openAI-math-dataset / lean /OAI /GroupTheory /SimpleAmenable /RandomFields /PolygonApproxIdentity.lean
| import Mathlib | |
| import OAI.GroupTheory.SimpleAmenable.RandomFields.SignalGoodRows | |
| namespace OAI | |
| section | |
| section | |
| open scoped symmDiff | |
| namespace SimpleAmenable | |
| open scoped commutatorElement | |
| open scoped commutatorElement | |
| section PolygonApproxIdentity | |
| open Classical | |
| theorem polygon_signal_approx_identity {a m D : ℕ} {v : ℝ×ℝ} (hD : 0<D) | |
| (g : polygonFullGroup a m) (ψ χ : (ℝ×ℝ) → ℝ) | |
| (hψ : ContDiff ℝ 2 ψ) (hsψ : HasCompactSupport ψ) (q K L : ℝ) | |
| (hL : 0<L) (hKL : K+2 ≤ L) | |
| (hcompact : ∀x,K ≤ ‖x‖ → ψ x=0) | |
| (hχ : ∀x,χ x∈Set.Icc (0:ℝ) 1) | |
| (hχ₁ : ∀x,‖x‖ ≤ K+2 → χ x=1) | |
| (hχc : ∀x,L ≤ ‖x‖ → χ x=0) : | |
| ∃R Q Cv Ca : ℝ,0 ≤ R ∧ 0 ≤ Q ∧ 0 ≤ Cv ∧ 0 ≤ Ca ∧ | |
| ∀(N r η κ : ℝ) (hN : 1 ≤ N),Q ≤ N → 1 ≤ r → r ≤ N → | |
| 0<η → η ≤ 1 → 0<κ → 36 ≤ r*η → 1 ≤ 2*κ/(r/N) → R ≤ κ/(r/N) → | |
| let J := flagBoxFinset (a:=a) (m:=m) (v:=v) hD | |
| (mul_pos (lt_of_lt_of_le zero_lt_one hN) hL) | |
| let h := flagSignalDifference (v:=v) hD g (fun x => q+ψ x) N | |
| let e := fun z => (∑w∈J,flagAveragingMatrix χ ((D:ℝ)^4/5) N r η κ z w*h w)-h z | |
| let δ := 38880/((D:ℝ)*Real.sqrt (r*η)) | |
| (∀z,z∉J → e z=0) ∧ | |
| ∑z∈J,(e z)^2 ≤ flagVolumeConstant m D L*(Cv*η*(1+δ)+Ca*δ)^2+ | |
| flagExceptionalConstant a m D L*(κ+r/N)*(flagRowSumConstant D*Ca)^2 := by | |
| obtain ⟨R,Q₁,Cv,Ca,hR,hQ₁,hCv,hCa,hamp,hgood⟩ := | |
| flagSignal_good_row_constants (v:=v) hD g ψ χ hψ hsψ q K hcompact hχ₁ | |
| obtain ⟨Q₂,hQ₂,hsupp⟩ := flagSignal_box_support (v:=v) hD g ψ q hL (by linarith) hcompact | |
| refine ⟨R,max Q₁ Q₂,Cv,Ca,hR,(hQ₁.trans (le_max_left _ _)),hCv,hCa, | |
| fun N r η κ hN hQN hr hrN hη hη₁ hκ hscale hline hRf => ?_⟩ | |
| have hN' : 0<N := lt_of_lt_of_le zero_lt_one hN | |
| have hr' : 0<r := lt_of_lt_of_le zero_lt_one hr | |
| have hd : (0:ℝ)<D := by exact_mod_cast hD | |
| have hρ : 0<(D:ℝ)^4/5 := by positivity | |
| let J := flagBoxFinset (a:=a) (m:=m) (v:=v) hD (mul_pos hN' hL) | |
| let h := flagSignalDifference (v:=v) hD g (fun x => q+ψ x) N | |
| let e := fun z => (∑w∈J,flagAveragingMatrix χ ((D:ℝ)^4/5) N r η κ z w*h w)-h z | |
| let δ := 38880/((D:ℝ)*Real.sqrt (r*η)) | |
| change (∀z,z∉J → e z=0) ∧ _ | |
| have hprod (z w : FlagSite a m D v) (hw : w∉J) : | |
| flagAveragingMatrix χ ((D:ℝ)^4/5) N r η κ z w*h w=0 := by | |
| rw [flagAveragingMatrix_box_support hD χ hL hN' hχc _ _ _ _ z w (Or.inr hw),zero_mul] | |
| refine ⟨?_,?_⟩ | |
| · intro z hz | |
