| import OAI.Combinatorics.InfiniteMatroid.LocalBases |
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| namespace OAI |
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| section |
| open Set Filter |
| open scoped Topology symmDiff |
| namespace InfiniteMatroidCounterexample |
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| lemma sameClass_compl {T X : Set E0} (h : SameClass T X) : SameClass Tᶜ Xᶜ := by |
| have he : Xᶜ ∆ Tᶜ = X ∆ T := by ext x; simp only [mem_symmDiff, mem_compl_iff]; tauto |
| simpa only [SameClass, he] using h |
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| lemma sameClass_add {T X P : Set E0} (h : SameClass T X) (hP : Small P) : SameClass T (X ∪ P) := by |
| apply sameClass_trans h |
| apply small_subset hP |
| intro x hx |
| simp only [mem_symmDiff, Set.mem_union] at hx |
| tauto |
|
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| lemma sameClass_delete {T X P : Set E0} (h : SameClass T X) (hP : Small P) : SameClass T (X \ P) := by |
| apply sameClass_trans h |
| apply small_subset hP |
| intro x hx |
| simp only [mem_symmDiff, Set.mem_sdiff] at hx |
| tauto |
|
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| lemma positive_sameClass {T X : Set E0} (hT : ¬ Small T) (h : SameClass T X) : ¬ Small X := by |
| intro hX |
| apply hT |
| apply small_subset (small_union hX h) |
| intro x hx |
| simp only [Set.mem_union, mem_symmDiff] |
| tauto |
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| lemma positive_pool_delete {T X G : Set E0} (hX : SameClass T X) |
| (hG : ¬ Small G) (hGX : G ⊆ X) : |
| ∃ P, P ⊆ G ∧ Small P ∧ Tendsto (fun n ↦ fProbe T n (X \ P)) atTop atBot := by |
| obtain ⟨P,hPG,hP,ht⟩ := positive_pool_add (sameClass_compl hX) hG (by simpa using hGX) |
| refine ⟨P,hPG,hP,?_⟩ |
| have he (n : ℕ) : fProbe T n (X \ P) = -fProbe Tᶜ n (Xᶜ ∪ P) := by |
| have hc : (X \ P)ᶜ = Xᶜ ∪ P := by ext x; simp only [mem_compl_iff, Set.mem_sdiff, Set.mem_union]; tauto |
| rw [← hc, fProbe_compl, neg_neg] |
| simpa only [he, Function.comp_def] using (tendsto_neg_atTop_atBot.comp ht) |
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| lemma localLower_self (T : Set E0) : T ∈ LocalLower T := by |
| refine ⟨sameClass_refl _,?_⟩ |
| have hz (n : ℕ) : fProbe T n T = 0 := by |
| have hd : dProbe T T = fun _ ↦ 0 := funext fun _ ↦ sub_self _ |
| have ht (n : ℕ) : tailSup (fun _ ↦ (0:ℝ)) n = 0 ∧ tailInf (fun _ ↦ (0:ℝ)) n = 0 := by |
| simp [tailSup, tailInf, (nonempty_Ici (a := n)).image_const] |
| simp [fProbe, pProbe, hd, (ht n).1, (ht n).2] |
| simp [LowerBasis, probeLiminf, hz] |
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| lemma localLower_swap {T B : Set E0} (hB : B ∈ LocalLower T) |
