awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.Padic (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
x : ℤ_[p]
hx : Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])}) x
pos : ∀ (hn : n = 0), ‖(↑n ! : ℚ_[p])⁻¹ʳ * (↑x : ℚ_[p]) ^ n‖ ≤ 1
⊢ ‖(↑n ! : ℚ_[p])⁻¹ʳ * (↑x : ℚ_[p]) ^ n‖ ≤ 1
have neg : ∀ (hn : ¬n = 0), ‖(↑n ! : ℚ_[p])⁻¹ʳ * (↑x : ℚ_[p]) ^ n‖ ≤ 1
Tactic: by_cases hn : n = 0
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.Padic (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
x : ℤ_[p]
hx : Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])}) x
hn : ¬n = 0
⊢ ‖(↑n ! : ℚ_[p])⁻¹ʳ * (↑x : ℚ_[p]) ^ n‖ ≤ 1
have neg : LE.le (α := ℝ) (↑p : ℝ)⁻¹ 1
Tactic: apply le_trans (dpow'_norm_le_of_ne_zero p hn hx)
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.Padic (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
x : ℤ_[p]
hx : Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])}) x
hn : ¬n = 0
⊢ LE.le (α := ℝ) (↑p : ℝ)⁻¹ 1
have neg : HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-1) ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) 0
Tactic: rw [← zpow_neg_one, ← zpow_zero ↑p]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.Padic (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
x : ℤ_[p]
hx : Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])}) x
hn : ¬n = 0
⊢ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-1) ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) 0
have neg : LE.le (α := ℝ) 1 (↑p : ℝ)
Tactic: gcongr
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.Padic (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
x : ℤ_[p]
hx : Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])}) x
hn : ¬n = 0
neg : LE.le (α := ℝ) 1 (↑p : ℝ)
⊢ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-1) ≤ HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) 0
have neg₁ : LE.le (α := ℤ) (-1) 0
Tactic: gcongr
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.Padic (hypothesis)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
x : ℤ_[p]
hm : n ≠ 0
hx : Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])}) x
⊢ Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])})
    (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) (dpow' p n (↑x : ℚ_[p])) ⋯)
have hiff : norm (E := ℤ_[p]) (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) (dpow' p n (↑x : ℚ_[p])) sorry) ≤
    HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑1 : ℤ)) ↔
  Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {HPow.hPow (α := ℤ_[p]) (β := ℕ) (↑p : ℤ_[p]) 1})
    (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) (dpow' p n (↑x : ℚ_[p])) sorry)
Tactic: have hiff := PadicInt.norm_le_pow_iff_mem_span_pow ⟨dpow' p n x, dpow'_int p n hx⟩ 1
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.Padic (hypothesis)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
x : ℤ_[p]
hm : n ≠ 0
hx : Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])}) x
hiff :
  norm (E := ℤ_[p]) (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) (dpow' p n (↑x : ℚ_[p])) ⋯) ≤
      HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑1 : ℤ)) ↔
    Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {HPow.hPow (α := ℤ_[p]) (β := ℕ) (↑p : ℤ_[p]) 1})
      (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) (dpow' p n (↑x : ℚ_[p])) ⋯)
⊢ Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])})
    (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) (dpow' p n (↑x : ℚ_[p])) ⋯)
have hiff₁ : norm (E := ℤ_[p]) (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) (dpow' p n (↑x : ℚ_[p])) sorry) ≤
    HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑1 : ℤ)) ↔
  Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])})
    (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) (dpow' p n (↑x : ℚ_[p])) sorry)
Tactic: rw [pow_one] at hiff
Cleared hypotheses: hiff
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.Padic (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
x : ℤ_[p]
hm : n ≠ 0
hx : Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])}) x
hiff :
  norm (E := ℤ_[p]) (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) (dpow' p n (↑x : ℚ_[p])) ⋯) ≤
      HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑1 : ℤ)) ↔
    Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])})
      (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) (dpow' p n (↑x : ℚ_[p])) ⋯)
⊢ Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])})
    (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) (dpow' p n (↑x : ℚ_[p])) ⋯)
have rw : norm (E := ℤ_[p]) (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) (dpow' p n (↑x : ℚ_[p])) sorry) ≤
  HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑1 : ℤ))
Tactic: rw [← hiff]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.Padic (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
x : ℤ_[p]
hm : n ≠ 0
hx : Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])}) x
hiff :
  norm (E := ℤ_[p]) (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) (dpow' p n (↑x : ℚ_[p])) ⋯) ≤
      HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑1 : ℤ)) ↔
    Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])})
      (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) (dpow' p n (↑x : ℚ_[p])) ⋯)
⊢ norm (E := ℤ_[p]) (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) (dpow' p n (↑x : ℚ_[p])) ⋯) ≤
    HPow.hPow (α := ℝ) (β := ℤ) (↑p : ℝ) (-(↑1 : ℤ))
have simp : norm (E := ℤ_[p]) (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) (dpow' p n (↑x : ℚ_[p])) sorry) ≤ (↑p : ℝ)⁻¹
Tactic: simp only [cast_one, zpow_neg_one]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.Padic (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
⊢ DividedPowers (A := ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])})
have refine_1 : Eq (α := Ideal ℚ_[p]) (Ideal.map (R := ℤ_[p]) (F := ℤ_[p] →+* ℚ_[p]) Coe.ringHom (Ideal.span {(↑p : ℤ_[p])})) ⊤
Tactic: refine ofInjective (Ideal.span {(p : ℤ_[p])}) (⊤)
    PadicInt.Coe.ringHom ((Set.injective_codRestrict Subtype.property).mp fun ⦃a₁ a₂⦄ a ↦ a)
