l3lab/ntp-mathlib huggingface dataset:
/-- A strict vector subspace has measure zero. -/
theorem addHaar_submodule {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E]
    [BorelSpace E] [FiniteDimensional ℝ E] (μ : Measure E) [IsAddHaarMeasure μ] (s : Submodule ℝ E)
    (hs : s ≠ ⊤) : μ s = 0 := by
  obtain ⟨x, hx⟩ : ∃ x, x ∉ s := by
    simpa only [Submodule.eq_top_iff', not_exists, Ne.def, not_forall] using hs
  obtain ⟨c, cpos, cone⟩ : ∃ c : ℝ, 0 < c ∧ c < 1 := ⟨1 / 2, by 

Proof state:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
s : Submodule ℝ E
hs : s ≠ ⊤
x : E
hx : x ∉ s
⊢ 0 < 1 / 2

Next tactic:
norm_num
/-- A strict vector subspace has measure zero. -/
theorem addHaar_submodule {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E]
    [BorelSpace E] [FiniteDimensional ℝ E] (μ : Measure E) [IsAddHaarMeasure μ] (s : Submodule ℝ E)
    (hs : s ≠ ⊤) : μ s = 0 := by
  obtain ⟨x, hx⟩ : ∃ x, x ∉ s := by
    simpa only [Submodule.eq_top_iff', not_exists, Ne.def, not_forall] using hs
  obtain ⟨c, cpos, cone⟩ : ∃ c : ℝ, 0 < c ∧ c < 1 := ⟨1 / 2, by norm_num, by 

Proof state:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
s : Submodule ℝ E
hs : s ≠ ⊤
x : E
hx : x ∉ s
⊢ 1 / 2 < 1

Next tactic:
norm_num
/-- A strict vector subspace has measure zero. -/
theorem addHaar_submodule {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E]
    [BorelSpace E] [FiniteDimensional ℝ E] (μ : Measure E) [IsAddHaarMeasure μ] (s : Submodule ℝ E)
    (hs : s ≠ ⊤) : μ s = 0 := by
  obtain ⟨x, hx⟩ : ∃ x, x ∉ s := by
    simpa only [Submodule.eq_top_iff', not_exists, Ne.def, not_forall] using hs
  obtain ⟨c, cpos, cone⟩ : ∃ c : ℝ, 0 < c ∧ c < 1 := ⟨1 / 2, by norm_num, by norm_num⟩
  

Proof state:
case intro.intro.intro
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
s : Submodule ℝ E
hs : s ≠ ⊤
x : E
hx : x ∉ s
c : ℝ
cpos : 0 < c
cone : c < 1
⊢ ↑↑μ ↑s = 0

Next tactic:
have A : IsBounded (range fun n : ℕ => c ^ n • x) :=
    have : Tendsto (fun n : ℕ => c ^ n • x) atTop (𝓝 ((0 : ℝ) • x)) :=
      (tendsto_pow_atTop_nhds_0_of_lt_1 cpos.le cone).smul_const x
    isBounded_range_of_tendsto _ this
/-- A strict vector subspace has measure zero. -/
theorem addHaar_submodule {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E]
    [BorelSpace E] [FiniteDimensional ℝ E] (μ : Measure E) [IsAddHaarMeasure μ] (s : Submodule ℝ E)
    (hs : s ≠ ⊤) : μ s = 0 := by
  obtain ⟨x, hx⟩ : ∃ x, x ∉ s := by
    simpa only [Submodule.eq_top_iff', not_exists, Ne.def, not_forall] using hs
  obtain ⟨c, cpos, cone⟩ : ∃ c : ℝ, 0 < c ∧ c < 1 := ⟨1 / 2, by norm_num, by norm_num⟩
  have A : IsBounded (range fun n : ℕ => c ^ n • x) :=
    have : Tendsto (fun n : ℕ => c ^ n • x) atTop (𝓝 ((0 : ℝ) • x)) :=
      (tendsto_pow_atTop_nhds_0_of_lt_1 cpos.le cone).smul_const x
    isBounded_range_of_tendsto _ this
  

Proof state:
case intro.intro.intro
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
s : Submodule ℝ E
hs : s ≠ ⊤
x : E
hx : x ∉ s
c : ℝ
cpos : 0 < c
cone : c < 1
A : Bornology.IsBounded (range fun n => c ^ n • x)
⊢ ↑↑μ ↑s = 0

