| import Mathlib |
| import OAI.Probability.SKGap.Localization.LocalCoefficient |
| import OAI.Probability.SKGap.Localization.Operation |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators Matrix.Norms.Frobenius |
| namespace SKGapCutoff.Recipe |
| open Primary Matrix |
| variable {n : ℕ} {ι κ : Type*} [Fintype ι] [Fintype κ] |
|
|
| lemma row_vectorNorm_le_opNorm (D : Interaction n) (i : Fin n) : |
| vectorNorm (D i) ≤ SKGap.opNorm D := by |
| have h0 : 0≤SKGap.opNorm D := norm_nonneg _ |
| apply nonneg_le_nonneg_of_sq_le_sq h0 |
| simp only [←sq,vectorNorm_sq] |
| exact SKGap.SourceError.matrix_row_square_le D i |
|
|
| lemma local_argument_row_bound (H : ι→VectorFields n) (θ : κ→Spin n→ℝ) |
| (x : Spin n) (i : Fin n) : |
| vectorNorm (fun k=>‖flipHalfDiff k (fun y=>localArgs H θ y i) x‖) ≤ |
| (∑l,SKGap.opNorm (derivativeMatrix (H l) x))+(∑α,‖derivativeVector (θ α) x‖) := by |
| have hQ (l : ι) : vectorNorm (fun k=>|derivativeMatrix (H l) x i k|)≤SKGap.opNorm (derivativeMatrix (H l) x) := by |
| rw [vectorNorm_abs]; exact row_vectorNorm_le_opNorm _ _ |
| have hT (α : κ) : vectorNorm (fun k=>|halfDiff k (θ α) x|)=‖derivativeVector (θ α) x‖ := by |
| rw [vectorNorm_abs]; rfl |
| calc |
| _ ≤ vectorNorm (fun k=>(∑l,|derivativeMatrix (H l) x i k|)+(∑α,|halfDiff k (θ α) x|)) := |
| vectorNorm_mono fun k=>by |
| have hp : 0≤(∑l,|derivativeMatrix (H l) x i k|)+(∑α,|halfDiff k (θ α) x|) := |
| add_nonneg (Finset.sum_nonneg fun _ _=>abs_nonneg _) (Finset.sum_nonneg fun _ _=>abs_nonneg _) |
| simpa only [abs_of_nonneg (norm_nonneg _),abs_of_nonneg hp] using local_argument_bound H θ x i k |
| _ ≤ vectorNorm (fun k=>∑l,|derivativeMatrix (H l) x i k|)+ |
| vectorNorm (fun k=>∑α,|halfDiff k (θ α) x|) := vectorNorm_add _ _ |
| _ ≤ (∑l,vectorNorm (fun k=>|derivativeMatrix (H l) x i k|))+ |
| ∑α,vectorNorm (fun k=>|halfDiff k (θ α) x|) := by |
| apply add_le_add |
| · convert! vectorNorm_sum (fun l k=>|derivativeMatrix (H l) x i k|) using 1 |
| congr 1; ext k; simp |
| · convert! vectorNorm_sum (fun α k=>|halfDiff k (θ α) x|) using 1 |
| congr 1; ext k; simp |
| _ ≤ _ := add_le_add (Finset.sum_le_sum fun l _=>hQ l) (by simp only [hT]; rfl) |
|
|
| lemma coefficient_row_bound (H : ι→VectorFields n) (θ : κ→Spin n→ℝ) |
| (F : Fin n→Args (ι:=ι) (κ:=κ)→ℝ) |
| (F' : Fin n→Args (ι:=ι) (κ:=κ)→Args (ι:=ι) (κ:=κ)→L[ℝ]ℝ) |
| (x : Spin n) (i : Fin n) {L : ℝ} (hL : 0≤L) |
| (hF : ∀k t, t∈Set.Icc (0:ℝ) 1→HasFDerivAt (F i) |
| (F' i (localArgs H θ x i+t • (localArgs H θ (flip x k) i-localArgs H θ x i))) |
| (localArgs H θ x i+t • (localArgs H θ (flip x k) i-localArgs H θ x i))) |
| (hBound : ∀k t, t∈Set.Icc (0:ℝ) 1→ |
| ‖F' i (localArgs H θ x i+t • (localArgs H θ (flip x k) i-localArgs H θ x i))‖≤L) : |
| vectorNorm (derivativeMatrix (coefficient H θ F) x i) ≤ |
| L*((∑l,SKGap.opNorm (derivativeMatrix (H l) x))+(∑α,‖derivativeVector (θ α) x‖)) := by |
| calc |
| _ ≤ vectorNorm (fun k=>L*‖flipHalfDiff k (fun y=>localArgs H θ y i) x‖) := |
| vectorNorm_mono fun k=>by |
| rw [abs_of_nonneg (mul_nonneg hL (norm_nonneg _))] |
