ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- EuclideanSpace.coe_measurableEquiv_symm
theorem coe_measurableEquiv_symm :
    ⇑(EuclideanSpace.measurableEquiv ι).symm = (WithLp.equiv 2 _).symm 
Arguments: (ι : Type u_1)
Goal: ⇑(EuclideanSpace.measurableEquiv ι).symm = ⇑(WithLp.equiv 2 (ι → ℝ)).symm
-- MeasureTheory.Measure.haar.index_empty
@[to_additive addIndex_empty]
theorem index_empty {V : Set G} : index ∅ V = 0 
Arguments: {G : Type u_1} [Group G] {V : Set G}
Goal: MeasureTheory.Measure.haar.index ∅ V = 0
-- MeasureTheory.Measure.haar.addIndex_empty
@[to_additive addIndex_empty]
theorem index_empty {V : Set G} : index ∅ V = 0 
Arguments: {G : Type u_1} [AddGroup G] {V : Set G}
Goal: MeasureTheory.Measure.haar.addIndex ∅ V = 0
-- MeasureTheory.Measure.haar.addPrehaar_empty
@[to_additive]
theorem prehaar_empty (K₀ : PositiveCompacts G) {U : Set G} : prehaar (K₀ : Set G) U ⊥ = 0 
Arguments: {G : Type u_1} [AddGroup G] [TopologicalSpace G] (K₀ : TopologicalSpace.PositiveCompacts G) {U : Set G}
Goal: MeasureTheory.Measure.haar.addPrehaar (↑K₀) U ⊥ = 0
-- MeasureTheory.Measure.haar.prehaar_empty
@[to_additive]
theorem prehaar_empty (K₀ : PositiveCompacts G) {U : Set G} : prehaar (K₀ : Set G) U ⊥ = 0 
Arguments: {G : Type u_1} [Group G] [TopologicalSpace G] (K₀ : TopologicalSpace.PositiveCompacts G) {U : Set G}
Goal: MeasureTheory.Measure.haar.prehaar (↑K₀) U ⊥ = 0
-- MeasureTheory.Measure.haar.addPrehaar_nonneg
@[to_additive]
theorem prehaar_nonneg (K₀ : PositiveCompacts G) {U : Set G} (K : Compacts G) :
    0 ≤ prehaar (K₀ : Set G) U K 
Arguments: {G : Type u_1} [AddGroup G] [TopologicalSpace G] (K₀ : TopologicalSpace.PositiveCompacts G) {U : Set G} (K : TopologicalSpace.Compacts G)
Goal: 0 ≤ MeasureTheory.Measure.haar.addPrehaar (↑K₀) U K
-- MeasureTheory.Measure.haar.prehaar_nonneg
@[to_additive]
theorem prehaar_nonneg (K₀ : PositiveCompacts G) {U : Set G} (K : Compacts G) :
    0 ≤ prehaar (K₀ : Set G) U K 
Arguments: {G : Type u_1} [Group G] [TopologicalSpace G] (K₀ : TopologicalSpace.PositiveCompacts G) {U : Set G} (K : TopologicalSpace.Compacts G)
Goal: 0 ≤ MeasureTheory.Measure.haar.prehaar (↑K₀) U K
-- MeasureTheory.Measure.haar.mem_addPrehaar_empty
@[to_additive (attr := simp)]
theorem mem_prehaar_empty {K₀ : Set G} {f : Compacts G → ℝ} :
    f ∈ haarProduct K₀ ↔ ∀ K : Compacts G, f K ∈ Icc (0 : ℝ) (index (K : Set G) K₀) 
Arguments: {G : Type u_1} [AddGroup G] [TopologicalSpace G] {K₀ : Set G} {f : TopologicalSpace.Compacts G → ℝ}
Goal: f ∈ MeasureTheory.Measure.haar.addHaarProduct K₀ ↔
  ∀ (K : TopologicalSpace.Compacts G), f K ∈ Set.Icc 0 ↑(MeasureTheory.Measure.haar.addIndex (↑K) K₀)
-- MeasureTheory.Measure.haar.mem_prehaar_empty
@[to_additive (attr := simp)]
theorem mem_prehaar_empty {K₀ : Set G} {f : Compacts G → ℝ} :
    f ∈ haarProduct K₀ ↔ ∀ K : Compacts G, f K ∈ Icc (0 : ℝ) (index (K : Set G) K₀) 
Arguments: {G : Type u_1} [Group G] [TopologicalSpace G] {K₀ : Set G} {f : TopologicalSpace.Compacts G → ℝ}
Goal: f ∈ MeasureTheory.Measure.haar.haarProduct K₀ ↔
  ∀ (K : TopologicalSpace.Compacts G), f K ∈ Set.Icc 0 ↑(MeasureTheory.Measure.haar.index (↑K) K₀)
-- MeasureTheory.Measure.haar.addIndex_defined
@[to_additive addIndex_defined
"If `K` is compact and `V` has nonempty interior, then the index `(K : V)` is well-defined, there is
a finite set `t` satisfying the desired properties."]
