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)
x : ℚ
⊢ norm (E := ℚ_[p]) (↑x : ℚ_[p]) ≤ 1 ↔ ¬p ∣ x.den
have refine : ∀ (h : norm (E := ℚ_[p]) (↑x : ℚ_[p]) ≤ 1), ¬p ∣ x.den
Tactic: refine ⟨fun h ↦ ?_, fun h ↦ Padic.norm_rat_le_one h⟩
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.WithVal (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
⊢ IsClosed (X := Completion (WithVal (Rat.padicValuation p)))
    {a : Completion (WithVal (Rat.padicValuation p)) |
      norm (E := ℚ_[p]) ((withValUniformEquiv : (a : (Rat.padicValuation p).Completion) → ℚ_[p]) a : ℚ_[p]) ≤ 1 ↔
        LE.le (α := WithZero (Multiplicative ℤ))
          ((Valued.v : (a : (Rat.padicValuation p).Completion) → WithZero (Multiplicative ℤ)) a :
            WithZero (Multiplicative ℤ))
          (1 : WithZero (Multiplicative ℤ))}
have hp : IsClosed (X := Completion (WithVal (Rat.padicValuation p)))
  {x : Completion (WithVal (Rat.padicValuation p)) |
    LE.le (α := WithZero (Multiplicative ℤ))
        ((Valued.v : (a : (Rat.padicValuation p).Completion) → WithZero (Multiplicative ℤ)) x :
          WithZero (Multiplicative ℤ))
        (1 : WithZero (Multiplicative ℤ)) ↔
      norm (E := ℚ_[p]) ((withValUniformEquiv : (a : (Rat.padicValuation p).Completion) → ℚ_[p]) x : ℚ_[p]) ≤ 1}
Tactic: rw [Set.ext fun _ ↦ Iff.comm]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.WithVal (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
⊢ IsClosed (X := Completion (WithVal (Rat.padicValuation p)))
    {x : Completion (WithVal (Rat.padicValuation p)) |
      LE.le (α := WithZero (Multiplicative ℤ))
          ((Valued.v : (a : (Rat.padicValuation p).Completion) → WithZero (Multiplicative ℤ)) x :
            WithZero (Multiplicative ℤ))
          (1 : WithZero (Multiplicative ℤ)) ↔
        norm (E := ℚ_[p]) ((withValUniformEquiv : (a : (Rat.padicValuation p).Completion) → ℚ_[p]) x : ℚ_[p]) ≤ 1}
have hp : IsClosed (X := Completion (WithVal (Rat.padicValuation p)))
  {x : Completion (WithVal (Rat.padicValuation p)) |
    LE.le (α :=
        MonoidWithZeroHom.ValueGroup₀ (F := (Rat.padicValuation p).Completion →*₀ WithZero (Multiplicative ℤ))
          (↑Valued.v : (Rat.padicValuation p).Completion →*₀ WithZero (Multiplicative ℤ)))
        ((Valuation.restrict Valued.v :
              (a : (Rat.padicValuation p).Completion) →
                MonoidWithZeroHom.ValueGroup₀
                  (↑Valued.v : (Rat.padicValuation p).Completion →*₀ WithZero (Multiplicative ℤ)))
            x :
          MonoidWithZeroHom.ValueGroup₀ (F := (Rat.padicValuation p).Completion →*₀ WithZero (Multiplicative ℤ))
            (↑Valued.v : (Rat.padicValuation p).Completion →*₀ WithZero (Multiplicative ℤ)))
        (1 :
          MonoidWithZeroHom.ValueGroup₀ (F := (Rat.padicValuation p).Completion →*₀ WithZero (Multiplicative ℤ))
            (↑Valued.v : (Rat.padicValuation p).Completion →*₀ WithZero (Multiplicative ℤ))) ↔
