l3lab/ntp-mathlib huggingface dataset:
/-- The **Minkowski Convex Body Theorem**. If `s` is a convex symmetric domain of `E` whose volume
is large enough compared to the covolume of a lattice `L` of `E`, then it contains a non-zero
lattice point of `L`.  -/
theorem exists_ne_zero_mem_lattice_of_measure_mul_two_pow_lt_measure [NormedAddCommGroup E]
    [NormedSpace ℝ E] [BorelSpace E] [FiniteDimensional ℝ E] [IsAddHaarMeasure μ]
    {L : AddSubgroup E} [Countable L] (fund : IsAddFundamentalDomain L F μ)
    (h_symm : ∀ x ∈ s, -x ∈ s) (h_conv : Convex ℝ s) (h : μ F * 2 ^ finrank ℝ E < μ s) :
    ∃ x ≠ 0, ((x : L) : E) ∈ s := by
  have h_vol : μ F < μ ((2⁻¹ : ℝ) • s) := by
    rw [addHaar_smul_of_nonneg μ (by norm_num : 0 ≤ (2 : ℝ)⁻¹) s, ←
      mul_lt_mul_right (pow_ne_zero (finrank ℝ E) (two_ne_zero' _)) (pow_ne_top two_ne_top),
      mul_right_comm, ofReal_pow (by 

Proof state:
E : Type u_1
L✝ : Type u_2
inst✝⁶ : MeasurableSpace E
μ : Measure E
F s : Set E
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : IsAddHaarMeasure μ
L : AddSubgroup E
inst✝ : Countable ↥L
fund : IsAddFundamentalDomain (↥L) F
h_symm : ∀ x ∈ s, -x ∈ s
h_conv : Convex ℝ s
h : ↑↑μ F * 2 ^ finrank ℝ E < ↑↑μ s
⊢ 0 ≤ 2⁻¹

Next tactic:
norm_num
/-- The **Minkowski Convex Body Theorem**. If `s` is a convex symmetric domain of `E` whose volume
is large enough compared to the covolume of a lattice `L` of `E`, then it contains a non-zero
lattice point of `L`.  -/
theorem exists_ne_zero_mem_lattice_of_measure_mul_two_pow_lt_measure [NormedAddCommGroup E]
    [NormedSpace ℝ E] [BorelSpace E] [FiniteDimensional ℝ E] [IsAddHaarMeasure μ]
    {L : AddSubgroup E} [Countable L] (fund : IsAddFundamentalDomain L F μ)
    (h_symm : ∀ x ∈ s, -x ∈ s) (h_conv : Convex ℝ s) (h : μ F * 2 ^ finrank ℝ E < μ s) :
    ∃ x ≠ 0, ((x : L) : E) ∈ s := by
  have h_vol : μ F < μ ((2⁻¹ : ℝ) • s) := by
    rw [addHaar_smul_of_nonneg μ (by norm_num : 0 ≤ (2 : ℝ)⁻¹) s, ←
      mul_lt_mul_right (pow_ne_zero (finrank ℝ E) (two_ne_zero' _)) (pow_ne_top two_ne_top),
      mul_right_comm, ofReal_pow (by norm_num : 0 ≤ (2 : ℝ)⁻¹), ← ofReal_inv_of_pos zero_lt_two]
    

Proof state:
E : Type u_1
L✝ : Type u_2
inst✝⁶ : MeasurableSpace E
μ : Measure E
F s : Set E
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : IsAddHaarMeasure μ
L : AddSubgroup E
inst✝ : Countable ↥L
fund : IsAddFundamentalDomain (↥L) F
h_symm : ∀ x ∈ s, -x ∈ s
h_conv : Convex ℝ s
h : ↑↑μ F * 2 ^ finrank ℝ E < ↑↑μ s
⊢ ↑↑μ F * 2 ^ finrank ℝ E < (ENNReal.ofReal 2)⁻¹ ^ finrank ℝ E * 2 ^ finrank ℝ E * ↑↑μ s

Next tactic:
norm_num
/-- The **Minkowski Convex Body Theorem**. If `s` is a convex symmetric domain of `E` whose volume
is large enough compared to the covolume of a lattice `L` of `E`, then it contains a non-zero
lattice point of `L`.  -/
theorem exists_ne_zero_mem_lattice_of_measure_mul_two_pow_lt_measure [NormedAddCommGroup E]
    [NormedSpace ℝ E] [BorelSpace E] [FiniteDimensional ℝ E] [IsAddHaarMeasure μ]
    {L : AddSubgroup E} [Countable L] (fund : IsAddFundamentalDomain L F μ)
    (h_symm : ∀ x ∈ s, -x ∈ s) (h_conv : Convex ℝ s) (h : μ F * 2 ^ finrank ℝ E < μ s) :
    ∃ x ≠ 0, ((x : L) : E) ∈ s := by
  have h_vol : μ F < μ ((2⁻¹ : ℝ) • s) := by
    rw [addHaar_smul_of_nonneg μ (by norm_num : 0 ≤ (2 : ℝ)⁻¹) s, ←
      mul_lt_mul_right (pow_ne_zero (finrank ℝ E) (two_ne_zero' _)) (pow_ne_top two_ne_top),
      mul_right_comm, ofReal_pow (by norm_num : 0 ≤ (2 : ℝ)⁻¹), ← ofReal_inv_of_pos zero_lt_two]
    norm_num
    