    (RatAlgebra.dividedPowers (⊤ : Ideal ℚ_[p])) ?_ ?_
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.Padic (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
refine_1 :
  Eq (α := Ideal ℚ_[p]) (Ideal.map (R := ℤ_[p]) (F := ℤ_[p] →+* ℚ_[p]) Coe.ringHom (Ideal.span {(↑p : ℤ_[p])})) ⊤
⊢ DividedPowers (A := ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])})
have refine_2 : ∀ (n : ℕ) {x : ℤ_[p]},
  Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])}) x →
    ∃ (y : ℤ_[p]) (_ : n ≠ 0 → Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])}) y),
      Eq (α := ℚ_[p]) ((Coe.ringHom : (a : ℤ_[p]) → ℚ_[p]) y : ℚ_[p])
        (DividedPowers.dpow (I := ⊤) (RatAlgebra.dividedPowers ⊤) n ((Coe.ringHom : (a : ℤ_[p]) → ℚ_[p]) x) : ℚ_[p])
Tactic: refine ofInjective (Ideal.span {(p : ℤ_[p])}) (⊤)
    PadicInt.Coe.ringHom ((Set.injective_codRestrict Subtype.property).mp fun ⦃a₁ a₂⦄ a ↦ a)
    (RatAlgebra.dividedPowers (⊤ : Ideal ℚ_[p])) ?_ ?_
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.Padic (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
⊢ Eq (α := Ideal ℚ_[p]) (Ideal.map (R := ℤ_[p]) (F := ℤ_[p] →+* ℚ_[p]) Coe.ringHom (Ideal.span {(↑p : ℤ_[p])})) ⊤
have refine_1 : Eq (α := Ideal ℚ_[p]) (Ideal.span {(↑p : ℚ_[p])}) ⊤
Tactic: rw [Ideal.map_span, Set.image_singleton, map_natCast]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.Padic (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
⊢ Eq (α := Ideal ℚ_[p]) (Ideal.span {(↑p : ℚ_[p])}) ⊤
have refine_1 : ¬p = 0
Tactic: simp only [Ideal.span_singleton_eq_top, isUnit_iff_ne_zero, ne_eq, cast_eq_zero]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.Padic (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
⊢ DividedPowers (A := ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])})
have refine_1 : Eq (α := Ideal ℚ_[p]) (Ideal.map (R := ℤ_[p]) (F := ℤ_[p] →+* ℚ_[p]) Coe.ringHom (Ideal.span {(↑p : ℤ_[p])})) ⊤
Tactic: classical
  refine ofInjective (Ideal.span {(p : ℤ_[p])}) (⊤)
    PadicInt.Coe.ringHom ((Set.injective_codRestrict Subtype.property).mp fun ⦃a₁ a₂⦄ a ↦ a)
    (RatAlgebra.dividedPowers (⊤ : Ideal ℚ_[p])) ?_ ?_
  · rw [Ideal.map_span, Set.image_singleton, map_natCast]
    simp only [Ideal.span_singleton_eq_top, isUnit_iff_ne_zero, ne_eq, cast_eq_zero]
    exact Nat.Prime.ne_zero hp.elim
  · intro n x hx
    exact ⟨⟨dpow' p n x, dpow'_int p n hx⟩, fun hn ↦ dpow'_mem p hn hx, by
      simp [dpow', inverse_eq_inv', Coe.ringHom_apply, RatAlgebra.dpow_apply,
        Submodule.mem_top, ↓reduceIte]⟩
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.Padic (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
refine_1 :
  Eq (α := Ideal ℚ_[p]) (Ideal.map (R := ℤ_[p]) (F := ℤ_[p] →+* ℚ_[p]) Coe.ringHom (Ideal.span {(↑p : ℤ_[p])})) ⊤
⊢ DividedPowers (A := ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])})
have refine_2 : ∀ (n : ℕ) {x : ℤ_[p]},
  Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])}) x →
    ∃ (y : ℤ_[p]) (_ : n ≠ 0 → Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])}) y),
