File size: 3,117 Bytes
c270adf | 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 | 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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