theorem index_defined {K V : Set G} (hK : IsCompact K) (hV : (interior V).Nonempty) :
    ∃ n : ℕ, n ∈ Finset.card '' { t : Finset G | K ⊆ ⋃ g ∈ t, (fun h => g * h) ⁻¹' V } 
Arguments: {G : Type u_1} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] {K V : Set G} (hK : IsCompact K) (hV : (interior V).Nonempty)
Goal: ∃ n, n ∈ Finset.card '' {t | K ⊆ ⋃ g ∈ t, (fun h => g + h) ⁻¹' V}
-- MeasureTheory.Measure.haar.index_defined
@[to_additive addIndex_defined
"If `K` is compact and `V` has nonempty interior, then the index `(K : V)` is well-defined, there is
a finite set `t` satisfying the desired properties."]
theorem index_defined {K V : Set G} (hK : IsCompact K) (hV : (interior V).Nonempty) :
    ∃ n : ℕ, n ∈ Finset.card '' { t : Finset G | K ⊆ ⋃ g ∈ t, (fun h => g * h) ⁻¹' V } 
Arguments: {G : Type u_1} [Group G] [TopologicalSpace G] [TopologicalGroup G] {K V : Set G} (hK : IsCompact K) (hV : (interior V).Nonempty)
Goal: ∃ n, n ∈ Finset.card '' {t | K ⊆ ⋃ g ∈ t, (fun h => g * h) ⁻¹' V}
-- MeasureTheory.Measure.haar.addIndex_elim
@[to_additive addIndex_elim]
theorem index_elim {K V : Set G} (hK : IsCompact K) (hV : (interior V).Nonempty) :
    ∃ t : Finset G, (K ⊆ ⋃ g ∈ t, (fun h => g * h) ⁻¹' V) ∧ Finset.card t = index K V 
Arguments: {G : Type u_1} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] {K V : Set G} (hK : IsCompact K) (hV : (interior V).Nonempty)
Goal: ∃ t, K ⊆ ⋃ g ∈ t, (fun h => g + h) ⁻¹' V ∧ t.card = MeasureTheory.Measure.haar.addIndex K V
-- MeasureTheory.Measure.haar.index_elim
@[to_additive addIndex_elim]
theorem index_elim {K V : Set G} (hK : IsCompact K) (hV : (interior V).Nonempty) :
    ∃ t : Finset G, (K ⊆ ⋃ g ∈ t, (fun h => g * h) ⁻¹' V) ∧ Finset.card t = index K V 
Arguments: {G : Type u_1} [Group G] [TopologicalSpace G] [TopologicalGroup G] {K V : Set G} (hK : IsCompact K) (hV : (interior V).Nonempty)
Goal: ∃ t, K ⊆ ⋃ g ∈ t, (fun h => g * h) ⁻¹' V ∧ t.card = MeasureTheory.Measure.haar.index K V
-- MeasureTheory.Measure.haar.le_addIndex_mul
@[to_additive le_addIndex_mul]
theorem le_index_mul (K₀ : PositiveCompacts G) (K : Compacts G) {V : Set G}
    (hV : (interior V).Nonempty) :
    index (K : Set G) V ≤ index (K : Set G) K₀ * index (K₀ : Set G) V 
Arguments: {G : Type u_1} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] (K₀ : TopologicalSpace.PositiveCompacts G) (K : TopologicalSpace.Compacts G) {V : Set G} (hV : (interior V).Nonempty)
Goal: MeasureTheory.Measure.haar.addIndex (↑K) V ≤
  MeasureTheory.Measure.haar.addIndex ↑K ↑K₀ * MeasureTheory.Measure.haar.addIndex (↑K₀) V
-- MeasureTheory.Measure.haar.le_index_mul
@[to_additive le_addIndex_mul]
theorem le_index_mul (K₀ : PositiveCompacts G) (K : Compacts G) {V : Set G}
    (hV : (interior V).Nonempty) :
    index (K : Set G) V ≤ index (K : Set G) K₀ * index (K₀ : Set G) V 
Arguments: {G : Type u_1} [Group G] [TopologicalSpace G] [TopologicalGroup G] (K₀ : TopologicalSpace.PositiveCompacts G) (K : TopologicalSpace.Compacts G) {V : Set G} (hV : (interior V).Nonempty)
Goal: MeasureTheory.Measure.haar.index (↑K) V ≤
  MeasureTheory.Measure.haar.index ↑K ↑K₀ * MeasureTheory.Measure.haar.index (↑K₀) V
