awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
refine_1 : IsClosed (X := ℂ) fun (z : ℂ) ↦ Real.le✝ 0 z.im
refine_2 : IsClosed (X := ℂ) fun (z : ℂ) ↦ Real.le✝ 1 ‖z‖
⊢ IsClosed (X := ℂ) {z : ℂ | 0 ≤ z.im ∧ 1 ≤ ‖z‖ ∧ |z.re| ≤ 1 / 2}
have refine_3 : IsClosed (X := ℂ) fun (z : ℂ) ↦ Real.le✝ |z.re| (1 / 2)
Tactic: refine .inter ?_ (.inter ?_ ?_)
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (hypothesis)
Goal:
⊢ IsClosed (X := ℂ) {z : ℂ | 0 < z.im ∧ 1 ≤ ‖z‖ ∧ |z.re| ≤ 1 / 2}
have this : IsClosed (X := ℂ) {z : ℂ | 0 ≤ z.im ∧ 1 ≤ ‖z‖ ∧ |z.re| ≤ 1 / 2}
Tactic: have : IsClosed {z : ℂ | 0 ≤ z.im ∧ 1 ≤ ‖z‖ ∧ |z.re| ≤ 1/2} := by
    refine .inter ?_ (.inter ?_ ?_)
    · exact isClosed_le continuous_const Complex.continuous_im
    · exact isClosed_le continuous_const continuous_norm
    · exact isClosed_le (continuous_abs.comp Complex.continuous_re) continuous_const
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
this : IsClosed (X := ℂ) {z : ℂ | 0 ≤ z.im ∧ 1 ≤ ‖z‖ ∧ |z.re| ≤ 1 / 2}
⊢ IsClosed (X := ℂ) {z : ℂ | 0 < z.im ∧ 1 ≤ ‖z‖ ∧ |z.re| ≤ 1 / 2}
have h : Eq (α := Set ℂ) {z : ℂ | 0 < z.im ∧ 1 ≤ ‖z‖ ∧ |z.re| ≤ 1 / 2} {z : ℂ | 0 ≤ z.im ∧ 1 ≤ ‖z‖ ∧ |z.re| ≤ 1 / 2}
Tactic: convert! this using 1
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
this : IsClosed (X := ℂ) {z : ℂ | 0 ≤ z.im ∧ 1 ≤ ‖z‖ ∧ |z.re| ≤ 1 / 2}
⊢ Eq (α := Set ℂ) {z : ℂ | 0 < z.im ∧ 1 ≤ ‖z‖ ∧ |z.re| ≤ 1 / 2} {z : ℂ | 0 ≤ z.im ∧ 1 ≤ ‖z‖ ∧ |z.re| ≤ 1 / 2}
have h : ∀ (x : ℂ),
  Membership.mem (γ := Set ℂ) {z : ℂ | 0 < z.im ∧ 1 ≤ ‖z‖ ∧ |z.re| ≤ 1 / 2} x ↔
    Membership.mem (γ := Set ℂ) {z : ℂ | 0 ≤ z.im ∧ 1 ≤ ‖z‖ ∧ |z.re| ≤ 1 / 2} x
Tactic: ext x
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
this : IsClosed (X := ℂ) {z : ℂ | 0 ≤ z.im ∧ 1 ≤ ‖z‖ ∧ |z.re| ≤ 1 / 2}
x : ℂ
⊢ Membership.mem (γ := Set ℂ) {z : ℂ | 0 < z.im ∧ 1 ≤ ‖z‖ ∧ |z.re| ≤ 1 / 2} x ↔
    Membership.mem (γ := Set ℂ) {z : ℂ | 0 ≤ z.im ∧ 1 ≤ ‖z‖ ∧ |z.re| ≤ 1 / 2} x
have h : Membership.mem (γ := Set ℂ) {z : ℂ | 0 ≤ z.im ∧ 1 ≤ ‖z‖ ∧ |z.re| ≤ 1 / 2} x →
  ∀ (him : 0 ≤ x.im) (hre : 1 ≤ ‖x‖) (hnorm : |x.re| ≤ 1 / 2), 0 < x.im
Tactic: refine ⟨fun ⟨him, hre, hnorm⟩ ↦ ⟨him.le, hre, hnorm⟩, fun ⟨him, hre, hnorm⟩ ↦ ⟨?_, hre, hnorm⟩⟩
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
x : ℍ
hxnorm : 1 < ‖(↑x : ℂ)‖
hxre : |x.re| ≤ 1 / 2
⊢ x ∈ closure 𝒟ᵒ
have h : ∃ᶠ (x_1 : ℝ) in 𝓝[<] 1, (↑ofComplex : ℂ → ℍ) ((↑x_1 : ℂ) * (↑x : ℂ)) ∈ 𝒟ᵒ
Tactic: apply mem_closure_of_frequently_of_tendsto (α := ℝ)
      (b := 𝓝[<] 1) (f := fun t ↦ ofComplex (t * x))
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
x : ℍ
hxnorm : 1 < ‖(↑x : ℂ)‖
hxre : |x.re| ≤ 1 / 2
h : ∃ᶠ (x_1 : ℝ) in 𝓝[<] 1, (↑ofComplex : ℂ → ℍ) ((↑x_1 : ℂ) * (↑x : ℂ)) ∈ 𝒟ᵒ
⊢ x ∈ closure 𝒟ᵒ
have hf : Filter.Tendsto (fun (t : ℝ) ↦ (↑ofComplex : ℂ → ℍ) ((↑t : ℂ) * (↑x : ℂ))) (𝓝[<] 1) (𝓝 x)
Tactic: apply mem_closure_of_frequently_of_tendsto (α := ℝ)
      (b := 𝓝[<] 1) (f := fun t ↦ ofComplex (t * x))