Proof state:
E : Type u_1
L✝ : Type u_2
inst✝⁶ : MeasurableSpace E
μ : Measure E
F s : Set E
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : IsAddHaarMeasure μ
L : AddSubgroup E
inst✝ : Countable ↥L
fund : IsAddFundamentalDomain (↥L) F
h_symm : ∀ x ∈ s, -x ∈ s
h_conv : Convex ℝ s
h : ↑↑μ F * 2 ^ finrank ℝ E < ↑↑μ s
⊢ ↑↑μ F * 2 ^ finrank ℝ E < 2⁻¹ ^ finrank ℝ E * 2 ^ finrank ℝ E * ↑↑μ s

Next tactic:
rwa [← mul_pow, ENNReal.inv_mul_cancel two_ne_zero two_ne_top, one_pow, one_mul]
/-- The **Minkowski Convex Body Theorem**. If `s` is a convex symmetric domain of `E` whose volume
is large enough compared to the covolume of a lattice `L` of `E`, then it contains a non-zero
lattice point of `L`.  -/
theorem exists_ne_zero_mem_lattice_of_measure_mul_two_pow_lt_measure [NormedAddCommGroup E]
    [NormedSpace ℝ E] [BorelSpace E] [FiniteDimensional ℝ E] [IsAddHaarMeasure μ]
    {L : AddSubgroup E} [Countable L] (fund : IsAddFundamentalDomain L F μ)
    (h_symm : ∀ x ∈ s, -x ∈ s) (h_conv : Convex ℝ s) (h : μ F * 2 ^ finrank ℝ E < μ s) :
    ∃ x ≠ 0, ((x : L) : E) ∈ s := by
  have h_vol : μ F < μ ((2⁻¹ : ℝ) • s) := by
    rw [addHaar_smul_of_nonneg μ (by norm_num : 0 ≤ (2 : ℝ)⁻¹) s, ←
      mul_lt_mul_right (pow_ne_zero (finrank ℝ E) (two_ne_zero' _)) (pow_ne_top two_ne_top),
      mul_right_comm, ofReal_pow (by norm_num : 0 ≤ (2 : ℝ)⁻¹), ← ofReal_inv_of_pos zero_lt_two]
    norm_num
    rwa [← mul_pow, ENNReal.inv_mul_cancel two_ne_zero two_ne_top, one_pow, one_mul]
  

Proof state:
E : Type u_1
L✝ : Type u_2
inst✝⁶ : MeasurableSpace E
μ : Measure E
F s : Set E
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : IsAddHaarMeasure μ
L : AddSubgroup E
inst✝ : Countable ↥L
fund : IsAddFundamentalDomain (↥L) F
h_symm : ∀ x ∈ s, -x ∈ s
h_conv : Convex ℝ s
h : ↑↑μ F * 2 ^ finrank ℝ E < ↑↑μ s
h_vol : ↑↑μ F < ↑↑μ (2⁻¹ • s)
⊢ ∃ x, x ≠ 0 ∧ ↑x ∈ s

Next tactic:
obtain ⟨x, y, hxy, h⟩ :=
    exists_pair_mem_lattice_not_disjoint_vadd fund ((h_conv.smul _).nullMeasurableSet _) h_vol
/-- The **Minkowski Convex Body Theorem**. If `s` is a convex symmetric domain of `E` whose volume
is large enough compared to the covolume of a lattice `L` of `E`, then it contains a non-zero
lattice point of `L`.  -/
theorem exists_ne_zero_mem_lattice_of_measure_mul_two_pow_lt_measure [NormedAddCommGroup E]
    [NormedSpace ℝ E] [BorelSpace E] [FiniteDimensional ℝ E] [IsAddHaarMeasure μ]
    {L : AddSubgroup E} [Countable L] (fund : IsAddFundamentalDomain L F μ)
    (h_symm : ∀ x ∈ s, -x ∈ s) (h_conv : Convex ℝ s) (h : μ F * 2 ^ finrank ℝ E < μ s) :
    ∃ x ≠ 0, ((x : L) : E) ∈ s := by
  have h_vol : μ F < μ ((2⁻¹ : ℝ) • s) := by
    rw [addHaar_smul_of_nonneg μ (by norm_num : 0 ≤ (2 : ℝ)⁻¹) s, ←
      mul_lt_mul_right (pow_ne_zero (finrank ℝ E) (two_ne_zero' _)) (pow_ne_top two_ne_top),
      mul_right_comm, ofReal_pow (by norm_num : 0 ≤ (2 : ℝ)⁻¹), ← ofReal_inv_of_pos zero_lt_two]
    norm_num
    rwa [← mul_pow, ENNReal.inv_mul_cancel two_ne_zero two_ne_top, one_pow, one_mul]
  obtain ⟨x, y, hxy, h⟩ :=
    exists_pair_mem_lattice_not_disjoint_vadd fund ((h_conv.smul _).nullMeasurableSet _) h_vol
  

