| import Mathlib |
| import OAI.Probability.LatticeSurface.Coercivity.SparseAmbient |
| import OAI.Probability.LatticeSurface.Coercivity.SparseColor |
| import OAI.Probability.LatticeSurface.Coercivity.SparseModel |
| import OAI.Probability.LatticeSurface.Coercivity.SparseParameters |
| import OAI.Probability.LatticeSurface.Convexity.BoundedBase |
| import OAI.Probability.LatticeSurface.Convexity.BrunnMinkowski |
| import OAI.Probability.LatticeSurface.Convexity.CubePrekopa |
| import OAI.Probability.LatticeSurface.Convexity.GradientWindows |
| import OAI.Probability.LatticeSurface.Convexity.QuantileTransport |
| import OAI.Probability.LatticeSurface.Convexity.WeightedPrekopa |
| import OAI.Probability.LatticeSurface.Dynamics.DisplacementInequality |
| import OAI.Probability.LatticeSurface.Geometry.BlockVolume |
| import OAI.Probability.LatticeSurface.Geometry.CapCentral |
| import OAI.Probability.LatticeSurface.Geometry.CapCompact |
| import OAI.Probability.LatticeSurface.Geometry.CapContract |
| import OAI.Probability.LatticeSurface.Geometry.CapCoordinates |
| import OAI.Probability.LatticeSurface.Geometry.CapMatching |
| import OAI.Probability.LatticeSurface.Geometry.CapOrient |
| import OAI.Probability.LatticeSurface.Geometry.CapReflection |
| import OAI.Probability.LatticeSurface.Geometry.CapSmall |
| import OAI.Probability.LatticeSurface.Geometry.CapSplit |
| import OAI.Probability.LatticeSurface.Geometry.CapUnpin |
| import OAI.Probability.LatticeSurface.Geometry.CapWeld |
| import OAI.Probability.LatticeSurface.Geometry.TriangularCover |
| import OAI.Probability.LatticeSurface.Geometry.TriangularGeometry |
| import OAI.Probability.LatticeSurface.Moments.ControlledGradients |
| import OAI.Probability.LatticeSurface.Moments.OrderedGradients |
|
|
| namespace OAI |
|
|
| open Set MeasureTheory Filter |
|
|
| open scoped Topology |
|
|
| noncomputable section |
|
|
| namespace FacetVolume |
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| end FacetVolume |
|
|
| namespace ExtremalEdges |
|
|
| open scoped ENNReal |
|
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| end ExtremalEdges |
|
|
| noncomputable section |
|
|
| open MeasureTheory Set |
|
|
| open scoped ENNReal |
|
|
| namespace ExtremalEdges |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| end ExtremalEdges |
|
|
| namespace ExtremalEdges |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| variable {V : Type*} |
|
|
| end ExtremalEdges |
|
|
| namespace ExtremalEdges |
|
|
| attribute [local instance] Classical.propDecidable |
|
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| end ExtremalEdges |
|
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| namespace ExtremalEdges |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| variable {V : Type*} |
|
|
| end ExtremalEdges |
|
|
| namespace ExtremalEdges |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| variable {V : Type*} |
|
|
| end ExtremalEdges |
|
|
| namespace ExtremalEdges |
|
|
| attribute [local instance] Classical.propDecidable |
|
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| variable {V : Type*} |
|
|
| end ExtremalEdges |
|
|
| namespace ExtremalEdges |
|
|
| attribute [local instance] Classical.propDecidable |
|
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| variable {V : Type*} |
|
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| end ExtremalEdges |
|
|
| namespace ExtremalEdges |
|
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| attribute [local instance] Classical.propDecidable |
|
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| variable {V : Type*} |