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
x : ℍ
hxnorm : 1 < ‖(↑x : ℂ)‖
hxre : |x.re| ≤ 1 / 2
⊢ ∃ᶠ (x_1 : ℝ) in 𝓝[<] 1, (↑ofComplex : ℂ → ℍ) ((↑x_1 : ℂ) * (↑x : ℂ)) ∈ 𝒟ᵒ
have h : ∀ᶠ (x_1 : ℝ) in 𝓝[<] 1, (↑ofComplex : ℂ → ℍ) ((↑x_1 : ℂ) * (↑x : ℂ)) ∈ 𝒟ᵒ
Tactic: apply Filter.Eventually.frequently
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
x : ℍ
hxnorm : 1 < ‖(↑x : ℂ)‖
hxre : |x.re| ≤ 1 / 2
⊢ ∀ᶠ (x_1 : ℝ) in 𝓝[<] 1, (↑ofComplex : ℂ → ℍ) ((↑x_1 : ℂ) * (↑x : ℂ)) ∈ 𝒟ᵒ
have h : (∀ᶠ (x_1 : ℝ) in 𝓝[<] 1, 1 < ‖(↑((↑ofComplex : ℂ → ℍ) ((↑x_1 : ℂ) * (↑x : ℂ))) : ℂ)‖) ∧
  ∀ᶠ (x_1 : ℝ) in 𝓝[<] 1, |((↑ofComplex : ℂ → ℍ) ((↑x_1 : ℂ) * (↑x : ℂ))).re| < 1 / 2
Tactic: simp only [fdo, Set.mem_setOf, Filter.eventually_and, one_lt_normSq_iff]
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
x : ℍ
hxnorm : 1 < ‖(↑x : ℂ)‖
hxre : |x.re| ≤ 1 / 2
⊢ (∀ᶠ (x_1 : ℝ) in 𝓝[<] 1, 1 < ‖(↑((↑ofComplex : ℂ → ℍ) ((↑x_1 : ℂ) * (↑x : ℂ))) : ℂ)‖) ∧
    ∀ᶠ (x_1 : ℝ) in 𝓝[<] 1, |((↑ofComplex : ℂ → ℍ) ((↑x_1 : ℂ) * (↑x : ℂ))).re| < 1 / 2
have h : Filter.Tendsto (fun (x_1 : ℝ) ↦ ‖(↑((↑ofComplex : ℂ → ℍ) ((↑x_1 : ℂ) * (↑x : ℂ))) : ℂ)‖) (𝓝 1) (𝓝 ‖(↑x : ℂ)‖)
Tactic: refine ⟨Filter.Tendsto.eventually_const_lt hxnorm (.mono_left ?_ nhdsWithin_le_nhds), ?_⟩
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
x : ℍ
hxnorm : 1 < ‖(↑x : ℂ)‖
hxre : |x.re| ≤ 1 / 2
h : Filter.Tendsto (fun (x_1 : ℝ) ↦ ‖(↑((↑ofComplex : ℂ → ℍ) ((↑x_1 : ℂ) * (↑x : ℂ))) : ℂ)‖) (𝓝 1) (𝓝 ‖(↑x : ℂ)‖)
⊢ (∀ᶠ (x_1 : ℝ) in 𝓝[<] 1, 1 < ‖(↑((↑ofComplex : ℂ → ℍ) ((↑x_1 : ℂ) * (↑x : ℂ))) : ℂ)‖) ∧
    ∀ᶠ (x_1 : ℝ) in 𝓝[<] 1, |((↑ofComplex : ℂ → ℍ) ((↑x_1 : ℂ) * (↑x : ℂ))).re| < 1 / 2
have h₁ : ∀ᶠ (x_1 : ℝ) in 𝓝[<] 1, |((↑ofComplex : ℂ → ℍ) ((↑x_1 : ℂ) * (↑x : ℂ))).re| < 1 / 2
Tactic: refine ⟨Filter.Tendsto.eventually_const_lt hxnorm (.mono_left ?_ nhdsWithin_le_nhds), ?_⟩
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
x : ℍ
hxnorm : 1 < ‖(↑x : ℂ)‖
hxre : |x.re| ≤ 1 / 2
⊢ ContinuousAt (Y := ℂ) (fun (a : ℝ) ↦ (↑((↑ofComplex : ℂ → ℍ) ((↑a : ℂ) * (↑x : ℂ))) : ℂ)) 1
have refine : (↑1 : ℂ) * (↑x : ℂ) ∈ ofComplex.source
Tactic: refine .comp (by fun_prop) ((OpenPartialHomeomorph.continuousAt _ ?_).comp (by fun_prop))
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (hypothesis)
Goal:
x : ℍ
hxnorm : 1 < ‖(↑x : ℂ)‖
hxre : |x.re| ≤ 1 / 2
⊢ Filter.Tendsto (fun (x_1 : ℝ) ↦ ‖(↑((↑ofComplex : ℂ → ℍ) ((↑x_1 : ℂ) * (↑x : ℂ))) : ℂ)‖) (𝓝 1) (𝓝 ‖(↑x : ℂ)‖)
have this : ContinuousAt (Y := ℂ) (fun (a : ℝ) ↦ (↑((↑ofComplex : ℂ → ℍ) ((↑a : ℂ) * (↑x : ℂ))) : ℂ)) 1
Tactic: have : ContinuousAt (fun a : ℝ ↦ (ofComplex (a * x : ℂ) : ℂ)) 1 := by
        refine .comp (by fun_prop) ((OpenPartialHomeomorph.continuousAt _ ?_).comp (by fun_prop))
        simpa [ofComplex] using x.coe_im_pos
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
x : ℍ
hxnorm : 1 < ‖(↑x : ℂ)‖
hxre : |x.re| ≤ 1 / 2
⊢ ∀ᶠ (x_1 : ℝ) in 𝓝[<] 1, |((↑ofComplex : ℂ → ℍ) ((↑x_1 : ℂ) * (↑x : ℂ))).re| < 1 / 2
