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openAI-math-dataset / lean /OAI /Analysis /IntegralMeans /TransportBoxes.lean
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roseking
Add lean source and binary files (part 19)
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import OAI.Analysis.IntegralMeans.GoodPairs
namespace OAI
noncomputable section
open Set MeasureTheory Filter Function InnerProductSpace
open scoped Topology ComplexConjugate Manifold NNReal ENNReal InnerProductSpace Classical
open MeasureTheory Function
open Set Filter
open Set MeasureTheory Filter Function
open Set MeasureTheory Filter Function InnerProductSpace
open TopologicalSpace
open scoped CompactlySupported
open scoped ENNReal
open scoped Manifold
open scoped Topology CompactlySupported ComplexConjugate
open scoped Topology ComplexConjugate Manifold NNReal ENNReal InnerProductSpace Classical
open scoped Topology ENNReal NNReal
namespace Brennan
attribute [local irreducible] classWeight
attribute [local irreducible] classFun
def displacementWindow (N : ℝ) (z w : ℂ) : Prop :=
z.im ≤ N*w.im ∧ w.im ≤ N*z.im ∧ |w.re-z.re| ≤ N*z.im
def transportWindow (k N : ℝ) : Set (DiskClass × halfPlane) :=
{q | q ∈ pairingDomain k ∧
displacementWindow N q.2 (criticalPair (classFun q.1) k q.2)}
lemma measurableSet_transportWindow {k : ℝ} (hk : 0 < k) (N : ℝ) :
MeasurableSet (transportWindow k N) := by
have hr := measurable_criticalPair hk
have hz : Measurable (fun q : pairingDomain k => (q.val.2 : ℂ)) :=
measurable_subtype_coe.comp (measurable_subtype_coe.snd)
have hm : MeasurableSet {q : pairingDomain k |
displacementWindow N q.val.2 (criticalPair (classFun q.val.1) k q.val.2)} :=
(measurableSet_le (Complex.measurable_im.comp hz)
(measurable_const.mul (Complex.measurable_im.comp hr))).inter
((measurableSet_le (Complex.measurable_im.comp hr)
(measurable_const.mul (Complex.measurable_im.comp hz))).inter
(measurableSet_le (((Complex.measurable_re.comp hr).sub (Complex.measurable_re.comp hz)).abs)
(measurable_const.mul (Complex.measurable_im.comp hz))))
have h := (measurableSet_pairingDomain hk.le).subtype_image hm
convert h using 1
ext q
simp only [transportWindow,mem_ofPred_eq,mem_image,Subtype.exists,exists_prop]
aesop
lemma displacementWindow_affine (N : ℝ) {t : ℂ} (ht : t ∈ halfPlane) (z w : ℂ) :
displacementWindow N (affine t z) (affine t w) ↔ displacementWindow N z w := by
have him (v : ℂ) : (affine t v).im = t.im*v.im := by simp [affine]
have hre : (affine t w).re-(affine t z).re = t.im*(w.re-z.re) := by simp [affine]; ring
simp only [displacementWindow,him,hre,abs_mul,abs_of_pos (show 0<t.im from ht)]
simp only [show N*(t.im*w.im) = t.im*(N*w.im) from by ring,
show N*(t.im*z.im) = t.im*(N*z.im) from by ring,
mul_le_mul_iff_right₀ (show 0<t.im from ht)]
lemma transportWindow_root (g : DiskClass) {k : ℝ} (hk : 0 < k) (N : ℝ) (z : halfPlane) :
(rerootClass g z,halfOne) ∈ transportWindow k N ↔ (g,z) ∈ transportWindow k N := by
have hJ := class_normalizedJacobian_root g hk z
have hG := class_goodPair_root g hk z
change (0 < normalizedJacobian (classFun (rerootClass g z)) k Complex.I ∧
GoodPair (classFun (rerootClass g z)) k (criticalMap (classFun (rerootClass g z)) k Complex.I)) ∧ _ ↔
(0 < normalizedJacobian (classFun g) k z ∧ GoodPair (classFun g) k (criticalMap (classFun g) k z)) ∧ _
rw [hJ,hG]
apply and_congr_right
intro h
have he := class_criticalPair_reroot g hk z halfOne (by simpa only [halfOne,affine_I] using h.1)
(by simpa only [halfOne,affine_I] using h.2)
simp only [halfOne,affine_I] at he
change displacementWindow N Complex.I (criticalPair (classFun (rerootClass g z)) k Complex.I) ↔
