l3lab/ntp-mathlib huggingface dataset:
/-- A family `f` of mutually-orthogonal elements of `E` is summable, if and only if
`(fun i ↦ ‖f i‖ ^ 2)` is summable. -/
theorem OrthogonalFamily.summable_iff_norm_sq_summable [CompleteSpace E] (f : ∀ i, G i) :
    (Summable fun i => V i (f i)) ↔ Summable fun i => ‖f i‖ ^ 2 := by
  classical
    clear dec_ι
    simp only [summable_iff_cauchySeq_finset, NormedAddCommGroup.cauchySeq_iff, Real.norm_eq_abs]
    constructor
    · intro hf ε hε
      obtain ⟨a, H⟩ := hf _ (sqrt_pos.mpr hε)
      use a
      intro s₁ hs₁ s₂ hs₂
      

Proof state:
case h
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
dec_E : DecidableEq E
ι : Type u_4
G : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
dec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)
inst✝ : CompleteSpace E
f : (i : ι) → G i
hf :
  ∀ ε > 0, ∃ N, ∀ (m : Finset ι), N ≤ m → ∀ (n : Finset ι), N ≤ n → ‖∑ i in m, (V i) (f i) - ∑ i in n, (V i) (f i)‖ < ε
ε : ℝ
hε : ε > 0
a : Finset ι
H : ∀ (m : Finset ι), a ≤ m → ∀ (n : Finset ι), a ≤ n → ‖∑ i in m, (V i) (f i) - ∑ i in n, (V i) (f i)‖ < sqrt ε
s₁ : Finset ι
hs₁ : a ≤ s₁
s₂ : Finset ι
hs₂ : a ≤ s₂
⊢ |∑ i in s₁, ‖f i‖ ^ 2 - ∑ i in s₂, ‖f i‖ ^ 2| < ε

Next tactic:
rw [← Finset.sum_sdiff_sub_sum_sdiff]
/-- A family `f` of mutually-orthogonal elements of `E` is summable, if and only if
`(fun i ↦ ‖f i‖ ^ 2)` is summable. -/
theorem OrthogonalFamily.summable_iff_norm_sq_summable [CompleteSpace E] (f : ∀ i, G i) :
    (Summable fun i => V i (f i)) ↔ Summable fun i => ‖f i‖ ^ 2 := by
  classical
    clear dec_ι
    simp only [summable_iff_cauchySeq_finset, NormedAddCommGroup.cauchySeq_iff, Real.norm_eq_abs]
    constructor
    · intro hf ε hε
      obtain ⟨a, H⟩ := hf _ (sqrt_pos.mpr hε)
      use a
      intro s₁ hs₁ s₂ hs₂
      rw [← Finset.sum_sdiff_sub_sum_sdiff]
      

Proof state:
case h
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
dec_E : DecidableEq E
ι : Type u_4
G : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
dec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)
inst✝ : CompleteSpace E
f : (i : ι) → G i
hf :
  ∀ ε > 0, ∃ N, ∀ (m : Finset ι), N ≤ m → ∀ (n : Finset ι), N ≤ n → ‖∑ i in m, (V i) (f i) - ∑ i in n, (V i) (f i)‖ < ε
ε : ℝ
hε : ε > 0
a : Finset ι
H : ∀ (m : Finset ι), a ≤ m → ∀ (n : Finset ι), a ≤ n → ‖∑ i in m, (V i) (f i) - ∑ i in n, (V i) (f i)‖ < sqrt ε
s₁ : Finset ι
hs₁ : a ≤ s₁
s₂ : Finset ι
hs₂ : a ≤ s₂
⊢ |∑ x in s₁ \ s₂, ‖f x‖ ^ 2 - ∑ x in s₂ \ s₁, ‖f x‖ ^ 2| < ε

