| import OAI.Combinatorics.InfiniteMatroid.Basic |
|
|
| namespace OAI |
|
|
| section |
| open Set Matroid |
| namespace InfiniteMatroidCounterexample |
|
|
| / |
| def bundleAt (n : ℤ) (b : Bool) : ℤ := |
| if n < 0 then (if b then n - 1 else n) else (if b then n else n - 1) |
|
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| lemma bundleAt_bounds (n : ℤ) (b : Bool) : n - 1 ≤ bundleAt n b ∧ bundleAt n b ≤ n := by |
| unfold bundleAt |
| split_ifs <;> omega |
| lemma bundleAt_injective (n : ℤ) : Function.Injective (bundleAt n) := by |
| intro b c h |
| cases b <;> cases c <;> simp_all [bundleAt] <;> split_ifs at h <;> omega |
| lemma bundleAt_right (n : ℤ) : ∃ b, bundleAt n b = n := by |
| by_cases h : n < 0 |
| · exact ⟨false,by simp [bundleAt,h]⟩ |
| · exact ⟨true,by simp [bundleAt,h]⟩ |
| lemma bundleAt_left (n : ℤ) : ∃ b, bundleAt n b = n - 1 := by |
| by_cases h : n < 0 |
| · exact ⟨true,by simp [bundleAt,h]⟩ |
| · exact ⟨false,by simp [bundleAt,h]⟩ |
|
|
| def parityBit (p : Bool) : ℤ := if p then 1 else 0 |
| abbrev Vertex (p : Bool) := {n : ℤ // n % 2 = parityBit p} |
|
|
| def componentMap {α : Type*} {p : Bool} (v : Vertex p) (x : α × Bool) : ℤ × α := |
| (bundleAt v x.2,x.1) |
|
|
| def globalComponentMap (α : Type*) (p : Bool) (x : Vertex p × (α × Bool)) : ℤ × α := |
| componentMap x.1 x.2 |
|
|
| lemma globalComponentMap_injective (α : Type*) (p : Bool) : |
| Function.Injective (globalComponentMap α p) := by |
| rintro ⟨⟨n,hn⟩,d,b⟩ ⟨⟨m,hm⟩,e,c⟩ h |
| have hnm : bundleAt n b = bundleAt m c := congrArg Prod.fst h |
| have hde : d = e := congrArg Prod.snd h |
| have hb := bundleAt_bounds n b |
| have hc := bundleAt_bounds m c |
| have hnm' : n = m := by omega |
| subst m |
| have hbc := bundleAt_injective n hnm |
| subst e; subst c |
| rfl |
|
|
| lemma globalComponentMap_surjective (α : Type*) (p : Bool) : |
| Function.Surjective (globalComponentMap α p) := by |
| rintro ⟨n,d⟩ |
| by_cases hn : n % 2 = parityBit p |
| · obtain ⟨b,hb⟩ := bundleAt_right n |
| exact ⟨(⟨n,hn⟩,d,b),by simp [globalComponentMap,componentMap,hb]⟩ |
| · have hn' : (n+1) % 2 = parityBit p := by |
| cases p <;> simp only [parityBit,Bool.false_eq_true,ite_false,ite_true] at hn ⊢ <;> omega |
| obtain ⟨b,hb⟩ := bundleAt_left (n+1) |
| exact ⟨(⟨n+1,hn'⟩,d,b),by simp [globalComponentMap,componentMap,hb]⟩ |
|
|
| noncomputable def componentEquiv (α : Type*) (p : Bool) : (Vertex p × (α × Bool)) ≃ (ℤ × α) := |
| Equiv.ofBijective (globalComponentMap α p) (by |
| have bundleAt_bounds (n : ℤ) (b : Bool) : n - 1 ≤ bundleAt n b ∧ bundleAt n b ≤ n := by |
| unfold bundleAt |
| split_ifs <;> omega |
| have bundleAt_injective (n : ℤ) : Function.Injective (bundleAt n) := by |
| intro b c h |
| cases b <;> cases c <;> simp_all [bundleAt] <;> split_ifs at h <;> omega |
| have bundleAt_right (n : ℤ) : ∃ b, bundleAt n b = n := by |
| by_cases h : n < 0 |
| · exact ⟨false,by simp [bundleAt,h]⟩ |
| · exact ⟨true,by simp [bundleAt,h]⟩ |
| have bundleAt_left (n : ℤ) : ∃ b, bundleAt n b = n - 1 := by |
| by_cases h : n < 0 |
| · exact ⟨true,by simp [bundleAt,h]⟩ |
| · exact ⟨false,by simp [bundleAt,h]⟩ |
|
|
| have globalComponentMap_injective : |
| Function.Injective (globalComponentMap α p) := by |
| rintro ⟨⟨n,hn⟩,d,b⟩ ⟨⟨m,hm⟩,e,c⟩ h |
| have hnm : bundleAt n b = bundleAt m c := congrArg Prod.fst h |
| have hde : d = e := congrArg Prod.snd h |
| have hb := bundleAt_bounds n b |
| have hc := bundleAt_bounds m c |
| have hnm' : n = m := by omega |
| subst m |
| have hbc := bundleAt_injective n hnm |
| subst e; subst c |
| rfl |
|
|
| have globalComponentMap_surjective : |