have h : ∀ᶠ (x_1 : ℝ) in 𝓝 1,
  Membership.mem (γ := Set ℝ) (Set.Iio 1) x_1 → |((↑ofComplex : ℂ → ℍ) ((↑x_1 : ℂ) * (↑x : ℂ))).re| < 1 / 2
Tactic: simp only [eventually_nhdsWithin_iff]
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
x : ℍ
hxnorm : 1 < ‖(↑x : ℂ)‖
hxre : |x.re| ≤ 1 / 2
⊢ ∀ᶠ (x_1 : ℝ) in 𝓝 1,
    Membership.mem (γ := Set ℝ) (Set.Iio 1) x_1 → |((↑ofComplex : ℂ → ℍ) ((↑x_1 : ℂ) * (↑x : ℂ))).re| < 1 / 2
have h : ∀ (a : ℝ) (ha : 0 < a) (ha' : Membership.mem (γ := Set ℝ) (Set.Iio 1) a),
  |((↑ofComplex : ℂ → ℍ) ((↑a : ℂ) * (↑x : ℂ))).re| < 1 / 2
Tactic: filter_upwards [eventually_gt_nhds zero_lt_one] with a ha ha'
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
x : ℍ
hxnorm : 1 < ‖(↑x : ℂ)‖
hxre : |x.re| ≤ 1 / 2
a : ℝ
ha : 0 < a
ha' : Membership.mem (γ := Set ℝ) (Set.Iio 1) a
⊢ |((↑ofComplex : ℂ → ℍ) ((↑a : ℂ) * (↑x : ℂ))).re| < 1 / 2
have h : |(↑{ coe := (↑a : ℂ) * (↑x : ℂ), coe_im_pos := sorry : ℍ } : ℂ).re| < 1 / 2
Tactic: rw [← coe_re, ofComplex_apply_of_im_pos (by simpa using mul_pos ha x.coe_im_pos)]
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
x : ℍ
hxnorm : 1 < ‖(↑x : ℂ)‖
hxre : |x.re| ≤ 1 / 2
a : ℝ
ha : 0 < a
ha' : Membership.mem (γ := Set ℝ) (Set.Iio 1) a
⊢ |(↑{ coe := (↑a : ℂ) * (↑x : ℂ), coe_im_pos := ⋯ : ℍ } : ℂ).re| < 1 / 2
have h : a * |x.re| < 1 / 2
Tactic: suffices a * |x.re| < 1 / 2 by simpa [abs_of_pos ha]
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
x : ℍ
hxnorm : 1 < ‖(↑x : ℂ)‖
hxre : |x.re| ≤ 1 / 2
a : ℝ
ha : 0 < a
ha' : Membership.mem (γ := Set ℝ) (Set.Iio 1) a
⊢ a * |x.re| < 1 / 2
have nlinarith : 1 / 2 ≤ a * |x.re| → 2 / 2 = 1
Tactic: nlinarith [Set.mem_Iio.mp ha']
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
x : ℍ
hxnorm : 1 < ‖(↑x : ℂ)‖
hxre : |x.re| ≤ 1 / 2
a : ℝ
ha : 0 < a
ha' : Membership.mem (γ := Set ℝ) (Set.Iio 1) a
nlinarith : 1 / 2 ≤ a * |x.re| → 2 / 2 = 1
⊢ a * |x.re| < 1 / 2
have nlinarith₁ : 1 / 2 ≤ a * |x.re| → HMul.hMul (α := ℝ) 1 2 = 2
Tactic: nlinarith [Set.mem_Iio.mp ha']
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
x : ℍ
hxnorm : 1 < ‖(↑x : ℂ)‖
hxre : |x.re| ≤ 1 / 2
a : ℝ
ha : 0 < a
ha' : Membership.mem (γ := Set ℝ) (Set.Iio 1) a
nlinarith : 1 / 2 ≤ a * |x.re| → 2 / 2 = 1
nlinarith₁ nlinarith₂ : 1 / 2 ≤ a * |x.re| → HMul.hMul (α := ℝ) 1 2 = 2
⊢ a * |x.re| < 1 / 2
have nlinarith₃ : 1 / 2 ≤ a * |x.re| → GT.gt (α := ℝ) 2 0
Tactic: nlinarith [Set.mem_Iio.mp ha']
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
x : ℍ
hxnorm : 1 < ‖(↑x : ℂ)‖
hxre : |x.re| ≤ 1 / 2
a : ℝ
ha : 0 < a
ha' : Membership.mem (γ := Set ℝ) (Set.Iio 1) a
nlinarith : 1 / 2 ≤ a * |x.re| → 2 / 2 = 1
nlinarith₁ nlinarith₂ : 1 / 2 ≤ a * |x.re| → HMul.hMul (α := ℝ) 1 2 = 2
nlinarith₃ : 1 / 2 ≤ a * |x.re| → GT.gt (α := ℝ) 2 0
nlinarith₄ : 1 / 2 ≤ a * |x.re| → 2 / 2 = 1
nlinarith₅ : 1 / 2 ≤ a * |x.re| → HMul.hMul (α := ℝ) 1 2 = 2
nlinarith₆ : 1 / 2 ≤ a * |x.re| → GT.gt (α := ℝ) 2 0
⊢ a * |x.re| < 1 / 2
