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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 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 | import OAI.Combinatorics.BinaryEdit.FiniteTreeInstantiation
import OAI.Combinatorics.BinaryEdit.FiniteWordPrimeSeparation
import OAI.Combinatorics.BinaryEdit.FiniteTreeTags
import OAI.Combinatorics.BinaryEdit.FiniteSeparationArithmetic
import OAI.Combinatorics.BinaryEdit.TaggedArrays
namespace OAI
/-!
# Geometric separation for the concrete finite tree
The prime construction supplies the half-child condition in the
geometric induction.
-/
noncomputable section
namespace TreeEdit.BinaryLower.FiniteTree
open scoped BigOperators
/-- Literal row/block array of a concrete node word. -/
def finiteBlocks {m k : ℕ} (primes : Fin k → Fin m → ℕ) (i₀ : Fin m)
(T h : ℕ) (p : List (Fin m)) (x : State (wordPeriods primes)) :
Fin T → Fin m → Word (Symbol (wordPeriods primes)) ((m * T) ^ h) :=
fun a i => ⟨finiteWord primes i₀ T h (p ++ [i]) (x + a.val • wordU primes i₀ p),
finiteWord_length primes i₀ T h _ _⟩
theorem finiteWord_succ_eq_rows {m k : ℕ} (primes : Fin k → Fin m → ℕ)
(i₀ : Fin m) (T h : ℕ) (p : List (Fin m)) (x : State (wordPeriods primes)) :
finiteWord primes i₀ T (h + 1) p x =
TaggedExtraction.rows (finiteBlocks primes i₀ T h p x) := by
have hrange : (List.finRange T).map Fin.val = List.range T := by
apply List.ext_getElem
· simp
· intro j hj hj'
simp
unfold finiteWord nodeWord
rw [← hrange, List.flatMap_map]
rfl
/-- The relative coefficient comparing two rows in a child. -/
def finiteRowShift {m k T : ℕ} (primes : Fin k → Fin m → ℕ)
(p : List (Fin m)) (hp : p.length < k) (a b : Fin T) (t : ℤ) : ℤ :=
(b.val : ℤ) - (a.val : ℤ) + t *
(FiniteVectors.levelProduct primes ⟨p.length, hp⟩ : ℤ)
/-- Children whose prescribed period does not divide the relative coefficient. -/
def separatedChildren {m k T : ℕ} (primes : Fin k → Fin m → ℕ)
(i₀ : Fin m) (p : List (Fin m)) (hp : p.length < k) (a b : Fin T) (t : ℤ) :
Finset (Fin m) :=
Finset.univ.filter fun i => ¬ (wordPeriod primes i₀ (p ++ [i]) : ℤ) ∣
finiteRowShift primes p hp a b t
/-- Ordered-tag alignment and half-child nondivisibility imply separation
for arbitrary common-subsequence matchings. -/
theorem finiteWord_separation_of_half_children {m k T : ℕ}
(primes : Fin k → Fin m → ℕ) (i₀ : Fin m) (hm : 2 ≤ m) (hT : 0 < T)
(hhalf : ∀ p (hp : p.length < k) (a b : Fin T) (t : ℤ),
¬ (wordPeriod primes i₀ p : ℤ) ∣ t →
m ≤ 2 * (separatedChildren primes i₀ p hp a b t).card)
(h : ℕ) (p : List (Fin m)) (hlen : p.length + h = k)
(x : State (wordPeriods primes)) (t : ℤ)
(ht : ¬ (wordPeriod primes i₀ p : ℤ) ∣ t) :
(((m * T) ^ h : ℕ) : ℝ) / (4 : ℝ) ^ h ≤
(deficit ((m * T) ^ h) (finiteWord primes i₀ T h p x)
(finiteWord primes i₀ T h p (x + t • wordW primes i₀ p)) : ℝ) := by
classical
induction h generalizing p x t with
| zero =>
have hp : p.length = k := by simpa using hlen
simpa using (finiteWord_leaf_deficit primes i₀ T p hp x t ht).symm.le
| succ h ih =>
have hp : p.length < k := by omega
let B : ℕ := (m * T) ^ h
let q : ℝ := (B : ℝ) / (4 : ℝ) ^ h
let d : ℝ := (m : ℝ) * q / 2
let X := finiteBlocks primes i₀ T h p x
let Y := finiteBlocks primes i₀ T h p (x + t • wordW primes i₀ p)
have hB : 0 < B := by dsimp [B]; exact pow_pos (Nat.mul_pos (by omega) hT) _
have hq : 0 ≤ q := by dsimp [q]; positivity
have hd : 0 ≤ d := by dsimp [d]; positivity
have hlocal (a b : Fin T) : d ≤ ∑ i, (deficit B (X a i).val (Y b i).val : ℝ) := by
apply row_deficit_charge (separatedChildren primes i₀ p hp a b t) _ q hq
(fun i => Nat.cast_nonneg _) _ (hhalf p hp a b t ht)
intro i hi
have hnot : ¬ (wordPeriod primes i₀ (p ++ [i]) : ℤ) ∣
finiteRowShift primes p hp a b t := (Finset.mem_filter.mp hi).2
change q ≤ (deficit B
(finiteWord primes i₀ T h (p ++ [i]) (x + a.val • wordU primes i₀ p))
(finiteWord primes i₀ T h (p ++ [i])
