| import Mathlib |
| import OAI.Probability.LatticeSurface.Geometry.TriangularCover |
| import OAI.Probability.LatticeSurface.Geometry.TriangularGeometry |
|
|
| namespace OAI |
|
|
| open MeasureTheory |
|
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| open scoped ENNReal |
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| noncomputable section |
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| namespace TriangularLipschitz |
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| @[simp] lemma mem_interior {L : ℕ} {x : Vertex} : |
| x ∈ interior L ↔ (1 ≤ x.1 ∧ x.1 ≤ (L : ℤ) - 1) ∧ |
| (1 ≤ x.2 ∧ x.2 ≤ (L : ℤ) - 1) := by |
| simp [interior] |
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|
| lemma interior_subset_box (L : ℕ) : interior L ⊆ box L := by |
| intro x hx |
| simp only [mem_interior, mem_box] at hx ⊢ |
| omega |
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|
| @[simp] lemma origin_not_interior (L : ℕ) : (0,0) ∉ interior L := by |
| simp [mem_interior] |
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| @[simp] lemma filledHeight_free (L : ℕ) (s : Vertex → ℝ) (a : Filling L) |
| (x : {x : Vertex // x ∈ interior L}) : filledHeight L s a x = a x := |
| dite_eq_left x.property |
|
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| @[simp] lemma filledHeight_fixed (L : ℕ) (s : Vertex → ℝ) (a : Filling L) |
| {x : Vertex} (hx : x ∉ interior L) : filledHeight L s a x = s x := |
| dite_eq_right hx |
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| lemma continuous_filledHeight (L : ℕ) (s : Vertex → ℝ) (x : Vertex) : |
| Continuous (fun a : Filling L => filledHeight L s a x) := by |
| unfold filledHeight |
| split_ifs with hx |
| · exact continuous_apply (⟨x,hx⟩ : {x : Vertex // x ∈ interior L}) |
| · exact continuous_const |
|
|
| lemma isClosed_fillingPolytope (L : ℕ) (s : Vertex → ℝ) : |
| IsClosed (fillingPolytope L s) := by |
| simp only [fillingPolytope, Set.ofPred_forall] |
| exact isClosed_iInter fun x => isClosed_iInter fun _ => |
| isClosed_iInter fun y => isClosed_iInter fun _ => isClosed_iInter fun _ => |
| isClosed_le ((continuous_filledHeight L s x).sub |
| (continuous_filledHeight L s y)).abs continuous_const |
|
|
| lemma measurableSet_fillingPolytope (L : ℕ) (s : Vertex → ℝ) : |
| MeasurableSet (fillingPolytope L s) := (isClosed_fillingPolytope L s).measurableSet |
|
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| def filledHeightAffine (L : ℕ) (s : Vertex → ℝ) (x : Vertex) : |
| Filling L →ᵃ[ℝ] ℝ := |
| if hx : x ∈ interior L then (LinearMap.proj ⟨x,hx⟩).toAffineMap |
| else AffineMap.const ℝ _ (s x) |
|
|
| @[simp] lemma filledHeightAffine_apply (L : ℕ) (s : Vertex → ℝ) (x : Vertex) |
| (a : Filling L) : filledHeightAffine L s x a = filledHeight L s a x := by |
| unfold filledHeightAffine filledHeight |
| split_ifs <;> rfl |
|
|
| lemma convex_fillingPolytope (L : ℕ) (s : Vertex → ℝ) : |
| Convex ℝ (fillingPolytope L s) := by |
| intro a ha b hb α β hα hβ hsum x hx y hy hxy |
| let φ := filledHeightAffine L s x - filledHeightAffine L s y |
| have hφ : ∀ c, φ c = filledHeight L s c x - filledHeight L s c y := by |
| intro c |
| simp only [φ, AffineMap.coe_sub, Pi.sub_apply, filledHeightAffine_apply] |
| have h := ((convex_Icc (-1 : ℝ) 1).affine_preimage φ) |
| (show φ a ∈ Set.Icc (-1) 1 by rw [hφ]; exact abs_le.mp (ha x hx y hy hxy)) |
| (show φ b ∈ Set.Icc (-1) 1 by rw [hφ]; exact abs_le.mp (hb x hx y hy hxy)) |
| hα hβ hsum |
| rw [Set.mem_preimage, hφ] at h |
| exact abs_le.mpr h |
|
|
| lemma latticeDist_origin_le {L : ℕ} {x : Vertex} (hx : x ∈ box L) : |
