| import OAI.Combinatorics.InfiniteMatroid.Basic |
|
|
| namespace OAI |
|
|
| section |
|
|
| open Set Filter |
| open scoped Topology |
|
|
| namespace InfiniteMatroidCounterexample |
|
|
| /-- Two subsets of ℕ agree on the first `m` labels. -/ |
| def AgreeBelow (X Y : Set ℕ) (m : ℕ) : Prop := |
| ∀ x < m, (x ∈ X ↔ x ∈ Y) |
|
|
| def CylinderContinuous (f : ℕ → Set ℕ → ℝ) (I Y : Set ℕ) : Prop := |
| ∀ j ε, 0 < ε → ∃ m, ∀ X Z, |
| I ⊆ X → X ⊆ Y → I ⊆ Z → Z ⊆ Y → AgreeBelow X Z m → |
| ∀ n < j, |f n X - f n Z| < ε |
|
|
| /-- Delete, in the usual order of ℕ, the labels not reserved by the |
| lower endpoint or by the finite prefix. -/ |
| def deletePrefix (I X : Set ℕ) (m k : ℕ) : Set ℕ := |
| {x | x ∈ X ∧ (x ∈ I ∨ x < m ∨ k ≤ x)} |
|
|
| lemma deletePrefix_zero (I X : Set ℕ) (m : ℕ) : deletePrefix I X m 0 = X := by |
| ext x |
| simp [deletePrefix] |
|
|
| lemma subset_deletePrefix {I X : Set ℕ} (hIX : I ⊆ X) (m k : ℕ) : |
| I ⊆ deletePrefix I X m k := by |
| intro x hx |
| exact ⟨hIX hx, Or.inl hx⟩ |
|
|
| lemma deletePrefix_subset (I X : Set ℕ) (m k : ℕ) : |
| deletePrefix I X m k ⊆ X := fun _ h ↦ h.1 |
|
|
| lemma deletePrefix_antitone (I X : Set ℕ) (m : ℕ) : |
| Antitone (deletePrefix I X m) := by |
| intro a b hab x hx |
| rcases hx with ⟨hx, hi | hm | hb⟩ |
| · exact ⟨hx, Or.inl hi⟩ |
| · exact ⟨hx, Or.inr (Or.inl hm)⟩ |
| · exact ⟨hx, Or.inr (Or.inr (hab.trans hb))⟩ |
|
|
| lemma deletePrefix_agree (I X : Set ℕ) (m k : ℕ) : |
| AgreeBelow X (deletePrefix I X m k) m := by |
| intro x hx |
| exact ⟨fun h ↦ ⟨h, Or.inr (Or.inl hx)⟩, fun h ↦ h.1⟩ |
|
|
| lemma deletePrefix_step (I X : Set ℕ) (m k : ℕ) : |
| deletePrefix I X m k ⊆ insert k (deletePrefix I X m (k+1)) := by |
| intro x hx |
| by_cases h : x = k |
| · exact Or.inl h |
| · right |
| refine ⟨hx.1, ?_⟩ |
| rcases hx.2 with hi | hm | hk |
| · exact Or.inl hi |
| · exact Or.inr (Or.inl hm) |
| · exact Or.inr (Or.inr (by omega)) |
|
|
| lemma deletePrefix_finite (I X : Set ℕ) (m k : ℕ) : |
| (X \ deletePrefix I X m k).Finite := by |
| apply (finite_Iio k).subset |
| intro x hx |
| simp only [mem_Iio] |
| by_contra hn |
| exact hx.2 ⟨hx.1, Or.inr (Or.inr (by omega))⟩ |
|
|
| /-- The limit after every eligible label has been deleted. -/ |
| def deleteTail (I X : Set ℕ) (m : ℕ) : Set ℕ := |
| {x | x ∈ X ∧ (x ∈ I ∨ x < m)} |
|
|
| lemma subset_deleteTail {I X : Set ℕ} (hIX : I ⊆ X) (m : ℕ) : |
| I ⊆ deleteTail I X m := fun _ hx ↦ ⟨hIX hx, Or.inl hx⟩ |
|
|
| lemma deleteTail_subset (I X : Set ℕ) (m : ℕ) : |
| deleteTail I X m ⊆ X := fun _ h ↦ h.1 |
|
|
| lemma deleteTail_finite (I X : Set ℕ) (m : ℕ) : |
| (deleteTail I X m \ I).Finite := by |