-- MeasureTheory.Measure.haar.addIndex_pos
@[to_additive addIndex_pos]
theorem index_pos (K : PositiveCompacts G) {V : Set G} (hV : (interior V).Nonempty) :
    0 < index (K : Set G) V 
Arguments: {G : Type u_1} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] (K : TopologicalSpace.PositiveCompacts G) {V : Set G} (hV : (interior V).Nonempty)
Goal: 0 < MeasureTheory.Measure.haar.addIndex (↑K) V
-- MeasureTheory.Measure.haar.index_pos
@[to_additive addIndex_pos]
theorem index_pos (K : PositiveCompacts G) {V : Set G} (hV : (interior V).Nonempty) :
    0 < index (K : Set G) V 
Arguments: {G : Type u_1} [Group G] [TopologicalSpace G] [TopologicalGroup G] (K : TopologicalSpace.PositiveCompacts G) {V : Set G} (hV : (interior V).Nonempty)
Goal: 0 < MeasureTheory.Measure.haar.index (↑K) V
-- MeasureTheory.Measure.haar.index_mono
@[to_additive addIndex_mono]
theorem index_mono {K K' V : Set G} (hK' : IsCompact K') (h : K ⊆ K') (hV : (interior V).Nonempty) :
    index K V ≤ index K' V 
Arguments: {G : Type u_1} [Group G] [TopologicalSpace G] [TopologicalGroup G] {K K' V : Set G} (hK' : IsCompact K') (h : K ⊆ K') (hV : (interior V).Nonempty)
Goal: MeasureTheory.Measure.haar.index K V ≤ MeasureTheory.Measure.haar.index K' V
-- MeasureTheory.Measure.haar.addIndex_mono
@[to_additive addIndex_mono]
theorem index_mono {K K' V : Set G} (hK' : IsCompact K') (h : K ⊆ K') (hV : (interior V).Nonempty) :
    index K V ≤ index K' V 
Arguments: {G : Type u_1} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] {K K' V : Set G} (hK' : IsCompact K') (h : K ⊆ K') (hV : (interior V).Nonempty)
Goal: MeasureTheory.Measure.haar.addIndex K V ≤ MeasureTheory.Measure.haar.addIndex K' V
-- MeasureTheory.Measure.haar.addIndex_union_le
@[to_additive addIndex_union_le]
theorem index_union_le (K₁ K₂ : Compacts G) {V : Set G} (hV : (interior V).Nonempty) :
    index (K₁.1 ∪ K₂.1) V ≤ index K₁.1 V + index K₂.1 V 
Arguments: {G : Type u_1} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] (K₁ K₂ : TopologicalSpace.Compacts G) {V : Set G} (hV : (interior V).Nonempty)
Goal: MeasureTheory.Measure.haar.addIndex (K₁.carrier ∪ K₂.carrier) V ≤
  MeasureTheory.Measure.haar.addIndex K₁.carrier V + MeasureTheory.Measure.haar.addIndex K₂.carrier V
-- MeasureTheory.Measure.haar.index_union_le
@[to_additive addIndex_union_le]
theorem index_union_le (K₁ K₂ : Compacts G) {V : Set G} (hV : (interior V).Nonempty) :
    index (K₁.1 ∪ K₂.1) V ≤ index K₁.1 V + index K₂.1 V 
Arguments: {G : Type u_1} [Group G] [TopologicalSpace G] [TopologicalGroup G] (K₁ K₂ : TopologicalSpace.Compacts G) {V : Set G} (hV : (interior V).Nonempty)
Goal: MeasureTheory.Measure.haar.index (K₁.carrier ∪ K₂.carrier) V ≤
  MeasureTheory.Measure.haar.index K₁.carrier V + MeasureTheory.Measure.haar.index K₂.carrier V
-- MeasureTheory.Measure.haar.addIndex_union_eq
@[to_additive addIndex_union_eq]
theorem index_union_eq (K₁ K₂ : Compacts G) {V : Set G} (hV : (interior V).Nonempty)
    (h : Disjoint (K₁.1 * V⁻¹) (K₂.1 * V⁻¹)) :
    index (K₁.1 ∪ K₂.1) V = index K₁.1 V + index K₂.1 V 
