| import OAI.Combinatorics.InfiniteMatroid.ProbeLimits |
|
|
| namespace OAI |
|
|
| section |
| open Set Filter Finset |
| open scoped Topology symmDiff |
| namespace InfiniteMatroidCounterexample |
|
|
| lemma exists_strictMono_choice (P : ℕ → ℕ → Prop) |
| (hP : ∀ j M, ∃ m, M ≤ m ∧ P j m) : |
| ∃ m : ℕ → ℕ, StrictMono m ∧ ∀ j, P j (m j) := by |
| classical |
| choose pick hpick using hP |
| let m : ℕ → ℕ := Nat.rec (pick 0 0) (fun j prev ↦ pick (j+1) (prev+1)) |
| have he0 : m 0 = pick 0 0 := rfl |
| have hes (j : ℕ) : m (j+1) = pick (j+1) (m j+1) := rfl |
| refine ⟨m, strictMono_nat_of_lt_succ (fun j ↦ ?_), ?_⟩ |
| · rw [hes] |
| exact Nat.lt_of_lt_of_le (Nat.lt_succ_self _) (hpick (j+1) (m j+1)).1 |
| · intro j |
| cases j with |
| | zero => exact (hpick 0 0).2 |
| | succ j => exact (hpick (j+1) (m j+1)).2 |
|
|
| lemma block_card_atTop : Tendsto (fun m ↦ ((wBlock m).card : ℝ)) atTop atTop := by |
| apply tendsto_atTop_mono (fun m ↦ ?_) prefix_card_atTop |
| have h := block_prefix_growth (le_refl m) |
| exact_mod_cast (show (blockPrefix m).card ≤ (wBlock m).card by omega) |
|
|
| /-- A sparse collection of finite block selections with prescribed lower spikes. -/ |
| lemma exists_small_spikes {G : Set E0} {δ : ℝ} (_hδ : 0 < δ) |
| (hG : ∀ N, ∃ m, N ≤ m ∧ δ ≤ density G m) |
| (b : ℕ → ℝ) (hbpos : ∀ j, 0 < b j) (hble : ∀ j, b j ≤ δ) |
| (hbzero : Tendsto b atTop (𝓝 0)) (N : ℕ → ℕ) : |
| ∃ P : Set E0, P ⊆ G ∧ Small P ∧ ∀ j, ∃ m, N j ≤ m ∧ b j ≤ density P m := by |
| classical |
| have hwzero : Tendsto (fun m ↦ (1:ℝ)/(wBlock m).card) atTop (𝓝 0) := |
| tendsto_const_nhds.div_atTop block_card_atTop |
| have hchoose : ∀ j M, ∃ m, M ≤ m ∧ N j ≤ m ∧ δ ≤ density G m ∧ |
| (1:ℝ)/(wBlock m).card ≤ b j := by |
| intro j M |
| obtain ⟨L,hL⟩ := eventually_atTop.mp (hwzero.eventually (gt_mem_nhds (hbpos j))) |
| obtain ⟨m,hm,hGm⟩ := hG (max M (max (N j) L)) |
| refine ⟨m,(le_max_left _ _).trans hm,(le_max_left _ _).trans ((le_max_right _ _).trans hm),hGm,?_⟩ |
| exact (hL m ((le_max_right _ _).trans ((le_max_right _ _).trans hm))).le |
| obtain ⟨m,hm,hmp⟩ := exists_strictMono_choice |
| (fun j m ↦ N j ≤ m ∧ δ ≤ density G m ∧ (1:ℝ)/(wBlock m).card ≤ b j) hchoose |
| have hwpos (j : ℕ) : (0:ℝ) < (wBlock (m j)).card := by rw [card_wBlock]; positivity |
| have hselect (j : ℕ) : ∃ s : Finset E0, s ⊆ traceFinset (wBlock (m j)) G ∧ |
| s.card = ⌈b j * (wBlock (m j)).card⌉₊ := by |
| apply Finset.exists_subset_card_eq |
| apply Nat.ceil_le.mpr |
| have hd := (hmp j).2.1 |
