| 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 |
| |