      norm (E := ℚ_[p]) ((withValUniformEquiv : (a : (Rat.padicValuation p).Completion) → ℚ_[p]) x : ℚ_[p]) ≤ 1}
Tactic: simp_rw [← Valuation.restrict_le_one_iff Valued.v]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.WithVal (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
⊢ IsClosed (X := Completion (WithVal (Rat.padicValuation p)))
    {x : Completion (WithVal (Rat.padicValuation p)) |
      LE.le (α :=
          MonoidWithZeroHom.ValueGroup₀ (F := (Rat.padicValuation p).Completion →*₀ WithZero (Multiplicative ℤ))
            (↑Valued.v : (Rat.padicValuation p).Completion →*₀ WithZero (Multiplicative ℤ)))
          ((Valuation.restrict Valued.v :
                (a : (Rat.padicValuation p).Completion) →
                  MonoidWithZeroHom.ValueGroup₀
                    (↑Valued.v : (Rat.padicValuation p).Completion →*₀ WithZero (Multiplicative ℤ)))
              x :
            MonoidWithZeroHom.ValueGroup₀ (F := (Rat.padicValuation p).Completion →*₀ WithZero (Multiplicative ℤ))
              (↑Valued.v : (Rat.padicValuation p).Completion →*₀ WithZero (Multiplicative ℤ)))
          (1 :
            MonoidWithZeroHom.ValueGroup₀ (F := (Rat.padicValuation p).Completion →*₀ WithZero (Multiplicative ℤ))
              (↑Valued.v : (Rat.padicValuation p).Completion →*₀ WithZero (Multiplicative ℤ))) ↔
        norm (E := ℚ_[p]) ((withValUniformEquiv : (a : (Rat.padicValuation p).Completion) → ℚ_[p]) x : ℚ_[p]) ≤ 1}
have hp : IsClopen (X := ℚ_[p]) {y : ℚ_[p] | ‖y‖ ≤ 1}
Tactic: apply withValUniformEquiv.toHomeomorph.isClosed_setOf_iff (q := fun x ↦ ‖x‖ ≤ 1)
      (Valued.isClopen_closedBall _ one_ne_zero)
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.WithVal (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
a : WithVal (Rat.padicValuation p)
⊢ norm (E := ℚ_[p])
        ((withValUniformEquiv : (a : (Rat.padicValuation p).Completion) → ℚ_[p])
            (↑a : Completion (WithVal (Rat.padicValuation p))) :
          ℚ_[p]) ≤
      1 ↔
    LE.le (α := WithZero (Multiplicative ℤ))
      ((Valued.v : (a : (Rat.padicValuation p).Completion) → WithZero (Multiplicative ℤ))
          (↑a : Completion (WithVal (Rat.padicValuation p))) :
        WithZero (Multiplicative ℤ))
      (1 : WithZero (Multiplicative ℤ))
have ih : norm (E := ℚ_[p]) (↑((WithVal.equiv (Rat.padicValuation p) : (a : WithVal (Rat.padicValuation p)) → ℚ) a) : ℚ_[p]) ≤ 1 ↔
  LE.le (α := WithZero (Multiplicative ℤ))
    ((Rat.padicValuation p : (a : ℚ) → WithZero (Multiplicative ℤ)) a.ofVal : WithZero (Multiplicative ℤ))
    (1 : WithZero (Multiplicative ℤ))
Tactic: rw [Valued.valuedCompletion_apply, ← WithVal.apply_ofVal, withValUniformEquiv_cast_apply]
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.WithVal (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
x : (Rat.padicValuation p).Completion
⊢ norm (E := ℚ_[p]) ((withValUniformEquiv : (a : (Rat.padicValuation p).Completion) → ℚ_[p]) x : ℚ_[p]) ≤ 1 ↔
    LE.le (α := WithZero (Multiplicative ℤ))
