.hMul (β := ℚ_[p]) x ((algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) (↑x_2.2 : ℤ_[p]) : ℚ_[p]) =
      (algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) x_2.1
Tactic: by_cases hx : ‖x‖ ≤ 1
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x_1 y : ℤ_[p]
x : ℚ_[p]
pos :
  ∀ (hx : ‖x‖ ≤ 1),
    ∃ (x_2 : ℤ_[p] × ↥(nonZeroDivisors ℤ_[p])),
      HMul.hMul (β := ℚ_[p]) x ((algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) (↑x_2.2 : ℤ_[p]) : ℚ_[p]) =
        (algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) x_2.1
⊢ ∃ (x_2 : ℤ_[p] × ↥(nonZeroDivisors ℤ_[p])),
    HMul.hMul (β := ℚ_[p]) x ((algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) (↑x_2.2 : ℤ_[p]) : ℚ_[p]) =
      (algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) x_2.1
have neg : ∀ (hx : ¬‖x‖ ≤ 1),
  ∃ (x_2 : ℤ_[p] × ↥(nonZeroDivisors ℤ_[p])),
    HMul.hMul (β := ℚ_[p]) x ((algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) (↑x_2.2 : ℤ_[p]) : ℚ_[p]) =
      (algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) x_2.1
Tactic: by_cases hx : ‖x‖ ≤ 1
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x_1 y : ℤ_[p]
x : ℚ_[p]
hx : ‖x‖ ≤ 1
⊢ ∃ (x_2 : ℤ_[p] × ↥(nonZeroDivisors ℤ_[p])),
    HMul.hMul (β := ℚ_[p]) x ((algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) (↑x_2.2 : ℤ_[p]) : ℚ_[p]) =
      (algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) x_2.1
have h : x * (algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) (↑(Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) x hx, 1).2 : ℤ_[p]) =
  (algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) x hx, 1).1
Tactic: use (⟨x, hx⟩, 1)
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x_1 y : ℤ_[p]
x : ℚ_[p]
hx : ‖x‖ ≤ 1
⊢ x *
      (algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p])
        (↑(Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) x hx, 1).2 : ℤ_[p]) =
    (algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) x hx, 1).1
have h : x * (algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) 1 =
  (algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) x hx, 1).1
Tactic: rw [Submonoid.coe_one, map_one, mul_one, PadicInt.algebraMap_apply, Subtype.coe_mk]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x_1 y : ℤ_[p]
x : ℚ_[p]
hx : ‖x‖ ≤ 1
h :
  x * (algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) 1 =
    (algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) x hx, 1).1
⊢ x *
      (algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p])
        (↑(Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) x hx, 1).2 : ℤ_[p]) =
    (algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) x hx, 1).1
have h₁ : x * 1 = (algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) x hx, 1).1
Tactic: rw [Submonoid.coe_one, map_one, mul_one, PadicInt.algebraMap_apply, Subtype.coe_mk]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x_1 y : ℤ_[p]
x : ℚ_[p]
hx : ‖x‖ ≤ 1
h :
  x * (algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) 1 =
    (algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) x hx, 1).1
h₁ : x * 1 = (algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) x hx, 1).1
⊢ x *
      (algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p])
        (↑(Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) x hx, 1).2 : ℤ_[p]) =
    (algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) x hx, 1).1
have h₂ : x = (algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) x hx, 1).1
Tactic: rw [Submonoid.coe_one, map_one, mul_one, PadicInt.algebraMap_apply, Subtype.coe_mk]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x_1 y : ℤ_[p]
x : ℚ_[p]
hx : ‖x‖ ≤ 1
h :
  x * (algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) 1 =
