l3lab/ntp-mathlib huggingface dataset:
@[to_additive addCHaar_sup_eq]
theorem chaar_sup_eq [T2Space G] {K₀ : PositiveCompacts G} {K₁ K₂ : Compacts G}
    (h : Disjoint K₁.1 K₂.1) : chaar K₀ (K₁ ⊔ K₂) = chaar K₀ K₁ + chaar K₀ K₂ := by
  rcases isCompact_isCompact_separated K₁.2 K₂.2 h with ⟨U₁, U₂, h1U₁, h1U₂, h2U₁, h2U₂, hU⟩
  rcases compact_open_separated_mul_right K₁.2 h1U₁ h2U₁ with ⟨L₁, h1L₁, h2L₁⟩
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen U₁
h1U₂ : IsOpen U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
L₁ : Set G
h1L₁ : L₁ ∈ 𝓝 1
h2L₁ : K₁.carrier * L₁ ⊆ U₁
⊢ chaar K₀ (K₁ ⊔ K₂) = chaar K₀ K₁ + chaar K₀ K₂

Next tactic:
rcases mem_nhds_iff.mp h1L₁ with ⟨V₁, h1V₁, h2V₁, h3V₁⟩
@[to_additive addCHaar_sup_eq]
theorem chaar_sup_eq [T2Space G] {K₀ : PositiveCompacts G} {K₁ K₂ : Compacts G}
    (h : Disjoint K₁.1 K₂.1) : chaar K₀ (K₁ ⊔ K₂) = chaar K₀ K₁ + chaar K₀ K₂ := by
  rcases isCompact_isCompact_separated K₁.2 K₂.2 h with ⟨U₁, U₂, h1U₁, h1U₂, h2U₁, h2U₂, hU⟩
  rcases compact_open_separated_mul_right K₁.2 h1U₁ h2U₁ with ⟨L₁, h1L₁, h2L₁⟩
  rcases mem_nhds_iff.mp h1L₁ with ⟨V₁, h1V₁, h2V₁, h3V₁⟩
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen U₁
h1U₂ : IsOpen U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
L₁ : Set G
h1L₁ : L₁ ∈ 𝓝 1
h2L₁ : K₁.carrier * L₁ ⊆ U₁
V₁ : Set G
h1V₁ : V₁ ⊆ L₁
h2V₁ : IsOpen V₁
h3V₁ : 1 ∈ V₁
⊢ chaar K₀ (K₁ ⊔ K₂) = chaar K₀ K₁ + chaar K₀ K₂

Next tactic:
replace h2L₁ := Subset.trans (mul_subset_mul_left h1V₁) h2L₁
@[to_additive addCHaar_sup_eq]
theorem chaar_sup_eq [T2Space G] {K₀ : PositiveCompacts G} {K₁ K₂ : Compacts G}
    (h : Disjoint K₁.1 K₂.1) : chaar K₀ (K₁ ⊔ K₂) = chaar K₀ K₁ + chaar K₀ K₂ := by
  rcases isCompact_isCompact_separated K₁.2 K₂.2 h with ⟨U₁, U₂, h1U₁, h1U₂, h2U₁, h2U₂, hU⟩
  rcases compact_open_separated_mul_right K₁.2 h1U₁ h2U₁ with ⟨L₁, h1L₁, h2L₁⟩
  rcases mem_nhds_iff.mp h1L₁ with ⟨V₁, h1V₁, h2V₁, h3V₁⟩
  replace h2L₁ := Subset.trans (mul_subset_mul_left h1V₁) h2L₁
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen U₁
h1U₂ : IsOpen U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
L₁ : Set G
h1L₁ : L₁ ∈ 𝓝 1
V₁ : Set G
h1V₁ : V₁ ⊆ L₁
h2V₁ : IsOpen V₁
h3V₁ : 1 ∈ V₁
h2L₁ : K₁.carrier * V₁ ⊆ U₁
⊢ chaar K₀ (K₁ ⊔ K₂) = chaar K₀ K₁ + chaar K₀ K₂

