| import OAI.Geometry.Convex.GeneralMahler.Threshold |
|
|
| namespace OAI |
| /-! Divided difference evaluated on two possibly different frames. -/ |
| noncomputable section |
| open Set Filter MeasureTheory MeasureTheory.Measure Matrix Real Metric |
| open scoped Topology NNReal ENNReal MatrixOrder Matrix.Norms.L2Operator RealInnerProductSpace Interval |
| namespace GeneralMahler |
| open Layers Profile |
| variable {m:ℕ} |
|
|
| -- averages on segment |
| def along (u:Plane) (t:ℝ) := (1-t)*u.1+t*u.2 |
| def bav (f:ℝ→ℝ) (u:Plane) := ∫ t in (0:ℝ)..1,f (along u t) |
| lemma bav_con {f:ℝ→ℝ} (hf:Continuous f) : Continuous (bav f) := by |
| have hm : Continuous fun p:Plane × ℝ=> f (along p.1 p.2) := by unfold along; fun_prop |
| exact intervalIntegral.continuous_parametric_intervalIntegral_of_continuous' hm 0 1 |
| lemma bav_same (f:ℝ→ℝ) (x:ℝ) : bav f (x,x)=f x := by |
| have he (t) : along (x,x) t=x := by unfold along; ring |
| unfold bav; simp_rw [he]; simp |
| lemma delta_eq {f} (hf:TestF f) (x y:ℝ) : |
| f x-f y=bav (deriv f) (x,y)*(x-y) := by |
| have he (t:ℝ) : HasDerivAt (fun t=>along (x,y) t) (y-x) t := by |
| convert ((((hasDerivAt_const t 1).sub (hasDerivAt_id' t)).mul_const x).fun_add |
| ((hasDerivAt_id' t).mul_const y)) using 1 |
| all_goals first | rfl | ring |
| have hd (t:ℝ) : HasDerivAt (fun t=> f (along (x,y) t)) (deriv f (along (x,y) t)*(y-x)) t := |
| ((hf.diff _).hasDerivAt).comp t (he t) |
| have hc : Continuous (fun t:ℝ=> deriv f (along (x,y) t)*(y-x)) := |
| ((hf.der.cont.comp (by unfold along; fun_prop)).mul continuous_const) |
| have hh := intervalIntegral.integral_eq_sub_of_hasDerivAt (fun t _=>hd t) |
| (hc.intervalIntegrable 0 1) |
| rw [intervalIntegral.integral_mul_const,show along (x,y) 1=y from by unfold along; ring, |
| show along (x,y) 0=x from by unfold along; ring] at hh |
| unfold bav; linarith |
|
|
| lemma bav_bound {f} (hf:PolyBound f) : PolyBound (bav f) := by |
| obtain ⟨C,n,hc,h⟩ := hf |
| refine ⟨C,n,hc,fun u=>?_⟩ |
| have he := intervalIntegral.norm_integral_le_of_norm_le_const (f:=fun t=>f (along u t)) |
| (C:=C*(1+‖u‖)^n) (a:=0) (b:=1) ?_ |
| · unfold bav; convert he using 1; first | rfl | simp |
| intro z hz |
| have ha : 0≤z ∧ z≤1 := by rw [uIoc_of_le zero_le_one] at hz; exact ⟨le_of_lt hz.1,hz.2⟩ |
| apply (h _).trans |
| have hb : ‖along u z‖≤‖u‖ := by |
| change |(1-z)*u.1+z*u.2| ≤ max |u.1| |u.2| |
| apply (abs_add_le ..).trans |
| rw [abs_mul,abs_mul,abs_of_nonneg ha.1,abs_of_nonneg (by linarith)] |
| calc |
| _ ≤ _ := add_le_add |
| (mul_le_mul_of_nonneg_left (le_max_left |u.1| |u.2|) (sub_nonneg.mpr ha.2)) |
| (mul_le_mul_of_nonneg_left (le_max_right |u.1| |u.2|) ha.1) |
| _ = _ := by ring |
| gcongr |
|
|
| namespace Frm |
| variable (u v w:Frm m) |
| def showM (A:Mat m) := star u.val*A*v.val |
| def hideM (A:Mat m) := u.val*A*star v.val |
| lemma reveal (A:Mat m) : u.showM v (u.hideM v A)=A := by |
| unfold showM hideM |
| simp only [mul_assoc]; rw [v.m₁,mul_one,← mul_assoc,u.m₁,one_mul] |
| lemma combine (C:Mat m) (v w:Frm m) : |
| u.showM w C=(u.showM v C)*(v.showM w 1) := by |
| unfold showM; simp only [mul_one,mul_assoc] |
| rw [show v.val*(star v.val*w.val)=w.val from by rw [← mul_assoc,v.m₂,one_mul] ] |
|
|
