| import OAI.Probability.LatticePassage.Paths |
| import Mathlib.Probability.Distributions.Exponential |
| import Mathlib.Probability.ProductMeasure |
| import Mathlib.Algebra.Order.Algebra |
| import Mathlib.Analysis.Normed.Group.Basic |
| import Mathlib.Data.EReal.Operations |
| import Mathlib.Topology.Algebra.InfiniteSum.Order |
| import Mathlib.Topology.MetricSpace.Bounded |
|
|
| namespace OAI |
|
|
| namespace CompanionFPP |
| open MeasureTheory ProbabilityTheory Filter Topology |
|
|
| abbrev Vertex := ℤ × ℤ |
| abbrev Plane := ℝ × ℝ |
|
|
| def lattice : SimpleGraph Vertex where |
| Adj x y := |x.1 - y.1| + |x.2 - y.2| = 1 |
| symm := ⟨by intros x y h; simpa only [abs_sub_comm] using h⟩ |
| loopless := ⟨by intro x; simp⟩ |
|
|
| abbrev Edge := lattice.edgeSet |
| abbrev Environment := Weights lattice |
|
|
| noncomputable instance expOneProbability : IsProbabilityMeasure (expMeasure 1) := |
| isProbabilityMeasure_expMeasure (by norm_num) |
|
|
| noncomputable def environmentLaw : Measure Environment := |
| Measure.infinitePi (fun _ : Edge => expMeasure 1) |
|
|
| def embed (x : Vertex) : Plane := ((x.1 : ℝ), (x.2 : ℝ)) |
|
|
| def scaleVertex (n : ℕ) (x : Vertex) : Vertex := ((n : ℤ) * x.1, (n : ℤ) * x.2) |
|
|
| structure TimeNorm where |
| seminorm : Seminorm ℝ Plane |
| definite : ∀ x, seminorm x = 0 → x = 0 |
|
|
| def HasTimeConstant (μ : TimeNorm) : Prop := |
| ∀ᵐ ω ∂environmentLaw, ∀ x : Vertex, |
| Tendsto (fun n : ℕ => passageTime ω (0, 0) (scaleVertex n x) / (n : ℝ)) |
| atTop (𝓝 (μ.seminorm (embed x))) |
|
|
| def StrictlyConvexUnitSphere (μ : TimeNorm) : Prop := |
| ∀ x y : Plane, μ.seminorm x = 1 → μ.seminorm y = 1 → x ≠ y → |
| ∀ t : ℝ, 0 < t → t < 1 → μ.seminorm ((1 - t) • x + t • y) < 1 |
|
|
| def StrictlyConvexLimitShape (μ : TimeNorm) : Prop := |
| ∀ x y : Plane, x ∈ frontier {z | μ.seminorm z ≤ 1} → |
| y ∈ frontier {z | μ.seminorm z ≤ 1} → x ≠ y → |
| ∀ t : ℝ, 0 < t → t < 1 → μ.seminorm ((1 - t) • x + t • y) < 1 |
|
|
| def MainTarget : Prop := (∃ μ : TimeNorm, HasTimeConstant μ) ∧ |
| ∀ μ : TimeNorm, HasTimeConstant μ → StrictlyConvexLimitShape μ |
|
|
| end CompanionFPP |
|
|
| end OAI |
|
|
