| import OAI.MeasureTheory.DyadicAvoidance.CenterFirstDefault |
| import OAI.MeasureTheory.DyadicAvoidance.FirstDefaultSelection |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace Problem310.FiniteTableModel |
|
|
| /-- Exact no-default probability for the actual depth-`d` spatial router |
| under the finite selector/terminal outcome law. All randomness is instantiated. -/ |
| theorem outcomeLaw_routed_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) |
| {ω : SelectorTable bS × TerminalTable bT | |
| Fin.last M ∉ RoutingPath.routeFrom |
| (fun P y => OrderedRouting.chooseChild (fun i => selectorValue bS ω.1 P i y)) |
| d [] x} = (1 - ((2 : ENNReal)⁻¹) ^ M) ^ d := by |
| have heq : {ω : SelectorTable bS × TerminalTable bT | |
| Fin.last M ∉ RoutingPath.routeFrom |
| (fun P y => OrderedRouting.chooseChild (fun i => selectorValue bS ω.1 P i y)) |
| d [] x} = |
| ⋃ 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} := by |
| ext ω |
| exact FirstDefault.spatial_default_not_mem_iff |
| (fun ω P i y => selectorValue bS ω.1 P i y) ω x |
| rw [heq] |
| exact outcomeLaw_center_no_default bS bT p hp0 hp1 x |
|
|
| /-- The same exact law stated with the step-by-step absence of a default. |
| This form pairs directly with first-default decomposition of the exposure atoms. -/ |
| theorem outcomeLaw_no_default_steps {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) |
| {ω : SelectorTable bS × TerminalTable bT | |
| ∀ k < d, |
| OrderedRouting.chooseChild (fun i => selectorValue bS ω.1 |
| (RoutingPath.routeFrom |
| (fun P y => OrderedRouting.chooseChild (fun i => selectorValue bS ω.1 P i y)) |
| k [] x) i x) ≠ Fin.last M} = |
| (1 - ((2 : ENNReal)⁻¹) ^ M) ^ d := by |
| convert outcomeLaw_routed_no_default bS bT p hp0 hp1 x using 1 |
| congr 1 |
| ext ω |
| exact RoutingPath.noDefault_iff_not_mem |
| (fun P y => OrderedRouting.chooseChild (fun i => selectorValue bS ω.1 P i y)) (Fin.last M) d x |
|
|
| end Problem310.FiniteTableModel |
|
|
| end |
|
|
| end OAI |
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|