| (e : E0) (he : e ∈ B) (f : E0) (hf : f ∉ B) : insert f (B \ {e}) ∈ LocalLower T := by |
| have heq : B \ (B \ {e}) = {e} := by ext x; simp only [Set.mem_sdiff, Set.mem_singleton_iff]; aesop |
| have hfq : insert f (B \ {e}) \ (B \ {e}) = {f} := by |
| ext x |
| simp only [Set.mem_sdiff, Set.mem_insert_iff, Set.mem_singleton_iff] |
| have : f ∉ B \ {e} := fun h ↦ hf h.1 |
| aesop |
| have h1 := finite_change_eventually (T := T) (Set.sdiff_subset : B \ {e} ⊆ B) |
| (show (B \ (B \ {e})).Finite by rw [heq]; exact finite_singleton e) |
| have h2 := finite_change_eventually (T := T) (subset_insert f (B \ {e})) |
| (show (insert f (B \ {e}) \ (B \ {e})).Finite by rw [hfq]; exact finite_singleton f) |
| simp only [heq, hfq, ncard_singleton, Nat.cast_one] at h1 h2 |
| have hevent : ∀ᶠ n in atTop, fProbe T n (insert f (B \ {e})) = fProbe T n B + 0 := by |
| filter_upwards [h1,h2] with n hn1 hn2 |
| rw [hn2, hn1, add_zero] |
| have hlim := probeLiminf_finite_change hevent |
| simp only [EReal.coe_zero, add_zero] at hlim |
| refine ⟨?_,by simpa only [LowerBasis,hlim] using hB.2⟩ |
| have hclass := sameClass_add (sameClass_delete hB.1 (small_finite (finite_singleton e))) |
| (small_finite (finite_singleton f)) |
| simpa only [Set.union_singleton] using hclass |
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| lemma localLower_correct_up {T X G : Set E0} (hX : SameClass T X) |
| (hG : ¬ Small G) (hGX : G ⊆ Xᶜ) (hlo : probeLiminf (fProbe T) X ≤ -1) : |
| ∃ B ∈ LocalLower T, X ⊆ B ∧ B ⊆ X ∪ G := by |
| obtain ⟨P,hPG,hP,ht⟩ := positive_pool_add hX hG hGX |
| have hlim : probeLiminf (fProbe T) (X ∪ P) = ⊤ := (EReal.tendsto_coe_atTop.comp ht).liminf_eq |
| obtain ⟨B,hXB,hBP,hBT,hB⟩ := fProbe_interpolation hX (sameClass_add hX hP) |
| (Set.subset_union_left) hlo (by rw [hlim]; exact EReal.coe_lt_top 0) |
| exact ⟨B,⟨hBT,hB⟩,hXB,hBP.trans (union_subset_union_right X hPG)⟩ |
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| lemma localLower_correct_down {T X G : Set E0} (hX : SameClass T X) |
| (hG : ¬ Small G) (hGX : G ⊆ X) (hhi : 0 < probeLiminf (fProbe T) X) : |
| ∃ B ∈ LocalLower T, X \ G ⊆ B ∧ B ⊆ X := by |
| obtain ⟨P,hPG,hP,ht⟩ := positive_pool_delete hX hG hGX |
| have hlim : probeLiminf (fProbe T) (X \ P) = ⊥ := (EReal.tendsto_coe_atBot.comp ht).liminf_eq |
| obtain ⟨B,hPB,hBX,hBT,hB⟩ := fProbe_interpolation (sameClass_delete hX hP) hX |
| sdiff_subset (by rw [hlim]; exact bot_le) hhi |
| exact ⟨B,⟨hBT,hB⟩,(Set.sdiff_subset_sdiff_right hPG).trans hPB,hBX⟩ |
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|
| /-- Upper rule with its complement identity built into the definition. -/ |
