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a44267b | 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 | import OAI.NumberTheory.DukePrimeDegree.Entropy.LayeredObservations
namespace OAI
/-! # Finite integer codes for families of real cuts -/
noncomputable section
open MeasureTheory
namespace DukePrimeDegree
def finiteCutCode {Y : Type*} {N : ℕ} (f : Fin N → Y → ℝ) (c : Fin N → ℝ)
(x : Y) : Fin (Fintype.card (Fin N → Bool)) :=
Fintype.equivFin (Fin N → Bool) (fun i => decide (f i x < c i))
lemma finiteCutCode_measurable {Y : Type*} [MeasurableSpace Y] {N : ℕ}
{f : Fin N → Y → ℝ} (hf : ∀ i, Measurable (f i)) (c : Fin N → ℝ) :
Measurable (finiteCutCode f c) := by
apply (measurable_of_finite (Fintype.equivFin (Fin N → Bool))).comp
apply measurable_pi_iff.mpr
intro i
apply measurable_to_bool
have he : (fun x => decide (f i x < c i)) ⁻¹' {true} = {x | f i x < c i} := by
ext x
simp
rw [he]
exact measurableSet_lt (hf i) measurable_const
lemma finiteCutCode_eq_of_iff {Y : Type*} {N : ℕ}
(f : Fin N → Y → ℝ) (c : Fin N → ℝ) {x y : Y}
(h : ∀ i, f i y < c i ↔ f i x < c i) : finiteCutCode f c y=finiteCutCode f c x := by
apply congrArg (Fintype.equivFin (Fin N → Bool))
funext i
simp only [h i]
lemma finiteCutCode_eq_imp_iff {Y : Type*} {N : ℕ}
(f : Fin N → Y → ℝ) (c : Fin N → ℝ) {x y : Y}
(h : finiteCutCode f c y=finiteCutCode f c x) (i : Fin N) :
f i y < c i ↔ f i x < c i := by
have he := congrFun ((Fintype.equivFin (Fin N → Bool)).injective h) i
constructor
· intro hy
have hx : decide (f i x < c i)=true := by
rw [←he]
exact decide_eq_true hy
exact of_decide_eq_true hx
· intro hx
have hy : decide (f i y < c i)=true := by
rw [he]
exact decide_eq_true hx
exact of_decide_eq_true hy
end DukePrimeDegree
end
end OAI
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