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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 | import OAI.Combinatorics.InfiniteMatroid.Zones
namespace OAI
section
open Set Filter
open scoped Topology symmDiff
namespace InfiniteMatroidCounterexample
namespace RankModel
variable (R : RankModel D) (hK : R.ideal = KIdeal)
include hK
lemma insert_prototype {ι : Type} (hι : Cardinal.mk ι < Cardinal.continuum)
(T : ι → Set E0) {A Z : Set E0} (hAZ : A ⊆ Z)
(hA : ¬ R.Upper A) (hZ : ¬ R.Lower Z)
(hTA : ∀ i, ¬ Small (T i \ A)) (hZT : ∀ i, ¬ Small (Z \ T i)) :
∃ B, A ⊆ B ∧ B ⊆ Z ∧ R.Admissible B ∧
∀ i, ¬ Small (B \ T i) ∧ ¬ Small (T i \ B) := by
classical
let G := Z \ A
let S0 (i : ι) : Set E0 := if Small (A \ T i) then G \ T i else Z \ T i
let S1 (i : ι) : Set E0 := if Small (T i \ Z) then T i ∩ G else T i \ A
have h0 (i : ι) : ¬ Small (S0 i) := by
dsimp only [S0]
split_ifs with hs
· intro hg
apply hZT i
apply small_subset (small_union hs hg)
intro x hx
change x ∈ (A \ T i) ∪ ((Z \ A) \ T i)
simp only [Set.mem_union,Set.mem_sdiff] at hx ⊢
tauto
· exact hZT i
have h1 (i : ι) : ¬ Small (S1 i) := by
dsimp only [S1]
split_ifs with hs
· intro hg
apply hTA i
apply small_subset (small_union hs hg)
intro x hx
change x ∈ (T i \ Z) ∪ (T i ∩ (Z \ A))
simp only [Set.mem_union,Set.mem_sdiff,Set.mem_inter_iff] at hx ⊢
tauto
· exact hTA i
let traces (p : ι × Bool) : Set E0 := if p.2 then S1 p.1 else S0 p.1
have ht : ∀ p, ¬ Small (traces p) := by
rintro ⟨i,b⟩; cases b
· exact h0 i
· exact h1 i
obtain ⟨t,hres⟩ := simultaneous_reservations (cardinal_prod_bool_lt hι) traces ht
have hr0 (i : ι) : ¬ Small (S0 i ∩ HCell t false) := hres (i,false) false
have hr1 (i : ι) : ¬ Small (S1 i ∩ HCell t true) := hres (i,true) true
let A' := A ∪ (HCell t false ∩ G)
let Z' := Z \ (HCell t true ∩ G)
have hAA : A ⊆ A' := Set.subset_union_left
have hZZ : Z' ⊆ Z := Set.sdiff_subset
have hAZ' : A' ⊆ Z' := by
intro x hx
refine ⟨?_,?_⟩
· rcases hx with hx | hx
· exact hAZ hx
· exact hx.2.1
· intro hh
rcases hx with hx | hx
· exact hh.2.2 hx
· exact Set.disjoint_left.mp (hCell_disjoint t) hx.1 hh.1
have haInv : KSmall0 (A ∆ A') := by
apply kSmall0_mono (hCell_kSmall0 t false)
intro x hx
change x ∈ A ∆ (A ∪ (HCell t false ∩ G)) at hx
simp only [Set.mem_symmDiff,Set.mem_union,Set.mem_inter_iff] at hx
tauto
have hzInv : KSmall0 (Z ∆ Z') := by
apply kSmall0_mono (hCell_kSmall0 t true)
intro x hx
change x ∈ Z ∆ (Z \ (HCell t true ∩ G)) at hx
simp only [Set.mem_symmDiff,Set.mem_sdiff,Set.mem_inter_iff] at hx
tauto
have haU : ¬ R.Upper A' := ((R.status_invariant hK haInv).1).not.mp hA
have hzL : ¬ R.Lower Z' := ((R.status_invariant hK hzInv).2.1).not.mp hZ
obtain ⟨B,hAB,hBZ,hB⟩ := R.exists_admissible_between hAZ' haU hzL
refine ⟨B,hAA.trans hAB,hBZ.trans hZZ,hB,fun i ↦ ⟨?_,?_⟩⟩
· by_cases hs : Small (A \ T i)
· have hr : ¬ Small ((G \ T i) ∩ HCell t false) := by simpa only [S0,ite_eq_left hs] using hr0 i
intro hbad
apply hr
apply small_subset hbad
intro x hx
exact ⟨hAB (Or.inr ⟨hx.2,hx.1.1⟩),hx.1.2⟩
· exact fun hb ↦ hs (small_subset hb (Set.sdiff_subset_sdiff_left (hAA.trans hAB)))
· by_cases hs : Small (T i \ Z)
· have hr : ¬ Small ((T i ∩ G) ∩ HCell t true) := by simpa only [S1,ite_eq_left hs] using hr1 i
intro hbad
apply hr
apply small_subset hbad
intro x hx
exact ⟨hx.1.1,fun hxB ↦ (hBZ hxB).2 ⟨hx.2,hx.1.2⟩⟩
· exact fun hb ↦ hs (small_subset hb (Set.sdiff_subset_sdiff_right (hBZ.trans hZZ)))
end RankModel
end InfiniteMatroidCounterexample
end
end OAI
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