ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
<VAR>α : Type u_1<VAR>inst✝³ : Fintype α<VAR>inst✝² : DecidableEq α<VAR>P : Finpartition univ<VAR>hP : P.IsEquipartition<VAR>G : SimpleGraph α<VAR>inst✝¹ : DecidableRel G.Adj<VAR>ε : ℝ<VAR>U : Finset α<VAR>hU✝ : U ∈ P.parts<VAR>V : Finset α<VAR>𝒜 : Finset (Finset α)<VAR>s✝ : Finset α<VAR>inst✝ : Nonempty α<VAR>hPα : P.parts.card * 16 ^ P.parts.card ≤ Fintype.card α<VAR>hPε : 100 ≤ 4 ^ P.parts.card * ε ^ 5<VAR>hε₁ : ε ≤ 1<VAR>hU : U ∈ P.parts<VAR>hV : V ∈ P.parts<VAR>hUVne : U ≠ V<VAR>hUV : ¬G.IsUniform ε U V<VAR>p : ℝ := (∑ ab ∈ (star hP G ε hU V).product (star hP G ε hV U), ↑(G.edgeDensity ab.1 ab.2)) / (↑(star hP G ε hU V).card * ↑(star hP G ε hV U).card)<VAR>q : ℝ := (∑ ab ∈ (chunk hP G ε hU).parts.product (chunk hP G ε hV).parts, ↑(G.edgeDensity ab.1 ab.2)) / (4 ^ P.parts.card * 4 ^ P.parts.card)<VAR>r : ℝ := ↑(G.edgeDensity ((star hP G ε hU V).biUnion id) ((star hP G ε hV U).biUnion id))<VAR>s : ℝ := ↑(G.edgeDensity (G.nonuniformWitness ε U V) (G.nonuniformWitness ε V U))<VAR>t : ℝ := ↑(G.edgeDensity U V)<VAR>hrs : |r - s| ≤ ε / 5<VAR>hst : ε ≤ |s - t|<VAR>hpr : |p - r| ≤ ε ^ 5 / 49<VAR>hqt : |q - t| ≤ ε ^ 5 / 49<VAR>hε' : ε ^ 5 ≤ ε<GOAL>3 / 4 * ε ≤ |p - q|
<VAR>α : Type u_1<VAR>inst✝³ : Fintype α<VAR>inst✝² : DecidableEq α<VAR>P : Finpartition univ<VAR>hP : P.IsEquipartition<VAR>G : SimpleGraph α<VAR>inst✝¹ : DecidableRel G.Adj<VAR>ε : ℝ<VAR>U : Finset α<VAR>hU✝ : U ∈ P.parts<VAR>V : Finset α<VAR>𝒜 : Finset (Finset α)<VAR>s✝ : Finset α<VAR>inst✝ : Nonempty α<VAR>hPα : P.parts.card * 16 ^ P.parts.card ≤ Fintype.card α<VAR>hPε : 100 ≤ 4 ^ P.parts.card * ε ^ 5<VAR>hε₁ : ε ≤ 1<VAR>hU : U ∈ P.parts<VAR>hV : V ∈ P.parts<VAR>hUVne : U ≠ V<VAR>hUV : ¬G.IsUniform ε U V<VAR>p : ℝ := (∑ ab ∈ (star hP G ε hU V).product (star hP G ε hV U), ↑(G.edgeDensity ab.1 ab.2)) / (↑(star hP G ε hU V).card * ↑(star hP G ε hV U).card)<VAR>q : ℝ := (∑ ab ∈ (chunk hP G ε hU).parts.product (chunk hP G ε hV).parts, ↑(G.edgeDensity ab.1 ab.2)) / (4 ^ P.parts.card * 4 ^ P.parts.card)<VAR>r : ℝ := ↑(G.edgeDensity ((star hP G ε hU V).biUnion id) ((star hP G ε hV U).biUnion id))<VAR>s : ℝ := ↑(G.edgeDensity (G.nonuniformWitness ε U V) (G.nonuniformWitness ε V U))<VAR>t : ℝ := ↑(G.edgeDensity U V)<VAR>hrs : r - s ≤ ε / 5 ∧ s - r ≤ ε / 5<VAR>hst : ε ≤ |s - t|<VAR>hpr : p - r ≤ ε ^ 5 / 49 ∧ r - p ≤ ε ^ 5 / 49<VAR>hqt : q - t ≤ ε ^ 5 / 49 ∧ t - q ≤ ε ^ 5 / 49<VAR>hε' : ε ^ 5 ≤ ε<GOAL>3 / 4 * ε ≤ |p - q|
<VAR>α : Type u_1<VAR>inst✝³ : Fintype α<VAR>inst✝² : DecidableEq α<VAR>P : Finpartition univ<VAR>hP : P.IsEquipartition<VAR>G : SimpleGraph α<VAR>inst✝¹ : DecidableRel G.Adj<VAR>ε : ℝ<VAR>U : Finset α<VAR>hU✝ : U ∈ P.parts<VAR>V : Finset α<VAR>𝒜 : Finset (Finset α)<VAR>s✝ : Finset α<VAR>inst✝ : Nonempty α<VAR>hPα : P.parts.card * 16 ^ P.parts.card ≤ Fintype.card α<VAR>hPε : 100 ≤ 4 ^ P.parts.card * ε ^ 5<VAR>hε₁ : ε ≤ 1<VAR>hU : U ∈ P.parts<VAR>hV : V ∈ P.parts<VAR>hUVne : U ≠ V<VAR>hUV : ¬G.IsUniform ε U V<VAR>p : ℝ := (∑ ab ∈ (star hP G ε hU V).product (star hP G ε hV