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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 | import OAI.MeasureTheory.DyadicAvoidance.StableRoutingBridge
import OAI.MeasureTheory.DyadicAvoidance.FiniteLocalPredicate
import OAI.MeasureTheory.DyadicAvoidance.RoutedSetDensity
namespace OAI
noncomputable section
namespace Problem310.FiniteLocalHit
open FiniteTableModel StableRouting RoutingPath OrderedRouting FiniteLocalPredicate
variable {M d : ℕ}
/-- A successful concrete child-local test is a successful terminal lookup on the
actual root route. Only the earlier center-path selectors must be stable. -/
theorem local_test_implies_root_hit
(bS : Node M d → Fin M → ℕ) (bT : Leaf M d → ℕ)
(ωS : SelectorTable bS) (ωT : TerminalTable bT)
(k r : ℕ) (U : Node M d) (i : Fin M) (x y : ℝ)
(hdepth : (U.val ++ [i.castSucc]).length + r = d)
(hU : routeFrom (select (selectorValue bS ωS)) k [] x = U.val)
(hnd : ∀ l < k, select (selectorValue bS ωS)
(routeFrom (select (selectorValue bS ωS)) l [] x) x ≠ Fin.last M)
(hprefix : ∀ l < k, ∀ j : Fin M,
j.castSucc ≤ select (selectorValue bS ωS)
(routeFrom (select (selectorValue bS ωS)) l [] x) x →
selectorValue bS ωS (routeFrom (select (selectorValue bS ωS)) l [] x) j x =
selectorValue bS ωS (routeFrom (select (selectorValue bS ωS)) l [] x) j y)
(hdefault : select (selectorValue bS ωS) U.val x = Fin.last M)
(hearlier : ∀ j < i, selectorValue bS ωS U.val j x = selectorValue bS ωS U.val j y)
(hlocal : (ωS (selectorAddress bS U i y) &&
ωT (terminalAddress bT
(localLeaf (select (selectorValue bS ωS)) r (U.val ++ [i.castSucc]) hdepth y) y)) = true) :
ωT (terminalAddress bT
(localLeaf (select (selectorValue bS ωS)) d [] (by simp) y) y) = true := by
have hklength : k = U.val.length := by
have h := congrArg List.length hU
simpa using h
have htotal : k + 1 + r = d := by
simpa only [List.length_append, List.length_singleton, ← hklength] using hdepth
simp only [Bool.and_eq_true] at hlocal
obtain ⟨hselector, hterminal⟩ := hlocal
have hi : select (selectorValue bS ωS) U.val y = i.castSucc := by
apply chooseChild_of_local_success
· exact rejects_earlier_at_default (selectorValue bS ωS) U.val x y i hdefault hearlier
· simpa using hselector
have hroute : routeFrom (select (selectorValue bS ωS)) d [] y =
routeFrom (select (selectorValue bS ωS)) r (U.val ++ [i.castSucc]) y := by
apply (congrArg (fun n => routeFrom (select (selectorValue bS ωS)) n [] y) htotal).symm.trans
apply routeFrom_eq_local_route _ k r [] U.val x y i.castSucc hU ?_ hi
intro l hl
exact select_eq_of_earlier_agreement (selectorValue bS ωS) _ x y
(hnd l hl) (hprefix l hl)
have hleaf : localLeaf (select (selectorValue bS ωS)) d [] (by simp) y =
localLeaf (select (selectorValue bS ωS)) r (U.val ++ [i.castSucc]) hdepth y :=
Subtype.ext hroute
rw [hleaf]
exact hterminal
/-- Final canonical-set form of the local-hit implication. This uses precisely the
concrete `selectorChoice`, `localLeaf`, and `routedSet` definitions of the model. -/
theorem local_test_mem_routedSet
(bS : Node M d → Fin M → ℕ) (bT : Leaf M d → ℕ)
(ωS : SelectorTable bS) (ωT : TerminalTable bT)
(k r : ℕ) (U : Node M d) (i : Fin M) (x y : ℝ)
(hdepth : (U.val ++ [i.castSucc]).length + r = d)
(hU : routeFrom (RoutedSetDensity.selectorChoice bS ωS) k [] x = U.val)
(hnd : ∀ l < k, RoutedSetDensity.selectorChoice bS ωS
(routeFrom (RoutedSetDensity.selectorChoice bS ωS) l [] x) x ≠ Fin.last M)
(hprefix : ∀ l < k, ∀ j : Fin M,
j.castSucc ≤ RoutedSetDensity.selectorChoice bS ωS
(routeFrom (RoutedSetDensity.selectorChoice bS ωS) l [] x) x →
selectorValue bS ωS (routeFrom (RoutedSetDensity.selectorChoice bS ωS) l [] x) j x =
selectorValue bS ωS (routeFrom (RoutedSetDensity.selectorChoice bS ωS) l [] x) j y)
(hdefault : RoutedSetDensity.selectorChoice bS ωS U.val x = Fin.last M)
(hearlier : ∀ j < i, selectorValue bS ωS U.val j x = selectorValue bS ωS U.val j y)
(hlocal : (ωS (selectorAddress bS U i y) &&
ωT (terminalAddress bT
(localLeaf (RoutedSetDensity.selectorChoice bS ωS) r (U.val ++ [i.castSucc]) hdepth y) y)) = true) :
y ∈ RoutedSetDensity.routedSet bS bT (ωS, ωT) := by
exact local_test_implies_root_hit bS bT ωS ωT k r U i x y hdepth hU hnd
hprefix hdefault hearlier hlocal
end Problem310.FiniteLocalHit
end
end OAI
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