boundary_exception

lemma boundary_exception {u : 𝔓 X} :    volume (⋃ i ∈ 𝓛 (X := X) n u, (i : Set X)) ≤ C5_2_9 X n * volume (𝓘 u : Set X) := by  by_cases  h_𝓛_n_u_non_empty : Set.Nonempty (𝓛 (X := X) n u)  · set X_u := { x ∈ GridStructure.coeGrid (𝓘 u) | Metric.infEDist x (GridStructure.coeGrid (𝓘 u))ᶜ ≤ 12 * (D ^ (𝔰 u - Z * (n + 1) - 1 : ℤ) : ℝ≥0∞)} with h_X_u -- 5.2.25    calc volume (⋃ i ∈ 𝓛 (X := X) n u, (i : Set X))      _ ≤ volume X_u := by          have i_subset_X_u : ∀ i ∈ 𝓛 (X := X) n u, GridStructure.coeGrid i ⊆ X_u := by            intro i ⟨⟨i_subset_I_u, _⟩, s_i_eq_stuff, I_not_contain_8_ball⟩ ipt hipt            refine ⟨i_subset_I_u hipt, ?_⟩            rw [show 𝔰 u - Z * (n + 1) - 1 = s i by norm_cast; linarith]            obtain ⟨bpt, hbpt, h_bpt_not_in_I_u⟩ : ∃ b ∈ ball (c i) (8 * ↑D ^ s i), b ∉ ↑(𝓘 u) := not_subset.mp I_not_contain_8_ball            -- triangle inequality between ipt, bpt, c i            have ipt_bpt_triangle_ineq : dist ipt bpt ≤ (12 * D ^ s i : ℝ) :=              calc dist ipt bpt                _ ≤ dist ipt (c i) + dist (c i) bpt := dist_triangle ipt (c i) bpt                _ ≤ 4 * D ^ s i + dist (c i) bpt := by                  have dist_ipt_c_i_le : dist ipt (c i) < 4 * D ^ s i := by                    have ipt_in_ball_4 : ipt ∈ ball (c i) (4 * D ^ s i) := Grid_subset_ball hipt                    simp_all                  rel [dist_ipt_c_i_le]                _ ≤ 4 * D ^ s i + dist bpt (c i) := by rw [dist_comm]                _ ≤ 4 * D ^ s i + 8 * D ^ s i := by                    have dist_bpt_c_i_le : dist bpt (c i) < 8 * D ^ s i := by simp_all                    rel [dist_bpt_c_i_le]                _ ≤ 12 * D ^ s i := by linarith            -- show the the triangle inequality implies distance between ipt and (𝓘 u)ᶜ <= 12 * D ^ s i            calc Metric.infEDist ipt (GridStructure.coeGrid (𝓘 u))ᶜ              _ ≤ edist ipt bpt := Metric.infEDist_le_edist_of_mem <| Set.mem_compl h_bpt_not_in_I_u              _ ≤ ENNReal.ofReal (12 * D ^ s i) := by                rw [edist_dist]                exact ENNReal.ofReal_le_ofReal ipt_bpt_triangle_ineq              _ ≤ ENNReal.ofNNReal (12 * D ^ s i) := le_of_eq <|                congr_arg (ENNReal.ofNNReal) <| NNReal.coe_injective <| by                    simpa using zpow_nonneg (by simp) (s i)              _ ≤ 12 * (D ^ (s i : ℤ) :  ℝ≥0∞) := by                  push_cast                  rw [ENNReal.coe_zpow]                  · push_cast                    rfl                  · simp          exact measure_mono (Set.iUnion₂_subset i_subset_X_u)      _ ≤ 2 * (12 * D ^ (- Z * (n + 1) - 1 : ℤ) : ℝ≥0) ^ κ * volume (𝓘 u : Set X) := by          have small_boundary_observation : ∀ i ∈ 𝓛 (X := X) n u, volume X_u ≤ 2 * (12 * D ^ (- Z * (n + 1) - 1 : ℤ) : ℝ≥0) ^ κ * volume (𝓘 u : Set X) := by            intro i ⟨_, s_i_eq_stuff, _⟩            -- choose t for small boundary property            set t := 12 * (D ^ (- Z * (n + 1) - 1 : ℤ) : ℝ≥0) with ht            -- algebra useful in multiple steps of the proof            have D_pow_algebra : 12 * (D ^ (- Z * (n + 1) - 1 : ℤ) : ℝ≥0)  * (D ^ (𝔰 u : ℤ) : ℝ≥0) = 12 * (D ^ ( 𝔰 u - Z * (n + 1) - 1 : ℤ) : ℝ≥0) := by              