ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.Conservative.measure_mem_forall_ge_image_not_mem_eq_zero
theorem measure_mem_forall_ge_image_not_mem_eq_zero (hf : Conservative f μ)
    (hs : NullMeasurableSet s μ) (n : ℕ) :
    μ ({ x ∈ s | ∀ m ≥ n, f^[m] x ∉ s }) = 0 
Arguments: {α : Type u_2} [MeasurableSpace α] {f : α → α} {s : Set α} {μ : MeasureTheory.Measure α} (hf : MeasureTheory.Conservative f μ) (hs : MeasureTheory.NullMeasurableSet s μ) (n : ℕ)
Goal: μ {x | x ∈ s ∧ ∀ m ≥ n, f^[m] x ∉ s} = 0
-- MeasureTheory.Conservative.ae_mem_imp_frequently_image_mem
theorem ae_mem_imp_frequently_image_mem (hf : Conservative f μ) (hs : NullMeasurableSet s μ) :
    ∀ᵐ x ∂μ, x ∈ s → ∃ᶠ n in atTop, f^[n] x ∈ s 
Arguments: {α : Type u_2} [MeasurableSpace α] {f : α → α} {s : Set α} {μ : MeasureTheory.Measure α} (hf : MeasureTheory.Conservative f μ) (hs : MeasureTheory.NullMeasurableSet s μ)
Goal: ∀ᵐ (x : α) ∂μ, x ∈ s → ∃ᶠ (n : ℕ) in Filter.atTop, f^[n] x ∈ s
-- MeasureTheory.Conservative.inter_frequently_image_mem_ae_eq
theorem inter_frequently_image_mem_ae_eq (hf : Conservative f μ) (hs : NullMeasurableSet s μ) :
    (s ∩ { x | ∃ᶠ n in atTop, f^[n] x ∈ s } : Set α) =ᵐ[μ] s 
Arguments: {α : Type u_2} [MeasurableSpace α] {f : α → α} {s : Set α} {μ : MeasureTheory.Measure α} (hf : MeasureTheory.Conservative f μ) (hs : MeasureTheory.NullMeasurableSet s μ)
Goal: s ∩ {x | ∃ᶠ (n : ℕ) in Filter.atTop, f^[n] x ∈ s} =ᵐ[μ] s
-- MeasureTheory.Conservative.measure_inter_frequently_image_mem_eq
theorem measure_inter_frequently_image_mem_eq (hf : Conservative f μ) (hs : NullMeasurableSet s μ) :
    μ (s ∩ { x | ∃ᶠ n in atTop, f^[n] x ∈ s }) = μ s 
Arguments: {α : Type u_2} [MeasurableSpace α] {f : α → α} {s : Set α} {μ : MeasureTheory.Measure α} (hf : MeasureTheory.Conservative f μ) (hs : MeasureTheory.NullMeasurableSet s μ)
Goal: μ (s ∩ {x | ∃ᶠ (n : ℕ) in Filter.atTop, f^[n] x ∈ s}) = μ s
-- MeasureTheory.Conservative.ae_forall_image_mem_imp_frequently_image_mem
theorem ae_forall_image_mem_imp_frequently_image_mem (hf : Conservative f μ)
    (hs : NullMeasurableSet s μ) : ∀ᵐ x ∂μ, ∀ k, f^[k] x ∈ s → ∃ᶠ n in atTop, f^[n] x ∈ s 
Arguments: {α : Type u_2} [MeasurableSpace α] {f : α → α} {s : Set α} {μ : MeasureTheory.Measure α} (hf : MeasureTheory.Conservative f μ) (hs : MeasureTheory.NullMeasurableSet s μ)
Goal: ∀ᵐ (x : α) ∂μ, ∀ (k : ℕ), f^[k] x ∈ s → ∃ᶠ (n : ℕ) in Filter.atTop, f^[n] x ∈ s
-- MeasureTheory.Conservative.frequently_ae_mem_and_frequently_image_mem
theorem frequently_ae_mem_and_frequently_image_mem (hf : Conservative f μ)
    (hs : NullMeasurableSet s μ) (h0 : μ s ≠ 0) : ∃ᵐ x ∂μ, x ∈ s ∧ ∃ᶠ n in atTop, f^[n] x ∈ s 
Arguments: {α : Type u_2} [MeasurableSpace α] {f : α → α} {s : Set α} {μ : MeasureTheory.Measure α} (hf : MeasureTheory.Conservative f μ) (hs : MeasureTheory.NullMeasurableSet s μ) (h0 : μ s ≠ 0)
Goal: ∃ᵐ (x : α) ∂μ, x ∈ s ∧ ∃ᶠ (n : ℕ) in Filter.atTop, f^[n] x ∈ s
-- MeasureTheory.Conservative.ae_frequently_mem_of_mem_nhds
theorem ae_frequently_mem_of_mem_nhds [TopologicalSpace α] [SecondCountableTopology α]