Next tactic:
rcases compact_open_separated_mul_right K₂.2 h1U₂ h2U₂ with ⟨L₂, h1L₂, h2L₂⟩
@[to_additive addCHaar_sup_eq]
theorem chaar_sup_eq [T2Space G] {K₀ : PositiveCompacts G} {K₁ K₂ : Compacts G}
    (h : Disjoint K₁.1 K₂.1) : chaar K₀ (K₁ ⊔ K₂) = chaar K₀ K₁ + chaar K₀ K₂ := by
  rcases isCompact_isCompact_separated K₁.2 K₂.2 h with ⟨U₁, U₂, h1U₁, h1U₂, h2U₁, h2U₂, hU⟩
  rcases compact_open_separated_mul_right K₁.2 h1U₁ h2U₁ with ⟨L₁, h1L₁, h2L₁⟩
  rcases mem_nhds_iff.mp h1L₁ with ⟨V₁, h1V₁, h2V₁, h3V₁⟩
  replace h2L₁ := Subset.trans (mul_subset_mul_left h1V₁) h2L₁
  rcases compact_open_separated_mul_right K₂.2 h1U₂ h2U₂ with ⟨L₂, h1L₂, h2L₂⟩
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen U₁
h1U₂ : IsOpen U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
L₁ : Set G
h1L₁ : L₁ ∈ 𝓝 1
V₁ : Set G
h1V₁ : V₁ ⊆ L₁
h2V₁ : IsOpen V₁
h3V₁ : 1 ∈ V₁
h2L₁ : K₁.carrier * V₁ ⊆ U₁
L₂ : Set G
h1L₂ : L₂ ∈ 𝓝 1
h2L₂ : K₂.carrier * L₂ ⊆ U₂
⊢ chaar K₀ (K₁ ⊔ K₂) = chaar K₀ K₁ + chaar K₀ K₂

Next tactic:
rcases mem_nhds_iff.mp h1L₂ with ⟨V₂, h1V₂, h2V₂, h3V₂⟩
@[to_additive addCHaar_sup_eq]
theorem chaar_sup_eq [T2Space G] {K₀ : PositiveCompacts G} {K₁ K₂ : Compacts G}
    (h : Disjoint K₁.1 K₂.1) : chaar K₀ (K₁ ⊔ K₂) = chaar K₀ K₁ + chaar K₀ K₂ := by
  rcases isCompact_isCompact_separated K₁.2 K₂.2 h with ⟨U₁, U₂, h1U₁, h1U₂, h2U₁, h2U₂, hU⟩
  rcases compact_open_separated_mul_right K₁.2 h1U₁ h2U₁ with ⟨L₁, h1L₁, h2L₁⟩
  rcases mem_nhds_iff.mp h1L₁ with ⟨V₁, h1V₁, h2V₁, h3V₁⟩
  replace h2L₁ := Subset.trans (mul_subset_mul_left h1V₁) h2L₁
  rcases compact_open_separated_mul_right K₂.2 h1U₂ h2U₂ with ⟨L₂, h1L₂, h2L₂⟩
  rcases mem_nhds_iff.mp h1L₂ with ⟨V₂, h1V₂, h2V₂, h3V₂⟩
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen U₁
h1U₂ : IsOpen U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
L₁ : Set G
h1L₁ : L₁ ∈ 𝓝 1
V₁ : Set G
h1V₁ : V₁ ⊆ L₁
h2V₁ : IsOpen V₁
h3V₁ : 1 ∈ V₁
h2L₁ : K₁.carrier * V₁ ⊆ U₁
L₂ : Set G
h1L₂ : L₂ ∈ 𝓝 1
h2L₂ : K₂.carrier * L₂ ⊆ U₂
V₂ : Set G
h1V₂ : V₂ ⊆ L₂
h2V₂ : IsOpen V₂
h3V₂ : 1 ∈ V₂
⊢ chaar K₀ (K₁ ⊔ K₂) = chaar K₀ K₁ + chaar K₀ K₂

