| import Mathlib |
| import OAI.MeasureTheory.Falconer.Model |
| import OAI.MeasureTheory.Falconer.Recursion.MinimalDescriptorLength |
| import OAI.MeasureTheory.Falconer.Recursion.PolicyDescriptors |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.SharedBinSupportCollision |
|
|
| namespace OAI |
|
|
| /-! Recursive policies and quantitative depth estimates. -/ |
|
|
| open _root_.OAI.PlanarFalconer |
|
|
| section |
| section |
| open Set MeasureTheory |
| open scoped ComplexConjugate RealInnerProductSpace |
| namespace GraphLiteralSupport |
| namespace CurrentFalconer.TreeND |
| open Set Metric MeasureTheory PlanarFalconer |
| open scoped ENNReal RealInnerProductSpace |
| variable {d : ℕ} |
|
|
| noncomputable def nominalRadius (N : ℕ) (sig : ℝ) (l : ℕ) : ℝ := |
| min 1 (1/(2:ℝ)^l*((2:ℝ)^N)^(50*sig)) |
|
|
| lemma testWidth_gap (N : ℕ) (sig : ℝ) {J : ℕ} (hJ : 0<J) (i n : ℕ) : |
| testWidth N J i n sig-testWidth N J (i+1) n sig = |
| ((2:ℝ)^N)^(500*sig)/((J:ℝ)*(2:ℝ)^n) := by |
| have hj : (J:ℝ) ≠ 0 := by exact_mod_cast hJ.ne' |
| simp only [testWidth,Nat.cast_add,Nat.cast_one,Real.rpow_neg (by norm_num : (0:ℝ)≤2), |
| Real.rpow_natCast] |
| field_simp |
| ring |
|
|
| lemma uniform_testWidth_gap {sig : ℝ} (hsig : 0<sig) {J : ℕ} (hJ : 0<J) |
| (C : ℝ) (hC : 0≤C) : |
| ∃ N₀ : ℕ, ∀ N ≥ N₀, ∀ g n : ℕ, g≤n → ∀ i : ℕ, |
| testWidth N J (i+1) n sig + (C/(2:ℝ)^g)*nominalRadius N sig (n-g) ≤ |
| testWidth N J i n sig := by |
| obtain ⟨N₀,h₀⟩ := fixed_constant_absorbed (by positivity : 0<450*sig) (C*J) |
| refine ⟨N₀,?_⟩ |
| intro N hN g n hgn i |
| have hj : 0<(J:ℝ) := by exact_mod_cast hJ |
| have hRp : 0<(2:ℝ)^N := by positivity |
| have hc := mul_le_mul_of_nonneg_right (h₀ N hN) |
| (by positivity : 0≤((2:ℝ)^N)^(50*sig)) |
| have he : ((2:ℝ)^N)^(450*sig)*((2:ℝ)^N)^(50*sig)=((2:ℝ)^N)^(500*sig) := by |
| rw [← Real.rpow_add hRp]; congr 1; ring |
| rw [he] at hc |
| have hp : (2:ℝ)^g*(2:ℝ)^(n-g)=(2:ℝ)^n := by rw [← pow_add,Nat.add_sub_of_le hgn] |
| have hb : (C/(2:ℝ)^g)*nominalRadius N sig (n-g) ≤ |
| ((2:ℝ)^N)^(500*sig)/((J:ℝ)*(2:ℝ)^n) := by |
| calc |
| _ ≤ (C/(2:ℝ)^g)*(1/(2:ℝ)^(n-g)*((2:ℝ)^N)^(50*sig)) := |
| mul_le_mul_of_nonneg_left (min_le_right _ _) (by positivity) |
| _ = (C*(J:ℝ)*((2:ℝ)^N)^(50*sig))/((J:ℝ)*(2:ℝ)^n) := by |
| rw [← hp]; field_simp |
| _ ≤ _ := div_le_div_of_nonneg_right hc (by positivity) |
| rw [← testWidth_gap N sig hJ i n] at hb |