| def secDiag (A B:Mat m) (g:Plane→ℝ) (C:Mat m) : Mat m := |
| Matrix.of (fun i j=> g (u.eig A i,v.eig B j)*u.showM v C i j) |
| def sec (A B:Mat m) (g:Plane→ℝ) (C:Mat m) := |
| u.hideM v (u.secDiag v A B g C) |
| lemma sec_eq (A B C:Mat m) (g:Plane→ℝ) (i j) : |
| u.showM v (u.sec v A B g C) i j = g (u.eig A i,v.eig B j)*u.showM v C i j := by |
| rw [sec,reveal]; rfl |
|
|
| lemma sec_add (A B:Mat m) (g:Plane→ℝ) (C D:Mat m) : |
| u.sec v A B g (C+D)=u.sec v A B g C+u.sec v A B g D := by |
| have he : u.secDiag v A B g (C+D) = u.secDiag v A B g C+u.secDiag v A B g D := by |
| ext i j; simp [secDiag,showM,mul_add,add_mul] |
| unfold sec; rw [he]; simp only [hideM,mul_add,add_mul] |
| lemma sec_smul (A B:Mat m) (g:Plane→ℝ) (a:ℝ) (C:Mat m) : |
| u.sec v A B g (a•C)=a•u.sec v A B g C := by |
| unfold sec |
| have he : u.secDiag v A B g (a•C)=a•u.secDiag v A B g C := by |
| ext i j; simp [secDiag,showM,mul_left_comm] |
| rw [he] |
| simp only [hideM,smul_mul_assoc,mul_smul_comm] |
|
|
| lemma Aentry (A:Mat m) (h:u.Dgn A) (i j) : |
| u.showM v A i j= u.eig A i * u.showM v 1 i j := by |
| unfold showM; rw [mul_one,u.fix_mul_cross v A,show u.loc A=_ from h,Matrix.diagonal_mul] |
| lemma Bentry (A:Mat m) (h:v.Dgn A) (i j) : |
| u.showM v A i j=u.showM v 1 i j*v.eig A j := by |
| unfold showM; rw [mul_one,u.fix_mul_cross' v A,show v.loc A=_ from h,Matrix.mul_diagonal] |
| lemma Aeval (A:Mat m) (h:A.IsHermitian) (hh:u.Dgn A) (f:ℝ→ℝ) (i j): |
| u.showM v (cfc f A) i j=f (u.eig A i)*u.showM v 1 i j := by |
| unfold showM; rw [mul_one,u.fix_mul_cross v,u.loc_cfc h hh f,Matrix.diagonal_mul] |
| lemma Beval (A:Mat m) (h:A.IsHermitian) (hh:v.Dgn A) (f:ℝ→ℝ) (i j): |
| u.showM v (cfc f A) i j=u.showM v 1 i j*f (v.eig A j) := by |
| unfold showM; rw [mul_one,u.fix_mul_cross' v,v.loc_cfc h hh f,Matrix.mul_diagonal] |
| lemma show_sub (A B:Mat m) : u.showM v (A-B)=u.showM v A-u.showM v B := by |
| simp only [showM,mul_sub,sub_mul] |
|
|
| lemma exact_sec {A B:Mat m} (ha:A.IsHermitian) (hb:B.IsHermitian) |
| (hu:u.Dgn A) (hv:v.Dgn B) {f:ℝ→ℝ} (hf:TestF f) : |
| cfc f A-cfc f B=u.sec v A B (bav (deriv f)) (A-B) := by |
| apply u.cross_cancel v |
| change u.showM v _ = u.showM v _ |
| ext i j |
| rw [sec_eq,u.show_sub,u.show_sub]; simp only [Matrix.sub_apply] |
| rw [u.Aentry v A hu,u.Bentry v B hv,u.Aeval v A ha hu,u.Beval v B hb hv] |
| have hh := delta_eq hf (u.eig A i) (v.eig B j) |
| linear_combination u.showM v 1 i j*hh |
|
|
| -- Right frame independence for all test kernels on entries |
| lemma sec_ind (A B C:Mat m) (g:Plane→ℝ) (hv:v.Dgn B) (hw:w.Dgn B) : |
| u.sec v A B g C=u.sec w A B g C := by |
| apply u.cross_cancel w |
| change u.showM w _= u.showM w _ |
| ext i j |
| rw [sec_eq,combine u (u.sec v A B g C) v w, combine u C v w] |
| simp_rw [Matrix.mul_apply, Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro k _ |
| rw [sec_eq] |
| by_cases h : v.showM w 1 k j=0 |
| · rw [h]; ring |
| have he : v.eig B k=w.eig B j := by |
| have he := (v.Aentry w B hv k j).symm.trans (v.Bentry w B hw k j) |
| exact mul_right_cancel₀ h (he.trans (mul_comm ..)) |
| rw [he]; ring |
| end Frm |
| end GeneralMahler |
|
|
| end |
|
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| end OAI |
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