      ((Valued.v : (a : (Rat.padicValuation p).Completion) → WithZero (Multiplicative ℤ)) x :
        WithZero (Multiplicative ℤ))
      (1 : WithZero (Multiplicative ℤ))
have hp : IsClosed (X := Completion (WithVal (Rat.padicValuation p)))
  {a : Completion (WithVal (Rat.padicValuation p)) |
    norm (E := ℚ_[p]) ((withValUniformEquiv : (a : (Rat.padicValuation p).Completion) → ℚ_[p]) a : ℚ_[p]) ≤ 1 ↔
      LE.le (α := WithZero (Multiplicative ℤ))
        ((Valued.v : (a : (Rat.padicValuation p).Completion) → WithZero (Multiplicative ℤ)) a :
          WithZero (Multiplicative ℤ))
        (1 : WithZero (Multiplicative ℤ))}
Tactic: induction x using UniformSpace.Completion.induction_on with
  | hp =>
    rw [Set.ext fun _ ↦ Iff.comm]
    simp_rw [← Valuation.restrict_le_one_iff Valued.v]
    apply withValUniformEquiv.toHomeomorph.isClosed_setOf_iff (q := fun x ↦ ‖x‖ ≤ 1)
      (Valued.isClopen_closedBall _ one_ne_zero)
    simpa [Metric.closedBall] using IsUltrametricDist.isClopen_closedBall (0 : ℚ_[p]) one_ne_zero
  | ih a =>
    rw [Valued.valuedCompletion_apply, ← WithVal.apply_ofVal, withValUniformEquiv_cast_apply]
    exact (norm_rat_le_one_iff_padicValuation_le_one p)
Cleared hypotheses: x
-- Lean 4 proof step from Mathlib.NumberTheory.Padics.WithVal (goal)
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
x : (Rat.padicValuation p).Completion
hp :
  IsClosed (X := Completion (WithVal (Rat.padicValuation p)))
    {a : Completion (WithVal (Rat.padicValuation p)) |
      norm (E := ℚ_[p]) ((withValUniformEquiv : (a : (Rat.padicValuation p).Completion) → ℚ_[p]) a : ℚ_[p]) ≤ 1 ↔
        LE.le (α := WithZero (Multiplicative ℤ))
          ((Valued.v : (a : (Rat.padicValuation p).Completion) → WithZero (Multiplicative ℤ)) a :
            WithZero (Multiplicative ℤ))
          (1 : WithZero (Multiplicative ℤ))}
⊢ norm (E := ℚ_[p]) ((withValUniformEquiv : (a : (Rat.padicValuation p).Completion) → ℚ_[p]) x : ℚ_[p]) ≤ 1 ↔
    LE.le (α := WithZero (Multiplicative ℤ))
      ((Valued.v : (a : (Rat.padicValuation p).Completion) → WithZero (Multiplicative ℤ)) x :
        WithZero (Multiplicative ℤ))
      (1 : WithZero (Multiplicative ℤ))
have ih : ∀ (a : WithVal (Rat.padicValuation p)),
  norm (E := ℚ_[p])
        ((withValUniformEquiv : (a : (Rat.padicValuation p).Completion) → ℚ_[p])
            (↑a : Completion (WithVal (Rat.padicValuation p))) :
          ℚ_[p]) ≤
      1 ↔
    LE.le (α := WithZero (Multiplicative ℤ))
      ((Valued.v : (a : (Rat.padicValuation p).Completion) → WithZero (Multiplicative ℤ))
          (↑a : Completion (WithVal (Rat.padicValuation p))) :
        WithZero (Multiplicative ℤ))
      (1 : WithZero (Multiplicative ℤ))
Tactic: induction x using UniformSpace.Completion.induction_on with
  | hp =>
    rw [Set.ext fun _ ↦ Iff.comm]
    simp_rw [← Valuation.restrict_le_one_iff Valued.v]
    apply withValUniformEquiv.toHomeomorph.isClosed_setOf_iff (q := fun x ↦ ‖x‖ ≤ 1)