    [OpensMeasurableSpace α] {f : α → α} {μ : Measure α} (h : Conservative f μ) :
    ∀ᵐ x ∂μ, ∀ s ∈ 𝓝 x, ∃ᶠ n in atTop, f^[n] x ∈ s 
Arguments: {α : Type u_2} [MeasurableSpace α] [TopologicalSpace α] [SecondCountableTopology α] [OpensMeasurableSpace α] {f : α → α} {μ : MeasureTheory.Measure α} (h : MeasureTheory.Conservative f μ)
Goal: ∀ᵐ (x : α) ∂μ, ∀ s ∈ 𝓝 x, ∃ᶠ (n : ℕ) in Filter.atTop, f^[n] x ∈ s
-- MeasureTheory.Conservative.iterate
protected theorem iterate (hf : Conservative f μ) (n : ℕ) : Conservative f^[n] μ 
Arguments: {α : Type u_2} [MeasurableSpace α] {f : α → α} {μ : MeasureTheory.Measure α} (hf : MeasureTheory.Conservative f μ) (n : ℕ)
Goal: MeasureTheory.Conservative f^[n] μ
-- omegaLimit_def
theorem omegaLimit_def : ω f ϕ s = ⋂ u ∈ f, closure (image2 ϕ u s) 
Arguments: {τ : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace β] (f : Filter τ) (ϕ : τ → α → β) (s : Set α)
Goal: omegaLimit f ϕ s = ⋂ u ∈ f, closure (Set.image2 ϕ u s)
-- omegaLimit_subset_of_tendsto
theorem omegaLimit_subset_of_tendsto {m : τ → τ} {f₁ f₂ : Filter τ} (hf : Tendsto m f₁ f₂) :
    ω f₁ (fun t x ↦ ϕ (m t) x) s ⊆ ω f₂ ϕ s 
Arguments: {τ : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace β] (ϕ : τ → α → β) (s : Set α) {m : τ → τ} {f₁ f₂ : Filter τ} (hf : Filter.Tendsto m f₁ f₂)
Goal: omegaLimit f₁ (fun t x => ϕ (m t) x) s ⊆ omegaLimit f₂ ϕ s
-- omegaLimit_mono_left
theorem omegaLimit_mono_left {f₁ f₂ : Filter τ} (hf : f₁ ≤ f₂) : ω f₁ ϕ s ⊆ ω f₂ ϕ s 
Arguments: {τ : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace β] (ϕ : τ → α → β) (s : Set α) {f₁ f₂ : Filter τ} (hf : f₁ ≤ f₂)
Goal: omegaLimit f₁ ϕ s ⊆ omegaLimit f₂ ϕ s
-- omegaLimit_mono_right
theorem omegaLimit_mono_right {s₁ s₂ : Set α} (hs : s₁ ⊆ s₂) : ω f ϕ s₁ ⊆ ω f ϕ s₂ 
Arguments: {τ : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace β] (f : Filter τ) (ϕ : τ → α → β) {s₁ s₂ : Set α} (hs : s₁ ⊆ s₂)
Goal: omegaLimit f ϕ s₁ ⊆ omegaLimit f ϕ s₂
-- isClosed_omegaLimit
theorem isClosed_omegaLimit : IsClosed (ω f ϕ s) 
Arguments: {τ : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace β] (f : Filter τ) (ϕ : τ → α → β) (s : Set α)
Goal: IsClosed (omegaLimit f ϕ s)
-- mapsTo_omegaLimit'
theorem mapsTo_omegaLimit' {α' β' : Type*} [TopologicalSpace β'] {f : Filter τ} {ϕ : τ → α → β}
    {ϕ' : τ → α' → β'} {ga : α → α'} {s' : Set α'} (hs : MapsTo ga s s') {gb : β → β'}
    (hg : ∀ᶠ t in f, EqOn (gb ∘ ϕ t) (ϕ' t ∘ ga) s) (hgc : Continuous gb) :
    MapsTo gb (ω f ϕ s) (ω f ϕ' s') 
Arguments: {τ : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace β] (s : Set α) {α' : Type u_5} {β' : Type u_6} [TopologicalSpace β'] {f : Filter τ} {ϕ : τ → α → β} {ϕ' : τ → α' → β'} {ga : α → α'} {s' : Set α'} (hs : Set.MapsTo ga s s') {gb : β → β'} (hg : ∀ᶠ (t : τ) in f, Set.EqOn (gb ∘ ϕ t) (ϕ' t ∘ ga) s) (hgc : Continuous gb)
Goal: Set.MapsTo gb (omegaLimit f ϕ s) (omegaLimit f ϕ' s')
-- mapsTo_omegaLimit
theorem mapsTo_omegaLimit {α' β' : Type*} [TopologicalSpace β'] {f : Filter τ} {ϕ : τ → α → β}