Next tactic:
replace h2L₂ := Subset.trans (mul_subset_mul_left h1V₂) h2L₂
@[to_additive addCHaar_sup_eq]
theorem chaar_sup_eq [T2Space G] {K₀ : PositiveCompacts G} {K₁ K₂ : Compacts G}
    (h : Disjoint K₁.1 K₂.1) : chaar K₀ (K₁ ⊔ K₂) = chaar K₀ K₁ + chaar K₀ K₂ := by
  rcases isCompact_isCompact_separated K₁.2 K₂.2 h with ⟨U₁, U₂, h1U₁, h1U₂, h2U₁, h2U₂, hU⟩
  rcases compact_open_separated_mul_right K₁.2 h1U₁ h2U₁ with ⟨L₁, h1L₁, h2L₁⟩
  rcases mem_nhds_iff.mp h1L₁ with ⟨V₁, h1V₁, h2V₁, h3V₁⟩
  replace h2L₁ := Subset.trans (mul_subset_mul_left h1V₁) h2L₁
  rcases compact_open_separated_mul_right K₂.2 h1U₂ h2U₂ with ⟨L₂, h1L₂, h2L₂⟩
  rcases mem_nhds_iff.mp h1L₂ with ⟨V₂, h1V₂, h2V₂, h3V₂⟩
  replace h2L₂ := Subset.trans (mul_subset_mul_left h1V₂) h2L₂
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen U₁
h1U₂ : IsOpen U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
L₁ : Set G
h1L₁ : L₁ ∈ 𝓝 1
V₁ : Set G
h1V₁ : V₁ ⊆ L₁
h2V₁ : IsOpen V₁
h3V₁ : 1 ∈ V₁
h2L₁ : K₁.carrier * V₁ ⊆ U₁
L₂ : Set G
h1L₂ : L₂ ∈ 𝓝 1
V₂ : Set G
h1V₂ : V₂ ⊆ L₂
h2V₂ : IsOpen V₂
h3V₂ : 1 ∈ V₂
h2L₂ : K₂.carrier * V₂ ⊆ U₂
⊢ chaar K₀ (K₁ ⊔ K₂) = chaar K₀ K₁ + chaar K₀ K₂

Next tactic:
let eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
@[to_additive addCHaar_sup_eq]
theorem chaar_sup_eq [T2Space G] {K₀ : PositiveCompacts G} {K₁ K₂ : Compacts G}
    (h : Disjoint K₁.1 K₂.1) : chaar K₀ (K₁ ⊔ K₂) = chaar K₀ K₁ + chaar K₀ K₂ := by
  rcases isCompact_isCompact_separated K₁.2 K₂.2 h with ⟨U₁, U₂, h1U₁, h1U₂, h2U₁, h2U₂, hU⟩
  rcases compact_open_separated_mul_right K₁.2 h1U₁ h2U₁ with ⟨L₁, h1L₁, h2L₁⟩
  rcases mem_nhds_iff.mp h1L₁ with ⟨V₁, h1V₁, h2V₁, h3V₁⟩
  replace h2L₁ := Subset.trans (mul_subset_mul_left h1V₁) h2L₁
  rcases compact_open_separated_mul_right K₂.2 h1U₂ h2U₂ with ⟨L₂, h1L₂, h2L₂⟩
  rcases mem_nhds_iff.mp h1L₂ with ⟨V₂, h1V₂, h2V₂, h3V₂⟩
  replace h2L₂ := Subset.trans (mul_subset_mul_left h1V₂) h2L₂
  let eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen U₁
h1U₂ : IsOpen U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
L₁ : Set G
h1L₁ : L₁ ∈ 𝓝 1
V₁ : Set G
h1V₁ : V₁ ⊆ L₁
h2V₁ : IsOpen V₁
h3V₁ : 1 ∈ V₁
h2L₁ : K₁.carrier * V₁ ⊆ U₁
L₂ : Set G
h1L₂ : L₂ ∈ 𝓝 1
V₂ : Set G
h1V₂ : V₂ ⊆ L₂
h2V₂ : IsOpen V₂
h3V₂ : 1 ∈ V₂
h2L₂ : K₂.carrier * V₂ ⊆ U₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
⊢ chaar K₀ (K₁ ⊔ K₂) = chaar K₀ K₁ + chaar K₀ K₂