| def LocalUpper (T : Set E0) : Set (Set E0) := {X | Xᶜ ∈ LocalLower Tᶜ} |
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| lemma localUpper_iff (T X : Set E0) : X ∈ LocalUpper T ↔ |
| SameClass T X ∧ 0 ≤ limsup (fun n ↦ (fProbe T n X : EReal)) atTop ∧ |
| limsup (fun n ↦ (fProbe T n X : EReal)) atTop < 1 := by |
| have hc : SameClass Tᶜ Xᶜ ↔ SameClass T X := |
| ⟨fun h ↦ by simpa using sameClass_compl h, sameClass_compl⟩ |
| have he : probeLiminf (fProbe Tᶜ) Xᶜ = -limsup (fun n ↦ (fProbe T n X : EReal)) atTop := by |
| simp only [probeLiminf, fProbe_compl, EReal.coe_neg] |
| exact EReal.liminf_neg |
| simp only [LocalUpper, Set.mem_ofPred_eq, LocalLower, LowerBasis, hc, he] |
| simp only [EReal.neg_le_zero, EReal.lt_neg_comm, neg_neg] |
| tauto |
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| def LocalRule (upper : Bool) (T : Set E0) : Set (Set E0) := |
| if upper then LocalUpper T else LocalLower T |
|
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| lemma localRule_class {s : Bool} {T X : Set E0} (h : X ∈ LocalRule s T) : SameClass T X := by |
| cases s with |
| | false => exact h.1 |
| | true => |
| change Xᶜ ∈ LocalLower Tᶜ at h |
| simpa using sameClass_compl h.1 |
|
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| lemma localRule_self (s : Bool) (T : Set E0) : T ∈ LocalRule s T := by |
| cases s with |
| | false => exact localLower_self _ |
| | true => |
| change Tᶜ ∈ LocalLower Tᶜ |
| exact localLower_self Tᶜ |
|
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| lemma localRule_compl {s : Bool} {T X : Set E0} (h : X ∈ LocalRule s T) : |
| Xᶜ ∈ LocalRule (!s) Tᶜ := by |
| cases s with |
| | false => simpa [LocalRule,LocalUpper] using h |
| | true => exact h |
|
|
| lemma localRule_antichain (s : Bool) (T : Set E0) : |
| ∀ ⦃X Y⦄, X ∈ LocalRule s T → Y ∈ LocalRule s T → X ⊆ Y → X = Y := by |
| cases s with |
| | false => exact localLower_antichain T |
| | true => |
| intro X Y hX hY hXY |
| have he := localLower_antichain Tᶜ hY hX (compl_subset_compl.mpr hXY) |
| exact compl_injective he.symm |
|
|
| lemma localRule_swap {s : Bool} {T B : Set E0} (hB : B ∈ LocalRule s T) |
| (e : E0) (he : e ∈ B) (f : E0) (hf : f ∉ B) : insert f (B \ {e}) ∈ LocalRule s T := by |
| cases s with |
| | false => exact localLower_swap hB e he f hf |
| | true => |
| have hc : (insert f (B \ {e}))ᶜ = insert e (Bᶜ \ {f}) := by |
| ext x |
| simp only [mem_compl_iff, Set.mem_insert_iff, Set.mem_sdiff, Set.mem_singleton_iff] |
| have hx1 : x = e → x ∈ B := fun h ↦ h ▸ he |
| have hx2 : x = f → x ∉ B := fun h ↦ h ▸ hf |
| tauto |
| change (insert f (B \ {e}))ᶜ ∈ LocalLower Tᶜ |
| rw [hc] |
