l3lab/ntp-mathlib huggingface dataset:
@[to_additive]
theorem prehaar_pos (K₀ : PositiveCompacts G) {U : Set G} (hU : (interior U).Nonempty) {K : Set G}
    (h1K : IsCompact K) (h2K : (interior K).Nonempty) : 0 < prehaar (K₀ : Set G) U ⟨K, h1K⟩ := by
  apply div_pos <;> norm_cast; apply index_pos ⟨⟨K, h1K⟩, h2K⟩ hU; 

Proof state:
case hb
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
hU : Set.Nonempty (interior U)
K : Set G
h1K : IsCompact K
h2K : Set.Nonempty (interior K)
⊢ 0 < index (↑K₀) U

Next tactic:
exact index_pos K₀ hU
@[to_additive]
theorem prehaar_mono {K₀ : PositiveCompacts G} {U : Set G} (hU : (interior U).Nonempty)
    {K₁ K₂ : Compacts G} (h : (K₁ : Set G) ⊆ K₂.1) :
    prehaar (K₀ : Set G) U K₁ ≤ prehaar (K₀ : Set G) U K₂ := by
  

Proof state:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
hU : Set.Nonempty (interior U)
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ K₂.carrier
⊢ prehaar (↑K₀) U K₁ ≤ prehaar (↑K₀) U K₂

Next tactic:
simp only [prehaar]
@[to_additive]
theorem prehaar_mono {K₀ : PositiveCompacts G} {U : Set G} (hU : (interior U).Nonempty)
    {K₁ K₂ : Compacts G} (h : (K₁ : Set G) ⊆ K₂.1) :
    prehaar (K₀ : Set G) U K₁ ≤ prehaar (K₀ : Set G) U K₂ := by
  simp only [prehaar]; 

Proof state:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
hU : Set.Nonempty (interior U)
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ K₂.carrier
⊢ ↑(index (↑K₁) U) / ↑(index (↑K₀) U) ≤ ↑(index (↑K₂) U) / ↑(index (↑K₀) U)

Next tactic:
rw [div_le_div_right]
@[to_additive]
theorem prehaar_mono {K₀ : PositiveCompacts G} {U : Set G} (hU : (interior U).Nonempty)
    {K₁ K₂ : Compacts G} (h : (K₁ : Set G) ⊆ K₂.1) :
    prehaar (K₀ : Set G) U K₁ ≤ prehaar (K₀ : Set G) U K₂ := by
  simp only [prehaar]; rw [div_le_div_right]; 

Proof state:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
hU : Set.Nonempty (interior U)
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ K₂.carrier
⊢ ↑(index (↑K₁) U) ≤ ↑(index (↑K₂) U)
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
hU : Set.Nonempty (interior U)
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ K₂.carrier
⊢ 0 < ↑(index (↑K₀) U)

Next tactic:
exact mod_cast index_mono K₂.2 h hU
@[to_additive]
theorem prehaar_mono {K₀ : PositiveCompacts G} {U : Set G} (hU : (interior U).Nonempty)
    {K₁ K₂ : Compacts G} (h : (K₁ : Set G) ⊆ K₂.1) :
    prehaar (K₀ : Set G) U K₁ ≤ prehaar (K₀ : Set G) U K₂ := by
  simp only [prehaar]; rw [div_le_div_right]; exact mod_cast index_mono K₂.2 h hU
  

Proof state:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
hU : Set.Nonempty (interior U)
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ K₂.carrier
⊢ 0 < ↑(index (↑K₀) U)

Next tactic:
exact mod_cast index_pos K₀ hU
@[to_additive]
theorem prehaar_sup_le {K₀ : PositiveCompacts G} {U : Set G} (K₁ K₂ : Compacts G)
    (hU : (interior U).Nonempty) :
    prehaar (K₀ : Set G) U (K₁ ⊔ K₂) ≤ prehaar (K₀ : Set G) U K₁ + prehaar (K₀ : Set G) U K₂ := by
  

Proof state:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
K₁ K₂ : Compacts G
hU : Set.Nonempty (interior U)
⊢ prehaar (↑K₀) U (K₁ ⊔ K₂) ≤ prehaar (↑K₀) U K₁ + prehaar (↑K₀) U K₂

