ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- denseRange_vadd
@[to_additive]
theorem denseRange_smul [IsMinimal M α] (x : α) : DenseRange fun c : M ↦ c • x 
Arguments: (M : Type u_1) {α : Type u_3} [AddMonoid M] [TopologicalSpace α] [AddAction M α] [AddAction.IsMinimal M α] (x : α)
Goal: DenseRange fun c => c +ᵥ x
-- denseRange_smul
@[to_additive]
theorem denseRange_smul [IsMinimal M α] (x : α) : DenseRange fun c : M ↦ c • x 
Arguments: (M : Type u_1) {α : Type u_3} [Monoid M] [TopologicalSpace α] [MulAction M α] [MulAction.IsMinimal M α] (x : α)
Goal: DenseRange fun c => c • x
-- IsOpen.exists_smul_mem
@[to_additive]
theorem IsOpen.exists_smul_mem [IsMinimal M α] (x : α) {U : Set α} (hUo : IsOpen U)
    (hne : U.Nonempty) : ∃ c : M, c • x ∈ U 
Arguments: (M : Type u_1) {α : Type u_3} [Monoid M] [TopologicalSpace α] [MulAction M α] [MulAction.IsMinimal M α] (x : α) {U : Set α} (hUo : IsOpen U) (hne : U.Nonempty)
Goal: ∃ c, c • x ∈ U
-- IsOpen.exists_vadd_mem
@[to_additive]
theorem IsOpen.exists_smul_mem [IsMinimal M α] (x : α) {U : Set α} (hUo : IsOpen U)
    (hne : U.Nonempty) : ∃ c : M, c • x ∈ U 
Arguments: (M : Type u_1) {α : Type u_3} [AddMonoid M] [TopologicalSpace α] [AddAction M α] [AddAction.IsMinimal M α] (x : α) {U : Set α} (hUo : IsOpen U) (hne : U.Nonempty)
Goal: ∃ c, c +ᵥ x ∈ U
-- IsOpen.iUnion_preimage_smul
@[to_additive]
theorem IsOpen.iUnion_preimage_smul [IsMinimal M α] {U : Set α} (hUo : IsOpen U)
    (hne : U.Nonempty) : ⋃ c : M, (c • ·) ⁻¹' U = univ 
Arguments: (M : Type u_1) {α : Type u_3} [Monoid M] [TopologicalSpace α] [MulAction M α] [MulAction.IsMinimal M α] {U : Set α} (hUo : IsOpen U) (hne : U.Nonempty)
Goal: ⋃ c, (fun x => c • x) ⁻¹' U = Set.univ
-- IsOpen.iUnion_preimage_vadd
@[to_additive]
theorem IsOpen.iUnion_preimage_smul [IsMinimal M α] {U : Set α} (hUo : IsOpen U)
    (hne : U.Nonempty) : ⋃ c : M, (c • ·) ⁻¹' U = univ 
Arguments: (M : Type u_1) {α : Type u_3} [AddMonoid M] [TopologicalSpace α] [AddAction M α] [AddAction.IsMinimal M α] {U : Set α} (hUo : IsOpen U) (hne : U.Nonempty)
Goal: ⋃ c, (fun x => c +ᵥ x) ⁻¹' U = Set.univ
-- IsOpen.iUnion_vadd
@[to_additive]
theorem IsOpen.iUnion_smul [IsMinimal G α] {U : Set α} (hUo : IsOpen U) (hne : U.Nonempty) :
    ⋃ g : G, g • U = univ 
Arguments: (G : Type u_2) {α : Type u_3} [AddGroup G] [TopologicalSpace α] [AddAction G α] [AddAction.IsMinimal G α] {U : Set α} (hUo : IsOpen U) (hne : U.Nonempty)
Goal: ⋃ g, g +ᵥ U = Set.univ
-- IsOpen.iUnion_smul
@[to_additive]
theorem IsOpen.iUnion_smul [IsMinimal G α] {U : Set α} (hUo : IsOpen U) (hne : U.Nonempty) :
    ⋃ g : G, g • U = univ 
Arguments: (G : Type u_2) {α : Type u_3} [Group G] [TopologicalSpace α] [MulAction G α] [MulAction.IsMinimal G α] {U : Set α} (hUo : IsOpen U) (hne : U.Nonempty)
Goal: ⋃ g, g • U = Set.univ
-- IsCompact.exists_finite_cover_smul
@[to_additive]
theorem IsCompact.exists_finite_cover_smul [IsMinimal G α] [ContinuousConstSMul G α]
