| import OAI.Geometry.Convex.GeneralMahler.DefectLayer |
|
|
| namespace OAI |
| /-! Scalar layer δ integration identities (including Eq 12). -/ |
| noncomputable section |
| open MeasureTheory Filter Set Real Metric |
| open scoped Topology ENNReal NNReal |
| namespace GeneralMahler |
| open Profile Layers |
| lemma rapid_ftc {f g:ℝ→ℝ} (hf:rapid f) (hd:∀ x,HasDerivAt f (g x) x) (hi:Integrable g) : |
| (∫ x,g x)=0 := by |
| simpa using integral_of_hasDerivAt_of_tendsto hd hi hf.tail_limit_bot hf.tail_limit |
| lemma MomentF.int_test {f g:ℝ→ℝ} (hf:MomentF f) (hg:TestF g) : |
| Integrable fun x=> f x*g x := hf g hg.cont.measurable hg.poly |
| lemma N2_test {f:ℝ→ℝ} (hf:TestF f) : TestF (N2 f) := |
| ((TestF.id.mul hf.der).sub hf.der.der).add ((TestF.const 2).mul hf) |
| variable {m:ℕ} |
| namespace ProjField |
| variable (q:ProjField m) |
| def pb (x:ℝ) := p x*q.dB x |
| def ab (x:ℝ) := a x*q.dB x |
| def rk (x:ℝ) := p x*q.dr x+q.hDel x |
| lemma Bt_d (x:ℝ) : HasDerivAt q.Bt (-q.r1 x) x := by |
| have h := (q.Bt_cd.differentiable (by norm_num)).differentiableAt (x:=x) |
| simp only [r1,neg_neg]; exact h.hasDerivAt |
| lemma dB_d (x:ℝ) : HasDerivAt q.dB (-q.dr x) x := by |
| convert ((q.Bt_d x).sub (((d_a (-x)).comp x (hasDerivAt_neg' x)).const_mul q.s0)) using 1 |
| all_goals first | rfl | (rw [neg_p]; unfold dr; ring) |
| lemma hd_d (x:ℝ) : HasDerivAt q.hDel (q.dm x) x := (q.H_deriv x).sub ((d_p x).const_mul _) |
| lemma dm_eq (x:ℝ) : q.dm x=q.pb x+a x*q.dr x := by |
| unfold dm pb layerW dr dB |
| rw [← a0_id x]; ring |
| lemma ab_d (x:ℝ) : HasDerivAt q.ab (q.pb x-a x*q.dr x) x := by |
| convert (d_a x).mul (q.dB_d x) using 1 |
| all_goals first | rfl | (unfold pb; ring) |
| lemma pb_d (x:ℝ) : HasDerivAt q.pb (phi x*q.dB x-p x*q.dr x) x := by |
| convert (d_p x).mul (q.dB_d x) using 1 |
| all_goals first | rfl | ring |
| lemma dB_c : Continuous q.dB := continuous_iff_continuousAt.mpr fun x=>(q.dB_d x).continuousAt |
| lemma hd_c : Continuous q.hDel := continuous_iff_continuousAt.mpr fun x=>(q.hd_d x).continuousAt |
|
|
| variable [NeZero m] |
|
|
| lemma pb_m : MomentF q.pb := .of_rapid q.pB_tail (cp.mul q.dB_c).measurable |
| lemma ab_m : MomentF q.ab := .of_rapid q.aB_tail (ca.mul q.dB_c).measurable |
| lemma hd_m : MomentF q.hDel := .of_rapid q.hd_tail q.hd_c.measurable |
| lemma dm_m : MomentF q.dm := by |
| rw [show q.dm=_ from funext q.dm_eq] |
| exact q.pb_m.add q.arDelta_moment |
| lemma rk_m : MomentF q.rk := q.prDelta_moment.add q.hd_m |
|
|
| lemma mass : (∫ x,q.pb x)=0 ∧ (∫ x,a x*q.dr x)=0 := by |