have nlinarith₇ : 1 / 2 ≤ a * |x.re| →
  HMul.hMul (α := ℝ) 1 1 - 1 * a * (2 * |x.re|) + (a - 1) + (0 - (2 * |x.re| - HMul.hMul (α := ℝ) 1 1) * (0 - a)) = 0
Tactic: nlinarith [Set.mem_Iio.mp ha']
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
x : ℍ
hxnorm : 1 < ‖(↑x : ℂ)‖
hxre : |x.re| ≤ 1 / 2
⊢ Filter.Tendsto (fun (t : ℝ) ↦ (↑ofComplex : ℂ → ℍ) ((↑t : ℂ) * (↑x : ℂ))) (𝓝[<] 1) (𝓝 x)
have hf : Filter.Tendsto (fun (t : ℝ) ↦ (↑ofComplex : ℂ → ℍ) ((↑t : ℂ) * (↑x : ℂ))) (𝓝 1) (𝓝 x)
Tactic: refine .mono_left ?_ nhdsWithin_le_nhds
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
x : ℍ
hxnorm : 1 < ‖(↑x : ℂ)‖
hxre : |x.re| ≤ 1 / 2
⊢ Filter.Tendsto (fun (t : ℝ) ↦ (↑ofComplex : ℂ → ℍ) ((↑t : ℂ) * (↑x : ℂ))) (𝓝 1) (𝓝 x)
have hf : Filter.Tendsto (fun (x_1 : ℝ) ↦ (↑((↑ofComplex : ℂ → ℍ) ((↑x_1 : ℂ) * (↑x : ℂ))) : ℂ)) (𝓝 1) (𝓝 (↑x : ℂ))
Tactic: rw [isOpenEmbedding_coe.tendsto_nhds_iff, Function.comp_def]
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
x : ℍ
hxnorm : 1 < ‖(↑x : ℂ)‖
hxre : |x.re| ≤ 1 / 2
⊢ Filter.Tendsto (fun (t : ℝ) ↦ (↑t : ℂ) * (↑x : ℂ)) (𝓝 1) (𝓝 (↑x : ℂ))
have rw : Filter.Tendsto (fun (t : ℝ) ↦ (↑t : ℂ) * (↑x : ℂ)) (𝓝 1) (𝓝 ((↑1 : ℂ) * (↑x : ℂ)))
Tactic: rw [show 𝓝 (x : ℂ) = 𝓝 ((1 : ℝ) * (x : ℂ)) by simp]
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (hypothesis)
Goal:
x : ℍ
hxnorm : 1 < ‖(↑x : ℂ)‖
hxre : |x.re| ≤ 1 / 2
⊢ Filter.Tendsto (fun (x_1 : ℝ) ↦ (↑((↑ofComplex : ℂ → ℍ) ((↑x_1 : ℂ) * (↑x : ℂ))) : ℂ)) (𝓝 1) (𝓝 (↑x : ℂ))
have this : Filter.Tendsto (fun (t : ℝ) ↦ (↑t : ℂ) * (↑x : ℂ)) (𝓝 1) (𝓝 (↑x : ℂ))
Tactic: have : Filter.Tendsto (fun t : ℝ ↦ t * (x : ℂ)) (𝓝 1) (𝓝 (x : ℂ)) := by
      rw [show 𝓝 (x : ℂ) = 𝓝 ((1 : ℝ) * (x : ℂ)) by simp]
      exact Continuous.tendsto (by fun_prop) _
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
x : ℍ
hxnorm : 1 < ‖(↑x : ℂ)‖
hxre : |x.re| ≤ 1 / 2
this : Filter.Tendsto (fun (t : ℝ) ↦ (↑t : ℂ) * (↑x : ℂ)) (𝓝 1) (𝓝 (↑x : ℂ))
⊢ Filter.Tendsto (fun (x_1 : ℝ) ↦ (↑((↑ofComplex : ℂ → ℍ) ((↑x_1 : ℂ) * (↑x : ℂ))) : ℂ)) (𝓝 1) (𝓝 (↑x : ℂ))
have hf : Filter.EventuallyEq (β := ℂ) (𝓝 1) (fun (t : ℝ) ↦ (↑t : ℂ) * (↑x : ℂ)) fun (x_1 : ℝ) ↦
  (↑((↑ofComplex : ℂ → ℍ) ((↑x_1 : ℂ) * (↑x : ℂ))) : ℂ)
Tactic: refine this.congr' ?_
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
x : ℍ
hxnorm : 1 < ‖(↑x : ℂ)‖
hxre : |x.re| ≤ 1 / 2
this : Filter.Tendsto (fun (t : ℝ) ↦ (↑t : ℂ) * (↑x : ℂ)) (𝓝 1) (𝓝 (↑x : ℂ))
⊢ Filter.EventuallyEq (β := ℂ) (𝓝 1) (fun (t : ℝ) ↦ (↑t : ℂ) * (↑x : ℂ)) fun (x_1 : ℝ) ↦
    (↑((↑ofComplex : ℂ → ℍ) ((↑x_1 : ℂ) * (↑x : ℂ))) : ℂ)
have h : ∀ (a : ℝ) (ha : 0 < a), (↑a : ℂ) * (↑x : ℂ) = (↑((↑ofComplex : ℂ → ℍ) ((↑a : ℂ) * (↑x : ℂ))) : ℂ)
Tactic: filter_upwards [eventually_gt_nhds zero_lt_one] with a ha
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
x : ℍ
hxnorm : 1 < ‖(↑x : ℂ)‖
hxre : |x.re| ≤ 1 / 2
this : Filter.Tendsto (fun (t : ℝ) ↦ (↑t : ℂ) * (↑x : ℂ)) (𝓝 1) (𝓝 (↑x : ℂ))
a : ℝ
ha : 0 < a
