| import OAI.NumberTheory.DukePrimeDegree.Entropy.LayeredObservations |
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| namespace OAI |
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| /-! # Finite integer codes for families of real cuts -/ |
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|
| noncomputable section |
| open MeasureTheory |
| namespace DukePrimeDegree |
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|
| 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)) |
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| 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 |
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| 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] |
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| 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 |
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| end DukePrimeDegree |
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| end |
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| end OAI |
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