| import OAI.Geometry.Convex.GeneralMahler.Segment.Disp |
|
|
| namespace OAI |
| /-! Combine chord and endpoint variation using pointwise U star. -/ |
| noncomputable section |
| open Set Filter Real MeasureTheory |
| open scoped Interval |
| namespace GeneralMahler.SCal.SE |
| open Tag Grid Profile Jet Segment |
| variable (m:ℝ){h:ℝ} |
| def alongD (m h:ℝ)(f:ℝ→ℝ):=fun x=> DotF f (loc m h x) |
| def alongN (m h:ℝ)(f:ℝ→ℝ):=fun x=> NNf f (loc m h x) |
| lemma hoD {f:ℝ→ℝ} (hf:TestF f) : |
| anti (seg m h) f=2*h*mean0 (alongD m h f) := by |
| let g:= (liftF f) ∘ (loc m h) |
| have hd(x): HasDerivAt g (h*alongD m h f x) x:=by |
| rw [mul_comm]; exact (liftD hf _).comp x (loct ..) |
| have hc : Continuous (alongD m h f) := (Dt_cont hf).comp (cLoc ..) |
| have hi:= intervalIntegral.integral_eq_sub_of_hasDerivAt (fun t _=>hd t) |
| ((continuous_const.mul hc).intervalIntegrable (-1:ℝ) 1) |
| have he:g 1=f (seg m h).2:=by simp [g,Function.comp_apply,liftF,seg,right,loc] |
| have hh:g (-1)=f (seg m h).1:=by simp [g,Function.comp_apply,liftF,seg,left,loc,sub_eq_add_neg] |
| rw [intervalIntegral.integral_const_mul,he,hh] at hi |
| unfold mean0 anti; linarith |
|
|
| def GammaB (f g:ℝ):=(12*(8/10:ℝ)/mstar^2)*(125/100*(f+tc*g)^2+5*Profile.tr^2*g^2) |
| lemma gmBound : |
| sgamma (seg m h) ≤ h^2/3*mean0 (fun x=>GammaB (alongD m h Kp x) (alongD m h Cp x)):= by |
| let f:=alongD m h Kp |
| let g:=alongD m h Cp |
| have hc:Continuous f:=(Dt_cont testK).comp (cLoc ..) |
| have hd:Continuous g:=(Dt_cont testC).comp (cLoc ..) |
| let l:=fun x:ℝ=>f x+tc*g x |
| have hi:Continuous l:=hc.add (continuous_const.mul hd) |
| have hm:mean0 l=mean0 f+tc*mean0 g:=by rw [show mean0 l= _ from mean_add hc |
| (show Continuous (fun x=>tc*g x) from continuous_const.mul hd),mean_scale] |
| let c:ℝ:=12*(8/10)/mstar^2 |
| let a:ℝ:=125/100;let b:=5*Profile.tr^2 |
| have he:=mean_cs hi; have hh:=mean_cs hd |
| unfold sgamma |
| rw [hoD m testK,hoD m testC] |
| change _ ≤ h^2/3*mean0 (fun x=> c * (a * (l x)^2+b*(g x)^2)) |
| rw [mean_scale,mean_add (show Continuous (fun x=>a*(l x)^2) from continuous_const.mul (hi.pow _)) |
| (show Continuous (fun x=>b*(g x)^2) from continuous_const.mul (hd.pow _)),mean_scale,mean_scale] |
| have hx : a*(mean0 l)^2+b*(mean0 g)^2 ≤ a*mean0 _+b*mean0 _:= |
| add_le_add (mul_le_mul_of_nonneg_left he (by unfold a;norm_num)) |
| (mul_le_mul_of_nonneg_left hh (by unfold b;positivity)) |
| apply le_trans _ (mul_le_mul_of_nonneg_left (mul_le_mul_of_nonneg_left hx (by unfold c; positivity)) |
| (by positivity)) |
| rw [hm]; unfold c a b f g;apply le_of_eq;ring |
|
|
| def negN (m h:ℝ):= fun x=> -(omega m h x*alongN m h Kp x) |
| lemma ekB (hh:0<h): |