(x + t • wordW primes i₀ p + b.val • wordU primes i₀ p)) : ℝ)
rw [finiteWord_child_relative primes i₀ i T h p hlen x t a.val b.val]
exact ih (p ++ [i]) (by simp; omega) _ _ hnot
have hX : ∀ (a : Fin T) (i : Fin m) (s : Symbol (wordPeriods primes)),
s ∈ (X a i).val → s ∈ subtreeAlphabet (wordPeriods primes) (p ++ [i]) := by
intro a i s hs
exact mem_subtreeAlphabet_of_mem (wordPeriods primes)
⟨List.replicate k i₀, by simp⟩ (wordU primes i₀) h (p ++ [i])
(by simp; omega) _ s hs
have hY : ∀ (a : Fin T) (i : Fin m) (s : Symbol (wordPeriods primes)),
s ∈ (Y a i).val → s ∈ subtreeAlphabet (wordPeriods primes) (p ++ [i]) := by
intro a i s hs
exact mem_subtreeAlphabet_of_mem (wordPeriods primes)
⟨List.replicate k i₀, by simp⟩ (wordU primes i₀) h (p ++ [i])
(by simp; omega) _ s hs
have hdg : d ≤ (((m - 1) * B : ℕ) : ℝ) := by
have hh := finite_charge_le_gap m h hm (B : ℝ) (Nat.cast_nonneg _)
simpa [d, q, Nat.cast_mul, Nat.cast_sub (show 1 ≤ m by omega)] using hh
have hsep := (Tagged.rows_with_ordered_tags hT hm hB
(fun i => subtreeAlphabet (wordPeriods primes) (p ++ [i]))
(subtreeAlphabet_children_pairwise_disjoint (wordPeriods primes) p)
X Y hX hY d hd hlocal).2.2 hdg
have hN : T * m * B = (m * T) ^ (h + 1) := by dsimp [B]; rw [pow_succ]; ring
have hcharge : (T : ℝ) * d / 2 = (((m * T) ^ (h + 1) : ℕ) : ℝ) / (4 : ℝ) ^ (h + 1) := by
dsimp [d, q, B]
push_cast
rw [pow_succ, pow_succ]
ring
rw [hcharge, hN] at hsep
simpa only [X, Y, ← finiteWord_succ_eq_rows] using hsep
/-- Uniform separation of the root half-period shift. -/
theorem finiteWord_root_deficit_of_half_children {m k T : ℕ}
(primes : Fin k → Fin m → ℕ) (i₀ : Fin m) (hm : 2 ≤ m) (hT : 0 < T)
(hhalf : ∀ p (hp : p.length < k) (a b : Fin T) (t : ℤ),
¬ (wordPeriod primes i₀ p : ℤ) ∣ t →
m ≤ 2 * (separatedChildren primes i₀ p hp a b t).card)
(x : State (wordPeriods primes)) :
(((m * T) ^ k : ℕ) : ℝ) / (4 : ℝ) ^ k ≤
(deficit ((m * T) ^ k) (finiteWord primes i₀ T k [] x)
(finiteWord primes i₀ T k [] (x + wordW primes i₀ [])) : ℝ) := by
have h := finiteWord_separation_of_half_children primes i₀ hm hT hhalf
k [] (by simp) x 1 (by simp)
simpa using h
/-- Source finite-tree separation for the actual prime periods and words. -/
theorem finiteWord_separation {k : ℕ} (hk : 2 ≤ k)
(primes : Fin k → Fin (20 * k) → ℕ) (i₀ : Fin (20 * k))
(hinj : Function.Injective
(fun z : Fin k × Fin (20 * k) => primes z.1 z.2))
(hprime : ∀ q i, Nat.Prime (primes q i))
(hlarge : ∀ q i, k ^ 4 ≤ primes q i)
(h : ℕ) (p : List (Fin (20 * k))) (hlen : p.length + h = k)
(x : State (wordPeriods primes)) (t : ℤ)
(ht : ¬ (wordPeriod primes i₀ p : ℤ) ∣ t) :
(((20 * k * k ^ (20 * k)) ^ h : ℕ) : ℝ) / (4 : ℝ) ^ h ≤
(deficit ((20 * k * k ^ (20 * k)) ^ h)
(finiteWord primes i₀ (k ^ (20 * k)) h p x)
(finiteWord primes i₀ (k ^ (20 * k)) h p
(x + t • wordW primes i₀ p)) : ℝ) := by
apply finiteWord_separation_of_half_children primes i₀ (by omega)
(pow_pos (by omega) _) _ h p hlen x t ht
intro p hp a b t ht
exact finiteWord_half_children hk primes i₀ p hp hinj hprime hlarge a b t ht
/-- Distinct large primes give the uniform root deficit for the
prescribed half-period shift. -/
theorem finiteWord_root_deficit {k : ℕ} (hk : 2 ≤ k)
(primes : Fin k → Fin (20 * k) → ℕ) (i₀ : Fin (20 * k))
(hinj : Function.Injective
(fun z : Fin k × Fin (20 * k) => primes z.1 z.2))
(hprime : ∀ q i, Nat.Prime (primes q i))
(hlarge : ∀ q i, k ^ 4 ≤ primes q i)
(x : State (wordPeriods primes)) :
(((20 * k * k ^ (20 * k)) ^ k : ℕ) : ℝ) / (4 : ℝ) ^ k ≤
(deficit ((20 * k * k ^ (20 * k)) ^ k)
(finiteWord primes i₀ (k ^ (20 * k)) k [] x)
(finiteWord primes i₀ (k ^ (20 * k)) k []
(x + wordW primes i₀ [])) : ℝ) := by
have h := finiteWord_separation hk primes i₀ hinj hprime hlarge k []
(by simp) x 1 (by simp)
simpa using h
end TreeEdit.BinaryLower.FiniteTree
end
end OAI
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