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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 | import OAI.Combinatorics.InfiniteMatroid.PrototypeExtension
import OAI.Combinatorics.InfiniteMatroid.Matroid
namespace OAI
section
open Set Filter
open scoped Topology symmDiff
namespace InfiniteMatroidCounterexample
def stagePrototypes : Option (Set E0) → Set Prototype
| none => ∅
| some B => prototypePair B
lemma stagePrototypes_finite (o : Option (Set E0)) : (stagePrototypes o).Finite := by
cases o with
| none => exact finite_empty
| some B => exact prototypePair_finite B
def pastPrototypes (f : CIndex → Option (Set E0)) (a : CIndex) : Set Prototype :=
⋃ b : Iio a, stagePrototypes (f b)
def initialPrototypes (f : CIndex → Option (Set E0)) (a : CIndex) : Set Prototype :=
pastPrototypes f a ∪ stagePrototypes (f a)
def allPrototypes (f : CIndex → Option (Set E0)) : Set Prototype :=
⋃ a, stagePrototypes (f a)
lemma stage_subset_past {f : CIndex → Option (Set E0)} {a b : CIndex} (h : b < a) :
stagePrototypes (f b) ⊆ pastPrototypes f a := by
intro p hp
exact mem_iUnion.mpr ⟨⟨b,h⟩,hp⟩
lemma stage_subset_initial {f : CIndex → Option (Set E0)} {a b : CIndex} (h : b ≤ a) :
stagePrototypes (f b) ⊆ initialPrototypes f a := by
rcases h.lt_or_eq with h|rfl
· exact (stage_subset_past h).trans subset_union_left
· exact subset_union_right
lemma initial_subset_past {f : CIndex → Option (Set E0)} {a b : CIndex} (h : b < a) :
initialPrototypes f b ⊆ pastPrototypes f a := by
rintro p (hp|hp)
· obtain ⟨c,hc⟩ := mem_iUnion.mp hp
exact stage_subset_past (c.2.trans h) hc
· exact stage_subset_past h hp
lemma initial_subset_all (f : CIndex → Option (Set E0)) (a : CIndex) :
initialPrototypes f a ⊆ allPrototypes f := by
rintro p (hp|hp)
· obtain ⟨c,hc⟩ := mem_iUnion.mp hp
exact mem_iUnion.mpr ⟨c,hc⟩
· exact mem_iUnion.mpr ⟨a,hp⟩
lemma past_cardinal (f : CIndex → Option (Set E0)) (a : CIndex) :
Cardinal.mk (pastPrototypes f a) < Cardinal.continuum :=
finite_union_small (cindex_predecessors a) _ (fun b ↦ stagePrototypes_finite (f b))
lemma exists_interval_enumeration :
∃ e : CIndex → (Set E0 × Set E0), Function.Surjective e := by
classical
obtain ⟨e,he⟩ := exists_set_enumeration (E0 × Bool)
refine ⟨fun a ↦ ({x | (x,false) ∈ e a},{x | (x,true) ∈ e a}),?_⟩
rintro ⟨I,Y⟩
obtain ⟨a,ha⟩ := he {p | if p.2 then p.1 ∈ Y else p.1 ∈ I}
exact ⟨a,by simp [ha]⟩
namespace RankModel
variable (R : RankModel D)
lemma good_past_of_initial (f : CIndex → Option (Set E0)) (a : CIndex)
(h : ∀ b < a, R.GoodPrototypes (initialPrototypes f b)) :
R.GoodPrototypes (pastPrototypes f a) := by
constructor
· intro p hp
obtain ⟨b,hb⟩ := mem_iUnion.mp hp
exact (h b b.2).admissible p (Or.inr hb)
· intro p hp
obtain ⟨b,hb⟩ := mem_iUnion.mp hp
exact initial_subset_past b.2 ((h b b.2).flip_closed p (Or.inr hb))
· intro p hp q hq hne
obtain ⟨b,hb⟩ := mem_iUnion.mp hp
obtain ⟨c,hc⟩ := mem_iUnion.mp hq
have hg := h (max b.1 c.1) (max_lt b.2 c.2)
exact hg.separated p (stage_subset_initial (le_max_left _ _) hb)
q (stage_subset_initial (le_max_right _ _) hc) hne
lemma good_all_of_initial (f : CIndex → Option (Set E0))
(h : ∀ a, R.GoodPrototypes (initialPrototypes f a)) :
R.GoodPrototypes (allPrototypes f) := by
constructor
· intro p hp
obtain ⟨a,ha⟩ := mem_iUnion.mp hp
exact (h a).admissible p (Or.inr ha)
· intro p hp
obtain ⟨a,ha⟩ := mem_iUnion.mp hp
exact initial_subset_all f a ((h a).flip_closed p (Or.inr ha))
· intro p hp q hq hne
obtain ⟨a,ha⟩ := mem_iUnion.mp hp
obtain ⟨b,hb⟩ := mem_iUnion.mp hq
exact (h (max a b)).separated p (stage_subset_initial (le_max_left _ _) ha)
q (stage_subset_initial (le_max_right _ _) hb) hne
variable (hK : R.ideal = KIdeal)
include hK
lemma exists_stage (P : Set Prototype) (hP : R.GoodPrototypes P)