      Eq (α := ℚ_[p]) ((Coe.ringHom : (a : ℤ_[p]) → ℚ_[p]) y : ℚ_[p])
        (DividedPowers.dpow (I := ⊤) (RatAlgebra.dividedPowers ⊤) n ((Coe.ringHom : (a : ℤ_[p]) → ℚ_[p]) x) : ℚ_[p])
Tactic: classical
  refine ofInjective (Ideal.span {(p : ℤ_[p])}) (⊤)
    PadicInt.Coe.ringHom ((Set.injective_codRestrict Subtype.property).mp fun ⦃a₁ a₂⦄ a ↦ a)
    (RatAlgebra.dividedPowers (⊤ : Ideal ℚ_[p])) ?_ ?_
  · rw [Ideal.map_span, Set.image_singleton, map_natCast]
    simp only [Ideal.span_singleton_eq_top, isUnit_iff_ne_zero, ne_eq, cast_eq_zero]
    exact Nat.Prime.ne_zero hp.elim
  · intro n x hx
    exact ⟨⟨dpow' p n x, dpow'_int p n hx⟩, fun hn ↦ dpow'_mem p hn hx, by
      simp [dpow', inverse_eq_inv', Coe.ringHom_apply, RatAlgebra.dpow_apply,
        Submodule.mem_top, ↓reduceIte]⟩
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.Padic (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
x : ℤ_[p]
⊢ Eq (α := ℤ_[p])
    (if hx : Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])}) x then
      Exists.choose (p := fun (a : ℤ_[p]) ↦
        ∃ (_ : n ≠ 0 → Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])}) a),
          Eq (α := ℚ_[p]) ((Coe.ringHom : (a : ℤ_[p]) → ℚ_[p]) a : ℚ_[p])
            (DividedPowers.dpow (I := ⊤) (RatAlgebra.dividedPowers ⊤) n ((Coe.ringHom : (a : ℤ_[p]) → ℚ_[p]) x) :
              ℚ_[p]))
        ⋯
    else 0)
    (if hx : Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])}) x then
      Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) (dpow' p n (↑x : ℚ_[p])) ⋯
    else 0)
have pos : ∀ (hx : Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])}) x),
  Eq (α := ℤ_[p])
    (Exists.choose (p := fun (a : ℤ_[p]) ↦
      ∃ (_ : n ≠ 0 → Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])}) a),
        Eq (α := ℚ_[p]) ((Coe.ringHom : (a : ℤ_[p]) → ℚ_[p]) a : ℚ_[p])
          (DividedPowers.dpow (I := ⊤) (RatAlgebra.dividedPowers ⊤) n ((Coe.ringHom : (a : ℤ_[p]) → ℚ_[p]) x) : ℚ_[p]))
      sorry)
    (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) (dpow' p n (↑x : ℚ_[p])) sorry)
Tactic: split_ifs with hx
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.Padic (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
x : ℤ_[p]
pos :
  ∀ (hx : Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])}) x),
    Eq (α := ℤ_[p])
      (Exists.choose (p := fun (a : ℤ_[p]) ↦
        ∃ (_ : n ≠ 0 → Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])}) a),
          Eq (α := ℚ_[p]) ((Coe.ringHom : (a : ℤ_[p]) → ℚ_[p]) a : ℚ_[p])
            (DividedPowers.dpow (I := ⊤) (RatAlgebra.dividedPowers ⊤) n ((Coe.ringHom : (a : ℤ_[p]) → ℚ_[p]) x) :
              ℚ_[p]))
        ⋯)
      (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) (dpow' p n (↑x : ℚ_[p])) ⋯)
⊢ Eq (α := ℤ_[p])
    (if hx : Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])}) x then
      Exists.choose (p := fun (a : ℤ_[p]) ↦
        ∃ (_ : n ≠ 0 → Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])}) a),
          Eq (α := ℚ_[p]) ((Coe.ringHom : (a : ℤ_[p]) → ℚ_[p]) a : ℚ_[p])