have : 12 * (D ^ (- Z * (n + 1) - 1 : ℤ) : ℝ≥0)  * (D ^ (𝔰 u : ℤ) : ℝ≥0) = 12 * (D ^ (- Z * (n + 1) - 1 + 𝔰 u : ℤ) : ℝ≥0) := by                rw [zpow_add₀ (show (D : ℝ≥0) ≠ 0 by norm_num) _ _]                ring              rw [this]              rw [show - Z * (n + 1) - 1 + 𝔰 u = 𝔰 u - Z * (n + 1) - 1 by linarith]            -- small boundary property assumption for 𝓘 u            have small_boundary_h : D ^ ((- S - s (𝓘 u)) : ℤ) ≤ t := by              have one_le_nnreal_D : 1 ≤ (D : ℝ≥0) := by                have h1 : 1 ≤ (D : ℝ) := one_le_realD _                assumption_mod_cast              have small_boundary_h_intermediate : D ^ (- S : ℤ) ≤ t * D ^ (𝔰 u: ℤ) := by                rw [ht, D_pow_algebra,                    show 𝔰 u - Z * (n + 1) - 1 = s i by rw [← s_i_eq_stuff]; norm_cast; linarith]                have bound_i_neg_S : -S ≤ s i := (mem_Icc.mp (range_s_subset ⟨i, rfl⟩)).1                exact le_mul_of_one_le_of_le (by simp) <| zpow_le_zpow_right₀ (one_le_nnreal_D) bound_i_neg_S              apply (mul_inv_le_iff₀ <| by positivity).mpr at small_boundary_h_intermediate              rw [← NNReal.rpow_neg_one] at small_boundary_h_intermediate              have : (D ^ (𝔰 u : ℤ) : ℝ≥0) ^ (-1 : ℝ) = (D ^ (𝔰 u * (-1)) : ℝ≥0) := by                rw [show (D ^ (𝔰 u : ℤ) : ℝ≥0) = (D ^ (𝔰 u : ℝ) : ℝ≥0) by norm_cast, ← NNReal.rpow_mul]                norm_cast              rwa [this, mul_neg_one, ← zpow_add₀ (show (D : ℝ≥0) ≠ 0 by norm_num),                   show 𝔰 u = s (𝓘 u) from rfl, add_comm,                   neg_add_eq_sub] at small_boundary_h_intermediate            have small_b := GridStructure.small_boundary small_boundary_h            have X_u_in_terms_of_t : X_u = { x ∈ GridStructure.coeGrid (𝓘 u) | Metric.infEDist x (GridStructure.coeGrid (𝓘 u))ᶜ ≤ ((t * D ^ (s (𝓘 u))):ℝ≥0∞)} := by              rw [ht, show s (𝓘 u) = 𝔰 u from rfl,                  show (D ^ 𝔰 u : ℝ≥0∞) = (D ^ 𝔰 u : ℝ≥0) by simp]              rw_mod_cast [D_pow_algebra, h_X_u]              have : 12 * (D ^ (𝔰 u - (Z * (n + 1) : ℤ) - 1) : ℝ≥0∞) = ((12 * (D ^ (𝔰 u - (Z * (n + 1)) - 1) : ℝ≥0)) : ℝ≥0∞) := by                simp              rw_mod_cast [this]            rw [show s (𝓘 u) = GridStructure.s (𝓘 u) from rfl] at X_u_in_terms_of_t            rw [← X_u_in_terms_of_t, measureReal_def, measureReal_def] at small_b            rw [← ENNReal.toReal_le_toReal] -- this requires showing everything is finite            · rw [ENNReal.toReal_mul]              have : (2 * (t ^ κ : ℝ≥0∞)).toReal = 2 * t ^ κ  := by                norm_cast                rw [ENNReal.toReal_mul, ← ENNReal.toReal_rpow]                rfl              rwa [this]            · apply LT.lt.ne              rw [h_X_u]              apply lt_of_le_of_lt <| volume.mono inter_subset_left              simp [volume_coeGrid_lt_top]            · apply LT.lt.ne              have t_k_lt_top : 2 * (t : ℝ≥0∞) ^ κ < ⊤ := by                rw [ht]                exact WithTop.mul_lt_top (by apply WithTop.coe_lt_top) <|                  (ENNReal.rpow_lt_top_of_nonneg κ_nonneg) (lt_top_iff_ne_top.mp (by apply WithTop.coe_lt_top))              exact WithTop.mul_lt_top t_k_lt_top