      (Valued.isClopen_closedBall _ one_ne_zero)
    simpa [Metric.closedBall] using IsUltrametricDist.isClopen_closedBall (0 : ℚ_[p]) one_ne_zero
  | ih a =>
    rw [Valued.valuedCompletion_apply, ← WithVal.apply_ofVal, withValUniformEquiv_cast_apply]
    exact (norm_rat_le_one_iff_padicValuation_le_one p)
Cleared hypotheses: x
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a : ℤ√d
⊢ HPow.hPow (β := ℕ) a.re 2 - d * HPow.hPow (β := ℕ) a.im 2 = 1 ↔
    Membership.mem (γ := Submonoid (ℤ√d)) (unitary (ℤ√d)) a
have rw : HPow.hPow (β := ℕ) a.re 2 - d * HPow.hPow (β := ℕ) a.im 2 = 1 ↔ a.norm = 1
Tactic: rw [← norm_eq_one_iff_mem_unitary, norm_def, sq, sq, ← mul_assoc]
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a : ℤ√d
rw : HPow.hPow (β := ℕ) a.re 2 - d * HPow.hPow (β := ℕ) a.im 2 = 1 ↔ a.norm = 1
⊢ HPow.hPow (β := ℕ) a.re 2 - d * HPow.hPow (β := ℕ) a.im 2 = 1 ↔
    Membership.mem (γ := Submonoid (ℤ√d)) (unitary (ℤ√d)) a
have rw₁ : HPow.hPow (β := ℕ) a.re 2 - d * HPow.hPow (β := ℕ) a.im 2 = 1 ↔ a.re * a.re - d * a.im * a.im = 1
Tactic: rw [← norm_eq_one_iff_mem_unitary, norm_def, sq, sq, ← mul_assoc]
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a : ℤ√d
rw : HPow.hPow (β := ℕ) a.re 2 - d * HPow.hPow (β := ℕ) a.im 2 = 1 ↔ a.norm = 1
rw₁ : HPow.hPow (β := ℕ) a.re 2 - d * HPow.hPow (β := ℕ) a.im 2 = 1 ↔ a.re * a.re - d * a.im * a.im = 1
⊢ HPow.hPow (β := ℕ) a.re 2 - d * HPow.hPow (β := ℕ) a.im 2 = 1 ↔
    Membership.mem (γ := Submonoid (ℤ√d)) (unitary (ℤ√d)) a
have rw₂ : a.re * a.re - d * HPow.hPow (β := ℕ) a.im 2 = 1 ↔ a.re * a.re - d * a.im * a.im = 1
Tactic: rw [← norm_eq_one_iff_mem_unitary, norm_def, sq, sq, ← mul_assoc]
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a : ℤ√d
rw : HPow.hPow (β := ℕ) a.re 2 - d * HPow.hPow (β := ℕ) a.im 2 = 1 ↔ a.norm = 1
rw₁ : HPow.hPow (β := ℕ) a.re 2 - d * HPow.hPow (β := ℕ) a.im 2 = 1 ↔ a.re * a.re - d * a.im * a.im = 1
rw₂ : a.re * a.re - d * HPow.hPow (β := ℕ) a.im 2 = 1 ↔ a.re * a.re - d * a.im * a.im = 1
⊢ HPow.hPow (β := ℕ) a.re 2 - d * HPow.hPow (β := ℕ) a.im 2 = 1 ↔
    Membership.mem (γ := Submonoid (ℤ√d)) (unitary (ℤ√d)) a
have rw₃ : a.re * a.re - d * (a.im * a.im) = 1 ↔ a.re * a.re - d * a.im * a.im = 1
Tactic: rw [← norm_eq_one_iff_mem_unitary, norm_def, sq, sq, ← mul_assoc]
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a : ℤ√d
rw : HPow.hPow (β := ℕ) a.re 2 - d * HPow.hPow (β := ℕ) a.im 2 = 1 ↔ a.norm = 1
rw₁ : HPow.hPow (β := ℕ) a.re 2 - d * HPow.hPow (β := ℕ) a.im 2 = 1 ↔ a.re * a.re - d * a.im * a.im = 1
rw₂ : a.re * a.re - d * HPow.hPow (β := ℕ) a.im 2 = 1 ↔ a.re * a.re - d * a.im * a.im = 1
rw₃ : a.re * a.re - d * (a.im * a.im) = 1 ↔ a.re * a.re - d * a.im * a.im = 1
⊢ HPow.hPow (β := ℕ) a.re 2 - d * HPow.hPow (β := ℕ) a.im 2 = 1 ↔
    Membership.mem (γ := Submonoid (ℤ√d)) (unitary (ℤ√d)) a
have rw₄ : a.re * a.re - d * a.im * a.im = 1 ↔ a.re * a.re - d * a.im * a.im = 1
Tactic: rw [← norm_eq_one_iff_mem_unitary, norm_def, sq, sq, ← mul_assoc]