    {K U : Set α} (hK : IsCompact K) (hUo : IsOpen U) (hne : U.Nonempty) :
    ∃ I : Finset G, K ⊆ ⋃ g ∈ I, g • U 
Arguments: (G : Type u_2) {α : Type u_3} [Group G] [TopologicalSpace α] [MulAction G α] [MulAction.IsMinimal G α] [ContinuousConstSMul G α] {K U : Set α} (hK : IsCompact K) (hUo : IsOpen U) (hne : U.Nonempty)
Goal: ∃ I, K ⊆ ⋃ g ∈ I, g • U
-- IsCompact.exists_finite_cover_vadd
@[to_additive]
theorem IsCompact.exists_finite_cover_smul [IsMinimal G α] [ContinuousConstSMul G α]
    {K U : Set α} (hK : IsCompact K) (hUo : IsOpen U) (hne : U.Nonempty) :
    ∃ I : Finset G, K ⊆ ⋃ g ∈ I, g • U 
Arguments: (G : Type u_2) {α : Type u_3} [AddGroup G] [TopologicalSpace α] [AddAction G α] [AddAction.IsMinimal G α] [ContinuousConstVAdd G α] {K U : Set α} (hK : IsCompact K) (hUo : IsOpen U) (hne : U.Nonempty)
Goal: ∃ I, K ⊆ ⋃ g ∈ I, g +ᵥ U
-- dense_of_nonempty_vadd_invariant
@[to_additive]
theorem dense_of_nonempty_smul_invariant [IsMinimal M α] {s : Set α} (hne : s.Nonempty)
    (hsmul : ∀ c : M, c • s ⊆ s) : Dense s 
Arguments: (M : Type u_1) {α : Type u_3} [AddMonoid M] [TopologicalSpace α] [AddAction M α] [AddAction.IsMinimal M α] {s : Set α} (hne : s.Nonempty) (hsmul : ∀ (c : M), c +ᵥ s ⊆ s)
Goal: Dense s
-- dense_of_nonempty_smul_invariant
@[to_additive]
theorem dense_of_nonempty_smul_invariant [IsMinimal M α] {s : Set α} (hne : s.Nonempty)
    (hsmul : ∀ c : M, c • s ⊆ s) : Dense s 
Arguments: (M : Type u_1) {α : Type u_3} [Monoid M] [TopologicalSpace α] [MulAction M α] [MulAction.IsMinimal M α] {s : Set α} (hne : s.Nonempty) (hsmul : ∀ (c : M), c • s ⊆ s)
Goal: Dense s
-- eq_empty_or_univ_of_vadd_invariant_closed
@[to_additive]
theorem eq_empty_or_univ_of_smul_invariant_closed [IsMinimal M α] {s : Set α} (hs : IsClosed s)
    (hsmul : ∀ c : M, c • s ⊆ s) : s = ∅ ∨ s = univ 
Arguments: (M : Type u_1) {α : Type u_3} [AddMonoid M] [TopologicalSpace α] [AddAction M α] [AddAction.IsMinimal M α] {s : Set α} (hs : IsClosed s) (hsmul : ∀ (c : M), c +ᵥ s ⊆ s)
Goal: s = ∅ ∨ s = Set.univ
-- eq_empty_or_univ_of_smul_invariant_closed
@[to_additive]
theorem eq_empty_or_univ_of_smul_invariant_closed [IsMinimal M α] {s : Set α} (hs : IsClosed s)
    (hsmul : ∀ c : M, c • s ⊆ s) : s = ∅ ∨ s = univ 
Arguments: (M : Type u_1) {α : Type u_3} [Monoid M] [TopologicalSpace α] [MulAction M α] [MulAction.IsMinimal M α] {s : Set α} (hs : IsClosed s) (hsmul : ∀ (c : M), c • s ⊆ s)
Goal: s = ∅ ∨ s = Set.univ
-- isMinimal_iff_closed_smul_invariant
@[to_additive]
theorem isMinimal_iff_closed_smul_invariant [ContinuousConstSMul M α] :
    IsMinimal M α ↔ ∀ s : Set α, IsClosed s → (∀ c : M, c • s ⊆ s) → s = ∅ ∨ s = univ 
Arguments: (M : Type u_1) {α : Type u_3} [Monoid M] [TopologicalSpace α] [MulAction M α] [ContinuousConstSMul M α]
Goal: MulAction.IsMinimal M α ↔ ∀ (s : Set α), IsClosed s → (∀ (c : M), c • s ⊆ s) → s = ∅ ∨ s = Set.univ
-- isMinimal_iff_closed_vadd_invariant
@[to_additive]
theorem isMinimal_iff_closed_smul_invariant [ContinuousConstSMul M α] :