U), ↑(G.edgeDensity ab.1 ab.2)) / (↑(star hP G ε hU V).card * ↑(star hP G ε hV U).card)<VAR>q : ℝ := (∑ ab ∈ (chunk hP G ε hU).parts.product (chunk hP G ε hV).parts, ↑(G.edgeDensity ab.1 ab.2)) / (4 ^ P.parts.card * 4 ^ P.parts.card)<VAR>r : ℝ := ↑(G.edgeDensity ((star hP G ε hU V).biUnion id) ((star hP G ε hV U).biUnion id))<VAR>s : ℝ := ↑(G.edgeDensity (G.nonuniformWitness ε U V) (G.nonuniformWitness ε V U))<VAR>t : ℝ := ↑(G.edgeDensity U V)<VAR>hrs : r - s ≤ ε / 5 ∧ s - r ≤ ε / 5<VAR>hst : ε ≤ s - t ∨ ε ≤ -(s - t)<VAR>hpr : p - r ≤ ε ^ 5 / 49 ∧ r - p ≤ ε ^ 5 / 49<VAR>hqt : q - t ≤ ε ^ 5 / 49 ∧ t - q ≤ ε ^ 5 / 49<VAR>hε' : ε ^ 5 ≤ ε<GOAL>3 / 4 * ε ≤ p - q ∨ 3 / 4 * ε ≤ -(p - q)
<VAR>ι : Type u_1<VAR>α : Type u_2<VAR>inst✝² : Preorder α<VAR>inst✝¹ : LocallyFiniteOrder α<VAR>a a₁ a₂ b b₁ b₂ c x : α<VAR>inst✝ : LocallyFiniteOrderTop α<GOAL>Ioc a b ⊆ Ioi a
<VAR>R : Type u<VAR>S : Type v<VAR>a✝ b c d : R<VAR>n✝ m : ℕ<VAR>inst✝¹ : Ring R<VAR>inst✝ : Nontrivial R<VAR>n : ℕ<VAR>hn : 0 < n<VAR>a : R<GOAL>(X ^ n - C a).degree = ↑n
<VAR>y : ℝ<VAR>h : 0 < y<GOAL>0 ^ y = 0
<VAR>y : ℝ<VAR>h : 0 < y<GOAL>some 0 ^ y = some 0
<VAR>y : ℝ<VAR>h : 0 < y<GOAL>(if 0 = 0 ∧ y < 0 then ⊤ else ↑(NNReal.rpow 0 y)) = some 0
<VAR>α : Type u_1<VAR>m : MeasurableSpace α<VAR>μ : Measure α<VAR>f : α → ℝ<VAR>hf : AEStronglyMeasurable f μ<VAR>A : ℝ<GOAL>AEStronglyMeasurable (ProbabilityTheory.truncation f A) μ
<VAR>α : Type u_1<VAR>m : MeasurableSpace α<VAR>μ : Measure α<VAR>f : α → ℝ<VAR>hf : AEStronglyMeasurable f μ<VAR>A : ℝ<GOAL>AEStronglyMeasurable ((Set.Ioc (-A) A).indicator id) (Measure.map f μ)
<VAR>X : Type u_1<VAR>α : Type u_2<VAR>ι : Sort u_3<VAR>inst✝² : TopologicalSpace X<VAR>inst✝¹ : BaireSpace X<VAR>inst✝ : Countable ι<VAR>s : Set X<VAR>hs : IsGδ s<VAR>hd : Dense s<VAR>f : ι → Set X<VAR>hc : ∀ (i : ι), IsClosed (f i)<VAR>hU : s ⊆ ⋃ i, f i<VAR>g : ι → Set X := fun i => (frontier (f i))ᶜ<GOAL>Dense (⋃ i, interior (f i))
<VAR>X : Type u_1<VAR>α : Type u_2<VAR>ι : Sort u_3<VAR>inst✝² : TopologicalSpace X<VAR>inst✝¹ : BaireSpace X<VAR>inst✝ : Countable ι<VAR>s : Set X<VAR>hs : IsGδ s<VAR>hd : Dense s<VAR>f : ι → Set X<VAR>hc : ∀ (i : ι), IsClosed (f i)<VAR>hU : s ⊆ ⋃ i, f i<VAR>g : ι → Set X := fun i => (frontier (f i))ᶜ<VAR>hgo : ∀ (i : ι), IsOpen (g i)<GOAL>Dense (⋃ i, interior (f i))
<VAR>X : Type u_1<VAR>α : Type u_2<VAR>ι : Sort u_3<VAR>inst✝² : TopologicalSpace X<VAR>inst✝¹ : BaireSpace X<VAR>inst✝ : Countable ι<VAR>s : Set X<VAR>hs : IsGδ s<VAR>hd : Dense s<VAR>f : ι → Set X<VAR>hc : ∀ (i : ι), IsClosed (f i)<VAR>hU : s ⊆ ⋃ i, f i<VAR>g : ι → Set X := fun i => (frontier (f i))ᶜ<VAR>hgo : ∀ (i : ι), IsOpen (g i)<VAR>hgd : Dense (⋂ i, g i)<GOAL>Dense (⋃ i, interior (f i))
<VAR>X : Type u_1<VAR>α : Type u_2<VAR>ι : Sort u_3<VAR>inst✝² : TopologicalSpace X<VAR>inst✝¹ : BaireSpace X<VAR>inst✝ : Countable ι<VAR>s : Set X<VAR>hs : IsGδ s<VAR>hd : Dense s<VAR>f : ι → Set X<VAR>hc : ∀ (i : ι), IsClosed (f i)<VAR>hU : s ⊆ ⋃ i, f i<VAR>g : ι → Set X := fun i => (frontier (f i))ᶜ<VAR>hgo : ∀ (i : ι), IsOpen (g i)<VAR>hgd : Dense (⋂ i, g i)<GOAL>s ∩ ⋂ i, g i ⊆ ⋃ i, interior (f i)
