| import OAI.Geometry.Convex.GeneralMahler.InTail |
|
|
| namespace OAI |
| /-! Diagonal balance, with separate estimates at the two ends of the coordinate box. -/ |
| noncomputable section |
| open Set Filter MeasureTheory MeasureTheory.Measure Real Metric Matrix |
| open scoped ENNReal NNReal Topology MatrixOrder Matrix.Norms.L2Operator RealInnerProductSpace |
| namespace GeneralMahler |
| open Layers ProjField |
| variable {m : ℕ} [NeZero m] |
|
|
| -- Polynomial coefficient D as in angle_est. |
| def ek (j : ℝ) := Real.exp (-(j^2)/4) |
|
|
| lemma ekpos (j:ℝ) : 0 ≤ ek j := Real.exp_nonneg _ |
| lemma ekrapid : rapid ek := by |
| convert rapid_exp_sq (show 0<(1/4:ℝ) by norm_num) using 1 |
| ext x |
| unfold ek; ring_nf |
| lemma small_minus_bound (m : ℕ) [NeZero m] {D:ℝ} (hd:0≤D) : |
| ∃ W≥0, ∀ q:ProjField m, ∀ v:Rn m, ‖v‖=1 → ∀ R:ℝ, 2≤R → |
| (∀ z j:ℝ, 1≤j → q.dimH v z ≤ D*(j/R*a z + ek j)) → |
| q.minus v ≤ W := by |
| let b := fun z:ℝ => (phi z * (1+‖z‖) + ek z) |
| let c := fun z:ℝ => D*b z |
| have hb : PolyBound (fun z:ℝ => 1+‖z‖) := |
| (PolyBound.const _).add PolyBound.id.norm |
| have hm : Measurable ek := by unfold ek; fun_prop |
| have hf : Integrable b := ((rapid_phi.product hb).integrable_real |
| (by have h := (by unfold phi; fun_prop : Measurable phi); fun_prop)).add |
| (ekrapid.integrable_real hm) |
| refine ⟨∫ z,c z,?_, fun q v hv R hr h => ?_⟩ |
| · exact integral_nonneg fun z=>mul_nonneg hd |
| (add_nonneg (mul_nonneg (phi_pos z).le (by positivity)) (ekpos _)) |
| change ∫ z in Iio (0:ℝ), q.dimH v z ≤ _ |
| have hI : Integrable c := hf.const_mul _ |
| apply le_trans _ (setIntegral_le_integral (s:=Iio 0) hI |
| (ae_of_all _ fun z=>mul_nonneg hd |
| (add_nonneg (mul_nonneg (phi_pos z).le (by positivity)) (ekpos _)))) |
| apply setIntegral_mono_on (q.minus_int v hv) hI.integrableOn measurableSet_Iio |
| intro z hz |
| apply le_trans (h z (1+‖z‖) (by have h := norm_nonneg z; linarith)) |
| apply mul_le_mul_of_nonneg_left _ hd |
| apply add_le_add |
| · rw [mul_comm (phi z)] |
| have hA := a_high (-z) (by change z<0 at hz; linarith) |
| rw [neg_neg, neg_phi] at hA |
| exact mul_le_mul (div_le_self (by positivity) (show 1 ≤ R by linarith)) hA (a_pos z).le (by positivity) |
| apply Real.exp_le_exp.mpr |
| rw [Real.norm_eq_abs,abs_of_neg (mem_Iio.mp hz)] |
| change z<0 at hz |
| nlinarith |
|
|
| lemma small_plus_lower {D : ℝ} (q:ProjField m) (v:Rn m) (hv:‖v‖=1) |
| {L R N : ℝ} (hd:0≤D) (hr:0<R) (hN : 1 ≤ N) (hl:0≤L) |
| (h:∀ z:ℝ, q.dimH v z ≤ D*(N/R*a z+ek N)) |
| (hh:D*(N/R*a L+ek N) ≤ 1/2) : |
| L*(1/2) ≤ q.plus v := by |
| let s := Icc 0 L |
| let f := fun z=>1-q.dimH v z |
| have hi : IntegrableOn f (Ici 0) := q.plus_int v hv |
| have hs : s ⊆ Ici 0 := Icc_subset_Ici_self |
| apply le_trans _ (setIntegral_mono_set hi |
| (ae_of_all _ fun z=> sub_nonneg.mpr (q.h_between v hv z).2) |
| (Filter.Eventually.of_forall hs)) |
| have hL : (∫ _ : ℝ in s,(1/2:ℝ))=L*(1/2) := by |
| rw [setIntegral_const] |
| simp [s,Real.volume_real_Icc_of_le hl] |
| rw [← hL] |
| refine setIntegral_mono_on (integrableOn_const (hs := isCompact_Icc.measure_lt_top.ne)) |
| (hi.mono_set hs) measurableSet_Icc (fun z hz=>?_) |
| have he : q.dimH v z ≤ 1/2 := le_trans (h z) |
| (le_trans (show D*(N/R*a z+ek N) ≤ _ from |