    {ϕ' : τ → α' → β'} {ga : α → α'} {s' : Set α'} (hs : MapsTo ga s s') {gb : β → β'}
    (hg : ∀ t x, gb (ϕ t x) = ϕ' t (ga x)) (hgc : Continuous gb) :
    MapsTo gb (ω f ϕ s) (ω f ϕ' s') 
Arguments: {τ : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace β] (s : Set α) {α' : Type u_5} {β' : Type u_6} [TopologicalSpace β'] {f : Filter τ} {ϕ : τ → α → β} {ϕ' : τ → α' → β'} {ga : α → α'} {s' : Set α'} (hs : Set.MapsTo ga s s') {gb : β → β'} (hg : ∀ (t : τ) (x : α), gb (ϕ t x) = ϕ' t (ga x)) (hgc : Continuous gb)
Goal: Set.MapsTo gb (omegaLimit f ϕ s) (omegaLimit f ϕ' s')
-- omegaLimit_image_eq
theorem omegaLimit_image_eq {α' : Type*} (ϕ : τ → α' → β) (f : Filter τ) (g : α → α') :
    ω f ϕ (g '' s) = ω f (fun t x ↦ ϕ t (g x)) s 
Arguments: {τ : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace β] (s : Set α) {α' : Type u_5} (ϕ : τ → α' → β) (f : Filter τ) (g : α → α')
Goal: omegaLimit f ϕ (g '' s) = omegaLimit f (fun t x => ϕ t (g x)) s
-- omegaLimit_preimage_subset
theorem omegaLimit_preimage_subset {α' : Type*} (ϕ : τ → α' → β) (s : Set α') (f : Filter τ)
    (g : α → α') : ω f (fun t x ↦ ϕ t (g x)) (g ⁻¹' s) ⊆ ω f ϕ s 
Arguments: {τ : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace β] {α' : Type u_5} (ϕ : τ → α' → β) (s : Set α') (f : Filter τ) (g : α → α')
Goal: omegaLimit f (fun t x => ϕ t (g x)) (g ⁻¹' s) ⊆ omegaLimit f ϕ s
-- mem_omegaLimit_iff_frequently
theorem mem_omegaLimit_iff_frequently (y : β) :
    y ∈ ω f ϕ s ↔ ∀ n ∈ 𝓝 y, ∃ᶠ t in f, (s ∩ ϕ t ⁻¹' n).Nonempty 
Arguments: {τ : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace β] (f : Filter τ) (ϕ : τ → α → β) (s : Set α) (y : β)
Goal: y ∈ omegaLimit f ϕ s ↔ ∀ n ∈ 𝓝 y, ∃ᶠ (t : τ) in f, (s ∩ ϕ t ⁻¹' n).Nonempty
-- mem_omegaLimit_iff_frequently₂
theorem mem_omegaLimit_iff_frequently₂ (y : β) :
    y ∈ ω f ϕ s ↔ ∀ n ∈ 𝓝 y, ∃ᶠ t in f, (ϕ t '' s ∩ n).Nonempty 
Arguments: {τ : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace β] (f : Filter τ) (ϕ : τ → α → β) (s : Set α) (y : β)
Goal: y ∈ omegaLimit f ϕ s ↔ ∀ n ∈ 𝓝 y, ∃ᶠ (t : τ) in f, (ϕ t '' s ∩ n).Nonempty
-- mem_omegaLimit_singleton_iff_map_cluster_point
theorem mem_omegaLimit_singleton_iff_map_cluster_point (x : α) (y : β) :
    y ∈ ω f ϕ {x} ↔ MapClusterPt y f fun t ↦ ϕ t x 
Arguments: {τ : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace β] (f : Filter τ) (ϕ : τ → α → β) (x : α) (y : β)
Goal: y ∈ omegaLimit f ϕ {x} ↔ MapClusterPt y f fun t => ϕ t x
-- omegaLimit_inter
theorem omegaLimit_inter : ω f ϕ (s₁ ∩ s₂) ⊆ ω f ϕ s₁ ∩ ω f ϕ s₂ 
Arguments: {τ : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace β] (f : Filter τ) (ϕ : τ → α → β) (s₁ s₂ : Set α)
Goal: omegaLimit f ϕ (s₁ ∩ s₂) ⊆ omegaLimit f ϕ s₁ ∩ omegaLimit f ϕ s₂
-- omegaLimit_iInter
theorem omegaLimit_iInter (p : ι → Set α) : ω f ϕ (⋂ i, p i) ⊆ ⋂ i, ω f ϕ (p i) 
Arguments: {τ : Type u_1} {α : Type u_2} {β : Type u_3} {ι : Type u_4} [TopologicalSpace β] (f : Filter τ) (ϕ : τ → α → β) (p : ι → Set α)
Goal: omegaLimit f ϕ (⋂ i, p i) ⊆ ⋂ i, omegaLimit f ϕ (p i)
-- omegaLimit_union
theorem omegaLimit_union : ω f ϕ (s₁ ∪ s₂) = ω f ϕ s₁ ∪ ω f ϕ s₂ 
