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8769036 | 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 | import OAI.NumberTheory.TwoPoint.Walks.TupleResidueLabels
import OAI.NumberTheory.TwoPoint.Bounds.ObservedReciprocalBound
import OAI.NumberTheory.TwoPoint.Walks.TuplePerfectRows
namespace OAI
/-! Actual tuple slots have the same equality pattern in the ambient prime pool and column coordinates. -/
namespace TwoPointCorrelations
open Finset
open scoped Classical
lemma columnPrimeValue_injective {J : ℕ} {P : Fin J → Finset ℕ}
(hdisjoint : ∀ j l, l ≠ j → Disjoint (P j) (P l)) :
Function.Injective (fun c : (j : Fin J) × P j => c.2.val) := by
rintro ⟨j, p⟩ ⟨l, q⟩ hpq
change p.val = q.val at hpq
have hjl : j = l := by
by_contra hne
exact disjoint_left.mp (hdisjoint j l (Ne.symm hne)) p.property (hpq.symm ▸ q.property)
subst l
exact congrArg (Sigma.mk j) (Subtype.ext hpq)
lemma tuple_label_eq_iff {ι : Type*} [Fintype ι] [DecidableEq ι]
{J R : ℕ} {P : Fin J → Finset ℕ} (w : ColumnPrimeAssignment J R P)
(hdisjoint : ∀ j l, l ≠ j → Disjoint (P j) (P l))
(p : ι → ℕ) (hinj : Function.Injective p) (label : Fin R × Fin J → ι)
(hlabel : ∀ i j, p (label (i, j)) = (w j i).val) (t u : Fin R × Fin J) :
label t = label u ↔ tupleSlotLabel w t = tupleSlotLabel w u := by
rw [← hinj.eq_iff, ← (columnPrimeValue_injective hdisjoint).eq_iff]
change p (label t) = p (label u) ↔ (w t.2 t.1).val = (w u.2 u.1).val
rw [show p (label t) = (w t.2 t.1).val from hlabel t.1 t.2,
show p (label u) = (w u.2 u.1).val from hlabel u.1 u.2]
lemma tuple_label_singletonSlots {ι : Type*} [Fintype ι] [DecidableEq ι]
{J R : ℕ} {P : Fin J → Finset ℕ} (w : ColumnPrimeAssignment J R P)
(hdisjoint : ∀ j l, l ≠ j → Disjoint (P j) (P l))
(p : ι → ℕ) (hinj : Function.Injective p) (label : Fin R × Fin J → ι)
(hlabel : ∀ i j, p (label (i, j)) = (w j i).val) :
singletonSlots label = singletonSlots (tupleSlotLabel w) := by
have he (t : Fin R × Fin J) : labelOccurrences label (label t) =
labelOccurrences (tupleSlotLabel w) (tupleSlotLabel w t) := by
ext u
simp only [labelOccurrences, mem_filter, mem_univ, true_and]
exact tuple_label_eq_iff w hdisjoint p hinj label hlabel u t
ext t
simp only [singletonSlots, singletonLabels, mem_filter, mem_univ, true_and, he]
lemma tuple_label_singleton_count {ι : Type*} [Fintype ι] [DecidableEq ι]
{J R : ℕ} {P : Fin J → Finset ℕ} (w : ColumnPrimeAssignment J R P)
(hdisjoint : ∀ j l, l ≠ j → Disjoint (P j) (P l))
(p : ι → ℕ) (hinj : Function.Injective p) (label : Fin R × Fin J → ι)
(hlabel : ∀ i j, p (label (i, j)) = (w j i).val) :
(singletonLabels label).card = columnSingletonCount w := by
calc
_ = (singletonSlots label).card := (singleton_slot_count label).symm
_ = (singletonSlots (tupleSlotLabel w)).card :=
congrArg Finset.card (tuple_label_singletonSlots w hdisjoint p hinj label hlabel)
_ = _ := (singleton_slot_count (tupleSlotLabel w)).trans (tupleSlotLabel_singletons w)
lemma tuple_label_perfectRows {ι : Type*} [Fintype ι] [DecidableEq ι]
{J R : ℕ} {P : Fin J → Finset ℕ} (w : ColumnPrimeAssignment J R P)
(hdisjoint : ∀ j l, l ≠ j → Disjoint (P j) (P l))
(p : ι → ℕ) (hinj : Function.Injective p) (label : Fin R × Fin J → ι)
(hlabel : ∀ i j, p (label (i, j)) = (w j i).val) (U : Finset (Fin R × Fin J)) :
perfectRows label U = perfectRows (tupleSlotLabel w) U := by
simp only [perfectRows, tuple_label_singletonSlots w hdisjoint p hinj label hlabel]
lemma tuple_label_reciprocal {ι : Type*} [Fintype ι] [DecidableEq ι]
{J R : ℕ} {P : Fin J → Finset ℕ} (w : ColumnPrimeAssignment J R P)
(hdisjoint : ∀ j l, l ≠ j → Disjoint (P j) (P l))
(p : ι → ℕ) (hinj : Function.Injective p) (label : Fin R × Fin J → ι)
(hlabel : ∀ i j, p (label (i, j)) = (w j i).val) :
(∏ i ∈ univ.image label, (p i : ℝ)⁻¹) = columnReciprocalWeight w := by
have he : (univ.image label).image p = tuplePrimeSupport w := by
ext q
simp only [mem_image, mem_univ, true_and, mem_tuplePrimeSupport]
constructor
· rintro ⟨i, ⟨t, rfl⟩, rfl⟩
exact ⟨t.2, t.1, (hlabel t.1 t.2).symm⟩
· rintro ⟨j, i, rfl⟩
exact ⟨label (i, j), ⟨(i, j), rfl⟩, hlabel i j⟩
rw [← tuplePrimeSupport_reciprocal w hdisjoint, ← he]
exact (prod_image (g := p) (f := fun q : ℕ => (q : ℝ)⁻¹)
(fun _ _ _ _ h => hinj h)).symm
theorem designatedReciprocal_le_tuple {ι : Type*} [Fintype ι] [DecidableEq ι]
{J R : ℕ} {P : Fin J → Finset ℕ} (w : ColumnPrimeAssignment J R P)
(hdisjoint : ∀ j l, l ≠ j → Disjoint (P j) (P l))
(p : ι → ℕ) (hinj : Function.Injective p) (hp : ∀ i, 0 < p i)
(label : Fin R × Fin J → ι)
(hlabel : ∀ i j, p (label (i, j)) = (w j i).val)
(U : Finset (Fin R × Fin J)) (hU : U ⊆ nonsingletonSlots label) :
designatedReciprocal p (singletonLabels label) (nonsingletonSlots label \ U) U label ≤
columnReciprocalWeight w := by
have hb := full_word_designated_reciprocal_bound label U hU p 1
(by norm_num) (fun i => by exact_mod_cast hp i)
rw [designation_sdiff_decidable (fun a b => Classical.propDecidable (a = b))
(inferInstance : DecidableEq (Fin R × Fin J)) _ _] at hb
simpa only [inv_one, one_pow, mul_one, tuple_label_reciprocal w hdisjoint p hinj label hlabel]
using hb
end TwoPointCorrelations
end OAI
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