    (algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) x hx, 1).1
h₁ : x * 1 = (algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) x hx, 1).1
h₂ : x = (algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) x hx, 1).1
⊢ x *
      (algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p])
        (↑(Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) x hx, 1).2 : ℤ_[p]) =
    (algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) x hx, 1).1
have h₃ : x = (↑(Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) x hx, 1).1 : ℚ_[p])
Tactic: rw [Submonoid.coe_one, map_one, mul_one, PadicInt.algebraMap_apply, Subtype.coe_mk]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x_1 y : ℤ_[p]
x : ℚ_[p]
hx : ‖x‖ ≤ 1
h :
  x * (algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) 1 =
    (algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) x hx, 1).1
h₁ : x * 1 = (algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) x hx, 1).1
h₂ : x = (algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) x hx, 1).1
h₃ : x = (↑(Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) x hx, 1).1 : ℚ_[p])
⊢ x *
      (algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p])
        (↑(Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) x hx, 1).2 : ℤ_[p]) =
    (algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) x hx, 1).1
have h₄ : x = x
Tactic: rw [Submonoid.coe_one, map_one, mul_one, PadicInt.algebraMap_apply, Subtype.coe_mk]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (hypothesis)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x_1 y : ℤ_[p]
x : ℚ_[p]
hx : ¬‖x‖ ≤ 1
⊢ ∃ (x_2 : ℤ_[p] × ↥(nonZeroDivisors ℤ_[p])),
    HMul.hMul (β := ℚ_[p]) x ((algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) (↑x_2.2 : ℤ_[p]) : ℚ_[p]) =
      (algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) x_2.1
have hn : let n := (-x.valuation).toNat;
n = (-x.valuation).toNat
Tactic: set n := Int.toNat (-x.valuation) with hn
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x_1 y : ℤ_[p]
x : ℚ_[p]
hx : ¬‖x‖ ≤ 1
n : ℕ := (-x.valuation).toNat
hn : n = (-x.valuation).toNat
⊢ (↑n : ℤ) = -x.valuation
have rw : 0 ≤ -x.valuation
Tactic: rw [hn, Int.toNat_of_nonneg]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x_1 y : ℤ_[p]
x : ℚ_[p]
hx : ¬‖x‖ ≤ 1
n : ℕ := (-x.valuation).toNat
hn : n = (-x.valuation).toNat
⊢ 0 ≤ -x.valuation
have rw : x.valuation ≤ 0
Tactic: rw [Right.nonneg_neg_iff]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (hypothesis)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x_1 y : ℤ_[p]
x : ℚ_[p]
hx : ¬‖x‖ ≤ 1
n : ℕ := (-x.valuation).toNat
hn : n = (-x.valuation).toNat
⊢ x.valuation ≤ 0
have hx₁ : x.valuation < 0
Tactic: rw [Padic.norm_le_one_iff_val_nonneg, not_le] at hx
Cleared hypotheses: hx, n, hn
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (hypothesis)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x_1 y : ℤ_[p]
x : ℚ_[p]
hx : ¬‖x‖ ≤ 1
n : ℕ := (-x.valuation).toNat
hn : n = (-x.valuation).toNat
⊢ ∃ (x_2 : ℤ_[p] × ↥(nonZeroDivisors ℤ_[p])),
    HMul.hMul (β := ℚ_[p]) x ((algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) (↑x_2.2 : ℤ_[p]) : ℚ_[p]) =
      (algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) x_2.1
have hn_coe : (↑n : ℤ) = -x.valuation
Tactic: have hn_coe : (n : ℤ) = -x.valuation := by
        rw [hn, Int.toNat_of_nonneg]
        rw [Right.nonneg_neg_iff]
        rw [Padic.norm_le_one_iff_val_nonneg, not_le] at hx
        exact hx.le
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (hypothesis)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x_1 y : ℤ_[p]
x : ℚ_[p]
hx : ¬‖x‖ ≤ 1
n : ℕ := (-x.valuation).toNat
hn : n = (-x.valuation).toNat
hn_coe : (↑n : ℤ) = -x.valuation
⊢ ∃ (x_2 : ℤ_[p] × ↥(nonZeroDivisors ℤ_[p])),