Arguments: {G : Type u_1} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] (K₁ K₂ : TopologicalSpace.Compacts G) {V : Set G} (hV : (interior V).Nonempty) (h : Disjoint (K₁.carrier + -V) (K₂.carrier + -V))
Goal: MeasureTheory.Measure.haar.addIndex (K₁.carrier ∪ K₂.carrier) V =
  MeasureTheory.Measure.haar.addIndex K₁.carrier V + MeasureTheory.Measure.haar.addIndex K₂.carrier V
-- MeasureTheory.Measure.haar.index_union_eq
@[to_additive addIndex_union_eq]
theorem index_union_eq (K₁ K₂ : Compacts G) {V : Set G} (hV : (interior V).Nonempty)
    (h : Disjoint (K₁.1 * V⁻¹) (K₂.1 * V⁻¹)) :
    index (K₁.1 ∪ K₂.1) V = index K₁.1 V + index K₂.1 V 
Arguments: {G : Type u_1} [Group G] [TopologicalSpace G] [TopologicalGroup G] (K₁ K₂ : TopologicalSpace.Compacts G) {V : Set G} (hV : (interior V).Nonempty) (h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹))
Goal: MeasureTheory.Measure.haar.index (K₁.carrier ∪ K₂.carrier) V =
  MeasureTheory.Measure.haar.index K₁.carrier V + MeasureTheory.Measure.haar.index K₂.carrier V
-- MeasureTheory.Measure.haar.add_left_addIndex_le
@[to_additive add_left_addIndex_le]
theorem mul_left_index_le {K : Set G} (hK : IsCompact K) {V : Set G} (hV : (interior V).Nonempty)
    (g : G) : index ((fun h => g * h) '' K) V ≤ index K V 
Arguments: {G : Type u_1} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] {K : Set G} (hK : IsCompact K) {V : Set G} (hV : (interior V).Nonempty) (g : G)
Goal: MeasureTheory.Measure.haar.addIndex ((fun h => g + h) '' K) V ≤ MeasureTheory.Measure.haar.addIndex K V
-- MeasureTheory.Measure.haar.mul_left_index_le
@[to_additive add_left_addIndex_le]
theorem mul_left_index_le {K : Set G} (hK : IsCompact K) {V : Set G} (hV : (interior V).Nonempty)
    (g : G) : index ((fun h => g * h) '' K) V ≤ index K V 
Arguments: {G : Type u_1} [Group G] [TopologicalSpace G] [TopologicalGroup G] {K : Set G} (hK : IsCompact K) {V : Set G} (hV : (interior V).Nonempty) (g : G)
Goal: MeasureTheory.Measure.haar.index ((fun h => g * h) '' K) V ≤ MeasureTheory.Measure.haar.index K V
-- MeasureTheory.Measure.haar.is_left_invariant_addIndex
@[to_additive is_left_invariant_addIndex]
theorem is_left_invariant_index {K : Set G} (hK : IsCompact K) (g : G) {V : Set G}
    (hV : (interior V).Nonempty) : index ((fun h => g * h) '' K) V = index K V 
Arguments: {G : Type u_1} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] {K : Set G} (hK : IsCompact K) (g : G) {V : Set G} (hV : (interior V).Nonempty)
Goal: MeasureTheory.Measure.haar.addIndex ((fun h => g + h) '' K) V = MeasureTheory.Measure.haar.addIndex K V
-- MeasureTheory.Measure.haar.is_left_invariant_index
@[to_additive is_left_invariant_addIndex]
theorem is_left_invariant_index {K : Set G} (hK : IsCompact K) (g : G) {V : Set G}
    (hV : (interior V).Nonempty) : index ((fun h => g * h) '' K) V = index K V 
Arguments: {G : Type u_1} [Group G] [TopologicalSpace G] [TopologicalGroup G] {K : Set G} (hK : IsCompact K) (g : G) {V : Set G} (hV : (interior V).Nonempty)
Goal: MeasureTheory.Measure.haar.index ((fun h => g * h) '' K) V = MeasureTheory.Measure.haar.index K V
-- MeasureTheory.Measure.haar.add_prehaar_le_addIndex
@[to_additive add_prehaar_le_addIndex]