Next tactic:
simp only [prehaar]
@[to_additive]
theorem prehaar_sup_le {K₀ : PositiveCompacts G} {U : Set G} (K₁ K₂ : Compacts G)
    (hU : (interior U).Nonempty) :
    prehaar (K₀ : Set G) U (K₁ ⊔ K₂) ≤ prehaar (K₀ : Set G) U K₁ + prehaar (K₀ : Set G) U K₂ := by
  simp only [prehaar]; 

Proof state:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
K₁ K₂ : Compacts G
hU : Set.Nonempty (interior U)
⊢ ↑(index (↑(K₁ ⊔ K₂)) U) / ↑(index (↑K₀) U) ≤ ↑(index (↑K₁) U) / ↑(index (↑K₀) U) + ↑(index (↑K₂) U) / ↑(index (↑K₀) U)

Next tactic:
rw [div_add_div_same, div_le_div_right]
@[to_additive]
theorem prehaar_sup_le {K₀ : PositiveCompacts G} {U : Set G} (K₁ K₂ : Compacts G)
    (hU : (interior U).Nonempty) :
    prehaar (K₀ : Set G) U (K₁ ⊔ K₂) ≤ prehaar (K₀ : Set G) U K₁ + prehaar (K₀ : Set G) U K₂ := by
  simp only [prehaar]; rw [div_add_div_same, div_le_div_right]
  

Proof state:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
K₁ K₂ : Compacts G
hU : Set.Nonempty (interior U)
⊢ ↑(index (↑(K₁ ⊔ K₂)) U) ≤ ↑(index (↑K₁) U) + ↑(index (↑K₂) U)
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
K₁ K₂ : Compacts G
hU : Set.Nonempty (interior U)
⊢ 0 < ↑(index (↑K₀) U)

Next tactic:
exact mod_cast index_union_le K₁ K₂ hU
@[to_additive]
theorem prehaar_sup_le {K₀ : PositiveCompacts G} {U : Set G} (K₁ K₂ : Compacts G)
    (hU : (interior U).Nonempty) :
    prehaar (K₀ : Set G) U (K₁ ⊔ K₂) ≤ prehaar (K₀ : Set G) U K₁ + prehaar (K₀ : Set G) U K₂ := by
  simp only [prehaar]; rw [div_add_div_same, div_le_div_right]
  exact mod_cast index_union_le K₁ K₂ hU; 

Proof state:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
K₁ K₂ : Compacts G
hU : Set.Nonempty (interior U)
⊢ 0 < ↑(index (↑K₀) U)

Next tactic:
exact mod_cast index_pos K₀ hU
@[to_additive]
theorem prehaar_sup_eq {K₀ : PositiveCompacts G} {U : Set G} {K₁ K₂ : Compacts G}
    (hU : (interior U).Nonempty) (h : Disjoint (K₁.1 * U⁻¹) (K₂.1 * U⁻¹)) :
    prehaar (K₀ : Set G) U (K₁ ⊔ K₂) = prehaar (K₀ : Set G) U K₁ + prehaar (K₀ : Set G) U K₂ := by
  

Proof state:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
K₁ K₂ : Compacts G
hU : Set.Nonempty (interior U)
h : Disjoint (K₁.carrier * U⁻¹) (K₂.carrier * U⁻¹)
⊢ prehaar (↑K₀) U (K₁ ⊔ K₂) = prehaar (↑K₀) U K₁ + prehaar (↑K₀) U K₂

Next tactic:
simp only [prehaar]
@[to_additive]
theorem prehaar_sup_eq {K₀ : PositiveCompacts G} {U : Set G} {K₁ K₂ : Compacts G}
    (hU : (interior U).Nonempty) (h : Disjoint (K₁.1 * U⁻¹) (K₂.1 * U⁻¹)) :
    prehaar (K₀ : Set G) U (K₁ ⊔ K₂) = prehaar (K₀ : Set G) U K₁ + prehaar (K₀ : Set G) U K₂ := by
  simp only [prehaar]; 