<VAR>X : Type u_1<VAR>α : Type u_2<VAR>ι : Sort u_3<VAR>inst✝² : TopologicalSpace X<VAR>inst✝¹ : BaireSpace X<VAR>inst✝ : Countable ι<VAR>s : Set X<VAR>hs : IsGδ s<VAR>hd : Dense s<VAR>f : ι → Set X<VAR>hc : ∀ (i : ι), IsClosed (f i)<VAR>hU : s ⊆ ⋃ i, f i<VAR>g : ι → Set X := fun i => (frontier (f i))ᶜ<VAR>hgo : ∀ (i : ι), IsOpen (g i)<VAR>hgd : Dense (⋂ i, g i)<VAR>x : X<VAR>hxs : x ∈ s<VAR>hxg : x ∈ ⋂ i, g i<GOAL>x ∈ ⋃ i, interior (f i)
<VAR>X : Type u_1<VAR>α : Type u_2<VAR>ι : Sort u_3<VAR>inst✝² : TopologicalSpace X<VAR>inst✝¹ : BaireSpace X<VAR>inst✝ : Countable ι<VAR>s : Set X<VAR>hs : IsGδ s<VAR>hd : Dense s<VAR>f : ι → Set X<VAR>hc : ∀ (i : ι), IsClosed (f i)<VAR>hU : s ⊆ ⋃ i, f i<VAR>g : ι → Set X := fun i => (frontier (f i))ᶜ<VAR>hgo : ∀ (i : ι), IsOpen (g i)<VAR>hgd : Dense (⋂ i, g i)<VAR>x : X<VAR>hxs : x ∈ s<VAR>hxg : ∀ (i : ι), x ∈ g i<GOAL>x ∈ ⋃ i, interior (f i)
<VAR>X : Type u_1<VAR>α : Type u_2<VAR>ι : Sort u_3<VAR>inst✝² : TopologicalSpace X<VAR>inst✝¹ : BaireSpace X<VAR>inst✝ : Countable ι<VAR>s : Set X<VAR>hs : IsGδ s<VAR>hd : Dense s<VAR>f : ι → Set X<VAR>hc : ∀ (i : ι), IsClosed (f i)<VAR>hU : s ⊆ ⋃ i, f i<VAR>g : ι → Set X := fun i => (frontier (f i))ᶜ<VAR>hgo : ∀ (i : ι), IsOpen (g i)<VAR>hgd : Dense (⋂ i, g i)<VAR>x : X<VAR>hxs : x ∈ s<VAR>hxg : ∀ (i : ι), x ∈ g i<VAR>i : ι<VAR>hi : x ∈ f i<GOAL>x ∈ ⋃ i, interior (f i)
<VAR>M : Type u_1<VAR>α : Type u_2<VAR>inst✝¹ : Group M<VAR>inst✝ : MulAction M α<VAR>s : Set α<VAR>m : M<GOAL>m ∈ fixingSubgroup M sᶜ ↔ (fixedBy α m)ᶜ ⊆ s
<VAR>C : Type u₁<VAR>D : Type u₂<VAR>inst✝⁶ : Category.{v₁, u₁} C<VAR>inst✝⁵ : Category.{v₂, u₂} D<VAR>E : Type u₃<VAR>inst✝⁴ : Category.{v₃, u₃} E<VAR>L : C ⥤ D<VAR>W : MorphismProperty C<VAR>inst✝³ : L.IsLocalization W<VAR>A : Type w<VAR>inst✝² : AddMonoid A<VAR>inst✝¹ : HasShift C A<VAR>inst✝ : W.IsCompatibleWithShift A<VAR>X Y : C<VAR>f : X ⟶ Y<VAR>a : A<GOAL>W ((shiftFunctor C a).map f) ↔ W f
<VAR>C : Type u₁<VAR>D : Type u₂<VAR>inst✝⁶ : Category.{v₁, u₁} C<VAR>inst✝⁵ : Category.{v₂, u₂} D<VAR>E : Type u₃<VAR>inst✝⁴ : Category.{v₃, u₃} E<VAR>L : C ⥤ D<VAR>W : MorphismProperty C<VAR>inst✝³ : L.IsLocalization W<VAR>A : Type w<VAR>inst✝² : AddMonoid A<VAR>inst✝¹ : HasShift C A<VAR>inst✝ : W.IsCompatibleWithShift A<VAR>X Y : C<VAR>f : X ⟶ Y<VAR>a : A<GOAL>W ((shiftFunctor C a).map f) ↔ W.inverseImage (shiftFunctor C a) f
<VAR>α : Type u_1<VAR>ι : Sort u_3<VAR>β : Type u_2<VAR>S : ι → Set α<VAR>f : (i : ι) → ↑(S i) → β<VAR>hf : ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j), f i ⟨x, hxi⟩ = f j ⟨x, hxj⟩<VAR>T : Set α<VAR>hT : T ⊆ iUnion S<VAR>hT' : T = iUnion S<VAR>t : Set β<VAR>x : ↑T<GOAL>x ∈ iUnionLift S f hf T hT ⁻¹' t ↔ x ∈ inclusion hT ⁻¹' ⋃ i, inclusion ⋯ '' (f i ⁻¹' t)
<VAR>α : Type u_1<VAR>ι : Sort u_3<VAR>β : Type u_2<VAR>S : ι → Set α<VAR>f : (i : ι) → ↑(S i) → β<VAR>hf : ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j), f i ⟨x, hxi⟩ = f j ⟨x, hxj⟩<VAR>T : Set α<VAR>hT : T ⊆ iUnion S<VAR>hT' : T = iUnion S<VAR>t : Set β<VAR>x : ↑T<GOAL>iUnionLift S f hf T hT x ∈ t ↔ ∃ i x_1, f i x_1 ∈ t ∧ inclusion ⋯ x_1 = inclusion hT x