    IsMinimal M α ↔ ∀ s : Set α, IsClosed s → (∀ c : M, c • s ⊆ s) → s = ∅ ∨ s = univ 
Arguments: (M : Type u_1) {α : Type u_3} [AddMonoid M] [TopologicalSpace α] [AddAction M α] [ContinuousConstVAdd M α]
Goal: AddAction.IsMinimal M α ↔ ∀ (s : Set α), IsClosed s → (∀ (c : M), c +ᵥ s ⊆ s) → s = ∅ ∨ s = Set.univ
-- Function.isFixedPt_id
theorem isFixedPt_id (x : α) : IsFixedPt id x 
Arguments: {α : Type u} (x : α)
Goal: Function.IsFixedPt id x
-- Function.IsFixedPt.eq
protected theorem eq (hf : IsFixedPt f x) : f x = x 
Arguments: {α : Type u} {f : α → α} {x : α} (hf : Function.IsFixedPt f x)
Goal: f x = x
-- Function.IsFixedPt.comp
protected theorem comp (hf : IsFixedPt f x) (hg : IsFixedPt g x) : IsFixedPt (f ∘ g) x 
Arguments: {α : Type u} {f g : α → α} {x : α} (hf : Function.IsFixedPt f x) (hg : Function.IsFixedPt g x)
Goal: Function.IsFixedPt (f ∘ g) x
-- Function.IsFixedPt.iterate
protected theorem iterate (hf : IsFixedPt f x) (n : ℕ) : IsFixedPt f^[n] x 
Arguments: {α : Type u} {f : α → α} {x : α} (hf : Function.IsFixedPt f x) (n : ℕ)
Goal: Function.IsFixedPt f^[n] x
-- Function.IsFixedPt.left_of_comp
theorem left_of_comp (hfg : IsFixedPt (f ∘ g) x) (hg : IsFixedPt g x) : IsFixedPt f x 
Arguments: {α : Type u} {f g : α → α} {x : α} (hfg : Function.IsFixedPt (f ∘ g) x) (hg : Function.IsFixedPt g x)
Goal: Function.IsFixedPt f x
-- Function.IsFixedPt.to_leftInverse
theorem to_leftInverse (hf : IsFixedPt f x) (h : LeftInverse g f) : IsFixedPt g x 
Arguments: {α : Type u} {f g : α → α} {x : α} (hf : Function.IsFixedPt f x) (h : Function.LeftInverse g f)
Goal: Function.IsFixedPt g x
-- Function.IsFixedPt.map
protected theorem map {x : α} (hx : IsFixedPt fa x) {g : α → β} (h : Semiconj g fa fb) :
    IsFixedPt fb (g x) 
Arguments: {α : Type u} {β : Type v} {fa : α → α} {fb : β → β} {x : α} (hx : Function.IsFixedPt fa x) {g : α → β} (h : Function.Semiconj g fa fb)
Goal: Function.IsFixedPt fb (g x)
-- Function.IsFixedPt.apply
protected theorem apply {x : α} (hx : IsFixedPt f x) : IsFixedPt f (f x) 
Arguments: {α : Type u} {f : α → α} {x : α} (hx : Function.IsFixedPt f x)
Goal: Function.IsFixedPt f (f x)
-- Function.IsFixedPt.preimage_iterate
theorem preimage_iterate {s : Set α} (h : IsFixedPt (Set.preimage f) s) (n : ℕ) :
    IsFixedPt (Set.preimage f^[n]) s 
Arguments: {α : Type u} {f : α → α} {s : Set α} (h : Function.IsFixedPt (Set.preimage f) s) (n : ℕ)
Goal: Function.IsFixedPt (Set.preimage f^[n]) s
-- Function.IsFixedPt.image_iterate
lemma image_iterate {s : Set α} (h : IsFixedPt (Set.image f) s) (n : ℕ) :
    IsFixedPt (Set.image f^[n]) s 
Arguments: {α : Type u} {f : α → α} {s : Set α} (h : Function.IsFixedPt (Set.image f) s) (n : ℕ)
Goal: Function.IsFixedPt (Set.image f^[n]) s
-- Function.IsFixedPt.equiv_symm
protected theorem equiv_symm (h : IsFixedPt e x) : IsFixedPt e.symm x 
Arguments: {α : Type u} {x : α} {e : Equiv.Perm α} (h : Function.IsFixedPt (⇑e) x)
Goal: Function.IsFixedPt (⇑(Equiv.symm e)) x
-- Function.IsFixedPt.perm_inv