| have hh := hsupp N hN' ((le_max_right _ _).trans hQN) z hz | |
| dsimp [e,h] | |
| rw [hh,sub_zero] | |
| apply Finset.sum_eq_zero | |
| intro w hw | |
| rw [flagAveragingMatrix_box_support hD χ hL hN' hχc _ _ _ _ z w (Or.inl hz),zero_mul] | |
| · apply flag_good_bad_energy hD hL hκ (div_pos hr' hN') ((div_le_one hN').mpr hrN) hN | |
| (by rwa [div_mul_cancel₀ _ hN'.ne']) | |
| (show 0 ≤ Cv*η*(1+δ)+Ca*δ by dsimp [δ]; positivity) | |
| (show 0 ≤ flagRowSumConstant D*Ca by dsimp [flagRowSumConstant]; positivity) J | |
| (fun z hz => by simpa only [mul_comm N L] using (mem_flagBoxFinset _ _ _).mp hz) e | |
| · intro z hz hzg | |
| have hh := hgood N r η κ hN' ((le_max_left _ _).trans hQN) hr' hη hη₁ hκ | |
| hscale hline hRf z J hzg (hprod z) | |
| convert hh using 1 | |
| dsimp [e,h,δ] | |
| ring | |
| · intro z hz | |
| have hh := flagAveragingMatrix_crude_error hD χ hχ h hρ hN' hr' hη hκ | |
| (show 1 ≤ r*η by linarith) (div_nonneg hCa hN'.le) (hamp N hN') z J | |
| convert hh using 1 | |
| dsimp [e,flagRowSumConstant] | |
| ring | |
| end PolygonApproxIdentity | |
| section FiniteResolvent | |
| open Classical Matrix | |
| noncomputable def finiteL2Norm {ι : Type*} [Fintype ι] (x : ι → ℝ) : ℝ := | |
| ‖WithLp.toLp 2 x‖ | |
| theorem finiteL2Norm_nonneg {ι : Type*} [Fintype ι] (x : ι → ℝ) : 0 ≤ finiteL2Norm x := norm_nonneg _ | |
| theorem finiteL2Norm_sq {ι : Type*} [Fintype ι] (x : ι → ℝ) : | |
| finiteL2Norm x^2=∑i,x i^2 := by | |
| simp [finiteL2Norm,PiLp.norm_sq_eq_of_L2,Real.norm_eq_abs,sq_abs] | |
| theorem finiteL2Norm_add {ι : Type*} [Fintype ι] (x y : ι → ℝ) : | |
| finiteL2Norm (x+y) ≤ finiteL2Norm x+finiteL2Norm y := by | |
| simpa only [finiteL2Norm,WithLp.toLp_add] using norm_add_le (WithLp.toLp 2 x) (WithLp.toLp 2 y) | |
| theorem finiteL2Norm_sub {ι : Type*} [Fintype ι] (x y : ι → ℝ) : | |
| finiteL2Norm (x-y) ≤ finiteL2Norm x+finiteL2Norm y := by | |
| simpa only [finiteL2Norm,WithLp.toLp_sub] using norm_sub_le (WithLp.toLp 2 x) (WithLp.toLp 2 y) | |
| theorem finiteL2Norm_smul {ι : Type*} [Fintype ι] (c : ℝ) (x : ι → ℝ) : | |
| finiteL2Norm (c • x)=|c| * finiteL2Norm x := by | |
| simp only [finiteL2Norm,WithLp.toLp_smul,norm_smul,Real.norm_eq_abs] | |
| theorem finiteL2Norm_one_add_le {ι : Type*} [Fintype ι] [DecidableEq ι] | |
| (B : Matrix ι ι ℝ) (hB : B.PosSemidef) (x : ι → ℝ) : | |
| finiteL2Norm x ≤ finiteL2Norm ((1+B)*ᵥx) := by | |
| let y := (1+B)*ᵥx | |
| let S := ∑i,x i^2 | |
| let T := ∑i,y i^2 | |
| let P := ∑i,x i*y i | |
| have hS : 0 ≤ S := Finset.sum_nonneg (fun i _ => sq_nonneg _) | |