⊢ (↑a : ℂ) * (↑x : ℂ) = (↑((↑ofComplex : ℂ → ℍ) ((↑a : ℂ) * (↑x : ℂ))) : ℂ)
have h : (↑a : ℂ) * (↑x : ℂ) = (↑{ coe := (↑a : ℂ) * (↑x : ℂ), coe_im_pos := sorry } : ℂ)
Tactic: rw [ofComplex_apply_of_im_pos (by simpa using mul_pos ha x.coe_im_pos)]
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
x : ℍ
hxnorm : ‖(↑x : ℂ)‖ = 1
hxre : |x.re| ≤ 1 / 2
⊢ x ∈ closure 𝒟ᵒ
have rw : x ∈ closure (closure 𝒟ᵒ)
Tactic: rw [← closure_closure]
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
x : ℍ
hxnorm : ‖(↑x : ℂ)‖ = 1
hxre : |x.re| ≤ 1 / 2
⊢ x ∈ closure (closure 𝒟ᵒ)
have h : ∃ᶠ (x_1 : ℝ≥0) in 𝓝[>] 0, { coe := (↑x : ℂ) + (↑(↑x_1 : ℝ) : ℂ) * Complex.I, coe_im_pos := sorry : ℍ } ∈ closure 𝒟ᵒ
Tactic: apply mem_closure_of_frequently_of_tendsto (b := 𝓝[>] 0)
    (f := fun t : ℝ≥0 ↦ ⟨x + t * Complex.I, by
      simpa using add_pos_of_pos_of_nonneg x.coe_im_pos t.property⟩)
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
x : ℍ
hxnorm : ‖(↑x : ℂ)‖ = 1
hxre : |x.re| ≤ 1 / 2
h : ∃ᶠ (x_1 : ℝ≥0) in 𝓝[>] 0, { coe := (↑x : ℂ) + (↑(↑x_1 : ℝ) : ℂ) * Complex.I, coe_im_pos := ⋯ : ℍ } ∈ closure 𝒟ᵒ
⊢ x ∈ closure (closure 𝒟ᵒ)
have hf : Filter.Tendsto (fun (t : ℝ≥0) ↦ { coe := (↑x : ℂ) + (↑(↑t : ℝ) : ℂ) * Complex.I, coe_im_pos := sorry }) (𝓝[>] 0) (𝓝 x)
Tactic: apply mem_closure_of_frequently_of_tendsto (b := 𝓝[>] 0)
    (f := fun t : ℝ≥0 ↦ ⟨x + t * Complex.I, by
      simpa using add_pos_of_pos_of_nonneg x.coe_im_pos t.property⟩)
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
x : ℍ
hxnorm : ‖(↑x : ℂ)‖ = 1
hxre : |x.re| ≤ 1 / 2
⊢ ∃ᶠ (x_1 : ℝ≥0) in 𝓝[>] 0, { coe := (↑x : ℂ) + (↑(↑x_1 : ℝ) : ℂ) * Complex.I, coe_im_pos := ⋯ : ℍ } ∈ closure 𝒟ᵒ
have h : ∀ᶠ (x_1 : ℝ≥0) in 𝓝[>] 0, { coe := (↑x : ℂ) + (↑(↑x_1 : ℝ) : ℂ) * Complex.I, coe_im_pos := sorry : ℍ } ∈ closure 𝒟ᵒ
Tactic: apply Filter.Eventually.frequently
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
x : ℍ
hxnorm : ‖(↑x : ℂ)‖ = 1
hxre : |x.re| ≤ 1 / 2
a : ℝ≥0
⊢ Membership.mem (γ := Set ℝ≥0) (Set.Ioi 0) a →
    { coe := (↑x : ℂ) + (↑(↑a : ℝ) : ℂ) * Complex.I, coe_im_pos := ⋯ : ℍ } ∈ closure 𝒟ᵒ
have h : ∀ (ha : 0 < a), { coe := (↑x : ℂ) + (↑(↑a : ℝ) : ℂ) * Complex.I, coe_im_pos := sorry : ℍ } ∈ closure 𝒟ᵒ
Tactic: (ha : 0 < a)
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
x : ℍ
hxnorm : ‖(↑x : ℂ)‖ = 1
hxre : |x.re| ≤ 1 / 2
⊢ ∀ᶠ (x_1 : ℝ≥0) in 𝓝[>] 0, { coe := (↑x : ℂ) + (↑(↑x_1 : ℝ) : ℂ) * Complex.I, coe_im_pos := ⋯ : ℍ } ∈ closure 𝒟ᵒ
have h : ∀ (a : ℝ≥0) (ha : 0 < a), { coe := (↑x : ℂ) + (↑(↑a : ℝ) : ℂ) * Complex.I, coe_im_pos := sorry : ℍ } ∈ closure 𝒟ᵒ
Tactic: filter_upwards [self_mem_nhdsWithin] with a (ha : 0 < a)
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
x : ℍ
hxnorm : ‖(↑x : ℂ)‖ = 1
hxre : |x.re| ≤ 1 / 2
a : ℝ≥0
ha : 0 < a
⊢ { coe := (↑x : ℂ) + (↑(↑a : ℝ) : ℂ) * Complex.I, coe_im_pos := ⋯ : ℍ } ∈ closure 𝒟ᵒ
have h : 1 < ‖(↑{ coe := (↑x : ℂ) + (↑(↑a : ℝ) : ℂ) * Complex.I, coe_im_pos := sorry : ℍ } : ℂ)‖
