| import OAI.Combinatorics.InfiniteMatroid.DensityIdeal |
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| namespace OAI |
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| section |
| open Set Filter Finset |
| open scoped Topology symmDiff |
| namespace InfiniteMatroidCounterexample |
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| def liftD (X : Set D) : Set E0 := Prod.fst ⁻¹' X |
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| lemma density_liftD (X : Set D) (m : ℕ) : density (liftD X) m = dDensity X m := by |
| classical |
| have he : traceFinset (wBlock m) (liftD X) = (traceFinset (dBlock m) X).product univ := by |
| ext x |
| simp [wBlock, liftD] |
| rw [density, finiteDensity, he] |
| simp [dDensity, finiteDensity, wBlock] |
| ring |
|
|
| lemma small_liftD_iff (X : Set D) : Small (liftD X) ↔ DSmall X := by |
| have he : density (liftD X) = dDensity X := funext (density_liftD X) |
| rw [Small, DSmall, he] |
|
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| def DCell (t : CIndex) (k : Bool) : Set D := |
| {x | x.2 (prefixString (columnString (tagA t)) x.1) = false ∧ |
| x.2 (prefixString (columnString (tagB t)) x.1) = k} |
| def HCell (t : CIndex) (k : Bool) : Set E0 := liftD (DCell t k) |
|
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| lemma dCell_subset_Acompl (t : CIndex) (k : Bool) : DCell t k ⊆ (AColumn t)ᶜ := by |
| intro x hx ha |
| exact Bool.false_ne_true (hx.1.symm.trans ha) |
|
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| lemma hCell_disjoint (t : CIndex) : Disjoint (HCell t false) (HCell t true) := by |
| apply Set.disjoint_left.mpr |
| intro x hx hy |
| exact Bool.false_ne_true (hx.2.symm.trans hy.2) |
|
|
| lemma card_avoid_cells {A : Type} [Fintype A] (k : A → Bool) : |
| Nat.card {v : A × Bool → Bool // ∀ t, ¬ (v (t,false) = false ∧ v (t,true) = k t)} = |
| 3 ^ Fintype.card A := by |
| classical |
| let e := ((Equiv.curry A Bool Bool).subtypeEquiv |
| (p := fun v ↦ ∀ t, ¬(v (t,false) = false ∧ v (t,true) = k t)) |
| (q := fun v ↦ ∀ t, ¬(v t false = false ∧ v t true = k t)) (fun _ ↦ Iff.rfl)).trans |
| (Equiv.subtypePiEquivPi (α := A) (β := fun _ ↦ Bool → Bool) |
| (p := fun t v ↦ ¬(v false = false ∧ v true = k t))) |
| rw [Nat.card_congr e, Nat.card_eq_fintype_card, Fintype.card_pi] |
| have hc (b : Bool) : Fintype.card {v : Bool → Bool // ¬(v false = false ∧ v true = b)} = 3 := by |
| cases b <;> decide |
| simp only [hc, Finset.prod_const, Finset.card_univ] |
|
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| def avoidCells (F : Finset CIndex) (k : F → Bool) : Set D := |
| {x | ∀ t : F, x ∉ DCell t (k t)} |
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| lemma eventually_avoidCells (F : Finset CIndex) (k : F → Bool) : |
| ∀ᶠ m in atTop, dDensity (avoidCells F k) m = (3/4 : ℝ)^F.card := by |
| classical |
| let e : F × Bool ↪ ColumnTag := ⟨fun p ↦ if p.2 then tagB p.1 else tagA p.1, by |
| rintro ⟨t,b⟩ ⟨s,c⟩ he |
| have ht : t = s := by |
| apply Subtype.ext |
| cases b <;> cases c <;> exact congrArg Prod.fst he |
| subst s |
| congr 1 |
| cases b <;> cases c <;> simp_all [tagA,tagB]⟩ |
| have h := columns_uniform e (fun v ↦ ∀ t : F, ¬(v (t,false) = false ∧ v (t,true) = k t)) |
| rw [card_avoid_cells] at h |
| filter_upwards [h] with m hm |
| have he : {x : D | ∀ t : F, ¬(x.2 (prefixString (columnString (e (t,false))) x.1) = false ∧ |
