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f0f05de | 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 90 91 92 93 94 95 96 97 | import OAI.Combinatorics.BinaryEdit.FiniteVectors
import OAI.Combinatorics.BinaryEdit.SparseCharacterPeriod
import OAI.Combinatorics.BinaryEdit.FiniteActivityBridge
namespace OAI
/-!
# Actual sparse-node character tests
Short-period detection holds for the finite-tree vectors in both the
function-indexed and list-indexed presentations.
-/
noncomputable section
open scoped BigOperators
open Classical
namespace TreeEdit.BinaryLower.FiniteVectors
/-- The actual finite-tree cheap vector detects any active distinguished
vector whose children have support smaller than `k`. Exact orders and
primality are unnecessary here: the proved annihilating periods suffice. -/
theorem sparse_node_detection {m k : ℕ}
(p : Fin k → Fin m → ℕ) (hk : 1 ≤ k)
(hppos : ∀ q i, 0 < p q i) (hp : ∀ q i, p q i ≤ k ^ 5)
(j : ℕ) (hj : j < k) (a : Leaf m k) (χ : AddChar (State p) ℂ)
(hactive : χ (w p j a) ≠ 1)
(hsparse : (Finset.univ.filter fun i : Fin m =>
χ (w p (j + 1) (Function.update a ⟨j, hj⟩ i)) ≠ 1).card < k) :
((4 * k ^ (10 * k) : ℕ) : ℝ) ≤
∑ s : Fin (4 * k ^ (10 * k)), ‖1 - χ ((s.val + 1) • u p j a)‖ ^ 2 := by
let : Finite (State p) := state_finite p hppos
let child : Fin m → State p := fun i => w p (j + 1) (Function.update a ⟨j, hj⟩ i)
let S : Finset (Fin m) := Finset.univ.filter fun i => χ (child i) ≠ 1
have hsum : u p j a = ∑ i, child i := u_eq_sum_child_w p j hj a
have hroot : χ (levelProduct p ⟨j, hj⟩ • ∑ i, child i) ≠ 1 := by
rw [← hsum, ← w_eq_level_nsmul_u p j hj a]
exact hactive
have hchild : ∀ i ∈ S, (prefixPeriod p j a * p ⟨j, hj⟩ i) • child i = 0 := by
intro i _
dsimp [child]
rw [← prefixPeriod_child p j hj a i]
exact prefixPeriod_nsmul_w p (j + 1) (Function.update a ⟨j, hj⟩ i)
have houtside : ∀ i, i ∉ S → χ (child i) = 1 := by
intro i hi
simpa only [S, Finset.mem_filter, Finset.mem_univ, true_and, not_not] using hi
have havg := FinitePeriod.sparse_character_detection χ S child (p ⟨j, hj⟩)
hk hj hsparse (prefixPeriod_pos p hppos j a) (prefixPeriod_le p hp j hj.le a)
(fun i _ => hppos _ i) (fun i _ => hp _ i) houtside hchild hroot
have hJ : (0 : ℝ) < ((4 * k ^ (10 * k) : ℕ) : ℝ) := by positivity
have hraw := (le_div_iff₀ hJ).mp havg
rw [Fin.sum_univ_eq_sum_range (fun s : ℕ => ‖1 - χ ((s + 1) • u p j a)‖ ^ 2)]
simpa only [one_mul, ← hsum] using hraw
end TreeEdit.BinaryLower.FiniteVectors
namespace TreeEdit.BinaryLower.FiniteActivity
/-- The exact unnormalized finite-sample detector required by
`finite_tree_spectral_bound`, with the sparse predicate and child list
indices unchanged. All hypotheses are local vector/period identities. -/
theorem characterSparse_detection {α A : Type*}
[Fintype α] [AddCommGroup A] [Finite A] {k : ℕ}
(w : List α → A) (v : InternalNode α k) (u : A)
(Q R : ℕ) (p : α → ℕ) (hk : 1 ≤ k)
(hQpos : 0 < Q) (hQ : Q ≤ 2 * k ^ (5 * v.1.val))
(hppos : ∀ i, 0 < p i) (hp : ∀ i, p i ≤ k ^ 5)
(hroot : w (List.ofFn v.2) = R • u)
(hsum : u = ∑ i, w (List.ofFn v.2 ++ [i]))
(hchild : ∀ i, (Q * p i) • w (List.ofFn v.2 ++ [i]) = 0)
(χ : AddChar A ℂ) (hsparse : characterSparse w k v χ) :
((4 * k ^ (10 * k) : ℕ) : ℝ) ≤
∑ s : Fin (4 * k ^ (10 * k)), ‖1 - χ ((s.val + 1) • u)‖ ^ 2 := by
let S : Finset α := Finset.univ.filter fun i => χ (w (List.ofFn v.2 ++ [i])) ≠ 1
have hcard : S.card < k := by
simpa only [S, Branching.activeChildCount] using hsparse.2
have hactive : χ (R • ∑ i, w (List.ofFn v.2 ++ [i])) ≠ 1 := by
rw [← hsum, ← hroot]
exact hsparse.1
have houtside : ∀ i, i ∉ S → χ (w (List.ofFn v.2 ++ [i])) = 1 := by
intro i hi
simpa only [S, Finset.mem_filter, Finset.mem_univ, true_and, not_not] using hi
have havg := FinitePeriod.sparse_character_detection χ S
(fun i => w (List.ofFn v.2 ++ [i])) p hk v.1.isLt hcard hQpos hQ
(fun i _ => hppos i) (fun i _ => hp i) houtside (fun i _ => hchild i) hactive
have hJ : (0 : ℝ) < ((4 * k ^ (10 * k) : ℕ) : ℝ) := by positivity
have hraw := (le_div_iff₀ hJ).mp havg
rw [Fin.sum_univ_eq_sum_range (fun s : ℕ => ‖1 - χ ((s + 1) • u)‖ ^ 2)]
simpa only [one_mul, ← hsum] using hraw
end TreeEdit.BinaryLower.FiniteActivity
end
end OAI
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