Proof state:
case intro.intro.intro
E : Type u_1
L✝ : Type u_2
inst✝⁶ : MeasurableSpace E
μ : Measure E
F s : Set E
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : IsAddHaarMeasure μ
L : AddSubgroup E
inst✝ : Countable ↥L
fund : IsAddFundamentalDomain (↥L) F
h_symm : ∀ x ∈ s, -x ∈ s
h_conv : Convex ℝ s
h✝ : ↑↑μ F * 2 ^ finrank ℝ E < ↑↑μ s
h_vol : ↑↑μ F < ↑↑μ (2⁻¹ • s)
x y : ↥L
hxy : x ≠ y
h : ¬Disjoint (x +ᵥ 2⁻¹ • s) (y +ᵥ 2⁻¹ • s)
⊢ ∃ x, x ≠ 0 ∧ ↑x ∈ s

Next tactic:
obtain ⟨_, ⟨v, hv, rfl⟩, w, hw, hvw⟩ := Set.not_disjoint_iff.mp h
/-- The **Minkowski Convex Body Theorem**. If `s` is a convex symmetric domain of `E` whose volume
is large enough compared to the covolume of a lattice `L` of `E`, then it contains a non-zero
lattice point of `L`.  -/
theorem exists_ne_zero_mem_lattice_of_measure_mul_two_pow_lt_measure [NormedAddCommGroup E]
    [NormedSpace ℝ E] [BorelSpace E] [FiniteDimensional ℝ E] [IsAddHaarMeasure μ]
    {L : AddSubgroup E} [Countable L] (fund : IsAddFundamentalDomain L F μ)
    (h_symm : ∀ x ∈ s, -x ∈ s) (h_conv : Convex ℝ s) (h : μ F * 2 ^ finrank ℝ E < μ s) :
    ∃ x ≠ 0, ((x : L) : E) ∈ s := by
  have h_vol : μ F < μ ((2⁻¹ : ℝ) • s) := by
    rw [addHaar_smul_of_nonneg μ (by norm_num : 0 ≤ (2 : ℝ)⁻¹) s, ←
      mul_lt_mul_right (pow_ne_zero (finrank ℝ E) (two_ne_zero' _)) (pow_ne_top two_ne_top),
      mul_right_comm, ofReal_pow (by norm_num : 0 ≤ (2 : ℝ)⁻¹), ← ofReal_inv_of_pos zero_lt_two]
    norm_num
    rwa [← mul_pow, ENNReal.inv_mul_cancel two_ne_zero two_ne_top, one_pow, one_mul]
  obtain ⟨x, y, hxy, h⟩ :=
    exists_pair_mem_lattice_not_disjoint_vadd fund ((h_conv.smul _).nullMeasurableSet _) h_vol
  obtain ⟨_, ⟨v, hv, rfl⟩, w, hw, hvw⟩ := Set.not_disjoint_iff.mp h
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro
E : Type u_1
L✝ : Type u_2
inst✝⁶ : MeasurableSpace E
μ : Measure E
F s : Set E
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : IsAddHaarMeasure μ
L : AddSubgroup E
inst✝ : Countable ↥L
fund : IsAddFundamentalDomain (↥L) F
h_symm : ∀ x ∈ s, -x ∈ s
h_conv : Convex ℝ s
h✝ : ↑↑μ F * 2 ^ finrank ℝ E < ↑↑μ s
h_vol : ↑↑μ F < ↑↑μ (2⁻¹ • s)
x y : ↥L
hxy : x ≠ y
h : ¬Disjoint (x +ᵥ 2⁻¹ • s) (y +ᵥ 2⁻¹ • s)
v : E
hv : v ∈ 2⁻¹ • s
w : E
hw : w ∈ 2⁻¹ • s
hvw : (fun x => y +ᵥ x) w = (fun x_1 => x +ᵥ x_1) v
⊢ ∃ x, x ≠ 0 ∧ ↑x ∈ s

