import Mathlib import OAI.Probability.SKFluctuations.ComparisonCapTravel import OAI.Probability.SKFluctuations.InverseConstraintNonempty namespace OAI /-! Clock Retain Two. -/ noncomputable section namespace SKFluctuations.Hierarchy open MeasureTheory ProbabilityTheory Filter Set open scoped Topology NNReal theorem MassOrdered.le_of_mem {a : ℝ} {as : List PositiveMass} (ha : MassOrdered a as) {b : PositiveMass} (hb : b ∈ as) : a≤b.val := by induction as generalizing a with | nil => simp at hb | cons c cs ih => rcases List.mem_cons.mp hb with rfl|hb · exact ha.2.1 · exact ha.2.1.trans (ih ha.2.2 hb) theorem partitionSample_at_start (q : ℝ → ℝ) {a : ℝ} (as : List PositiveMass) (ha : MassOrdered a as) : partitionSample q a as a=q a := by induction as generalizing a with | nil => rfl | cons b bs ih => rw [partitionSample_cons] split_ifs with hab · rfl · have he : a=b.val := le_antisymm ha.2.1 (le_of_not_gt hab) subst a exact ih ha.2.2 theorem partitionSample_at_node (q : ℝ → ℝ) {a : ℝ} (as : List PositiveMass) (ha : MassOrdered a as) {u : PositiveMass} (hu : u ∈ as) : partitionSample q a as u.val=q u.val := by induction as generalizing a with | nil => simp at hu | cons b bs ih => rcases List.mem_cons.mp hu with rfl|hu · rw [partitionSample_cons,ite_eq_right (lt_irrefl _)] exact partitionSample_at_start q _ ha.2.2 · rw [partitionSample_cons,ite_eq_right (not_lt.mpr (ha.2.2.le_of_mem hu))] exact ih ha.2.2 hu def insertMass (u : PositiveMass) (as : List PositiveMass) : List PositiveMass := (splitMassPartition u as).1++(splitMassPartition u as).2 theorem insertMass_ordered (u : PositiveMass) (hu : u.val≤1) (as : List PositiveMass) (ha : MassOrdered 0 as) : MassOrdered 0 (insertMass u as) := by obtain ⟨h₁,h₂⟩ := splitMassPartition_ordered u hu as ha u.property.le apply h₁.append_end simpa only [splitMassPartition_end] using h₂ theorem mem_insertMass_self (u : PositiveMass) (as : List PositiveMass) : u ∈ insertMass u as := by induction as with | nil => simp [insertMass,splitMassPartition] | cons b bs ih => unfold insertMass splitMassPartition split_ifs · exact List.mem_cons_of_mem _ ih · simp theorem mem_insertMass_of_mem (u : PositiveMass) {as : List PositiveMass} {v : PositiveMass} (hv : v ∈ as) : v ∈ insertMass u as := by induction as with | nil => simp at hv | cons b bs ih => unfold insertMass splitMassPartition split_ifs · rcases List.mem_cons.mp hv with rfl|hv · exact List.mem_cons_self · exact List.mem_cons_of_mem _ (ih hv) · exact List.mem_cons_of_mem _ hv theorem insertMass_end (u : PositiveMass) (as : List PositiveMass) (hu : u.val≤1) (he : partitionEnd 0 as=1) : partitionEnd 0 (insertMass u as)=1 := by have hgen : ∀ {a : ℝ} (bs : List PositiveMass), a≤u.val → u.val≤partitionEnd a bs → partitionEnd a (insertMass u bs)=partitionEnd a bs := by intro a bs induction bs generalizing a with | nil => intro hau hua exact le_antisymm hua hau | cons b bs ih => intro hau hue unfold insertMass splitMassPartition split_ifs with hbu · exact ih hbu.le hue · rfl exact (hgen as u.property.le (he ▸ hu)).trans he def twoCutMasses (u v : PositiveMass) (n : ℕ) : List PositiveMass := insertMass u (insertMass v (uniformMasses n)) theorem twoCutMasses_ordered (u v : PositiveMass) (hu : u.val≤1) (hv : v.val≤1) (n : ℕ) : MassOrdered 0 (twoCutMasses u v n) := insertMass_ordered u hu _ (insertMass_ordered v hv _ (uniformMasses_ordered n)) theorem twoCutMasses_mesh (u v : PositiveMass) (hu : u.val≤1) (hv : v.val≤1) (n : ℕ) : PartitionMesh (((n+1:ℕ):ℝ)⁻¹) 0 (twoCutMasses u v n) := splitMassPartition_mesh u hu _ (insertMass_ordered v hv _ (uniformMasses_ordered n)) u.property.le (splitMassPartition_mesh v hv _ (uniformMasses_ordered n) v.property.le (uniformMasses_mesh n)) theorem twoCutMasses_nodes (u v : PositiveMass) (n : ℕ) : u ∈ twoCutMasses u v n ∧ v ∈ twoCutMasses u v n := ⟨mem_insertMass_self u _,mem_insertMass_of_mem u (mem_insertMass_self v _)⟩ end SKFluctuations.Hierarchy namespace SKFluctuations.Hierarchy open MeasureTheory ProbabilityTheory Filter Set open scoped Topology NNReal BigOperators variable {S ι : Type*} [Fintype S] [Nonempty S] [Fintype ι] [Nonempty ι] attribute [local instance] Classical.propDecidable theorem retainedMoment_tendsto {w : S → ℝ} (hw : ∀ σ, 0 q u i) (Icc (0:ℝ) 1)) (as : ℕ → List PositiveMass) (h : ℕ → ℝ) (ho : ∀ n, MassOrdered 0 (as n)) (hm : ∀ n, PartitionMesh (h n) 0 (as n)) (he : ∀ n, partitionEnd 0 (as n)=1) (hh : Tendsto h atTop (𝓝 0)) (u : PositiveMass) (hu : u.val≤1) (hnode : ∀ n, u ∈ as n) (P : S → S → ℝ) {K : ℝ} (hK : 0≤K) (hP : ∀ σ τ, |P σ τ|≤K) : Tendsto (fun n => stepPairMoment w φ (q 0) (sampleClockSteps q 0 (as n)) (sampleClockSteps_ordered q _ (ho n)).mem_pos P u.val) atTop (𝓝 (pathPairMoment w φ q P u.val)) := by have ht (n : ℕ) : clockCoordinateDistance D (stepClockFunction (fun i => (q 0 i:ℝ)) (sampleClockSteps q 0 (as n)) u.val) (fun i => (q u.val i:ℝ))=0 := by unfold clockCoordinateDistance simp_rw [sampleClockSteps_function q hq _ (ho n),partitionSample_at_node _ _ (ho n) (hnode n)] simp have hl : Tendsto (fun n => stepPathClockDistance D (q 0) (sampleClockSteps q 0 (as n)) q) atTop (𝓝 0) := squeeze_zero (fun _ => stepPathClockDistance_nonneg _ _ _ _) (fun n => sample_stepPathClockDistance_le D q hq _ (ho n) (hm n)) (by simpa using hh.mul_const (clockVariationBound D q)) rw [tendsto_iff_norm_sub_tendsto_zero] simp only [Real.norm_eq_abs] apply squeeze_zero (fun _ => abs_nonneg _) (fun n => stepPairMoment_path_comparison hw φ D hD hφ (q 0) _ (sampleClockSteps_ordered q _ (ho n)) q hq P hK hP ⟨u.property,hu⟩) simp_rw [ht,sampleClockSteps_terminal q hq _ _ (ho _),he] simpa [clockCoordinateDistance] using (hl.const_mul 2).const_mul (ClockTree.errorJetVariation 2 0 0*K) theorem retainedMoment_tendsto_all {w : S → ℝ} (hw : ∀ σ, 0 q u i) (Icc (0:ℝ) 1)) (as : ℕ → List PositiveMass) (h : ℕ → ℝ) (ho : ∀ n, MassOrdered 0 (as n)) (hm : ∀ n, PartitionMesh (h n) 0 (as n)) (he : ∀ n, partitionEnd 0 (as n)=1) (hh : Tendsto h atTop (𝓝 0)) (u : PositiveMass) (hu : u.val≤1) (hnode : ∀ n, u ∈ as n) (P : S → S → ℝ) : Tendsto (fun n => stepPairMoment w φ (q 0) (sampleClockSteps q 0 (as n)) (sampleClockSteps_ordered q _ (ho n)).mem_pos P u.val) atTop (𝓝 (pathPairMoment w φ q P u.val)) := by let D : ι → ℝ := fun i => ∑ σ, |φ σ (Pi.single i 1)| have hD : ∀ i, 0≤D i := by intro i; exact Finset.sum_nonneg fun σ _ => abs_nonneg _ have hφ : ∀ σ i, |φ σ (Pi.single i 1)|≤D i := by intro σ i exact Finset.single_le_sum (fun τ _ => abs_nonneg (φ τ (Pi.single i 1))) (Finset.mem_univ σ) let K : ℝ := ∑ σ, ∑ τ, |P σ τ| have hK : 0≤K := by dsimp [K]; positivity have hP (σ τ : S) : |P σ τ|≤K := by exact (Finset.single_le_sum (fun ρ _ => abs_nonneg (P σ ρ)) (Finset.mem_univ τ)).trans (Finset.single_le_sum (fun ρ _ => Finset.sum_nonneg fun η _ => abs_nonneg (P ρ η)) (Finset.mem_univ σ)) exact retainedMoment_tendsto hw φ D hD hφ q hq as h ho hm he hh u hu hnode P hK hP end SKFluctuations.Hierarchy namespace SKFluctuations.Hierarchy open MeasureTheory ProbabilityTheory Filter Set open scoped Topology NNReal BigOperators variable {S ι : Type*} [Fintype S] [Nonempty S] [Fintype ι] theorem rootLevelPairMean_product_monotone {w : S → ℝ} (hw : ∀ σ, 0 rootLevelPairMean w φ q (fun σ τ => A σ*A τ) steps j) := by intro j k hjk apply integral_mono ((levelPairMean_smooth hw φ _ steps ha j 0).integrable_zero_shift 0) ((levelPairMean_smooth hw φ _ steps ha k 0).integrable_zero_shift 0) intro x exact levelPairMean_product_monotone hw φ A steps ha (0+x) hjk theorem cumulativePair_product_monotone {w : S → ℝ} (hw : ∀ σ, 0 cumulativePair w φ as q j (fun σ τ => A σ*A τ)) := by have he (j : ℕ) : cumulativePair w φ as q j (fun σ τ => A σ*A τ)= rootLevelPairMean w φ (fun i => (q 0 i).toNNReal) (fun σ τ => A σ*A τ) (diagonalBoundarySteps as (q 0) (fun k => q (k+1))) j := ClockTree.pairDisorder_eq_rootLevelPairMean hw φ as ha q j (fun σ τ => A σ*A τ) simp_rw [he] apply rootLevelPairMean_product_monotone hw φ rw [←clockSteps_cumulative] exact clockSteps_ordered ha _ omit [Fintype ι] in theorem splitClockSteps_length_monotone [Fintype ι] (L : List (GaussianStep ι)) : Monotone (fun u : ℝ => (splitClockSteps u L).1.length) := by