| linarith |
|
|
| lemma fullParallelCount_transfer (ρ : Measure (EuclideanSpace ℝ (Fin d))) |
| (g p : ℕ) (P : CellPath d) {W W' ε : ℝ} {v w : EuclideanSpace ℝ (Fin d)} |
| (hv : ‖v‖=1) (hw : ‖w‖=1) (hd : dist v w ≤ ε) |
| (hW : W'+(20*(d+1:ℝ)/(2:ℝ)^g)*ε ≤ W) : |
| fullParallelCount ρ g p P W' w ≤ fullParallelCount ρ g p P W v := by |
| apply Finset.card_le_card |
| intro Q hQ |
| obtain ⟨hQ,hpos,hr,hperp⟩ := Finset.mem_filter.mp hQ |
| refine Finset.mem_filter.mpr ⟨hQ,hpos,hr,?_⟩ |
| have hε : 0≤ε := dist_nonneg.trans hd |
| have hchange := graphPerpND_change_direction w v (treeCenter Q-treeCenter P) hw hv |
| have hnorm : ‖graphPerpND v (treeCenter Q-treeCenter P)‖ ≤ |
| ‖graphPerpND w (treeCenter Q-treeCenter P)‖+ |
| 2*‖treeCenter Q-treeCenter P‖*dist v w := by |
| rw [dist_comm w v] at hchange |
| exact (norm_le_norm_sub_add _ _).trans (by linarith) |
| have hr' : ‖treeCenter Q-treeCenter P‖ ≤ 10*(d+1:ℝ)/(2:ℝ)^g := by |
| simpa only [dist_eq_norm] using hr.le |
| have htotal := mul_le_mul (mul_le_mul_of_nonneg_left hr' (by norm_num : (0:ℝ)≤2)) hd |
| dist_nonneg (by positivity : 0≤2*(10*(d+1:ℝ)/(2:ℝ)^g)) |
| simp only [div_eq_mul_inv] at htotal hW |
| nlinarith |
|
|
| lemma classConditional_ae_mem (ρ : Measure (EuclideanSpace ℝ (Fin d))) (P : CellPath d) : |
| ∀ᵐ x ∂classConditional ρ P, x ∈ treeCell P := |
| Measure.ae_smul_measure (ae_restrict_mem (measurableSet_treeCell P)) _ |
|
|
| lemma fullProjectionCollision_transfer (ρ : Measure (EuclideanSpace ℝ (Fin d))) |
| [IsFiniteMeasure ρ] (P : CellPath d) {W W' ε : ℝ} {v w : EuclideanSpace ℝ (Fin d)} |
| (hd : dist v w ≤ ε) |
| (hW : W'+((d+1:ℝ)/(2:ℝ)^P.length)*ε ≤ W) : |
| fullProjectionCollision ρ P W' w ≤ fullProjectionCollision ρ P W v := by |
| apply measure_mono_ae |
| filter_upwards [Measure.quasiMeasurePreserving_fst.ae (classConditional_ae_mem ρ P), |
| Measure.quasiMeasurePreserving_snd.ae (classConditional_ae_mem ρ P)] with z hz hz' |
| intro hproj |
| change |inner ℝ (z.1-z.2) w| ≤ W' at hproj |
| have hdiam : ‖z.1-z.2‖ ≤ (d+1:ℝ)/(2:ℝ)^P.length := by |
| simpa only [dist_eq_norm] using ambientGridCell_dist_le hz' hz |
| have hn : ‖v-w‖ ≤ ε := by simpa only [dist_eq_norm] using hd |
| have hb := abs_real_inner_le_norm (z.1-z.2) (v-w) |
| rw [inner_sub_right] at hb |
| have hb' := hb.trans (mul_le_mul hdiam hn (norm_nonneg _) (by positivity)) |
| have ht := abs_add_le (inner ℝ (z.1-z.2) v-inner ℝ (z.1-z.2) w) |