Next tactic:
refine' ⟨x - y, sub_ne_zero.2 hxy, _⟩
/-- The **Minkowski Convex Body Theorem**. If `s` is a convex symmetric domain of `E` whose volume
is large enough compared to the covolume of a lattice `L` of `E`, then it contains a non-zero
lattice point of `L`.  -/
theorem exists_ne_zero_mem_lattice_of_measure_mul_two_pow_lt_measure [NormedAddCommGroup E]
    [NormedSpace ℝ E] [BorelSpace E] [FiniteDimensional ℝ E] [IsAddHaarMeasure μ]
    {L : AddSubgroup E} [Countable L] (fund : IsAddFundamentalDomain L F μ)
    (h_symm : ∀ x ∈ s, -x ∈ s) (h_conv : Convex ℝ s) (h : μ F * 2 ^ finrank ℝ E < μ s) :
    ∃ x ≠ 0, ((x : L) : E) ∈ s := by
  have h_vol : μ F < μ ((2⁻¹ : ℝ) • s) := by
    rw [addHaar_smul_of_nonneg μ (by norm_num : 0 ≤ (2 : ℝ)⁻¹) s, ←
      mul_lt_mul_right (pow_ne_zero (finrank ℝ E) (two_ne_zero' _)) (pow_ne_top two_ne_top),
      mul_right_comm, ofReal_pow (by norm_num : 0 ≤ (2 : ℝ)⁻¹), ← ofReal_inv_of_pos zero_lt_two]
    norm_num
    rwa [← mul_pow, ENNReal.inv_mul_cancel two_ne_zero two_ne_top, one_pow, one_mul]
  obtain ⟨x, y, hxy, h⟩ :=
    exists_pair_mem_lattice_not_disjoint_vadd fund ((h_conv.smul _).nullMeasurableSet _) h_vol
  obtain ⟨_, ⟨v, hv, rfl⟩, w, hw, hvw⟩ := Set.not_disjoint_iff.mp h
  refine' ⟨x - y, sub_ne_zero.2 hxy, _⟩
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro
E : Type u_1
L✝ : Type u_2
inst✝⁶ : MeasurableSpace E
μ : Measure E
F s : Set E
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : IsAddHaarMeasure μ
L : AddSubgroup E
inst✝ : Countable ↥L
fund : IsAddFundamentalDomain (↥L) F
h_symm : ∀ x ∈ s, -x ∈ s
h_conv : Convex ℝ s
h✝ : ↑↑μ F * 2 ^ finrank ℝ E < ↑↑μ s
h_vol : ↑↑μ F < ↑↑μ (2⁻¹ • s)
x y : ↥L
hxy : x ≠ y
h : ¬Disjoint (x +ᵥ 2⁻¹ • s) (y +ᵥ 2⁻¹ • s)
v : E
hv : v ∈ 2⁻¹ • s
w : E
hw : w ∈ 2⁻¹ • s
hvw : (fun x => y +ᵥ x) w = (fun x_1 => x +ᵥ x_1) v
⊢ ↑(x - y) ∈ s

Next tactic:
rw [Set.mem_inv_smul_set_iff₀ (two_ne_zero' ℝ)] at hv hw
/-- The **Minkowski Convex Body Theorem**. If `s` is a convex symmetric domain of `E` whose volume
is large enough compared to the covolume of a lattice `L` of `E`, then it contains a non-zero
lattice point of `L`.  -/
theorem exists_ne_zero_mem_lattice_of_measure_mul_two_pow_lt_measure [NormedAddCommGroup E]
    [NormedSpace ℝ E] [BorelSpace E] [FiniteDimensional ℝ E] [IsAddHaarMeasure μ]
    {L : AddSubgroup E} [Countable L] (fund : IsAddFundamentalDomain L F μ)
    (h_symm : ∀ x ∈ s, -x ∈ s) (h_conv : Convex ℝ s) (h : μ F * 2 ^ finrank ℝ E < μ s) :
    ∃ x ≠ 0, ((x : L) : E) ∈ s := by
  have h_vol : μ F < μ ((2⁻¹ : ℝ) • s) := by
    rw [addHaar_smul_of_nonneg μ (by norm_num : 0 ≤ (2 : ℝ)⁻¹) s, ←
      mul_lt_mul_right (pow_ne_zero (finrank ℝ E) (two_ne_zero' _)) (pow_ne_top two_ne_top),
      mul_right_comm, ofReal_pow (by norm_num : 0 ≤ (2 : ℝ)⁻¹), ← ofReal_inv_of_pos zero_lt_two]
    norm_num
    rwa [← mul_pow, ENNReal.inv_mul_cancel two_ne_zero two_ne_top, one_pow, one_mul]
  obtain ⟨x, y, hxy, h⟩ :=
    exists_pair_mem_lattice_not_disjoint_vadd fund ((h_conv.smul _).nullMeasurableSet _) h_vol
  obtain ⟨_, ⟨v, hv, rfl⟩, w, hw, hvw⟩ := Set.not_disjoint_iff.mp h
  refine' ⟨x - y, sub_ne_zero.2 hxy, _⟩
  rw [Set.mem_inv_smul_set_iff₀ (two_ne_zero' ℝ)] at hv hw
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro
E : Type u_1
L✝ : Type u_2
inst✝⁶ : MeasurableSpace E
μ : Measure E
F s : Set E
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : IsAddHaarMeasure μ
L : AddSubgroup E
inst✝ : Countable ↥L
fund : IsAddFundamentalDomain (↥L) F
h_symm : ∀ x ∈ s, -x ∈ s
h_conv : Convex ℝ s
h✝ : ↑↑μ F * 2 ^ finrank ℝ E < ↑↑μ s
h_vol : ↑↑μ F < ↑↑μ (2⁻¹ • s)
x y : ↥L
hxy : x ≠ y
h : ¬Disjoint (x +ᵥ 2⁻¹ • s) (y +ᵥ 2⁻¹ • s)
v : E
hv : 2 • v ∈ s
w : E
hw : 2 • w ∈ s
hvw : (fun x => y +ᵥ x) w = (fun x_1 => x +ᵥ x_1) v
⊢ ↑(x - y) ∈ s

