ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
<VAR>E : Type u_1<VAR>inst✝⁴ : NormedAddCommGroup E<VAR>inst✝³ : NormedSpace ℝ E<VAR>inst✝² : FiniteDimensional ℝ E<VAR>inst✝¹ : MeasurableSpace E<VAR>inst✝ : BorelSpace E<VAR>D : ℝ<VAR>x : E<VAR>Dpos : 0 < D<VAR>A : (w D ⋆[lsmul ℝ ℝ, μ] φ) x ≤ (w D ⋆[lsmul ℝ ℝ, μ] 1) x<GOAL>y D x ≤ 1
<VAR>E : Type u_1<VAR>inst✝⁴ : NormedAddCommGroup E<VAR>inst✝³ : NormedSpace ℝ E<VAR>inst✝² : FiniteDimensional ℝ E<VAR>inst✝¹ : MeasurableSpace E<VAR>inst✝ : BorelSpace E<VAR>D : ℝ<VAR>x : E<VAR>Dpos : 0 < D<VAR>A : (w D ⋆[lsmul ℝ ℝ, μ] φ) x ≤ (w D ⋆[lsmul ℝ ℝ, μ] 1) x<VAR>B : (w D ⋆[lsmul ℝ ℝ, μ] fun x => 1) x = 1<GOAL>y D x ≤ 1
<VAR>R S : Type u_1<VAR>inst✝¹ : CommRing R<VAR>inst✝ : CommRing S<VAR>f : R →+* S<VAR>s : Set R<VAR>hs : Ideal.span s = ⊤<VAR>H : ∀ (r : ↑s), (fun {X Y} [CommRing X] [CommRing Y] f => Function.Surjective ⇑f) (Localization.awayMap f ↑r)<VAR>this : Algebra R S := f.toAlgebra<GOAL>Function.Surjective ⇑(Algebra.ofId R S)
<VAR>R S : Type u_1<VAR>inst✝¹ : CommRing R<VAR>inst✝ : CommRing S<VAR>f : R →+* S<VAR>s : Set R<VAR>hs : Ideal.span s = ⊤<VAR>H : ∀ (r : ↑s), (fun {X Y} [CommRing X] [CommRing Y] f => Function.Surjective ⇑f) (Localization.awayMap f ↑r)<VAR>this : Algebra R S := f.toAlgebra<GOAL>⊤ ≤ (Algebra.ofId R S).range
<VAR>R S : Type u_1<VAR>inst✝¹ : CommRing R<VAR>inst✝ : CommRing S<VAR>f : R →+* S<VAR>s : Set R<VAR>hs : Ideal.span s = ⊤<VAR>H : ∀ (r : ↑s), (fun {X Y} [CommRing X] [CommRing Y] f => Function.Surjective ⇑f) (Localization.awayMap f ↑r)<VAR>this : Algebra R S := f.toAlgebra<VAR>x : S<GOAL>x ∈ (Algebra.ofId R S).range
<VAR>R S : Type u_1<VAR>inst✝¹ : CommRing R<VAR>inst✝ : CommRing S<VAR>f : R →+* S<VAR>s : Set R<VAR>hs : Ideal.span s = ⊤<VAR>H : ∀ (r : ↑s), (fun {X Y} [CommRing X] [CommRing Y] f => Function.Surjective ⇑f) (Localization.awayMap f ↑r)<VAR>this : Algebra R S := f.toAlgebra<VAR>x : S<VAR>l : ↑s →₀ R<VAR>hl : (Finsupp.total (↑s) R R Subtype.val) l = 1<GOAL>x ∈ (Algebra.ofId R S).range
<VAR>R S : Type u_1<VAR>inst✝¹ : CommRing R<VAR>inst✝ : CommRing S<VAR>f : R →+* S<VAR>s : Set R<VAR>hs : Ideal.span s = ⊤<VAR>H : ∀ (r : ↑s), (fun {X Y} [CommRing X] [CommRing Y] f => Function.Surjective ⇑f) (Localization.awayMap f ↑r)<VAR>this : Algebra R S := f.toAlgebra<VAR>x : S<VAR>l : ↑s →₀ R<VAR>hl : (Finsupp.total (↑s) R R Subtype.val) l = 1<GOAL>∑ i ∈ l.support, f (l i) * f ↑i = 1
<VAR>R S : Type u_1<VAR>inst✝¹ : CommRing R<VAR>inst✝ : CommRing S<VAR>f : R →+* S<VAR>s : Set R<VAR>hs : Ideal.span s = ⊤<VAR>H : ∀ (r : ↑s), (fun {X Y} [CommRing X] [CommRing Y] f => Function.Surjective ⇑f) (Localization.awayMap f ↑r)<VAR>this : Algebra R S := f.toAlgebra<VAR>x : S<VAR>l : ↑s →₀ R<VAR>hl : (Finsupp.total (↑s) R R Subtype.val) l = 1<GOAL>∀ (i : ↑s), f ↑i ∈ (Algebra.ofId R S).range
<VAR>R S : Type u_1<VAR>inst✝¹ : CommRing R<VAR>inst✝ : CommRing S<VAR>f : R →+* S<VAR>s : Set R<VAR>hs : Ideal.span s = ⊤<VAR>H : ∀ (r : ↑s), (fun {X Y} [CommRing X] [CommRing Y] f => Function.Surjective ⇑f) (Localization.awayMap f ↑r)<VAR>this : Algebra R S := f.toAlgebra<VAR>x : S<VAR>l : ↑s →₀ R<VAR>hl : (Finsupp.total (↑s) R R Subtype.val) l = 1<GOAL>∀ (i : ↑s), f (l i) ∈ (Algebra.ofId R S).range