theorem prehaar_le_index (K₀ : PositiveCompacts G) {U : Set G} (K : Compacts G)
    (hU : (interior U).Nonempty) : prehaar (K₀ : Set G) U K ≤ index (K : Set G) K₀ 
Arguments: {G : Type u_1} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] (K₀ : TopologicalSpace.PositiveCompacts G) {U : Set G} (K : TopologicalSpace.Compacts G) (hU : (interior U).Nonempty)
Goal: MeasureTheory.Measure.haar.addPrehaar (↑K₀) U K ≤ ↑(MeasureTheory.Measure.haar.addIndex ↑K ↑K₀)
-- MeasureTheory.Measure.haar.prehaar_le_index
@[to_additive add_prehaar_le_addIndex]
theorem prehaar_le_index (K₀ : PositiveCompacts G) {U : Set G} (K : Compacts G)
    (hU : (interior U).Nonempty) : prehaar (K₀ : Set G) U K ≤ index (K : Set G) K₀ 
Arguments: {G : Type u_1} [Group G] [TopologicalSpace G] [TopologicalGroup G] (K₀ : TopologicalSpace.PositiveCompacts G) {U : Set G} (K : TopologicalSpace.Compacts G) (hU : (interior U).Nonempty)
Goal: MeasureTheory.Measure.haar.prehaar (↑K₀) U K ≤ ↑(MeasureTheory.Measure.haar.index ↑K ↑K₀)
-- MeasureTheory.Measure.haar.prehaar_pos
@[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⟩ 
Arguments: {G : Type u_1} [Group G] [TopologicalSpace G] [TopologicalGroup G] (K₀ : TopologicalSpace.PositiveCompacts G) {U : Set G} (hU : (interior U).Nonempty) {K : Set G} (h1K : IsCompact K) (h2K : (interior K).Nonempty)
Goal: 0 < MeasureTheory.Measure.haar.prehaar (↑K₀) U { carrier := K, isCompact' := h1K }
-- MeasureTheory.Measure.haar.addPrehaar_pos
@[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⟩ 
Arguments: {G : Type u_1} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] (K₀ : TopologicalSpace.PositiveCompacts G) {U : Set G} (hU : (interior U).Nonempty) {K : Set G} (h1K : IsCompact K) (h2K : (interior K).Nonempty)
Goal: 0 < MeasureTheory.Measure.haar.addPrehaar (↑K₀) U { carrier := K, isCompact' := h1K }
-- MeasureTheory.Measure.haar.prehaar_mono
@[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 
Arguments: {G : Type u_1} [Group G] [TopologicalSpace G] [TopologicalGroup G] {K₀ : TopologicalSpace.PositiveCompacts G} {U : Set G} (hU : (interior U).Nonempty) {K₁ K₂ : TopologicalSpace.Compacts G} (h : ↑K₁ ⊆ K₂.carrier)
Goal: MeasureTheory.Measure.haar.prehaar (↑K₀) U K₁ ≤ MeasureTheory.Measure.haar.prehaar (↑K₀) U K₂
-- MeasureTheory.Measure.haar.addPrehaar_mono
@[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 
Arguments: {G : Type u_1} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] {K₀ : TopologicalSpace.PositiveCompacts G} {U : Set G} (hU : (interior U).Nonempty) {K₁ K₂ : TopologicalSpace.Compacts G} (h : ↑K₁ ⊆ K₂.carrier)
Goal: MeasureTheory.Measure.haar.addPrehaar (↑K₀) U K₁ ≤ MeasureTheory.Measure.haar.addPrehaar (↑K₀) U K₂
-- MeasureTheory.Measure.haar.prehaar_self
@[to_additive]
theorem prehaar_self {K₀ : PositiveCompacts G} {U : Set G} (hU : (interior U).Nonempty) :
    prehaar (K₀ : Set G) U K₀.toCompacts = 1 
Arguments: {G : Type u_1} [Group G] [TopologicalSpace G] [TopologicalGroup G] {K₀ : TopologicalSpace.PositiveCompacts G} {U : Set G} (hU : (interior U).Nonempty)
Goal: MeasureTheory.Measure.haar.prehaar (↑K₀) U K₀.toCompacts = 1