Next tactic:
simp_rw [AddSubgroup.vadd_def, vadd_eq_add, add_comm _ w, ← sub_eq_sub_iff_add_eq_add, ←
    AddSubgroup.coe_sub] at hvw
/-- The **Minkowski Convex Body Theorem**. If `s` is a convex symmetric domain of `E` whose volume
is large enough compared to the covolume of a lattice `L` of `E`, then it contains a non-zero
lattice point of `L`.  -/
theorem exists_ne_zero_mem_lattice_of_measure_mul_two_pow_lt_measure [NormedAddCommGroup E]
    [NormedSpace ℝ E] [BorelSpace E] [FiniteDimensional ℝ E] [IsAddHaarMeasure μ]
    {L : AddSubgroup E} [Countable L] (fund : IsAddFundamentalDomain L F μ)
    (h_symm : ∀ x ∈ s, -x ∈ s) (h_conv : Convex ℝ s) (h : μ F * 2 ^ finrank ℝ E < μ s) :
    ∃ x ≠ 0, ((x : L) : E) ∈ s := by
  have h_vol : μ F < μ ((2⁻¹ : ℝ) • s) := by
    rw [addHaar_smul_of_nonneg μ (by norm_num : 0 ≤ (2 : ℝ)⁻¹) s, ←
      mul_lt_mul_right (pow_ne_zero (finrank ℝ E) (two_ne_zero' _)) (pow_ne_top two_ne_top),
      mul_right_comm, ofReal_pow (by norm_num : 0 ≤ (2 : ℝ)⁻¹), ← ofReal_inv_of_pos zero_lt_two]
    norm_num
    rwa [← mul_pow, ENNReal.inv_mul_cancel two_ne_zero two_ne_top, one_pow, one_mul]
  obtain ⟨x, y, hxy, h⟩ :=
    exists_pair_mem_lattice_not_disjoint_vadd fund ((h_conv.smul _).nullMeasurableSet _) h_vol
  obtain ⟨_, ⟨v, hv, rfl⟩, w, hw, hvw⟩ := Set.not_disjoint_iff.mp h
  refine' ⟨x - y, sub_ne_zero.2 hxy, _⟩
  rw [Set.mem_inv_smul_set_iff₀ (two_ne_zero' ℝ)] at hv hw
  simp_rw [AddSubgroup.vadd_def, vadd_eq_add, add_comm _ w, ← sub_eq_sub_iff_add_eq_add, ←
    AddSubgroup.coe_sub] at hvw
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro
E : Type u_1
L✝ : Type u_2
inst✝⁶ : MeasurableSpace E
μ : Measure E
F s : Set E
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : IsAddHaarMeasure μ
L : AddSubgroup E
inst✝ : Countable ↥L
fund : IsAddFundamentalDomain (↥L) F
h_symm : ∀ x ∈ s, -x ∈ s
h_conv : Convex ℝ s
h✝ : ↑↑μ F * 2 ^ finrank ℝ E < ↑↑μ s
h_vol : ↑↑μ F < ↑↑μ (2⁻¹ • s)
x y : ↥L
hxy : x ≠ y
h : ¬Disjoint (x +ᵥ 2⁻¹ • s) (y +ᵥ 2⁻¹ • s)
v : E
hv : 2 • v ∈ s
w : E
hw : 2 • w ∈ s
hvw : w - v = ↑(x - y)
⊢ ↑(x - y) ∈ s