Next tactic:
have : Continuous eval :=
    ((continuous_apply K₁).add (continuous_apply K₂)).sub (continuous_apply (K₁ ⊔ K₂))
@[to_additive addCHaar_sup_eq]
theorem chaar_sup_eq [T2Space G] {K₀ : PositiveCompacts G} {K₁ K₂ : Compacts G}
    (h : Disjoint K₁.1 K₂.1) : chaar K₀ (K₁ ⊔ K₂) = chaar K₀ K₁ + chaar K₀ K₂ := by
  rcases isCompact_isCompact_separated K₁.2 K₂.2 h with ⟨U₁, U₂, h1U₁, h1U₂, h2U₁, h2U₂, hU⟩
  rcases compact_open_separated_mul_right K₁.2 h1U₁ h2U₁ with ⟨L₁, h1L₁, h2L₁⟩
  rcases mem_nhds_iff.mp h1L₁ with ⟨V₁, h1V₁, h2V₁, h3V₁⟩
  replace h2L₁ := Subset.trans (mul_subset_mul_left h1V₁) h2L₁
  rcases compact_open_separated_mul_right K₂.2 h1U₂ h2U₂ with ⟨L₂, h1L₂, h2L₂⟩
  rcases mem_nhds_iff.mp h1L₂ with ⟨V₂, h1V₂, h2V₂, h3V₂⟩
  replace h2L₂ := Subset.trans (mul_subset_mul_left h1V₂) h2L₂
  let eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
  have : Continuous eval :=
    ((continuous_apply K₁).add (continuous_apply K₂)).sub (continuous_apply (K₁ ⊔ K₂))
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen U₁
h1U₂ : IsOpen U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
L₁ : Set G
h1L₁ : L₁ ∈ 𝓝 1
V₁ : Set G
h1V₁ : V₁ ⊆ L₁
h2V₁ : IsOpen V₁
h3V₁ : 1 ∈ V₁
h2L₁ : K₁.carrier * V₁ ⊆ U₁
L₂ : Set G
h1L₂ : L₂ ∈ 𝓝 1
V₂ : Set G
h1V₂ : V₂ ⊆ L₂
h2V₂ : IsOpen V₂
h3V₂ : 1 ∈ V₂
h2L₂ : K₂.carrier * V₂ ⊆ U₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
⊢ chaar K₀ (K₁ ⊔ K₂) = chaar K₀ K₁ + chaar K₀ K₂

Next tactic:
rw [eq_comm, ← sub_eq_zero]
@[to_additive addCHaar_sup_eq]
theorem chaar_sup_eq [T2Space G] {K₀ : PositiveCompacts G} {K₁ K₂ : Compacts G}
    (h : Disjoint K₁.1 K₂.1) : chaar K₀ (K₁ ⊔ K₂) = chaar K₀ K₁ + chaar K₀ K₂ := by
  rcases isCompact_isCompact_separated K₁.2 K₂.2 h with ⟨U₁, U₂, h1U₁, h1U₂, h2U₁, h2U₂, hU⟩
  rcases compact_open_separated_mul_right K₁.2 h1U₁ h2U₁ with ⟨L₁, h1L₁, h2L₁⟩
  rcases mem_nhds_iff.mp h1L₁ with ⟨V₁, h1V₁, h2V₁, h3V₁⟩
  replace h2L₁ := Subset.trans (mul_subset_mul_left h1V₁) h2L₁
  rcases compact_open_separated_mul_right K₂.2 h1U₂ h2U₂ with ⟨L₂, h1L₂, h2L₂⟩
  rcases mem_nhds_iff.mp h1L₂ with ⟨V₂, h1V₂, h2V₂, h3V₂⟩
  replace h2L₂ := Subset.trans (mul_subset_mul_left h1V₂) h2L₂
  let eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
  have : Continuous eval :=
    ((continuous_apply K₁).add (continuous_apply K₂)).sub (continuous_apply (K₁ ⊔ K₂))
  rw [eq_comm, ← sub_eq_zero]; 

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen U₁
h1U₂ : IsOpen U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
L₁ : Set G
h1L₁ : L₁ ∈ 𝓝 1
V₁ : Set G
h1V₁ : V₁ ⊆ L₁
h2V₁ : IsOpen V₁
h3V₁ : 1 ∈ V₁
h2L₁ : K₁.carrier * V₁ ⊆ U₁
L₂ : Set G
h1L₂ : L₂ ∈ 𝓝 1
V₂ : Set G
h1V₂ : V₂ ⊆ L₂
h2V₂ : IsOpen V₂
h3V₂ : 1 ∈ V₂
h2L₂ : K₂.carrier * V₂ ⊆ U₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
⊢ chaar K₀ K₁ + chaar K₀ K₂ - chaar K₀ (K₁ ⊔ K₂) = 0