|
|
| end ExtremalEdges |
|
|
| namespace ExtremalEdges |
|
|
| attribute [local instance] Classical.propDecidable |
|
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| variable {V : Type*} |
|
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| end ExtremalEdges |
|
|
| noncomputable section |
|
|
| namespace ExtremalEdges |
|
|
| open Finset |
|
|
| variable {V I : Type*} [Fintype V] [Fintype I] |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| end ExtremalEdges |
|
|
| namespace ExtremalEdges |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| variable {V I : Type*} [Fintype V] [Fintype I] |
|
|
| end ExtremalEdges |
|
|
| namespace ExtremalEdges |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| end ExtremalEdges |
|
|
| namespace ExtremalEdges |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| variable {V : Type*} [Fintype V] |
|
|
| end ExtremalEdges |
|
|
| open MeasureTheory |
|
|
| open scoped ENNReal |
|
|
| noncomputable section |
|
|
| namespace TriangularLipschitz |
|
|
| end TriangularLipschitz |
|
|
| namespace SparseBlocks |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| end SparseBlocks |
|
|
| namespace SparseBlocks |
|
|
| attribute [local instance] Classical.propDecidable |
|
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| end SparseBlocks |
|
|
| namespace SparseBlocks |
|
|
| attribute [local instance] Classical.propDecidable |
|
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| end SparseBlocks |
|
|
| namespace ExtremalEdges |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| end ExtremalEdges |
|
|
| namespace SparseBlocks |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| lemma volume_body_positive (V : Finset Vertex) : 0 < volume (body V) := by |
| obtain ⟨B,hVB,hcollar,hc,hp⟩ := exists_ambient V |
| rw [← volume_body_coordinates B V hVB hcollar] |
| exact ExtremalEdges.volume_zeroBody_pos _ _ |
|
|
| lemma volume_body_finite (V : Finset Vertex) : volume (body V) ≠ ⊤ := by |
| obtain ⟨B,hVB,hcollar,hc,hp⟩ := exists_ambient V |
| rw [← volume_body_coordinates B V hVB hcollar] |
| exact (ExtremalEdges.compact_zeroBody _ hc _ hp).measure_ne_top |
|
|
| def law (V : Finset Vertex) : Measure (V → ℝ) := |
| (volume (body V))⁻¹ • volume.restrict (body V) |
| |
| instance law_probability (V : Finset Vertex) : IsProbabilityMeasure (law V) := by |
| constructor |
| simp only [law, Measure.smul_apply, Measure.restrict_apply MeasurableSet.univ, |
| Set.univ_inter, smul_eq_mul] |
| exact ENNReal.inv_mul_cancel (ne_of_gt (volume_body_positive V)) (volume_body_finite V) |
| |
| lemma law_real (V : Finset Vertex) (S : Set (V → ℝ)) (hS : MeasurableSet S) : |
| (law V).real S = (volume (body V ∩ S)).toReal / (volume (body V)).toReal := by |
| simp only [Measure.real,law,Measure.smul_apply,Measure.restrict_apply hS, |
| smul_eq_mul,ENNReal.toReal_mul,ENNReal.toReal_inv] |
| rw [Set.inter_comm,div_eq_mul_inv,mul_comm] |
| |
| def window (b : Bool) (u t : ℝ) : Prop := if b then 1-u ≤ |t| else |t| ≤ u |
| |
| def edgeEvent (V : Finset Vertex) (b : Bool) (u : ℝ) (e : Vertex × Vertex) : Set (V → ℝ) := |
| {h | window b u (height V h e.1-height V h e.2)} |
| |
| def gradientEvent (V : Finset Vertex) (b : Bool) (u : ℝ) |
| (F : Finset (Vertex × Vertex)) : Set (V → ℝ) := |
| {h | ∀ e ∈ F, h ∈ edgeEvent V b u e} |
| |
| lemma closed_edgeEvent (V : Finset Vertex) (b : Bool) (u : ℝ) (e : Vertex × Vertex) : |
| IsClosed (edgeEvent V b u e) := by |
| cases b |
| · exact isClosed_le ((continuous_height V e.1).sub (continuous_height V e.2)).abs continuous_const |