| (inner ℝ (z.1-z.2) w) |
| simp only [sub_add_cancel] at ht |
| exact ht.trans (by linarith) |
|
|
| def ValidDescriptor (N : ℕ) : PolicyDescriptor → Prop |
| | .parallel g p => g ≤ p ∧ p ≤ N |
| | .projection p t => p ≤ t ∧ t ≤ N |
|
|
| lemma policyDescriptor_valid (A : ℕ → ℝ) {N q a t : ℕ} (hq : 0 < q) (ht : t ≤ N) |
| {b : PolicyDescriptor} (hb : b ∈ policyDescriptorList A N q a t) : ValidDescriptor N b := by |
| obtain ⟨s,u,_,hut,_,_,hpos,_,hbound⟩ := policyDescriptor_origin A hq hb |
| cases b <;> simp only [descriptorLength,descriptorAnchor,ValidDescriptor] at * <;> omega |
|
|
| lemma policyDescriptor_anchor_le_end (A : ℕ → ℝ) {N q a t : ℕ} |
| (hq : 0 < q) {b : PolicyDescriptor} (hb : b ∈ policyDescriptorList A N q a t) : |
| descriptorAnchor b ≤ t := by |
| obtain ⟨_,u,_,hut,_,_,_,_,hu⟩ := policyDescriptor_origin A hq hb |
| omega |
|
|
| lemma uniform_descriptorGood_transfer {sig : ℝ} (hsig : 0<sig) {J : ℕ} (hJ : 0<J) : |
| ∃ N₀ : ℕ, ∀ N ≥ N₀, ∀ (ρ : Measure (EuclideanSpace ℝ (Fin d))), IsFiniteMeasure ρ → |
| ∀ (m : ℝ → ℝ) (C : ℝ) (i : ℕ) (b : PolicyDescriptor), ValidDescriptor N b → |
| ∀ P : CellPath d, P.length=descriptorAnchor b → |
| ∀ v w : EuclideanSpace ℝ (Fin d), ‖w‖=1 → |
| dist v w ≤ nominalRadius N sig (descriptorLength b) → |
| v ∈ descriptorGood ρ m N sig C J i P b → |
| w ∈ descriptorGood ρ m N sig C J (i+1) P b := by |
| obtain ⟨N₁,h₁⟩ := uniform_testWidth_gap hsig hJ (20*(d+1)) (by positivity) |
| obtain ⟨N₂,h₂⟩ := uniform_testWidth_gap hsig hJ (d+1) (by positivity) |
| refine ⟨max N₁ N₂,?_⟩ |
| intro N hN ρ hf m C i b hb P hP v w hw hd hg |
| have := hf |
| cases b with |
| | parallel g p => |
| exact ⟨hw,(Nat.cast_le.mpr (fullParallelCount_transfer ρ g p P hg.1 hw hd |
| (h₁ N ((le_max_left _ _).trans hN) g p hb.1 i))).trans hg.2⟩ |
| | projection p t => |
| apply And.intro hw |
| apply (fullProjectionCollision_transfer ρ P hd ?_).trans hg.2 |
| simp only [descriptorAnchor] at hP |
| rw [hP] |
| exact h₂ N ((le_max_right _ _).trans hN) p t hb.1 i |
|
|
| lemma dyadic_ratio_nominal {a p l N : ℕ} {sig : ℝ} (hsig : 0 ≤ sig) (hl : a+l≤p) : |
| (2:ℝ)^a/(2:ℝ)^p ≤ nominalRadius N sig l := by |
| have hb : (2:ℝ)^a/(2:ℝ)^p ≤ 1/(2:ℝ)^l := by |
| apply (div_le_div_iff₀ (by positivity) (by positivity)).mpr |
| rw [one_mul,← pow_add] |
| exact pow_le_pow_right₀ (by norm_num) hl |
| have hR : 1 ≤ ((2:ℝ)^N)^(50*sig) := |
