| import OAI.MeasureTheory.DyadicAvoidance.LocalPredicateFactor |
| import OAI.MeasureTheory.DyadicAvoidance.FiniteTableModel |
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|
| namespace OAI |
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| noncomputable section |
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|
| namespace Problem310.FiniteLocalPredicate |
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| open RoutingPath GridSeparation LocalPredicateFactor FiniteTableModel |
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| variable {M d : ℕ} |
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| / |
| def localLeaf (choose : List (Child M) → ℝ → Child M) |
| (r : ℕ) (P : List (Child M)) (hdepth : P.length + r = d) (x : ℝ) : Leaf M d := |
| ⟨routeFrom choose r P x, (length_routeFrom choose r P x).trans hdepth⟩ |
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| / |
| theorem selector_lookup_eq {b : Node M d → Fin M → ℕ} (ω : SelectorTable b) |
| (Q : Node M d) (i : Fin M) {B : ℕ} (hb : b Q i ≤ B) {x y : ℝ} |
| (hxy : ⌊(2 : ℝ) ^ (B + 2) * Int.fract x⌋ = |
| ⌊(2 : ℝ) ^ (B + 2) * Int.fract y⌋) : |
| ω (selectorAddress b Q i x) = ω (selectorAddress b Q i y) := by |
| exact table_eq_of_finest_key hb (fun k => ω ⟨(Q, i), k⟩) hxy |
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|
| / |
| theorem terminal_lookup_eq {b : Leaf M d → ℕ} (ω : TerminalTable b) |
| (L : Leaf M d) {B : ℕ} (hb : b L ≤ B) {x y : ℝ} |
| (hxy : ⌊(2 : ℝ) ^ (B + 2) * Int.fract x⌋ = |
| ⌊(2 : ℝ) ^ (B + 2) * Int.fract y⌋) : |
| ω (terminalAddress b L x) = ω (terminalAddress b L y) := by |
| exact table_eq_of_finest_key hb (fun k => ω ⟨L, k⟩) hxy |
|
|
| / |
| the largest resolution in the child subtree. `rule` may be the canonical |
| first-success/default rule; no assumptions on that rule are needed. -/ |
| theorem finite_local_predicate_factors |
| (bS : Node M d → Fin M → ℕ) (bT : Leaf M d → ℕ) |
| (ωS : SelectorTable bS) (ωT : TerminalTable bT) |
| (choose : List (Child M) → ℝ → Child M) |
| (rule : List (Child M) → (Fin M → Bool) → Child M) |
| (hchoose : ∀ Q x, choose Q x = rule Q (fun i => selectorValue bS ωS Q i x)) |
| (r : ℕ) (P : List (Child M)) (hdepth : P.length + r = d) |
| (U : Node M d) (i : Fin M) (B : ℕ) |
| (hbS : ∀ Q : Node M d, P.IsPrefix Q.val → ∀ j, bS Q j ≤ B) |
| (hbT : ∀ L : Leaf M d, P.IsPrefix L.val → bT L ≤ B) |
| (hIn : bS U i ≤ B) {x y : ℝ} |
| (hxy : ⌊(2 : ℝ) ^ (B + 2) * Int.fract x⌋ = |
| ⌊(2 : ℝ) ^ (B + 2) * Int.fract y⌋) : |
| (ωS (selectorAddress bS U i x) && |
| ωT (terminalAddress bT (localLeaf choose r P hdepth x) x)) = |
| (ωS (selectorAddress bS U i y) && |
| ωT (terminalAddress bT (localLeaf choose r P hdepth y) y)) := by |
| have hchoice : ∀ Q : List (Child M), P.IsPrefix Q → Q.length < P.length + r → |
| ∀ u v : ℝ, ⌊(2 : ℝ) ^ (B + 2) * Int.fract u⌋ = |
| ⌊(2 : ℝ) ^ (B + 2) * Int.fract v⌋ → choose Q u = choose Q v := by |
| intro Q hPQ hlen u v huv |
| have hQ : Q.length < d := hlen.trans_eq hdepth |
| rw [hchoose Q u, hchoose Q v] |
| apply congrArg (rule Q) |
| funext j |
| have hlook := selector_lookup_eq ωS (⟨Q, hQ⟩ : Node M d) j |
| (hbS ⟨Q, hQ⟩ hPQ j) huv |
| simpa only [selectorValue, dite_eq_left hQ] using hlook |
| have hroute := route_eq_of_key_eq choose |
| (fun z : ℝ => ⌊(2 : ℝ) ^ (B + 2) * Int.fract z⌋) r P hchoice hxy |
| have hleaf : localLeaf choose r P hdepth x = localLeaf choose r P hdepth y := |
| Subtype.ext hroute |
| have hprefix : P.IsPrefix (localLeaf choose r P hdepth y).val := |
| prefix_routeFrom choose r P y |
| rw [selector_lookup_eq ωS U i hIn hxy, hleaf, |
| terminal_lookup_eq ωT (localLeaf choose r P hdepth y) (hbT _ hprefix) hxy] |
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| end Problem310.FiniteLocalPredicate |
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| end |
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| end OAI |
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