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a : Solution₁ d
⊢ HPow.hPow (β := ℕ) a.x 2 = 1 + d * HPow.hPow (β := ℕ) a.y 2
have rw : HPow.hPow (β := ℕ) a.x 2 = HPow.hPow (β := ℕ) a.x 2 - d * HPow.hPow (β := ℕ) a.y 2 + d * HPow.hPow (β := ℕ) a.y 2
Tactic: rw [← a.prop]
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a : Solution₁ d
⊢ d * HPow.hPow (β := ℕ) a.y 2 = HPow.hPow (β := ℕ) a.x 2 - 1
have rw : d * HPow.hPow (β := ℕ) a.y 2 = HPow.hPow (β := ℕ) a.x 2 - (HPow.hPow (β := ℕ) a.x 2 - d * HPow.hPow (β := ℕ) a.y 2)
Tactic: rw [← a.prop]
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a b : Solution₁ d
⊢ (a * b).x = a.x * b.x + d * (a.y * b.y)
have rw : (a * b).x = a.x * b.x + d * a.y * b.y
Tactic: rw [← mul_assoc]
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (hypothesis)
Goal:
d : ℤ
h₀ : d < 0
a : Solution₁ d
⊢ a.x = 0 ∨ a.y = 0
have h : HPow.hPow (β := ℕ) a.x 2 - d * HPow.hPow (β := ℕ) a.y 2 = 1
Tactic: have h := a.prop
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
h₀ : d < 0
a : Solution₁ d
h : HPow.hPow (β := ℕ) a.x 2 - d * HPow.hPow (β := ℕ) a.y 2 = 1
⊢ a.x = 0 ∨ a.y = 0
have contrapose! : ∀ (h : a.x ≠ 0 ∧ a.y ≠ 0), HPow.hPow (β := ℕ) a.x 2 - d * HPow.hPow (β := ℕ) a.y 2 ≠ 1
Tactic: contrapose! h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (hypothesis)
Goal:
d : ℤ
h₀ : d < 0
a : Solution₁ d
h : a.x ≠ 0 ∧ a.y ≠ 0
⊢ HPow.hPow (β := ℕ) a.x 2 - d * HPow.hPow (β := ℕ) a.y 2 ≠ 1
have h1 : 0 < HPow.hPow (β := ℕ) a.x 2
Tactic: have h1 := sq_pos_of_ne_zero h.1
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (hypothesis)
Goal:
d : ℤ
h₀ : d < 0
a : Solution₁ d
h : a.x ≠ 0 ∧ a.y ≠ 0
h1 : 0 < HPow.hPow (β := ℕ) a.x 2
⊢ HPow.hPow (β := ℕ) a.x 2 - d * HPow.hPow (β := ℕ) a.y 2 ≠ 1
have h2 : 0 < HPow.hPow (β := ℕ) a.y 2
Tactic: have h2 := sq_pos_of_ne_zero h.2
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
h₀ : d < 0
a : Solution₁ d
h : a.x ≠ 0 ∧ a.y ≠ 0
h1 : 0 < HPow.hPow (β := ℕ) a.x 2
h2 : 0 < HPow.hPow (β := ℕ) a.y 2
⊢ HPow.hPow (β := ℕ) a.x 2 - d * HPow.hPow (β := ℕ) a.y 2 ≠ 1
have nlinarith : HPow.hPow (β := ℕ) a.x 2 - d * HPow.hPow (β := ℕ) a.y 2 = 1 →
  -1 + (HAdd.hAdd (α := ℤ) 0 1 - HPow.hPow (β := ℕ) a.x 2) + (HAdd.hAdd (α := ℤ) 0 1 - HPow.hPow (β := ℕ) a.y 2) +
        (HPow.hPow (β := ℕ) a.x 2 - d * HPow.hPow (β := ℕ) a.y 2 - 1) +
      (0 - (0 - HPow.hPow (β := ℕ) a.y 2) * (d + 1 - 0)) =
    0
Tactic: nlinarith
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (hypothesis)
Goal:
d : ℤ
h₀ : 0 ≤ d
a : Solution₁ d
hx : a.x = 0
⊢ False
have h : 0 ≤ d * HPow.hPow (β := ℕ) a.y 2
Tactic: have h : 0 ≤ d * a.y ^ 2 := mul_nonneg h₀ (sq_nonneg _)
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (hypothesis)
Goal:
d : ℤ
h₀ : 0 ≤ d
a : Solution₁ d
hx : a.x = 0
h : 0 ≤ d * HPow.hPow (β := ℕ) a.y 2
⊢ False
have h₁ : LE.le (α := ℤ) 0 (-1)
Tactic: rw [a.prop_y, hx, sq, zero_mul, zero_sub] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (hypothesis)
Goal:
d : ℤ