displacementWindow N z (criticalPair (classFun g) k z)
rw [← displacementWindow_affine N z.property Complex.I,affine_I,he]
lemma displacementWindow_mono {N M : ℝ} (hNM : N ≤ M) {z w : ℂ}
(hz : z ∈ halfPlane) (hw : w ∈ halfPlane) (h : displacementWindow N z w) :
displacementWindow M z w :=
⟨h.1.trans (mul_le_mul_of_nonneg_right hNM hw.le),
h.2.1.trans (mul_le_mul_of_nonneg_right hNM hz.le),
h.2.2.trans (mul_le_mul_of_nonneg_right hNM hz.le)⟩
lemma exists_displacementWindow {z w : ℂ} (hz : z ∈ halfPlane) (hw : w ∈ halfPlane) :
∃ n : ℕ, displacementWindow (n+1) z w := by
obtain ⟨n,hn⟩ := exists_nat_gt (max (z.im/w.im) (max (w.im/z.im) (|w.re-z.re|/z.im)))
refine ⟨n,?_,?_,?_⟩
· exact (div_le_iff₀ hw).mp ((le_max_left _ _).trans (by exact_mod_cast hn.le.trans (le_add_of_nonneg_right zero_le_one)))
· exact (div_le_iff₀ hz).mp (((le_max_left _ _).trans (le_max_right _ _)).trans (by exact_mod_cast hn.le.trans (le_add_of_nonneg_right zero_le_one)))
· exact (div_le_iff₀ hz).mp (((le_max_right _ _).trans (le_max_right _ _)).trans (by exact_mod_cast hn.le.trans (le_add_of_nonneg_right zero_le_one)))
lemma transportWindow_mono {k : ℝ} (hk : 0 < k) {N M : ℝ} (hNM : N ≤ M) :
transportWindow k N ⊆ transportWindow k M := by
rintro ⟨g,z⟩ ⟨hp,hw⟩
exact ⟨hp,displacementWindow_mono hNM z.property
(criticalPair_spec (classFun_schlicht g).1 hk z.property hp.1 hp.2).1 hw⟩
lemma iUnion_transportWindow {k : ℝ} (hk : 0 < k) :
(⋃ n : ℕ, transportWindow k (n+1)) = pairingDomain k := by
ext q
constructor
· rintro ⟨_,⟨n,rfl⟩,hq⟩
exact hq.1
· intro hq
obtain ⟨n,hn⟩ := exists_displacementWindow q.2.property
(criticalPair_spec (classFun_schlicht q.1).1 hk q.2.property hq.1 hq.2).1
exact mem_iUnion.mpr ⟨n,hq,hn⟩
def transportSlice (g : DiskClass) (k N : ℝ) : Set ℂ :=
{z | z ∈ halfPlane ∧ 0 < normalizedJacobian (classFun g) k z ∧
GoodPair (classFun g) k (criticalMap (classFun g) k z) ∧
displacementWindow N z (criticalPair (classFun g) k z)}
lemma transportSlice_iff (g : DiskClass) (k N : ℝ) (z : halfPlane) :
z.val ∈ transportSlice g k N ↔ (g,z) ∈ transportWindow k N := by
simp only [transportSlice,transportWindow,pairingDomain,mem_ofPred_eq,z.property,true_and,and_assoc]
lemma measurableSet_transportSlice (g : DiskClass) {k : ℝ} (hk : 0 < k) (N : ℝ) :
MeasurableSet (transportSlice g k N) := by
apply measurableSet_halfPlane_pred
have hm : Measurable (fun z : halfPlane => (g,z)) := measurable_const.prodMk measurable_id
have h := (measurableSet_transportWindow hk N).preimage hm
convert h using 1
ext z
exact and_assoc.symm
def transportBox (R a b : ℝ) : Set ℂ := {z | |z.re| ≤ R ∧ a ≤ z.im ∧ z.im ≤ b}
lemma measurableSet_transportBox (R a b : ℝ) : MeasurableSet (transportBox R a b) :=
(measurableSet_le Complex.measurable_re.abs measurable_const).inter
((measurableSet_le measurable_const Complex.measurable_im).inter
(measurableSet_le Complex.measurable_im measurable_const))
lemma pair_mem_enlargedBox {N R Y : ℝ} (hN : 1 ≤ N) {z w : ℂ}
(hz : z ∈ transportBox R 1 Y) (hw : displacementWindow N z w) :
w ∈ transportBox (R+N*Y) (1/N) (N*Y) := by
have hn : 0 < N := lt_of_lt_of_le zero_lt_one hN
have hab : |w.re| ≤ |z.re|+|w.re-z.re| := by
calc |w.re| = |z.re+(w.re-z.re)| := by congr 1; ring
_ ≤ |z.re|+|w.re-z.re| := abs_add_le _ _
refine ⟨hab.trans (add_le_add hz.1 (hw.2.2.trans (mul_le_mul_of_nonneg_left hz.2.2 hn.le))),?_,
hw.2.1.trans (mul_le_mul_of_nonneg_left hz.2.2 hn.le)⟩
rw [div_le_iff₀ hn]
nlinarith [hw.1,hz.2.1]
lemma fiberSum_mul_indicator.{u_1, u_2} {X : Type u_1} {Y : Type u_2} (f : X → Y) (s t : Set X)
(u : X → ℝ≥0∞) (y : Y) :
fiberSum f s (fun x => u x * t.indicator 1 x) y = fiberSum f (s ∩ t) u y := by
classical