| latticeDist x (0,0) ≤ 2 * (L : ℝ) := by |
| obtain ⟨⟨hx0,hxL⟩,⟨hy0,hyL⟩⟩ := mem_box.mp hx |
| have hx0' : (0 : ℝ) ≤ x.1 := by exact_mod_cast hx0 |
| have hy0' : (0 : ℝ) ≤ x.2 := by exact_mod_cast hy0 |
| have hxL' : (x.1 : ℝ) ≤ L := by exact_mod_cast hxL |
| have hyL' : (x.2 : ℝ) ≤ L := by exact_mod_cast hyL |
| simp only [latticeDist, Int.cast_zero, sub_zero, abs_of_nonneg hx0', |
| abs_of_nonneg hy0', abs_of_nonneg (add_nonneg hx0' hy0'), max_le_iff] |
| constructor |
| · constructor <;> linarith |
| · linarith |
|
|
| lemma fillingPolytope_subset_closedBall (L : ℕ) (s : Vertex → ℝ) : |
| fillingPolytope L s ⊆ Metric.closedBall (fun _ => s (0,0) : Filling L) (2*L) := by |
| intro a ha |
| rw [Metric.mem_closedBall, dist_pi_le_iff (by positivity : (0 : ℝ) ≤ 2*L)] |
| intro x |
| have h := near_global_bound ha (interior_subset_box L x.property) |
| (show (0,0) ∈ box L by simp) |
| rw [filledHeight_free, filledHeight_fixed _ _ _ (origin_not_interior L), one_mul] at h |
| exact h.trans (latticeDist_origin_le (interior_subset_box L x.property)) |
|
|
| lemma fillingVolume_lt_top (L : ℕ) (s : Vertex → ℝ) : fillingVolume L s < ⊤ := |
| (measure_mono (fillingPolytope_subset_closedBall L s)).trans_lt measure_closedBall_lt_top |
|
|
| lemma isCompact_fillingPolytope (L : ℕ) (s : Vertex → ℝ) : |
| IsCompact (fillingPolytope L s) := |
| (isCompact_closedBall _ _).of_isClosed_subset (isClosed_fillingPolytope L s) |
| (fillingPolytope_subset_closedBall L s) |
|
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| lemma filledHeight_center (L : ℕ) (h : Vertex → ℝ) (x : Vertex) : |
| filledHeight L h (fun x => h x) x = h x := by |
| unfold filledHeight |
| split_ifs <;> rfl |
|
|
| lemma filledHeight_perturbation (L : ℕ) (h : Vertex → ℝ) (a : Filling L) (x : Vertex) : |
| |filledHeight L h a x - h x| ≤ ‖a - (fun x => h x : Filling L)‖ := by |
| by_cases hx : x ∈ interior L |
| · rw [filledHeight, dite_eq_left hx] |
| exact norm_le_pi_norm (a - (fun x => h x : Filling L)) ⟨x,hx⟩ |
| · rw [filledHeight_fixed _ _ _ hx, sub_self, abs_zero] |
| exact norm_nonneg _ |
|
|
| lemma ball_subset_fillingPolytope {L : ℕ} {h : Vertex → ℝ} {δ : ℝ} |
| (hne : ∀ x ∈ box L, ∀ y ∈ box L, Adj x y → |h x-h y| ≤ 1-δ) : |
| Metric.ball (fun x => h x : Filling L) (δ/4) ⊆ fillingPolytope L h := by |
| intro a ha x hx y hy hxy |
| have hnorm : ‖a - (fun x => h x : Filling L)‖ < δ/4 := by |
| simpa only [Metric.mem_ball, dist_eq_norm] using ha |
| have hxerr := filledHeight_perturbation L h a x |
| have hyerr := filledHeight_perturbation L h a y |
| have hed := hne x hx y hy hxy |
| have htri : |filledHeight L h a x-filledHeight L h a y| ≤ |
| |filledHeight L h a x-h x| + |h x-h y| + |filledHeight L h a y-h y| := by |
| calc |
| _ ≤ |filledHeight L h a x-h x| + |h x-filledHeight L h a y| := abs_sub_le _ _ _ |
| _ ≤ _ := by |
| have ht := abs_sub_le (h x) (h y) (filledHeight L h a y) |
| rw [abs_sub_comm (h y)] at ht |
| linarith |
| have hn := norm_nonneg (a - (fun x => h x : Filling L)) |
| linarith |
|
|
| lemma fillingVolume_pos_of_strict {L : ℕ} {h : Vertex → ℝ} {δ : ℝ} (hδ : 0 < δ) |
| (hne : ∀ x ∈ box L, ∀ y ∈ box L, Adj x y → |h x-h y| ≤ 1-δ) : |
| 0 < fillingVolume L h := |
| (Metric.measure_ball_pos volume _ (by positivity : 0 < δ/4)).trans_le |
| (measure_mono (ball_subset_fillingPolytope hne)) |
|
|
| end TriangularLipschitz |
|
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| end |
|
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| end OAI |
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