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0dd3af4 | 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 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 | import Mathlib
import OAI.GroupTheory.SimpleAmenable.RandomFields.ThresholdField
namespace OAI
section
section
open scoped symmDiff
namespace SimpleAmenable
open scoped commutatorElement
open scoped commutatorElement
section FieldSignalData
open Classical Set
open scoped ContDiff
structure FieldSignalData where
ψ : (ℝ×ℝ) → ℝ
χ : (ℝ×ℝ) → ℝ
smooth : ContDiff ℝ 2 ψ
compact : HasCompactSupport ψ
q : ℝ
K : ℝ
L : ℝ
Λ : ℝ
L_pos : 0<L
gap : K+2≤L
ψ_zero : ∀x,K≤‖x‖ → ψ x=0
χ_range : ∀x,χ x∈Icc (0:ℝ) 1
χ_one : ∀x,‖x‖≤K+2 → χ x=1
χ_zero : ∀x,L≤‖x‖ → χ x=0
Λ_nonneg : 0≤Λ
χ_lipschitz : ∀x y,|χ x-χ y|≤Λ*(|x.1-y.1|+|x.2-y.2|)
theorem exists_fieldSignalData (ψ : (ℝ×ℝ) → ℝ) (hs : ContDiff ℝ 2 ψ)
(hc : HasCompactSupport ψ) (q : ℝ) :
∃S : FieldSignalData,S.ψ=ψ ∧ S.q=q := by
obtain ⟨K,hK,hzero⟩ := hc.exists_pos_le_norm
let χ : ContDiffBump (0 : ℝ×ℝ) := ⟨K+2,K+3,by linarith,by linarith⟩
obtain ⟨Λ,hΛ⟩ := ContDiff.lipschitzWith_of_hasCompactSupport χ.hasCompactSupport (χ.contDiff : ContDiff ℝ 1 (χ : (ℝ×ℝ) → ℝ)) (by simp)
refine ⟨⟨ψ,χ,hs,hc,q,K,K+3,Λ,by linarith,by linarith,hzero,
fun x => ⟨χ.nonneg,χ.le_one⟩,?_,?_,Λ.coe_nonneg,?_⟩,rfl,rfl⟩
· intro x hx
exact χ.one_of_mem_closedBall (by simpa only [Metric.mem_closedBall,dist_zero_right,χ] using hx)
· intro x hx
exact χ.zero_of_le_dist (by simpa only [dist_zero_right,χ] using hx)
· intro x y
have hh := hΛ.dist_le_mul x y
rw [Real.dist_eq,dist_eq_norm,Prod.norm_def] at hh
apply hh.trans
apply mul_le_mul_of_nonneg_left _ Λ.coe_nonneg
simp only [Prod.fst_sub,Prod.snd_sub,Real.norm_eq_abs]
exact max_le (le_add_of_nonneg_right (abs_nonneg _)) (le_add_of_nonneg_left (abs_nonneg _))
namespace FieldSignalData
noncomputable def signal (S : FieldSignalData) (x : ℝ×ℝ) : ℝ := S.q+S.ψ x
noncomputable def realLaw (S : FieldSignalData) {a m D : ℕ} {v : ℝ×ℝ} (hD : 0<D)
(η κ : ℝ) (n : ℕ) : MeasureTheory.Measure (FlagSite a m D v → ℝ) :=
sourceSmoothFieldLaw hD S.signal S.χ S.L_pos ((D:ℝ)^4/5) η κ n
instance realLaw_probability (S : FieldSignalData) {a m D : ℕ} {v : ℝ×ℝ} (hD : 0<D)
(η κ : ℝ) (n : ℕ) : MeasureTheory.IsProbabilityMeasure (S.realLaw (a:=a) (m:=m) (v:=v) hD η κ n) :=
inferInstanceAs (MeasureTheory.IsProbabilityMeasure (sourceSmoothFieldLaw hD S.signal S.χ S.L_pos ((D:ℝ)^4/5) η κ n))
noncomputable def bitLaw (S : FieldSignalData) {a m D : ℕ} {v : ℝ×ℝ} (hD : 0<D)
(η κ : ℝ) (n : ℕ) : MeasureTheory.Measure (FlagSite a m D v → Bool) :=
sourceBitFieldLaw hD S.signal S.χ S.L_pos ((D:ℝ)^4/5) η κ n
instance bitLaw_probability (S : FieldSignalData) {a m D : ℕ} {v : ℝ×ℝ} (hD : 0<D)
(η κ : ℝ) (n : ℕ) : MeasureTheory.IsProbabilityMeasure (S.bitLaw (a:=a) (m:=m) (v:=v) hD η κ n) :=
inferInstanceAs (MeasureTheory.IsProbabilityMeasure (sourceBitFieldLaw hD S.signal S.χ S.L_pos ((D:ℝ)^4/5) η κ n))
end FieldSignalData
end FieldSignalData
end SimpleAmenable
end
end
end OAI
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