protected theorem perm_inv (h : IsFixedPt e x) : IsFixedPt (⇑e⁻¹) x 
Arguments: {α : Type u} {x : α} {e : Equiv.Perm α} (h : Function.IsFixedPt (⇑e) x)
Goal: Function.IsFixedPt (⇑e⁻¹) x
-- Function.IsFixedPt.perm_pow
protected theorem perm_pow (h : IsFixedPt e x) (n : ℕ) : IsFixedPt (⇑(e ^ n)) x 
Arguments: {α : Type u} {x : α} {e : Equiv.Perm α} (h : Function.IsFixedPt (⇑e) x) (n : ℕ)
Goal: Function.IsFixedPt (⇑(e ^ n)) x
-- Function.IsFixedPt.perm_zpow
protected theorem perm_zpow (h : IsFixedPt e x) : ∀ n : ℤ, IsFixedPt (⇑(e ^ n)) x
  
Arguments: {α : Type u} {x : α} {e : Equiv.Perm α} (h : Function.IsFixedPt (⇑e) x) (n : ℤ)
Goal: Function.IsFixedPt (⇑(e ^ n)) x
-- Function.Injective.isFixedPt_apply_iff
@[simp]
theorem Injective.isFixedPt_apply_iff (hf : Injective f) {x : α} :
    IsFixedPt f (f x) ↔ IsFixedPt f x 
Arguments: {α : Type u} {f : α → α} (hf : Function.Injective f) {x : α}
Goal: Function.IsFixedPt f (f x) ↔ Function.IsFixedPt f x
-- Function.mem_fixedPoints
@[simp]
theorem mem_fixedPoints : x ∈ fixedPoints f ↔ IsFixedPt f x 
Arguments: {α : Type u} {f : α → α} {x : α}
Goal: x ∈ Function.fixedPoints f ↔ Function.IsFixedPt f x
-- Function.mem_fixedPoints_iff
theorem mem_fixedPoints_iff {α : Type*} {f : α → α} {x : α} : x ∈ fixedPoints f ↔ f x = x 
Arguments: {α : Type u_1} {f : α → α} {x : α}
Goal: x ∈ Function.fixedPoints f ↔ f x = x
-- Function.fixedPoints_id
@[simp]
theorem fixedPoints_id : fixedPoints (@id α) = Set.univ 
Arguments: {α : Type u}
Goal: Function.fixedPoints id = Set.univ
-- Function.fixedPoints_subset_range
theorem fixedPoints_subset_range : fixedPoints f ⊆ Set.range f 
Arguments: {α : Type u} {f : α → α}
Goal: Function.fixedPoints f ⊆ Set.range f
-- Function.Semiconj.mapsTo_fixedPoints
theorem Semiconj.mapsTo_fixedPoints {g : α → β} (h : Semiconj g fa fb) :
    Set.MapsTo g (fixedPoints fa) (fixedPoints fb) 
Arguments: {α : Type u} {β : Type v} {fa : α → α} {fb : β → β} {g : α → β} (h : Function.Semiconj g fa fb)
Goal: Set.MapsTo g (Function.fixedPoints fa) (Function.fixedPoints fb)
-- Function.invOn_fixedPoints_comp
theorem invOn_fixedPoints_comp (f : α → β) (g : β → α) :
    Set.InvOn f g (fixedPoints <| f ∘ g) (fixedPoints <| g ∘ f) 
Arguments: {α : Type u} {β : Type v} (f : α → β) (g : β → α)
Goal: Set.InvOn f g (Function.fixedPoints (f ∘ g)) (Function.fixedPoints (g ∘ f))
-- Function.mapsTo_fixedPoints_comp
theorem mapsTo_fixedPoints_comp (f : α → β) (g : β → α) :
    Set.MapsTo f (fixedPoints <| g ∘ f) (fixedPoints <| f ∘ g) 
Arguments: {α : Type u} {β : Type v} (f : α → β) (g : β → α)
Goal: Set.MapsTo f (Function.fixedPoints (g ∘ f)) (Function.fixedPoints (f ∘ g))
-- Function.bijOn_fixedPoints_comp
theorem bijOn_fixedPoints_comp (f : α → β) (g : β → α) :
    Set.BijOn g (fixedPoints <| f ∘ g) (fixedPoints <| g ∘ f) 
Arguments: {α : Type u} {β : Type v} (f : α → β) (g : β → α)
Goal: Set.BijOn g (Function.fixedPoints (f ∘ g)) (Function.fixedPoints (g ∘ f))
-- Function.Commute.invOn_fixedPoints_comp
