theorem extracted_formal_statement_0 {m : ℕ+} (x : EuclideanSpace ℝ (Fin ↑m)) (h_original_goal : ‖x‖ ^ 2 = x ⬝ᵥ x) :
  0 ≤ ∑ i, ‖x i‖ ^ 2 := sorry

State before:
m : ℕ+
x : EuclideanSpace ℝ (Fin ↑m)
⊢ ‖x‖ ^ 2 = x ⬝ᵥ x

Tactic: rw [EuclideanSpace.norm_eq, Real.sq_sqrt, dotProduct]
    rw [Finset.sum_congr]
    simp

State after:
[case a
 m : ℕ+
 x : EuclideanSpace ℝ (Fin ↑m)
 ⊢ ∀ x_1 ∈ Finset.univ, ‖x x_1‖ ^ 2 = x x_1 * x x_1,
 m : ℕ+
 x : EuclideanSpace ℝ (Fin ↑m)
 ⊢ 0 ≤ ∑ i, ‖x i‖ ^ 2]

Relevant lemmas:
Finset.sum_congr : ∀ {α : Type u_3} {β : Type u_4} {s₁ s₂ : Finset α} {f g : α → β} [inst : AddCommMonoid β],
  s₁ = s₂ → (∀ x ∈ s₂, f x = g x) → s₁.sum f = s₂.sum g
Real.sq_sqrt : ∀ {x : ℝ}, 0 ≤ x → √x ^ 2 = x
EuclideanSpace.norm_eq : ∀ {𝕜 : Type u_7} [inst : RCLike 𝕜] {n : Type u_8} [inst_1 : Fintype n]
  (x : EuclideanSpace 𝕜 n), ‖x‖ = √(∑ i, ‖x i‖ ^ 2)