| import Mathlib |
| import OAI.Combinatorics.IndependentSets.Reduction.SubchainView |
| import OAI.Combinatorics.IndependentSets.Reduction.AuxiliaryJointMeasurable |
|
|
| namespace OAI |
|
|
| namespace LargeIndependentSets |
| open MeasureTheory |
| open scoped Classical BigOperators NNReal |
| open PhaseTest ProductAveraging |
|
|
| noncomputable instance mixedTupleFintype {A B : Type*} [Fintype A] [Fintype B] (n i : ℕ) : |
| Fintype (MixedTuple A B n i) := by |
| classical |
| unfold MixedTuple |
| infer_instance |
|
|
| noncomputable def subchainBase {n : ℕ} (I : Finset (Fin (n+1))) (hI : I.Nonempty) : I := |
| ⟨I.min' hI, I.min'_mem hI⟩ |
|
|
| lemma subchainBase_le {n : ℕ} (I : Finset (Fin (n+1))) (hI : I.Nonempty) (j : I) : |
| (subchainBase I hI).val ≤ j.val := Finset.min'_le _ _ j.property |
|
|
| noncomputable def viewFunction {U V L R : Type} {n : ℕ} |
| (F : ∀ q : LayerQuestion U V n, (LayerAnswer L R q → Circle) → ℝ) |
| {I : Finset (Fin (n+1))} (hI : I.Nonempty) (v : SubchainView U V L R n I) |
| {m : ℕ} (t : I → Fin (m+1)) (θ : SubchainCoord L R n I → ℝ) : ℝ := |
| F ⟨(subchainBase I hI).val, v.question (subchainBase I hI)⟩ |
| (phase (fun j => v.projection (subchainBase I hI) j (subchainBase_le I hI j)) |
| (fun j => Coefficient.value (t j)) θ) |
|
|
| theorem canonical_subchain_rules (Lips : ℝ≥0) (s : ℕ) {γ : ℝ} (hγ : 0 < γ) : |
| ∃ J : ℕ, 1 ≤ J ∧ |
| ∀ (U V A B : Type) [Fintype A] [Fintype B] (n : ℕ), |
| ∀ (F : ∀ q : LayerQuestion U V n, (LayerAnswer A B q → Circle) → ℝ), |
| (∀ q, LipschitzWith Lips (F q)) → (∀ q x, |F q x| ≤ 1) → |
| ∀ m : ℕ, 0 < m → |
| ∃ rule : ∀ (I : Finset (Fin (n+1))), I.Nonempty ∧ I.card ≤ s → |
| SubchainView U V A B n I → (I → Fin (m+1)) → Finset (SubchainCoord A B n I), |
| ∀ I hI v, |
| (∀ t, (rule I hI v t).card ≤ J ∧ |
| (∫ θ, (viewFunction F hI.1 v t θ - |
| ProductAveraging.average rotationLaw (rule I hI v t) (viewFunction F hI.1 v t) θ)^2 |
| ∂Measure.pi (fun _ : SubchainCoord A B n I => rotationLaw)) < γ^2) ∧ |
| (Finset.univ.biUnion (rule I hI v)).card ≤ J*(m+1)^s := by |
| obtain ⟨J,hJ,HJ⟩ := subchain_juntas Lips s hγ |
| refine ⟨J,hJ,?_⟩ |
| intro U V A B _ _ n F hF hb m hm |
| have H (I : Finset (Fin (n+1))) (hI : I.Nonempty ∧ I.card ≤ s) |
| (v : SubchainView U V A B n I) : |
| ∃ S : (I → Fin (m+1)) → Finset (SubchainCoord A B n I), |
| (∀ t, (S t).card ≤ J ∧ |
| (∫ θ, (viewFunction F hI.1 v t θ - |
| ProductAveraging.average rotationLaw (S t) (viewFunction F hI.1 v t) θ)^2 |
| ∂Measure.pi (fun _ : SubchainCoord A B n I => rotationLaw)) < γ^2) ∧ |
| (Finset.univ.biUnion S).card ≤ J*(m+1)^s := by |
| let b := subchainBase I hI.1 |
| exact HJ I (by simpa using hI.2) (fun j => MixedTuple A B n j.val.val) |
| (MixedTuple A B n b.val.val) (fun j => v.projection b j (subchainBase_le I hI.1 j)) |
| (F ⟨b.val,v.question b⟩) (hF ⟨b.val,v.question b⟩) (hb ⟨b.val,v.question b⟩) m hm |
| choose rule hrule using H |
| exact ⟨rule,hrule⟩ |
|
|
| end LargeIndependentSets |
|
|
| end OAI |
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|