| apply (finite_Iio m).subset |
| intro x hx |
| exact hx.1.2.resolve_left hx.2 |
|
|
| lemma deletePrefix_agree_tail (I X : Set ℕ) (m k : ℕ) : |
| AgreeBelow (deletePrefix I X m k) (deleteTail I X m) k := by |
| intro x hx |
| constructor |
| · rintro ⟨hX, hi | hm | hk⟩ |
| · exact ⟨hX, Or.inl hi⟩ |
| · exact ⟨hX, Or.inr hm⟩ |
| · omega |
| · rintro ⟨hX, hi | hm⟩ |
| · exact ⟨hX, Or.inl hi⟩ |
| · exact ⟨hX, Or.inr (Or.inl hm)⟩ |
|
|
| /-- One stage of the infinite-endpoint interpolation: the stopping time |
| is finite, all late probes are reset above -1, and earlier probes lose |
| less than the prescribed tolerance. -/ |
| lemma deletion_stage |
| {f : ℕ → Set ℕ → ℝ} {I Y X : Set ℕ} |
| (hcc : CylinderContinuous f I Y) |
| (hone : ∀ n A B, I ⊆ A → A ⊆ B → B ⊆ Y → |
| (∃ e, B ⊆ insert e A) → f n B ≤ f n A + 1) |
| (hlow : ∀ Z, I ⊆ Z → Z ⊆ Y → (Z \ I).Finite → |
| ∀ j, ∃ n ≥ j, f n Z < 0) |
| (hIX : I ⊆ X) (hXY : X ⊆ Y) (j : ℕ) (ε : ℝ) (hε : 0 < ε) : |
| ∃ Z, I ⊆ Z ∧ Z ⊆ X ∧ (X \ Z).Finite ∧ |
| (∃ n ≥ j, f n Z ≤ 0) ∧ |
| (∀ n < j, |f n X - f n Z| < ε) ∧ |
| (Z = X ∨ ∀ n ≥ j, -1 ≤ f n Z) := by |
| classical |
| by_cases hex : ∃ n ≥ j, f n X ≤ 0 |
| · refine ⟨X, hIX, Subset.rfl, ?_, hex, ?_, Or.inl rfl⟩ |
| · simp |
| · intro n hn |
| simpa using hε |
| have hpos : ∀ n ≥ j, 0 < f n X := by |
| intro n hn |
| exact lt_of_not_ge (fun h ↦ hex ⟨n, hn, h⟩) |
| obtain ⟨m, hm⟩ := hcc j ε hε |
| let T := deleteTail I X m |
| have hIT : I ⊆ T := subset_deleteTail hIX m |
| have hTY : T ⊆ Y := (deleteTail_subset I X m).trans hXY |
| obtain ⟨n, hnj, hn⟩ := hlow T hIT hTY (deleteTail_finite I X m) j |
| obtain ⟨k, hk⟩ := hcc (n+1) (-f n T) (neg_pos.mpr hn) |
| have happrox := hk (deletePrefix I X m k) T (subset_deletePrefix hIX m k) |
| ((deletePrefix_subset I X m k).trans hXY) hIT hTY |
| (deletePrefix_agree_tail I X m k) n (Nat.lt_succ_self n) |
| have hstop : ∃ k, ∃ n ≥ j, f n (deletePrefix I X m k) ≤ 0 := by |
| refine ⟨k, n, hnj, ?_⟩ |
| have := (abs_lt.mp happrox).2 |
| linarith |
| let k := Nat.find hstop |
| have hkstop : ∃ n ≥ j, f n (deletePrefix I X m k) ≤ 0 := Nat.find_spec hstop |
| have hk0 : k ≠ 0 := by |
| intro hk0 |
| obtain ⟨n, hnj, hn⟩ := hkstop |
| rw [hk0, deletePrefix_zero] at hn |
| exact (hpos n hnj).not_ge hn |
| obtain ⟨r, hr⟩ := Nat.exists_eq_succ_of_ne_zero hk0 |
| have hprev : ∀ n ≥ j, 0 < f n (deletePrefix I X m r) := by |
| intro n hnj |
| apply lt_of_not_ge |
| intro hn |
| have hrk : r < k := by omega |
| exact Nat.find_min hstop hrk ⟨n, hnj, hn⟩ |
| refine ⟨deletePrefix I X m k, subset_deletePrefix hIX m k, |
| deletePrefix_subset I X m k, deletePrefix_finite I X m k, hkstop, ?_, Or.inr ?_⟩ |