Next tactic:
show chaar K₀ ∈ eval ⁻¹' {(0 : ℝ)}
@[to_additive addCHaar_sup_eq]
theorem chaar_sup_eq [T2Space G] {K₀ : PositiveCompacts G} {K₁ K₂ : Compacts G}
    (h : Disjoint K₁.1 K₂.1) : chaar K₀ (K₁ ⊔ K₂) = chaar K₀ K₁ + chaar K₀ K₂ := by
  rcases isCompact_isCompact_separated K₁.2 K₂.2 h with ⟨U₁, U₂, h1U₁, h1U₂, h2U₁, h2U₂, hU⟩
  rcases compact_open_separated_mul_right K₁.2 h1U₁ h2U₁ with ⟨L₁, h1L₁, h2L₁⟩
  rcases mem_nhds_iff.mp h1L₁ with ⟨V₁, h1V₁, h2V₁, h3V₁⟩
  replace h2L₁ := Subset.trans (mul_subset_mul_left h1V₁) h2L₁
  rcases compact_open_separated_mul_right K₂.2 h1U₂ h2U₂ with ⟨L₂, h1L₂, h2L₂⟩
  rcases mem_nhds_iff.mp h1L₂ with ⟨V₂, h1V₂, h2V₂, h3V₂⟩
  replace h2L₂ := Subset.trans (mul_subset_mul_left h1V₂) h2L₂
  let eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
  have : Continuous eval :=
    ((continuous_apply K₁).add (continuous_apply K₂)).sub (continuous_apply (K₁ ⊔ K₂))
  rw [eq_comm, ← sub_eq_zero]; show chaar K₀ ∈ eval ⁻¹' {(0 : ℝ)}
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen U₁
h1U₂ : IsOpen U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
L₁ : Set G
h1L₁ : L₁ ∈ 𝓝 1
V₁ : Set G
h1V₁ : V₁ ⊆ L₁
h2V₁ : IsOpen V₁
h3V₁ : 1 ∈ V₁
h2L₁ : K₁.carrier * V₁ ⊆ U₁
L₂ : Set G
h1L₂ : L₂ ∈ 𝓝 1
V₂ : Set G
h1V₂ : V₂ ⊆ L₂
h2V₂ : IsOpen V₂
h3V₂ : 1 ∈ V₂
h2L₂ : K₂.carrier * V₂ ⊆ U₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
⊢ chaar K₀ ∈ eval ⁻¹' {0}

Next tactic:
let V := V₁ ∩ V₂
@[to_additive addCHaar_sup_eq]
theorem chaar_sup_eq [T2Space G] {K₀ : PositiveCompacts G} {K₁ K₂ : Compacts G}
    (h : Disjoint K₁.1 K₂.1) : chaar K₀ (K₁ ⊔ K₂) = chaar K₀ K₁ + chaar K₀ K₂ := by
  rcases isCompact_isCompact_separated K₁.2 K₂.2 h with ⟨U₁, U₂, h1U₁, h1U₂, h2U₁, h2U₂, hU⟩
  rcases compact_open_separated_mul_right K₁.2 h1U₁ h2U₁ with ⟨L₁, h1L₁, h2L₁⟩
  rcases mem_nhds_iff.mp h1L₁ with ⟨V₁, h1V₁, h2V₁, h3V₁⟩
  replace h2L₁ := Subset.trans (mul_subset_mul_left h1V₁) h2L₁
  rcases compact_open_separated_mul_right K₂.2 h1U₂ h2U₂ with ⟨L₂, h1L₂, h2L₂⟩
  rcases mem_nhds_iff.mp h1L₂ with ⟨V₂, h1V₂, h2V₂, h3V₂⟩
  replace h2L₂ := Subset.trans (mul_subset_mul_left h1V₂) h2L₂
  let eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
  have : Continuous eval :=
    ((continuous_apply K₁).add (continuous_apply K₂)).sub (continuous_apply (K₁ ⊔ K₂))
  rw [eq_comm, ← sub_eq_zero]; show chaar K₀ ∈ eval ⁻¹' {(0 : ℝ)}
  let V := V₁ ∩ V₂
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen U₁
h1U₂ : IsOpen U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
L₁ : Set G
h1L₁ : L₁ ∈ 𝓝 1
V₁ : Set G
h1V₁ : V₁ ⊆ L₁
h2V₁ : IsOpen V₁
h3V₁ : 1 ∈ V₁
h2L₁ : K₁.carrier * V₁ ⊆ U₁
L₂ : Set G
h1L₂ : L₂ ∈ 𝓝 1
V₂ : Set G
h1V₂ : V₂ ⊆ L₂
h2V₂ : IsOpen V₂
h3V₂ : 1 ∈ V₂
h2L₂ : K₂.carrier * V₂ ⊆ U₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
V : Set G := V₁ ∩ V₂
⊢ chaar K₀ ∈ eval ⁻¹' {0}

