| 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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|