            (DividedPowers.dpow (I := ⊤) (RatAlgebra.dividedPowers ⊤) n ((Coe.ringHom : (a : ℤ_[p]) → ℚ_[p]) x) :
              ℚ_[p]))
        ⋯
    else 0)
    (if hx : Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])}) x then
      Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) (dpow' p n (↑x : ℚ_[p])) ⋯
    else 0)
have neg : ∀ (hx : x ∉ Ideal.span {(↑p : ℤ_[p])}), Eq (α := ℤ_[p]) 0 0
Tactic: split_ifs with hx
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.Padic (hypothesis)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
x : ℤ_[p]
hx : Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])}) x
⊢ Eq (α := ℤ_[p])
    (Exists.choose (p := fun (a : ℤ_[p]) ↦
      ∃ (_ : n ≠ 0 → Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])}) a),
        Eq (α := ℚ_[p]) ((Coe.ringHom : (a : ℤ_[p]) → ℚ_[p]) a : ℚ_[p])
          (DividedPowers.dpow (I := ⊤) (RatAlgebra.dividedPowers ⊤) n ((Coe.ringHom : (a : ℤ_[p]) → ℚ_[p]) x) : ℚ_[p]))
      ⋯)
    (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) (dpow' p n (↑x : ℚ_[p])) ⋯)
have hinj : Injective (α := ℤ_[p]) (β := ℚ_[p]) (⇑Coe.ringHom : (a : ℤ_[p]) → ℚ_[p])
Tactic: have hinj : Injective (PadicInt.Coe.ringHom (p := p)) :=
      (Set.injective_codRestrict Subtype.property).mp fun ⦃a₁ a₂⦄ a ↦ a
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.Padic (hypothesis)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
x : ℤ_[p]
hx : Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])}) x
hinj : Injective (α := ℤ_[p]) (β := ℚ_[p]) (⇑Coe.ringHom : (a : ℤ_[p]) → ℚ_[p])
⊢ Eq (α := ℤ_[p])
    (Exists.choose (p := fun (a : ℤ_[p]) ↦
      ∃ (_ : n ≠ 0 → Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])}) a),
        Eq (α := ℚ_[p]) ((Coe.ringHom : (a : ℤ_[p]) → ℚ_[p]) a : ℚ_[p])
          (DividedPowers.dpow (I := ⊤) (RatAlgebra.dividedPowers ⊤) n ((Coe.ringHom : (a : ℤ_[p]) → ℚ_[p]) x) : ℚ_[p]))
      ⋯)
    (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) (dpow' p n (↑x : ℚ_[p])) ⋯)
have heq : (Coe.ringHom : (a : ℤ_[p]) → ℚ_[p]) (Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) (dpow' p n (↑x : ℚ_[p])) sorry) =
  (↑n ! : ℚ_[p])⁻¹ʳ * HPow.hPow (α := ℚ_[p]) ((Coe.ringHom : (a : ℤ_[p]) → ℚ_[p]) x : ℚ_[p]) n
Tactic: have heq : Coe.ringHom ⟨dpow' p n x, dpow'_int p n hx⟩ =
        inverse (n ! : ℚ_[p]) * Coe.ringHom x ^ n := by
      simp [dpow', inverse_eq_inv', Coe.ringHom_apply]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.Padic (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
x : ℤ_[p]
⊢ (↑(DividedPowers.dpow (dividedPowers p) n x) : ℚ_[p]) =
    if x_1 : Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])}) x then (↑n ! : ℚ_[p])⁻¹ʳ * (↑x : ℚ_[p]) ^ n
    else 0
have simp : (↑(if hx : Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])}) x then
        Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) ((↑n ! : ℚ_[p])⁻¹ * (↑x : ℚ_[p]) ^ n) sorry
      else 0) :
    ℚ_[p]) =
  if Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])}) x then (↑n ! : ℚ_[p])⁻¹ * (↑x : ℚ_[p]) ^ n else 0