Next tactic:
apply
    mem_of_subset_of_mem _
      (chaar_mem_clPrehaar K₀
        ⟨⟨V⁻¹, (h2V₁.inter h2V₂).preimage continuous_inv⟩, by
          simp only [mem_inv, inv_one, h3V₁, h3V₂, mem_inter_iff, true_and_iff]⟩)
@[to_additive addCHaar_sup_eq]
theorem chaar_sup_eq [T2Space G] {K₀ : PositiveCompacts G} {K₁ K₂ : Compacts G}
    (h : Disjoint K₁.1 K₂.1) : chaar K₀ (K₁ ⊔ K₂) = chaar K₀ K₁ + chaar K₀ K₂ := by
  rcases isCompact_isCompact_separated K₁.2 K₂.2 h with ⟨U₁, U₂, h1U₁, h1U₂, h2U₁, h2U₂, hU⟩
  rcases compact_open_separated_mul_right K₁.2 h1U₁ h2U₁ with ⟨L₁, h1L₁, h2L₁⟩
  rcases mem_nhds_iff.mp h1L₁ with ⟨V₁, h1V₁, h2V₁, h3V₁⟩
  replace h2L₁ := Subset.trans (mul_subset_mul_left h1V₁) h2L₁
  rcases compact_open_separated_mul_right K₂.2 h1U₂ h2U₂ with ⟨L₂, h1L₂, h2L₂⟩
  rcases mem_nhds_iff.mp h1L₂ with ⟨V₂, h1V₂, h2V₂, h3V₂⟩
  replace h2L₂ := Subset.trans (mul_subset_mul_left h1V₂) h2L₂
  let eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
  have : Continuous eval :=
    ((continuous_apply K₁).add (continuous_apply K₂)).sub (continuous_apply (K₁ ⊔ K₂))
  rw [eq_comm, ← sub_eq_zero]; show chaar K₀ ∈ eval ⁻¹' {(0 : ℝ)}
  let V := V₁ ∩ V₂
  apply
    mem_of_subset_of_mem _
      (chaar_mem_clPrehaar K₀
        ⟨⟨V⁻¹, (h2V₁.inter h2V₂).preimage continuous_inv⟩, by
          

Proof state:
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen U₁
h1U₂ : IsOpen U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
L₁ : Set G
h1L₁ : L₁ ∈ 𝓝 1
V₁ : Set G
h1V₁ : V₁ ⊆ L₁
h2V₁ : IsOpen V₁
h3V₁ : 1 ∈ V₁
h2L₁ : K₁.carrier * V₁ ⊆ U₁
L₂ : Set G
h1L₂ : L₂ ∈ 𝓝 1
V₂ : Set G
h1V₂ : V₂ ⊆ L₂
h2V₂ : IsOpen V₂
h3V₂ : 1 ∈ V₂
h2L₂ : K₂.carrier * V₂ ⊆ U₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
V : Set G := V₁ ∩ V₂
⊢ 1 ∈ { carrier := V⁻¹, is_open' := (_ : IsOpen ((fun a => a⁻¹) ⁻¹' (V₁ ∩ V₂))) }.carrier

Next tactic:
simp only [mem_inv, inv_one, h3V₁, h3V₂, mem_inter_iff, true_and_iff]
@[to_additive addCHaar_sup_eq]
theorem chaar_sup_eq [T2Space G] {K₀ : PositiveCompacts G} {K₁ K₂ : Compacts G}
    (h : Disjoint K₁.1 K₂.1) : chaar K₀ (K₁ ⊔ K₂) = chaar K₀ K₁ + chaar K₀ K₂ := by
  rcases isCompact_isCompact_separated K₁.2 K₂.2 h with ⟨U₁, U₂, h1U₁, h1U₂, h2U₁, h2U₂, hU⟩
  rcases compact_open_separated_mul_right K₁.2 h1U₁ h2U₁ with ⟨L₁, h1L₁, h2L₁⟩
  rcases mem_nhds_iff.mp h1L₁ with ⟨V₁, h1V₁, h2V₁, h3V₁⟩
  replace h2L₁ := Subset.trans (mul_subset_mul_left h1V₁) h2L₁
  rcases compact_open_separated_mul_right K₂.2 h1U₂ h2U₂ with ⟨L₂, h1L₂, h2L₂⟩
  rcases mem_nhds_iff.mp h1L₂ with ⟨V₂, h1V₂, h2V₂, h3V₂⟩
  replace h2L₂ := Subset.trans (mul_subset_mul_left h1V₂) h2L₂
  let eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
  have : Continuous eval :=
    ((continuous_apply K₁).add (continuous_apply K₂)).sub (continuous_apply (K₁ ⊔ K₂))
  rw [eq_comm, ← sub_eq_zero]; show chaar K₀ ∈ eval ⁻¹' {(0 : ℝ)}
  let V := V₁ ∩ V₂
  apply
    mem_of_subset_of_mem _
      (chaar_mem_clPrehaar K₀
        ⟨⟨V⁻¹, (h2V₁.inter h2V₂).preimage continuous_inv⟩, by
          simp only [mem_inv, inv_one, h3V₁, h3V₂, mem_inter_iff, true_and_iff]⟩)
  