| mul_le_mul_of_nonneg_left (add_le_add |
| (mul_le_mul_of_nonneg_left (a_mono.monotone hz.2) (by positivity)) le_rfl) hd) hh) |
| unfold f; linarith |
|
|
| lemma large_minus_lower {D R N : ℝ} (hr:0<R) (x:Param m R) (q₀:ProjField m) |
| (v:Rn m) (hv:‖v‖=1) (hd:0≤D) (hN : 1 ≤ N) |
| (hh:D*(N/R*(4/a (-1))+ek N) ≤ 1/2) : |
| let q := x.instanceQ hr q₀ |
| (∀ z:ℝ, 1-q.dimH v z ≤ D*(N/R*q.Bt z+ek N)) → |
| 1/2 ≤ q.minus v := by |
| intro q h |
| let s := Ico (-1:ℝ) 0 |
| have hi := q.minus_int v hv |
| have hs : s ⊆ Iio 0 := Ico_subset_Iio_self |
| apply le_trans _ (setIntegral_mono_set hi (ae_of_all _ fun z=>(q.h_between v hv z).1) |
| (Filter.Eventually.of_forall hs)) |
| have hL : (∫ _ : ℝ in s,(1/2:ℝ))=1/2 := by |
| rw [setIntegral_const] |
| norm_num [s,Real.volume_real_Ico] |
| rw [← hL] |
| refine setIntegral_mono_on (integrableOn_const |
| (hs := (lt_of_le_of_lt (measure_mono Ico_subset_Icc_self) |
| isCompact_Icc.measure_lt_top).ne)) (hi.mono_set hs) measurableSet_Ico ?_ |
| intro z hz |
| have he : 1-q.dimH v z ≤ 1/2 := le_trans (h z) |
| (le_trans (show D*(N/R*q.Bt z+ek N) ≤ _ from |
| mul_le_mul_of_nonneg_left (add_le_add |
| (mul_le_mul_of_nonneg_left (x.B_segment hr q₀ z hz.1) (by positivity)) le_rfl) hd) hh) |
| linarith |
|
|
| lemma large_core_upper {D R L N:ℝ} (hr:0<R) (x:Param m R) (q₀:ProjField m) |
| (v:Rn m) (hv:‖v‖=1) (hd:0≤D) (hN:1≤N) (hl:0≤L) : |
| let q := x.instanceQ hr q₀ |
| (∀ z, 1-q.dimH v z ≤ D*(N/R*q.Bt z+ek N)) → |
| (∫ z in Icc (0:ℝ) L, (1-q.dimH v z)) ≤ D*N/R*8 + D*ek N*L := by |
| intro q hh |
| have h := x.area_B hr q₀ |
| have hi : IntegrableOn q.Bt (Ici 0) := h.1 |
| let s := Icc 0 L |
| have hs : s ⊆ Ici 0 := Icc_subset_Ici_self |
| have ha : IntegrableOn (fun z : ℝ=>D*N/R*q.Bt z) s := (hi.mono_set hs).const_mul _ |
| have hb : IntegrableOn (fun _ : ℝ=>D*ek N) s := integrableOn_const |
| (hs := isCompact_Icc.measure_lt_top.ne) |
| have he : (∫ z in s,q.Bt z) ≤ 8 := le_trans (setIntegral_mono_set hi |
| (ae_of_all _ q.Bt_nonneg) (Filter.Eventually.of_forall hs)) h.2 |
| let f := fun z=>1-q.dimH v z |
| have hf : IntegrableOn f s := (q.plus_int v hv).mono_set hs |
| calc |
| _ ≤ ∫ z in s,(D*N/R*q.Bt z+D*ek N) := |
| setIntegral_mono_on hf (ha.add hb) measurableSet_Icc |
| (fun z _=> (hh z).trans_eq (by ring)) |
| _ = D*N/R*(∫ z in s,q.Bt z)+D*ek N*L := by |
| rw [integral_add ha hb, integral_const_mul,setIntegral_const] |
| simp [s,Real.volume_real_Icc_of_le hl,mul_comm] |
| _ ≤ _ := by gcongr |
|
|
| lemma plus_split (q:ProjField m) (v:Rn m) (hv:‖v‖=1) {L : ℝ} (hl:0≤L) : |
| let f := fun z=>1-q.dimH v z |
| q.plus v = (∫ z in Icc 0 L,f z)+(∫ z in Ioi L, f z) := by |
| intro f |
| have he : Icc 0 L ∪ Ioi L = Ici 0 := by |
| ext z |
| change ((0≤z ∧ z≤L) ∨ L<z) ↔ _ |
| simp only [mem_Ici] |
| grind |
| have hi : IntegrableOn f (Ici 0) := q.plus_int v hv |
| rw [← he] at hi |
| change (∫ z in Ici 0,f z) = _ |
| rw [← he] |
| refine setIntegral_union ?_ measurableSet_Ioi (hi.mono_set subset_union_left) |
| (hi.mono_set subset_union_right) |
| exact disjoint_left.mpr fun z hz hh=> not_lt_of_ge hz.2 hh |
| end GeneralMahler |
|
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| end |
|
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| end OAI |
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