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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 | import OAI.Combinatorics.InfiniteMatroid.CorollaryModel
import OAI.Combinatorics.InfiniteMatroid.Matroid
import OAI.Combinatorics.InfiniteMatroid.DoubleRay
namespace OAI
namespace InfiniteMatroidCounterexample
open Set Matroid
lemma BasisSystem.isUniform {α : Type*} (S : BasisSystem α) : IsUniform S.matroid := by
intro X hX
obtain ⟨I, hI⟩ := S.matroid.exists_isBasis X
by_cases hb : S.matroid.IsBase I
· exact Or.inr (hb.spanning_of_superset hI.subset hX)
· left
have hXI : X ⊆ I := fun e he ↦ hI.mem_of_insert_indep he
(S.insert_indep_of_not_base hI.indep hb e)
exact (Subset.antisymm hXI hI.subset) ▸ hI.indep
lemma mapEquiv_trans {α β γ : Type*} (M : Matroid α) (e : α ≃ β) (f : β ≃ γ) :
(M.mapEquiv e).mapEquiv f = M.mapEquiv (e.trans f) := by
apply Matroid.ext_indep
· simp only [Matroid.mapEquiv_ground_eq, image_image, Equiv.trans_apply]
rfl
· intro I _
simp only [Matroid.mapEquiv_indep_iff, image_image]
rfl
lemma rayMatroid_isPartitional {α : Type} (Q : Matroid (α × Bool))
(hQ : IsUniform Q) (p : Bool) : IsPartitional (rayMatroid Q p) := by
refine ⟨Vertex p, fun _ ↦ α × Bool, fun _ ↦ Q,
(Equiv.sigmaEquivProd (Vertex p) (α × Bool)).trans (componentEquiv α p),
fun _ ↦ hQ, ?_⟩
exact mapEquiv_trans _ _ _
end InfiniteMatroidCounterexample
end OAI
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