Proof state:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
K₁ K₂ : Compacts G
hU : Set.Nonempty (interior U)
h : Disjoint (K₁.carrier * U⁻¹) (K₂.carrier * U⁻¹)
⊢ ↑(index (↑(K₁ ⊔ K₂)) U) / ↑(index (↑K₀) U) = ↑(index (↑K₁) U) / ↑(index (↑K₀) U) + ↑(index (↑K₂) U) / ↑(index (↑K₀) U)

Next tactic:
rw [div_add_div_same]
@[to_additive]
theorem prehaar_sup_eq {K₀ : PositiveCompacts G} {U : Set G} {K₁ K₂ : Compacts G}
    (hU : (interior U).Nonempty) (h : Disjoint (K₁.1 * U⁻¹) (K₂.1 * U⁻¹)) :
    prehaar (K₀ : Set G) U (K₁ ⊔ K₂) = prehaar (K₀ : Set G) U K₁ + prehaar (K₀ : Set G) U K₂ := by
  simp only [prehaar]; rw [div_add_div_same]
  -- Porting note: Here was `congr`, but `to_additive` failed to generate a theorem.
  

Proof state:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
K₁ K₂ : Compacts G
hU : Set.Nonempty (interior U)
h : Disjoint (K₁.carrier * U⁻¹) (K₂.carrier * U⁻¹)
⊢ ↑(index (↑(K₁ ⊔ K₂)) U) / ↑(index (↑K₀) U) = (↑(index (↑K₁) U) + ↑(index (↑K₂) U)) / ↑(index (↑K₀) U)

Next tactic:
refine congr_arg (fun x : ℝ => x / index K₀ U) ?_
@[to_additive]
theorem prehaar_sup_eq {K₀ : PositiveCompacts G} {U : Set G} {K₁ K₂ : Compacts G}
    (hU : (interior U).Nonempty) (h : Disjoint (K₁.1 * U⁻¹) (K₂.1 * U⁻¹)) :
    prehaar (K₀ : Set G) U (K₁ ⊔ K₂) = prehaar (K₀ : Set G) U K₁ + prehaar (K₀ : Set G) U K₂ := by
  simp only [prehaar]; rw [div_add_div_same]
  -- Porting note: Here was `congr`, but `to_additive` failed to generate a theorem.
  refine congr_arg (fun x : ℝ => x / index K₀ U) ?_
  

Proof state:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
K₁ K₂ : Compacts G
hU : Set.Nonempty (interior U)
h : Disjoint (K₁.carrier * U⁻¹) (K₂.carrier * U⁻¹)
⊢ ↑(index (↑(K₁ ⊔ K₂)) U) = ↑(index (↑K₁) U) + ↑(index (↑K₂) U)

Next tactic:
exact mod_cast index_union_eq K₁ K₂ hU h
@[to_additive]
theorem is_left_invariant_prehaar {K₀ : PositiveCompacts G} {U : Set G} (hU : (interior U).Nonempty)
    (g : G) (K : Compacts G) :
    prehaar (K₀ : Set G) U (K.map _ <| continuous_mul_left g) = prehaar (K₀ : Set G) U K := by
  

Proof state:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
hU : Set.Nonempty (interior U)
g : G
K : Compacts G
⊢ prehaar (↑K₀) U (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) = prehaar (↑K₀) U K

Next tactic:
simp only [prehaar, Compacts.coe_map, is_left_invariant_index K.isCompact _ hU]
@[to_additive]
theorem prehaar_mem_haarProduct (K₀ : PositiveCompacts G) {U : Set G} (hU : (interior U).Nonempty) :
    prehaar (K₀ : Set G) U ∈ haarProduct (K₀ : Set G) := by
    

Proof state:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
hU : Set.Nonempty (interior U)
⊢ prehaar (↑K₀) U ∈ haarProduct ↑K₀

Next tactic:
rintro ⟨K, hK⟩ _
@[to_additive]
theorem prehaar_mem_haarProduct (K₀ : PositiveCompacts G) {U : Set G} (hU : (interior U).Nonempty) :
    prehaar (K₀ : Set G) U ∈ haarProduct (K₀ : Set G) := by
    rintro ⟨K, hK⟩ _; 

