| import OAI.Combinatorics.MatchingPSD.TableGeometry |
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| namespace OAI |
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| namespace MatchingSmoothing |
| open scoped BigOperators |
| attribute [local instance] Classical.propDecidable |
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|
| lemma simultaneous_cross_approximation (K : ℕ) : |
| ∃ d H : ℕ, 0 < d ∧ ∀ h : ℕ, H ≤ h → |
| ∀ (V G I T : Type) [Fintype V] [Fintype G] [Group G] [Fintype I] [Fintype T] |
| (_e : V ≃ Vertices h) (z : I → V → Fin 3), (∀ i c, 0 < histogram (z i) c) → |
| ∀ (M : I → T → Pairing V) |
| (lift : I → Equiv.Perm V → G) |
| (g : I → G → Pairing V → ℝ), |
| (∀ i a p, g i (a * lift i p) = fun N => g i a (Pairing.relabel p N)) → |
| ∀ B : ℝ, (∀ i a N, (g i a N)^2 ≤ B^2) → |
| ∃ a : G, ∃ P : I → (Fin 3 → ℝ) → ℝ, |
| (∀ i, P i ∈ lowDegree (fun c (v : Fin 3 → ℝ) => v c) d) ∧ |
| (∑ i, ∑ t : T, |
| (predicateAverage (fun N => crossVector (z i) N = crossVector (z i) (M i t)) (g i a) - |
| P i (crossVector (z i) (M i t)))^2) |
| ≤ (Fintype.card I : ℝ) * Fintype.card T * ((2*h : ℝ)^(2*K))⁻¹ * B^2 := by |
| classical |
| obtain ⟨d,H,hd,hmain⟩ := simultaneous_table_approximation K |
| refine ⟨d,H,hd,?_⟩ |
| intro h hh V G I T _ _ _ _ _ e z hz M lift g hcov B hB |
| let ev := Pairing.transportEquiv e.symm |
| let z' := fun i => z i ∘ e.symm |
| let M' := fun i t => Pairing.transport e (M i t) |
| let lift' := fun i (p : Equiv.Perm (Vertices h)) => lift i (e.trans (p.trans e.symm)) |
| let g' := fun i a N => g i a (ev N) |
| have hc (i : I) (a : G) (p : Equiv.Perm (Vertices h)) : |
| g' i (a * lift' i p) = fun N => g' i a (Pairing.relabel p N) := by |
| funext N |
| change g i (a * lift' i p) (ev N) = _ |
| rw [hcov] |
| change g i a (Pairing.relabel (e.trans (p.trans e.symm)) (Pairing.transport e.symm N)) = |
| g i a (Pairing.transport e.symm (Pairing.relabel p N)) |
| rw [Pairing.transport_relabel] |
| rfl |
| obtain ⟨a,P,hP,hbound⟩ := hmain h hh G I T z' |
| (fun i c => by |
| have he : histogram (z' i) = histogram (z i) := histogram_comp_equiv (z i) e.symm |
| rw [he] |
| exact hz i c) |
| M' lift' g' hc B (fun i a N => hB i a (ev N)) |
| refine ⟨a,P,hP,?_⟩ |
| have hvec (i : I) (t : T) : crossVector (z' i) (M' i t) = crossVector (z i) (M i t) := by |
| rw [crossVector_transport] |
| have he : z' i ∘ e = z i := by funext v; simp [z'] |
| rw [he] |
| have hav (i : I) (t : T) : |
| predicateAverage (fun N => N.table (z' i) = (M' i t).table (z' i)) (g' i a) = |
| predicateAverage (fun N => crossVector (z i) N = crossVector (z i) (M i t)) (g i a) := by |
| apply predicateAverage_equiv ev |
| intro N |
| rw [table_eq_iff_cross, hvec] |
| change _ ↔ crossVector (z i) (Pairing.transport e.symm N) = _ |
| rw [crossVector_transport] |
| have htvec (i : I) (t : T) : |
| (fun c => ((M' i t).table (z' i) (threeEdgeType (crossPair c)) : ℝ)) = |
| crossVector (z i) (M i t) := by |
| rw [← hvec] |
| funext c |
| exact (crossCoordinate_eq_table _ _ _).symm |
| simpa only [hav, htvec] using hbound |
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| end MatchingSmoothing |
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| end OAI |
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