File size: 4,991 Bytes
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 | import OAI.Combinatorics.InfiniteMatroid.SparsePerturbation
import OAI.Combinatorics.InfiniteMatroid.Interpolation
namespace OAI
section
open Set Filter
open scoped Topology symmDiff
namespace InfiniteMatroidCounterexample
instance d_infinite : Infinite D :=
Infinite.of_injective (fun m : ℕ ↦ (⟨m, fun _ ↦ false⟩ : D))
(fun _ _ h ↦ congrArg Sigma.fst h)
instance e0_infinite : Infinite E0 := inferInstance
instance d_countable : Countable D := inferInstance
instance e0_countable : Countable E0 := inferInstance
noncomputable def labelEquiv : E0 ≃ ℕ := by
letI := Classical.choice (nonempty_denumerable E0)
exact Denumerable.eqv E0
noncomputable def natProbe (T : Set E0) (n : ℕ) (A : Set ℕ) : ℝ :=
fProbe T n (labelEquiv ⁻¹' A)
lemma natProbe_cylinder {T I Y : Set E0} (hI : SameClass T I) (hY : SameClass T Y) :
CylinderContinuous (natProbe T) (labelEquiv.symm ⁻¹' I) (labelEquiv.symm ⁻¹' Y) := by
classical
intro j ε hε
obtain ⟨N, _, hN⟩ := fProbe_block_continuity hI hY j hε
refine ⟨(blockPrefix N).sup labelEquiv + 1, ?_⟩
intro X Z hIX hXY hIZ hZY hagree n hnj
apply hN (labelEquiv ⁻¹' X) (labelEquiv ⁻¹' Z)
· intro x hx
exact hIX (by simpa using hx)
· intro x hx
simpa using hXY hx
· intro x hx
exact hIZ (by simpa using hx)
· intro x hx
simpa using hZY hx
· intro x hx
apply hagree (labelEquiv x)
have hle := Finset.le_sup (f := labelEquiv) ((mem_prefix x N).mpr hx)
omega
· exact hnj.le
lemma natProbe_shift (T : Set E0) (I Y : Set ℕ) :
IntervalFiniteShift (natProbe T) I Y := by
intro A B _ hAB _ hfin
have hf : (labelEquiv ⁻¹' B \ labelEquiv ⁻¹' A).Finite := by
rw [← Set.preimage_sdiff]
exact Set.Finite.preimage labelEquiv.injective.injOn hfin
have he := finite_change_eventually (T := T) (Set.preimage_mono hAB) hf
have hc : (labelEquiv ⁻¹' B \ labelEquiv ⁻¹' A).ncard = (B \ A).ncard := by
rw [← Set.preimage_sdiff]
exact Set.ncard_preimage_of_injective_subset_range labelEquiv.injective
(by rw [labelEquiv.surjective.range_eq]; exact subset_univ _)
rw [hc] at he
simpa only [probeLiminf, natProbe, EReal.coe_natCast] using probeLiminf_finite_change he
lemma fProbe_interpolation {T I Y : Set E0} (hI : SameClass T I) (hY : SameClass T Y)
(hIY : I ⊆ Y) (hlow : probeLiminf (fProbe T) I ≤ -1)
(hhigh : 0 < probeLiminf (fProbe T) Y) :
∃ B, I ⊆ B ∧ B ⊆ Y ∧ SameClass T B ∧ LowerBasis (fProbe T) B := by
have hpre (X : Set E0) : labelEquiv ⁻¹' (labelEquiv.symm ⁻¹' X) = X := by
ext x
simp
have hlim (X : Set E0) : probeLiminf (natProbe T) (labelEquiv.symm ⁻¹' X) =
probeLiminf (fProbe T) X := by
simp only [probeLiminf, natProbe, hpre]
obtain ⟨B, hIB, hBY, hB⟩ := local_interpolation (Set.preimage_mono (f := labelEquiv.symm) hIY)
(natProbe_cylinder hI hY)
(fun n A B _ hAB _ ↦ fProbe_mono T (Set.preimage_mono hAB) n)
(fun n A B _ _ _ h ↦ by
obtain ⟨e, he⟩ := h
calc
natProbe T n B ≤ fProbe T n (insert (labelEquiv.symm e) (labelEquiv ⁻¹' A)) := by
apply fProbe_mono
intro x hx
rcases he hx with heq | hxA
· left
exact labelEquiv.injective (by simpa using heq)
· exact Or.inr hxA
_ ≤ natProbe T n A + 1 := fProbe_insert_le _ _ _ _)
(natProbe_shift T _ _) (by rw [hlim]; exact hlow) (by rw [hlim]; exact hhigh)
have hi : I ⊆ labelEquiv ⁻¹' B := by
intro x hx
exact hIB (by simpa using hx)
have hy : labelEquiv ⁻¹' B ⊆ Y := by
intro x hx
simpa using hBY hx
exact ⟨labelEquiv ⁻¹' B, hi, hy, sameClass_interval hI hY hi hy, hB⟩
def LocalLower (T : Set E0) : Set (Set E0) :=
{X | SameClass T X ∧ LowerBasis (fProbe T) X}
lemma localLower_antichain (T : Set E0) :
∀ ⦃X Y⦄, X ∈ LocalLower T → Y ∈ LocalLower T → X ⊆ Y → X = Y := by
intro X Y hX hY hXY
by_contra hne
obtain ⟨e, heY, heX⟩ := Set.not_subset.mp (fun hYX ↦ hne (Subset.antisymm hXY hYX))
have hs : probeLiminf (fProbe T) (insert e X) = probeLiminf (fProbe T) X + 1 := by
have heq : insert e X \ X = {e} := by ext x; simp [heX]
have he := finite_change_eventually (T := T) (subset_insert e X)
(show (insert e X \ X).Finite by rw [heq]; exact finite_singleton e)
simp only [heq, ncard_singleton, Nat.cast_one] at he
simpa using probeLiminf_finite_change he
have hm := probeLiminf_mono (fun n ↦ fProbe_mono T (Set.insert_subset heY hXY) n)
rw [hs] at hm
have : (0 : EReal) < probeLiminf (fProbe T) X + 1 := by
have hh := EReal.add_lt_add_right_coe hX.2.1 1
have hz : (-1 : EReal) + 1 = 0 := by
change ((-1 : ℝ) : EReal) + ((1 : ℝ) : EReal) = 0
rw [← EReal.coe_add]
norm_num
simpa only [EReal.coe_one, hz] using hh
exact (this.trans_le hm).not_ge hY.2.2
end InfiniteMatroidCounterexample
end
end OAI
|