| import OAI.MeasureTheory.DyadicAvoidance.OrderedRouting |
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|
| namespace OAI |
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| open MeasureTheory Set |
| open scoped BigOperators |
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| namespace Problem310.OrderedRouting |
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| variable {M : ℕ} (ν : Measure Bool) [IsProbabilityMeasure ν] |
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| /-- All nondefault selectors fail precisely on one product cylinder. -/ |
| theorem measure_chooseChild_default : |
| Measure.pi (fun _ : Fin M => ν) {S | chooseChild S = Fin.last M} = |
| (ν {false}) ^ M := by |
| have heq : {S : Fin M → Bool | chooseChild S = Fin.last M} = |
| Set.pi Set.univ (fun _ : Fin M => ({false} : Set Bool)) := by |
| ext S |
| simp [chooseChild_eq_default_iff, Set.mem_pi] |
| rw [heq, Measure.pi_pi] |
| simp |
|
|
| /-- The first-success choice has the geometric child law. -/ |
| theorem measure_chooseChild_nondefault (i : Fin M) : |
| Measure.pi (fun _ : Fin M => ν) {S | chooseChild S = i.castSucc} = |
| ν {true} * (ν {false}) ^ i.val := by |
| classical |
| let f : Fin M → Set Bool := fun j => if j = i then {true} else {false} |
| have heq : {S : Fin M → Bool | chooseChild S = i.castSucc} = |
| Set.pi (↑(Finset.Iic i) : Set (Fin M)) f := by |
| ext S |
| change (chooseChild S = i.castSucc) ↔ ∀ j ∈ Finset.Iic i, S j ∈ f j |
| rw [chooseChild_eq_nondefault_iff] |
| constructor |
| · rintro ⟨hi, hprev⟩ j hj |
| have hji : j ≤ i := Finset.mem_Iic.mp hj |
| by_cases h : j = i |
| · subst j |
| simpa [f] using hi |
| · simpa [f, h] using hprev j (lt_of_le_of_ne hji h) |
| · intro h |
| constructor |
| · simpa [f] using h i (Finset.mem_Iic.mpr le_rfl) |
| · intro j hj |
| simpa [f, ne_of_lt hj] using h j (Finset.mem_Iic.mpr (le_of_lt hj)) |
| rw [heq, Measure.pi_pi_finset, ← Finset.Iio_insert] |
| rw [Finset.prod_insert (by simp)] |
| have hprod : (∏ j ∈ Finset.Iio i, ν (f j)) = (ν {false}) ^ i.val := by |
| calc |
| (∏ j ∈ Finset.Iio i, ν (f j)) = ∏ _j ∈ Finset.Iio i, ν {false} := by |
| apply Finset.prod_congr rfl |
| intro j hj |
| simp [f, ne_of_lt (Finset.mem_Iio.mp hj)] |
| _ = (ν {false}) ^ i.val := by simp |
| simpa [f] using congrArg (fun z => ν {true} * z) hprod |
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| end Problem310.OrderedRouting |
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| end OAI |
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