intro u v huv induction L with | nil => exact le_rfl | cons s L ih => simp only [splitClockSteps] split_ifs with hsu hsv hsv · simpa only [List.length_cons] using Nat.succ_le_succ ih · exact False.elim (hsv (hsu.trans huv)) · exact Nat.zero_le _ · exact le_rfl theorem stepPairMoment_product_monotone [Nonempty ι] {w : S → ℝ} (hw : ∀ σ, 0 A σ*A τ)) := by intro u v huv unfold stepPairMoment rw [massStepFunction_split_index,massStepFunction_split_index] exact cumulativePair_product_monotone hw φ _ (positiveStepMasses_ordered ho _) _ A (splitClockSteps_length_monotone L huv) end SKFluctuations.Hierarchy namespace SKFluctuations.Hierarchy open MeasureTheory ProbabilityTheory Filter Set open scoped Topology NNReal BigOperators variable {S ι : Type*} [Fintype S] [Nonempty S] [Fintype ι] [Nonempty ι] theorem pathPairMoment_product_monotone {w : S → ℝ} (hw : ∀ σ, 0 q u i) (Icc (0:ℝ) 1)) (A : S → ℝ) : MonotoneOn (pathPairMoment w φ q (fun σ τ => A σ*A τ)) (Ioc (0:ℝ) 1) := by intro u hu v hv huv let u' : PositiveMass := ⟨u,hu.1⟩ let v' : PositiveMass := ⟨v,hv.1⟩ let as := twoCutMasses u' v' have ho (n : ℕ) : MassOrdered 0 (as n) := twoCutMasses_ordered u' v' hu.2 hv.2 n have hm (n : ℕ) : PartitionMesh (((n+1:ℕ):ℝ)⁻¹) 0 (as n) := twoCutMasses_mesh u' v' hu.2 hv.2 n have he (n : ℕ) : partitionEnd 0 (as n)=1 := insertMass_end u' _ hu.2 (insertMass_end v' _ hv.2 (uniformMasses_end n)) have h₁ := retainedMoment_tendsto_all hw φ q hq as _ ho hm he uniformMasses_mesh_tendsto u' hu.2 (fun n => (twoCutMasses_nodes u' v' n).1) (fun σ τ => A σ*A τ) have h₂ := retainedMoment_tendsto_all hw φ q hq as _ ho hm he uniformMasses_mesh_tendsto v' hv.2 (fun n => (twoCutMasses_nodes u' v' n).2) (fun σ τ => A σ*A τ) exact le_of_tendsto_of_tendsto h₁ h₂ (Filter.Eventually.of_forall fun n => stepPairMoment_product_monotone hw φ (q 0) _ (sampleClockSteps_ordered q _ (ho n)) A huv) end SKFluctuations.Hierarchy namespace SKFluctuations open Hierarchy MeasureTheory Filter Set open scoped Topology NNReal BigOperators variable {ι : Type*} [Fintype ι] [Nonempty ι] theorem pathPairMoment_overlap_monotone (n : ℕ) {w : Spin n → ℝ} (hw : ∀ σ, 0 q u i) (Icc (0:ℝ) 1)) : MonotoneOn (pathPairMoment w φ q (orderedOverlap n)) (Ioc (0:ℝ) 1) := by have he : orderedOverlap n = fun σ τ => (1/(n:ℝ))*∑ i, spinValue (σ i)*spinValue (τ i) := by funext σ τ unfold orderedOverlap ring intro u hu v hv huv rw [he,pathPairMoment_const_mul hw φ q hq _ _ hu, pathPairMoment_const_mul hw φ q hq _ _ hv, pathPairMoment_finsetSum hw φ q hq _ _ hu,pathPairMoment_finsetSum hw φ q hq _ _ hv] exact mul_le_mul_of_nonneg_left (Finset.sum_le_sum fun i _ => pathPairMoment_product_monotone hw φ q hq (fun σ => spinValue (σ i)) hu hv huv) (by