| import Mathlib |
| import OAI.Probability.LatticeSurface.Convexity.RadialExposure |
| import OAI.Probability.LatticeSurface.Geometry.BlockVolume |
| import OAI.Probability.LatticeSurface.Geometry.FacetExposure |
| import OAI.Probability.LatticeSurface.Geometry.TriangularCover |
| import OAI.Probability.LatticeSurface.Geometry.TriangularGeometry |
|
|
| namespace OAI |
|
|
| open MeasureTheory |
|
|
| open scoped ENNReal |
|
|
| noncomputable section |
|
|
| namespace TriangularLipschitz |
|
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| end TriangularLipschitz |
|
|
| namespace FacetExposure |
|
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| end FacetExposure |
|
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| namespace TriangularLipschitz |
|
|
| end TriangularLipschitz |
|
|
| namespace TriangularLipschitz |
|
|
| lemma shell_eq_box_not_interior {L : ℕ} {x : Vertex} : |
| x ∈ shell L ↔ x ∈ box L ∧ x ∉ interior L := Finset.mem_sdiff |
|
|
| lemma latticeDist_box_le {L : ℕ} {x y : Vertex} (hx : x ∈ box L) (hy : y ∈ box L) : |
| latticeDist x y ≤ 2*(L:ℝ) := by |
| obtain ⟨⟨hx0,hxL⟩,⟨hx20,hx2L⟩⟩ := mem_box.mp hx |
| obtain ⟨⟨hy0,hyL⟩,⟨hy20,hy2L⟩⟩ := mem_box.mp hy |
| have hx' : 0 ≤ (x.1:ℝ) ∧ (x.1:ℝ) ≤ L := by exact_mod_cast And.intro hx0 hxL |
| have hx2' : 0 ≤ (x.2:ℝ) ∧ (x.2:ℝ) ≤ L := by exact_mod_cast And.intro hx20 hx2L |
| have hy' : 0 ≤ (y.1:ℝ) ∧ (y.1:ℝ) ≤ L := by exact_mod_cast And.intro hy0 hyL |
| have hy2' : 0 ≤ (y.2:ℝ) ∧ (y.2:ℝ) ≤ L := by exact_mod_cast And.intro hy20 hy2L |
| simp only [latticeDist, max_le_iff, abs_le] |
| constructor |
| · constructor <;> constructor <;> linarith |
| · constructor <;> linarith |
|
|
| lemma endpoint_shell_difference {L : ℕ} {h χ : Vertex → ℝ} |
| (hplus : 0 < fillingVolume L (fun x => h x+χ x)) |
| (hminus : 0 < fillingVolume L (fun x => h x-χ x)) |
| {b c : Vertex} (hb : b ∈ shell L) (hc : c ∈ shell L) : |
| |χ b-χ c| ≤ latticeDist b c := by |
| obtain ⟨ap,hap⟩ : (fillingPolytope L (fun x => h x+χ x)).Nonempty := |
| nonempty_of_measure_ne_zero hplus.ne' |
| obtain ⟨am,ham⟩ : (fillingPolytope L (fun x => h x-χ x)).Nonempty := |
| nonempty_of_measure_ne_zero hminus.ne' |
| rw [shell_eq_box_not_interior] at hb hc |
| have hp := near_global_bound hap hb.1 hc.1 |
| have hm := near_global_bound ham hb.1 hc.1 |
| simp only [filledHeight_fixed _ _ _ hb.2, filledHeight_fixed _ _ _ hc.2, one_mul] at hp hm |
| obtain ⟨hp₁,hp₂⟩ := abs_le.mp hp |
| obtain ⟨hm₁,hm₂⟩ := abs_le.mp hm |