Next tactic:
refine' (abs_sub _ _).trans_lt _
/-- A family `f` of mutually-orthogonal elements of `E` is summable, if and only if
`(fun i ↦ ‖f i‖ ^ 2)` is summable. -/
theorem OrthogonalFamily.summable_iff_norm_sq_summable [CompleteSpace E] (f : ∀ i, G i) :
    (Summable fun i => V i (f i)) ↔ Summable fun i => ‖f i‖ ^ 2 := by
  classical
    clear dec_ι
    simp only [summable_iff_cauchySeq_finset, NormedAddCommGroup.cauchySeq_iff, Real.norm_eq_abs]
    constructor
    · intro hf ε hε
      obtain ⟨a, H⟩ := hf _ (sqrt_pos.mpr hε)
      use a
      intro s₁ hs₁ s₂ hs₂
      rw [← Finset.sum_sdiff_sub_sum_sdiff]
      refine' (abs_sub _ _).trans_lt _
      

Proof state:
case h
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
dec_E : DecidableEq E
ι : Type u_4
G : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
dec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)
inst✝ : CompleteSpace E
f : (i : ι) → G i
hf :
  ∀ ε > 0, ∃ N, ∀ (m : Finset ι), N ≤ m → ∀ (n : Finset ι), N ≤ n → ‖∑ i in m, (V i) (f i) - ∑ i in n, (V i) (f i)‖ < ε
ε : ℝ
hε : ε > 0
a : Finset ι
H : ∀ (m : Finset ι), a ≤ m → ∀ (n : Finset ι), a ≤ n → ‖∑ i in m, (V i) (f i) - ∑ i in n, (V i) (f i)‖ < sqrt ε
s₁ : Finset ι
hs₁ : a ≤ s₁
s₂ : Finset ι
hs₂ : a ≤ s₂
⊢ |∑ x in s₁ \ s₂, ‖f x‖ ^ 2| + |∑ x in s₂ \ s₁, ‖f x‖ ^ 2| < ε

Next tactic:
have : ∀ i, 0 ≤ ‖f i‖ ^ 2 := fun i : ι => sq_nonneg _
/-- A family `f` of mutually-orthogonal elements of `E` is summable, if and only if
`(fun i ↦ ‖f i‖ ^ 2)` is summable. -/
theorem OrthogonalFamily.summable_iff_norm_sq_summable [CompleteSpace E] (f : ∀ i, G i) :
    (Summable fun i => V i (f i)) ↔ Summable fun i => ‖f i‖ ^ 2 := by
  classical
    clear dec_ι
    simp only [summable_iff_cauchySeq_finset, NormedAddCommGroup.cauchySeq_iff, Real.norm_eq_abs]
    constructor
    · intro hf ε hε
      obtain ⟨a, H⟩ := hf _ (sqrt_pos.mpr hε)
      use a
      intro s₁ hs₁ s₂ hs₂
      rw [← Finset.sum_sdiff_sub_sum_sdiff]
      refine' (abs_sub _ _).trans_lt _
      have : ∀ i, 0 ≤ ‖f i‖ ^ 2 := fun i : ι => sq_nonneg _
      

Proof state:
case h
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
dec_E : DecidableEq E
ι : Type u_4
G : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
dec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)
inst✝ : CompleteSpace E
f : (i : ι) → G i
hf :
  ∀ ε > 0, ∃ N, ∀ (m : Finset ι), N ≤ m → ∀ (n : Finset ι), N ≤ n → ‖∑ i in m, (V i) (f i) - ∑ i in n, (V i) (f i)‖ < ε
ε : ℝ
hε : ε > 0
a : Finset ι
H : ∀ (m : Finset ι), a ≤ m → ∀ (n : Finset ι), a ≤ n → ‖∑ i in m, (V i) (f i) - ∑ i in n, (V i) (f i)‖ < sqrt ε
s₁ : Finset ι
hs₁ : a ≤ s₁
s₂ : Finset ι
hs₂ : a ≤ s₂
this : ∀ (i : ι), 0 ≤ ‖f i‖ ^ 2
⊢ |∑ x in s₁ \ s₂, ‖f x‖ ^ 2| + |∑ x in s₂ \ s₁, ‖f x‖ ^ 2| < ε