| · exact hm X (deletePrefix I X m k) hIX hXY (subset_deletePrefix hIX m k) |
| ((deletePrefix_subset I X m k).trans hXY) (deletePrefix_agree I X m k) |
| · intro n hnj |
| have h1 := hone n (deletePrefix I X m (r+1)) (deletePrefix I X m r) |
| (subset_deletePrefix hIX m (r+1)) |
| (deletePrefix_antitone I X m (Nat.le_succ r)) |
| ((deletePrefix_subset I X m r).trans hXY) |
| ⟨r, deletePrefix_step I X m r⟩ |
| rw [hr] |
| have := hprev n hnj |
| linarith |
|
|
| lemma antitone_agree_iInter {X : ℕ → Set ℕ} (hX : Antitone X) (m : ℕ) : |
| ∃ k, ∀ l ≥ k, AgreeBelow (X l) (⋂ j, X j) m := by |
| classical |
| induction m with |
| | zero => exact ⟨0, fun _ _ _ hx ↦ by omega⟩ |
| | succ m ih => |
| obtain ⟨k, hk⟩ := ih |
| by_cases hm : m ∈ ⋂ j, X j |
| · refine ⟨k, fun l hl x hx ↦ ?_⟩ |
| by_cases hxm : x < m |
| · exact hk l hl x hxm |
| · have hxm : x = m := by omega |
| subst x |
| exact ⟨fun _ ↦ hm, fun _ ↦ mem_iInter.mp hm l⟩ |
| · simp only [mem_iInter, not_forall] at hm |
| obtain ⟨k', hk'⟩ := hm |
| refine ⟨max k k', fun l hl x hx ↦ ?_⟩ |
| by_cases hxm : x < m |
| · exact hk l ((le_max_left _ _).trans hl) x hxm |
| · have hxm : x = m := by omega |
| subst x |
| have hml : m ∉ X l := fun h ↦ hk' (hX ((le_max_right _ _).trans hl) h) |
| exact ⟨fun h ↦ (hml h).elim, fun h ↦ mem_iInter.mp h l⟩ |
|
|
| lemma CylinderContinuous.tendsto_iInter {f : ℕ → Set ℕ → ℝ} {I Y : Set ℕ} |
| (hcc : CylinderContinuous f I Y) {X : ℕ → Set ℕ} |
| (hX : Antitone X) (hIX : ∀ j, I ⊆ X j) (hXY : ∀ j, X j ⊆ Y) (n : ℕ) : |
| Tendsto (fun j ↦ f n (X j)) atTop (𝓝 (f n (⋂ j, X j))) := by |
| rw [Metric.tendsto_atTop] |
| intro ε hε |
| obtain ⟨m, hm⟩ := hcc (n+1) ε hε |
| obtain ⟨k, hk⟩ := antitone_agree_iInter hX m |
| refine ⟨k, fun j hj ↦ ?_⟩ |
| rw [Real.dist_eq] |
| exact hm (X j) (⋂ l, X l) (hIX j) (hXY j) |
| (subset_iInter hIX) ((iInter_subset X 0).trans (hXY 0)) (hk j hj) |
| n (Nat.lt_succ_self n) |
|
|
| lemma infinite_endpoint_interpolation |
| {f : ℕ → Set ℕ → ℝ} {I Y : Set ℕ} (hIY : I ⊆ Y) |
| (hcc : CylinderContinuous f I Y) |
| (hmono : ∀ n A B, I ⊆ A → A ⊆ B → B ⊆ Y → f n A ≤ f n B) |
| (hone : ∀ n A B, I ⊆ A → A ⊆ B → B ⊆ Y → |
| (∃ e, B ⊆ insert e A) → f n B ≤ f n A + 1) |
| (hlow : ∀ Z, I ⊆ Z → Z ⊆ Y → (Z \ I).Finite → |
| ∀ j, ∃ n ≥ j, f n Z < 0) |
| (hhigh : ∃ j, ∀ n ≥ j, 0 < f n Y) : |
| ∃ Z, I ⊆ Z ∧ Z ⊆ Y ∧ |
| (∀ j, ∃ n ≥ j, f n Z ≤ 0) ∧ |
| (∀ ε : ℝ, 0 < ε → ∃ j, ∀ n ≥ j, -1 - ε ≤ f n Z) := by |
| classical |
| let S := {X : Set ℕ // I ⊆ X ∧ X ⊆ Y} |
| have hchoose : ∀ j (X : S), ∃ Z : S, |
| Z.1 ⊆ X.1 ∧ (X.1 \ Z.1).Finite ∧ |
| (∃ n ≥ j, f n Z.1 ≤ 0) ∧ |
| (∀ n < j, |f n X.1 - f n Z.1| < (1/2:ℝ)^j) ∧ |
| (Z.1 = X.1 ∨ ∀ n ≥ j, -1 ≤ f n Z.1) := by |