| change δ ≤ ((traceFinset (wBlock (m j)) G).card : ℝ) / (wBlock (m j)).card at hd |
| exact (mul_le_mul_of_nonneg_right (hble j) (hwpos j).le).trans ((le_div_iff₀ (hwpos j)).mp hd) |
| choose s hs hcard using hselect |
| let P : Set E0 := ⋃ j, (s j : Set E0) |
| have hP : P ⊆ G := by |
| rintro x ⟨_,⟨j,rfl⟩,hx⟩ |
| exact ((mem_traceFinset _ _ _).mp (hs j hx)).2 |
| have hsm (j : ℕ) {x : E0} (hx : x ∈ s j) : x.1.1 = m j := |
| (mem_wBlock _ _).mp (((mem_traceFinset _ _ _).mp (hs j hx)).1) |
| have htrace (j : ℕ) : traceFinset (wBlock (m j)) P = s j := by |
| ext x |
| constructor |
| · intro hx |
| obtain ⟨hxw,hxP⟩ := mem_traceFinset _ _ _ |>.mp hx |
| obtain ⟨i,hxi⟩ := mem_iUnion.mp hxP |
| have he : i = j := hm.injective ((hsm i hxi).symm.trans ((mem_wBlock _ _).mp hxw)) |
| exact he ▸ hxi |
| · intro hx |
| exact (mem_traceFinset _ _ _).mpr ⟨(mem_wBlock _ _).mpr (hsm j hx), mem_iUnion.mpr ⟨j,hx⟩⟩ |
| have htrace0 (k : ℕ) (hk : k ∉ Set.range m) : traceFinset (wBlock k) P = ∅ := by |
| apply Finset.eq_empty_iff_forall_notMem.mpr |
| intro x hx |
| obtain ⟨hxw,hxP⟩ := mem_traceFinset _ _ _ |>.mp hx |
| obtain ⟨j,hxj⟩ := mem_iUnion.mp hxP |
| exact hk ⟨j,(hsm j hxj).symm.trans ((mem_wBlock _ _).mp hxw)⟩ |
| have hden (j : ℕ) : b j ≤ density P (m j) ∧ density P (m j) ≤ 2 * b j := by |
| change b j ≤ ((traceFinset (wBlock (m j)) P).card : ℝ)/(wBlock (m j)).card ∧ |
| ((traceFinset (wBlock (m j)) P).card : ℝ)/(wBlock (m j)).card ≤ 2 * b j |
| rw [htrace, hcard] |
| constructor |
| · exact (le_div_iff₀ (hwpos j)).mpr (Nat.le_ceil _) |
| · have hc := Nat.ceil_lt_add_one (mul_nonneg (hbpos j).le (hwpos j).le) |
| have hi := (div_le_iff₀ (hwpos j)).mp (hmp j).2.2 |
| apply (div_le_iff₀ (hwpos j)).mpr |
| nlinarith |
| have hsmall : Small P := by |
| apply Metric.tendsto_atTop.mpr |
| intro ε hε |
| have hb2 : Tendsto (fun j ↦ 2 * b j) atTop (𝓝 0) := by simpa using hbzero.const_mul 2 |
| obtain ⟨J,hJ⟩ := eventually_atTop.mp (hb2.eventually (gt_mem_nhds hε)) |
| refine ⟨m J,fun k hk ↦ ?_⟩ |
| rw [Real.dist_eq, sub_zero, abs_of_nonneg (show 0 ≤ density P k from finiteDensity_nonneg _ _)] |
| by_cases hkr : k ∈ Set.range m |
| · obtain ⟨j,rfl⟩ := hkr |
| exact (hden j).2.trans_lt (hJ j (hm.le_iff_le.mp hk)) |
| · have hz : density P k = 0 := by simp [density,finiteDensity,htrace0 k hkr] |
| rwa [hz] |
| exact ⟨P,hP,hsmall,fun j ↦ ⟨m j,(hmp j).1,(hden j).1⟩⟩ |
|
|
| lemma strictMono_intervals {N : ℕ → ℕ} (hN : StrictMono N) {n : ℕ} (hn : N 0 ≤ n) : |