Next tactic:
simp only [Finset.abs_sum_of_nonneg' this]
/-- A family `f` of mutually-orthogonal elements of `E` is summable, if and only if
`(fun i ↦ ‖f i‖ ^ 2)` is summable. -/
theorem OrthogonalFamily.summable_iff_norm_sq_summable [CompleteSpace E] (f : ∀ i, G i) :
    (Summable fun i => V i (f i)) ↔ Summable fun i => ‖f i‖ ^ 2 := by
  classical
    clear dec_ι
    simp only [summable_iff_cauchySeq_finset, NormedAddCommGroup.cauchySeq_iff, Real.norm_eq_abs]
    constructor
    · intro hf ε hε
      obtain ⟨a, H⟩ := hf _ (sqrt_pos.mpr hε)
      use a
      intro s₁ hs₁ s₂ hs₂
      rw [← Finset.sum_sdiff_sub_sum_sdiff]
      refine' (abs_sub _ _).trans_lt _
      have : ∀ i, 0 ≤ ‖f i‖ ^ 2 := fun i : ι => sq_nonneg _
      simp only [Finset.abs_sum_of_nonneg' this]
      

Proof state:
case h
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
dec_E : DecidableEq E
ι : Type u_4
G : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
dec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)
inst✝ : CompleteSpace E
f : (i : ι) → G i
hf :
  ∀ ε > 0, ∃ N, ∀ (m : Finset ι), N ≤ m → ∀ (n : Finset ι), N ≤ n → ‖∑ i in m, (V i) (f i) - ∑ i in n, (V i) (f i)‖ < ε
ε : ℝ
hε : ε > 0
a : Finset ι
H : ∀ (m : Finset ι), a ≤ m → ∀ (n : Finset ι), a ≤ n → ‖∑ i in m, (V i) (f i) - ∑ i in n, (V i) (f i)‖ < sqrt ε
s₁ : Finset ι
hs₁ : a ≤ s₁
s₂ : Finset ι
hs₂ : a ≤ s₂
this : ∀ (i : ι), 0 ≤ ‖f i‖ ^ 2
⊢ ∑ x in s₁ \ s₂, ‖f x‖ ^ 2 + ∑ x in s₂ \ s₁, ‖f x‖ ^ 2 < ε

Next tactic:
have : ((∑ i in s₁ \ s₂, ‖f i‖ ^ 2) + ∑ i in s₂ \ s₁, ‖f i‖ ^ 2) < sqrt ε ^ 2 := by
        rw [← hV.norm_sq_diff_sum, sq_lt_sq, abs_of_nonneg (sqrt_nonneg _),
          abs_of_nonneg (norm_nonneg _)]
        exact H s₁ hs₁ s₂ hs₂
/-- A family `f` of mutually-orthogonal elements of `E` is summable, if and only if
`(fun i ↦ ‖f i‖ ^ 2)` is summable. -/
theorem OrthogonalFamily.summable_iff_norm_sq_summable [CompleteSpace E] (f : ∀ i, G i) :
    (Summable fun i => V i (f i)) ↔ Summable fun i => ‖f i‖ ^ 2 := by
  classical
    clear dec_ι
    simp only [summable_iff_cauchySeq_finset, NormedAddCommGroup.cauchySeq_iff, Real.norm_eq_abs]
    constructor
    · intro hf ε hε
      obtain ⟨a, H⟩ := hf _ (sqrt_pos.mpr hε)
      use a
      intro s₁ hs₁ s₂ hs₂
      rw [← Finset.sum_sdiff_sub_sum_sdiff]
      refine' (abs_sub _ _).trans_lt _
      have : ∀ i, 0 ≤ ‖f i‖ ^ 2 := fun i : ι => sq_nonneg _
      simp only [Finset.abs_sum_of_nonneg' this]
      have : ((∑ i in s₁ \ s₂, ‖f i‖ ^ 2) + ∑ i in s₂ \ s₁, ‖f i‖ ^ 2) < sqrt ε ^ 2 := by
        

