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dd14c26 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 | 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
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