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⟩
  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]
  · rintro _ ⟨U, ⟨h1U, h2U, h3U⟩, rfl⟩
    

Proof state:
case 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
U : Set G
h1U :
  U ⊆
    ↑{ toOpens := { carrier := V⁻¹, is_open' := (_ : IsOpen ((fun a => a⁻¹) ⁻¹' (V₁ ∩ V₂))) },
          mem' := (_ : 1 ∈ (V₁ ∩ V₂)⁻¹) }.toOpens
h2U : IsOpen U
h3U : 1 ∈ U
⊢ prehaar (↑K₀) U ∈ eval ⁻¹' {0}

Next tactic:
simp only [mem_preimage, sub_eq_zero, mem_singleton_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]
  · rintro _ ⟨U, ⟨h1U, h2U, h3U⟩, rfl⟩
    simp only [mem_preimage, sub_eq_zero, mem_singleton_iff]; 

Proof state:
case 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
U : Set G
h1U :
  U ⊆
    ↑{ toOpens := { carrier := V⁻¹, is_open' := (_ : IsOpen ((fun a => a⁻¹) ⁻¹' (V₁ ∩ V₂))) },
          mem' := (_ : 1 ∈ (V₁ ∩ V₂)⁻¹) }.toOpens
h2U : IsOpen U
h3U : 1 ∈ U
⊢ prehaar (↑K₀) U K₁ + prehaar (↑K₀) U K₂ = prehaar (↑K₀) U (K₁ ⊔ K₂)

Next tactic:
rw [eq_comm]
@[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]
  · rintro _ ⟨U, ⟨h1U, h2U, h3U⟩, rfl⟩
    simp only [mem_preimage, sub_eq_zero, mem_singleton_iff]; rw [eq_comm]
    

Proof state:
case 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
U : Set G
h1U :
  U ⊆
    ↑{ toOpens := { carrier := V⁻¹, is_open' := (_ : IsOpen ((fun a => a⁻¹) ⁻¹' (V₁ ∩ V₂))) },
          mem' := (_ : 1 ∈ (V₁ ∩ V₂)⁻¹) }.toOpens
h2U : IsOpen U
h3U : 1 ∈ U
⊢ prehaar (↑K₀) U (K₁ ⊔ K₂) = prehaar (↑K₀) U K₁ + prehaar (↑K₀) U K₂

Next tactic:
apply prehaar_sup_eq
@[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]
  · rintro _ ⟨U, ⟨h1U, h2U, h3U⟩, rfl⟩
    simp only [mem_preimage, sub_eq_zero, mem_singleton_iff]; rw [eq_comm]
    apply prehaar_sup_eq
    · 

Proof state:
case intro.intro.intro.intro.hU
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
U : Set G
h1U :
  U ⊆
    ↑{ toOpens := { carrier := V⁻¹, is_open' := (_ : IsOpen ((fun a => a⁻¹) ⁻¹' (V₁ ∩ V₂))) },
          mem' := (_ : 1 ∈ (V₁ ∩ V₂)⁻¹) }.toOpens
h2U : IsOpen U
h3U : 1 ∈ U
⊢ Set.Nonempty (interior U)

Next tactic:
rw [h2U.interior_eq]
@[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]
  · rintro _ ⟨U, ⟨h1U, h2U, h3U⟩, rfl⟩
    simp only [mem_preimage, sub_eq_zero, mem_singleton_iff]; rw [eq_comm]
    apply prehaar_sup_eq
    · rw [h2U.interior_eq]; 

Proof state:
case intro.intro.intro.intro.hU
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
U : Set G
h1U :
  U ⊆
    ↑{ toOpens := { carrier := V⁻¹, is_open' := (_ : IsOpen ((fun a => a⁻¹) ⁻¹' (V₁ ∩ V₂))) },
          mem' := (_ : 1 ∈ (V₁ ∩ V₂)⁻¹) }.toOpens
h2U : IsOpen U
h3U : 1 ∈ U
⊢ Set.Nonempty U

Next tactic:
exact ⟨1, h3U⟩
@[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]
  · rintro _ ⟨U, ⟨h1U, h2U, h3U⟩, rfl⟩
    simp only [mem_preimage, sub_eq_zero, mem_singleton_iff]; rw [eq_comm]
    apply prehaar_sup_eq
    · rw [h2U.interior_eq]; exact ⟨1, h3U⟩
    · 

Proof state:
case intro.intro.intro.intro.h
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
U : Set G
h1U :
  U ⊆
    ↑{ toOpens := { carrier := V⁻¹, is_open' := (_ : IsOpen ((fun a => a⁻¹) ⁻¹' (V₁ ∩ V₂))) },
          mem' := (_ : 1 ∈ (V₁ ∩ V₂)⁻¹) }.toOpens
h2U : IsOpen U
h3U : 1 ∈ U
⊢ Disjoint (K₁.carrier * U⁻¹) (K₂.carrier * U⁻¹)