Proof state:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
dec_E : DecidableEq E
ι : Type u_4
G : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
dec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)
inst✝ : CompleteSpace E
f : (i : ι) → G i
hf :
  ∀ ε > 0, ∃ N, ∀ (m : Finset ι), N ≤ m → ∀ (n : Finset ι), N ≤ n → ‖∑ i in m, (V i) (f i) - ∑ i in n, (V i) (f i)‖ < ε
ε : ℝ
hε : ε > 0
a : Finset ι
H : ∀ (m : Finset ι), a ≤ m → ∀ (n : Finset ι), a ≤ n → ‖∑ i in m, (V i) (f i) - ∑ i in n, (V i) (f i)‖ < sqrt ε
s₁ : Finset ι
hs₁ : a ≤ s₁
s₂ : Finset ι
hs₂ : a ≤ s₂
this : ∀ (i : ι), 0 ≤ ‖f i‖ ^ 2
⊢ ∑ i in s₁ \ s₂, ‖f i‖ ^ 2 + ∑ i in s₂ \ s₁, ‖f i‖ ^ 2 < sqrt ε ^ 2

Next tactic:
rw [← hV.norm_sq_diff_sum, sq_lt_sq, abs_of_nonneg (sqrt_nonneg _),
          abs_of_nonneg (norm_nonneg _)]
/-- A family `f` of mutually-orthogonal elements of `E` is summable, if and only if
`(fun i ↦ ‖f i‖ ^ 2)` is summable. -/
theorem OrthogonalFamily.summable_iff_norm_sq_summable [CompleteSpace E] (f : ∀ i, G i) :
    (Summable fun i => V i (f i)) ↔ Summable fun i => ‖f i‖ ^ 2 := by
  classical
    clear dec_ι
    simp only [summable_iff_cauchySeq_finset, NormedAddCommGroup.cauchySeq_iff, Real.norm_eq_abs]
    constructor
    · intro hf ε hε
      obtain ⟨a, H⟩ := hf _ (sqrt_pos.mpr hε)
      use a
      intro s₁ hs₁ s₂ hs₂
      rw [← Finset.sum_sdiff_sub_sum_sdiff]
      refine' (abs_sub _ _).trans_lt _
      have : ∀ i, 0 ≤ ‖f i‖ ^ 2 := fun i : ι => sq_nonneg _
      simp only [Finset.abs_sum_of_nonneg' this]
      have : ((∑ i in s₁ \ s₂, ‖f i‖ ^ 2) + ∑ i in s₂ \ s₁, ‖f i‖ ^ 2) < sqrt ε ^ 2 := by
        rw [← hV.norm_sq_diff_sum, sq_lt_sq, abs_of_nonneg (sqrt_nonneg _),
          abs_of_nonneg (norm_nonneg _)]
        

Proof state:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
dec_E : DecidableEq E
ι : Type u_4
G : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
dec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)
inst✝ : CompleteSpace E
f : (i : ι) → G i
hf :
  ∀ ε > 0, ∃ N, ∀ (m : Finset ι), N ≤ m → ∀ (n : Finset ι), N ≤ n → ‖∑ i in m, (V i) (f i) - ∑ i in n, (V i) (f i)‖ < ε
ε : ℝ
hε : ε > 0
a : Finset ι
H : ∀ (m : Finset ι), a ≤ m → ∀ (n : Finset ι), a ≤ n → ‖∑ i in m, (V i) (f i) - ∑ i in n, (V i) (f i)‖ < sqrt ε
s₁ : Finset ι
hs₁ : a ≤ s₁
s₂ : Finset ι
hs₂ : a ≤ s₂
this : ∀ (i : ι), 0 ≤ ‖f i‖ ^ 2
⊢ ‖∑ i in s₁, (V i) (f i) - ∑ i in s₂, (V i) (f i)‖ < sqrt ε