positivity) theorem pathPairMoment_overlap_sq_monotone (n : ℕ) {w : Spin n → ℝ} (hw : ∀ σ, 0 q u i) (Icc (0:ℝ) 1)) : MonotoneOn (pathPairMoment w φ q (fun σ τ => (orderedOverlap n σ τ)^2)) (Ioc (0:ℝ) 1) := by have he : (fun σ τ => (orderedOverlap n σ τ)^2) = fun σ τ => (1/(n:ℝ)^2)*∑ i, ∑ j, (spinValue (σ i)*spinValue (σ j))*(spinValue (τ i)*spinValue (τ j)) := by funext σ τ unfold orderedOverlap rw [div_pow,pow_two,Finset.sum_mul_sum,one_div_mul_eq_div] congr 1 apply Finset.sum_congr rfl intro i hi apply Finset.sum_congr rfl intro j hj ring intro u hu v hv huv rw [he,pathPairMoment_const_mul hw φ q hq _ _ hu, pathPairMoment_const_mul hw φ q hq _ _ hv, pathPairMoment_finsetSum hw φ q hq _ _ hu,pathPairMoment_finsetSum hw φ q hq _ _ hv] apply mul_le_mul_of_nonneg_left _ (by positivity) apply Finset.sum_le_sum intro i hi rw [pathPairMoment_finsetSum hw φ q hq _ _ hu,pathPairMoment_finsetSum hw φ q hq _ _ hv] exact Finset.sum_le_sum fun j _ => pathPairMoment_product_monotone hw φ q hq (fun σ => spinValue (σ i)*spinValue (σ j)) hu hv huv theorem orderedPath_SK_monotone (n : ℕ) (hn : 0 comparisonFloor_lower_on (δ:=δ) hN hU hr₀.1 hri let hm := comparisonCap_monotoneOn (δ:=δ) hN hU.le hL hE hd let hM := comparisonCap_nonneg (δ:=δ) hN hU.le hL hE hd (show (0:ℝ)≤1 by norm_num) ∀ (α : inverseIntegralConstraints r₀ u₀ k M) (hg : ContDiff ℝ 1 g) (_hgpos : ∀ r ∈ Icc r₀ 1, 0 b u-omittedExtension hu.2 (constrainedCoefficient hr hc hk hkc hm hM g hg.continuous β) u) (Icc (0:ℝ) 1)) (_hh : ∀ β : inverseIntegralConstraints r₀ u₀ k M, MonotoneOn (fun u => h u+omittedExtension hu.2 (constrainedCoefficient hr hc hk hkc hm hM (fun r => g r*r) (hg.continuous.mul continuous_id) β) u) (Icc (0:ℝ) 1)) (_hbn : ∀ β : inverseIntegralConstraints r₀ u₀ k M, 0 ≤ b 0-omittedExtension hu.2 (constrainedCoefficient hr hc hk hkc hm hM g hg.continuous β) 0) (_hhn : ∀ β : inverseIntegralConstraints r₀ u₀ k M, 0 ≤ h 0+omittedExtension hu.2 (constrainedCoefficient hr hc hk hkc hm hM (fun r => g r*r) (hg.continuous.mul continuous_id) β) 0) (_hmin : ∀ β : inverseIntegralConstraints r₀ u₀ k M, inverseCompensatedPressure hr hu.2 hc hk hkc hm hM g hg.continuous n b h 0 α ≤ inverseCompensatedPressure hr hu.2 hc hk hkc hm hM g hg.continuous n b h 0 β), let b' := fun u => (b u-omittedExtension hu.2 (constrainedCoefficient hr hc hk hkc hm hM g hg.continuous α) u).toNNReal let h' := fun u => (h u+omittedExtension hu.2 (constrainedCoefficient hr hc hk hkc hm hM (fun r => g r*r) (hg.continuous.mul continuous_id) α) u).toNNReal ∀ r ∈ Ioc r₀ 1, ∀ s ∈ Icc r 1, N^(-δ)*U < α.val.val s-α.val.val r → orderedPathS n b' h' (α.val.val r)≤ s+(L/N^δ)*Real.log ((1+U)/U) := by dsimp only intro α hg hgpos hp b h hb hh hbn hhn hmin let