Tactic: refine mem_closure_of_one_lt_norm ?_ (by simpa using hxre)
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
x : ℍ
hxnorm : ‖(↑x : ℂ)‖ = 1
hxre : |x.re| ≤ 1 / 2
a : ℝ≥0
ha : 0 < a
this : 1 < HPow.hPow (β := ℕ) ‖(↑x : ℂ)‖ 2 + HPow.hPow (β := ℕ) (↑a : ℝ) 2 + 2 * (↑a : ℝ) * x.im
⊢ 1 < ‖(↑{ coe := (↑x : ℂ) + (↑(↑a : ℝ) : ℂ) * Complex.I, coe_im_pos := ⋯ : ℍ } : ℂ)‖
have rw : LT.lt (α := ℝ) (1 : ℝ)
  ((normSq : (a : ℂ) → ℝ) (↑{ coe := (↑x : ℂ) + (↑(↑a : ℝ) : ℂ) * Complex.I, coe_im_pos := sorry : ℍ } : ℂ) : ℝ)
Tactic: rw [← one_lt_normSq_iff]
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
x : ℍ
hxnorm : ‖(↑x : ℂ)‖ = 1
hxre : |x.re| ≤ 1 / 2
a : ℝ≥0
ha : 0 < a
this : 1 < HPow.hPow (β := ℕ) ‖(↑x : ℂ)‖ 2 + HPow.hPow (β := ℕ) (↑a : ℝ) 2 + 2 * (↑a : ℝ) * x.im
⊢ LT.lt (α := ℝ) (1 : ℝ)
    ((normSq : (a : ℂ) → ℝ) (↑{ coe := (↑x : ℂ) + (↑(↑a : ℝ) : ℂ) * Complex.I, coe_im_pos := ⋯ : ℍ } : ℂ) : ℝ)
have h : (normSq : (a : ℂ) → ℝ) (↑{ coe := (↑x : ℂ) + (↑(↑a : ℝ) : ℂ) * Complex.I, coe_im_pos := sorry : ℍ } : ℂ) =
  HPow.hPow (β := ℕ) ‖(↑x : ℂ)‖ 2 + HPow.hPow (β := ℕ) (↑a : ℝ) 2 + 2 * (↑a : ℝ) * x.im
Tactic: convert! this
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
x : ℍ
hxnorm : ‖(↑x : ℂ)‖ = 1
hxre : |x.re| ≤ 1 / 2
a : ℝ≥0
ha : 0 < a
this : 1 < HPow.hPow (β := ℕ) ‖(↑x : ℂ)‖ 2 + HPow.hPow (β := ℕ) (↑a : ℝ) 2 + 2 * (↑a : ℝ) * x.im
⊢ (normSq : (a : ℂ) → ℝ) (↑{ coe := (↑x : ℂ) + (↑(↑a : ℝ) : ℂ) * Complex.I, coe_im_pos := ⋯ : ℍ } : ℂ) =
    HPow.hPow (β := ℕ) ‖(↑x : ℂ)‖ 2 + HPow.hPow (β := ℕ) (↑a : ℝ) 2 + 2 * (↑a : ℝ) * x.im
have h : x.re * x.re + (x.im + (↑a : ℝ)) * (x.im + (↑a : ℝ)) =
  x.re * x.re + x.im * x.im + HPow.hPow (β := ℕ) (↑a : ℝ) 2 + 2 * (↑a : ℝ) * x.im
Tactic: simp [← normSq_eq_norm_sq, normSq_apply]
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
x : ℍ
hxnorm : ‖(↑x : ℂ)‖ = 1
hxre : |x.re| ≤ 1 / 2
a : ℝ≥0
ha : 0 < a
⊢ 1 < ‖(↑{ coe := (↑x : ℂ) + (↑(↑a : ℝ) : ℂ) * Complex.I, coe_im_pos := ⋯ : ℍ } : ℂ)‖
have h : 1 < HPow.hPow (β := ℕ) ‖(↑x : ℂ)‖ 2 + HPow.hPow (β := ℕ) (↑a : ℝ) 2 + 2 * (↑a : ℝ) * x.im
Tactic: suffices 1 < ‖(x : ℂ)‖ ^ 2 + a ^ 2 + 2 * a * x.im by
      rw [← one_lt_normSq_iff]
      convert! this
      simp [← normSq_eq_norm_sq, normSq_apply]
      ring
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
x : ℍ
hxnorm : ‖(↑x : ℂ)‖ = 1
hxre : |x.re| ≤ 1 / 2
a : ℝ≥0
ha : 0 < a
⊢ 1 < HPow.hPow (β := ℕ) ‖(↑x : ℂ)‖ 2 + HPow.hPow (β := ℕ) (↑a : ℝ) 2 + 2 * (↑a : ℝ) * x.im
have h : 0 < HPow.hPow (β := ℕ) (↑a : ℝ) 2 + 2 * (↑a : ℝ) * x.im
Tactic: rw [hxnorm, one_pow, add_assoc, lt_add_iff_pos_right]
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
x : ℍ
hxnorm : ‖(↑x : ℂ)‖ = 1
hxre : |x.re| ≤ 1 / 2
⊢ Filter.Tendsto (fun (t : ℝ≥0) ↦ { coe := (↑x : ℂ) + (↑(↑t : ℝ) : ℂ) * Complex.I, coe_im_pos := ⋯ }) (𝓝[>] 0) (𝓝 x)
have hf : Filter.Tendsto (fun (t : ℝ≥0) ↦ { coe := (↑x : ℂ) + (↑(↑t : ℝ) : ℂ) * Complex.I, coe_im_pos := sorry }) (𝓝 0) (𝓝 x)
