| import OAI.Geometry.IsometricImmersion.Coordinates.RoundDisk |
| import Mathlib.Analysis.SpecificLimits.Basic |
| import Mathlib.Topology.LocallyFinite |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped ContDiff Topology BigOperators Matrix |
| open Filter Set Metric |
|
|
| namespace SmoothLocal.Geometry |
|
|
| def accumulatingScale (n : ℕ) : ℝ := (1 / 2 : ℝ) ^ (n + 1) |
|
|
| def accumulatingCenter (n : ℕ) : Coord := ![accumulatingScale n, 0] |
|
|
| def accumulatingRadius (n : ℕ) : ℝ := accumulatingScale n / 8 |
|
|
| def accumulatingDisk (n : ℕ) : Set Coord := |
| roundClosedDisk (accumulatingCenter n) (accumulatingRadius n) |
|
|
| def enlargedAccumulatingDisk (n : ℕ) : Set Coord := |
| roundClosedDisk (accumulatingCenter n) (2 * accumulatingRadius n) |
|
|
| theorem accumulatingScale_pos (n : ℕ) : 0 < accumulatingScale n := by |
| exact pow_pos (by norm_num) _ |
|
|
| theorem accumulatingScale_le_half (n : ℕ) : accumulatingScale n ≤ (1 / 2 : ℝ) := by |
| exact pow_le_of_le_one (by norm_num) (by norm_num) (Nat.succ_ne_zero n) |
|
|
| theorem accumulatingScale_succ (n : ℕ) : |
| accumulatingScale (n + 1) = accumulatingScale n / 2 := by |
| simp only [accumulatingScale, pow_succ] |
| ring |
|
|
| theorem accumulatingScale_antitone : Antitone accumulatingScale := by |
| intro m n hmn |
| exact pow_le_pow_of_le_one (by norm_num) (by norm_num) (Nat.add_le_add_right hmn 1) |
|
|
| theorem accumulatingScale_le_half_of_lt {m n : ℕ} (hmn : m < n) : |
| accumulatingScale n ≤ accumulatingScale m / 2 := by |
| rw [← accumulatingScale_succ] |
| exact accumulatingScale_antitone (Nat.succ_le_of_lt hmn) |
|
|
| theorem accumulatingRadius_pos (n : ℕ) : 0 < accumulatingRadius n := |
| div_pos (accumulatingScale_pos n) (by norm_num) |
|
|
| theorem roundClosedDisk_coord_abs_le (c : Coord) {r : ℝ} (hr : 0 ≤ r) |
| {p : Coord} (hp : p ∈ roundClosedDisk c r) (i : Fin 2) : |p i - c i| ≤ r := by |
| have hsq : (p i - c i) ^ 2 ≤ r ^ 2 := by |
| change (p 0 - c 0) ^ 2 + (p 1 - c 1) ^ 2 ≤ r ^ 2 at hp |
| fin_cases i <;> simp only [Fin.mk_zero, Fin.mk_one] |
| · nlinarith [sq_nonneg (p 1 - c 1)] |
| · nlinarith [sq_nonneg (p 0 - c 0)] |
| apply (sq_le_sq₀ (abs_nonneg _) hr).mp |
| simpa only [sq_abs] using hsq |
|
|
| theorem accumulatingDisk_subset_enlarged (n : ℕ) : |
| accumulatingDisk n ⊆ enlargedAccumulatingDisk n := by |
| intro p hp |
| change roundRadiusSq (accumulatingCenter n) p ≤ (accumulatingRadius n) ^ 2 at hp |
| change roundRadiusSq (accumulatingCenter n) p ≤ (2 * accumulatingRadius n) ^ 2 |
| nlinarith [sq_nonneg (accumulatingRadius n)] |
|
|
| theorem accumulatingDisk_subset_enlarged_open (n : ℕ) : |
| accumulatingDisk n ⊆ roundOpenDisk (accumulatingCenter n) (2 * accumulatingRadius n) := by |
| intro p hp |
| change roundRadiusSq (accumulatingCenter n) p ≤ (accumulatingRadius n) ^ 2 at hp |
| change roundRadiusSq (accumulatingCenter n) p < (2 * accumulatingRadius n) ^ 2 |
| nlinarith [sq_pos_of_pos (accumulatingRadius_pos n)] |
|
|
| theorem enlargedAccumulatingDisk_bounds (n : ℕ) {p : Coord} |
| (hp : p ∈ enlargedAccumulatingDisk n) : |
| (3 / 4 : ℝ) * accumulatingScale n ≤ p 0 ∧ |
| p 0 ≤ (5 / 4 : ℝ) * accumulatingScale n ∧ |
| |p 1| ≤ accumulatingScale n / 4 := by |
| have hr : 0 ≤ 2 * accumulatingRadius n := |
| (mul_pos (by norm_num) (accumulatingRadius_pos n)).le |
| have hx := roundClosedDisk_coord_abs_le (accumulatingCenter n) hr hp 0 |