Next tactic:
apply addHaar_eq_zero_of_disjoint_translates μ _ A _
    (Submodule.closed_of_finiteDimensional s).measurableSet
/-- A strict vector subspace has measure zero. -/
theorem addHaar_submodule {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E]
    [BorelSpace E] [FiniteDimensional ℝ E] (μ : Measure E) [IsAddHaarMeasure μ] (s : Submodule ℝ E)
    (hs : s ≠ ⊤) : μ s = 0 := by
  obtain ⟨x, hx⟩ : ∃ x, x ∉ s := by
    simpa only [Submodule.eq_top_iff', not_exists, Ne.def, not_forall] using hs
  obtain ⟨c, cpos, cone⟩ : ∃ c : ℝ, 0 < c ∧ c < 1 := ⟨1 / 2, by norm_num, by norm_num⟩
  have A : IsBounded (range fun n : ℕ => c ^ n • x) :=
    have : Tendsto (fun n : ℕ => c ^ n • x) atTop (𝓝 ((0 : ℝ) • x)) :=
      (tendsto_pow_atTop_nhds_0_of_lt_1 cpos.le cone).smul_const x
    isBounded_range_of_tendsto _ this
  apply addHaar_eq_zero_of_disjoint_translates μ _ A _
    (Submodule.closed_of_finiteDimensional s).measurableSet
  

Proof state:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
s : Submodule ℝ E
hs : s ≠ ⊤
x : E
hx : x ∉ s
c : ℝ
cpos : 0 < c
cone : c < 1
A : Bornology.IsBounded (range fun n => c ^ n • x)
⊢ Pairwise (Disjoint on fun n => {c ^ n • x} + ↑s)

Next tactic:
intro m n hmn
/-- A strict vector subspace has measure zero. -/
theorem addHaar_submodule {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E]
    [BorelSpace E] [FiniteDimensional ℝ E] (μ : Measure E) [IsAddHaarMeasure μ] (s : Submodule ℝ E)
    (hs : s ≠ ⊤) : μ s = 0 := by
  obtain ⟨x, hx⟩ : ∃ x, x ∉ s := by
    simpa only [Submodule.eq_top_iff', not_exists, Ne.def, not_forall] using hs
  obtain ⟨c, cpos, cone⟩ : ∃ c : ℝ, 0 < c ∧ c < 1 := ⟨1 / 2, by norm_num, by norm_num⟩
  have A : IsBounded (range fun n : ℕ => c ^ n • x) :=
    have : Tendsto (fun n : ℕ => c ^ n • x) atTop (𝓝 ((0 : ℝ) • x)) :=
      (tendsto_pow_atTop_nhds_0_of_lt_1 cpos.le cone).smul_const x
    isBounded_range_of_tendsto _ this
  apply addHaar_eq_zero_of_disjoint_translates μ _ A _
    (Submodule.closed_of_finiteDimensional s).measurableSet
  intro m n hmn
  

Proof state:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
s : Submodule ℝ E
hs : s ≠ ⊤
x : E
hx : x ∉ s
c : ℝ
cpos : 0 < c
cone : c < 1
A : Bornology.IsBounded (range fun n => c ^ n • x)
m n : ℕ
hmn : m ≠ n
⊢ (Disjoint on fun n => {c ^ n • x} + ↑s) m n

Next tactic:
simp only [Function.onFun, image_add_left, singleton_add, disjoint_left, mem_preimage,
    SetLike.mem_coe]
/-- A strict vector subspace has measure zero. -/
theorem addHaar_submodule {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E]
    [BorelSpace E] [FiniteDimensional ℝ E] (μ : Measure E) [IsAddHaarMeasure μ] (s : Submodule ℝ E)
    (hs : s ≠ ⊤) : μ s = 0 := by
  obtain ⟨x, hx⟩ : ∃ x, x ∉ s := by
    simpa only [Submodule.eq_top_iff', not_exists, Ne.def, not_forall] using hs
  obtain ⟨c, cpos, cone⟩ : ∃ c : ℝ, 0 < c ∧ c < 1 := ⟨1 / 2, by norm_num, by norm_num⟩
  have A : IsBounded (range fun n : ℕ => c ^ n • x) :=
    have : Tendsto (fun n : ℕ => c ^ n • x) atTop (𝓝 ((0 : ℝ) • x)) :=
      (tendsto_pow_atTop_nhds_0_of_lt_1 cpos.le cone).smul_const x
    isBounded_range_of_tendsto _ this
  apply addHaar_eq_zero_of_disjoint_translates μ _ A _
    (Submodule.closed_of_finiteDimensional s).measurableSet
  intro m n hmn
  simp only [Function.onFun, image_add_left, singleton_add, disjoint_left, mem_preimage,
    SetLike.mem_coe]
  