a : Solution₁ d
ha : 1 < a.x
hy : a.y = 0
⊢ False
have prop : HPow.hPow (β := ℕ) a.x 2 - d * HPow.hPow (β := ℕ) a.y 2 = 1
Tactic: have prop := a.prop
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (hypothesis)
Goal:
d : ℤ
a : Solution₁ d
ha : 1 < a.x
hy : a.y = 0
prop : HPow.hPow (β := ℕ) a.x 2 - d * HPow.hPow (β := ℕ) a.y 2 = 1
⊢ False
have prop₁ : HPow.hPow (β := ℕ) a.x 2 = 1
Tactic: rw [hy, sq (0 : ℤ), zero_mul, mul_zero, sub_zero] at prop
Cleared hypotheses: prop
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a : Solution₁ d
ha : 1 < a.x
⊢ 0 < d
have refine : 0 < d * HPow.hPow (β := ℕ) a.y 2
Tactic: refine pos_of_mul_pos_left ?_ (sq_nonneg a.y)
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a : Solution₁ d
ha : 1 < a.x
⊢ 0 < d * HPow.hPow (β := ℕ) a.y 2
have rw : 1 < HPow.hPow (β := ℕ) a.x 2
Tactic: rw [a.prop_y, sub_pos]
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (hypothesis)
Goal:
d : ℤ
a : Solution₁ d
ha : 1 < a.x
⊢ ¬IsSquare d
have hp : HPow.hPow (β := ℕ) a.x 2 - d * HPow.hPow (β := ℕ) a.y 2 = 1
Tactic: have hp := a.prop
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (hypothesis)
Goal:
b : ℤ
a : Solution₁ (b * b)
ha : 1 < a.x
hp : HPow.hPow (β := ℕ) a.x 2 - b * b * HPow.hPow (β := ℕ) a.y 2 = 1
⊢ False
have hp₁ : a.x + b * a.y = 1 ∧ a.x - b * a.y = 1 ∨ a.x + b * a.y = -1 ∧ a.x - b * a.y = -1
Tactic: simp_rw [← sq, ← mul_pow, sq_sub_sq, Int.mul_eq_one_iff_eq_one_or_neg_one] at hp
Cleared hypotheses: hp
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (hypothesis)
Goal:
d : ℤ
h₀ : d ≠ 0
a : Solution₁ d
ha : a.x = 1
⊢ a = 1
have prop : d * HPow.hPow (β := ℕ) a.y 2 = HPow.hPow (β := ℕ) a.x 2 - 1
Tactic: have prop := a.prop_y
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (hypothesis)
Goal:
d : ℤ
h₀ : d ≠ 0
a : Solution₁ d
ha : a.x = 1
prop : d * HPow.hPow (β := ℕ) a.y 2 = HPow.hPow (β := ℕ) a.x 2 - 1
⊢ a = 1
have prop₁ : a.y = 0
Tactic: rw [ha, one_pow, sub_self, mul_eq_zero, or_iff_right h₀, sq_eq_zero_iff] at prop
Cleared hypotheses: prop
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a : Solution₁ d
⊢ a = 1 ∨ a = -1 ↔ a.y = 0
have refine : ∀ (H : a.y = 0), a = 1 ∨ a = -1
Tactic: refine ⟨fun H => H.elim (fun h => by simp [h]) fun h => by simp [h], fun H => ?_⟩
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (hypothesis)
Goal:
d : ℤ
a : Solution₁ d
H : a.y = 0
⊢ a = 1 ∨ a = -1
have prop : HPow.hPow (β := ℕ) a.x 2 - d * HPow.hPow (β := ℕ) a.y 2 = 1
Tactic: have prop := a.prop
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (hypothesis)
Goal:
d : ℤ
a : Solution₁ d
H : a.y = 0
prop : HPow.hPow (β := ℕ) a.x 2 - d * HPow.hPow (β := ℕ) a.y 2 = 1
⊢ a = 1 ∨ a = -1
have prop₁ : a.x = 1 ∨ a.x = -1
Tactic: rw [H, sq (0 : ℤ), mul_zero, mul_zero, sub_zero, sq_eq_one_iff] at prop
Cleared hypotheses: prop
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a b : Solution₁ d
ha : 0 < a.x
hb : 0 < b.x
⊢ 0 < (a * b).x
have simp : 0 < a.x * b.x + d * (a.y * b.y)