Proof state:
case mk
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
hU : Set.Nonempty (interior U)
K : Set G
hK : IsCompact K
a✝ : { carrier := K, isCompact' := hK } ∈ univ
⊢ prehaar (↑K₀) U { carrier := K, isCompact' := hK } ∈
    (fun K => Icc 0 ↑(index ↑K ↑K₀)) { carrier := K, isCompact' := hK }

Next tactic:
rw [mem_Icc]
@[to_additive]
theorem prehaar_mem_haarProduct (K₀ : PositiveCompacts G) {U : Set G} (hU : (interior U).Nonempty) :
    prehaar (K₀ : Set G) U ∈ haarProduct (K₀ : Set G) := by
    rintro ⟨K, hK⟩ _; rw [mem_Icc]; 

Proof state:
case mk
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
hU : Set.Nonempty (interior U)
K : Set G
hK : IsCompact K
a✝ : { carrier := K, isCompact' := hK } ∈ univ
⊢ 0 ≤ prehaar (↑K₀) U { carrier := K, isCompact' := hK } ∧
    prehaar (↑K₀) U { carrier := K, isCompact' := hK } ≤ ↑(index ↑{ carrier := K, isCompact' := hK } ↑K₀)

Next tactic:
exact ⟨prehaar_nonneg K₀ _, prehaar_le_index K₀ _ hU⟩
@[to_additive]
theorem nonempty_iInter_clPrehaar (K₀ : PositiveCompacts G) :
    (haarProduct (K₀ : Set G) ∩ ⋂ V : OpenNhdsOf (1 : G), clPrehaar K₀ V).Nonempty := by
  

Proof state:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
⊢ Set.Nonempty (haarProduct ↑K₀ ∩ ⋂ V, clPrehaar (↑K₀) V)

Next tactic:
have : IsCompact (haarProduct (K₀ : Set G)) := by
    apply isCompact_univ_pi; intro K; apply isCompact_Icc
@[to_additive]
theorem nonempty_iInter_clPrehaar (K₀ : PositiveCompacts G) :
    (haarProduct (K₀ : Set G) ∩ ⋂ V : OpenNhdsOf (1 : G), clPrehaar K₀ V).Nonempty := by
  have : IsCompact (haarProduct (K₀ : Set G)) := by
    

Proof state:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
⊢ IsCompact (haarProduct ↑K₀)

Next tactic:
apply isCompact_univ_pi
@[to_additive]
theorem nonempty_iInter_clPrehaar (K₀ : PositiveCompacts G) :
    (haarProduct (K₀ : Set G) ∩ ⋂ V : OpenNhdsOf (1 : G), clPrehaar K₀ V).Nonempty := by
  have : IsCompact (haarProduct (K₀ : Set G)) := by
    apply isCompact_univ_pi; 

Proof state:
case h
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
⊢ ∀ (i : Compacts G), IsCompact (Icc 0 ↑(index ↑i ↑K₀))

Next tactic:
intro K
@[to_additive]
theorem nonempty_iInter_clPrehaar (K₀ : PositiveCompacts G) :
    (haarProduct (K₀ : Set G) ∩ ⋂ V : OpenNhdsOf (1 : G), clPrehaar K₀ V).Nonempty := by
  have : IsCompact (haarProduct (K₀ : Set G)) := by
    apply isCompact_univ_pi; intro K; 

Proof state:
case h
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K : Compacts G
⊢ IsCompact (Icc 0 ↑(index ↑K ↑K₀))

Next tactic:
apply isCompact_Icc
@[to_additive]
theorem nonempty_iInter_clPrehaar (K₀ : PositiveCompacts G) :
    (haarProduct (K₀ : Set G) ∩ ⋂ V : OpenNhdsOf (1 : G), clPrehaar K₀ V).Nonempty := by
  have : IsCompact (haarProduct (K₀ : Set G)) := by
    apply isCompact_univ_pi; intro K; apply isCompact_Icc
  