| change Bᶜ ∈ LocalLower Tᶜ at hB |
| exact localLower_swap hB f hf e (by simpa using he) |
|
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| lemma localLower_interval {T I Y : Set E0} (hT : ¬ Small T) (hTc : ¬ Small Tᶜ) |
| (hIY : I ⊆ Y) (hI : SameClass T I) (hY : SameClass T Y) : |
| ∃ B ∈ LocalLower T, B ⊆ I ∨ (I ⊆ B ∧ B ⊆ Y) ∨ Y ⊆ B := by |
| by_cases hlo : -1 < probeLiminf (fProbe T) I |
| · by_cases hhi : probeLiminf (fProbe T) I ≤ 0 |
| · exact ⟨I,⟨hI,hlo,hhi⟩,Or.inl Subset.rfl⟩ |
| · obtain ⟨B,hB,_,hBI⟩ := localLower_correct_down hI (positive_sameClass hT hI) |
| Subset.rfl (lt_of_not_ge hhi) |
| exact ⟨B,hB,Or.inl hBI⟩ |
| · by_cases hhi : 0 < probeLiminf (fProbe T) Y |
| · obtain ⟨B,hIB,hBY,hBT,hB⟩ := fProbe_interpolation hI hY hIY (le_of_not_gt hlo) hhi |
| exact ⟨B,⟨hBT,hB⟩,Or.inr (Or.inl ⟨hIB,hBY⟩)⟩ |
| · by_cases hyl : -1 < probeLiminf (fProbe T) Y |
| · exact ⟨Y,⟨hY,hyl,le_of_not_gt hhi⟩,Or.inr (Or.inr Subset.rfl)⟩ |
| · obtain ⟨B,hB,hYB,_⟩ := localLower_correct_up hY |
| (positive_sameClass hTc (sameClass_compl hY)) Subset.rfl (le_of_not_gt hyl) |
| exact ⟨B,hB,Or.inr (Or.inr hYB)⟩ |
|
|
| lemma localLower_lower_endpoint {T I Y : Set E0} (hT : ¬ Small T) |
| (hIY : I ⊆ Y) (hI : SameClass T I) (hgap : ¬ Small (Y \ I)) : |
| ∃ B ∈ LocalLower T, B ⊆ I ∨ (I ⊆ B ∧ B ⊆ Y) := by |
| by_cases hlo : -1 < probeLiminf (fProbe T) I |
| · by_cases hhi : probeLiminf (fProbe T) I ≤ 0 |
| · exact ⟨I,⟨hI,hlo,hhi⟩,Or.inl Subset.rfl⟩ |
| · obtain ⟨B,hB,_,hBI⟩ := localLower_correct_down hI (positive_sameClass hT hI) |
| Subset.rfl (lt_of_not_ge hhi) |
| exact ⟨B,hB,Or.inl hBI⟩ |
| · obtain ⟨B,hB,hIB,hBY⟩ := localLower_correct_up hI hgap (fun _ hx ↦ hx.2) (le_of_not_gt hlo) |
| exact ⟨B,hB,Or.inr ⟨hIB,hBY.trans (union_subset hIY sdiff_subset)⟩⟩ |
|
|
| lemma localLower_upper_endpoint {T I Y : Set E0} (hTc : ¬ Small Tᶜ) |
| (hIY : I ⊆ Y) (hY : SameClass T Y) (hgap : ¬ Small (Y \ I)) : |
| ∃ B ∈ LocalLower T, (I ⊆ B ∧ B ⊆ Y) ∨ Y ⊆ B := by |
| by_cases hhi : 0 < probeLiminf (fProbe T) Y |
| · obtain ⟨B,hB,hIB,hBY⟩ := localLower_correct_down hY hgap sdiff_subset hhi |
| exact ⟨B,hB,Or.inl ⟨(by intro x hx; exact hIB ⟨hIY hx,fun h ↦ h.2 hx⟩),hBY⟩⟩ |
| · by_cases hlo : -1 < probeLiminf (fProbe T) Y |
| · exact ⟨Y,⟨hY,hlo,le_of_not_gt hhi⟩,Or.inr Subset.rfl⟩ |
| · obtain ⟨B,hB,hYB,_⟩ := localLower_correct_up hY |
| (positive_sameClass hTc (sameClass_compl hY)) Subset.rfl (le_of_not_gt hlo) |
| exact ⟨B,hB,Or.inr hYB⟩ |
|
|
| lemma localRule_interval (s : Bool) {T I Y : Set E0} (hT : ¬ Small T) (hTc : ¬ Small Tᶜ) |
| (hIY : I ⊆ Y) (hI : SameClass T I) (hY : SameClass T Y) : |
| ∃ B ∈ LocalRule s T, B ⊆ I ∨ (I ⊆ B ∧ B ⊆ Y) ∨ Y ⊆ B := by |