| intro j X |
| obtain ⟨Z, hIZ, hZX, hfin, hneg, hclose, hreset⟩ := |
| deletion_stage hcc hone hlow X.2.1 X.2.2 j ((1/2:ℝ)^j) (by positivity) |
| exact ⟨⟨Z, hIZ, hZX.trans X.2.2⟩, hZX, hfin, hneg, hclose, hreset⟩ |
| choose step hstep using hchoose |
| let seq : ℕ → S := Nat.rec ⟨Y, hIY, Subset.rfl⟩ (fun j X ↦ step (j+1) X) |
| let X : ℕ → Set ℕ := fun j ↦ (seq j).1 |
| have hX0 : X 0 = Y := rfl |
| have hIX : ∀ j, I ⊆ X j := fun j ↦ (seq j).2.1 |
| have hXY : ∀ j, X j ⊆ Y := fun j ↦ (seq j).2.2 |
| have hstage : ∀ j, |
| X (j+1) ⊆ X j ∧ (X j \ X (j+1)).Finite ∧ |
| (∃ n ≥ j+1, f n (X (j+1)) ≤ 0) ∧ |
| (∀ n < j+1, |f n (X j) - f n (X (j+1))| < (1/2:ℝ)^(j+1)) ∧ |
| (X (j+1) = X j ∨ ∀ n ≥ j+1, -1 ≤ f n (X (j+1))) := by |
| intro j |
| exact hstep (j+1) (seq j) |
| have hanti : Antitone X := antitone_nat_of_succ_le fun j ↦ (hstage j).1 |
| let Z := ⋂ j, X j |
| have hIZ : I ⊆ Z := subset_iInter hIX |
| have hZY : Z ⊆ Y := (iInter_subset X 0).trans (hXY 0) |
| have hneg : ∀ j, ∃ n ≥ j, f n Z ≤ 0 := by |
| intro j |
| obtain ⟨n, hn, hfn⟩ := (hstage j).2.2.1 |
| exact ⟨n, by omega, |
| (hmono n Z (X (j+1)) hIZ (iInter_subset X (j+1)) (hXY (j+1))).trans hfn⟩ |
| have hreset : ∃ j, ∀ n ≥ j+1, -1 ≤ f n (X (j+1)) := by |
| by_contra hn |
| push Not at hn |
| have heq : ∀ j, X (j+1) = X j := by |
| intro j |
| rcases (hstage j).2.2.2.2 with h | h |
| · exact h |
| · obtain ⟨n, hnj, hlt⟩ := hn j |
| exact (hlt.not_ge (h n hnj)).elim |
| have hconst : ∀ j, X j = Y := by |
| intro j |
| induction j with |
| | zero => exact hX0 |
| | succ j ih => exact (heq j).trans ih |
| obtain ⟨j, hj⟩ := hhigh |
| obtain ⟨n, hn, hfn⟩ := (hstage j).2.2.1 |
| rw [hconst (j+1)] at hfn |
| exact (hj n (by omega)).not_ge hfn |
| obtain ⟨j₀, hj₀⟩ := hreset |
| have hreset_range : ∀ k, j₀+1 ≤ k → ∀ n ≥ k, -1 ≤ f n (X k) := by |
| intro k hk |
| induction k, hk using Nat.le_induction with |
| | base => exact hj₀ |
| | succ k hk ih => |
| intro n hn |
| rcases (hstage k).2.2.2.2 with h | h |
| · rw [h] |
| exact ih n (by omega) |
| · exact h n hn |
| have hlower : ∀ n, j₀+1 ≤ n → ∀ k ≥ n, |
| -1 - (1/2:ℝ)^n + (1/2:ℝ)^k ≤ f n (X k) := by |
| intro n hn k hk |
| induction k, hk using Nat.le_induction with |
| | base => simpa using hreset_range n hn n le_rfl |
| | succ k hk ih => |
| have hloss := (abs_lt.mp ((hstage k).2.2.2.1 n (by omega))).2 |
| have hpower : (1/2:ℝ)^(k+1) = (1/2:ℝ)^k / 2 := by rw [pow_succ]; ring |
| rw [hpower] at hloss ⊢ |
| linarith |
| have hlower_Z : ∀ n, j₀+1 ≤ n → -1 - (1/2:ℝ)^n ≤ f n Z := by |
| intro n hn |
| apply ge_of_tendsto (hcc.tendsto_iInter hanti hIX hXY n) |
| filter_upwards [eventually_ge_atTop n] with k hk |
