/-- Characterization of minimizers for the projection on a convex set in a real inner product
space. -/
theorem norm_eq_iInf_iff_real_inner_le_zero {K : Set F} (h : Convex ℝ K) {u : F} {v : F}
    (hv : v ∈ K) : (‖u - v‖ = ⨅ w : K, ‖u - w‖) ↔ ∀ w ∈ K, ⟪u - v, w - v⟫_ℝ ≤ 0 := by
  letI : Nonempty K := ⟨⟨v, hv⟩⟩
  constructor
  · intro eq w hw
    let δ := ⨅ w : K, ‖u - w‖
    let p := ⟪u - v, w - v⟫_ℝ
    let q := ‖w - v‖ ^ 2
    have δ_le (w : K) : δ ≤ ‖u - w‖ := ciInf_le ⟨0, fun _ ⟨_, h⟩ => h ▸ norm_nonneg _⟩ _
    have δ_le' (w) (hw : w ∈ K) : δ ≤ ‖u - w‖ := δ_le ⟨w, hw⟩
    have (θ : ℝ) (hθ₁ : 0 < θ) (hθ₂ : θ ≤ 1) : 2 * p ≤ θ * q := by
      have : ‖u - v‖ ^ 2 ≤ ‖u - v‖ ^ 2 - 2 * θ * ⟪u - v, w - v⟫_ℝ + θ * θ * ‖w - v‖ ^ 2 :=
        calc ‖u - v‖ ^ 2
          _ ≤ ‖u - (θ • w + (1 - θ) • v)‖ ^ 2 := by
            simp only [sq]; apply mul_self_le_mul_self (norm_nonneg _)
            rw [eq]; apply δ_le'
            apply h hw hv
            exacts [le_of_lt hθ₁, sub_nonneg.2 hθ₂, add_sub_cancel'_right _ _]
          _ = ‖u - v - θ • (w - v)‖ ^ 2 := by
            have : u - (θ • w + (1 - θ) • v) = u - v - θ • (w - v) := by
              rw [smul_sub, sub_smul, one_smul]
              simp only [sub_eq_add_neg, add_comm, add_left_comm, add_assoc, neg_add_rev]
            rw [this]
          _ = ‖u - v‖ ^ 2 - 2 * θ * inner (u - v) (w - v) + θ * θ * ‖w - v‖ ^ 2 := by
            rw [@norm_sub_sq ℝ, inner_smul_right, norm_smul]
            simp only [sq]
            show
              ‖u - v‖ * ‖u - v‖ - 2 * (θ * inner (u - v) (w - v)) +
                absR θ * ‖w - v‖ * (absR θ * ‖w - v‖) =
              ‖u - v‖ * ‖u - v‖ - 2 * θ * inner (u - v) (w - v) + θ * θ * (‖w - v‖ * ‖w - v‖)
            rw [abs_of_pos hθ₁]; ring
      have eq₁ :
        ‖u - v‖ ^ 2 - 2 * θ * inner (u - v) (w - v) + θ * θ * ‖w - v‖ ^ 2 =
          ‖u - v‖ ^ 2 + (θ * θ * ‖w - v‖ ^ 2 - 2 * θ * inner (u - v) (w - v)) := by
        abel
      rw [eq₁, le_add_iff_nonneg_right] at this
      have eq₂ :
        θ * θ * ‖w - v‖ ^ 2 - 2 * θ * inner (u - v) (w - v) =
          θ * (θ * ‖w - v‖ ^ 2 - 2 * inner (u - v) (w - v))
      ring
      rw [eq₂] at this
      have := le_of_sub_nonneg (nonneg_of_mul_nonneg_right this hθ₁)
      exact this
    by_cases hq : q = 0
    · rw [hq] at this
      have : p ≤ 0 := by
        have := this (1 : ℝ) (by norm_num) (by norm_num)
        linarith
      exact this
    · have q_pos : 0 < q := lt_of_le_of_ne (sq_nonneg _) fun h ↦ hq h.symm
      by_contra hp
      rw [not_le] at hp
      let θ := min (1 : ℝ) (p / q)
      have eq₁ : θ * q ≤ p :=
        calc
          θ * q ≤ p / q * q := mul_le_mul_of_nonneg_right (min_le_right _ _) (sq_nonneg _)
          _ = p := div_mul_cancel _ hq
      have : 2 * p ≤ p :=
        calc
          2 * p ≤ θ * q := by
            refine' this θ (lt_min (by norm_num) (div_pos hp q_pos)) (by norm_num)
          _ ≤ p := eq₁
      linarith
  · intro h
    apply le_antisymm
    · apply le_ciInf
      intro w
      apply nonneg_le_nonneg_of_sq_le_sq (norm_nonneg _)
      have := h w w.2
      calc
        ‖u - v‖ * ‖u - v‖ ≤ ‖u - v‖ * ‖u - v‖ - 2 * inner (u - v) ((w : F) - v) := by linarith
        _ ≤ ‖u - v‖ ^ 2 - 2 * inner (u - v) ((w : F) - v) + ‖(w : F) - v‖ ^ 2 := by
          rw [sq]
          refine' le_add_of_nonneg_right _
          exact sq_nonneg _
        _ = ‖u - v - (w - v)‖ ^ 2 := (@norm_sub_sq ℝ _ _ _ _ _ _).symm
        _ = ‖u - w‖ * ‖u - w‖ := by
          have : u - v - (w - v) = u - w := by abel
          rw [this, sq]
    · show ⨅ w : K, ‖u - w‖ ≤ (fun w : K => ‖u - w‖) ⟨v, hv⟩
      apply ciInf_le
      use 0
      rintro y ⟨z, rfl⟩
      

Proof state:
case h.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : InnerProductSpace ℝ F
K : Set F
h✝ : Convex ℝ K
u v : F
hv : v ∈ K
this : Nonempty ↑K := Nonempty.intro { val := v, property := hv }
h : ∀ w ∈ K, inner (u - v) (w - v) ≤ 0
z : ↑K
⊢ 0 ≤ (fun w => ‖u - ↑w‖) z

Next tactic:
exact norm_nonneg _