Proof state:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
s : Submodule ℝ E
hs : s ≠ ⊤
x : E
hx : x ∉ s
c : ℝ
cpos : 0 < c
cone : c < 1
A : Bornology.IsBounded (range fun n => c ^ n • x)
m n : ℕ
hmn : m ≠ n
⊢ ∀ ⦃a : E⦄, -(c ^ m • x) + a ∈ s → -(c ^ n • x) + a ∉ s

Next tactic:
intro y hym hyn
/-- A strict vector subspace has measure zero. -/
theorem addHaar_submodule {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E]
    [BorelSpace E] [FiniteDimensional ℝ E] (μ : Measure E) [IsAddHaarMeasure μ] (s : Submodule ℝ E)
    (hs : s ≠ ⊤) : μ s = 0 := by
  obtain ⟨x, hx⟩ : ∃ x, x ∉ s := by
    simpa only [Submodule.eq_top_iff', not_exists, Ne.def, not_forall] using hs
  obtain ⟨c, cpos, cone⟩ : ∃ c : ℝ, 0 < c ∧ c < 1 := ⟨1 / 2, by norm_num, by norm_num⟩
  have A : IsBounded (range fun n : ℕ => c ^ n • x) :=
    have : Tendsto (fun n : ℕ => c ^ n • x) atTop (𝓝 ((0 : ℝ) • x)) :=
      (tendsto_pow_atTop_nhds_0_of_lt_1 cpos.le cone).smul_const x
    isBounded_range_of_tendsto _ this
  apply addHaar_eq_zero_of_disjoint_translates μ _ A _
    (Submodule.closed_of_finiteDimensional s).measurableSet
  intro m n hmn
  simp only [Function.onFun, image_add_left, singleton_add, disjoint_left, mem_preimage,
    SetLike.mem_coe]
  intro y hym hyn
  

Proof state:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
s : Submodule ℝ E
hs : s ≠ ⊤
x : E
hx : x ∉ s
c : ℝ
cpos : 0 < c
cone : c < 1
A : Bornology.IsBounded (range fun n => c ^ n • x)
m n : ℕ
hmn : m ≠ n
y : E
hym : -(c ^ m • x) + y ∈ s
hyn : -(c ^ n • x) + y ∈ s
⊢ False

Next tactic:
have A : (c ^ n - c ^ m) • x ∈ s := by
    convert s.sub_mem hym hyn using 1
    simp only [sub_smul, neg_sub_neg, add_sub_add_right_eq_sub]
/-- A strict vector subspace has measure zero. -/
theorem addHaar_submodule {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E]
    [BorelSpace E] [FiniteDimensional ℝ E] (μ : Measure E) [IsAddHaarMeasure μ] (s : Submodule ℝ E)
    (hs : s ≠ ⊤) : μ s = 0 := by
  obtain ⟨x, hx⟩ : ∃ x, x ∉ s := by
    simpa only [Submodule.eq_top_iff', not_exists, Ne.def, not_forall] using hs
  obtain ⟨c, cpos, cone⟩ : ∃ c : ℝ, 0 < c ∧ c < 1 := ⟨1 / 2, by norm_num, by norm_num⟩
  have A : IsBounded (range fun n : ℕ => c ^ n • x) :=
    have : Tendsto (fun n : ℕ => c ^ n • x) atTop (𝓝 ((0 : ℝ) • x)) :=
      (tendsto_pow_atTop_nhds_0_of_lt_1 cpos.le cone).smul_const x
    isBounded_range_of_tendsto _ this
  apply addHaar_eq_zero_of_disjoint_translates μ _ A _
    (Submodule.closed_of_finiteDimensional s).measurableSet
  intro m n hmn
  simp only [Function.onFun, image_add_left, singleton_add, disjoint_left, mem_preimage,
    SetLike.mem_coe]
  intro y hym hyn
  have A : (c ^ n - c ^ m) • x ∈ s := by
    