<VAR>R S : Type u_1<VAR>inst✝¹ : CommRing R<VAR>inst✝ : CommRing S<VAR>f : R →+* S<VAR>s : Set R<VAR>hs : Ideal.span s = ⊤<VAR>H : ∀ (r : ↑s), (fun {X Y} [CommRing X] [CommRing Y] f => Function.Surjective ⇑f) (Localization.awayMap f ↑r)<VAR>this : Algebra R S := f.toAlgebra<VAR>x : S<VAR>l : ↑s →₀ R<VAR>hl : (Finsupp.total (↑s) R R Subtype.val) l = 1<GOAL>∀ (i : ↑s), ∃ n, f ↑i ^ n • x ∈ (Algebra.ofId R S).range
<VAR>C : Type u_2<VAR>inst✝⁸ : Category.{u_1, u_2} C<VAR>inst✝⁷ : HasZeroMorphisms C<VAR>S₁ S₂ S₃ S₄ : ShortComplex C<VAR>inst✝⁶ : S₁.HasHomology<VAR>inst✝⁵ : S₂.HasHomology<VAR>inst✝⁴ : S₃.HasHomology<VAR>inst✝³ : S₄.HasHomology<VAR>φ : S₁ ⟶ S₂<VAR>inst✝² : Epi φ.τ₁<VAR>inst✝¹ : IsIso φ.τ₂<VAR>inst✝ : Mono φ.τ₃<GOAL>QuasiIso φ
<VAR>C : Type u_2<VAR>inst✝⁸ : Category.{u_1, u_2} C<VAR>inst✝⁷ : HasZeroMorphisms C<VAR>S₁ S₂ S₃ S₄ : ShortComplex C<VAR>inst✝⁶ : S₁.HasHomology<VAR>inst✝⁵ : S₂.HasHomology<VAR>inst✝⁴ : S₃.HasHomology<VAR>inst✝³ : S₄.HasHomology<VAR>φ : S₁ ⟶ S₂<VAR>inst✝² : Epi φ.τ₁<VAR>inst✝¹ : IsIso φ.τ₂<VAR>inst✝ : Mono φ.τ₃<GOAL>IsIso (LeftHomologyMapData.ofEpiOfIsIsoOfMono φ S₁.leftHomologyData).φH
<VAR>α : Type u_1<VAR>β : Type u_2<VAR>inst✝² : LinearOrderedField α<VAR>inst✝¹ : Ring β<VAR>abv : β → α<VAR>inst✝ : IsAbsoluteValue abv<VAR>f g : ℕ → β<VAR>a : ℕ → α<VAR>ha : IsCauSeq abs fun m => ∑ n ∈ range m, abv (f n)<VAR>hb : IsCauSeq abv fun m => ∑ n ∈ range m, g n<VAR>ε : α<VAR>ε0 : 0 < ε<GOAL>∃ i, ∀ j ≥ i, abv ((∑ k ∈ range j, f k) * ∑ k ∈ range j, g k - ∑ n ∈ range j, ∑ m ∈ range (n + 1), f m * g (n - m)) < ε
<VAR>α : Type u_1<VAR>β : Type u_2<VAR>inst✝² : LinearOrderedField α<VAR>inst✝¹ : Ring β<VAR>abv : β → α<VAR>inst✝ : IsAbsoluteValue abv<VAR>f g : ℕ → β<VAR>a : ℕ → α<VAR>ha : IsCauSeq abs fun m => ∑ n ∈ range m, abv (f n)<VAR>hb : IsCauSeq abv fun m => ∑ n ∈ range m, g n<VAR>ε : α<VAR>ε0 : 0 < ε<VAR>P : α<VAR>hP : ∀ (i : ℕ), |∑ n ∈ range i, abv (f n)| < P<GOAL>∃ i, ∀ j ≥ i, abv ((∑ k ∈ range j, f k) * ∑ k ∈ range j, g k - ∑ n ∈ range j, ∑ m ∈ range (n + 1), f m * g (n - m)) < ε
<VAR>α : Type u_1<VAR>β : Type u_2<VAR>inst✝² : LinearOrderedField α<VAR>inst✝¹ : Ring β<VAR>abv : β → α<VAR>inst✝ : IsAbsoluteValue abv<VAR>f g : ℕ → β<VAR>a : ℕ → α<VAR>ha : IsCauSeq abs fun m => ∑ n ∈ range m, abv (f n)<VAR>hb : IsCauSeq abv fun m => ∑ n ∈ range m, g n<VAR>ε : α<VAR>ε0 : 0 < ε<VAR>P : α<VAR>hP : ∀ (i : ℕ), |∑ n ∈ range i, abv (f n)| < P<VAR>Q : α<VAR>hQ : ∀ (i : ℕ), abv (∑ n ∈ range i, g n) < Q<GOAL>∃ i, ∀ j ≥ i, abv ((∑ k ∈ range j, f k) * ∑ k ∈ range j, g k - ∑ n ∈ range j, ∑ m ∈ range (n + 1), f m * g (n - m)) < ε
