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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 | import OAI.MeasureTheory.DyadicAvoidance.AddressInjectivity
import OAI.MeasureTheory.DyadicAvoidance.RoutingPath
namespace OAI
universe u_I u_A u_X u_K
noncomputable section
namespace Problem310.AddressInjectivity
/-- The terminal-address bridge specialized to the canonical recursive router.
This applies separately to every selector outcome, with no probabilistic
independence assumption on the selected leaves. -/
theorem routedTerminalAddress_injective {I : Type u_I} {A : Type u_A} {X : Type u_X} {K : Type u_K}
(choose : List A → X → A) (r : ℕ) (U : List A)
(child : I → A) (point : I → X)
(base : A → ℕ) (level : List A → ℕ) (key : ℕ → X → K)
(hlevel : ∀ i, base (child i) ≤
level (RoutingPath.routeFrom choose r (U ++ [child i]) (point i)))
(hrefine : ∀ {b B : ℕ} {x y : X}, b ≤ B →
key B x = key B y → key b x = key b y)
(hseparate : ∀ {i j : I}, child i = child j → i ≠ j →
key (base (child i)) (point i) ≠ key (base (child i)) (point j)) :
Function.Injective (fun i =>
(RoutingPath.routeFrom choose r (U ++ [child i]) (point i),
key (level (RoutingPath.routeFrom choose r (U ++ [child i]) (point i)))
(point i))) := by
apply terminalAddress_injective U child point
(fun i => RoutingPath.routeFrom choose r (U ++ [child i]) (point i))
base level key
· intro i
exact RoutingPath.prefix_routeFrom choose r (U ++ [child i]) (point i)
· exact hlevel
· exact hrefine
· exact hseparate
/-- In particular, routes starting in distinct child subtrees always end at
distinct leaves, even at different spatial points. -/
theorem routed_leaves_ne_of_child_ne {A : Type u_A} {X : Type u_X}
(choose : List A → X → A) (r s : ℕ) (U : List A)
{i j : A} (hij : i ≠ j) (x y : X) :
RoutingPath.routeFrom choose r (U ++ [i]) x ≠
RoutingPath.routeFrom choose s (U ++ [j]) y := by
intro h
exact hij (child_eq_of_common_descendant
(RoutingPath.prefix_routeFrom choose r (U ++ [i]) x)
(h.symm ▸ RoutingPath.prefix_routeFrom choose s (U ++ [j]) y))
end Problem310.AddressInjectivity
end
end OAI
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