k := comparisonFloor N δ U let M := comparisonCap N δ U L E d let c := k 1 let hr : r₀≤1 := le_trans hr₀.2 (by norm_num) let hu : u₀≤1 := le_trans hu₀.2 (by norm_num) let hc : 0 comparisonFloor_lower_on (δ:=δ) hN hU hr₀.1 hri let hm := comparisonCap_monotoneOn (δ:=δ) hN hU.le hL hE hd let hM := comparisonCap_nonneg (δ:=δ) hN hU.le hL hE hd (show (0:ℝ)≤1 by norm_num) let b' := fun u => (b u-omittedExtension hu (constrainedCoefficient hr hc hk hkc hm hM g hg.continuous α) u).toNNReal let h' := fun u => (h u+omittedExtension hu (constrainedCoefficient hr hc hk hkc hm hM (fun r => g r*r) (hg.continuous.mul continuous_id) α) u).toNNReal let S := fun r => orderedPathS n b' h' (α.val.val r) let B := fun r => orderedPathB n b' h' (α.val.val r) let J := inverseControlFactor r₀ (fun r => inverseControlWeight α.val.val k M r* comparisonCapSlope N δ U L E d (α.val.val r)) obtain ⟨H,hH,hD,hbang,hzero⟩ := comparison_minimum_contact_potential hN hU hL hLU hE hd hq hr₀ hu₀ n hn g α hg (fun r hr => hgpos r ⟨hr.1.le,hr.2⟩) hp b h hb hh hbn hhn hmin have hα := inverseConstraint_strictMonoOn hc hk hkc α.property have hαL := inverseIntegralConstraints_lipschitz hm hM α.property have hαr (r : ℝ) (hri : r ∈ Ioc r₀ 1) : α.val.val r ∈ Ioc (0:ℝ) 1 := by refine ⟨?_,(α.val.property.1 r).2⟩ have ht := hα ⟨le_rfl,hr⟩ ⟨hri.1.le,hri.2⟩ hri.1 rw [α.property.1] at ht exact hu₀.1.trans_lt ht have hb' : MonotoneOn b' (Icc (0:ℝ) 1) := fun u hu v hv huv => Real.toNNReal_mono ((hb α) hu hv huv) have hh' : MonotoneOn h' (Icc (0:ℝ) 1) := fun u hu v hv huv => Real.toNNReal_mono ((hh α) hu hv huv) have hSm : MonotoneOn S (Ioc r₀ 1) := by intro r hr s hs hrs exact (orderedPath_SK_monotone n hn b' h' hb' hh').1 (hαr r hr) (hαr s hs) (α.val.property.2 hrs) have hbounds (r : ℝ) (hr : r ∈ Ioc r₀ 1) := orderedPath_SK_bounds n hn b' h' hb' hh' (hαr r hr) have hD' : ∀ᵐ r, r ∈ Icc r₀ 1 → HasDerivAt H ((J r*g r)*comparisonBracket (comparisonLogSlope g r) r (S r) (B r)) r := by filter_upwards [hD,volume.ae_ne r₀] with r hDr hrne hri have hrpos : 0 mul_pos (inverseControlFactor_pos _ _ _) (hgpos t ht)) hD' (hbang.mono fun t ht hti => (ht hti).1) (hbang.mono fun t ht hti => (ht hti).2) (hzero.mono fun t ht hti hh => (ht hti hh).2.1) (fun t ht => (hp t ht).1) hSm (fun t ht => (hbounds t ht).2.1) (fun t ht => (hbounds t ht).2.2.1) hαL hα (fun t _ => α.val.property.1 t) (comparisonCap_continuous N δ U L E d).continuousOn (fun u hu => (comparisonCap_first_term hN hU hL hE hd hu.1).1) (comparisonFloor_continuousOn (δ:=δ) hN hU (show (0:ℝ)≤0 by rfl)).integrableOn_Icc (fun t ht => (comparisonFloor_pos (δ:=δ) hN hU ht.1).le) ((comparisonFloor_total_mass hN hU).trans_lt hinc) have hcT := comparisonCap_log_travel (δ:=δ) hN hU hL