| ∃ j, N j ≤ n ∧ n < N (j+1) := by |
| classical |
| let j := Nat.findGreatest (fun j ↦ N j ≤ n) n |
| refine ⟨j,Nat.findGreatest_spec (P := fun j ↦ N j ≤ n) (n := n) (Nat.zero_le n) hn,?_⟩ |
| by_contra! h |
| exact Nat.findGreatest_is_greatest (show j < j+1 by omega) |
| ((hN.id_le (j+1)).trans h) h |
|
|
| lemma spike_probe_bound {T X P : Set E0} (hXP : Disjoint X P) {n m : ℕ} (hnm : n ≤ m) : |
| pProbe T X n + (blockPrefix n).card * (density P m - 2 * errorTail T X n) ≤ |
| fProbe T n (X ∪ P) := by |
| have hsup := le_tailSup (dProbe_bddAbove T (X ∪ P)) hnm |
| have heq : dProbe T (X ∪ P) m = dProbe T X m + density P m := by |
| rw [dProbe, density, finiteDensity_union _ hXP] |
| unfold dProbe density |
| ring |
| rw [heq] at hsup |
| have hlow := (abs_le.mp ((abs_dProbe_le_errorTail T X hnm))).1 |
| have hinf : -errorTail T X n ≤ tailInf (dProbe T (X ∪ P)) n := by |
| change -errorTail T X n ≤ sInf (dProbe T (X ∪ P) '' Ici n) |
| apply le_csInf (tail_nonempty (dProbe T (X ∪ P)) n) |
| rintro _ ⟨k,hk,rfl⟩ |
| exact ((abs_le.mp (abs_dProbe_le_errorTail T X hk)).1).trans |
| (dProbe_mono T (Set.subset_union_left (s := X) (t := P)) k) |
| have hp := pProbe_mono T (show X ⊆ X ∪ P from subset_union_left) n |
| have ht : density P m - 2 * errorTail T X n ≤ |
| tailSup (dProbe T (X ∪ P)) n + tailInf (dProbe T (X ∪ P)) n := by linarith |
| exact add_le_add hp (mul_le_mul_of_nonneg_left ht (Nat.cast_nonneg _)) |
|
|
| lemma positive_pool_add {T X G : Set E0} (hX : SameClass T X) |
| (hG : ¬ Small G) (hGX : G ⊆ Xᶜ) : |
| ∃ P, P ⊆ G ∧ Small P ∧ Tendsto (fun n ↦ fProbe T n (X ∪ P)) atTop atTop := by |
| classical |
| obtain ⟨δ,hδ,hδG⟩ := small_iff_density hG |
| let b (j : ℕ) : ℝ := δ / (j+1:ℕ) |
| have hbpos : ∀ j, 0 < b j := by intro j; dsimp [b]; positivity |
| have hble : ∀ j, b j ≤ δ := by |
| intro j |
| apply (div_le_iff₀ (by positivity : (0:ℝ) < (j+1:ℕ))).mpr |
| have : (1:ℝ) ≤ (j+1:ℕ) := by exact_mod_cast Nat.succ_le_succ (Nat.zero_le j) |
| nlinarith |
| have hbzero : Tendsto b atTop (𝓝 0) := by |
| apply tendsto_const_nhds.div_atTop |
| exact tendsto_natCast_atTop_atTop.comp (tendsto_add_atTop_nat 1) |
| let c (n : ℕ) : ℝ := (blockPrefix n).card |
| let err (n : ℕ) : ℝ := |pProbe T X n| / c n + 2 * errorTail T X n + 1 / Real.sqrt (c n) |
| have hE : Tendsto err atTop (𝓝 0) := by |
| have hab : Tendsto (fun n ↦ |pProbe T X n| / c n) atTop (𝓝 0) := by |
| simpa [c, abs_div] using (pProbe_normalized_zero hX).abs |
| have hi : Tendsto (fun n ↦ (1:ℝ)/Real.sqrt (c n)) atTop (𝓝 0) := |