-- MeasureTheory.Measure.haar.addPrehaar_self
@[to_additive]
theorem prehaar_self {K₀ : PositiveCompacts G} {U : Set G} (hU : (interior U).Nonempty) :
    prehaar (K₀ : Set G) U K₀.toCompacts = 1 
Arguments: {G : Type u_1} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] {K₀ : TopologicalSpace.PositiveCompacts G} {U : Set G} (hU : (interior U).Nonempty)
Goal: MeasureTheory.Measure.haar.addPrehaar (↑K₀) U K₀.toCompacts = 1
-- MeasureTheory.Measure.haar.prehaar_sup_le
@[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 
Arguments: {G : Type u_1} [Group G] [TopologicalSpace G] [TopologicalGroup G] {K₀ : TopologicalSpace.PositiveCompacts G} {U : Set G} (K₁ K₂ : TopologicalSpace.Compacts G) (hU : (interior U).Nonempty)
Goal: MeasureTheory.Measure.haar.prehaar (↑K₀) U (K₁ ⊔ K₂) ≤
  MeasureTheory.Measure.haar.prehaar (↑K₀) U K₁ + MeasureTheory.Measure.haar.prehaar (↑K₀) U K₂
-- MeasureTheory.Measure.haar.addPrehaar_sup_le
@[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 
Arguments: {G : Type u_1} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] {K₀ : TopologicalSpace.PositiveCompacts G} {U : Set G} (K₁ K₂ : TopologicalSpace.Compacts G) (hU : (interior U).Nonempty)
Goal: MeasureTheory.Measure.haar.addPrehaar (↑K₀) U (K₁ ⊔ K₂) ≤
  MeasureTheory.Measure.haar.addPrehaar (↑K₀) U K₁ + MeasureTheory.Measure.haar.addPrehaar (↑K₀) U K₂
-- MeasureTheory.Measure.haar.prehaar_sup_eq
@[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 
Arguments: {G : Type u_1} [Group G] [TopologicalSpace G] [TopologicalGroup G] {K₀ : TopologicalSpace.PositiveCompacts G} {U : Set G} {K₁ K₂ : TopologicalSpace.Compacts G} (hU : (interior U).Nonempty) (h : Disjoint (K₁.carrier * U⁻¹) (K₂.carrier * U⁻¹))
Goal: MeasureTheory.Measure.haar.prehaar (↑K₀) U (K₁ ⊔ K₂) =
  MeasureTheory.Measure.haar.prehaar (↑K₀) U K₁ + MeasureTheory.Measure.haar.prehaar (↑K₀) U K₂
-- MeasureTheory.Measure.haar.addPrehaar_sup_eq
@[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 
Arguments: {G : Type u_1} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] {K₀ : TopologicalSpace.PositiveCompacts G} {U : Set G} {K₁ K₂ : TopologicalSpace.Compacts G} (hU : (interior U).Nonempty) (h : Disjoint (K₁.carrier + -U) (K₂.carrier + -U))
Goal: MeasureTheory.Measure.haar.addPrehaar (↑K₀) U (K₁ ⊔ K₂) =
  MeasureTheory.Measure.haar.addPrehaar (↑K₀) U K₁ + MeasureTheory.Measure.haar.addPrehaar (↑K₀) U K₂
-- MeasureTheory.Measure.haar.is_left_invariant_prehaar
@[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 
Arguments: {G : Type u_1} [Group G] [TopologicalSpace G] [TopologicalGroup G] {K₀ : TopologicalSpace.PositiveCompacts G} {U : Set G} (hU : (interior U).Nonempty) (g : G) (K : TopologicalSpace.Compacts G)
Goal: MeasureTheory.Measure.haar.prehaar (↑K₀) U (TopologicalSpace.Compacts.map (fun b => g * b) ⋯ K) =
  MeasureTheory.Measure.haar.prehaar (↑K₀) U K
-- MeasureTheory.Measure.haar.is_left_invariant_addPrehaar
@[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 
Arguments: {G : Type u_1} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] {K₀ : TopologicalSpace.PositiveCompacts G} {U : Set G} (hU : (interior U).Nonempty) (g : G) (K : TopologicalSpace.Compacts G)
