| import OAI.Combinatorics.InfiniteMatroid.CorollaryModel |
|
|
| namespace OAI |
|
|
| namespace InfiniteMatroidCounterexample |
| open Set Matroid |
|
|
| def Wave {α ι : Type*} (M : ι → Matroid α) (F : Set α) |
| (W : Set α × (ι → Set α)) : Prop := |
| W.1 ⊆ F ∧ (∀ i, W.2 i ⊆ W.1) ∧ |
| (Pairwise fun i j ↦ Disjoint (W.2 i) (W.2 j)) ∧ |
| ∀ i, W.1 ⊆ (M i).closure (W.2 i) |
|
|
| theorem exists_maximal_wave {α ι : Type*} (M : ι → Matroid α) (F : Set α) : |
| ∃ W, Maximal (Wave M F) W := by |
| classical |
| apply zorn_le₀ {W | Wave M F W} |
| intro c hc hchain |
| let U : Set α × (ι → Set α) := |
| (⋃ W : c, W.1.1, fun i ↦ ⋃ W : c, W.1.2 i) |
| have hle : ∀ W ∈ c, W ≤ U := by |
| intro W hW |
| refine ⟨?_, fun i ↦ ?_⟩ |
| · exact subset_iUnion_of_subset ⟨W, hW⟩ Subset.rfl |
| · exact subset_iUnion_of_subset ⟨W, hW⟩ Subset.rfl |
| refine ⟨U, ?_, hle⟩ |
| refine ⟨?_, ?_, ?_, ?_⟩ |
| · intro x hx |
| obtain ⟨W, hx⟩ := mem_iUnion.mp hx |
| exact (hc W.2).1 hx |
| · intro i x hx |
| obtain ⟨W, hx⟩ := mem_iUnion.mp hx |
| exact (hle W W.2).1 ((hc W.2).2.1 i hx) |
| · intro i j hij |
| apply disjoint_left.mpr |
| intro x hxi hxj |
| obtain ⟨W, hxW⟩ := mem_iUnion.mp hxi |
| obtain ⟨V, hxV⟩ := mem_iUnion.mp hxj |
| rcases hchain.total W.2 V.2 with hWV | hVW |
| · exact disjoint_left.mp ((hc V.2).2.2.1 hij) (hWV.2 i hxW) hxV |
| · exact disjoint_left.mp ((hc W.2).2.2.1 hij) hxW (hVW.2 j hxV) |
| · intro i x hx |
| obtain ⟨W, hx⟩ := mem_iUnion.mp hx |
| exact (M i).closure_mono ((hle W W.2).2 i) ((hc W.2).2.2.2 i hx) |
|
|
| lemma contractOnto_ground_sdiff {α : Type*} (M : Matroid α) (P : Set α) : |
| contractOnto M (M.E \ P) = M ／ P := rfl |
|
|
| @[simp] lemma contractOnto_ground {α : Type*} (M : Matroid α) (C : Set α) : |
| (contractOnto M C).E = C := rfl |
|
|
| lemma Wave.lift_contract {α ι : Type*} {M : ι → Matroid α} {F : Set α} |
| (hE : ∀ i, (M i).E = F) {W V : Set α × (ι → Set α)} |
| (hW : Wave M F W) (hV : Wave (fun i ↦ M i ／ W.1) (F \ W.1) V) : |
| Wave M F (W.1 ∪ V.1, fun i ↦ W.2 i ∪ V.2 i) := by |
| refine ⟨union_subset hW.1 (hV.1.trans sdiff_subset), ?_, ?_, ?_⟩ |
| · intro i |
| exact union_subset_union (hW.2.1 i) (hV.2.1 i) |
| · intro i j hij |
| apply disjoint_left.mpr |
| rintro x (hxi | hxi) (hxj | hxj) |
| · exact disjoint_left.mp (hW.2.2.1 hij) hxi hxj |
| · exact (hV.1 (hV.2.1 j hxj)).2 (hW.2.1 i hxi) |
