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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 | import OAI.Combinatorics.InfiniteMatroid.Basic
namespace OAI
section
open Set Matroid
namespace InfiniteMatroidCounterexample
/-- Inward (false) and outward (true) bundles with the central edge at -1. -/
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)
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))
/-- The independent partition is assigned to vertices by parity. -/
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}
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
end OAI
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