File size: 4,105 Bytes
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 | import OAI.Combinatorics.InfiniteMatroid.BlockDensity
import OAI.Combinatorics.InfiniteMatroid.Cesaro
namespace OAI
section
open Set Filter Finset
open scoped Topology symmDiff
namespace InfiniteMatroidCounterexample
lemma finiteDensity_abs_sub {α : Type*} (s : Finset α) (X Y : Set α) :
|finiteDensity s X - finiteDensity s Y| ≤ finiteDensity s (X ∆ Y) := by
apply abs_le.mpr
constructor
· have h := (finiteDensity_mono s (show Y ⊆ X ∪ (X ∆ Y) by
intro x hy; by_cases hx : x ∈ X; exact Or.inl hx
exact Or.inr (by simp [symmDiff_def, hy, hx]))).trans (finiteDensity_union_le s X (X ∆ Y))
linarith
· have h := (finiteDensity_mono s (show X ⊆ Y ∪ (X ∆ Y) by
intro x hx; by_cases hy : x ∈ Y; exact Or.inl hy
exact Or.inr (by simp [symmDiff_def, hy, hx]))).trans (finiteDensity_union_le s Y (X ∆ Y))
linarith
lemma dProbe_zero {T X : Set E0} (hX : SameClass T X) :
Tendsto (dProbe T X) atTop (𝓝 0) := by
apply (tendsto_zero_iff_abs_tendsto_zero _).mpr
exact squeeze_zero (fun _ ↦ abs_nonneg _) (fun m ↦ finiteDensity_abs_sub (wBlock m) X T) hX
lemma trace_prefix_card (X : Set E0) (n : ℕ) :
(traceFinset (blockPrefix n) X).card = ∑ m ∈ range n, (traceFinset (wBlock m) X).card := by
classical
have he : traceFinset (blockPrefix n) X = (range n).biUnion (fun m ↦ traceFinset (wBlock m) X) := by
ext x
simp [mem_prefix]
rw [he]
apply Finset.card_biUnion
intro i _ j _ hij
exact (wBlock_disjoint hij).mono (traceFinset_subset _ _) (traceFinset_subset _ _)
lemma prefix_card_atTop : Tendsto (fun n ↦ ((blockPrefix n).card : ℝ)) atTop atTop := by
have h : ∀ n, n ≤ (blockPrefix n).card := by
intro n
rw [card_prefix]
induction n with
| zero => simp
| succ n ih =>
rw [Finset.sum_range_succ]
have : 1 ≤ (wBlock n).card := by apply Nat.succ_le_of_lt; rw [card_wBlock]; positivity
omega
apply tendsto_atTop_mono (fun n ↦ ?_) tendsto_natCast_atTop_atTop
exact_mod_cast h n
lemma pProbe_sum (T X : Set E0) (n : ℕ) :
pProbe T X n = ∑ m ∈ range n, ((wBlock m).card : ℝ) * dProbe T X m := by
rw [pProbe, trace_prefix_card, trace_prefix_card]
push_cast
rw [← Finset.sum_sub_distrib]
apply Finset.sum_congr rfl
intro m _
have hc : ((wBlock m).card : ℝ) ≠ 0 := by simp
simp only [dProbe, density, finiteDensity]
field_simp
lemma pProbe_normalized_zero {T X : Set E0} (hX : SameClass T X) :
Tendsto (fun n ↦ pProbe T X n / (blockPrefix n).card) atTop (𝓝 0) := by
have hc : Tendsto (fun n ↦ ∑ m ∈ range n, ((wBlock m).card : ℝ)) atTop atTop := by
simpa only [card_prefix, Nat.cast_sum] using prefix_card_atTop
simpa only [pProbe_sum, card_prefix, Nat.cast_sum] using
weighted_mean_zero (fun m ↦ ((wBlock m).card : ℝ)) (dProbe T X)
(fun _ ↦ Nat.cast_nonneg _) hc (dProbe_zero hX)
noncomputable def errorTail (T X : Set E0) (n : ℕ) : ℝ := tailSup (fun m ↦ |dProbe T X m|) n
lemma dProbe_abs_bdd (T X : Set E0) : BddAbove (Set.range (fun m ↦ |dProbe T X m|)) := by
refine ⟨1, ?_⟩
rintro _ ⟨m, rfl⟩
exact abs_le.mpr (dProbe_bounds T X m)
lemma errorTail_nonneg (T X : Set E0) (n : ℕ) : 0 ≤ errorTail T X n :=
(abs_nonneg _).trans (le_tailSup (dProbe_abs_bdd T X) le_rfl)
lemma abs_dProbe_le_errorTail (T X : Set E0) {n m : ℕ} (hnm : n ≤ m) :
|dProbe T X m| ≤ errorTail T X n := le_tailSup (dProbe_abs_bdd T X) hnm
lemma errorTail_zero {T X : Set E0} (hX : SameClass T X) :
Tendsto (errorTail T X) atTop (𝓝 0) :=
tail_absSup_zero _ (dProbe_abs_bdd T X) (dProbe_zero hX)
lemma small_iff_density {G : Set E0} (hG : ¬ Small G) :
∃ δ : ℝ, 0 < δ ∧ ∀ N : ℕ, ∃ m, N ≤ m ∧ δ ≤ density G m := by
by_contra! hn
apply hG
apply Metric.tendsto_atTop.mpr
intro ε hε
obtain ⟨N, hN⟩ := hn ε hε
refine ⟨N, fun m hm ↦ ?_⟩
rw [Real.dist_eq, sub_zero, abs_of_nonneg (show 0 ≤ density G m from finiteDensity_nonneg _ _)]
exact hN m hm
end InfiniteMatroidCounterexample
end
end OAI
|