Tactic: simp only [x_mul]
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a b : Solution₁ d
ha : 0 < a.x
hb : 0 < b.x
⊢ 0 < a.x * b.x + d * (a.y * b.y)
have refine : |d * (a.y * b.y)| < a.x * b.x
Tactic: refine neg_lt_iff_pos_add'.mp (abs_lt.mp ?_).1
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a b : Solution₁ d
ha : 0 < a.x
hb : 0 < b.x
⊢ |d * (a.y * b.y)| < a.x * b.x
have rw : 0 < (1 + d * HPow.hPow (β := ℕ) a.y 2) * (1 + d * HPow.hPow (β := ℕ) b.y 2) - HPow.hPow (β := ℕ) (d * (a.y * b.y)) 2
Tactic: rw [← abs_of_pos ha, ← abs_of_pos hb, ← abs_mul, ← sq_lt_sq, mul_pow a.x, a.prop_x, b.prop_x, ←
    sub_pos]
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a b : Solution₁ d
ha : 0 < a.x
hb : 0 < b.x
⊢ 0 < (1 + d * HPow.hPow (β := ℕ) a.y 2) * (1 + d * HPow.hPow (β := ℕ) b.y 2) - HPow.hPow (β := ℕ) (d * (a.y * b.y)) 2
have ring_nf : 0 < 1 + d * HPow.hPow (β := ℕ) a.y 2 + d * HPow.hPow (β := ℕ) b.y 2
Tactic: ring_nf
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a b : Solution₁ d
ha : 0 < a.x
hb : 0 < b.x
⊢ 0 < 1 + d * HPow.hPow (β := ℕ) a.y 2 + d * HPow.hPow (β := ℕ) b.y 2
have inl : ∀ (h : 0 ≤ d), 0 < 1 + d * HPow.hPow (β := ℕ) a.y 2 + d * HPow.hPow (β := ℕ) b.y 2
Tactic: rcases le_or_gt 0 d with h | h
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a b : Solution₁ d
ha : 0 < a.x
hb : 0 < b.x
inl : ∀ (h : 0 ≤ d), 0 < 1 + d * HPow.hPow (β := ℕ) a.y 2 + d * HPow.hPow (β := ℕ) b.y 2
⊢ 0 < 1 + d * HPow.hPow (β := ℕ) a.y 2 + d * HPow.hPow (β := ℕ) b.y 2
have inr : ∀ (h : d < 0), 0 < 1 + d * HPow.hPow (β := ℕ) a.y 2 + d * HPow.hPow (β := ℕ) b.y 2
Tactic: rcases le_or_gt 0 d with h | h
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a b : Solution₁ d
ha : 0 < a.x
hb : 0 < b.x
h : d < 0
⊢ 0 < 1 + d * HPow.hPow (β := ℕ) a.y 2 + d * HPow.hPow (β := ℕ) b.y 2
have inr : 0 < 1 + d * HPow.hPow (α := ℤ) (β := ℕ) 0 2 + d * HPow.hPow (α := ℤ) (β := ℕ) 0 2
Tactic: rw [(eq_zero_of_d_neg h a).resolve_left ha.ne', (eq_zero_of_d_neg h b).resolve_left hb.ne']
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a b : Solution₁ d
hax : 0 < a.x
hay : 0 < a.y
hbx : 0 < b.x
hby : 0 < b.y
⊢ 0 < (a * b).y
have simp : 0 < a.x * b.y + a.y * b.x
Tactic: simp only [y_mul]
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a : Solution₁ d
hax : 0 < a.x
n : ℕ
⊢ 0 < (a ^ n).x
have zero : 0 < (HPow.hPow (β := ℕ) a 0).x
Tactic: induction n with
  | zero => simp only [pow_zero, x_one, zero_lt_one]
  | succ n ih => rw [pow_succ]; exact x_mul_pos ih hax
Cleared hypotheses: n
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a : Solution₁ d
hax : 0 < a.x
n : ℕ
zero : 0 < (HPow.hPow (β := ℕ) a 0).x
⊢ 0 < (a ^ n).x
have succ : ∀ (n : ℕ) (ih : 0 < (a ^ n).x), 0 < (a ^ (n + 1)).x
Tactic: induction n with
  | zero => simp only [pow_zero, x_one, zero_lt_one]
  | succ n ih => rw [pow_succ]; exact x_mul_pos ih hax
Cleared hypotheses: n
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a : Solution₁ d
hax : 0 < a.x
