| import OAI.Combinatorics.InfiniteMatroid.DoubleRay |
|
|
| namespace OAI |
|
|
| section |
| open Set Filter Matroid |
| namespace InfiniteMatroidCounterexample |
|
|
| def raySlice {α : Type*} (I : Set (ℤ × α)) (n : ℤ) (b : Bool) : Set α := |
| {d | (d,b) ∈ assignment I n} |
|
|
| lemma raySlice_central {α : Type*} (I : Set (ℤ × α)) : |
| raySlice I (-1) false = (raySlice I 0 false)ᶜ := by |
| ext d |
| simp [raySlice,assignment,bundleAt] |
|
|
| lemma raySlice_right {α : Type*} (I : Set (ℤ × α)) (n : ℤ) (hn : 0 ≤ n) : |
| raySlice I (n+1) false = (raySlice I n true)ᶜ := by |
| have hn0 : ¬ n < 0 := not_lt.mpr hn |
| have hn1 : ¬ n+1 < 0 := by omega |
| by_cases hp : n % 2 = 0 |
| · have hp' : ¬ (n+1) % 2 = 0 := by omega |
| ext d |
| simp [raySlice,assignment,bundleAt,hn0,hn1,hp,hp'] |
| · have hp' : (n+1) % 2 = 0 := by omega |
| ext d |
| simp [raySlice,assignment,bundleAt,hn0,hn1,hp,hp'] |
|
|
| lemma raySlice_left {α : Type*} (I : Set (ℤ × α)) (n : ℤ) (hn : n < 0) : |
| raySlice I (n-1) false = (raySlice I n true)ᶜ := by |
| have hn1 : n-1 < 0 := by omega |
| by_cases hp : n % 2 = 0 |
| · have hp' : ¬ (n-1) % 2 = 0 := by omega |
| ext d |
| simp [raySlice,assignment,bundleAt,hn,hn1,hp,hp'] |
| · have hp' : (n-1) % 2 = 0 := by omega |
| ext d |
| simp [raySlice,assignment,bundleAt,hn,hn1,hp,hp'] |
|
|
| def rayVertex (left : Bool) (k : ℕ) : ℤ := if left then -(k : ℤ)-1 else (k : ℤ) |
|
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| lemma raySlice_step {α : Type*} (I : Set (ℤ × α)) (left : Bool) (k : ℕ) : |
| raySlice I (rayVertex left (k+1)) false = (raySlice I (rayVertex left k) true)ᶜ := by |
| cases left with |
| | false => simpa only [rayVertex,Bool.false_eq_true,ite_false,Nat.cast_add,Nat.cast_one] using |
| raySlice_right I (k : ℤ) (Int.natCast_nonneg k) |
| | true => |
| have he : rayVertex true (k+1) = rayVertex true k - 1 := by simp [rayVertex]; ring |
| rw [he] |
| exact raySlice_left I _ (by simp [rayVertex]; omega) |
|
|
| section Descent |
| variable {α β : Type*} [LinearOrder β] [WellFoundedLT β] |
| variable (Q : Matroid (α × Bool)) (U : Ultrafilter α) (ρ : Set α → β) |
| variable (hanti : ∀ ⦃X Y : Set α⦄, X ∈ U → Y ∈ U → X ⊆ Y → ρ Y ≤ ρ X) |
| variable (hbasis : ∀ B, Q.IsBase B → (Prod.mk · false) ⁻¹' B ∈ U → |
| ((Prod.mk · true) ⁻¹' B)ᶜ ∈ U ∧ ρ ((Prod.mk · true) ⁻¹' B)ᶜ < ρ ((Prod.mk · false) ⁻¹' B)) |
| include hanti hbasis |
|
|
| omit [WellFoundedLT β] in |
| lemma independent_rank_step {X : Set (α × Bool)} (hX : Q.Indep X) |
| (hl : (Prod.mk · false) ⁻¹' X ∈ U) : |
| ((Prod.mk · true) ⁻¹' X)ᶜ ∈ U ∧ ρ ((Prod.mk · true) ⁻¹' X)ᶜ < ρ ((Prod.mk · false) ⁻¹' X) := by |
| obtain ⟨B,hB,hXB⟩ := hX.exists_isBase_superset |
| have hlu : (Prod.mk · false) ⁻¹' B ∈ U := mem_of_superset hl (preimage_mono hXB) |
| obtain ⟨hy,hlt⟩ := hbasis B hB hlu |
| have hyx : ((Prod.mk · true) ⁻¹' B)ᶜ ⊆ ((Prod.mk · true) ⁻¹' X)ᶜ := |
| compl_subset_compl.mpr (preimage_mono hXB) |
| have hr : ((Prod.mk · true) ⁻¹' X)ᶜ ∈ U := mem_of_superset hy hyx |
| refine ⟨hr,?_⟩ |
| exact lt_of_le_of_lt (hanti hy hr hyx) |
| (lt_of_lt_of_le hlt (hanti hl hlu (preimage_mono hXB))) |
|
|
| lemma no_independent_partition {I : Set (ℤ × α)} (h0 : (rayMatroid Q false).Indep I) |
| (h1 : (rayMatroid Q true).Indep Iᶜ) : False := by |
| have hstart : ∃ left : Bool, raySlice I (rayVertex left 0) false ∈ U := by |
| by_cases hc : raySlice I 0 false ∈ U |
| · exact ⟨false,hc⟩ |
| · refine ⟨true,?_⟩ |
| change raySlice I (-1) false ∈ U |
| rw [raySlice_central] |
| exact Ultrafilter.compl_mem_iff_notMem.mpr hc |
| obtain ⟨left,hstart⟩ := hstart |
| let x (k : ℕ) := raySlice I (rayVertex left k) false |
| have hstep (k : ℕ) (hk : x k ∈ U) : x (k+1) ∈ U ∧ ρ (x (k+1)) < ρ (x k) := by |
| have hi := assignment_indep Q h0 h1 (rayVertex left k) |
| have hs := independent_rank_step Q U ρ hanti hbasis hi hk |
| change (raySlice I (rayVertex left k) true)ᶜ ∈ U ∧ |
| ρ (raySlice I (rayVertex left k) true)ᶜ < ρ (x k) at hs |
| simpa only [x,raySlice_step] using hs |
| have hx : ∀ k, x k ∈ U := by |
| intro k |
| induction k with |
| | zero => exact hstart |
| | succ k ih => exact (hstep k ih).1 |
| have hd : ∀ k, ρ (x (k+1)) < ρ (x k) := fun k ↦ (hstep k (hx k)).2 |
| exact (wellFounded_iff_isEmpty_descending_chain.mp (wellFounded_lt (α := β))).false ⟨fun k ↦ ρ (x k),hd⟩ |
|
|
| lemma no_independent_cover (I J : Set (ℤ × α)) |
| (hI : (rayMatroid Q false).Indep I) (hJ : (rayMatroid Q true).Indep J) : I ∪ J ≠ univ := by |
| intro hcover |
| have hIJ : Iᶜ ⊆ J := by |
| intro x hx |
| have hm : x ∈ I ∪ J := hcover.symm ▸ mem_univ x |
| exact hm.resolve_left hx |
| exact no_independent_partition Q U ρ hanti hbasis hI (hJ.subset hIJ) |
|
|
| end Descent |
| end InfiniteMatroidCounterexample |
| end |
|
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| end OAI |
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