theorem Commute.invOn_fixedPoints_comp (h : Commute f g) :
    Set.InvOn f g (fixedPoints <| f ∘ g) (fixedPoints <| f ∘ g) 
Arguments: {α : Type u} {f g : α → α} (h : Function.Commute f g)
Goal: Set.InvOn f g (Function.fixedPoints (f ∘ g)) (Function.fixedPoints (f ∘ g))
-- Function.Commute.left_bijOn_fixedPoints_comp
theorem Commute.left_bijOn_fixedPoints_comp (h : Commute f g) :
    Set.BijOn f (fixedPoints <| f ∘ g) (fixedPoints <| f ∘ g) 
Arguments: {α : Type u} {f g : α → α} (h : Function.Commute f g)
Goal: Set.BijOn f (Function.fixedPoints (f ∘ g)) (Function.fixedPoints (f ∘ g))
-- Function.Commute.right_bijOn_fixedPoints_comp
theorem Commute.right_bijOn_fixedPoints_comp (h : Commute f g) :
    Set.BijOn g (fixedPoints <| f ∘ g) (fixedPoints <| f ∘ g) 
Arguments: {α : Type u} {f g : α → α} (h : Function.Commute f g)
Goal: Set.BijOn g (Function.fixedPoints (f ∘ g)) (Function.fixedPoints (f ∘ g))
-- MvPolynomial.C_dvd_iff_zmod
theorem C_dvd_iff_zmod (n : ℕ) (φ : MvPolynomial σ ℤ) :
    C (n : ℤ) ∣ φ ↔ map (Int.castRingHom (ZMod n)) φ = 0 
Arguments: {σ : Type u_1} (n : ℕ) (φ : MvPolynomial σ ℤ)
Goal: MvPolynomial.C ↑n ∣ φ ↔ (MvPolynomial.map (Int.castRingHom (ZMod n))) φ = 0
-- MvPolynomial.frobenius_zmod
theorem frobenius_zmod (f : MvPolynomial σ (ZMod p)) : frobenius _ p f = expand p f 
Arguments: {σ : Type u_1} {p : ℕ} [Fact (Nat.Prime p)] (f : MvPolynomial σ (ZMod p))
Goal: (frobenius (MvPolynomial σ (ZMod p)) p) f = (MvPolynomial.expand p) f
-- MvPolynomial.expand_zmod
theorem expand_zmod (f : MvPolynomial σ (ZMod p)) : expand p f = f ^ p 
Arguments: {σ : Type u_1} {p : ℕ} [Fact (Nat.Prime p)] (f : MvPolynomial σ (ZMod p))
Goal: (MvPolynomial.expand p) f = f ^ p
-- MvPolynomial.eval_indicator_apply_eq_one
theorem eval_indicator_apply_eq_one (a : σ → K) : eval a (indicator a) = 1 
Arguments: {K : Type u_1} {σ : Type u_2} [Fintype K] [Fintype σ] [CommRing K] (a : σ → K)
Goal: (MvPolynomial.eval a) (MvPolynomial.indicator a) = 1
-- MvPolynomial.degrees_indicator
theorem degrees_indicator (c : σ → K) :
    degrees (indicator c) ≤ ∑ s : σ, (Fintype.card K - 1) • {s} 
Arguments: {K : Type u_1} {σ : Type u_2} [Fintype K] [Fintype σ] [CommRing K] (c : σ → K)
Goal: (MvPolynomial.indicator c).degrees ≤ ∑ s : σ, (Fintype.card K - 1) • {s}
-- MvPolynomial.indicator_mem_restrictDegree
theorem indicator_mem_restrictDegree (c : σ → K) :
    indicator c ∈ restrictDegree σ K (Fintype.card K - 1) 
Arguments: {K : Type u_1} {σ : Type u_2} [Fintype K] [Fintype σ] [CommRing K] (c : σ → K)
Goal: MvPolynomial.indicator c ∈ MvPolynomial.restrictDegree σ K (Fintype.card K - 1)
-- MvPolynomial.eval_indicator_apply_eq_zero
theorem eval_indicator_apply_eq_zero (a b : σ → K) (h : a ≠ b) : eval a (indicator b) = 0 
Arguments: {K : Type u_1} {σ : Type u_2} [Fintype K] [Fintype σ] [Field K] (a b : σ → K) (h : a ≠ b)
Goal: (MvPolynomial.eval a) (MvPolynomial.indicator b) = 0
-- MvPolynomial.evalₗ_apply
@[simps]
def evalₗ [CommSemiring K] : MvPolynomial σ K →ₗ[K] (σ → K) → K 