Proof state:
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen U₁
h1U₂ : IsOpen U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
L₁ : Set G
h1L₁ : L₁ ∈ 𝓝 1
V₁ : Set G
h1V₁ : V₁ ⊆ L₁
h2V₁ : IsOpen V₁
h3V₁ : 1 ∈ V₁
h2L₁ : K₁.carrier * V₁ ⊆ U₁
L₂ : Set G
h1L₂ : L₂ ∈ 𝓝 1
V₂ : Set G
h1V₂ : V₂ ⊆ L₂
h2V₂ : IsOpen V₂
h3V₂ : 1 ∈ V₂
h2L₂ : K₂.carrier * V₂ ⊆ U₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
V : Set G := V₁ ∩ V₂
⊢ clPrehaar ↑K₀
      { toOpens := { carrier := V⁻¹, is_open' := (_ : IsOpen ((fun a => a⁻¹) ⁻¹' (V₁ ∩ V₂))) },
        mem' := (_ : 1 ∈ (V₁ ∩ V₂)⁻¹) } ⊆
    eval ⁻¹' {0}

Next tactic:
unfold clPrehaar
@[to_additive addCHaar_sup_eq]
theorem chaar_sup_eq [T2Space G] {K₀ : PositiveCompacts G} {K₁ K₂ : Compacts G}
    (h : Disjoint K₁.1 K₂.1) : chaar K₀ (K₁ ⊔ K₂) = chaar K₀ K₁ + chaar K₀ K₂ := by
  rcases isCompact_isCompact_separated K₁.2 K₂.2 h with ⟨U₁, U₂, h1U₁, h1U₂, h2U₁, h2U₂, hU⟩
  rcases compact_open_separated_mul_right K₁.2 h1U₁ h2U₁ with ⟨L₁, h1L₁, h2L₁⟩
  rcases mem_nhds_iff.mp h1L₁ with ⟨V₁, h1V₁, h2V₁, h3V₁⟩
  replace h2L₁ := Subset.trans (mul_subset_mul_left h1V₁) h2L₁
  rcases compact_open_separated_mul_right K₂.2 h1U₂ h2U₂ with ⟨L₂, h1L₂, h2L₂⟩
  rcases mem_nhds_iff.mp h1L₂ with ⟨V₂, h1V₂, h2V₂, h3V₂⟩
  replace h2L₂ := Subset.trans (mul_subset_mul_left h1V₂) h2L₂
  let eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
  have : Continuous eval :=
    ((continuous_apply K₁).add (continuous_apply K₂)).sub (continuous_apply (K₁ ⊔ K₂))
  rw [eq_comm, ← sub_eq_zero]; show chaar K₀ ∈ eval ⁻¹' {(0 : ℝ)}
  let V := V₁ ∩ V₂
  apply
    mem_of_subset_of_mem _
      (chaar_mem_clPrehaar K₀
        ⟨⟨V⁻¹, (h2V₁.inter h2V₂).preimage continuous_inv⟩, by
          simp only [mem_inv, inv_one, h3V₁, h3V₂, mem_inter_iff, true_and_iff]⟩)
  unfold clPrehaar; 