| have hy := roundClosedDisk_coord_abs_le (accumulatingCenter n) hr hp 1 |
| simp only [accumulatingCenter, Matrix.cons_val_zero, Matrix.cons_val_one, |
| sub_zero, accumulatingRadius] at hx hy |
| obtain ⟨hxlo, hxhi⟩ := abs_le.mp hx |
| refine ⟨?_, ?_, ?_⟩ <;> linarith |
|
|
| theorem enlargedAccumulatingDisk_disjoint_of_lt {m n : ℕ} (hmn : m < n) : |
| Disjoint (enlargedAccumulatingDisk m) (enlargedAccumulatingDisk n) := by |
| apply Set.disjoint_left.mpr |
| intro p hpm hpn |
| have hm := enlargedAccumulatingDisk_bounds m hpm |
| have hn := enlargedAccumulatingDisk_bounds n hpn |
| have hscale := accumulatingScale_le_half_of_lt hmn |
| have hpos := accumulatingScale_pos m |
| linarith |
|
|
| theorem enlargedAccumulatingDisks_pairwise_disjoint : |
| Pairwise (fun m n => Disjoint (enlargedAccumulatingDisk m) (enlargedAccumulatingDisk n)) := by |
| intro m n hmn |
| rcases lt_or_gt_of_ne hmn with hlt | hgt |
| · exact enlargedAccumulatingDisk_disjoint_of_lt hlt |
| · exact (enlargedAccumulatingDisk_disjoint_of_lt hgt).symm |
|
|
| theorem accumulatingDisks_pairwise_disjoint : |
| Pairwise (fun m n => Disjoint (accumulatingDisk m) (accumulatingDisk n)) := by |
| intro m n hmn |
| exact (enlargedAccumulatingDisks_pairwise_disjoint hmn).mono |
| (accumulatingDisk_subset_enlarged m) (accumulatingDisk_subset_enlarged n) |
|
|
| theorem enlargedAccumulatingDisk_subset_square (n : ℕ) : enlargedAccumulatingDisk n ⊆ square := by |
| intro p hp i |
| have hb := enlargedAccumulatingDisk_bounds n hp |
| have hpos := accumulatingScale_pos n |
| have hhalf := accumulatingScale_le_half n |
| fin_cases i |
| · change -1 < p 0 ∧ p 0 < 1 |
| constructor <;> linarith |
| · change -1 < p 1 ∧ p 1 < 1 |
| obtain ⟨hylo, hyhi⟩ := abs_le.mp hb.2.2 |
| constructor <;> linarith |
|
|
| theorem accumulatingDisk_subset_square (n : ℕ) : accumulatingDisk n ⊆ square := |
| (accumulatingDisk_subset_enlarged n).trans (enlargedAccumulatingDisk_subset_square n) |
|
|
| theorem zero_not_mem_enlargedAccumulatingDisk (n : ℕ) : |
| (0 : Coord) ∉ enlargedAccumulatingDisk n := by |
| intro h0 |
| have hb := (enlargedAccumulatingDisk_bounds n h0).1 |
| have hpos := accumulatingScale_pos n |
| change (3 / 4 : ℝ) * accumulatingScale n ≤ 0 at hb |
| linarith |
|
|
| theorem zero_not_mem_accumulatingDisk (n : ℕ) : (0 : Coord) ∉ accumulatingDisk n := |
| fun h => zero_not_mem_enlargedAccumulatingDisk n (accumulatingDisk_subset_enlarged n h) |
|
|
| theorem accumulatingDisk_isCompact (n : ℕ) : IsCompact (accumulatingDisk n) := |
| isCompact_roundClosedDisk (accumulatingCenter n) (accumulatingRadius_pos n) |
|
|
| theorem enlargedAccumulatingDisk_isCompact (n : ℕ) : IsCompact (enlargedAccumulatingDisk n) := |
| isCompact_roundClosedDisk (accumulatingCenter n) (mul_pos (by norm_num) (accumulatingRadius_pos n)) |
|
|
| theorem accumulatingDisk_interior_nonempty (n : ℕ) : (interior (accumulatingDisk n)).Nonempty := |
| interior_roundClosedDisk_nonempty (accumulatingCenter n) (accumulatingRadius_pos n) |
|
|
| theorem tendsto_accumulatingScale : Tendsto accumulatingScale atTop (𝓝 0) := by |
| have hpow := tendsto_pow_atTop_nhds_zero_of_lt_one |
| (show (0 : ℝ) ≤ 1 / 2 by norm_num) (show (1 / 2 : ℝ) < 1 by norm_num) |
| convert hpow.mul (tendsto_const_nhds (x := (1 / 2 : ℝ))) using 1 <;> |
| first | rfl | simp only [zero_mul] |
|
|
