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a97875e | 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 | 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
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