| cases s with |
| | false => exact localLower_interval hT hTc hIY hI hY |
| | true => |
| obtain ⟨B,hB,hBI | ⟨hIB,hBY⟩ | hYB⟩ := localLower_interval hTc (by simpa using hT) |
| (compl_subset_compl.mpr hIY) (sameClass_compl hY) (sameClass_compl hI) |
| · refine ⟨Bᶜ,?_,Or.inr (Or.inr ?_)⟩ |
| · simpa [LocalRule,LocalUpper] using hB |
| · simpa using compl_subset_compl.mpr hBI |
| · refine ⟨Bᶜ,?_,Or.inr (Or.inl ⟨?_,?_⟩)⟩ |
| · simpa [LocalRule,LocalUpper] using hB |
| · simpa using compl_subset_compl.mpr hBY |
| · simpa using compl_subset_compl.mpr hIB |
| · refine ⟨Bᶜ,?_,Or.inl ?_⟩ |
| · simpa [LocalRule,LocalUpper] using hB |
| · simpa using compl_subset_compl.mpr hYB |
|
|
| lemma localRule_lower_endpoint (s : Bool) {T I Y : Set E0} (hT : ¬ Small T) (_hTc : ¬ Small Tᶜ) |
| (hIY : I ⊆ Y) (hI : SameClass T I) (hgap : ¬ Small (Y \ I)) : |
| ∃ B ∈ LocalRule s T, B ⊆ I ∨ (I ⊆ B ∧ B ⊆ Y) := by |
| cases s with |
| | false => exact localLower_lower_endpoint hT hIY hI hgap |
| | true => |
| have hg : ¬ Small (Iᶜ \ Yᶜ) := by |
| have he : Iᶜ \ Yᶜ = Y \ I := by ext x; simp only [Set.mem_sdiff,mem_compl_iff]; tauto |
| rwa [he] |
| obtain ⟨B,hB,⟨hIB,hBY⟩ | hYB⟩ := localLower_upper_endpoint (T := Tᶜ) (I := Yᶜ) (Y := Iᶜ) |
| (by simpa only [compl_compl] using hT) |
| (compl_subset_compl.mpr hIY) (sameClass_compl hI) hg |
| · refine ⟨Bᶜ,?_,Or.inr ⟨?_,?_⟩⟩ |
| · simpa [LocalRule,LocalUpper] using hB |
| · simpa using compl_subset_compl.mpr hBY |
| · simpa using compl_subset_compl.mpr hIB |
| · refine ⟨Bᶜ,?_,Or.inl ?_⟩ |
| · simpa [LocalRule,LocalUpper] using hB |
| · simpa using compl_subset_compl.mpr hYB |
|
|
| lemma localRule_upper_endpoint (s : Bool) {T I Y : Set E0} (_hT : ¬ Small T) (hTc : ¬ Small Tᶜ) |
| (hIY : I ⊆ Y) (hY : SameClass T Y) (hgap : ¬ Small (Y \ I)) : |
| ∃ B ∈ LocalRule s T, (I ⊆ B ∧ B ⊆ Y) ∨ Y ⊆ B := by |
| cases s with |
| | false => exact localLower_upper_endpoint hTc hIY hY hgap |
| | true => |
| have hg : ¬ Small (Iᶜ \ Yᶜ) := by |
| have he : Iᶜ \ Yᶜ = Y \ I := by ext x; simp only [Set.mem_sdiff,mem_compl_iff]; tauto |
| rwa [he] |
| obtain ⟨B,hB,hBI | ⟨hIB,hBY⟩⟩ := localLower_lower_endpoint hTc |
| (compl_subset_compl.mpr hIY) (sameClass_compl hY) hg |
| · refine ⟨Bᶜ,?_,Or.inr ?_⟩ |
| · simpa [LocalRule,LocalUpper] using hB |
| · simpa using compl_subset_compl.mpr hBI |
| · refine ⟨Bᶜ,?_,Or.inl ⟨?_,?_⟩⟩ |
| · simpa [LocalRule,LocalUpper] using hB |
| · simpa using compl_subset_compl.mpr hBY |
| · simpa using compl_subset_compl.mpr hIB |
|
|
| end InfiniteMatroidCounterexample |
| end |
|
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| end OAI |
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|