| Function.Surjective (globalComponentMap α p) := by |
| rintro ⟨n,d⟩ |
| by_cases hn : n % 2 = parityBit p |
| · obtain ⟨b,hb⟩ := bundleAt_right n |
| exact ⟨(⟨n,hn⟩,d,b),by simp [globalComponentMap,componentMap,hb]⟩ |
| · have hn' : (n+1) % 2 = parityBit p := by |
| cases p <;> simp only [parityBit,Bool.false_eq_true,ite_false,ite_true] at hn ⊢ <;> omega |
| obtain ⟨b,hb⟩ := bundleAt_left (n+1) |
| exact ⟨(⟨n+1,hn'⟩,d,b),by simp [globalComponentMap,componentMap,hb]⟩ |
| exact ⟨globalComponentMap_injective, globalComponentMap_surjective⟩) |
|
|
| noncomputable def rayMatroid {α : Type*} (Q : Matroid (α × Bool)) (p : Bool) : Matroid (ℤ × α) := |
| (Matroid.sum' (fun _ : Vertex p ↦ Q)).mapEquiv (componentEquiv α p) |
|
|
| lemma rayMatroid_ground {α : Type*} (Q : Matroid (α × Bool)) (hQ : Q.E = univ) (p : Bool) : |
| (rayMatroid Q p).E = univ := by |
| have hg : (Matroid.sum' (fun _ : Vertex p ↦ Q)).E = univ := by |
| ext ⟨v,x⟩ |
| simp only [Matroid.sum'_ground_eq,hQ,mem_iUnion,mem_image,mem_univ,true_and,iff_true] |
| exact ⟨v,x,rfl⟩ |
| rw [rayMatroid,Matroid.mapEquiv_ground_eq,hg] |
| rw [Set.image_univ] |
| exact Set.range_eq_univ.mpr (componentEquiv α p).surjective |
|
|
| lemma rayMatroid_indep_iff {α : Type*} (Q : Matroid (α × Bool)) (p : Bool) (I : Set (ℤ × α)) : |
| (rayMatroid Q p).Indep I ↔ ∀ v : Vertex p, Q.Indep (componentMap v ⁻¹' I) := by |
| rw [rayMatroid,Matroid.mapEquiv_indep_iff,Equiv.image_symm_eq_preimage,Matroid.sum'_indep_iff] |
| rfl |
| lemma rayMatroid_isBase_iff {α : Type*} (Q : Matroid (α × Bool)) (p : Bool) (B : Set (ℤ × α)) : |
| (rayMatroid Q p).IsBase B ↔ ∀ v : Vertex p, Q.IsBase (componentMap v ⁻¹' B) := by |
| rw [rayMatroid,Matroid.mapEquiv_isBase_iff,Equiv.image_symm_eq_preimage,Matroid.sum'_isBase_iff] |
| rfl |
|
|
| lemma selfDual_compl_base_iff {α : Type*} (Q : Matroid α) (hE : Q.E = univ) (hQ : Q.dual = Q) |
| (B : Set α) : Q.IsBase Bᶜ ↔ Q.IsBase B := by |
| have h : Q.dual.IsBase B ↔ Q.IsBase Bᶜ := by |
| rw [Matroid.dual_isBase_iff,hE,←Set.compl_eq_univ_sdiff] |
| rw [hQ] at h |
| exact h.symm |
|
|
| lemma rayMatroid_selfDual {α : Type*} (Q : Matroid (α × Bool)) (hE : Q.E = univ) (hQ : Q.dual = Q) |
| (p : Bool) : (rayMatroid Q p).dual = rayMatroid Q p := by |
| apply Matroid.ext_isBase Matroid.dual_ground |
| intro B _ |
| rw [Matroid.dual_isBase_iff,rayMatroid_ground Q hE,←Set.compl_eq_univ_sdiff, |
| rayMatroid_isBase_iff,rayMatroid_isBase_iff] |
| exact forall_congr' (fun v ↦ selfDual_compl_base_iff Q hE hQ (componentMap v ⁻¹' B)) |
|
|
| / |
| def assignment {α : Type*} (I : Set (ℤ × α)) (n : ℤ) : Set (α × Bool) := |
| {x | if n % 2 = 0 then (bundleAt n x.2,x.1) ∈ I else (bundleAt n x.2,x.1) ∉ I} |
|
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| lemma assignment_indep {α : Type*} (Q : Matroid (α × Bool)) {I : Set (ℤ × α)} |
| (h0 : (rayMatroid Q false).Indep I) (h1 : (rayMatroid Q true).Indep Iᶜ) (n : ℤ) : |
| Q.Indep (assignment I n) := by |
| by_cases hn : n % 2 = 0 |
| · have hv : n % 2 = parityBit false := hn |
| have hi := (rayMatroid_indep_iff Q false I).mp h0 ⟨n,hv⟩ |
| change Q.Indep {x | (bundleAt n x.2,x.1) ∈ I} at hi |
| simpa only [assignment,hn,ite_true] using hi |
| · have hv : n % 2 = parityBit true := by change n % 2 = 1; omega |
| have hi := (rayMatroid_indep_iff Q true Iᶜ).mp h1 ⟨n,hv⟩ |
| change Q.Indep {x | (bundleAt n x.2,x.1) ∉ I} at hi |
| simpa only [assignment,hn,ite_false] using hi |
|
|
| end InfiniteMatroidCounterexample |
| end |
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| end OAI |
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