Proof state:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
this : IsCompact (haarProduct ↑K₀)
⊢ Set.Nonempty (haarProduct ↑K₀ ∩ ⋂ V, clPrehaar (↑K₀) V)

Next tactic:
refine' this.inter_iInter_nonempty (clPrehaar K₀) (fun s => isClosed_closure) fun t => _
@[to_additive]
theorem nonempty_iInter_clPrehaar (K₀ : PositiveCompacts G) :
    (haarProduct (K₀ : Set G) ∩ ⋂ V : OpenNhdsOf (1 : G), clPrehaar K₀ V).Nonempty := by
  have : IsCompact (haarProduct (K₀ : Set G)) := by
    apply isCompact_univ_pi; intro K; apply isCompact_Icc
  refine' this.inter_iInter_nonempty (clPrehaar K₀) (fun s => isClosed_closure) fun t => _
  

Proof state:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
this : IsCompact (haarProduct ↑K₀)
t : Finset (OpenNhdsOf 1)
⊢ Set.Nonempty (haarProduct ↑K₀ ∩ ⋂ i ∈ t, clPrehaar (↑K₀) i)

Next tactic:
let V₀ := ⋂ V ∈ t, (V : OpenNhdsOf (1 : G)).carrier
@[to_additive]
theorem nonempty_iInter_clPrehaar (K₀ : PositiveCompacts G) :
    (haarProduct (K₀ : Set G) ∩ ⋂ V : OpenNhdsOf (1 : G), clPrehaar K₀ V).Nonempty := by
  have : IsCompact (haarProduct (K₀ : Set G)) := by
    apply isCompact_univ_pi; intro K; apply isCompact_Icc
  refine' this.inter_iInter_nonempty (clPrehaar K₀) (fun s => isClosed_closure) fun t => _
  let V₀ := ⋂ V ∈ t, (V : OpenNhdsOf (1 : G)).carrier
  

Proof state:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
this : IsCompact (haarProduct ↑K₀)
t : Finset (OpenNhdsOf 1)
V₀ : Set G := ⋂ V ∈ t, V.carrier
⊢ Set.Nonempty (haarProduct ↑K₀ ∩ ⋂ i ∈ t, clPrehaar (↑K₀) i)

Next tactic:
have h1V₀ : IsOpen V₀ := isOpen_biInter_finset $ by rintro ⟨⟨V, hV₁⟩, hV₂⟩ _; exact hV₁
@[to_additive]
theorem nonempty_iInter_clPrehaar (K₀ : PositiveCompacts G) :
    (haarProduct (K₀ : Set G) ∩ ⋂ V : OpenNhdsOf (1 : G), clPrehaar K₀ V).Nonempty := by
  have : IsCompact (haarProduct (K₀ : Set G)) := by
    apply isCompact_univ_pi; intro K; apply isCompact_Icc
  refine' this.inter_iInter_nonempty (clPrehaar K₀) (fun s => isClosed_closure) fun t => _
  let V₀ := ⋂ V ∈ t, (V : OpenNhdsOf (1 : G)).carrier
  have h1V₀ : IsOpen V₀ := isOpen_biInter_finset $ by 

Proof state:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
this : IsCompact (haarProduct ↑K₀)
t : Finset (OpenNhdsOf 1)
V₀ : Set G := ⋂ V ∈ t, V.carrier
⊢ ∀ i ∈ t, IsOpen i.carrier

Next tactic:
rintro ⟨⟨V, hV₁⟩, hV₂⟩ _
@[to_additive]
theorem nonempty_iInter_clPrehaar (K₀ : PositiveCompacts G) :
    (haarProduct (K₀ : Set G) ∩ ⋂ V : OpenNhdsOf (1 : G), clPrehaar K₀ V).Nonempty := by
  have : IsCompact (haarProduct (K₀ : Set G)) := by
    apply isCompact_univ_pi; intro K; apply isCompact_Icc
  refine' this.inter_iInter_nonempty (clPrehaar K₀) (fun s => isClosed_closure) fun t => _
  let V₀ := ⋂ V ∈ t, (V : OpenNhdsOf (1 : G)).carrier
  have h1V₀ : IsOpen V₀ := isOpen_biInter_finset $ by rintro ⟨⟨V, hV₁⟩, hV₂⟩ _; 