Next tactic:
rw [← hvw, ← inv_smul_smul₀ (two_ne_zero' ℝ) (_ - _), smul_sub, sub_eq_add_neg, smul_add]
/-- The **Minkowski Convex Body Theorem**. If `s` is a convex symmetric domain of `E` whose volume
is large enough compared to the covolume of a lattice `L` of `E`, then it contains a non-zero
lattice point of `L`.  -/
theorem exists_ne_zero_mem_lattice_of_measure_mul_two_pow_lt_measure [NormedAddCommGroup E]
    [NormedSpace ℝ E] [BorelSpace E] [FiniteDimensional ℝ E] [IsAddHaarMeasure μ]
    {L : AddSubgroup E} [Countable L] (fund : IsAddFundamentalDomain L F μ)
    (h_symm : ∀ x ∈ s, -x ∈ s) (h_conv : Convex ℝ s) (h : μ F * 2 ^ finrank ℝ E < μ s) :
    ∃ x ≠ 0, ((x : L) : E) ∈ s := by
  have h_vol : μ F < μ ((2⁻¹ : ℝ) • s) := by
    rw [addHaar_smul_of_nonneg μ (by norm_num : 0 ≤ (2 : ℝ)⁻¹) s, ←
      mul_lt_mul_right (pow_ne_zero (finrank ℝ E) (two_ne_zero' _)) (pow_ne_top two_ne_top),
      mul_right_comm, ofReal_pow (by norm_num : 0 ≤ (2 : ℝ)⁻¹), ← ofReal_inv_of_pos zero_lt_two]
    norm_num
    rwa [← mul_pow, ENNReal.inv_mul_cancel two_ne_zero two_ne_top, one_pow, one_mul]
  obtain ⟨x, y, hxy, h⟩ :=
    exists_pair_mem_lattice_not_disjoint_vadd fund ((h_conv.smul _).nullMeasurableSet _) h_vol
  obtain ⟨_, ⟨v, hv, rfl⟩, w, hw, hvw⟩ := Set.not_disjoint_iff.mp h
  refine' ⟨x - y, sub_ne_zero.2 hxy, _⟩
  rw [Set.mem_inv_smul_set_iff₀ (two_ne_zero' ℝ)] at hv hw
  simp_rw [AddSubgroup.vadd_def, vadd_eq_add, add_comm _ w, ← sub_eq_sub_iff_add_eq_add, ←
    AddSubgroup.coe_sub] at hvw
  rw [← hvw, ← inv_smul_smul₀ (two_ne_zero' ℝ) (_ - _), smul_sub, sub_eq_add_neg, smul_add]
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro
E : Type u_1
L✝ : Type u_2
inst✝⁶ : MeasurableSpace E
μ : Measure E
F s : Set E
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : IsAddHaarMeasure μ
L : AddSubgroup E
inst✝ : Countable ↥L
fund : IsAddFundamentalDomain (↥L) F
h_symm : ∀ x ∈ s, -x ∈ s
h_conv : Convex ℝ s
h✝ : ↑↑μ F * 2 ^ finrank ℝ E < ↑↑μ s
h_vol : ↑↑μ F < ↑↑μ (2⁻¹ • s)
x y : ↥L
hxy : x ≠ y
h : ¬Disjoint (x +ᵥ 2⁻¹ • s) (y +ᵥ 2⁻¹ • s)
v : E
hv : 2 • v ∈ s
w : E
hw : 2 • w ∈ s
hvw : w - v = ↑(x - y)
⊢ 2⁻¹ • 2 • w + 2⁻¹ • -(2 • v) ∈ s

