openAI-math-dataset / lean /OAI /Computability /UniqueGames /Inverse /KMSAnalyticCharacterInduction.lean
| import Mathlib.FieldTheory.Finiteness | |
| import OAI.Computability.UniqueGames.Inverse.KMSAffineRestrictionComplementLemmas | |
| import OAI.Computability.UniqueGames.Inverse.KMSAnalyticBlockFrequenciesLemmas | |
| import OAI.Computability.UniqueGames.Inverse.KMSFourthMomentMixedBaseLemmas | |
| namespace OAI | |
| /-! The full fixed-character induction for the actual KMS small component. | |
| The only local hypothesis bounds squared norms of homogeneous restrictions; | |
| all Fourier recursions, multiplicities and coordinate changes are proved. | |
| -/ | |
| namespace UniqueGamesTheorem.Inverse.KMSAnalytic | |
| noncomputable section | |
| open scoped BigOperators Classical | |
| open UniqueGamesTheorem.Integration.BinaryLinear (F2) | |
| open UniqueGamesTheorem.Inverse.KMSBasisInvariant | |
| universe u v w x | |
| local instance mapFintype {V : Type*} {W : Type*} | |
| [AddCommGroup V] [Module F2 V] [AddCommGroup W] [Module F2 W] | |
| [Fintype V] [Fintype W] : Fintype (V →ₗ[F2] W) := | |
| Fintype.ofInjective (fun L : V →ₗ[F2] W => (L : V → W)) DFunLike.coe_injective | |
| variable {E : Type u} [AddCommGroup E] [Module F2 E] | |
| [FiniteDimensional F2 E] [Fintype E] | |
| private theorem fixedFrequencyEnergy_bound_induction : | |
| ∀ (n : ℕ) (F : Type v) [AddCommGroup F] [Module F2 F] | |
| [FiniteDimensional F2 F] [Fintype F] | |
| (I : Type w) [AddCommGroup I] [Module F2 I] | |
| [FiniteDimensional F2 I] [Fintype I] | |
| (J : Type x) [AddCommGroup J] [Module F2 J] | |
| [FiniteDimensional F2 J] [Fintype J], | |
| Module.finrank F2 J = n → | |
| ∀ (ι : I →ₗ[F2] E), Function.Injective ι → | |
| ∀ (f : (E →ₗ[F2] F) → ℝ), IsBasisInvariant f → | |
| ∀ (ε : ℝ), HomogeneousRestrictionBound (Module.finrank F2 I) ε f → | |
| ∀ (π : I →ₗ[F2] J), Function.Surjective π → ∀ A : F →ₗ[F2] J, | |
| (2 : ℝ) ^ ((Module.finrank F2 I + Module.finrank F2 J) * Module.finrank F2 E) * | |
| fixedFrequencyEnergy ι f π A ≤ | |
| (2 : ℝ) ^ (4 * Module.finrank F2 I * Module.finrank F2 I) * ε := by | |
| intro n | |
| induction n using Nat.strong_induction_on with | |
| | h n ih => | |
| intro F _ _ _ _ I _ _ _ _ J _ _ _ _ hJn ι hι f hf ε hg π hπ A | |
| let : Finite (KMSKernelFiberCard.KernelClass F I) := | |
| Finite.of_surjective (KMSKernelFiberCard.fullKernel (F := F) (I := I)) | |
| (KMSKernelFiberCard.fullKernel_surjective (F := F) (I := I)) | |
| let : Fintype (KMSKernelFiberCard.KernelClass F I) := Fintype.ofFinite _ | |
| have hε : 0 ≤ ε := (Finset.expect_nonneg (fun _ _ => sq_nonneg _)).trans | |
| (HomogeneousRestrictionBound.energy_le _ ε f hg) | |
| by_cases hn : n = 0 | |
| · exact fixedFrequencyEnergy_zero_dim_bound ι hι f hf ε hg π A (hJn.trans hn) | |
| by_cases hA : Function.Surjective A | |
| · have hp : 0 < Module.finrank F2 J := by omega | |
| let s := KMSAnalyticCompatibleSplitting.chooseSplit π A hπ hA hp | |
| let : Fintype (s.B →ₗ[F2] s.J) := mapFintype | |
| let ι' : (s.J × F2) →ₗ[F2] E := ι.comp s.leftEquiv.symm.toLinearMap | |
| let f' : (E →ₗ[F2] (s.B × F2)) → ℝ := codomainTransport s.rightEquiv f | |
| have hι' : Function.Injective ι' := hι.comp s.leftEquiv.symm.injective | |
| have hf' : IsBasisInvariant f' := basisInvariant_codomainTransport s.rightEquiv f hf | |
| have hg' : HomogeneousRestrictionBound (Module.finrank F2 s.J + 1) ε f' := by | |