Proof state:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
s : Submodule ℝ E
hs : s ≠ ⊤
x : E
hx : x ∉ s
c : ℝ
cpos : 0 < c
cone : c < 1
A : Bornology.IsBounded (range fun n => c ^ n • x)
m n : ℕ
hmn : m ≠ n
y : E
hym : -(c ^ m • x) + y ∈ s
hyn : -(c ^ n • x) + y ∈ s
⊢ (c ^ n - c ^ m) • x ∈ s

Next tactic:
convert s.sub_mem hym hyn using 1
/-- A strict vector subspace has measure zero. -/
theorem addHaar_submodule {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E]
    [BorelSpace E] [FiniteDimensional ℝ E] (μ : Measure E) [IsAddHaarMeasure μ] (s : Submodule ℝ E)
    (hs : s ≠ ⊤) : μ s = 0 := by
  obtain ⟨x, hx⟩ : ∃ x, x ∉ s := by
    simpa only [Submodule.eq_top_iff', not_exists, Ne.def, not_forall] using hs
  obtain ⟨c, cpos, cone⟩ : ∃ c : ℝ, 0 < c ∧ c < 1 := ⟨1 / 2, by norm_num, by norm_num⟩
  have A : IsBounded (range fun n : ℕ => c ^ n • x) :=
    have : Tendsto (fun n : ℕ => c ^ n • x) atTop (𝓝 ((0 : ℝ) • x)) :=
      (tendsto_pow_atTop_nhds_0_of_lt_1 cpos.le cone).smul_const x
    isBounded_range_of_tendsto _ this
  apply addHaar_eq_zero_of_disjoint_translates μ _ A _
    (Submodule.closed_of_finiteDimensional s).measurableSet
  intro m n hmn
  simp only [Function.onFun, image_add_left, singleton_add, disjoint_left, mem_preimage,
    SetLike.mem_coe]
  intro y hym hyn
  have A : (c ^ n - c ^ m) • x ∈ s := by
    convert s.sub_mem hym hyn using 1
    

Proof state:
case h.e'_4
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
s : Submodule ℝ E
hs : s ≠ ⊤
x : E
hx : x ∉ s
c : ℝ
cpos : 0 < c
cone : c < 1
A : Bornology.IsBounded (range fun n => c ^ n • x)
m n : ℕ
hmn : m ≠ n
y : E
hym : -(c ^ m • x) + y ∈ s
hyn : -(c ^ n • x) + y ∈ s
⊢ (c ^ n - c ^ m) • x = -(c ^ m • x) + y - (-(c ^ n • x) + y)

Next tactic:
simp only [sub_smul, neg_sub_neg, add_sub_add_right_eq_sub]
/-- A strict vector subspace has measure zero. -/
theorem addHaar_submodule {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E]
    [BorelSpace E] [FiniteDimensional ℝ E] (μ : Measure E) [IsAddHaarMeasure μ] (s : Submodule ℝ E)
    (hs : s ≠ ⊤) : μ s = 0 := by
  obtain ⟨x, hx⟩ : ∃ x, x ∉ s := by
    simpa only [Submodule.eq_top_iff', not_exists, Ne.def, not_forall] using hs
  obtain ⟨c, cpos, cone⟩ : ∃ c : ℝ, 0 < c ∧ c < 1 := ⟨1 / 2, by norm_num, by norm_num⟩
  have A : IsBounded (range fun n : ℕ => c ^ n • x) :=
    have : Tendsto (fun n : ℕ => c ^ n • x) atTop (𝓝 ((0 : ℝ) • x)) :=
      (tendsto_pow_atTop_nhds_0_of_lt_1 cpos.le cone).smul_const x
    isBounded_range_of_tendsto _ this
  apply addHaar_eq_zero_of_disjoint_translates μ _ A _
    (Submodule.closed_of_finiteDimensional s).measurableSet
  intro m n hmn
  simp only [Function.onFun, image_add_left, singleton_add, disjoint_left, mem_preimage,
    SetLike.mem_coe]
  intro y hym hyn
  have A : (c ^ n - c ^ m) • x ∈ s := by
    convert s.sub_mem hym hyn using 1
    simp only [sub_smul, neg_sub_neg, add_sub_add_right_eq_sub]
  

Proof state:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
s : Submodule ℝ E
hs : s ≠ ⊤
x : E
hx : x ∉ s
c : ℝ
cpos : 0 < c
cone : c < 1
A✝ : Bornology.IsBounded (range fun n => c ^ n • x)
m n : ℕ
hmn : m ≠ n
y : E
hym : -(c ^ m • x) + y ∈ s
hyn : -(c ^ n • x) + y ∈ s
A : (c ^ n - c ^ m) • x ∈ s
⊢ False