Tactic: simp only [dividedPowers_eq, dpow', inverse_eq_inv', dite_eq_ite]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.Padic (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
x : ℤ_[p]
⊢ (↑(if hx : Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])}) x then
          Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) ((↑n ! : ℚ_[p])⁻¹ * (↑x : ℚ_[p]) ^ n) ⋯
        else 0) :
      ℚ_[p]) =
    if Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])}) x then (↑n ! : ℚ_[p])⁻¹ * (↑x : ℚ_[p]) ^ n else 0
have pos : ∀ (h : Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])}) x),
  (↑(Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) ((↑n ! : ℚ_[p])⁻¹ * (↑x : ℚ_[p]) ^ n) sorry) : ℚ_[p]) =
    (↑n ! : ℚ_[p])⁻¹ * (↑x : ℚ_[p]) ^ n
Tactic: split_ifs
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.Padic (goal)
Goal:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
x : ℤ_[p]
pos :
  ∀ (h : Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])}) x),
    (↑(Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) ((↑n ! : ℚ_[p])⁻¹ * (↑x : ℚ_[p]) ^ n) ⋯) : ℚ_[p]) =
      (↑n ! : ℚ_[p])⁻¹ * (↑x : ℚ_[p]) ^ n
⊢ (↑(if hx : Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])}) x then
          Subtype.mk (p := fun (x : ℚ_[p]) ↦ ‖x‖ ≤ 1) ((↑n ! : ℚ_[p])⁻¹ * (↑x : ℚ_[p]) ^ n) ⋯
        else 0) :
      ℚ_[p]) =
    if Membership.mem (γ := Ideal ℤ_[p]) (Ideal.span {(↑p : ℤ_[p])}) x then (↑n ! : ℚ_[p])⁻¹ * (↑x : ℚ_[p]) ^ n else 0
have neg : x ∉ Ideal.span {(↑p : ℤ_[p])} → Eq (α := ℚ_[p]) (↑0 : ℚ_[p]) 0
Tactic: split_ifs
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.RatAlgebra (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
inst_1 : DecidablePred fun (x : A) ↦ x ∈ I
m : ℕ
hm : m ≠ 0
x : A
hx : x ∈ I
⊢ dpow I m x ∈ I
have rw : (↑m ! : A)⁻¹ʳ * x ^ m ∈ I
Tactic: rw [dpow_eq_of_mem hx]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.RatAlgebra (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
inst_1 : DecidablePred fun (x : A) ↦ x ∈ I
n : ℕ
hn_fac : IsUnit.{u_1} (M := A) (↑(n - 1)! : A)
m : ℕ
hmn : m < n
x y : A
hx : x ∈ I
hy : y ∈ I
⊢ dpow I m (x + y) = ∑ k ∈ @Finset.antidiagonal _ instAddMonoid _ m, dpow I k.1 x * dpow I k.2 y
have rw : (↑m ! : A)⁻¹ʳ * (x + y) ^ m = ∑ k ∈ @Finset.antidiagonal _ instAddMonoid _ m, dpow I k.1 x * dpow I k.2 y
Tactic: rw [dpow_eq_of_mem (Ideal.add_mem I hx hy)]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.RatAlgebra (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
inst_1 : DecidablePred fun (x : A) ↦ x ∈ I
n : ℕ
hn_fac : IsUnit.{u_1} (M := A) (↑(n - 1)! : A)
m : ℕ
hmn : m < n
x y : A
hx : x ∈ I
hy : y ∈ I
⊢ (↑m ! : A)⁻¹ʳ * (x + y) ^ m = ∑ k ∈ @Finset.antidiagonal _ instAddMonoid _ m, dpow I k.1 x * dpow I k.2 y
have simp : (↑m ! : A)⁻¹ʳ * (x + y) ^ m =
  ∑ x_1 ∈ @Finset.antidiagonal _ instAddMonoid _ m,
    (if x ∈ I then (↑x_1.1! : A)⁻¹ʳ * x ^ x_1.1 else 0) * if y ∈ I then (↑x_1.2! : A)⁻¹ʳ * y ^ x_1.2 else 0
Tactic: simp only [dpow]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.RatAlgebra (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
inst_1 : DecidablePred fun (x : A) ↦ x ∈ I