| · exact (hV.1 (hV.2.1 i hxi)).2 (hW.2.1 j hxj) |
| · exact disjoint_left.mp (hV.2.2.1 hij) hxi hxj |
| · intro i |
| have hP : W.1 ⊆ (M i).closure (W.2 i ∪ V.2 i) := |
| (hW.2.2.2 i).trans ((M i).closure_mono subset_union_left) |
| have hT : V.2 i ⊆ (M i).closure (W.2 i ∪ V.2 i) := by |
| intro x hx |
| refine (M i).mem_closure_of_mem' (Or.inr hx) ?_ |
| rw [hE i] |
| exact (hV.1 (hV.2.1 i hx)).1 |
| refine union_subset hP ?_ |
| have hv := hV.2.2.2 i |
| rw [Matroid.contract_closure_eq] at hv |
| exact (hv.trans sdiff_subset).trans |
| ((M i).closure_subset_closure_of_subset_closure (union_subset hT hP)) |
|
|
| lemma residual_wave_empty {α ι : Type*} {M : ι → Matroid α} {F : Set α} |
| (hE : ∀ i, (M i).E = F) {W V : Set α × (ι → Set α)} |
| (hW : Maximal (Wave M F) W) |
| (hV : Wave (fun i ↦ M i ／ W.1) (F \ W.1) V) : V.1 = ∅ := by |
| have hLift := hW.1.lift_contract hE hV |
| have hle : W ≤ (W.1 ∪ V.1, fun i ↦ W.2 i ∪ V.2 i) := |
| ⟨subset_union_left, fun _ ↦ subset_union_left⟩ |
| have hsub : V.1 ⊆ W.1 := subset_union_right.trans (hW.2 hLift hle).1 |
| exact eq_empty_iff_forall_notMem.mpr fun x hx ↦ (hV.1 hx).2 (hsub hx) |
|
|
| lemma wave_of_restriction_packing {α ι : Type*} {M : ι → Matroid α} {F Y : Set α} |
| (hE : ∀ i, (M i).E = F) (hY : Y ⊆ F) |
| (h : FamilyPacking (fun i ↦ M i ↾ Y)) : |
| ∃ S, Wave M F (Y, S) := by |
| obtain ⟨S, hdisj, hS⟩ := h |
| have hSY : ∀ i, S i ⊆ Y := fun i ↦ (hS i).subset_ground |
| refine ⟨S, hY, hSY, hdisj, fun i ↦ ?_⟩ |
| exact ((M i).restrict_spanning_iff (hSY i) (by rw [hE]; exact hY)).mp (hS i) |
|
|
| lemma FamilyCovering.empty {α ι : Type*} (M : ι → Matroid α) : |
| FamilyCovering ∅ (fun i ↦ M i ↾ ∅) := by |
| refine ⟨fun _ ↦ ∅, fun _ ↦ Matroid.empty_indep _, ?_⟩ |
| simp |
|
|
| lemma residual_covering_condition {α ι : Type*} {M : ι → Matroid α} {F : Set α} |
| (hE : ∀ i, (M i).E = F) {W : Set α × (ι → Set α)} |
| (hW : Maximal (Wave M F) W) : |
| ∀ Y ⊆ F \ W.1, FamilyPacking (fun i ↦ (M i ／ W.1) ↾ Y) → |
| FamilyCovering Y (fun i ↦ (M i ／ W.1) ↾ Y) := by |
| intro Y hY hpack |
| have hg : ∀ i, (M i ／ W.1).E = F \ W.1 := by simp [hE] |
| obtain ⟨S, hS⟩ := wave_of_restriction_packing hg hY hpack |
| have hzero : Y = ∅ := residual_wave_empty hE hW hS |
| subst Y |
| exact FamilyCovering.empty _ |
|
|
| end InfiniteMatroidCounterexample |
|
|
| end OAI |
|
|