Next tactic:
have H : c ^ n - c ^ m ≠ 0 := by
    simpa only [sub_eq_zero, Ne.def] using (pow_right_strictAnti cpos cone).injective.ne hmn.symm
/-- A strict vector subspace has measure zero. -/
theorem addHaar_submodule {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E]
    [BorelSpace E] [FiniteDimensional ℝ E] (μ : Measure E) [IsAddHaarMeasure μ] (s : Submodule ℝ E)
    (hs : s ≠ ⊤) : μ s = 0 := by
  obtain ⟨x, hx⟩ : ∃ x, x ∉ s := by
    simpa only [Submodule.eq_top_iff', not_exists, Ne.def, not_forall] using hs
  obtain ⟨c, cpos, cone⟩ : ∃ c : ℝ, 0 < c ∧ c < 1 := ⟨1 / 2, by norm_num, by norm_num⟩
  have A : IsBounded (range fun n : ℕ => c ^ n • x) :=
    have : Tendsto (fun n : ℕ => c ^ n • x) atTop (𝓝 ((0 : ℝ) • x)) :=
      (tendsto_pow_atTop_nhds_0_of_lt_1 cpos.le cone).smul_const x
    isBounded_range_of_tendsto _ this
  apply addHaar_eq_zero_of_disjoint_translates μ _ A _
    (Submodule.closed_of_finiteDimensional s).measurableSet
  intro m n hmn
  simp only [Function.onFun, image_add_left, singleton_add, disjoint_left, mem_preimage,
    SetLike.mem_coe]
  intro y hym hyn
  have A : (c ^ n - c ^ m) • x ∈ s := by
    convert s.sub_mem hym hyn using 1
    simp only [sub_smul, neg_sub_neg, add_sub_add_right_eq_sub]
  have H : c ^ n - c ^ m ≠ 0 := by
    

Proof state:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
s : Submodule ℝ E
hs : s ≠ ⊤
x : E
hx : x ∉ s
c : ℝ
cpos : 0 < c
cone : c < 1
A✝ : Bornology.IsBounded (range fun n => c ^ n • x)
m n : ℕ
hmn : m ≠ n
y : E
hym : -(c ^ m • x) + y ∈ s
hyn : -(c ^ n • x) + y ∈ s
A : (c ^ n - c ^ m) • x ∈ s
⊢ c ^ n - c ^ m ≠ 0

Next tactic:
simpa only [sub_eq_zero, Ne.def] using (pow_right_strictAnti cpos cone).injective.ne hmn.symm
/-- A strict vector subspace has measure zero. -/
theorem addHaar_submodule {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E]
    [BorelSpace E] [FiniteDimensional ℝ E] (μ : Measure E) [IsAddHaarMeasure μ] (s : Submodule ℝ E)
    (hs : s ≠ ⊤) : μ s = 0 := by
  obtain ⟨x, hx⟩ : ∃ x, x ∉ s := by
    simpa only [Submodule.eq_top_iff', not_exists, Ne.def, not_forall] using hs
  obtain ⟨c, cpos, cone⟩ : ∃ c : ℝ, 0 < c ∧ c < 1 := ⟨1 / 2, by norm_num, by norm_num⟩
  have A : IsBounded (range fun n : ℕ => c ^ n • x) :=
    have : Tendsto (fun n : ℕ => c ^ n • x) atTop (𝓝 ((0 : ℝ) • x)) :=
      (tendsto_pow_atTop_nhds_0_of_lt_1 cpos.le cone).smul_const x
    isBounded_range_of_tendsto _ this
  apply addHaar_eq_zero_of_disjoint_translates μ _ A _
    (Submodule.closed_of_finiteDimensional s).measurableSet
  intro m n hmn
  simp only [Function.onFun, image_add_left, singleton_add, disjoint_left, mem_preimage,
    SetLike.mem_coe]
  intro y hym hyn
  have A : (c ^ n - c ^ m) • x ∈ s := by
    convert s.sub_mem hym hyn using 1
    simp only [sub_smul, neg_sub_neg, add_sub_add_right_eq_sub]
  have H : c ^ n - c ^ m ≠ 0 := by
    simpa only [sub_eq_zero, Ne.def] using (pow_right_strictAnti cpos cone).injective.ne hmn.symm
  