Proof state:
case mk.mk
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
this : IsCompact (haarProduct ↑K₀)
t : Finset (OpenNhdsOf 1)
V₀ : Set G := ⋂ V ∈ t, V.carrier
V : Set G
hV₁ : IsOpen V
hV₂ : 1 ∈ { carrier := V, is_open' := hV₁ }.carrier
a✝ : { toOpens := { carrier := V, is_open' := hV₁ }, mem' := hV₂ } ∈ t
⊢ IsOpen { toOpens := { carrier := V, is_open' := hV₁ }, mem' := hV₂ }.toOpens.carrier

Next tactic:
exact hV₁
@[to_additive]
theorem nonempty_iInter_clPrehaar (K₀ : PositiveCompacts G) :
    (haarProduct (K₀ : Set G) ∩ ⋂ V : OpenNhdsOf (1 : G), clPrehaar K₀ V).Nonempty := by
  have : IsCompact (haarProduct (K₀ : Set G)) := by
    apply isCompact_univ_pi; intro K; apply isCompact_Icc
  refine' this.inter_iInter_nonempty (clPrehaar K₀) (fun s => isClosed_closure) fun t => _
  let V₀ := ⋂ V ∈ t, (V : OpenNhdsOf (1 : G)).carrier
  have h1V₀ : IsOpen V₀ := isOpen_biInter_finset $ by rintro ⟨⟨V, hV₁⟩, hV₂⟩ _; exact hV₁
  

Proof state:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
this : IsCompact (haarProduct ↑K₀)
t : Finset (OpenNhdsOf 1)
V₀ : Set G := ⋂ V ∈ t, V.carrier
h1V₀ : IsOpen V₀
⊢ Set.Nonempty (haarProduct ↑K₀ ∩ ⋂ i ∈ t, clPrehaar (↑K₀) i)

Next tactic:
have h2V₀ : (1 : G) ∈ V₀ := by simp only [mem_iInter]; rintro ⟨⟨V, hV₁⟩, hV₂⟩ _; exact hV₂
@[to_additive]
theorem nonempty_iInter_clPrehaar (K₀ : PositiveCompacts G) :
    (haarProduct (K₀ : Set G) ∩ ⋂ V : OpenNhdsOf (1 : G), clPrehaar K₀ V).Nonempty := by
  have : IsCompact (haarProduct (K₀ : Set G)) := by
    apply isCompact_univ_pi; intro K; apply isCompact_Icc
  refine' this.inter_iInter_nonempty (clPrehaar K₀) (fun s => isClosed_closure) fun t => _
  let V₀ := ⋂ V ∈ t, (V : OpenNhdsOf (1 : G)).carrier
  have h1V₀ : IsOpen V₀ := isOpen_biInter_finset $ by rintro ⟨⟨V, hV₁⟩, hV₂⟩ _; exact hV₁
  have h2V₀ : (1 : G) ∈ V₀ := by 

Proof state:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
this : IsCompact (haarProduct ↑K₀)
t : Finset (OpenNhdsOf 1)
V₀ : Set G := ⋂ V ∈ t, V.carrier
h1V₀ : IsOpen V₀
⊢ 1 ∈ V₀

Next tactic:
simp only [mem_iInter]
@[to_additive]
theorem nonempty_iInter_clPrehaar (K₀ : PositiveCompacts G) :
    (haarProduct (K₀ : Set G) ∩ ⋂ V : OpenNhdsOf (1 : G), clPrehaar K₀ V).Nonempty := by
  have : IsCompact (haarProduct (K₀ : Set G)) := by
    apply isCompact_univ_pi; intro K; apply isCompact_Icc
  refine' this.inter_iInter_nonempty (clPrehaar K₀) (fun s => isClosed_closure) fun t => _
  let V₀ := ⋂ V ∈ t, (V : OpenNhdsOf (1 : G)).carrier
  have h1V₀ : IsOpen V₀ := isOpen_biInter_finset $ by rintro ⟨⟨V, hV₁⟩, hV₂⟩ _; exact hV₁
  have h2V₀ : (1 : G) ∈ V₀ := by simp only [mem_iInter]; 