Arguments: (K : Type u_1) (σ : Type u_2) [CommSemiring K] (p : MvPolynomial σ K) (e : σ → K)
Goal: (MvPolynomial.evalₗ K σ) p e = (MvPolynomial.eval e) p
-- MvPolynomial.map_restrict_dom_evalₗ
theorem map_restrict_dom_evalₗ : (restrictDegree σ K (Fintype.card K - 1)).map (evalₗ K σ) = ⊤ 
Arguments: (K : Type u_1) (σ : Type u_2) [Field K] [Fintype K] [Finite σ]
Goal: Submodule.map (MvPolynomial.evalₗ K σ) (MvPolynomial.restrictDegree σ K (Fintype.card K - 1)) = ⊤
-- MvPolynomial.rank_R
theorem rank_R [Fintype σ] : Module.rank K (R σ K) = Fintype.card (σ → K) 
Arguments: (σ K : Type u) [Fintype K] [Field K] [Fintype σ]
Goal: Module.rank K (MvPolynomial.R σ K) = ↑(Fintype.card (σ → K))
-- MvPolynomial.finrank_R
theorem finrank_R [Fintype σ] : FiniteDimensional.finrank K (R σ K) = Fintype.card (σ → K) 
Arguments: (σ K : Type u) [Fintype K] [Field K] [Fintype σ]
Goal: FiniteDimensional.finrank K (MvPolynomial.R σ K) = Fintype.card (σ → K)
-- MvPolynomial.range_evalᵢ
theorem range_evalᵢ [Finite σ] : range (evalᵢ σ K) = ⊤ 
Arguments: (σ K : Type u) [Fintype K] [Field K] [Finite σ]
Goal: LinearMap.range (MvPolynomial.evalᵢ σ K) = ⊤
-- MvPolynomial.ker_evalₗ
theorem ker_evalₗ [Finite σ] : ker (evalᵢ σ K) = ⊥ 
Arguments: (σ K : Type u) [Fintype K] [Field K] [Finite σ]
Goal: LinearMap.ker (MvPolynomial.evalᵢ σ K) = ⊥
-- MvPolynomial.eq_zero_of_eval_eq_zero
theorem eq_zero_of_eval_eq_zero [Finite σ] (p : MvPolynomial σ K) (h : ∀ v : σ → K, eval v p = 0)
    (hp : p ∈ restrictDegree σ K (Fintype.card K - 1)) : p = 0 
Arguments: (σ K : Type u) [Fintype K] [Field K] [Finite σ] (p : MvPolynomial σ K) (h : ∀ (v : σ → K), (MvPolynomial.eval v) p = 0) (hp : p ∈ MvPolynomial.restrictDegree σ K (Fintype.card K - 1))
Goal: p = 0
-- Function.IsFixedPt.tendsto_birkhoffAverage
theorem Function.IsFixedPt.tendsto_birkhoffAverage
    (R : Type*) [DivisionSemiring R] [CharZero R]
    [AddCommMonoid E] [TopologicalSpace E] [Module R E]
    {f : α → α} {x : α} (h : f.IsFixedPt x) (g : α → E) :
    Tendsto (birkhoffAverage R f g · x) atTop (𝓝 (g x)) 
Arguments: {α : Type u_1} {E : Type u_2} (R : Type u_3) [DivisionSemiring R] [CharZero R] [AddCommMonoid E] [TopologicalSpace E] [Module R E] {f : α → α} {x : α} (h : Function.IsFixedPt f x) (g : α → E)
Goal: Filter.Tendsto (fun x_1 => birkhoffAverage R f g x_1 x) Filter.atTop (𝓝 (g x))
-- dist_birkhoffSum_apply_birkhoffSum
theorem dist_birkhoffSum_apply_birkhoffSum (f : α → α) (g : α → E) (n : ℕ) (x : α) :
    dist (birkhoffSum f g n (f x)) (birkhoffSum f g n x) = dist (g (f^[n] x)) (g x) 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] (f : α → α) (g : α → E) (n : ℕ) (x : α)
Goal: dist (birkhoffSum f g n (f x)) (birkhoffSum f g n x) = dist (g (f^[n] x)) (g x)
-- dist_birkhoffSum_birkhoffSum_le
theorem dist_birkhoffSum_birkhoffSum_le (f : α → α) (g : α → E) (n : ℕ) (x y : α) :
    dist (birkhoffSum f g n x) (birkhoffSum f g n y) ≤