| tendsto_const_nhds.div_atTop (Real.tendsto_sqrt_atTop.comp prefix_card_atTop) |
| simpa only [zero_add,mul_zero,add_zero] using (hab.add ((errorTail_zero hX).const_mul 2)).add hi |
| have hcut : ∀ j M, ∃ N, M ≤ N ∧ 1 ≤ N ∧ ∀ n, N ≤ n → err n ≤ b j := by |
| intro j M |
| obtain ⟨L,hL⟩ := eventually_atTop.mp (hE.eventually (gt_mem_nhds (hbpos j))) |
| refine ⟨max M (max 1 L),le_max_left _ _,(le_max_left _ _).trans (le_max_right _ _),?_⟩ |
| exact fun n hn ↦ (hL n ((le_max_right _ _).trans ((le_max_right _ _).trans hn))).le |
| obtain ⟨N,hN,hNcut⟩ := exists_strictMono_choice |
| (fun j N ↦ 1 ≤ N ∧ ∀ n, N ≤ n → err n ≤ b j) hcut |
| obtain ⟨P,hPG,hP,hspike⟩ := exists_small_spikes hδ hδG b hbpos hble hbzero (fun j ↦ N (j+1)) |
| refine ⟨P,hPG,hP,?_⟩ |
| have hXP : Disjoint X P := Set.disjoint_left.mpr (fun _ hx hp ↦ hGX (hPG hp) hx) |
| have hbound : ∀ᶠ n in atTop, Real.sqrt (c n) ≤ fProbe T n (X ∪ P) := by |
| filter_upwards [eventually_ge_atTop (N 0)] with n hn |
| obtain ⟨j,hjn,hnj⟩ := strictMono_intervals hN hn |
| obtain ⟨m,hm,hbm⟩ := hspike j |
| have hs := spike_probe_bound (T := T) hXP (hnj.le.trans hm) |
| have hcn : 0 < c n := by |
| have hn1 : 1 ≤ n := (hNcut 0).1.trans hn |
| have hprefix : 1 ≤ (blockPrefix n).card := by |
| have hx : (⟨⟨0,fun _ ↦ false⟩,false⟩ : E0) ∈ blockPrefix n := (mem_prefix _ _).mpr (by change 0 < n; omega) |
| exact Finset.card_pos.mpr ⟨_,hx⟩ |
| change (0:ℝ) < ((blockPrefix n).card : ℝ) |
| exact_mod_cast (Nat.lt_of_lt_of_le Nat.zero_lt_one hprefix) |
| have he := (hNcut j).2 n hjn |
| change |pProbe T X n| / c n + 2 * errorTail T X n + 1 / Real.sqrt (c n) ≤ b j at he |
| have hsqrt : 0 < Real.sqrt (c n) := Real.sqrt_pos.2 hcn |
| have hid : c n / Real.sqrt (c n) = Real.sqrt (c n) := by |
| apply (div_eq_iff hsqrt.ne').mpr |
| simpa only [pow_two] using (Real.sq_sqrt hcn.le).symm |
| have hem := mul_le_mul_of_nonneg_right he hcn.le |
| have hcne := hcn.ne' |
| change pProbe T X n + c n * (density P m - 2 * errorTail T X n) ≤ _ at hs |
| have hpabs := neg_abs_le (pProbe T X n) |
| have hbmult := mul_le_mul_of_nonneg_left hbm hcn.le |
| rw [add_mul, add_mul, div_mul_cancel₀ _ hcne] at hem |
| have hm2 : (1 / Real.sqrt (c n)) * c n = Real.sqrt (c n) := by |
| rw [one_div, inv_mul_eq_div, hid] |
| rw [hm2] at hem |
| nlinarith |
| exact tendsto_atTop_mono' atTop hbound (Real.tendsto_sqrt_atTop.comp prefix_card_atTop) |
| |
| end InfiniteMatroidCounterexample |
| end |
| |
| end OAI |
| |
