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caec230 | 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 82 83 84 85 86 87 88 89 90 91 | import OAI.MeasureTheory.DyadicAvoidance.FiniteTableModel
import OAI.MeasureTheory.DyadicAvoidance.FiniteTableFirstDefault
import OAI.MeasureTheory.DyadicAvoidance.ConcreteLabelMeasures
namespace OAI
universe u_T
noncomputable section
open MeasureTheory ProbabilityTheory
namespace Problem310.FiniteTableModel
/-- Center-table coordinate in the bounded-address convention used by the
first-default probability theorem. -/
def centerTableAddress {M d : ℕ} (b : Node M d → Fin M → ℕ) (x : ℝ)
(a : FirstDefault.BoundedAddress M d) : SelectorAddress b :=
selectorAddress b ⟨a.1.1, a.2⟩ a.1.2 x
lemma centerTableAddress_injective {M d : ℕ} (b : Node M d → Fin M → ℕ) (x : ℝ) :
Function.Injective (centerTableAddress b x) := by
intro a c h
have hp := congrArg Sigma.fst h
apply Subtype.ext
apply Prod.ext
· exact congrArg (fun z : Node M d × Fin M => z.1.val) hp
· exact congrArg (fun z : Node M d × Fin M => z.2) hp
/-- Exact no-default probability for the canonical spatial selector tables,
at every real center and every choice of deterministic grid resolutions. -/
theorem measure_center_no_default {M d : ℕ} (b : Node M d → Fin M → ℕ)
(ν : Measure Bool) [IsProbabilityMeasure ν]
(hfair : ∀ bit, ν {bit} = (2 : ENNReal)⁻¹) (x : ℝ) :
(Measure.pi (fun _ : SelectorAddress b => ν))
(⋃ f : FirstDefault.Path M d, {ω | ∀ i : FirstDefault.Tests f,
selectorValue b ω (FirstDefault.address f i).1 (FirstDefault.address f i).2 x =
FirstDefault.requiredBit f i}) =
(1 - ((2 : ENNReal)⁻¹) ^ M) ^ d := by
have h := FirstDefault.measure_no_default_finite_table ν hfair
(centerTableAddress b x) (centerTableAddress_injective b x)
simpa only [centerTableAddress, FirstDefault.boundedAddress,
selectorValue, dite_eq_left (FirstDefault.address_length_lt _ _)] using h
/-- Adding the independent terminal tables does not change the center-path
survival probability. This is the full outcome-space form used by repair. -/
theorem measure_center_no_default_prod {M d : ℕ}
{T : Type u_T} [MeasurableSpace T] (τ : Measure T) [IsProbabilityMeasure τ]
(b : Node M d → Fin M → ℕ)
(ν : Measure Bool) [IsProbabilityMeasure ν]
(hfair : ∀ bit, ν {bit} = (2 : ENNReal)⁻¹) (x : ℝ) :
((Measure.pi (fun _ : SelectorAddress b => ν)).prod τ)
(⋃ f : FirstDefault.Path M d, {ω : SelectorTable b × T |
∀ i : FirstDefault.Tests f,
selectorValue b ω.1 (FirstDefault.address f i).1 (FirstDefault.address f i).2 x =
FirstDefault.requiredBit f i}) =
(1 - ((2 : ENNReal)⁻¹) ^ M) ^ d := by
let E : Set (SelectorTable b) := ⋃ f : FirstDefault.Path M d,
{ω | ∀ i : FirstDefault.Tests f,
selectorValue b ω (FirstDefault.address f i).1 (FirstDefault.address f i).2 x =
FirstDefault.requiredBit f i}
have hE : MeasurableSet E := (Set.to_countable E).measurableSet
have heq : (⋃ f : FirstDefault.Path M d, {ω : SelectorTable b × T |
∀ i : FirstDefault.Tests f,
selectorValue b ω.1 (FirstDefault.address f i).1 (FirstDefault.address f i).2 x =
FirstDefault.requiredBit f i}) = Prod.fst ⁻¹' E := by
ext ω
simp only [E, Set.mem_iUnion, Set.mem_ofPred_eq, Set.mem_preimage]
rw [heq, measurePreserving_fst.measure_preimage hE.nullMeasurableSet]
exact measure_center_no_default b ν hfair x
/-- The survival estimate with all probability measures instantiated, including
the Bernoulli terminal labels in the canonical complete outcome space. -/
theorem outcomeLaw_center_no_default {M d : ℕ}
(bS : Node M d → Fin M → ℕ) (bT : Leaf M d → ℕ)
(p : ℝ) (hp0 : 0 ≤ p) (hp1 : p ≤ 1) (x : ℝ) :
(ConcreteLabels.outcomeLaw (SelectorAddress bS) (TerminalAddress bT) p hp0 hp1)
(⋃ f : FirstDefault.Path M d, {ω : SelectorTable bS × TerminalTable bT |
∀ i : FirstDefault.Tests f,
selectorValue bS ω.1 (FirstDefault.address f i).1 (FirstDefault.address f i).2 x =
FirstDefault.requiredBit f i}) =
(1 - ((2 : ENNReal)⁻¹) ^ M) ^ d := by
exact measure_center_no_default_prod
(ConcreteLabels.terminalLaw (TerminalAddress bT) p hp0 hp1)
bS ConcreteLabels.fairLaw ConcreteLabels.fairLaw_singleton x
end Problem310.FiniteTableModel
end
end OAI
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