<VAR>ι : Sort u_1<VAR>f : ι → ℕ<VAR>s : Set ℕ<GOAL>⨅ i, ↑(f i) ≠ ⊤ ↔ Nonempty ι
<VAR>X : Type u_1<VAR>M : Type u_2<VAR>R : Type u_3<VAR>𝕜 : Type u_4<VAR>inst✝⁴ : TopologicalSpace X<VAR>inst✝³ : NormedField 𝕜<VAR>inst✝² : NormedDivisionRing R<VAR>inst✝¹ : Algebra 𝕜 R<VAR>inst✝ : CompleteSpace R<VAR>f : C(X, R)<GOAL>IsUnit f ↔ ∀ (x : X), f x ≠ 0
<VAR>α : Type u_1<VAR>inst✝¹ : MeasurableSpace α<VAR>β : Type u_2<VAR>inst✝ : MeasurableSpace β<VAR>μ : FiniteMeasure α<VAR>ν : FiniteMeasure β<GOAL>μ.prod 0 = 0
<VAR>R : Type u_1<VAR>L : Type u_2<VAR>M : Type u_3<VAR>inst✝⁵ : CommRing R<VAR>inst✝⁴ : LieRing L<VAR>inst✝³ : LieAlgebra R L<VAR>inst✝² : AddCommGroup M<VAR>inst✝¹ : Module R M<VAR>inst✝ : LieRingModule L M<VAR>Φ : LinearMap.BilinForm R M<VAR>hΦ_nondeg : Φ.Nondegenerate<VAR>hΦ_inv : LinearMap.BilinForm.lieInvariant L Φ<VAR>N : LieSubmodule R L M<VAR>y : M<GOAL>y ∈ orthogonal Φ hΦ_inv N ↔ ∀ x ∈ N, (Φ x) y = 0
<VAR>C : Type u_1<VAR>inst✝³ : Category.{u_2, u_1} C<VAR>inst✝² : Preadditive C<VAR>inst✝¹ : HasZeroObject C<VAR>inst✝ : HasBinaryBiproducts C<VAR>X₁ X₂ X₃ : CochainComplex C ℤ<VAR>f : X₁ ⟶ X₂<VAR>g : X₂ ⟶ X₃<VAR>Y₁ Y₂ Y₃ : CochainComplex C ℤ<VAR>f' : Y₁ ⟶ Y₂<VAR>g' : Y₂ ⟶ Y₃<VAR>φ : mk₂ f g ⟶ mk₂ f' g'<VAR>n : ℤ<GOAL>(map g g' (φ.app 1) (φ.app 2) ⋯ ≫ (mappingConeCompTriangle f' g').mor₃).f n = ((mappingConeCompTriangle f g).mor₃ ≫ (CategoryTheory.shiftFunctor (CochainComplex C ℤ) 1).map (map f f' (φ.app 0) (φ.app 1) ⋯)).f n
<VAR>I : Type u_1<VAR>J : Type u_2<VAR>K : Type u_3<VAR>C : Type u_4<VAR>inst✝⁴ : Category.{u_5, u_4} C<VAR>X Y Z : GradedObject I C<VAR>φ : X ⟶ Y<VAR>e : X ≅ Y<VAR>ψ : Y ⟶ Z<VAR>p : I → J<VAR>j : J<VAR>inst✝³ : X.HasMap p<VAR>inst✝² : Y.HasMap p<VAR>inst✝¹ : Z.HasMap p<VAR>c : X.CofanMapObjFun p j<VAR>hc : IsColimit c<VAR>inst✝ : X.HasMap p<VAR>i : I<VAR>hi : p i = j<GOAL>Cofan.inj c ⟨i, hi⟩ ≫ (iso hc).hom = X.ιMapObj p i j hi
<VAR>E : Type u_1<VAR>F : Type u_2<VAR>inst✝⁵ : NormedAddCommGroup E<VAR>inst✝⁴ : NormedSpace ℝ E<VAR>inst✝³ : FiniteDimensional ℝ E<VAR>inst✝² : NormedAddCommGroup F<VAR>inst✝¹ : NormedSpace ℝ F<VAR>s✝ : Set E<VAR>f✝ : E → E<VAR>f'✝ : E → E →L[ℝ] E<VAR>inst✝ : SecondCountableTopology F<VAR>f : E → F<VAR>s : Set E<VAR>f' : E → E →L[ℝ] F<VAR>hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x<VAR>r : (E →L[ℝ] F) → ℝ≥0<VAR>rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0<GOAL>∃ t A, (∀ (n : ℕ), IsClosed (t n)) ∧ s ⊆ ⋃ n, t n ∧ (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧ (s.Nonempty → ∀ (n : ℕ), ∃ y ∈ s, A n = f' y)
<VAR>E : Type u_1<VAR>F : Type u_2<VAR>inst✝⁵ : NormedAddCommGroup E<VAR>inst✝⁴ : NormedSpace ℝ E<VAR>inst✝³ : FiniteDimensional ℝ E<VAR>inst✝² : NormedAddCommGroup F<VAR>inst✝¹ : NormedSpace ℝ F<VAR>s : Set E<VAR>f✝ : E → E<VAR>f'✝ : E → E →L[ℝ] E<VAR>inst✝ : SecondCountableTopology F<VAR>f : E → F<VAR>f' : E → E →L[ℝ] F<VAR>r : (E →L[ℝ] F) → ℝ≥0<VAR>rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0<VAR>hf' : ∀ x ∈ ∅, HasFDerivWithinAt f (f' x) ∅ x<GOAL>∃ t A, (∀ (n : ℕ), IsClosed (t n)) ∧ ∅ ⊆ ⋃ n, t n ∧ (∀ (n : ℕ), ApproximatesLinearOn f (A n) (∅ ∩ t n) (r (A n))) ∧ (∅.Nonempty → ∀ (n : ℕ), ∃ y ∈ ∅, A n = f' y)