| Real.one_le_rpow (one_le_pow₀ (by norm_num)) (by positivity) |
| apply le_min |
| · exact (div_le_one (by positivity)).mpr (pow_le_pow_right₀ (by norm_num) (by omega)) |
| · exact hb.trans (by simpa only [mul_one] using |
| mul_le_mul_of_nonneg_left hR (by positivity : 0≤1/(2:ℝ)^l)) |
|
|
| lemma same_tree_ancestor {P A : CellPath d} {a p : ℕ} (hP : P.length=p) |
| (hA : A.length=a) (hap : a≤p) {x x' : EuclideanSpace ℝ (Fin d)} |
| (hxA : x ∈ treeCell A) (hx : x ∈ treeCell P) (hx' : x' ∈ treeCell P) : |
| x' ∈ treeCell A := by |
| have hlen : (P.take a).length=A.length := by simp [List.length_take,hP,hA,hap] |
| have he : P.take a=A := by |
| by_contra hn |
| exact Set.disjoint_left.mp (treeCell_disjoint hlen hn) (treeCell_subset_take _ _ hx) hxA |
| simpa only [he] using treeCell_subset_take P a hx' |
|
|
| lemma fine_radial_dist {X Y P Q : CellPath d} {a p : ℕ} {K : ℝ} |
| (hK : 1000*(d+1:ℝ)<K) (hX : X.length=a) (hY : Y.length=a) |
| (hsep : CentreSeparated X Y a K) (hP : P.length=p) (hQ : Q.length=p) |
| (hPX : P.take a=X) (hQY : Q.take a=Y) |
| {x y x' y' : EuclideanSpace ℝ (Fin d)} (hx : x ∈ treeCell P) (hy : y ∈ treeCell Q) |
| (hx' : x' ∈ treeCell P) (hy' : y' ∈ treeCell Q) : |
| dist (radialVector x y) (radialVector x' y') ≤ (2:ℝ)^a/(2:ℝ)^p := by |
| have hPx : treeCell P ⊆ treeCell X := by simpa only [hPX] using treeCell_subset_take P a |
| have hQy : treeCell Q ⊆ treeCell Y := by simpa only [hQY] using treeCell_subset_take Q a |
| have hd := separated_parent_points hK hX hY hsep (hPx hx) (hQy hy) |
| have hd' := separated_parent_points hK hX hY hsep (hPx hx') (hQy hy') |
| have hpos : 0<100*(d+1:ℝ)/(2:ℝ)^a := by positivity |
| rw [dist_eq_norm] at hd hd' |
| have hmove : ‖(x-y)-(x'-y')‖ ≤ 2*(d+1:ℝ)/(2:ℝ)^p := by |
| have he : (x-y)-(x'-y')=(x-x')-(y-y') := by abel |
| rw [he] |
| have hxmove := ambientGridCell_dist_le hx' hx |
| have hymove := ambientGridCell_dist_le hy' hy |
| change dist x x' ≤ (d+1:ℝ)/(2:ℝ)^P.length at hxmove |
| change dist y y' ≤ (d+1:ℝ)/(2:ℝ)^Q.length at hymove |
| rw [hP,dist_eq_norm] at hxmove |
| rw [hQ,dist_eq_norm] at hymove |
| exact (norm_sub_le _ _).trans (by simp only [div_eq_mul_inv] at hxmove hymove ⊢; linarith) |
| apply (normalized_direction_distND (x-y) (x'-y') (hpos.trans_le hd) (hpos.trans_le hd')).trans |
| apply (div_le_div₀ (by positivity) (mul_le_mul_of_nonneg_left hmove (by norm_num : (0:ℝ) ≤ 2)) hpos hd).trans |