| exact abs_le.mpr ⟨by linarith, by linarith⟩ |
|
|
| lemma filledHeight_constant_shift (L : ℕ) (g : Vertex → ℝ) (r : ℝ) (x : Vertex) : |
| filledHeight L (fun z => g z+r) (fun z => g z+r) x = g x+r := by |
| apply filledHeight_center |
|
|
| lemma filledHeight_base_difference {L : ℕ} {s g : Vertex → ℝ} {r η : ℝ} (hη : 0 ≤ η) |
| (hs : ∀ b ∈ shell L, |s b-(g b+r)| ≤ η) {x : Vertex} (hx : x ∈ box L) : |
| |filledHeight L s (fun z => g z+r) x-(g x+r)| ≤ η := by |
| by_cases hi : x ∈ interior L |
| · simp only [filledHeight, dite_eq_left hi, sub_self, abs_zero] |
| exact hη |
| · rw [filledHeight_fixed _ _ _ hi] |
| exact hs x (shell_eq_box_not_interior.mpr ⟨hx,hi⟩) |
|
|
| lemma filledHeight_change_bound {L : ℕ} {s : Vertex → ℝ} (a a₀ : Filling L) (x : Vertex) : |
| |filledHeight L s a x-filledHeight L s a₀ x| ≤ ‖a-a₀‖ := by |
| by_cases hi : x ∈ interior L |
| · simp only [filledHeight, dite_eq_left hi] |
| exact norm_le_pi_norm (a-a₀) ⟨x,hi⟩ |
| · simp only [filledHeight_fixed _ _ _ hi, sub_self, abs_zero] |
| exact norm_nonneg _ |
|
|
| lemma filledHeight_edge_change_bound {L : ℕ} {s : Vertex → ℝ} (a a₀ : Filling L) (x y : Vertex) : |
| |(filledHeight L s a x-filledHeight L s a y)- |
| (filledHeight L s a₀ x-filledHeight L s a₀ y)| ≤ 2*‖a-a₀‖ := by |
| calc |
| _ = |(filledHeight L s a x-filledHeight L s a₀ x)- |
| (filledHeight L s a y-filledHeight L s a₀ y)| := by congr 1; ring |
| _ ≤ |filledHeight L s a x-filledHeight L s a₀ x|+ |
| |filledHeight L s a y-filledHeight L s a₀ y| := abs_sub _ _ |
| _ ≤ ‖a-a₀‖+‖a-a₀‖ := add_le_add (filledHeight_change_bound a a₀ x) (filledHeight_change_bound a a₀ y) |
| _ = 2*‖a-a₀‖ := by ring |
|
|
| lemma shifted_exposure {L : ℕ} {g s : Vertex → ℝ} {x y : Vertex} {β r : ℝ} |
| (hβ : 0 < β) |
| (hs : ∀ b ∈ shell L, |s b-(g b+r)| ≤ β/4) |
| (hex : ∀ v ∈ box L, ∀ w ∈ box L, Adj v w → s(v,w) ≠ s(x,y) → |g v-g w| ≤ 1-β) : |
| ∀ v ∈ box L, ∀ w ∈ box L, Adj v w → s(v,w) ≠ s(x,y) → |
| |filledHeight L s (fun z => g z+r) v-filledHeight L s (fun z => g z+r) w| ≤ 1-β/2 := by |
| intro v hv w hw hvw hne |
| have hgv := filledHeight_base_difference (by linarith : 0 ≤ β/4) hs hv |
| have hgw := filledHeight_base_difference (by linarith : 0 ≤ β/4) hs hw |
| have hbase : |(g v+r)-(g w+r)| ≤ 1-β := by simpa only [add_sub_add_right_eq_sub] using hex v hv w hw hvw hne |
| have htri := abs_add_le (filledHeight L s (fun z => g z+r) v-(g v+r)) |
| (((g v+r)-(g w+r))+((g w+r)-filledHeight L s (fun z => g z+r) w)) |