| have hpow : 0 ≤ (1/2:ℝ)^k := by positivity |
| exact (le_add_of_nonneg_right hpow).trans (hlower n hn k hk) |
| refine ⟨Z, hIZ, hZY, hneg, ?_⟩ |
| intro ε hε |
| have hpowlim : Tendsto (fun n : ℕ ↦ (1/2:ℝ)^n) atTop (𝓝 0) := |
| tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num) (by norm_num) |
| obtain ⟨N, hN⟩ := eventually_atTop.mp ((tendsto_order.mp hpowlim).2 ε hε) |
| refine ⟨max (j₀+1) N, fun n hn ↦ ?_⟩ |
| have h1 := hlower_Z n ((le_max_left _ _).trans hn) |
| have h2 := hN n ((le_max_right _ _).trans hn) |
| linarith |
|
|
| /-- Extended lower limiting value of the real probes. -/ |
| noncomputable def probeLiminf {α : Type*} (f : ℕ → Set α → ℝ) (X : Set α) : EReal := |
| liminf (fun n ↦ (f n X : EReal)) atTop |
|
|
| lemma ereal_liminf_add_real (u : ℕ → EReal) (b : ℝ) : |
| liminf (fun n ↦ u n + (b : EReal)) atTop = liminf u atTop + (b : EReal) := by |
| apply le_antisymm |
| · have h := EReal.liminf_add_le (u := fun _ : ℕ ↦ (b : EReal)) (v := u) |
| (f := atTop) (Or.inl (by simp)) (Or.inl (by simp)) |
| simpa [Pi.add_def, add_comm] using h |
| · simpa [Pi.add_def] using |
| EReal.le_liminf_add (u := u) (v := fun _ : ℕ ↦ (b : EReal)) (f := atTop) |
|
|
| lemma probeLiminf_finite_change {α : Type*} {f : ℕ → Set α → ℝ} |
| {X Y : Set α} {b : ℝ} (h : ∀ᶠ n in atTop, f n Y = f n X + b) : |
| probeLiminf f Y = probeLiminf f X + (b : EReal) := by |
| unfold probeLiminf |
| rw [← ereal_liminf_add_real] |
| exact liminf_congr (h.mono fun n hn ↦ by simp [hn, EReal.coe_add]) |
|
|
| lemma probeLiminf_mono {α : Type*} {f : ℕ → Set α → ℝ} |
| {X Y : Set α} (h : ∀ n, f n X ≤ f n Y) : |
| probeLiminf f X ≤ probeLiminf f Y := by |
| exact liminf_le_liminf (Eventually.of_forall fun n ↦ EReal.coe_le_coe_iff.mpr (h n)) |
|
|
| lemma probeLiminf_le_zero {α : Type*} {f : ℕ → Set α → ℝ} {X : Set α} |
| (h : ∀ j, ∃ n ≥ j, f n X ≤ 0) : probeLiminf f X ≤ 0 := by |
| apply liminf_le_of_frequently_le' |
| rw [frequently_atTop] |
| simpa using h |
|
|
| lemma probeLiminf_ge_neg_one {α : Type*} {f : ℕ → Set α → ℝ} {X : Set α} |
| (h : ∀ ε : ℝ, 0 < ε → ∃ j, ∀ n ≥ j, -1 - ε ≤ f n X) : |
| -1 ≤ probeLiminf f X := by |
| by_contra hbad |
| obtain ⟨r, hr1, hr2⟩ := EReal.exists_between_coe_real (lt_of_not_ge hbad) |
| have hr : r < -1 := EReal.coe_lt_coe_iff.mp (by simpa using hr2) |
| let ε : ℝ := (-1-r)/2 |
| have hε : 0 < ε := by dsimp [ε]; linarith |
| obtain ⟨j, hj⟩ := h ε hε |
| have hbound : ((-1-ε : ℝ) : EReal) ≤ probeLiminf f X := by |
| apply le_liminf_of_le (h := ?_) |
| filter_upwards [eventually_ge_atTop j] with n hn |
| exact EReal.coe_le_coe_iff.mpr (hj n hn) |
| have hstrict : (r : EReal) < ((-1-ε : ℝ) : EReal) := by |