Proof state:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
s : Submodule ℝ E
hs : s ≠ ⊤
x : E
hx : x ∉ s
c : ℝ
cpos : 0 < c
cone : c < 1
A✝ : Bornology.IsBounded (range fun n => c ^ n • x)
m n : ℕ
hmn : m ≠ n
y : E
hym : -(c ^ m • x) + y ∈ s
hyn : -(c ^ n • x) + y ∈ s
A : (c ^ n - c ^ m) • x ∈ s
H : c ^ n - c ^ m ≠ 0
⊢ False

Next tactic:
have : x ∈ s := by
    convert s.smul_mem (c ^ n - c ^ m)⁻¹ A
    rw [smul_smul, inv_mul_cancel H, one_smul]
/-- A strict vector subspace has measure zero. -/
theorem addHaar_submodule {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E]
    [BorelSpace E] [FiniteDimensional ℝ E] (μ : Measure E) [IsAddHaarMeasure μ] (s : Submodule ℝ E)
    (hs : s ≠ ⊤) : μ s = 0 := by
  obtain ⟨x, hx⟩ : ∃ x, x ∉ s := by
    simpa only [Submodule.eq_top_iff', not_exists, Ne.def, not_forall] using hs
  obtain ⟨c, cpos, cone⟩ : ∃ c : ℝ, 0 < c ∧ c < 1 := ⟨1 / 2, by norm_num, by norm_num⟩
  have A : IsBounded (range fun n : ℕ => c ^ n • x) :=
    have : Tendsto (fun n : ℕ => c ^ n • x) atTop (𝓝 ((0 : ℝ) • x)) :=
      (tendsto_pow_atTop_nhds_0_of_lt_1 cpos.le cone).smul_const x
    isBounded_range_of_tendsto _ this
  apply addHaar_eq_zero_of_disjoint_translates μ _ A _
    (Submodule.closed_of_finiteDimensional s).measurableSet
  intro m n hmn
  simp only [Function.onFun, image_add_left, singleton_add, disjoint_left, mem_preimage,
    SetLike.mem_coe]
  intro y hym hyn
  have A : (c ^ n - c ^ m) • x ∈ s := by
    convert s.sub_mem hym hyn using 1
    simp only [sub_smul, neg_sub_neg, add_sub_add_right_eq_sub]
  have H : c ^ n - c ^ m ≠ 0 := by
    simpa only [sub_eq_zero, Ne.def] using (pow_right_strictAnti cpos cone).injective.ne hmn.symm
  have : x ∈ s := by
    

Proof state:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
s : Submodule ℝ E
hs : s ≠ ⊤
x : E
hx : x ∉ s
c : ℝ
cpos : 0 < c
cone : c < 1
A✝ : Bornology.IsBounded (range fun n => c ^ n • x)
m n : ℕ
hmn : m ≠ n
y : E
hym : -(c ^ m • x) + y ∈ s
hyn : -(c ^ n • x) + y ∈ s
A : (c ^ n - c ^ m) • x ∈ s
H : c ^ n - c ^ m ≠ 0
⊢ x ∈ s

Next tactic:
convert s.smul_mem (c ^ n - c ^ m)⁻¹ A
/-- A strict vector subspace has measure zero. -/
theorem addHaar_submodule {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E]
    [BorelSpace E] [FiniteDimensional ℝ E] (μ : Measure E) [IsAddHaarMeasure μ] (s : Submodule ℝ E)
    (hs : s ≠ ⊤) : μ s = 0 := by
  obtain ⟨x, hx⟩ : ∃ x, x ∉ s := by
    simpa only [Submodule.eq_top_iff', not_exists, Ne.def, not_forall] using hs
  obtain ⟨c, cpos, cone⟩ : ∃ c : ℝ, 0 < c ∧ c < 1 := ⟨1 / 2, by norm_num, by norm_num⟩
  have A : IsBounded (range fun n : ℕ => c ^ n • x) :=
    have : Tendsto (fun n : ℕ => c ^ n • x) atTop (𝓝 ((0 : ℝ) • x)) :=
      (tendsto_pow_atTop_nhds_0_of_lt_1 cpos.le cone).smul_const x
    isBounded_range_of_tendsto _ this
  apply addHaar_eq_zero_of_disjoint_translates μ _ A _
    (Submodule.closed_of_finiteDimensional s).measurableSet
  intro m n hmn
  simp only [Function.onFun, image_add_left, singleton_add, disjoint_left, mem_preimage,
    SetLike.mem_coe]
  intro y hym hyn
  have A : (c ^ n - c ^ m) • x ∈ s := by
    convert s.sub_mem hym hyn using 1
    simp only [sub_smul, neg_sub_neg, add_sub_add_right_eq_sub]
  have H : c ^ n - c ^ m ≠ 0 := by
    simpa only [sub_eq_zero, Ne.def] using (pow_right_strictAnti cpos cone).injective.ne hmn.symm
  have : x ∈ s := by
    convert s.smul_mem (c ^ n - c ^ m)⁻¹ A
    