| have h := HomogeneousRestrictionBound.codomain_equiv _ ε f hg s.rightEquiv | |
| simpa only [s.left_finrank] using h | |
| have hs : Module.finrank F2 s.K0 < n := by | |
| have hdim : Module.finrank F2 J = Module.finrank F2 s.K0 + 1 := s.target_finrank | |
| omega | |
| have hgfirst : HomogeneousRestrictionBound (Module.finrank F2 s.J) ε | |
| (fun X : E →ₗ[F2] s.B => f' ((LinearMap.inl F2 s.B F2).comp X)) := | |
| HomogeneousRestrictionBound.codomain_pullback _ 1 ε f' hg' | |
| (LinearMap.inl F2 s.B F2) (by intro x y h; exact congrArg Prod.fst h) | |
| (by simp only [Module.finrank_prod, Module.finrank_self]; omega) | |
| have hglower : HomogeneousRestrictionBound (Module.finrank F2 s.J) ε f' := | |
| HomogeneousRestrictionBound.mono (Nat.le_succ _) ε f' hg' | |
| have hfirst := ih (Module.finrank F2 s.K0) hs s.B s.J s.K0 rfl | |
| (leftEmbedding ι') (leftEmbedding_injective ι' hι') | |
| (fun X : E →ₗ[F2] s.B => f' ((LinearMap.inl F2 s.B F2).comp X)) | |
| (basisInvariant_codomain_pullback f' hf' (LinearMap.inl F2 s.B F2)) | |
| ε hgfirst s.leftMap s.leftMap_surjective s.rightMap | |
| have hlower (w : s.J) := ih (Module.finrank F2 s.K0) hs (s.B × F2) s.J s.K0 rfl | |
| (leftEmbedding ι') (leftEmbedding_injective ι' hι') f' hf' | |
| ε hglower s.leftMap s.leftMap_surjective (hyperplaneExtend s.rightMap (s.leftMap w)) | |
| have hrec := fixedFrequencyEnergy_product_bound ι' hι' f' hf' | |
| s.leftMap s.rightMap ε hε hfirst hlower | |
| have htransport : fixedFrequencyEnergy ι' f' | |
| (s.leftMap.prodMap (LinearMap.id : F2 →ₗ[F2] F2)) | |
| (s.rightMap.prodMap (LinearMap.id : F2 →ₗ[F2] F2)) = | |
| fixedFrequencyEnergy ι f π A := by | |
| rw [← s.left_conjugation, ← s.right_conjugation, | |
| fixedFrequencyEnergy_postcomp_injective _ _ _ _ s.targetEquiv.toLinearMap | |
| s.targetEquiv.injective] | |
| change fixedFrequencyEnergy (ι.comp s.leftEquiv.symm.toLinearMap) | |
| (codomainTransport s.rightEquiv f) (π.comp s.leftEquiv.symm.toLinearMap) | |
| (A.comp s.rightEquiv.symm.toLinearMap) = _ | |
| rw [fixedFrequencyEnergy_codomainTransport, fixedFrequencyEnergy_transport] | |
| rw [htransport, ← s.leftEquiv.finrank_eq, ← s.targetEquiv.finrank_eq] at hrec | |
| exact hrec | |
| · rw [fixedFrequencyEnergy_eq_zero_of_not_surjective ι f π hπ A hA, mul_zero] | |
| positivity | |
| /-- KMS's fixed-character estimate, with actual homogeneous restriction | |
| control. The injection is the only ambient-dimension assumption. -/ | |
| theorem fixedFrequencyEnergy_le_of_homogeneous | |
| {F : Type v} [AddCommGroup F] [Module F2 F] | |
| [FiniteDimensional F2 F] [Fintype F] | |
| {I : Type w} [AddCommGroup I] [Module F2 I] | |
| [FiniteDimensional F2 I] [Fintype I] | |
| {J : Type x} [AddCommGroup J] [Module F2 J] | |
| [FiniteDimensional F2 J] [Fintype J] | |
| (ι : I →ₗ[F2] E) (hι : Function.Injective ι) | |
| (f : (E →ₗ[F2] F) → ℝ) (hf : IsBasisInvariant f) | |
| (ε : ℝ) (hg : HomogeneousRestrictionBound (Module.finrank F2 I) ε f) | |
| (π : I →ₗ[F2] J) (hπ : Function.Surjective π) (A : F →ₗ[F2] J) : | |
| (2 : ℝ) ^ ((Module.finrank F2 I + Module.finrank F2 J) * Module.finrank F2 E) * | |
| fixedFrequencyEnergy ι f π A ≤ | |
| (2 : ℝ) ^ (4 * Module.finrank F2 I * Module.finrank F2 I) * ε := | |
| fixedFrequencyEnergy_bound_induction (Module.finrank F2 J) F I J rfl ι hι f hf ε hg π hπ A | |
| end | |
| end UniqueGamesTheorem.Inverse.KMSAnalytic | |
| end OAI | |