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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 | 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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