<VAR>α : Type u_1<VAR>β : Type u_2<VAR>inst✝² : LinearOrderedField α<VAR>inst✝¹ : Ring β<VAR>abv : β → α<VAR>inst✝ : IsAbsoluteValue abv<VAR>f g : ℕ → β<VAR>a : ℕ → α<VAR>ha : IsCauSeq abs fun m => ∑ n ∈ range m, abv (f n)<VAR>hb : IsCauSeq abv fun m => ∑ n ∈ range m, g n<VAR>ε : α<VAR>ε0 : 0 < ε<VAR>P : α<VAR>hP : ∀ (i : ℕ), |∑ n ∈ range i, abv (f n)| < P<VAR>Q : α<VAR>hQ : ∀ (i : ℕ), abv (∑ n ∈ range i, g n) < Q<VAR>hP0 : 0 < P<GOAL>∃ i, ∀ j ≥ i, abv ((∑ k ∈ range j, f k) * ∑ k ∈ range j, g k - ∑ n ∈ range j, ∑ m ∈ range (n + 1), f m * g (n - m)) < ε
<VAR>α : Type u_1<VAR>β : Type u_2<VAR>inst✝² : LinearOrderedField α<VAR>inst✝¹ : Ring β<VAR>abv : β → α<VAR>inst✝ : IsAbsoluteValue abv<VAR>f g : ℕ → β<VAR>a : ℕ → α<VAR>ha : IsCauSeq abs fun m => ∑ n ∈ range m, abv (f n)<VAR>hb : IsCauSeq abv fun m => ∑ n ∈ range m, g n<VAR>ε : α<VAR>ε0 : 0 < ε<VAR>P : α<VAR>hP : ∀ (i : ℕ), |∑ n ∈ range i, abv (f n)| < P<VAR>Q : α<VAR>hQ : ∀ (i : ℕ), abv (∑ n ∈ range i, g n) < Q<VAR>hP0 : 0 < P<VAR>hPε0 : 0 < ε / (2 * P)<GOAL>∃ i, ∀ j ≥ i, abv ((∑ k ∈ range j, f k) * ∑ k ∈ range j, g k - ∑ n ∈ range j, ∑ m ∈ range (n + 1), f m * g (n - m)) < ε
<VAR>α : Type u_1<VAR>β : Type u_2<VAR>inst✝² : LinearOrderedField α<VAR>inst✝¹ : Ring β<VAR>abv : β → α<VAR>inst✝ : IsAbsoluteValue abv<VAR>f g : ℕ → β<VAR>a : ℕ → α<VAR>ha : IsCauSeq abs fun m => ∑ n ∈ range m, abv (f n)<VAR>hb : IsCauSeq abv fun m => ∑ n ∈ range m, g n<VAR>ε : α<VAR>ε0 : 0 < ε<VAR>P : α<VAR>hP : ∀ (i : ℕ), |∑ n ∈ range i, abv (f n)| < P<VAR>Q : α<VAR>hQ : ∀ (i : ℕ), abv (∑ n ∈ range i, g n) < Q<VAR>hP0 : 0 < P<VAR>hPε0 : 0 < ε / (2 * P)<VAR>N : ℕ<VAR>hN : ∀ j ≥ N, ∀ k ≥ N, abv (∑ n ∈ range j, g n - ∑ n ∈ range k, g n) < ε / (2 * P)<GOAL>∃ i, ∀ j ≥ i, abv ((∑ k ∈ range j, f k) * ∑ k ∈ range j, g k - ∑ n ∈ range j, ∑ m ∈ range (n + 1), f m * g (n - m)) < ε
<VAR>α : Type u_1<VAR>β : Type u_2<VAR>inst✝² : LinearOrderedField α<VAR>inst✝¹ : Ring β<VAR>abv : β → α<VAR>inst✝ : IsAbsoluteValue abv<VAR>f g : ℕ → β<VAR>a : ℕ → α<VAR>ha : IsCauSeq abs fun m => ∑ n ∈ range m, abv (f n)<VAR>hb : IsCauSeq abv fun m => ∑ n ∈ range m, g n<VAR>ε : α<VAR>ε0 : 0 < ε<VAR>P : α<VAR>hP : ∀ (i : ℕ), |∑ n ∈ range i, abv (f n)| < P<VAR>Q : α<VAR>hQ : ∀ (i : ℕ), abv (∑ n ∈ range i, g n) < Q<VAR>hP0 : 0 < P<VAR>hPε0 : 0 < ε / (2 * P)<VAR>N : ℕ<VAR>hN : ∀ j ≥ N, ∀ k ≥ N, abv (∑ n ∈ range j, g n - ∑ n ∈ range k, g n) < ε / (2 * P)<VAR>hQε0 : 0 < ε / (4 * Q)<GOAL>∃ i, ∀ j ≥ i, abv ((∑ k ∈ range j, f k) * ∑ k ∈ range j, g k - ∑ n ∈ range j, ∑ m ∈ range (n + 1), f m * g (n - m)) < ε
<VAR>α : Type u_1<VAR>β : Type u_2<VAR>inst✝² : LinearOrderedField α<VAR>inst✝¹ : Ring β<VAR>abv : β → α<VAR>inst✝ : IsAbsoluteValue abv<VAR>f g : ℕ → β<VAR>a : ℕ → α<VAR>ha : IsCauSeq abs fun m => ∑ n ∈ range m, abv (f n)<VAR>hb : IsCauSeq abv fun m => ∑ n ∈ range m, g n<VAR>ε : α<VAR>ε0 : 0 < ε<VAR>P : α<VAR>hP : ∀ (i : ℕ), |∑ n ∈ range i, abv (f n)| < P<VAR>Q : α<VAR>hQ : ∀ (i : ℕ), abv (∑ n ∈ range i, g n) < Q<VAR>hP0 : 0 < P<VAR>hPε0 : 0 < ε / (2 * P)<VAR>N : ℕ<VAR>hN : ∀ j ≥ N, ∀ k ≥ N, abv (∑ n ∈ range j, g n - ∑ n ∈ range k, g n) < ε / (2 * P)<VAR>hQε0 : 0 < ε / (4 * Q)<VAR>M : ℕ<VAR>hM : ∀ j ≥ M, ∀ k ≥ M, |∑ n ∈ range j, abv (f n) - ∑ n ∈ range k, abv (f n)| < ε / (4 * Q)<GOAL>∃ i, ∀ j ≥ i, abv ((∑ k ∈ range j, f k) * ∑ k ∈ range j, g k - ∑ n ∈ range j, ∑ m ∈ range (n + 1), f m * g (n - m)) < ε
