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c270adf | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 | import OAI.Combinatorics.InfiniteMatroid.DensityIdeal
namespace OAI
section
open Set Filter Finset
open scoped Topology symmDiff
namespace InfiniteMatroidCounterexample
def liftD (X : Set D) : Set E0 := Prod.fst ⁻¹' X
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]
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)
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)
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]
def avoidCells (F : Finset CIndex) (k : F → Bool) : Set D :=
{x | ∀ t : F, x ∉ DCell t (k t)}
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
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)))
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
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⟩⟩
end InfiniteMatroidCounterexample
end
end OAI
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