| exact coefficient_lipschitz k (fun y=>localArgs H θ y i) x (F i) (F' i) hL (hF k) (hBound k) |
| _ = L*vectorNorm (fun k=>‖flipHalfDiff k (fun y=>localArgs H θ y i) x‖) := by |
| convert! vectorNorm_smul L (fun k=>‖flipHalfDiff k (fun y=>localArgs H θ y i) x‖) using 1 |
| rw [abs_of_nonneg hL] |
| _ ≤ _ := mul_le_mul_of_nonneg_left (local_argument_row_bound H θ x i) hL |
|
|
| end SKGapCutoff.Recipe |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators Matrix.Norms.Frobenius |
| namespace SKGapCutoff.Recipe |
| open Primary Matrix |
| variable {n : ℕ} |
|
|
| structure SmallBound (U : VectorFields n) (x : Spin n) (C : ℝ) : Prop where |
| nonneg : 0≤C |
| size : vectorNorm (U x)≤C |
| derivative : ‖derivativeMatrix U x‖≤C |
|
|
| lemma derivativeMatrix_add (U V : VectorFields n) (x : Spin n) : |
| derivativeMatrix (fun x i=>U x i+V x i) x=derivativeMatrix U x+derivativeMatrix V x := by |
| ext i k; simp only [derivativeMatrix,halfDiff,Matrix.add_apply]; ring |
|
|
| lemma derivativeMatrix_sum {ι : Type*} [Fintype ι] (U : ι→VectorFields n) (x : Spin n) : |
| derivativeMatrix (fun x i=>∑l,U l x i) x=∑l,derivativeMatrix (U l) x := by |
| ext i k; simp only [Matrix.sum_apply,derivativeMatrix,halfDiff_sum] |
|
|
| lemma SmallBound.mono {U : VectorFields n} {x : Spin n} {C D : ℝ} |
| (h : SmallBound U x C) (hCD : C≤D) : SmallBound U x D := |
| ⟨h.nonneg.trans hCD,h.size.trans hCD,h.derivative.trans hCD⟩ |
|
|
| lemma SmallBound.seed (s : Fin n→ℝ) (x : Spin n) : |
| SmallBound (fun _=>s) x (vectorNorm s) := by |
| refine ⟨vectorNorm_nonneg _,le_rfl,?_⟩ |
| have he : derivativeMatrix (fun _=>s) x=0 := by ext i k; simp [derivativeMatrix,halfDiff] |
| rw [he,norm_zero]; exact vectorNorm_nonneg _ |
|
|
| lemma SmallBound.add {U V : VectorFields n} {x : Spin n} {C D : ℝ} |
| (hU : SmallBound U x C) (hV : SmallBound V x D) : |
| SmallBound (fun x i=>U x i+V x i) x (C+D) := by |
| refine ⟨add_nonneg hU.nonneg hV.nonneg,(vectorNorm_add _ _).trans (add_le_add hU.size hV.size),?_⟩ |
| rw [derivativeMatrix_add] |
| exact (norm_add_le _ _).trans (add_le_add hU.derivative hV.derivative) |
|
|
| lemma SmallBound.smul {U : VectorFields n} {x : Spin n} {C : ℝ} |
| (hU : SmallBound U x C) (c : ℝ) : SmallBound (fun x i=>c*U x i) x (|c| *C) := by |
| refine ⟨mul_nonneg (abs_nonneg _) hU.nonneg,?_,?_⟩ |
| · calc |
| _ = |c| *vectorNorm (U x) := vectorNorm_smul _ _ |
| _ ≤ _ := mul_le_mul_of_nonneg_left hU.size (abs_nonneg _) |
| · rw [derivativeMatrix_smul,norm_smul,Real.norm_eq_abs] |
| exact mul_le_mul_of_nonneg_left hU.derivative (abs_nonneg _) |
|
|
| lemma SmallBound.sub {U V : VectorFields n} {x : Spin n} {C D : ℝ} |
| (hU : SmallBound U x C) (hV : SmallBound V x D) : |
| SmallBound (fun x i=>U x i-V x i) x (C+D) := by |
| simpa only [abs_neg,abs_one,one_mul,neg_one_mul,sub_eq_add_neg] using hU.add (hV.smul (-1)) |
|
|
| lemma SmallBound.sum {ι : Type*} [Fintype ι] {U : ι→VectorFields n} {x : Spin n} {C : ι→ℝ} |
| (hU : ∀l,SmallBound (U l) x (C l)) : |