| apply EReal.coe_lt_coe_iff.mpr |
| dsimp [ε] |
| linarith |
| exact (hstrict.trans_le hbound).not_ge hr1.le |
|
|
| /-- The half-closed interval used by the lower-limit rule. -/ |
| def LowerBasis {α : Type*} (f : ℕ → Set α → ℝ) (X : Set α) : Prop := |
| -1 < probeLiminf f X ∧ probeLiminf f X ≤ 0 |
|
|
| lemma infinite_endpoint_liminf_interpolation |
| {f : ℕ → Set ℕ → ℝ} {I Y : Set ℕ} (hIY : I ⊆ Y) |
| (hcc : CylinderContinuous f I Y) |
| (hmono : ∀ n A B, I ⊆ A → A ⊆ B → B ⊆ Y → f n A ≤ f n B) |
| (hone : ∀ n A B, I ⊆ A → A ⊆ B → B ⊆ Y → |
| (∃ e, B ⊆ insert e A) → f n B ≤ f n A + 1) |
| (hlow : ∀ Z, I ⊆ Z → Z ⊆ Y → (Z \ I).Finite → probeLiminf f Z = ⊥) |
| (hhigh : 0 < probeLiminf f Y) : |
| ∃ Z, I ⊆ Z ∧ Z ⊆ Y ∧ -1 ≤ probeLiminf f Z ∧ probeLiminf f Z ≤ 0 := by |
| have hlow' : ∀ Z, I ⊆ Z → Z ⊆ Y → (Z \ I).Finite → |
| ∀ j, ∃ n ≥ j, f n Z < 0 := by |
| intro Z hIZ hZY hfin |
| have hlim : probeLiminf f Z < 0 := by rw [hlow Z hIZ hZY hfin]; exact EReal.bot_lt_coe 0 |
| have hh := frequently_lt_of_liminf_lt (u := fun n ↦ (f n Z : EReal)) (h := hlim) |
| simpa [frequently_atTop] using hh |
| have hhigh' : ∃ j, ∀ n ≥ j, 0 < f n Y := by |
| have hh := eventually_lt_of_lt_liminf hhigh |
| simpa [eventually_atTop] using hh |
| obtain ⟨Z, hIZ, hZY, hneg, hbd⟩ := |
| infinite_endpoint_interpolation hIY hcc hmono hone hlow' hhigh' |
| exact ⟨Z, hIZ, hZY, probeLiminf_ge_neg_one hbd, probeLiminf_le_zero hneg⟩ |
|
|
| /-- Exact finite-cardinality translation on a fixed interval. -/ |
| def IntervalFiniteShift {α : Type*} (f : ℕ → Set α → ℝ) (I Y : Set α) : Prop := |
| ∀ A B, I ⊆ A → A ⊆ B → B ⊆ Y → (B \ A).Finite → |
| probeLiminf f B = probeLiminf f A + ((B \ A).ncard : EReal) |
|
|
| lemma finite_lower_interpolation {α : Type*} {f : ℕ → Set α → ℝ} {I Y : Set α} |
| (hIY : I ⊆ Y) (hshift : IntervalFiniteShift f I Y) |
| (hlow : probeLiminf f I ≤ -1) (hhigh : 0 < probeLiminf f Y) |
| (hbot : probeLiminf f I ≠ ⊥) : |
| ∃ B, I ⊆ B ∧ B ⊆ Y ∧ LowerBasis f B := by |
| classical |
| have htop : probeLiminf f I ≠ ⊤ := ne_top_of_le_ne_top (by simpa only [EReal.coe_neg, EReal.coe_one] using EReal.coe_ne_top (-1)) hlow |
| let a := (probeLiminf f I).toReal |
| have ha : (a : EReal) = probeLiminf f I := EReal.coe_toReal htop hbot |
| have halow : a ≤ -1 := EReal.coe_le_coe_iff.mp (by simpa [ha] using hlow) |
| let k : ℕ := ⌊-a⌋₊ |
| have hk_le : (k : ℝ) ≤ -a := Nat.floor_le (by linarith) |
| have hk_gt : -a < (k : ℝ) + 1 := Nat.lt_floor_add_one (-a) |
| have hk : -1 < a + k ∧ a + k ≤ 0 := by constructor <;> linarith |
| have hchoose : ∃ H : Finset α, (H : Set α) ⊆ Y \ I ∧ H.card = k := by |
| by_cases hfin : (Y \ I).Finite |