| have htri₂ := abs_add_le ((g v+r)-(g w+r)) ((g w+r)-filledHeight L s (fun z => g z+r) w) |
| rw [abs_sub_comm (g w+r)] at htri₂ |
| have heq : filledHeight L s (fun z => g z+r) v-(g v+r)+ |
| ((g v+r-(g w+r))+(g w+r-filledHeight L s (fun z => g z+r) w)) = |
| filledHeight L s (fun z => g z+r) v-filledHeight L s (fun z => g z+r) w := by ring |
| rw [heq] at htri |
| linarith |
|
|
| lemma filling_of_small_perturbation_on_facet {L : ℕ} {s : Vertex → ℝ} {x y : Vertex} |
| {a a₀ : Filling L} {β : ℝ} |
| (hfacet : |filledHeight L s a x-filledHeight L s a y| ≤ 1) |
| (hother : ∀ v ∈ box L, ∀ w ∈ box L, Adj v w → s(v,w) ≠ s(x,y) → |
| |filledHeight L s a₀ v-filledHeight L s a₀ w| ≤ 1-β) |
| (hsmall : 2*‖a-a₀‖ ≤ β) : a ∈ fillingPolytope L s := by |
| intro v hv w hw hvw |
| by_cases he : s(v,w)=s(x,y) |
| · rcases Sym2.eq_iff.mp he with ⟨rfl,rfl⟩ | ⟨rfl,rfl⟩ |
| · exact hfacet |
| · simpa only [abs_sub_comm] using hfacet |
| · have hdiff := filledHeight_edge_change_bound (s:=s) a a₀ v w |
| have h₀ := hother v hv w hw hvw he |
| have htri := abs_add_le ((filledHeight L s a v-filledHeight L s a w)- |
| (filledHeight L s a₀ v-filledHeight L s a₀ w)) |
| (filledHeight L s a₀ v-filledHeight L s a₀ w) |
| rw [sub_add_cancel] at htri |
| linarith |
|
|
| end TriangularLipschitz |
|
|
| namespace TriangularLipschitz |
|
|
| lemma ball_subset_polytope_of_slack {L : ℕ} {s : Vertex → ℝ} {a₀ : Filling L} {δ : ℝ} |
| (hh : ∀ x ∈ box L, ∀ y ∈ box L, Adj x y → |
| |filledHeight L s a₀ x-filledHeight L s a₀ y| ≤ 1-δ) : |
| Metric.ball a₀ (δ/4) ⊆ fillingPolytope L s := by |
| intro a ha x hx y hy hxy |
| have hnorm : ‖a-a₀‖ < δ/4 := by simpa only [Metric.mem_ball, dist_eq_norm] using ha |
| have hdiff := filledHeight_edge_change_bound (s:=s) a a₀ x y |
| have htri := abs_add_le ((filledHeight L s a x-filledHeight L s a y)- |
| (filledHeight L s a₀ x-filledHeight L s a₀ y)) |
| (filledHeight L s a₀ x-filledHeight L s a₀ y) |
| rw [sub_add_cancel] at htri |
| have h := hh x hx y hy hxy |
| linarith [norm_nonneg (a-a₀)] |
|
|
| lemma fillingVolume_pos_of_slack {L : ℕ} {s : Vertex → ℝ} {a₀ : Filling L} {δ : ℝ} |
| (hδ : 0 < δ) |
| (hh : ∀ x ∈ box L, ∀ y ∈ box L, Adj x y → |
| |filledHeight L s a₀ x-filledHeight L s a₀ y| ≤ 1-δ) : |
| 0 < fillingVolume L s := |
| (Metric.measure_ball_pos volume a₀ (by linarith : 0 < δ/4)).trans_le |
| (measure_mono (ball_subset_polytope_of_slack hh)) |
|
|
| lemma filledHeight_mix (L : ℕ) (s t : Vertex → ℝ) (a b : Filling L) (α γ : ℝ) (x : Vertex) : |