| have he := rapid_ftc (f:=q.ab) q.aB_tail q.ab_d |
| (q.pb_m.inte.sub q.arDelta_moment.inte) |
| have hi := rapid_ftc q.hd_tail q.hd_d q.dm_m.inte |
| rw [integral_sub q.pb_m.inte q.arDelta_moment.inte] at he |
| simp_rw [q.dm_eq] at hi |
| rw [integral_add q.pb_m.inte q.arDelta_moment.inte] at hi |
| constructor <;> linarith |
| |
| lemma hd_test {f:ℝ→ℝ} (hf:TestF f) : |
| (∫ x,q.dm x*f x)= -(∫ x,q.hDel x*deriv f x) := by |
| have h (x:ℝ) : HasDerivAt (fun x=>q.hDel x*f x) (q.dm x*f x+q.hDel x*deriv f x) x := |
| (q.hd_d x).fun_mul (hf.diff x).hasDerivAt |
| have he := rapid_ftc (q.hd_tail.product hf.poly) h |
| ((q.dm_m.int_test hf).add (q.hd_m.int_test hf.der)) |
| rw [integral_add (q.dm_m.int_test hf) (q.hd_m.int_test hf.der)] at he |
| linarith |
|
|
| lemma eq12 {f:ℝ→ℝ} (hf:TestF f) : |
| (∫ x,q.pb x*N2 f x+q.rk x*deriv f x)=0 := by |
| let g := fun x=>(q.ab x+q.hDel x)*f x-q.pb x*deriv f x |
| have hr : rapid g := |
| ((q.aB_tail.rfadd q.hd_tail).product hf.poly).rfsub (q.pB_tail.product hf.der.poly) |
| have hd (x:ℝ) : HasDerivAt g (q.pb x*N2 f x+q.rk x*deriv f x) x := by |
| convert ((((q.ab_d x).fun_add (q.hd_d x)).fun_mul (hf.diff x).hasDerivAt).fun_sub |
| ((q.pb_d x).fun_mul (hf.der.diff x).hasDerivAt)) using 1 |
| rw [q.dm_eq] |
| unfold rk N2 N ab pb a |
| ring |
| apply rapid_ftc hr hd |
| exact (q.pb_m.int_test (N2_test hf)).add (q.rk_m.int_test hf.der) |
|
|
| def shiftD (f:ℝ→ℝ) := fun x=>2*x*f x-deriv f x |
| lemma delta_eq {f:ℝ→ℝ} (hf:TestF f) : |
| q.delt f = -(2*(∫ x,q.pb x*f x)+∫ x,q.rk x*shiftD f x) := by |
| have he : TestF (shiftD f) := |
| (((TestF.const 2).mul TestF.id).mul hf).sub hf.der |
| have hi := q.rk_m.int_test he |
| have hu : (fun x=>2*p x*q.dB x*f x)=fun x=>2*(q.pb x*f x) := by ext; unfold pb; ring |
| unfold delt; simp_rw [congrFun hu] |
| change -(∫ x,2*(q.pb x*f x)+q.rk x*shiftD f x)=_ |
| rw [integral_add,integral_const_mul] |
| · exact (q.pb_m.int_test hf).const_mul _ |
| exact hi |
| |
| lemma delta_d : q.delt Profile.d= -(∫ x,q.rk x*deriv Profile.v x) := by |
| have h := q.eq12 g_test |
| simp_rw [Profile.n_identity] at h |
| have hi := q.pb_m.int_test Profile.d_test |
| have hh := q.rk_m.int_test g_test.der |
| have hv := q.rk_m.int_test v_test.der |
| rw [integral_add hi hh] at h |
| rw [q.delta_eq d_test] |
| have he (x) : q.rk x*shiftD d x=q.rk x*deriv Profile.v x+2*(q.rk x*deriv g x) := by |
| unfold shiftD; rw [(Profile.dd x).deriv]; ring |
| simp_rw [he]; rw [integral_add hv (hh.const_mul _),integral_const_mul] |
| linarith |
| end ProjField |
| end GeneralMahler |
| |
| end |
| |
| end OAI |
| |