Arguments: {τ : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace β] (f : Filter τ) (ϕ : τ → α → β) (s₁ s₂ : Set α)
Goal: omegaLimit f ϕ (s₁ ∪ s₂) = omegaLimit f ϕ s₁ ∪ omegaLimit f ϕ s₂
-- omegaLimit_iUnion
theorem omegaLimit_iUnion (p : ι → Set α) : ⋃ i, ω f ϕ (p i) ⊆ ω f ϕ (⋃ i, p i) 
Arguments: {τ : Type u_1} {α : Type u_2} {β : Type u_3} {ι : Type u_4} [TopologicalSpace β] (f : Filter τ) (ϕ : τ → α → β) (p : ι → Set α)
Goal: ⋃ i, omegaLimit f ϕ (p i) ⊆ omegaLimit f ϕ (⋃ i, p i)
-- omegaLimit_eq_iInter
theorem omegaLimit_eq_iInter : ω f ϕ s = ⋂ u : ↥f.sets, closure (image2 ϕ u s) 
Arguments: {τ : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace β] (f : Filter τ) (ϕ : τ → α → β) (s : Set α)
Goal: omegaLimit f ϕ s = ⋂ u, closure (Set.image2 ϕ (↑u) s)
-- omegaLimit_eq_biInter_inter
theorem omegaLimit_eq_biInter_inter {v : Set τ} (hv : v ∈ f) :
    ω f ϕ s = ⋂ u ∈ f, closure (image2 ϕ (u ∩ v) s) 
Arguments: {τ : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace β] (f : Filter τ) (ϕ : τ → α → β) (s : Set α) {v : Set τ} (hv : v ∈ f)
Goal: omegaLimit f ϕ s = ⋂ u ∈ f, closure (Set.image2 ϕ (u ∩ v) s)
-- omegaLimit_eq_iInter_inter
theorem omegaLimit_eq_iInter_inter {v : Set τ} (hv : v ∈ f) :
    ω f ϕ s = ⋂ u : ↥f.sets, closure (image2 ϕ (u ∩ v) s) 
Arguments: {τ : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace β] (f : Filter τ) (ϕ : τ → α → β) (s : Set α) {v : Set τ} (hv : v ∈ f)
Goal: omegaLimit f ϕ s = ⋂ u, closure (Set.image2 ϕ (↑u ∩ v) s)
-- omegaLimit_subset_closure_fw_image
theorem omegaLimit_subset_closure_fw_image {u : Set τ} (hu : u ∈ f) :
    ω f ϕ s ⊆ closure (image2 ϕ u s) 
Arguments: {τ : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace β] (f : Filter τ) (ϕ : τ → α → β) (s : Set α) {u : Set τ} (hu : u ∈ f)
Goal: omegaLimit f ϕ s ⊆ closure (Set.image2 ϕ u s)
-- eventually_closure_subset_of_isCompact_absorbing_of_isOpen_of_omegaLimit_subset'
theorem eventually_closure_subset_of_isCompact_absorbing_of_isOpen_of_omegaLimit_subset' {c : Set β}
    (hc₁ : IsCompact c) (hc₂ : ∃ v ∈ f, closure (image2 ϕ v s) ⊆ c) {n : Set β} (hn₁ : IsOpen n)
    (hn₂ : ω f ϕ s ⊆ n) : ∃ u ∈ f, closure (image2 ϕ u s) ⊆ n 
Arguments: {τ : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace β] (f : Filter τ) (ϕ : τ → α → β) (s : Set α) {c : Set β} (hc₁ : IsCompact c) (hc₂ : ∃ v ∈ f, closure (Set.image2 ϕ v s) ⊆ c) {n : Set β} (hn₁ : IsOpen n) (hn₂ : omegaLimit f ϕ s ⊆ n)
Goal: ∃ u ∈ f, closure (Set.image2 ϕ u s) ⊆ n
-- eventually_closure_subset_of_isCompact_absorbing_of_isOpen_of_omegaLimit_subset
theorem eventually_closure_subset_of_isCompact_absorbing_of_isOpen_of_omegaLimit_subset [T2Space β]
    {c : Set β} (hc₁ : IsCompact c) (hc₂ : ∀ᶠ t in f, MapsTo (ϕ t) s c) {n : Set β} (hn₁ : IsOpen n)
    (hn₂ : ω f ϕ s ⊆ n) : ∃ u ∈ f, closure (image2 ϕ u s) ⊆ n 
Arguments: {τ : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace β] (f : Filter τ) (ϕ : τ → α → β) (s : Set α) [T2Space β] {c : Set β} (hc₁ : IsCompact c) (hc₂ : ∀ᶠ (t : τ) in f, Set.MapsTo (ϕ t) s c) {n : Set β} (hn₁ : IsOpen n) (hn₂ : omegaLimit f ϕ s ⊆ n)