hay : 0 < a.y
n : ℕ
ih : 0 < (a ^ n.succ).y
⊢ 0 < (a ^ (n + 1).succ).y
have succ : 0 < (a * a ^ (n + 1)).y
Tactic: rw [pow_succ']
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a : Solution₁ d
hax : 0 < a.x
hay : 0 < a.y
n : ℕ
⊢ 0 < (a ^ n.succ).y
have zero : 0 < (a ^ Nat.succ 0).y
Tactic: induction n with
  | zero => simp only [pow_one, hay]
  | succ n ih => rw [pow_succ']; exact y_mul_pos hax hay (x_pow_pos hax _) ih
Cleared hypotheses: n
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a : Solution₁ d
hax : 0 < a.x
hay : 0 < a.y
n : ℕ
zero : 0 < (a ^ Nat.succ 0).y
⊢ 0 < (a ^ n.succ).y
have succ : ∀ (n : ℕ) (ih : 0 < (a ^ n.succ).y), 0 < (a ^ (n + 1).succ).y
Tactic: induction n with
  | zero => simp only [pow_one, hay]
  | succ n ih => rw [pow_succ']; exact y_mul_pos hax hay (x_pow_pos hax _) ih
Cleared hypotheses: n
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a : Solution₁ d
hax : 0 < a.x
hay : 0 < a.y
n : ℤ
hn : 0 < n
⊢ 0 < (a ^ n).y
have lift : ∀ (n : ℕ) (hn : LT.lt (α := ℤ) 0 (↑n : ℤ)), 0 < (HPow.hPow (β := ℤ) a (↑n : ℤ)).y
Tactic: lift n to ℕ using hn.le
Cleared hypotheses: n, hn
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a : Solution₁ d
hax : 0 < a.x
hay : 0 < a.y
n : ℕ
hn : LT.lt (α := ℤ) 0 (↑n : ℤ)
⊢ 0 < (HPow.hPow (β := ℤ) a (↑n : ℤ)).y
have norm_cast : ∀ (hn : 0 < n), 0 < (a ^ n).y
Tactic: norm_cast at hn ⊢
Cleared hypotheses: hn
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a : Solution₁ d
hax : 0 < a.x
hay : 0 < a.y
n : ℕ
hn : 0 < n
⊢ 0 < (a ^ n).y
have rw : 0 < (a ^ n.pred.succ).y
Tactic: rw [← Nat.succ_pred_eq_of_pos hn]
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a : Solution₁ d
hax : 0 < a.x
n : ℕ
⊢ 0 < (a ^ Int.ofNat n).x
have ofNat : 0 < (a ^ n).x
Tactic: rw [Int.ofNat_eq_natCast, zpow_natCast]
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a : Solution₁ d
hax : 0 < a.x
n : ℕ
⊢ 0 < (a ^ Int.negSucc n).x
have negSucc : 0 < (a ^ (n + 1))⁻¹.x
Tactic: rw [zpow_negSucc]
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a : Solution₁ d
hax : 0 < a.x
n : ℤ
⊢ 0 < (a ^ n).x
have ofNat : ∀ (n : ℕ), 0 < (a ^ Int.ofNat n).x
Tactic: cases n with
  | ofNat n =>
    rw [Int.ofNat_eq_natCast, zpow_natCast]
    exact x_pow_pos hax n
  | negSucc n =>
    rw [zpow_negSucc]
    exact x_pow_pos hax (n + 1)
Cleared hypotheses: n
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a : Solution₁ d
hax : 0 < a.x
n : ℤ
ofNat : ∀ (n : ℕ), 0 < (a ^ Int.ofNat n).x
⊢ 0 < (a ^ n).x
have negSucc : ∀ (n : ℕ), 0 < (a ^ Int.negSucc n).x
Tactic: cases n with
  | ofNat n =>
    rw [Int.ofNat_eq_natCast, zpow_natCast]
    exact x_pow_pos hax n
  | negSucc n =>
    rw [zpow_negSucc]
    exact x_pow_pos hax (n + 1)
Cleared hypotheses: n
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a : Solution₁ d
hax : 0 < a.x
hay : 0 < a.y
n : ℤ
⊢ (a ^ n).y.sign = n.sign
have ofNat : (a ^ Int.ofNat 0).y.sign = (Int.ofNat 0).sign
Tactic: rcases n with ((_ | n) | n)
Cleared hypotheses: n