<VAR>E : Type u_1<VAR>F : Type u_2<VAR>inst✝⁵ : NormedAddCommGroup E<VAR>inst✝⁴ : NormedSpace ℝ E<VAR>inst✝³ : FiniteDimensional ℝ E<VAR>inst✝² : NormedAddCommGroup F<VAR>inst✝¹ : NormedSpace ℝ F<VAR>s✝ : Set E<VAR>f✝ : E → E<VAR>f'✝ : E → E →L[ℝ] E<VAR>inst✝ : SecondCountableTopology F<VAR>f : E → F<VAR>s : Set E<VAR>f' : E → E →L[ℝ] F<VAR>hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x<VAR>r : (E →L[ℝ] F) → ℝ≥0<VAR>rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0<VAR>hs : s.Nonempty<GOAL>∃ t A, (∀ (n : ℕ), IsClosed (t n)) ∧ s ⊆ ⋃ n, t n ∧ (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧ (s.Nonempty → ∀ (n : ℕ), ∃ y ∈ s, A n = f' y)
<VAR>E : Type u_1<VAR>F : Type u_2<VAR>inst✝⁵ : NormedAddCommGroup E<VAR>inst✝⁴ : NormedSpace ℝ E<VAR>inst✝³ : FiniteDimensional ℝ E<VAR>inst✝² : NormedAddCommGroup F<VAR>inst✝¹ : NormedSpace ℝ F<VAR>s✝ : Set E<VAR>f✝ : E → E<VAR>f'✝ : E → E →L[ℝ] E<VAR>inst✝ : SecondCountableTopology F<VAR>f : E → F<VAR>s : Set E<VAR>f' : E → E →L[ℝ] F<VAR>hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x<VAR>r : (E →L[ℝ] F) → ℝ≥0<VAR>rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0<VAR>hs : s.Nonempty<VAR>T : Set ↑s<VAR>T_count : T.Countable<VAR>hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x))<GOAL>∃ t A, (∀ (n : ℕ), IsClosed (t n)) ∧ s ⊆ ⋃ n, t n ∧ (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧ (s.Nonempty → ∀ (n : ℕ), ∃ y ∈ s, A n = f' y)
<VAR>E : Type u_1<VAR>F : Type u_2<VAR>inst✝⁵ : NormedAddCommGroup E<VAR>inst✝⁴ : NormedSpace ℝ E<VAR>inst✝³ : FiniteDimensional ℝ E<VAR>inst✝² : NormedAddCommGroup F<VAR>inst✝¹ : NormedSpace ℝ F<VAR>s✝ : Set E<VAR>f✝ : E → E<VAR>f'✝ : E → E →L[ℝ] E<VAR>inst✝ : SecondCountableTopology F<VAR>f : E → F<VAR>s : Set E<VAR>f' : E → E →L[ℝ] F<VAR>hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x<VAR>r : (E →L[ℝ] F) → ℝ≥0<VAR>rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0<VAR>hs : s.Nonempty<VAR>T : Set ↑s<VAR>T_count : T.Countable<VAR>hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x))<VAR>u : ℕ → ℝ<VAR>left✝ : StrictAnti u<VAR>u_pos : ∀ (n : ℕ), 0 < u n<VAR>u_lim : Tendsto u atTop (𝓝 0)<GOAL>∃ t A, (∀ (n : ℕ), IsClosed (t n)) ∧ s ⊆ ⋃ n, t n ∧ (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧ (s.Nonempty → ∀ (n : ℕ), ∃ y ∈ s, A n = f' y)
<VAR>E : Type u_1<VAR>F : Type u_2<VAR>inst✝⁵ : NormedAddCommGroup E<VAR>inst✝⁴ : NormedSpace ℝ E<VAR>inst✝³ : FiniteDimensional ℝ E<VAR>inst✝² : NormedAddCommGroup F<VAR>inst✝¹ : NormedSpace ℝ F<VAR>s✝ : Set E<VAR>f✝ : E → E<VAR>f'✝ : E → E →L[ℝ] E<VAR>inst✝ : SecondCountableTopology F<VAR>f : E → F<VAR>s : Set E<VAR>f' : E → E →L[ℝ] F<VAR>hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x<VAR>r : (E →L[ℝ] F) → ℝ≥0<VAR>rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0<VAR>hs : s.Nonempty<VAR>T : Set ↑s<VAR>T_count : T.Countable<VAR>hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x))<VAR>u : ℕ → ℝ<VAR>left✝ : StrictAnti u<VAR>u_pos : ∀ (n : ℕ), 0 < u n<VAR>u_lim : Tendsto u atTop (𝓝 0)<VAR>M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ y ∈ s ∩ ball x (u n), ‖f y - f x - (f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}<GOAL>∃ t A, (∀ (n : ℕ), IsClosed (t n)) ∧ s ⊆ ⋃ n, t n ∧ (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧ (s.Nonempty → ∀ (n : ℕ), ∃ y ∈ s, A n = f' y)
<VAR>E : Type u_1<VAR>F : Type u_2<VAR>inst✝⁵ : NormedAddCommGroup E<VAR>inst✝⁴ : NormedSpace ℝ E<VAR>inst✝³ : FiniteDimensional ℝ E<VAR>inst✝² : NormedAddCommGroup F<VAR>inst✝¹ : NormedSpace ℝ F<VAR>s✝ : Set E<VAR>f✝ : E → E<VAR>f'✝ : E → E →L[ℝ] E<VAR>inst✝ : SecondCountableTopology F<VAR>f : E → F<VAR>s : Set E<VAR>f' : E → E →L[ℝ] F<VAR>hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x<VAR>r : (E →L[ℝ] F) → ℝ≥0<VAR>rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0<VAR>hs : s.Nonempty<VAR>T : Set ↑s<VAR>T_count : T.Countable<VAR>hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x))<VAR>u : ℕ → ℝ<VAR>left✝ : StrictAnti u<VAR>u_pos : ∀ (n : ℕ), 0 < u n<VAR>u_lim : Tendsto u atTop (𝓝 0)<VAR>M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ y ∈ s ∩ ball x (u n), ‖f y - f x - (f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}<VAR>s_subset : ∀ x ∈ s, ∃ n z, x ∈ M n z<GOAL>∃ t A, (∀ (n : ℕ), IsClosed (t n)) ∧ s ⊆ ⋃ n, t n ∧ (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧ (s.Nonempty → ∀ (n : ℕ), ∃ y ∈ s, A n = f' y)
<VAR>E : Type u_1<VAR>F : Type u_2<VAR>inst✝⁵ : NormedAddCommGroup E<VAR>inst✝⁴ : NormedSpace ℝ E<VAR>inst✝³ : FiniteDimensional ℝ E<VAR>inst✝² : NormedAddCommGroup F<VAR>inst✝¹ : NormedSpace ℝ F<VAR>s✝ : Set E<VAR>f✝ : E → E<VAR>f'✝ : E → E →L[ℝ] E<VAR>inst✝ : SecondCountableTopology F<VAR>f : E → F<VAR>s : Set E<VAR>f' : E → E →L[ℝ] F<VAR>hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x<VAR>r : (E →L[ℝ] F) → ℝ≥0<VAR>rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0<VAR>hs : s.Nonempty<VAR>T : Set ↑s<VAR>T_count : T.Countable<VAR>hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x))<VAR>u : ℕ → ℝ<VAR>left✝ : StrictAnti u<VAR>u_pos : ∀ (n : ℕ), 0 < u n<VAR>u_lim : Tendsto u atTop (𝓝 0)<VAR>M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ y ∈ s ∩ ball x (u n), ‖f y - f x - (f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}<VAR>s_subset : ∀ x ∈ s, ∃ n z, x ∈ M n z<VAR>closure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z<GOAL>∃ t A, (∀ (n : ℕ), IsClosed (t n)) ∧ s ⊆ ⋃ n, t n ∧ (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧ (s.Nonempty → ∀ (n : ℕ), ∃ y ∈ s, A n = f' y)