Tactic: refine .mono_left ?_ nhdsWithin_le_nhds
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
⊢ 𝒟 = closure 𝒟ᵒ
have refine : 𝒟 ⊆ closure 𝒟ᵒ
Tactic: refine subset_antisymm ?_ (isClosed_fd.closure_subset_iff.mpr fdo_subset_fd)
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
x : ℍ
⊢ x ∈ 𝒟 → x ∈ closure 𝒟ᵒ
have this : ∀ (hx : LE.le (α := ℝ) (1 : ℝ) ((normSq : (a : ℂ) → ℝ) (↑x : ℂ) : ℝ)) (hx' : |x.re| ≤ 1 / 2), x ∈ closure 𝒟ᵒ
Tactic: ⟨hx, hx'⟩
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (hypothesis)
Goal:
x : ℍ
hx : LE.le (α := ℝ) (1 : ℝ) ((normSq : (a : ℂ) → ℝ) (↑x : ℂ) : ℝ)
hx' : |x.re| ≤ 1 / 2
⊢ x ∈ closure 𝒟ᵒ
have hx₁ : 1 ≤ ‖(↑x : ℂ)‖
Tactic: rw [one_le_normSq_iff] at hx
Cleared hypotheses: hx, hx'
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
x : ℍ
hx : 1 ≤ ‖(↑x : ℂ)‖
hx' : |x.re| ≤ 1 / 2
⊢ x ∈ closure 𝒟ᵒ
have inl : ∀ (hx : 1 < ‖(↑x : ℂ)‖), x ∈ closure 𝒟ᵒ
Tactic: rcases lt_or_eq_of_le hx with hx | hx
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
x : ℍ
hx : 1 ≤ ‖(↑x : ℂ)‖
hx' : |x.re| ≤ 1 / 2
inl : ∀ (hx : 1 < ‖(↑x : ℂ)‖), x ∈ closure 𝒟ᵒ
⊢ x ∈ closure 𝒟ᵒ
have inr : ∀ (hx : 1 = ‖(↑x : ℂ)‖), x ∈ closure 𝒟ᵒ
Tactic: rcases lt_or_eq_of_le hx with hx | hx
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
⊢ 𝒟ᵒ = interior 𝒟
have refine : interior 𝒟 ⊆ 𝒟ᵒ
Tactic: refine subset_antisymm (isOpen_fdo.subset_interior_iff.mpr fdo_subset_fd) ?_
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (hypothesis)
Goal:
⊢ interior 𝒟 ⊆ 𝒟ᵒ
have ho1 : UpperHalfPlane.re '' interior 𝒟 ⊆ interior (UpperHalfPlane.re '' 𝒟)
Tactic: have ho1 := isOpenMap_re.image_interior_subset 𝒟
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (hypothesis)
Goal:
ho1 : UpperHalfPlane.re '' interior 𝒟 ⊆ interior (UpperHalfPlane.re '' 𝒟)
⊢ interior 𝒟 ⊆ 𝒟ᵒ
have ho2 : Subset (α := Set ℝ) ((fun (τ : ℍ) ↦ ‖(↑τ : ℂ)‖) '' interior 𝒟) (interior ((fun (τ : ℍ) ↦ ‖(↑τ : ℂ)‖) '' 𝒟))
Tactic: have ho2 := isOpenMap_norm.image_interior_subset 𝒟
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
ho1 : UpperHalfPlane.re '' interior 𝒟 ⊆ interior (UpperHalfPlane.re '' 𝒟)
ho2 : Subset (α := Set ℝ) ((fun (τ : ℍ) ↦ ‖(↑τ : ℂ)‖) '' interior 𝒟) (interior ((fun (τ : ℍ) ↦ ‖(↑τ : ℂ)‖) '' 𝒟))
x : ℍ
hx : x ∈ interior 𝒟
⊢ x ∈ 𝒟ᵒ
have rw : ∀ (ho1 : interior 𝒟 ⊆ UpperHalfPlane.re ⁻¹' interior (UpperHalfPlane.re '' 𝒟))
  (ho2 : interior 𝒟 ⊆ Set.preimage (β := ℝ) (fun (τ : ℍ) ↦ ‖(↑τ : ℂ)‖) (interior ((fun (τ : ℍ) ↦ ‖(↑τ : ℂ)‖) '' 𝒟)))
  ⦃x : ℍ⦄ (hx : x ∈ interior 𝒟), x ∈ 𝒟ᵒ
Tactic: rw [Set.image_subset_iff] at *
Cleared hypotheses: ho1, ho2, x, hx
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
ho1 : interior 𝒟 ⊆ UpperHalfPlane.re ⁻¹' interior (UpperHalfPlane.re '' 𝒟)
ho2 : interior 𝒟 ⊆ Set.preimage (β := ℝ) (fun (τ : ℍ) ↦ ‖(↑τ : ℂ)‖) (interior ((fun (τ : ℍ) ↦ ‖(↑τ : ℂ)‖) '' 𝒟))
x : ℍ
hx : x ∈ interior 𝒟
⊢ x ∈ 𝒟ᵒ
have left : LT.lt (α := ℝ) (1 : ℝ) ((normSq : (a : ℂ) → ℝ) (↑x : ℂ) : ℝ)
Tactic: constructor