      ∑ k ∈ Finset.range n, dist (g (f^[k] x)) (g (f^[k] y)) 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] (f : α → α) (g : α → E) (n : ℕ) (x y : α)
Goal: dist (birkhoffSum f g n x) (birkhoffSum f g n y) ≤ ∑ k ∈ Finset.range n, dist (g (f^[k] x)) (g (f^[k] y))
-- dist_birkhoffAverage_birkhoffAverage
theorem dist_birkhoffAverage_birkhoffAverage (f : α → α) (g : α → E) (n : ℕ) (x y : α) :
    dist (birkhoffAverage 𝕜 f g n x) (birkhoffAverage 𝕜 f g n y) =
      dist (birkhoffSum f g n x) (birkhoffSum f g n y) / n 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] (𝕜 : Type u_3) [RCLike 𝕜] [Module 𝕜 E] [BoundedSMul 𝕜 E] (f : α → α) (g : α → E) (n : ℕ) (x y : α)
Goal: dist (birkhoffAverage 𝕜 f g n x) (birkhoffAverage 𝕜 f g n y) = dist (birkhoffSum f g n x) (birkhoffSum f g n y) / ↑n
-- dist_birkhoffAverage_birkhoffAverage_le
theorem dist_birkhoffAverage_birkhoffAverage_le (f : α → α) (g : α → E) (n : ℕ) (x y : α) :
    dist (birkhoffAverage 𝕜 f g n x) (birkhoffAverage 𝕜 f g n y) ≤
      (∑ k ∈ Finset.range n, dist (g (f^[k] x)) (g (f^[k] y))) / n 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] (𝕜 : Type u_3) [RCLike 𝕜] [Module 𝕜 E] [BoundedSMul 𝕜 E] (f : α → α) (g : α → E) (n : ℕ) (x y : α)
Goal: dist (birkhoffAverage 𝕜 f g n x) (birkhoffAverage 𝕜 f g n y) ≤
  (∑ k ∈ Finset.range n, dist (g (f^[k] x)) (g (f^[k] y))) / ↑n
-- dist_birkhoffAverage_apply_birkhoffAverage
theorem dist_birkhoffAverage_apply_birkhoffAverage (f : α → α) (g : α → E) (n : ℕ) (x : α) :
    dist (birkhoffAverage 𝕜 f g n (f x)) (birkhoffAverage 𝕜 f g n x) =
      dist (g (f^[n] x)) (g x) / n 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] (𝕜 : Type u_3) [RCLike 𝕜] [Module 𝕜 E] [BoundedSMul 𝕜 E] (f : α → α) (g : α → E) (n : ℕ) (x : α)
Goal: dist (birkhoffAverage 𝕜 f g n (f x)) (birkhoffAverage 𝕜 f g n x) = dist (g (f^[n] x)) (g x) / ↑n
-- tendsto_birkhoffAverage_apply_sub_birkhoffAverage
theorem tendsto_birkhoffAverage_apply_sub_birkhoffAverage {f : α → α} {g : α → E} {x : α}
    (h : Bornology.IsBounded (range (g <| f^[·] x))) :
    Tendsto (fun n ↦ birkhoffAverage 𝕜 f g n (f x) - birkhoffAverage 𝕜 f g n x) atTop (𝓝 0) 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] (𝕜 : Type u_3) [RCLike 𝕜] [Module 𝕜 E] [BoundedSMul 𝕜 E] {f : α → α} {g : α → E} {x : α} (h : Bornology.IsBounded (Set.range fun x_1 => g (f^[x_1] x)))
Goal: Filter.Tendsto (fun n => birkhoffAverage 𝕜 f g n (f x) - birkhoffAverage 𝕜 f g n x) Filter.atTop (𝓝 0)
-- tendsto_birkhoffAverage_apply_sub_birkhoffAverage'
theorem tendsto_birkhoffAverage_apply_sub_birkhoffAverage' {g : α → E}
    (h : Bornology.IsBounded (range g)) (f : α → α) (x : α) :
    Tendsto (fun n ↦ birkhoffAverage 𝕜 f g n (f x) - birkhoffAverage 𝕜 f g n x) atTop (𝓝 0) 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] (𝕜 : Type u_3) [RCLike 𝕜] [Module 𝕜 E] [BoundedSMul 𝕜 E] {g : α → E} (h : Bornology.IsBounded (Set.range g)) (f : α → α) (x : α)
Goal: Filter.Tendsto (fun n => birkhoffAverage 𝕜 f g n (f x) - birkhoffAverage 𝕜 f g n x) Filter.atTop (𝓝 0)