| · exact isClosed_le continuous_const ((continuous_height V e.1).sub (continuous_height V e.2)).abs |
| |
| lemma closed_gradientEvent (V : Finset Vertex) (b : Bool) (u : ℝ) (F : Finset (Vertex × Vertex)) : |
| IsClosed (gradientEvent V b u F) := by |
| simp only [gradientEvent,Set.ofPred_forall] |
| exact isClosed_iInter fun e => isClosed_iInter fun _ => closed_edgeEvent V b u e |
| |
| theorem joint_gradients (V : Finset Vertex) (F : Finset (Vertex × Vertex)) |
| (he : ∀ e ∈ F, Adj e.1 e.2) (hf : ∀ e ∈ F, e.1 ∈ V ∧ e.2 ∈ V) |
| (b : Bool) (u : ℝ) (hu : 0 < u) (hu8 : u ≤ 1/8) : |
| (law V).real (gradientEvent V b u F) ≤ |
| ((ExtremalEdges.gradientConstant 6*u)^(1/ExtremalEdges.gradientConstant 6))^F.card := by |
| classical |
| obtain ⟨B,hVB,hcollar,hc,hp⟩ := exists_ambient V |
| let ee : Fin F.card → F := (Finset.equivFin F).symm |
| let e : Fin F.card → B × B := fun i => |
| (⟨(ee i).val.1,hVB (hf _ (ee i).property).1⟩,⟨(ee i).val.2,hVB (hf _ (ee i).property).2⟩) |
| have hei : Function.Injective e := by |
| intro i j hij |
| have hh : ee i = ee j := Subtype.ext (Prod.ext |
| (congrArg (fun p : B × B => p.1.val) hij) (congrArg (fun p : B × B => p.2.val) hij)) |
| exact (Finset.equivFin F).symm.injective hh |
| have heG : ∀ i, (finiteGraph B).Adj (e i).1 (e i).2 := fun i => he _ (ee i).property |
| have hfree : ∀ i, ¬ ((e i).1 ∈ fixed B V ∧ (e i).2 ∈ fixed B V) := by |
| intro i hh |
| exact hh.1 (hf _ (ee i).property).1 |
| let S : Set (ExtremalEdges.Free (fixed B V) → ℝ) := |
| {h | ∀ i, window b u (ExtremalEdges.height (fixed B V) h (e i).1- |
| ExtremalEdges.height (fixed B V) h (e i).2)} |
| have hS : IsClosed S := by |
| simp only [S,Set.ofPred_forall] |
| apply isClosed_iInter |
| intro i |
| have hh := ((ExtremalEdges.continuous_restore (fixed B V) (fun _ => 0) (e i).1).sub |
| (ExtremalEdges.continuous_restore (fixed B V) (fun _ => 0) (e i).2)).abs |
| cases b |
| · exact isClosed_le hh continuous_const |
| · exact isClosed_le continuous_const hh |
| have hpre : (coordinates B V hVB) ⁻¹' S = gradientEvent V b u F := by |
| ext h |
| simp only [Set.mem_preimage, S, Set.mem_ofPred_eq, coordinates_height, gradientEvent, edgeEvent] |
| constructor |
| · intro hh z hz |
| have hh' := hh ((Finset.equivFin F) ⟨z,hz⟩) |
| simpa only [e,ee,Equiv.symm_apply_apply] using hh' |
| · intro hh i |
| exact hh _ (ee i).property |
| have hvol : volume (ExtremalEdges.body (finiteGraph B) (fixed B V) ∩ S) = |
| volume (body V ∩ gradientEvent V b u F) := by |
| rw [← body_coordinates B V hVB hcollar, ← hpre, ← Set.preimage_inter] |
| exact ((preserving_coordinates B V hVB).measure_preimage |
| ((ExtremalEdges.closed_conditionalBody (finiteGraph B) (fixed B V) (fun _ => 0)).inter hS).measurableSet.nullMeasurableSet).symm |
| have hb : (volume (ExtremalEdges.body (finiteGraph B) (fixed B V) ∩ S)).toReal / |
| (volume (ExtremalEdges.body (finiteGraph B) (fixed B V))).toReal ≤ |
| (ExtremalEdges.gradientConstant 6*u)^((F.card:ℝ)/ExtremalEdges.gradientConstant 6) := by |
| cases b |
| · exact ExtremalEdges.near_zero_gradients_ordered 6 (finiteGraph B) hc (finiteGraph_degree B) |
| (fixed B V) hp F.card e heG hei hfree u hu |
| · exact ExtremalEdges.controlled_gradients_ordered 6 (finiteGraph B) hc (finiteGraph_degree B) |
| (fixed B V) hp F.card e heG hei u hu hu8 |
| rw [hvol,volume_body_coordinates B V hVB hcollar,← law_real V _ (closed_gradientEvent V b u F).measurableSet] at hb |
| convert hb using 1 |
| rw [← Real.rpow_mul_natCast (mul_pos (ExtremalEdges.gradientConstant_pos 6) hu).le] |
| congr 1 |
| ring |
| |
| end SparseBlocks |
| |
| end |
| end |
| end |
| end |
| |
| end OAI |
| |