<VAR>α : Type u_1<VAR>β : Type u_2<VAR>inst✝² : LinearOrderedField α<VAR>inst✝¹ : Ring β<VAR>abv : β → α<VAR>inst✝ : IsAbsoluteValue abv<VAR>f g : ℕ → β<VAR>a : ℕ → α<VAR>ha : IsCauSeq abs fun m => ∑ n ∈ range m, abv (f n)<VAR>hb : IsCauSeq abv fun m => ∑ n ∈ range m, g n<VAR>ε : α<VAR>ε0 : 0 < ε<VAR>P : α<VAR>hP : ∀ (i : ℕ), |∑ n ∈ range i, abv (f n)| < P<VAR>Q : α<VAR>hQ : ∀ (i : ℕ), abv (∑ n ∈ range i, g n) < Q<VAR>hP0 : 0 < P<VAR>hPε0 : 0 < ε / (2 * P)<VAR>N : ℕ<VAR>hN : ∀ j ≥ N, ∀ k ≥ N, abv (∑ n ∈ range j, g n - ∑ n ∈ range k, g n) < ε / (2 * P)<VAR>hQε0 : 0 < ε / (4 * Q)<VAR>M : ℕ<VAR>hM : ∀ j ≥ M, ∀ k ≥ M, |∑ n ∈ range j, abv (f n) - ∑ n ∈ range k, abv (f n)| < ε / (4 * Q)<VAR>K : ℕ<VAR>hK : K ≥ 2 * (max N M + 1)<GOAL>abv ((∑ k ∈ range K, f k) * ∑ k ∈ range K, g k - ∑ n ∈ range K, ∑ m ∈ range (n + 1), f m * g (n - m)) < ε
<VAR>α : Type u_1<VAR>β : Type u_2<VAR>inst✝² : LinearOrderedField α<VAR>inst✝¹ : Ring β<VAR>abv : β → α<VAR>inst✝ : IsAbsoluteValue abv<VAR>f g : ℕ → β<VAR>a : ℕ → α<VAR>ha : IsCauSeq abs fun m => ∑ n ∈ range m, abv (f n)<VAR>hb : IsCauSeq abv fun m => ∑ n ∈ range m, g n<VAR>ε : α<VAR>ε0 : 0 < ε<VAR>P : α<VAR>hP : ∀ (i : ℕ), |∑ n ∈ range i, abv (f n)| < P<VAR>Q : α<VAR>hQ : ∀ (i : ℕ), abv (∑ n ∈ range i, g n) < Q<VAR>hP0 : 0 < P<VAR>hPε0 : 0 < ε / (2 * P)<VAR>N : ℕ<VAR>hN : ∀ j ≥ N, ∀ k ≥ N, abv (∑ n ∈ range j, g n - ∑ n ∈ range k, g n) < ε / (2 * P)<VAR>hQε0 : 0 < ε / (4 * Q)<VAR>M : ℕ<VAR>hM : ∀ j ≥ M, ∀ k ≥ M, |∑ n ∈ range j, abv (f n) - ∑ n ∈ range k, abv (f n)| < ε / (4 * Q)<VAR>K : ℕ<VAR>hK : K ≥ 2 * (max N M + 1)<VAR>h₁ : ∑ m ∈ range K, ∑ k ∈ range (m + 1), f k * g (m - k) = ∑ m ∈ range K, ∑ n ∈ range (K - m), f m * g n<GOAL>abv ((∑ k ∈ range K, f k) * ∑ k ∈ range K, g k - ∑ n ∈ range K, ∑ m ∈ range (n + 1), f m * g (n - m)) < ε
<VAR>α : Type u_1<VAR>β : Type u_2<VAR>inst✝² : LinearOrderedField α<VAR>inst✝¹ : Ring β<VAR>abv : β → α<VAR>inst✝ : IsAbsoluteValue abv<VAR>f g : ℕ → β<VAR>a : ℕ → α<VAR>ha : IsCauSeq abs fun m => ∑ n ∈ range m, abv (f n)<VAR>hb : IsCauSeq abv fun m => ∑ n ∈ range m, g n<VAR>ε : α<VAR>ε0 : 0 < ε<VAR>P : α<VAR>hP : ∀ (i : ℕ), |∑ n ∈ range i, abv (f n)| < P<VAR>Q : α<VAR>hQ : ∀ (i : ℕ), abv (∑ n ∈ range i, g n) < Q<VAR>hP0 : 0 < P<VAR>hPε0 : 0 < ε / (2 * P)<VAR>N : ℕ<VAR>hN : ∀ j ≥ N, ∀ k ≥ N, abv (∑ n ∈ range j, g n - ∑ n ∈ range k, g n) < ε / (2 * P)<VAR>hQε0 : 0 < ε / (4 * Q)<VAR>M : ℕ<VAR>hM : ∀ j ≥ M, ∀ k ≥ M, |∑ n ∈ range j, abv (f n) - ∑ n ∈ range k, abv (f n)| < ε / (4 * Q)<VAR>K : ℕ<VAR>hK : K ≥ 2 * (max N M + 1)<VAR>h₁ : ∑ m ∈ range K, ∑ k ∈ range (m + 1), f k * g (m - k) = ∑ m ∈ range K, ∑ n ∈ range (K - m), f m * g n<VAR>h₂ : (fun i => ∑ k ∈ range (K - i), f i * g k) = fun i => f i * ∑ k ∈ range (K - i), g k<GOAL>abv ((∑ k ∈ range K, f k) * ∑ k ∈ range K, g k - ∑ n ∈ range K, ∑ m ∈ range (n + 1), f m * g (n - m)) < ε
