| import Mathlib | |
| namespace OAI | |
| open Set MeasureTheory intervalIntegral | |
| universe u v | |
| namespace FeasibleShift | |
| variable {V : Type u} [Fintype V] [DecidableEq V] | |
| noncomputable def component (G : SimpleGraph V) (x : V) : Finset V := by | |
| classical | |
| exact Finset.univ.filter (G.Reachable x) | |
| noncomputable def contactGraph (G : SimpleGraph V) (h : V → ℝ) (s : ℝ) : SimpleGraph V where | |
| Adj x y := G.Adj x y ∧ 1 - |h x - h y| ≤ s | |
| symm := ⟨by intro x y hxy; simpa only [abs_sub_comm] using And.intro hxy.1.symm hxy.2⟩ | |
| loopless := ⟨by intro x hx; exact G.irrefl hx.1⟩ | |
| noncomputable def prescription (G : SimpleGraph V) (ground : Finset V) | |
| (f : V → ℝ) (x : V) : ℝ := | |
| if Disjoint (component G x) ground then | |
| (∑ y ∈ component G x, f y) / (component G x).card else 0 | |
| noncomputable def averagedShift (G : SimpleGraph V) (ground : Finset V) | |
| (f h : V → ℝ) (s₀ s₁ : ℝ) (x : V) : ℝ := | |
| (s₁ - s₀)⁻¹ * ∫ s in s₀..s₁, prescription (contactGraph G h s) ground f x | |
| abbrev FreeSites (pins : Finset V) := {x : V // x ∉ pins} | |
| noncomputable def extendHeight (pins : Finset V) (β : V → ℝ) | |
| (h : FreeSites pins → ℝ) (x : V) : ℝ := | |
| if hx : x ∈ pins then β x else h ⟨x, hx⟩ | |
| noncomputable def freeShift (G : SimpleGraph V) (ground pins : Finset V) | |
| (β f : V → ℝ) (s₀ s₁ : ℝ) (h : FreeSites pins → ℝ) (x : FreeSites pins) : ℝ := | |
| averagedShift G ground f (extendHeight pins β h) s₀ s₁ x | |
| noncomputable def heightPolytope (G : SimpleGraph V) (pins walls : Finset V) | |
| (β : V → ℝ) (I : V → Set ℝ) : Set (FreeSites pins → ℝ) := | |
| {h | (∀ x y, G.Adj x y → |extendHeight pins β h x - extendHeight pins β h y| ≤ 1) ∧ | |
| ∀ x ∈ walls, extendHeight pins β h x ∈ I x} | |
| noncomputable def neighbors (G : SimpleGraph V) (x : V) : Finset V := by | |
| classical | |
| exact Finset.univ.filter (G.Adj x) | |
| noncomputable def shiftGraph (G : SimpleGraph V) (ground : Finset V) : SimpleGraph V where | |
| Adj x y := G.Adj x y ∧ ¬ (x ∈ ground ∧ y ∈ ground) | |
| symm := ⟨by intro x y h; exact ⟨h.1.symm, fun hh => h.2 ⟨hh.2, hh.1⟩⟩⟩ | |
| loopless := ⟨fun x h => G.irrefl h.1⟩ | |
| noncomputable def localSites (G : SimpleGraph V) (ground : Finset V) : Finset V := by | |
| classical | |
| exact Finset.univ.filter fun x => x ∉ ground ∨ ∃ y, y ∉ ground ∧ G.Adj x y | |
| structure ShiftData (W : Type v) [Fintype W] [DecidableEq W] | |
| (n D Db B : ℕ) (ρ : ENNReal) (sstar L M N : ℝ) where | |
| graph : SimpleGraph V | |
| ground : Finset V | |
| pins : Finset V | |
| walls : Finset V | |
| pins_ground : pins ⊆ ground | |
| walls_ground : walls ⊆ ground | |
| boundary : V → ℝ | |
| intervals : V → Set ℝ | |
| prescribed : V → ℝ | |
| prescribed_ground : ∀ x ∈ ground, prescribed x = 0 | |
| prescribed_bound : ∀ x, |prescribed x| ≤ M | |
| prescribed_gradient : ∀ x y, graph.Adj x y → |prescribed x - prescribed y| ≤ L / n | |
| vertex_count : ((localSites graph ground).card : ℝ) ≤ N * (n : ℝ)^2 | |
| degree : ∀ x, (neighbors graph x).card ≤ D | |
| polytope_measurable : MeasurableSet (heightPolytope graph pins walls boundary intervals) | |
| polytope_positive : volume (heightPolytope graph pins walls boundary intervals) ≠ 0 | |
| polytope_finite : volume (heightPolytope graph pins walls boundary intervals) ≠ ⊤ | |
| boxGraph : SimpleGraph W | |
| box_degree : ∀ w, (neighbors boxGraph w).card ≤ Db | |
| boxMap : V → W | |
| box_fiber : ∀ w, (Finset.univ.filter fun x => boxMap x = w).card ≤ B | |
| box_adjacent : ∀ x y, graph.Adj x y → boxMap x = boxMap y ∨ boxGraph.Adj (boxMap x) (boxMap y) | |
| badBoxes : (FreeSites pins → ℝ) → Set W | |
| contact_bad : ∀ h x y, (contactGraph (shiftGraph graph ground) | |
| (extendHeight pins boundary h) sstar).Adj x y → boxMap x ∈ badBoxes h ∧ boxMap y ∈ badBoxes h | |
| sparse : ∀ A : Finset W, (ProbabilityTheory.cond volume | |
| (heightPolytope graph pins walls boundary intervals)) {h | ∀ w ∈ A, w ∈ badBoxes h} ≤ ρ^A.card | |
| namespace ShiftData | |
| variable {W : Type v} [Fintype W] [DecidableEq W] | |
| variable {n D Db B : ℕ} {ρ : ENNReal} {sstar L M N : ℝ} | |
| variable (S : ShiftData (V := V) W n D Db B ρ sstar L M N) | |
| noncomputable def polytope := heightPolytope S.graph S.pins S.walls S.boundary S.intervals | |
| noncomputable def law := ProbabilityTheory.cond volume S.polytope | |
| noncomputable def shift (s₀ s₁ : ℝ) := | |
| freeShift (shiftGraph S.graph S.ground) S.ground S.pins S.boundary S.prescribed s₀ s₁ | |
| noncomputable def map (s₀ s₁ : ℝ) (h : FreeSites S.pins → ℝ) := h + S.shift s₀ s₁ h | |
| end ShiftData | |
| end FeasibleShift | |
| end OAI | |