Rel Close To Code iff rel Close To Codeword of min Dist

lemma relCloseToCode_iff_relCloseToCodeword_of_minDist [Nonempty ι] [DecidableEq F]    {C : Set (ι → F)} (u : ι → F) (δ : ℝ≥0) :    δᵣ(u, C) ≤ δ ↔ ∃ v ∈ C, δᵣ(u, v) ≤ δ := by  constructor  · -- Direction 1: (→)    -- Assume: δᵣ(u, C) ≤ ↑δ    -- Goal: ∃ v ∈ C, δᵣ(u, v) ≤ δ    intro h_dist_le_e    -- We need to handle two cases: the code C being empty or non-empty.    by_cases hC_empty : C = ∅    · -- Case 1: C is empty      -- The goal is `∃ v ∈ ∅, ...`, which is `False`.      -- We must show the assumption `h_dist_le_e` is also `False`.      rw [hC_empty] at h_dist_le_e      rw [relDistFromCode_of_empty] at h_dist_le_e      -- h_dist_le_e is now `⊤ ≤ ↑e`.      -- Since `e : ℕ`, `↑e` is finite (i.e., `↑e ≠ ⊤`).      have h_e_ne_top : (δ : ENNReal) ≠ ⊤ := ENNReal.coe_ne_top (r := δ)      -- `⊤ ≤ ↑e` is only true if `↑e = ⊤`, so this is a contradiction.      simp only [top_le_iff, ENNReal.coe_ne_top] at h_dist_le_e    · -- Case 2: C is non-empty      have hC_nonempty : Set.Nonempty C := Set.nonempty_iff_ne_empty.mpr hC_empty      have hC_nonempty_instance : Nonempty C := Set.Nonempty.to_subtype hC_nonempty      let v := pickRelClosestCodeword_of_Nonempty_Code C u      use v; constructor      · simp only [Subtype.coe_prop]      · rw [relDistFromPickRelClosestCodeword_of_Nonempty_Code] at h_dist_le_e        rw [←ENNReal.coe_le_coe]        exact h_dist_le_e  · -- Direction 2: (←)    -- Assume: `∃ v ∈ C, δᵣ(u, v) ≤ e`    -- Goal: `δᵣ(u, C) ≤ ↑e`    intro h_exists    -- Unpack the assumption    rcases h_exists with ⟨v, hv_mem, h_dist_le_e⟩    -- Goal is `sInf {d | ∃ w ∈ C, ↑(δᵣ(u, w)) ≤ d} ≤ ↑e`    -- We can use the lemma `ENat.sInf_le` (or `sInf_le` for complete linear orders)    -- which says `sInf S ≤ x` if `x ∈ S`.    have h_sInf_le: δᵣ(u, C) ≤ δᵣ(u, v) := by      apply sInf_le      simp only [Set.mem_setOf_eq]      use v    calc δᵣ(u, C) ≤ δᵣ(u, v) := h_sInf_le    _ ≤ δ := by exact ENNReal.coe_le_coe.mpr h_dist_le_e