    HMul.hMul (β := ℚ_[p]) x ((algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) (↑x_2.2 : ℤ_[p]) : ℚ_[p]) =
      (algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) x_2.1
have ha : let a := x * HPow.hPow (α := ℚ_[p]) (↑p : ℚ_[p]) n;
a = x * HPow.hPow (α := ℚ_[p]) (↑p : ℚ_[p]) n
Tactic: set a := x * (p : ℚ_[p]) ^ n with ha
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (hypothesis)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x_1 y : ℤ_[p]
x : ℚ_[p]
hx : ¬‖x‖ ≤ 1
n : ℕ := (-x.valuation).toNat
hn : n = (-x.valuation).toNat
hn_coe : (↑n : ℤ) = -x.valuation
a : ℚ_[p] := x * HPow.hPow (α := ℚ_[p]) (↑p : ℚ_[p]) n
ha : a = x * HPow.hPow (α := ℚ_[p]) (↑p : ℚ_[p]) n
h0 : x = 0
⊢ False
have hx₁ : ¬LE.le (α := ℝ) 0 1
Tactic: rw [h0, norm_zero] at hx
Cleared hypotheses: hx, n, hn, hn_coe, a, ha, h0
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (hypothesis)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x_1 y : ℤ_[p]
x : ℚ_[p]
hx : ¬‖x‖ ≤ 1
n : ℕ := (-x.valuation).toNat
hn : n = (-x.valuation).toNat
hn_coe : (↑n : ℤ) = -x.valuation
a : ℚ_[p] := x * HPow.hPow (α := ℚ_[p]) (↑p : ℚ_[p]) n
ha : a = x * HPow.hPow (α := ℚ_[p]) (↑p : ℚ_[p]) n
⊢ ‖a‖ = 1
have hx₁ : x ≠ 0
Tactic: have hx : x ≠ 0 := by
          intro h0
          rw [h0, norm_zero] at hx
          exact hx zero_le_one
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x_1 y : ℤ_[p]
x : ℚ_[p]
hx_1 : ¬‖x‖ ≤ 1
n : ℕ := (-x.valuation).toNat
hn : n = (-x.valuation).toNat
hn_coe : (↑n : ℤ) = -x.valuation
a : ℚ_[p] := x * HPow.hPow (α := ℚ_[p]) (↑p : ℚ_[p]) n
ha : a = x * HPow.hPow (α := ℚ_[p]) (↑p : ℚ_[p]) n
hx : x ≠ 0
⊢ ‖a‖ = 1
have rw : Ne (α := ℝ) (↑p : ℝ) 0 ∨ -x.valuation + -(↑n : ℤ) ≠ 0 ∨ -x.valuation = 0 ∧ Eq (α := ℤ) (-(↑n : ℤ)) 0
Tactic: rw [ha, padicNormE.mul, Padic.norm_p_pow, Padic.norm_eq_zpow_neg_valuation hx,
          ← zpow_add', hn_coe, neg_neg, neg_add_cancel, zpow_zero]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (hypothesis)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x_1 y : ℤ_[p]
x : ℚ_[p]
hx : ¬‖x‖ ≤ 1
n : ℕ := (-x.valuation).toNat
hn : n = (-x.valuation).toNat
hn_coe : (↑n : ℤ) = -x.valuation
a : ℚ_[p] := x * HPow.hPow (α := ℚ_[p]) (↑p : ℚ_[p]) n
ha : a = x * HPow.hPow (α := ℚ_[p]) (↑p : ℚ_[p]) n
⊢ ∃ (x_2 : ℤ_[p] × ↥(nonZeroDivisors ℤ_[p])),
    HMul.hMul (β := ℚ_[p]) x ((algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) (↑x_2.2 : ℤ_[p]) : ℚ_[p]) =
      (algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) x_2.1
have ha_norm : ‖a‖ = 1
Tactic: have ha_norm : ‖a‖ = 1 := by
        have hx : x ≠ 0 := by
          intro h0
          rw [h0, norm_zero] at hx
          exact hx zero_le_one
        rw [ha, padicNormE.mul, Padic.norm_p_pow, Padic.norm_eq_zpow_neg_valuation hx,
          ← zpow_add', hn_coe, neg_neg, neg_add_cancel, zpow_zero]
        exact Or.inl (Nat.cast_ne_zero.mpr (NeZero.ne p))
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x_1 y : ℤ_[p]
x : ℚ_[p]
hx : ¬‖x‖ ≤ 1
n : ℕ := (-x.valuation).toNat
hn : n = (-x.valuation).toNat
hn_coe : (↑n : ℤ) = -x.valuation
a : ℚ_[p] := x * HPow.hPow (α := ℚ_[p]) (↑p : ℚ_[p]) n
ha : a = x * HPow.hPow (α := ℚ_[p]) (↑p : ℚ_[p]) n
ha_norm : ‖a‖ = 1
⊢ ∃ (x_2 : ℤ_[p] × ↥(nonZeroDivisors ℤ_[p])),
    HMul.hMul (β := ℚ_[p]) x ((algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) (↑x_2.2 : ℤ_[p]) : ℚ_[p]) =
      (algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) x_2.1
have h : HMul.hMul (β := ℚ_[p]) x
    ((algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p])
        (↑(Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) a sorry,
                Subtype.mk (p := fun (x : ℤ_[p]) ↦ Membership.mem (γ := Submonoid ℤ_[p]) (nonZeroDivisors ℤ_[p]) x)
                  (HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n) sorry).2 :
          ℤ_[p]) :
      ℚ_[p]) =