Proof state:
case h.e'_4
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
s : Submodule ℝ E
hs : s ≠ ⊤
x : E
hx : x ∉ s
c : ℝ
cpos : 0 < c
cone : c < 1
A✝ : Bornology.IsBounded (range fun n => c ^ n • x)
m n : ℕ
hmn : m ≠ n
y : E
hym : -(c ^ m • x) + y ∈ s
hyn : -(c ^ n • x) + y ∈ s
A : (c ^ n - c ^ m) • x ∈ s
H : c ^ n - c ^ m ≠ 0
⊢ x = (c ^ n - c ^ m)⁻¹ • (c ^ n - c ^ m) • x

Next tactic:
rw [smul_smul, inv_mul_cancel H, one_smul]
/-- A strict vector subspace has measure zero. -/
theorem addHaar_submodule {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E]
    [BorelSpace E] [FiniteDimensional ℝ E] (μ : Measure E) [IsAddHaarMeasure μ] (s : Submodule ℝ E)
    (hs : s ≠ ⊤) : μ s = 0 := by
  obtain ⟨x, hx⟩ : ∃ x, x ∉ s := by
    simpa only [Submodule.eq_top_iff', not_exists, Ne.def, not_forall] using hs
  obtain ⟨c, cpos, cone⟩ : ∃ c : ℝ, 0 < c ∧ c < 1 := ⟨1 / 2, by norm_num, by norm_num⟩
  have A : IsBounded (range fun n : ℕ => c ^ n • x) :=
    have : Tendsto (fun n : ℕ => c ^ n • x) atTop (𝓝 ((0 : ℝ) • x)) :=
      (tendsto_pow_atTop_nhds_0_of_lt_1 cpos.le cone).smul_const x
    isBounded_range_of_tendsto _ this
  apply addHaar_eq_zero_of_disjoint_translates μ _ A _
    (Submodule.closed_of_finiteDimensional s).measurableSet
  intro m n hmn
  simp only [Function.onFun, image_add_left, singleton_add, disjoint_left, mem_preimage,
    SetLike.mem_coe]
  intro y hym hyn
  have A : (c ^ n - c ^ m) • x ∈ s := by
    convert s.sub_mem hym hyn using 1
    simp only [sub_smul, neg_sub_neg, add_sub_add_right_eq_sub]
  have H : c ^ n - c ^ m ≠ 0 := by
    simpa only [sub_eq_zero, Ne.def] using (pow_right_strictAnti cpos cone).injective.ne hmn.symm
  have : x ∈ s := by
    convert s.smul_mem (c ^ n - c ^ m)⁻¹ A
    rw [smul_smul, inv_mul_cancel H, one_smul]
  

Proof state:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
s : Submodule ℝ E
hs : s ≠ ⊤
x : E
hx : x ∉ s
c : ℝ
cpos : 0 < c
cone : c < 1
A✝ : Bornology.IsBounded (range fun n => c ^ n • x)
m n : ℕ
hmn : m ≠ n
y : E
hym : -(c ^ m • x) + y ∈ s
hyn : -(c ^ n • x) + y ∈ s
A : (c ^ n - c ^ m) • x ∈ s
H : c ^ n - c ^ m ≠ 0
this : x ∈ s
⊢ False

Next tactic:
exact hx this
/-- A strict affine subspace has measure zero. -/
theorem addHaar_affineSubspace {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : Measure E) [IsAddHaarMeasure μ]
    (s : AffineSubspace ℝ E) (hs : s ≠ ⊤) : μ s = 0 := by
  

Proof state:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
s : AffineSubspace ℝ E
hs : s ≠ ⊤
⊢ ↑↑μ ↑s = 0

Next tactic:
rcases s.eq_bot_or_nonempty with (rfl | hne)