Next tactic:
exact H s₁ hs₁ s₂ hs₂
/-- A family `f` of mutually-orthogonal elements of `E` is summable, if and only if
`(fun i ↦ ‖f i‖ ^ 2)` is summable. -/
theorem OrthogonalFamily.summable_iff_norm_sq_summable [CompleteSpace E] (f : ∀ i, G i) :
    (Summable fun i => V i (f i)) ↔ Summable fun i => ‖f i‖ ^ 2 := by
  classical
    clear dec_ι
    simp only [summable_iff_cauchySeq_finset, NormedAddCommGroup.cauchySeq_iff, Real.norm_eq_abs]
    constructor
    · intro hf ε hε
      obtain ⟨a, H⟩ := hf _ (sqrt_pos.mpr hε)
      use a
      intro s₁ hs₁ s₂ hs₂
      rw [← Finset.sum_sdiff_sub_sum_sdiff]
      refine' (abs_sub _ _).trans_lt _
      have : ∀ i, 0 ≤ ‖f i‖ ^ 2 := fun i : ι => sq_nonneg _
      simp only [Finset.abs_sum_of_nonneg' this]
      have : ((∑ i in s₁ \ s₂, ‖f i‖ ^ 2) + ∑ i in s₂ \ s₁, ‖f i‖ ^ 2) < sqrt ε ^ 2 := by
        rw [← hV.norm_sq_diff_sum, sq_lt_sq, abs_of_nonneg (sqrt_nonneg _),
          abs_of_nonneg (norm_nonneg _)]
        exact H s₁ hs₁ s₂ hs₂
      

Proof state:
case h
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
dec_E : DecidableEq E
ι : Type u_4
G : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
dec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)
inst✝ : CompleteSpace E
f : (i : ι) → G i
hf :
  ∀ ε > 0, ∃ N, ∀ (m : Finset ι), N ≤ m → ∀ (n : Finset ι), N ≤ n → ‖∑ i in m, (V i) (f i) - ∑ i in n, (V i) (f i)‖ < ε
ε : ℝ
hε : ε > 0
a : Finset ι
H : ∀ (m : Finset ι), a ≤ m → ∀ (n : Finset ι), a ≤ n → ‖∑ i in m, (V i) (f i) - ∑ i in n, (V i) (f i)‖ < sqrt ε
s₁ : Finset ι
hs₁ : a ≤ s₁
s₂ : Finset ι
hs₂ : a ≤ s₂
this✝ : ∀ (i : ι), 0 ≤ ‖f i‖ ^ 2
this : ∑ i in s₁ \ s₂, ‖f i‖ ^ 2 + ∑ i in s₂ \ s₁, ‖f i‖ ^ 2 < sqrt ε ^ 2
⊢ ∑ x in s₁ \ s₂, ‖f x‖ ^ 2 + ∑ x in s₂ \ s₁, ‖f x‖ ^ 2 < ε

Next tactic:
have hη := sq_sqrt (le_of_lt hε)
/-- A family `f` of mutually-orthogonal elements of `E` is summable, if and only if
`(fun i ↦ ‖f i‖ ^ 2)` is summable. -/
theorem OrthogonalFamily.summable_iff_norm_sq_summable [CompleteSpace E] (f : ∀ i, G i) :
    (Summable fun i => V i (f i)) ↔ Summable fun i => ‖f i‖ ^ 2 := by
  classical
    clear dec_ι
    simp only [summable_iff_cauchySeq_finset, NormedAddCommGroup.cauchySeq_iff, Real.norm_eq_abs]
    constructor
    · intro hf ε hε
      obtain ⟨a, H⟩ := hf _ (sqrt_pos.mpr hε)
      use a
      intro s₁ hs₁ s₂ hs₂
      rw [← Finset.sum_sdiff_sub_sum_sdiff]
      refine' (abs_sub _ _).trans_lt _
      have : ∀ i, 0 ≤ ‖f i‖ ^ 2 := fun i : ι => sq_nonneg _
      simp only [Finset.abs_sum_of_nonneg' this]
      have : ((∑ i in s₁ \ s₂, ‖f i‖ ^ 2) + ∑ i in s₂ \ s₁, ‖f i‖ ^ 2) < sqrt ε ^ 2 := by
        rw [← hV.norm_sq_diff_sum, sq_lt_sq, abs_of_nonneg (sqrt_nonneg _),
          abs_of_nonneg (norm_nonneg _)]
        exact H s₁ hs₁ s₂ hs₂
      have hη := sq_sqrt (le_of_lt hε)
      