  (algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p])
    (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) a sorry,
        Subtype.mk (p := fun (x : ℤ_[p]) ↦ Membership.mem (γ := Submonoid ℤ_[p]) (nonZeroDivisors ℤ_[p]) x)
          (HPow.hPow (α := ℤ_[p]) (↑p : ℤ_[p]) n) sorry).1
Tactic: use
        (⟨a, le_of_eq ha_norm⟩,
          ⟨(p ^ n : ℤ_[p]), mem_nonZeroDivisors_iff_ne_zero.mpr (NeZero.ne _)⟩)
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x y : ℤ_[p]
⊢ ∀ {x y : ℤ_[p]},
    Eq (α := ℚ_[p]) ((algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) x : ℚ_[p])
        ((algebraMap ℤ_[p] ℚ_[p] : (a : ℤ_[p]) → ℚ_[p]) y : ℚ_[p]) →
      ∃ (c : ↥(nonZeroDivisors ℤ_[p])), (↑c : ℤ_[p]) * x = (↑c : ℤ_[p]) * y
have simp_rw : ∀ {x y : ℤ_[p]}, x = y → ∃ (c : ↥(nonZeroDivisors ℤ_[p])), (↑c : ℤ_[p]) * x = (↑c : ℤ_[p]) * y
Tactic: simp_rw [algebraMap_apply, Subtype.coe_inj]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicIntegers (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
x y x_1 y_1 : ℤ_[p]
h : x_1 = y_1
⊢ (↑1 : ℤ_[p]) * x_1 = (↑1 : ℤ_[p]) * y_1
have rw : (↑1 : ℤ_[p]) * y_1 = (↑1 : ℤ_[p]) * y_1
Tactic: rw [h]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNorm (goal)
Goal:
p : ℕ
q : ℚ
hq : ¬q = 0
⊢ 0 ≤ if q = 0 then 0 else HPow.hPow (α := ℚ) (↑p : ℚ) (-padicValRat p q)
have split_ifs : 0 ≤ HPow.hPow (α := ℚ) (↑p : ℚ) (-padicValRat p q)
Tactic: split_ifs
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNorm (goal)
Goal:
p : ℕ
q : ℚ
hq : ¬q = 0
⊢ 0 ≤ HPow.hPow (α := ℚ) (↑p : ℚ) (-padicValRat p q)
have ha : LE.le (α := ℚ) 0 (↑p : ℚ)
Tactic: apply zpow_nonneg
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNorm (hypothesis)
Goal:
p q : ℕ
p_prime : Fact (Nat.Prime p)
q_prime : Fact (Nat.Prime q)
ne : p ≠ q
⊢ padicNorm p (↑q : ℚ) = 1
have p₁ : padicValRat p (↑q : ℚ) = 0
Tactic: have p : padicValRat p q = 0 := mod_cast padicValNat_primes ne
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNorm (goal)
Goal:
p_1 q : ℕ
p_prime : Fact (Nat.Prime p_1)
q_prime : Fact (Nat.Prime q)
ne : p_1 ≠ q
p : padicValRat p_1 (↑q : ℚ) = 0
⊢ padicNorm p_1 (↑q : ℚ) = 1
have rw : (if Eq (α := ℚ) (↑q : ℚ) 0 then 0 else HPow.hPow (α := ℚ) (β := ℤ) (↑p_1 : ℚ) (-0)) = 1
Tactic: rw [padicNorm, p]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNorm (goal)
Goal:
p : ℕ
hp : 1 < p
⊢ padicNorm p (↑p : ℚ) < 1
have rw : LE.le (α := ℚ) (↑p : ℚ) 0 ∨ LT.lt (α := ℚ) 1 (↑p : ℚ)
Tactic: rw [padicNorm_p hp, inv_lt_one_iff₀]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNorm (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
q : ℚ
hq : q ≠ 0
⊢ padicNorm p q ≠ 0
have rw : HPow.hPow (α := ℚ) (↑p : ℚ) (-padicValRat p q) ≠ 0
Tactic: rw [padicNorm.eq_zpow_of_nonzero hq]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNorm (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
q : ℚ
hq : q ≠ 0
⊢ HPow.hPow (α := ℚ) (↑p : ℚ) (-padicValRat p q) ≠ 0
have a : Ne (α := ℚ) (↑p : ℚ) 0
Tactic: apply zpow_ne_zero
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNorm (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
q : ℚ
h : padicNorm p q = 0
⊢ q = 0
have a : ¬q = 0 → False
Tactic: apply by_contradiction
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNorm (hypothesis)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
q : ℚ
h : (if q = 0 then 0 else HPow.hPow (α := ℚ) (↑p : ℚ) (-padicValRat p q)) = 0
hq : ¬q = 0
⊢ False
have h₁ : HPow.hPow (α := ℚ) (↑p : ℚ) (-padicValRat p q) = 0
Tactic: rw [if_neg hq] at h
Cleared hypotheses: h, hq
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.PadicNorm (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
q : ℚ
h : HPow.hPow (α := ℚ) (↑p : ℚ) (-padicValRat p q) = 0
hq : ¬q = 0
⊢ False
have a : ¬HPow.hPow (α := ℚ) (↑p : ℚ) (-padicValRat p q) = 0
Tactic: apply absurd h