<VAR>α : Type u_1<VAR>β : Type u_2<VAR>inst✝² : LinearOrderedField α<VAR>inst✝¹ : Ring β<VAR>abv : β → α<VAR>inst✝ : IsAbsoluteValue abv<VAR>f g : ℕ → β<VAR>a : ℕ → α<VAR>ha : IsCauSeq abs fun m => ∑ n ∈ range m, abv (f n)<VAR>hb : IsCauSeq abv fun m => ∑ n ∈ range m, g n<VAR>ε : α<VAR>ε0 : 0 < ε<VAR>P : α<VAR>hP : ∀ (i : ℕ), |∑ n ∈ range i, abv (f n)| < P<VAR>Q : α<VAR>hQ : ∀ (i : ℕ), abv (∑ n ∈ range i, g n) < Q<VAR>hP0 : 0 < P<VAR>hPε0 : 0 < ε / (2 * P)<VAR>N : ℕ<VAR>hN : ∀ j ≥ N, ∀ k ≥ N, abv (∑ n ∈ range j, g n - ∑ n ∈ range k, g n) < ε / (2 * P)<VAR>hQε0 : 0 < ε / (4 * Q)<VAR>M : ℕ<VAR>hM : ∀ j ≥ M, ∀ k ≥ M, |∑ n ∈ range j, abv (f n) - ∑ n ∈ range k, abv (f n)| < ε / (4 * Q)<VAR>K : ℕ<VAR>hK : K ≥ 2 * (max N M + 1)<VAR>h₁ : ∑ m ∈ range K, ∑ k ∈ range (m + 1), f k * g (m - k) = ∑ m ∈ range K, ∑ n ∈ range (K - m), f m * g n<VAR>h₂ : (fun i => ∑ k ∈ range (K - i), f i * g k) = fun i => f i * ∑ k ∈ range (K - i), g k<VAR>h₃ : ∑ i ∈ range K, f i * ∑ k ∈ range (K - i), g k = ∑ i ∈ range K, f i * (∑ k ∈ range (K - i), g k - ∑ k ∈ range K, g k) + ∑ i ∈ range K, f i * ∑ k ∈ range K, g k<GOAL>abv ((∑ k ∈ range K, f k) * ∑ k ∈ range K, g k - ∑ n ∈ range K, ∑ m ∈ range (n + 1), f m * g (n - m)) < ε
<VAR>α : Type u_1<VAR>β : Type u_2<VAR>inst✝² : LinearOrderedField α<VAR>inst✝¹ : Ring β<VAR>abv : β → α<VAR>inst✝ : IsAbsoluteValue abv<VAR>f g : ℕ → β<VAR>a : ℕ → α<VAR>ha : IsCauSeq abs fun m => ∑ n ∈ range m, abv (f n)<VAR>hb : IsCauSeq abv fun m => ∑ n ∈ range m, g n<VAR>ε : α<VAR>ε0 : 0 < ε<VAR>P : α<VAR>hP : ∀ (i : ℕ), |∑ n ∈ range i, abv (f n)| < P<VAR>Q : α<VAR>hQ : ∀ (i : ℕ), abv (∑ n ∈ range i, g n) < Q<VAR>hP0 : 0 < P<VAR>hPε0 : 0 < ε / (2 * P)<VAR>N : ℕ<VAR>hN : ∀ j ≥ N, ∀ k ≥ N, abv (∑ n ∈ range j, g n - ∑ n ∈ range k, g n) < ε / (2 * P)<VAR>hQε0 : 0 < ε / (4 * Q)<VAR>M : ℕ<VAR>hM : ∀ j ≥ M, ∀ k ≥ M, |∑ n ∈ range j, abv (f n) - ∑ n ∈ range k, abv (f n)| < ε / (4 * Q)<VAR>K : ℕ<VAR>hK : K ≥ 2 * (max N M + 1)<VAR>h₁ : ∑ m ∈ range K, ∑ k ∈ range (m + 1), f k * g (m - k) = ∑ m ∈ range K, ∑ n ∈ range (K - m), f m * g n<VAR>h₂ : (fun i => ∑ k ∈ range (K - i), f i * g k) = fun i => f i * ∑ k ∈ range (K - i), g k<VAR>h₃ : ∑ i ∈ range K, f i * ∑ k ∈ range (K - i), g k = ∑ i ∈ range K, f i * (∑ k ∈ range (K - i), g k - ∑ k ∈ range K, g k) + ∑ i ∈ range K, f i * ∑ k ∈ range K, g k<VAR>two_mul_two : 4 = 2 * 2<GOAL>abv ((∑ k ∈ range K, f k) * ∑ k ∈ range K, g k - ∑ n ∈ range K, ∑ m ∈ range (n + 1), f m * g (n - m)) < ε
<VAR>α : Type u_1<VAR>β : Type u_2<VAR>inst✝² : LinearOrderedField α<VAR>inst✝¹ : Ring β<VAR>abv : β → α<VAR>inst✝ : IsAbsoluteValue abv<VAR>f g : ℕ → β<VAR>a : ℕ → α<VAR>ha : IsCauSeq abs fun m => ∑ n ∈ range m, abv (f n)<VAR>hb : IsCauSeq abv fun m => ∑ n ∈ range m, g n<VAR>ε : α<VAR>ε0 : 0 < ε<VAR>P : α<VAR>hP : ∀ (i : ℕ), |∑ n ∈ range i, abv (f n)| < P<VAR>Q : α<VAR>hQ : ∀ (i : ℕ), abv (∑ n ∈ range i, g n) < Q<VAR>hP0 : 0 < P<VAR>hPε0 : 0 < ε / (2 * P)<VAR>N : ℕ<VAR>hN : ∀ j ≥ N, ∀ k ≥ N, abv (∑ n ∈ range j, g n - ∑ n ∈ range k, g n) < ε / (2 * P)<VAR>hQε0 : 0 < ε / (4 * Q)<VAR>M : ℕ<VAR>hM : ∀ j ≥ M, ∀ k ≥ M, |∑ n ∈ range j, abv (f n) - ∑ n ∈ range