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c270adf | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 | import OAI.Combinatorics.InfiniteMatroid.Basic
namespace OAI
section
open Set Matroid
namespace InfiniteMatroidCounterexample
def IntervalAlternative {α : Type*} (Bases : Set (Set α)) : Prop :=
∀ I Y, I ⊆ Y → (Y \ I).Infinite →
∃ B ∈ Bases, B ⊆ I ∨ (I ⊆ B ∧ B ⊆ Y) ∨ Y ⊆ B
/-- The purely set-theoretic assumptions supplied by the prototype assembly. -/
structure BasisSystem (α : Type*) where
bases : Set (Set α)
nonempty : bases.Nonempty
antichain : ∀ ⦃B C⦄, B ∈ bases → C ∈ bases → B ⊆ C → B = C
compl_closed : ∀ ⦃B⦄, B ∈ bases → Bᶜ ∈ bases
swap : ∀ ⦃B⦄, B ∈ bases → ∀ e ∈ B, ∀ f ∉ B, insert f (B \ {e}) ∈ bases
intervals : IntervalAlternative bases
namespace BasisSystem
variable {α : Type*} (S : BasisSystem α)
def Independent (I : Set α) : Prop := ∃ B ∈ S.bases, I ⊆ B
lemma independent_mono {I J : Set α} (hJ : S.Independent J) (hIJ : I ⊆ J) :
S.Independent I := by
obtain ⟨B, hB, hJB⟩ := hJ
exact ⟨B, hB, hIJ.trans hJB⟩
lemma maximal_of_base {B : Set α} (hB : B ∈ S.bases) {Y : Set α} (hBY : B ⊆ Y) :
Maximal (fun K ↦ S.Independent K ∧ K ⊆ Y) B := by
refine ⟨⟨⟨B, hB, Subset.rfl⟩, hBY⟩, ?_⟩
intro C hC hBC
obtain ⟨B', hB', hCB'⟩ := hC.1
have heq := S.antichain hB hB' (hBC.trans hCB')
exact heq ▸ hCB'
lemma exists_maximal_finite_gap {I Y : Set α} (hI : S.Independent I) (hIY : I ⊆ Y)
(hgap : (Y \ I).Finite) :
∃ J, I ⊆ J ∧ Maximal (fun K ↦ S.Independent K ∧ K ⊆ Y) J := by
let F : Set (Set α) := {J | I ⊆ J ∧ S.Independent J ∧ J ⊆ Y}
have hF : F.Finite := by
apply ((hgap.finite_subsets).image (fun H ↦ I ∪ H)).subset
intro J hJ
refine ⟨J \ I, (fun x hx ↦ ⟨hJ.2.2 hx.1, hx.2⟩), ?_⟩
ext x
simp only [Set.mem_union, Set.mem_sdiff]
have := @hJ.1 x
tauto
obtain ⟨J, hJ, hmax⟩ := hF.exists_maximal ⟨I, Subset.rfl, hI, hIY⟩
refine ⟨J, hJ.1, ⟨hJ.2, ?_⟩⟩
intro K hK hJK
exact hmax ⟨hJ.1.trans hJK, hK⟩ hJK
lemma exists_maximal {I Y : Set α} (hI : S.Independent I) (hIY : I ⊆ Y) :
∃ J, I ⊆ J ∧ Maximal (fun K ↦ S.Independent K ∧ K ⊆ Y) J := by
by_cases hgap : (Y \ I).Finite
· exact S.exists_maximal_finite_gap hI hIY hgap
obtain ⟨B, hB, hBI | ⟨hIB, hBY⟩ | hYB⟩ := S.intervals I Y hIY hgap
· obtain ⟨B', hB', hIB'⟩ := hI
have heq := S.antichain hB hB' (hBI.trans hIB')
have hIB : I ⊆ B := heq ▸ hIB'
exact ⟨B, hIB, S.maximal_of_base hB (hBI.trans hIY)⟩
· exact ⟨B, hIB, S.maximal_of_base hB hBY⟩
· refine ⟨Y, hIY, ⟨⟨⟨B, hB, hYB⟩, Subset.rfl⟩, ?_⟩⟩
exact fun _ h _ ↦ h.2
lemma exchange : Matroid.ExchangeProperty (· ∈ S.bases) := by
intro B C hB hC e he
have hn : ¬ C ⊆ B := by
intro hCB
have heq := S.antichain hC hB hCB
exact he.2 (heq ▸ he.1)
obtain ⟨f, hfC, hfB⟩ := Set.not_subset.mp hn
exact ⟨f, ⟨hfC, hfB⟩, S.swap hB e he.1 f hfB⟩
noncomputable def matroid : Matroid α :=
Matroid.ofBase univ (· ∈ S.bases) S.nonempty S.exchange
(fun _ _ _ hI hIY ↦ S.exists_maximal hI hIY)
(fun _ _ ↦ subset_univ _)
@[simp] lemma ground : S.matroid.E = univ := rfl
@[simp] lemma isBase_iff (B : Set α) : S.matroid.IsBase B ↔ B ∈ S.bases := Iff.rfl
@[simp] lemma indep_iff (I : Set α) : S.matroid.Indep I ↔ S.Independent I := Iff.rfl
lemma self_dual : S.matroid.dual = S.matroid := by
apply Matroid.ext_isBase Matroid.dual_ground
intro B _
rw [Matroid.dual_isBase_iff, ground, ← Set.compl_eq_univ_sdiff, isBase_iff, isBase_iff]
exact ⟨fun h ↦ by simpa using S.compl_closed h, fun h ↦ S.compl_closed h⟩
lemma insert_indep_of_not_base {I : Set α} (hI : S.matroid.Indep I)
(hnot : ¬ S.matroid.IsBase I) (e : α) : S.matroid.Indep (insert e I) := by
obtain ⟨B, hB, hIB⟩ := hI
by_cases heB : e ∈ B
· exact ⟨B, hB, Set.insert_subset heB hIB⟩
have hn : ¬ B ⊆ I := by
intro hBI
exact hnot ((Subset.antisymm hIB hBI) ▸ hB)
obtain ⟨f, hfB, hfI⟩ := Set.not_subset.mp hn
refine ⟨insert e (B \ {f}), S.swap hB f hfB e heB, ?_⟩
intro x hx
rcases hx with rfl | hx
· exact Set.mem_insert _ _
· right
exact ⟨hIB hx, fun hxf ↦ hfI (Set.mem_singleton_iff.mp hxf ▸ hx)⟩
end BasisSystem
end InfiniteMatroidCounterexample
end
end OAI
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