Next tactic:
refine' h_conv hw (h_symm _ hv) _ _ _
/-- The **Minkowski Convex Body Theorem**. If `s` is a convex symmetric domain of `E` whose volume
is large enough compared to the covolume of a lattice `L` of `E`, then it contains a non-zero
lattice point of `L`.  -/
theorem exists_ne_zero_mem_lattice_of_measure_mul_two_pow_lt_measure [NormedAddCommGroup E]
    [NormedSpace ℝ E] [BorelSpace E] [FiniteDimensional ℝ E] [IsAddHaarMeasure μ]
    {L : AddSubgroup E} [Countable L] (fund : IsAddFundamentalDomain L F μ)
    (h_symm : ∀ x ∈ s, -x ∈ s) (h_conv : Convex ℝ s) (h : μ F * 2 ^ finrank ℝ E < μ s) :
    ∃ x ≠ 0, ((x : L) : E) ∈ s := by
  have h_vol : μ F < μ ((2⁻¹ : ℝ) • s) := by
    rw [addHaar_smul_of_nonneg μ (by norm_num : 0 ≤ (2 : ℝ)⁻¹) s, ←
      mul_lt_mul_right (pow_ne_zero (finrank ℝ E) (two_ne_zero' _)) (pow_ne_top two_ne_top),
      mul_right_comm, ofReal_pow (by norm_num : 0 ≤ (2 : ℝ)⁻¹), ← ofReal_inv_of_pos zero_lt_two]
    norm_num
    rwa [← mul_pow, ENNReal.inv_mul_cancel two_ne_zero two_ne_top, one_pow, one_mul]
  obtain ⟨x, y, hxy, h⟩ :=
    exists_pair_mem_lattice_not_disjoint_vadd fund ((h_conv.smul _).nullMeasurableSet _) h_vol
  obtain ⟨_, ⟨v, hv, rfl⟩, w, hw, hvw⟩ := Set.not_disjoint_iff.mp h
  refine' ⟨x - y, sub_ne_zero.2 hxy, _⟩
  rw [Set.mem_inv_smul_set_iff₀ (two_ne_zero' ℝ)] at hv hw
  simp_rw [AddSubgroup.vadd_def, vadd_eq_add, add_comm _ w, ← sub_eq_sub_iff_add_eq_add, ←
    AddSubgroup.coe_sub] at hvw
  rw [← hvw, ← inv_smul_smul₀ (two_ne_zero' ℝ) (_ - _), smul_sub, sub_eq_add_neg, smul_add]
  refine' h_conv hw (h_symm _ hv) _ _ _ <;> 

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1
E : Type u_1
L✝ : Type u_2
inst✝⁶ : MeasurableSpace E
μ : Measure E
F s : Set E
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : IsAddHaarMeasure μ
L : AddSubgroup E
inst✝ : Countable ↥L
fund : IsAddFundamentalDomain (↥L) F
h_symm : ∀ x ∈ s, -x ∈ s
h_conv : Convex ℝ s
h✝ : ↑↑μ F * 2 ^ finrank ℝ E < ↑↑μ s
h_vol : ↑↑μ F < ↑↑μ (2⁻¹ • s)
x y : ↥L
hxy : x ≠ y
h : ¬Disjoint (x +ᵥ 2⁻¹ • s) (y +ᵥ 2⁻¹ • s)
v : E
hv : 2 • v ∈ s
w : E
hw : 2 • w ∈ s
hvw : w - v = ↑(x - y)
⊢ 0 ≤ 2⁻¹

Next tactic:
norm_num
/-- The **Minkowski Convex Body Theorem**. If `s` is a convex symmetric domain of `E` whose volume
is large enough compared to the covolume of a lattice `L` of `E`, then it contains a non-zero
lattice point of `L`.  -/
theorem exists_ne_zero_mem_lattice_of_measure_mul_two_pow_lt_measure [NormedAddCommGroup E]
    [NormedSpace ℝ E] [BorelSpace E] [FiniteDimensional ℝ E] [IsAddHaarMeasure μ]
    {L : AddSubgroup E} [Countable L] (fund : IsAddFundamentalDomain L F μ)
    (h_symm : ∀ x ∈ s, -x ∈ s) (h_conv : Convex ℝ s) (h : μ F * 2 ^ finrank ℝ E < μ s) :
    ∃ x ≠ 0, ((x : L) : E) ∈ s := by
  have h_vol : μ F < μ ((2⁻¹ : ℝ) • s) := by
    rw [addHaar_smul_of_nonneg μ (by norm_num : 0 ≤ (2 : ℝ)⁻¹) s, ←
      mul_lt_mul_right (pow_ne_zero (finrank ℝ E) (two_ne_zero' _)) (pow_ne_top two_ne_top),
      mul_right_comm, ofReal_pow (by norm_num : 0 ≤ (2 : ℝ)⁻¹), ← ofReal_inv_of_pos zero_lt_two]
    norm_num
    rwa [← mul_pow, ENNReal.inv_mul_cancel two_ne_zero two_ne_top, one_pow, one_mul]
  obtain ⟨x, y, hxy, h⟩ :=
    exists_pair_mem_lattice_not_disjoint_vadd fund ((h_conv.smul _).nullMeasurableSet _) h_vol
  obtain ⟨_, ⟨v, hv, rfl⟩, w, hw, hvw⟩ := Set.not_disjoint_iff.mp h
  refine' ⟨x - y, sub_ne_zero.2 hxy, _⟩
  rw [Set.mem_inv_smul_set_iff₀ (two_ne_zero' ℝ)] at hv hw
  simp_rw [AddSubgroup.vadd_def, vadd_eq_add, add_comm _ w, ← sub_eq_sub_iff_add_eq_add, ←
    AddSubgroup.coe_sub] at hvw
  rw [← hvw, ← inv_smul_smul₀ (two_ne_zero' ℝ) (_ - _), smul_sub, sub_eq_add_neg, smul_add]
  refine' h_conv hw (h_symm _ hv) _ _ _ <;> 