hE hd (show (0:ℝ)≤0 by rfl) (show (0:ℝ)≤1 by norm_num) simp only [zero_add] at hcT exact hf.trans (by linarith) end SKFluctuations namespace SKFluctuations open Set theorem finite_dyadic_future_cover {x B μ : ℝ} (hx : 0 < x) (hB : 0 < B) (hμ : 0 < μ) (hμB : 3*B*μ ≤ 1) : ∃ vs : Finset ℝ, (∀ v ∈ vs, 0 < v ∧ 3*B*v ≤ 1) ∧ ∀ y : ℝ, x ≤ y → μ ≤ y ∨ ∃ v ∈ vs, v ≤ y ∧ y < 2*v := by obtain ⟨N,hN⟩ := exists_nat_gt (max 0 (μ/x)) rw [max_eq_right (div_pos hμ hx).le] at hN classical let vs := ((Finset.range N).image (fun j : ℕ => (2:ℝ)^j*x)).filter (fun v => 3*B*v ≤ 1) refine ⟨vs,?_,?_⟩ · intro v hv obtain ⟨hj,hv⟩ := Finset.mem_filter.mp hv obtain ⟨j,hj,rfl⟩ := Finset.mem_image.mp hj exact ⟨by positivity,hv⟩ · intro y hxy by_cases hmy : μ ≤ y · exact Or.inl hmy · right have hym : y < μ := lt_of_not_ge hmy have hrat : 1 ≤ y/x := (le_div_iff₀ hx).mpr (by simpa using hxy) obtain ⟨j,hlo,hhi⟩ := exists_nat_pow_near hrat (show (1:ℝ) < 2 by norm_num) have hvy : (2:ℝ)^j*x ≤ y := (le_div_iff₀ hx).mp hlo have hyv : y < 2*((2:ℝ)^j*x) := by have hh := (div_lt_iff₀ hx).mp hhi rw [pow_succ] at hh nlinarith have hjpow : (j:ℝ) < (2:ℝ)^j := by exact_mod_cast (Nat.lt_two_pow_self (n := j)) have hjN : j < N := by have hyx : y/x < μ/x := div_lt_div_of_pos_right hym hx have hh : (j:ℝ) < N := lt_trans (hjpow.trans_le hlo) (hyx.trans hN) exact_mod_cast hh refine ⟨(2:ℝ)^j*x,Finset.mem_filter.mpr ⟨Finset.mem_image.mpr ⟨j,Finset.mem_range.mpr hjN,rfl⟩,?_⟩,hvy,hyv⟩ exact (mul_le_mul_of_nonneg_left (hvy.trans hym.le) (by positivity : 0 ≤ 3*B)).trans hμB theorem future_window_radius_geometry {α : ℝ → ℝ} {x y v B cl cu R a b A : ℝ} (hx : 0 < x) (hxy : x ≤ y) (hvy : v ≤ y) (hyv : y < 2*v) (hcl : 0 < cl) (hcu : 0 < cu) (hB : 2 ≤ B) (hA : 1 ≤ A) (hBA : 2*cu*A ≤ B) (hyR : B*y ≤ cl*R) (ha : α a=B*v) (hb : α b=2*B*v) (hregA : cl*a ≤ α a ∧ α a ≤ cu*a) (hregB : cl*b ≤ α b ∧ α b ≤ cu*b) : 0 < v ∧ A*x ≤ a ∧ a ≤ R ∧ b ≤ (2*cu/cl)*a ∧ cl*a ≤ B*v ∧ B*v ≤ cu*a ∧ y-x < B*v := by have hv : 0 < v := by linarith have hB0 : 0 < B := by linarith have hAx : 0 < A*x := mul_pos (by linarith) hx have hAv : A*x < 2*A*v := by nlinarith have hBv : 2*cu*A*v ≤ B*v := mul_le_mul_of_nonneg_right hBA hv.le rw [ha] at hregA rw [hb] at hregB have hAa : A*x ≤ a := by nlinarith refine ⟨hv,hAa,?_,?_,hregA.1,hregA.2,?_⟩ · have hh := mul_le_mul_of_nonneg_left hvy hB0.le nlinarith · rw [div_mul_eq_mul_div] apply (le_div_iff₀ hcl).mpr have hh := mul_le_mul_of_nonneg_left hregA.2 (show (0:ℝ) ≤ 2 by norm_num) nlinarith [hregB.1] · nlinarith end SKFluctuations namespace SKFluctuations open Set Hierarchy theorem radius_in_regular_region {α : ℝ → ℝ} {r₀ z a p cl cu : ℝ} (hα : Monotone α) (ha : a ∈ Icc r₀ 1) (hz : z ∈ Icc r₀ 1) (heq : α a=p) (hp : cu*z