| · have hgap : k ≤ (Y \ I).ncard := by |
| by_contra hn |
| have hn' : ((Y \ I).ncard : ℝ) < k := by exact_mod_cast lt_of_not_ge hn |
| have hy := hshift I Y Subset.rfl hIY Subset.rfl hfin |
| have hval : probeLiminf f Y = ((a + (Y \ I).ncard : ℝ) : EReal) := by |
| rw [hy, ← ha, EReal.coe_add, EReal.coe_natCast] |
| have hpos : 0 < a + (Y \ I).ncard := EReal.coe_lt_coe_iff.mp (by simpa [hval] using hhigh) |
| linarith |
| have hc : k ≤ hfin.toFinset.card := by simpa only [Set.ncard_eq_toFinset_card _ hfin] using hgap |
| obtain ⟨H, hH, hcH⟩ := Finset.exists_subset_card_eq hc |
| exact ⟨H, fun x hx ↦ by simpa using hH hx, hcH⟩ |
| · exact Set.Infinite.exists_subset_card_eq hfin k |
| obtain ⟨H, hH, hcard⟩ := hchoose |
| let B := I ∪ (H : Set α) |
| have hIB : I ⊆ B := subset_union_left |
| have hBY : B ⊆ Y := union_subset hIY (hH.trans sdiff_subset) |
| have hdiff : B \ I = (H : Set α) := by |
| ext x |
| have hx := @hH x |
| simp only [B, Set.mem_sdiff, mem_union] at hx ⊢ |
| tauto |
| have hval := hshift I B Subset.rfl hIB hBY (by rw [hdiff]; exact H.finite_toSet) |
| have hval' : probeLiminf f B = ((a+k : ℝ) : EReal) := by |
| simpa [hdiff, hcard, ← ha, EReal.coe_add] using hval |
| refine ⟨B, hIB, hBY, ?_⟩ |
| rw [LowerBasis, hval'] |
| constructor |
| · exact EReal.coe_lt_coe_iff.mpr hk.1 |
| · exact EReal.coe_le_coe_iff.mpr hk.2 |
|
|
| lemma finite_upper_interpolation {α : Type*} {f : ℕ → Set α → ℝ} {I Y : Set α} |
| (hIY : I ⊆ Y) (hshift : IntervalFiniteShift f I Y) |
| (hlow : probeLiminf f I = ⊥) (hhigh : 0 < probeLiminf f Y) |
| (htop : probeLiminf f Y ≠ ⊤) : |
| ∃ B, I ⊆ B ∧ B ⊆ Y ∧ LowerBasis f B := by |
| classical |
| have hbot : probeLiminf f Y ≠ ⊥ := ne_bot_of_gt hhigh |
| let b := (probeLiminf f Y).toReal |
| have hb : (b : EReal) = probeLiminf f Y := EReal.coe_toReal htop hbot |
| have hbpos : 0 < b := EReal.coe_lt_coe_iff.mp (by simpa [hb] using hhigh) |
| let k : ℕ := ⌈b⌉₊ |
| have hk_le : b ≤ (k : ℝ) := Nat.le_ceil b |
| have hk_gt : (k : ℝ) < b + 1 := Nat.ceil_lt_add_one hbpos.le |
| have hk : -1 < b - k ∧ b - k ≤ 0 := by constructor <;> linarith |
| have hgap : (Y \ I).Infinite := by |
| intro hfin |
| have hh := hshift I Y Subset.rfl hIY Subset.rfl hfin |
| rw [hlow, EReal.bot_add] at hh |
| exact hbot hh |
| obtain ⟨H, hH, hcard⟩ := hgap.exists_subset_card_eq k |
| let B := Y \ (H : Set α) |
| have hIB : I ⊆ B := by |
| intro x hx |
| exact ⟨hIY hx, fun h ↦ (hH h).2 hx⟩ |
| have hBY : B ⊆ Y := sdiff_subset |
| have hdiff : Y \ B = (H : Set α) := by |
| ext x |
| have hx := @hH x |
| simp only [B, Set.mem_sdiff] at hx ⊢ |
| tauto |
| have hval := hshift B Y hIB hBY Subset.rfl (by rw [hdiff]; exact H.finite_toSet) |