Goal: MeasureTheory.Measure.haar.addPrehaar (↑K₀) U (TopologicalSpace.Compacts.map (fun b => g + b) ⋯ K) =
  MeasureTheory.Measure.haar.addPrehaar (↑K₀) U K
-- MeasureTheory.Measure.haar.addPrehaar_mem_addHaarProduct
@[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) 
Arguments: {G : Type u_1} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] (K₀ : TopologicalSpace.PositiveCompacts G) {U : Set G} (hU : (interior U).Nonempty)
Goal: MeasureTheory.Measure.haar.addPrehaar (↑K₀) U ∈ MeasureTheory.Measure.haar.addHaarProduct ↑K₀
-- MeasureTheory.Measure.haar.prehaar_mem_haarProduct
@[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) 
Arguments: {G : Type u_1} [Group G] [TopologicalSpace G] [TopologicalGroup G] (K₀ : TopologicalSpace.PositiveCompacts G) {U : Set G} (hU : (interior U).Nonempty)
Goal: MeasureTheory.Measure.haar.prehaar (↑K₀) U ∈ MeasureTheory.Measure.haar.haarProduct ↑K₀
-- MeasureTheory.Measure.haar.nonempty_iInter_clAddPrehaar
@[to_additive]
theorem nonempty_iInter_clPrehaar (K₀ : PositiveCompacts G) :
    (haarProduct (K₀ : Set G) ∩ ⋂ V : OpenNhdsOf (1 : G), clPrehaar K₀ V).Nonempty 
Arguments: {G : Type u_1} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] (K₀ : TopologicalSpace.PositiveCompacts G)
Goal: (MeasureTheory.Measure.haar.addHaarProduct ↑K₀ ∩ ⋂ V, MeasureTheory.Measure.haar.clAddPrehaar (↑K₀) V).Nonempty
-- MeasureTheory.Measure.haar.nonempty_iInter_clPrehaar
@[to_additive]
theorem nonempty_iInter_clPrehaar (K₀ : PositiveCompacts G) :
    (haarProduct (K₀ : Set G) ∩ ⋂ V : OpenNhdsOf (1 : G), clPrehaar K₀ V).Nonempty 
Arguments: {G : Type u_1} [Group G] [TopologicalSpace G] [TopologicalGroup G] (K₀ : TopologicalSpace.PositiveCompacts G)
Goal: (MeasureTheory.Measure.haar.haarProduct ↑K₀ ∩ ⋂ V, MeasureTheory.Measure.haar.clPrehaar (↑K₀) V).Nonempty
-- MeasureTheory.Measure.haar.chaar_mem_haarProduct
@[to_additive addCHaar_mem_addHaarProduct]
theorem chaar_mem_haarProduct (K₀ : PositiveCompacts G) : chaar K₀ ∈ haarProduct (K₀ : Set G) 
Arguments: {G : Type u_1} [Group G] [TopologicalSpace G] [TopologicalGroup G] (K₀ : TopologicalSpace.PositiveCompacts G)
Goal: MeasureTheory.Measure.haar.chaar K₀ ∈ MeasureTheory.Measure.haar.haarProduct ↑K₀
-- MeasureTheory.Measure.haar.addCHaar_mem_addHaarProduct
@[to_additive addCHaar_mem_addHaarProduct]
theorem chaar_mem_haarProduct (K₀ : PositiveCompacts G) : chaar K₀ ∈ haarProduct (K₀ : Set G) 
Arguments: {G : Type u_1} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] (K₀ : TopologicalSpace.PositiveCompacts G)
Goal: MeasureTheory.Measure.haar.addCHaar K₀ ∈ MeasureTheory.Measure.haar.addHaarProduct ↑K₀
-- MeasureTheory.Measure.haar.chaar_mem_clPrehaar
@[to_additive addCHaar_mem_clAddPrehaar]
theorem chaar_mem_clPrehaar (K₀ : PositiveCompacts G) (V : OpenNhdsOf (1 : G)) :
    chaar K₀ ∈ clPrehaar (K₀ : Set G) V 
Arguments: {G : Type u_1} [Group G] [TopologicalSpace G] [TopologicalGroup G] (K₀ : TopologicalSpace.PositiveCompacts G) (V : TopologicalSpace.OpenNhdsOf 1)
Goal: MeasureTheory.Measure.haar.chaar K₀ ∈ MeasureTheory.Measure.haar.clPrehaar (↑K₀) V