k, abv (f n)| < ε / (4 * Q)<VAR>K : ℕ<VAR>hK : K ≥ 2 * (max N M + 1)<VAR>h₁ : ∑ m ∈ range K, ∑ k ∈ range (m + 1), f k * g (m - k) = ∑ m ∈ range K, ∑ n ∈ range (K - m), f m * g n<VAR>h₂ : (fun i => ∑ k ∈ range (K - i), f i * g k) = fun i => f i * ∑ k ∈ range (K - i), g k<VAR>h₃ : ∑ i ∈ range K, f i * ∑ k ∈ range (K - i), g k = ∑ i ∈ range K, f i * (∑ k ∈ range (K - i), g k - ∑ k ∈ range K, g k) + ∑ i ∈ range K, f i * ∑ k ∈ range K, g k<VAR>two_mul_two : 4 = 2 * 2<VAR>hQ0 : Q ≠ 0<GOAL>abv ((∑ k ∈ range K, f k) * ∑ k ∈ range K, g k - ∑ n ∈ range K, ∑ m ∈ range (n + 1), f m * g (n - m)) < ε
<VAR>α : Type u_1<VAR>β : Type u_2<VAR>inst✝² : LinearOrderedField α<VAR>inst✝¹ : Ring β<VAR>abv : β → α<VAR>inst✝ : IsAbsoluteValue abv<VAR>f g : ℕ → β<VAR>a : ℕ → α<VAR>ha : IsCauSeq abs fun m => ∑ n ∈ range m, abv (f n)<VAR>hb : IsCauSeq abv fun m => ∑ n ∈ range m, g n<VAR>ε : α<VAR>ε0 : 0 < ε<VAR>P : α<VAR>hP : ∀ (i : ℕ), |∑ n ∈ range i, abv (f n)| < P<VAR>Q : α<VAR>hQ : ∀ (i : ℕ), abv (∑ n ∈ range i, g n) < Q<VAR>hP0 : 0 < P<VAR>hPε0 : 0 < ε / (2 * P)<VAR>N : ℕ<VAR>hN : ∀ j ≥ N, ∀ k ≥ N, abv (∑ n ∈ range j, g n - ∑ n ∈ range k, g n) < ε / (2 * P)<VAR>hQε0 : 0 < ε / (4 * Q)<VAR>M : ℕ<VAR>hM : ∀ j ≥ M, ∀ k ≥ M, |∑ n ∈ range j, abv (f n) - ∑ n ∈ range k, abv (f n)| < ε / (4 * Q)<VAR>K : ℕ<VAR>hK : K ≥ 2 * (max N M + 1)<VAR>h₁ : ∑ m ∈ range K, ∑ k ∈ range (m + 1), f k * g (m - k) = ∑ m ∈ range K, ∑ n ∈ range (K - m), f m * g n<VAR>h₂ : (fun i => ∑ k ∈ range (K - i), f i * g k) = fun i => f i * ∑ k ∈ range (K - i), g k<VAR>h₃ : ∑ i ∈ range K, f i * ∑ k ∈ range (K - i), g k = ∑ i ∈ range K, f i * (∑ k ∈ range (K - i), g k - ∑ k ∈ range K, g k) + ∑ i ∈ range K, f i * ∑ k ∈ range K, g k<VAR>two_mul_two : 4 = 2 * 2<VAR>hQ0 : Q ≠ 0<VAR>h2Q0 : 2 * Q ≠ 0<GOAL>abv ((∑ k ∈ range K, f k) * ∑ k ∈ range K, g k - ∑ n ∈ range K, ∑ m ∈ range (n + 1), f m * g (n - m)) < ε
<VAR>α : Type u_1<VAR>β : Type u_2<VAR>inst✝² : LinearOrderedField α<VAR>inst✝¹ : Ring β<VAR>abv : β → α<VAR>inst✝ : IsAbsoluteValue abv<VAR>f g : ℕ → β<VAR>a : ℕ → α<VAR>ha : IsCauSeq abs fun m => ∑ n ∈ range m, abv (f n)<VAR>hb : IsCauSeq abv fun m => ∑ n ∈ range m, g n<VAR>ε : α<VAR>ε0 : 0 < ε<VAR>P : α<VAR>hP : ∀ (i : ℕ), |∑ n ∈ range i, abv (f n)| < P<VAR>Q : α<VAR>hQ : ∀ (i : ℕ), abv (∑ n ∈ range i, g n) < Q<VAR>hP0 : 0 < P<VAR>hPε0 : 0 < ε / (2 * P)<VAR>N : ℕ<VAR>hN : ∀ j ≥ N, ∀ k ≥ N, abv (∑ n ∈ range j, g n - ∑ n ∈ range k, g n) < ε / (2 * P)<VAR>hQε0 : 0 < ε / (4 * Q)<VAR>M : ℕ<VAR>hM : ∀ j ≥ M, ∀ k ≥ M, |∑ n ∈ range j, abv (f n) - ∑ n ∈ range k, abv (f n)| < ε / (4 * Q)<VAR>K : ℕ<VAR>hK : K ≥ 2 * (max N M + 1)<VAR>h₁ : ∑ m ∈ range K, ∑ k ∈ range (m + 1), f k * g (m - k) = ∑ m ∈ range K, ∑ n ∈ range (K - m), f m * g n<VAR>h₂ : (fun i => ∑ k ∈ range (K - i), f i * g k) = fun i => f i * ∑ k ∈ range (K - i), g k<VAR>h₃ : ∑ i ∈ range K, f i * ∑ k ∈ range (K - i), g k = ∑ i ∈ range K, f i * (∑ k ∈ range (K - i), g k - ∑ k ∈ range K, g k) + ∑ i ∈ range K, f i * ∑ k ∈ range K, g k<VAR>two_mul_two : 4 = 2 * 2<VAR>hQ0 : Q ≠ 0<VAR>h2Q0 : 2 * Q ≠ 0<VAR>hε : ε / (2 * P) * P + ε / (4 * Q) * (2 * Q) = ε<GOAL>abv ((∑ k ∈ range K, f k) * ∑ k ∈ range K, g k - ∑ n ∈ range K, ∑ m ∈ range (n + 1), f m * g (n - m)) < ε