<VAR>E : Type u_1<VAR>F : Type u_2<VAR>inst✝⁵ : NormedAddCommGroup E<VAR>inst✝⁴ : NormedSpace ℝ E<VAR>inst✝³ : FiniteDimensional ℝ E<VAR>inst✝² : NormedAddCommGroup F<VAR>inst✝¹ : NormedSpace ℝ F<VAR>s✝ : Set E<VAR>f✝ : E → E<VAR>f'✝ : E → E →L[ℝ] E<VAR>inst✝ : SecondCountableTopology F<VAR>f : E → F<VAR>s : Set E<VAR>f' : E → E →L[ℝ] F<VAR>hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x<VAR>r : (E →L[ℝ] F) → ℝ≥0<VAR>rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0<VAR>hs : s.Nonempty<VAR>T : Set ↑s<VAR>T_count : T.Countable<VAR>hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x))<VAR>u : ℕ → ℝ<VAR>left✝ : StrictAnti u<VAR>u_pos : ∀ (n : ℕ), 0 < u n<VAR>u_lim : Tendsto u atTop (𝓝 0)<VAR>M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ y ∈ s ∩ ball x (u n), ‖f y - f x - (f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}<VAR>s_subset : ∀ x ∈ s, ∃ n z, x ∈ M n z<VAR>closure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z<VAR>d : ℕ → E<VAR>hd : DenseRange d<GOAL>∃ t A, (∀ (n : ℕ), IsClosed (t n)) ∧ s ⊆ ⋃ n, t n ∧ (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧ (s.Nonempty → ∀ (n : ℕ), ∃ y ∈ s, A n = f' y)
<VAR>E : Type u_1<VAR>F : Type u_2<VAR>inst✝⁵ : NormedAddCommGroup E<VAR>inst✝⁴ : NormedSpace ℝ E<VAR>inst✝³ : FiniteDimensional ℝ E<VAR>inst✝² : NormedAddCommGroup F<VAR>inst✝¹ : NormedSpace ℝ F<VAR>s✝ : Set E<VAR>f✝ : E → E<VAR>f'✝ : E → E →L[ℝ] E<VAR>inst✝ : SecondCountableTopology F<VAR>f : E → F<VAR>s : Set E<VAR>f' : E → E →L[ℝ] F<VAR>hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x<VAR>r : (E →L[ℝ] F) → ℝ≥0<VAR>rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0<VAR>hs : s.Nonempty<VAR>T : Set ↑s<VAR>T_count : T.Countable<VAR>hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x))<VAR>u : ℕ → ℝ<VAR>left✝ : StrictAnti u<VAR>u_pos : ∀ (n : ℕ), 0 < u n<VAR>u_lim : Tendsto u atTop (𝓝 0)<VAR>M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ y ∈ s ∩ ball x (u n), ‖f y - f x - (f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}<VAR>s_subset : ∀ x ∈ s, ∃ n z, x ∈ M n z<VAR>closure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z<VAR>d : ℕ → E<VAR>hd : DenseRange d<VAR>K : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)<GOAL>∃ t A, (∀ (n : ℕ), IsClosed (t n)) ∧ s ⊆ ⋃ n, t n ∧ (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧ (s.Nonempty → ∀ (n : ℕ), ∃ y ∈ s, A n = f' y)
<VAR>E : Type u_1<VAR>F : Type u_2<VAR>inst✝⁵ : NormedAddCommGroup E<VAR>inst✝⁴ : NormedSpace ℝ E<VAR>inst✝³ : FiniteDimensional ℝ E<VAR>inst✝² : NormedAddCommGroup F<VAR>inst✝¹ : NormedSpace ℝ F<VAR>s✝ : Set E<VAR>f✝ : E → E<VAR>f'✝ : E → E →L[ℝ] E<VAR>inst✝ : SecondCountableTopology F<VAR>f : E → F<VAR>s : Set E<VAR>f' : E → E →L[ℝ] F<VAR>hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x<VAR>r : (E →L[ℝ] F) → ℝ≥0<VAR>rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0<VAR>hs : s.Nonempty<VAR>T : Set ↑s<VAR>T_count : T.Countable<VAR>hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x))<VAR>u : ℕ → ℝ<VAR>left✝ : StrictAnti u<VAR>u_pos : ∀ (n : ℕ), 0 < u n<VAR>u_lim : Tendsto u atTop (𝓝 0)<VAR>M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ y ∈ s ∩ ball x (u n), ‖f y - f x - (f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}<VAR>s_subset : ∀ x ∈ s, ∃ n z, x ∈ M n z<VAR>closure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z<VAR>d : ℕ → E<VAR>hd : DenseRange d<VAR>K : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)<VAR>K_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z))<GOAL>∃ t A, (∀ (n : ℕ), IsClosed (t n)) ∧ s ⊆ ⋃ n, t n ∧ (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧ (s.Nonempty → ∀ (n : ℕ), ∃ y ∈ s, A n = f' y)