| SmallBound (fun x i=>∑l,U l x i) x (∑l,C l) := by |
| refine ⟨Finset.sum_nonneg fun l _=>(hU l).nonneg,?_,?_⟩ |
| · have h:= (vectorNorm_sum (fun l=>U l x)).trans (Finset.sum_le_sum fun l _=>(hU l).size) |
| convert! h using 1 |
| congr 1; ext i; simp |
| · rw [derivativeMatrix_sum] |
| exact (norm_sum_le _ _).trans (Finset.sum_le_sum fun l _=>(hU l).derivative) |
|
|
| lemma SmallBound.coefficient {U : VectorFields n} {x : Spin n} {C : ℝ} |
| (hU : SmallBound U x C) (a : VectorFields n) {A R : ℝ} |
| (hA : 0≤A) (hR : 0≤R) (ha : ∀i,|a x i|≤A) |
| (hda : ∀i,∑k,(derivativeMatrix a x i k)^2≤R^2) : |
| SmallBound (fun x i=>a x i*U x i) x (C*(A+3*R)) := by |
| refine ⟨mul_nonneg hU.nonneg (by positivity),?_,?_⟩ |
| · exact ((vectorNorm_mul _ _ hA ha).trans (mul_le_mul_of_nonneg_left hU.size hA)).trans |
| (by nlinarith [mul_nonneg hU.nonneg hR]) |
| · calc |
| _ ≤ A*‖derivativeMatrix U x‖+R*vectorNorm (U x)+2*R*‖derivativeMatrix U x‖ := |
| source_frobenius a U x hA hR ha hda |
| _ ≤ A*C+R*C+2*R*C := add_le_add (add_le_add |
| (mul_le_mul_of_nonneg_left hU.derivative hA) (mul_le_mul_of_nonneg_left hU.size hR)) |
| (mul_le_mul_of_nonneg_left hU.derivative (by positivity)) |
| _ = _ := by ring |
|
|
| lemma SmallBound.linear {U : VectorFields n} {x : Spin n} {C : ℝ} |
| (hU : SmallBound U x C) (J : Interaction n) : |
| SmallBound (fun x=>J.mulVec (U x)) x (SKGap.opNorm J*C) := by |
| refine ⟨mul_nonneg (norm_nonneg _) hU.nonneg, |
| (vectorNorm_matrix_mul _ _).trans (mul_le_mul_of_nonneg_left hU.size (norm_nonneg _)),?_⟩ |
| have he : derivativeMatrix (fun x=>J.mulVec (U x)) x=J*derivativeMatrix U x := by |
| convert! derivativeMatrix_linear J U (fun _=>0) x using 1 |
| simp only [zero_add]; rfl |
| rw [he] |
| exact (SKGap.frobenius_mul_le_opNorm _ _).trans (mul_le_mul_of_nonneg_left hU.derivative (norm_nonneg _)) |
|
|
| lemma SmallBound.primaryCorrection {U : VectorFields n} {x : Spin n} {C P : ℝ} |
| (hU : SmallBound U x C) (m : VectorFields n) (hn : 0<n) (hP : 0≤P) |
| (hm : ∀i,|m x i|≤1) (hdm : SKGap.opNorm (derivativeMatrix m x)≤P) : |
| SmallBound (fun y i=>siteMean U y*m y i) x (C*(1+3*P)) := by |
| refine ⟨mul_nonneg hU.nonneg (by positivity),?_, |
| primary_mean_frobenius U m x hn hU.nonneg hP hU.size hU.derivative hm hdm⟩ |
| exact ((primary_mean_size U m x hn hm).trans hU.size).trans (by nlinarith [mul_nonneg hU.nonneg hP]) |
|
|
| lemma SmallBound.auxiliaryCorrection {U : VectorFields n} {x : Spin n} {C : ℝ} |
| (hU : SmallBound U x C) (a : VectorFields n) {A R : ℝ} |
| (hA : 0≤A) (hR : 0≤R) (ha : ∀i,|a x i|≤A) |
| (hda : ∀i,∑k,(derivativeMatrix a x i k)^2≤R^2) : |
| SmallBound (fun y i=>siteMean a y*U y i) x (C*(A+3*R)) := by |
| apply hU.coefficient (fun x _=>siteMean a x) hA hR (fun _=>coefficient_mean_value a x hA ha) |
| intro i |
| have H:=pow_le_pow_left₀ (norm_nonneg _) (coefficient_mean_difference a x hR hda) 2 |
| simpa only [derivativeMatrix,derivativeVector,EuclideanSpace.real_norm_sq_eq,WithLp.ofLp_toLp] using H |
|
|
| end SKGapCutoff.Recipe |
|
|
| end |
| end |
|
|
| end OAI |
|
|