Proof state:
case h
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
dec_E : DecidableEq E
ι : Type u_4
G : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
dec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)
inst✝ : CompleteSpace E
f : (i : ι) → G i
hf :
  ∀ ε > 0, ∃ N, ∀ (m : Finset ι), N ≤ m → ∀ (n : Finset ι), N ≤ n → ‖∑ i in m, (V i) (f i) - ∑ i in n, (V i) (f i)‖ < ε
ε : ℝ
hε : ε > 0
a : Finset ι
H : ∀ (m : Finset ι), a ≤ m → ∀ (n : Finset ι), a ≤ n → ‖∑ i in m, (V i) (f i) - ∑ i in n, (V i) (f i)‖ < sqrt ε
s₁ : Finset ι
hs₁ : a ≤ s₁
s₂ : Finset ι
hs₂ : a ≤ s₂
this✝ : ∀ (i : ι), 0 ≤ ‖f i‖ ^ 2
this : ∑ i in s₁ \ s₂, ‖f i‖ ^ 2 + ∑ i in s₂ \ s₁, ‖f i‖ ^ 2 < sqrt ε ^ 2
hη : sqrt ε ^ 2 = ε
⊢ ∑ x in s₁ \ s₂, ‖f x‖ ^ 2 + ∑ x in s₂ \ s₁, ‖f x‖ ^ 2 < ε

Next tactic:
linarith
/-- A family `f` of mutually-orthogonal elements of `E` is summable, if and only if
`(fun i ↦ ‖f i‖ ^ 2)` is summable. -/
theorem OrthogonalFamily.summable_iff_norm_sq_summable [CompleteSpace E] (f : ∀ i, G i) :
    (Summable fun i => V i (f i)) ↔ Summable fun i => ‖f i‖ ^ 2 := by
  classical
    clear dec_ι
    simp only [summable_iff_cauchySeq_finset, NormedAddCommGroup.cauchySeq_iff, Real.norm_eq_abs]
    constructor
    · intro hf ε hε
      obtain ⟨a, H⟩ := hf _ (sqrt_pos.mpr hε)
      use a
      intro s₁ hs₁ s₂ hs₂
      rw [← Finset.sum_sdiff_sub_sum_sdiff]
      refine' (abs_sub _ _).trans_lt _
      have : ∀ i, 0 ≤ ‖f i‖ ^ 2 := fun i : ι => sq_nonneg _
      simp only [Finset.abs_sum_of_nonneg' this]
      have : ((∑ i in s₁ \ s₂, ‖f i‖ ^ 2) + ∑ i in s₂ \ s₁, ‖f i‖ ^ 2) < sqrt ε ^ 2 := by
        rw [← hV.norm_sq_diff_sum, sq_lt_sq, abs_of_nonneg (sqrt_nonneg _),
          abs_of_nonneg (norm_nonneg _)]
        exact H s₁ hs₁ s₂ hs₂
      have hη := sq_sqrt (le_of_lt hε)
      linarith
    · 

Proof state:
case mpr
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
dec_E : DecidableEq E
ι : Type u_4
G : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
dec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)
inst✝ : CompleteSpace E
f : (i : ι) → G i
⊢ (∀ ε > 0, ∃ N, ∀ (m : Finset ι), N ≤ m → ∀ (n : Finset ι), N ≤ n → |∑ i in m, ‖f i‖ ^ 2 - ∑ i in n, ‖f i‖ ^ 2| < ε) →
    ∀ ε > 0,
      ∃ N, ∀ (m : Finset ι), N ≤ m → ∀ (n : Finset ι), N ≤ n → ‖∑ i in m, (V i) (f i) - ∑ i in n, (V i) (f i)‖ < ε