<VAR>α : Type u_1<VAR>β : Type u_2<VAR>inst✝² : LinearOrderedField α<VAR>inst✝¹ : Ring β<VAR>abv : β → α<VAR>inst✝ : IsAbsoluteValue abv<VAR>f g : ℕ → β<VAR>a : ℕ → α<VAR>ha : IsCauSeq abs fun m => ∑ n ∈ range m, abv (f n)<VAR>hb : IsCauSeq abv fun m => ∑ n ∈ range m, g n<VAR>ε : α<VAR>ε0 : 0 < ε<VAR>P : α<VAR>hP : ∀ (i : ℕ), |∑ n ∈ range i, abv (f n)| < P<VAR>Q : α<VAR>hQ : ∀ (i : ℕ), abv (∑ n ∈ range i, g n) < Q<VAR>hP0 : 0 < P<VAR>hPε0 : 0 < ε / (2 * P)<VAR>N : ℕ<VAR>hN : ∀ j ≥ N, ∀ k ≥ N, abv (∑ n ∈ range j, g n - ∑ n ∈ range k, g n) < ε / (2 * P)<VAR>hQε0 : 0 < ε / (4 * Q)<VAR>M : ℕ<VAR>hM : ∀ j ≥ M, ∀ k ≥ M, |∑ n ∈ range j, abv (f n) - ∑ n ∈ range k, abv (f n)| < ε / (4 * Q)<VAR>K : ℕ<VAR>hK : K ≥ 2 * (max N M + 1)<VAR>h₁ : ∑ m ∈ range K, ∑ k ∈ range (m + 1), f k * g (m - k) = ∑ m ∈ range K, ∑ n ∈ range (K - m), f m * g n<VAR>h₂ : (fun i => ∑ k ∈ range (K - i), f i * g k) = fun i => f i * ∑ k ∈ range (K - i), g k<VAR>h₃ : ∑ i ∈ range K, f i * ∑ k ∈ range (K - i), g k = ∑ i ∈ range K, f i * (∑ k ∈ range (K - i), g k - ∑ k ∈ range K, g k) + ∑ i ∈ range K, f i * ∑ k ∈ range K, g k<VAR>two_mul_two : 4 = 2 * 2<VAR>hQ0 : Q ≠ 0<VAR>h2Q0 : 2 * Q ≠ 0<VAR>hε : ε / (2 * P) * P + ε / (4 * Q) * (2 * Q) = ε<VAR>hNMK : max N M + 1 < K<GOAL>abv ((∑ k ∈ range K, f k) * ∑ k ∈ range K, g k - ∑ n ∈ range K, ∑ m ∈ range (n + 1), f m * g (n - m)) < ε
<VAR>α : Type u_1<VAR>β : Type u_2<VAR>inst✝² : LinearOrderedField α<VAR>inst✝¹ : Ring β<VAR>abv : β → α<VAR>inst✝ : IsAbsoluteValue abv<VAR>f g : ℕ → β<VAR>a : ℕ → α<VAR>ha : IsCauSeq abs fun m => ∑ n ∈ range m, abv (f n)<VAR>hb : IsCauSeq abv fun m => ∑ n ∈ range m, g n<VAR>ε : α<VAR>ε0 : 0 < ε<VAR>P : α<VAR>hP : ∀ (i : ℕ), |∑ n ∈ range i, abv (f n)| < P<VAR>Q : α<VAR>hQ : ∀ (i : ℕ), abv (∑ n ∈ range i, g n) < Q<VAR>hP0 : 0 < P<VAR>hPε0 : 0 < ε / (2 * P)<VAR>N : ℕ<VAR>hN : ∀ j ≥ N, ∀ k ≥ N, abv (∑ n ∈ range j, g n - ∑ n ∈ range k, g n) < ε / (2 * P)<VAR>hQε0 : 0 < ε / (4 * Q)<VAR>M : ℕ<VAR>hM : ∀ j ≥ M, ∀ k ≥ M, |∑ n ∈ range j, abv (f n) - ∑ n ∈ range k, abv (f n)| < ε / (4 * Q)<VAR>K : ℕ<VAR>hK : K ≥ 2 * (max N M + 1)<VAR>h₁ : ∑ m ∈ range K, ∑ k ∈ range (m + 1), f k * g (m - k) = ∑ m ∈ range K, ∑ n ∈ range (K - m), f m * g n<VAR>h₂ : (fun i => ∑ k ∈ range (K - i), f i * g k) = fun i => f i * ∑ k ∈ range (K - i), g k<VAR>h₃ : ∑ i ∈ range K, f i * ∑ k ∈ range (K - i), g k = ∑ i ∈ range K, f i * (∑ k ∈ range (K - i), g k - ∑ k ∈ range K, g k) + ∑ i ∈ range K, f i * ∑ k ∈ range K, g k<VAR>two_mul_two : 4 = 2 * 2<VAR>hQ0 : Q ≠ 0<VAR>h2Q0 : 2 * Q ≠ 0<VAR>hε : ε / (2 * P) * P + ε / (4 * Q) * (2 * Q) = ε<VAR>hNMK : max N M + 1 < K<VAR>hKN : N < K<GOAL>abv ((∑ k ∈ range K, f k) * ∑ k ∈ range K, g k - ∑ n ∈ range K, ∑ m ∈ range (n + 1), f m * g (n - m)) < ε