Goal: ∃ u ∈ f, closure (Set.image2 ϕ u s) ⊆ n
-- eventually_mapsTo_of_isCompact_absorbing_of_isOpen_of_omegaLimit_subset
theorem eventually_mapsTo_of_isCompact_absorbing_of_isOpen_of_omegaLimit_subset [T2Space β]
    {c : Set β} (hc₁ : IsCompact c) (hc₂ : ∀ᶠ t in f, MapsTo (ϕ t) s c) {n : Set β} (hn₁ : IsOpen n)
    (hn₂ : ω f ϕ s ⊆ n) : ∀ᶠ t in f, MapsTo (ϕ t) s n 
Arguments: {τ : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace β] (f : Filter τ) (ϕ : τ → α → β) (s : Set α) [T2Space β] {c : Set β} (hc₁ : IsCompact c) (hc₂ : ∀ᶠ (t : τ) in f, Set.MapsTo (ϕ t) s c) {n : Set β} (hn₁ : IsOpen n) (hn₂ : omegaLimit f ϕ s ⊆ n)
Goal: ∀ᶠ (t : τ) in f, Set.MapsTo (ϕ t) s n
-- eventually_closure_subset_of_isOpen_of_omegaLimit_subset
theorem eventually_closure_subset_of_isOpen_of_omegaLimit_subset [CompactSpace β] {v : Set β}
    (hv₁ : IsOpen v) (hv₂ : ω f ϕ s ⊆ v) : ∃ u ∈ f, closure (image2 ϕ u s) ⊆ v 
Arguments: {τ : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace β] (f : Filter τ) (ϕ : τ → α → β) (s : Set α) [CompactSpace β] {v : Set β} (hv₁ : IsOpen v) (hv₂ : omegaLimit f ϕ s ⊆ v)
Goal: ∃ u ∈ f, closure (Set.image2 ϕ u s) ⊆ v
-- eventually_mapsTo_of_isOpen_of_omegaLimit_subset
theorem eventually_mapsTo_of_isOpen_of_omegaLimit_subset [CompactSpace β] {v : Set β}
    (hv₁ : IsOpen v) (hv₂ : ω f ϕ s ⊆ v) : ∀ᶠ t in f, MapsTo (ϕ t) s v 
Arguments: {τ : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace β] (f : Filter τ) (ϕ : τ → α → β) (s : Set α) [CompactSpace β] {v : Set β} (hv₁ : IsOpen v) (hv₂ : omegaLimit f ϕ s ⊆ v)
Goal: ∀ᶠ (t : τ) in f, Set.MapsTo (ϕ t) s v
-- nonempty_omegaLimit_of_isCompact_absorbing
theorem nonempty_omegaLimit_of_isCompact_absorbing [NeBot f] {c : Set β} (hc₁ : IsCompact c)
    (hc₂ : ∃ v ∈ f, closure (image2 ϕ v s) ⊆ c) (hs : s.Nonempty) : (ω f ϕ s).Nonempty 
Arguments: {τ : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace β] (f : Filter τ) (ϕ : τ → α → β) (s : Set α) [f.NeBot] {c : Set β} (hc₁ : IsCompact c) (hc₂ : ∃ v ∈ f, closure (Set.image2 ϕ v s) ⊆ c) (hs : s.Nonempty)
Goal: (omegaLimit f ϕ s).Nonempty
-- nonempty_omegaLimit
theorem nonempty_omegaLimit [CompactSpace β] [NeBot f] (hs : s.Nonempty) : (ω f ϕ s).Nonempty 
Arguments: {τ : Type u_1} {α : Type u_2} {β : Type u_3} [TopologicalSpace β] (f : Filter τ) (ϕ : τ → α → β) (s : Set α) [CompactSpace β] [f.NeBot] (hs : s.Nonempty)
Goal: (omegaLimit f ϕ s).Nonempty
-- Flow.isInvariant_omegaLimit
theorem isInvariant_omegaLimit (hf : ∀ t, Tendsto (t + ·) f f) : IsInvariant ϕ (ω f ϕ s) 
Arguments: {τ : Type u_1} [TopologicalSpace τ] [AddMonoid τ] [ContinuousAdd τ] {α : Type u_2} [TopologicalSpace α] (f : Filter τ) (ϕ : Flow τ α) (s : Set α) (hf : ∀ (t : τ), Filter.Tendsto (fun x => t + x) f f)
Goal: IsInvariant ϕ.toFun (omegaLimit f ϕ.toFun s)
-- Flow.omegaLimit_image_subset
theorem omegaLimit_image_subset (t : τ) (ht : Tendsto (· + t) f f) :
    ω f ϕ (ϕ t '' s) ⊆ ω f ϕ s 