n : ℕ
hn_fac : IsUnit.{u_1} (M := A) (↑(n - 1)! : A)
m : ℕ
hmn : m < n
x y : A
hx : x ∈ I
hy : y ∈ I
⊢ (↑m ! : A)⁻¹ʳ * (x + y) ^ m =
    ∑ x_1 ∈ @Finset.antidiagonal _ instAddMonoid _ m,
      (if x ∈ I then (↑x_1.1! : A)⁻¹ʳ * x ^ x_1.1 else 0) * if y ∈ I then (↑x_1.2! : A)⁻¹ʳ * y ^ x_1.2 else 0
have rw : Eq.{u_1 + 1} (α := A) (∑ m_1 ∈ @Finset.antidiagonal _ instAddMonoid _ m, m.choose m_1.1 • (x ^ m_1.1 * y ^ m_1.2))
  (∑ i ∈ @Finset.antidiagonal _ instAddMonoid _ m,
    (↑m ! : A) * ((if x ∈ I then (↑i.1! : A)⁻¹ʳ * x ^ i.1 else 0) * if y ∈ I then (↑i.2! : A)⁻¹ʳ * y ^ i.2 else 0))
Tactic: rw [inverse_mul_eq_iff_eq_mul _ _ _ (hn_fac.natCast_factorial_of_lt hmn),
    Finset.mul_sum, Commute.add_pow' (Commute.all _ _)]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.RatAlgebra (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
inst_1 : DecidablePred fun (x : A) ↦ x ∈ I
n : ℕ
hn_fac : IsUnit.{u_1} (M := A) (↑(n - 1)! : A)
m : ℕ
hmn : m < n
x y : A
hx : x ∈ I
hy : y ∈ I
⊢ Eq.{u_1 + 1} (α := A) (∑ m_1 ∈ @Finset.antidiagonal _ instAddMonoid _ m, m.choose m_1.1 • (x ^ m_1.1 * y ^ m_1.2))
    (∑ i ∈ @Finset.antidiagonal _ instAddMonoid _ m,
      (↑m ! : A) * ((if x ∈ I then (↑i.1! : A)⁻¹ʳ * x ^ i.1 else 0) * if y ∈ I then (↑i.2! : A)⁻¹ʳ * y ^ i.2 else 0))
have apply : ∀ x_1 ∈ @Finset.antidiagonal _ instAddMonoid _ m,
  m.choose x_1.1 • (x ^ x_1.1 * y ^ x_1.2) =
    (↑m ! : A) *
      ((if x ∈ I then (↑x_1.1! : A)⁻¹ʳ * x ^ x_1.1 else 0) * if y ∈ I then (↑x_1.2! : A)⁻¹ʳ * y ^ x_1.2 else 0)
Tactic: apply Finset.sum_congr rfl
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.RatAlgebra (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
inst_1 : DecidablePred fun (x : A) ↦ x ∈ I
n : ℕ
hn_fac : IsUnit.{u_1} (M := A) (↑(n - 1)! : A)
m : ℕ
hmn : m < n
x y : A
hx : x ∈ I
hy : y ∈ I
k : ℕ × ℕ
hk : k ∈ @Finset.antidiagonal _ instAddMonoid _ m
⊢ m.choose k.1 • (x ^ k.1 * y ^ k.2) =
    (↑m ! : A) * ((if x ∈ I then (↑k.1! : A)⁻¹ʳ * x ^ k.1 else 0) * if y ∈ I then (↑k.2! : A)⁻¹ʳ * y ^ k.2 else 0)
have rw : m.choose k.1 • (x ^ k.1 * y ^ k.2) = (↑m ! : A) * ((↑k.1! : A)⁻¹ʳ * x ^ k.1 * ((↑k.2! : A)⁻¹ʳ * y ^ k.2))
Tactic: rw [if_pos hx, if_pos hy]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.RatAlgebra (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
inst_1 : DecidablePred fun (x : A) ↦ x ∈ I
n : ℕ
hn_fac : IsUnit.{u_1} (M := A) (↑(n - 1)! : A)
m : ℕ
hmn : m < n
x y : A
hx : x ∈ I
hy : y ∈ I
k : ℕ × ℕ
hk : k ∈ @Finset.antidiagonal _ instAddMonoid _ m
⊢ m.choose k.1 • (x ^ k.1 * y ^ k.2) = (↑m ! : A) * ((↑k.1! : A)⁻¹ʳ * x ^ k.1 * ((↑k.2! : A)⁻¹ʳ * y ^ k.2))
have ring_nf : x ^ k.1 * y ^ k.2 * (↑(m.choose k.1) : A) = x ^ k.1 * y ^ k.2 * (↑m ! : A) * (↑k.1! : A)⁻¹ʳ * (↑k.2! : A)⁻¹ʳ
Tactic: ring_nf
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.RatAlgebra (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
inst_1 : DecidablePred fun (x : A) ↦ x ∈ I
n : ℕ
hn_fac : IsUnit.{u_1} (M := A) (↑(n - 1)! : A)
m : ℕ
hmn : m < n
x y : A
hx : x ∈ I
hy : y ∈ I
k : ℕ × ℕ
hk : k ∈ @Finset.antidiagonal _ instAddMonoid _ m
