| import OAI.Geometry.Convex.GeneralMahler.Middle.Bound2 |
| import OAI.Geometry.Convex.GeneralMahler.Segment.Phase |
|
|
| namespace OAI |
| /-! Center profiles for direct evaluation in compact region. -/ |
| open Set Filter Real |
| namespace GeneralMahler.SCal.Mid |
| open Tag Grid Jet Profile Segment SE |
| inductive Ft |
| | K|C|Q|J|V|M |
| deriving DecidableEq |
|
|
| instance : Fintype Ft := |
| Fintype.ofList [Ft.K, Ft.C, Ft.Q, Ft.J, Ft.V, Ft.M] |
| (by intro p; cases p <;> simp) |
|
|
| noncomputable section |
| def k0:ℝ:=1/100 |
| def c0:ℝ:=-(42/100) |
| def q0:ℝ:=16/100 |
| namespace Ft |
| def f: Ft→ℝ→ℝ |
| | K=> fun x=> Kp x-k0 |
| | C=> fun x=> Cp x-c0 |
| | Q=> fun x=> qu x-q0 |
| | M=> fun x=> (qu x-q0)^2 |
| | J=>uc |
| | V=>us |
| def g(p:Ft):= liftF p.f |
| def cK:Cap:= ⟨18/1000,66/1000,26/100⟩ |
| def cC:Cap:=⟨81/1000,247/1000,95/100⟩ |
| def cP:Cap:=⟨96/1000,193/1000,56/100⟩ |
| def cJ:Cap:=⟨22/100,1012/1000,466/100⟩ |
| def cap:Ft→Cap |
| | K=>cK |
| | C=>cC |
| | Q=>cP |
| | M=> Cap.times cP cP |
| | _=> cJ |
| lemma test (h:NG)(p:Ft):TestF p.f:= by |
| have hf:=hSeg01 h |
| cases p |
| · exact testK.sub (TestF.const _) |
| · exact testC.sub (TestF.const _) |
| · exact hf.q_test.sub (TestF.const _) |
| · exact hf.c_test |
| · exact hf.s_test |
| exact sq_test (hf.q_test.sub (TestF.const _)) |
|
|
| lemma shiftGuard {f:ℝ→ℝ} (hf:TestF f) (c:ℝ) (A:Cap) |
| (he:∀ x, |f (xs x)-c|≤A.a) |
| (hb:∀ x, |DotF f x|≤A.b) |
| (hc:∀ x, |DDot f x|≤A.c): |
| Guard (fun x=>f (xs x)-c) A:= by |
| have hv : Continuous (DDot f) := by |
| have hh : DDot f= fun x=> xs x* deriv f (xs x) + ast x*(ast x*deriv (deriv f) (xs x)) := |
| funext fun x=> (dd_lift hf x).deriv |
| have ha : Continuous xs:=by unfold xs; fun_prop |
| have he:Continuous ast:=by unfold ast;fun_prop |
| rw [hh]; exact (ha.mul (hf.der.cont.comp ha)).add (he.mul (he.mul (hf.der.der.cont.comp ha))) |
| apply mkGuard (g:=DotF f) (h:=DDot f) _ _ hv he hb hc |
| · intro x;exact (liftD hf x).sub_const c |
| intro x; exact (dd_lift hf x).differentiableAt.hasDerivAt |
|
|
| lemma guardQ (h:NG):Guard (g Q) cP:=by |
| apply shiftGuard (hqt h) q0 cP |
| · intro x;unfold cP; dsimp; rw [abs_le] |
| unfold q0; have he:=hQ h x;constructor<;>linarith [he.1,he.2] |
| · intro x;unfold cP;dsimp;rw [abs_le];have he:=SE.hdQ h x; norm_num at *;exact he |
| intro x;unfold cP;dsimp;rw [abs_le];have he:=hddQ h x; norm_num at *;exact he |
|
|
| lemma guardCS:Guard lc cJ∧ Guard ls cJ:=by |
