| 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 |
|
|