<VAR>E : Type u_1<VAR>F : Type u_2<VAR>inst✝⁵ : NormedAddCommGroup E<VAR>inst✝⁴ : NormedSpace ℝ E<VAR>inst✝³ : FiniteDimensional ℝ E<VAR>inst✝² : NormedAddCommGroup F<VAR>inst✝¹ : NormedSpace ℝ F<VAR>s✝ : Set E<VAR>f✝ : E → E<VAR>f'✝ : E → E →L[ℝ] E<VAR>inst✝ : SecondCountableTopology F<VAR>f : E → F<VAR>s : Set E<VAR>f' : E → E →L[ℝ] F<VAR>hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x<VAR>r : (E →L[ℝ] F) → ℝ≥0<VAR>rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0<VAR>hs : s.Nonempty<VAR>T : Set ↑s<VAR>T_count : T.Countable<VAR>hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x))<VAR>u : ℕ → ℝ<VAR>left✝ : StrictAnti u<VAR>u_pos : ∀ (n : ℕ), 0 < u n<VAR>u_lim : Tendsto u atTop (𝓝 0)<VAR>M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ y ∈ s ∩ ball x (u n), ‖f y - f x - (f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}<VAR>s_subset : ∀ x ∈ s, ∃ n z, x ∈ M n z<VAR>closure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z<VAR>d : ℕ → E<VAR>hd : DenseRange d<VAR>K : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)<VAR>K_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z))<VAR>K_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed (K n z p)<GOAL>∃ t A, (∀ (n : ℕ), IsClosed (t n)) ∧ s ⊆ ⋃ n, t n ∧ (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧ (s.Nonempty → ∀ (n : ℕ), ∃ y ∈ s, A n = f' y)
<VAR>E : Type u_1<VAR>F✝ : Type u_2<VAR>inst✝⁵ : NormedAddCommGroup E<VAR>inst✝⁴ : NormedSpace ℝ E<VAR>inst✝³ : FiniteDimensional ℝ E<VAR>inst✝² : NormedAddCommGroup F✝<VAR>inst✝¹ : NormedSpace ℝ F✝<VAR>s✝ : Set E<VAR>f✝ : E → E<VAR>f'✝ : E → E →L[ℝ] E<VAR>inst✝ : SecondCountableTopology F✝<VAR>f : E → F✝<VAR>s : Set E<VAR>f' : E → E →L[ℝ] F✝<VAR>hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x<VAR>r : (E →L[ℝ] F✝) → ℝ≥0<VAR>rpos : ∀ (A : E →L[ℝ] F✝), r A ≠ 0<VAR>hs : s.Nonempty<VAR>T : Set ↑s<VAR>T_count : T.Countable<VAR>hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x))<VAR>u : ℕ → ℝ<VAR>left✝ : StrictAnti u<VAR>u_pos : ∀ (n : ℕ), 0 < u n<VAR>u_lim : Tendsto u atTop (𝓝 0)<VAR>M : ℕ → ↑T → Set E := fun n z => {x | x ∈ s ∧ ∀ y ∈ s ∩ ball x (u n), ‖f y - f x - (f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}<VAR>s_subset : ∀ x ∈ s, ∃ n z, x ∈ M n z<VAR>closure_M_subset : ∀ (n : ℕ) (z : ↑T), s ∩ closure (M n z) ⊆ M n z<VAR>d : ℕ → E<VAR>hd : DenseRange d<VAR>K : ℕ → ↑T → ℕ → Set E := fun n z p => closure (M n z) ∩ closedBall (d p) (u n / 3)<VAR>K_approx : ∀ (n : ℕ) (z : ↑T) (p : ℕ), ApproximatesLinearOn f (f' ↑↑z) (s ∩ K n z p) (r (f' ↑↑z))<VAR>K_closed : ∀ (n : ℕ) (z : ↑T) (p : ℕ), IsClosed (K n z p)<VAR>F : ℕ → ℕ × ↑T × ℕ<VAR>hF : Function.Surjective F<GOAL>∃ t A, (∀ (n : ℕ), IsClosed (t n)) ∧ s ⊆ ⋃ n, t n ∧ (∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧ (s.Nonempty → ∀ (n : ℕ), ∃ y ∈ s, A n = f' y)