| have he : 2*(2*(d+1:ℝ)/(2:ℝ)^p)/(100*(d+1:ℝ)/(2:ℝ)^a) = |
| (1/25:ℝ)*((2:ℝ)^a/(2:ℝ)^p) := by |
| field_simp |
| ring_nf |
| rw [he] |
| have hg : 0≤(2:ℝ)^a/(2:ℝ)^p := by positivity |
| linarith |
|
|
| namespace PositiveDistanceState |
| lemma uniform_suffix_good_transport {sig : ℝ} (hsig : 0<sig) {J : ℕ} (hJ : 0<J) : |
| ∃ N₀ : ℕ, ∀ e : PositiveDistanceState d, e.sig=sig → e.J=J → N₀≤e.N → |
| IsFiniteMeasure e.ρ → ∀ K : ℝ, 1000*(d+1:ℝ)<K → CentreSeparated e.X e.Y e.a K → |
| e.X.length=e.a → e.Y.length=e.a → 0<e.q → |
| ∀ p : ℕ, e.a≤p → p≤e.t → e.t≤e.N → |
| ∀ P Q : CellPath d, P.length=p → Q.length=p → P.take e.a=e.X → Q.take e.a=e.Y → |
| ∀ x ∈ treeCell P, ∀ y ∈ treeCell Q, (x,y) ∈ e.good → |
| ∀ x' ∈ treeCell P, ∀ y' ∈ treeCell Q, |
| (x',y') ∈ ({e with t := p,i := e.i+1} : PositiveDistanceState d).good := by |
| obtain ⟨N₀,h₀⟩ := uniform_descriptorGood_transfer (d := d) hsig hJ |
| refine ⟨N₀,?_⟩ |
| intro e hsig' hJ' hN hf K hK hsep hX hY hq p hap hpt htN P Q hP hQ hPX hQY x hx y hy hg x' hx' y' hy' |
| have hmove := fine_radial_dist hK hX hY hsep hP hQ hPX hQY hx hy hx' hy' |
| have hPx : treeCell P ⊆ treeCell e.X := by simpa only [hPX] using treeCell_subset_take P e.a |
| have hQy : treeCell Q ⊆ treeCell e.Y := by simpa only [hQY] using treeCell_subset_take Q e.a |
| have hne : x' ≠ y' := by |
| have hh := separated_parent_points hK hX hY hsep (hPx hx') (hQy hy') |
| exact dist_pos.mp ((by positivity : 0<100*(d+1:ℝ)/(2:ℝ)^e.a).trans_le hh) |
| intro b hb |
| change b ∈ policyDescriptorList (adjustedProfile (d := d) e.mass e.N) e.N e.q e.a p at hb |
| have hb' := policyDescriptorList_mono (adjustedProfile (d := d) e.mass e.N) le_rfl hpt hb |
| have hbg := hg b hb' |
| have ha := policyDescriptor_anchor_le_end (adjustedProfile (d := d) e.mass e.N) hq hb |
| have hl := policyDescriptor_twice_length (adjustedProfile (d := d) e.mass e.N) hq hb |
| have hv := policyDescriptor_valid (adjustedProfile (d := d) e.mass e.N) hq (hpt.trans htN) hb |
| have hrad := hmove.trans (dyadic_ratio_nominal (N := e.N) hsig.le (by omega : e.a+descriptorLength b≤p)) |
| have htransfer (A : CellPath d) (hA : A.length=descriptorAnchor b) |
| (hw : radialVector x y ∈ descriptorGood e.ρ e.mass e.N e.sig e.C e.J e.i A b) : |
| radialVector x' y' ∈ descriptorGood e.ρ e.mass e.N e.sig e.C e.J (e.i+1) A b := by |
| rw [hsig',hJ'] at hw ⊢ |
| exact h₀ e.N hN e.ρ hf e.mass e.C e.i b hv A hA _ _ (norm_radialVector hne) hrad hw |
