| import OAI.Analysis.CircleDomains.Transfer.OrderedGridCompatibility |
|
|
| namespace OAI |
|
|
| /-! Ordered grid selection for a finite list containing both real and |
| masked period coordinates. It uses their actual closed level graphs. -/ |
|
|
| open Set Function Filter MeasureTheory |
|
|
| namespace CircleDomainRigidity.FiniteTransfer |
|
|
| def reciprocalGraphLevel (R : Set (Sphere × UnitAddCircle)) |
| (θ : ℝ) (N : ℕ) (k : Fin (N + 1)) : Set Sphere := |
| {x | (x, ((θ + (k.val : ℝ) * reciprocalGridStep N : ℝ) : UnitAddCircle)) ∈ R} |
|
|
| def OrderedGraphGridCompatibility {S : Type*} {n : ℕ} |
| (R : Fin n → Set (Sphere × UnitAddCircle)) (θ : Fin n → ℝ) (N : ℕ) |
| (A : Set Sphere) (σ : S → Sphere) : Prop := |
| ∀ x ∈ A, x ∉ range σ → ∀ j : Fin n, |
| x ∈ ⋃ k, reciprocalGraphLevel (R j) (θ j) N k → |
| ∀ I : Finset (Fin n), (∀ i ∈ I, i.val < j.val) → ∀ k : Fin n → Fin (N + 1), |
| ¬ (connectedComponentIn |
| (⋃ i ∈ I, reciprocalGraphLevel (R i) (θ i) N (k i)) x).Nontrivial |
|
|
| theorem isClosed_reciprocalGraphLevel |
| {R : Set (Sphere × UnitAddCircle)} (hR : IsClosed R) |
| (θ : ℝ) (N : ℕ) (k : Fin (N + 1)) : |
| IsClosed (reciprocalGraphLevel R θ N k) := |
| hR.preimage (continuous_id.prodMk continuous_const) |
|
|
| theorem ae_ordered_graph_grid_compatibility |
| {S : Type*} {n : ℕ} |
| (R : Fin n → Set (Sphere × UnitAddCircle)) (B : Set Sphere) (σ : S → Sphere) |
| (hnull : ∀ (j : Fin n) (d : ℕ) (labels : Fin d → Fin n), |
| (∀ i, (labels i).val < j.val) → |
| (volume.prod (Measure.pi (fun _ : Fin d => (volume : Measure UnitAddCircle)))) |
| (freeComponentFailure (B \ range σ) (R j) |
| R labels) = 0) |
| (N : ℕ) : |
| ∀ᵐ θ ∂reciprocalGridLaw n N, OrderedGraphGridCompatibility R θ N B σ := by |
| classical |
| let E (d : ℕ) (labels : Fin (d + 1) → Fin n) : |
| Set (UnitAddCircle × (Fin d → UnitAddCircle)) := |
| if h : ∀ i : Fin d, (labels i.succ).val < (labels 0).val then |
| freeComponentFailure (B \ range σ) (R (labels 0)) |
| R (fun i => labels i.succ) |
| else ∅ |
| have hE (d : ℕ) (labels : Fin (d + 1) → Fin n) (_ : Injective labels) : |
| (volume.prod (Measure.pi (fun _ : Fin d => (volume : Measure UnitAddCircle)))) |
| (E d labels) = 0 := by |
| dsimp only [E] |
| split_ifs with h |
| · exact hnull _ _ _ h |
| · exact measure_empty |
| filter_upwards [ae_reciprocal_grid_tuples_avoid_null_events N E hE] with θ hθ |
| intro x hxB hxσ j hxj I hIj k hnon |
| obtain ⟨kj, hxj⟩ := mem_iUnion.mp hxj |
| let e := I.orderEmbOfFin rfl |
| have hmem (i : Fin I.card) : e i ∈ I := I.orderEmbOfFin_mem rfl i |
| have honto (l : Fin n) (hl : l ∈ I) : ∃ i : Fin I.card, e i = l := by |
| let l' : I := ⟨l, hl⟩ |
| exact ⟨(I.orderIsoOfFin rfl).symm l', |
| congrArg Subtype.val ((I.orderIsoOfFin rfl).apply_symm_apply l')⟩ |
| let labels : Fin (I.card + 1) → Fin n := Fin.cons j e |
| have hjnot : j ∉ range e := by |
| rintro ⟨i, hi⟩ |
| have hh := hIj (e i) (hmem i) |
| rw [hi] at hh |
| exact (lt_irrefl _ hh) |
| have hlabels : Injective labels := Fin.cons_injective_of_injective hjnot e.injective |
| let offsets : Fin (I.card + 1) → ℤ := Fin.cons (kj.val : ℤ) (fun i => (k (e i)).val) |
| let t : UnitAddCircle := ((θ j + (kj.val : ℝ) * reciprocalGridStep N : ℝ) : UnitAddCircle) |
| let τ : Fin I.card → UnitAddCircle := fun i => |
| ((θ (e i) + ((k (e i)).val : ℝ) * reciprocalGridStep N : ℝ) : UnitAddCircle) |
| have hlt : ∀ i : Fin I.card, (labels i.succ).val < (labels 0).val := by |
| intro i |
| simpa only [labels, Fin.cons_succ, Fin.cons_zero] using hIj (e i) (hmem i) |
| have hav := hθ I.card labels hlabels offsets |
| have hnot : (t, τ) ∉ freeComponentFailure (B \ range σ) (R j) |
| R e := by |
| simpa only [E, dite_eq_left hlt, labels, offsets, Fin.cons_zero, Fin.cons_succ, |
| Int.cast_natCast, t, τ] using hav |
| apply hnot |
| refine ⟨(x, t, τ), ⟨⟨hxB, hxσ⟩, hxj, ?_⟩, rfl⟩ |
| change (connectedComponentIn |
| {z | (z, τ) ∈ finiteLevelUnionGraph R e} x).Nontrivial |
| have hset : {z | (z, τ) ∈ finiteLevelUnionGraph R e} = |
| ⋃ i ∈ I, reciprocalGraphLevel (R i) (θ i) N (k i) := by |
| ext z |
| simp only [finiteLevelUnionGraph, mem_iUnion, mem_preimage, mem_ofPred_eq] |
| constructor |
| · rintro ⟨i, hi⟩ |
| exact ⟨e i, hmem i, hi⟩ |
| · rintro ⟨l, hl, hz⟩ |
| obtain ⟨i, rfl⟩ := honto l hl |
| exact ⟨i, hz⟩ |
| rwa [hset] |
|
|
| end CircleDomainRigidity.FiniteTransfer |
|
|
| end OAI |
|
|