| have hT : 0 ≤ T := Finset.sum_nonneg (fun i _ => sq_nonneg _) | |
| have hp : 0 ≤ ∑i,x i*(B*ᵥx) i := by | |
| simpa only [dotProduct,star_trivial] using hB.dotProduct_mulVec_nonneg x | |
| have he : P=S+(∑i,x i*(B*ᵥx) i) := by | |
| simp only [P,S,y,Matrix.add_mulVec,Matrix.one_mulVec,Pi.add_apply,mul_add,Finset.sum_add_distrib,pow_two] | |
| have hSP : S ≤ P := by linarith | |
| have hP : 0 ≤ P := hS.trans hSP | |
| have hcs : P^2 ≤ S*T := Finset.sum_mul_sq_le_sq_mul_sq Finset.univ x y | |
| have hST : S ≤ T := by | |
| by_cases h : S=0 | |
| · simpa only [h] using hT | |
| · have hS' : 0<S := lt_of_le_of_ne hS (Ne.symm h) | |
| have hsq := pow_le_pow_left₀ hS hSP 2 | |
| nlinarith | |
| apply (sq_le_sq₀ (finiteL2Norm_nonneg _) (finiteL2Norm_nonneg _)).mp | |
| simpa only [finiteL2Norm_sq] using hST | |
| theorem one_add_psd_isUnit_det {ι : Type*} [Fintype ι] [DecidableEq ι] | |
| (B : Matrix ι ι ℝ) (hB : B.PosSemidef) : IsUnit (1+B).det := | |
| (Matrix.isUnit_iff_isUnit_det _).mp (Matrix.PosDef.one.add_posSemidef hB).isUnit | |
| theorem finiteL2Norm_inverse_contraction {ι : Type*} [Fintype ι] [DecidableEq ι] | |
| (B : Matrix ι ι ℝ) (hB : B.PosSemidef) (x : ι → ℝ) : | |
| finiteL2Norm ((1+B)⁻¹*ᵥx) ≤ finiteL2Norm x := by | |
| have hh := finiteL2Norm_one_add_le B hB ((1+B)⁻¹*ᵥx) | |
| rwa [Matrix.mulVec_mulVec,Matrix.mul_nonsing_inv _ (one_add_psd_isUnit_det B hB),Matrix.one_mulVec] at hh | |
| theorem finiteL2Norm_inverse_signal {ι : Type*} [Fintype ι] [DecidableEq ι] | |
| (B : Matrix ι ι ℝ) (hB : B.PosSemidef) (W : ℝ) (hW : 0 ≤ W) (h : ι → ℝ) : | |
| finiteL2Norm ((1+B)⁻¹*ᵥh) ≤ | |
| (finiteL2Norm h+finiteL2Norm (B*ᵥh-W • h))/(1+W) := by | |
| let C := (1+B)⁻¹ | |
| let e := B*ᵥh-W • h | |
| have hid : C*ᵥ((1+B)*ᵥh)=h := by | |
| rw [Matrix.mulVec_mulVec,Matrix.nonsing_inv_mul _ (one_add_psd_isUnit_det B hB),Matrix.one_mulVec] | |
| have he : (1+B)*ᵥh=(1+W) • h+e := by | |
| ext i | |
| simp only [Matrix.add_mulVec,Matrix.one_mulVec,Pi.add_apply,Pi.smul_apply,smul_eq_mul,e,Pi.sub_apply] | |
| ring | |
| rw [he,Matrix.mulVec_add,Matrix.mulVec_smul] at hid | |
| have he' : (1+W) • (C*ᵥh)=h-C*ᵥe := by | |
| exact eq_sub_iff_add_eq.mpr hid | |
| have hb := finiteL2Norm_sub h (C*ᵥe) | |
| rw [← he',finiteL2Norm_smul,abs_of_pos (show 0<1+W by linarith)] at hb | |
| have hc := finiteL2Norm_inverse_contraction B hB e | |
| apply (le_div_iff₀ (show 0<1+W by linarith)).mpr | |
| change finiteL2Norm (C*ᵥh)*(1+W) ≤ finiteL2Norm h+finiteL2Norm e | |
| nlinarith | |
| end FiniteResolvent | |
| end SimpleAmenable | |
| end | |
| end | |
| end OAI | |