Proof state:
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen U₁
h1U₂ : IsOpen U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
L₁ : Set G
h1L₁ : L₁ ∈ 𝓝 1
V₁ : Set G
h1V₁ : V₁ ⊆ L₁
h2V₁ : IsOpen V₁
h3V₁ : 1 ∈ V₁
h2L₁ : K₁.carrier * V₁ ⊆ U₁
L₂ : Set G
h1L₂ : L₂ ∈ 𝓝 1
V₂ : Set G
h1V₂ : V₂ ⊆ L₂
h2V₂ : IsOpen V₂
h3V₂ : 1 ∈ V₂
h2L₂ : K₂.carrier * V₂ ⊆ U₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
V : Set G := V₁ ∩ V₂
⊢ closure
      (prehaar ↑K₀ ''
        {U |
          U ⊆
              ↑{ toOpens := { carrier := V⁻¹, is_open' := (_ : IsOpen ((fun a => a⁻¹) ⁻¹' (V₁ ∩ V₂))) },
                    mem' := (_ : 1 ∈ (V₁ ∩ V₂)⁻¹) }.toOpens ∧
            IsOpen U ∧ 1 ∈ U}) ⊆
    eval ⁻¹' {0}

Next tactic:
rw [IsClosed.closure_subset_iff]
@[to_additive addCHaar_sup_eq]
theorem chaar_sup_eq [T2Space G] {K₀ : PositiveCompacts G} {K₁ K₂ : Compacts G}
    (h : Disjoint K₁.1 K₂.1) : chaar K₀ (K₁ ⊔ K₂) = chaar K₀ K₁ + chaar K₀ K₂ := by
  rcases isCompact_isCompact_separated K₁.2 K₂.2 h with ⟨U₁, U₂, h1U₁, h1U₂, h2U₁, h2U₂, hU⟩
  rcases compact_open_separated_mul_right K₁.2 h1U₁ h2U₁ with ⟨L₁, h1L₁, h2L₁⟩
  rcases mem_nhds_iff.mp h1L₁ with ⟨V₁, h1V₁, h2V₁, h3V₁⟩
  replace h2L₁ := Subset.trans (mul_subset_mul_left h1V₁) h2L₁
  rcases compact_open_separated_mul_right K₂.2 h1U₂ h2U₂ with ⟨L₂, h1L₂, h2L₂⟩
  rcases mem_nhds_iff.mp h1L₂ with ⟨V₂, h1V₂, h2V₂, h3V₂⟩
  replace h2L₂ := Subset.trans (mul_subset_mul_left h1V₂) h2L₂
  let eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
  have : Continuous eval :=
    ((continuous_apply K₁).add (continuous_apply K₂)).sub (continuous_apply (K₁ ⊔ K₂))
  rw [eq_comm, ← sub_eq_zero]; show chaar K₀ ∈ eval ⁻¹' {(0 : ℝ)}
  let V := V₁ ∩ V₂
  apply
    mem_of_subset_of_mem _
      (chaar_mem_clPrehaar K₀
        ⟨⟨V⁻¹, (h2V₁.inter h2V₂).preimage continuous_inv⟩, by
          simp only [mem_inv, inv_one, h3V₁, h3V₂, mem_inter_iff, true_and_iff]⟩)
  unfold clPrehaar; rw [IsClosed.closure_subset_iff]
  · 

Proof state:
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen U₁
h1U₂ : IsOpen U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
L₁ : Set G
h1L₁ : L₁ ∈ 𝓝 1
V₁ : Set G
h1V₁ : V₁ ⊆ L₁
h2V₁ : IsOpen V₁
h3V₁ : 1 ∈ V₁
h2L₁ : K₁.carrier * V₁ ⊆ U₁
L₂ : Set G
h1L₂ : L₂ ∈ 𝓝 1
V₂ : Set G
h1V₂ : V₂ ⊆ L₂
h2V₂ : IsOpen V₂
h3V₂ : 1 ∈ V₂
h2L₂ : K₂.carrier * V₂ ⊆ U₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
V : Set G := V₁ ∩ V₂
⊢ prehaar ↑K₀ ''
      {U |
        U ⊆
            ↑{ toOpens := { carrier := V⁻¹, is_open' := (_ : IsOpen ((fun a => a⁻¹) ⁻¹' (V₁ ∩ V₂))) },
                  mem' := (_ : 1 ∈ (V₁ ∩ V₂)⁻¹) }.toOpens ∧
          IsOpen U ∧ 1 ∈ U} ⊆
    eval ⁻¹' {0}

Next tactic:
rintro _ ⟨U, ⟨h1U, h2U, h3U⟩, rfl⟩