Next tactic:
intro hf ε hε
/-- A family `f` of mutually-orthogonal elements of `E` is summable, if and only if
`(fun i ↦ ‖f i‖ ^ 2)` is summable. -/
theorem OrthogonalFamily.summable_iff_norm_sq_summable [CompleteSpace E] (f : ∀ i, G i) :
    (Summable fun i => V i (f i)) ↔ Summable fun i => ‖f i‖ ^ 2 := by
  classical
    clear dec_ι
    simp only [summable_iff_cauchySeq_finset, NormedAddCommGroup.cauchySeq_iff, Real.norm_eq_abs]
    constructor
    · intro hf ε hε
      obtain ⟨a, H⟩ := hf _ (sqrt_pos.mpr hε)
      use a
      intro s₁ hs₁ s₂ hs₂
      rw [← Finset.sum_sdiff_sub_sum_sdiff]
      refine' (abs_sub _ _).trans_lt _
      have : ∀ i, 0 ≤ ‖f i‖ ^ 2 := fun i : ι => sq_nonneg _
      simp only [Finset.abs_sum_of_nonneg' this]
      have : ((∑ i in s₁ \ s₂, ‖f i‖ ^ 2) + ∑ i in s₂ \ s₁, ‖f i‖ ^ 2) < sqrt ε ^ 2 := by
        rw [← hV.norm_sq_diff_sum, sq_lt_sq, abs_of_nonneg (sqrt_nonneg _),
          abs_of_nonneg (norm_nonneg _)]
        exact H s₁ hs₁ s₂ hs₂
      have hη := sq_sqrt (le_of_lt hε)
      linarith
    · intro hf ε hε
      

Proof state:
case mpr
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
dec_E : DecidableEq E
ι : Type u_4
G : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
dec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)
inst✝ : CompleteSpace E
f : (i : ι) → G i
hf : ∀ ε > 0, ∃ N, ∀ (m : Finset ι), N ≤ m → ∀ (n : Finset ι), N ≤ n → |∑ i in m, ‖f i‖ ^ 2 - ∑ i in n, ‖f i‖ ^ 2| < ε
ε : ℝ
hε : ε > 0
⊢ ∃ N, ∀ (m : Finset ι), N ≤ m → ∀ (n : Finset ι), N ≤ n → ‖∑ i in m, (V i) (f i) - ∑ i in n, (V i) (f i)‖ < ε

Next tactic:
have hε' : 0 < ε ^ 2 / 2 := half_pos (sq_pos_of_pos hε)
/-- A family `f` of mutually-orthogonal elements of `E` is summable, if and only if
`(fun i ↦ ‖f i‖ ^ 2)` is summable. -/
theorem OrthogonalFamily.summable_iff_norm_sq_summable [CompleteSpace E] (f : ∀ i, G i) :
    (Summable fun i => V i (f i)) ↔ Summable fun i => ‖f i‖ ^ 2 := by
  classical
    clear dec_ι
    simp only [summable_iff_cauchySeq_finset, NormedAddCommGroup.cauchySeq_iff, Real.norm_eq_abs]
    constructor
    · intro hf ε hε
      obtain ⟨a, H⟩ := hf _ (sqrt_pos.mpr hε)
      use a
      intro s₁ hs₁ s₂ hs₂
      rw [← Finset.sum_sdiff_sub_sum_sdiff]
      refine' (abs_sub _ _).trans_lt _
      have : ∀ i, 0 ≤ ‖f i‖ ^ 2 := fun i : ι => sq_nonneg _
      simp only [Finset.abs_sum_of_nonneg' this]
      have : ((∑ i in s₁ \ s₂, ‖f i‖ ^ 2) + ∑ i in s₂ \ s₁, ‖f i‖ ^ 2) < sqrt ε ^ 2 := by
        rw [← hV.norm_sq_diff_sum, sq_lt_sq, abs_of_nonneg (sqrt_nonneg _),
          abs_of_nonneg (norm_nonneg _)]
        exact H s₁ hs₁ s₂ hs₂
      have hη := sq_sqrt (le_of_lt hε)
      linarith
    · intro hf ε hε
      have hε' : 0 < ε ^ 2 / 2 := half_pos (sq_pos_of_pos hε)
      