⊢ x ^ k.1 * y ^ k.2 * (↑(m.choose k.1) : A) = x ^ k.1 * y ^ k.2 * (↑m ! : A) * (↑k.1! : A)⁻¹ʳ * (↑k.2! : A)⁻¹ʳ
have simp : x ^ k.1 * (y ^ k.2 * (↑(m.choose k.1) : A)) =
  x ^ k.1 * (y ^ k.2 * ((↑m ! : A) * HMul.hMul (α := A) (↑k.1! : A)⁻¹ʳ (↑k.2! : A)⁻¹ʳ))
Tactic: simp only [mul_assoc]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.RatAlgebra (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
inst_1 : DecidablePred fun (x : A) ↦ x ∈ I
n : ℕ
hn_fac : IsUnit.{u_1} (M := A) (↑(n - 1)! : A)
m : ℕ
hmn : m < n
x y : A
hx : x ∈ I
hy : y ∈ I
k : ℕ × ℕ
hk : k ∈ @Finset.antidiagonal _ instAddMonoid _ m
⊢ x ^ k.1 * (y ^ k.2 * (↑(m.choose k.1) : A)) =
    x ^ k.1 * (y ^ k.2 * ((↑m ! : A) * HMul.hMul (α := A) (↑k.1! : A)⁻¹ʳ (↑k.2! : A)⁻¹ʳ))
have e_a : (↑(m.choose k.1) : A) = (↑m ! : A) * HMul.hMul (α := A) (↑k.1! : A)⁻¹ʳ (↑k.2! : A)⁻¹ʳ
Tactic: congr
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.RatAlgebra (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
inst_1 : DecidablePred fun (x : A) ↦ x ∈ I
n : ℕ
hn_fac : IsUnit.{u_1} (M := A) (↑(n - 1)! : A)
m : ℕ
hmn : m < n
x y : A
hx : x ∈ I
hy : y ∈ I
k : ℕ × ℕ
hk : k ∈ @Finset.antidiagonal _ instAddMonoid _ m
⊢ (↑(m.choose k.1) : A) = (↑m ! : A) * HMul.hMul (α := A) (↑k.1! : A)⁻¹ʳ (↑k.2! : A)⁻¹ʳ
have e_a : (↑(m.choose k.1) : A) = HMul.hMul (α := A) (↑m ! : A) (↑k.1! : A)⁻¹ʳ * (↑k.2! : A)⁻¹ʳ
Tactic: rw [← mul_assoc]
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.RatAlgebra (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
inst_1 : DecidablePred fun (x : A) ↦ x ∈ I
n : ℕ
hn_fac : IsUnit.{u_1} (M := A) (↑(n - 1)! : A)
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
⊢ dpow I m (x + y) = ∑ k ∈ @Finset.antidiagonal _ instAddMonoid _ m, dpow I k.1 x * dpow I k.2 y
have pos : ∀ (hmn : m < n), dpow I m (x + y) = ∑ k ∈ @Finset.antidiagonal _ instAddMonoid _ m, dpow I k.1 x * dpow I k.2 y
Tactic: by_cases! hmn : m < n
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.RatAlgebra (goal)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
inst_1 : DecidablePred fun (x : A) ↦ x ∈ I
n : ℕ
hn_fac : IsUnit.{u_1} (M := A) (↑(n - 1)! : A)
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
pos : ∀ (hmn : m < n), dpow I m (x + y) = ∑ k ∈ @Finset.antidiagonal _ instAddMonoid _ m, dpow I k.1 x * dpow I k.2 y
⊢ dpow I m (x + y) = ∑ k ∈ @Finset.antidiagonal _ instAddMonoid _ m, dpow I k.1 x * dpow I k.2 y
have neg : ∀ (hmn : n ≤ m), dpow I m (x + y) = ∑ k ∈ @Finset.antidiagonal _ instAddMonoid _ m, dpow I k.1 x * dpow I k.2 y
Tactic: by_cases! hmn : m < n
-- Lean 4 proof step from Mathlib.RingTheory.DividedPowers.RatAlgebra (hypothesis)
Goal:
A : Type u_1
inst : CommSemiring A
I : Ideal A
inst_1 : DecidablePred fun (x : A) ↦ x ∈ I
n : ℕ
hn_fac : IsUnit.{u_1} (M := A) (↑(n - 1)! : A)
hnI : I ^ n = 0
m : ℕ
x : A
hx : x ∈ I
y : A
hy : y ∈ I
hmn : n ≤ m
⊢ dpow I m (x + y) = ∑ k ∈ @Finset.antidiagonal _ instAddMonoid _ m, dpow I k.1 x * dpow I k.2 y
have h_sub : I ^ m ≤ I ^ n
Tactic: have h_sub : I ^ m ≤ I ^ n := Ideal.pow_le_pow_right hmn