| import OAI.Geometry.Convex.GeneralMahler.Budget.Eq23 |
|
|
| namespace OAI |
| /-! §06 Gaussian spectral averaging functional. Eq24 in normalized trace pairing. -/ |
| noncomputable section |
| open Set Filter MeasureTheory MeasureTheory.Measure Matrix Real Metric |
| open scoped Topology NNReal ENNReal RealInnerProductSpace MatrixOrder Matrix.Norms.L2Operator Interval |
| namespace GeneralMahler |
| open HMode Profile Layers |
| variable {m:ℕ} |
| lemma reg_constM (W:Mat m): regular (fun _:Rn m=>W) := |
| ⟨PolyBound.const _,aestronglyMeasurable_const⟩ |
| lemma int_Pj (W:Mat m) {f g:Rn m→Mat m} (hf:regular f) (hg:regular g) : |
| Integrable (fun x=>Pj W (f x) (g x)) (normal m) := by |
| simp_rw [← pJ_eq]; apply ProjField.proj_int _ hf hg |
| |
| namespace ProjField |
| variable [NeZero m] (q:ProjField m) (W:Mat m) |
| def sumS : Mat m := ∑ i:Fin m,q.M i*q.M i |
| def locR (l:Fin m) (f:ℝ→ℝ) (x:Rn m) := |
| Pj W (q.M l) (q.DLMat f x l) - |
| Pj W (q.M l*q.M l) (q.FL.eval f x) |
| def RM (f:ℝ→ℝ) := |
| ∑ l:Fin m,∫ x,q.locR W l f x ∂normal m |
|
|
| lemma DRi (l:Fin m) {f:ℝ→ℝ} (hf:TestF f) : |
| Integrable (q.locR W l f) (normal m) := |
| (int_Pj _ (reg_constM _) (q.LD_reg l (pl_test hf))).sub |
| (int_Pj _ (reg_constM _) (q.FL.eval_reg hf)) |
|
|
| lemma dl_again (i:Fin m) {f:ℝ→ℝ} (hf:TestF f) (x:Rn m) : |
| q.DLMat (deriv f) x i=fderiv ℝ (q.FL.eval f) x (e i) := by |
| unfold DLMat |
| apply (q.FL.Fr x).loc_inj |
| ext j k |
| rw [q.mder_frame _ _ _ hf x,q.mder_frame _ _ _ (pl_test hf.der) x,pl_d hf.der] |
|
|
| lemma xf_reg (l:Fin m) (f:ℝ→ℝ) (hf:TestF f) : |
| regular (fun x:Rn m=> x l • q.FL.eval f x) := |
| (q.FL.eval_reg hf).smul |
| (show regular (coord l) from ⟨PolyBound.clm _,(coord l).continuous.aestronglyMeasurable⟩) |
| lemma dl_identity (i:Fin m) {f:ℝ→ℝ} (hf:TestF f) : |
| (∫ x,q.DLMat (deriv f) x i ∂normal m)= ∫ x:Rn m,x i • q.FL.eval f x ∂normal m := by |
| let g := q.FL.eval f |
| let h := fun (x:Rn m)=> fderiv ℝ g x (e i) |
| have hg : regular g := q.FL.eval_reg hf |
| have he : Differentiable ℝ g := fun x=> (q.hasdl x hf).differentiableAt |
| have hh : regular h := q.LD_reg i hf |
| have ht := coef_raise hg.p (PLip.fd he (q.dPoly hf)) he.continuous |
| (ae_of_all _ fun x=> (he x).hasFDerivAt) i hh.p hh.meas 0 |
| simp_rw [q.dl_again i hf] |
| unfold cof at ht |
| simpa only [Mzero, Mi1, one_smul, Pi.zero_apply,Nat.zero_add,sn,Nat.cast_one,Real.sqrt_one] using ht |
|
|
| lemma coord_pb (l:Fin m) {f:ℝ→ℝ} (hf:TestF f) : |
| (∫ x,Pj W (q.M l) (q.DLMat (deriv f) x l) ∂normal m) = |
| ∫ x:Rn m,Pj W (q.M l) (x l • q.FL.eval f x) ∂normal m := by |
| simp_rw [← pJ_eq] |
| let g := PJ W (q.M l) |
| let D := fun x=>q.DLMat (deriv f) x l |
| have hi : regular D := q.LD_reg l (pl_test hf.der) |
| change (∫ x,g (D x) ∂normal m)= ∫ x:Rn m,g _ ∂normal m |
| rw [g.integral_comp_comm hi.ig,g.integral_comp_comm (q.xf_reg l f hf).ig, |
| show (∫ x, D x ∂normal m) =_ from q.dl_identity l hf] |
|
|
| -- the evaluation weighted by W |
| lemma L_identity {f:ℝ→ℝ} (hf:TestF f) : |