| ek (seg m h)=h^2/3*mean0 (negN m h):=by |
| have hi:=ch_err m h hh testK |
| have he (f:ℝ→ℝ): mean0 (fun x=> -f x) = -mean0 f:=by |
| unfold mean0;rw [intervalIntegral.integral_neg];ring |
| unfold negN alongN ek |
| have hEq := he (fun x=>omega m h x*NNf Kp (loc m h x)) |
| rw [hEq] |
| dsimp only at hi |
| linarith |
|
|
| lemma smallU (H:NG)(hh:0<h)(he:h≤16/100): |
| ek (seg m h)+sgamma (seg m h) ≤ 54/100*ubase h := by |
| let l:= fun x=>GammaB (alongD m h Kp x) (alongD m h Cp x) |
| let S:=negN m h |
| have hc:Continuous S:= ((omCont m h).mul ((nn_cont testK).comp (cLoc ..))).neg |
| have hd: Continuous l:= by |
| have he:Continuous (alongD m h Kp):=(Dt_cont testK).comp (cLoc ..) |
| have hv:Continuous (alongD m h Cp):=(Dt_cont testC).comp (cLoc ..) |
| unfold l GammaB;fun_prop |
| have hp:=mean_mono (hc.add hd) continuous_const |
| (f:=fun x=>S x+l x) (g:=fun _=>Ur) ?_ |
| · rw [mean_add hc hd, mean_const] at hp |
| rw [ekB m hh] |
| apply le_trans (add_le_add le_rfl (gmBound m)); rw [← mul_add] |
| apply le_trans (mul_le_mul_of_nonneg_left hp (by positivity)) |
| unfold Ur ubase wp; apply le_of_eq;ring |
| intro x hx |
| let t:=loc m h x |
| let D:=NNf Kp t |
| have hi: 1507/1000*max 0 (-D)+l x≤Ur:= (GlobalCorr H t).2.2.2 |
| have hh:= (Wshr m hh he).1 x hx |
| have hz :0 ≤ omega m h x := hh.1 |
| change -(omega m h x * D)+l x ≤ Ur |
| apply le_trans _ hi |
| apply add_le_add _ le_rfl |
| rw [← mul_neg] |
| calc |
| _ ≤ omega m h x*max 0 (-D):=mul_le_mul_of_nonneg_left (le_max_right ..) hz |
| _ ≤ _:=mul_le_mul_of_nonneg_right hh.2 (le_max_left ..) |
| lemma chAbs (hh:0<h)(he:h≤16/100) {f:ℝ→ℝ} (hf:TestF f){a:ℝ} |
| (ha:0≤a) (hp:∀ x,|NNf f x| ≤ a): |
| let v:=seg m h |
| |bav f v-(f v.1+f v.2)/2| ≤h^2/3*((1012/1000:ℝ)*a) :=by |
| intro v; rw [ch_err m h hh hf,abs_mul,abs_neg,abs_of_nonneg (by positivity)] |
| let g:=fun x=>omega m h x*NNf f (loc m h x) |
| apply mul_le_mul_of_nonneg_left _ (by positivity) |
| change |mean0 g|≤_ |
| obtain ⟨he,hb⟩:=Wshr m hh he |
| have hc:=omCont m h |
| have hg:Continuous g:= hc.mul ((nn_cont hf).comp (cLoc ..)) |
| have hx(t:ℝ)(ht:t∈Icc (-1:ℝ) 1): -a*omega m h t≤g t ∧ g t≤a*omega m h t:=by |
| obtain ⟨ha,hc⟩:=abs_le.mp (hp (loc m h t)) |
| have h' :=(he t ht).1 |
| unfold g; constructor<;>nlinarith |
| have hh:= mean_mono (show Continuous (fun x=> -a*omega m h x) from continuous_const.mul hc) hg |
| (fun t ht=>(hx t ht).1) |
| have hj:= mean_mono hg (show Continuous (fun x=>a*omega m h x) from continuous_const.mul hc) |
| (fun t ht=>(hx t ht).2) |
| rw [mean_scale] at hh hj |
| apply abs_le.mpr; constructor <;>nlinarith |
| end GeneralMahler.SCal.SE |
| |
| end |
| |
| end OAI |
| |