rw [fiberSum,fiberSum,tsum_subtype {x | x ∈ s ∧ f x = y} (fun x => u x * t.indicator 1 x),
tsum_subtype {x | x ∈ s ∩ t ∧ f x = y} u]
apply tsum_congr
intro x
by_cases hs : x ∈ s <;> by_cases ht : x ∈ t <;> by_cases hy : f x = y <;>
simp [hs,ht,hy]
lemma transport_box (g : DiskClass) {β k N R Y : ℝ} (hk : 0 < k)
(hb : β-1 = 4*(k-1)) (hN : 1 ≤ N) :
(∫⁻ z in halfPlane, transportDensity g β z * ENNReal.ofReal (normalizedJacobian (classFun g) k z) *
(transportSlice g k N ∩ transportBox R 1 Y).indicator 1 z) ≤
∫⁻ z in halfPlane, transportDensity g β z * ENNReal.ofReal (-normalizedJacobian (classFun g) k z) *
(transportBox (R+N*Y) (1/N) (N*Y)).indicator 1 z := by
let S := transportSlice g k N ∩ transportBox R 1 Y
let T := transportBox (R+N*Y) (1/N) (N*Y)
have hs : MeasurableSet S := (measurableSet_transportSlice g hk N).inter (measurableSet_transportBox R 1 Y)
have ht : MeasurableSet T := measurableSet_transportBox _ _ _
have hp := transport_area g β k 1 (by norm_num) (measurable_const.indicator hs : Measurable (S.indicator (1 : ℂ → ℝ≥0∞)))
have hn := transport_area g β k (-1) (by norm_num) (measurable_const.indicator ht : Measurable (T.indicator (1 : ℂ → ℝ≥0∞)))
simp only [one_mul] at hp
simp only [neg_one_mul] at hn
apply le_trans hp.le
apply le_trans ?_ hn.symm.le
apply lintegral_mono
intro ξ
apply le_trans (fiberSum_mul_indicator (criticalMap (classFun g) k) (jacobianRegion g k 1) S (targetDensity g β) ξ).le
apply le_trans ?_ (fiberSum_mul_indicator (criticalMap (classFun g) k) (jacobianRegion g k (-1)) T (targetDensity g β) ξ).symm.le
apply fiberSum_mono_injective (criticalPair (classFun g) k) ξ
· intro z hz hG
have hsz := hz.2.1
have hpair := criticalPair_spec (classFun_schlicht g).1 hk hsz.1 hsz.2.1 hsz.2.2.1
exact ⟨⟨⟨hpair.1,by simpa only [neg_one_mul] using neg_pos.mpr hpair.2.2.1⟩,
pair_mem_enlargedBox hN hz.2.2 hsz.2.2.2⟩,hpair.2.1.trans hG⟩
· apply (criticalPair_injective (classFun_schlicht g).1 hk).mono
intro z hz
exact ⟨hz.1.2.1.1,hz.1.2.1.2.1,hz.1.2.1.2.2.1⟩
· intro z hz _
exact targetDensity_le_pair g hk hb hz.2.1.1 hz.2.1.2.1 hz.2.1.2.2.1
lemma measurableSet_exceptionSlice (g : DiskClass) {k : ℝ} (hk : 0 < k) :
MeasurableSet {z : ℂ | z ∈ halfPlane ∧ ¬GoodPair (classFun g) k (criticalMap (classFun g) k z)} := by
apply measurableSet_halfPlane_pred
have hm : Measurable (fun z : halfPlane => (g,criticalMap (classFun g) k z)) :=
(continuous_const.prodMk ((continuous_class_criticalMap k).comp
(show Continuous (fun z : halfPlane => (g,z)) from continuous_const.prodMk continuous_id))).measurable
exact ((measurableSet_class_goodPairs hk.le).preimage hm).compl
lemma exceptional_transport_zero (g : DiskClass) {β k : ℝ} (hk : 0 < k)
(hg : ∀ᵐ ξ : ℂ, GoodPair (classFun g) k ξ) :
(∫⁻ z in halfPlane, transportDensity g β z * ENNReal.ofReal (normalizedJacobian (classFun g) k z) *
{z : ℂ | z ∈ halfPlane ∧ ¬GoodPair (classFun g) k (criticalMap (classFun g) k z)}.indicator 1 z) = 0 := by
have hs := measurableSet_exceptionSlice g hk
have he := transport_area g β k 1 (by norm_num)
(measurable_const.indicator hs : Measurable ({z : ℂ | z ∈ halfPlane ∧ ¬GoodPair (classFun g) k (criticalMap (classFun g) k z)}.indicator (1 : ℂ → ℝ≥0∞)))
simp only [one_mul] at he
refine he.trans ?_
apply lintegral_eq_zero_of_ae_eq_zero
filter_upwards [hg] with ξ hξ
unfold fiberSum
apply ENNReal.tsum_eq_zero.mpr
intro z
have hn : z.val ∉ {z : ℂ | z ∈ halfPlane ∧ ¬GoodPair (classFun g) k (criticalMap (classFun g) k z)} := by
rintro ⟨_,hbad⟩
exact hbad (z.property.2.symm ▸ hξ)
exact (congrArg (targetDensity g β z * ·) (indicator_of_notMem (f := (1 : ℂ → ℝ≥0∞)) hn)).trans (mul_zero _)
end Brennan
end
end OAI