Next tactic:
refine' disjoint_of_subset _ _ hU
@[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]
  · rintro _ ⟨U, ⟨h1U, h2U, h3U⟩, rfl⟩
    simp only [mem_preimage, sub_eq_zero, mem_singleton_iff]; rw [eq_comm]
    apply prehaar_sup_eq
    · rw [h2U.interior_eq]; exact ⟨1, h3U⟩
    · refine' disjoint_of_subset _ _ hU
      · 

Proof state:
case intro.intro.intro.intro.h.refine'_1
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
U : Set G
h1U :
  U ⊆
    ↑{ toOpens := { carrier := V⁻¹, is_open' := (_ : IsOpen ((fun a => a⁻¹) ⁻¹' (V₁ ∩ V₂))) },
          mem' := (_ : 1 ∈ (V₁ ∩ V₂)⁻¹) }.toOpens
h2U : IsOpen U
h3U : 1 ∈ U
⊢ K₁.carrier * U⁻¹ ⊆ U₁

Next tactic:
refine' Subset.trans (mul_subset_mul Subset.rfl _) 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
  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]
  · rintro _ ⟨U, ⟨h1U, h2U, h3U⟩, rfl⟩
    simp only [mem_preimage, sub_eq_zero, mem_singleton_iff]; rw [eq_comm]
    apply prehaar_sup_eq
    · rw [h2U.interior_eq]; exact ⟨1, h3U⟩
    · refine' disjoint_of_subset _ _ hU
      · refine' Subset.trans (mul_subset_mul Subset.rfl _) h2L₁
        

Proof state:
case intro.intro.intro.intro.h.refine'_1
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
U : Set G
h1U :
  U ⊆
    ↑{ toOpens := { carrier := V⁻¹, is_open' := (_ : IsOpen ((fun a => a⁻¹) ⁻¹' (V₁ ∩ V₂))) },
          mem' := (_ : 1 ∈ (V₁ ∩ V₂)⁻¹) }.toOpens
h2U : IsOpen U
h3U : 1 ∈ U
⊢ U⁻¹ ⊆ V₁

Next tactic:
exact Subset.trans (inv_subset.mpr h1U) (inter_subset_left _ _)
@[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]
  · rintro _ ⟨U, ⟨h1U, h2U, h3U⟩, rfl⟩
    simp only [mem_preimage, sub_eq_zero, mem_singleton_iff]; rw [eq_comm]
    apply prehaar_sup_eq
    · rw [h2U.interior_eq]; exact ⟨1, h3U⟩
    · refine' disjoint_of_subset _ _ hU
      · refine' Subset.trans (mul_subset_mul Subset.rfl _) h2L₁
        exact Subset.trans (inv_subset.mpr h1U) (inter_subset_left _ _)
      · 

Proof state:
case intro.intro.intro.intro.h.refine'_2
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
U : Set G
h1U :
  U ⊆
    ↑{ toOpens := { carrier := V⁻¹, is_open' := (_ : IsOpen ((fun a => a⁻¹) ⁻¹' (V₁ ∩ V₂))) },
          mem' := (_ : 1 ∈ (V₁ ∩ V₂)⁻¹) }.toOpens
h2U : IsOpen U
h3U : 1 ∈ U
⊢ K₂.carrier * U⁻¹ ⊆ U₂

Next tactic:
refine' Subset.trans (mul_subset_mul Subset.rfl _) 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
  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]
  · rintro _ ⟨U, ⟨h1U, h2U, h3U⟩, rfl⟩
    simp only [mem_preimage, sub_eq_zero, mem_singleton_iff]; rw [eq_comm]
    apply prehaar_sup_eq
    · rw [h2U.interior_eq]; exact ⟨1, h3U⟩
    · refine' disjoint_of_subset _ _ hU
      · refine' Subset.trans (mul_subset_mul Subset.rfl _) h2L₁
        exact Subset.trans (inv_subset.mpr h1U) (inter_subset_left _ _)
      · refine' Subset.trans (mul_subset_mul Subset.rfl _) h2L₂
        

Proof state:
case intro.intro.intro.intro.h.refine'_2
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
U : Set G
h1U :
  U ⊆
    ↑{ toOpens := { carrier := V⁻¹, is_open' := (_ : IsOpen ((fun a => a⁻¹) ⁻¹' (V₁ ∩ V₂))) },
          mem' := (_ : 1 ∈ (V₁ ∩ V₂)⁻¹) }.toOpens
h2U : IsOpen U
h3U : 1 ∈ U
⊢ U⁻¹ ⊆ V₂

Next tactic:
exact Subset.trans (inv_subset.mpr h1U) (inter_subset_right _ _)