| import OAI.Combinatorics.InfiniteMatroid.Zones |
|
|
| namespace OAI |
|
|
| section |
| open Set Filter |
| open scoped Topology symmDiff |
| namespace InfiniteMatroidCounterexample |
| namespace RankModel |
|
|
| variable (R : RankModel D) (hK : R.ideal = KIdeal) |
| include hK |
|
|
| lemma insert_prototype {ι : Type} (hι : Cardinal.mk ι < Cardinal.continuum) |
| (T : ι → Set E0) {A Z : Set E0} (hAZ : A ⊆ Z) |
| (hA : ¬ R.Upper A) (hZ : ¬ R.Lower Z) |
| (hTA : ∀ i, ¬ Small (T i \ A)) (hZT : ∀ i, ¬ Small (Z \ T i)) : |
| ∃ B, A ⊆ B ∧ B ⊆ Z ∧ R.Admissible B ∧ |
| ∀ i, ¬ Small (B \ T i) ∧ ¬ Small (T i \ B) := by |
| classical |
| let G := Z \ A |
| let S0 (i : ι) : Set E0 := if Small (A \ T i) then G \ T i else Z \ T i |
| let S1 (i : ι) : Set E0 := if Small (T i \ Z) then T i ∩ G else T i \ A |
| have h0 (i : ι) : ¬ Small (S0 i) := by |
| dsimp only [S0] |
| split_ifs with hs |
| · intro hg |
| apply hZT i |
| apply small_subset (small_union hs hg) |
| intro x hx |
| change x ∈ (A \ T i) ∪ ((Z \ A) \ T i) |
| simp only [Set.mem_union,Set.mem_sdiff] at hx ⊢ |
| tauto |
| · exact hZT i |
| have h1 (i : ι) : ¬ Small (S1 i) := by |
| dsimp only [S1] |
| split_ifs with hs |
| · intro hg |
| apply hTA i |
| apply small_subset (small_union hs hg) |
| intro x hx |
| change x ∈ (T i \ Z) ∪ (T i ∩ (Z \ A)) |
| simp only [Set.mem_union,Set.mem_sdiff,Set.mem_inter_iff] at hx ⊢ |
| tauto |
| · exact hTA i |
| let traces (p : ι × Bool) : Set E0 := if p.2 then S1 p.1 else S0 p.1 |
| have ht : ∀ p, ¬ Small (traces p) := by |
| rintro ⟨i,b⟩; cases b |
| · exact h0 i |
| · exact h1 i |
| obtain ⟨t,hres⟩ := simultaneous_reservations (cardinal_prod_bool_lt hι) traces ht |
| have hr0 (i : ι) : ¬ Small (S0 i ∩ HCell t false) := hres (i,false) false |
| have hr1 (i : ι) : ¬ Small (S1 i ∩ HCell t true) := hres (i,true) true |
| let A' := A ∪ (HCell t false ∩ G) |
| let Z' := Z \ (HCell t true ∩ G) |
| have hAA : A ⊆ A' := Set.subset_union_left |
| have hZZ : Z' ⊆ Z := Set.sdiff_subset |
| have hAZ' : A' ⊆ Z' := by |
| intro x hx |
| refine ⟨?_,?_⟩ |
| · rcases hx with hx | hx |
| · exact hAZ hx |
| · exact hx.2.1 |
| · intro hh |
| rcases hx with hx | hx |
| · exact hh.2.2 hx |
| · exact Set.disjoint_left.mp (hCell_disjoint t) hx.1 hh.1 |
| have haInv : KSmall0 (A ∆ A') := by |
| apply kSmall0_mono (hCell_kSmall0 t false) |
| intro x hx |
| change x ∈ A ∆ (A ∪ (HCell t false ∩ G)) at hx |
| simp only [Set.mem_symmDiff,Set.mem_union,Set.mem_inter_iff] at hx |
| tauto |
| have hzInv : KSmall0 (Z ∆ Z') := by |
| apply kSmall0_mono (hCell_kSmall0 t true) |
| intro x hx |
| change x ∈ Z ∆ (Z \ (HCell t true ∩ G)) at hx |
| simp only [Set.mem_symmDiff,Set.mem_sdiff,Set.mem_inter_iff] at hx |
| tauto |
| have haU : ¬ R.Upper A' := ((R.status_invariant hK haInv).1).not.mp hA |
| have hzL : ¬ R.Lower Z' := ((R.status_invariant hK hzInv).2.1).not.mp hZ |
| obtain ⟨B,hAB,hBZ,hB⟩ := R.exists_admissible_between hAZ' haU hzL |
| refine ⟨B,hAA.trans hAB,hBZ.trans hZZ,hB,fun i ↦ ⟨?_,?_⟩⟩ |
| · by_cases hs : Small (A \ T i) |
| · have hr : ¬ Small ((G \ T i) ∩ HCell t false) := by simpa only [S0,ite_eq_left hs] using hr0 i |
| intro hbad |
| apply hr |
| apply small_subset hbad |
| intro x hx |
| exact ⟨hAB (Or.inr ⟨hx.2,hx.1.1⟩),hx.1.2⟩ |
| · exact fun hb ↦ hs (small_subset hb (Set.sdiff_subset_sdiff_left (hAA.trans hAB))) |
| · by_cases hs : Small (T i \ Z) |
| · have hr : ¬ Small ((T i ∩ G) ∩ HCell t true) := by simpa only [S1,ite_eq_left hs] using hr1 i |
| intro hbad |
| apply hr |
| apply small_subset hbad |
| intro x hx |
| exact ⟨hx.1.1,fun hxB ↦ (hBZ hxB).2 ⟨hx.2,hx.1.2⟩⟩ |
| · exact fun hb ↦ hs (small_subset hb (Set.sdiff_subset_sdiff_right (hBZ.trans hZZ))) |
|
|
| end RankModel |
| end InfiniteMatroidCounterexample |
| end |
|
|
| end OAI |
|
|