| constructor |
| · intro A hA hxA |
| have hlen := (mem_dyadicPaths _ _).mp hA |
| apply htransfer A hlen |
| apply hbg.1 A hA |
| exact same_tree_ancestor hP hlen ha hxA hx' hx |
| · intro A hA hyA |
| have hlen := (mem_dyadicPaths _ _).mp hA |
| apply htransfer A hlen |
| apply hbg.2 A hA |
| exact same_tree_ancestor hQ hlen ha hyA hy' hy |
| end PositiveDistanceState |
|
|
| end CurrentFalconer.TreeND |
| end GraphLiteralSupport |
| end |
| end |
|
|
| section |
| section |
| open Set MeasureTheory |
| open scoped ComplexConjugate RealInnerProductSpace |
| namespace GraphLiteralSupport |
| namespace CurrentFalconer |
| open Filter Real |
| open scoped Topology |
|
|
| noncomputable def sourceMaskScaleND (l : ℕ) (R σ : ℝ) : ℝ := |
| max 1 ((2:ℝ)^l*R^(-50*σ)) |
|
|
| noncomputable def sourceMaskRadiusND (d l : ℕ) (R σ : ℝ) : ℝ := |
| (sourceMaskScaleND l R σ)⁻¹/(100*(d+1)^2) |
|
|
| noncomputable def sourceTestWidthND (J i n : ℕ) (R σ : ℝ) : ℝ := |
| (2-(i:ℝ)/J)*((2:ℝ)^n)⁻¹*R^(500*σ) |
|
|
| lemma sourceMaskScaleND_one_le (l : ℕ) (R σ : ℝ) : 1 ≤ sourceMaskScaleND l R σ := le_max_left _ _ |
|
|
| lemma sourceMaskScaleND_pos (l : ℕ) (R σ : ℝ) : 0<sourceMaskScaleND l R σ := |
| zero_lt_one.trans_le (sourceMaskScaleND_one_le l R σ) |
|
|
| lemma sourceMaskRadiusND_pos (d l : ℕ) (R σ : ℝ) : 0<sourceMaskRadiusND d l R σ := |
| div_pos (inv_pos.mpr (sourceMaskScaleND_pos l R σ)) (by positivity) |
|
|
| lemma sourceMaskScaleND_mono {l L : ℕ} (h : l ≤ L) {R σ : ℝ} (hR : 0 ≤ R) : |
| sourceMaskScaleND l R σ ≤ sourceMaskScaleND L R σ := by |
| apply max_le_max_left |
| exact mul_le_mul_of_nonneg_right (pow_le_pow_right₀ (by norm_num) h) (Real.rpow_nonneg hR _) |
|
|
| lemma sourceMaskScaleND_inv_depth {g p : ℕ} (hgp : g ≤ p) {R σ : ℝ} (hR : 0<R) : |
| ((2:ℝ)^g)⁻¹*(sourceMaskScaleND (p-g) R σ)⁻¹ ≤ ((2:ℝ)^p)⁻¹*R^(50*σ) := by |
| have hpos : 0<(2:ℝ)^(p-g)*R^(-50*σ) := by positivity |
| have hh := inv_anti₀ hpos (le_max_right (1:ℝ) ((2:ℝ)^(p-g)*R^(-50*σ))) |
| apply (mul_le_mul_of_nonneg_left hh (by positivity : (0:ℝ) ≤ ((2:ℝ)^g)⁻¹)).trans_eq |
| rw [mul_inv_rev,show -50*σ=-(50*σ) by ring,Real.rpow_neg hR.le,inv_inv] |
| calc |
| _ = (((2:ℝ)^g)*((2:ℝ)^(p-g)))⁻¹*R^(50*σ) := by rw [mul_inv_rev]; ring |
| _ = _ := by rw [← pow_add,Nat.add_sub_of_le hgp] |
|
|
| end CurrentFalconer |
| end GraphLiteralSupport |
| end |
| end |
|
|
| end OAI |
|
|
