| import OAI.NumberTheory.TwoPoint.Walks.ColumnWordGeometry |
|
|
| namespace OAI |
|
|
| /-! The perfect-block geometry for the actual labels of a fixed prime column. -/ |
|
|
| namespace TwoPointCorrelations.ColumnWordPattern |
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|
| open Finset |
|
|
| variable {α : Type*} |
| |
| /-- Disjoint column supplies turn numerical prime intervals into intervals |
| of the abstract labels used by the decoder. -/ |
| theorem label_intervals_of_prime_intervals (w : ColumnWordPattern α) (value : α → ℕ) |
| (hvalue : Function.Injective value) (hprime : ∀ z, (value z).Prime) |
| (hother : ∀ z t, t < w.length → ¬value z ∣ w.otherColumns t) |
| (hinterval : TuplePrimeIntervals (w.word value)) : |
| ∀ i j k : Fin w.length, i ≤ j → j ≤ k → w.label i = w.label k → w.label j = w.label i := by |
| have hmem (i : Fin w.length) : TuplePrimeAt (w.word value) (value (w.label i)) i := by |
| apply (tuplePrimeAt_iff_getElem _ _ i (by simp)).mpr |
| refine ⟨hprime _, ?_⟩ |
| simp [word, step] |
| intro i j k hij hjk heq |
| have hpk : TuplePrimeAt (w.word value) (value (w.label i)) k := by |
| simpa only [heq] using hmem k |
| have hpj := hinterval (value (w.label i)) i j k hij hjk (by simp) (hmem i) hpk |
| have hdiv := ((tuplePrimeAt_iff_getElem _ _ j (by simp)).mp hpj).2 |
| simp only [word, List.getElem_ofFn, step] at hdiv |
| have hval : value (w.label i) ∣ value (w.label j) := |
| ((hprime _).dvd_mul.mp hdiv).resolve_right (hother _ j j.isLt) |
| have he : value (w.label i) = value (w.label j) := |
| (Nat.dvd_prime (hprime _)).mp hval |>.resolve_left (hprime _).ne_one |
| exact hvalue he.symm |
|
|
| /-- All block geometry needed for coding follows from the genuine surviving |
| positive-word lemma, with only the disjoint prime-supply data added. -/ |
| theorem positive_block_geometry (w : ColumnWordPattern α) (value : α → ℕ) |
| (hvalue : Function.Injective value) (hprime : ∀ z, (value z).Prime) |
| (hother : ∀ z t, t < w.length → ¬value z ∣ w.otherColumns t) |
| {h s J : ℕ} {supply : ℕ → ℕ → Prop} {x : ℤ} |
| (hw : PositiveWord h x (w.word value)) (hh : 0 < h) |
| (hlen : w.length ≤ s) |
| (heligible : ∀ a ∈ w.word value, supply a.tuple a.padding) |
| (hq : ∀ a ∈ w.word value, 0 < a.padding) |
| (hsq : ∀ a ∈ w.word value, Squarefree a.tuple) |
| (hcard : ∀ a ∈ w.word value, a.tuple.primeFactors.card = J) |
| (hchain : (w.word value).IsChain (fun a b => a.tuple ≠ b.tuple)) |
| (hsupport : ∀ p j, TuplePrimeAt (w.word value) p j → |
| ¬p ∣ h ∧ ∀ a ∈ w.word value, ¬p ∣ a.padding) |
| (hsurvive : ∀ y, WordVertex h x (w.word value) y → ¬ProhibitedSite h s supply y) : |
| (∀ i j k : Fin w.length, i ≤ j → j ≤ k → w.label i = w.label k → w.label j = w.label i) ∧ |
| ∀ a b z, a < b → b ≤ w.length → |
| (∀ t ∈ Ico a b, w.label t = z) → |
| (∑ t ∈ Ico a b, (w.coefficient h t : ℝ)) ≠ 0 := by |
| have hg := hw.block_geometry hh (by simpa using hlen) heligible hq hsq hcard hchain hsupport hsurvive |
| refine ⟨w.label_intervals_of_prime_intervals value hvalue hprime hother hg.1, ?_⟩ |
| intro a b z hab hb hconst |
| exact w.constant_run_real_coefficient_ne_zero h value a b hab.le hb z hconst |
| (hg.2 a b hab (by simpa using hb)) |
|
|
| end TwoPointCorrelations.ColumnWordPattern |
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| end OAI |
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|