| let w:= Profile.omegaP |
| let a:= fun x=> -w*ls x |
| let b:=fun x=>w*lc x |
| have hh:0≤w:=by norm_num [w,Profile.omegaP] |
| have h: 0≤ Profile.r:=by norm_num [Profile.r] |
| have h1(x): |lc x| ≤r ∧ |ls x|≤ r:=by |
| rw [lc_e,ls_e,abs_mul,abs_mul,abs_of_nonneg h] |
| exact ⟨mul_le_of_le_one_right h (abs_cos_le_one _),mul_le_of_le_one_right h (abs_sin_le_one _)⟩ |
| have h₁ :Continuous lc:=continuous_iff_continuousAt.mpr fun x=>(dlc x).continuousAt |
| have h₂ :Continuous ls:=continuous_iff_continuousAt.mpr fun x=>(dls x).continuousAt |
| have ha(x): |a x|≤ w*r ∧ |b x|≤ w*r:=by |
| unfold a b; rw [abs_mul,abs_neg,abs_mul,abs_of_nonneg hh]; constructor<;> |
| apply mul_le_mul_of_nonneg_left _ hh |
| exact (h1 _).2;exact (h1 _).1 |
| have he:w*r=(1012/1000:ℝ):=by norm_num [w,omegaP,r] |
| have hp:w*w*r ≤ (466/100:ℝ):=by norm_num [w,omegaP,r] |
| have H1(x): HasDerivAt a (-w*b x) x:= (dls x).const_mul (-w) |
| have H2(x): HasDerivAt b (w*a x) x:= (dlc x).const_mul w |
| constructor |
| · apply mkGuard dlc H1 (continuous_const.mul (continuous_const.mul h₁)) |
| · exact fun x=> (h1 _).1 |
| · intro x;change |a x|≤ 1012/1000;exact he ▸ (ha x).1 |
| intro x |
| rw [abs_mul,abs_neg,abs_of_nonneg hh];change _ ≤ 466/100 |
| rw [mul_assoc] at hp |
| exact le_trans (mul_le_mul_of_nonneg_left (ha x).2 hh) hp |
| apply mkGuard dls H2 (continuous_const.mul (continuous_const.mul h₂)) |
| · exact fun x=>(h1 _).2 |
| · intro x; change |b x|≤1012/1000 |
| exact he ▸ (ha x).2 |
| intro x |
| rw [abs_mul,abs_of_nonneg hh];change _ ≤466/100 |
| rw [mul_assoc] at hp |
| exact le_trans (mul_le_mul_of_nonneg_left (ha x).1 hh) hp |
|
|
| lemma guarded (h:NG)(p:Ft):Guard p.g p.cap:=by |
| cases p |
| · apply shiftGuard testK k0 cK |
| · intro x |
| unfold k0;have he:=hK h x;rw [abs_le];change _∧_≤18/1000; dsimp [cK];constructor<;>linarith [he.1,he.2] |
| · intro x;have he:=hdK h x; unfold cK;dsimp; rw [abs_le];norm_num at *;exact he |
| intro x;have he:=hddK h x; unfold cK;dsimp;rw [abs_le];norm_num at *;exact he |
| · apply shiftGuard testC c0 cC |
| · intro x |
| unfold c0; have he:=hC h x;rw [abs_le];dsimp [cC]; constructor<;>linarith [he.1,he.2] |
| · intro x;have he:=hdC h x; unfold cC;dsimp; rw [abs_le];norm_num at *;exact he |
| intro x;have he:=hddC h x; unfold cC;dsimp;rw [abs_le];norm_num at *;exact he |
| · exact guardQ h |
| · exact guardCS.1 |
| · exact guardCS.2 |
| exact (guardQ h).sq |
| end Ft |
| end |
| end GeneralMahler.SCal.Mid |
|
|
| end OAI |
|
|