Proof state:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
this : IsCompact (haarProduct ↑K₀)
t : Finset (OpenNhdsOf 1)
V₀ : Set G := ⋂ V ∈ t, V.carrier
h1V₀ : IsOpen V₀
⊢ ∀ i ∈ t, 1 ∈ i.carrier

Next tactic:
rintro ⟨⟨V, hV₁⟩, hV₂⟩ _
@[to_additive]
theorem nonempty_iInter_clPrehaar (K₀ : PositiveCompacts G) :
    (haarProduct (K₀ : Set G) ∩ ⋂ V : OpenNhdsOf (1 : G), clPrehaar K₀ V).Nonempty := by
  have : IsCompact (haarProduct (K₀ : Set G)) := by
    apply isCompact_univ_pi; intro K; apply isCompact_Icc
  refine' this.inter_iInter_nonempty (clPrehaar K₀) (fun s => isClosed_closure) fun t => _
  let V₀ := ⋂ V ∈ t, (V : OpenNhdsOf (1 : G)).carrier
  have h1V₀ : IsOpen V₀ := isOpen_biInter_finset $ by rintro ⟨⟨V, hV₁⟩, hV₂⟩ _; exact hV₁
  have h2V₀ : (1 : G) ∈ V₀ := by simp only [mem_iInter]; rintro ⟨⟨V, hV₁⟩, hV₂⟩ _; 

Proof state:
case mk.mk
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
this : IsCompact (haarProduct ↑K₀)
t : Finset (OpenNhdsOf 1)
V₀ : Set G := ⋂ V ∈ t, V.carrier
h1V₀ : IsOpen V₀
V : Set G
hV₁ : IsOpen V
hV₂ : 1 ∈ { carrier := V, is_open' := hV₁ }.carrier
i✝ : { toOpens := { carrier := V, is_open' := hV₁ }, mem' := hV₂ } ∈ t
⊢ 1 ∈ { toOpens := { carrier := V, is_open' := hV₁ }, mem' := hV₂ }.toOpens.carrier

Next tactic:
exact hV₂
@[to_additive]
theorem nonempty_iInter_clPrehaar (K₀ : PositiveCompacts G) :
    (haarProduct (K₀ : Set G) ∩ ⋂ V : OpenNhdsOf (1 : G), clPrehaar K₀ V).Nonempty := by
  have : IsCompact (haarProduct (K₀ : Set G)) := by
    apply isCompact_univ_pi; intro K; apply isCompact_Icc
  refine' this.inter_iInter_nonempty (clPrehaar K₀) (fun s => isClosed_closure) fun t => _
  let V₀ := ⋂ V ∈ t, (V : OpenNhdsOf (1 : G)).carrier
  have h1V₀ : IsOpen V₀ := isOpen_biInter_finset $ by rintro ⟨⟨V, hV₁⟩, hV₂⟩ _; exact hV₁
  have h2V₀ : (1 : G) ∈ V₀ := by simp only [mem_iInter]; rintro ⟨⟨V, hV₁⟩, hV₂⟩ _; exact hV₂
  

Proof state:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
this : IsCompact (haarProduct ↑K₀)
t : Finset (OpenNhdsOf 1)
V₀ : Set G := ⋂ V ∈ t, V.carrier
h1V₀ : IsOpen V₀
h2V₀ : 1 ∈ V₀
⊢ Set.Nonempty (haarProduct ↑K₀ ∩ ⋂ i ∈ t, clPrehaar (↑K₀) i)

