File size: 3,270 Bytes
180bc03 | 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 | import OAI.NumberTheory.TwoPoint.Walks.ColumnWordGeometry
namespace OAI
/-! The perfect-block geometry for the actual labels of a fixed prime column. -/
namespace TwoPointCorrelations.ColumnWordPattern
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
end OAI
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