| theorem tendsto_accumulatingCenter : Tendsto accumulatingCenter atTop (𝓝 (0 : Coord)) := by |
| apply tendsto_pi_nhds.mpr |
| intro i |
| fin_cases i |
| · change Tendsto accumulatingScale atTop (𝓝 (0 : ℝ)) |
| exact tendsto_accumulatingScale |
| · change Tendsto (fun _ : ℕ => (0 : ℝ)) atTop (𝓝 0) |
| exact tendsto_const_nhds |
|
|
| theorem enlargedAccumulatingDisk_norm_le (n : ℕ) {p : Coord} |
| (hp : p ∈ enlargedAccumulatingDisk n) : ‖p‖ ≤ (5 / 4 : ℝ) * accumulatingScale n := by |
| have hb := enlargedAccumulatingDisk_bounds n hp |
| have hpos := accumulatingScale_pos n |
| apply (pi_norm_le_iff_of_nonneg (by positivity)).mpr |
| intro i |
| fin_cases i |
| · change |p 0| ≤ (5 / 4 : ℝ) * accumulatingScale n |
| rw [abs_of_nonneg (by linarith : 0 ≤ p 0)] |
| exact hb.2.1 |
| · change |p 1| ≤ (5 / 4 : ℝ) * accumulatingScale n |
| linarith [hb.2.2] |
|
|
| theorem eventually_enlargedAccumulatingDisk_subset {V : Set Coord} (hV : V ∈ 𝓝 (0 : Coord)) : |
| ∀ᶠ n in atTop, enlargedAccumulatingDisk n ⊆ V := by |
| obtain ⟨ε, hε, hball⟩ := Metric.mem_nhds_iff.mp hV |
| have ht : Tendsto (fun n => (5 / 4 : ℝ) * accumulatingScale n) atTop (𝓝 0) := by |
| simpa only [mul_zero] using |
| (tendsto_const_nhds (x := (5 / 4 : ℝ))).mul tendsto_accumulatingScale |
| filter_upwards [ht.eventually (Iio_mem_nhds hε)] with n hn |
| intro p hp |
| apply hball |
| change dist p (0 : Coord) < ε |
| simpa only [dist_zero_right] using lt_of_le_of_lt (enlargedAccumulatingDisk_norm_le n hp) hn |
|
|
| theorem eventually_accumulatingDisk_subset {V : Set Coord} (hV : V ∈ 𝓝 (0 : Coord)) : |
| ∀ᶠ n in atTop, accumulatingDisk n ⊆ V := |
| (eventually_enlargedAccumulatingDisk_subset hV).mono |
| (fun n hn => (accumulatingDisk_subset_enlarged n).trans hn) |
|
|
| theorem eventually_all_accumulatingDisks_in_open {V : Set Coord} |
| (hV : IsOpen V) (h0 : (0 : Coord) ∈ V) : |
| ∃ N : ℕ, ∀ n ≥ N, accumulatingDisk n ⊆ V := |
| eventually_atTop.mp (eventually_accumulatingDisk_subset (hV.mem_nhds h0)) |
|
|
| theorem enlargedAccumulatingDisks_locallyFinite_away_zero {p : Coord} (hp : p ≠ 0) : |
| ∃ W : Set Coord, IsOpen W ∧ p ∈ W ∧ |
| {n : ℕ | (enlargedAccumulatingDisk n ∩ W).Nonempty}.Finite := by |
| obtain ⟨W, V, hWo, hVo, hpW, h0V, hdisj⟩ := t2_separation hp |
| obtain ⟨N, hN⟩ := eventually_atTop.mp |
| (eventually_enlargedAccumulatingDisk_subset (hVo.mem_nhds h0V)) |
| refine ⟨W, hWo, hpW, (Finset.range N).finite_toSet.subset ?_⟩ |
| intro n hn |
| change n ∈ Finset.range N |
| rw [Finset.mem_range] |
| by_contra hlt |
| obtain ⟨q, hqD, hqW⟩ := hn |
| exact (Set.disjoint_left.mp hdisj) hqW (hN n (Nat.le_of_not_gt hlt) hqD) |
|
|
| theorem enlargedAccumulatingDisks_locallyFinite_punctured : |
| LocallyFinite (fun n : ℕ => |
| {p : {p : Coord // p ≠ 0} | p.val ∈ enlargedAccumulatingDisk n}) := by |
| intro p |
| obtain ⟨W, hWo, hpW, hfin⟩ := enlargedAccumulatingDisks_locallyFinite_away_zero p.property |
| refine ⟨Subtype.val ⁻¹' W, |
| (hWo.preimage continuous_subtype_val).mem_nhds hpW, hfin.subset ?_⟩ |
| intro n hn |
| obtain ⟨q, hqD, hqW⟩ := hn |
| exact ⟨q.val, hqD, hqW⟩ |
|
|
| theorem accumulatingDisks_locallyFinite_punctured : |
| LocallyFinite (fun n : ℕ => |
| {p : {p : Coord // p ≠ 0} | p.val ∈ accumulatingDisk n}) := |
| enlargedAccumulatingDisks_locallyFinite_punctured.subset |
| (fun index _ hp => accumulatingDisk_subset_enlarged index hp) |
|
|
| end SmoothLocal.Geometry |
|
|
| end |
|
|
| end OAI |
|
|