Next tactic:
refine' ⟨prehaar K₀ V₀, _⟩
@[to_additive]
theorem nonempty_iInter_clPrehaar (K₀ : PositiveCompacts G) :
    (haarProduct (K₀ : Set G) ∩ ⋂ V : OpenNhdsOf (1 : G), clPrehaar K₀ V).Nonempty := by
  have : IsCompact (haarProduct (K₀ : Set G)) := by
    apply isCompact_univ_pi; intro K; apply isCompact_Icc
  refine' this.inter_iInter_nonempty (clPrehaar K₀) (fun s => isClosed_closure) fun t => _
  let V₀ := ⋂ V ∈ t, (V : OpenNhdsOf (1 : G)).carrier
  have h1V₀ : IsOpen V₀ := isOpen_biInter_finset $ by rintro ⟨⟨V, hV₁⟩, hV₂⟩ _; exact hV₁
  have h2V₀ : (1 : G) ∈ V₀ := by simp only [mem_iInter]; rintro ⟨⟨V, hV₁⟩, hV₂⟩ _; exact hV₂
  refine' ⟨prehaar K₀ V₀, _⟩
  

Proof state:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
this : IsCompact (haarProduct ↑K₀)
t : Finset (OpenNhdsOf 1)
V₀ : Set G := ⋂ V ∈ t, V.carrier
h1V₀ : IsOpen V₀
h2V₀ : 1 ∈ V₀
⊢ prehaar (↑K₀) V₀ ∈ haarProduct ↑K₀ ∩ ⋂ i ∈ t, clPrehaar (↑K₀) i

Next tactic:
constructor
@[to_additive]
theorem nonempty_iInter_clPrehaar (K₀ : PositiveCompacts G) :
    (haarProduct (K₀ : Set G) ∩ ⋂ V : OpenNhdsOf (1 : G), clPrehaar K₀ V).Nonempty := by
  have : IsCompact (haarProduct (K₀ : Set G)) := by
    apply isCompact_univ_pi; intro K; apply isCompact_Icc
  refine' this.inter_iInter_nonempty (clPrehaar K₀) (fun s => isClosed_closure) fun t => _
  let V₀ := ⋂ V ∈ t, (V : OpenNhdsOf (1 : G)).carrier
  have h1V₀ : IsOpen V₀ := isOpen_biInter_finset $ by rintro ⟨⟨V, hV₁⟩, hV₂⟩ _; exact hV₁
  have h2V₀ : (1 : G) ∈ V₀ := by simp only [mem_iInter]; rintro ⟨⟨V, hV₁⟩, hV₂⟩ _; exact hV₂
  refine' ⟨prehaar K₀ V₀, _⟩
  constructor
  · 

Proof state:
case left
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
this : IsCompact (haarProduct ↑K₀)
t : Finset (OpenNhdsOf 1)
V₀ : Set G := ⋂ V ∈ t, V.carrier
h1V₀ : IsOpen V₀
h2V₀ : 1 ∈ V₀
⊢ prehaar (↑K₀) V₀ ∈ haarProduct ↑K₀

Next tactic:
apply prehaar_mem_haarProduct K₀
@[to_additive]
theorem nonempty_iInter_clPrehaar (K₀ : PositiveCompacts G) :
    (haarProduct (K₀ : Set G) ∩ ⋂ V : OpenNhdsOf (1 : G), clPrehaar K₀ V).Nonempty := by
  have : IsCompact (haarProduct (K₀ : Set G)) := by
    apply isCompact_univ_pi; intro K; apply isCompact_Icc
  refine' this.inter_iInter_nonempty (clPrehaar K₀) (fun s => isClosed_closure) fun t => _
  let V₀ := ⋂ V ∈ t, (V : OpenNhdsOf (1 : G)).carrier
  have h1V₀ : IsOpen V₀ := isOpen_biInter_finset $ by rintro ⟨⟨V, hV₁⟩, hV₂⟩ _; exact hV₁
  have h2V₀ : (1 : G) ∈ V₀ := by simp only [mem_iInter]; rintro ⟨⟨V, hV₁⟩, hV₂⟩ _; exact hV₂
  refine' ⟨prehaar K₀ V₀, _⟩
  constructor
  · apply prehaar_mem_haarProduct K₀; 

Proof state:
case left
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
this : IsCompact (haarProduct ↑K₀)
t : Finset (OpenNhdsOf 1)
V₀ : Set G := ⋂ V ∈ t, V.carrier
h1V₀ : IsOpen V₀
h2V₀ : 1 ∈ V₀
⊢ Set.Nonempty (interior V₀)

Next tactic:
use 1