| have hval' : probeLiminf f B = ((b-k : ℝ) : EReal) := by |
| have hh := congrArg (fun x : EReal ↦ x - ((k : ℝ) : EReal)) hval |
| simpa [hdiff, hcard, ← hb, ← EReal.coe_natCast, EReal.add_sub_cancel_right, |
| ← EReal.coe_sub] using hh.symm |
| refine ⟨B, hIB, hBY, ?_⟩ |
| rw [LowerBasis, hval'] |
| exact ⟨EReal.coe_lt_coe_iff.mpr hk.1, EReal.coe_le_coe_iff.mpr hk.2⟩ |
|
|
| /-- The complete local interpolation argument, including both finite |
| rounding cases and the opposite-infinite-endpoint case. -/ |
| lemma local_interpolation |
| {f : ℕ → Set ℕ → ℝ} {I Y : Set ℕ} (hIY : I ⊆ Y) |
| (hcc : CylinderContinuous f I Y) |
| (hmono : ∀ n A B, I ⊆ A → A ⊆ B → B ⊆ Y → f n A ≤ f n B) |
| (hone : ∀ n A B, I ⊆ A → A ⊆ B → B ⊆ Y → |
| (∃ e, B ⊆ insert e A) → f n B ≤ f n A + 1) |
| (hshift : IntervalFiniteShift f I Y) |
| (hlow : probeLiminf f I ≤ -1) (hhigh : 0 < probeLiminf f Y) : |
| ∃ B, I ⊆ B ∧ B ⊆ Y ∧ LowerBasis f B := by |
| classical |
| by_cases hbot : probeLiminf f I = ⊥ |
| · by_cases htop : probeLiminf f Y = ⊤ |
| · have hlow' : ∀ Z, I ⊆ Z → Z ⊆ Y → (Z \ I).Finite → probeLiminf f Z = ⊥ := by |
| intro Z hIZ hZY hfin |
| rw [hshift I Z Subset.rfl hIZ hZY hfin, hbot, EReal.bot_add] |
| obtain ⟨Z, hIZ, hZY, hZlo, hZhi⟩ := |
| infinite_endpoint_liminf_interpolation hIY hcc hmono hone hlow' hhigh |
| rcases lt_or_eq_of_le hZlo with hZlo | hZeq |
| · exact ⟨Z, hIZ, hZY, hZlo, hZhi⟩ |
| · have hgap : (Y \ Z).Nonempty := by |
| by_contra hn |
| have hyz : Y ⊆ Z := sdiff_eq_empty.mp (not_nonempty_iff_eq_empty.mp hn) |
| have hZYeq : Z = Y := Subset.antisymm hZY hyz |
| rw [hZYeq, htop] at hZhi |
| exact (EReal.coe_lt_top 0).not_ge hZhi |
| obtain ⟨e, heY, heZ⟩ := hgap |
| have hIBe : I ⊆ insert e Z := hIZ.trans (subset_insert _ _) |
| have hBeY : insert e Z ⊆ Y := insert_subset heY hZY |
| have hdiff : insert e Z \ Z = {e} := by ext x; simp [heZ] |
| have hval := hshift Z (insert e Z) hIZ (subset_insert _ _) hBeY |
| (by rw [hdiff]; exact finite_singleton e) |
| have hzero : probeLiminf f (insert e Z) = 0 := by |
| rw [hdiff, ncard_singleton, ← hZeq] at hval |
| have hc : (-1 : EReal) + 1 = 0 := by |
| have hh := congrArg (fun r : ℝ ↦ (r : EReal)) (show (-1 : ℝ) + 1 = 0 by ring) |
| simpa only [EReal.coe_add, EReal.coe_neg, EReal.coe_one, EReal.coe_zero] using hh |
| simpa only [Nat.cast_one, hc] using hval |
| refine ⟨insert e Z, hIBe, hBeY, ?_⟩ |
| simp [LowerBasis, hzero] |
| · exact finite_upper_interpolation hIY hshift hbot hhigh htop |
| · exact finite_lower_interpolation hIY hshift hlow hhigh hbot |
|
|
| end InfiniteMatroidCounterexample |
| end |
|
|
| end OAI |
|
|