(hcard : Cardinal.mk P < Cardinal.continuum) (I Y : Set E0) :
∃ o : Option (Set E0), R.GoodPrototypes (P ∪ stagePrototypes o) ∧
(I ⊆ Y → Meets (PrototypeBases (P ∪ stagePrototypes o)) I Y) := by
classical
by_cases hIY : I ⊆ Y
· by_cases hmet : Meets (PrototypeBases P) I Y
· exact ⟨none,by simpa [stagePrototypes],by simpa [stagePrototypes] using (fun _ : I ⊆ Y ↦ hmet)⟩
· obtain ⟨B,hB,hmet⟩ := R.extend_unmet hK hP hcard hIY hmet
exact ⟨some B,hB,fun _ ↦ hmet⟩
· exact ⟨none,by simpa [stagePrototypes],fun h ↦ (hIY h).elim⟩
noncomputable def assemblyStep (e : CIndex → (Set E0 × Set E0)) (a : CIndex)
(f : Iio a → Option (Set E0)) : Option (Set E0) := by
classical
let P := ⋃ b : Iio a, stagePrototypes (f b)
exact if h : R.GoodPrototypes P then
Classical.choose (R.exists_stage hK P h
(finite_union_small (cindex_predecessors a) _ (fun b ↦ stagePrototypes_finite (f b))) (e a).1 (e a).2)
else none
noncomputable def assembled (e : CIndex → (Set E0 × Set E0)) : CIndex → Option (Set E0) :=
wellFounded_lt.fix (fun a f ↦ R.assemblyStep hK e a (fun b ↦ f b b.2))
lemma assembled_eq (e : CIndex → (Set E0 × Set E0)) (a : CIndex) :
R.assembled hK e a = R.assemblyStep hK e a (fun b ↦ R.assembled hK e b) :=
WellFounded.fix_eq _ _ _
lemma assemblyStep_spec (e : CIndex → (Set E0 × Set E0)) (f : CIndex → Option (Set E0)) (a : CIndex)
(h : R.GoodPrototypes (pastPrototypes f a)) :
R.GoodPrototypes (pastPrototypes f a ∪ stagePrototypes (R.assemblyStep hK e a (fun b ↦ f b))) ∧
((e a).1 ⊆ (e a).2 →
Meets (PrototypeBases (pastPrototypes f a ∪ stagePrototypes (R.assemblyStep hK e a (fun b ↦ f b))))
(e a).1 (e a).2) := by
classical
unfold pastPrototypes at h ⊢
simp only [assemblyStep,dite_eq_left h]
exact Classical.choose_spec (R.exists_stage hK _ h
(finite_union_small (cindex_predecessors a) _ (fun b ↦ stagePrototypes_finite (f b))) _ _)
lemma assembled_good_initial (e : CIndex → (Set E0 × Set E0)) (a : CIndex) :
R.GoodPrototypes (initialPrototypes (R.assembled hK e) a) := by
induction a using (wellFounded_lt (α := CIndex)).induction with
| h a ih =>
have hg := R.good_past_of_initial (R.assembled hK e) a ih
have hs := (R.assemblyStep_spec hK e (R.assembled hK e) a hg).1
change R.GoodPrototypes (pastPrototypes (R.assembled hK e) a ∪ stagePrototypes (R.assembled hK e a))
rw [R.assembled_eq hK e a]
exact hs
lemma assembled_meets (e : CIndex → (Set E0 × Set E0)) (a : CIndex)
(hIY : (e a).1 ⊆ (e a).2) :
Meets (PrototypeBases (allPrototypes (R.assembled hK e))) (e a).1 (e a).2 := by
have hg := R.good_past_of_initial (R.assembled hK e) a
(fun b _ ↦ R.assembled_good_initial hK e b)
have hs := (R.assemblyStep_spec hK e (R.assembled hK e) a hg).2 hIY
rw [← R.assembled_eq hK e a] at hs
exact meets_mono (prototypeBases_mono (initial_subset_all _ _)) hs
/-- The continuum recursion meets every interval and preserves all old classes. -/
lemma exists_complete_prototypes : ∃ P : Set Prototype,
R.GoodPrototypes P ∧ ∀ I Y, I ⊆ Y → Meets (PrototypeBases P) I Y := by
obtain ⟨e,he⟩ := exists_interval_enumeration
refine ⟨allPrototypes (R.assembled hK e),R.good_all_of_initial _ (R.assembled_good_initial hK e),?_⟩
intro I Y hIY
obtain ⟨a,ha⟩ := he (I,Y)
have hs := R.assembled_meets hK e a (by simpa [ha] using hIY)
simpa [ha] using hs
lemma exists_basisSystem : ∃ S : BasisSystem E0, ∀ B ∈ S.bases, R.Admissible B := by
obtain ⟨P,hP,hmeet⟩ := R.exists_complete_prototypes hK
obtain ⟨B,hB,_⟩ := hmeet ∅ univ (empty_subset _)
refine ⟨⟨PrototypeBases P,⟨B,hB⟩,R.bases_antichain hP,R.bases_compl_closed hP,
bases_swap,fun I Y hIY _ ↦ hmeet I Y hIY⟩,?_⟩
intro B hB
exact R.bases_admissible hK hP hB
end RankModel
end InfiniteMatroidCounterexample
end
end OAI
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