Arguments: {τ : Type u_1} [TopologicalSpace τ] [AddMonoid τ] [ContinuousAdd τ] {α : Type u_2} [TopologicalSpace α] (f : Filter τ) (ϕ : Flow τ α) (s : Set α) (t : τ) (ht : Filter.Tendsto (fun x => x + t) f f)
Goal: omegaLimit f ϕ.toFun (ϕ.toFun t '' s) ⊆ omegaLimit f ϕ.toFun s
-- Flow.omegaLimit_image_eq
@[simp]
theorem omegaLimit_image_eq (hf : ∀ t, Tendsto (· + t) f f) (t : τ) : ω f ϕ (ϕ t '' s) = ω f ϕ s 
Arguments: {τ : Type u_1} [TopologicalSpace τ] [AddCommGroup τ] [TopologicalAddGroup τ] {α : Type u_2} [TopologicalSpace α] (f : Filter τ) (ϕ : Flow τ α) (s : Set α) (hf : ∀ (t : τ), Filter.Tendsto (fun x => x + t) f f) (t : τ)
Goal: omegaLimit f ϕ.toFun (ϕ.toFun t '' s) = omegaLimit f ϕ.toFun s
-- Flow.omegaLimit_omegaLimit
theorem omegaLimit_omegaLimit (hf : ∀ t, Tendsto (t + ·) f f) : ω f ϕ (ω f ϕ s) ⊆ ω f ϕ s 
Arguments: {τ : Type u_1} [TopologicalSpace τ] [AddCommGroup τ] [TopologicalAddGroup τ] {α : Type u_2} [TopologicalSpace α] (f : Filter τ) (ϕ : Flow τ α) (s : Set α) (hf : ∀ (t : τ), Filter.Tendsto (fun x => t + x) f f)
Goal: omegaLimit f ϕ.toFun (omegaLimit f ϕ.toFun s) ⊆ omegaLimit f ϕ.toFun s
-- FiniteField.card_image_polynomial_eval
theorem card_image_polynomial_eval [DecidableEq R] [Fintype R] {p : R[X]} (hp : 0 < p.degree) :
    Fintype.card R ≤ natDegree p * (univ.image fun x => eval x p).card 
Arguments: {R : Type u_2} [CommRing R] [IsDomain R] [DecidableEq R] [Fintype R] {p : R[X]} (hp : 0 < p.degree)
Goal: Fintype.card R ≤ p.natDegree * (Finset.image (fun x => Polynomial.eval x p) Finset.univ).card
-- FiniteField.exists_root_sum_quadratic
theorem exists_root_sum_quadratic [Fintype R] {f g : R[X]} (hf2 : degree f = 2) (hg2 : degree g = 2)
    (hR : Fintype.card R % 2 = 1) : ∃ a b, f.eval a + g.eval b = 0 
Arguments: {R : Type u_2} [CommRing R] [IsDomain R] [Fintype R] {f g : R[X]} (hf2 : f.degree = 2) (hg2 : g.degree = 2) (hR : Fintype.card R % 2 = 1)
Goal: ∃ a b, Polynomial.eval a f + Polynomial.eval b g = 0
-- FiniteField.prod_univ_units_id_eq_neg_one
theorem prod_univ_units_id_eq_neg_one [CommRing K] [IsDomain K] [Fintype Kˣ] :
    ∏ x : Kˣ, x = (-1 : Kˣ) 
Arguments: {K : Type u_1} [CommRing K] [IsDomain K] [Fintype Kˣ]
Goal: ∏ x : Kˣ, x = -1
-- FiniteField.card_cast_subgroup_card_ne_zero
theorem card_cast_subgroup_card_ne_zero [Ring K] [NoZeroDivisors K] [Nontrivial K]
    (G : Subgroup Kˣ) [Fintype G] : (Fintype.card G : K) ≠ 0 
Arguments: {K : Type u_1} [Ring K] [NoZeroDivisors K] [Nontrivial K] (G : Subgroup Kˣ) [Fintype ↥G]
Goal: ↑(Fintype.card ↥G) ≠ 0
-- FiniteField.sum_subgroup_units_eq_zero
theorem sum_subgroup_units_eq_zero [Ring K] [NoZeroDivisors K]
    {G : Subgroup Kˣ} [Fintype G] (hg : G ≠ ⊥) :
    ∑ x : G, (x.val : K) = 0 
Arguments: {K : Type u_1} [Ring K] [NoZeroDivisors K] {G : Subgroup Kˣ} [Fintype ↥G] (hg : G ≠ ⊥)
Goal: ∑ x : ↥G, ↑↑x = 0
-- FiniteField.sum_subgroup_units
@[simp]
theorem sum_subgroup_units [Ring K] [NoZeroDivisors K]
    {G : Subgroup Kˣ} [Fintype G] [Decidable (G = ⊥)] :
    ∑ x : G, (x.val : K) = if G = ⊥ then 1 else 0 
Arguments: {K : Type u_1} [Ring K] [NoZeroDivisors K] {G : Subgroup Kˣ} [Fintype ↥G] [Decidable (G = ⊥)]
