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8e4d533 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 | 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
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