volume_coeGrid_lt_top          obtain ⟨i, hi⟩ := h_𝓛_n_u_non_empty          exact small_boundary_observation i hi      _ ≤ C5_2_9 X n * volume (𝓘 u : Set X) := by -- choosing the right k and D        have coeff_lt : 2 * (12 * D ^ (-Z * (n + 1) - 1 : ℝ)) ^ κ            ≤ (D ^ (1 - κ * Z * (n + 1)) : ℝ≥0) := by          have twelve_le_D : 12 ≤ D := by            apply le_trans (by norm_num) (hundred_lt_D X).le          have two_time_twelve_over_D_to_the_k_le_D : 2 * (12 / D) ^ κ ≤ (D : ℝ≥0) := by            have two_le_D : 2 ≤ D := by linarith            have : 2 * (12 / D) ^ κ ≤ (2 : ℝ≥0) := by              apply (MulLECancellable.mul_le_iff_le_one_right ?_).mpr              · apply NNReal.rpow_le_one                · apply div_le_one_of_le₀ <| by norm_cast                  simp only [zero_le]                · apply κ_nonneg              · simp [MulLECancellable]            exact this.trans <| by norm_cast          have two_times_twelve_k_D_minus_k_le_D : 2 * 12 ^ κ * D ^ (-κ) ≤ (D : ℝ≥0) := by            rwa [← inv_mul_eq_div, NNReal.mul_rpow, NNReal.inv_rpow,                ← NNReal.rpow_neg, mul_comm _ (12 ^ κ), ← mul_assoc] at two_time_twelve_over_D_to_the_k_le_D          have mul_by_D_to_the_k_Z : 2 * 12 ^ κ * D ^ (-1*κ)  * D ^ (-1* κ  * Z * (n + 1)) ≤ (D : ℝ≥0) * D ^ (-κ * Z * (n + 1)) := by            have : 2 * 12 ^ κ * D ^ (-κ) * D ^ (-κ * Z * (n + 1)) ≤ (D : ℝ≥0) * D ^ (-κ * Z * (n + 1)) :=              mul_le_mul_of_nonneg_right two_times_twelve_k_D_minus_k_le_D (by positivity)            rwa [← neg_eq_neg_one_mul]          have rearrange_exponents : 2 * (12 : ℝ≥0) ^ κ * (D ^ (-(1 : ℝ))) ^ κ * (D ^ (-(1 : ℝ) * Z * (n + 1)) : ℝ≥0) ^ κ ≤ (D : ℝ≥0) ^ (1 : ℝ) * D ^ (-κ * Z * (n + 1)) := by            have : (-1* κ  * Z * (n + 1) : ℝ) = (-1 * Z * (n + 1)) * κ := by ring            rw [this, NNReal.rpow_mul, NNReal.rpow_mul] at mul_by_D_to_the_k_Z            rwa [NNReal.rpow_one]          rwa [mul_assoc, ← NNReal.mul_rpow, mul_assoc, ← NNReal.mul_rpow,              ← NNReal.rpow_add (by positivity), ← NNReal.rpow_add (by positivity), add_comm,              ← neg_eq_neg_one_mul, ← Ring.sub_eq_add_neg,              show  1 + -κ * Z * (n + 1) = 1 - κ * Z * (n + 1) by ring] at rearrange_exponents        rw [C5_2_9]        apply ENNReal.coe_le_coe.mpr at coeff_lt        norm_cast        have : 12 * (D ^ (-Z * (n + 1) - 1: ℤ ) : ℝ≥0) ≠ 0 := by          simp only [defaultD, Nat.cast_pow, Nat.cast_ofNat, defaultZ, neg_mul, ne_eq, mul_eq_zero,            OfNat.ofNat_ne_zero, false_or]          positivity        rw [← ENNReal.coe_rpow_of_ne_zero (by exact this)]        exact_mod_cast mul_le_mul_left coeff_lt (volume (𝓘 u : Set X))  · have : volume (⋃ i ∈ 𝓛 (X := X) n u, (i : Set X)) = 0 := by      have h1 : volume (⋃ i ∈ 𝓛 (X := X) n u, (i : Set X)) ≤        ∑' i : 𝓛 (X := X) n u, volume (i : Set X) := measure_biUnion_le _ (𝓛 n u).to_countable _      have h2 : ∑' i : 𝓛 (X := X) n u, volume (i : Set X) = 0 := by        have : IsEmpty (𝓛 (X := X) n u) := by          rw [Set.isEmpty_coe_sort]          exact Set.not_nonempty_iff_eq_empty.mp h_𝓛_n_u_non_empty        exact tsum_empty      exact (le_of_le_of_eq h1 h2).antisymm (by simp)    simp only [this, zero_le]