Proof state:
case mpr
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
dec_E : DecidableEq E
ι : Type u_4
G : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
dec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)
inst✝ : CompleteSpace E
f : (i : ι) → G i
hf : ∀ ε > 0, ∃ N, ∀ (m : Finset ι), N ≤ m → ∀ (n : Finset ι), N ≤ n → |∑ i in m, ‖f i‖ ^ 2 - ∑ i in n, ‖f i‖ ^ 2| < ε
ε : ℝ
hε : ε > 0
hε' : 0 < ε ^ 2 / 2
⊢ ∃ N, ∀ (m : Finset ι), N ≤ m → ∀ (n : Finset ι), N ≤ n → ‖∑ i in m, (V i) (f i) - ∑ i in n, (V i) (f i)‖ < ε

Next tactic:
obtain ⟨a, H⟩ := hf _ hε'
/-- A family `f` of mutually-orthogonal elements of `E` is summable, if and only if
`(fun i ↦ ‖f i‖ ^ 2)` is summable. -/
theorem OrthogonalFamily.summable_iff_norm_sq_summable [CompleteSpace E] (f : ∀ i, G i) :
    (Summable fun i => V i (f i)) ↔ Summable fun i => ‖f i‖ ^ 2 := by
  classical
    clear dec_ι
    simp only [summable_iff_cauchySeq_finset, NormedAddCommGroup.cauchySeq_iff, Real.norm_eq_abs]
    constructor
    · intro hf ε hε
      obtain ⟨a, H⟩ := hf _ (sqrt_pos.mpr hε)
      use a
      intro s₁ hs₁ s₂ hs₂
      rw [← Finset.sum_sdiff_sub_sum_sdiff]
      refine' (abs_sub _ _).trans_lt _
      have : ∀ i, 0 ≤ ‖f i‖ ^ 2 := fun i : ι => sq_nonneg _
      simp only [Finset.abs_sum_of_nonneg' this]
      have : ((∑ i in s₁ \ s₂, ‖f i‖ ^ 2) + ∑ i in s₂ \ s₁, ‖f i‖ ^ 2) < sqrt ε ^ 2 := by
        rw [← hV.norm_sq_diff_sum, sq_lt_sq, abs_of_nonneg (sqrt_nonneg _),
          abs_of_nonneg (norm_nonneg _)]
        exact H s₁ hs₁ s₂ hs₂
      have hη := sq_sqrt (le_of_lt hε)
      linarith
    · intro hf ε hε
      have hε' : 0 < ε ^ 2 / 2 := half_pos (sq_pos_of_pos hε)
      obtain ⟨a, H⟩ := hf _ hε'
      

Proof state:
case mpr.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
dec_E : DecidableEq E
ι : Type u_4
G : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
dec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)
inst✝ : CompleteSpace E
f : (i : ι) → G i
hf : ∀ ε > 0, ∃ N, ∀ (m : Finset ι), N ≤ m → ∀ (n : Finset ι), N ≤ n → |∑ i in m, ‖f i‖ ^ 2 - ∑ i in n, ‖f i‖ ^ 2| < ε
ε : ℝ
hε : ε > 0
hε' : 0 < ε ^ 2 / 2
a : Finset ι
H : ∀ (m : Finset ι), a ≤ m → ∀ (n : Finset ι), a ≤ n → |∑ i in m, ‖f i‖ ^ 2 - ∑ i in n, ‖f i‖ ^ 2| < ε ^ 2 / 2
⊢ ∃ N, ∀ (m : Finset ι), N ≤ m → ∀ (n : Finset ι), N ≤ n → ‖∑ i in m, (V i) (f i) - ∑ i in n, (V i) (f i)‖ < ε

Next tactic:
use a