Goal: ∑ x : ↥G, ↑↑x = if G = ⊥ then 1 else 0
-- FiniteField.sum_subgroup_pow_eq_zero
@[simp]
theorem sum_subgroup_pow_eq_zero [CommRing K] [NoZeroDivisors K]
    {G : Subgroup Kˣ} [Fintype G] {k : ℕ} (k_pos : k ≠ 0) (k_lt_card_G : k < Fintype.card G) :
    ∑ x : G, ((x : Kˣ) : K) ^ k = 0 
Arguments: {K : Type u_1} [CommRing K] [NoZeroDivisors K] {G : Subgroup Kˣ} [Fintype ↥G] {k : ℕ} (k_pos : k ≠ 0) (k_lt_card_G : k < Fintype.card ↥G)
Goal: ∑ x : ↥G, ↑↑x ^ k = 0
-- FiniteField.pow_card_sub_one_eq_one
theorem pow_card_sub_one_eq_one (a : K) (ha : a ≠ 0) : a ^ (q - 1) = 1 
Arguments: {K : Type u_1} [GroupWithZero K] [Fintype K] (a : K) (ha : a ≠ 0)
Goal: a ^ (q - 1) = 1
-- FiniteField.pow_card
theorem pow_card (a : K) : a ^ q = a 
Arguments: {K : Type u_1} [GroupWithZero K] [Fintype K] (a : K)
Goal: a ^ q = a
-- FiniteField.pow_card_pow
theorem pow_card_pow (n : ℕ) (a : K) : a ^ q ^ n = a 
Arguments: {K : Type u_1} [GroupWithZero K] [Fintype K] (n : ℕ) (a : K)
Goal: a ^ q ^ n = a
-- FiniteField.card
theorem card (p : ℕ) [CharP K p] : ∃ n : ℕ+, Nat.Prime p ∧ q = p ^ (n : ℕ) 
Arguments: (K : Type u_1) [Field K] [Fintype K] (p : ℕ) [CharP K p]
Goal: ∃ n, Nat.Prime p ∧ q = p ^ ↑n
-- FiniteField.card'
theorem card' : ∃ (p : ℕ) (n : ℕ+), Nat.Prime p ∧ Fintype.card K = p ^ (n : ℕ) 
Arguments: (K : Type u_1) [Field K] [Fintype K]
Goal: ∃ p n, Nat.Prime p ∧ q = p ^ ↑n
-- FiniteField.cast_card_eq_zero
theorem cast_card_eq_zero : (q : K) = 0 
Arguments: (K : Type u_1) [Field K] [Fintype K]
Goal: ↑q = 0
-- FiniteField.forall_pow_eq_one_iff
theorem forall_pow_eq_one_iff (i : ℕ) : (∀ x : Kˣ, x ^ i = 1) ↔ q - 1 ∣ i 
Arguments: (K : Type u_1) [Field K] [Fintype K] (i : ℕ)
Goal: (∀ (x : Kˣ), x ^ i = 1) ↔ q - 1 ∣ i
-- FiniteField.sum_pow_units
theorem sum_pow_units [DecidableEq K] (i : ℕ) :
    (∑ x : Kˣ, (x ^ i : K)) = if q - 1 ∣ i then -1 else 0 
Arguments: (K : Type u_1) [Field K] [Fintype K] [DecidableEq K] (i : ℕ)
Goal: ∑ x : Kˣ, ↑x ^ i = if q - 1 ∣ i then -1 else 0
-- FiniteField.sum_pow_lt_card_sub_one
theorem sum_pow_lt_card_sub_one (i : ℕ) (h : i < q - 1) : ∑ x : K, x ^ i = 0 
Arguments: (K : Type u_1) [Field K] [Fintype K] (i : ℕ) (h : i < q - 1)
Goal: ∑ x : K, x ^ i = 0
-- FiniteField.X_pow_card_sub_X_natDegree_eq
theorem X_pow_card_sub_X_natDegree_eq (hp : 1 < p) : (X ^ p - X : K'[X]).natDegree = p 
Arguments: (K' : Type u_3) [Field K'] {p : ℕ} (hp : 1 < p)
Goal: (Polynomial.X ^ p - Polynomial.X).natDegree = p
-- FiniteField.X_pow_card_pow_sub_X_natDegree_eq
theorem X_pow_card_pow_sub_X_natDegree_eq (hn : n ≠ 0) (hp : 1 < p) :
    (X ^ p ^ n - X : K'[X]).natDegree = p ^ n 
Arguments: (K' : Type u_3) [Field K'] {p n : ℕ} (hn : n ≠ 0) (hp : 1 < p)
Goal: (Polynomial.X ^ p ^ n - Polynomial.X).natDegree = p ^ n
-- FiniteField.X_pow_card_sub_X_ne_zero
theorem X_pow_card_sub_X_ne_zero (hp : 1 < p) : (X ^ p - X : K'[X]) ≠ 0 
Arguments: (K' : Type u_3) [Field K'] {p : ℕ} (hp : 1 < p)
Goal: Polynomial.X ^ p - Polynomial.X ≠ 0
-- FiniteField.X_pow_card_pow_sub_X_ne_zero