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
ho1 : interior 𝒟 ⊆ UpperHalfPlane.re ⁻¹' interior (UpperHalfPlane.re '' 𝒟)
ho2 : interior 𝒟 ⊆ Set.preimage (β := ℝ) (fun (τ : ℍ) ↦ ‖(↑τ : ℂ)‖) (interior ((fun (τ : ℍ) ↦ ‖(↑τ : ℂ)‖) '' 𝒟))
x : ℍ
hx : x ∈ interior 𝒟
left : LT.lt (α := ℝ) (1 : ℝ) ((normSq : (a : ℂ) → ℝ) (↑x : ℂ) : ℝ)
⊢ x ∈ 𝒟ᵒ
have right : |x.re| < 1 / 2
Tactic: constructor
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
ho1 : interior 𝒟 ⊆ UpperHalfPlane.re ⁻¹' interior (UpperHalfPlane.re '' 𝒟)
ho2 : interior 𝒟 ⊆ Set.preimage (β := ℝ) (fun (τ : ℍ) ↦ ‖(↑τ : ℂ)‖) (interior ((fun (τ : ℍ) ↦ ‖(↑τ : ℂ)‖) '' 𝒟))
x : ℍ
hx : x ∈ interior 𝒟
⊢ LT.lt (α := ℝ) (1 : ℝ) ((normSq : (a : ℂ) → ℝ) (↑x : ℂ) : ℝ)
have left : Membership.mem (γ := Set ℝ) (interior (Set.Ici 1)) ‖(↑x : ℂ)‖
Tactic: rw [one_lt_normSq_iff, ← Set.mem_Ioi, ← interior_Ici]
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
ho1 : interior 𝒟 ⊆ UpperHalfPlane.re ⁻¹' interior (UpperHalfPlane.re '' 𝒟)
ho2 : interior 𝒟 ⊆ Set.preimage (β := ℝ) (fun (τ : ℍ) ↦ ‖(↑τ : ℂ)‖) (interior ((fun (τ : ℍ) ↦ ‖(↑τ : ℂ)‖) '' 𝒟))
x : ℍ
hx : x ∈ interior 𝒟
⊢ Subset (α := Set ℝ) ((fun (τ : ℍ) ↦ ‖(↑τ : ℂ)‖) '' 𝒟) (Set.Ici 1)
have rw : 𝒟 ⊆ Set.preimage (β := ℝ) (fun (τ : ℍ) ↦ ‖(↑τ : ℂ)‖) (Set.Ici 1)
Tactic: rw [Set.image_subset_iff]
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
ho1 : interior 𝒟 ⊆ UpperHalfPlane.re ⁻¹' interior (UpperHalfPlane.re '' 𝒟)
ho2 : interior 𝒟 ⊆ Set.preimage (β := ℝ) (fun (τ : ℍ) ↦ ‖(↑τ : ℂ)‖) (interior ((fun (τ : ℍ) ↦ ‖(↑τ : ℂ)‖) '' 𝒟))
x : ℍ
hx : x ∈ interior 𝒟
⊢ |x.re| < 1 / 2
have right : x.re ∈ interior (Set.Icc (-(1 / 2)) (1 / 2))
Tactic: rw [abs_lt, ← Set.mem_Ioo, ← interior_Icc]
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
ho1 : interior 𝒟 ⊆ UpperHalfPlane.re ⁻¹' interior (UpperHalfPlane.re '' 𝒟)
ho2 : interior 𝒟 ⊆ Set.preimage (β := ℝ) (fun (τ : ℍ) ↦ ‖(↑τ : ℂ)‖) (interior ((fun (τ : ℍ) ↦ ‖(↑τ : ℂ)‖) '' 𝒟))
x : ℍ
hx : x ∈ interior 𝒟
⊢ x.re ∈ interior (Set.Icc (-(1 / 2)) (1 / 2))
have apply : UpperHalfPlane.re '' 𝒟 ⊆ Set.Icc (-(1 / 2)) (1 / 2)
Tactic: apply Set.mem_of_mem_of_subset ((Set.mem_preimage.mp (ho1 hx))) (interior_mono ?_)
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
ho1 : interior 𝒟 ⊆ UpperHalfPlane.re ⁻¹' interior (UpperHalfPlane.re '' 𝒟)
ho2 : interior 𝒟 ⊆ Set.preimage (β := ℝ) (fun (τ : ℍ) ↦ ‖(↑τ : ℂ)‖) (interior ((fun (τ : ℍ) ↦ ‖(↑τ : ℂ)‖) '' 𝒟))
x : ℍ
hx : x ∈ interior 𝒟
⊢ UpperHalfPlane.re '' 𝒟 ⊆ Set.Icc (-(1 / 2)) (1 / 2)
have rw : 𝒟 ⊆ UpperHalfPlane.re ⁻¹' Set.Icc (-(1 / 2)) (1 / 2)
Tactic: rw [Set.image_subset_iff]
-- Lean 4 proof step from Mathlib.NumberTheory.Modular (goal)
Goal:
y : ℝ
⊢ UpperHalfPlane.coe '' truncatedFundamentalDomain y = {z : ℂ | 0 ≤ z.im ∧ z.im ≤ y ∧ |z.re| ≤ 1 / 2 ∧ 1 ≤ ‖z‖}
have h : ∀ (z : ℂ),
  z ∈ UpperHalfPlane.coe '' truncatedFundamentalDomain y ↔
    Membership.mem (γ := Set ℂ) {z : ℂ | 0 ≤ z.im ∧ z.im ≤ y ∧ |z.re| ≤ 1 / 2 ∧ 1 ≤ ‖z‖} z
Tactic: ext z