| import OAI.MeasureTheory.DyadicAvoidance.AddressInjectivity |
| import OAI.MeasureTheory.DyadicAvoidance.RoutingPath |
|
|
| namespace OAI |
|
|
| universe u_I u_A u_X u_K |
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|
| noncomputable section |
|
|
| namespace Problem310.AddressInjectivity |
|
|
| / |
| 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 |
|
|
| / |
| 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 |
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| end OAI |
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|