| x.2 (prefixString (columnString (e (t,true))) x.1) = k t)} = avoidCells F k := rfl |
| rw [he] at hm |
| rw [hm] |
| simp only [Fintype.card_fun, Fintype.card_prod, Fintype.card_bool, |
| Fintype.card_coe, Nat.cast_pow, Nat.cast_ofNat] |
| rw [mul_comm F.card 2, pow_mul, div_pow] |
| norm_num |
|
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| lemma eventually_avoidHCells (F : Finset CIndex) (k : F → Bool) : |
| ∀ᶠ m in atTop, density {x | ∀ t : F, x ∉ HCell t (k t)} m = (3/4 : ℝ)^F.card := by |
| have h := eventually_avoidCells F k |
| filter_upwards [h] with m hm |
| change density (liftD (avoidCells F k)) m = _ |
| rwa [density_liftD] |
|
|
| lemma small_biUnion {ι : Type} (F : Finset ι) (X : ι → Set E0) (hX : ∀ t ∈ F, Small (X t)) : |
| Small (⋃ t ∈ F, X t) := by |
| classical |
| induction F using Finset.induction_on with |
| | empty => simpa using small_empty |
| | @insert t F ht ih => |
| simpa only [Finset.mem_insert, iUnion_iUnion_eq_or_left] using |
| small_union (hX t (Finset.mem_insert_self _ _)) |
| (ih (fun s hs ↦ hX s (Finset.mem_insert_of_mem hs))) |
|
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| lemma bad_cells_finite {S : Set E0} (hS : ¬ Small S) : |
| {t | ∃ k : Bool, Small (S ∩ HCell t k)}.Finite := by |
| classical |
| by_contra hinf |
| apply hS |
| apply tendsto_order.mpr |
| constructor |
| · intro a ha |
| exact Eventually.of_forall fun m ↦ ha.trans_le (finiteDensity_nonneg _ _) |
| · intro ε hε |
| have hp := (tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num : (0:ℝ) ≤ 3/4) |
| (by norm_num : (3/4:ℝ) < 1)).eventually (gt_mem_nhds (half_pos hε)) |
| obtain ⟨q,hq⟩ := hp.exists |
| obtain ⟨F,hF,hFfin,hFcard⟩ := Set.Infinite.exists_subset_ncard_eq hinf q |
| let Q := hFfin.toFinset |
| have hchoose : ∀ t : Q, ∃ k : Bool, Small (S ∩ HCell t k) := by |
| intro t |
| exact hF (hFfin.mem_toFinset.mp t.property) |
| choose k hk using hchoose |
| let N := ⋃ t : Q, S ∩ HCell t (k t) |
| have hN : Small N := by |
| have := small_biUnion (Finset.univ : Finset Q) (fun t ↦ S ∩ HCell t (k t)) (fun t _ ↦ hk t) |
| simpa only [Finset.mem_univ, iUnion_true] using this |
| have hsub : S ⊆ N ∪ {x | ∀ t : Q, x ∉ HCell t (k t)} := by |
| intro x hx |
| by_cases hn : x ∈ N |
| · exact Or.inl hn |
| · right |
| intro t ht |
| exact hn (mem_iUnion.mpr ⟨t,hx,ht⟩) |
| have hQ : Q.card = q := (Set.ncard_eq_toFinset_card F hFfin).symm.trans hFcard |
| filter_upwards [eventually_avoidHCells Q k, hN.eventually (gt_mem_nhds (half_pos hε))] with m hm hn |
| have hle := (finiteDensity_mono (wBlock m) hsub).trans |
| (finiteDensity_union_le (wBlock m) N {x | ∀ t : Q, x ∉ HCell t (k t)}) |
| change density S m ≤ density N m + density _ m at hle |
| rw [hm, hQ] at hle |
| linarith |
|
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| lemma simultaneous_reservations {ι : Type} (hι : Cardinal.mk ι < Cardinal.continuum) |
| (S : ι → Set E0) (hS : ∀ i, ¬ Small (S i)) : |
| ∃ t : CIndex, ∀ i k, ¬ Small (S i ∩ HCell t k) := by |
| obtain ⟨t,ht⟩ := exists_avoid_finite_family hι |
| (fun i ↦ {t | ∃ k : Bool, Small (S i ∩ HCell t k)}) (fun i ↦ bad_cells_finite (hS i)) |
| exact ⟨t,fun i k h ↦ ht i ⟨k,h⟩⟩ |
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| end InfiniteMatroidCounterexample |
| end |
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| end OAI |
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