theorem X_pow_card_pow_sub_X_ne_zero (hn : n ≠ 0) (hp : 1 < p) : (X ^ p ^ n - X : K'[X]) ≠ 0 
Arguments: (K' : Type u_3) [Field K'] {p n : ℕ} (hn : n ≠ 0) (hp : 1 < p)
Goal: Polynomial.X ^ p ^ n - Polynomial.X ≠ 0
-- FiniteField.roots_X_pow_card_sub_X
theorem roots_X_pow_card_sub_X : roots (X ^ q - X : K[X]) = Finset.univ.val 
Arguments: (K : Type u_1) [Field K] [Fintype K]
Goal: (Polynomial.X ^ q - Polynomial.X).roots = Finset.univ.val
-- FiniteField.frobenius_pow
theorem frobenius_pow {p : ℕ} [Fact p.Prime] [CharP K p] {n : ℕ} (hcard : q = p ^ n) :
    frobenius K p ^ n = 1 
Arguments: {K : Type u_1} [Field K] [Fintype K] {p : ℕ} [Fact (Nat.Prime p)] [CharP K p] {n : ℕ} (hcard : q = p ^ n)
Goal: frobenius K p ^ n = 1
-- FiniteField.expand_card
theorem expand_card (f : K[X]) : expand K q f = f ^ q 
Arguments: {K : Type u_1} [Field K] [Fintype K] (f : K[X])
Goal: (Polynomial.expand K q) f = f ^ q
-- ZMod.sq_add_sq
theorem sq_add_sq (p : ℕ) [hp : Fact p.Prime] (x : ZMod p) : ∃ a b : ZMod p, a ^ 2 + b ^ 2 = x 
Arguments: (p : ℕ) [hp : Fact (Nat.Prime p)] (x : ZMod p)
Goal: ∃ a b, a ^ 2 + b ^ 2 = x
-- Nat.sq_add_sq_zmodEq
theorem Nat.sq_add_sq_zmodEq (p : ℕ) [Fact p.Prime] (x : ℤ) :
    ∃ a b : ℕ, a ≤ p / 2 ∧ b ≤ p / 2 ∧ (a : ℤ) ^ 2 + (b : ℤ) ^ 2 ≡ x [ZMOD p] 
Arguments: (p : ℕ) [Fact (Nat.Prime p)] (x : ℤ)
Goal: ∃ a b, a ≤ p / 2 ∧ b ≤ p / 2 ∧ ↑a ^ 2 + ↑b ^ 2 ≡ x [ZMOD ↑p]
-- Nat.sq_add_sq_modEq
theorem Nat.sq_add_sq_modEq (p : ℕ) [Fact p.Prime] (x : ℕ) :
    ∃ a b : ℕ, a ≤ p / 2 ∧ b ≤ p / 2 ∧ a ^ 2 + b ^ 2 ≡ x [MOD p] 
Arguments: (p : ℕ) [Fact (Nat.Prime p)] (x : ℕ)
Goal: ∃ a b, a ≤ p / 2 ∧ b ≤ p / 2 ∧ a ^ 2 + b ^ 2 ≡ x [MOD p]
-- CharP.sq_add_sq
theorem sq_add_sq (R : Type*) [CommRing R] [IsDomain R] (p : ℕ) [NeZero p] [CharP R p] (x : ℤ) :
    ∃ a b : ℕ, ((a : R) ^ 2 + (b : R) ^ 2) = x 
Arguments: (R : Type u_3) [CommRing R] [IsDomain R] (p : ℕ) [NeZero p] [CharP R p] (x : ℤ)
Goal: ∃ a b, ↑a ^ 2 + ↑b ^ 2 = ↑x
-- ZMod.pow_totient
@[simp]
theorem ZMod.pow_totient {n : ℕ} (x : (ZMod n)ˣ) : x ^ φ n = 1 
Arguments: {n : ℕ} (x : (ZMod n)ˣ)
Goal: x ^ φ n = 1
-- Nat.ModEq.pow_totient
theorem Nat.ModEq.pow_totient {x n : ℕ} (h : Nat.Coprime x n) : x ^ φ n ≡ 1 [MOD n] 
Arguments: {x n : ℕ} (h : x.Coprime n)
Goal: x ^ φ n ≡ 1 [MOD n]
-- card_eq_pow_finrank
theorem card_eq_pow_finrank [Fintype V] : Fintype.card V = q ^ FiniteDimensional.finrank K V 
Arguments: {K : Type u_1} {V : Type u_3} [Fintype K] [DivisionRing K] [AddCommGroup V] [Module K V] [Fintype V]
Goal: Fintype.card V = q ^ FiniteDimensional.finrank K V
-- ZMod.pow_card
@[simp]
theorem pow_card {p : ℕ} [Fact p.Prime] (x : ZMod p) : x ^ p = x 
Arguments: {p : ℕ} [Fact (Nat.Prime p)] (x : ZMod p)
Goal: x ^ p = x
-- ZMod.pow_card_pow
@[simp]
theorem pow_card_pow {n p : ℕ} [Fact p.Prime] (x : ZMod p) : x ^ p ^ n = x 
Arguments: {n p : ℕ} [Fact (Nat.Prime p)] (x : ZMod p)
Goal: x ^ p ^ n = x
-- ZMod.frobenius_zmod
@[simp]
theorem frobenius_zmod (p : ℕ) [Fact p.Prime] : frobenius (ZMod p) p = RingHom.id _ 
Arguments: (p : ℕ) [Fact (Nat.Prime p)]
Goal: frobenius (ZMod p) p = RingHom.id (ZMod p)