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2
E : Type u_1
L✝ : Type u_2
inst✝⁶ : MeasurableSpace E
μ : Measure E
F s : Set E
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : IsAddHaarMeasure μ
L : AddSubgroup E
inst✝ : Countable ↥L
fund : IsAddFundamentalDomain (↥L) F
h_symm : ∀ x ∈ s, -x ∈ s
h_conv : Convex ℝ s
h✝ : ↑↑μ F * 2 ^ finrank ℝ E < ↑↑μ s
h_vol : ↑↑μ F < ↑↑μ (2⁻¹ • s)
x y : ↥L
hxy : x ≠ y
h : ¬Disjoint (x +ᵥ 2⁻¹ • s) (y +ᵥ 2⁻¹ • s)
v : E
hv : 2 • v ∈ s
w : E
hw : 2 • w ∈ s
hvw : w - v = ↑(x - y)
⊢ 0 ≤ 2⁻¹

Next tactic:
norm_num
/-- The **Minkowski Convex Body Theorem**. If `s` is a convex symmetric domain of `E` whose volume
is large enough compared to the covolume of a lattice `L` of `E`, then it contains a non-zero
lattice point of `L`.  -/
theorem exists_ne_zero_mem_lattice_of_measure_mul_two_pow_lt_measure [NormedAddCommGroup E]
    [NormedSpace ℝ E] [BorelSpace E] [FiniteDimensional ℝ E] [IsAddHaarMeasure μ]
    {L : AddSubgroup E} [Countable L] (fund : IsAddFundamentalDomain L F μ)
    (h_symm : ∀ x ∈ s, -x ∈ s) (h_conv : Convex ℝ s) (h : μ F * 2 ^ finrank ℝ E < μ s) :
    ∃ x ≠ 0, ((x : L) : E) ∈ s := by
  have h_vol : μ F < μ ((2⁻¹ : ℝ) • s) := by
    rw [addHaar_smul_of_nonneg μ (by norm_num : 0 ≤ (2 : ℝ)⁻¹) s, ←
      mul_lt_mul_right (pow_ne_zero (finrank ℝ E) (two_ne_zero' _)) (pow_ne_top two_ne_top),
      mul_right_comm, ofReal_pow (by norm_num : 0 ≤ (2 : ℝ)⁻¹), ← ofReal_inv_of_pos zero_lt_two]
    norm_num
    rwa [← mul_pow, ENNReal.inv_mul_cancel two_ne_zero two_ne_top, one_pow, one_mul]
  obtain ⟨x, y, hxy, h⟩ :=
    exists_pair_mem_lattice_not_disjoint_vadd fund ((h_conv.smul _).nullMeasurableSet _) h_vol
  obtain ⟨_, ⟨v, hv, rfl⟩, w, hw, hvw⟩ := Set.not_disjoint_iff.mp h
  refine' ⟨x - y, sub_ne_zero.2 hxy, _⟩
  rw [Set.mem_inv_smul_set_iff₀ (two_ne_zero' ℝ)] at hv hw
  simp_rw [AddSubgroup.vadd_def, vadd_eq_add, add_comm _ w, ← sub_eq_sub_iff_add_eq_add, ←
    AddSubgroup.coe_sub] at hvw
  rw [← hvw, ← inv_smul_smul₀ (two_ne_zero' ℝ) (_ - _), smul_sub, sub_eq_add_neg, smul_add]
  refine' h_conv hw (h_symm _ hv) _ _ _ <;> 

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3
E : Type u_1
L✝ : Type u_2
inst✝⁶ : MeasurableSpace E
μ : Measure E
F s : Set E
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : IsAddHaarMeasure μ
L : AddSubgroup E
inst✝ : Countable ↥L
fund : IsAddFundamentalDomain (↥L) F
h_symm : ∀ x ∈ s, -x ∈ s
h_conv : Convex ℝ s
h✝ : ↑↑μ F * 2 ^ finrank ℝ E < ↑↑μ s
h_vol : ↑↑μ F < ↑↑μ (2⁻¹ • s)
x y : ↥L
hxy : x ≠ y
h : ¬Disjoint (x +ᵥ 2⁻¹ • s) (y +ᵥ 2⁻¹ • s)
v : E
hv : 2 • v ∈ s
w : E
hw : 2 • w ∈ s
hvw : w - v = ↑(x - y)
⊢ 2⁻¹ + 2⁻¹ = 1

Next tactic:
norm_num