<VAR>α : Type u_1<VAR>β : Type u_2<VAR>inst✝² : LinearOrderedField α<VAR>inst✝¹ : Ring β<VAR>abv : β → α<VAR>inst✝ : IsAbsoluteValue abv<VAR>f g : ℕ → β<VAR>a : ℕ → α<VAR>ha : IsCauSeq abs fun m => ∑ n ∈ range m, abv (f n)<VAR>hb : IsCauSeq abv fun m => ∑ n ∈ range m, g n<VAR>ε : α<VAR>ε0 : 0 < ε<VAR>P : α<VAR>hP : ∀ (i : ℕ), |∑ n ∈ range i, abv (f n)| < P<VAR>Q : α<VAR>hQ : ∀ (i : ℕ), abv (∑ n ∈ range i, g n) < Q<VAR>hP0 : 0 < P<VAR>hPε0 : 0 < ε / (2 * P)<VAR>N : ℕ<VAR>hN : ∀ j ≥ N, ∀ k ≥ N, abv (∑ n ∈ range j, g n - ∑ n ∈ range k, g n) < ε / (2 * P)<VAR>hQε0 : 0 < ε / (4 * Q)<VAR>M : ℕ<VAR>hM : ∀ j ≥ M, ∀ k ≥ M, |∑ n ∈ range j, abv (f n) - ∑ n ∈ range k, abv (f n)| < ε / (4 * Q)<VAR>K : ℕ<VAR>hK : K ≥ 2 * (max N M + 1)<VAR>h₁ : ∑ m ∈ range K, ∑ k ∈ range (m + 1), f k * g (m - k) = ∑ m ∈ range K, ∑ n ∈ range (K - m), f m * g n<VAR>h₂ : (fun i => ∑ k ∈ range (K - i), f i * g k) = fun i => f i * ∑ k ∈ range (K - i), g k<VAR>h₃ : ∑ i ∈ range K, f i * ∑ k ∈ range (K - i), g k = ∑ i ∈ range K, f i * (∑ k ∈ range (K - i), g k - ∑ k ∈ range K, g k) + ∑ i ∈ range K, f i * ∑ k ∈ range K, g k<VAR>two_mul_two : 4 = 2 * 2<VAR>hQ0 : Q ≠ 0<VAR>h2Q0 : 2 * Q ≠ 0<VAR>hε : ε / (2 * P) * P + ε / (4 * Q) * (2 * Q) = ε<VAR>hNMK : max N M + 1 < K<VAR>hKN : N < K<VAR>hsumlesum : ∑ i ∈ range (max N M + 1), abv (f i) * abv (∑ k ∈ range (K - i), g k - ∑ k ∈ range K, g k) ≤ ∑ i ∈ range (max N M + 1), abv (f i) * (ε / (2 * P))<GOAL>abv ((∑ k ∈ range K, f k) * ∑ k ∈ range K, g k - ∑ n ∈ range K, ∑ m ∈ range (n + 1), f m * g (n - m)) < ε
<VAR>α : Type u_1<VAR>β : Type u_2<VAR>inst✝² : LinearOrderedField α<VAR>inst✝¹ : Ring β<VAR>abv : β → α<VAR>inst✝ : IsAbsoluteValue abv<VAR>f g : ℕ → β<VAR>a : ℕ → α<VAR>ha : IsCauSeq abs fun m => ∑ n ∈ range m, abv (f n)<VAR>hb : IsCauSeq abv fun m => ∑ n ∈ range m, g n<VAR>ε : α<VAR>ε0 : 0 < ε<VAR>P : α<VAR>hP : ∀ (i : ℕ), |∑ n ∈ range i, abv (f n)| < P<VAR>Q : α<VAR>hQ : ∀ (i : ℕ), abv (∑ n ∈ range i, g n) < Q<VAR>hP0 : 0 < P<VAR>hPε0 : 0 < ε / (2 * P)<VAR>N : ℕ<VAR>hN : ∀ j ≥ N, ∀ k ≥ N, abv (∑ n ∈ range j, g n - ∑ n ∈ range k, g n) < ε / (2 * P)<VAR>hQε0 : 0 < ε / (4 * Q)<VAR>M : ℕ<VAR>hM : ∀ j ≥ M, ∀ k ≥ M, |∑ n ∈ range j, abv (f n) - ∑ n ∈ range k, abv (f n)| < ε / (4 * Q)<VAR>K : ℕ<VAR>hK : K ≥ 2 * (max N M + 1)<VAR>h₁ : ∑ m ∈ range K, ∑ k ∈ range (m + 1), f k * g (m - k) = ∑ m ∈ range K, ∑ n ∈ range (K - m), f m * g n<VAR>h₂ : (fun i => ∑ k ∈ range (K - i), f i * g k) = fun i => f i * ∑ k ∈ range (K - i), g k<VAR>h₃ : ∑ i ∈ range K, f i * ∑ k ∈ range (K - i), g k = ∑ i ∈ range K, f i * (∑ k ∈ range (K - i), g k - ∑ k ∈ range K, g k) + ∑ i ∈ range K, f i * ∑ k ∈ range K, g k<VAR>two_mul_two : 4 = 2 * 2<VAR>hQ0 : Q ≠ 0<VAR>h2Q0 : 2 * Q ≠ 0<VAR>hε : ε / (2 * P) * P + ε / (4 * Q) * (2 * Q) = ε<VAR>hNMK : max N M + 1 < K<VAR>hKN : N < K<VAR>hsumlesum : ∑ i ∈ range (max N M + 1), abv (f i) * abv (∑ k ∈ range (K - i), g k - ∑ k ∈ range K, g k) ≤ ∑ i ∈ range (max N M + 1), abv (f i) * (ε / (2 * P))<VAR>hsumltP : ∑ n ∈ range (max N M + 1), abv (f n) < P<GOAL>abv ((∑ k ∈ range K, f k) * ∑ k ∈ range K, g k - ∑ n ∈ range K, ∑ m ∈ range (n + 1), f m * g (n - m)) < ε