-- uniformEquicontinuous_birkhoffAverage
theorem uniformEquicontinuous_birkhoffAverage (hf : LipschitzWith 1 f) (hg : UniformContinuous g) :
    UniformEquicontinuous (birkhoffAverage 𝕜 f g) 
Arguments: (𝕜 : Type u_1) {X : Type u_2} {E : Type u_3} [PseudoEMetricSpace X] [RCLike 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : X → X} {g : X → E} (hf : LipschitzWith 1 f) (hg : UniformContinuous g)
Goal: UniformEquicontinuous (birkhoffAverage 𝕜 f g)
-- isClosed_setOf_tendsto_birkhoffAverage
theorem isClosed_setOf_tendsto_birkhoffAverage
    (hf : LipschitzWith 1 f) (hg : UniformContinuous g) (hl : Continuous l) :
    IsClosed {x | Tendsto (birkhoffAverage 𝕜 f g · x) atTop (𝓝 (l x))} 
Arguments: (𝕜 : Type u_1) {X : Type u_2} {E : Type u_3} [PseudoEMetricSpace X] [RCLike 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : X → X} {g l : X → E} (hf : LipschitzWith 1 f) (hg : UniformContinuous g) (hl : Continuous l)
Goal: IsClosed {x | Filter.Tendsto (fun x_1 => birkhoffAverage 𝕜 f g x_1 x) Filter.atTop (𝓝 (l x))}
-- isFixedPt_of_tendsto_iterate
theorem isFixedPt_of_tendsto_iterate {x y : α} (hy : Tendsto (fun n => f^[n] x) atTop (𝓝 y))
    (hf : ContinuousAt f y) : IsFixedPt f y 
Arguments: {α : Type u_1} [TopologicalSpace α] [T2Space α] {f : α → α} {x y : α} (hy : Filter.Tendsto (fun n => f^[n] x) Filter.atTop (𝓝 y)) (hf : ContinuousAt f y)
Goal: Function.IsFixedPt f y
-- isClosed_fixedPoints
theorem isClosed_fixedPoints (hf : Continuous f) : IsClosed (fixedPoints f) 
Arguments: {α : Type u_1} [TopologicalSpace α] [T2Space α] {f : α → α} (hf : Continuous f)
Goal: IsClosed (Function.fixedPoints f)
-- IntermediateField.inv_mem'
structure IntermediateField extends Subalgebra K L where
  inv_mem' : ∀ x ∈ carrier, x⁻¹ ∈ carrier
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (self : IntermediateField K L) (x : L)
Goal: x ∈ self.carrier → x⁻¹ ∈ self.carrier
-- IntermediateField.neg_mem
protected theorem neg_mem {x : L} (hx : x ∈ S) : -x ∈ S 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : IntermediateField K L) {x : L} (hx : x ∈ S)
Goal: -x ∈ S
-- IntermediateField.mem_carrier
theorem mem_carrier {s : IntermediateField K L} {x : L} : x ∈ s.carrier ↔ x ∈ s 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] {s : IntermediateField K L} {x : L}
Goal: x ∈ s.carrier ↔ x ∈ s
-- IntermediateField.ext
@[ext]
theorem ext {S T : IntermediateField K L} (h : ∀ x, x ∈ S ↔ x ∈ T) : S = T 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] {S T : IntermediateField K L} (h : ∀ (x : L), x ∈ S ↔ x ∈ T)
Goal: S = T
-- IntermediateField.coe_toSubalgebra
@[simp]
theorem coe_toSubalgebra : (S.toSubalgebra : Set L) = S 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : IntermediateField K L)
Goal: ↑S.toSubalgebra = ↑S
-- IntermediateField.coe_toSubfield
@[simp]
theorem coe_toSubfield : (S.toSubfield : Set L) = S 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [Field L] [Algebra K L] (S : IntermediateField K L)
Goal: ↑S.toSubfield = ↑S