| filledHeight L (fun x => α*s x+γ*t x) (α • a+γ • b) x = |
| α*filledHeight L s a x+γ*filledHeight L t b x := by |
| unfold filledHeight |
| split_ifs <;> rfl |
|
|
| lemma fillingPolytope_mix {L : ℕ} {s t : Vertex → ℝ} {a b : Filling L} |
| {α γ : ℝ} (ha : a ∈ fillingPolytope L s) (hb : b ∈ fillingPolytope L t) |
| (hα : 0 ≤ α) (hγ : 0 ≤ γ) (hsum : α+γ=1) : |
| α • a+γ • b ∈ fillingPolytope L (fun x => α*s x+γ*t x) := by |
| intro x hx y hy hxy |
| rw [filledHeight_mix, filledHeight_mix] |
| calc |
| _ = |α*(filledHeight L s a x-filledHeight L s a y)+ |
| γ*(filledHeight L t b x-filledHeight L t b y)| := by congr 1; ring |
| _ ≤ |α*(filledHeight L s a x-filledHeight L s a y)|+ |
| |γ*(filledHeight L t b x-filledHeight L t b y)| := abs_add_le _ _ |
| _ = α*|filledHeight L s a x-filledHeight L s a y|+ |
| γ*|filledHeight L t b x-filledHeight L t b y| := by rw [abs_mul,abs_mul,abs_of_nonneg hα,abs_of_nonneg hγ] |
| _ ≤ α*1+γ*1 := add_le_add (mul_le_mul_of_nonneg_left (ha x hx y hy hxy) hα) |
| (mul_le_mul_of_nonneg_left (hb x hx y hy hxy) hγ) |
| _ = 1 := by linarith |
|
|
| lemma fillingVolume_interpolated_pos {L : ℕ} {h t : Vertex → ℝ} {δ α : ℝ} |
| (hδ : 0 < δ) (hα : 0 ≤ α) (hα1 : α < 1) |
| (hne : ∀ x ∈ box L, ∀ y ∈ box L, Adj x y → |h x-h y| ≤ 1-δ) |
| (ht : (fillingPolytope L t).Nonempty) : |
| 0 < fillingVolume L (fun x => (1-α)*h x+α*t x) := by |
| obtain ⟨b,hb⟩ := ht |
| let a₀ : Filling L := (1-α) • (fun x : {x : Vertex // x ∈ interior L} => h x)+α • b |
| apply fillingVolume_pos_of_slack (a₀:=a₀) (δ:=(1-α)*δ) (mul_pos (by linarith) hδ) |
| intro x hx y hy hxy |
| change |filledHeight L (fun z => (1-α)*h z+α*t z) ((1-α) • (fun z : {x : Vertex // x ∈ interior L} => h z)+α • b) x- |
| filledHeight L (fun z => (1-α)*h z+α*t z) ((1-α) • (fun z : {x : Vertex // x ∈ interior L} => h z)+α • b) y| ≤ _ |
| rw [filledHeight_mix, filledHeight_mix, filledHeight_center, filledHeight_center] |
| calc |
| _ = |(1-α)*(h x-h y)+α*(filledHeight L t b x-filledHeight L t b y)| := by congr 1; ring |
| _ ≤ |(1-α)*(h x-h y)|+|α*(filledHeight L t b x-filledHeight L t b y)| := abs_add_le _ _ |
| _ = (1-α)*|h x-h y|+α*|filledHeight L t b x-filledHeight L t b y| := by |
| rw [abs_mul,abs_mul,abs_of_nonneg (by linarith : 0 ≤ 1-α),abs_of_nonneg hα] |
| _ ≤ (1-α)*(1-δ)+α*1 := add_le_add |
| (mul_le_mul_of_nonneg_left (hne x hx y hy hxy) (by linarith)) |
| (mul_le_mul_of_nonneg_left (hb x hx y hy hxy) hα) |
| _ = 1-(1-α)*δ := by ring |
|
|
| end TriangularLipschitz |
|
|
| end |
|
|
| end OAI |
|
|