| etw W (fun x:Rn m=> q.Lmat x*q.FL.eval f x) = |
| ∑ i:Fin m, (∫ x,Pj W (q.M i) (q.DLMat (deriv f) x i) ∂normal m) := by |
| have he (x:Rn m) : trN (W*(q.Lmat x*q.FL.eval f x)) = |
| ∑ i:Fin m, Pj W (q.M i) (x i • q.FL.eval f x) := by |
| let A := q.FL.eval f x |
| have hh : jprod (q.Lmat x) A=q.Lmat x*A := by |
| have hx := (q.FL.ccomm x f (fun z=>z)).eq |
| rw [q.evId] at hx |
| change A*q.Lmat x=_ at hx; unfold jprod |
| rw [hx]; module |
| rw [show trN (W*(q.Lmat x*q.FL.eval f x))=Pj W (q.Lmat x) A from |
| (congrArg (fun z=>trN (W*z)) hh).symm] |
| unfold Lmat; simp_rw [← pJ_eq] |
| rw [_root_.map_sum,_root_.sum_apply] |
| congr 1 |
| ext i |
| rw [pJ_eq,pJ_eq,pJ_smul_left,pJ_smul_right] |
| unfold etw et; simp_rw [he,q.coord_pb W _ hf] |
| rw [integral_finsetSum] |
| intro i _; exact int_Pj W (reg_constM _) (q.xf_reg i f hf) |
| |
| lemma RM_Form {g:ℝ→ℝ} (hg:TestF g) : |
| q.RM W (deriv g)= etw W (fun x=> q.Lmat x*q.FL.eval g x)- |
| Pj W q.sumS (q.expL (deriv g)) := by |
| unfold RM sumS |
| simp_rw [q.L_identity W hg,locR, integral_sub (int_Pj _ |
| (reg_constM _) (show regular (fun x=>q.DLMat (deriv g) x _) from q.LD_reg _ (pl_test hg.der))) |
| (int_Pj _ (reg_constM _) (q.FL.eval_reg hg.der))] |
| rw [Finset.sum_sub_distrib] |
| congr 1 |
| simp_rw [← pJ_eq] |
| rw [_root_.map_sum,_root_.sum_apply] |
| apply Finset.sum_congr rfl |
| intro i _ |
| exact (PJ W _).integral_comp_comm (q.FL.eval_reg hg.der).ig |
|
|
| lemma W_int {f:ℝ→ℝ} (hf:TestF f) : |
| etw W (q.FL.eval f)=trN (W*q.expL f) := by |
| unfold etw et |
| simp_rw [← pj_I, pj_sym, ← pJ_eq] |
| exact (PJ W _).integral_comp_comm (q.FL.eval_reg hf).ig |
|
|
| lemma eq24 {f:ℝ→ℝ} (hf:TestF f) : |
| LDel q.FL W (N f)=Pj W (q.sumS-1) (q.expL (deriv (deriv f))) + |
| q.RM W (deriv (deriv f)) := by |
| have hh : ga (N f)=0 := by |
| have he := Nr_ga hf |
| change ga (fun x=>f x+N f x)=ga f at he |
| rw [ga_add hf (N_test hf)] at he |
| linarith |
| rw [show LDel q.FL W _= etw W (q.FL.eval (N f)) from by |
| have hi := L_formula q.FL W (N_test hf) |
| rw [hh,mul_zero,add_zero] at hi; exact hi.symm] |
| have he : etw W (q.FL.eval (N f))= |
| etw W (fun x=>q.Lmat x*q.FL.eval (deriv f) x) - |
| trN (W*q.expL (deriv (deriv f))) := by |
| rw [← q.W_int W hf.der.der] |
| let h := fun x:ℝ=> x*deriv f x |
| have H := TestF.id.mul hf.der |
| have he (x) : q.FL.eval (N f) x=q.FL.eval h x-q.FL.eval (deriv (deriv f)) x := |
| q.FL.eval_sub H hf.der.der _ |
| have hx (x) : q.Lmat x*q.FL.eval (deriv f) x=q.FL.eval h x := by |
| rw [show q.FL.eval h x=_ from q.FL.eval_mul TestF.id hf.der _,q.evId] |
| unfold etw et; simp_rw [he,hx,mul_sub,trN_sub] |
| exact integral_sub (etwReg _ (q.FL.eval_reg H)) (etwReg _ (q.FL.eval_reg hf.der.der)) |
| rw [he,q.RM_Form W hf.der] |
| let A := q.expL (deriv (deriv f)) |
| have h : Pj W (q.sumS-1) A=Pj W q.sumS A-trN (W*A) := by |
| simp_rw [← pj_I,pj_sym W A 1, ← pJ_eq,_root_.map_sub,_root_.sub_apply] |
| rw [show Pj W (q.sumS-1) _=_ from h] |
| unfold A; ring |
| end ProjField |
| end GeneralMahler |
| |
| end |
| |
| end OAI |
| |
